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\section*{Introduction}
The singularities in General Relativity can be avoided only if the stress-energy tensor in the right hand side of Einstein's equation satisfies some particular conditions. One way to avoid them was proposed by the authors of \cite{corda2010removingBHsingularities}, who have shown that the singularities can be removed by constructing the stress-energy tensor with non-linear electrodynamics. On the other hand, Einstein's
equation leads to singularities in general conditions \cite{Pen65,Haw66i,Haw66ii,Haw67iii,HP70,HE95}, and there the time evolution breaks down. Is this a problem of the theory itself or of the way it is formulated?
This paper proposes a version of Einstein's equation which is equivalent to the standard version at the points of spacetime where the metric is non-singular. But unlike Einstein's equation, in many cases it can be extended at and beyond the singular points.
Let $(M,g)$ be a Riemannian or a {semi{-\penalty0\hskip0pt\relax}Riemannian} manifold of dimension $n$.
It is useful to recall the definition of the \textit{Kulkarni-Nomizu product} of two symmetric bilinear forms $h$ and $k$,
\begin{equation}
\label{eq_kulkarni_nomizu}
(h\circ k)_{abcd} := h_{ac}k_{bd} - h_{ad}k_{bc} + h_{bd}k_{ac} - h_{bc}k_{ad}.
\end{equation}
The Riemann curvature tensor can be decomposed algebraically as
\begin{equation}
\label{eq_ricci_decomposition}
R_{abcd} = S_{abcd} + E_{abcd} + C_{abcd}.
\end{equation}
where
\begin{equation}
\label{eq_ricci_part_S}
S_{abcd} = \dsfrac{1}{2n(n-1)}R(g\circ g)_{abcd}
\end{equation}
is the scalar part of the Riemann curvature and
\begin{equation}
\label{eq_ricci_part_E}
E_{abcd} = \dsfrac{1}{n-2}(S \circ g)_{abcd}
\end{equation}
is the \textit{semi-traceless part} of the Riemann curvature. Here
\begin{equation}
\label{eq_ricci_traceless}
S_{ab} := R_{ab} - \dsfrac{1}{n}Rg_{ab}
\end{equation}
is the traceless part of the Ricci curvature.
The \textit{Weyl curvature tensor} is defined as the \textit{traceless part} of the Riemann curvature
\begin{equation}
\label{eq_weyl_curvature}
C_{abcd} = R_{abcd} - S_{abcd} - E_{abcd}.
\end{equation}
The Einstein equation is
\begin{equation}
\label{eq_einstein}
G_{ab} + \Lambda g_{ab} = \kappa T_{ab},
\end{equation}
where $T_{ab}$ is the stress-energy tensor of the matter, the constant $\kappa$ is defined as $\kappa:=\dsfrac{8\pi \mc G}{c^4}$, where $\mc G$ and $c$ are the gravitational constant and the speed of light, and $\Lambda$ is the \textit{cosmological constant}.
The term
\begin{equation}
\label{eq_einstein_tensor}
G_{ab}:=R_{ab}-\frac 1 2 R g_{ab}
\end{equation}
is the Einstein tensor, constructed from the \textit{Ricci curvature} $R_{ab} := g^{st}R_{asbt}$ and the \textit{scalar curvature} $R := g^{st}R_{st}$.
As it is understood, the Einstein equation establishes the connection between curvature and stress-energy. The curvature contributes to the equation in the form of the Ricci tensor $R_{ab}$ and the scalar curvature. In the proposed equation, the curvature contributes in the form of the semi-traceless and scalar parts of the Riemann tensor, $E_{abcd}$ \eqref{eq_ricci_part_E} and $S_{abcd}$ \eqref{eq_ricci_part_S}, which are tensors of the same order and have the same symmetries as $R_{abcd}$.
The Ricci tensor $R_{ab}$ is obtained by contracting the tensor $E_{abcd}+S_{abcd}$, and has the same information (if the metric is {non{-\penalty0\hskip0pt\relax}degenerate}). One can move from the fourth-order tensors $E_{abcd}+S_{abcd}$ to $R_{ab}$ by contraction, and one can move back to them by taking the Kulkarni-Nomizu product \eqref{eq_kulkarni_nomizu}, but they are equivalent. Yet, if the metric $g_{ab}$ is degenerate, then $g^{ab}$ and the contraction $R_{ab}=g^{st}(E_{asbt}+S_{asbt})$ become divergent, even if $g_{ab}$, $E_{abcd}$, and $S_{abcd}$ are smooth. This suggests the possibility that $E_{abcd}$ and $S_{abcd}$ are more fundamental that the Ricci and scalar curvatures.
This suggestion is in agreement with the following observation. In the case of \textit{electrovac} solutions, where $F_{ab}$ is the electromagnetic tensor,
\begin{equation}
\label{eq_stress_energy_maxwell}
T_{ab}=\frac{1}{4\pi}\(\frac 1 4 g_{ab} F_{st}F^{st} - F_{as} F_b{}^s\)=-\frac{1}{8\pi}\(F_{ac}F_b{}^c + {}^\ast F_{ac} {}^\ast F_b{}^c\),
\end{equation}
where ${}^\ast$ is the Hodge duality operation. It can be obtained by contracting the semi-traceless part of the Riemann tensor
\begin{equation}
\label{eq_stress_energy_maxwell_expanded}
E_{abcd}=-\frac{\kappa}{8\pi}\(F_{ab}F_{cd} + {}^\ast F_{ab} {}^\ast F_{cd}\).
\end{equation}
Therefore it is natural to at least consider an equation in terms of these fourth-order tensors, rather than the Ricci and scalar curvatures.
The main advantage of this method is that there are singularities in which the new formulation of the Einstein equation is not singular (although the original Einstein equation exhibits singularities, obtained when contracting with the singular tensor $g^{ab}$). The expanded Einstein equation is written in terms of the smooth geometric objects $E_{abcd}$ and $S_{abcd}$. Because of this the solutions can be extended at singularities where the original Einstein equation diverges. This doesn't mean that the singularities are removed; for example the Kretschmann scalar $R_{abcd}R^{abcd}$ is still divergent at some of these singularities. But this is not a problem, since the Kretschmann scalar is not part of the evolution equation. It is normally used as an indicator that there is a singularity, for example to prove that the {Schwarzschild} singularity at $r=0$ cannot be removed by coordinate changes, as the event horizon singularity can. While a singularity of the Kretschmann scalar indicates the presence of a singularity of the curvature, it doesn't have implications on whether the singularity can be resolved or not. In the proposed equation we use $R_{abcd}$ which is smooth at the studied singularities, and we don't use $R^{abcd}$ which is singular and causes the singularity of the Kretschmann scalar.
A second reason to consider the expanded version of the Einstein equation and the {quasi{-\penalty0\hskip0pt\relax}regular} singularities at which it is smooth is that at these singularities the Weyl curvature tensor vanishes. The implications of this feature will be explored in \cite{Sto12c}.
It will be seen that there are some important examples of singularities which turn out to be {quasi{-\penalty0\hskip0pt\relax}regular}. While singularities still exist, our approach provides a description in terms of smooth geometric objects which remain finite at singularities. By this we hope to improve our understanding of singularities and to distinguish those to which our resolution applies.
The \textit{expanded Einstein equations} and the {quasi{-\penalty0\hskip0pt\relax}regular} spacetimes on which they hold are introduced in section \sref{s_einstein_exp_qreg}. They are obtained by taking the Kulkarni-Nomizu product between Einstein's equation and the metric tensor. In a {quasi{-\penalty0\hskip0pt\relax}regular} spacetime the metric tensor becomes degenerate at singularities in a way which cancels them and makes the equations smooth.
The situations when the new version of Einstein's equation extends at singularities include isotropic singularities (section \sref{s_qreg_examples_isotropic}) and a class of warped product singularities (section \sref{s_qreg_examples_warped}). It also contains the {Schwarzschild} singularity (section \sref{s_qreg_examples_schw}) and the {FLRW} Big Bang singularity (section \sref{s_qreg_examples_flrw}).
\section{Expanded Einstein equation and {quasi{-\penalty0\hskip0pt\relax}regular} spacetimes}
\label{s_einstein_exp_qreg}
\subsection{The expanded Einstein equation}
\label{s_einstein_exp}
An equation which is equivalent to Einstein's equation whenever the metric tensor $g_{ab}$ is {non{-\penalty0\hskip0pt\relax}degenerate}, but is valid also in a class of situations when $g_{ab}$ becomes degenerate and Einstein's tensor is not defined will be discussed in this section. Later it will be shown that the proposed version of Einstein's equation remains smooth in various important situations such as the FLRW Big-Bang singularity, isotropic singularities, and at the singularity of the {Schwarzschild} black hole.
We introduce the \textit{expanded Einstein equation}
\begin{equation}
\label{eq_einstein_expanded}
(G\circ g)_{abcd} + \Lambda (g\circ g)_{abcd} = \kappa (T\circ g)_{abcd}.
\end{equation}
If the metric is {non{-\penalty0\hskip0pt\relax}degenerate} then the Einstein equation and its expanded version are equivalent. This can be seen by contracting the expanded Einstein equation, for instance in the indices $b$ and $d$. From \eqref{eq_kulkarni_nomizu} the contraction in $b$ and $d$ of a Kulkarni-Nomizu product $(h\circ g)_{abcd}$ is
\begin{equation}
\hat h_{ac}:=(h\circ g)_{asct}g^{st} = h_{ac}g^s_s - h_{at}\delta^t_c + h^s_sg_{ac} - h_{sc}\delta^s_a = 2h_{ac} + h^s_sg_{ac}.
\end{equation}
From $\hat h_{ac}$ the original tensor $h_{ac}$ can be obtained again by
\begin{equation}
\label{eq_expanded_to_standard}
h_{ac}=\frac 1 2 \hat h_{ac} - \frac 1{12}\hat h{}^s_s g_{ac}.
\end{equation}
By this procedure the terms $G_{ab}$, $T_{ab}$, and $\Lambda g_{ab}$ can be recovered from the equation \eqref{eq_einstein_expanded}, thus obtaining the Einstein equation \eqref{eq_einstein}
. Hence, the Einstein equation and its expanded version are equivalent for a {non{-\penalty0\hskip0pt\relax}degenerate} metric.
If the metric becomes degenerate its inverse becomes singular, and in general the Riemann, Ricci, and scalar curvatures, and consequently the Einstein tensor $G_{ab}$, diverge. For certain cases the metric term from the Kulkarni-Nomizu product $G\circ g$ tends to $0$ fast enough to cancel the divergence of the Einstein tensor. The {quasi{-\penalty0\hskip0pt\relax}regular} singularities satisfy the condition that the divergence of $G$ is compensated by the degeneracy of the metric, so that $G\circ g$ is smooth.
This cancellation allows us to weaken the condition that the metric tensor is {non{-\penalty0\hskip0pt\relax}degenerate}, to some cases when it can be degenerate. It will be seen that these cases include some important singularities.
\subsection{A more explicit form of the expanded Einstein equation}
\label{s_einstein_exp_explicit}
To give a more explicit form of the expanded Einstein equation, the \textit{Ricci decomposition} of the Riemann curvature tensor is used (see \textit{e.g.} \cite{ST69,BESS87,GHLF04}).
By using the equations \eqref{eq_einstein_tensor}
and \eqref{eq_ricci_traceless} in dimension $n=4$, the Einstein tensor in terms of the traceless part of the Ricci tensor and the scalar curvature can be written:
\begin{equation}
G_{ab} = S_{ab} - \dsfrac{1}{4}R g_{ab}.
\end{equation}
This equation can be used to calculate the \textit{expanded Einstein tensor}:
\begin{equation}
\label{eq_einstein_tensor_expanded}
\begin{array}{lrl}
G_{abcd} &:=& (G\circ g)_{abcd} \\
&=& (S \circ g)_{abcd} - \dsfrac{1}{4}R (g\circ g)_{abcd}\\
&=& 2 E_{abcd} - 6 S_{abcd}.
\end{array}
\end{equation}
The expanded Einstein equation now takes the form
\begin{equation}
\label{eq_einstein_expanded_explicit}
2 E_{abcd} - 6 S_{abcd} + \Lambda (g\circ g)_{abcd} = \kappa (T\circ g)_{abcd}.
\end{equation}
\subsection{Quasi-regular spacetimes}
\label{s_qreg_spacetimes}
We are interested in singular spacetimes on which the expanded Einstein equation \eqref{eq_einstein_expanded} can be written and is smooth. From \eqref{eq_einstein_expanded_explicit} it can be seen that this requires the smoothness of the tensors $E_{abcd}$ and $S_{abcd}$.
In addition we are interested to have the nice properties of the {semi{-\penalty0\hskip0pt\relax}regular} spacetimes.
As showed in \cite{Sto11a}, the {semi{-\penalty0\hskip0pt\relax}regular} manifolds are a class of singular {semi{-\penalty0\hskip0pt\relax}Riemannian} manifolds which are nice for several reasons, one of them being that the Riemann tensor $R_{abcd}$ is smooth.
First, a contraction between covariant indices is needed. This is in general prohibited by the fact that when the metric tensor $g_{ab}$ becomes degenerate it doesn't admit a reciprocal $g^{ab}$.
Although the metric $g_{ab}$ can't induce an invariant inner product on the cotangent space $T_p^*M$, it induces one on its subspace $\flat(T_pM)$, where $\flat:T_pM\to T_p^*M$ is the vector space morphism defined by $X^\flat(Y):=\metric{X,Y}$, for any $X,Y\in T_pM$. Equivalently, $\flat(T_pM)$ is the space of $1$-forms $\omega$ on $T_pM$ so that $\omega|_{\ker\flat}=0$. The morphism $\flat$ is isomorphism if and only if $g$ is {non{-\penalty0\hskip0pt\relax}degenerate}; in this case its inverse is denoted by $\sharp$.
The inner product on $\flat(T_pM)$ is then defined by ${g}_{\bullet}(X^\flat,Y^\flat):=\metric{X,Y}$ and it is invariant.
This allows us to define a contraction between covariant slots of a tensor $T$, which vanishes when vectors from $\ker\flat$ are plugged in those slots.
This will turn out to be enough for our needs.
We denote the contractions between covariant indices of a tensor $T$ by $T(\omega_1,\ldots,\omega_r,v_1,\ldots,{{}_\bullet},\ldots,{{}_\bullet},\ldots,v_s)$.
A degenerate metric also prohibits in general the construction of a Levi-Civita connection. For vector fields we use instead of $\nabla_XY$, the \textit{Koszul form}, defined as:
\begin{equation*}
\mc K:\fivect M^3\to\mathbb{R},
\end{equation*}
\begin{equation}
\label{eq_Koszul_form}
\mc K(X,Y,Z) :=\displaystyle{\frac 1 2} \{ X \metric{Y,Z} + Y \metric{Z,X} - Z \metric{X,Y}
- \metric{X,[Y,Z]} + \metric{Y, [Z,X]} + \metric{Z, [X,Y]}\}
\end{equation}
which defines the Levi-Civita connection by $\nabla_XY=\mc K(X,Y,\_)^\sharp$ for a {non{-\penalty0\hskip0pt\relax}degenerate} metric, but not when the metric becomes degenerate.
We define now {semi{-\penalty0\hskip0pt\relax}regular} manifolds, on which we can define covariant derivatives for a large class of differential forms and tensors. We can also define a generalization of the Riemann curvature $R_{abcd}$, which turns out to be smooth and non-singular.
\begin{definition}
\label{def_semi_regular}
A singular {semi{-\penalty0\hskip0pt\relax}Riemannian} manifold satisfying the condition that $\mc K(X,Y,\_)\in\flat(T_pM)$, and that the contraction $\mc K(X,Y,{{}_\bullet})\mc K(Z,T,{{}_\bullet})$ is smooth for any local vector fields $X,Y,Z,T$, is named \textit{{semi{-\penalty0\hskip0pt\relax}regular} manifold}, and its metric is called \textit{{semi{-\penalty0\hskip0pt\relax}regular} metric}.
A $4$-dimensional {semi{-\penalty0\hskip0pt\relax}regular} manifold with metric having the signature at each point $(r,s,t)$, $s\leq 3$, $t\leq 1$, but which is {non{-\penalty0\hskip0pt\relax}degenerate} on a dense subset, is called \textit{{semi{-\penalty0\hskip0pt\relax}regular} spacetime} \cite{Sto11a}.
\end{definition}
In \cite{Sto11a} we defined the Riemann curvature $R_{abcd}$ for {semi{-\penalty0\hskip0pt\relax}regular} metrics, even for {non{-\penalty0\hskip0pt\relax}degenerate} metrics, in a way which avoids the undefined $\nabla_XY$, but relies on the defined and smooth $\mc K(X,Y,Z)$, by
\begin{equation}
\label{eq_riemann_curvature_tensor_coord}
R_{abcd}= \partial_a \Gamma_{bcd} - \partial_b \Gamma_{acd} + \Gamma_{ac{{}_\bullet}}\Gamma_{bd{{}_\bullet}} - \Gamma_{bc{{}_\bullet}}\Gamma_{ad{{}_\bullet}},
\end{equation}
where $\Gamma_{abc}=\mc K(\partial_a,\partial_b,\partial_c)$ are the Christoffel's symbols of the first kind. From Definition \ref{def_semi_regular}, $R_{abcd}$ is smooth.
More details on the {semi{-\penalty0\hskip0pt\relax}regular} manifolds can be found in \cite{Sto11a,Sto11b,Sto12e}.
In a {semi{-\penalty0\hskip0pt\relax}regular} spacetime, since $R_{abcd}$ is smooth, the densitized Einstein tensor $G_{ab}\det g$ is smooth \cite{Sto11a}, and a densitized version of the Einstein equation can be written, which is equivalent to the usual version when the metric is {non{-\penalty0\hskip0pt\relax}degenerate}:
\begin{equation}
\label{eq_einstein_idx:densitized}
G_{ab}\sqrt{-g}^W + \Lambda g_{ab}\sqrt{-g}^W = \kappa T_{ab}\sqrt{-g}^W,
\end{equation}
where it is enough to take the weight $W\leq 2$.
Although the {semi{-\penalty0\hskip0pt\relax}regular} approach is more general, here is explored the {quasi{-\penalty0\hskip0pt\relax}regular} one, which is more strict. Consequently, these results are stronger.
\begin{definition}
\label{def_quasi_regular}
We say that a {semi{-\penalty0\hskip0pt\relax}regular} manifold $(M,g_{ab})$ is \textit{{quasi{-\penalty0\hskip0pt\relax}regular}}, and that $g_{ab}$ is a \textit{{quasi{-\penalty0\hskip0pt\relax}regular} metric}, if:
\begin{enumerate}
\item
$g_{ab}$ is {non{-\penalty0\hskip0pt\relax}degenerate} on a subset dense in $M$
\item
the tensors $S_{abcd}$ and $E_{abcd}$ defined at the points where the metric is {non{-\penalty0\hskip0pt\relax}degenerate} extend smoothly to the entire manifold $M$.
\end{enumerate}
If the {quasi{-\penalty0\hskip0pt\relax}regular} manifold $M$ is a {semi{-\penalty0\hskip0pt\relax}regular} spacetime, we call it \textit{{quasi{-\penalty0\hskip0pt\relax}regular} spacetime}. Singularities of {quasi{-\penalty0\hskip0pt\relax}regular} manifolds are called {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{definition}
It can be seen that on an {quasi{-\penalty0\hskip0pt\relax}regular} spacetime the expanded Einstein tensor can be extended at the points where the metric is degenerate, and the extension is smooth. This is in fact the motivation of Definition \ref{def_quasi_regular}.
\section{Examples of {quasi{-\penalty0\hskip0pt\relax}regular} spacetimes}
\label{s_qreg_examples}
The {quasi{-\penalty0\hskip0pt\relax}regular} spacetimes are more general than the regular ones (those with {non{-\penalty0\hskip0pt\relax}degenerate} metric), containing them as a particular case. The question is, are they general enough to cover the singularities which plagued General Relativity? In the following it will be seen that at least for some relevant cases the answer is positive. It will be seen that the class of {quasi{-\penalty0\hskip0pt\relax}regular} singularities contain isotropic singularities \sref{s_qreg_examples_isotropic}, singularities obtained as warped products \sref{s_qreg_examples_warped} (including the {Friedmann-Lema\^itre-Robertson-Walker} spacetime \sref{s_qreg_examples_flrw}), and even the {Schwarzschild} singularity \sref{s_qreg_examples_schw}. The existence of these examples which are extensively researched justifies the study of the more general {quasi{-\penalty0\hskip0pt\relax}regular} singularities and of the extended Einstein equations.
\subsection{Isotropic singularities}
\label{s_qreg_examples_isotropic}
\textit{Isotropic singularities} occur in conformal rescalings of {non{-\penalty0\hskip0pt\relax}degenerate} metrics, when the scaling function cancels. They were extensively studied by Tod \cite{Tod87,Tod90,Tod91,Tod92,Tod02,Tod03}, Claudel \& Newman \cite{CN98}, Anguige \& Tod \cite{AT99i,AT99ii}, in connection with cosmological models. The following theorem shows that the isotropic singularities are {quasi{-\penalty0\hskip0pt\relax}regular}.
\begin{theorem}[Isotropic singularities]
\label{thm_quasireg_example_conformal}
Let $(M,g_{ab})$ be a regular spacetime (we assume therefore that the metric $g_{ab}$ is {non{-\penalty0\hskip0pt\relax}degenerate}). Then, if $\Omega:M\to\mathbb{R}$ is a smooth function which is non-zero on a dense subset of $M$, the spacetime $(M,\widetilde g_{ab} :=\Omega^2 g_{ab})$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{theorem}
\begin{proof}
From \cite{Sto11a} is known that $(M,\widetilde g_{ab})$ is {semi{-\penalty0\hskip0pt\relax}regular}.
The Ricci and the scalar curvatures take the following forms (\cite{HE95}, p. 42.):
\begin{equation}
\label{eq_conformal_ricci_curv_ud}
\widetilde R^a{}_b = \Omega^{-2}R^a{}_b + 2\Omega^{-1}(\Omega^{-1})_{;bs}g^{as}-\dsfrac 1 2\Omega^{-4}(\Omega^2)_{;st}g^{st}\delta^a{}_b
\end{equation}
\begin{equation}
\label{eq_conformal_scalar_curv}
\widetilde R=\Omega^{-2}R-6\Omega^{-3}\Omega_{;st}g^{st}
\end{equation}
where the covariant derivatives correspond to the metric $g$.
From equation \eqref{eq_conformal_ricci_curv_ud} follows that
\begin{equation}
\widetilde R_{ab}=\Omega^2 g_{as} \widetilde R^s{}_b=R_{ab} + 2\Omega(\Omega^{-1})_{;ab}-\dsfrac 1 2\Omega^{-2}(\Omega^2)_{;st}g^{st}g_{ab},
\end{equation}
which tends to infinity when $\Omega\to 0$. But we are interested to prove the smoothness of the Kulkarni-Nomizu product $\widetilde\textnormal{Ric}\circ \widetilde g$. We notice that the term $\widetilde g$ contributes with a factor $\Omega^2$, and it is enough to prove the smoothness of
\begin{equation}
\Omega^2\widetilde R_{ab}=\Omega^2 R_{ab} + 2\Omega^3(\Omega^{-1})_{;ab}-\dsfrac 1 2(\Omega^2)_{;st}g^{st}g_{ab},
\end{equation}
which follows from
\begin{equation}
\begin{array}{lll}
\Omega^3(\Omega^{-1})_{;ab} &=& \Omega^3\((\Omega^{-1})_{;a}\)_{;b} = \Omega^3\(-\Omega^{-2}\Omega_{;a}\)_{;b} \\
&=& \Omega^3\(2\Omega^{-3}\Omega_{;b}\Omega_{;a} - \Omega^{-2}\Omega_{;ab}\) \\
&=& 2\Omega_{;a}\Omega_{;b} - \Omega\Omega_{;ab} \\
\end{array}
\end{equation}
Hence, the tensor $\widetilde\textnormal{Ric}\circ \widetilde g$ is smooth. The fact that $\widetilde R \widetilde g\circ \widetilde g$ is smooth follows from the observation that $\widetilde g\circ \widetilde g$ contributes with $\Omega^4$, and the least power in which $\Omega$ appears in the expression \eqref{eq_conformal_scalar_curv} of $\widetilde R$ is $-3$.
From the above follows that $\widetilde E_{abcd}$ and $\widetilde S_{abcd}$ are smooth. Hence the spacetime $(M,\widetilde g_{ab})$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{proof}
\subsection{{Quasi{-\penalty0\hskip0pt\relax}regular} warped products}
\label{s_qreg_examples_warped}
Another example useful in cosmology is the following, which is a generalization of the warped products. Warped products are extensively researched, since they allow the construction of {semi{-\penalty0\hskip0pt\relax}Riemannian} spacetimes, having applications to GR. But when the warping function becomes $0$, singularities occur (see \textit{e.g.} \citep{ONe83}{ 204}). Fortunately, in the cases of interest for General Relativity, these singularities are {quasi{-\penalty0\hskip0pt\relax}regular}. We will allow the warped function $f$ to become $0$ (generalizing the standard definition \cite{ONe83}, where it is not allowed to vanish because it leads to degenerate metrics), and prove that what the resulting singularities are {quasi{-\penalty0\hskip0pt\relax}regular}.
\begin{definition}
\label{def_wp}
Let $(B,\textnormal{d} s_B^2)$ and $(F,\textnormal{d} s_F^2)$ be two {semi{-\penalty0\hskip0pt\relax}Riemannian} manifolds, and $f: B\to\mathbb{R}$ a smooth function on $B$. The \textit{degenerate warped product} of $B$ and $F$ with \textit{warping function} $f$ is the manifold $B\times_f F:=\(B\times F,\textnormal{d} s_{B\times F}^2\)$, with the metric
\begin{equation}
\textnormal{d} s_{B\times F}^2 = \textnormal{d} s_B^2 + f^2\textnormal{d} s_F^2
\end{equation}
\end{definition}
\begin{theorem}[{Quasi{-\penalty0\hskip0pt\relax}regular} warped product]
\label{thm_quasireg_example_wp}
A degenerate warped product $B\times_f F$ with $\dim B=1$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{theorem}
\begin{proof}
From \cite{Sto11b}, $B\times_f F$ is {semi{-\penalty0\hskip0pt\relax}regular}.
Let's denote by $g_B$, $g_F$ and $g$ the metrics on $B$, $F$ and $B\times_f F$.
It is known (\cite{ONe83}, p. 211) that for horizontal vector fields $X,Y\in\fivectlift{B \times F,B}$ and vertical vector fields $V,W\in\fivectlift{B \times F,F}$,
\begin{enumerate}
\item $\tn{Ric}(X,Y) = \tn{Ric}_B(X,Y) + \dsfrac{\dim F}{f}H^f(X,Y)$
\item $\tn{Ric}(X,V) = 0$
\item $\tn{Ric}(V,W) = \tn{Ric}_F(V,W) + \(f\Delta f + (\dim F-1)g_B(\textnormal{grad } f,\textnormal{grad } f)\)g_F(V,W)$
\end{enumerate}
where $\Delta f$ is the Laplacian, $H^f$ the Hessian, and $\textnormal{grad } f$ the gradient.
It follows that $\tn{Ric}(X,V)$ and $\tn{Ric}(V,W)$ are smooth, but $\tn{Ric}(X,Y)$ in general is not, because of the term containing $f^{-1}$. But since $\dim B=1$, the only terms in the Kulkarni-Nomizu product $\textnormal{Ric}\circ g$ containing $\textnormal{Ric}(X,Y)$ are of the form
\begin{equation*}
\textnormal{Ric}(X,Y)g(V,W)=f^2\textnormal{Ric}(X,Y)g_F(V,W).
\end{equation*}
Hence, $\textnormal{Ric}\circ g$ is smooth.
From the expression of the scalar curvature
\begin{equation}
\label{eq_scalar_curv_wp}
R = R_B + \frac {R_F}{f^2} + 2\dim F\dsfrac{\Delta f}{f} + \dim F(\dim F - 1)\dsfrac{g_B(\textnormal{grad } f,\textnormal{grad } f)}{f^2}
\end{equation}
can be concluded that $S_{abcd}$ is smooth too, because $g\circ g$ contains at least one factor of $f^2$. Hence, $B\times_f F$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{proof}
The following example important in cosmology is a direct application of this result.
\begin{proposition}[{Semi{-\penalty0\hskip0pt\relax}regular} manifold which is not {quasi{-\penalty0\hskip0pt\relax}regular}]
Let $B=\mathbb{R}^k$, $k>1$, be an Euclidean space, with the canonical metric $g_B$, and $f:B\to\mathbb{R}$ a linear function $f\neq 0$. Let $F=\mathbb{R}^l$, $l>1$, with the canonical metric $g_F$.
Then the degenerate warped product $B\times_f F$ is {semi{-\penalty0\hskip0pt\relax}regular}, but it isn't {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{proposition}
\begin{proof}
Because $f$ is linear but not constant, $\textnormal{grad } f\neq 0$ is constant, and $\Delta f=0$.
The scalar curvature \eqref{eq_scalar_curv_wp} becomes $R=l(l - 1)\dsfrac{g_B(\textnormal{grad } f,\textnormal{grad } f)}{f^2}$, which is singular at $0$. Because $k>1$, $g_B\circ g_B$ doesn't vanish, hence it doesn't cancel the denominator $f^2$ of $R$ in the term $R g_B\circ g_B$. Also, the term $R g_B\circ g_B$ is not canceled by other terms composing $S_{abcd}$, because they are all smooth, containing at least one $g_F$. Hence, $S_{abcd}$ is singular, and the degenerate warped product $B\times_f F$ isn't {quasi{-\penalty0\hskip0pt\relax}regular}. On the other hand, according to \cite{Sto11b}, because $B$ and $F$ are {non{-\penalty0\hskip0pt\relax}degenerate}, $B\times_f F$ is {semi{-\penalty0\hskip0pt\relax}regular}.
\end{proof}
\subsection{The {Friedmann-Lema\^itre-Robertson-Walker} spacetime}
\label{s_qreg_examples_flrw}
The {Friedmann-Lema\^itre-Robertson-Walker} ({FLRW}) spacetime is defined as the warped product $I\times_a \Sigma$, where
\begin{enumerate}
\item
$I\subseteq \mathbb{R}$ is an interval representing the time, which is viewed as a {semi{-\penalty0\hskip0pt\relax}Riemannian} space with the negative definite metric $-c^2\textnormal{d} t^2$.
\item
$(\Sigma,\textnormal{d}\Sigma^2)$ is a three-dimensional Riemannian space, usually one of the homogeneous spaces $S^3$, $\mathbb{R}^3$, and $H^3$ (to model the homogeneity and isotropy conditions at large scale). Then the metric on $\Sigma$ is, in spherical coordinates $(r,\theta,\phi)$,
\begin{equation}
\label{eq_flrw_sigma_metric}
\textnormal{d}\Sigma^2 = \dsfrac{\textnormal{d} r^2}{1-k r^2} + r^2\(\textnormal{d}\theta^2 + \sin^2\theta\textnormal{d}\phi^2\),
\end{equation}
where $k=1,0,-1$, for the $3$-sphere $S^3$, the Euclidean space $\mathbb{R}^3$, or hyperbolic space $H^3$ respectively.
\item
$a: I\to \mathbb{R}$ is a function of time.
\end{enumerate}
The {FLRW} metric is
\begin{equation}
\label{eq_flrw_metric}
\textnormal{d} s^2 = -c^2\textnormal{d} t^2 + a^2(t)\textnormal{d}\Sigma^2.
\end{equation}
At any moment of time $t\in I$ the space is $\Sigma_t=(\Sigma,a^2(t)g_\Sigma)$.
For a {FLRW} universe filled with a fluid with mass density $\rho(t)$ and pressure density $p(t)$, the stress-energy tensor is defined as
\begin{equation}
\label{eq_friedmann_stress_energy}
T^{ab} = \(\rho + \dsfrac{p}{c^2}\)u^a u^b + p g^{ab},
\end{equation}
where $g(u,u)=-c^2$.
From Einstein's equation with the stress-energy tensor \eqref{eq_friedmann_stress_energy} follow the \textit{Friedmann equation}
\begin{equation}
\label{eq_friedmann_density}
\rho = \kappa^{-1}\(3\dsfrac{\dot{a}^2 + kc^2}{c^2 a^2} - \Lambda \),
\end{equation}
which gives the mass density $\rho(t)$ in terms of $a(t)$, and the \textit{acceleration equation}
\begin{equation}
\label{eq_acceleration}
\dsfrac{p}{c^2} = \dsfrac{2}{\kappa c^2}\(\dsfrac{\Lambda}{3}-\dsfrac{1}{c^2} \dsfrac{\ddot{a}}{a}\) - \dsfrac \rho 3,
\end{equation}
giving the pressure density $p(t)$.
A question that may arise is what happens with the densities $\rho$ and $p$. Equations \eqref{eq_friedmann_density} and \eqref{eq_acceleration} show that $\rho$ and $p$ may diverge in most cases for $a\to 0$. As explained in \cite{Sto11h}, $\rho$ and $p$ are calculated considering orthonormal frames. If the frame is not necessarily orthonormal (because there is no orthonormal frame at the point where the metric is degenerate), then the volume element is not necessarily equal to $1$, and it has to be included in the equations. The scalars $\rho$ and $p$ are replaced by the differential $4$-forms which have the components $\rho\sqrt{-g}$ and $p\sqrt{-g}$. It can be seen by calculation that these forms are smooth.
If the metric on the manifold $\Sigma$ is denoted by $g_{\Sigma}$, then the Friedmann equation \eqref{eq_friedmann_density} becomes
\begin{equation}
\label{eq_friedmann_density_tilde}
\rho\sqrt{-g} = \dsfrac{3}{\kappa}a\(\dot a^2 + k\) \sqrt{g_{\Sigma}},
\end{equation}
and the acceleration equation \eqref{eq_acceleration} becomes
\begin{equation}
\label{eq_acceleration_tilde}
\rho\sqrt{-g} + 3p\sqrt{-g} = -\dsfrac{6}{\kappa}a^2\ddot{a} \sqrt{g_{\Sigma}},
\end{equation}
hence $\rho\sqrt{-g}$ and $p\sqrt{-g}$ are smooth.
As $a\to 0$, the metric becomes degenerate, $\rho$ and $p$ diverge, and therefore the stress-energy tensor \eqref{eq_friedmann_stress_energy} diverges too. Because of this, the Ricci tensor also diverges. But, from Theorem \ref{thm_quasireg_example_wp}, $R_{abcd}$, $E_{abcd}$, and $S_{abcd}$ are smooth. What can be said about the expanded stress-energy tensor $(T \circ g)_{abcd}$? The following corollary shows that the metric is {quasi{-\penalty0\hskip0pt\relax}regular}, hence the expanded stress-energy tensor is smooth.
\begin{corollary}
\label{thm_flrw}
The {FLRW} spacetime with smooth $a: I\to \mathbb{R}$ is {quasi{-\penalty0\hskip0pt\relax}regular}.
\end{corollary}
\begin{proof}
Since the {FLRW} spacetime is a warped product between a $1$-dimensional and a $3$-dimensional manifold with warping function $a$, this is a direct consequence of Theorem \ref{thm_quasireg_example_wp}.
\end{proof}
\begin{remark}
Corollary \ref{thm_flrw} applies not only to a {FLRW} universe filled with a fluid, but to more general ones. For this particular case a direct proof was given in \cite{Sto12a}, showing explicitly how the expected infinities of the physical fields cancel out.
\end{remark}
While the expanded Einstein equation for the {FLRW} spacetime with smooth $a$ is written in terms of smooth objects like $E_{abcd}$, $S_{abcd}$, and $T_{abcd}:=(T \circ g)_{abcd}$, a question arises, as to why use these objects, instead of $R_{ab}$, $S$, and $T_{ab}$? It is true that the expanded objects remain smooth, while the standard ones don't, but is there other, more fundamental reason? It can be said that $E_{abcd}$ and $S_{abcd}$ are more fundamental, since $R_{ab}$ and $R$ are obtained from them by contractions. But for $T_{abcd}$, unfortunately, at this time we don't know an interpretation. The stress-energy tensor $T_{ab}$ can be obtained from a Lagrangian, but we don't know yet a way to obtain directly $T_{abcd}$ from a Lagrangian. One hint that, at least for some fields, $T_{abcd}$ seems more fundamental is that, for electrovac solutions, it is given by $T_{abcd}=-\frac{1}{8\pi}\(F_{ab}F_{cd} + {}^\ast F_{ab} {}^\ast F_{cd}\)$ \eqref{eq_stress_energy_maxwell_expanded}, while $T_{ab}$ by contracting it \eqref{eq_stress_energy_maxwell}. Similar form has the stress-energy tensor for Yang-Mills fields.
Another question that may appear is what is obtained, given that the solution can be extended beyond the moment when $a(t)=0$? Say that $a(0)=0$. The extended solution will describe two universes, both originating from the same Big-Bang at the same moment $t=0$, one of them expanding toward the direction in which $t$ increases and the other one toward the direction in which $t$ decreases. The parameter $t$ is just a coordinate, and the physical laws are symmetric with respect to time reversal in General Relativity (if one wants to consider quantum fields, the combined symmetry $CPT$ should be considered instead of $T$ alone).
\subsection{{Schwarzschild} black hole}
\label{s_qreg_examples_schw}
The {Schwarzschild} solution describing a black hole of mass $m$ is given in the {Schwarzschild} coordinates by the metric tensor:
\begin{equation}
\label{eq_schw_schw}
\textnormal{d} s^2 = -\(1-\dsfrac{2m}{r}\)\textnormal{d} t^2 + \(1-\dsfrac{2m}{r}\)^{-1}\textnormal{d} r^2 + r^2\textnormal{d}\sigma^2,
\end{equation}
where
\begin{equation}
\label{eq_sphere}
\textnormal{d}\sigma^2 = \textnormal{d}\theta^2 + \sin^2\theta \textnormal{d} \phi^2
\end{equation}
is the metric of the unit sphere $S^2$. The units were chosen so that $c=1$ and $G=1$ (see \textit{e.g.} \citep{HE95}{149}).
Apparently the metric is singular at $r=2m$, on the event horizon. As it is known from the work of Eddington \cite{eddington1924comparison} and Finkelstein \cite{finkelstein1958past} appropriate coordinate changes make the metric {non{-\penalty0\hskip0pt\relax}degenerate} on the event horizon, showing that the singularity is apparent, being due to the coordinates. The coordinate change is singular, but it can be said that the proper coordinates around the event horizon are those of Eddington and Finkelstein, and the {Schwarzschild} coordinates are the singular coordinates.
Can we apply a similar method for the singularity at $r=0$? It can be checked that the Kretschmann scalar $R_{abcd}R^{abcd}$ is singular at $r=0$, and since scalars are invariant at any coordinate changes (including the singular ones), it is usually correctly concluded that the singularity at $r=0$ cannot be removed. Although it cannot be removed, it can be improved by finding coordinates making the metric analytic at $r=0$. As shown in \cite{Sto11e} the singularity $r=0$ in the {Schwarzschild} metric \eqref{eq_schw_schw} has two origins -- it is a combination of degenerate metric and singular coordinates. Firstly, the {Schwarzschild} coordinates are singular at $r=0$, but they can be desingularized by applying the coordinate transformations from equation \eqref{eq_coordinate_semireg} which necessarily have the Jacobian equal to zero at $r=0$. It is not possible to desingularize a coordinate system, by using transformations that have non-vanishing Jacobian at the singularity, because such transformations preserve the regularity of the metric. Secondly, after the transformation the singularity is not completely removed, because the metric remains degenerate. However, the metric remains {semi{-\penalty0\hskip0pt\relax}regular}, as shown in \cite{Sto11e}. Here will be shown that it is also {quasi{-\penalty0\hskip0pt\relax}regular}.
In \cite{Sto11e} we showed that the {Schwarzschild} solution can be made analytic at the singularity by a coordinate transformation of the form
\begin{equation}
\label{eq_coordinate_change}
\left\{
\begin{array}{ll}
r &= \tau^S \\
t &= \xi\tau^T \\
\end{array}
\right.
\end{equation}
As it turns out,
\begin{equation}
\label{eq_coordinate_semireg}
\left\{
\begin{array}{ll}
r &= \tau^2 \\
t &= \xi\tau^4 \\
\end{array}
\right.
\end{equation}
is the only choice which makes analytic at the singularity not only the metric, but also the Riemann curvature $R_{abcd}$. In the new coordinates the metric has the form
\begin{equation}
\label{eq_schw_semireg}
\textnormal{d} s^2 = -\dsfrac{4\tau^4}{2m-\tau^2}\textnormal{d} \tau^2 + (2m-\tau^2)\tau^4\(4\xi\textnormal{d}\tau + \tau\textnormal{d}\xi\)^2 + \tau^4\textnormal{d}\sigma^2.
\end{equation}
\begin{corollary}
\label{thm_schw_quasireg}
The {Schwarzschild} spacetime is {quasi{-\penalty0\hskip0pt\relax}regular} (in any atlas compatible with the coordinates \eqref{eq_coordinate_semireg}).
\end{corollary}
\begin{proof}
We know from \cite{Sto11e} that the {Schwarzschild} spacetime is {semi{-\penalty0\hskip0pt\relax}regular}. Since it is also Ricci flat, \textit{i.e.} $R_{ab}=0$, it follows that $S_{ab}=1$ and $R=0$, hence $S_{abcd}= \dsfrac{1}{24}R(g\circ g)_{abcd}=0$, and $E_{abcd}\dsfrac{1}{2}(S \circ g)_{abcd}=0$. Therefore, $S_{abcd}$ and $E_{abcd}$ are smooth. Consequently, the only non-vanishing part of the curvature in the Ricci decomposition \eqref{eq_ricci_decomposition} is the Weyl tensor $C_{abcd}$, which in this case is equal to $R_{abcd}$, so it is smooth too.
\end{proof}
\begin{remark}
It has been seen that even if the {Schwarzschild} metric $g_{ab}$ is singular at $r=0$ there is a coordinate system in which it becomes {quasi{-\penalty0\hskip0pt\relax}regular}. Because the metric becomes {quasi{-\penalty0\hskip0pt\relax}regular} at $r=0$, the expanded Einstein equations are valid at $r=0$ too. But also Einstein's equation can be extended at $r=0$, because in this special case it becomes $G_{ab}=0$, the {Schwarzschild} solution being a vacuum solution.
Hence, in this case we can just use the standard Einstein equations, of course in coordinates compatible with the coordinates \eqref{eq_coordinate_semireg}. Corollary \ref{thm_schw_quasireg} shows that the {Schwarzschild} singularity is {quasi{-\penalty0\hskip0pt\relax}regular} in any such coordinates. Since $S_{abcd}=E_{abcd}=0$, the only non-vanishing part of $R_{abcd}$ is the Weyl curvature $C_{abcd}=R_{abcd}$, which is smooth because $R_{abcd}$ is smooth.
\end{remark}
\begin{remark}
In the limit $m=0$, the {Schwarzschild} solution \eqref{eq_schw_schw} coincides with the Minkowski metric, which is regular at $r=0$. The event horizon singularity $r=2m$ merges with the $r=0$ singularity, and cancel one another. Because the {Schwarzschild} radius becomes $0$, the false singularity $r=0$ is not spacelike as in the case $m> 0$, but timelike.
In the case $m=0$, because there is no singularity at $r=0$, our coordinates \eqref{eq_coordinate_semireg}, rather than removing a (non-existent) singularity, introduce one. The new coordinates provide a double covering for the Minkowski spacetime, because $\tau$ extends beyond $r=0$ to negative values, in a way similar to the case described in \cite{Sto11f}.
\end{remark}
\begin{openproblem}
What can be said about the other stationary black hole solutions?
In \cite{Sto11f} and \cite{Sto11g} we showed that there are coordinate transformations which make the {Reissner-Nordstr\"om} metric and the {Kerr-Newman} metric analytic at the singularity. This is already a big step, because it allows us to foliate with Cauchy hypersurfaces these spacetimes. Is it possible to find coordinate transformations which make them {quasi{-\penalty0\hskip0pt\relax}regular} too?
\end{openproblem}
\section{Conclusions}
An important problem in General Relativity is that of singularities. At singularities some of the quantities involved in the Einstein equation become infinite. But there are other quantities which are also invariant and in addition remain finite at a large class of singularities. In this paper it has been seen that translating the Einstein equation in terms of such quantities allows it to be extended at such singularities.
The Riemann tensor is, from geometric and linear-algebraic viewpoints, more fundamental than the Ricci tensor $R_{ab}$, which is just its trace. This suggests that the scalar part $S_{abcd}$ \eqref{eq_ricci_part_S} and the Ricci part $E_{abcd}$ \eqref{eq_ricci_part_E} of the Riemann curvature may be more fundamental than the Ricci tensor. Consequently, this justifies the study of an equation equivalent to Einstein's, but in terms of $E_{abcd}$ and $S_{abcd}$, instead of $R_{ab}$ and $R$. This is the expanded Einstein equation \eqref{eq_einstein_expanded}. The idea that $E_{abcd}$ is more fundamental than $R_{ab}$ seems to be suggested also by the electrovac solution, with the expanded Einstein equation \eqref{eq_stress_energy_maxwell_expanded}, and from which the electrovac Einstein equation is obtained by contraction.
To go from Einstein's equation to its expanded version we use the Kulkarni-Nomizu product \eqref{eq_kulkarni_nomizu}. To go back, we use contraction \eqref{eq_expanded_to_standard}. When the metric is {non{-\penalty0\hskip0pt\relax}degenerate}, these operations establish an equivalence between the standard and the expanded Einstein equations.
The question of whether the Ricci part of the Riemann tensor is more fundamental than the Ricci tensor may be irrelevant, or the answer may be debatable. But an important feature is that $E_{abcd}$ and $S_{abcd}$ can be defined in more general situations than $R_{ab}$ and $R$. Hence, the expanded Einstein equation is more general than the Einstein equation -- it makes sense even when the metric is degenerate, at least for a class of singularities named {quasi{-\penalty0\hskip0pt\relax}regular}.
A brief investigation revealed that the class of {quasi{-\penalty0\hskip0pt\relax}regular} singularities is rich enough to contain some known singularities, which were already considered by researchers, but now can be understood in a unified framework. Among these there are the isotropic singularities, which are obtained by multiplying a regular metric with a scaling factor which is allowed to vanish. Another class is given by the {Friedmann-Lema\^itre-Robertson-Walker} singularities \cite{Sto12a}, and other warped product singularities. Even the {Schwarzschild} singularity (in proper coordinates which make the metric analytic \cite{Sto11e}) turns out to be quasi{-\penalty0\hskip0pt\relax}regular.
The fact that these apparently unrelated types of singularities turn out to be {quasi{-\penalty0\hskip0pt\relax}regular} suggests the following open question:
\begin{openproblem}
Are {quasi{-\penalty0\hskip0pt\relax}regular} singularities general enough to cover all possible singularities of General Relativity?
\end{openproblem}
\subsection*{Acknowledgments}
The author thanks the anonymous referees for the valuable comments and suggestions to improve the clarity and the quality of this paper.
|
{
"timestamp": "2014-01-27T02:05:38",
"yymm": "1203",
"arxiv_id": "1203.2140",
"language": "en",
"url": "https://arxiv.org/abs/1203.2140"
}
|
\section{Introduction}
It is generally believed that inflation can be a solution to the
problems of standard cosmology such as the horizon, flatness and
monopole problem. In addition to these achievements, inflation's
predictions are compatible with the large scale structure and CMB
fluctuations which is strong evidence in favour of inflation. The
idea of inflation is the existence of an exponentially expanding
universe at early times. But identifying a unique theoretical
realization of this period is challenging. Many theoretical models
are compatible with the observational data. For example, they are in
agreement with adiabatic, nearly Gaussian fluctuations in the CMB
fluctuations. To potentially discriminate between them more accurate
observations, such as PLANCK, are needed. This fact is a starting
point for a huge amount of work on studying non-Gaussianity of
primordial fluctuations. In this field the effective field theory
approach to inflation has been used to study the general possible
interaction terms in the single field models
\cite{paolo,weinberg,eftsingle} and in the multi-field context
\cite{senatore,eftmulti}.
The advantage of using effective theories can be seen in two
regimes. Sometimes a full theory exists for an energy domain of
interest. In this case the effective field theory may be performed
to simplify calculations in a special sub-domain of energy. In the
second case the full theory is not known for the energy scales of
interest. Here, by imposing the symmetries of the full theory one
can still build an effective field theory. In this situation the
most general form of the allowed theory, e.g. a general Lagrangian,
is constructed; by comparison to observations unspecified
coefficients can be fixed. Eventually the deduced effective theory
may shed some lights on the real theory which is beyond our current
understanding. In the special case of inflation in addition to the
above reasons, the effective field theory approach can be used to
justify the use of scalar fields as inflatons, as well as to provide
a systematic classification of non-Gaussianities \cite{baumann}
among other properties.
In \cite{paolo} the effective field theory has been developed for an
inflationary single field model. In their approach the Lagrangian is
determined by all spatially diffeomorphism invariant operators. Then
the broken time invariance is reproduced by a scalar field which
transforms in a definite form under diffeomorphism transformation.
This scalar field is well-known as the Goldstone boson. It is shown
that this scalar field represents the curvature perturbation in the
validity regime of the effective field theory. In \cite{senatore}
the generalization to multi-field inflation is studied. The
existence of more than one field in the early universe is not
unnatural and the extra fields may have observable consequences. For
example entropy modes (a property of multi-field models) can affect
the curvature mode which is for example in the CMB. In
\cite{weinberg} an alternative approach to \cite{paolo} has been
given for the effective field theory of single field inflation. In
this approach all the possible terms containing up to the fourth
derivative of a scalar field and the metric enter the effective
Lagrangian. The final result is in agreement with \cite{paolo},
except some additional fourth ordered contributions tracing back to
geometrical terms. In this approach, due to the presence of metric
perturbations, it is possible to study the gravitational wave
behavior which differs in the propagation of waves with different
helicities.
In this work we are going to generalize Weinberg's approach
\cite{weinberg} to multi-field models. In the following we will
avoid the scalar metric perturbations by an appropriate gauge
choice. However it is mentioned that for energy scales of interest
the existence of them has no observable effects on non-Gaussianity
\cite{baumann,paolo1}. Note that in \cite{weinberg,paolo,senatore},
the additional correction terms arise via space-time derivatives of
the perturbations. However one can extend the effective field theory
formalism to include correction terms corresponding to the potential
terms. It is mentioned in \cite{baumann-green} that they have no
significant contribution to non-Gaussianities since they are highly
restricted by the effectiveness of the inflationary era. But it is
well-known that in the context of multi-field inflation the
non-Gaussianity window becomes wider and maybe observable by the
future data \cite{gpmulti}. This fact also has been considered in
the context of effective field theory for multi-field inflation in
different aspects \cite{eftmulti}.
In the next section we briefly review the main results of
\cite{weinberg}. In the third section we generalize the idea of
\cite{weinberg} to illustrate the perturbations in the most general
multi-field model. This section is based on the first appendix where
we find the most Lagrangian for multi-field models. Then in the
fourth section we concentrate on a two-field case, studying the
evolution of adiabatic and entropy modes in details. In this section
we will discuss on the amplitude and shape of non-Gaussianity in our model
and illustrate a specific example. At the end of this section we
infer to some differences between this approach and Senatore and Zaldarriaga
\cite{senatore}. In the second appendix we
compare our results with \cite{gordon} as a check. Finally we
conclude in the last section.
\section{Briefly Review of Weinberg's Approach \cite{weinberg}}
To generalize Einstein-Hilbert action in the presence of matter
field it is possible to add the terms containing higher order
derivatives in the Lagrangian in addition to the standard second
order ones. In principle these additional terms can be important in
the larger energy scales. As discussed in \cite{weinberg} this
situation occurs naturally in the inflation era before horizon
crossing. According to the observations; the Hubble parameter, $H$,
and physical momentum, $k/a$, are equal (at horizon crossing) and
much less than $M_P$ and even unification scale. But due to
denominator of physical momentum in a period before horizon crossing
the physical momentum has had larger value. As a consequence,
considering the correction terms will help us to understand better
the inflationary predictions.
Due to the above discussions Weinberg in \cite{weinberg} has studied
the effects of the fourth order derivative terms in
the Einstein-Hilbert
Lagrangian in the presence of one scalar field. We are going to
generalize this model by adding more than one scalar field which is interesting
for its well-known observational consequences. Before that let us
review very briefly\footnote{Here we report Weinberg's idea very
quickly without any details. But in the following when we are going
to study its generalization we will do it in details in Appendix \ref{appendixA}.}
the main results of \cite{weinberg}. The starting point is the
Einstein-Hilbert Lagrangian which includes the leading term
\begin{eqnarray}\label{E-H action}
{\cal{L}}_0=\sqrt{g}\bigg[-\frac{M_P^2}{2}R-\frac{M^2}{2}g^{\mu\nu}\partial_\mu
\varphi\partial_\nu\varphi-M_P^2U(\varphi)\bigg]
\end{eqnarray}
where dimensionless $\varphi\equiv\varphi_c/M$ is defined such that
the kinetic term of $\varphi_c$ has the canonical form.
Obviously $\varphi_c$ has dimension of mass. It is now easier to define
the hierarchy of different derivative terms as an advantage of introduction
of the scale $M$ explicitly\footnote{Just remember that the ${\cal{L}}$
has dimension of $M^4$ and in natural unit $\partial_\mu$ has dimension
of $M^{-1}$.}. The leading correction terms are satisfied general
covariance and contain four derivatives. These term can be reduced to
the following form
\begin{eqnarray}\label{correction terms}
\Delta{\cal{L}}=\sqrt{g}f(\vp)\bigg(g^{\mu\nu}\vp_{,\mu}\vp_{,\nu}\bigg)^2+\sqrt{g}
h_1(\vp)C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}+\sqrt{g}
h_2(\vp)\varepsilon^{\kappa\lambda\mu\nu}C_{\kappa\lambda}^{\hspace{3mm}\rho\sigma}C_{\mu\nu\rho\sigma}
\end{eqnarray}
where $f$, $h_1$ and $h_2$ are some dimensionless arbitrary
functions which are assumed to be in order
one\footnote{\label{footnote7}Actually it is the second term of an
expansion with respect to the inverse of $M^2$ i.e.
``$M^2,1,M^{-2},...$". The first term is
``$-\frac{M^2}{2}g^{\mu\nu}\partial_\mu \varphi\partial_\nu\varphi$"
in (\ref{E-H action}).} and $C_{\mu\nu\rho\sigma}$ is the Weyl
tensor. It is noteworthy that the above terms are not all the
generally covariant terms containing four derivatives. But all the
allowed terms except the above terms are transformed to these terms
by employing the equation of motion for the leading term as well as
ignoring the surface terms (for details see Appendix
\ref{appendixA}). For the scalar perturbations it is convenient to
assume a gauge in which metric scalar perturbations vanish. In this
gauge by splitting the scalar field to its background and perturbed
parts as $\vp=\vpb+\delvp$ the Lagrangian becomes
\begin{eqnarray}\label{scalar-perturbation}\nonumber
{\cal{L}}&=&\sqrt{g}\bigg[-\frac{M^2}{2}g^{\mu\nu}\partial_\mu
\varphi\partial_\nu\varphi-M_P^2U(\varphi)+f(\vp)\bigg(g^{\mu\nu}\vp_{,\mu}\vp_{,\nu}\bigg)^2\bigg]\\
&=&\bar{\cal{L}}-\frac{1}{2}a^3\bigg(M^2+4f(\vpb)\dot{\vpb}^2\bigg)\times\bigg(-\dot{\delvp}^2+a^{-2}(\vec{\nabla}\delvp)^2\bigg)\\\nonumber
&+&4 a^3
f(\vpb)\dot{\vpb}^2\bigg(\dot\delvp^2+\dot\delvp^3/\dot\vpb-a^{-2}\dot\delvp(\vec{\nabla}\delvp)^2/\dot\vpb+\frac{1}{4}\dot\delvp^4/\dot\vpb^2
-\frac{1}{2}a^{-2}\dot\delvp^2(\vec{\nabla}\delvp)^2/\dot\vpb^2+\frac{1}{4}a^{-4}(\vec{\nabla}\delvp)^4/\dot\vpb^2\bigg)
\end{eqnarray}
which reduces to the Lagrangian $(19)$ in \cite{weinberg} with
$\pi\equiv\delvp/\dot\vpb$ and $\dot
H=-\dot\vpb^2(M^2+4f(\vpb)\dot{\vpb}^2)/2M_P^2$. This result is
compatible with \cite{paolo} with a minor disagreement. This
disagreement is in the presence of quartic terms as well as
quadratic and cubic terms. Due to the above Lagrangian obviously
ignoring the correction term, i.e. setting $f(\vp)=0$, results in a
model with $c_s=1$, where $c_s$ is the speed of sound. But in the
presence of the correction term the speed of sound is not one and
may cause large non-Gaussianity. In addition the terms in the third
line of (\ref{scalar-perturbation}) infer to the possible shapes of
non-Gaussianities as well as their amplitude.
In this section we very briefly reviewed the idea of \cite{weinberg}
for scalar perturbations in the context of effective field theory
for inflation. In addition to scalar perturbation in \cite{weinberg}
the tensor perturbations have been considered. It is concluded in
\cite{weinberg} that the propagation of gravitational wave depends
on the helicity of the wave in this model. In the next section we
are going to generalize the above idea for a multi-field theory of
inflation without considering the tensor perturbations. Since
existence of multi-scalar-field has no effect on the tensor
perturbations and consequently gravitational wave. The detailed
calculations for the next section is in appendix \ref{appendixA}
which is also useful for clarifying the case of one field studying
very briefly in this section.
\section{Effective Field Theory of Multi-Field Inflation}
The corresponding Lagrangian to (\ref{scalar-perturbation}) for
multi-field inflation can be written as follow, which has been deduced in
details in the appendix \ref{appendixA},
\begin{eqnarray}\label{most-general-lagrangian-simplified}
{\cal{L}}=\sqrt{g}&\bigg\{&b_3^{IJKL}(\vec\vp)\nabla_\mu\vp_I\nabla^\mu\vp_J\nabla_\nu\vp_K\nabla^\nu\vp_L-\frac{M^2}{2}\delta^{IJ}\nabla_\mu\vp_I\nabla^\mu\vp_J
-M_P^2U(\vec\vp)\\\nonumber
&+&a_1(\vec\vp)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+a_2(\vec\vp)R_{\mu\nu}R^{\mu\nu}-\frac{M_P^2}{2}R\bigg\}
\end{eqnarray}
which exactly reduces to (\ref{scalar-perturbation}) for one field
case\footnote{\label{footnote8}In \cite{weinberg} instead of Riemann
and Ricci tensors in (\ref{most-general-lagrangian-simplified}),
Weyl tensor has been used.}. Now, by splitting the scalar fields to
their background and perturbed parts $\vp_I=\bar\vp_I+\delvp$ we are
going to study the Lagrangian for the perturbations as well as the
background. To do this we start with
(\ref{most-general-lagrangian-simplified}) without worrying about
the tensor perturbations. The above Lagrangian can be written as
${\cal{L}}={\cal{L}}_0+\Delta{\cal{L}}$ such that
\begin{eqnarray}\label{most-general-lagrangian-simplified+background}\nonumber
a^{-3}{\cal{L}}_0&=&b_3^{IJKL}(\vpb){\dot\vpb}_I{\dot\vpb}_J{\dot\vpb}_K{\dot{\bar\vp}}_L
+\frac{M^2}{2}\delta^{IJ}{\dot\vpb}_I{\dot\vpb}_J-M_P^2U(\vpb),
\end{eqnarray}
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations}\nonumber
a^{-3}\Delta{\cal{L}}&=&\bigg[\sum_{n=1}\frac{1}{n!}\frac{\partial^n
b_3^{IJKL}(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]{\dot\vpb}_I{\dot\vpb}_J{\dot\vpb}_K{\dot{\bar\vp}}_L\\\nonumber
&-& \bigg[\sum_{n=0}\frac{1}{n!}\frac{\partial^n
b_3^{IJKL}(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]\dot\vpb_I\dot\vpb_J\bigg[-\dot\vpb_K\dot\delvp_L-\dot\vpb_L\dot\delvp_K-\dot\delvp_K\dot\delvp_L+
a^{-2}\partial_i\delvp_K\partial^i\delvp_L\bigg]
\\\nonumber
&-& \bigg[\sum_{n=0}\frac{1}{n!}\frac{\partial^n
b_3^{IJKL}(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]\bigg[-\dot\vpb_I\dot\delvp_J-\dot\vpb_J\dot\delvp_I-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]\dot\vpb_K\dot\vpb_L\\\nonumber
&+&\bigg[\sum_{n=0}\frac{1}{n!}\frac{\partial^n
b_3^{IJKL}(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]\times\\\nonumber&&\bigg[-\dot\vpb_I\dot\delvp_J-\dot\vpb_J\dot\delvp_I-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]\bigg[-\dot\vpb_K\dot\delvp_L-\dot\vpb_L\dot\delvp_K-\dot\delvp_K\dot\delvp_L+
a^{-2}\partial_i\delvp_K\partial^i\delvp_L\bigg]\\\nonumber
&-&\frac{M^2}{2}\delta^{IJ}\bigg[-2\dot\vpb_I\dot\delvp_J-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]
-M_P^2\bigg[\sum_{n=1}\frac{1}{n!}\frac{\partial^n
U(\vpb)}{\partial(\vpb_M)^n}(\delvp_M)^n\bigg]
\end{eqnarray}
where $a=a(t)$ is the scale factor of the FRW metric and
$\partial_i$ are spatial derivatives. Note that the terms containing
$\delvp_I$ without any differentiations do not show themselves in
the Lagrangian effectively. Since the $n^{th}$ equation of motion
causes vanishing of the coefficients of $(n+1)^{th}$ terms without
any differentiation. Also the linear perturbation terms even
including differentiation vanish because of the same reason. So
effectively the Lagrangian for the perturbations is
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations-effectively}\nonumber
a^{-3}\Delta{\cal{L}}&=&
b_3^{IJKL}(\vpb)\bigg\{\bigg[-\dot\vpb_I\dot\delvp_J-\dot\vpb_J\dot\delvp_I-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]\bigg[-\dot\vpb_K\dot\delvp_L-\dot\vpb_L\dot\delvp_K-\dot\delvp_K\dot\delvp_L+
a^{-2}\partial_i\delvp_K\partial^i\delvp_L\bigg]\\\nonumber
&-&\dot\vpb_I\dot\vpb_J\bigg[-\dot\delvp_K\dot\delvp_L+
a^{-2}\partial_i\delvp_K\partial^i\delvp_L\bigg]-\bigg[-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg]\dot\vpb_K\dot\vpb_L\bigg\}\\\nonumber
&-&\frac{M^2}{2}\delta^{IJ}\bigg[-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg].
\end{eqnarray}
The second, third and fourth order of perturbations respectively can
be represented as follows
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations-effectively-L2}\nonumber
a^{-3}\Delta{\cal{L}}^{(2)}&=&
b_3^{IJKL}(\vpb)\bigg\{\dot\vpb_I\dot\vpb_J\dot\delvp_K\dot\delvp_L+\dot\vpb_I\dot\vpb_K\dot\delvp_J\dot\delvp_L+
\dot\vpb_I\dot\vpb_L\dot\delvp_K\dot\delvp_J+\dot\vpb_K\dot\vpb_J\dot\delvp_I\dot\delvp_L+
\dot\vpb_L\dot\vpb_J\dot\delvp_I\dot\delvp_L+\dot\vpb_K\dot\vpb_L\dot\delvp_I\dot\delvp_J\\&-&
a^{-2}\dot\vpb_I\dot\vpb_J\partial_i\delvp_K\partial^i\delvp_L-a^{-2}\dot\vpb_K\dot\vpb_L\partial_i\delvp_I\partial^i\delvp_J
\bigg\}-\frac{M^2}{2}\delta^{IJ}\bigg[-\dot\delvp_I\dot\delvp_J+
a^{-2}\partial_i\delvp_I\partial^i\delvp_J\bigg],
\end{eqnarray}
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations-effectively-L3}
a^{-3}\Delta{\cal{L}}^{(3)}&=&
b_3^{IJKL}(\vpb)\bigg\{\dot\vpb_I\dot\delvp_J\dot\delvp_K\dot\delvp_L+\dot\vpb_J\dot\delvp_I\dot\delvp_K\dot\delvp_L
+\dot\vpb_K\dot\delvp_L\dot\delvp_I\dot\delvp_J+\dot\vpb_L\dot\delvp_K\dot\delvp_I\dot\delvp_J\\\nonumber
&-&a^{-2}\bigg(\dot\vpb_I\dot\delvp_J\partial_i\delvp_K\partial^i\delvp_L+\dot\vpb_J\dot\delvp_I\partial_i\delvp_K\partial^i\delvp_L
+\dot\vpb_K\dot\delvp_L\partial_i\delvp_I\partial^i\delvp_J+\dot\vpb_L\dot\delvp_K\partial_i\delvp_I\partial^i\delvp_J\bigg)\bigg\},\\\nonumber&&
\end{eqnarray}
\begin{eqnarray}\label{most-general-lagrangian-simplified+perturbations-effectively-L4}
a^{-3}\Delta{\cal{L}}^{(4)}=
b_3^{IJKL}(\vpb)\bigg\{\dot\delvp_I\dot\delvp_J\dot\delvp_K\dot\delvp_L&-&a^{-2}\bigg(\dot\delvp_I\dot\delvp_J\partial_i\delvp_K\partial^i\delvp_L+
\dot\delvp_K\dot\delvp_L\partial_i\delvp_I\partial^i\delvp_J\bigg)\\\nonumber
&+&a^{-4}\partial_i\delvp_I\partial^i\delvp_J\partial_j\delvp_K\partial^j\delvp_L\bigg\}.
\end{eqnarray}
Note that the above result exactly reduces to single field result in
(\ref{scalar-perturbation}) with
$b_3^{IJKL}(\vpb)=f(\vpb)$\footnote{To do this one should set
$I=J=K=L=1$.}. It is obvious from
(\ref{most-general-lagrangian-simplified+perturbations-effectively-L2})
that the speed of sound is not one in the presence of non-vanishing
$b_3^{IJKL}(\vpb)$. Note that due to $b_3^{IJKL}(\vpb)$ the cubic
and quartic terms are appeared. This feature is in disagreement with
\cite{paolo,senatore}. In their work the coefficient which displays
$c_s$ just connects to the cubic term. But here it connects to the
fourth order term too. In the next section we restrict the model to
a two-field model. This makes it possible to study the adiabatic and
entropy perturbations in more details without loss of generality in
the main results.
\section{A Specific Case: Adiabatic versus Entropy Perturbation}
In this section we re-write the above formalism in the language of
adiabatic and entropy perturbations for a two-field model. This is
crucial in this approach since in contrast to \cite{senatore} here
the adiabatic perturbation is not initially supposed. In
\cite{senatore} the additional perturbations are added to a model
already containing the adiabatic perturbation i.e. \cite{paolo}. In
\cite{senatore} the Goldstone boson, introduced in \cite{paolo},
plays the role of the adiabatic perturbation and the additional
fields are employed as the entropy perturbations. But in our model
there is no initially difference between $\vp_I$'s and consequently
$\delvp_I$'s. So it is critical to distinguish between adiabatic and
entropy modes to manifest their different physical meanings.
\subsection{The Most General Two-Field Model}
In this subsection we re-do perturbation calculations for a two-field model.
To do this we start
with (\ref{most-general-lagrangian-simplified}) for two fields named
$\vp$ and $\chi$
\begin{eqnarray}\label{two-field-lagrangian-simplified}
{\cal{L}}=-a^3\bigg\{&-&\frac{M_1^2}{2}\partial_\mu\vp\partial^\mu\vp-\frac{M_2^2}{2}\partial_\mu\chi\partial^\mu\chi
-M_P^2U(\vp,\chi)+g_1(\vp,\chi)\big(\partial_\mu\vp\partial^\mu\vp\big)^2+g_2(\vp,\chi)\big(\partial_\mu\chi\partial^\mu\chi\big)^2\\\nonumber
&+&g_3(\vp,\chi)\big(\partial_\mu\vp\partial^\mu\vp\big)\big(\partial_\nu\vp\partial^\nu\chi\big)
+g_4(\vp,\chi)\big(\partial_\mu\chi\partial^\mu\chi\big)\big(\partial_\nu\chi\partial^\nu\vp\big)\\\nonumber&+&g_5(\vp,\chi)\big(\partial_\mu\vp\partial^\mu\vp\big)\big(\partial_\nu\chi\partial^\nu\chi\big)
+g_6(\vp,\chi)\big(\partial_\mu\vp\partial^\mu\chi\big)\big(\partial_\nu\vp\partial^\nu\chi\big)\bigg\}
\end{eqnarray}
where $a=a(t)$ is the scale factor and $g_i$'s are some arbitrary
dimensionless and order one functions as mentioned before. By assuming
$\vp=\vpb+\delvp$ and $\chi=\bar\chi+\delchi$ the above Lagrangian
reduces to
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-0}
a^{-3}{\cal{L}}_0&=&\bigg\{-\frac{M_1^2}{2}\partial_\mu\vpb\partial^\mu\vpb-\frac{M_2^2}{2}\partial_\mu\chib\partial^\mu\chib
-M_P^2U(\vpb,\chib)+g_1\big(\partial_\mu\vpb\partial^\mu\vpb\big)^2+g_2\big(\partial_\mu\chib\partial^\mu\chib\big)^2
\\\nonumber&+&g_3\big(\partial_\mu\vpb\partial^\mu\vpb\big)\big(\partial_\nu\vpb\partial^\nu\chib\big)
+g_4\big(\partial_\mu\chib\partial^\mu\chib\big)\big(\partial_\nu\chib\partial^\nu\vpb\big)+
g_5\big(\partial_\mu\vpb\partial^\mu\vpb\big)\big(\partial_\nu\chib\partial^\nu\chib\big)
+g_6\big(\partial_\mu\vpb\partial^\mu\chib\big)\big(\partial_\nu\vpb\partial^\nu\chib\big)\bigg\}\\\nonumber
&=&\frac{M_1}{2}\dot\vpb^2+\frac{M_2}{2}\dot\chib^2-M_P^2U(\vpb,\chib)+g_1\dot\vpb^4+g_2\dot\chib^4+g_3\dot\vpb^3\dot\chib+g_4\dot\vpb\dot\chib^3
+(g_5+g_6)\dot\vpb^2\dot\chib^2
\end{eqnarray}
for the background part. The corresponding equations of motion for $\vpb$ reads as
\begin{eqnarray}\label{eq-mo-background}\nonumber
&&\frac{d}{dt}\bigg[M_1\dot\vpb+4g_1\dot\vpb^3+3g_3\dot\vpb^2\dot\chib+g_4\dot\chib^3
+2(g_5+g_6)\dot\vpb\dot\chib^2\bigg]+3H\bigg[M_1\dot\vpb+4g_1\dot\vpb^3+3g_3\dot\vpb^2\dot\chib+g_4\dot\chib^3
+2(g_5+g_6)\dot\vpb\dot\chib^2\bigg]\\\nonumber
&&+M_P^2U'-\bigg(g'_1\dot\vpb^4+g'_2\dot\chib^4+g'_3\dot\vpb^3\dot\chib+g'_4\dot\vpb\dot\chib^3
+(g'_5+g'_6)\dot\vpb^2\dot\chib^2\bigg)=0
\end{eqnarray}
where $'$ is the differentiation with respect to $\vpb$ and the
similar equation is true for $\chib$. It is straightforward but
messy to show that the above equation of motion (as well as
$\chib$'s) for the background causes the Lagrangian of the first
order perturbation becomes vanishing. So the non-trivial terms start
from the second order perturbations succeeding with the third and
the fourth order terms\footnote{It is obvious if one expands the
correction terms in (\ref{most-general-lagrangian-simplified}) or
(\ref{most-general-lagrangian}) for more than four derivative terms
then the higher order perturbations show themselves.} for
(\ref{two-field-lagrangian-simplified}) as the following
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2}
a^{-3}\Delta{\cal{L}}^{(2)}&=&\dot\delvp^2\big[\frac{M^2_1}{2}+6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\big]
+\dot\delchi^2\big[\frac{M^2_2}{2}+6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\big]\\\nonumber&+&
\dot\delvp\dot\delchi\big[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\big]\\\nonumber
&-&a^{-2}\bigg(\partial_i\delvp\partial^i\delvp\big[\frac{M_1}{2}+2g_1\dot\vpb^2+g_3\dot\vpb\dot\chib+g_5\dot\chib^2\big]
+\partial_i\delchi\partial^i\delchi\big[\frac{M_2}{2}+2g_2\dot\chib^2+g_4\dot\vpb\dot\chib+g_5\dot\vpb^2\big]\\\nonumber
&&\hspace{1.2cm}+\partial_i\delvp\partial^i\delchi\big[g_3\dot\vpb^2+g_4\dot\chib^2+2g_6\dot\vpb\dot\chib\big]\bigg),
\end{eqnarray}
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-3}
a^{-3}\Delta{\cal{L}}^{(3)}&=&\dot\delvp^3\big[4g_1\dot\vpb+g_3\dot\chib\big]+\dot\delchi^3\big[4g_2\dot\chib+g_4\dot\vpb\big]+
\dot\delvp^2\dot\delchi\big[3g_3\dot\vpb+2(g_5+g_6)\dot\chib\big]+\dot\delvp\dot\delchi^2\big[3g_4\dot\chib+2(g_5+g_6)\dot\vpb\big]\\\nonumber
&-&a^{-2}\bigg(\dot\delvp\partial_i\delvp\partial^i\delvp\big[4g_1\dot\vpb+g_3\dot\chib\big]+
\dot\delchi\partial_i\delchi\partial^i\delchi\big[4g_2\dot\chib+g_4\dot\vpb\big]+
\dot\delvp\partial_i\delchi\partial^i\delchi\big[g_4\dot\chib+2g_5\dot\vpb\big]\\\nonumber&&\hspace{1.5cm}
+\dot\delchi\partial_i\delvp\partial^i\delvp\big[g_3\dot\vpb+2g_5\dot\chib\big]
+\dot\delvp\partial_i\delvp\partial^i\delchi\big[2g_3\dot\vpb+2g_6\dot\chib\big]+
\dot\delchi\partial_i\delvp\partial^i\delchi\big[2g_4\dot\chib+2g_6\dot\vpb\big]\bigg),
\end{eqnarray}
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-4}
a^{-3}\Delta{\cal{L}}^{(4)}&=&g_1\dot\delvp^4+g_2\dot\delchi^4+g_3\dot\delvp^3\dot\delchi+g_4\dot\delchi^3\dot\delvp+(g_5+g_6)\dot\delvp^2\dot\delchi^2\\\nonumber
&-&a^{-2}\bigg(2g_1\dot\delvp^2\partial_i\delvp\partial^i\delvp+2g_2\dot\delchi^2\partial_i\delchi\partial^i\delchi+
g_3\dot\delvp^2\partial_i\delvp\partial^i\delchi+g_4\dot\delchi^2\partial_i\delchi\partial^i\delvp\\\nonumber
&&\vspace{2.5cm}+g_3\dot\delvp\dot\delchi\partial_i\delvp\partial^i\delvp+g_4\dot\delchi\dot\delvp\partial_i\delchi\partial^i\delchi+
g_5(\dot\delchi^2\partial_i\delvp\partial^i\delvp+\dot\delvp^2\partial_i\delchi\partial^i\delchi)+2g_6\dot\delvp
\dot\delchi\partial_i\delvp\partial^i\delchi\bigg)\\\nonumber
&+&a^{-4}\bigg(g_1(\partial_i\delvp\partial^i\delvp)^2+g_2(\partial_i\delchi\partial^i\delchi)^2
+g_3\partial_i\delvp\partial^i\delvp\partial_j\delchi\partial^j\delvp+g_4\partial_i\delchi\partial^i\delchi\partial_j\delchi\partial^j\delvp
\\\nonumber&&\vspace{2.5cm}+g_5\partial_i\delvp\partial^i\delvp\partial_j\delchi\partial^j\delchi+g_6(\partial_i\delvp\partial^i\delchi)^2\bigg).
\end{eqnarray}
It is interesting to mention that for $\vp=\chi$, i.e. going back to one
field case, all the above relations reduce to
(\ref{scalar-perturbation}) with $f=g_1+g_2+g_3+g_4+g_5+g_6$ as it was expected.
\subsection{Adiabatic vs. Entropy Modes}
Now let us re-write the above terms in the language of adiabatic and
entropy perturbations. The adiabatic perturbation is along the classical
path and the entropy perturbation
is orthogonal to it. Due to \cite{gordon} they can be defined as
follows
\begin{eqnarray}\label{adi-ent-perturbations}
\delta\sigma\equiv\vec{T}.\vec\delta,\hspace{2cm}\delta s\equiv
\vec{N}.\vec\delta
\end{eqnarray}
where $\delta \sigma$ and $\delta s$ are the adiabatic and entropy
modes respectively and
\begin{eqnarray}\label{tangent-normal-vector}
\vec\delta\equiv\left(\delvp,\delchi\right),\hspace{2cm}\vec
T=\left(\cos\theta,\sin\theta\right)\equiv\left(\dot\vp/\dot\sigma,\dot\chi/\dot\sigma\right),\hspace{2cm}
\vec N\equiv\left(\sin\theta,-\cos\theta\right)
\end{eqnarray}
where $\dot\sigma^2=\dot\vp^2+\dot\chi^2$. One can show easily
\begin{eqnarray}\label{adi-ent-perturbations-timederivative-original-perturbations}\nonumber
\dot\delvp=\cos\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}+\sin\theta\hspace{2mm}\vec{N}.\dot{\vec\delta},\hspace{2cm}
\dot\delchi=\sin\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}-\cos\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}
\end{eqnarray}
and it is easy to see that
\begin{eqnarray}\label{adi-ent-perturbations-timederivative}\nonumber
\dot{\delta\sigma}=\dot\theta\delta
s+\vec{T}.\dot{\vec\delta},\hspace{2cm}\dot{\delta s}=-\dot\theta
\delta\sigma+ \vec{N}.\dot{\vec\delta}
\end{eqnarray}
and due to above relations
\begin{eqnarray}\label{adi-ent-perturbations-spatialderivative-original-perturbations}\nonumber
\partial_i\delvp=\cos\theta\hspace{2mm}\partial_i\delta\sigma+\sin\theta\hspace{2mm}\partial_i\delta s,\hspace{2cm}
\partial_i\delchi=\sin\theta\hspace{2mm}\partial_i\delta\sigma-\cos\theta\hspace{2mm}\partial_i\delta
s.
\end{eqnarray}
Now by the above definitions we re-write the results of previous
sub-section by plugging $\delta\sigma$ and $\delta s$ in. Let's
start with the kinetic terms for leading order term in
(\ref{two-field-lagrangian-simplified-order-2})
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2-M1-M2-adi-ent}
&&\frac{M^2_1}{2}\dot\delvp^2+\frac{M^2_2}{2}\dot\delchi^2=\\\nonumber
&&\frac{M^2_1+M_2^2}{4}\left[\dot\delvp^2+\dot\delchi^2\right]+\frac{M_1^2-M^2_2}{4}\left[\dot\delvp^2-\dot\delchi^2\right]=\\\nonumber&&
\frac{M^2_1+M_2^2}{4}\left[(\vec T.\dot{\vec\delta})^2+(\vec
N.\dot{\vec\delta})^2\right]+\frac{M_1^2-M^2_2}{4}\left[(\cos^2\theta-\sin^2\theta)\bigg((\vec
T.\dot{\vec\delta})^2-(\vec
N.\dot{\vec\delta})^2\bigg)+4\sin\theta\cos\theta\hspace{2mm}
T.\dot{\vec\delta}\hspace{2mm}N.\dot{\vec\delta}\right]\\\nonumber&&
\end{eqnarray}
the same procedure is applicable for
$a^{-2}\left(\frac{M^2_1}{2}\partial_i\delvp\partial^i\delvp
+\frac{M^2_2}{2}\partial_i\delchi\partial^i\delchi\right)$ in
(\ref{two-field-lagrangian-simplified-order-2}). Now let us assume
$M_1=M_2=M$ to make it comparable to results in \cite{gordon}. In
this case
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2-M1=M2=1-adi-ent}
\frac{1}{M^2}a^{-3}{\cal{L}}&=&\frac{1}{2}\dot\delvp^2+\frac{1}{2}\dot\delchi^2-a^{-2}\left(\frac{1}{2}\partial_i\delvp\partial^i\delvp
+\frac{1}{2}\partial_i\delchi\partial^i\delchi\right)=\\\nonumber &&
\frac{1}{2}\left[(\vec T.\dot{\vec\delta})^2+(\vec
N.\dot{\vec\delta})^2\right]-\frac{1}{2}a^{-2}\left(\partial_i\delta\sigma\partial^i\delta\sigma
+\partial_i\delta s\partial^i\delta s\right)=\\\nonumber&&
\frac{1}{2}\left[(\dot{\delta\sigma}-\dot\theta\delta s)^2+(\dot{\delta s}+\dot\theta\delta\sigma)^2\right]
-\frac{1}{2}a^{-2}\left(\partial_i\delta\sigma\partial^i\delta\sigma
+\partial_i\delta s\partial^i\delta s\right)=\\\nonumber&&
\frac{1}{2}\left[\dot{\delta\sigma}^2+\dot\theta^2\delta s^2-2\dot\theta\delta s\dot{\delta\sigma}+\dot{\delta s}^2
+\dot\theta^2\delta\sigma^2+2\dot\theta\delta\sigma\dot{\delta s}\right]-\frac{1}{2}a^{-2}\left(\partial_i\delta\sigma\partial^i\delta\sigma
+\partial_i\delta s\partial^i\delta s\right)
\end{eqnarray}
According to the above Lagrangian the equations of motion
for $\delta\sigma$ and $\delta s$ become\footnote{The potential term in the second order of
perturbations should be added to
(\ref{two-field-lagrangian-simplified-order-2-M1=M2=1-adi-ent}) to
make our results comparable with \cite{gordon}. This term is
$-\frac{1}{2}\big(V_{\sigma\sigma}\delta\sigma^2+V_{\sigma s}\delta
\sigma \delta s+V_{ss}\delta s^2\big)$.}
\begin{eqnarray}\label{eq-mo-pert}
&&\ddot{\delta\sigma}+3H\dot{\delta\sigma}-a^{-2}\partial^i\partial_i\delta\sigma+(V_{\sigma\sigma}-\dot\theta^2)\delta\sigma=2\dot\theta\dot{\delta
s} +(\ddot\theta+3H\dot\theta-V_{\sigma s})\delta s\\\nonumber
&&\ddot{\delta s}+3H\dot{\delta s}-a^{-2}\partial^i\partial_i\delta
s+(V_{ss}-\dot\theta^2)\delta s=-2\dot\theta\dot{\delta \sigma}
-(\ddot\theta+3H\dot\theta+V_{\sigma s})\delta \sigma
\end{eqnarray}
where $V_{\sigma s}=(\cos^2\theta-\sin^2\theta)
V_{\vp\chi}+\sin\theta\cos\theta(V_{\chi\chi}-V_{\vp\vp})$. The
above results are exactly same as (47) and (48) in \cite{gordon}
when ignoring metric perturbations, see Appendix B. Now let us do
the same procedure for the second order perturbations due to the
first order correction term in
(\ref{two-field-lagrangian-simplified-order-2}). At the first the
terms containing time derivative
\begin{eqnarray}\label{adi-ent-pert-correction-order-2}\nonumber
&&\dot\delvp^2\big[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\big]+
\dot\delchi^2\big[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\big]+
\dot\delvp\dot\delchi\big[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\big]\\\nonumber&=&
\bigg(\cos\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}+\sin\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}\bigg)^2
\bigg[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\bigg]+
\bigg(\sin\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}-\cos\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}\bigg)^2
\bigg[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\bigg]\\\nonumber&+&
\bigg(\cos\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}+\sin\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}\bigg)
\bigg(\sin\theta\hspace{2mm}\vec{T}.\dot{\vec\delta}-\cos\theta\hspace{2mm}\vec{N}.\dot{\vec\delta}\bigg)
\bigg[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\bigg]\\\nonumber&=&
\big(\vec{T}.\dot{\vec\delta}\big)^2\times\bigg\{\cos^2\theta\bigg[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\bigg]
+\sin^2\theta\bigg[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\bigg]\\\nonumber&&\hspace{2cm}+\sin\theta\cos\theta
\bigg[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\bigg]\bigg\}\\\nonumber&+&
\big(\vec{N}.\dot{\vec\delta}\big)^2\times\bigg\{\sin^2\theta\bigg[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\bigg]
+\cos^2\theta\bigg[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\bigg]\\\nonumber&&\hspace{2cm}-\sin\theta\cos\theta
\bigg[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\bigg]\bigg\}\\\nonumber&+&
\big(\vec{T}.\dot{\vec\delta}\big)\big(\vec{N}.\dot{\vec\delta}\big)\times
\bigg\{2\cos\theta\sin\theta\bigg(\big[6g_1\dot\vpb^2+3g_3\dot\vpb\dot\chib+(g_5+g_6)\dot\chib^2\big]
-\big[6g_2\dot\chib^2+3g_4\dot\vpb\dot\chib+(g_5+g_6)\dot\vpb^2\big]\bigg)\\\nonumber&&\hspace{2.5cm}+\bigg(\sin^2\theta-\cos^2\theta\bigg)
\bigg[3g_3\dot\vpb^2+3g_4\dot\chib^2+4(g_5+g_6)\dot\vpb\dot\chib\bigg]\bigg\}
\end{eqnarray}
which can be re-written as the following\footnote{Here we do not
expand $\vec{T}.\dot{\vec\delta}$ and $\vec{N}.\dot{\vec\delta}$
since they contain no common terms to factorize. So their expansion
may cause just messy stuffs without any physical interests.}
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2-correction-terms}
&&6\dot\sigma^2\big(\vec{T}.\dot{\vec\delta}\big)^2\times\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\dot\sigma^2\big(\vec{N}.\dot{\vec\delta}\big)^2\times\\\nonumber&&\vspace{0cm}\bigg[(g_5+g_6)\bigg(\cos^4\theta+\sin^4\theta\bigg)+3(g_4-g_3)\bigg(\cos^3\theta\sin\theta
-\cos\theta\sin^3\theta\bigg)+2\bigg(3(g_1+g_2)+2(g_5+g_6)\bigg)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
3\dot\sigma^2\big(\vec{T}.\dot{\vec\delta}\big)\big(\vec{N}.\dot{\vec\delta}\big)\times\\\nonumber&&
\bigg[-g_3\cos^4\theta+g_4\sin^4\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos^3\theta\sin\theta-2\bigg(2g_2+(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_3-g_4)\cos^2\theta\sin^2\theta\bigg]
\end{eqnarray}
and similarly for the spatial differentiation
\begin{eqnarray}\label{two-field-lagrangian-simplified-order-2-correction-terms-spatial}
&-&2a^{-2}\dot\sigma^2(\partial_i\delta\sigma)^2\times\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+2(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&-&
a^{-2}\dot\sigma^2(\partial_i\delta
s)^2\times\vspace{0cm}\bigg[g_5\bigg(\cos^4\theta+\sin^4\theta\bigg)+(g_4-g_3)\bigg(\cos^3\theta\sin\theta
-\cos\theta\sin^3\theta\bigg)+2(g_1+g_2-g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&-&
a^{-2}\dot\sigma^2\partial_i\delta\sigma\partial^i \delta
s\times\\\nonumber&&
\bigg[-g_3\cos^4\theta+g_4\sin^4\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos^3\theta\sin\theta-2\bigg(2g_2+(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_3-g_4)\cos^2\theta\sin^2\theta\bigg].
\end{eqnarray}
Up to now we fully considered the second order perturbation terms in
the language of adiabatic and entropy modes. In quadratic level the
speed of sound is a matter of interest hence it is noteworthy to
take a look at it. It is obvious to the above relations that $\delta
\sigma$ and $\delta s$ can have different speeds of sound generally.
To have a sense about it let's consider a special case that $\delta
\sigma$ has $c_s=1$ and $\delta s$ has $c_s\neq 1$. To see this,
assume the special case with $g_1=g_2=g_3=g_4=0$ and $g_5+g_6=0$. In
this case the coefficients of $\dot{\delta\sigma}^2$ and
$(\partial_i\delta\sigma)^2$ are same and result in $c_s=1$ for
$\delta\sigma$\footnote{Note that we employ the standard definition
of $c_s$. It means we skip the interaction terms between
$\delta\sigma$ and $\delta s$ which exist even in quadratic level.}.
But the coefficient of $\dot{\delta s}^2$ is $\frac{M^2}{2}$ and for
$(\partial_i\delta s)^2$ is
$-\frac{1}{2}a^{-2}(M^2-2g_5\dot\sigma^2)$ that means
$c_s^2=1-2\frac{g_5\dot\sigma^2}{M^2}$. Note that here we write the
$M$ explicitly to make the comparison of the terms easier. A
characteristic property of $c_s$ is its $\dot\sigma^2$-dependence
which seems interesting. However the effective field theory is valid
where the correction terms are smaller than the leading terms in
(\ref{most-general-lagrangian-simplified}) to have an acceptable
expansion i.e. $\frac{\vert
b_3^{IJKL}(\vpb)\vert\dot{\sigma}^2}{M^2}<1$. Even more,
$\frac{\vert b_3^{IJKL}(\vpb)\vert\dot{\sigma}^2}{M^2}<<1$ should be
satisfied to make skipping higher order correction terms in
(\ref{most-general-lagrangian-simplified}) acceptable. So the speed
of sound in this model is almost one. This fact shows that for the
single field model the large non-Gaussinity is not expected. But in
the following we will discuss on the case of multi-field models. In
multi-field models even with $c_s\simeq 1$ the large non-Gaussianity
can be occurred in some specific circumstances.
One can do this procedure for the higher order perturbation terms
(\ref{two-field-lagrangian-simplified-order-3}) and
(\ref{two-field-lagrangian-simplified-order-4}) which are the
fundaments of studying non-Gaussianity for multi-field models. The
results are as follows\footnote{We here just consider the terms
containing time derivatives and not any spatial derivatives. However
the procedure is same.} for $a^{-3}\Delta{\cal{L}}^{(3)}$
\begin{eqnarray}\label{adi-ent-pert-correction-order-3}\nonumber
&&4\dot\sigma\big(\vec{T}.\dot{\vec\delta}\big)^3\times\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\dot\sigma\big(\vec{N}.\dot{\vec\delta}\big)^3\times\\\nonumber&&\vspace{0cm}
\bigg[-g_4\cos^4\theta+g_3\sin^4\theta-2\bigg(2g_2-(g_5+g_6)\bigg)\cos^3\theta\sin\theta
+2\bigg(2g_1-(g_5+g_6)\bigg)\cos\theta\sin^3\theta+3(g_4-g_3)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
3\dot\sigma\big(\vec{T}.\dot{\vec\delta}\big)^2\big(\vec{N}.\dot{\vec\delta}\big)\times\\\nonumber&&
\bigg[-g_3\cos^4\theta+g_4\sin^4\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos^3\theta\sin\theta-2\bigg(2g_2-(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_3-g_4)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
2\dot\sigma\big(\vec{T}.\dot{\vec\delta}\big)\big(\vec{N}.\dot{\vec\delta}\big)^2\times\\\nonumber&&
\bigg[(g_5+g_6)\bigg(\cos^4\theta+\sin^4\theta\bigg)+3(g_4-g_3)\bigg(\cos^3\theta\sin\theta-\cos\theta\sin^3\theta\bigg)
+\bigg(3(g_1+g_2)-2(g_5+g_6)\bigg)\cos^2\theta\sin^2\theta\bigg]
\end{eqnarray}
and for $a^{-3}\Delta{\cal{L}}^{(4)}$ it becomes
\begin{eqnarray}\label{adi-ent-pert-correction-order-4}\nonumber
&&\big(\vec{T}.\dot{\vec\delta}\big)^4\times\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\big(\vec{N}.\dot{\vec\delta}\big)^4\times
\bigg[g_2\cos^4\theta+g_1\sin^4\theta-g_4\cos^3\theta\sin\theta
-g_3\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\big(\vec{T}.\dot{\vec\delta}\big)^3\big(\vec{N}.\dot{\vec\delta}\big)\times\\\nonumber&&
\bigg[-g_3\cos^4\theta+g_4\sin^4\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos^3\theta\sin\theta-2\bigg(2g_2-(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_3-g_4)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\big(\vec{T}.\dot{\vec\delta}\big)\big(\vec{N}.\dot{\vec\delta}\big)^3\times\\\nonumber&&
\bigg[-g_4\cos^4\theta+g_3\sin^4\theta-2\bigg(2g_2-(g_5+g_6)\bigg)\cos^3\theta\sin\theta+2\bigg(2g_1-(g_5+g_6)\bigg)\cos\theta\sin^3\theta
+3(g_4-g_3)\cos^2\theta\sin^2\theta\bigg]\\\nonumber&+&
\big(\vec{T}.\dot{\vec\delta}\big)^2\big(\vec{N}.\dot{\vec\delta}\big)^2\times\\\nonumber&&
\bigg[(g_5+g_6)\bigg(\cos^4\theta+\sin^4\theta\bigg)+3(g_4-g_3)\bigg(\cos^3\theta\sin\theta-\cos\theta\sin^3\theta\bigg)
+2\bigg(3(g_1+g_2)-2(g_5+g_6)\bigg)\cos^2\theta\sin^2\theta\bigg].
\end{eqnarray}
Now let us just consider the terms containing
$(\vec{T}.\dot{\vec\delta}\big)$ and write them together
as\footnote{Note that the coefficient of
$\big(\vec{T}.\dot{\vec\delta}\big)^2$ in
(\ref{two-field-lagrangian-simplified-order-2-correction-terms}) is
$6$. But by comparison to
(\ref{two-field-lagrangian-simplified-order-2-correction-terms-spatial}),
$2$ of $6$ appear in definition of the speed of sound, $c_s$,
(exactly same as the second line in (\ref{scalar-perturbation})) and
what remains is $4\big(\vec{T}.\dot{\vec\delta}\big)^2$.}
\begin{eqnarray}\label{Tdelta}\nonumber
4\dot\sigma^2\bigg[g_1\cos^4\theta+g_2\sin^4\theta+g_3\cos^3\theta\sin\theta
+g_4\cos\theta\sin^3\theta+(g_5+g_6)\cos^2\theta\sin^2\theta\bigg]\times
\bigg(\big(\vec{T}.\dot{\vec\delta}\big)^2+
\big(\vec{T}.\dot{\vec\delta}\big)^3/\dot\sigma+\frac{1}{4}\big(\vec{T}.\dot{\vec\delta}\big)^4/\dot\sigma^2\bigg)
\end{eqnarray}
Comparison the above relation with the relation in
(\ref{scalar-perturbation}) manifests that $\sigma$, $[...]$ and
$\big(\vec{T}.\dot{\vec\delta}\big)$ play the role of $\vpb$,
$f(\vpb)$ and $\dot\delvp$ respectively. The significant property of
this model is the appearance of $\big(\vec{T}.\dot{\vec\delta}\big)$
and $\big(\vec{N}.\dot{\vec\delta}\big)$ or equivalently
$(\dot{\delta\sigma}-\dot\theta\delta s)$ and $(\dot{\delta
s}+\dot\theta\delta \sigma)$ respectively. This means that
$\dot{\delta\sigma}$ and $\delta s$ are always together and the same
for $\dot{\delta s}$ and $\delta\sigma$. This characteristic feature
of this model has some observational consequences which are
discussed in the following.
\subsubsection{The Amplitude of Non-Gaussianity}
Now we are going to estimate the non-Gaussianity amplitude. To do
this one procedure is comparison between the non-linear terms and
the linear ones. Mathematically, the amplitude of non-Gaussianity
$f_{NL}$, bi-spectrum, can be estimated as
$\frac{{\cal{L}}^{(3)}}{{\cal{L}}^{(2)}}\times\zeta^{-1}$ where
$\zeta$ is the curvature perturbation \cite{baumann}. Note that the
dominant amplitude of the terms containing time derivatives comes
from their amplitude at horizon crossing. At this time
$\frac{d}{dt}\sim H$ where $H$ is the Hubble constant. Hence for the
second order perturbations the Lagrangian ${\cal{L}}^{(2)}$ can be
written in an abstract form as $\{H^2, H \dot{\theta},
\dot{\theta}^2\}\times M^2 \times \delta\sigma^2$. The same analyze
for ${{\cal{L}}^{(3)}}$ results in $\{H^3, H^2 \dot{\theta}, H
\dot{\theta}^2, \dot{\theta}^3\}\times f(g_i) \times \dot\sigma
\times \delta\sigma^3$. So in an abstract form
\begin{eqnarray}\label{fNL}
\frac{{\cal{L}}^{(3)}}{{\cal{L}}^{(2)}}=\frac{\{H^3, H^2
\dot{\theta}, H \dot{\theta}^2, \dot{\theta}^3\}}{\{H^2, H
\dot{\theta}, \dot{\theta}^2\}}\times\bigg(\frac{f(g_i)}{M^2}\dot\sigma^2\bigg)\times
\frac{\delta\sigma}{\dot\sigma}=\frac{\{H^3, H^2 \dot{\theta}, H
\dot{\theta}^2, \dot{\theta}^3\}}{H\times\{H^2, H \dot{\theta},
\dot{\theta}^2\}}\times\bigg(\frac{f(g_i)}{M^2}\dot\sigma^2\bigg)\times \zeta
\end{eqnarray}
where $\zeta\sim \frac{H\delta\sigma}{\dot\sigma}$ is interpreted as
curvature perturbation. Now we consider two different regimes
$\dot{\theta}<<H$ and $\dot{\theta}>>H$. The first regime,
$\dot{\theta}<<H$, physically means that the model is a single field model effectively. In
this case the amplitude of bi-spectrum can be approximated by
\begin{eqnarray}\label{fNL-thetadot<<H}
f_{NL}\sim\frac{{\cal{L}}^{(3)}}{{\cal{L}}^{(2)}}\zeta^{-1}\sim\frac{f(g_i)}{M^2}\dot\sigma^2
\end{eqnarray}
but remember that the validity of the effective
field theory imposes $\frac{f(g_i)}{M^2}\dot\sigma^2<1$. So in this case as mentioned before
there is no significant non-Gaussianity which is in agreement with
the single field models of inflation \cite{chen}. But the other case,
$\dot{\theta}>>H$, means that the classical path in the
phase space is highly curved \cite{ana}. In other words it means the classical path
in the phase-space is far from a straight line ($\dot\theta=0$).
So the existence of the entropic field
is unavoidable. For this case the amplitude of $f_{NL}$ is
\begin{eqnarray}\label{fNL-thetadot>>H}
f_{NL}\sim\frac{{\cal{L}}^{(3)}}{{\cal{L}}^{(2)}}\zeta^{-1}
\sim\frac{\dot\theta}{H}\times\bigg(\frac{f(g_i)}{M^2}\dot\sigma^2\bigg).
\end{eqnarray}
Now the factor
$\frac{\dot\theta}{H}\times(\frac{f(g_i)}{M^2}\dot\sigma^2)$ can be
large and results in large non-Gaussianity consequently. So the
large curvature of the classical path in the phase-space results in
the large non-Gaussianity. Though this result can be compared to the
other works in the literature \cite{gpmulti} and in the effective
field theory context \cite{eftmulti} but the curvature of the
classical path is restricted due to observed scale invariant power
spectrum \cite{max}.
For a moment let us relax the constraint on the
$\frac{f(g_i)}{M^2}\dot\sigma^2$. Consequently the correction terms
in (\ref{two-field-lagrangian-simplified}) causes the large
non-Gaussinity. However the relaxation of the constraint can be
justified by assuming that our model is completely described by
(\ref{two-field-lagrangian-simplified}) without any higher order
correction terms. This needs fine tuning which is not impossible but
it is not natural. However there is another method to rationalize
this assumption. Instead of fine tuning the model automatically
shows this property via for example Vainshtein mechanism
\cite{derham,vain}.
The same is applicable for tri-spectrum by an estimation as
$\tau_{NL}\sim\frac{{\cal{L}}^{(4)}}{{\cal{L}}^{(2)}}\times\zeta^{-2}$.
The fourth order Lagrangian, ${{\cal{L}}^{(4)}}$, can be written in
the abstract form as $\{H^4,H^3\dot\theta, H^2 \dot{\theta}^2, H
\dot{\theta}^3,H \dot{\theta}^3,\dot\theta^4\}\times f(g_i) \times
\delta\sigma^4$ and then
\begin{eqnarray}\label{tNL-thetadot<<H}
\tau_{NL}\sim\frac{{\cal{L}}^{(4)}}{{\cal{L}}^{(2)}}\zeta^{-2}\sim\frac{f(g_i)}{M^2}\dot\sigma^2
\end{eqnarray}
for $\dot\theta<<H$ and
\begin{eqnarray}\label{tNL-thetadot>>H}
\tau_{NL}\sim\frac{{\cal{L}}^{(4)}}{{\cal{L}}^{(2)}}\zeta^{-2}
\sim\left(\frac{\dot\theta}{H}\right)^2\times\bigg(\frac{f(g_i)}{M^2}\dot\sigma^2\bigg).
\end{eqnarray}
for $\dot\theta>>H$.
\subsubsection{The Shape of Non-Gaussianity}
Now let us focus on the shape of possible non-Gaussinity predicted
by our model. In principle all the possible interaction terms
between $\delta\sigma$, $\delta s$ and their derivatives exist in
our model. This fact means all the non-Gaussianity shape can be
produced. However the amplitude of different shapes are different.
As a general argument it can be emphasized that for different
regimes of $\frac{\dot\theta}{H}$ different shapes are dominant. For
the case $\dot\theta<<H$ the terms containing time derivatives
become dominated. This means in this limit the equilateral shape is
the main one among the others. But it does not mean the other shapes
do not exist i.e. the ``Cosine" between different shapes are not
zero. In the other limit, $\dot\theta>>H$, the terms without
derivative become dominant and then the local shape is dominant.
This result is in agreement with the prediction for multi-field
inflation models \cite{chen}. Note that in \cite{senatore} since the
additional entropy perturbations are added by symmetry then they do
not have any term without derivative in their Lagrangian. In this
sense they do not predict a dominant local shape for their model
which is in disagreement with our result.
A characteristic feature of this model is appearance of just two
combinations of the fields i.e. $\vec{T}.\dot{\vec{\delta}}$ and
$\vec{N}.\dot{\vec{\delta}}$ in all the terms including the second,
third and fourth orders. To explain what is the physical result of
this fact let us concentrate on
$\vec{T}.\dot{\vec{\delta}}=\dot{\delta\sigma}-\dot\theta\delta s$,
as an example. The third order term of this combination is
$(\vec{T}.\dot{\vec{\delta}})^3=\dot{\delta\sigma^3}-
3\dot\theta\dot{\delta\sigma^2} \delta
s+3\dot\theta^2\dot{\delta\sigma} \delta s^2 -\dot\theta^3\delta
s^3$. Without worrying about the amplitude in this part let us focus
on the first and the last term. The definite prediction of this
model is that if any equilateral non-Gaussianity due to the first
term, i.e. $\dot{\delta\sigma^3}$, be observed then it has to be
observed a local non-Gaussianity due to the last term\footnote{ Note
that due to the first equation of motion in (\ref{eq-mo-pert}) the
$\delta s$ sources $\delta \sigma$.}, i.e. $\delta s^3$. So the
non-Gaussianity predicted by this model cannot be pure e.g. pure
equilateral shape. Hence mathematically, the ``Cosine" between two
shapes cannot be zero and more the ``Cosine" depends on the
$\dot\theta$ and is fixed by the model. This argument is true for
the other third order terms as well as fourth order ones. To
conclude, it seems that this model predicts a definite combination
of different shapes for the non-Gaussianity if the amplitude allows
to observe them.
\subsection{An Example}
In this subsection we are going to show how the general statements
mentioned before do work in a simple example. Here we assume that
all the $g_i$'s vanish except $g_1(\vp,\chi)$\footnote{Note that
except here in the rest of the paper we assumed that $g_i$'s are
constant as a matter of simplification. But here we would like to
show how the functionality of $g_i$'s may affect the final result.}
which is a generalization of the model in \cite{paolo1}. In addition
we assume there is no potential term\footnote{Note that the most
general form of the potential term can be supposed. But as mentioned
in \cite{baumann-green}, in the slow-roll regime there is no
interesting non-Gaussianity prediction for single field models.
However for multi-field models the potential term can result in
large non-Gaussianity which considered in \cite{gpmulti}. Here, we
restrict our calculations to kinetic terms.}. According to the
background Lagrangian
(\ref{two-field-lagrangian-simplified-order-0}) the equations of
motion for our special case become
\begin{eqnarray}\label{special-case-eq.of.motion}
&&\ddot{\vp}\big(1+12
\frac{g_1}{M^2}\dot\vp^2\big)+3H\dot\vp\big(1+4
\frac{g_1}{M^2}\dot\vp^2\big)+\frac{1}{M^2}\dot\vp^3\bigg(3\dot\vp\frac{\partial
g_1}{\partial \vp}+4\dot\chi\frac{\partial g_1}{\partial
\chi}\bigg)=0\\\nonumber
&&\ddot{\chi}+3H\dot\chi-\frac{1}{M^2}\dot\vp^4\frac{\partial
g_1}{\partial\chi}=0
\end{eqnarray}
where $M=M_1=M_2$ is assumed. On the other hand, what can cause the
significant non-Gaussianity is $\dot\theta$ as mentioned before. In
general due to the definition of $\theta$ in
(\ref{tangent-normal-vector}), $\dot\theta$ can be read as
\begin{eqnarray}\label{dot-theta}\nonumber
\dot\theta=\frac{-\ddot\vp\dot\chi+\dot\vp\ddot\chi}{\dot\vp^2+\dot\chi^2}
\end{eqnarray}
and in our special case by considering
(\ref{special-case-eq.of.motion}) it becomes (up to the first order
of $g_1\frac{\dot\vp^2}{M^2}$)
\begin{eqnarray}\label{dot-theta-special-case}\nonumber
\frac{\dot\theta}{H}\simeq24 \bigg(-g_1
\frac{\dot\vp^2}{M^2}\bigg)\frac{\dot\vp\dot\chi}{\dot\vp^2+\dot\chi^2}+\frac{1}{H}
\frac{1}{M^2}\frac{\dot\vp^3}{\dot\vp^2+\dot\chi^2}\bigg[3\frac{\partial
g_1}{\partial\vp}\dot\vp\dot\chi+\frac{\partial
g_1}{\partial\chi}\big(\dot\vp^2+4\dot\chi^2\big)\bigg].
\end{eqnarray}
The condition $g_1 \frac{\dot\vp^2}{M_1^2}<1$ ensures the validity
of the effective field theory. So it is not bad to estimate $24g_1
\frac{\dot\vp^2}{M_1^2}\sim 1$. Then due to the first term
$\frac{\dot\theta}{H}\sim
\frac{\dot\vp\dot\chi}{\dot\vp^2+\dot\chi^2}$ which means the
maximum of $f_{NL}$ in (\ref{fNL-thetadot>>H}) is less than one. A
successful inflation in the slow-roll regime restricts the value of
the field velocities which may restrict more the above estimation.
To discuss on the second term let us divide $g_1(\vp,\chi)$ to its
amplitude and functionality as $g_1(\vp,\chi)=\mid g_1\mid\times
f(\vp,\chi)$ such that $\mid g\mid$ is the amplitude of the
$g_1(\vp,\chi)$ and $f(\vp,\chi)$ represents its functional form. So
the second term can be estimated as (by assuming
$\dot\vp\sim\dot\chi$)
\begin{eqnarray}\label{dot-theta-special-case-second-term}\nonumber
\frac{\dot\theta}{H}\simeq \bigg(\mid
g_1\mid\frac{\dot\vp^2}{M^2}\bigg)\frac{\dot\vp}{2H}\bigg[3\frac{\partial
f}{\partial\vp}+5\frac{\partial f}{\partial\chi}\bigg],
\end{eqnarray}
where $\mid g_1\mid \frac{\dot\vp^2}{M^2}<1$ has to be satisfied. On
the other hand one of the Friedmann equations (in the absence of the
potential) is
$H^2=\frac{M^2}{2}\dot\vp^2+\frac{M^2}{2}\dot\chi^2\sim
M^2\dot\vp^2$ for the zeroth order of $g_1 \frac{\dot\vp^2}{M^2}$.
Now if $\frac{\partial f}{\partial\vp}$ or
$\frac{\partial f}{\partial\chi}$ have the significant
amplitude with respect to $M$ then a large amplitude of
non-Gaussianity would be expected. This can be realized by
assuming sharp features in the functionality of $g_1(\vp,\chi)$
maybe due to a phase transition.
\subsection{Some Clarifications on Differences with Senatore and
Zaldarriaga \cite{senatore}} The significant difference is the
existence of the terms containing the adiabatic and entropy
perturbations themselves not just their derivatives. The reason for
this difference is in how the effective field theory is constructed
in \cite{senatore}. As mentioned before in their model the adiabatic
mode is borrowed from \cite{paolo} which satisfies a shift symmetry.
Then the entropy modes are added and satisfy the shift symmetry too.
Consequently, in their formalism they have just derivative of
perturbations. But in contrast we do not start with distinguishable
fields then we do not have any difference between the perturbations
initially. So by this starting point we had to define the adiabatic
and entropy perturbations. This is what has been done in this
section in details. Now the question is that is there any special
transformation for adiabatic and entropy perturbation in our model?
Yes, it is locally rotated shift transformation i.e.
\begin{eqnarray}\label{locally-rotated-shift-symm.}
\delta\sigma\rightarrow\delta\sigma+(c_1
\cos\theta+c_2\sin\theta)\\\nonumber \delta s\rightarrow\delta
s+(c_1 \sin\theta-c_2\cos\theta)
\end{eqnarray}
where $\theta=\arctan(\dot\chi/\dot\vp)$. Note that the rotational
angle depends on the background fields time evolution. To achieve
this result, the starting point is the Lagrangian for two fields
i.e. the relations (\ref{two-field-lagrangian-simplified-order-2}),
(\ref{two-field-lagrangian-simplified-order-3}) and
(\ref{two-field-lagrangian-simplified-order-4}). By looking at these
relations it is obvious that $\delvp\rightarrow\delvp+c_1$ and
$\delchi\rightarrow\delchi+c_2$ is a symmetry of the model where
$c_1$ and $c_2$ are two independent arbitrary constants. So due to
(\ref{adi-ent-perturbations}) one can get the above relation
(\ref{locally-rotated-shift-symm.}) as the corresponding
transformation of $\delta \sigma$ and $\delta s$. According to this
relation the invariant terms corresponding to $\dot{\delvp}$ and
$\dot{\delta \chi}$ are not $\dot{\delta\sigma}$ and $\dot{\delta
s}$ but
\begin{eqnarray}\label{invariant-combination}
\dot{\delta\sigma}-\dot\theta\delta
s\rightarrow\dot{\delta\sigma}-\dot\theta\delta
s\\\nonumber\dot{\delta s}+\dot\theta\delta
\sigma\rightarrow\dot{\delta s}+\dot\theta\delta \sigma
\end{eqnarray}
which are $\vec T.\dot{\vec\delta}$ and $\vec N.\dot{\vec\delta}$
respectively. Not surprisingly, these terms construct whole
Lagrangian in adiabatic and entropy perturbations language
as seen previously.
So it seems initially supposed adiabatic perturbation in
\cite{senatore} results in lack of all possible terms in the
effective Lagrangian. Our proposition to solve this problem is based
on the discussion in this subsection. The main building blocks for
an effective field theory of multi-field inflation are not
$\dot{\delta\sigma}$ and $\dot{\delta s}$ but they are $\vec
T.\dot{\vec\delta}$ and $\vec N.\dot{\vec\delta}$. So the most
general Lagrangian for the perturbations in the multi-field context
should be written as\footnote{Note that here we just consider the
time derivative since in the discussion of this section there is no
difference between our model and \cite{senatore} for the terms
containing spatial derivatives. This is because the background is
not spatial dependent. Remember that the angle of rotation just
depends on time.}
\begin{eqnarray}\label{general-perturbation-lag.-adi-ent.}
\Delta{\cal{L}}\propto \sum_{m,n} c_{mn} \bigg(\vec
T.\dot{\vec\delta}\bigg)^m\bigg(\vec N.\dot{\vec\delta}\bigg)^n
\end{eqnarray}
for arbitrary $c_{mn}$. The above Lagrangian can be considered as
the effective field theory for the two-field inflation in the
language of \cite{senatore} but with additional terms. Note that the
above result can be generalized to multi-field inflation as
\begin{eqnarray}\label{general-perturbation-lag.-adi-ent.-multi}
\Delta{\cal{L}}\propto \sum c_{{n_0},{n_1},...,{n_N}} \bigg(\vec
T.\dot{\vec\delta}\bigg)^{n_0}\bigg(\vec
N_1.\dot{\vec\delta}\bigg)^{n_1}\bigg(\vec
N_2.\dot{\vec\delta}\bigg)^{n_2}...\bigg(\vec
N_N.\dot{\vec\delta}\bigg)^{n_N}
\end{eqnarray}
where $\vec T$ and $\vec N_i$'s are a set of orthonormal vectors for
an $(N+1)$-field model.
\section{Conclusions}
In this work the effective field theory of multi-field inflation has
been studied as a generalization of Weinberg's idea \cite{weinberg}
for a single field. In this approach the most general Lagrangian is
built by using all the covariant terms of the fields. Though
effectively the terms with higher order derivatives are interested
in the higher energy scales. In this work we restricted the model to
the first correction terms. They results in up to fourth order terms
in perturbations. Then due to the physical interests we switched to
the adiabatic and entropy formalism. It has been shown that
generally these modes can have different speeds of sound. By
considering the non-linear terms we studied the non-Gaussinity in
this model. It has been shown that the amplitude of non-Gaussianity
can be significant when the curvature of the classical path in the
phase-space becomes large. For example a sharp turn in the classical
path can realize it. However it seems that existence of the higher
order derivative terms in the Lagrangian cannot produce large
non-Gaussinity. The bottom line for this fact is the strong
constraint on the coefficients to keep the effective field theory
valid. But there is an idea that it is possible to take the higher
order correction terms under control automatically, e.g. by
Vainshtein mechanism. This relaxes the constraint on the coefficient
of the correction terms and results in large non-Gaussianity. On the
other hand the structure of the interacting terms in the Lagrangian
predicts the existence of all the shapes of non-Gaussinity with the
different amplitude for different cases. But the characteristic
feature of the model is that the non-Gaussinities are correlated.
That means if there is a local non-Gaussinity due to the entropy
mode then certainly there is a non-Gaussinity in adiabatic mode
which is equilateral. The amplitude of these different types of
non-Gaussinities are not independent.
In contrast to \cite{senatore}, the adiabatic and entropy
perturbations are not distinguishable initially. This fact results
in the existence of the perturbations as well as their derivatives.
In other words the adiabatic and entropy perturbations are not
invariant under the shift symmetry of original fields. However a
combination of them is invariant under such a symmetry. These
combinations are ``$\dot{\delta\sigma}-\dot\theta\delta s$" and
``$\dot{\delta s}+\dot\theta\delta \sigma$" or in other form $\vec
T.\dot{\vec\delta}$ and $\vec N.\dot{\vec\delta}$ respectively. This
result is important for constructing the effective field theory for
multi-field inflation and causes the additional terms with respect
to what is considered in \cite{senatore}.
\begin{acknowledgments}
We would like to thank B. A. Bassett, H. Firouzjahi, J. Fonseca, H.
R. Sepnagi, N. Sivanandam and M. M. Sheikh-Jabbari for their
comments. We are grateful of T. Battefeld for his very useful
comments and careful reading of the manuscript. We also specially
thank P. Creminelli for very fruitful discussions and comments. We
would like to thank ICTP for their warm hospitality and support when
this work was initiated.
\end{acknowledgments}
|
{
"timestamp": "2012-03-13T01:01:21",
"yymm": "1203",
"arxiv_id": "1203.2266",
"language": "en",
"url": "https://arxiv.org/abs/1203.2266"
}
|
\section{Introduction}
Periodically ordered arrays of nanoparticles, colloidal crystals,
crystalline mesophases formed from surfactant molecules or block
copolymers, etc. are all examples of complex periodic structures that
can occur in soft matter systems. Since often the interactions between
the constituent particles of these structures are to a large degree tunable, one
has the possibility of producing materials with ``tailored'' properties
which have potential applications in nanotechnological devices
\cite{1,2,3,4,5}. When seeking to provide theoretical guidance for
understanding structure-property relations in such complex soft matter
systems, a basic issue is how to judge the relative stability of
competing candidate structures, i.e. to distinguish the stable structure
(having the lowest free energy) from the metastable ones. For standard
crystals formed from atoms or small molecules, this question can be
answered by comparing ground state energies of the competing structures
(and --if necessary-- also taking entropic contributions from lattice
vibrations into account, within the framework of the harmonic
approximation). In soft matter systems, disorder in the structure and
thermally driven entropic effects rule out such an approach, and hence
there is a need for computer simulation methods that compute the free
energy of the various complex structures. However, as is well known, the
free energy of a model system is not a direct output of either
Molecular Dynamics or Monte Carlo simulations, and special techniques
have to be used \cite{6,7,8,9,10,11}.
In principle, one can obtain the absolute free energy of a structure by
linking it to some reference state of known free energy by means of
thermodynamic integration (TI)
\cite{6,7,8,9,10,11,12,13,14,15,16}. The strengths of TI are that
it is both conceptually simple and often straightforward to implement.
Its principal drawback is that the quantity of interest, namely the free
energy {\em difference} between candidate structures is typically
orders of magnitude smaller than the absolute free energies of the
individual structures which TI measures. Essentially, therefore, TI
estimates a small number by taking the difference of two large ones; As
a consequence, the precision of the method is limited and an enormous
(even sometimes wasteful) investment of computer resources may be needed
to resolve free energy difference accurately \cite{9}.
A much more elegant approach, albeit one which is not quite so easy to
implement as TI, is the ``phase switch Monte Carlo'' \cite{17a,17,18,19,20,21,21a}
technique. This method is potentially more powerful than TI because it
focuses directly on the small free energy difference between the
structures to be compared, rather than their absolute free energies. In
previous work, the precision of the method was demonstrated in the
context of measurements of the free energy difference between fcc and
hcp structures of hard spheres \cite{19}, the phase behaviour of
Lennard-Jones crystals \cite{19} and as a means of studying liquid-solid
phase transitions \cite{17}. In the latter case, simple model systems
containing only a few hundred particles could be studied, while for the
study of the fcc-hcp free energy difference \cite{17a,20} larger systems
of up to $1728$ particles could be studied by virtue of the fact that
these crystals only differ in their packing sequence of close-packed
triangular defect-free lattice planes. However, it is an open question
as to what system sizes one can attain with the phase switch method for
more general crystalline systems, including -- as in the present work --
ones which exhibit considerable structural disorder (``soliton
staircases'', see below). Furthermore, there have hitherto been no
like-for-like comparisons of the TI and phase switch methods on the same
system, so whilst their are good reasons for {\em presuming} the
superiority of phase switch (in terms of precision delivered for a given
computational investment), this has never actually been quantified.
In the present paper, we address these matters, considering as a generic
example a two-dimensional colloidal crystal in varying geometrical
confinement \cite{22,23,24,25,26}. As is well-known, two dimensional
colloidal crystals are experimentally much studied model systems
\cite{27,28,29,30,31,32,33,34,35,36,37,38} comprising, for example,
polystyrene spheres containing a superparamagnetic core adsorbed at the
air-water-interface. Applying a magnetic field oriented perpendicular to
this interface creates a repulsive interaction that scales like $r
^{-3}$, ($r$ being the particle separation), whose magnitude is
controlled by the magnetic field strength \cite{27}. Lateral
confinement of such two-dimensional crystals can be effected
mechanically or by laser fields (if the latter are also applied in the
bulk of such a crystal, one can study laser-induced melting and/or
freezing \cite{39,40,41,42}). Of course, there exist many related
problems in rather different physical contexts (``dusty plasmas''
\cite{43,44}, i.e. negatively charged SiO$_2$ fine particles with 10$\mu
m$ diameter in weakly ionized $rf$ discharges; lattices of confined
spherical block copolymer micelles \cite{45}; vortex matter in slit
channels \cite{46}, etc.). However, our study does not address a
specific system, rather we focus on the methodological aspects of
how one can study such problems by computer simulation.
The outline of the present paper is a follows. In Sec. 2, we summarize
the key facts about our model, namely strips of two-dimensional crystals
confined between two walls where structural phase transitions may occur
when the distance between the (corrugated) rigid boundaries is varied
\cite{23,24,25,26,47,48,49} (i.e., a succession of transitions in the
number of crystal rows $n$ parallel to the walls occur, $n \rightarrow
n-1 \rightarrow n-2$, with increasing compression, accompanied by the
formation of a ``soliton staircase'' at the walls \cite{23,24,25,26}). In
Sec. 3, the methods that are used are briefly described: the
thermodynamic integration method of Schmid and Schilling \cite{15,16} is
used as a baseline, while the main emphasis is on the phase switch Monte
Carlo method (implementation details of which are deferred to an
Appendix). In Sec. 4 we describe the results of the application of these
techniques to the model of Sec. 2. We show that phase switch Monte Carlo
\cite{17,18,19} can accurately locate the phase transitions despite the
need to deal with thousands of particles, and is orders of magnitude
more efficient than thermodynamic integration. Sec. 5 summarizes some
conclusions.
\section{Structural Transitions in Crystalline Strips confined by corrugated boundaries: Phenomenology}
Here we introduce the model for which our methodology is exemplified,
and recall briefly the main findings concerning the rather unconventional
transitions that have been detected \cite{23,24,25,26}, as far as they
are relevant for the present study.
We consider monodisperse colloidal particles in a strictly
two-dimensional geometry, which then are treated like point particles in
a plane interacting with a suitable effective potential $V(r)$ that
depends only on the interparticle distance $r$. In the real systems
\cite{27,29,30,31,32,33} this potential is purely repulsive, but due to
the magnetostatic dipole-dipole interaction (whose strength is
controlled by the external magnetic field) it is very slowly decaying,
$V(r) \propto r ^{-3}$. Since we here are not concerned with
quantitative comparisons with real experimental data on such systems, we
simplify the problem by adopting a computationally more efficient
$r^{-12}$ potential, in accord with previous work \cite{23,24,25,26}.
Moreover, to render it strictly short-ranged, we introduce a cutoff
$r_c$, such that $V(r \geq r_c)\equiv 0$, and employ a smoothing function
to make $V(r)$ differentiable at $r=r_c$. Thus, the model potential used
is
\begin{equation} \label{eq1}
V(r) = \varepsilon\Big[(\sigma/r)^{12} - (\sigma/r_c)^{12} \Big] \Big[\frac{(r-r_c)^4}{h^4 + (r-r_c)^4}\Big] \quad ,
\end{equation}
with parameters $r_c=2.5\sigma$ and $h=0.01 \sigma$. Henceforth, the
particle diameter $\sigma=1$ defines the length units in our model, and
for the energy scale, $\varepsilon=1$ is taken, while Boltzmann's constant
$k_B=1$. It is known that at $T=0$ the ground state of this model is a
perfect triangular lattice, with a lattice spacing $a$ related to
the choice of number density $\rho=N/V$ (with $N$ the particle number and $V$ the
(two-dimensional) ``volume'' of the system) via
\begin{equation} \label{eq2}
a^2 =2 / (\sqrt{3} \rho) \quad .
\end{equation}
Assuming the physical effect of truncating the potential can be
neglected, only the choice of the combination $X=\rho
(\varepsilon/k_BT)^{1/6}$ controls the phase behavior \cite{49a}.
Thus, following previous work in the NVT-ensemble it
suffices to choose a single density when the temperature variation is
considered \cite{23,50}. For the particular choice $\rho=1.05$, the melting
transition of this model is known to occur at about $T=T_m \approx 1.35$
\cite{50}. Note that here we are not at all concerned with the
peculiarities of melting in two dimensions \cite{51}, and hence we focus
on a temperature deep within the crystalline phase, $T=1$. Although it
is known that the density of vacancies and interstitials in $d=2$ for
any nonzero temperature is also nonzero in thermal equilibrium
\cite{51,52}, for the chosen particle number $N= 3240$ the system is
essentially defect free, since the densities of these point defects at
$T=1$ are extremely small \cite{23,50}.
\begin{figure}
\includegraphics[scale=0.28, clip=true]{fig1.eps}
\caption{\label{fig1} Sketch of the system geometry, showing the fixed
wall particles (black spheres) and the mobile particles (gray spheres).
The orientation of the coordinate axes is indicated, as well as the
lattice spacing of the triangular lattice ($a$) and the linear dimensions
$L_x,D$ of the system.}
\end{figure}
The geometry of the present system is a $D \times L_x$ slit, confined in
the y-direction and periodic in the x-direction. In the y-direction
there are $n_y=30$ rows of the triangular lattice, each containing
$n_x=108$ particles, stacked upon each other. The $x$-direction
coincides with a lattice direction so that $L_x=n_xa$. The
confining boundaries (one at the top and one at the bottom of the
system) each take the form of a double rows of particles in which the
particles are rigidly fixed at the sites of a perfect triangular lattice
(Fig.~\ref{fig1}). These rows of fixed particles represent rigid
corrugated walls, essentially acting as a periodic wall potential on the
mobile particles. While the distance of the first row at the upper wall
from the first row of mobile particles in the ideal stress-free crystal is
simply $D=n_y a \sqrt{3}/2$, in the following we are interested in the response of the
system when the walls occur at a smaller distance, caused by a misfit
$\Delta$, defined via \cite{53}
\begin{equation} \label{eq3}
D=(n_y - \Delta) a \sqrt{3} / 2 \quad .
\end{equation}
As described in the previous work \cite{23,24,25,26}, standard
Monte Carlo simulation \cite{6,7} allows one to study this model at
various values of $\Delta$, and also sample the stress
$\sigma=\sigma_{yy} - \sigma_{xx}$ ($\sigma_{\alpha \beta}$ are the
Cartesian components of the pressure tensor) applying the virial formula
\cite{6,7}. Fig.~\ref{fig2} shows that when one starts out with the
perfect crystal $(n_y=30)$ with no misfit, the crystal already shows a
small finite stress, because the rigid wall particles somewhat hinder
the vibrations of the mobile particles in their potential wells, but
this effect is of no importance here. Rather we focus on the (slightly
nonlinear) increase of the stress up to about $\Delta =\Delta_c \approx
2$, followed by the (almost) discontinuous decrease, and the subsequent
increases again with further enhancement of the misfit. A previous
structural analysis has revealed \cite{23,24,25,26} that the sudden
decrease of stress is due to a transition in the number of rows in the
crystal, $n_y \rightarrow n_y -1=29$. However, since in the NVT ensemble
the particle number is conserved, the $n_x=108$ particles of the row
that disappears must be redistributed among the remaining rows. A
closer examination of the structure revealed that none of these
particles enter the two rows adjacent to the rigid walls, instead they
all go into the $n_y-3=27$ rows of the system that are further away from the
walls. Thus, in the present case, the particle number per row becomes
$n'_x+n_x/(n_y-3)=n_x+4$, and this leads to a new lattice spacing in the
$x$-direction of $a'=a/(1+4/n_x)$, which is no longer commensurate with
the spacing between the particles forming the rigid walls (or the two immediately adjacent
layers which remain commensurate with them). While for the rows in the
center of the system (near $n_y/2)$ this compression of the lattice
spacing occurs uniformly along the $x$-direction, this is not the case
close to the walls, which provide a periodic potential (with periodicity
$a$) that acts on the row of mobile particles a little further inside
the slit. The fact that on the scale $L_x$ the effective wall potential
exhibits $n_x$ minima but that $n'_x=n_x+4$ particles need to be
accommodated, leads to the formation of a lattice of solitons close to
both walls (``soliton staircase'') \cite{54,55}, as depicted for an
idealized case in Fig.~\ref{fig3}.
\begin{figure}
\includegraphics[scale=0.32, clip=true]{fig2.eps}
\caption{\label{fig2} Stress $\sigma$ plotted vs. misfit $\Delta$, for a
system of $N=3240$ particles, and using different starting
configurations having $n_y=30$, $n_y=29$, and $n_y=28$, as indicated in
the figure. Note the huge hysteresis of the $n_y=30 \rightarrow n_y=29$
and $n_y=29 \rightarrow n_y=28$ transitions. For further explanations
see the main text.}
\end{figure}
\begin{figure}
\includegraphics[scale=0.1, clip=true]{fig3a.eps}\\
\includegraphics[scale=0.18, clip=true]{fig3b.eps}\\
\includegraphics[scale=0.18, clip=true]{fig3c.eps}
\caption{\label{fig3} a) Putting $n+1$ particles in a periodic potential with $n$ minima creates a soliton configuration,
i.e. over a range of several lattice spacings particles are displaced from the potential minima (schematic) b) Superimposed snapshot pictures of 750 configurations of the particle positions, where for a system of $n_y=30$ rows and a large misfit ($\Delta=2.6$) a transition to $n_y-1=29$ rows has occured ($n_x=108$ and $T=1.0$ were chosen). The $4$ solitons at each wall are visible due to the larger lateral displacements of the particles, leading to a darker region in the snapshot. Part (c) shows a close-up of the structure near the upper wall. Numbers shown along the axes indicate the Cartesian coordinates of the particles. Parts (b) and (c) have been adapted from Chui et al. \cite{23}.}
\end{figure}
In practice, the actual structure having $n_y-1=29$ rows that is formed
in the simulations on increasing the misfit $\Delta$ beyond the critical
value $\Delta_c$, is generally less regular than the 'idealized' one
shown in Fig.~\ref{fig3}: specifically, the relative distance between
neighboring solitons showed a considerable variation. This comes about
because (i) the solitons are formed from the stressed crystal with
$n_y=30$ rows via random defect nucleation events \cite{24}, and (ii)
the mutual interaction between neighboring solitons, which is the
thermodynamic driving force towards a regular soliton arrangement, is
very small \cite{25}. Despite this, it is nevertheless reasonable to
construct ``by hand'' the expected regular structure of $n_x/(n_y-3) \,
(=4)$ solitons near each wall as a starting configuration for a system
with $29$ rows, which can subsequently be equilibrated \cite{23}. Of
course, there is no guarantee that this guessed structure actually is
the one of lowest free energy; but it does exhibit slightly less stress
than all other structures that had been tested, for misfits in the range
$1.5 \leq \Delta \leq 3$, and hence has been used as a starting point
for studies in which $\Delta$ was varied in this range.
Starting from this idealized $29$ row structure and decreasing the
misfit one finds that the $29$ row structure is stable down to about
$\Delta'_c\approx 1.3$, at which point the soliton lattice disappears
and the system spontaneously transforms into a defect free structure
with $n_y=30$ rows again (Fig.~\ref{fig2}). This value of $\Delta$ is to
be compared with that for the reverse transition from $30$ to $29$ rows
which we recall occurs at $\Delta_c \approx 2.0$. Thus, with the standard
Monte Carlo approach there is considerable hysteresis which precludes
the accurate location of the transition point. Clearly, therefore a
method is needed from which one can locate where the transition occurs
in equilibrium.
\begin{figure}
\includegraphics[scale=0.4, clip=true]{fig4a.eps}\\
\includegraphics[scale=0.4, clip=true]{fig4b.eps}\\
\includegraphics[scale=0.4, clip=true]{fig4c.eps}\\
\includegraphics[scale=0.4, clip=true]{fig4d.eps}\\
\caption{\label{fig4} Configurations with $N=3240$ particles and $n_y-2=28$ rows, but different configurations of the solitons. In the text, they are referenced as ``configuration nr.~1,~2,~3,~4'' from top to bottom. For a clear identification of the positions of the solitons, the method described in \cite{25} was used.}
\end{figure}
Similar hysteresis is observed if one starts out from the $29$ row
structure but increases the misfit beyond $\Delta =3$ (a case that has
not been studied previously). As Fig.~\ref{fig2} shows, a transition occurs
to structures with $n_y-2=28$ rows (at about $\Delta \approx 4.1$).
Unfortunately, there seem to be no unique candidates for stable
structures having $n_y-2=28$. Fig.~\ref{fig4} displays four candidate
structures that we have identified, each of which is at least metastable
on simulation timescales. Depending on which of these $28$ row
candidates one takes, the transition from $28$ to $29$ rows on reducing
the misfit occurs at anything between $\Delta=3.2$ and $3.75$. As
regards the nature of the candidate structures, in each case $2n_x=216$
extra particles have to be distributed across the system. If we again
keep the rows adjacent to the walls free of extra particles, the
particle number per inner row becomes $n'_x=n_x + 2 n_x/(n_y-4)\approx
n_x + 8.3$, i.e. is non-integer. If we kept two rows adjacent to the wall
rows free of extra particles, we would have $9$ extra particles per row,
and thus this structure has been tried (this is configuration number $1$
in Fig.~\ref{fig4}). Another structure was obtained if we place $4$ extra
particles in the rows directly adjacent to the walls and $8$ extra
particles in each of the $26$ inner rows (configuration number $2$). By
energy minimization of a somewhat disordered structure resulting from a
transition from $29$ to $28$ rows a structure was obtained which had $9$
solitons on one wall but only $8$ on the other wall (configuration
number $3$). Finally another configuration with $8$ solitons on each
wall (configuration number $4$) was found. Note that the configurations shown in Fig.~\ref{fig4}
are not the actual structures at $T=1.0$ but the corresponding ``inherent structures''
found from the actual structures by cooling to $T=0$, to clearly display where the solitons occur.
Clearly, it again is a
problem to (i) identify which of these $4$ configurations with $28$ rows
is the stable one (at $T=1.0$), and (ii) determine at which misfit the transition to the
structure with $29$ rows occurs. As we shall demonstrate below, both
problems can be elegantly dealt with by employing the phase switch Monte Carlo
method.
\section{Free energy based simulation methodologies to locate
transitions between imperfectly ordered crystal structures}
\subsection{Thermodynamic Integration}
The general strategy of thermodynamic integration is to consider a
Hamiltonian $\mathcal{H} (\lambda)$ that depends on a parameter
$\lambda$ that can be varied from a reference state (characterized by
$\lambda_0$) whose free energy is known, to the state of interest
$(\lambda_1)$, without encountering phase transitions. The free energy
difference $\Delta F$ can then be written as
\begin{equation} \label{eq4}
\Delta F= F (\lambda_1) - F (\lambda_0) = \int\limits_{\lambda_0}^{\lambda_1} d \lambda' \langle \partial \mathcal{H}(\lambda')
/\partial \lambda' \rangle_{\lambda'} \quad .
\end{equation}
For a dense disordered system (fluid or a solid containing defects),
Schilling and Schmid \cite{15,16} proposed to take as a reference state a
configuration chosen at random from a well equilibrated simulation of
the structure of interest, at values of the external control parameters
for which one wishes to determine the free energy. Particles can be
held rigidly in the reference configuration $\{\vec{r}_i \,^ {\rm ref}\}$
by means of a suitable external potentials. (We recall that a somewhat related
thermodynamic integration scheme for disordered systems known as the
``Tethered spheres method'' has already been proposed by Speedy
\cite{55a}.) When these external potentials act, the internal
interactions can be switched off. In practice, one can use the following
pinning potential $U_{\rm ref} (\lambda)$ to create the reference state,
where $r_{\rm cut}$ is a parameter discussed below.
\begin{equation} \label{eq5}
U_{\rm ref} (\lambda)= \lambda \sum\limits_i \phi (|\vec{r}_i - \vec{r}\;^{\rm ref}_i |/r_{\rm cut}) \quad {\rm with}\,
\phi\, (x)=x-1 \quad .
\end{equation}
Here it is to be understood that particle $i$ is only pinned by well $i$ at
$\vec{r}\;^{\rm ref}_{i}$, and not by other wells. However, identity swaps
need to be carried out to ensure the indistinguishability of particles.
The free energy of this non-interacting reference system then is
\begin{equation} \label{eq6}
F_{\rm ref} (\lambda) =\ln (N/V) -\ln [1+ (V_0/V) g_\phi (\beta\lambda)]\:,
\end{equation}
where $\beta=(k_B T)^{-1}$, $V_0$ (in $d=2$ dimensions) is $V_0= \pi r^2 _{\rm cut} $ and
\begin{eqnarray} \label{eq7}
&& g_\phi (a) = \frac{2}{\lambda^2} [\exp (a) - \sum\limits_{k=0}^2 e^k / k!]\:,\nonumber\\
\end{eqnarray}
for the choice of $\phi(x)$ written in Eq.~(\ref{eq5}).
Then intermediate models $\mathcal{H}(\lambda)$ to be used in Eq.~(\ref{eq4}) are chosen as
\begin{equation} \label{eq8}
\mathcal{H}' (\lambda) = \mathcal{H}_{\rm int} + U_{\rm ref} (\lambda) \quad ,
\end{equation}
where $\mathcal{H}_{\rm int}$ describes interactions in the system,
which then are switched on (if necessary, in several steps). The free
energy contribution of switching on these interactions can easily be
determined by a Monte Carlo simulation which includes a move that
switches the interactions on and off. The logarithm of the ratio of how
many times the states with and without interactions were visited gives
the free energy contribution. The free energy difference between the
intermediate model where particle interactions are turned on and
potential wells are also turned on, and the target system with particle
interactions but without potential wells, then is computed by
thermodynamic integration, for which
\begin{equation} \label{eq9}
\langle \partial \mathcal{H}_{\rm ref} (\lambda) / \partial \lambda \rangle = \langle \sum_i \phi (
|\vec{r}_i - \vec{r}_i\;^{\rm ref}|/ r_{\rm cut}) \rangle
\end{equation}
needs to be sampled \cite{15,16}. This method has been tested for hard spheres \cite{15,16},
including also systems confined by walls from which wall excess free energies could be
sampled \cite{56}.
\subsection{Phase Switch Monte Carlo}
The phase switch method \cite{17,18,19,20,21,21a} computes directly the
relative probabilities of two phases, by switching between them and
recording the ratio of the simulation time spent in each. This ratio
directly yields their free energy difference $\Delta F$ via $\Delta F=
\ln(A^{(1)}/A^{(2)})$. Here $A^{(1)}$ and $A^{(2)}$ are the times spent
in the respective phases which are proportional to the statistical
weight of each phase \cite{9}.
\begin{figure}
\includegraphics[scale=0.32, clip=true]{fig5.eps}
\caption{\label{fig5} Schematic comparison of (a) the standard method
for linking phases via a sampling path and (b) The phase switch method.
The blobs represent the set of values of some macroscopic property (eg
order parameter or energy) associated with configurations belonging to
two distinct phases $(\alpha=1,2)$. These pure phase states (having high
probability) are separated by a ``deep valley'' in the free energy
landscape corresponding to interfacial states having a very low
probability. (a) In the standard strategy one uses extended sampling to
negotiate the valley, by climbing down into it from one side and climbing
up out of it on the other. (b) The idea of phase switch Monte Carlo is
to ``jump over the valley''.}
\end{figure}
The power of the phase switch method derives from its ability to leap
directly from configurations of one pure phase to those of another pure
phase (Fig.~\ref{fig5}), avoiding the mixed phase states which -- when
one or both phases are crystalline -- can be computationally problematic
(see appendix A). The leap is implemented as a suitable global Monte
Carlo move. One starts out by specifying for each of the two phases of
interest (labeled by index $\alpha=1,2$), a reference configuration. This
can be expressed as a set of $i=1\ldots N$ particle positions $\{
\vec{R}_i^{\,(\alpha)}\}$. Note that the specific choice of a reference
configuration for phase $\alpha$ does not matter (at least in principle, see
Appendix), it need only be a member of the set of pure phase
configurations that ``belong'' to phase $\alpha$. Thus for example in the
present case, a suitable reference configuration for the $n=30$ row
defect-free structure could simply be a typical configuration chosen
from a simulation run on this structure. However, it could equally be a
configuration in which all particles are at the lattice sites of this
structure.
Given the two reference configurations, one can express the
position vectors $\vec{r}_i^{\,(\alpha)}$ of each particle $i$ in phase $\alpha$ as
\begin{equation}
\vec{r}_i^{\,(\alpha)}= \vec{R}_i^{(\alpha)} + \vec{u}_i\:.
\end{equation}
where $\{\vec{u}_i\}$ is a set of displacement vectors which measure the
deviation of each particle from the reference site to which it is
nominally associated. Note that while there is a separate reference
configuration for each phase, the single set of displacements is
common to both phases.
Let us suppose the simulation is currently in phase $\alpha=1$. Now the phase
switch idea is to a map the current configuration $\{\vec{r}_i^{\,(1)}\}$ of this
phase on to a configuration of phase $\alpha=2$ by switching the sets of
reference sites from $\{\vec{R}_i^{\,(1)}\}$ to $\{\vec{R}_i^{\,(2)}\}$ but
keeping the set of displacements $\{\vec{u}_i\}$ {\em fixed}. This
switch can be incorporated in a global Monte Carlo move. Of
course, in general the set displacements that are typical for phase
$\alpha=1$ will not be typical displacements for phase $\alpha=2$.
As a consequence, in a naive implementation such a global move will
almost always be rejected by the Monte Carlo lottery. This problem is
circumvented by employing extended sampling methods \cite{9,10,56a} that
create a bias which enhances the occurrence of displacements
$\{\vec{u}_i\}$ for which the switch operation does have a sufficiently
high Monte Carlo acceptance probability. Such states are called
``gateway states'' \cite{17,18,19,20,21}: crucially, they do not need to
be specified beforehand - the system autonomously guides itself to them
in the course of the biased sampling.
In practice, the bias is administered with respect to an ``order
parameter'' $M$ whose instantaneous value is closely related to the
energy cost of implementing the phase switch. One then introduces a
weight function $\eta(M)$ into the sampling of the effective Hamiltonian
which enhances the probability of the system sampling configurations for
which the energy cost of the phase switch is low, thereby increasing the
switch acceptance rate. Of course, the weight function $\eta(M)$ to be
used is not known beforehand, and thus needs to be iteratively
constructed in the course of the Monte Carlo sampling. One has a choice
of ways of doing so: we have used the transition matrix Monte Carlo
method \cite{56a,57,58} (see also the Appendix for implementation
details). Alternative methods such as Wang-Landau sampling \cite{59} or
successive umbrella sampling \cite{73} could also be applied.
Once a suitable form for the weight function $\eta(M)$ has been found, a
long Monte Carlo run is performed, in the course of which both phases
are visited many times. The statistics of the switching between phases
is monitored by accumulating the histogram of $M$, which (as in all
extended sampling methods) is corrected for the imposed bias at the end
of the simulation. Doing so yields an estimate of the true equilibrium
distribution $P(M)$, which in general exhibits a double peaked form (one
peak for each phase). The free energy difference between the two phases
is simply the logarithm of the ratio of the peak weights as described at
the start of this subsection.
Of course, the above description was only intended to outline the
phase switch strategy; more extensive implementation details are given
in the appendix.
\section{Results}
\subsection{Free energy differences and computational efficiency}
Fig.~\ref{fig6} shows the absolute free energies in the NVT ensemble for
the phase with 30 rows (and no defects) and the phase with 29 rows and
the ``soliton staircases'' (Fig.~\ref{fig3}b) as a function of the
misfit $\Delta$, as obtained from the thermodynamic integration method
(Sec. III.1). One sees that these free energies are very large (note the
ordinate scale) and vary rather strongly with $\Delta$. However, the
free energy curves with these two structures are barely distinct from
each other, and hence a very substantial computational effort is needed
to locate, with meaningful accuracy, the intersection point marking the
equilibrium transition between $n=30$ and $n=29$ rows.
\begin{figure}
\includegraphics[scale=0.32, clip=true]{fig6.eps}
\caption{\label{fig6}Absolute free energy $F$ of systems of $N=3240$
particles interacting with the potential given in Eq.~(\ref{eq1}) in $L
\times D$ geometry with $L=108 a$, $a$ being the lattice spacing, and
periodic boundaries in $x$-direction, confined by two rows of fixed
particles on either side in $y$-direction (Fig.~\ref{fig1}, as a
function of the misfit $\Delta$ \(Eq.~(\ref{eq3})\). Two structures are
compared:(i) a (compressed) triangular lattice with $n_y=30$ rows
containing $n_x=108$ particles per row; (ii) a lattice with $n_y=29$
rows and corresponding soliton staircases (Fig.~\ref{fig3}b).}
\end{figure}
\begin{figure}
\includegraphics[scale=0.32, clip=true]{fig7.eps}
\caption{\label{fig7} Free energy differences between structures with 29 and 30 rows plotted versus the misfit $\Delta$. Both results obtained from thermodynamic integration and from the phase switch method are shown, as indicated.}
\end{figure}
Fig.~\ref{fig7} plots the free energy difference $\Delta F$ versus the
misfit, comparing the results from the thermodynamic integration method
(points with error bars) with the results from the phase switch method,
and focusing on the region near the transition. One can see that within
the errors the results of both methods agree very well with each other,
although for the thermodynamic integration method the error is at least
an order of magnitude larger than that of phase switch. We note that the
predicted equilibrium value of the misfit at the transition point ($\Delta_t
\approx 1.7)$ falls well within the hysteresis loop of Fig.~\ref{fig2}.
Since the absolute free energies are of the order of 20000 (for our
system with $N=3240$ particles) but, in the region of interest, free
energy differences are of order $\pm 40$ only, we have that the relative
error $\delta F/F$ is of order $1/500$. Thus for thermodynamic
integration, it would be difficult to bring the error bars down further
in Fig.~\ref{fig7}. The error bars for the phase switch simulation were
computed from the results of four independent runs for each value of the
misfit, and are hardly visible on the scale of Fig.~\ref{fig7}.
In addition to this significant difference with respect to the size of
the statistical errors, phase switch Monte Carlo also outperformed the
thermodynamic integration method with respect to the necessary
investment of computer resources. In order to obtain a suitable weight
function for our system, at a certain value of the misfit, we let the
simulation run for about 15 million steps (each step consisting of one
sweep of local moves and one attempt to switch the phases). On the ZDV
cluster of the University of Mainz, this takes about $4.5$ days on a
single core (though in hindsight we could have got away with a less
smooth weight function, further reducing the computing time of this
step). Having determined the weight function, we initiated four
production runs for every value of the misfit. These runs needed again
10 million steps each (i.e. about 3 days each) in order to perform a
sufficient number of phase switches to yield results of the desired
precision. Overall, then, computing each point of the free energy difference
curve of Fig.~\ref{fig7} by phase switch took about $16.5$ days of CPU time.
In contrast to this, the thermodynamic integration method required a
calculation not only of the free energy difference in which we are
interested, but of the free energy difference along the path of the
thermodynamic integration, gradually switching off the wells of
attraction used there, and of the free energy difference between the
state where the particle interactions were turned on and the state where
they were turned off. This needs to be done for both phases separately.
It is therefore not surprising, that considerably more CPU time was
needed: roughly $250$ days of CPU time were invested for each phase and
for each value of the misfit to obtain the absolute free energy (again
converting units to a single core). Thus, each of the 12 values of free
energy differences needed for Fig.~\ref{fig7} required 500 days (rather
than $16.5$ days), i.e. a factor of $30$ more computational effort!
However, if we were to bring the statistical errors of the
thermodynamic integration method a factor of 10 down (to make it
comparable to the phase switch method), we would need another factor of
100 in computer time; the benefit of using the (clearly much more
powerful) phase switch approach hence amounts to a gain of the
order of 10$^3$ in computational resources! Of course, this is no
surprise when we remember that the free energy differences of interest
are only of the order of (1/500) of the total free energies for the
present model system.
\subsection{Ensemble inequivalence}
\begin{figure}
\includegraphics[scale=0.25, clip=true]{fig8.eps}
\caption{\label{fig8} Schematic description of phase transitions in thin films of thickness $D$ in the conjugate NpT
(left) and NVT (right) ensembles, for the case of a vapor to liquid transition (a) and the present transition where the number of rows is reduced $(n \rightarrow n -1)$ when either the (normal) pressure $p$ increases (left) or the thickness decreases (right). Note that in the latter case two-phase coexistence is possible for the vapor-liquid transition, but
not for the transition where the number of rows parallel to the boundaries change. For further explanations cf text.}
\end{figure}
We turn now to a discussion of a puzzling aspect of the physics, namely
the fact that we treat here a first-order structural phase transition
obtained by variation of the distance $D$ between the walls formed by
the rigidly fixed particles, i.e. an {\it extensive} rather than an {\it
intensive} thermodynamic variable. If we were concerned with the study
of a vapor to liquid transition of a fluid in such a geometry, the
proper way to locate a discontinuous transition is the variation of the
intensive variable thermodynamically conjugate to $D$, which is the
normal pressure $p_N$ (force per area acting on the walls; in the
following the index $N$ will be omitted. Of course, at fixed lateral
dimensions $L$ a variation of $D$ is equivalent to a variation of the
volume $V$).
\begin{figure}[h!]
\includegraphics[scale=0.3, clip=true]{fig9a.eps}\\
\includegraphics[scale=0.3, clip=true]{fig9b.eps}\\
\includegraphics[scale=0.3, clip=true]{fig9c.eps}
\caption{\label{fig9} a) Free energy difference $\Delta F$ for the transition from $n=30$ to $n=29$ rows as a
function of pressure. (b) The distribution of the internal energy difference between the two phases $p(E_{30 rows} -E_{29 rows})$ at fixed $\{\vec{u}\}$. Curves for $4$ pressures near and at the transition pressure $p_t=22.146 \pm 0.015$ are shown, as generated via histogram reweighting. The simulation was run at a pressure of $p=22.13$. (c) System length $D$ as a function of pressure. Clearly, the curve for the stable phase exhibits a jump at the transition pressure. Statistical errors are smaller than the symbol sizes.}
\end{figure}
To fix ideas, we remind the reader about this classical vapor-liquid
problem in Fig.~\ref{fig8}a): In the NpT ensemble, we would have a jump
in volume $V=LD$ from $V_v=LD_v$ (density of the vapor $\rho_v=N/V_v)$
to $V_\ell=LD_\ell$ (density of the liquid $\rho_\ell=N/V_\ell)$ at the
transition pressure $p_t$. If we work in the conjugate NVT ensemble, of
course, the behavior simply follows from a Legendre transform, the
volume jump from $V_v$ to $V_\ell$ translates into a horizontal plateau
at $p=p_t$, and any state of this plateau is a situation of two-phase
coexistence, as schematically indicated in Fig.~\ref{fig8}a).
Of course, it is also possible to consider the present transition
between a state of $n$ rows to $n-1$ rows in the NpT ensemble
(Fig.~\ref{fig8}b and Fig.~\ref{fig9}c). Then it is clear that the transition will show up as
a jump in the thickness $D$ from $D_n(=na_n)$ to $D_{n-1}\, (=(n-1)
a_{n-1})$, where $a_n$, $a_{n-1}$ are the (average) distances between
the lattice rows (or lattice planes, in three dimensional films,
respectively). The corresponding phases of the $n$-layer state and
$(n-1)$ layer state are indicated below the isotherm in the $(p-D)$ plane
schematically.
However, one simply cannot construct a state of two-phase coexistence
out of these two ``pure phases'' at a value of $D$ intermediate between
$D_{n-1}$ and $D_n$: locally the $n$-layer state requires a thickness
$D_n$, the $(n-1)$ layer state a thickness $D_{n-1}$, so one would have
to ``break'' the walls. Of course, it is not just sufficient to have a
state with $n$ layers separated by a grain boundary from a state with
$(n-1)$ layers at the same value of $D$: these domains are {\it not} the
coexisting pure phases in the NpT ensemble!
So the phase coexistence drawn (horizontal broken curve) in
Fig.~\ref{fig8}b) is unphysical, it requires a state where the
constraining walls were broken. Requesting the integrity of the walls is
a global constraint which makes phase coexistence in the standard sense
impossible for the present transitions! Thus, the rule that the
different ensembles of statistical mechanics yield equivalent results in
the thermodynamic limit is not true for the present system; in the
transition region $D_{n-1} < D < D_n$ the NVT ensemble and the NpT
ensemble are {\it not equivalent}.
Actually this is not the first time that such an ensemble inequivalence
has been pointed out. A case much discussed in the literature is the
``escape transition'' of a single polymer chain of $N$ beads grafted at
a planar surface underneath a piston held at a distance $D$ above the
surface to compress the polymer \cite{61,62,63,64,65,66,67}. For pressures
$p<p_t$ (where the piston is at distance $D_{t,1}$) the chain is
completely confined underneath the piston (which has the cross section
of a circle in the directions parallel to the surface) while for $p >
p_t$ the chain is (partially) escaped into the region outside of where
the piston acts (the piston distance at $p_T$ jumps to a smaller value
$D_{t,2}$). When we use instead $D$ as the control variable, again a
sharp transition occurs (for $N \rightarrow \infty$) at some
intermediate value $D_t$ $(D_{t,2}< D_t <D_{t,1})$, since obviously it is
simply inconceivable to have within a single chain phase coexistence
between states ``partially escaped'' and ``fully confined'', since these
states are only defined via a global description of the whole polymer
chain.
Another case where transitions of the number $n$ of layers in layered
structures in thin films occurs is the confinement of symmetric block
copolymer melts (which may form a lamellar mesophase of period
$\lambda_0$ in the bulk) in thin films between identical walls
\cite{68,69,70,71}. When then the thickness $D$ of such films is varied,
one observes experimentally discontinuous transitions in the number $n$
of lamellae parallel to the film \cite{69,70}. However, when one
considers block copolymer films on a substrate and does not impose the
constraint of a uniform thickness but rather allows the upper surface to
be free, then indeed mixed phase configurations of a region where $n-1$
layers occur (and take a thickness $D_{n-1})$ and of a region where $n$
layers occur (and take a thickness $D_n$) are conceivable \cite{71} and
have been observed, see e.g. \cite{72}. In summary of these remarks, we
note that it is not uncommon that global geometric constraints may
destroy the possibility of phase coexistence.
In view of the above discussion, it is of interest also in the present
case to investigate the use of the (normal) pressure $p$ (instead of the
strip width $D$) as the control variable. Taking, in the spirit of the
general remarks on the phase switch method, the appropriate phase switch
energy cost as an order parameter $M$, we can sample the probability
distribution function $p(M)$ which exhibits two well separated peaks of generally different
weights. These peaks are even more clearly visible in the distribution of the energy difference $p(E_{30 rows} -E_{29 rows})$ at fixed $\{\vec{u}\}$ as the order parameter $M$ is related to this energy difference via a logarithmic function (cf. eq.~\ref{def_M}). The transition pressure $p_t$ is that for which the peaks have
equal weight (Fig.~\ref{fig9}) and can be determined accurately via histogram
reweighting. From this we estimate that $p_t=22.146
\pm 0.015$. At the transition, the measured misfit $\Delta$ jumps from
$\Delta_1=1.913 \pm 0.043$ (for $n=30$) to $\Delta_2=1.503 \pm 0.046$
(for $n=29$). Interestingly, the misfit where the transition in the NVT
ensemble occurs ($\Delta_t \approx 1.71)$ is just the average of these
two values.
\subsection{Comparison of competing candidate stable structures}
\begin{figure}[h!]
\includegraphics[scale=0.32, clip=true]{fig10.eps}
\caption{\label{fig10} Free energy differences between various structures with $n=28$ rows and the structure with $n=29$ plotted vs. the misfit $\Delta$. As configurations nr. 2 and nr. 4 turned out to be the same, their free energy curves fall on top of each other.}
\end{figure}
Returning again to the NVT ensemble, we now consider the transition from
states with 29 layers to states with 28 layers. We recall
(Fig.~\ref{fig4}) that several different candidate structures do exist,
and it is not at all clear {\em a-priori}, which of them should be
favored. Again, the phase switch Monte Carlo is a convenient tool to
solve such a problem: we utilize reference states from all four of the
candidate structures having $n=28$ (as shown in Fig.~\ref{fig4}) and
calculate the free energy difference $\Delta F$ between the (unique)
structure with $n=29$ and these four candidates.
The results (Fig.~\ref{fig10}) clearly show that configurations number
$1$ and number $3$ are metastable, because they have distinctly higher
free energy differences throughout the range of $\Delta$ than
configurations number $2$ and $4$ which practically coincide. In fact,
this coincidence between the free energies of configurations nr. $2$ and $4$ is not accidental: a closer evaluation of
their time evolution shows that they transform into each other via
sequences of ``easy'' local moves, and although the instantaneous
snapshot pictures reproduced in Fig.~\ref{fig4} were different, they do
not belong to different phases in a thermodynamic sense.
It is also interesting to note that the conclusion that structure number
2 is the stable one would not have been obtained by a simply comparison
of the internal energies of the four structures: indeed configuration
number 2 has the highest energy of all four structures.
Thus, entropy matters in soft crystals, such as those studied here.
\section{Concluding remarks}
The principle findings of our study are two-fold: (i) We have
performed a thorough test of the suitability of the phase switch Monte
Carlo method for the task of determining the relative stability of
imperfectly ordered structures of typical soft-matter systems, where one
must deal with systems which have at least one very large linear
dimension. For such a test, it is crucial to provide full information on
the model that is studied, and to give a careful description of the
method and its implementation. Moreover we have studied precisely the
same model system by a thermodynamic integration method thereby allowing
the first like-for-like comparison between the two approaches. We find
that the results from both methods are compatible, but the accuracy that
can be achieved using phase switch MC is at least an order of magnitude
better (Fig.~\ref{fig7}), despite requiring a factor of $30$ less
computational time.
The reasons for this efficiency gain can be appreciated from a glance at
Fig.~\ref{fig6}: the absolute free energies of our system of $3240$
particles vary from about $22000$ to $24000$ (in suitably scaled units),
for a misfit parameter $\Delta$ varying from $1$ to $2$, while the free
energy difference between the two states that we wish to compare vary
only from $-60$ to $+60$ in the same range. These numbers illustrate
vividly the basic concept of phase switch Monte Carlo: one does better
in focusing directly on the small free energy difference between the
states that one wishes to compare, rather than extracting them
indirectly by subtracting two measurements of large absolute free
energies. Thus (in the present context at least) phase switch Monte
Carlo seems a much more powerful approach than thermodynamic
integration. In fact, if one were to try to bring the errors of the
thermodynamic integration method down by an order of magnitude -- to make
the error bars of both methods in Fig.~\ref{fig7} comparable -- one would
have to invest a factor of 3000 more computational time. We feel that
the case of relatively small free energy differences between competing
phases and/or structures is rather typical for soft matter systems.
Indeed for many soft matter systems, such as block copolymer mesophases,
the relative magnitude of free energy differences is much less than the
factor of about $1/500$ encountered here, and hence such problems could
never be tackled successfully with thermodynamic integration methods
since the computational effort to reach the requisite accuracy would be
prohibitive.
The first problem to which phase switch Monte Carlo was applied (in the
form of the "Lattice-switch" method), evaluated the free energy
difference of perfectly ordered face-centered cubic and hexagonal close
packed crystals. Such an application might be regarded as a somewhat
special case due to the perfect long-range order in these defect-free
crystals. However, the present work shows that the method can equally be
applied to imperfectly ordered crystals. Here, due to the confinement by
structured walls together with a misfit between the distance between the
walls and the appropriate multiple of the distance between the lattice
rows, somewhat irregular long range defect structures form along the
walls (``soliton staircase''). Additionally several similarly
ill-crystallized structures can present themselves as candidates for the
optimal structure (Fig.~\ref{fig4}). It would be absolutely impossible
to identify which is the equilibrium structure and which structures are
only metastable without the phase switch Monte Carlo method
(Fig.~\ref{fig10}).
We note that the model system that we have chosen to study
(Fig.~\ref{fig1}) could also be experimentally realized in colloidal
dispersions, though with some effort: colloids coated with polymer
brushes experience a short ranged, almost hard-sphere-like, repulsive
effective potential, and bringing them to an interface where water is on
top and air is below, rather perfect two-dimensional crystals with
triangular lattice structure form. Interference of strong laser fields
can be used to create a periodic confining potential, through which the
misfit and thus the crystal structure can be manipulated. We hope that
our study will solicit some corresponding experimental studies to show
that the proposed transitions in the number of rows in these crystalline
strips actually occur.
(ii) Our second main finding is that this type of system has an
interesting physical property, namely the inequivalence between
conjugate ensembles of statistical mechanics. When we fix the distance
$D$ between the confining ``walls'', the total particle number $N$ and
the total (two-dimensional) ``volume'' $V$ of the system, we realize the
NVT ensemble. When one studies first order transitions in the bulk using
such an ensemble containing two extensive variables ($N$, $V$), a first
order transition normally shows up as a two-phase coexistence region
(e.g., at fixed $N$ the two-phase coexistence extends from $V_I$ to
$V_{II}$). However, here such a two-phase coexistence is not possible
(Fig.~\ref{fig8}), and thus one has the unusual behaviour that at the
equilibrium in the ``constant $D$''-ensemble the conjugate intensive
variable (the normal pressure $p_N$, as well as the stress $\sigma$, cf.
Fig.~\ref{fig2}) exhibit jumps (in Fig.~\ref{fig2}, we display the
hysteresis loops, but the positions of the jumps in equilibrium can be
inferred from $\Delta F=0$ in Figs.~\ref{fig7} and \ref{fig10},
respectively). When we use a ``constant $p$''-ensemble (which is
physically reasonable if the confinement of the crystal is effected
mechanically in a Surface Force Apparatus), it is the ``volume'' (i.e.,
the distance between the walls $D$) which jumps from $D_I$ to $D_{II}$
at a well-defined transition pressure, cf. Figs.~\ref{fig8},~\ref{fig9}.
One should not confuse this ensemble inequivalence with the well-known
ensemble inequivalence between NVT and NpT ensembles in systems where
$N$ is finite: in the latter case, the ensemble inequivalence is
dominated by interfacial contributions (in the NVT-ensemble, when $V_I <
V < V_{II}$, the system is in a two-phase configuration, as suggested
for $V \rightarrow \infty$ by the ``lever rule'', but for finite $V$ the
relative contribution due to the interface between the coexisting phases
dominate the finite size effects). But for $V \rightarrow \infty$ these
interfacial effects become negligible, the properties in the two
conjugate ensembles are just related by the appropriate Legendre
transformation. This equivalence between the ensembles holds also for
liquid-vapor or liquid-liquid unmixing under confinement in a thin film
geometry: when $D$ is finite and the particle number $N \rightarrow
\infty$, i.e. the lateral linear dimensions become macroscopic, we still
have ordinary two-phase coexistence in the thin films (cf.
Fig.~\ref{fig8}). The ensemble inequivalence in the present system
arises from the lack of commensurability between the thickness $D$ of
the slit and the appropriate multiple of the lattice distance. At a
transition pressure $p_t$ in the NpT ensemble we inevitably have
different distances $D_I$, $D_{II}$ between the walls for the two phases
$I$, $II$. Thus, they cannot coexist for any uniform value of $D$.
Similar phenomena (where the number of layers of a layered lamellar
structure confined between walls exhibits jump discontinuities when $D$
is varied) are already known, both experimentally and theoretically, for
block copolymer mesophases, but the aspect of ensemble inequivalence has
not been addressed, to our knowledge, in these systems studied here.
\section{Acknowledgements}
One of us (D.W.) acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG) under grant number TR6/C4 and from the
Graduate School of Excellence ``Material Science in Mainz (MAINZ)''. She
is also grateful to the Department of Physics, University of Bath
(UK), for its hospitality during an extended research stay under the
auspices of the visiting postgraduate scholar scheme. We thank P.
Virnau, T. Schilling, F. Schmid and I.M. Snook for helpful discussions
and advice.
\clearpage
\section{Appendix}
|
{
"timestamp": "2012-03-09T02:02:54",
"yymm": "1203",
"arxiv_id": "1203.1794",
"language": "en",
"url": "https://arxiv.org/abs/1203.1794"
}
|
\section*{Introduction}
Let $X$ be an algebraic variety of general type, over the complex field.
The dominant rational maps of finite degree
$X \dasharrow Y$ to varieties of general type,
up to birational isomorphisms $Y \dasharrow Y'$,
form a finite set. We call this the {\em finiteness theorem for rational maps
on a variety of general type}. The proof follows from the approach of
Maehara \cite{M} joined with some recent advances in the theory of
pluricanonical maps, due to Hacon and McKernan \cite{HK}
and to Takayama \cite{Tak}, \cite{Tak2}.
In our paper \cite{GP}, motivated by the wish of some effective estimate
for the finite number of maps in the theorem, we provided some update
and refinement in the treatment of the subject.
We brought the rigidity theorem to a general form,
avoiding certain technical restrictions, we pointed out the role of
the canonical volume ${\rm vol}(K_X)$ in bounding the
rational maps in the finiteness theorem, and we proposed
a new argument leading to a refined version of the theorem.
However, something still not satisfactory was the use of a certain bunch
of subvarieties of Chow varieties as a parameter space for rational maps,
as in Maehara's approach is too. The most natural and simple parameter space
should be the space of linear projections in a suitable projective space,
already appearing for instance in the work
of Kobayashi and Ochiai \cite{KO}.
In the present paper we are able to replace the Chow parametrization
with the natural parametrization, and this leads to some new insight
into the geometry of the finiteness theorem. The main result concerns the
structure of the special birational equivalence classes of maps viewed
as unions of connected components of a certain space of linear rational maps,
see Theorem \ref{connectedcomponent}.
This has as an immediate consequence a better refined finiteness theorem,
see Theorem \ref{finiteness}.
\bigskip
\small \noindent {\em Acknowledgements.}
The first author is partially supported by:
Finanziamento Ricerca di Base 2008 Univ. Perugia.
The second author is partially supported by:
1) INdAM (GNSAGA);
2) FAR 2010 (PV):{\em ``Variet\`{a} algebriche, calcolo
algebrico, grafi orientati e topologici"}.
\normalsize
\section{Preliminary material}
\subsection*{a. Results on pluricanonical maps}
A recent achievement in the theory of pluricanonical maps is the following
theorem of uniform pluricanonical birational embedding, due to
Hacon and McKernan \cite{HK} and to Takayama \cite{Tak}.
\begin{thm} \label{HKT}
For any dimension $n$ there is some positive integer $r_n$ such that: for
every $n$-dimensional variety $V$ of general type the multicanonical divisor
$r_nK_V$ defines a birational embedding $V \dashrightarrow V' \subset
\mathbb P^M$.
\end{thm}
A basic tool is the canonical volume of a variety,
the invariant arising in the
asymptotic theory of divisors, see Lazarsfeld's book \cite{L}.
In terms of the canonical volume we have a bound
\begin{equation} \label{degvol}
\deg V' \leq {\rm vol} (r_{n}K_{V}),
\end{equation}
see \cite{HK}, Lemma 2.2.
Moreover from elementary geometry we have a bound
\begin{equation} \label{embdim}
M \leq \deg V' +n-1.
\end{equation}
Note that the embedded variety $V'$ needs not be smooth.
Intimately related to the theorem above is the following
result, proved in \cite{HK} and in \cite{Tak}.
\begin{thm} \label{HK}
For any dimension $n$ there is some positive number $\epsilon_n$
such that every $n$-dimensional variety $V$ of general type has
${\rm vol} (K_{V}) \geq \epsilon_n$.
\end{thm}
For instance, concerning the minimum $r_n$ we know from the classical theory
that $r_1=3$ and $r_2=5$, and a recent result is that $r_3 \leq 73$, while
concerning the maximum $\epsilon_n$
it is clear that $\epsilon_1 = 2$ and $\epsilon_2 = 1$ and a recent
result is $\epsilon_3 \geq 1/2660$, see J. A. Chen and M. Chen \cite{CC}.
Note that \cite{HK} and \cite{Tak} do not give
explicit bounds for $r_n$ and $\epsilon_n$ in the theorems above.
\subsection*{b. Bounds for the degree of a rational map}
Let $f: X \dasharrow Y$ be a rational map of finite degree
between varieties of general type.
Because of Theorem \ref{HKT}, taking the $r_n$-canonical
birational models $X'$ and $Y'$ in $\mathbb P^M$
(note that $Y'$ lies within the embedding space of $X'$),
the map $f$ is identified with a {\em linear rational map} $X' \dasharrow Y'$,
a rational map which is the restriction of a linear projection
$\mathbb P^M \dasharrow \mathbb P^M$.
For a linear map of finite degree the inequality
$\deg f \, \deg Y' \leq \deg X'$ holds.
Using (\ref{degvol}) it follows that
\begin{equation} \label{deg1}
\deg f \leq \deg X' \leq (r_n)^n\, {\rm vol} (K_X).
\end{equation}
A more precise estimate is as follows. For any rational map of finite degree
the inequality $\deg f \; {\rm vol} (K_Y) \leq {\rm vol} (K_X)$ holds,
see \cite{GP}, Proposition 3.2. Using Theorem \ref{HK} it follows that
\begin{equation} \label{deg2}
\deg f \leq \dfrac{1}{\epsilon_n}\, {\rm vol} (K_X).
\end{equation}
This bound is sharp for curves, and in this case it reduces
to the usual bound from the Hurwitz formula.
\subsection*{c. Families of rational maps}
Let $T$ be a smooth variety.
If $X \rightarrow T$ is a relative scheme over $T$,
we denote by $X(t)$ the scheme fibre over $t$, and
by $X_t$ the associated reduced scheme.
A {\em family of varieties}, parametrized by a smooth variety $T$,
is a surjective morphism $X \rightarrow T$, with $X$ a variety,
such that every scheme fibre $X(t)$ is: $(i)$ irreducible, $(ii)$ generically smooth
(in order to be assigned multiplicity one in the associated algebraic cycle,
see Fulton \cite{F}, Chap. 10),
and $(iii)$ of dimension equal to the relative dimension of $X$ over $T$, of course.
When the structure morphism is projective or smooth,
we speak of a family of projective varieties or a family of smooth varieties.
A {\em family of rational maps} is the datum of
a family of varieties $X \rightarrow T$ and
a relative scheme $X' \rightarrow T$, over the same smooth variety $T$,
and a rational map $f: X \dasharrow X'$, commuting with the structural
projections, which for every $t \in T$
restricts to a rational map $f_t: X_t \dasharrow X'_t$.
\subsection*{d. The rigidity theorem}
A family of rational maps {\em on a fixed variety} $X$ is the datum of a
relative scheme $Y \rightarrow T$, with $T$ smooth, and a rational map
$$f: X\times T \dasharrow Y$$
which is a family of rational maps $f_t: X \dasharrow Y_t$
in the sense of the previous definition.
A {\em trivial family} is one which is obtained as follows.
Let $h: X \dasharrow U$ be a rational map and
let $g: T \times U \dasharrow Y$ be a birational isomorphism
which is a family of birational isomorphisms $g_t: U \dasharrow Y_t$.
Then the composite map
$$T \times X \overset{1 \times h}{\dasharrow} T \times U
\overset{g}{\dasharrow} Y$$
is a trivial family, because
all maps $g_t \circ h$ are birationally equivalent.
Recall that two dominant rational maps
$f: X \dasharrow Y$ and $f': X \dasharrow Y'$,
defined on the same variety, are {\em birationally equivalent} if there is
a birational isomorphism $g:Y \dasharrow Y'$ such that $f' = g \circ f$.
For projective varieties of general type and
dominant rational maps of finite degree
there are results of rigidity.
\begin{thm} \label{rigidity}
Let $X$ be a smooth projective variety of general type.
Let $T$ be a smooth variety, let $Y \rightarrow T$ be a family of
smooth projective varieties of general type, and let
$f: X\times T \dasharrow Y$ be a family of rational maps of finite degree.
Then $f$ is a trivial family, so all maps $f_t$ are birationally equivalent.
\end{thm}
The rigidity theorem above was proved by Maehara \cite{M}
with some technical restrictions, and has been brought to the
present form in our previous paper \cite{GP}, Theorem 2.1.
More generally, if the family of image varieties is not known
to be a smooth family, one has the following.
\begin{cor} \label{weakrigidity}
Let $X$ be a projective variety of general type.
Let $T$ be a smooth variety, let $Y \rightarrow T$ be a family of
projective varieties of general type, and let
$f: X\times T \dasharrow Y$ be a family of rational maps of finite degree.
There is a nonempty open subset $T'$ of $T$ such that the restriction
$f|_{T'}: X\times T' \dasharrow Y|_{T'}$ is a trivial family.
\end{cor}
\section{Graphs and images in a family of maps}
Let $f: X \dasharrow X'$ be a family of rational maps
parametrized by a smooth variety $T$, as in \S 1.c.
Consider the relative product $X \times_{T} X'$
and call $p$ and $p'$ the projections to $X$ and $X'$.
{\em Assume now that $X \rightarrow T$ is a projective morphism}.
Thus $p'$ is a closed map. Then define: \medskip
\begin{tabular}{rl}
$\Gamma$ & the closed graph of $f$ in $X \times_{T} X'$, \\
$Y$ & the closed image of $X$ in $X'$, \\
$C$ & any closed subscheme of $X$ such that
$X \smallsetminus C \rightarrow T$ is surjective \\
& and $f$ is a regular map $X \smallsetminus C \rightarrow Y$, \\
$E$ & the inverse image of $C$ in $\Gamma$.
\end{tabular}
\medskip
\noindent Note that $p'(\Gamma) = Y$, as $p'$ is a closed map.
A natural question is whether $\Gamma \rightarrow T$
is the family of closed graphs for the given family of maps,
more precisely: whether $\Gamma \rightarrow T$ is a family of varieties,
as in \S 1.c, and every reduced fibre $\Gamma_t$
coincides with the {\em closed graph} $\Gamma(f_t)$.
A related question is whether $Y \rightarrow T$
is the family of closed images $\overline{f_t(X_t)}$, that is: whether
$Y \rightarrow T$ is a family of varieties and
every reduced fibre $Y_t$ coincides with the {\em closed image} $\overline{f_t(X_t)}$.
The following equality of reduced schemes holds:
$$\Gamma_t = \Gamma(f_t) \cup E_t$$
and from this, applying $p'$, a description of $Y_t$ follows.
\begin{prop} \label{familyofgraphs}
In the setting above, assume that $T$ is a smooth curve.
$(1)$ There is a nonempty open subset $T'$ of $T$ such that
$\Gamma|_{T'} \rightarrow T'$ is the family of closed graphs
for the restricted family $f|_{T'}$.
$(2)$ There is a nonempty open subset $T''$ of $T'$ such that moreover
$Y|_{T''} \rightarrow T''$ is the family of closed images
for the family $f|_{T''}$.
\end{prop}
\begin{proof}
We start with an easy remark.
Let $V \rightarrow T$ be a surjective morphism of varieties,
with irreducible fibres, all of the same dimension.
Then there is a nonempty open subset $T'$ of $T$ such that
the restriction $V|_{T'} \rightarrow T'$ is a family of varieties.
Now we apply this to the relative varieties $\Gamma$
and $Y$ over the curve $T$.
In order to prove the statement we only need to identify the
reduced fibres $\Gamma_t$ and $Y_t$ for sufficiently general $t$.
This is what we do in the following.
(1) First, we show that $\Gamma_t = \Gamma(f_t)$ holds for every $t$
if $E \rightarrow T$ is a flat morphism.
Recall that this happens if and only if
every irreducible component of $E$ dominates $T$.
Write $\dim X =: n+1$. We have $\Gamma_t = \Gamma(f_t) \cup E_t$. Remark that
$\dim E < n+1$. Then $\dim E_t < n$ for every $t$, because of flatness.
But all components of $\Gamma_t$ must have dimension $=n$ for every $t$.
Thus $E_t$ is not a component and $\Gamma_t = \Gamma(f_t)$, for every $t$.
In particular, every $\Gamma_t$ is irreducible of dimension $n$.
In the present situation, the statement follows from the remark in the beginning.
In the general case, by generic flatness, we have that
$E|_{T'} \rightarrow T'$ is flat for some $T'$ and then,
because of the remark, the statement follows.
(2) We know that $Y_t = p'(\Gamma_t)$, and for $t \in T'$
we have from (1) that $\Gamma_t = \Gamma(f_t)$
and hence $Y_t = \overline{f_t(X_t)}$.
In particular every such $Y_t$ is irreducible,
and necessarily of dimension $= \dim Y -1$.
Because of the remark above, the statement follows.
\end{proof}
In general, the family of graphs needs not exist for the full family of maps,
as is seen later on in Remark \ref{example}.
\section{The varieties of general type in a family}
Using the technique of extension of differentials, from a special fibre
to the total space of the family, we gave in \cite{GP}, \S 1.4, a proof
of the assertion that the property of being a variety of general type
is invariant in a 1-dimensional small deformation, where small refers
to the Zariski topology. Here we point out that the same proof shows
indeed a slightly stronger assertion, to the effect that the same property
'propagates' from a component of a fibre.
\begin{thm}[] \label{generaltype}
Let $T$ be a smooth irreducible curve,
let $Y$ be a variety and let $Y \rightarrow T$ be a
projective morphism. Assume that some fibre
$Y_a$ has an irreducible component $Z$ which
is a variety of general type, and that the restriction
$Y \smallsetminus Y_a \rightarrow T \smallsetminus \{a\}$
is a family of varieties, as in \S {\rm 1.c}.
Then there is a nonempty open subset $T'$ of $T$
such that $Y_t$ is a variety of general type for $t \in T'$.
\end{thm}
\begin{proof}
Let $V \rightarrow Y$ be a resolution of singularities
such that the strict transform $Z'$ of $Z$ is smooth.
So $Z'$ is of general type, and \
$\dim H^{0}(Z',mK_{Z'}) \geq cm^{n}$ for $m \gg 0$.
Denote by $\pi$ the composite map $V \rightarrow Y \rightarrow T$.
Since $V \rightarrow T$ is generically smooth, and since
$Y \rightarrow T$ is generically a family of varieties,
restricting to some neighborhood of $a$,
we may assume that for every $t \neq a$
the induced map $V_{t} \rightarrow Y_{t}$ is a
resolution of singularities.
As the general $V_t$ is irreducible, it follows that
every $V_t$ is connected, by the Zariski connectedness theorem.
The extension theorem of Takayama \cite{Tak2} applies,
and gives us that there is a surjective restriction homomorphism
$$\begin{array}{ccc}
\pi_{*}\mathcal O_{V}(mK_{V}) \otimes k(a)
& \longrightarrow & H^{0}(Z', mK_{Z'})
\end{array}.$$
The image $\pi_{*}\mathcal O_{V}(mK_{V})$ is a
torsion free coherent sheaf on the smooth curve $T$,
hence it is a locally free sheaf.
So the dimension of
$\pi_{*}\mathcal O_{V}(mK_{V}) \otimes k(t)$
is constant.
For $t=a$ this dimension is $\geq cm^{n}$ for $m \gg 0$,
by what we have seen above.
\newcommand{\localmentelibero}
{let $f:Y \rightarrow S$ be flat, $\mathcal F$ on $Y$ be flat over $S$.
if $f_{*} \mathcal{F} \otimes k(t) \rightarrow H^{0}(Y_{t}, \mathcal{F}_{t})$
is surjective then it is an isomorphism, and the same holds in a neighborhood of $t$.
moreover $f_{*} \mathcal{F}$ is locally free in a neighborhood of $t$
[Hartshorne, p. 290] }
For $t \neq a$, since $mK_{V}|_{V_{t}} = m K_{V_{t}}$,
one has the restriction homomorphism
$$\begin{array}{ccc}
\pi_{*}\mathcal O_{V}(mK_{V}) \otimes k(t) & \longrightarrow &
H^{0}(V_{t}, \mathcal O_{V_{t}}(mK_{V}|_{V_{t}}))
= H^{0}(V_{t}, mK_{V_{t}})
\end{array}$$
and in a smaller neighborhood of $a$ we may assume that
this is an isomorphism for $t \neq a$.
It follows that
$\dim H^{0}(V_{t},mK_{V_{t}})
\geq cm^{n}$ for $m \gg 0$, hence $Y_{t}$
is of general type. This holds for every $t$
in a neighborhood of $a$.
\end{proof}
\section{Rigidity and limits}
Another key point in our treatment is a result about
limit maps in a generically trivial family of maps.
The result that we give here is only slightly more general than the
one in our previous paper, and the proof given here is more apparent.
Let $X$ be a projective variety.
Let $T$ be a smooth irreducible curve,
let $Y \rightarrow T$ be a projective morphism,
and let $f: T \times X \dasharrow Y$ be a family of rational maps on $X$,
as in \S 1.d. Assume that for every $t \in T$ the rational map
$f_t : X \dasharrow \overline{f_t(X)}$ is of finite degree $k$.
Assume moreover that the family is {\em generically trivial},
as in Corollary \ref{weakrigidity},
i.e. that there is a nonempty open subset $T'$ of $T$ such that
the restriction $f|_{T'}$ is obtained as
$$T' \times X \overset{1 \times h}{\dasharrow} T' \times U
\overset{g}{\dasharrow} Y|_{T'}$$
where $h: X \dasharrow U$ is a fixed dominant rational map, and where $g$
is a birational isomorphism which restricts to a birational isomorphism
$g_t: U \dasharrow Y_t$ for every $t \in T'$.
Then $f_{t} = g_{t} \circ h$ for $t \in T'$,
so all these maps are birationally equivalent,
of degree $\deg(f_{t}) = k = \deg(h)$.
\begin{prop} \label{rigidityandlimits}
Assume that $f: T \times X \dasharrow Y$ is a family
of rational maps of constant degree $\deg(f_t)=k$,
and assume that the family is generically trivial, as in the setting above.
Then all maps $f_t$ are in the same birational equivalence class.
\end{prop}
\begin{proof}
Let $a \in T$ be any point, and let us
prove that $f_a$ is in the birational equivalence class
of every $f_t$ with $t \in T'$.
We may assume that $U$ is a normal variety.
Recall that for a rational map of varieties over a base curve,
from a normal variety to a variety which is proper over the base,
the exceptional locus is of codimension $\geq 2$,
by the valuative criterion of properness for instance.
It follows that $g: T \times U \dasharrow Y$
restricts to a rational map $g_a: U \dasharrow Y_a$.
Since $f = g \circ (1 \times h)$ holds as an equality of
rational maps $T \times X \dasharrow Y$
then there is equality of restrictions $f_a = g_a \circ h$.
And since $\deg(f_a) = k = \deg(h)$ then $\deg(g_a) = 1$
and $f_a$ is birationally equivalent to $h$ and to every $f_t$.
\end{proof}
\section{Linear rational maps}
Let $\mathbb P^m = {\rm P}(V^{m+1})$ and let
$X \subseteq \mathbb P^m$ be a non degenerate subvariety,
of dimension $n$. The space of linear maps
$\mathbb P^m \dasharrow \mathbb P^m$ is the projective space
\begin{center}
$\mathbb P^N = {\rm P}({\rm End(V)})$ \ with $N = (m+1)^2-1$.
\end{center}
We denote by $\alpha = \overline\ell$ a point in $\mathbb P^N$
and by $x = \overline v$ a point in $\mathbb P^m$.
The evaluation homomorphism $(\ell,v) \mapsto \ell(v)$
determines a rational map
$$\mathbb P^N \times X \dasharrow \mathbb P^m$$
and this is the family of linear rational maps
$\alpha : X \dasharrow \mathbb P^m$.
We denote by $\overline{\alpha(X)}$ the closed image
and by $\Gamma(\alpha)$ the closed graph of the map $\alpha$.
The subscheme $C \subset \mathbb P^N \times X$
defined by $\ell(v)=0$
is the exceptional locus of the rational map above.
Consider the projection $C \rightarrow \mathbb P^N$.
The fibre $C_{\alpha}$ is the trace in $X$ of the center of
the linear projection $\alpha: \mathbb P^m \dasharrow \mathbb P^m$.
\begin{rem} \label{example} \em
The subscheme $\Gamma \subset \mathbb P^N \times X \times \mathbb P^m$
defined by $\ell(v) \wedge w =0$ is the closed graph of the rational map above.
Clearly $\Gamma$ contains $C \times \mathbb P^m$.
The projection $\Gamma \rightarrow \mathbb P^N$
does not define the family of graphs. The fibre is given by
$\Gamma_{\alpha} = \Gamma(\alpha) \cup\, C_{\alpha} \times \mathbb P^m$.
It is clear, just looking at dimensions, that
$\Gamma_\alpha = \Gamma(\alpha)$ if and only if
$C_{\alpha} = \emptyset$.
\end{rem}
In $\mathbb P^N$ define the following subsets:
\begin{itemize}
\item[]
$R$ \ \ the subset of all $\alpha$ such that
$\alpha: X \dasharrow \overline{\alpha(X)}$ is of finite degree,
\item[]
$R_k$ \ the subset of all $\alpha \in R$ with $\deg(\alpha)= k$,
\end{itemize}
for every integer $k > 0$.
\begin{prop} \label{constructible}
$(1)$ $R$ is an open subset. $(2)$ $R_k$ is a constructible subset for every $k > 0$.
\end{prop}
\begin{proof}
(1) In $(\mathbb P^N \times X) \smallsetminus C$ let $U$ be the subset
of pairs $(\alpha,x)$ such that $\dim_{x} \alpha^{-1}(\overline{\alpha(X)}) = 0$.
It is an open subset. In $\mathbb P^N$ the image of $U$ coincides with $R$.
In fact, if $\alpha$ admits some point $x \in X \smallsetminus C_{\alpha}$
which is isolated in its fibre, then its general fibre is of dimension $0$.
As the projection $\mathbb P^N \times X \rightarrow \mathbb P^N$
is an open map, $R$ is open in $\mathbb P^N$.
(2) In $\mathbb P^N \times X^{\times k}$ let $U_{k}$ be the subset of sequences
$(\alpha,x_1,\ldots,x_k) =: (\alpha, \bar x)$ such that every $(\alpha, x_i)$
belongs to $U$ and $\alpha(x_1) = \cdots = \alpha(x_k)$ while in the
sequence $(x_1,\ldots,x_k)$ there is no coincidence.
For every $\alpha \in R$
denote by $U_k(\alpha)$ the fibre of $U_k$ over $\alpha$.
Let $V_k$ be the subset such that $\dim_{(\alpha, \bar x)} U_k(\alpha) = n$.
This is a locally closed subset in $\mathbb P^N \times X^{\times k}$.
In $\mathbb P^N$ the image ${V_k}'$ of $V_k$
is the locus of $\alpha \in R$ with $\deg \alpha \geq k$.
In fact, if $\alpha$ admits some sequence $(x_1,\ldots,x_k)$
such that $\dim_{(\bar x)} U_k(\alpha) = n$, as the projection
$U_k(\alpha) \rightarrow X$ has 0-dimensional fibres, then $U_k(\alpha)$
dominates $X$, and hence for a general point $x_1$ the fibre of $\alpha$
contains at least $k$ distinct points $x_1,\ldots,x_k$.
It follows that $R_k$ coincides with ${V_k}' \smallsetminus {V_{k+1}}'$.
\end{proof}
\section{Refined finiteness theorem}
Let $X$ be a smooth projective variety of general type, of dimension $n$.
Let $X' \subset \mathbb P^M$ be the image of $X$ in the $r_n$-canonical
birational embedding, see Theorem \ref{HKT}.
Here $M = h^0(X,r_nK_X) -1$ is bounded above in (\ref{embdim}).
Every rational map of finite degree
$f: X \dasharrow Y$ to a smooth projective variety of general type,
taking the $r_n$-canonical model $Y' \subset \mathbb P^M$,
gives rise to a linear rational map $\alpha: X' \dasharrow \mathbb P^M$
with $\overline{\alpha(X')} = Y'$.
In this natural way
the set of birational equivalence classes of rational maps of finite degree
from $X$ to varieties of general type is injected into
the set of birational equivalence classes
of linear rational maps of finite degree from $X'$ to $\mathbb P^M$.
Our main result is concerned
with the geometric structure of these special equivalence classes.
\begin{thm} \label{connectedcomponent}
Let $X$ be a smooth projective variety of general type.
A birational equivalence class of rational maps of degree $k$
from $X$ to smooth projective varieties of general type
forms a union of connected components of $R_k$.
\end{thm}
\begin{proof}
Let $\alpha \in R_k$ be such that $\overline{\alpha(X')}$ is of general type.
Let $T$ be a smooth irreducible curve with a morphism
$T \rightarrow R_k$, that we write as $t \mapsto \alpha_t$,
and with some point $a \in T$ such that $a \mapsto \alpha$.
We claim that all maps $\alpha_t$ are birationally isomorphic to $\alpha$.
Consider the rational map $T \times X' \dasharrow T \times \mathbb P^M$
which represents the family of maps $\alpha_t$.
Let $Y$ be its closed image in $T \times \mathbb P^M$.
There is a nonempty open subset $T'$ of $T$ such that
$Y|_{T'} \rightarrow T'$ is the family of closed images,
by Proposition \ref{familyofgraphs}.
The fibre $Y_a$ contains $\overline{\alpha(X')}$, a variety of general type.
It follows from Theorem \ref{generaltype}
that, shrinking $T'$ if necessary, we may assume that
for every $t \in T'$ the variety $\overline{\alpha_t(X')}$ is of general type.
Then it follows from Corollary \ref{weakrigidity} to the rigidity theorem
that, shrinking $T'$ again, we may assume that
the restriction $T' \times X' \dasharrow Y|_{T'}$ is a trivial family.
And then it follows from Proposition \ref{rigidityandlimits} that
all maps $\alpha_t$ with $t \in T$ are birationally equivalent, as we claimed.
So we reach the conclusion.
Every irreducible curve through $\alpha$ in $R_k$ is the image
of a smooth irreducible curve $T$ as above, and therefore
is fully contained in the birational equivalence class of $\alpha$.
Therefore every connected curve through $\alpha$ in $R_k$
is fully contained in the birational equivalence class of $\alpha$.
Since $R_k$ is constructible, by Proposition \ref{constructible},
this means that the connected component of $\alpha$ in $R_k$
is contained in the birational equivalence class of $\alpha$.
\end{proof}
The space $R$ admits the stratification
$\bigsqcup R_k$, where the degree $k$ is bounded above in $(\ref{deg1})$
in terms of the function $r_n$, or in $(\ref{deg2})$ in terms of
the function $\epsilon_n$. As an immediate consequence of the previous result
we obtain the following refined version of the finiteness theorem,
which improves our previous result \cite{GP}, Theorem 4.3.
\begin{thm} \label{finiteness}
Let $X$ be a smooth projective variety of general type.
The number of birational equivalence classes of rational maps of finite degree
from $X$ to smooth projective varieties of general type
is bounded above by the number
of connected components of strata in the stratification
$R = \bigsqcup R_k$.
\end{thm}
We showed in \cite{GP} that the finite number of classes of maps
in the finiteness theorem has an upper bound of the form $B(n,v)$
where $n = \dim (X)$ and $v = {\rm vol}(K_X)$, and that such a function
$B$ can be explicitely computed in terms of the function $r_n$.
This is obtained by means of rather cumbersome computations
with the complexity of a certain bunch of subvarieties of Chow varieties,
that was used as a parameter space for rational maps. We believe that an
analogous computation working with the much simpler parametrization
that has been established in the present paper will lead to a simpler
procedure and to a better result for the function $B$.
|
{
"timestamp": "2012-03-13T01:01:04",
"yymm": "1203",
"arxiv_id": "1203.2246",
"language": "en",
"url": "https://arxiv.org/abs/1203.2246"
}
|
"\\section{Introduction}\n\nGalactic bars are believed to play a crucial role in galaxy evolution.\n(...TRUNCATED)
| {"timestamp":"2012-03-09T02:01:16","yymm":"1203","arxiv_id":"1203.1693","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\\label{Introduction}\n\\hl{Thermodynamic measurements with both good abso(...TRUNCATED)
| {"timestamp":"2012-03-12T01:01:19","yymm":"1203","arxiv_id":"1203.2049","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\r\n\r\nIt is proved in \\cite{TeTr} that a simplicial complex $\\Delta$ is (...TRUNCATED)
| {"timestamp":"2012-03-12T01:00:29","yymm":"1203","arxiv_id":"1203.1969","language":"en","url":"https(...TRUNCATED)
|
"\\section*{Methods}\n\\begin{small}\n\n\\begin{bfseries}\nLasersystem for the fundamental beams.\n\(...TRUNCATED)
| {"timestamp":"2012-03-12T01:02:01","yymm":"1203","arxiv_id":"1203.2121","language":"en","url":"https(...TRUNCATED)
|
"\\chapter*{\\sc \\textbf{Preface}}\n\\markboth{\\sc \\textbf{Preface}}{\\sc \\textbf{Preface}}\n\\a(...TRUNCATED)
| {"timestamp":"2012-03-12T01:02:34","yymm":"1203","arxiv_id":"1203.2159","language":"en","url":"https(...TRUNCATED)
|
"\\section{Introduction}\n\nIn his celebrated theorem \n\\citet{Nas:PNASUSA1950, Nas:AM1951} used fi(...TRUNCATED)
| {"timestamp":"2013-09-18T02:03:34","yymm":"1203","arxiv_id":"1203.2301","language":"en","url":"https(...TRUNCATED)
|
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