## Cosmology and Astrophysics from Relaxed Galaxy Clusters II

### Introduction

Massive, dynamically relaxed clusters of galaxies represent a minority of the cluster population, and are a precious resource for probing both the astrophysics of clusters and cosmology. They provide precise and minimally biased mass estimates to studies of scaling relations and the growth of cosmic structure, and enable complementary contraints on cosmic expansion. Additionally, they represent the most natural targets for studying thermodynamic features of the intracluster medium with minimal systematics from deprojection or non-equilibrium processes.

In this paper, we employ a sample of relaxed clusters to obtain constraints on cosmology and the expansion of the Universe. This work is enabled by the prediction, from hydrodynamical simulations of cluster formation, that the ratio of gas mass to total mass (fgas) in rich clusters is approximately independent of redshift and mass, and has small cluster-to-cluster variability. X-ray mass estimates of these clusters (assuming hydrostatic equilibrium) are expected to be minimally biased by non-thermal pressure, and can now be directly calibrated using robust gravitational lensing measurements. By measuring gas and total masses for a sample of relaxed clusters, and employing simulation results for the expected depletion of gas in clusters compared to the Universe as a whole, we can constrain the cosmic ratio of baryon to total matter density. Combining this result with external constraints on the present-day expansion rate of the Universe (H0) and baryon density, tight and robust constraints can be placed on the density of matter in the Universe (Ωm). At the same time, the wide range in redshift covered by the fgas data set provides additional information about the history of cosmic expansion. These data provide corroboration of the finding (from type Ia supernovae) that the cosmic expansion is accelerating, and place competitive constraints on models of the dark energy which is thought to cause this acceleration.

### Data and likelihood code

There are three ways to get the code. If you use any of these in your own work, you should cite this paper (and others in the series, if applicable).

• Data files.
• A stand-alone code library for evaluating the likelihood for a given cosmological model. Requires the GNU Scientific Library. The code is written in C++, but C and Fortran interfaces are also included.
• Additional files and code to use the data with COSMOMC. See instructions.
• (Partially complete interfaces for Python and the cosmological code MontePython also exist, and can be obtained from the author by request.)
For questions/problems related to these files, contact the authors.
2. As of June 2015 (version 1.2), cosmosis includes this code under the module named "cluster_fgas".
3. As of July 2014, the COSMOMC git has a clusters_fgas branch which includes the fgas code and data. You can email Antony Lewis, the developer of COSMOMC, for access. This is probably the easiest way to get started using our results with COSMOMC. The instructions below apply to an earlier version of COSMOMC, but will remain here just in case they're useful.

Important note: For the time being, we are witholding our gravitational lensing data, pending the acceptance of a paper specifically detailing the calibration of X-ray and lensing masses (Applegate et al.). The code is designed to work even when these additional data files are absent. The mass calibration result can be largely captured by a 0.9 ± 0.9 Gaussian prior on the ratio of lensing to X-ray masses, although this fails to account for a mild degeneracy with cosmological parameters. In COSMOMC, this is accomplished by adding the line prior[cl_cal] = 0.9 0.09 to the ini file, and fixing the remaining cl_ parameters (as done in the current download).

### Reference for these results

Cosmology and Astrophysics from Relaxed Galaxy Clusters II: Cosmological Constraints.
A. B. Mantz, S. W. Allen, R. G. Morris, D. A. Rapetti, D. E. Applegate, P. L. Kelly, A. von der Linden, and R. W. Schmidt. MNRAS 440:2077, 2014. ADS, arXiv, BibTeX.

### Instructions for using the source distribution with COSMOMC

Apart from compiling the package above, the installation consists of pasting a few short blocks of code into the COSMOMC source. These instructions work as of the December 2013 COSMOMC version and are not guaranteed in perpetuity. If you run into trouble or find errors in these instructions, feel free to contact me.

1. Go to your COSMOMC directory.
2. Unzip the download and run make in the fgas/src directory to compile the code. You may need to customize the Makefile.
3. Move or link fgas/cosmomc/clusters.ini to the COSMOMC batch1 directory (optional, for convenience).
4. Move or link fgas/cosmomc/clusters.f90 to the COSMOMC source directory (required).
5. In the COSMOMC source/DataLikelihoods.f90, add these lines in the obvious places.
use clusters

6. In the COSMOMC source/Makefile, make these additions as indicated:
### at the top is fine
# clusters module
CLUSTERS ?= ../fgas
ifneq ($(CLUSTERS),) CLUSTERSO = clusters.o CLUSTERSF = clusters.f90 endif ### add$(CLUSTERSO) to the end of the DATAMODULES definition

### can go anywhere after the OBJFILES and LINKFLAGS definitions
### note that you may need to specify the GSL path using -L in the 2nd statement
ifneq ($(CLUSTERS),) OBJFILES +=$(CLUSTERS)/src/fwrapper.o
endif

### can go anywhere after the LINKFLAGS definition
ifneq ($(CLUSTERS),) LINKFLAGS += -lstdc++ -L$(CLUSTERS)/src -lclusters
endif

7. Add the following to modules.f90 in the camb subdirectory of COSMOMC. Just after the usual AngularDiameterDistance definition is a logical place.
    function AngularDiameterDistance2(z1, z2) ! z1 < z2
real(dl) AngularDiameterDistance2
real(dl), intent(in) :: z1, z2
end function AngularDiameterDistance2

8. You should now be able to make COSMOMC. If you've previously compiled, do a make clean first to force CAMB to recompile completely as well.

Below is an example of a minimal top-level .ini file using the clusters module. Note that, due to a legacy "use_clusters" keyword that is no longer used in COSMOMC as distributed but nevertheless has a default value, the use_clusters=T instruction must be in the top-level file.

DEFAULT(batch1/common_batch1.ini)

use_clusters = T
DEFAULT(batch1/clusters.ini)

file_root = chains/test


The following is optional, and implements the cosmological parametrization described in the paper, which is convenient for the fgas data.

1. In source/params_CMB.f90, add the following code blocks in the indicated places:
! in the module definition
Type, extends(CosmologyParameterization) :: clBackgroundParameterization
logical :: flat = .false.
real(mcp) :: H0_prior_mean = 0._mcp, H0_prior_std = 0._mcp
real(mcp) :: ombh2_prior_mean = 0._mcp, ombh2_prior_std = 0._mcp
contains
procedure :: ParamArrayToTheoryParams => clBK_ParamArrayToTheoryParams
procedure :: NonBaseParameterPriors => clBK_NonBaseParameterPriors
procedure :: CalcDerivedParams => clBK_CalcDerivedParams
procedure :: Initialize => clBK_Init
end type clBackgroundParameterization

! at the top of SetTheoryParametrization
Type(clBackgroundParameterization), pointer :: clBackgroundParam
! and in the following branching statement
else if (paramtxt=='cluster_background') then
allocate(clBackgroundParam)
Parameterization => clBackgroundParam
call clBackgroundParam%Initialize(Ini,Names)

! the rest can go at the end of the module
!!! background parameterization with a little extra stuff (namely omb) for clusters
subroutine clBK_Init(this, Ini, Names)
Class(clBackgroundParameterization) :: this
Type(TIniFile) :: Ini
Type(TParamNames) :: Names
character(LEN=Ini_max_string_len) prior

call SetTheoryParameterNumbers(9,0)
this%late_time_only = .true.

! special flag to require a flat universe, since omegak is not a free parameter

! use the special H0 prior keys just because they already exist
if (prior/='') then
end if
! also honor the omegabh2 prior, if there is one
if (prior/='') then
end if

call this%Init(Ini,Names, 'params_clbackground.paramnames')

end subroutine clBK_Init

function clBK_NonBaseParameterPriors(this,CMB)
class(clBackgroundParameterization) :: this
class(TTheoryParams) :: CMB
real(mcp):: clBK_NonBaseParameterPriors

select type (CMB)
class is (CMBParams)
clBK_NonBaseParameterPriors = logZero
if (CMB%YHe < 0.0) return ! this catches ombh2 out of bounds with bbn_consistency (below)
clBK_NonBaseParameterPriors = 0
if (this%H0_prior_mean/=0._mcp) then
clBK_NonBaseParameterPriors = clBK_NonBaseParameterPriors + ((CMB%H0 - this%H0_prior_mean)/this%H0_prior_std)**2/2
end if
if (this%ombh2_prior_mean/=0._mcp) then
clBK_NonBaseParameterPriors = clBK_NonBaseParameterPriors + ((CMB%ombh2 - this%ombh2_prior_mean)/this%ombh2_prior_std)**2/2
end if
end select
end function clBK_NonBaseParameterPriors

subroutine clBK_ParamArrayToTheoryParams(this, Params, CMB)
use bbn
class(clBackgroundParameterization) :: this
real(mcp) Params(:)
class(TTheoryParams), target :: CMB
real(mcp) fbaryon, omegam, h2

select type (CMB)
class is (CMBParams)
fbaryon = Params(1)
omegam = Params(2)
CMB%H0 = 100.0 * Params(3)
if (this%flat) then
CMB%omv = 1.0 - omegam
else
CMB%omv = Params(4)
end if
CMB%omnuh2=Params(5)/neutrino_mass_fac*(3.046_mcp/3)**0.75_mcp
CMB%w =    Params(6)
CMB%wa =    Params(7)
CMB%nnu =    Params(8)

CMB%h=CMB%H0/100
h2 = CMB%h**2
CMB%omnu = CMB%omnuh2/h2
CMB%omb = fbaryon * omegam
CMB%ombh2 = CMB%omb*h2
CMB%omc = (1.0 - fbaryon) * omegam - CMB%omnu
CMB%omch2 = CMB%omc*h2
CMB%zre=0
CMB%tau=0
CMB%omdmh2 = CMB%omch2+ CMB%omnuh2
CMB%omdm = CMB%omdmh2/h2
CMB%omk = 1 - CMB%omv - CMB%omb - CMB%omdm
CMB%nufrac=CMB%omnuh2/CMB%omdmh2
CMB%reserved=0
CMB%fdm=0
CMB%iso_cdm_correlated=0
CMB%Alens=1
if (bbn_consistency) then
if (CMB%ombh2 > 0.005 .and. CMB%ombh2 < 0.040) then ! magic #s
CMB%YHe = yp_bbn(CMB%ombh2,CMB%nnu  - 3.046)
else
CMB%Yhe = -1.0
end if
else
CMB%YHe = Params(9)
end if
end select
end subroutine clBK_ParamArrayToTheoryParams

function clBK_CalcDerivedParams(this, P, Theory, derived) result (num_derived)
class(clBackgroundParameterization) :: this
Type(mc_real_pointer) :: derived
class(TTheoryPredictions) :: Theory
real(mcp) :: P(:)
Type(CMBParams) CMB
integer num_derived

num_derived = 3

allocate(Derived%P(num_derived))

call this%ParamArrayToTheoryParams(P,CMB)

derived%P(1) = CMB%omk
derived%P(2) = CMB%omb
derived%P(3) = CMB%Yhe

end function clBK_CalcDerivedParams

2. Put the following in params_clbackground.paramnames in your top-level COSMOMC directory:
fbaryon       f_b # cosmic baryon fraction
omegam        \Omega_m
h             h
omegal        \Omega_\Lambda
mnu           \Sigma m_\nu
w             w         #equation of state parameter for scalar field dark energy today
wa            w_a       #w_a variation
nnu           N_{eff}   #effective number of neutrinos (only clearly defined for massless)
yhe           Y_{He} # helium mass fraction
omegak*       \Omega_K
omegab*       \Omega_b
yheused*      Y_{He} # used value (if set from bbn_consistency)

3. Here is a generic .ini file with defaults for this parametrization:
parameterization = cluster_background
get_sigma8 = F
enforce_flatness = T

param[fbaryon] = 0.17 0.0 1.0 0.03 0.03
param[omegam] = 0.27 0.0 1.0 0.04 0.04
param[h] = 0.7 0.01 10.0 0.03 0.03
param[omegal] = 0.73 0.0 2.0 0 0
num_massive_neutrinos = 3
param[mnu] = 0 0 0 0 0
param[w] = -1 -5 0 0 0
param[wa] = 0 0 0 0 0
param[nnu] = 3.046 3.046 3.046 0 0
param[yhe] = 0.24 0.24 0.24 0 0