Constraints to cosmological parameters through clusters evolution
A. Del Popolo \& N. Ercan
In this paper, I revisit the constraints obtained by several authors (Reichart et al. 1999; Eke et al. 1998; Henry 2000) on the estimated values of $\Omega_{m}$, $n$ and $\sigma_8$ in the light of recent theoretical developments: 1) new theoretical mass functions (Sheth \& Tormen 1999, Sheth, Mo \& Tormen 2001, Del Popolo 2002b); 2) a more accurate mass-temperature relation, also determined for arbitrary $\Omega_{m}$ and $\Omega_{\Lambda}$ (Del Popolo 2002a). \begin{figure} \label{Fig. 1} \centerline{\hbox{(a) \psfig{file=fellis11.ps,width=5cm} (b) \psfig{file=fgenry20.ps,width=5cm} }} \centerline{\hbox{(a) \psfig{file=omsig11.ps,width=5cm} (c) }} %\includegraphics{file=fellis11.ps} %\includegraphics{file=fellis11.ps,width=12cm} %\resizebox{12cm}{!}{\includegraphics{fellis1.ps}} %\parbox[b]{9cm}{ \caption{ (1a) The $68\%$ confidence contours for the parameters $n$, $k_{o}$ and $\Omega_{m}$ for the open model. The dashed lines are lines of constant $\sigma_8$. (1b) $\Delta$(likelihood) %contours as a function of for the parameter $\Omega_{m}$. The solid line is obtained from the model of this paper while the dotted line is that calculated by Henry (2000). The dashed lines represent various confidence levels (65\%, 90\%, 95\%, 99\%). (1c) The $68\%$ confidence contours for the parameters $\sigma_8$, and $\Omega_{m}$ for the open model (see also Henry (2000), Fig. 9).} %} \end{figure} Firstly, using the quoted improvements, I re-derive an expression for the X-ray Luminosity Function (XLF), similarly to Reichart et al. (1999), and then I get some constraints to $\Omega_{m}$ and $n$, by using the {ROSAT} BCS and {EMSS} samples and maximum-likelihood analysis. Then I re-derive the X-ray Temperature Function (XTF), similarly to Henry (2000), re-obtaining the constraints on $\Omega_{m}$, $n$, $\sigma_8$. Both in the case of the XLF and XTF, the changes in the mass function and M-T relation produces an increase in $\Omega_{m}$ of $ \simeq 20\%$ and similar results in $\sigma_8$ and $n$.
contribb.tex
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\title[Constraints to cosmological parameters through clusters evolution]
      {Constraints to cosmological parameters through clusters evolution}

\author[A. Del Popolo \& N. Ercan]
{A. Del Popolo \& N. Ercan}

\affiliation{$^1$$^,$$^2$
Bo$\breve{g}azi$\c{c}i University, Physics Department,
     80815 Bebek, Istanbul, Turkey
}

\pubyear{2004}
\volume{195}
\pagerange{1--8}
\date{?? and in revised form ??}
\setcounter{page}{1}
\jname{Outskirts of Galaxy Clusters: intense life in the suburbs}
\editors{A. Diaferio, ed.}
\begin{document}

\maketitle

\begin{abstract}
In this paper, I revisit the constraints obtained
by several authors (Reichart et al. 1999; Eke et al. 1998; Henry 2000)
on the estimated values of $\Omega_{\rm m}$, $n$ and $\sigma_8$ in the light of recent theoretical developments: 1) new theoretical mass functions (Sheth \& Tormen 1999, Sheth, Mo \& Tormen 2001, Del Popolo 2002b); 2) a more accurate mass-temperature relation, also determined for arbitrary $\Omega_{\rm m}$ and $\Omega_{\rm \Lambda}$ (Del Popolo 2002a). 



\begin{figure}
\label{Fig. 1} 
\centerline{\hbox{(a)
\psfig{file=fellis11.ps,width=5cm} (b)
\psfig{file=fgenry20.ps,width=5cm}
}}
\centerline{\hbox{(a)
\psfig{file=omsig11.ps,width=5cm} (c)
}}
%\includegraphics{file=fellis11.ps}
%\includegraphics{file=fellis11.ps,width=12cm}
%\resizebox{12cm}{!}{\includegraphics{fellis1.ps}}
%\parbox[b]{9cm}{
\caption{ (1a) The $68\%$ confidence contours for the parameters $n$, $k_{\rm o}$ and $\Omega_{\rm m}$ for the open model. The dashed lines are lines of constant $\sigma_8$. (1b) 
$\Delta$(likelihood)
%contours as a function of
for
the parameter $\Omega_{\rm m}$. The solid line is obtained from the model of this paper while the dotted line is that calculated by Henry (2000). The dashed lines represent various confidence levels (65\%, 90\%, 95\%, 99\%). (1c) 
The $68\%$ confidence contours for the parameters $\sigma_8$, and $\Omega_{\rm m}$ for the open model (see also Henry (2000), Fig. 9).}
%}
\end{figure}

Firstly, using the quoted improvements, I re-derive an expression for the X-ray Luminosity Function (XLF), similarly to Reichart et al. (1999), and then I get some constraints to $\Omega_{\rm m}$ and $n$, by using the {\it ROSAT} BCS and {\rm EMSS} samples and maximum-likelihood analysis. Then I re-derive the X-ray Temperature Function (XTF), similarly to Henry (2000), re-obtaining the constraints on $\Omega_{\rm m}$, $n$, $\sigma_8$.
Both in the case of the XLF and XTF, the changes in the mass function and M-T relation produces an increase in $\Omega_{\rm m}$ of $ \simeq 20\%$ and similar results in $\sigma_8$ and $n$.
\end{abstract}
\section{Introduction}
%{\bf Introduction}
%\vskip 0.1cm
It is well known that clusters are strong X-ray emitters whose study
%and so cluster evolution can be inferred from the study of X-ray properties of distant clusters. 
%These objects are 
can put constraints on fundamental cosmological parameters.
There are different methods to trace the evolution of the cluster number density: 
a) The X-ray temperature function (XTF) has been presented for local (e.g. Henry \& Arnaud 1991) and distant clusters (Eke et al. 1998; Henry 2000). 
b) The evolution of the X-ray luminosity function (XLF).\\
The results obtained for $\Omega_{\rm m}$ and other cosmological parameters are in many cases discrepant the one with the other. Several studies in literature, show
that the parameters values span the entire range of acceptable solutions: $0.2 \leq \Omega_{\rm m} \leq 1$ (see Reichart et al. 1999).
The reasons leading to the quoted discrepancies has been studied in several papers (Eke et al. 1998; Borgani et al. 2001) 
( 1) The inadequate approximation given by the PS (e.g., Bryan \& Norman 1997). 2) Inadequacy in the structure formation as described by the spherical model leading to changes in the threshold parameter $\delta_{\rm c}$ (e.g., Governato et al. 1998). 3) Inadequacy in the M-T relation obtained from the virial theorem (see Del Popolo 2002a). 4) Effects of cowling flows. 5) Determination of the X-ray cluster catalog's selection function. 6) Evolution of the L-T relation. 7) Optimization methods used in the analysis.)
These reasons lead me to re-calculate the constraints on $\Omega_{\rm m}$, $n$ and $\sigma_8$, using the XLF and XTF.
\section{Constraints to $\Omega_{\rm m}$ and $n$ from the XLF}
%\vskip 0.1cm
%{\bf Constraints to $\Omega_{\rm m}$ and $n$ from the XLF}
%\vskip 0.1cm
Similarly to Reichart et al. (1999), I re-derived an expression for the XLF, using an improved version of the mass function and M-T relation, obtained in Del Popolo 2000a, Del Popolo 2000b, respectively, taking account of the effects of asphericity and tidal interaction with neighbors. Then I got some constraints to $\Omega_{\rm m}$ and $n$, by using the {\it ROSAT} BCS and {\rm EMSS} samples.

As described in Del Popolo (2000b), the
%``unconditional"
mass function can be approximated by:
%
%%\begin{equation}
%%f(\nu )d\nu \simeq 1.21\left( 1+\frac{0.06}{\left( a\nu \right) ^{0.585}}\right) \sqrt{\frac{a\nu }{2\pi }}\exp{\{-a\nu %%\left[ 1+\frac{0.57}{\left( a\nu \right) ^{0.585}}\right] ^{2}/2\}}
%%\end{equation}
%%where $a=0.707$
%
\begin{equation}
n(m,z)\simeq 1.21 \frac{\overline{\rho}}{m^{2}}\frac{d\log (\nu )}{d\log m}
\left( 1+\frac{0.06}{\left( a\nu \right) ^{0.585}}\right) \sqrt{\frac{a\nu }{2\pi }}\exp{\{-a\nu \left[ 1+\frac{0.57}{\left( a\nu \right) ^{0.585}}\right] ^{2}/2\}}
\label{eq:nmm}
\end{equation}
where $a=0.707$.
Eq. (\ref{eq:nmm}) can be converted from a mass function to a luminosity function using a L-T relation (I use that of  
Mathiesen \& Evrard (1998)), 
%and in order to remove the temperature dependence introduced by I'll use the 
and a T-M relation. This last is the one obtained in Del Popolo (2000a), and is based on the merging-halo formalism of Lacey \& Cole (1993), accounting for the fact that
massive clusters accrete matter quasi-continuously, and again take account of angular momentum acquisition by protostructures:
%In the following, I'll use a modified version of the M-T relation obtained
%improving V98, V2000, to take account of tidal interaction between clusters. This M-T relation is given by:
\begin{equation}
kT \simeq 8 keV \left(\frac{M^{\frac 23}}{10^{15}h^{-1} M_{\odot}}\right)
\frac{
\left[
\frac{1}{m_1}+\left( \frac{t_\Omega }t\right) ^{\frac 23}
+\frac{K(m_1,x)}{M^{8/3}}
\right]
}
{
\left[
\frac{1}{m_1}+\left( \frac{t_\Omega }{t_{0}}\right) ^{\frac 23}
 +\frac{K_0(m_1,x)}{M_{0}^{8/3}}
\right]
}
\label{eq:kTT1}
\end{equation}
(see Del Popolo 2000a for a derivation of the previous equation and the definition of the terms in it).
The luminosity function is obtained as in Reichart et al. (1999), using the mass function and the M-T relation previously introduced (see Del Popolo 2003 for a detailed analysis).
A Bayesian inference analysis used to constraint the model parameters  
shows that: $\Omega_{\rm m}=1.15^{+0.40}_{-0.33}$ and
$n=-1.55^{+0.42}_{-0.41}$.
The previous result shows that the change in the mass function and M-T relation gives rise to an increase of $\Omega_{\rm m}$ and $n$ of $\simeq 20\%$ with respect to Reichart's results.  
The lesson from the previous calculation is that taking account of non-sphericity in collapse and the fact that
massive clusters accrete matter quasi-continuously gives rise to a noteworthy change in the prediction of cosmological parameters, as $\Omega_{\rm m}$. In order to check the previous trend,
I have also estimated the value of $\Omega_{\rm m }$ following Borgani et al. (2001).
Analyzing the ROSAT Deep Cluster Survey (RDCS) and using the XLF to obtain constraints on cosmological parameters, Borgani et al. (2001), found that $\Omega_{\rm m}=0.35^{+0.13}_{-0.10}$. 
%In their study, they used the ST mass function instead of the usual PS and Eke et al. (1998) M-T relation.
Using their method and data, but our mass function and M-T relation, one obtains larger values of $\Omega_{\rm m}$ ($\Omega_{\rm m} \simeq 0.4 \pm 0.1$) that differently from the previous analysis (Reichart et al. 1999) exclude an Einstein-de Sitter model.
\section{Constraints to $\Omega_{\rm m}$, $n$, and $\sigma_8$ from the XTF}
%\vskip 0.1cm
%{\bf Constraints to $\Omega_{\rm m}$, $n$, and $\sigma_8$ from the XTF}
%\vskip 0.1cm
As previously reported, the mass function (MF) is a critical ingredient in putting strong constraints on
cosmological parameters (e.g., $\Omega_{\rm m}$).
In the following, I'll re-calculate the constraints obtained by Henry (2000), by using the mass function and the M-T relation modified as described in the previous section Eq. (\ref{eq:nmm}) and Eq. (\ref{eq:kTT1}) (see Del Popolo 2003).
I use a maximum likelihood fit to the unbinned data in order to determine various model parameters.
The method is described in Marshall et al (1983). The likelihood function is given by their Eq. (2),
adapted to our present situation.
At this point, we can fit the data described in Section. 2 of Henry (2000) to the theory previously described using
the quoted maximum likelihood method.
The most general description of the results requires the three parameters of the fit ($\Omega_{\rm m}$, $\sigma_8$ and n). 
These values shows that the correction introduced by the new form of the mass function and M-T relation gives rise to higher values of $\Omega_{\rm m}$ ($\Omega_{\rm m}=0.6 \pm 0.13$, while it is $\Omega_{\rm m}=0.49 \pm 0.12$ for Henry (2000)) and
$n=-1.5 \pm 0.32$ ($n=-1.72 \pm 0.34$ in Henry (2000)).
Constraints are relatively tight when
considering this single parameter. We find that $\Omega_{\rm m}=0.6^{+0.12}_{-0.11}$
at the 68\% confidence level and $\Omega_{\rm m}=0.6^{+0.23}_{-0.2}$ 
at the 95\% confidence level for the open model.

Concluding, our analysis shows that improvements in the mass function and M-T relation increases the value of $\Omega_{\rm m}$ and that even small correction in the physics of the collapse can induce noteworthy effects on the constraints obtained. 




%\baselineskip 0.0 cm
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%\baselineskip 0.0 cm
{\bf References}
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\acknowledgments

I would like to thank Prof. E. Recami, E. Nihal Ercan, A. Diaferio 
and Y. Ek\c{s}i for some useful comments and an anonymous referee whose
suggestions helped us to considerably improve
the quality of this paper. \\
Finally, I would like to thank Bo$\breve{g}azi$\c{c}i University
Research Foundation for the financial support through the project
code 01B304.


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%% delimit the year from the author name used in the citation.
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\newpage
\begin{flushleft}
{\bf Figure Caption}
\end{flushleft}
{\bf Fig. 1} Evolution of {\it isolated} homogeneous ellipsoidal
perturbations in an EdS universe with $H_0=50 {\rm km/s/Mpc}$ and
$\rho_{\rm e}/\rho_{\rm b}=1.003$. The solid lines represent
numerical solutions obtained using Bulirsh-Stoer algorithm while
the dotted ones the approximate analytical solution of WS. (a)-(c)
In these cases, the axial ratio is $1:1.25:1.5$, $1:1.5:3$ and
$1:1.5:5$, respectively. (d)-(f) Same as the previous figures
(1a)-(1c) but now the WS analytical approximate solution is
improved by using some free parameters fitted to the numerical
solution by means of the least-square method (see text for a
description).\\

{\bf Fig. 2} Evolution of {\it unisolated} homogeneous ellipsoidal
perturbations in an EdS universe with $H_0=50 {\rm km/s/Mpc}$ and
$\rho_{\rm e}/\rho_{\rm b}=1.003$. The solid lines represent
numerical solutions obtained using Bulirsh-Stoer algorithm while
the dotted ones the approximate analytical solution expressed by
equation (\ref{eq:predd1})-(\ref{eq:predd3}). (a)-(c) In these
cases, the axial ratio is $1:1.25:1.5$ while $b_{\rm
i}=(-0.5,-0.5,1)$, and $Q=1,2,4$. (d)-(f) Same as the previous
figures (2a)-(2c) but now the axial ratio is $1:1.25:1.5$,
$1:1.25:2$, $1:1.25:5$, respectively, while $Q=1$ and $b_{\rm
i}=(-0.5,-0.5,1)$.\\

{\bf Fig. 3} Evolution of the axial velocity. (a)-(c) The solid
lines represent, from up to bottom, the numerical result for the
peculiar velocity along the shortest, medium and longest axis. The
dotted line represents the velocity obtained from equation
(\ref{eq:vel}) using the approximation for the semiaxes (equations
(\ref{eq:predd1})-(\ref{eq:predd3})). Going from Fig. 3a to Fig.
3c, the value of the axial ratio is $1:1.25:1.5$, $1:1.25:5$,
$1:2:3$, respectively, while $Q=1$, and $b_{\rm
i}=(-0.5,-0.5,1)$. (d) The solid line represents the peculiar velocity
for an ellipsoid of axial ratio $1:1.25:1.5$ when no external
field is present, while the dotted and dashed lines represent the
case $b_{\rm i}=(-0.5,-0.5,1)$, and $b_{\rm i}=(-1,-1,2)$,
respectively. (e) Same as (d), but now the axial ratio is $1:2:3$.\\

{\bf Fig. 4} The evolution of the density contrast. (a) The axial
ratio of the ellipsoid is $1:1.25:1.5$, the lines from bottom to
top represent the case of an {\it isolated} ellipsoid ($b_{\rm
i}=(0,0,0)$), $b_{\rm i}=(-0.5,-0.5,1)$, $b_{\rm i}=(-1,-1,2)$,
respectively. (b) Same of the previous one but the configuration
is less spherical, with axial ratio $1:2:3$.\\

{\bf Fig. 5} The density contrast at turnaround for a prolate
spheroid for several values of the longest axis, $a_3$, (the other
two axes have fixed value $a_1:a_2=1:1$). The solid lines, from
top to bottom, represent numerical results for the density
contrast for an {\it isolated} spheroid ($b_{\rm i}=(0,0,0)$), and
for {\it unisolated} spheroids with $b_{\rm i}=(-0.5,-0.5,1)$ and
$(-1,-1,2)$, respectively. The dashed lines represent the
approximate solution (equation (\ref{eq:denstu})). The upper
dotted line represents the value of the density contrast at
turnaround for a spherical perturbation. \\

{\bf Fig. 6} Comparison between
density contrast predicted by spherical and ellipsoidal model. The
solid line represents the density contrast predicted by the
spherical model, while the dotted, short-dashed, long-dashed and
long-dashed--short-dashed lines represents the prediction of the
ellipsoid model with $Q=1$, $b_{\rm i}=(-0.5,-0.5,1)$ and initial
axial ratio $1:1.25:1.5$, $1:1.25:3$, $1:1.25:5$, and $1:1.25:8$
respectively.\\

{\bf Fig. 7} Comparison
between density contrast predicted by spherical and oblate
spheroidal models. The solid line represents the density contrast
predicted by the spherical model, while the dotted, short-dashed,
and long-dashed lines represents the prediction of the model with
$Q=1$, $b_{\rm i}=(-0.5,-0.5,1)$ and initial axial ratio $2:2:1$,
$4:4:1$, $8:8:1$, respectively.\\

{\bf Fig. 8} Density contrast at virialization. The solid line
refers to an ${\it isolated}$ prolate spheroid.\\

{\bf Fig. 9} Turnaround epoch for a prolate spheroid. The solid,
short-dashed and long-dashed lines, represent respectively the
time of turnaround for an {\it isolated} spheroid and for
spheroids having $b_{\rm i}=(-0.5,-0.5,1)$ and $(-1,-1,2)$. The
upper dotted line represents the value of the density contrast at
turnaround for a spherical perturbation. \\

{\bf Fig. 10} Axial velocity at collapse as function of the ratio of
the initial value of the axes, $a_3/a_1$. The solid lines
represent the numerical results of the collapse velocity for a
prolate spheroid, the dotted lines the result obtained from the
approximate solution. The top curve represents the velocity for an
{\it isolated} spheroid, the medium one the same quantity for a
spheroid having $Q=1$, $b_{\rm i}=(-0.5,-0.5,1)$, and the last
bottom curve the velocity for a spheroid having $Q=2$, and $b_{\rm
i}=(-0.5,-0.5,1)$.
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astro-ph0404382.extracted_bib
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