prompt stringlengths 0 256 | answer stringlengths 1 256 |
|---|---|
nd into several frequency
bins to capture the variation in sensitivity and velocity resolution
(which is in the range 7.8\,km\,s$^{-1}$ at 711.5\,MHz to
5.5\,km\,s$^{-1}$ at 1015.5\,MHz).
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{z | dist.pdf}
\caption{The distribution of CENSORS sources (\citealt{Brookes:2008})
brighter than 10\,mJy beyond a given redshift $z$. The red line
denotes the cumulative distribution calculated from the parametric
model of \citet{deZotti:2010}.}\label{f |
igure:zdist}
\end{figure}
In order to simulate a realistic survey of the southern sky we select
all radio sources south of $\delta = +10\degr$ from catalogues of the
National Radio Astronomy Observatory Very Large Array Sky Survey
(NVSS, $\nu = 1.4$\,GHz, | $S_{\rm src} \gtrsim 2.5$\,mJy;
\citealt{Condon:1998}), the Sydney University Molonglo SkySurvey ($\nu
= 843$\,MHz, $S_{\rm src} \gtrsim 10$\,mJy; \citealt{Mauch:2003}) and
the second epoch Molonglo Galactic Plane Survey ($\nu = 843$\,MHz,
$S_{\rm src} \g |
trsim 10$\,mJy; \citealt{Murphy:2007}). The source flux
densities, used to calculate the optical depth limit in
\autoref{equation:optical_depth_limit}, are estimated at the centre of
each frequency bin by extrapolating from the catalogue values and
assumin | g a canonical spectral index of $\alpha = -0.7$. In
\autoref{figure:nsources_flux}, we show the resulting cumulative
distribution of radio sources in our sample as a function of flux
density across the band.
\begin{figure}
\centering
\includegraphics[widt |
h=0.475\textwidth]{nsources_tau.pdf}
\caption{The number of sources in our simulated ASKAP survey with a
21\,cm opacity sensitivity greater than or equal to
$\tau_{5\sigma}$, as defined by
\autoref{equation:optical_depth_limit}. The grey region enclo | ses
opacity sensitivities for the 711.5 - 1011.5\,MHz band. Random
samples for the line FWHM and covering factor were drawn from the
distributions shown in \autoref{figure:width_dist} and
\autoref{figure:covfact_dist}.}\label{figure:nsources_tau}
\ |
end{figure}
For any given sight-line, the redshift interval over which absorption
may be detected is dependent upon the distance to the continuum
source. The lack of accurate spectroscopic redshift measurements for
most radio sources over the sky necess | itates the use of a statistical
approach based on a model for the source redshift distribution. We
therefore apply a statistical weighting to each comoving path element
$\delta{X}(z)$ such that the expected number of absorber detections is
now given by
\be |
gin{equation}\label{equation:weighted_sum_number}
\mu = \iint{f(N_{\rm HI},X)\,\mathcal{F}_{\rm src}(z^{\prime} \geq z)\,\mathrm{d}X\,\mathrm{d}N_{\rm HI}},
\end{equation}
where
\begin{equation}
\mathcal{F}_{\rm src}(z^{\prime} \geq z) = {\int_{z}^{\ | infty} \mathcal{N}_{\rm src}(z^{\prime})\mathrm{d}z^{\prime}\over\int_{0}^{\infty}\mathcal{N}_{\rm src} (z^{\prime})\mathrm{d}z^{\prime}},
\end{equation}
and $\mathcal{N}_{\rm src}(z)$ is the number of radio sources as a
function of redshift. To estimate $ |
\mathcal{N}_{\rm src}(z)$ we use
the Combined EIS-NVSS Survey Of Radio Sources (CENSORS;
\citealt{Brookes:2008}), which forms a complete sample of radio
sources brighter than 7.2\,mJy at 1.4\,GHz with spectroscopic
redshifts out to cosmological distances. | In \autoref{figure:zdist} we
show the distribution of CENSORS sources brighter than 10\,mJy beyond
a given redshift $z$, and the corresponding analytical function
derived from the model fit of \citet{deZotti:2010}, given by
\begin{equation}
\mathcal{N}_{ |
\rm src}(z) \approx 1.29 + 32.37z - 32.89z^{2} + 11.13z^{3} - 1.25z^{4},
\end{equation}
which we use in our analysis. For the redshifts spanned by our
simulated ASKAP survey, the fraction of background sources evolves
from 87\,per\,cent at $z = 0.4$ to 53\ | ,per\,cent at $z = 1.0$.
We assume that this redshift distribution applies to any sight-line
irrespective of the continuum flux density. However, this assumption
is only true if the source population in the target sample evolves
such that the effect of di |
stance is nullified by an increase in
luminosity. Given this criterion, and the sensitivity of our simulated
survey, we limit our sample to sources with flux densities between 10
and 1000\,mJy, which are dominated by the rapidly evolving population
of high | -excitation radio galaxies and quasars
(e.g. \citealt{Jackson:1999, Best:2012, Best:2014, Pracy:2016}) and
for which the redshift distribution is known to be almost independent
of flux density (e.g. \citealt{Condon:1984, Condon:1998}). In
\autoref{figure:n |
sources_tau}, we show the number of sources from this
sub-sample as a function of opacity sensitivity [as defined by
\autoref{equation:optical_depth_limit}], drawing random samples of the
line FWHM and covering factor from the distributions shown in
\autor | ef{figure:width_dist} and \autoref{figure:covfact_dist}. There
are approximately 190\,000 sightlines with sufficient sensitivity to
detect absorption of optical depth greater than $\tau_{5\sigma}
\approx 1.0$ and 25\,000 sensitive to optical depths greater |
than
$\tau_{5\sigma} \approx 0.1$. Since this distribution converges at
optical depth sensitivities greater than $\tau_{5\sigma} \approx 5$,
the population of sources fainter than 10\,mJy, which are excluded
from our simulated ASKAP survey, would not sign | ificantly contribute to
further detections of absorption. Similarly, while sources brighter
than 1\,Jy are good probes of low-column-density \mbox{H\,{\sc i}}
gas, they do not constitute a sufficiently large enough population to
significantly affect the to |
tal number of absorber detections expected
in the survey and can also be safely excluded.
Based on these assumptions, we can estimate the number of absorbers we
would expect to detect in our survey with ASKAP as a function of spin
temperature. In \autoref | {figure:ndetections_nhi}, we show the expected
detection yield as a cumulative function of column density. We show
results for two scenarios where the spin temperature is fixed at a
single value of either 100 or 1000\,K, and the line FWHM and covering
fact |
ors are drawn from the random distributions shown in
\autoref{figure:width_dist} and \autoref{figure:covfact_dist}. We find
that for both these cases the expected number of detections is not
sensitive to column densities below the DLA definition of $N_{\rm | HI}
= 2\times 10^{20}$\,cm$^{-2}$. We also show in
\autoref{figure:ndetections_total} the expected total detection yield
(integrated over all \mbox{H\,{\sc i}} column densities) as a function
of a single spin temperature $T_\mathrm{spin}$ and line FWHM
$\ |
Delta{v}_\mathrm{50}$. We find that for typical spin temperatures of
a few hundred kelvin (consistent with the typical fraction of CNM
observed in the local Universe) and a line FWHM of approximately
$20$\,km\,s$^{-1}$, a wide-field 21\,cm survey with ASKA | P is expected
to yield $\sim 1000$ detections. However, even moderate evolution to a
higher spin temperature in the DLA population should see significant
reduction in the detection yield from this survey.
\section{Inferring the average spin
temperature} |
\label{section:spin_temp}
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{ndetections_nhi.pdf}
\caption{The expected number of absorber detections (as a cumulative
function of column density) in our simulated ASKAP survey. We show
tw | o scenarios for a single spin temperature $T_{\rm spin} = 100$ and
$1000$\,K, where we have drawn random samples for the line FWHM and
covering factor from the distributions shown in
\autoref{figure:width_dist} and \autoref{figure:covfact_dist}. In
|
both cases we find that the expected number of detections is not
sensitive to column densities below $N_{\rm HI} = 2 \times
10^{20}$\,cm$^{-2}$, indicating that such a survey will only be
sensitive to DLA systems.}\label{figure:ndetections_nhi}
\end | {figure}
We cannot directly measure the spin temperatures of individual systems
without additional data from either 21\,cm emission or Lyman-$\alpha$
absorption. However, from \autoref{figure:ndetections_total} it is
evident that the total number of abso |
rbing systems expected to be
detected with a reasonably large 21\,cm survey is strongly dependent
on the assumed value for the spin temperature. Therefore, by comparing
the actual survey yield with that expected from the known
\mbox{H\,{\sc i}} distributio | n, we can infer the average spin
temperature of the atomic gas within the DLA population for a given
redshift interval.
Assuming that the total number of detections follows a Poisson
distribution, the probability of detecting $\mathcal{N}$ intervening
abs |
orbing systems is given by
\begin{equation} p(\mathcal{N}|\overline{\mu}) =
{\overline{\mu}^{\mathcal{N}} \over \mathcal{N}!}
\mathrm{e}^{-\overline{\mu}},
\end{equation}
where $\overline{\mu}$ is the expected total number of detections
given by the in | tegral
\begin{equation}
\overline{\mu} = \iiint{\mu(T_{\rm spin},\Delta{v}_{50},c_{\rm f})\rho(T_{\rm spin},\Delta{v}_{50},c_{\rm f})\mathrm{d}T_{\rm spin}\mathrm{d}\Delta{v}_{50}\mathrm{d}c_{\rm f}},
\end{equation}
and $\rho$ is the distribution of syst |
ems as a function of spin
temperature, line FWHM and covering factor. We assume that all three
of these variables are independent\footnote{In the case where thermal
broadening contributes significantly to the velocity dispersion, and
the spin temperatu | re is dominated by collisional excitation, the
assumption that these are independent may no longer hold. However,
given that collisional excitation dominates in the CNM, where
$T_{\rm spin} \sim 100\,\mathrm{K}$, the velocity dispersion would
have |
to satisfy $\Delta{v}_{50} \ll 10$\,km\,s$^{-1}$ (c.f. the
distribution shown in \autoref{figure:width_dist}).} so that $\rho$
factorizes into functions of each. We then marginalise over the
covering factor and line width distributions shown in
\autoref{ | figure:width_dist} and \autoref{figure:covfact_dist} so that
the expression for $\overline{\mu}$ reduces to
\begin{equation}\label{equation:harmonic_spin_temp}
\overline{\mu} = \int{\mu(T_{\rm spin})\rho(T_{\rm spin})\mathrm{d}T_{\rm spin}} = \mu(\overli |
ne{T}_{\rm spin}),
\end{equation}
where $\overline{T}_{\rm spin}$ is the harmonic mean of the unknown
spin temperature distribution, weighted by column density. This is
analogous to the spin temperature inferred from the detection of
absorption in a single | intervening galaxy averaged over several
gaseous components at different temperatures
(e.g. \citealt{Carilli:1996}).
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{ndetections_total.pdf}
\caption{The expected total number of detections |
in our simulated
ASKAP survey, as a function of a single spin temperature ($T_{\rm
spin}$) and line FWHM ($\Delta{v}_{50}$). The vertical dotted
lines enclose the velocity resolution across the observed frequency
band. We draw random samples for | the covering factor from a uniform
distribution between 0 and 1 (as shown in
\autoref{figure:covfact_dist}). The contours are truncated at $\mu <
1$ for clarity.}\label{figure:ndetections_total}
\end{figure}
In the event of the survey yielding $\ma |
thcal{N}$ detections, we can
calculate the posterior probability density of $\overline{T}_{\rm
spin}$ using the following relationship between conditional
probabilities
\begin{equation}
p(\overline{T}_{\rm spin}|\mathcal{N}) = {p(\mathcal{N}|\overline{ | T}_{\rm spin})p(\overline{T}_{\rm
spin})\over p(\mathcal{N})},
\end{equation}
where $p(\overline{T}_{\rm spin})$ is our prior probability density
for $\overline{T}_{\rm spin}$ and $p(\mathcal{N})$ is the marginal
probability of the number of detectio |
ns, which can be treated as a
normalizing constant. The minimally informative Jeffreys prior for the
mean value $\mu$ of a Poisson distribution is $1/\sqrt{\mu}$
(\citealt{Jeffreys:1946})\footnote{A suitable alternative choice for
the prior is the standa | rd scale-invariant form $1/\mu$
(e.g. \citealt{Jeffreys:1961, Novick:1965, Villegas:1977}). While we
find that our choice of non-informative prior has negligible effect
on the spin temperature posterior for the full \mbox{H\,{\sc i}}
absorption sur |
vey, as one would expect this choice becomes more
important for smaller surveys. For the early-science 1000\,deg$^{2}$
survey discussed in \autoref{section:tspin_results} we find that the
difference in these two priors produces a $\sim 2$ to 20\,per\ | ,cent
effect in the posterior. However, in all cases considered this
change is smaller than the 68.3\,per\,cent credible interval spanned
by the posterior.}. From \autoref{equation:harmonic_spin_temp} it
therefore follows that a suitable form for the |
non-informative spin
temperature prior is $p(\overline{T}_{\rm spin}) =
1/\sqrt{\overline{\mu}}$, so that
\begin{equation}\label{equation:tspin_prob}
p(\overline{T}_{\rm spin}|\mathcal{N}) = C^{-1}\,{\overline{\mu}^{(\mathcal{N}-1/2)} \over \mathcal{N}! | } \mathrm{e}^{-\overline{\mu}},
\end{equation}
where the distribution is normalised to unit total probability by
evaluating the integral
\begin{equation}
C = \int{{\overline{\mu}^{(\mathcal{N}-1/2)} \over \mathcal{N}!} \mathrm{e}^{-\overline{\mu}}}\,\ma |
thrm{d}\overline{T}_{\rm spin}.
\end{equation}
The probabilistic relationship given by \autoref{equation:tspin_prob}
and the expected detection yield derived in
\autoref{section:all_sky_survey} can be used as a frame-work for
inferring the harmonic-mean s | pin temperature using the results of any
homogeneous 21-cm survey. We have assumed that we can accurately
distinguish intervening absorbing systems from those associated with
the host galaxy of the radio source. However, any 21-cm survey will be
accompanie |
d by follow-up observations, at optical and sub-mm
wavelengths, which will aid identification. Furthermore, future
implementation of probabilistic techniques to either use photometric
redshift information or distinguish between line profiles should
provide | further disambiguation.
Of course we have not yet accounted for any error in our estimate of
$\overline{\mu}$, which will increase our uncertainty in
$\overline{T}_\mathrm{spin}$. In the following section we discuss
these possible sources of error and th |
eir effect on the result.
\section{Sources of error}\label{section:errors}
Our estimate of the expected number of 21\,cm absorbers is dependent
upon several distributions describing the properties of the foreground
absorbing gas and the background source | distribution. For future
large-scale 21\,cm surveys, the accuracy to which we can infer the
harmonic mean of the spin temperature distribution will eventually be
limited by the accuracy to which we can measure these other
distributions. In this section, w |
e describe these errors and their
propagation through to the estimate of $\overline{T}_\mathrm{spin}$,
summarizing our results in \autoref{table:tspin_uncertainties}.
\subsection{The covering factor}\label{section:covering_factor}
\subsubsection{Deviatio | n from a uniform distribution between 0 and 1}
The fraction $c_{\rm f}$ by which the foreground gas subtends the
background radiation source is difficult to measure directly and is
thereby a significant source of error for 21\,cm absorption
surveys. In th |
is work, we have assumed a uniform distribution for
$c_{\rm f}$, taking random values between 0 and 1. In
\autoref{section:expected_number}, we tested this assumption by
comparing it with the distribution of flux density core fractions in a
sample of 37 qu | asars, used by \cite{Kanekar:2014a} as a proxy for the
covering factor. By carrying out a two-tailed KS test, we found some
evidence (at the 0.05 level) that this quasar sample was inconsistent
with our assumption of a uniform distribution between 0 and
1. |
Noticeably there seems to be an under-representation of quasars in
the Kanekar et al. sample with estimated $c_{\rm f} \lesssim 0.2$. In
the low optical depth limit, the detection rate is dependent on the
ratio of spin temperature to covering factor, in w | hich case a
fractional deviation in $c_{\rm f}$ will propagate as an equal
fractional deviation in $\overline{T}_\mathrm{spin}$. Based on the
difference seen in the covering factor distribution of the Kanekar et
al. sample and the uniform distribution, we |
assume that the spin
temperature can deviate by as much as $\pm$10\,per\,cent.
\subsubsection{Evolution with redshift}
We also consider that the covering factor distribution may evolve with
redshift, which would mimic a perceived evolution in the average | spin
temperature. Such an effect was proposed by \cite{Curran:2006b} and
\cite{Curran:2012b}, who claimed that the relative change in
angular-scale behaviour of absorbers and radio sources between low-
and high-redshift samples could explain the apparent |
evolution of the
spin temperature found by \cite{Kanekar:2003b}. To test for this
effect in their larger DLA sample, \cite{Kanekar:2014a} considered a
sub-sample at redshifts greater than $z = 1$, for which the relative
evolution of the absorber and source | angular sizes should be
minimal. While the significance of their result was reduced by
removing the lower redshift absorbers from their sample, they still
found a difference at $3.5\,\sigma$ significance between spin
temperature distributions in the two D |
LA sub-samples separated by a
median redshift of $z = 2.683$.
Future surveys with ASKAP and the other SKA pathfinders will search
for \mbox{H\,{\sc i}} absorption at intermediate redshifts ($z \sim
1$), where the relative evolution of the absorber and sou | rce angular
sizes is expected to be more significant than for the higher redshift
DLA sample considered by \cite{Kanekar:2014a}. We therefore consider
the potential effect of this cosmological evolution on the inferred
value of $\overline{T}_\mathrm{spin}$ |
. We approximate the covering
factor using the following model of \cite{Curran:2006b}
\begin{equation}\label{equation:covering_factor}
c_{f} \approx
\begin{cases}
\left({\theta_{\rm abs}\over \theta_{\rm src}}\right)^{2}, & \text{if}\ \theta_{\rm abs} | < \theta_{\rm src}, \\
1, & \text{otherwise},
\end{cases}
\end{equation}
where $\theta_{\rm abs}$ and $\theta_{\rm src}$ are the angular sizes
of the absorber and background source, respectively. Under the
small-angle approximation $\theta_{\rm abs} \app |
rox {d_{\rm
abs}/D_{\rm abs}}$ and $\theta_{\rm abs} \approx {d_{\rm
src}/D_{\rm src}}$, where $d_{\rm abs}$ and $D_{\rm abs}$ are the
linear size and angular diameter distance of the absorber, and
likewise $d_{\rm src}$ and $D_{\rm src}$ are the l | inear size and
angular diameter distance of the background source. Assuming that the
ratio $d_{\rm abs}/d_{\rm src}$ is randomly distributed and
independent of redshift, any evolution in the covering factor is
therefore dominated by relative changes in the |
angular diameter
distances. We calculate the expected angular diameter distance ratio
at a redshift $z$ by
\begin{equation}
\left\langle{D_{\rm abs}\over D_{\rm src}}\right\rangle_{z} = D_{\rm abs}(z){\int_{z}^{\infty} \mathcal{N}_{\rm src}(z^{\prime})D | _{\rm src}(z^{\prime})^{-1}\mathrm{d}z^{\prime}\over\int_{z}^{\infty}\mathcal{N}_{\rm src} (z^{\prime})\mathrm{d}z^{\prime}},
\end{equation}
which, for the source redshift distribution model given by
\cite{deZotti:2010}, evolves from 0.7 at $z = 0.4$ to 1. |
0 at $z = 1.0$
(see \autoref{figure:dang_ratio}). We note that this is consistent
with the behaviour measured by \cite{Curran:2012b} for the total
sample of DLAs observed at 21\,cm wavelengths. By applying this as a
correction to the otherwise uniformly d | istributed covering factor
(using \autoref{equation:covering_factor}), we find that the inferred
value of $\overline{T}_{\rm spin}$ systematically increases by
approximately 30 per\,cent.
\begin{figure}
\centering
\includegraphics[width=0.475\textwidth]{d |
ang_ratio.pdf}
\caption{The expected redshift behaviour of $D_{\rm abs}/D_{\rm src}$
based on the \citet{deZotti:2010} model for the radio source redshift
distribution.}\label{figure:dang_ratio}
\end{figure}
\subsection{The $\bmath{N_{\rm HI}}$ freque | ncy distribution}
\subsubsection{Uncertainty in the measurement of $f(N_{\rm HI},X)$}
We assume that $f(N_{\rm HI}, X)$ is relatively well understood as a
function of redshift by interpolating between model gamma functions
fitted to the distributions at |
$z = 0$ and $3$. However, these
distributions were measured from finite samples of galaxies, which of
course have associated uncertainties that need to be considered. In
the case of the data presented by \cite{Zwaan:2005} and
\cite{Noterdaeme:2009}, both h | ave typical measurement uncertainties in
$f(N_{\rm HI}, X)$ of approximately 10\,per\,cent over the range of
column densities for which our simulated ASKAP survey is sensitive
(see \autoref{figure:ndetections_nhi}). This will propagate as a
10\,per\,cent f |
ractional error in the expected number of absorber
detections, and contribute a similar percentage uncertainty in the
inferred average spin temperature.
\subsubsection{Correcting for 21\,cm self-absorption}
In the local Universe, \cite{Braun:2012} showed | that self-absorption
from opaque \mbox{H\,{\sc i}} clouds identified in high-resolution
images of the Local Group galaxies M31, M33 and the Large Magellanic
Cloud may necessitate a correction to the local atomic mass density of
up to 30\,per\,cent. Althou |
gh it is not yet clear whether this small
sample of Local Group galaxies is representative of the low-redshift
population, it is useful to understand how this effect might propagate
through to our average spin temperature measurement. We therefore
replace | the gamma-function parametrization of the local $f(N_{\rm
HI})$ given by \cite{Zwaan:2005} with the non-parametric values
given in table\,2 of \cite{Braun:2012}, and recalculate
$\overline{T}_{\rm spin}$. For an all-sky survey with the full
36-antenna AS |
KAP we find that $\overline{T}_{\rm spin}$ increases by
$\sim$30 for 100 detections and $\sim$10\,per\,cent for 1000
detections. Note that the correction increases for low numbers of
detections, which are dominated by the highest column density systems.
\ | subsubsection{Dust obscuration bias in optically-selected DLAs}
At higher redshifts, it is possible that the number density of
optically-selected DLAs could be significantly underestimated as a
result of dust obscuration of the background quasar
(\citealt |
{Ostriker:1984}). This would cause a reduction in the
$f(\mbox{H\,{\sc i}}, X)$ measured from optical surveys, thereby
significantly underestimating the expected number of intervening
21\,cm absorbers at high redshifts. The issue is further compounded by
t | he expectation that the highest column density DLAs ($N_{\rm HI}
\gtrsim 10^{21}$\,cm$^{-2}$), for which future wide-field 21\,cm
surveys are most sensitive (see \autoref{figure:ndetections_nhi}), may
contain more dust than their less-dense counterparts.
|
This conclusion was supported by early analyses of the existing quasar
surveys at that time (e.g. \citealt{Fall:1993}), which indicated that
up to 70\,per\,cent of quasars could be missing from optical surveys
through the effect of dust obscuration, albeit | with large
uncertainties. However, subsequent optical and infrared observations
of radio-selected quasars (e.g. \citealt{Ellison:2001};
\citealt*{Ellison:2005}; \citealt{Jorgenson:2006}), which are free of
the potential selection biases associated with th |
ese optical surveys,
found that the severity of this issue was substantially over-estimated
and that there was minimal evidence in support of a correlation
between the presence of DLAs and dust reddening. Furthermore, the
\mbox{H\,{\sc i}} column density f | requency distribution measured by
\cite{Jorgenson:2006} was found to be consistent with the
optically-determined gamma-function parametrization of
\cite{Prochaska:2005}, with no evidence of DLA systems missing from
the SDSS sample at a sensitivity of $N_{\ |
rm HI} \lesssim 5 \times
10^{21}$\,cm$^{-2}$. Although radio-selected surveys of quasars are
free of the selection biases associated with optical surveys, they do
typically suffer from smaller sample sizes and are therefore less
sensitive to the rarer DLAs | with the highest column densities.
Another approach is to directly test whether optically-selected
quasars with intervening DLAs, selected from the SDSS sample, are
systematically more dust reddened than a control sample of non-DLA
quasars. Comparisons i |
n the literature are based on several different
colour indicators, which include the spectral index
(e.g. \citealt{Murphy:2004,Murphy:2016}), spectral stacking
(e.g. \citealt{Frank:2010, Khare:2012}) and direct photometry
(e.g. \citealt*{Vladilo:2008}; \ci | tealt{Fukugita:2015}). The current
status of these efforts is summarized by \citet{Murphy:2016}, showing
broad support for a missing DLA population at the level of
$\sim$5\,per\,cent but highlighting that tension still exists between
different dust measure |
ments. No substantial evidence has yet been
found to support a correlation between the dust reddening and
\mbox{H\,{\sc i}} column density in these optically selected DLA
surveys (e.g. \citealt{Vladilo:2008, Khare:2012, Murphy:2016}).
In an attempt to rec | oncile the differences and myriad biases
associated with these techniques, \cite{Pontzen:2009} carried out a
statistically-robust meta-analysis of the available optical and radio
data, using a Bayesian parameter estimation approach to model the dust
as a f |
unction of column density and metallicity. They found that the
expected fraction of DLAs missing from optical surveys is
7\,per\,cent, with fewer than 28\,per\,cent missing at 3\,$\sigma$
confidence. Based on this body of work we therefore assume that
appr | oximately 10\,per\,cent of DLAs are missing from the SDSS sample
of \cite{Noterdaeme:2009} and consider the affect on our estimate of
$\overline{T}_{\rm spin}$. We further assume that there is no
dependance on column density, an assumption which is support |
ed by the
aforementioned observational data for the range of column densities to
which our 21\,cm survey is sensitive. We find that increasing the
high-redshift column density frequency distribution by 10\,per\,cent
introduces a systematic increase of appr | oximately 3\,per\,cent in the
expected number of detections for the redshifts covered by our ASKAP
surveys. We note that this error will increase significantly for
21\,cm surveys at higher redshifts where the optically derived
$f(\mbox{H\,{\sc i}}, X)$ dom |
inates the calculation of the expected
detection rate.
\subsection{The radio source background}
As described in \autoref{section:all_sky_survey}, we weight the
comoving path-length for each sight-line by a statistical redshift
distribution in order to ac | count for evolution in the radio source
background. We use the parametric model of \cite{deZotti:2010}, which
is derived from fitting the measured redshifts of \cite{Brookes:2008}
for CENSORS sources brighter than 10\,mJy, and assume that this
applies to a |
ll sources in the range 10 - 1000\,mJy. In
\autoref{figure:zdist}, we show the cumulative distribution of sources
located behind a given redshift and the associated measurement
uncertainty given by the errorbars. For the intermediate redshifts
covered by t | he ASKAP survey, the fractional uncertainty in this
distribution increases from $\sigma_{\mathcal{F}_{\rm
src}}/\mathcal{F}_{\rm src} \approx 3.5$ to 8\,per\,cent between
$z = 0.4$ and 1.0, which propagates through to a similar fractional
uncertainty i |
n $\overline{T}_{\rm spin}$. However, for higher
redshifts this fractional uncertainty increases rapidly at $z > 2$, to
more than 50\,per\,cent at $z = 3$, reflecting the paucity of optical
spectroscopic data for the high-redshift radio source
population. | Understanding how the radio source population is
distributed at lower flux densities and at higher redshifts is
therefore a concern for the future 21\,cm absorption surveys
undertaken with the SKA mid- and low-frequency telescopes (see
\citealt{Kanekar:200 |
4} and \citealt*{Morganti:2015} for reviews).
\begin{table}
\begin{threeparttable}
\caption{An account of errors in our estimate of $\overline{T}_{\rm
spin}$ due to the accuracy to which we can determine the
expected number of absorber
| detections.}\label{table:tspin_uncertainties}
\begin{tabular}{l@{\hspace{0.05in}}l@{\hspace{0.05in}}l@{\hspace{0.05in}}l@{\hspace{0.05in}}l}
\hline
& Source of error & $\mathrm{err}(\overline{T}_{\rm spin})$ & Refs. \\
& & [per\,cent] & |
\\
\hline
Covering factor & Distribution uncertainty & $\pm10$ & $a$ \\
Covering factor & Systematic evolution & +30 & $a$, $b$\\
$f(N_{\rm HI}, X)$ & Measurement uncertainty & $\pm10$ & $c, d$\\
Low-$z$ $f(N_{\rm HI}, X)$ & Systemati | c self-absorption & $+(10-30)$ & $e$ \\
High-$z$ $f(N_{\rm HI}, X)$ & Systematic dust-obscuration & $+3$ & $f$, $g$ \\
$\mathcal{F}_{\rm src}(z^{\prime} \geq z)$ & Measurement uncertainty & $\pm 5$ & $h$, $i$ \\
\hline
\end{tabular}
\be |
gin{tablenotes}
\item[] References: $^{a}${\citet{Kanekar:2014a}},
$^{b}${\citet{Curran:2012b}}, $^{c}${\citet{Zwaan:2005}},
$^{d}${\citet{Noterdaeme:2009}} , $^{e}${\citet{Braun:2012}},
$^{f}${\citet{Pontzen:2009}}, $^{g}${\citet{Murphy | :2016}},
$^{h}${\citet{Brookes:2008}}, $^{i}${\citet{deZotti:2010}}.
\end{tablenotes}
\end{threeparttable}
\end{table}
\section{Expected results for future 21-cm absorption
surveys}\label{section:tspin_results}
\begin{figure}
\centering
\includ |
egraphics[width=0.475\textwidth]{tspin_prob.pdf}
\caption{The posterior probability density of the average spin
temperature, as a function of absorber detection yield
($\mathcal{N}$). We show results for our simulated all-southern-sky
survey with 2-h | per pointing using the full 36-antenna ASKAP (top
panel) and a smaller 1000\,deg$^{2}$ survey with 12-h per pointing
and 12 antennas of ASKAP (bottom panel). The dashed curves show the
cumulative effect of the systematic errors discussed in
\autor |
ef{section:errors}. $\overline{\mathcal{F}}_{\rm CNM}$ is the
average CNM fraction assuming a simple two-phase neutral ISM with
$T_{\rm spin,CNM} = 100$\,K and $T_{\rm spin,WNM} = 1800$\,K
(\citealt{Liszt:2001}).}\label{figure:tspin_prob}
\end{figure | }
In the top panel of \autoref{figure:tspin_prob} we show the results of
applying our method for inferring $\overline{T}_{\rm spin}$ to the
simulated all-southern-sky \mbox{H\,{\sc i}} absorption survey with
ASKAP described in \autoref{section:all_sky_sur |
vey}. We account for
the uncertainties in the expected detection rate $\overline{\mu}$,
discussed in \autoref{section:errors}, by using a Monte Carlo approach
and marginalizing over many realizations. A yield of 1000 absorbers
from such a survey would impl | y an average spin temperature of
$\overline{T}_\mathrm{spin} =
127^{+14}_{-14}\,(193^{+23}_{-23})$\,K\footnote{We give the
68.3\,per\,cent interval about the median value measured from the
posterior distributions shown in \autoref{figure:tspin_prob}.}, |
where values in parentheses denote the alternative posterior
probability resulting from the systematic errors discussed in
\autoref{section:errors}. This scenario would indicate that a large
fraction of the atomic gas in DLAs at these intermediate redshif | ts is
in the classical stable CNM phase. Conversely, a yield of only 100
detections would imply that $\overline{T}_\mathrm{spin} =
679^{+64}_{-65}\,(1184^{+116}_{-120})$\,K, indicating that less than
10\,per\,cent of the atomic gas is in the CNM and that t |
he bulk of the
neutral gas in galaxies is significantly different at intermediate
redshifts compared with the local Universe.
We also consider the effect of reducing the sky area and array size,
which is relevant for planned early science surveys with ASK | AP and
other SKA pathfinder telescopes. In the bottom panel of
\autoref{figure:tspin_prob}, we show the spin temperatures inferred
when observing a random 1000\,deg$^{2}$ field for 12\,h per pointing,
between $z_{\rm HI} = 0.4$ and $1.0$, using a 12-antenn |
a version of
ASKAP. We find that detection yields of 30 and 3 from such a survey
would give inferred spin temperatures of $\overline{T}_{\rm spin}
=134^{+23}_{-27}\,(209^{+40}_{-47})$ and
$848^{+270}_{-430}\,(1535^{+513}_{-837})$\,K, respectively. The
sign | ificant reduction in telescope sensitivity and sky-area,
compensated by the increase in integration time per pointing planned
for early-science, results in a factor of 30 decrease in the expected
number of detections and therefore an increase in the sample |
variance
and uncertainty in $\overline{T}_{\rm spin}$. However, this result
demonstrates that we expect to be able to distinguish between the
limiting cases of CNM-rich or deficient DLA populations even during
the early-science phases of the SKA pathfinde | rs. For example 30
detections with the early ASKAP survey rules out an average spin
temperature of 1000\,K at high probability.
\section{Conclusions}
We have demonstrated a statistical method for measuring the average
spin temperature of the neutral ISM |
in distant galaxies, using the
expected detection yields from future wide-field 21\,cm absorption
surveys. The spin temperature is a crucial property of the ISM that
can be used to determine the fraction of the cold ($T_{\rm k} \sim
100$\,K) and dense ($n | \sim 100$\,cm$^{-2}$) atomic gas that provides
sites for the future formation of cold molecular gas clouds and star
formation. Recent 21\,cm surveys for \mbox{H\,{\sc i}} absorption in
\mbox{Mg\,{\sc ii}} absorbers and DLAs towards distant quasars have
yie |
lded some evidence of an evolution in the average spin temperature
that might reveal a decrease in the fraction of cold dense atomic gas
at high redshift (e.g. \citealt{Gupta:2009, Kanekar:2014a}).
By combining recent specifications for ASKAP, with availa | ble
information for the population of background radio sources, we show
that strong statistical constraints (approximately $\pm10$\,per\,cent)
in the average spin temperature can be achieved by carrying out a
shallow 2-h per pointing survey of the southern |
sky between redshifts
of $z = 0.4$ and $1.0$. However, we find that the accuracy to which we
can measure the average spin temperature is ultimately limited by the
accuracy to which we can measure the distribution of the covering
factor, the $N_{\rm HI}$ f | requency distribution function and the
evolution of the radio source population as a function of redshift. By
improving our understanding of these distributions we will be able to
leverage the order-of-magnitude increases in sensitivity and redshift
covera |
ge of the future SKA telescope, allowing us to measure the
evolution of the average spin temperature to much higher redshifts.
\section*{Acknowledgements}
We thank Robert Allison, Elaine Sadler and Michael Pracy for useful
discussions, and the anonymous | referee for providing comments that
helped improve this paper. JRA acknowledges support from a Bolton
Fellowship. We have made use of \texttt{Astropy}, a
community-developed core \texttt{Python} package for astronomy
(\citealt{Astropy:2013}); NASA's Astro |
physics Data System
Bibliographic Services; and the VizieR catalogue acces | s tool, operated
at CDS, Strasbourg, France.
\bibliographystyle{mnras}
|
\section{Introduction}
Given $\rho>0$, we consider the problem
\begin{equation}\label{eq:main_prob_U}
\begin{cases}
-\Delta U + \lambda U = |U|^{p-1}U & \text{in }\Omega,\smallskip\\
\int_\Omega U^2\,dx = \rho, \quad U=0 & \text{on }\partial\Omega,
\end | {cases}
\end{equation}
where $\Omega\subset{\mathbb{R}}^N$ is a Lipschitz, bounded domain, $1<p<2^*-1$, $\rho>0$ is a fixed
parameter, and both $U\in H^1_0(\Omega)$ and $\lambda\in{\mathbb{R}}$ are unknown. More precisely,
we investigate conditions on $p$ |
and $\rho$ (and also $\Omega$) for the solvability of
the problem.
The main interest in \eqref{eq:main_prob_U} relies on the investigation of standing wave
solutions for the nonlinear Schr\"odinger equation
\[
i\frac{\partial \Phi}{\partial t}+\Delta \Phi | + |\Phi|^{p-1}\Phi=0,\qquad
(t,x)\in {\mathbb{R}}\times \Omega
\]
with Dirichlet boundary conditions on $\partial\Omega$. This equation appears in several
different physical models, both in the case $\Omega={\mathbb{R}}^N$ \cite{MR2002047}, and on bounded
|
domains \cite{MR1837207}. In particular, the latter case appears in nonlinear optics and in
the theory of Bose-Einstein condensation, also as a limiting case of the equation on ${\mathbb{R}}^N$
with confining potential. When searching for solutions having | the
wave function $\Phi$ factorized as $\Phi(x,t)=e^{i\lambda t} U(x)$, one obtains
that the real valued function $U$ must solve
\begin{equation}\label{eq:NLS}
-\Delta U + \lambda U = |U|^{p-1}U ,\qquad U\in H^1_0(\Omega),
\end{equation}
and two points of |
view are available. The first possibility is to assign the chemical
potential $\lambda\in{\mathbb{R}}$, and search for solutions of \eqref{eq:NLS} as critical points of the
related action functional. The literature concerning this approach is huge and we d | o not even
make an attempt to summarize it here. On the contrary, we focus on the second possibility,
which consists in considering $\lambda$ as part of the unknown and prescribing the mass (or
charge) $\|U\|_{L^2(\Omega)}^2$ as a natural additional condi |
tion. Up to our knowledge,
the only previous paper dealing with this case, in bounded domains, is \cite{MR3318740},
which we describe below. The problem of searching for normalized solutions in ${\mathbb{R}}^N$,
with non-homogeneous nonlinearities, is more | investigated \cite{MR3009665,MR1430506},
even though the methods used there can not be easily extended to bounded domains, where
dilations are not allowed. Very recently, also the case of partial confinement has been
considered \cite{BeBoJeVi_2016}.
Solu |
tions of \eqref{eq:main_prob_U} can be identified with critical points of the
associated energy functional
\[
\mathcal{E}(U) = \frac12\int_\Omega|\nabla U|^2\,dx - \frac{1}{p+1} \int_\Omega|U|^{p+1}\,dx
\]
restricted to the mass constraint
\[
{\mathcal{M} | }_\rho=\{U\in H_0^1(\Omega) : \|U\|_{L^2(\Omega)}=\rho\},
\]
with $\lambda$ playing the role of a Lagrange multiplier.
A cricial role in the discussion of the above problem is played by the Gagliardo-Nirenberg
inequality: for any $\Omega$ and for any $v\i |
n H^1_0(\Omega)$,
\begin{equation}
\label{sobest}
\|v\|^{p+1}_{L^{p+1}(\Omega)} \leq C_{N,p} \| \nabla v \|_{L^2(\Omega)}^{N(p-1)/2}
\| v \|_{L^2(\Omega)} ^{(p+1)-N(p-1)/2},
\end{equation}
the equality holding only when $\Omega={\mathbb{R}}^N$ and $v=Z_{N, | p}$, the positive solution of $-\Delta Z + Z = Z^{p}$ (which is unique up to
translations \cite{MR969899}).
Accordingly, the exponent $p$ can be classified in relation with the so called
\emph{$L^2$-critical exponent} $1+4/N$ (throughout all the paper, $p$ |
will be always
Sobolev-subcritical and its criticality will be understood in the $L^2$ sense).
Indeed we have that ${\mathcal{E}}$ is bounded below and coercive on ${\mathcal{M}}_\rho$ if and only if
either $p$ is subcritical, or it is critical and $\rho$ | is sufficiently small.
The recent paper \cite{MR3318740} deals with problem \eqref{eq:main_prob_U} in the case of the spherical domain $\Omega = B_1$, when searching for positive solutions $U$.
In particular, it is shown that the solvability of \eqref{eq |
:main_prob_U} is strongly influenced by the exponent $p$, indeed:
\begin{itemize}
\item in the subcritical case $1<p<1+4/N$, \eqref{eq:main_prob_U} admits a unique positive
solution for every $\rho>0$;
\item if $p=1+4/N$ then \eqref{eq:main_prob_U} admi | ts a unique
positive solution for
\[
0<\rho<\rho^*=\left(\frac{p+1}{2C_{N,p}}\right)^{N/2}=\|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)},
\]
and no positive solutions for $\rho\geq\rho^*$;
\item finally, in the supercritical regime $1+4/N<p<2^*-1$, \eqref{eq:m |
ain_prob_U} admits positive
solutions if and only if $0<\rho\leq\rho^*$ (the threshold $\rho^*$ depending on $p$), and such
solutions are at least two for $\rho<\rho^*$.
\end{itemize}
In this paper we carry on such analysis, dealing with a general domai | n $\Omega$ and with solutions which
are not necessarily positive.
More precisely, let us recall that for any $U$ solving \eqref{eq:main_prob_U} for some $\lambda$, it is well-defined the Morse index
\[
m(U) = \max\left\{k : \begin{array}{l}
\exists V\subse |
t H^1_0(\Omega),\,\dim(V)= k:\forall v\in V\setminus\{0\}\smallskip\\
\displaystyle\int_\Omega |\nabla v|^2 + \lambda v^2 - p|U|^{p-1}v^2\,dx<0
\end{array}
\right\}\in{\mathbb{N}}.
\]
Then, if $\Omega=B_1$, it is well known that a solution $U$ of \eqref{eq | :main_prob_U} is
positive if and only if $m(U)=1$. Under this perspective, the results in \cite{MR3318740}
can be read in terms of Morse index one--solutions, rather than positive ones: introducing the sets of admissible masses
\[
{\mathfrak{A}}_k ={\mathf |
rak{A}}_k(p,\Omega) := \left\{\rho>0 : \begin{array}{l} \eqref{eq:main_prob_U}
\text{ admits a solution $U$ (for some $\lambda$)}\\ \text{having Morse index }m(U)\leq k
\end{array} \right\},
\]
then \cite{MR3318740} implies that ${\mathfrak{A}}_1(p,B_1)$ i | s a bounded interval if
and only if $p$ is critical or supercritical, while ${\mathfrak{A}}_1(p,B_1)={\mathbb{R}}^+$ in the subcritical case.
On the contrary, when considering general domains and higher Morse index, the situation may become
much more compl |
icated. We collect some examples in the following remark.
\begin{remark}\label{rem:specialdomains}
In the case of a symmetric domain, one can use any solution as a building block to construct other solutions with
a more complex behavior, obtaining the so-c | alled necklace solitary waves. Such kind of solutions are constructed
in \cite{MR3426917}, even though in such paper the focus is on stability, rather than on normalization conditions.
For instance, by scaling argument, any Dirichlet solution of $-\Delta U |
+ \lambda U = |U|^{p-1}U$ in a rectangle $R=\prod_{i=1}^N(a_i,b_i)$ can
be scaled to a solution of $-\Delta U + k^2\lambda U = |U|^{p-1}U$ in $R/k$, $k\in{\mathbb{N}}_+$, and then $k^N$ copies of it can be
juxtaposed, with alternating sign. In this way on | e obtains a new solution on $R$ having $k^{4/(p-1)}$ times the mass of the starting one,
and eventually solutions in $R$ with arbitrarily high mass (but with higher Morse index) can be constructed even in the critical
and supercritical case. An analogous c |
onstruction can be performed in the disk, using solutions in circular sectors as building blocks,
even though in this case explicit bounds on the mass obtained are more delicate.
Also, instead of symmetric domains, singular perturbed ones can be considered | , such as dumbbell domains
\cite{MR949628}: for instance, using \cite[Theorem 3.5]{MR2997381}, one can show that for any $k$, there exists a domain $\Omega$,
which is close in a suitable sense to the disjoint union of $k$ domains, such that \eqref{eq:main_ |
prob_U} has a \emph{positive}
solution on $\Omega$ with Morse index $k$ and $\rho=\rho_k\to+\infty$ as $k\to+\infty$.
This kind of results justifies the choice of classifying the solutions in terms of their Morse index, rather
than in terms of their nodal | properties.
\end{remark}
Motivated by the previous remark, the first question we address in this paper concerns the boundedness of ${\mathfrak{A}}_k$.
We provide the following complete classification.
\begin{theorem}\label{thm:bbd_index}
For every $\Omega |
\subset{\mathbb{R}}^N$ bounded $C^1$ domain, $k\ge1$, $1<p<2^*-1$,
\[
\sup{\mathfrak{A}}_k(p,\Omega) < +\infty
\qquad\iff\qquad
p \ge 1+\frac{4}{N}.
\]
\end{theorem}
The proof of such result, which is outlined in Section \ref{sec:blow-up}, is obtained
by a | detailed blow-up analysis of sequences of solutions with bounded Morse index, via
suitable a priori pointwise estimates (see \cite{MR2063399}). In this respect, the
regularity assumption on $\partial\Omega$ simplifies the treatment of possible
concentrati |
on phenomena towards the boundary. The argument, which holds
for solutions which possibly change sign, is inspired by \cite{MR2825606}, where the
case of positive solutions is treated.
Once Theorem \ref{thm:bbd_index} is established, in case $p\geq 1 + 4/ | N$ two questions arise, namely:
\begin{enumerate}
\item is it possible to provide lower bounds for $\sup{\mathfrak{A}}_k$? Is it true that $\sup{\mathfrak{A}}_k$ is strictly increasing in $k$, or, at least, that $\sup{\mathfrak{A}}_k > \sup{\mathfrak{A}}_ |
1$ for some $k$?
\item is \eqref{eq:main_prob_U} solvable for every $\rho\in(0,\sup{\mathfrak{A}}_k)$, or at least can we characterize some subinterval of solvability?
\end{enumerate}
It is clear that both issues can be addressed by characterizing
values | of $\rho$ for which existence (and multiplicity) of solutions with
bounded Morse index can be guaranteed. To this aim, it can be useful to restate
problem \eqref{eq:main_prob_U} as
\begin{equation}\label{eq:main_prob_u}
\begin{cases}
-\Delta u + \lambda u |
= \mu|u|^{p-1}u & \text{in }\Omega,\\
\int_\Omega u^2\,dx = 1, \quad u=0 & \text{on }\partial\Omega,
\end{cases}
\qquad\text{where}\quad
\begin{cases}
U=\sqrt{\rho} u\\
\mu = \rho^{(p-1)/2},
\end{cases}
\end{equation}
where now $\mu>0$ is prescribed. Sin | ce
\begin{equation}
\label{Emu}
\text{both } \mathcal{E}_{\mu}(u) := \frac{1}{2}\int_{\Omega}|\nabla u|^2- \frac{\mu}{p+1}\int_{\Omega}| u|^{p+1}
\qquad
\text{and }{\mathcal{M}}={\mathcal{M}}_1=\{u : \|u\|_{L^2(\Omega)}=1\}
\end{equation}
are invariant und |
er the ${\mathbb{Z}}_2$-action of the involution $u\mapsto -u$, solutions of
\eqref{eq:main_prob_u} can be found via min-max principles in the framework of index
theories (see e.g. \cite[Ch. II.5]{St_2008}). Notice that in the supercritical case
${\mathcal | {E}}_\mu$ is not bounded from below on ${\mathcal{M}}$. Following \cite{MR3318740}, it can be
convenient to parameterize solutions to \eqref{eq:main_prob_u} with respect to the
$H^1_0$-norm, therefore we introduce the sets
\begin{equation}\label{eq:defBU} |
\mathcal{B}_\alpha:=\left\{u\in {\mathcal{M}}:\,\int_\Omega |\nabla u|^2\,dx<\alpha\right\},\quad\quad
\mathcal{U}_\alpha:=\left\{u\in {\mathcal{M}}:\,\int_\Omega |\nabla u|^2\,dx=\alpha\right\}.
\end{equation}
Introducing the first Dirichlet eigenvalue o | f $-\Delta$ in $H^1_0(\Omega)$,
$\lambda_1(\Omega)$, we have that the sets above are non-empty whenever $\alpha>
\lambda_1(\Omega)$.
Since we are interested in critical points
having Morse index bounded from above, following \cite{MR968487,MR954951,MR99126 |
4} we
introduce the following notion of genus.
\begin{definition}\label{def:genus}
Let $A\subset H^1_0(\Omega)$ be a closed set, symmetric with respect to the origin (i.e.
$-A=A$). We define the \emph{genus} $\gamma$ of a $A$ as
\[
\gamma(A) := \sup\{m : \ | exists h \in C({\mathbb{S}}^{m-1};A),\, h(-u)=-h(u)\}.
\]
Furthermore, we define
\[
\Sigma_{\alpha}=\{A\subset \overline{\mathcal{B}}_\alpha: A\text{ is closed and }-A=A\},
\qquad
\Sigma^{(k)}_{\alpha}=\{A\in \Sigma_{\alpha} : \gamma(A)\ge k\},
\]
\end{def |
inition}
We remark that this notion of genus is different from the
classical one of \emph{Krasnoselskii genus}, which is well suited for estimates
of the Morse index from below, rather than above. Nonetheless, $\gamma$ shares with
the Krasnoselskii genus m | ost of the main properties of an index
\cite{MR0163310,MR0065910}. In particular, by the Borsuk-Ulam Theorem, any set $A$
homeomorphic to the sphere ${\mathbb{S}}^{m-1} := \partial B_1 \subset {\mathbb{R}}^m$ has genus
$\gamma(A) = m$. Furthermore, we show |
in Section \ref{sec:2const} that $\Sigma^{(k)}_{\alpha}$ is not empty,
provided $\alpha>\lambda_k(\Omega)$ (the $k$-th Dirichlet eigenvalue of $-\Delta$ in $H^1_0(\Omega)$).
Equipped with this notion of genus we provide two different variational principl | es
for solutions of \eqref{eq:main_prob_u} (and thus of \eqref{eq:main_prob_U}). The first
one is based on a variational problem with \emph{two constraints}, which was exploited as the
main tool in proving the results in \cite{MR3318740}.
\begin{theorem}\l |
abel{thm:genus_2constr}
Let $k\geq1$ and $\alpha>\lambda_{k}(\Omega)$. Then
\begin{equation}
\label{maxmin}
M_{\alpha,\,k}:= \sup_{A\in\Sigma^{(k)}_{\alpha}}\inf_{u\in A}\int_{\Omega}|u|^{p+1}
\end{equation}
is achieved on ${\mathcal{U}}_\alpha$, and there | exists a critical point
$u_\alpha\in {\mathcal{M}}$ such
that, for some $\lambda_\alpha\in{\mathbb{R}}$ and $\mu_\alpha>0$,
\begin{equation}
\label{lagreq}
\int_\Omega|\nabla u_\alpha|^2 = \alpha\qquad\text{and}\qquad -\Delta u_\alpha+\lambda_\alpha\,u_\a |
lpha=\mu_\alpha |u_\alpha|^{p-1}u_\alpha\quad \text{in }\Omega.
\end{equation}
\end{theorem}
As a matter of fact, the results in \cite{MR3318740} were obtained by a detailed analysis of the map $\alpha \mapsto \mu_\alpha$ in the case $k=1$, i.e.
when deali | ng with
\[
M_{\alpha,1} = \max\left\{\|u\|_{L^{p+1}}^{p+1} : \|u\|_{L^2}^2=1,\,\|\nabla u\|_{L^2}^2=\alpha \right\}.
\]
In the present paper we do not investigate the properties of the map
$\alpha \mapsto \mu_\alpha$ for general $k$, but we rather prefer |
to exploit the
characterization of $M_{\alpha,k}$ in connection with
a second variational principle, which deals with only
\emph{one constraint}.
\begin{theorem}\label{thm:genus_1constr}
Let $1+{N}/{4}\leq p<2^*-1$. There exists a sequence $(\hat \mu_k)_k$ |
(depending on $\Omega$ and $p$) such that, for every $k\geq 1$ and $0<\mu<\hat \mu_k$, the
value
\begin{equation}
\label{infsuplev}
c_k:= \inf_{A\in\Sigma^{(k)}_{\alpha}}
\sup_{A}{\mathcal{E}}_\mu,
\end{equation}
is achieved in $\mathcal{B}_\alpha$, for a |
suitable $\alpha>\lambda_{k}(\Omega)$. Furthermore there exists a critical point $u_\mu\in {\mathcal{M}}$ such
that, for some $\lambda_\mu\in{\mathbb{R}}$,
\[
-\Delta u_\mu+\lambda_\mu\,u_\mu=\mu |u_\mu|^{p-1}u_\mu\quad \text{in }\Omega,
\]
$\|\nabla u\|_ | {L^2}^2<\alpha$, and $m(u_\mu)\le k$.
\end{theorem}
\begin{remark}
Of course, if $p<1+4/N$, the above theorem holds with $\hat\mu_k=+\infty$ for every $k$.
\end{remark}
\begin{corollary}
Let $\hat \rho_k := \hat \mu_k^{2/(p-1)}$. Then
\[
(0,\hat\rho_k) \su |
bset {\mathfrak{A}}_k.
\]
\end{corollary}
The link between Theorem \ref{thm:genus_2constr} and Theorem \ref{thm:genus_1constr} is that we can provide explicit estimates
of $\hat \mu_k$ (and hence of $\hat\rho_k$) in terms of the map $\alpha\mapsto M_{\alph | a,k}$ (see Section \ref{sec:1const}).
We stress that the above
results hold for any Lipschitz $\Omega$. As a first consequence, this allows to extend the
existence result in \cite{MR3318740} to non-radial domains.
\begin{theorem}\label{thm:intro_GS}
For e |
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