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Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 31
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 20072)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 31
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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\section{Introduction}
In this paper, all groups are finite. Given a group $G$, the commuting graph of $G$, denoted by $\Gamma(G)$, is defined as the graph whose vertex set is the noncentral elements of $G$ and two vertices are adjacent if and only if they commute in $G$. Giudici and Pope in \cite{GiuPo} state that commuting graphs were first studied by Brauer and Fowler in \cite{BrFo} in relation to the classification of simple groups. They go on to say that commuting graphs were first studied in their own right by Segev and Seitz in \cite{SeSe} in terms of the classical simple groups. In fact, much of the research regarding the commuting graph is related to simple groups. We are not going to try describe all of this research, but note that it culminates in the work of Solomon and Woldar in \cite{SoWo} were they complete the final step in proving that if $S$ is a simple group and $X$ is any group where $\Gamma (S) \cong \Gamma (X)$, then $S \cong X$.
In the course of this research, Iranmanesh and Jafarzedah conjecture in \cite{IraJaf} that there is a universal bound on the diameter of commuting graphs. Giudici and Pope in \cite{GiuPo} study the diameters of commuting graphs for a number of families of group. Surprisingly, Giudici and Parker produced in \cite{GiuPark} a family of $2$-groups of nilpotence class $2$ where there is no bound on the diameter of the commuting graphs. On the other hand, in the seminal 2013 paper \cite{parkerSoluble}, Parker proves that the commuting graph of a solvable group $G$ with a trivial center is disconnected if and only if $G$ is a Frobenius or $2$-Frobenius group. We will state the definition of $2$-Frobenius groups in Section 2. In addition, when the commuting graph of $G$ is connected, he proves that it has diameter at most $8$. He also provides an example of a solvable group where this bound is met. Parker and Morgan remove the solvability hypothesis on $G$ in \cite{parkerTrivialCenter} by proving for the commuting graph of any group with a trivial center that all the connected components have diameter at most $10$.
In this paper, we look to extend Parker's and Parker and Morgan's results. In particular, we would like to extend the class of groups where we can determine which commuting graphs are disconnected and determine further classes of groups where we can show that diameters of the commuting graph are bounded by the constants $8$ and $10$. In particular, we show that we can replace the hypothesis that $\centerof{G} = 1$ with the hypothesis that $G' \cap \centerof{G} = 1$.
\begin{mainthm}
Let $G$ be a group and suppose that $G' \cap \centerof{G} = 1$.
\begin{enumerate}
\item $\Gamma (G)$ is connected if and only if $\Gamma (G/\centerof{G})$ is connected.
\item Every connected component of $\Gamma (G)$ has diameter at most $10$.
\item If $G$ is solvable and $\Gamma (G)$ is connected, then $\Gamma (G)$ has diameter at most $8$.
\item If $G$ is solvable, then $\Gamma (G)$ is disconnected if and only if $G/Z$ is either a Frobenius group or a $2$-Frobenius group.
\end{enumerate}
\end{mainthm}
In fact, we will see that we can relax the hypothesis that $G' \cap \centerof{G} = 1$ even further. In particular, we show that it suffices to assume that $C (G) \cap \centerof{G} = \{ 1 \}$ where $C (G) = \{ [x,y] \mid x,y \in G \}$. I.e., we only need the set of commutators, not the whole commutator subgroup. (We note that we use $1$ to denote the trivial subgroup. Since $C(G)$ is not a subgroup, it is not appropriate to view $C(G) \cap \centerof{G}$ as a subgroup, so we use $\{ 1 \}$ to denote the set consisting only of the identity.) We will present examples of groups $G$ where $C (G) \cap \centerof{G} = \{ 1 \}$, but $G' \cap Z (G) > 1$, so this replacement does actually improve the result.
Following the literature, we say that a group $G$ is an {\it $A$-group} if every Sylow subgroup of $G$ is abelian. We show that if $G$ is an $A$-group then $G' \cap \centerof{G} = 1$. In particular, this shows that the results of the Main Theorem apply to $A$-groups; however, we would not be surprised if one could show that the diameter bounds can be lowered for $A$-groups, especially solvable $A$-groups, but we have not investigated this at this time.
One other consequence of $G' \cap \centerof{G} = 1$ is that $\centerof{G/\centerof{G}} = 1$, and so it makes sense to ask whether we can determine which groups satisfying $\centerof{G/\centerof{G}} = 1$ have a disconnected commuting graph and if we can bound the diameter of the commuting graphs of groups where the commuting graph is connected. At this point, we can find one class of groups where the commuting graph is disconnected as seen in the following theorem, but we will present examples of other groups satisfying this condition where the commuting graph is disconnected.
\begin{theorem}
If $G$ is a group where $G/\centerof{G}$ is either a Frobenius or a $2$-Frobenius group, then $\Gamma (G)$ is disconnected.
\end{theorem}
This research was conducted during a summer REU in 2020 at Kent State University with the funding of NSF Grant DMS-1653002.
We thank the NSF and Professor Soprunova for their support.
\section{Results}
Let $G$ be a finite group and $\Gamma(G)$ the commuting graph of $G$. When $x = y$ or $x$ and $y$ are adjacent in $\Gamma(G)$, we write $x \sim y$. In particular, writing $x \sim y$ emphasizes that $x,y \in G \setminus \centerof{G}$. Viewing $x,y$ as vertices of $\Gamma (G)$, we use $d(x,y)$ to denote the distance between $x$ and $y$. (I.e., the number of edges in the shortest path between $x$ and $y$.) For the remainder of the paper, we set $Z = \centerof{G}$, the center of the group, and let $Z_2 = \ithcenter{2}{G}$, the preimage of $\centerof{G/\centerof{G}}$. We also set $C = C(G)$. Note that $G'$ is the group generated by $C$, but $C$ itself is not always a group.
The following lemma addresses the relationship between adjacent elements in $\Gamma(G)$ and adjacent elements in $\Gamma(G/Z)$.
\begin{lemma} \label{three}
Let $G$ be a group and fix $x, y\in G \setminus Z_2$. If $x \sim y$ in $\Gamma(G)$, then $xZ \sim yZ$ in $\Gamma(G/Z)$. If $C \cap Z = \{ 1 \} $ and $xZ \sim yZ$ in $\Gamma(G/Z)$, then $x \sim y$ in $\Gamma(G)$.
\end{lemma}
\begin{proof}
Suppose that $x \sim y$ in $\Gamma(G)$. Then, we have that $xy=yx$, and hence, $xZ yZ = yZ xZ$ in $G/Z$. Thus, $xZ \sim yZ$.
Now, suppose $C \cap Z = \{ 1 \}$ and $xZ \sim yZ$. This implies that $[x,y]\in Z$. Also, $[x,y] \in C$, and since $C \cap Z = \{ 1 \}$, we obtain $[x,y] = 1$. Therefore, $xy = yx$ and so $x \sim y$.
\end{proof}
We now consider the relationship between $\Gamma (G)$ and $\Gamma (G/Z)$ when $Z_2 = Z$ and in particular, when $C \cap Z = \{ 1 \}$.
\begin{lemma} \label {four}
Let $G$ be a group and suppose that $Z_2 = Z$.
\begin{enumerate}
\item If $\Gamma (G)$ is connected, then $\Gamma (G/Z)$ is connected and the diameter of $\Gamma (G/Z)$ is less than or equal to the diameter of $\Gamma (G)$.
\item If $C \cap Z = \{ 1 \}$ and $\Gamma (G/Z)$ is connected, then $\Gamma (G)$ is connected and has the same diameter as $\Gamma (G/Z)$.
\item If $C \cap Z = \{ 1 \}$ and $\Gamma (G/Z)$ is disconnected, then there is a one-to-one correspondence between the connected components of $\Gamma (G/Z)$ and of $\Gamma (G)$ that preserves diameter except in the case that a connected component of $\Gamma (G/Z)$ consists of a single coset and $Z > 1$; in this case, the corresponding component of $\Gamma (G)$ will have diameter $1$.
\end{enumerate}
\end{lemma}
\begin{proof}
Suppose $x, y \in G \setminus Z$. Since $\Gamma (G)$ is connected, we can find $x = x_0, x_1, \dots, x_n = y \in G \setminus Z$ so that $x_i \sim x_{i+1}$ for $i = 0, \dots, n -1$. By Lemma \ref{three}, we have $x_i Z \sim x_{i+1}Z$. It follows that $xZ$ and $yZ$ are connected by a path of length of $n$. It follows that $d(xZ,yZ) \le d(x,y)$. We deduce that $\Gamma (G/Z)$ is connected and its diameter is at most the diameter of $\Gamma (G)$.
We now add the assumption that $C \cap Z = \{ 1 \}$. From Lemma \ref{three}, we see that $x \sim y$ in $\Gamma (G)$ if and only if $xZ \sim yZ$ in $\Gamma (G/Z)$. It follows that $x$ and $y$ lie in the same connected component of $\Gamma (G)$ if and only if $xZ$ and $yZ$ lie in the same connected component of $\Gamma (G/Z)$. In the case where they are in the same connected component, then $d (x,y) = d(xZ, yZ)$; except when $xZ = yZ$ and $x \ne y$, in this case $d(x,y) = 1$ and $d (xZ,yZ) = 0$. Notice that if $xZ \ne yZ$, then $x \ne y$, so a connected component of $\Gamma (G/Z)$ with more than one element will correspond to a connected component of $\Gamma (G)$ with more than one element and they will have the same diameter. Finally, if $\{ xZ \}$ is a connected component of $\Gamma (G/Z)$, then $\{ xz \mid z \in Z\}$ will be the corresponding connected component of $\Gamma (G)$, so the component in $\Gamma (G/Z)$ has diameter $0$, but the corresponding component in $\Gamma (G)$ will have diameter $1$.
\end{proof}
Following the literature, a group $G$ is a {\it Frobenius group} if it contains a proper, nontrivial subgroup $H$ so that $H \cap H^g = 1$ for all $g \in G \setminus H$. The subgroup $H$ is called a {\it Frobenius complement} for $H$. Frobenius proved that $N = (G \setminus \cup_{g \in G} H^g) \cup \{ 1 \}$ is a normal Hall subgroup of $G$ that satisfies $G = HN$ and $H \cap N = 1$; the subgroup $N$ is called the {\it Frobenius kernel} of $G$. We say $G$ is a {\it $2$-Frobenius group} if there exist normal subgroups $K \le L$ so that $L$ and $G/K$ are Frobenius groups with Frobenius kernels $K$ and $L/K$ respectively.
We now show that when $G/Z$ is either a Frobenius group or a $2$-Frobenius group, then $\Gamma (G)$ is disconnected.
\begin{corollary}
If $G$ is a group so that $G/Z$ is either a Frobenius group or a $2$-Frobenius group, then $\Gamma (G)$ is disconnected.
\end{corollary}
\begin{proof}
We know that the center of a Frobenius group or a $2$-Frobenius group is trivial, so $Z_2 (G) = Z$. Now, Parker has shown that $\Gamma (G/Z)$ is disconnected. We apply the contrapositive of Lemma \ref{four}(1) to see that $\Gamma (G)$ is disconnected.
\end{proof}
We next show that the condition that $C \cap Z = \{ 1 \}$ implies that $Z = Z_2$.
For $x \in G$, set $D_G (x) = \{ d \in G \mid [d, x] \in Z \}$. Thus, $D_G(x)/Z = C_{G/Z} (Zx)$.
\begin{lemma}\label{five}
If $G$ is a group such that $C \cap Z = \{ 1 \}$, then $C_G(x) = D_G(x)$ for each
$x \in G$. In particular, $Z = Z_2$.
\end{lemma}
\begin{proof}
Let $x \in G$. Of course, $C_G(x) \leq D_G(x)$. If $d \in D_G(x)$, then $[d,x] \in
C \cap Z = \{ 1 \}$ and so $d \in C_G(x)$. Thus, $D_G(x) \le C_G(x)$, and we conclude that
$C_G (x) = D_G (x)$. Finally, observe that $Z = \cap_{x \in G} C_G(x) = \cap_{x \in G}
D_G(x) = Z_2$, as wanted.
\end{proof}
We now obtain Parker and Morgan's and Parker's result for $G$ when $C \cap Z = \{ 1 \}$.
\begin{corollary} \label{six}
Let $G$ be a group and suppose that $C \cap Z = \{ 1 \}$.
\begin{enumerate}
\item $\Gamma (G)$ is connected if and only if $\Gamma (G/Z)$ is connected.
\item Every connected component of $\Gamma (G)$ has diameter at most $10$.
\item If $G$ is solvable and $\Gamma (G)$ is connected, then $\Gamma (G)$ has diameter at most $8$.
\item If $G$ is solvable, then $\Gamma (G)$ is disconnected if and only if $G/Z$ is either a Frobenius group or a $2$-Frobenius group.
\end{enumerate}
\end{corollary}
\begin{proof}
Conclusion (1) is an immediate consequence of Lemma \ref{four} (1) and (2) combined with Lemma \ref{five}. Since $Z(G/Z) = 1$, Parker's results apply to $G/Z$, and applying Parker's results in $G/Z$ with Lemma \ref{four} (2) and (3) yields (2), (3), and (4).
\end{proof}
The following result has appeared in the literature. See, for example, Corollary 4.5 in \cite{Bro} or Theorem 4.1 in \cite{Tau }.
\begin{lemma} \label{agroup}
Let $G$ be an $A$-group. Then $G' \cap Z = 1$.
\end{lemma}
Using Lemma \ref{agroup}, we see that if $G$ is an $A$-group, then $G$ satisfies the hypothesis of Corollary \ref{six}, and thus, $G$ satisfies the conclusions of Corollary \ref{six}.
We close by presenting some examples using the small groups library \cite{small} that can be accessed by the computer algebra systems GAP \cite{gap} or Magma \cite{magma} to illustrate various points. We first present examples of groups $G$ where $C \cap Z = \{ 1 \}$ but $G' \cap Z > 1$. Take $G$ to be one of SmallGroup(768,1083474), SmallGroup(768,1083475), or SmallGroup(768,1083476).
We next present examples of groups $G$ where $\Gamma (G/Z)$ is connected but $\Gamma (G)$ is not connected. Taking $G$ to be one of the Small Groups (72,22), (72,23), (120,11), (144,125), and (288,565) yield examples of groups $G$ with this property. We actually have many more examples in this category, but we have just pulled a few examples at random from the list. We include the graphs $\Gamma (G)$ and $\Gamma (G/Z)$ for $G = {\rm SmallGroup} (72,22)$ in Figures \ref{g7222} and \ref{g7222qc}.
\begin{figure}[h]
\centering
\begin{minipage}{0.8\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{EX-72-22-Graph}
\caption{\texttt{$\Gamma (G)$ for $G = $ SmallGroup(72,22)}}
\label{g7222}
\end{minipage}
\bigskip
\begin{minipage}{0.8\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{EX-72-22-Quotient}
\caption{\texttt{$\Gamma (G/Z)$ for $G = $ SmallGroup(72,22)}}
\label{g7222qc}
\end{minipage}
\end{figure}
The groups that we have considered all satisfy the condition that $\centerof{G/Z} = 1$. It makes sense to ask what can be said about groups that satisfy $\centerof{G/Z} = 1$ and do not satisfy $C \cap Z = \{ 1 \}$. Notice that if $G$ is a such group, then there will definitely exist elements $x,y \in G \setminus Z$ so that $xZ \sim yZ$ in $\Gamma (G/Z)$ but $x \not\sim y$ in $\Gamma (G)$; so the results of Lemma \ref{four} (2) and (3) are not guaranteed to hold. We will present some examples to show in fact conclusion (3) does not hold.
First, we claim that it is not hard to see that if $G$ is a Frobenius group with Frobenius kernel $N$, then $\Gamma (G)$ has $1 + |N|$ connected components and if $G$ is a $2$-Frobenius group with normal subgroups $K \le L$ as in the definition above, then $\Gamma (G)$ has $1 + |K|$ connected components. (Although neither of these computations are difficult, the second computation is done explicitly at the end of Section 3 of \cite{CoLe}.) Also, Thompson's celebrated theorem shows that a Frobenius kernel is always nilpotent and it known that a Frobenius complement always has a nontrivial center. Since the nonidentity elements of these subgroups form the connected components of $\Gamma (G)$ when $G$ is a Frobenius group, it follows that each connected component has diameter at most $2$.
When $G$ is a $2$-Frobenius group, there are $|K|$ connected components in $\Gamma (G)$ that each consist of the nonidentity elements of a Frobenius complement of $L$. Since these Frobenius complements are all cyclic, the corresponding connected components in $\Gamma (G)$ are complete graphs. The remaining nonidentity elements of $G$ form a single connected component in $\Gamma (G)$. We know that every element of prime order in $G \setminus L$ centralizes some nonidentity element in $K$. (See Lemma 3.8 of \cite{CoLe}.) It follows that every element outside of $K$ has distance at most $2$ to a nonidentity element of $K$, and since $K$ is nilpotent, we deduce that this remaining connected component of $\Gamma (G)$ has diameter at most $6$. (We would not be surprised if one could actually get a tighter bound for the diameter of this component.)
Take $P$ to be an extra special group of order $p^3$ and exponent $p$ for an odd prime $p$. Let $q$ be a prime divisor of $p - 1$. It is well-known that $P$ has an automorphism $\sigma$ of order $q$ that centralizes $\centerof{P}$. Let $C = \langle \sigma \rangle$ so that $C$ acts on $P$ via automorphisms, and let $G$ be the resulting semi-direct product. Observe that $\centerof{P} = Z$ and $G/Z$ is a Frobenius group of order $p^2q$. Also, we have $Z = C (P) \subseteq C (G)$, so we do not have $C \cap Z = \{ 1 \}$. We see that $\Gamma (G/Z)$ is disconnected and has $1 + p^2$ connected components. It follows that $\Gamma (G)$ is disconnected. It is not difficult to see that the $p^2$ connected components of $\Gamma (G/Z)$ that correspond to the Frobenius complements in $G/Z$ will correspond to $p^2$ connected components in $\Gamma (G)$.
On the other hand, since $Z (P) = Z (G)$, we see that the remaining connected component of $\Gamma (G/Z)$ consists of the nontrivial cosets in $P/Z (P)$. On the other hand, in the graph of $\Gamma (G)$, the elements in $P \setminus Z$ will have the same connected components as the graph $\Gamma (P)$. It is not difficult to see that every noncentral element of $P$ has a centralizer that is abelian of order $p^2$. This implies that $P$ is what is sometimes called a CA-group in the literature. In particular, it is not difficult to see that the elements in $P \setminus Z$ split into $p + 1$ different connected components each having $p^2 - p$ elements. In particular, $\Gamma (G)$ has $1 + p + p^2$ different connected components. This shows that the correspondence in Lemma \ref{four} (3) does not hold when we do not have $C \cap Z = \{ 1 \}$.
We next present ${\rm GL}_2 (3)$. In this case, $G/Z \cong S_4$ is a $2$-Frobenius group so that $\Gamma (G/Z)$ has $4$ connected components that are complete graphs and one connected component that has diameter $3$. On the other hand, $\Gamma (G)$ has $13$ connected components that are all complete graphs. We include these graphs as Figures \ref{gl23} and \ref{gl23qc}.
\begin{figure}[h]
\centering
\begin{minipage}{0.8\textwidth}
\centering
\includegraphics[width=0.5\linewidth]{GL23Graph}
\caption{\texttt{$\Gamma (G)$ for $G = \rm{GL}_2 (3)$ }}
\label{gl23}
\end{minipage}
\end{figure}
\bigskip
\begin{figure}
\begin{minipage}{0.8\textwidth}
\centering
\includegraphics[width=0.4\linewidth]{S4Graph}
\caption{\texttt{$\Gamma (G/Z)$ for $G = \rm{GL}_2 (3)$ }}
\label{gl23qc}
\end{minipage}
\end{figure}
\eject
The last examples we present are examples where $\Gamma (G)$ and $\Gamma (G/Z)$ are both connected and the diameters are different. The groups are SmallGroups (400,125), (400,126), and (400,127). In all three cases, we have that $\Gamma (G)$ has diameter $5$ and $\Gamma (G/Z)$ has diameter $3$.
|
{
"timestamp": "2021-03-15T01:19:02",
"yymm": "2103",
"arxiv_id": "2103.07355",
"language": "en",
"url": "https://arxiv.org/abs/2103.07355"
}
|
\section{Introduction}
Last year on May $24$ the SpaceX launched the first set of $60$ Starlink satellites and up to now the total number is approximately $400$ aiming to reach at least $12000$ \citep{starlink} at the end of a decade program announced by Elon Musk. Such a huge number of satellites, distributed over almost the whole surface of the Earth might be considered as the first prototype of a possible megastructure around the Earth, which in principal, might be visible from the cosmos. Similarly, one may search for techno-signatures of alien civilizations.
The discovery of the Tabby's star \citep{kic846} and "Oumuamua" \citep{oum1} have provoked the revival of the search for extraterrestrial intelligence (SETI). The idea to search for techno-signatures of advanced alien societies have been proposed by \cite{dyson}. Assuming that a civilization is advanced enough to build a megastructure around a host star to consume its whole energy, Dyson has concluded that such a huge (having the length-scale of the order of one AU) spherical construction - Dyson sphere (DS) - should be visible in the infrared (IR) spectrum. Civilizations harnessing the host star's total energy belong to the Type-II societies according to the classification by \cite{kardashev}. Type-I civilization is harnessing the total energy coming from the sun to the Earth. Our society is consuming less than the mentioned energy, therefore, an index, $K = log_{10}\left(P\right)/10-0.6$, introduced by \cite{shkl} for Earthlings is $0.7$, where $P$ denotes the average harnessed power in Watts. In the framework of the same classification Type-III is the alien high tech society which is able to use the total energy of the host galaxy.
It is clear that detection of Type-II and Type-III civilizations is much easier than Type-I because of the much higher total consumed energies. Therefore, a special interest in the Dysonian SETI projects deserve Type-II,III techno-signatures and a series of papers are dedicated to identification of DS candidates \citep{timofeev,carrigan,gaia}. Dyson's original idea has been extended to hot DSs \citep{paper3,paper4} and the megastructures around pulsars \citep{paper1,paper2}.
Despite high radiation intensity of Type-II,III technologies compared to Type-I techno-signatures, the latter still can be considered seriously in the SETI context. In particular, \cite{warming} studied effects of global warming as detectable biomarkers in Earth-like societies.
Our civilization consumes approximately $1.5\times 10^{20}$ ergs s$^{-1}$ \citep{warming}, which is less than for Type-I society, $1.7\times 10^{24}$ ergs s$^{-1}$. If one assumes $1\%$ of an average growth rate of industry and the subsequent energy consumption, one can straightforwardly show that our civilization might reach Type-I in $\sim1000$yrs. It is quite probable that in $1000$yrs the level of technology will differ from ours, likewise ours is different from the one of middle ages. Therefore, one can assume that Type-I alien society is able to cloak their planet by a sphere-like (or ring-like) structure to harness the total energy emitted from their host star toward the planet.
In this paper we consider the possible observational characteristics of a planetary megastructure partially or completely covering an Earth-like planet located in the habitable zone.
The paper is organized in the following way: in Sec. 2, we introduce main theory and study the techno-signatures of Type-I megaconstructions, obtaining major results and in Sec. 3 we outline them.
\section[]{Discussion and results}
At first we consider the question: energetically how feasible is launching material to cover an area comparable to the Earth's surface. As an example we examine the construction designed by means of Graphene as up to now the strongest material, our civilization is able to produce. Then, for the total mass of the megastructure we obtain
$$M\simeq 4\kappa\pi R_p^2\rho\Delta r\simeq$$
\begin{equation}
\label{mass}
\simeq 2\kappa\times 10^{17}\times\frac{\rho}{0.4g/cm^3}\times\frac{\Delta r}{1mm} \; g,
\end{equation}
where $\kappa\leq 1$ is a dimensionless geometric factor, $R_p$ is the Earth-like planet's radius ($R_{\oplus}\simeq 6400\;km$) density, $\rho$, is normalised by the Graphene density and it is assumed that since the Graphene is $10$ times stronger than steel the effective thickness of the construction, $\Delta r$, is normalised by $1$mm value. As it clear from here, the total mass is less than the total storage of Earth's Carbon, $1.85\times 10^{24}$ by seven orders of magnitude and thus one can find enough material to build a construction. Here we assume that the altitude is much less than $R_{\oplus}$. One can straightforwardly check that at geostationary orbit ($R\simeq 36000$km) the required mass becomes enormous compared to the above considered.
Another issue one should address, is to understand time scale required for launching such a huge mass of material. Roughly speaking, the civilization can start launching material from the times when its index was equal to ours, $0.7$ (See the introduction). Then, as we have already mentioned, to reach the Type-I level, it needs $1000$ yrs. The process will be feasible if the annual rate of growth of mass launching, $\mu$, is a small parameter. If one assumes that the first year's launched mass equals $M_0$, then in $1000$ yrs the total mass will be given by
\begin{equation}
\label{launch}
M=M_0\left(1+\mu\right)^{1000}.
\end{equation}
After taking into account Eq. (\ref{mass}) and the fact that up to date there are $895$ satellites each with the mass $260$ kg \citep{starlink}, one can straightforwardly show that $\mu\simeq 0.021$.
Generally speaking, the megastructure might be constructed by advanced Type-I civilization from extraterrestrial resources without launch costs \citep{haliki}, but let us consider the worst case: the Solar energy is stored and then utilized to launch material on the orbit.
In this case energy required to launch material on the altitude $H\sim 500$km is as follows
\begin{equation}
\label{energy}
E\simeq \frac{GMM_pH}{R_{\oplus}^2}\simeq 5.7\kappa\times 10^{27}\; ergs,
\end{equation}
where $M_p = M_E $ is Earth-like planet's mass and $M_E\simeq 6\times 10^{27}$ g is the Earth's mass.
On the other hand, power from the host star toward the planet is given by
\begin{equation}
\label{power}
P\simeq \frac{L}{4}\times\left(\frac{R_p}{r}\right)^2\simeq 1.7\kappa\times 10^{24}\; ergs\;s^{-1},
\end{equation}
where $L\simeq 3.8\times 10^{33}$ergs s$^{-1}$ is the solar-type star's luminosity and $r\simeq 1.5\times 10^{13}$ cm is the radius of the habitable zone. Whatever the propulsion mechanism (electromagnetic launch or some other mechanisms), it is evident from Eq. (\ref{power}) that the energy required to launch a thin shell around a planet can be extracted in approximately ten hours with $10\%$ of efficiency of energy conversion. If the megastructure is a web of many satellites orbiting the planet, then the kinetic energy should be added to the aforementioned value, leading to ten times more total energy, which, energetically is quite feasible for Type-I alien societies.
For a uniformly distributed spherical mass such a construction will not require any energy to stabilise it. Unlike this scenario, if instead of the sphere one uses a ring, the planet will be characterised by the out of plane stability, whereas for in-plane displacements the dynamics is unstable \citep{paper1}. In this work we have discussed that such rings may have sense only if the power required to maintain stability is small compared to the total received power, leading to the following condition for maximum displacement from equilibrium
\begin{equation}
\label{ksi}
\xi<<\frac{0.37}{R_p}\left(\frac{P}{M}\right)^{1/2}\left(\frac{2R_p^3}{GM_p}\right)^{3/4}\simeq 0.07,
\end{equation}
where $\xi\equiv d/R_p$ and $d$ denotes the displacement of the ring from the equilibrium position. We would like to emphasise that such a precision is not difficult to achieve: in the Lunar laser ranging experiment\footnote{Data is avalable from the Paris Observatory Lunar Analysis Center: http://polac.obspm.fr/llrdatae.htm}
the distance is measured with the precision of the order of $10^{-10}$.
By assuming that the whole emission incident on the megastructure is absorbed, one can easily show that the temperature is given by
\begin{equation}
\label{temp}
T\simeq\left(\frac{P}{16\sigma r^2}\right)^{1/4}\simeq 280 K,
\end{equation}
leading by means of the Wien's law to the emission peak in the IR spectral band with the following wavelength
\begin{equation}
\label{lambda}
\lambda = \frac{b}{T}\simeq 10 \mu m,
\end{equation}
where $\sigma$ denotes the Stefan-Boltzmann constant and $b\simeq 2898\mu$mK.
The average value of the flux can be estimated as follows
\begin{equation}
\label{flux}
F\simeq\frac{P}{4\pi D^2}\simeq 1.6\times10^{-17}\times\left(\frac{100 \;ly}{D}\right)^2 ergs\;s^{-1} cm^{-2},
\end{equation}
where the distance to the extrasolar system, $D$, is normalised by $100$ light years. The Very Large Telescope Interferometer (VLTI) could detect such fluxes. In particular, the limiting flux value of the VLTI instruments\footnote{https://www.eso.org/sci/facilities/paranal/telescopes/vlti} in $1$ hour is of the order of $2.3\times 10^{-18} ergs\;s^{-1} cm^{-2}$. Therefore, potentially one can monitor the spherical volume with radius of the order of $R_0 = 260\; ly$. In the Solar neighbourhood the stellar number density, $n$, is of the order of $0.138$ star pc$^{-3}$ \citep{dens}, which means that the total number of stars could be
\begin{equation}
\label{numb}
N\simeq\frac{4\pi R_0^3}{3}\; n\simeq 3\times 10^5,
\end{equation}
with approximately $1000$ G-type (Solar-type) stars. Although the flux method cannot distinguish emission from the planet and the megastructure, it allows to find objects in the habitable zone. For identifying the candidates of megaconstruction one can use the spatial resolution of the telescopes.
The VLTI telescope has the maximum angular resolution, $\theta_m$, of the order of $0.001$ mas (milliarcsecond - please see the technical characteristics of the VLTI instruments. By taking into account this value, one can obtain the maximum resolving distance
\begin{equation}
\label{dist}
D_m = \frac{2R}{\theta_m}\simeq 280\;ly.
\end{equation}
From the aforementioned two critical values, it is clear that planetary-scale megastructures inside $260\; ly$ might be detectable.
It is worth noting that a monolithic DS will be less stable compared to concentric rotating rings. In this case very interesting observational features might arise. By means of the non-relativistic Doppler effect, the observed wavelength is given by \citep{carroll}
\begin{equation}
\label{dopp1}
\lambda \simeq \lambda_s\left(1+v\cos\theta/c\right),
\end{equation}
where by $\lambda_s$ we represent the original wavelength emitted by the ring, $\theta$ is an angle of velocity direction measured in an observer's frame of reference, $v$ denotes the corresponding velocity and $c$ is the sped of light. Without going into geometric details, to make the orders of magnitude, we consider diametrically opposite sides of the ring relative to an observer moving with velocities $\upsilon_p \pm v_{_I}$, where $\upsilon_p$ denotes the orbital velocity of the planet around a host star and $v_{_I}=\sqrt{GM_p/R_p}$ is the orbital speed of the ring around a planet. Then, for the diametrically opposite sides of the ring from Eq. (\ref{dopp1}) one obtains
\begin{equation}
\label{dopp2}
\frac{\lambda_s}{\Delta\lambda}\simeq\frac{c}{2\upsilon_{_I}}\simeq 1.9\times 10^4\times \left(
\frac{R_p}{R_{\oplus}}\times \frac{M_{\oplus}}{M_p} \right)^{1/2}.
\end{equation}
In this regard, one should note that up to now the VLTI has the highest resolving power $RP\equiv\lambda/\Delta\lambda = 25000$ for $\lambda \approx 10 \mu$m (which corresponds to the blackbody radiation with $T=300$K) \footnote{www.eso.org/public/teles-instr/paranal-observatory/vlt/} and since $RP>\lambda_s/\Delta\lambda$, such orbital motions of the ring might be detected by the spectrographs of the VLTI instruments. Similarly, if instead of a solid mega-construction one uses a web of satellites, the same method might be used. It is worth noting that the observational pattern might be characterized by absorption lines of the matter the megastructure is made of. Therefore, in principle, the emission fingerprint might carry information about the matter, but this is not the scope of the current paper and we do not consider it.
One has to note that the SpaceX starlink satellites use radio communication. Therefore, a reasonable question might appear: is it possible to detect their radio signals by means of the China's FAST telescope? By taking into account that its system temperature is $T_{sys}\simeq 20$K and the illuminated aperture area, $A\simeq 7108$ cm$^2$ \citep{fast}, one can conclude that the minimum spectral flux density, $kT_{sys}/A$, which can be distinguished from noise is of the order of $6\times 10^{-24}$ ergs s$^{-1}$cm$^{-2}$Hz$^{-1}$. On the other hand, by assuming that only $1\%$ of the incident Solar energy is used for interstellar communication, one can find that if one uses the Hydrogen atom's frequency, 1420 Hz, the FAST telescope can detect such isotropic sources up to the distances of the order of $160$ pc.
Making use of simple geometry we have performed our calculations for solar-type stars with Earth-like planets and it is clear that the similar estimates can be straightforwardly performed for Type-I civilizations living in the systems with different parameters.
\section{Conclusion}
Extrapolating the idea of SpaceX's Starlink constellation we assume that an alien society with index $K = 0.7$ (our civilization) will reach Type-I in $1000$ years, which is enough to build a planetary megastructure for collecting the required material.
Launching has been analysed from the point of view energy costs and it has been shown that energy required to construct a megastructure is small compared to the total energy received from the star.
We have shown that the construction will be visible in the IR spectrum, which might be detected by VLTI instruments up to the distance $260$ light years, with $\sim 10^3$ Solar-type stars.
We have also emphasised that the spectral variability method might be an efficient tool to detect either orbital rotation of the solid megastructure or internal motions of small satellites.
It has been estimated the possibility to detect the radio emission operating on the Hydrogen atom?s frequency. It has been found that for reasonable parameters FAST can detect such radio sources from relatively large distances.
\section*{Acknowledgments}
The research was supported by the Shota Rustaveli National Science Foundation grant (NFR17-587).
|
{
"timestamp": "2021-03-15T01:15:08",
"yymm": "2103",
"arxiv_id": "2103.07227",
"language": "en",
"url": "https://arxiv.org/abs/2103.07227"
}
|
\section{Introduction}
In 5G and beyond, the number of wireless devices, and their rate requirements, will increase by orders of magnitude \cite{paperG6}. To cope with such demands, different methods are proposed to improve the capacity and spectral efficiency. Here, one of the promising techniques is network densification, i.e., the deployment of many base stations (BSs) of different types such that there are more resource blocks per unit area. The BSs, however, need to be connected to the operators' core network via a transport network, the problem which becomes challenging
as the number of the BSs/users increases. Particularly, the increase of backhaul traffic may lead to backhaul congestion which, in turn, leads to end-to-end latency increment. This is the main motivation for wireless caching schemes reducing the backhaul load.
Caching is defined as storing popular reusable information at intermediate nodes to reduce the backhauling load. Such a technique is of interest in delay- and/or backhaul-constrained applications such as D2D, V2X and integrated access and backhaul.
Not every type of information is cacheable, for example, interactive applications such as gaming and voice calls. However, most of the network traffic today, including trending tweets, breaking news and video, is cacheable. Here, particular attention is paid to video. This is because, according to \cite{video2022}, by 2022, $79\%$ of the world's mobile traffic will be video. For instance, Netflix and YouTube alone account for almost half of peak downstream traffic in USA \cite{youtube}. Also, video is typically long and pre-recorded, which makes planning, prediction and segmentation easy.
Specially, as demonstrated in Fig. 1, video has high variation of the daily traffic profile, and increases the backhaul peak rate during the night significantly. This is important because the wireless network is designed based on the peak traffic. Then, with a variant daily traffic profile, the network will be underutilized most of the time, which is not economically viable. Thus, predicting the videos of interest in the high-traffic (HT) periods and caching them in the access points close to the devices during the low-traffic (LT) periods will lead to considerable cost reduction. For these reasons, caching is currently used by different content providers where, for instance, caches serve approximately $60\%-80\%$ (resp. up to $90\%$) of the Netflix \cite{Netflix2} (resp. Facebook \cite{facebook}) content requests.
\begin{figure}
\vspace{-2mm}
\centering
\includegraphics[width=0.6\columnwidth]{Figtraffic-eps-converted-to.pdf}\vspace{-2mm}
\caption{Application traffic daily profile in western Europe \cite[Fig. A3.3]{ssitu}.}\label{figure111}
\vspace{-4mm}
\end{figure}
The initial (un-coded) caching schemes where based on distributing the same information between the cache nodes, and minimizing the cache miss probability defined as the probability of the event that the device's requested file is not in the cache node. Here, depending on the amount of the information available about the sequence of the future requested files, different, e.g., Belady, highest-popularity-first, least-frequently-used and least-recently-used, algorithms have been proposed \cite{7080842}. Minimizing the cache miss probability improves the average system performance. On the other hand, the fundamental work of M. A. Maddah-ali and U. Niesen \cite{6763007,6807823} exploited the multicasting opportunity of cache networks and network coding concept to introduce coded-cashing, minimizing the worst-case backhaul peak rate in HT period. Also, following \cite{6763007,6807823}, there have been several works on the performance analysis of coded-caching systems, e.g., \cite{7857805,8674819,cairecodedcaching}. In these works, it is mainly concentrated on proper partitioning and distribution of the sub-packets between the cache nodes while the wireless channel between the serve and the cache nodes has been rarely studied, e.g. \cite{9014575}.
In this letter, we study the effect of adaptive data transmission and different data decoding/buffering schemes on the performance of wireless coded-caching networks. We present adaptive rate and power allocation schemes between the sub-packets such that the network successful transmission probability (STP) is maximized. Here, STP is defined as the probability of the event that all cache nodes can decode their intended signals correctly. Moreover, we investigate the effect of different decoding and buffering schemes on the network STP. We concentrate on the worst-case peak backhaul traffic cases where the cache nodes request for different signals during the HT periods. As we show, the performance of coded-caching networks is considerably affected by the decoding scheme as well as rate/power allocation.
\section{System Model}
\begin{figure}
\vspace{-2mm}
\centering
\includegraphics[width=0.6\columnwidth]{xFigchache2.jpg}\vspace{-2mm}
\caption{Coded-caching concept. Subplot A: Placements phase (at 2 AM) where the cache nodes are filled in with different sub-packets. Subplot B: Delivery phase (at 9 PM) where a single subpacket is braodcasted to the cache nodes. }\label{figure222}
\vspace{-5mm}
\end{figure}
As illustrated in Fig. 2, consider the simplest case of coded-caching networks where a server connects to two cache nodes $\text{C}_1$ and $\text{C}_2$. However, it is straightforward to extend the results to the cases with different numbers of cache nodes. In general, coded-caching has two, namely, placement and delivery, phases. During the placement phase, performed in the LT period (say, at 2 AM), the server divides the packets, for instance, packets $X$ and $\tilde X$, to sub-packets $X_1, X_2, \tilde X_1, \tilde X_2$ with $X=[X_1, X_2]$, $\tilde X=[\tilde X_1 \tilde X_2],$ and fills in the caches with different sub-packets having no knowledge, or probably a rough estimation, of the cache nodes' data requests during the HT period (see Fig. 2a). Assuming the packets $X$ and $\tilde X$ to be of length $2L$, the sub-packets $X_1, X_2, \tilde X_1, \tilde X_2$ are of length $L$. In this way, during the LT period, the server sends separate signals $[X_1 \tilde X_1]$ and $[X_2 \tilde X_2]$, each of length $2L,$ to $\text{C}_1$ and $\text{C}_2$, respectively.
During the HT period (say, 9 PM), the server serves the cache nodes based on their instantaneous data requests. Let $\bigoplus$ be the superposition operator. Also, considering the worst-case scenario in terms of backhaul traffic, assume that the caches $\text{C}_1$ and $\text{C}_2$ request for different packets $X$ and $\tilde X$, respectively. Then, as demonstrated in Fig. 2b, the server broadcasts a single sub-packet $X_2\bigoplus \tilde X_1$ of length $L$. Also, using the accumulated signals, each cache node may use different methods to decode its message of interest (see Section III). In this way, as shown in \cite{6763007,6807823}, coded-caching reduces the peak HT backhaul traffic by $50\%$ because, unlike uncoded caching, only a single sub-packet is broadcasted at, say, 9 PM. Finally, it is interesting to note that the presented coded-caching approach is a specific combination of the orthogonal multiple access (OMA) and NOMA (N:non) schemes used in different time slots with proper packet partitioning and signal decoding at the receivers.
Let us denote the server-cache $i$ channel coefficient by $h_i, i=1,2,$ and define the channel gains as $g_i=|h_i|^2, i=1,2.$ We consider Rayleigh-fading conditions with channel probability density functions (PDFs) $f_{g_i}(u)=\lambda_ie^{-\lambda_iu}, i=1,2,$ where $\lambda_i, i=1,2,$ depends on the long-term channel quality. Then, the signals received by $\text{C}_1$ and $\text{C}_2$ during the LT period are given by
\begin{align}\label{eq:eqLT}
\left\{\begin{matrix}
\left[Y_{\text{C}_1}^\text{LT}(t)\, \tilde Y_{\text{C}_1}^\text{LT}(t)\right]=\sqrt{P}h_1^\text{LT}\left[X_1(t) \tilde X_1(t)\right]+\left[Z_{\text{C}_1,1}(t)\, Z_{\text{C}_1,2}(t)\right], t=1,\ldots,L
\\
\left[Y_{\text{C}_2}^\text{LT}(t)\, \tilde Y_{\text{C}_2}^\text{LT}(t)\right]=
\sqrt{P}h_2^\text{LT}\left[X_2(t) \tilde X_2(t)\right]+\left[Z_{\text{C}_2,1}(t)\, Z_{\text{C}_2,2}(t)\right], t=1,\ldots,L
\end{matrix}\right.
\end{align}
while, at the HT period, the received signals are \begin{align}\label{eq:eqHT}
\left\{\begin{matrix}
Y_{C_1}^\text{HT}(t)=\sqrt{P}h_{1}^\text{HT}S(t)+Z_{\text{C}_1,1}(t), t=1,\ldots,L
\\
Y_{C_2}^\text{HT}(t)=\sqrt{P}h_{2}^\text{HT}S(t)+Z_{\text{C}_2,1}(t), t=1,\ldots,L
\\
S(t)=\alpha X_{2}(t)+\sqrt{1-\alpha^2}\tilde X_{1}(t).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\end{matrix}\right.
\end{align}
Here, $h_i^\text{LT}$ and $h_i^\text{HT}, i=1,2,$ represent the channel realizations of $h_i$ at LT and HT periods, respectively. Then, $Z_{\text{C}_i,j},i,j=1,2,$ denotes the unit-variance additive Gaussian noise, and $P$ is the server maximum transmit power. Also, $X_i, \tilde X_i, i=1,2,$ are the unit-variance signals of the sub-packets, $S(t)$ represents the unit-variance superimposed signals of $X_2$ and $\tilde X_1$, and $\alpha\in [0,1]$ gives the power partitioning between these signals.
\vspace{-4mm}
\section{Analytical results}
Let us denote the number of information bits in the packets $X$ and $\tilde X$ by $K$ and $\tilde K$, respectively. Also, the information nats are divided between the sub-packets according to
\begin{align}\label{eq:eqbitpart}
\left\{\begin{matrix}
K=K_1+K_2
\\
\tilde K=\tilde K_1+\tilde K_2,
\end{matrix}\right.
\end{align}
which, defining the code rates $R=\frac{K}{2L}$ and $\tilde R=\frac{\tilde K}{2L},$ leads to \begin{align}\label{eq:eqratepart}
\left\{\begin{matrix}
2R=R_1+R_2
\\
2\tilde R=\tilde R_1+\tilde R_2.
\end{matrix}\right.
\end{align}
Our goal is to design proper rate allocation between sub-packets, i.e., $R_i,\tilde R_i, i=1,2,$ and power split parameter $\alpha$ in (\ref{eq:eqHT}) such that the network STP is maximized. STP is defined as
\begin{align}\label{eq:eqSTPdef}
\Pr(\text{ST})=\frac{1}{2}\left(\Pr\left(\text{C}_1 \,\text{successful}\right)+\Pr\left(\text{C}_2 \,\text{successful}\right)\right),
\end{align}
i.e., the probability of the event that both cache nodes can decode their intended packets correctly. Depending on the buffering and the coding schemes of the cache nodes, the network may lead to different STPs as follows.
\vspace{-3mm}
\subsection{Joint Decoding at HT Period using Successive Interference Cancellation}
In one approach, referred to as Method 1 in the following, the cache nodes buffer the signals received in the LT period, and use both maximum ratio combining (MRC) and successive interference cancellation (SIC) for message decoding at HT periods. Let us concentrate on cache node $\text{C}_1$. Receiving $Y_{\text{C}_1}^\text{HT}$ in (\ref{eq:eqHT}) during the HT period and with $\tilde Y_{\text{C}_1}^\text{LT}$ in (\ref{eq:eqLT}) already buffered, $\text{C}_1$ first uses MRC to decode $\tilde X_1(t)$. Then, with a successful decoding of $\tilde X_1(t)$, node $\text{C}_1$ uses the SIC concept to remove $\tilde X_1(t)$ from $Y_{\text{C}_1}^\text{HT}(t)$, leading to an interference-free signal
\vspace{-2mm}
\begin{align}\label{eq:Eqsic1}
\mathcal{Y}_{\text{C}_1}^\text{HT}=\sqrt{P}h_1^\text{HT}\alpha X_2(t)+Z_{\text{C}_1,1}(t).
\end{align}
Finally, the decoder generates the concatenated signal $[Y_{\text{C}_1}^\text{LT}(t) \,\mathcal{Y}_{\text{C}_1}^\text{HT}(t)]$, with $Y_{\text{C}_1}^\text{LT}(t)$ and $\mathcal{Y}_{\text{C}_1}^\text{HT}(t)$ given in (\ref{eq:eqLT}) and (\ref{eq:Eqsic1}), respectively, and decodes the packet $X$ in \emph{one-shot.} In this way, and considering the same procedure in $\text{C}_2$ by using (\ref{eq:eqLT})-(\ref{eq:eqHT}), STP in (\ref{eq:eqSTPdef}) is given by
\begin{align}\label{eq:eqSTPdef1}
&\Pr(\text{ST})=\frac{1}{2}(\eta_1\gamma_1+\eta_2\gamma_2),\nonumber\\&
\eta_1=\Pr\left(\log\left(1+\frac{(1-\alpha^2)Pg_1^\text{HT}}{1+\alpha^2Pg_1^\text{HT}}+Pg_1^\text{LT}\right)\ge \tilde R_1\right),
\nonumber\\&\gamma_1=\Pr\left(\log\left(1+Pg_1^\text{LT}\right)+\log\left(1+\alpha^2Pg_1^\text{HT}\right)\ge 2R\right)\nonumber\\&
\eta_2=\Pr\left(\log\left(1+\frac{\alpha^2Pg_2^\text{HT}}{1+(1-\alpha^2)Pg_2^\text{HT}}+Pg_2^\text{LT}\right)\ge R_2\right),\nonumber\\
&\gamma_2=\Pr\left(\log\left(1+Pg_2^\text{LT}\right)+\log\left(1+(1-\alpha^2)Pg_2^\text{HT}\right)\ge 2\tilde R\right).
\end{align}
Here, $\eta_1$ (resp. $\eta_2$) is the probability of successful decoding of $\tilde X_1(t)$ (resp. $X_2(t)$) at $\text{C}_1$ (resp. $\text{C}_2$) using MRC. Then, $\gamma_1$ and $\gamma_2$ give the probability that, removing the interference from the received signal in HT period, the caches can decode their intended signals correctly. Note that in $\gamma_1$ and $\gamma_2$ we have used the results on the maximum achievable rates of parallel Gaussian channels.
Considering Rayleigh-fading conditions, we have
\begin{align}\label{eq:eqeta11}
&\eta_1=1-\Pr\left(\frac{(1-\alpha^2)Pg_1^\text{HT}}{1+\alpha^2Pg_1^\text{HT}}+Pg_1^\text{LT}\le e^{\tilde R_1}-1\right)=
\nonumber\\&
1-\int_0^{\frac{e^{\tilde R_1}-1}{P}}f_{g_1}(y)\Pr\left(\frac{(1-\alpha^2)Pg_1^\text{HT}}{1+\alpha^2Pg_1^\text{HT}} \le {e^{\tilde R_1}-1}-Py \right)\text{d}y
\nonumber\\&
=e^{-\frac{\lambda_1\left({e^{\tilde R_1}-1}\right)}{P}}+
\int_{\frac{\alpha^2 e^{\tilde R_1}-1}{\alpha^2P}}^{\frac{e^{\tilde R_1}-1}{P}}e^{\lambda_1\left(y+\frac{{e^{\tilde R_1}-1}-Py}{\alpha^2P-(1-\alpha^2)P\left({e^{\tilde R_1}-1}-Py\right)}\right)}\text{d}y,
\end{align}
which can be calculated numerically. Also, following the same procedure, we have
\begin{align}\label{eq:Eqeta22}
&\eta_2=e^{-\frac{\lambda_2\left(e^{R_2}-1\right)}{P}}
+\int_{\frac{\left(1-\alpha^2\right)e^{R_2}-1}{\left(1-\alpha^2\right)P}}^{\frac{e^{R_2}-1}{P}}{\lambda_2e^{-\lambda_2\left(y+\frac{e^{R_2}-1-Py}{\alpha^2P-\left(1-\alpha^2\right)P\left(e^{R_2}-1-Py\right)}\right)}}\text{d}y.
\end{align}
The terms $\gamma_i,i=1,2,$ on the other hand, do not have closed-form or easy-to-deal integration expressions. Thus, we use the Jensen's inequality $\frac{1}{n}\sum_{i=1}^n\log(1+x_i)\le \log\left(1+\frac{1}{n}\sum_{i=1}^nx_i\right)$ \cite[Eq. (30)]{7445896} and $f_{g_i}(u)=\lambda_ie^{-\lambda_iu}, i=1,2,$ to rephrase $\gamma_i,i=1,2,$ as
\vspace{-2mm}
\begin{align}\label{eq:eqgamma1}
\gamma_1&\le 1-\Pr\left(g_1^\text{LT}+\alpha^2g_1^\text{HT}\le \frac{2}{P}\left(e^R-1\right)\right) \nonumber\\&
=1-\int_0^{\frac{2\left(e^R-1\right)}{P}}{f_{g_1}(x)\Pr\left(g_1^\text{HT}\le\frac{\frac{2\left(e^R-1\right)}{P}-x}{\alpha^2}\right)}\text{d}x\nonumber\\&
=e^{\frac{-2\lambda_1\theta}{P}}+\frac{\alpha^2 e^{\frac{-2\lambda_1\theta}{P\alpha^2}}}{\alpha^2-1}\left(1-e^{\frac{-2\lambda_1\left(\alpha^2-1\right)\theta}{P\alpha^2}}\right),
\end{align}
\vspace{-2mm}
\begin{align}\label{eq:eqgamma2}
\gamma_2&\le e^{\frac{-2\lambda_2\theta}{P}}+\frac{\left(\alpha^2-1\right) e^{\frac{-2\lambda_2\theta}{P\left(1-\alpha^2\right)}}}{\alpha^2}\left(1-e^{\frac{-2\lambda_2\alpha^2\theta}{P\left(\alpha^2-1\right)}}\right),
\end{align}
where (\ref{eq:eqgamma2}) follows the same procedure as in (\ref{eq:eqgamma1}). In this way, the optimal rate/power allocation maximizing STP is given by
\begin{align}\label{eq:eqoptprob}
&\max \frac{1}{2}\{\eta_1\gamma_1+\eta_2\gamma_2\}\nonumber\\
&\text{s.t. } \,\alpha\in[0,1], R_1+R_2=2R, \,\tilde R_1+\tilde R_2=2\tilde R,
\end{align}
which can be effectively solved by, e.g., exhaustive search or the machine-learning based scheme of \cite{8520925}.
\vspace{-2mm}
\subsection{Joint Decoding at HT Period without SIC}
Implementation of MRC and SIC, to decode and remove the interference, increases the decoding complexity/delay. Also, SIC suffers from error propagation problem, e.g., \cite{nomaharq}. For these reasons, in Method 2, each cache node decodes its intended packets in one-shot by considering the interference as an additive noise. Here, (\ref{eq:eqSTPdef1}) is rephrased as
\begin{align}\label{eq:eqSTPdef3}
&\Pr(\text{ST})=\frac{1}{2}(\bar \gamma_1+\bar \gamma_2),\nonumber\\&
\bar\gamma_1=\Pr\left(\log\left(1+Pg_1^\text{LT}\right)+\log\left(1+\frac{\alpha^2Pg_1^\text{HT}}{1+(1-\alpha^2)Pg_1^\text{HT}}\right)\ge 2R\right)\nonumber\\&
\bar\gamma_2=\Pr\left(\log\left(1+Pg_2^\text{LT}\right)+\log\left(1+\frac{(1-\alpha^2)Pg_2^\text{HT}}{1+\alpha^2Pg_2^\text{HT}}\right)\ge 2\tilde R\right),
\end{align}
where, using the Jensen's inequality and the same procedure as in (\ref{eq:eqgamma1}), we have
\begin{align}\label{eq:eqjensen2}
&\bar\gamma_1\le e^{-\frac{2\lambda_1\left(e^R-1\right)}{P}}+\int_{\frac{2\left(e^R-1\right)-\frac{\alpha^2}{1-\alpha^2}}{P}}^{\frac{2\left(e^R-1\right)}{P}}\lambda_1e^{-\lambda_1\left(x+\frac{2\left(e^R-1\right)-Px}{\alpha^2P+(1-\alpha^2)P(Px-2\left(e^R-1\right))}\right)}\text{d}x,\nonumber\\&
\bar\gamma_2\le e^{-\frac{2\lambda_2\left(e^{\tilde R}-1\right)}{P}}
+\int_{\frac{2\left(e^{\tilde R}-1\right)-\frac{1-\alpha^2}{\alpha^2}}{P}}^{\frac{2\left(e^{\tilde R}-1\right)}{P}}\lambda_2e^{-\lambda_2\left(x+\frac{2\left(e^{\tilde R}-1\right)-Px}{(1-\alpha^2)P+\alpha^2P(Px-2\left(e^{\tilde R}-1\right))}\right)}\text{d}x,
\end{align}
which can be calculated numerically.
Finally, note that, replacing (\ref{eq:eqSTPdef3}) into (\ref{eq:eqoptprob}), the optimal performance of the cache nodes in Method 2 is independent of the rate split between the sub-packets. This, although Method 1 gives the best performance in terms of the worst-case peak traffic,
may give an advantage to Method 2, compared to Method 1. This is because in Method 1 the rate split is performed by considering the worst-case condition with the cache nodes requesting for different signals during HT period. However, if the caches request for the same signals during HT period, the rate split scheme of Method 1 is not necessarily optimal. As opposed, in Method 2, the rate split is independent of the caches requested signals in HT periods.
\subsection{Separate Decoding using SIC}
In Methods 1-2, one needs to follow the coding schemes of incremental redundancy hybrid automatic repeat request (HARQ)-based protocols or Raptor codes, e.g., \cite{6164088}, where the message is decoded by concatenating different sub-packets. Alternatively, in Method 3, we consider the case where, while MRC and SIC are used to decode and remove the interference signal, respectively, each cache node decodes its sub-packets of interest separately. That is, considering $\text{C}_1,$ $X_1$ (resp. $X_2$) is decoded during the LT (resp. HT) period. In this case, the STP (\ref{eq:eqSTPdef1}) is changed to
\vspace{-1mm}
\begin{align}\label{eq:eqSTPdef2}
&\Pr\left(\text{ST}\right)=\frac{1}{2}\left(\eta_1\breve{\gamma}_{11}\breve{\gamma}_{12}+\eta_2\breve{\gamma}_{21}\breve{\gamma}_{22}\right),\nonumber\\&
\breve{\gamma}_{11}=\Pr\left(\log\left(1+Pg_1^\text{LT}\right)\ge R_1\right)=e^{-\frac{\lambda_1\left(e^{R_1}-1\right)}{P}},\nonumber\\&
\breve{\gamma}_{12}=\Pr\left(\log\left(1+\alpha^2Pg_1^\text{HT}\right)\ge R_2\right)=e^{-\frac{\lambda_1\left(e^{R_2}-1\right)}{\alpha^2P}}
\nonumber\\&
\breve{\gamma}_{21}=\Pr\left(\log\left(1+(1-\alpha^2)Pg_2^\text{HT}\right)\ge \tilde R_1\right)=e^{-\frac{\lambda_2\left(e^{\tilde R_1}-1\right)}{(1-\alpha^2)P}},\nonumber\\&
\breve{\gamma}_{22}=\Pr\left(\log\left(1+Pg_2^\text{LT}\right)\ge \tilde R_2\right)=e^{-\frac{\lambda_2\left(e^{\tilde R_2}-1\right)}{P}},
\end{align}
with $\eta_i,i=1,2,$ given in (\ref{eq:eqSTPdef1}), and (\ref{eq:eqoptprob}) is adapted correspondingly. In (\ref{eq:eqSTPdef2}), $\breve{\gamma}_{11}$ is the probability that $\text{C}_1$ decodes $X_1$ during the LT period. Also, $\breve{\gamma}_{12}$ gives the probability that, after decoding and removing $\tilde X_1$, the cache node $\text{C}_1$ correctly decodes $X_2$ in the HT period. Also, the same arguments hold for $\breve{\gamma}_{2i},i=1,2.$
Note that, although Method 1 maximizes the achievable rate/STP, Method 3 has a number of advantages including:
\begin{itemize}
\item \textbf{Low decoding complexity}: Because, as opposed to Methods 1-2 decoding long codewords of length $2L$, Method 3 is based on decoding sub-packets of length $L$.
\item \textbf{Efficient HARQ-based transmissions}: In Methods 1-2, all packets are decoded during the HT periods and, in case of decoding failure, the message is retransmitted at that period. Such HARQ-based retransmissions increase the backhauling load at HT period. As opposed, in Method 3, the decoding of the first received sub-packets and all their required HARQ-based retransmissions are performed during the LT period, which reduces the backhauling cost of HARQ.
\end{itemize}
Finally, depending on the considered method, the buffering scheme of the caches during the LT period may change. Particularly, in Methods 1-2 the caches buffer the signals received during LT period without decoding. In Method 3, however, the caches buffer the sub-packets successfully decoded during LT period.
\vspace{-1mm}
\subsection{Separate Decoding without SIC}
To further reduce the complexity of Method 3, one can consider the case where, while decoding the sub-packets separately, the cache nodes consider the interference as an additive noise (Method 4). In this case, where the sub-packets are decoded in different LT and HT periods without SIC, the STP is given by
\begin{align}\label{eq:eqSTPdef4}
&\Pr\left(\text{ST}\right)=\frac{1}{2}(\breve{\gamma}_{11}\hat{\gamma}_{12}+\hat{\gamma}_{21}\breve{\gamma}_{22}),\nonumber\\&
\hat{\gamma}_{12}=\Pr\left(\log\left(1+\frac{\alpha^2Pg_1^\text{HT}}{1+(1-\alpha^2)Pg_1^\text{HT}}\right)\ge R_2\right)\nonumber\\&\,\,\,\,\,\,\,\,=\left\{\begin{matrix}
0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \text{if } R_2\ge -\log\left(1-\alpha^2 \right )\\
e^{-\lambda_1\frac{\left(e^{R_2}-1 \right )}{\alpha^2P-(1-\alpha^2)P\left(e^{R_2}-1 \right )}} & \text{otherwise}
\end{matrix}\right.
\nonumber\\&
\hat{\gamma}_{21}=\Pr\left(\log\left(1+\frac{(1-\alpha^2)Pg_2^\text{HT}}{1+\alpha^2Pg_2^\text{HT}}\right)\ge \tilde R_1\right)\nonumber\\&=\left\{\begin{matrix}
0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \text{if } \tilde R_1\ge -2\log\alpha\\
e^{-\lambda_2\frac{\left(e^{\tilde R_1}-1 \right )}{(1-\alpha^2)P-\alpha^2P\left(e^{\tilde R_1}-1 \right )}} & \text{otherwise}
\end{matrix}\right.
\end{align}
with $\breve{\gamma}_{11}$ and $\breve{\gamma}_{22}$ given in (\ref{eq:eqSTPdef2}). Also, in (\ref{eq:eqSTPdef4}) we use Rayleigh channel PDFs and some manipulations to derive the probabilities. For further comparisons between Methods 1-4, see Section IV.
\section{Simulation Results}
The simulation results are presented for the cases with $\lambda_1=1$ and $\lambda_2=0.1$, i.e., with $10$ dB difference between the channel gains of the server-cache links, and we define the transmission signal-to-noise ratio (SNR) as $10\log_{10} P$, considering the additive noises to be unit-variance. Note that we have evaluated the results for different parameter settings, and they show the same qualitative conclusions as those presented in the following. In Figs. 3-4, the results are obtained by optimizing the rate and power allocation. Here, both exhaustive search and the genetic-algorithm based scheme of \cite{8520925} have been used which have ended up in the same results, indicating the accuracy of the optimization process. In Fig. 5, we study the effect of rate/power allocation.
\begin{figure}
\vspace{-3mm}
\centering
\includegraphics[width=0.6\columnwidth]{FigCodedcaching1-eps-converted-to.pdf}\\\vspace{-3mm}
\caption{Comparison between the STP of Methods 1-4, $\lambda_1=1,$ $\lambda_2=0.1,$ $R=\tilde R=1$ npcu. The results are obtained by optimal rate and power allocation.}
\label{fig:fig_ivd9}
\vspace{-2mm}
\end{figure}
Considering $R=\tilde R=1$ nats-per-channel-use (npcu), Fig. 3 compares the performance of Methods 1-4. Also, the figure verifies the tightness of the Jensen's inequality-based approximation results of (\ref{eq:eqgamma1})-(\ref{eq:eqgamma2}). Then, Fig. 4 shows the STP versus the data rates $R=\tilde R$ for the cases with different decoding/buffering methods and transmission SNRs.
Considering Methods 1 and 3, with joint and separate decoding on the sub-packets, respectively, Fig. 5 studies the effect of optimal rate and power allocation on the network STP. Particularly, the figure compares the optimal results obtained by (\ref{eq:eqoptprob}) with the cases using uniform power allocation, i.e., with $\alpha=\frac{\sqrt{2}}{2}$ in (\ref{eq:eqHT}), and/or equal rate split, i.e., $R_i=\tilde R_i, i=1,2$. According to the figures, the following conclusions can be drawn:
\begin{figure}
\vspace{-3mm}
\centering
\includegraphics[width=0.6\columnwidth]{FigCodedcachingrate-eps-converted-to.pdf}\\\vspace{-2mm}
\caption{Achievable STP versus the data rate for different methods. The results are obtained by optimal rate and power allocation and $\lambda_1=1, \lambda_2=0.1.$}
\label{fig:fig_ivd10}
\vspace{-0mm}
\end{figure}
\begin{figure}
\vspace{-3mm}
\centering
\includegraphics[width=0.6\columnwidth]{plottingcomparison-eps-converted-to.pdf}\\\vspace{-2mm}
\caption{On the effect of optimal rate and power allocation in Methods 1 and 3, $R=\tilde R=1$ npcu and $\lambda_1=1, \lambda_2=0.1$.}
\label{fig:fig_ivd11}
\vspace{-0mm}
\end{figure}
\begin{itemize}
\item The approximation results of (\ref{eq:eqgamma1})-(\ref{eq:eqgamma2}) properly approximate the probabilities $\gamma_i, i=1,2$ (Fig. 3. Also, the same point is observed for $\bar \gamma_i,i=1,2,$ in (\ref{eq:eqjensen2}) although not shown in the figure). Thus, the approximations can be well utilized for the performance evaluation of Methods 1-2, i.e., in the cases with joint decoding of the sub-packets.
\item Compared to the cases with separate decoding of sub-packets, i.e., Methods 3-4, considerable STP improvement is observed by joint decoding of the sub-packet, i.e., Methods 1-2 (Figs. 3-4). However, as explained in Section III.C, the STP increment of Methods 1-2 is at the cost of decoding complexity and possible HARQ-based retransmissions at HT periods. On the other hand, for both cases with joint and separate decoding of the sub-packets, using SIC-based interference cancellation leads to marginal performance improvement at low rates while its effect increases with the data rate (Fig. 4).
Finally, as the data rate increases, the performance gap between Methods 2 and 3 decreases, i.e., one can reach the same STP as in the cases with joint sub-packet decoding of interference-affected signals by separate sub-packets decoding if the interference signals are removed using SIC. (Fig. 4).
\item For all parameter settings, Method 1 leads to the highest STP, compared to Methods 2-4, at the cost of decoding delay/possible retransmissions at HT periods (Figs. 3-4). For instance, with the parameter settings of Fig. 4 and data rate 1.5 npcu, the implementation of Method 1 with transmit SNR 10 dB results in the same STP, $80\%$, as in the cases with Method 4 and SNR 15 dB, i.e., using advanced decoding methods leads to 5 dB gain in SNR (Fig. 4).
\item For both cases with and without interference cancellation (Methods 1 and 3), optimal rate allocation leads to considerable STP increment (Fig. 5. The same conclusion is observed in Methods 2 and 4, although not presented in the figure). Also, the relative performance gain of optimal rate split increases in the cases with interference cancellation. Finally, considering the interference as additive noise, optimal power allocation between sub-packets during HT period increases the STP. However, with interference cancellation and joint decoding of sub-packets, the effect of optimal power allocation between HT period sub-packets is marginal (Fig. 5).
\end{itemize}
\section{Conclusions}
This paper studied the performance coded-caching networks in the cases with adaptive rate/power allocation and different decoding/buffering schemes. As we showed, joint decoding of the sub-packets at HT periods leads to considerable performance improvement of coded-caching setups. Also, for different decoding schemes, optimal rate split between the sub-packets increases the STP considerably while optimal power allocation between the sub-packets of HT period only improves the STP if SIC-based receiver is not implemented and the sub-packets are decoded separately.
\vspace{-2mm}
\bibliographystyle{IEEEtran}
\section{Introduction}
In 5G and beyond, the number of wireless devices, and their rate requirements, will increase by orders of magnitude \cite{paperG6}. To cope with such demands, different methods are proposed to improve the capacity and spectral efficiency. Here, one of the promising techniques is network densification, i.e., the deployment of many base stations (BSs) of different types such that there are more resource blocks per unit area. The BSs, however, need to be connected to the operators' core network via a transport network, the problem which becomes challenging
as the number of the BSs/users increases. Particularly, the increase of backhaul traffic may lead to backhaul congestion which, in turn, leads to end-to-end latency increment. This is the main motivation for wireless caching schemes reducing the backhaul load.
Caching is defined as storing popular reusable information at intermediate nodes to reduce the backhauling load. Such a technique is of interest in delay- and/or backhaul-constrained applications such as D2D, V2X and integrated access and backhaul.
Not every type of information is cacheable, for example, interactive applications such as gaming and voice calls. However, most of the network traffic today, including trending tweets, breaking news and video, is cacheable. Here, particular attention is paid to video. This is because, according to \cite{video2022}, by 2022, $79\%$ of the world's mobile traffic will be video. For instance, Netflix and YouTube alone account for almost half of peak downstream traffic in USA \cite{youtube}. Also, video is typically long and pre-recorded, which makes planning, prediction and segmentation easy.
Specially, as demonstrated in Fig. 1, video has high variation of the daily traffic profile, and increases the backhaul peak rate during the night significantly. This is important because the wireless network is designed based on the peak traffic. Then, with a variant daily traffic profile, the network will be underutilized most of the time, which is not economically viable. Thus, predicting the videos of interest in the high-traffic (HT) periods and caching them in the access points close to the devices during the low-traffic (LT) periods will lead to considerable cost reduction. For these reasons, caching is currently used by different content providers where, for instance, caches serve approximately $60\%-80\%$ (resp. up to $90\%$) of the Netflix \cite{Netflix2} (resp. Facebook \cite{facebook}) content requests.
\begin{figure}
\vspace{-2mm}
\centering
\includegraphics[width=0.6\columnwidth]{Figtraffic-eps-converted-to.pdf}\vspace{-2mm}
\caption{Application traffic daily profile in western Europe \cite[Fig. A3.3]{ssitu}.}\label{figure111}
\vspace{-4mm}
\end{figure}
The initial (un-coded) caching schemes where based on distributing the same information between the cache nodes, and minimizing the cache miss probability defined as the probability of the event that the device's requested file is not in the cache node. Here, depending on the amount of the information available about the sequence of the future requested files, different, e.g., Belady, highest-popularity-first, least-frequently-used and least-recently-used, algorithms have been proposed \cite{7080842}. Minimizing the cache miss probability improves the average system performance. On the other hand, the fundamental work of M. A. Maddah-ali and U. Niesen \cite{6763007,6807823} exploited the multicasting opportunity of cache networks and network coding concept to introduce coded-cashing, minimizing the worst-case backhaul peak rate in HT period. Also, following \cite{6763007,6807823}, there have been several works on the performance analysis of coded-caching systems, e.g., \cite{7857805,8674819,cairecodedcaching}. In these works, it is mainly concentrated on proper partitioning and distribution of the sub-packets between the cache nodes while the wireless channel between the serve and the cache nodes has been rarely studied, e.g. \cite{9014575}.
In this letter, we study the effect of adaptive data transmission and different data decoding/buffering schemes on the performance of wireless coded-caching networks. We present adaptive rate and power allocation schemes between the sub-packets such that the network successful transmission probability (STP) is maximized. Here, STP is defined as the probability of the event that all cache nodes can decode their intended signals correctly. Moreover, we investigate the effect of different decoding and buffering schemes on the network STP. We concentrate on the worst-case peak backhaul traffic cases where the cache nodes request for different signals during the HT periods. As we show, the performance of coded-caching networks is considerably affected by the decoding scheme as well as rate/power allocation.
\section{System Model}
\begin{figure}
\vspace{-2mm}
\centering
\includegraphics[width=0.6\columnwidth]{xFigchache2.jpg}\vspace{-2mm}
\caption{Coded-caching concept. Subplot A: Placements phase (at 2 AM) where the cache nodes are filled in with different sub-packets. Subplot B: Delivery phase (at 9 PM) where a single subpacket is braodcasted to the cache nodes. }\label{figure222}
\vspace{-5mm}
\end{figure}
As illustrated in Fig. 2, consider the simplest case of coded-caching networks where a server connects to two cache nodes $\text{C}_1$ and $\text{C}_2$. However, it is straightforward to extend the results to the cases with different numbers of cache nodes. In general, coded-caching has two, namely, placement and delivery, phases. During the placement phase, performed in the LT period (say, at 2 AM), the server divides the packets, for instance, packets $X$ and $\tilde X$, to sub-packets $X_1, X_2, \tilde X_1, \tilde X_2$ with $X=[X_1, X_2]$, $\tilde X=[\tilde X_1 \tilde X_2],$ and fills in the caches with different sub-packets having no knowledge, or probably a rough estimation, of the cache nodes' data requests during the HT period (see Fig. 2a). Assuming the packets $X$ and $\tilde X$ to be of length $2L$, the sub-packets $X_1, X_2, \tilde X_1, \tilde X_2$ are of length $L$. In this way, during the LT period, the server sends separate signals $[X_1 \tilde X_1]$ and $[X_2 \tilde X_2]$, each of length $2L,$ to $\text{C}_1$ and $\text{C}_2$, respectively.
During the HT period (say, 9 PM), the server serves the cache nodes based on their instantaneous data requests. Let $\bigoplus$ be the superposition operator. Also, considering the worst-case scenario in terms of backhaul traffic, assume that the caches $\text{C}_1$ and $\text{C}_2$ request for different packets $X$ and $\tilde X$, respectively. Then, as demonstrated in Fig. 2b, the server broadcasts a single sub-packet $X_2\bigoplus \tilde X_1$ of length $L$. Also, using the accumulated signals, each cache node may use different methods to decode its message of interest (see Section III). In this way, as shown in \cite{6763007,6807823}, coded-caching reduces the peak HT backhaul traffic by $50\%$ because, unlike uncoded caching, only a single sub-packet is broadcasted at, say, 9 PM. Finally, it is interesting to note that the presented coded-caching approach is a specific combination of the orthogonal multiple access (OMA) and NOMA (N:non) schemes used in different time slots with proper packet partitioning and signal decoding at the receivers.
Let us denote the server-cache $i$ channel coefficient by $h_i, i=1,2,$ and define the channel gains as $g_i=|h_i|^2, i=1,2.$ We consider Rayleigh-fading conditions with channel probability density functions (PDFs) $f_{g_i}(u)=\lambda_ie^{-\lambda_iu}, i=1,2,$ where $\lambda_i, i=1,2,$ depends on the long-term channel quality. Then, the signals received by $\text{C}_1$ and $\text{C}_2$ during the LT period are given by
\begin{align}\label{eq:eqLT}
\left\{\begin{matrix}
\left[Y_{\text{C}_1}^\text{LT}(t)\, \tilde Y_{\text{C}_1}^\text{LT}(t)\right]=\sqrt{P}h_1^\text{LT}\left[X_1(t) \tilde X_1(t)\right]+\left[Z_{\text{C}_1,1}(t)\, Z_{\text{C}_1,2}(t)\right], t=1,\ldots,L
\\
\left[Y_{\text{C}_2}^\text{LT}(t)\, \tilde Y_{\text{C}_2}^\text{LT}(t)\right]=
\sqrt{P}h_2^\text{LT}\left[X_2(t) \tilde X_2(t)\right]+\left[Z_{\text{C}_2,1}(t)\, Z_{\text{C}_2,2}(t)\right], t=1,\ldots,L
\end{matrix}\right.
\end{align}
while, at the HT period, the received signals are \begin{align}\label{eq:eqHT}
\left\{\begin{matrix}
Y_{C_1}^\text{HT}(t)=\sqrt{P}h_{1}^\text{HT}S(t)+Z_{\text{C}_1,1}(t), t=1,\ldots,L
\\
Y_{C_2}^\text{HT}(t)=\sqrt{P}h_{2}^\text{HT}S(t)+Z_{\text{C}_2,1}(t), t=1,\ldots,L
\\
S(t)=\alpha X_{2}(t)+\sqrt{1-\alpha^2}\tilde X_{1}(t).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\end{matrix}\right.
\end{align}
Here, $h_i^\text{LT}$ and $h_i^\text{HT}, i=1,2,$ represent the channel realizations of $h_i$ at LT and HT periods, respectively. Then, $Z_{\text{C}_i,j},i,j=1,2,$ denotes the unit-variance additive Gaussian noise, and $P$ is the server maximum transmit power. Also, $X_i, \tilde X_i, i=1,2,$ are the unit-variance signals of the sub-packets, $S(t)$ represents the unit-variance superimposed signals of $X_2$ and $\tilde X_1$, and $\alpha\in [0,1]$ gives the power partitioning between these signals.
\vspace{-4mm}
\section{Analytical results}
Let us denote the number of information bits in the packets $X$ and $\tilde X$ by $K$ and $\tilde K$, respectively. Also, the information nats are divided between the sub-packets according to
\begin{align}\label{eq:eqbitpart}
\left\{\begin{matrix}
K=K_1+K_2
\\
\tilde K=\tilde K_1+\tilde K_2,
\end{matrix}\right.
\end{align}
which, defining the code rates $R=\frac{K}{2L}$ and $\tilde R=\frac{\tilde K}{2L},$ leads to \begin{align}\label{eq:eqratepart}
\left\{\begin{matrix}
2R=R_1+R_2
\\
2\tilde R=\tilde R_1+\tilde R_2.
\end{matrix}\right.
\end{align}
Our goal is to design proper rate allocation between sub-packets, i.e., $R_i,\tilde R_i, i=1,2,$ and power split parameter $\alpha$ in (\ref{eq:eqHT}) such that the network STP is maximized. STP is defined as
\begin{align}\label{eq:eqSTPdef}
\Pr(\text{ST})=\frac{1}{2}\left(\Pr\left(\text{C}_1 \,\text{successful}\right)+\Pr\left(\text{C}_2 \,\text{successful}\right)\right),
\end{align}
i.e., the probability of the event that both cache nodes can decode their intended packets correctly. Depending on the buffering and the coding schemes of the cache nodes, the network may lead to different STPs as follows.
\vspace{-3mm}
\subsection{Joint Decoding at HT Period using Successive Interference Cancellation}
In one approach, referred to as Method 1 in the following, the cache nodes buffer the signals received in the LT period, and use both maximum ratio combining (MRC) and successive interference cancellation (SIC) for message decoding at HT periods. Let us concentrate on cache node $\text{C}_1$. Receiving $Y_{\text{C}_1}^\text{HT}$ in (\ref{eq:eqHT}) during the HT period and with $\tilde Y_{\text{C}_1}^\text{LT}$ in (\ref{eq:eqLT}) already buffered, $\text{C}_1$ first uses MRC to decode $\tilde X_1(t)$. Then, with a successful decoding of $\tilde X_1(t)$, node $\text{C}_1$ uses the SIC concept to remove $\tilde X_1(t)$ from $Y_{\text{C}_1}^\text{HT}(t)$, leading to an interference-free signal
\vspace{-2mm}
\begin{align}\label{eq:Eqsic1}
\mathcal{Y}_{\text{C}_1}^\text{HT}=\sqrt{P}h_1^\text{HT}\alpha X_2(t)+Z_{\text{C}_1,1}(t).
\end{align}
Finally, the decoder generates the concatenated signal $[Y_{\text{C}_1}^\text{LT}(t) \,\mathcal{Y}_{\text{C}_1}^\text{HT}(t)]$, with $Y_{\text{C}_1}^\text{LT}(t)$ and $\mathcal{Y}_{\text{C}_1}^\text{HT}(t)$ given in (\ref{eq:eqLT}) and (\ref{eq:Eqsic1}), respectively, and decodes the packet $X$ in \emph{one-shot.} In this way, and considering the same procedure in $\text{C}_2$ by using (\ref{eq:eqLT})-(\ref{eq:eqHT}), STP in (\ref{eq:eqSTPdef}) is given by
\begin{align}\label{eq:eqSTPdef1}
&\Pr(\text{ST})=\frac{1}{2}(\eta_1\gamma_1+\eta_2\gamma_2),\nonumber\\&
\eta_1=\Pr\left(\log\left(1+\frac{(1-\alpha^2)Pg_1^\text{HT}}{1+\alpha^2Pg_1^\text{HT}}+Pg_1^\text{LT}\right)\ge \tilde R_1\right),
\nonumber\\&\gamma_1=\Pr\left(\log\left(1+Pg_1^\text{LT}\right)+\log\left(1+\alpha^2Pg_1^\text{HT}\right)\ge 2R\right)\nonumber\\&
\eta_2=\Pr\left(\log\left(1+\frac{\alpha^2Pg_2^\text{HT}}{1+(1-\alpha^2)Pg_2^\text{HT}}+Pg_2^\text{LT}\right)\ge R_2\right),\nonumber\\
&\gamma_2=\Pr\left(\log\left(1+Pg_2^\text{LT}\right)+\log\left(1+(1-\alpha^2)Pg_2^\text{HT}\right)\ge 2\tilde R\right).
\end{align}
Here, $\eta_1$ (resp. $\eta_2$) is the probability of successful decoding of $\tilde X_1(t)$ (resp. $X_2(t)$) at $\text{C}_1$ (resp. $\text{C}_2$) using MRC. Then, $\gamma_1$ and $\gamma_2$ give the probability that, removing the interference from the received signal in HT period, the caches can decode their intended signals correctly. Note that in $\gamma_1$ and $\gamma_2$ we have used the results on the maximum achievable rates of parallel Gaussian channels.
Considering Rayleigh-fading conditions, we have
\begin{align}\label{eq:eqeta11}
&\eta_1=1-\Pr\left(\frac{(1-\alpha^2)Pg_1^\text{HT}}{1+\alpha^2Pg_1^\text{HT}}+Pg_1^\text{LT}\le e^{\tilde R_1}-1\right)=
\nonumber\\&
1-\int_0^{\frac{e^{\tilde R_1}-1}{P}}f_{g_1}(y)\Pr\left(\frac{(1-\alpha^2)Pg_1^\text{HT}}{1+\alpha^2Pg_1^\text{HT}} \le {e^{\tilde R_1}-1}-Py \right)\text{d}y
\nonumber\\&
=e^{-\frac{\lambda_1\left({e^{\tilde R_1}-1}\right)}{P}}+
\int_{\frac{\alpha^2 e^{\tilde R_1}-1}{\alpha^2P}}^{\frac{e^{\tilde R_1}-1}{P}}e^{\lambda_1\left(y+\frac{{e^{\tilde R_1}-1}-Py}{\alpha^2P-(1-\alpha^2)P\left({e^{\tilde R_1}-1}-Py\right)}\right)}\text{d}y,
\end{align}
which can be calculated numerically. Also, following the same procedure, we have
\begin{align}\label{eq:Eqeta22}
&\eta_2=e^{-\frac{\lambda_2\left(e^{R_2}-1\right)}{P}}
+\int_{\frac{\left(1-\alpha^2\right)e^{R_2}-1}{\left(1-\alpha^2\right)P}}^{\frac{e^{R_2}-1}{P}}{\lambda_2e^{-\lambda_2\left(y+\frac{e^{R_2}-1-Py}{\alpha^2P-\left(1-\alpha^2\right)P\left(e^{R_2}-1-Py\right)}\right)}}\text{d}y.
\end{align}
The terms $\gamma_i,i=1,2,$ on the other hand, do not have closed-form or easy-to-deal integration expressions. Thus, we use the Jensen's inequality $\frac{1}{n}\sum_{i=1}^n\log(1+x_i)\le \log\left(1+\frac{1}{n}\sum_{i=1}^nx_i\right)$ \cite[Eq. (30)]{7445896} and $f_{g_i}(u)=\lambda_ie^{-\lambda_iu}, i=1,2,$ to rephrase $\gamma_i,i=1,2,$ as
\vspace{-2mm}
\begin{align}\label{eq:eqgamma1}
\gamma_1&\le 1-\Pr\left(g_1^\text{LT}+\alpha^2g_1^\text{HT}\le \frac{2}{P}\left(e^R-1\right)\right) \nonumber\\&
=1-\int_0^{\frac{2\left(e^R-1\right)}{P}}{f_{g_1}(x)\Pr\left(g_1^\text{HT}\le\frac{\frac{2\left(e^R-1\right)}{P}-x}{\alpha^2}\right)}\text{d}x\nonumber\\&
=e^{\frac{-2\lambda_1\theta}{P}}+\frac{\alpha^2 e^{\frac{-2\lambda_1\theta}{P\alpha^2}}}{\alpha^2-1}\left(1-e^{\frac{-2\lambda_1\left(\alpha^2-1\right)\theta}{P\alpha^2}}\right),
\end{align}
\vspace{-2mm}
\begin{align}\label{eq:eqgamma2}
\gamma_2&\le e^{\frac{-2\lambda_2\theta}{P}}+\frac{\left(\alpha^2-1\right) e^{\frac{-2\lambda_2\theta}{P\left(1-\alpha^2\right)}}}{\alpha^2}\left(1-e^{\frac{-2\lambda_2\alpha^2\theta}{P\left(\alpha^2-1\right)}}\right),
\end{align}
where (\ref{eq:eqgamma2}) follows the same procedure as in (\ref{eq:eqgamma1}). In this way, the optimal rate/power allocation maximizing STP is given by
\begin{align}\label{eq:eqoptprob}
&\max \frac{1}{2}\{\eta_1\gamma_1+\eta_2\gamma_2\}\nonumber\\
&\text{s.t. } \,\alpha\in[0,1], R_1+R_2=2R, \,\tilde R_1+\tilde R_2=2\tilde R,
\end{align}
which can be effectively solved by, e.g., exhaustive search or the machine-learning based scheme of \cite{8520925}.
\vspace{-2mm}
\subsection{Joint Decoding at HT Period without SIC}
Implementation of MRC and SIC, to decode and remove the interference, increases the decoding complexity/delay. Also, SIC suffers from error propagation problem, e.g., \cite{nomaharq}. For these reasons, in Method 2, each cache node decodes its intended packets in one-shot by considering the interference as an additive noise. Here, (\ref{eq:eqSTPdef1}) is rephrased as
\begin{align}\label{eq:eqSTPdef3}
&\Pr(\text{ST})=\frac{1}{2}(\bar \gamma_1+\bar \gamma_2),\nonumber\\&
\bar\gamma_1=\Pr\left(\log\left(1+Pg_1^\text{LT}\right)+\log\left(1+\frac{\alpha^2Pg_1^\text{HT}}{1+(1-\alpha^2)Pg_1^\text{HT}}\right)\ge 2R\right)\nonumber\\&
\bar\gamma_2=\Pr\left(\log\left(1+Pg_2^\text{LT}\right)+\log\left(1+\frac{(1-\alpha^2)Pg_2^\text{HT}}{1+\alpha^2Pg_2^\text{HT}}\right)\ge 2\tilde R\right),
\end{align}
where, using the Jensen's inequality and the same procedure as in (\ref{eq:eqgamma1}), we have
\begin{align}\label{eq:eqjensen2}
&\bar\gamma_1\le e^{-\frac{2\lambda_1\left(e^R-1\right)}{P}}+\int_{\frac{2\left(e^R-1\right)-\frac{\alpha^2}{1-\alpha^2}}{P}}^{\frac{2\left(e^R-1\right)}{P}}\lambda_1e^{-\lambda_1\left(x+\frac{2\left(e^R-1\right)-Px}{\alpha^2P+(1-\alpha^2)P(Px-2\left(e^R-1\right))}\right)}\text{d}x,\nonumber\\&
\bar\gamma_2\le e^{-\frac{2\lambda_2\left(e^{\tilde R}-1\right)}{P}}
+\int_{\frac{2\left(e^{\tilde R}-1\right)-\frac{1-\alpha^2}{\alpha^2}}{P}}^{\frac{2\left(e^{\tilde R}-1\right)}{P}}\lambda_2e^{-\lambda_2\left(x+\frac{2\left(e^{\tilde R}-1\right)-Px}{(1-\alpha^2)P+\alpha^2P(Px-2\left(e^{\tilde R}-1\right))}\right)}\text{d}x,
\end{align}
which can be calculated numerically.
Finally, note that, replacing (\ref{eq:eqSTPdef3}) into (\ref{eq:eqoptprob}), the optimal performance of the cache nodes in Method 2 is independent of the rate split between the sub-packets. This, although Method 1 gives the best performance in terms of the worst-case peak traffic,
may give an advantage to Method 2, compared to Method 1. This is because in Method 1 the rate split is performed by considering the worst-case condition with the cache nodes requesting for different signals during HT period. However, if the caches request for the same signals during HT period, the rate split scheme of Method 1 is not necessarily optimal. As opposed, in Method 2, the rate split is independent of the caches requested signals in HT periods.
\subsection{Separate Decoding using SIC}
In Methods 1-2, one needs to follow the coding schemes of incremental redundancy hybrid automatic repeat request (HARQ)-based protocols or Raptor codes, e.g., \cite{6164088}, where the message is decoded by concatenating different sub-packets. Alternatively, in Method 3, we consider the case where, while MRC and SIC are used to decode and remove the interference signal, respectively, each cache node decodes its sub-packets of interest separately. That is, considering $\text{C}_1,$ $X_1$ (resp. $X_2$) is decoded during the LT (resp. HT) period. In this case, the STP (\ref{eq:eqSTPdef1}) is changed to
\vspace{-1mm}
\begin{align}\label{eq:eqSTPdef2}
&\Pr\left(\text{ST}\right)=\frac{1}{2}\left(\eta_1\breve{\gamma}_{11}\breve{\gamma}_{12}+\eta_2\breve{\gamma}_{21}\breve{\gamma}_{22}\right),\nonumber\\&
\breve{\gamma}_{11}=\Pr\left(\log\left(1+Pg_1^\text{LT}\right)\ge R_1\right)=e^{-\frac{\lambda_1\left(e^{R_1}-1\right)}{P}},\nonumber\\&
\breve{\gamma}_{12}=\Pr\left(\log\left(1+\alpha^2Pg_1^\text{HT}\right)\ge R_2\right)=e^{-\frac{\lambda_1\left(e^{R_2}-1\right)}{\alpha^2P}}
\nonumber\\&
\breve{\gamma}_{21}=\Pr\left(\log\left(1+(1-\alpha^2)Pg_2^\text{HT}\right)\ge \tilde R_1\right)=e^{-\frac{\lambda_2\left(e^{\tilde R_1}-1\right)}{(1-\alpha^2)P}},\nonumber\\&
\breve{\gamma}_{22}=\Pr\left(\log\left(1+Pg_2^\text{LT}\right)\ge \tilde R_2\right)=e^{-\frac{\lambda_2\left(e^{\tilde R_2}-1\right)}{P}},
\end{align}
with $\eta_i,i=1,2,$ given in (\ref{eq:eqSTPdef1}), and (\ref{eq:eqoptprob}) is adapted correspondingly. In (\ref{eq:eqSTPdef2}), $\breve{\gamma}_{11}$ is the probability that $\text{C}_1$ decodes $X_1$ during the LT period. Also, $\breve{\gamma}_{12}$ gives the probability that, after decoding and removing $\tilde X_1$, the cache node $\text{C}_1$ correctly decodes $X_2$ in the HT period. Also, the same arguments hold for $\breve{\gamma}_{2i},i=1,2.$
Note that, although Method 1 maximizes the achievable rate/STP, Method 3 has a number of advantages including:
\begin{itemize}
\item \textbf{Low decoding complexity}: Because, as opposed to Methods 1-2 decoding long codewords of length $2L$, Method 3 is based on decoding sub-packets of length $L$.
\item \textbf{Efficient HARQ-based transmissions}: In Methods 1-2, all packets are decoded during the HT periods and, in case of decoding failure, the message is retransmitted at that period. Such HARQ-based retransmissions increase the backhauling load at HT period. As opposed, in Method 3, the decoding of the first received sub-packets and all their required HARQ-based retransmissions are performed during the LT period, which reduces the backhauling cost of HARQ.
\end{itemize}
Finally, depending on the considered method, the buffering scheme of the caches during the LT period may change. Particularly, in Methods 1-2 the caches buffer the signals received during LT period without decoding. In Method 3, however, the caches buffer the sub-packets successfully decoded during LT period.
\vspace{-1mm}
\subsection{Separate Decoding without SIC}
To further reduce the complexity of Method 3, one can consider the case where, while decoding the sub-packets separately, the cache nodes consider the interference as an additive noise (Method 4). In this case, where the sub-packets are decoded in different LT and HT periods without SIC, the STP is given by
\begin{align}\label{eq:eqSTPdef4}
&\Pr\left(\text{ST}\right)=\frac{1}{2}(\breve{\gamma}_{11}\hat{\gamma}_{12}+\hat{\gamma}_{21}\breve{\gamma}_{22}),\nonumber\\&
\hat{\gamma}_{12}=\Pr\left(\log\left(1+\frac{\alpha^2Pg_1^\text{HT}}{1+(1-\alpha^2)Pg_1^\text{HT}}\right)\ge R_2\right)\nonumber\\&\,\,\,\,\,\,\,\,=\left\{\begin{matrix}
0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \text{if } R_2\ge -\log\left(1-\alpha^2 \right )\\
e^{-\lambda_1\frac{\left(e^{R_2}-1 \right )}{\alpha^2P-(1-\alpha^2)P\left(e^{R_2}-1 \right )}} & \text{otherwise}
\end{matrix}\right.
\nonumber\\&
\hat{\gamma}_{21}=\Pr\left(\log\left(1+\frac{(1-\alpha^2)Pg_2^\text{HT}}{1+\alpha^2Pg_2^\text{HT}}\right)\ge \tilde R_1\right)\nonumber\\&=\left\{\begin{matrix}
0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \text{if } \tilde R_1\ge -2\log\alpha\\
e^{-\lambda_2\frac{\left(e^{\tilde R_1}-1 \right )}{(1-\alpha^2)P-\alpha^2P\left(e^{\tilde R_1}-1 \right )}} & \text{otherwise}
\end{matrix}\right.
\end{align}
with $\breve{\gamma}_{11}$ and $\breve{\gamma}_{22}$ given in (\ref{eq:eqSTPdef2}). Also, in (\ref{eq:eqSTPdef4}) we use Rayleigh channel PDFs and some manipulations to derive the probabilities. For further comparisons between Methods 1-4, see Section IV.
\section{Simulation Results}
The simulation results are presented for the cases with $\lambda_1=1$ and $\lambda_2=0.1$, i.e., with $10$ dB difference between the channel gains of the server-cache links, and we define the transmission signal-to-noise ratio (SNR) as $10\log_{10} P$, considering the additive noises to be unit-variance. Note that we have evaluated the results for different parameter settings, and they show the same qualitative conclusions as those presented in the following. In Figs. 3-4, the results are obtained by optimizing the rate and power allocation. Here, both exhaustive search and the genetic-algorithm based scheme of \cite{8520925} have been used which have ended up in the same results, indicating the accuracy of the optimization process. In Fig. 5, we study the effect of rate/power allocation.
\begin{figure}
\vspace{-3mm}
\centering
\includegraphics[width=0.6\columnwidth]{FigCodedcaching1-eps-converted-to.pdf}\\\vspace{-3mm}
\caption{Comparison between the STP of Methods 1-4, $\lambda_1=1,$ $\lambda_2=0.1,$ $R=\tilde R=1$ npcu. The results are obtained by optimal rate and power allocation.}
\label{fig:fig_ivd9}
\vspace{-2mm}
\end{figure}
Considering $R=\tilde R=1$ nats-per-channel-use (npcu), Fig. 3 compares the performance of Methods 1-4. Also, the figure verifies the tightness of the Jensen's inequality-based approximation results of (\ref{eq:eqgamma1})-(\ref{eq:eqgamma2}). Then, Fig. 4 shows the STP versus the data rates $R=\tilde R$ for the cases with different decoding/buffering methods and transmission SNRs.
Considering Methods 1 and 3, with joint and separate decoding on the sub-packets, respectively, Fig. 5 studies the effect of optimal rate and power allocation on the network STP. Particularly, the figure compares the optimal results obtained by (\ref{eq:eqoptprob}) with the cases using uniform power allocation, i.e., with $\alpha=\frac{\sqrt{2}}{2}$ in (\ref{eq:eqHT}), and/or equal rate split, i.e., $R_i=\tilde R_i, i=1,2$. According to the figures, the following conclusions can be drawn:
\begin{figure}
\vspace{-3mm}
\centering
\includegraphics[width=0.6\columnwidth]{FigCodedcachingrate-eps-converted-to.pdf}\\\vspace{-2mm}
\caption{Achievable STP versus the data rate for different methods. The results are obtained by optimal rate and power allocation and $\lambda_1=1, \lambda_2=0.1.$}
\label{fig:fig_ivd10}
\vspace{-0mm}
\end{figure}
\begin{figure}
\vspace{-3mm}
\centering
\includegraphics[width=0.6\columnwidth]{plottingcomparison-eps-converted-to.pdf}\\\vspace{-2mm}
\caption{On the effect of optimal rate and power allocation in Methods 1 and 3, $R=\tilde R=1$ npcu and $\lambda_1=1, \lambda_2=0.1$.}
\label{fig:fig_ivd11}
\vspace{-0mm}
\end{figure}
\begin{itemize}
\item The approximation results of (\ref{eq:eqgamma1})-(\ref{eq:eqgamma2}) properly approximate the probabilities $\gamma_i, i=1,2$ (Fig. 3. Also, the same point is observed for $\bar \gamma_i,i=1,2,$ in (\ref{eq:eqjensen2}) although not shown in the figure). Thus, the approximations can be well utilized for the performance evaluation of Methods 1-2, i.e., in the cases with joint decoding of the sub-packets.
\item Compared to the cases with separate decoding of sub-packets, i.e., Methods 3-4, considerable STP improvement is observed by joint decoding of the sub-packet, i.e., Methods 1-2 (Figs. 3-4). However, as explained in Section III.C, the STP increment of Methods 1-2 is at the cost of decoding complexity and possible HARQ-based retransmissions at HT periods. On the other hand, for both cases with joint and separate decoding of the sub-packets, using SIC-based interference cancellation leads to marginal performance improvement at low rates while its effect increases with the data rate (Fig. 4).
Finally, as the data rate increases, the performance gap between Methods 2 and 3 decreases, i.e., one can reach the same STP as in the cases with joint sub-packet decoding of interference-affected signals by separate sub-packets decoding if the interference signals are removed using SIC. (Fig. 4).
\item For all parameter settings, Method 1 leads to the highest STP, compared to Methods 2-4, at the cost of decoding delay/possible retransmissions at HT periods (Figs. 3-4). For instance, with the parameter settings of Fig. 4 and data rate 1.5 npcu, the implementation of Method 1 with transmit SNR 10 dB results in the same STP, $80\%$, as in the cases with Method 4 and SNR 15 dB, i.e., using advanced decoding methods leads to 5 dB gain in SNR (Fig. 4).
\item For both cases with and without interference cancellation (Methods 1 and 3), optimal rate allocation leads to considerable STP increment (Fig. 5. The same conclusion is observed in Methods 2 and 4, although not presented in the figure). Also, the relative performance gain of optimal rate split increases in the cases with interference cancellation. Finally, considering the interference as additive noise, optimal power allocation between sub-packets during HT period increases the STP. However, with interference cancellation and joint decoding of sub-packets, the effect of optimal power allocation between HT period sub-packets is marginal (Fig. 5).
\end{itemize}
\section{Conclusions}
This paper studied the performance coded-caching networks in the cases with adaptive rate/power allocation and different decoding/buffering schemes. As we showed, joint decoding of the sub-packets at HT periods leads to considerable performance improvement of coded-caching setups. Also, for different decoding schemes, optimal rate split between the sub-packets increases the STP considerably while optimal power allocation between the sub-packets of HT period only improves the STP if SIC-based receiver is not implemented and the sub-packets are decoded separately.
\vspace{-2mm}
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2021-03-15T01:15:14",
"yymm": "2103",
"arxiv_id": "2103.07234",
"language": "en",
"url": "https://arxiv.org/abs/2103.07234"
}
|
\section{Introduction}
Let $S$ be a scheme. Let $\mathfrak{G}$ be an
$S$--group scheme. It is natural to ask
whether $\mathfrak{G}$ is linear; that is, there exists a group monomorphism
$\mathfrak{G} \to \GL(\mathcal E)$ where
$\mathcal E$ is a locally free $\mathcal{O}_S$-module of finite rank. In particular, $\mathfrak{G}$ admits a faithful representation on $\mathcal E$.
This holds for affine algebraic groups over a field
\cite[II, \S 2.3.3]{DG}.
In the case $S$ is locally noetherian and $G$ is of multiplicative type
of constant type and of finite type, Grothendieck has shown that
$G$ is linear if and if $G$ is isotrivial, i.e. $G$ is split by a finite \'etale extension of $S$ \cite[XI.4.6]{SGA3}. In particular there exist non linear tori of rank $2$ over the local ring (at a node) of a nodal
algebraic curve ({\it ibid}, X.1.6).
Firstly we extend that criterion over an arbitrary base by using
Azumaya and separable $\mathcal{O}_S$--algebras (Theorem \ref{thm_torus}).
Secondly we deal with the case $\mathfrak{G}$ reductive; that
is, $\mathfrak{G}$ is smooth affine with reductive (connected) geometric fibers.
In this case a faithful representation is necessarily a closed immersion \cite[XVI.1.5]{SGA3}.
Positive results on the linearity question are due to
M. Raynaud \cite[VI$_B$]{SGA3} and R. Thomason \cite[3.1]{T}
which is essentially the implication $(i) \Longrightarrow (ii)$
in the theorem below.
We can restrict our attention to the case
when $\mathfrak{G}$ is of constant type (recall that the type is a locally
constant function on $S$);
this implies that there exists a Chevalley $\mathbb{Z}$--group scheme $G$
such that $\mathfrak{G}$ is locally isomorphic to $G_S$ for the \'etale topology
\cite[XXII.2.3, 2.5]{SGA3}.
A short version of our main result is the following.
\begin{stheorem}\label{thm_main}
The following are equivalent:
\smallskip
(i) The radical torus $\mathrm{rad}(\mathfrak{G})$ is isotrivial;
\smallskip
(ii) $\mathfrak{G}$ is linear.
\smallskip
\noindent Furthermore if $S$ is affine, the above are equivalent to
\smallskip
(ii') there exists a
closed immersion $i: \mathfrak{G} \to \GL_n$ with $n \geq 1$
which is a homomorphism.
\end{stheorem}
We recall that $\mathrm{rad}(\mathfrak{G})$ is the maximal central subtorus of $\mathfrak{G}$ \cite[XXIV.4.3.6]{SGA3} and that (i)
means that $\mathrm{rad}(\mathfrak{G})$ splits
after passing to a finite \'etale cover $S'\to S$.
In the noetherian setting, a variant of the implication $(i) \Longrightarrow (ii)$
has been shown by Margaux who furthermore provided an $\text{\rm{Aut}}(\mathfrak{G})$-equivariant representation \cite{M}.
Note that condition (i) depends only on the quasi-split form of
$\mathfrak{G}$ and also that it is always satisfied
if $\mathfrak{G}$ is semisimple or if $\mathrm{rad}(\mathfrak{G})$ is of rank one.
Furthermore, if $S$ is a semilocal scheme,
Demazure's characterization of isotrivial group schemes \cite[XXIV.3.5]{SGA3}
permits us to deduce that
$\mathfrak{G}$ is isotrivial if and only if $\mathfrak{G}$ is linear,
see section \ref{section_semilocal}.
Finally for $S=\mathop{\rm Spec}\nolimits(R)$ with $R$ noetherian, we complete Thomason's approach
by showing that linearity for $\mathfrak{G}$ is equivalent to the
resolution property (Th. \ref{thm_re}).
\medskip
\smallskip
\noindent{\bf Acknowledgements.} I thank Vladimir Chernousov, Laurent Moret-Bailly, Erhard Neher, Arturo Pianzola,
and Anastasia Stavrova for their valuable suggestions.
I thank the referee for a simplification of the proof of Proposition \ref{prop_torus}.
\section{Definitions and basic facts}
\subsection{Notation}
We use mainly the terminology and notation of Grothendieck-Dieudonn\'e \cite[\S 9.4 and 9.6]{EGA1}
which agrees with that of Demazure-Grothendieck used in \cite[Exp. I.4]{SGA3}.
\smallskip
\noindent (a) Let $S$ be a scheme and let $\mathcal E$ be a quasi-coherent sheaf over $S$.
For each morphism $f:T \to S$,
we denote by $\mathcal E_{T}=f^*(\mathcal E)$ the inverse image of $\mathcal E$
by the morphism $f$.
We denote by $\mathbf{V}(\mathcal E)$ the affine $S$--scheme defined by
$\mathbf{V}(\mathcal E)=\mathop{\rm Spec}\nolimits\bigl( \mathrm{Sym}^\bullet(\mathcal E)\bigr)$;
it represents the $S$--functor $Y \mapsto \mathop{\rm Hom}\nolimits_{\mathcal{O}_Y}(\mathcal E_{Y}, \mathcal{O}_Y)$
\cite[9.4.9]{EGA1}.
\smallskip
\noindent (b) We assume now that $\mathcal E$ is locally free of finite rank and denote by $\mathcal E^\vee$ its dual.
In this case the affine $S$--scheme $\mathbf{V}(\mathcal E)$ is of finite presentation
(ibid, 9.4.11); also
the $S$--functor $Y \mapsto H^0(Y, \mathcal E_{Y})=
\mathop{\rm Hom}\nolimits_{\mathcal{O}_Y}(\mathcal{O}_Y, \mathcal E_{Y} )$
is representable by the affine $S$--scheme $\mathbf{V}(\mathcal E^\vee)$
which is also denoted by $\mathbf{W}(\mathcal E)$ \cite[I.4.6]{SGA3}.
The above applies to the locally free quasi-coherent sheaf
${\mathcal E}nd(\mathcal E) = \mathcal E^\vee \otimes_{\mathcal{O}_S} \mathcal E$
over $S$ so that we can consider
the affine $S$--scheme $\mathbf{V}\bigl({\mathcal E}nd(\mathcal E)\bigr)$
which is an $S$--functor in associative and unital algebras
\cite[9.6.2]{EGA1}.
Now we consider the $S$--functor $Y \mapsto \text{\rm{Aut}}_{\mathcal{O}_Y}(\mathcal E_{Y})$.
It is representable by an open $S$--subscheme of $\mathbf{V}\bigl({\mathcal E}nd(\mathcal E)\bigr)$
which is denoted by $\GL(\mathcal E)$ ({\it loc. cit.}, 9.6.4).
\smallskip
\noindent (c) If $\mathcal B$ is a locally free $\mathcal{O}_S$--algebra (unital, associative) of finite rank,
we recall that the functor of invertible elements of $\mathcal B$ is representable
by an affine $S$-group scheme which is denoted by $\GL_1(\mathcal B)$ \cite[2.4.2.1]{CF}.
For separable and Azumaya algebras, we refer to \cite{KO}.
Note that in \cite[\S 2.5.1]{CF}, separable algebras are supposed furthermore to be locally free of finite rank.
If $\mathcal B$ is a separable $\mathcal{O}_S$--algebra which is a
locally free $\mathcal{O}_S$--algebra of finite rank,
then $\GL_1(\mathcal B)$ it is a reductive $S$--group scheme \cite[3.1.0.50]{CF}.
\smallskip
\noindent (d) We use the theory and terminology of tori and multiplicative group schemes of \cite{SGA3}; see also Oesterl\'e's survey \cite{O}.
\subsection{Finite \'etale covers}
The next lemma is a consequence of the equivalence of categories describing
finite \'etale $\mathcal{O}_S$--algebra of rank $N$
\cite[\S 2.5.2]{CF}; it admits a simple direct proof.
\begin{slemma} \label{lem_brian} Let $N$ be a positive integer and let
$\mathcal C$ be a finite \'etale $\mathcal{O}_S$--algebra of rank $N$. Then
there exists a finite \'etale cover $T$ of $S$
of degree $N!$ such that
$\mathcal C \otimes_{\mathcal{O}_S} {\mathcal{O}_T} \buildrel\sim\over\lgr (\mathcal{O}_T)^N $.
\end{slemma}
\begin{proof}
We proceed by induction on $N$, the case $N=1$ being obvious.
We put $S'= \mathop{\rm Spec}\nolimits(\mathcal{O}_C)$,
this is a finite \'etale cover of $S$ of degree $N$.
Since the diagonal map $S' \to S' \times_S S'$ is closed and
open \cite[$_4$.17.4.2]{EGA4}
there exists a decomposition $\mathcal C \otimes_{\mathcal{O}_S} \mathcal{O}_S'=\mathcal{O}_{S'} \times \mathcal C'$ where $\mathcal C'$ is a finite \'etale $\mathcal{O}_{S'}$--algebra of rank $N-1$.
Applying the induction process to $\mathcal C'$ provides
a finite \'etale cover $T$ of $S'$
of degree $(N-1)!$ such that
$\mathcal C' \otimes_{\mathcal{O}_{S'}} {\mathcal{O}_T} \cong (\mathcal{O}_T)^{N-1}$.
Thus $\mathcal C \otimes_{\mathcal{O}_S} \mathcal{O}_T=\mathcal{O}_{T} \times (\mathcal{O}_T)^{N-1}$
and $T$ is a finite \'etale cover of $S$ of degree $N!= N \times (N-1)!$.
\end{proof}
\subsection{Isotriviality} \cite[XXIV.4]{SGA3}
Let $\mathcal H$ be a fppf $S$--sheaf in groups and let
$\mathcal X$ be a $\mathcal H$--torsor. We say that $\mathcal X$ is {\it isotrivial}
if there exists a finite \'etale cover $S'$ of $S$
which trivializes $\mathcal X$; that is, satisfying $\mathcal X(S') \not = \emptyset$.
The notion of locally isotrivial (with respect to the Zariski topology) is then clear and
there is also the following variant of {\it semilocally isotrivial}.
We say that $\mathcal X$ is {\it semilocally isotrivial}
if for each subset $\{s_1,\dots, s_n\}$ of points of $S$
contained in an affine open subset of $S$, there exists
an open subscheme $U$ of $S$ containing $s_1, \dots, s_n$
such that $\mathcal X \times_S U$ is isotrivial over $U$.
\medskip
A reductive $S$--group scheme $\mathfrak{G}$ is {\it isotrivial} if it is split by
a finite \'etale cover $S'$ of $S$.
An isotrivial reductive $S$--group scheme $\mathfrak{G}$ is necessarily of constant type.
If $\mathfrak{G}$ is of constant type with underlying Chevalley group scheme $G$,
$\mathfrak{G}$ is isotrivial if and only if
the $\text{\rm{Aut}}(G)$--torsor $\mathrm{Isom}( G_S, \mathfrak{G})$ is isotrivial.
\subsection{Rank one tori}\label{subsec-rank1}
The simplest case is that of $G=\mathbb{G}_{m,S}$, the split
$S$--torus of rank $1$.
The $S$--functor $S' \mapsto \mathop{\rm Hom}\nolimits_{S'-gp}( \mathbb{G}_{m,S'}, \mathbb{G}_{m,S'})$
is representable by the constant $S$--group scheme $\mathbb{Z}_S$
\cite[VIII.1.5]{SGA3}. It follows that
the $S$--functor $S' \mapsto \mathop{\rm Isom}\nolimits_{S'-gp}( \mathbb{G}_{m,S'}, \mathbb{G}_{m,S'})$
is representable by the constant $S$--group scheme $(\mathbb{Z}/2\mathbb{Z})_S
=\text{\rm{Aut}}_{S-gp}(\mathbb{Z}_S)$.
On the other hand, $(\mathbb{Z}/2\mathbb{Z})_S$ is the automorphism group
of the split \'etale cover $S \sqcup S \to S$ of degree $2$.
By definition an $S$--torus of rank one is a form
of $\mathbb{G}_m$ for the fpqc topology;
in the other hand, a degree $2$ \'etale cover
is a form of $S \sqcup S$ for the finite \'etale topology
(see for example Lemma \ref{lem_brian}) and a fortiori
for the fpqc topology.
According to the faithfully flat descent technique (e.g. \cite[XXIV.1.17]{SGA3}),
there is then an equivalence of categories between
the groupoid of rank one tori over $S$ (resp.\ the groupoid of degree $2$ \'etale covers of $S$) and
the groupoid of $(\mathbb{Z}/2\mathbb{Z})_S$-torsors.
More precisely one associates to an $S$-torus $T$ of rank $1$
(resp.\ to an \'etale cover $E$ of degree $2$) the
$(\mathbb{Z}/2\mathbb{Z})_S$-torsor $\mathrm{Isom}_{gr}( \mathbb{G}_m, T)$
(resp. $\mathrm{Isom}( S \sqcup S , E)$).
The inverse map is given by twisting the split object
by a given $(\mathbb{Z}/2\mathbb{Z})_S$-torsor.
Let $T$ be an $S$--torus of rank one and let $S'\to T$ be its
associated \'etale cover of degree $2$.
Then $T$ is splits after base change to $S'$ and so is
isotrivial. Furthermore we claim that $T$ is isomorphic to the quotient
$Q=R_{S'/S}(\mathbb{G}_{m,S'})/ \mathbb{G}_m$ where $R_{S'/S}(\mathbb{G}_{m,S'})$
stands for the Weil restriction.
We consider the $(\mathbb{Z}/2\mathbb{Z})_S$--equivariant exact sequences
$$
1 \to \mathbb{G}_{m,S} \xrightarrow{\Delta} \mathbb{G}_{m,S} \times \mathbb{G}_{m,S}
\xrightarrow{\Pi} \mathbb{G}_{m,S} \to 1
$$
where $\Pi(x,y)=x \, y^{-1}$ and the $\mathbb{Z}/2\mathbb{Z}$-action on $\mathbb{G}_{m,S} \times \mathbb{G}_{m,S}$ is by $(x,y) \mapsto (y,x)$.
The $\mathbb{Z}/2\mathbb{Z}$-action on the last factor is then by $x \mapsto x^{-1}$.
Twisting this sequence by the $(\mathbb{Z}/2\mathbb{Z})_S$-torsor $\mathrm{Isom}( S \sqcup S , S')$ yields the desired exact sequence
$$
1 \to \mathbb{G}_m \to R_{S'/S}(\mathbb{G}_{m,S'}) \to Q \to 1
$$
where the identification of the second term is left to the reader
(it is similar to that of the proof of \cite[XXIX.3.13]{SGA3}).
\medskip
\subsection{Linear representations}
Let $\mathfrak{G}$ be an $S$--group scheme. We say that $\mathfrak{G}$ is {\it linear} if there exists
a locally free $\mathcal{O}_S$-module $\mathcal E$ is of finite rank
and a group homomorphism $\mathfrak{G} \to \GL(\mathcal E)$ which is a monomorphism.
The notion of locally linear $S$--group scheme is then clear and
there is also the following variant of {\it semilocally linear}.
We say that $\mathfrak{G}$ is {\it semilocally linear}
if for each subset $\{ s_1,\dots, s_n\}$ of points of $S$
contained in an affine open subset of $S$, there exists
an open subscheme $U$ of $S$ containing $s_1, \dots, s_n$
such that $\mathfrak{G} \times_S U$ is linear over $U$.
\begin{slemma}\label{lem_affine}
Assume that $S=\mathop{\rm Spec}\nolimits(R)$ and let $\mathcal E$ be a locally free $\mathcal{O}_S$--module of finite rank.
Then $\GL(\mathcal E)$ embeds as a closed $S$--subgroup scheme in $\GL_n$ for some $n \geq 1$.
\end{slemma}
\begin{proof} Since the rank of $\mathcal E$ is a locally constant function \cite[Ch. \, 0, 5.4.1]{EGA1},
we can assume that $\mathcal E$ is locally free of constant of rank $r$.
Then $E=H^0(R, \mathcal E)$ is a locally free $R$--module of rank $r$,
so is finitely generated projective \cite[Tag 00NX]{St}. It follows that there exists an integer
$n \geq 1$ and a decomposition $R^n =E \oplus E'$.
The homomorphism $\GL(\mathcal E) \to \GL_n$ is a closed immersion.
\end{proof}
\begin{slemma}\label{lem_weil}
Let $\mathfrak{G}$ be an $S$--group scheme
and let $S'$ be a finite locally free cover of $S$.
Then $\mathfrak{G}$ is linear if and only if $\mathfrak{G} \times_S S'$
is linear.
\end{slemma}
\begin{proof} We denote by $p: S' \to S$ the structure map.
If $\mathfrak{G}$ is linear, then $\mathfrak{G} \times_S S'$ is linear.
Conversely we assume that there exists a monomorphism
$i: \mathfrak{G} \times_S S' \to \GL(\mathcal E')$ where
$\mathcal E'$ is a locally free $\mathcal{O}_{S'}$--module of finite rank. We put $\mathcal E= p_*( \mathcal E')$, this is
a locally free $\mathcal{O}_S$--module of finite rank.
We consider the sequence of $S$--functors in $S$--groups
$$
\mathfrak{G} \to R_{S'/S}( \mathfrak{G} \times_S S') \xrightarrow{R_{S'/S}(i)} R_{S'/S}( \GL(\mathcal E') ) \to \GL( \mathcal E)
$$
where $R_{S'/S}$ stands for the Weil restriction
and the first map is the diagonal map which is a monomorphism.
Since the Weil restriction for $S'/S$ transforms monomorphisms into monomorphisms,
the map $R_{S'/S}(i)$ is also a monomorphism and
so is the last map since
$R_{S'/S}( \GL(\mathcal E') )(T) \subset \GL( \mathcal E)(T)$
corresponds to automorphisms of $\mathcal E \otimes_{\mathcal{O}_S} \mathcal{O}_T$
which are $\mathcal{O}_{S'} \otimes_{\mathcal{O}_S} \mathcal{O}_T$-linear.
Since all maps are monomorphisms, we conclude that $\mathfrak{G}$ is linear.
\end{proof}
\section{Tori and group of multiplicative type}\label{section_torus}
\subsection{Maximal tori of linear groups}
Let $\mathcal A$ be an Azumaya $\mathcal{O}_S$--algebra.
We consider the reductive $S$--group scheme $\GL_1(\mathcal A)$.
Let $\mathcal B \subset \mathcal A$ be a separable $\mathcal{O}_S$--subalgebra of $\mathcal A$
which is a locally free $\mathcal{O}_S$--module of finite rank and
which is locally a direct summand of $\mathcal A$ as $\mathcal{O}_S$--module.
We get a monomorphism of reductive $S$-group schemes
$\GL_1(\mathcal B) \to \GL_1(\mathcal A)$.
In particular, if $\mathcal B$ is commutative, $\mathcal B$ is a finite
\'etale $\mathcal{O}_S$--algebra and $\GL_1(\mathcal B)$ is a torus.
We come now to Grothendieck's definition of maximal \'etale subalgebras.
\begin{sdefinition} \cite[D\'ef. 5.6]{G}.
We say that a finite \'etale $\mathcal{O}_S$--subalgebra $\mathcal C \subset \mathcal A$ is
{\it maximal} if $\mathcal C$ is locally a direct summand of $\mathcal A$ as $\mathcal{O}_S$--module
and if the rank of $\mathcal C \otimes_{\mathcal{O}_S} \kappa(s)$ is the degree of $\mathcal A_s \otimes_{\mathcal{O}_S} \kappa(s)$
for each $s \in S$.
\end{sdefinition}
If $\mathcal C \subset \mathcal A$ is maximal finite \'etale $\mathcal{O}_S$--subalgebra of
$\mathcal A$, then the torus $\GL_1(\mathcal C)$ is a maximal torus of $\GL_1(\mathcal A)$
since it is the case on geometric fibers.
According to \cite[\S 7.5]{G}, all maximal $S$--tori of $\GL_1(\mathcal A)$ occur in that manner;
this is part (3) of the following enlarged statement.
\begin{sproposition} \label{prop_torus}
Let $\mathfrak{S}$ be a subgroup scheme of multiplicative type of $\GL_1(\mathcal A)$
and put $\mathcal B=\mathcal A^\mathfrak{S}$, the centralizer subalgebra of $\mathfrak{S}$.
\smallskip
\noindent (1) $\mathcal B$ is a separable $\mathcal{O}_S$--algebra which is locally a direct summand of $\mathcal A$ as $\mathcal{O}_S$--module.
\smallskip
\noindent (2) Let $\mathcal C$ be the center of $\mathcal B$; this is a
finite \'etale $\mathcal{O}_S$--algebra of positive rank which is locally a direct summand of $\mathcal B$ (and $\mathcal A$)
as $\mathcal{O}_S$--module.
We have the closed immersions
\[
\mathfrak{S} \, \subset \, \GL_1(\mathcal C) \, \subset \, \GL_1(\mathcal A).
\]
\smallskip
\noindent (3) If $\mathfrak{S}$ is a maximal torus of $\GL_1(\mathcal A)$,
then $\mathfrak{S}= \GL_1(\mathcal C)$ and $\mathcal C$ is a maximal finite \'etale $\mathcal{O}_S$--subalgebra of
$\mathcal A$.
\smallskip
\noindent (4) If $\mathfrak{S}$ is of constant type, then $\mathfrak{S}$ is isotrivial.
\end{sproposition}
\begin{proof}
According to \cite[IX.2.5]{SGA3}, the map $\mathfrak{S} \to \GL_1(\mathcal A)$ is a closed
immersion so that $\mathfrak{S}$ is of finite type over $S$.
As a preliminary observation we notice that
(1), (2), (3) are local for the \'etale topology.
We can assume that $S=\mathop{\rm Spec}\nolimits(R)$, $\mathcal A=\mathop{\rm M}\nolimits_n(R)$ and that $\mathfrak{S}=D(M)$ for
$M$ an abelian group. Note that $M$ is finitely generated since $\mathfrak{S}$ is
of finite type ({\it ibid}, VII.2.1.b).
\smallskip
\noindent (1) We consider the $M$--grading
$R^n= \bigoplus_{m \in M} R^n_m$.
The $R$--modules $(R^n_m)_{m \in M}$ are finitely generated projective
so locally free of finite rank.
There is a finite subset $M' \subset M$ such that
$R^n= \bigoplus_{m \in M'} R^n_m$.
Then $\mathcal B= \prod_{m \in M'} \mathop{\rm End}\nolimits_{R}( R^n_m)$
and each $\mathop{\rm End}\nolimits_{R}(R^n_m)$ is a separable $R$-algebra which is locally free of finite rank
\cite[III, example 2.8]{KO}.
Since a product of separable algebras is
a separable algebra ({\it ibid}, III, proposition 1.7),
it follows that $\mathcal B$ is a separable $R$-algebra.
Furthermore $\mathcal B$ is locally a direct summand of $\mathop{\rm M}\nolimits_n(R)$
as $R$-module.
\smallskip
\noindent (2) Let $\mathcal C$ be the center of $\mathcal B$.
We have $R$--monomorphisms of groups $\mathfrak{S} \, \subset \, \GL_1(\mathcal C) \, \subset \, \GL_1(\mathcal A)$.
\smallskip
\noindent (3) We assume that $\mathfrak{S}$ is a maximal torus of $\GL_1(\mathcal A)$
and want to establish that $\mathfrak{S}= \GL_1(\mathcal C)$. So $\GL_1(\mathcal C)$ is an $R$--torus containing $\goth T$ and since maximality
holds also in the naive sense \cite[XII.1.4]{SGA3},
we conclude that $\goth T=\GL_1(\mathcal C)$.
\smallskip
\noindent (4)
We come back to the initial setting (i.e. without localizing).
We want to show that $\mathfrak{S}$ is isotrivial.
According to \cite[ch.0, 5.4.1]{EGA1}, for each integer $l \geq 0$,
$S_l= {\bigl\{ s \in S \, \mid \, \mathop{\rm rank}\nolimits( \mathcal C_{\kappa(s)}) =l \bigr\}}$ is an open subset of $S$
so that we have a decomposition in clopen subschemes $S=\coprod\limits_{l \geq 0} S_l$.
Without loss of generality we can assume that $\mathcal C$ is locally free of rank $l$.
Lemma \ref{lem_brian} provides a finite \'etale cover $S'$ of $S$
such that $\mathcal C \otimes_{\mathcal{O}_S} \mathcal{O}_{S'} \cong (\mathcal{O}_{S'})^l$.
Hence $\GL_1(\mathcal C) \times_S S \cong (\mathbb{G}_m)^l \times_S S'$.
It follows that $\mathfrak{S} \times_S S'$ is a subgroup $S'$--scheme of $(\mathbb{G}_m)^l \times_S S'$
of multiplicative type.
According to \cite[IX.2.11.(i)]{SGA3} there exists a partition in clopen subsets
$S'= \sqcup_{i \in I} S'_i$ such that each $\mathfrak{S} \times_S S'_i$ is diagonalizable.
Since $\mathfrak{S} \times_S S'$ is of constant type we conclude that $\mathfrak{S} \times_S S'$
is diagonalizable.
\end{proof}
\subsection{Characterization of isotrivial groups of multiplicative type}
\begin{stheorem}\label{thm_torus} Let $\mathfrak{S}$ be an $S$-group scheme of multiplicative type of finite type and of
constant type. Then the following are equivalent:
\smallskip
(i) $\mathfrak{S}$ is isotrivial;
\smallskip
(ii) $\mathfrak{S}$ is linear;
\smallskip
(iii) $\mathfrak{S}$ is an $S$--subgroup scheme of an $S$--group scheme $\GL_1(\mathcal A)$
where $\mathcal A$ is an Azumaya $\mathcal{O}_S$--algebra.
\end{stheorem}
\begin{proof}
$(i) \Longrightarrow (ii)$. This follows from Lemma \ref{lem_weil}.
\smallskip
\noindent $(ii) \Longrightarrow (iii)$. By definition we have a monomorphism
$\mathfrak{S} \to \GL(\mathcal E)$ where $\mathcal E$ is a locally free $\mathcal{O}_{S}$--module of finite rank.
Since $\mathop{\rm End}\nolimits_{\mathcal{O}_S}(\mathcal E)$ is an Azumaya $\mathcal{O}_S$--algebra, we get (iii).
\smallskip
\noindent $(iii) \Longrightarrow (i)$. This follows of Proposition \ref{prop_torus}.(4).
\end{proof}
\begin{sexamples} \label{ex_groth} {\rm (a) Grothendieck constructed
a scheme $S$ and an $S$--torus $\goth T$ which is locally trivial of rank $2$
but which is not isotrivial \cite[\S X.1.6]{SGA3} (e.g. $S$ consists of
two copies of the projective line over a field pinched at $0$ and $\infty$).
Theorem \ref{thm_torus} shows that such an $S$--torus is not linear.
\newline
(b) Also there exists a local ring $R$ and an $R$--torus $\goth T$ of rank $2$ which is not isotrivial \cite[\S 1.6]{SGA3};
the ring $R$ can be taken as the local ring of an algebraic curve at a double point.
Theorem \ref{thm_torus} shows that such an $R$--torus is not linear.
}
\end{sexamples}
\section{Reductive case}
For stating the complete version of our main result,
we need more notation.
As in the introduction, $\mathfrak{G}$ is a reductive $S$--group
scheme of constant type and $G$ is the underlying Chevalley
$\mathbb{Z}$--group scheme.
We denote by $\text{\rm{Aut}}(G)$ the automorphism group scheme of $G$
and we have an exact sequence of $\mathbb{Z}$--group schemes \cite[th. XXIV.1.3]{SGA3}
$$
1 \to G_{ad} \to \text{\rm{Aut}}(G) \to \text{\rm{Out}}(G) \to 1.
$$
We remind the reader of the representability of the fppf sheaf
$\underline{\mathrm{Isom}}(G_S, \mathfrak{G})$ by
a $\text{\rm{Aut}}(G)_S$--torsor $\mathrm{Isom}(G_S, \mathfrak{G})$
defined in \cite[XXIV.1.8]{SGA3}. The contracted product
$$\mathrm{Isomext}(G_S, \mathfrak{G}) := \mathrm{Isom}(G_S, \mathfrak{G})
\wedge^{\text{\rm{Aut}}(G)_S} \text{\rm{Out}}(G)_S$$ is an $\text{\rm{Out}}(G)_S$--torsor ({\it ibid}, 1.10)
which encodes the isomorphism class of the quasi-split form of $\mathfrak{G}$.
\begin{stheorem}\label{thm_main_complete}
The following are equivalent:
\smallskip
(i) The torus $\mathrm{rad}(\mathfrak{G})$ is isotrivial;
\smallskip
(ii) the $\text{\rm{Out}}(G)_S$--torsor $\mathop{\rm Isomext}\nolimits(G_S, \mathfrak{G})$ is isotrivial;
\smallskip
(iii) $\mathfrak{G}$ is linear;
\smallskip
(iv) $\mathrm{rad}(\mathfrak{G})$ is linear.
\smallskip
\noindent Furthermore if $S$ is affine, we can take a faithful linear representation in some $\GL_n$ for $(iii)$ and $(iv)$.
\end{stheorem}
\begin{proof}$(i) \Longrightarrow (ii)$. We assume that $\mathrm{rad}(\mathfrak{G})$
is isotrivial and want to show that the $\text{\rm{Out}}(G)_S$--torsor $\mathcal F=\mathop{\rm Isomext}\nolimits(G_S, \mathfrak{G})$ is isotrivial.
In other words, we want to show that there exists a finite \'etale cover $S'$ of
$S$ such that $\mathfrak{G} \times_S S'$ is an inner form of $G$.
Without loss of generality we can assume that $\mathrm{rad}(\mathfrak{G})$ is a split torus.
We quote now \cite[XXIV.2.16]{SGA3} for the Chevalley group $G$ over $\mathbb{Z}$
which introduces the $\mathbb{Z}$-group scheme
$$H= \ker\bigl( \text{\rm{Aut}}(G) \to \text{\rm{Aut}}(\mathrm{rad}(G) \bigr);$$
furthermore there is an equivalence of categories
between the category of \break $H$-torsors over $S$
and the category of pairs $(\goth M, \phi)$ where $\goth M$ is an $S$--form
of $G$ and $\phi: \mathrm{rad}(G)_S \buildrel\sim\over\lgr \mathrm{rad}(\goth M)$.
Since $\mathrm{rad}(\mathfrak{G})$ is split, we choose an isomorphism $\phi: \mathrm{rad}(G)_S \buildrel\sim\over\lgr \mathrm{rad}(\mathfrak{G})$
and consider an $H$-torsor $\mathfrak{P}$ mapping to an object isomorphic to
$(\mathfrak{G},\phi)$. Furthermore the quoted reference provides an exact sequence
of $\mathbb{Z}$--group schemes
$$
1 \to G_{ad} \to H \xrightarrow{p} F \to 1
$$
where $F$ is finite \'etale over $\mathbb{Z}$, so is constant.
We denote by $S'= \goth M \wedge^{H_S} F_S$ the contracted product of $\goth M$ and $F_S$ with respect to $H_S$;
this is an $F$--torsor over $S$, hence is a finite \'etale cover of $S$.
It follows that $\mathfrak{P} \times_S S'$ admits a reduction to
a $G_{ad,S'}$--torsor $\mathfrak{Q}'$. Since the map $G_{ad} \to H \to \text{\rm{Aut}}(G)$
is the canonical map, we conclude that $\mathfrak{G}_{S'} \cong \, ^{\mathfrak{Q}'}\!\!G$ is an inner form of $G$.
\smallskip
\noindent $(ii) \Longrightarrow (iii)$. Our assumption is that there
exists a finite \'etale cover $S'/S$ which splits
the $\text{\rm{Out}}(G)$--torsor $\mathrm{Isomext}(G_S, \mathfrak{G})$.
Lemma \ref{lem_weil} permits us to replace $S$ by $S'$, so we can assume
that the $\text{\rm{Out}}(G)$--torsor $\mathrm{Isomext}(G_S, \mathfrak{G})$ is trivial;
that is, $\mathfrak{G}$ is an inner form of $G$.
There exists a $G_{ad}$--torsor $\mathfrak{Q}$ over $S$
such that $\mathfrak{G} \cong \, {^\mathfrak{Q} \! G}$.
Since $G \mathbin{{>}\!{\triangleleft}} G_{ad}$ is defined over $\mathbb{Z}$, it admits
a faithful representation $\rho: G \mathbin{{>}\!{\triangleleft}} G_{ad} \to \GL_n$ over $\mathbb{Z}$ \cite[\S 1.4.5]{BT}.
The map $\rho$ is then $G_{ad}$--equivariant and can be twisted
by the $G_{ad}$--torsor $\mathfrak{Q}$. We obtain a faithful representation
${^\mathfrak{Q} \! G} \mathbin{{>}\!{\triangleleft}} {^\mathfrak{Q} \! G_{ad}} \to {^\mathfrak{Q} \! \GL_n}= \GL(\mathcal E)$ where
$\mathcal E$ is the locally free $\mathcal{O}_S$--module of rank $n$
which is the twist of $(\mathcal{O}_S)^n$ by the $\GL_n$--torsor
$\mathfrak{Q} \wedge^{G_{ad}} \GL_n$. Thus ${^\mathfrak{Q} \! G}$ is linear.
\smallskip
\noindent $(iii) \Longrightarrow (iv)$. Obvious.
\smallskip
\noindent $(iv) \Longrightarrow (i)$.
Since $\mathrm{rad}(\mathfrak{G})$ is a form of $\mathrm{rad}(G)$, it is of constant rank
and Theorem \ref{thm_torus} shows that $\mathrm{rad}(G)$ is isotrivial.
\smallskip
Finally the refinement for $S$ affine follows from Lemma \ref{lem_affine}.
\end{proof}
\begin{scorollary}\label{cor_main}
Under the assumptions of $\mathfrak{G}$, let $\mathfrak{G}^{qs}$ be the quasi-split form of
$\mathfrak{G}$.
Then $\mathfrak{G}$ is linear if and only if $\mathfrak{G}^{qs}$ is
linear.
\end{scorollary}
The next corollary slightly generalizes a result by Thomason \cite[cor. 3.2]{T}.
\begin{scorollary}\label{cor_main2} Assume that either
\smallskip
\noindent (i) $S$ is locally noetherian and geometrically unibranch
(e.g. normal);
\smallskip
\noindent (ii) $\mathrm{rad}(\mathfrak{G})$ is of rank $\leq 1$ (in particular if $G$ is semisimple).
\smallskip
\noindent Then $\mathfrak{G}$ is linear.
\end{scorollary}
\begin{proof}
In case (i), the torus $\mathrm{rad}(\mathfrak{G})$ is isotrivial \cite[X.5.16]{SGA3}.
In case (ii), we have $\mathrm{rad}(G)=1$ or $\mathbb{G}_m$ (since $G$ is split), so
that $\mathrm{rad}(\mathfrak{G})$ is
split by a quadratic \'etale cover of $S$ (\S \ref{subsec-rank1}), hence is isotrivial.
Hence Theorem \ref{thm_main} implies that $\mathfrak{G}$ is linear.
\end{proof}
The next corollary extends
Demazure's characterization of locally isotrivial
reductive group schemes \cite[XXIV.4.1.5]{SGA3}.
\newpage
\begin{scorollary}\label{cor_main_complete}
The following are equivalent:
\smallskip
(i) $\mathfrak{G}$ is locally (resp.\ semilocally) isotrivial;
\smallskip
(ii) The torus $\mathrm{rad}(\mathfrak{G})$ is locally (resp.\ semilocally) isotrivial;
\smallskip
(iii) the $\text{\rm{Out}}(G)_S$--torsor $\mathop{\rm Isomext}\nolimits(G_S, \mathfrak{G})$ is locally (resp.\ semilocally) isotrivial;
\smallskip
(iv) $\mathfrak{G}$ is locally (resp.\ semilocally) linear;
\smallskip
(v) $\mathrm{rad}(\mathfrak{G})$ is locally (resp.\ semilocally) linear.
\end{scorollary}
\begin{proof}
In view of Theorem \ref{thm_main_complete}, it remains
to establish the equivalence $(i) \Longleftrightarrow (ii)$.
Since this is precisely the quoted result \cite[XXIV.3.5]{SGA3}, the proof is complete.
\end{proof}
\begin{scorollary}\label{cor_main_complete2}
Let $\goth H$ be a reductive $S$--subgroup scheme of $\mathfrak{G}$.
If $\mathfrak{G}$ is locally (resp.\ semilocally) isotrivial, then
$\goth H$ is locally (resp.\ semilocally) isotrivial.
\end{scorollary}
\begin{proof}
Corollary \ref{cor_main_complete} shows that $\mathfrak{G}$ is
locally (resp.\ semilocally) linear and so is $\goth H$.
Therefore $\goth H$ is locally (resp.\ semilocally) isotrivial.
\end{proof}
\smallskip
\section{The semilocal case} \label{section_semilocal}
We assume that $S=\mathop{\rm Spec}\nolimits(R)$ where $R$ is a semilocal ring
and continue to assume that the reductive $S$--group scheme
$\mathfrak{G}$ is of constant type.
We remind the reader that $\mathfrak{G}$ admits a maximal torus
(Grothendieck, \cite[XIV.3.20 and footnote]{SGA3}).
\begin{scorollary} \label{cor_semilocal} Let $\goth T$ be a maximal torus of $\mathfrak{G}$.
The following are equivalent:
\smallskip
(i) $\mathfrak{G}$ is isotrivial;
\smallskip
(ii) The torus $\mathrm{rad}(\mathfrak{G})$ is isotrivial;
\smallskip
(iii) the $\text{\rm{Out}}(G)_S$--torsor $\mathop{\rm Isomext}\nolimits(G_S, \mathfrak{G})$ is isotrivial;
\smallskip
(iv) $\mathfrak{G}$ is linear;
\smallskip
(v) $\mathrm{rad}(\mathfrak{G})$ is linear;
\smallskip
(vi) $\goth T$ is linear;
\smallskip
(vii) $\goth T$ is isotrivial.
\end{scorollary}
\begin{proof}
From Corollary \ref{cor_main_complete}, we have the equivalences
$(i) \Longleftrightarrow (ii) \Longleftrightarrow (iii) \Longleftrightarrow (iv)
\Longleftrightarrow (v)$. On the other hand,
the equivalence $(vi) \Longleftrightarrow (vii)$ holds
according to Theorem \ref{thm_torus}. Now we observe that
the implications $(iv) \Longrightarrow (vi)$ and
$(vi) \Longrightarrow (v)$ are obvious so the proof is complete.
\end{proof}
\section{Equivariant resolution property}
\begin{sdefinition} Let $\mathfrak{G}$ be a flat group scheme over $S$ acting on a locally noetherian $S$--scheme
$\goth X$. One says that $(\mathfrak{G}, S, \goth X)$ has the
resolution property ( $(RE)$ for short) if for every coherent $\mathfrak{G}$-module $\mathcal F$ on $\goth X$, there
is a locally free coherent $\mathfrak{G}$-module ( i.e. a $\mathfrak{G}$-vector bundle $\mathcal E$) and a
$\mathfrak{G}$-equivariant epimorphism $\mathcal E \to \mathcal F \to 0$.
\end{sdefinition}
We strengthen Thomason's results.
\begin{stheorem} \label{thm_re} Let $\mathfrak{G}$ be a reductive $S$--group scheme.
We assume that $S$ is separated noetherian and that
$(1,S,S)$ satisfies the resolution property, e.g. $S$ is affine or regular or
admits an ample family of line bundles.
Then the following are equivalent:
\smallskip
(i) $\mathfrak{G}$ is linear;
\smallskip
(ii) $\mathrm{rad}(\mathfrak{G})$ is isotrivial;
\smallskip
(iii) $\mathfrak{G}$ satisfies $(RE)$.
\end{stheorem}
\begin{proof}
$(i) \Longleftrightarrow (ii)$. This is a special case of Theorem \ref{thm_main_complete}.
\smallskip
\noindent $(ii) \Longrightarrow (iii)$. This is Thomason's result \cite[Theorem 2.18]{T}.
\smallskip
\noindent $(iii) \Longrightarrow (i)$. This is Thomason's result \cite[Theorem 3.1]{T}, see also \cite[VI$_B$.13.5]{SGA3}.
\end{proof}
\begin{sremark}{\rm Example \ref{ex_groth}.(b) is an example of
a local noetherian ring $R$ and of a rank two non-isotrivial torus $\goth T$.
Theorem \ref{thm_re} shows that $\goth T$ does not satisfy (RE).
This answers a question of Thomason \cite[\S 2.3]{T}.
}
\end{sremark}
\smallskip
|
{
"timestamp": "2021-04-23T02:15:50",
"yymm": "2103",
"arxiv_id": "2103.07305",
"language": "en",
"url": "https://arxiv.org/abs/2103.07305"
}
|
\section{Introduction}
\label{introduction}
The speed of sound, $c_s$, is a fundamental property of any substance. In fluids, it is the velocity of a longitudinal compression wave propagating through the medium, and its square is computed as the ratio of a change in the pressure, $P$, corresponding to a change in the energy density, $\mathcal{E}$. Therefore, it is directly related to the thermodynamic properties of the system, including its equation of state (EOS).
In dense nuclear matter, $c_s$ is of particular interest to neutron star research: its behavior as a function of baryon number density, $n_B$, influences the mass-radius relationship and, consequently, the maximum possible mass of neutron stars \cite{Ozel:2016oaf}. Current neutron star data suggest that $c_s$ rises significantly for $n_B$ larger than the nuclear saturation density, $n_0$, and that it perhaps exceeds
$c_s\sim1/\sqrt{3}$ at densities as low as few times that of normal nuclear matter. This possibility was first suggested in \cite{Bedaque:2014sqa}, followed by other studies, e.g.\ \cite{Tews:2018kmu,McLerran:2018hbz,Fujimoto:2019hxv}.
Presently, heavy-ion collisions are the only means of studying dense nuclear matter in a
laboratory. Experiments probing nuclear matter at high $n_B$, such as the Beam
Energy Scan (BES) program at RHIC, put special significance on the search for the QCD critical point (CP). Here, $c_s$ also conveys relevant information: it displays a local minimum at a crossover transition, whereas it vanishes at the CP and on the associated spinodal lines. Indeed, lattice QCD (LQCD) shows that at vanishing baryon chemical potential, $\mu_B=0$, a minimum in $c_s$ occurs at temperature $T_0=156.5\pm1.5$ MeV \cite{Bazavov:2018mes} (see also \cite{Borsanyi:2020fev}), corresponding to a crossover transition between hadron gas and quark-gluon plasma (QGP).
To date, a few attempts have been made to evaluate $c_s$ from heavy-ion collision data. In \cite{Gardim:2019xjs}, $c_s$ is estimated in ultra-relativistic collisions, where $\mu_B\approx n_B\approx0$, based on the proportionality of entropy density, $s$, and temperature, $T$, to charged particle multiplicity and mean transverse momentum, respectively. The estimated value agrees with LQCD results. At finite $\mu_B$, the Landau model and hybrid UrQMD simulations were utilized in \cite{Steinheimer:2012bp} to reproduce the widths of the negatively charged pion rapidity distribution. That study purports to locate a minimum in $c_s$ within the collision energy range $\sqrt{s_{NN}}=4\txt{-}9\ \txt{GeV}$.
In this work we suggest a novel approach to exploring the behavior of $c_s$ by utilizing cumulants of the baryon number distribution. The sensitivity of the cumulants to the EOS near the CP \cite{Asakawa:2009aj, Stephanov:2011pb}, making them central observables pursued in the BES, follows directly from their sensitivity to derivatives of the pressure with respect to $\mu_B$. The key observation in this paper is that besides the vicinity of the CP, cumulants provide rich information about the EOS at all points of the phase diagram, and in particular they allow a measurement of $c_s$ in matter created in heavy-ion collisions.
\section{Cumulants and the speed of sound}
\label{relationship_cumulants_cT2}
Cumulants of net baryon number $\kappa_j$ are defined as $\kappa_j=VT^{j-1}\left(\frac{d^jP}{d\mu_B^j}\right)_T$, where $V$ is the volume. Expressed in terms of derivatives with respect to $n_B$, the first three cumulants are given by
\begin{eqnarray}
&& \kappa_1 = V n_B ~, \label{cumulant_1} \hspace{8mm} \kappa_2 = \frac{VTn_B}{\left( \frac{dP}{dn_B} \right)_T}~, \label{cumulant_2} \nonumber \\
&& \kappa_3= \frac{VT^2n_B}{\left( \frac{dP}{dn_B} \right)_T^2} \left[ 1 - \frac{n_B}{\left( \frac{dP}{dn_B} \right)_T} \left(\frac{d^2P}{dn_B^2}\right)_T \right] \label{cumulant_3} ~.
\end{eqnarray}
Importantly, cumulants are related to moments of the baryon number distribution. In particular, for $j\leq3$, $\kappa_j\equiv\big\langle\big(N_B-\big\langle N_B\big\rangle\big)^j\big\rangle$.
The definition of $c_s$ requires specifying which properties of the system are
considered constant during the propagation of the compression wave. One often uses the speed of
sound at constant entropy $S$ per net baryon number $N_B$, $c_{\sigma}^2\equiv\left(
\frac{dP}{d\mathcal E}\right)_{\sigma}$, where $\sigma=S/N_B$. Similarly, the speed of sound at constant temperature is $c_T^2\equiv\left(\frac{dP}{d\mathcal E}\right)_{T}$. These variants have specific regions of applicability: For example, the propagation of sound in air is governed by adiabatic
compression, so that using $c_{\sigma}^2$ is appropriate. On the other hand, when
there is a temperature reservoir (e.g.\ in porous media) or when the cooling timescale is fast compared with the sound wave period (as is the case e.g.\ for an interstellar medium subject to radiative cooling), $c_T^2$ is applicable.
Explicitly,
$c_{\sigma}^2$ and $c_T^2$
can be written as
\begin{eqnarray}
c_{\sigma}^2 = \frac{ \Big( \frac{d P}{dn_B} \Big)_{T} \Big(\frac{d s}{d T} \Big)_{n_B} + \Big( \frac{ d P}{dT} \Big)_{n_B} \bigg[ \frac{s}{n_B} - \Big(\frac{d s}{d n_B}\Big)_T \bigg] }{ \Big(\frac{sT}{n_B} + \mu_B \Big) \Big( \frac{d s}{d T} \Big)_{n_B} }
\label{speed_isentropic}
\end{eqnarray}
and
\begin{eqnarray}
c_T^2 = \frac{\Big( \frac{dP}{dn_B} \Big)_T}{ T \Big(\frac{d s}{d n_B} \Big)_T + \mu_B }~.
\label{speed_isothermal}
\end{eqnarray}
In the limit $T\to0$, the above expressions both lead to
\begin{eqnarray}
c^2\Big|_{T=0} = \frac{1}{\mu_B} \bigg( \frac{dP}{dn_B} \bigg)_T ~.
\label{cT2_approx}
\end{eqnarray}
Consequently, for $\frac{\mu_B}{T}\gg1$ the values of $c_{\sigma}^2$ and $c_T^2$ should largely coincide. Moreover, Eq.\ (\ref{speed_isothermal}) can be transformed to express $c_T^2$ as a function of the cumulants, Eq.\ (\ref{cumulant_3}),
\begin{eqnarray}
c_T^{2} = \left[\bigg(\parr{\log \kappa_1}{\log T}\bigg)_{\mu_B} + \frac{\mu_B}{T} \frac{\kappa_2}{\kappa_1} \right]^{-1}~.
\label{cT2_as_function_of_cumulants}
\end{eqnarray}
The first term in Eq.\ (\ref{cT2_as_function_of_cumulants}) is challenging to estimate from experimental data, however, it can be shown to be negligible for a degenerate Fermi gas, $\frac{\mu_B}{T}\gg1$, where it constitutes an order $\left(T/\mu_B\right)^2$ correction; then
\begin{eqnarray}
c_T^2 \approx \frac{T \kappa_1}{\mu_B \kappa_2}~.
\label{magic_equation_1}
\end{eqnarray}
We note that Eq.\ (\ref{magic_equation_1}) provides an upper limit to the value of $c_T^2$ as long as $\Big(\parr{\log \kappa_1}{\log T}\Big)_{\mu_B} > 0$.
Using Eq.\ (\ref{speed_isothermal}), one can also calculate the logarithmic derivative of $c_T^2$,
\begin{eqnarray}
\hspace{-1mm}\bigg(\frac{d \ln c_T^2}{d \ln n_B} \bigg)_T
= \frac{n_B \Big( \frac{d^2P}{dn_B^2} \Big)_T}{ \Big( \frac{dP}{dn_B} \Big)_T} - \frac{ \Big( \frac{dP}{dn_B} \Big)_T + Tn_B \Big( \frac{d^2s}{dn_B^2} \Big)_T }{\mu_B + T \Big( \frac{ds}{dn_B} \big)_T} ~.
\end{eqnarray}
It is again possible to rewrite the above equation in terms of the cumulants,
\begin{eqnarray}
\bigg(\frac{d \ln c_T^2}{d \ln n_B} \bigg)_T + c_T^2
= 1 - \frac{\kappa_3 \kappa_1}{\kappa_2^2} - c_T^2 \bigg(\frac{d \ln \big(\frac{\kappa_2}{T}\big)}{d \ln T}\bigg)_{n_B} ~,
\end{eqnarray}
and neglecting the last term on the right-hand side yields
\begin{eqnarray}
\left(\frac{d \ln c_T^2}{d \ln n_B} \right)_T + c_T^2 \approx 1 - \frac{\kappa_3 \kappa_1}{\kappa_2^2} ~.
\label{magic_equation_2}
\end{eqnarray}
This approximation is again valid for $\frac{\mu_B}{T} \gg 1$, and the correction due to the neglected term is likewise of order $\left(T/\mu_B\right)^2$.
We note that in the opposite limit, $\mu_B \to 0$, Eq.\ (\ref{cT2_as_function_of_cumulants}) reveals a similarly simple form, $c_T^{2} = \left( \frac{d \ln \kappa_2}{d \ln T} \right)^{-1}_{\mu_B=0}$, suggesting that $c_T^2$ can be estimated in ultrarelativistic heavy-ion collisions, provided measurements of $\kappa_2$ are available at different temperatures. It might be possible to achieve this with data from a combination of centralities, energies, collision species, or rapidity ranges. In this work, however, we are interested in utilizing Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}) applied to collisions at medium and low energies.
\begin{figure*}[t]
\includegraphics[width = 0.99\textwidth]{model_comparison_combined.jpeg}
%
\caption{(Color online) Model study of regions of applicability of Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}). The left (right) panels show results obtained in the VDF (Walecka) model. The upper and lower panels show quantities entering Eq.\ (\ref{magic_equation_1}) and Eq.\ (\ref{magic_equation_2}), respectively. Results at $T=50,100,150,200\ \txt{MeV}$ are given by blue and green solid lines, dark and light purple long-dashed lines, red and pink short-dashed lines, and orange and brown dash-dotted lines, respectively. For each $T$, the thickest lines correspond to the exact results and the medium-thick lines correspond to the approximations, given by the right-hand sides of Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}). Additionally, on upper panels the thinnest lines correspond to $c_{\sigma}^2$. Upper panels: for both models, Eq.\ (\ref{magic_equation_1}) is valid for $T\lesssim100\ \txt{MeV}$ and $\mu_B\gtrsim600\ \txt{MeV}$. Lower panels: for both models, Eq.\ (\ref{magic_equation_2}) is valid for $\mu_B\gtrsim200\ \txt{MeV}$; the exception is the Walecka model at $T=200\ \txt{MeV}$, where a phase transition to an almost massless gas of nucleons dramatically decreases the applicability of both Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}).
}
\label{tests_of_formulas}
\end{figure*}
\section{Validation}
We are interested in finding the limitations of the low-temperature approximation used to derive Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}), and for this we use effective models. Anticipating applying our formulas in regions of the phase diagram described by hadronic degrees of freedom, we choose two models of dense nuclear matter: the vector density functional (VDF) model with two phase transitions \cite{Sorensen:2020ygf} and the Walecka model \cite{Walecka:1974qa}. The VDF model utilizes interactions of vector type, while the Walecka model employs both vector- and scalar-type interactions. Both models describe the nuclear liquid-gas phase transition, while the VDF model additionally describes a conjectured high-density, high-temperature phase transition modeling the QGP phase transition. In this work, the QGP-like phase transition is chosen to exhibit a CP at $T_c=100\ \txt{MeV}$ and $n_c=3\ n_0$, with the $T=0$ boundaries of the spinodal region in the $T$-$n_B$ plane given by $n_{B, \txt{left spinodal} } (T=0) \equiv \eta_L=2.5\ n_0$ and $n_{B, \txt{right spinodal}} (T=0) \equiv\eta_R=3.32\ n_0$, where $n_0=0.160\ \txt{fm}^{-3}$; this choice is arbitrary and serves as a plausible example.
We plot both sides of Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}) as functions of $\mu_B$ at a series of
temperatures in Fig.\ \ref{tests_of_formulas}. We note that the explored temperature range reaches beyond the region of the phase diagram where hadronic models plausibly describe matter created in heavy-ion collisions, nevertheless, it is instructive to test our approximations in this regime.
We utilize natural units in which the speed of light in vacuum is $c=1$. We note that in the VDF model, $c_s$ quickly becomes acausal for $\mu_B$ above the QGP-like phase transition. It is an expected behavior in models utilizing interactions dependent on high powers of $n_B$ \cite{Zeldovich:1962emp}, and while not ideal, it does not affect the current analysis.
On all panels in Fig.\ \ref{tests_of_formulas}, the exact model calculations show expected
features as functions of $\mu_B$. In the upper left panel, showing both $c_{T}^2$ and $c_{\sigma}^2$ in the VDF model, at small $\mu_B$ we see a softening of the EOS due to the influence of the nuclear CP, followed by an increase at densities of the order of $n_0$, then a dive in $c_s^2$ caused by the QGP-like phase transition, and finally a steep rise for high values of $\mu_B$. In the upper right panel, showing $c_T^2$ and $c_{\sigma}^2$ in the Walecka model, we similarly observe a soft EOS at small $\mu_B$, while the value of $c_s^2$ goes asymptotically to 1 for large $\mu_B$. Additionally, for $T=200\ \txt{MeV}$, the Walecka model shows effects due to a phase transition in the nucleon-antinucleon plasma, occurring around $T\approx190\ \txt{MeV}$ and $n_B=0$; above this transition, the model describes an almost non-interacting gas of nearly massless nucleons \cite{Theis:1984qc}. The behavior of the curves in the lower panels, showing $c_T^2+\big(\frac{d\ln c_T^2}{d\ln n_B}\big)_T$, can be directly traced to the behavior of the curves in the upper panels. In particular, for the VDF model we observe strong divergences due to the softening of the EOS in the QGP-like phase transition region.
Comparing the exact results with the approximations, we see that while Eq.\ (\ref{magic_equation_1}) is valid for $T\lesssim100\ \txt{MeV}$ and $\mu_B\gtrsim600\ \txt{MeV}$, it behaves poorly, both qualitatively and quantitatively, for $T$ and $\mu_B$ corresponding to regions of the phase diagram probed by moderately- to highly-energetic heavy-ion collisions (upper panels). On the other hand, the approximation introduced in Eq.\ (\ref{magic_equation_2}) is qualitatively valid for most of the probed $T$ and $\mu_B$, with the exception of regions characterized by $\mu_B\lesssim200\ \txt{MeV}$ (lower panels).
\begin{figure}[t]
\includegraphics[width = 0.99\columnwidth]{STAR+HADES_and_VDF_plots_combined_v2.jpeg}\\
%
\caption{(Color online) Comparison of the right-hand sides of Eq.\ (\ref{magic_equation_1}) (upper panel) and Eq.\ (\ref{magic_equation_2}) (lower panel) for experimental data (red triangles), ideal gas at the freeze-out (small grey circles), the VDF model at the freeze-out (light green stars), and the VDF model at a set of points chosen to reproduce the data (dark purple stars); exact results, that is left-hand sides of Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}), are shown for the two cases considered in the VDF model (green and purple circles). The data points for the matched VDF results (shown only for collisions at low energies, where using the model is justified) are chosen to reproduce experimental values of $1-\kappa_3\kappa_1/\kappa_2^2$ (see Fig.\ \ref{diagram}). We note that at $\sqrt{s}=2.4\ \txt{GeV}$, matching the value of $1-\kappa_3 \kappa_1/\kappa_2^2$ exactly is possible, but would place the matched point close to the nuclear liquid-gas CP, which we find unlikely.
}
\label{STAR_HADES_plots}
\end{figure}
\begin{figure}[t]
%
\includegraphics[width = 0.99\columnwidth]{Cumulants_diagram_1minusk3k1overk2k2_with_legend.jpeg}
\caption{(Color online) Contour plot of $1-\kappa_3\kappa_1/\kappa_2^2$ in the VDF model. Yellow and black lines correspond to the spinodal and coexistence lines, respectively; white contours signifiy regions where $1-\kappa_3\kappa_1/\kappa_2^2=1\pm0.03$. Light green stars denote experimentally measured freeze-out parameters $(T_{\txt{fo}},\mu_{\txt{fo}})$, while dark purple stars denote points where $1-\kappa_3\kappa_1/\kappa_2^2$, taken along lines informed by average phase diagram trajectories for STAR collision energies \cite{Shen:2020jwv}, matches the experimentally measured values for a given collision energy. The softening of the EOS, leading to negative values of $\big(\frac{d\ln c_T^2}{d\ln n_B}\big)_T$, occurs in two regions of the phase diagram, corresponding to the ordinary nuclear matter phase transition and to the conjectured QGP-like phase transition.
}
\label{diagram}
\end{figure}
\section{Experimental data and interpretation}
We consider cumulants of net proton number and chemical freeze-out parameters, $(T_{\txt{fo}},\mu_{\txt{fo}})$, in collisions at 0-5\% centrality, determined by the STAR \cite{Abdallah:2021fzj} and HADES \cite{Adamczewski-Musch:2020slf, HADES_MLorentz_talk} experiments, and we use them to plot Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}) (red triangles, upper and lower panel in Fig.\ \ref{STAR_HADES_plots}, respectively) against $\mu_B$.
Based on the previous section, we trust the results presented in the upper panel of Fig.\ \ref{STAR_HADES_plots}, approximating $c_T^2$, only for the lowest collision energy, $\sqrt{s}=2.4\ \txt{GeV}$ from the HADES experiment. Here, $c_T^2$ as obtained from Eq.\ (\ref{magic_equation_1}) is small: less than half of the ideal gas value. At the same time, the value of $1-\kappa_3\kappa_1/\kappa_2^2$ (shown in the lower panel in Fig.\ \ref{STAR_HADES_plots}), which we assume is dominated by $\big(\frac{d\ln c_T^2}{d\ln n_B}\big)_T$, drops with decreasing collision energy to reach a minimum at the lowest STAR point, $\sqrt{s}=7.7\ \txt{GeV}$, and then steeply rises for the HADES point. This could mean that in $\sqrt{s} = 7.7\ \txt{GeV}$ collisions, $c_T^2$ is approximately constant as a function of $n_B$, while in $\sqrt{s} = 2.4\ \txt{GeV}$ collisions the matter is characterized by a small $c^2_T$ which nevertheless has a large slope as a function of $n_B$.
To further understand this behavior, we study the dependence of $1-\kappa_3\kappa_1/\kappa_2^2$ on $\mu_{B}$ and $T$ within the VDF model, shown in Fig.\ \ref{diagram}. Comparing model results with experiment requires choosing at which $T$ and $\mu_B$ to take values of $1-\kappa_3 \kappa_1/\kappa_2^2$. A natural choice is to use values at $(T_{\txt{fo}},\mu_{\txt{fo}})$ (light green stars), but as shown in the lower panel of Fig.\ \ref{STAR_HADES_plots}, these values do not lead to an agreement with experimental data, with the biggest discrepancy for the HADES point. (We note that $(T_{\txt{fo}},\mu_{\txt{fo}})$ are established with hadron interactions neglected, and the degree to which this affects our results may vary across the phase diagram.) However, critical fluctuations exhibit a large relaxation time \cite{Berdnikov:1999ph,Stephanov:2017ghc,Du:2020bxp}, and their measured values could be affected by stages of the collision preceding the freeze-out. With this insight and taking guidance from average phase diagram trajectories of hybrid simulations of heavy-ion collisions \cite{Shen:2020jwv}, we consider $1-\kappa_3\kappa_1/\kappa_2^2$ at slightly earlier stages of the evolution. In this way we obtain values of $1-\kappa_3\kappa_1/\kappa_2^2$ (dark purple stars) that reproduce the experimental results for a given collision energy (lower panel in Fig.\ \ref{STAR_HADES_plots}); here the exception is the HADES point, for which we prioritized choosing a point in a reasonable vicinity of the measured freeze-out over obtaining a value equal to the experimental data. Comparing with the exact model results for $c_T^2$ and $c_T^2 + \big(\frac{d\ln c_T^2}{d\ln n_B}\big)_T$, also displayed in Fig.\ \ref{STAR_HADES_plots}, as well as with the upper left panel of Fig.\ \ref{tests_of_formulas}, we can confirm that at the point reproducing the experimental value of $1-\kappa_3\kappa_1/\kappa_2^2$ for the lowest STAR energy, $c_T^2$ is nearly constant as a function of $\mu_B$ (thick short-dashed line, $T=150\ \txt{MeV}$, at $\mu_B\approx410\ \txt{MeV}$ in Fig.\ \ref{tests_of_formulas}), while at the point reproducing the result for the HADES energy, $c_T^2$ increases sharply with $\mu_B$ (thick solid line, $T=50\ \txt{MeV}$, at $\mu_B\approx850\ \txt{MeV}$ in Fig.\ \ref{tests_of_formulas}).
Naturally, the choice of $T$ and $\mu_B$ at which we compare model calculations with STAR and HADES cumulant data is driven by the wish to match the experimental results, and it serves mainly to show that baryon number cumulants measured in heavy-ion collisions can be connected to the speed of sound in hot and dense nuclear matter. Whether values of higher order cumulants are indeed significantly affected by stages of the evolution preceding the freeze-out needs to be further investigated (for recent developments, see \cite{Nahrgang:2018afz,Jiang:2017sni}). Moreover, while experiments measure proton number cumulants, the VDF model provides baryon number cumulants, putting more strain on our interpretation. Baryon number conservation should likewise be important \cite{Bzdak:2012an,Vovchenko:2020gne}. Finally, our model results may not be applicable in regions of the phase diagram where quarks and gluons become increasingly relevant.
Nonetheless, hadronic models are well-justified for describing low-energy collisions whose evolution is dominated by the hadronic stage. The comparison between the experimental data and the VDF model suggests that collisions at the lowest STAR and HADES energies may be probing regions of the phase diagram affected most by the nuclear liquid-gas phase transition, and the corresponding cumulants of the baryon number may be telling us more about hadronic physics than the QCD CP. If this is true, it may be worthwhile to study the cumulants at even lower collision energies, starting from $0.1\ \txt{GeV}$ projectile kinetic energy, and obtain the speed of sound around the nuclear liquid-gas CP. Conversely, at higher energies it could be possible to utilize collisions at different centralities and different rapidity windows to estimate the neglected terms in Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}), and obtain a stronger estimate for the speed of sound in the respective regions of the phase diagram.
\section{Summary and conclusions}
In this work we utilize cumulants of the baryon number distribution to estimate the isothermal speed of sound squared and its logarithmic derivative with respect to the baryon number density. This result provides a new method for obtaining information about fundamental properties of nuclear matter studied in heavy-ion collisions, with consequences for both the search for the QCD CP and neutron star studies. While the approximations and the model comparison we considered apply to experiments at low energies, the approach itself can be used at any collision energy provided that measurements of cumulants of baryon number as well as their temperature dependence are available. Further studies of effects due to dynamics, in particular utilizing state-of-the-art simulations, will be absolutely essential in determining the extent to which the proposed method provides a reliable extraction of sound velocities and their derivatives.
{\hspace{10mm}}
\section{Acknowledgements}
A.S.\ thanks Chun Shen for providing average phase diagram trajectories of hybrid heavy-ion collision simulations at RHIC BES energies.
A.S.\ and V.K.\ received support through the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics, under contract number
DE-AC02-05CH11231231 and received support within the framework of the
Beam Energy Scan Theory (BEST) Topical Collaboration. D.O.\ and L.M.\ were supported by the U.S.\ DOE under Grant No.\ DE-FG02-00ER4113.
\section{Introduction}
\label{introduction}
The speed of sound, $c_s$, is a fundamental property of any substance. In fluids, it is the velocity of a longitudinal compression wave propagating through the medium, and its square is computed as the ratio of a change in the pressure, $P$, corresponding to a change in the energy density, $\mathcal{E}$. Therefore, it is directly related to the thermodynamic properties of the system, including its equation of state (EOS).
In dense nuclear matter, $c_s$ is of particular interest to neutron star research: its behavior as a function of baryon number density, $n_B$, influences the mass-radius relationship and, consequently, the maximum possible mass of neutron stars \cite{Ozel:2016oaf}. Current neutron star data suggest that $c_s$ rises significantly for an $n_B$ larger than the nuclear saturation density, $n_0$, and that it perhaps exceeds
$c_s\sim1/\sqrt{3}$ at densities as low as a few times that of normal nuclear matter. This possibility was first suggested in \cite{Bedaque:2014sqa}, followed by other studies, e.g.,\ \cite{Tews:2018kmu,McLerran:2018hbz,Fujimoto:2019hxv}.
Presently, heavy-ion collisions are the only means of studying dense nuclear matter in a
laboratory. Experiments probing nuclear matter at high $n_B$, such as the Beam
Energy Scan program at the Relativistic Heavy Ion Collider (RHIC), put special significance on the search for the QCD critical point (CP). Here, $c_s$ also conveys relevant information: it displays a local minimum at a crossover transition, whereas it vanishes at the CP and on the associated spinodal lines. Indeed, lattice QCD shows that at vanishing baryon chemical potential, $\mu_B=0$, a minimum in $c_s$ occurs at temperature $T_0=156.5\pm1.5$ MeV \cite{Bazavov:2018mes} (see also \cite{Borsanyi:2020fev}), corresponding to a crossover transition between hadron gas and quark-gluon plasma (QGP).
To date, a few attempts have been made to evaluate $c_s$ from heavy-ion collision data. In \cite{Gardim:2019xjs}, $c_s$ is estimated in ultrarelativistic collisions, where $\mu_B\approx n_B\approx0$, based on the proportionality of the entropy density, $s$, and the temperature, $T$, to the charged particle multiplicity and mean transverse momentum, respectively. The estimated value agrees with lattice QCD results. At finite $\mu_B$, the Landau model as well as hybrid hydrodynamics and hadronic transport simulations were used in \cite{Steinheimer:2012bp} to reproduce the widths of the negatively charged pion rapidity distribution. That study purports to locate a minimum in $c_s$ within the collision energy range $\sqrt{s_{NN}}=4\txt{-}9\ \txt{GeV}$.
In this Letter, we suggest a novel approach to exploring the behavior of $c_s$ by using cumulants of the baryon number distribution. The sensitivity of the cumulants to the EOS near the CP \cite{Asakawa:2009aj, Stephanov:2011pb}, which makes them central observables pursued in the Beam Energy Scan, follows directly from their sensitivity to derivatives of the pressure with respect to $\mu_B$. The key observation in this Letter is that, besides the vicinity of the CP, cumulants provide rich information about the EOS at all points of the phase diagram, and in particular they allow a measurement of $c_s$ in matter created in heavy-ion collisions.
\section{Cumulants and the speed of sound}
\label{relationship_cumulants_cT2}
Cumulants of net baryon number $\kappa_j$ are defined as $\kappa_j=VT^{j-1}\left(d^jP/d\mu_B^j\right)_T$, where $V$ is the volume. Expressed in terms of derivatives with respect to $n_B$, the first three cumulants are given by
\begin{eqnarray}
&& \kappa_1 = V n_B ~, \label{cumulant_1} \hspace{8mm} \kappa_2 = \frac{VTn_B}{\left( \frac{dP}{dn_B} \right)_T}~, \label{cumulant_2} \nonumber \\
&& \kappa_3= \frac{VT^2n_B}{\left( \frac{dP}{dn_B} \right)_T^2} \left[ 1 - \frac{n_B}{\left( \frac{dP}{dn_B} \right)_T} \left(\frac{d^2P}{dn_B^2}\right)_T \right] \label{cumulant_3} ~.
\end{eqnarray}
Importantly, cumulants are related to moments of the baryon number distribution. In particular, for $j\leq3$, $\kappa_j\equiv\big\langle\big(N_B-\big\langle N_B\big\rangle\big)^j\big\rangle$.
The definition of $c_s$ requires specifying which properties of the system are
considered constant during the propagation of the compression wave. One often uses the speed of
sound at constant entropy $S$ per net baryon number $N_B$, $c_{\sigma}^2\equiv\left(
dP/d\mathcal E\right)_{\sigma}$, where $\sigma=S/N_B$. Similarly, the speed of sound at constant temperature is $c_T^2\equiv\left(dP/d\mathcal E\right)_{T}$. These variants have specific regions of applicability. For example, the propagation of sound in air is governed by adiabatic
compression, so that using $c_{\sigma}^2$ is appropriate. On the other hand, when
there is a temperature reservoir (e.g.,\ in porous media) or when the cooling timescale is fast compared with the sound wave period (as is the case, e.g.,\ for an interstellar medium subject to radiative cooling), $c_T^2$ is applicable.
Explicitly,
$c_{\sigma}^2$ and $c_T^2$
can be written as
\begin{eqnarray}
c_{\sigma}^2 = \frac{ \Big( \frac{d P}{dn_B} \Big)_{T} \Big(\frac{d s}{d T} \Big)_{n_B} + \Big( \frac{ d P}{dT} \Big)_{n_B} \bigg[ \frac{s}{n_B} - \Big(\frac{d s}{d n_B}\Big)_T \bigg] }{ \Big(\frac{sT}{n_B} + \mu_B \Big) \Big( \frac{d s}{d T} \Big)_{n_B} }
\label{speed_isentropic}
\end{eqnarray}
and
\begin{eqnarray}
c_T^2 = \frac{\Big( \frac{dP}{dn_B} \Big)_T}{ T \Big(\frac{d s}{d n_B} \Big)_T + \mu_B }~.
\label{speed_isothermal}
\end{eqnarray}
In the limit $T\to0$, the above expressions both lead to
\begin{eqnarray}
c^2\Big|_{T=0} = \frac{1}{\mu_B} \bigg( \frac{dP}{dn_B} \bigg)_T ~.
\label{cT2_approx}
\end{eqnarray}
Consequently, for $(\mu_B/T)\gg1$, the values of $c_{\sigma}^2$ and $c_T^2$ should largely coincide. Moreover, Eq.\ (\ref{speed_isothermal}) can be transformed to express $c_T^2$ as a function of the Eq.\ (\ref{cumulant_3}) cumulants,
\begin{eqnarray}
c_T^{2} = \left[\bigg(\parr{\log \kappa_1}{\log T}\bigg)_{\mu_B} + \frac{\mu_B}{T} \frac{\kappa_2}{\kappa_1} \right]^{-1}~.
\label{cT2_as_function_of_cumulants}
\end{eqnarray}
The first term in Eq.\ (\ref{cT2_as_function_of_cumulants}) is challenging to estimate from experimental data, however, it can be shown to be negligible for a degenerate Fermi gas, $(\mu_B/T)\gg1$, where it constitutes an order $\left(T/\mu_B\right)^2$ correction; then
\begin{eqnarray}
c_T^2 \approx \frac{T \kappa_1}{\mu_B \kappa_2}~.
\label{magic_equation_1}
\end{eqnarray}
We note that Eq.\ (\ref{magic_equation_1}) provides an upper limit to the value of $c_T^2$ as long as $(\partial \log \kappa_1/\partial \log T)_{\mu_B} > 0$.
Using Eq.\ (\ref{speed_isothermal}), one can also calculate the logarithmic derivative of $c_T^2$,
\begin{eqnarray}
\hspace{-1mm}\bigg(\frac{d \ln c_T^2}{d \ln n_B} \bigg)_T
= \frac{n_B \Big( \frac{d^2P}{dn_B^2} \Big)_T}{ \Big( \frac{dP}{dn_B} \Big)_T} - \frac{ \Big( \frac{dP}{dn_B} \Big)_T + Tn_B \Big( \frac{d^2s}{dn_B^2} \Big)_T }{\mu_B + T \Big( \frac{ds}{dn_B} \big)_T} ~.
\end{eqnarray}
It is again possible to rewrite the above equation in terms of the cumulants,
\begin{eqnarray}
\bigg(\frac{d \ln c_T^2}{d \ln n_B} \bigg)_T + c_T^2
= 1 - \frac{\kappa_3 \kappa_1}{\kappa_2^2} - c_T^2 \bigg(\frac{d \ln (\kappa_2/T)}{d \ln T}\bigg)_{n_B} ~,
\end{eqnarray}
and neglecting the last term on the right-hand side yields
\begin{eqnarray}
\left(\frac{d \ln c_T^2}{d \ln n_B} \right)_T + c_T^2 \approx 1 - \frac{\kappa_3 \kappa_1}{\kappa_2^2} ~.
\label{magic_equation_2}
\end{eqnarray}
This approximation is again valid for $(\mu_B/T) \gg 1$, and the correction due to the neglected term is likewise of order $\left(T/\mu_B\right)^2$.
We note that in the opposite limit, $\mu_B \to 0$, Eq.\ (\ref{cT2_as_function_of_cumulants}) reveals a similarly simple form, $c_T^{2} = \left( d \ln \kappa_2/d \ln T \right)^{-1}_{\mu_B=0}$, suggesting that $c_T^2$ can be estimated in ultrarelativistic heavy-ion collisions, provided measurements of $\kappa_2$ are available at different temperatures. It might be possible to achieve this with data from a combination of centralities, energies, collision species, or rapidity ranges. In this work, however, we are interested in utilizing Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}) applied to collisions at medium and low energies.
\begin{figure*}[t]
\includegraphics[width = 0.99\textwidth]{model_comparison_combined.jpg}
%
\caption{Model study of regions of applicability of Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}). The left (right) panels show results obtained in the VDF (Walecka) model. The upper and lower panels show quantities entering Eq.\ (\ref{magic_equation_1}) and Eq.\ (\ref{magic_equation_2}), respectively. Results at $T=50,100,150,200\ \txt{MeV}$ are given by blue and green solid lines, dark and light purple long-dashed lines, red and pink short-dashed lines, and orange and brown dash-dotted lines, respectively. For each $T$, the thickest lines correspond to the exact results and the medium-thick lines correspond to the approximations, given by the right-hand sides of Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}). Additionally, in upper panels the thinnest lines correspond to $c_{\sigma}^2$. Upper panels: for both models, Eq.\ (\ref{magic_equation_1}) is valid for $T\lesssim100\ \txt{MeV}$ and $\mu_B\gtrsim600\ \txt{MeV}$. Lower panels: for both models, Eq.\ (\ref{magic_equation_2}) is valid for $\mu_B\gtrsim200\ \txt{MeV}$; the exception is the Walecka model at $T=200\ \txt{MeV}$, where a phase transition to an almost massless gas of nucleons dramatically decreases the applicability of both Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}).
}
\label{tests_of_formulas}
\end{figure*}
\section{Validation}
We are interested in finding the limitations of the low-temperature approximation used to derive Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}), and for this we use effective models. Anticipating applying our formulas in regions of the phase diagram described by hadronic degrees of freedom, we choose two models of dense nuclear matter: the vector density functional (VDF) model with two phase transitions \cite{Sorensen:2020ygf} and the Walecka model \cite{Walecka:1974qa}. The VDF model utilizes interactions of the vector type, while the Walecka model employs both vector- and scalar-type interactions. Both models describe the nuclear liquid-gas phase transition, while the VDF model additionally describes a conjectured high-density, high-temperature phase transition modeling the QGP phase transition. In this work, the QGP-like phase transition is chosen to exhibit a CP at $T_c=100\ \txt{MeV}$ and $n_c=3n_0$, with the $T=0$ boundaries of the spinodal region in the $T$-$n_B$ plane given by $n_{B, \txt{left spinodal} } (T=0) \equiv \eta_L=2.5 n_0$ and $n_{B, \txt{right spinodal}} (T=0) \equiv\eta_R=3.32 n_0$, where $n_0=0.160\ \txt{fm}^{-3}$; this choice is arbitrary and serves as a plausible example.
We plot both sides of Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}) as functions of $\mu_B$ at a series of
temperatures in Fig.\ \ref{tests_of_formulas}. We note that the explored temperature range reaches beyond the region of the phase diagram where hadronic models plausibly describe matter created in heavy-ion collisions; nevertheless, it is instructive to test our approximations in this regime.
We use natural units in which the speed of light in vacuum is $c=1$. We note that in the VDF model, $c_s$ quickly becomes acausal for $\mu_B$ above the QGP-like phase transition. It is an expected behavior in models using interactions dependent on high powers of $n_B$ \cite{Zeldovich:1962emp}, and while not ideal, it does not affect the current analysis.
In all panels in Fig.\ \ref{tests_of_formulas}, the exact model calculations show expected
features as functions of $\mu_B$. In the upper left panel, showing both $c_{T}^2$ and $c_{\sigma}^2$ in the VDF model, at small $\mu_B$ we see a softening of the EOS due to the influence of the nuclear CP, followed by an increase at densities of the order of $n_0$, then a dive in $c_s^2$ caused by the QGP-like phase transition, and finally a steep rise for high values of $\mu_B$. In the upper right panel, showing $c_T^2$ and $c_{\sigma}^2$ in the Walecka model, we similarly observe a soft EOS at small $\mu_B$, while the value of $c_s^2$ goes asymptotically to 1 for large $\mu_B$. Additionally, for $T=200\ \txt{MeV}$, the Walecka model shows effects due to a phase transition in the nucleon-antinucleon plasma, occurring around $T\approx190\ \txt{MeV}$ and $n_B=0$; above this transition, the model describes an almost noninteracting gas of nearly massless nucleons \cite{Theis:1984qc}. The behavior of the curves in the lower panels, showing $c_T^2+\big(\frac{d\ln c_T^2}{d\ln n_B}\big)_T$, can be directly traced to the behavior of the curves in the upper panels. In particular, for the VDF model we observe strong divergences due to the softening of the EOS in the QGP-like phase transition region.
Comparing the exact results to the approximations, we see that, while Eq.\ (\ref{magic_equation_1}) is valid for $T\lesssim100\ \txt{MeV}$ and $\mu_B\gtrsim600\ \txt{MeV}$, it behaves poorly, both qualitatively and quantitatively, for $T$ and $\mu_B$ corresponding to regions of the phase diagram probed by moderately to highly energetic heavy-ion collisions (upper panels). On the other hand, the approximation introduced in Eq.\ (\ref{magic_equation_2}) is qualitatively valid for most of the probed $T$ and $\mu_B$, with the exception of regions characterized by $\mu_B\lesssim200\ \txt{MeV}$ (lower panels).
\begin{figure}[t]
\includegraphics[width = 0.99\columnwidth]{STAR+HADES_and_VDF_plots_combined_v2.jpg}\\
%
\caption{Comparison of the right-hand sides of Eq.\ (\ref{magic_equation_1}) (upper panel) and Eq.\ (\ref{magic_equation_2}) (lower panel) for experimental data (red triangles), ideal gas at the freeze-out (small gray circles), the VDF model at the freeze-out (light green stars), and the VDF model at a set of points chosen to reproduce the data (dark purple stars); exact results, that is, the left-hand sides of Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}), are shown for the two cases considered in the VDF model (green and purple circles). The data points for the matched VDF results (shown only for collisions at low energies, where using the model is justified) are chosen to reproduce experimental values of $1-\kappa_3\kappa_1/\kappa_2^2$ (see Fig.\ \ref{diagram}). We note that at $\sqrt{s}=2.4\ \txt{GeV}$, matching the value of $1-\kappa_3 \kappa_1/\kappa_2^2$ exactly is possible, but would place the matched point close to the nuclear liquid-gas CP, which we find unlikely.
}
\label{STAR_HADES_plots}
\end{figure}
\begin{figure}[t]
%
\includegraphics[width = 0.99\columnwidth]{Cumulants_diagram_1minusk3k1overk2k2_with_legend.jpg}
\caption{Contour plot of $1-\kappa_3\kappa_1/\kappa_2^2$ in the VDF model. Yellow and black lines correspond to the spinodal and coexistence lines, respectively; white contours signify regions where $1-\kappa_3\kappa_1/\kappa_2^2=1\pm0.03$. Light green stars denote experimentally measured freeze-out parameters $(T_{\txt{fo}},\mu_{\txt{fo}})$, while dark purple stars denote points where $1-\kappa_3\kappa_1/\kappa_2^2$, taken along lines informed by average phase diagram trajectories for STAR collision energies \cite{Shen:2020jwv}, matches the experimentally measured values for a given collision energy. The softening of the EOS, leading to negative values of $\big(\frac{d\ln c_T^2}{d\ln n_B}\big)_T$, occurs in two regions of the phase diagram, corresponding to the ordinary nuclear matter phase transition and to the conjectured QGP-like phase transition.
}
\label{diagram}
\end{figure}
\section{Experimental data and interpretation}
We consider cumulants of the net proton number and chemical freeze-out parameters, $(T_{\txt{fo}},\mu_{\txt{fo}})$, in collisions at 0-5\% centrality, determined by the solenoidal tracker at RHIC (STAR) \cite{Abdallah:2021fzj} and high acceptance dielectron spectrometer (HADES) \cite{Adamczewski-Musch:2020slf, HADES_MLorentz_talk} experiments, and we use them to plot Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}) (red triangles, upper and lower panel in Fig.\ \ref{STAR_HADES_plots}, respectively) against $\mu_B$.
Based on the previous section, we trust the results presented in the upper panel of Fig.\ \ref{STAR_HADES_plots}, approximating $c_T^2$, only for the lowest collision energy, $\sqrt{s}=2.4\ \txt{GeV}$ from the HADES experiment. Here, $c_T^2$ as obtained from Eq.\ (\ref{magic_equation_1}) is small: less than half of the ideal gas value. At the same time, the value of $1-\kappa_3\kappa_1/\kappa_2^2$ (shown in the lower panel in Fig.\ \ref{STAR_HADES_plots}), which we assume is dominated by $\big(\frac{d\ln c_T^2}{d\ln n_B}\big)_T$, drops with decreasing collision energy to reach a minimum at the lowest STAR point, $\sqrt{s}=7.7\ \txt{GeV}$, and then steeply rises for the HADES point. This could mean that in $\sqrt{s} = 7.7\ \txt{GeV}$ collisions, $c_T^2$ is approximately constant as a function of $n_B$, while in $\sqrt{s} = 2.4\ \txt{GeV}$ collisions the matter is characterized by a small $c^2_T$ which nevertheless has a large slope as a function of $n_B$.
To further understand this behavior, we study the dependence of $1-\kappa_3\kappa_1/\kappa_2^2$ on $\mu_{B}$ and $T$ within the VDF model, shown in Fig.\ \ref{diagram}. Comparing model results to experiment requires choosing at which $T$ and $\mu_B$ to take values of $1-\kappa_3 \kappa_1/\kappa_2^2$. A natural choice is to use values at $(T_{\txt{fo}},\mu_{\txt{fo}})$ (light green stars), but as shown in the lower panel of Fig.\ \ref{STAR_HADES_plots}, these values do not lead to an agreement with experimental data, with the biggest discrepancy for the HADES point. (We note that $(T_{\txt{fo}},\mu_{\txt{fo}})$ are established with hadron interactions neglected, and the degree to which this affects our results may vary across the phase diagram.) However, critical fluctuations exhibit a large relaxation time \cite{Berdnikov:1999ph,Stephanov:2017ghc,Du:2020bxp}, and their measured values could be affected by stages of the collision preceding the freeze-out. With this insight and taking guidance from average phase diagram trajectories of hybrid simulations of heavy-ion collisions \cite{Shen:2020jwv}, we consider $1-\kappa_3\kappa_1/\kappa_2^2$ at slightly earlier stages of the evolution. In this way we obtain values of $1-\kappa_3\kappa_1/\kappa_2^2$ (dark purple stars) that reproduce the experimental results for a given collision energy (lower panel in Fig.\ \ref{STAR_HADES_plots}); here the exception is the HADES point, for which we prioritized choosing a point in a reasonable vicinity of the measured freeze-out over obtaining a value equal to the experimental data. Comparing to the exact model results for $c_T^2$ and $c_T^2 + \big(\frac{d\ln c_T^2}{d\ln n_B}\big)_T$, also displayed in Fig.\ \ref{STAR_HADES_plots}, as well as to the upper left panel of Fig.\ \ref{tests_of_formulas}, we can confirm that at the point reproducing the experimental value of $1-\kappa_3\kappa_1/\kappa_2^2$ for the lowest STAR energy, $c_T^2$ is nearly constant as a function of $\mu_B$ (thick short-dashed line, $T=150\ \txt{MeV}$, at $\mu_B\approx410\ \txt{MeV}$ in Fig.\ \ref{tests_of_formulas}), while at the point reproducing the result for the HADES energy, $c_T^2$ increases sharply with $\mu_B$ (thick solid line, $T=50\ \txt{MeV}$, at $\mu_B\approx850\ \txt{MeV}$ in Fig.\ \ref{tests_of_formulas}).
Naturally, the choice of $T$ and $\mu_B$ at which we compare model calculations with STAR and HADES cumulant data is driven by the wish to match the experimental results, and it serves mainly to show that baryon number cumulants measured in heavy-ion collisions can be connected to the speed of sound in hot and dense nuclear matter. Whether values of higher order cumulants are indeed significantly affected by stages of the evolution preceding the freeze-out needs to be further investigated (for recent developments, see \cite{Nahrgang:2018afz,Jiang:2017sni}). Moreover, while experiments measure proton number cumulants, the VDF model provides baryon number cumulants, putting more strain on our interpretation. Baryon number conservation should likewise be important \cite{Bzdak:2012an,Vovchenko:2020gne}. Finally, our model results may not be applicable in regions of the phase diagram where quarks and gluons become increasingly relevant.
Nonetheless, hadronic models are well-justified for describing low-energy collisions whose evolution is dominated by the hadronic stage. The comparison between the experimental data and the VDF model suggests that collisions at the lowest STAR and HADES energies may be probing regions of the phase diagram where the cumulants of the baryon number tell us more about hadronic physics than the QCD CP. In particular, the change in the sign of $\kappa_3$, predicted to take place in the vicinity of a critical point \cite{Asakawa:2009aj} and apparent in the HADES data (see lower panel of Fig.\ \ref{STAR_HADES_plots}), may mark the region of the phase diagram affected by the nuclear liquid-gas phase transition. If this is true, it may be worthwhile to study the cumulants at even lower collision energies, starting from $0.1\ \txt{GeV}$ projectile kinetic energy, and obtain the speed of sound around the nuclear liquid-gas CP. Conversely, at higher energies it could be possible to use collisions at different centralities and different rapidity windows to estimate the neglected terms in Eqs.\ (\ref{magic_equation_1}) and (\ref{magic_equation_2}), and obtain a stronger estimate for the speed of sound in the respective regions of the phase diagram.
\section{Summary and conclusions}
In this work, we use cumulants of the baryon number distribution to estimate the isothermal speed of sound squared and its logarithmic derivative with respect to the baryon number density. This result provides a new method for obtaining information about fundamental properties of nuclear matter studied in heavy-ion collisions, with consequences for both the search for the QCD CP and neutron star studies. While the approximations and the model comparison we considered apply to experiments at low energies, the approach itself can be used at any collision energy provided that measurements of cumulants of baryon number distribution as well as their temperature dependence are available. Further studies of effects due to dynamics, in particular using state-of-the-art simulations, will be absolutely essential in determining the extent to which the proposed method provides a reliable extraction of sound velocities and their derivatives.
{\hspace{10mm}}
\section{Acknowledgements}
A.S.\ thanks Chun Shen for providing average phase diagram trajectories of hybrid heavy-ion collision simulations at RHIC Beam Energy Scan energies.
A.S.\ and V.K.\ received support through the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics, under contract no.\
DE-AC02-05CH11231231 and received support within the framework of the
Beam Energy Scan Theory (BEST) Topical Collaboration. D.O.\ and L.M.\ were supported by the U.S.\ DOE under Grant No.\ DE-FG02-00ER4113.
|
{
"timestamp": "2021-07-28T02:01:13",
"yymm": "2103",
"arxiv_id": "2103.07365",
"language": "en",
"url": "https://arxiv.org/abs/2103.07365"
}
|
"\\section{Introduction}\n\nFood fraud has become a issue on a global scale for producers, consumer(...TRUNCATED)
| {"timestamp":"2021-03-15T01:17:41","yymm":"2103","arxiv_id":"2103.07315","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\nThis work is on the interface between Computer Science in the area of qu(...TRUNCATED)
| {"timestamp":"2021-09-29T02:01:03","yymm":"2103","arxiv_id":"2103.07264","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\\label{sec:introduction}\n\nModern technologies such as Internet of Things (...TRUNCATED)
| {"timestamp":"2021-10-20T02:04:03","yymm":"2103","arxiv_id":"2103.07368","language":"en","url":"http(...TRUNCATED)
|
"\\section{Motivation and significance}\n\\label{}\nDespite the wide use of open-source libraries fo(...TRUNCATED)
| {"timestamp":"2021-03-15T01:18:01","yymm":"2103","arxiv_id":"2103.07329","language":"en","url":"http(...TRUNCATED)
|
"\n\n\n\\section*{Acknowledgment}\r\nThe authors would like to thank Jasan Zughaibi, Michael Egli, M(...TRUNCATED)
| {"timestamp":"2021-03-15T01:20:21","yymm":"2103","arxiv_id":"2103.07379","language":"en","url":"http(...TRUNCATED)
|
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