# Measuring the baryon fraction in cluster of galaxies with

Kinematic Sunyeav Zeldovich and a Standard Candle

###### Abstract

We propose a new method to use the kinematic Sunyaev-Zeldovich for measuring the baryon fraction in cluster of galaxies. In this proposal we need a configuration in which a supernova Type Ia resides in a brightest cluster galaxy of low redshift clusters. We show this supernova Type Ia can be used to measure the bulk velocity of a galaxy cluster. In low redshifts the main contribution to the deviation of standard candles distance modulus from cosmological background prediction comes from peculiar velocity of the host galaxy. In this work we argue that by the knowledge of the bulk flow of the galaxy cluster and the cosmic microwave background photons temperature change due to kinematic Sunyaev-Zeldovich, we can constrain the baryon fraction of galaxy cluster. The probability of this configuration for clusters in low redshift is obtained. We estimate in a conservative parameter estimation the large synoptic survey telescope can find spectroscopically followed galaxy clusters in low redshift with a bright cluster galaxy which hosts a type Ia Supernova. Finally, we show the improving of the distance modulus measurement is the key improvement in future surveys which will be crucial to detect the baryon fraction of cluster with the proposed method.

## I Introduction

The baryon density of the Universe and its ratio with respect to the dark matter is fixed by Cosmic Microwave Background (CMB) radiation Ade:2015xua () and Big Bang Nucleosynthesis (BBN) Steigman:2007xt () to a fraction of . However this ratio is smaller by a factor of 2 or 3 in galaxies and cluster of galaxies. This is known as the missing baryon problem Bregman:2007ac (). It is believed that the missing baryons are in intergalactic medium in diffuse warm-hot plasma which is hard to detect in X-ray or/and they are not in virialized gravitationally bound objects and resides in voids and filaments Eckert:2015akf (); deGraaff:2017byg (). Also there is an idea that a considerable amount of missing baryon could be in cold transparent molecular clouds, which can be detected via optical scintillation method Habibi:2013sd ().
The search for the missing baryons is one of the ambitious quests of cosmology, which will shed light on the process of galaxy formation and evolution. One of the promising cosmological probes to address this problem is the study of the galaxy clusters as the largest gravitationally bound systems. It seems that the galaxy clusters suffer less from the missing baryon problem than the other structures, however they are very valuable structures in Universe to check the consistency of the universal baryon fraction McGaugh:2009mt ().
We can learn about the physics of baryons in galaxy clusters by studying the interaction of the galaxy cluster with CMB photons. Although less than one percent of the CMB photons passed through the galaxy clusters but the physics of the interaction is known and under control.
The inverse Compton scattering of CMB photons by hot intra-cluster gas of electrons change the intensity of the observed CMB. This effect is known as Sunyaev-Zeldovich (SZ) effect Zeldovich:1969ff (); Sunyaev1972 (); Birkinshaw:1998qp ().
The bulk motion of the galaxy cluster also introduce a Doppler shift effect on the CMB photons known as kinetic Sunyaev Zeldovich (kSZ) effect Sunyaev:1980nv (). The kSZ is a physical process of electron-photon scattering which keeps the CMB spectrum almost unchanged, while the thermal Sunyaev-Zeldovich (tSZ) is a process that changes the CMB spectrum. We should note that for a typical cluster
of galaxies the thermal velocities are higher than the bulk velocity and accordingly kSZ amplitude is an order of magnitude smaller than
the tSZ. Thermal SZ is observed via CMB temperature Das:2013zf (); George:2014oba () and also from individual cluster studiesPlagge:2009eu (); :2012vza (). The kSZ is more challenging to be detected because of the smaller amplitude and also the fact that it does not change the spectrum of CMB. Despite to its observational challenges, kSZ is a very valuable quantity to be measured in clusters where it can be used to address some astrophysical questions like missing baryon problem Hernandez-Monteagudo:2015cfa (); Schaan:2015uaa () and also some cosmological questions like the study of the growth of structures to constrain dark energy and modified gravity theories Haehnelt:1995dg (); Diaferio:1999ig (); Bhattacharya:2006ke (); Bhattacharya:2007sk (); Bianchini:2015iaa ().
It is worth to mention that there is a frequency band in CMB observations GHz where the tSZ change on the CMB is almost zero.
The first detection of kSZ is reported by Hand et al. (2012) Hand:2012ui () used the correlation of the Atacama Cosmology Telescope (ACT) data Swetz:2010fy () with the pair-wise velocity of the Baryon Oscillation Spectroscopic Survey (BOSS) spectroscopic catalogue Ahn:2012fh (). The kSZ signal is also detected in Planck and Sloan dataAde:2015lza (), South Pole Telescope and Dark Energy Survey cross correlation as well Soergel:2016mce ().

In this work we propose a novel idea to measure the baryon fraction in cluster of galaxies by using the kSZ effect. The temperature change due to the kSZ is proportional to the baryon fraction and the bulk velocity. In the case if we can find out the bulk velocity of the galaxy cluster then we can pin down the baryon fraction more accurately by adding the CMB-kSZ information. The main idea of this work is centered on the bulk velocity measurement. The bulk velocity is usually measured by using the X-ray catalog of clusters as a complimentary probe to kSZ Mody:2012rh () or it is obtained via reconstructed matter density field Li:2014mja (). In this direction we suggest to use standard candles such as Supernova (SNe) type Ia to measure the bulk velocity. Traditionally SNe Ia used as a probe of cosmic bulk flow Appleby:2014kea (). In this work we assume that the SN resides in the brightest cluster galaxy (BCG), accordingly this can be used as a probe of bulk velocity of cluster. Finally adding up the two independent observations of kSZ and SNe Ia, one can use it as a probe of baryon fraction in cluster. We also discuss the observational prospects of this proposal and the probability that this configuration can be observed.
The structure of this work is as follows: In Sec. (II) we discuss the theoretical framework of kSZ.
In Sec. (III) we discuss the idea of measuring the peculiar velocity via SNe type Ia. In Sec. (IV) we discuss the observational prospects of the idea raised in this work and finally we conclude in Sec.(V).

## Ii kSZ and the baryon fraction

The CMB temperature is changed due to kinetic Sunyaev-Zeldovich (kSZ) effect as

(1) |

where is the mean CMB temperature, is the comoving distance which is a function of redshift, is Thomson scattering cross section, is the optical depth, is the physical number density of free electrons, is the peculiar velocity of free electrons and is the direction defined from observer to source. The optical depth is defined as , where is the physical length of the ionized patch in the sky. Note that the optical depth is very small and we set hereafter.

This signal is an order of magnitude lower than the tSZ and also we should note that the kSZ does not change the black body spectrum of the CMB. This signal is extracted from the cross correlation with the late time tracers of gravitational potential.
In this framework the leading science is done by Atacama Cosmology Telescope(ACT), which detect the kSZ effect by correlating the signal with reconstructed velocity field obtained for the galaxies in LSS surveys such as Baryon Acoustic Oscillation Sky Survey (BOSS)Schaan:2015uaa ().
In a standard way of thinking about the kSZ effect, It seems that the signal on CMB plus our knowledge on properties of galaxy cluster can be used to obtain the peculiar velocity of the free electrons.
Now let us look to this problem in another point of view, if we can find the peculiar velocity of the galaxy cluster from an independent way, then we can use the kSZ signal to extract the information about the baryon fraction in a cluster of galaxies.
One way to reconstruct the peculiar velocity is by linear perturbation theory, the semi-local density contrast of matter perturbation can be used
as a probe of peculiar velocity due to the Euler equation Padmanabhan:2012hf (). The other method is to use the standard candles like SNe type Ia to extract the information.
We will discuss this method extensively in Sec.(III).

In what follows we reexpress the kSZ signal in terms of a) the physics of baryons, b) the physics of peculiar velocity.
With this point of view we write the physical number of electrons as

(2) |

where is the cosmic background number density of electrons and is the electron density contrast which is in order of for a virialized cluster. The background number density of electrons can be expressed in terms of cosmological parameters as

(3) |

where is the mean gas density in redshift z, is the proton mass and is the effective number of electrons per nucleon. Accordingly is the mean mass per electron. is the ionization fraction which is defined as , where is the primordial abundance of Helium and is the number of ionized electrons corresponding to Helium atoms ^{1}^{1}1 Helium atoms remain singly ionized until and below that, they are thought to be doubly ionized Shaw:2011sy ().
Now we can relate the gas density to the baryon fraction of universe and matter density of universe as below

(4) |

where is the gas fraction of baryons in a cluster and is our crucial parameter of the study, known as baryon fraction parameter. is the matter density parameter and is the Hubble parameter both defined in present time. Now by assuming that the evolution of all above parameters inside a cluster is negligible and position independent, the kSZ effect will become

(5) |

where is the size of the cluster labeled by superscript and . Now by using Eq.(3) and Eq.(4), it is straightforward to show that the signal can be written as

(6) |

where is a specific parameter for each cluster, which depends on the mean redshift, the physics of intra-cluster medium and the line of sight length of the cluster as

(7) |

Now can be written in the terms of cluster’s gas fraction, ionization factor and line of sight length

(8) |

Finally we can define a parameter for each cluster, which encapsulates in it the physics of baryonic matter and it is related to kSZ temperature change and the bulk flow of the cluster as below

(9) |

where the approximation works for , , , , and . This means that by knowing the temperature change of the CMB due to the kSZ effect and the peculiar velocity measurement by an independent method we can constrain the physics of free electrons and baryon fraction in a cluster. In the next section we will discuss that how SNe type Ia will be a great candidate in order to calculate the bulk flow parameter.

## Iii Standard Candles as a probe of peculiar velocity

In this section we will discuss how standard candles can be used to determine the peculiar velocity of the structures.
The idea is straightforward, the standard candles are used to establish a luminosity distance-redshift relation for a given background cosmology model.
However the deviation from the homogenous background will change this relation. One of the important modifications is due to the peculiar velocity of the host galaxy of a SN, which affects both the luminosity and redshift of the standard candle.
Accordingly, we can use the deviation of the luminosity distance of a SN type Ia as a probe of peculiar velocity.

For this task, we assume that we are living in a perturbed FRW universe with a Newtonian comoving gauge chosen metric

(10) |

where is the conformal time, and are the scalar perturbations of metric. If we assume that the General Relativity (GR) is the correct classical theory of gravity and also assume that the universe is filled with components that have no anisotropic pressure, then we get .

In a perturbed universe, the luminosity distance of a standard candle is corrected by the change in the space-time component of the light propagation induced by amount of the matter in line of sight (Sachs Wolfe effect and gravitational lensing). The luminosity distance is also corrected due to the peculiar velocity of the source and observer. These corrections can be formulated in the equation belowBonvin:2005ps (); Bacon:2014uja (); Baghram:2014qja ()

(11) |

where is the luminosity distance of a supernova in observed redshift of the source and direction (Note that is unit vector in the direction of observer toward source). is the comoving distance of the source in FRW universe. The parameter is the correction due to peculiar velocity of source. The , and are the lensing convergence, Sachs-Wolfe and Integrated Sachs-Wolfe correction terms. (These terms are defined and studied extensively in Bacon:2014uja ()). In the redshift range of the most contribution to the luminosity distance change comes from the peculiar velocity Habibi:2014cva ()

(12) |

where is the background luminosity distance and is luminosity correction due to peculiar velocity Bonvin:2005ps () defined as

(13) |

where is the Hubble parameter and where ( is the peculiar velocity). In low and intermediate redshifts (), the term in parentheses is negative, accordingly the objects moving toward us () introduce a where if we replace this in Eq.(12) we will get a dimmer SN. In the other hand when the host of a standard candle is moving away from us (), that introduces a and accordingly the source become brighter. These chain of conclusions are changed in higher redshifts.

The main idea of this work is that we can use SNe for finding the peculiar velocity (bulk flow) of the galaxy cluster.
Before proceed further, we should numerate the velocity contributions to our specific case of study. The velocities which can be assigned to a SN Ia are:

a) The peculiar velocity introduced from the progenitor of the SNIa, b) The peculiar velocity of the host galaxy due to the gravitational potential of the cluster and c) The peculiar velocity due to the bulk flow of the galaxy cluster.
In this work we are merely interested in bulk flow, accordingly in order to extract this term, which is the source of the kSZ effect, we assume, that the host galaxy of the supernova is the one which is located in the gravitational/ optical center of the galaxy cluster (bright cluster galaxy (BCG)).
The assumption that the SN host galaxy is in the center of the cluster is needed, as we want to assign the bulk velocity of the cluster to the peculiar velocity of the BCG. This means that bulk velocity of cluster, which is measured the velocity of the free electrons is the dominant term. This assumption comes from the idea that the central bright galaxies reside in the minimum of the gravitational potential well of the cluster and the peculiar velocity with respect to the center of the mass of a cluster is smaller than the total bulk motion of the whole system.

We assert that main contribution to the luminosity change of a SN in BCG is due to the peculiar velocity of the bulk motion of the cluster. This configuration is schematically shown in Fig.(1). However we should keep in mind that the effect of SNe Ia’s progenitor’s velocity may have a very significant velocity offset with respect to the peculiar velocity of the BCG host galaxy. This offset is included as a source of an error in our upcoming estimations. This can be done by the idea that the progenitor velocity introduce a Gaussian error to the velocity estimation related to the dynamical mass of BCG. The contribution of this error can be calculated by setting the velocity of the center of mass of a SNIa progenitor equal to the dispersion velocity inside a typical BCG using the fundamental plane relation of elliptical galaxies BoylanKolchin:2006rs (); Bernardi:2006qm (), for a specific example, we express the dispersion velocity in terms of magnitude Lauer:2014pfa () as where is the metric magnitude ^{2}^{2}2The metric magnitude obtained from measuring the Luminosity from the central region of BCG which is much smaller than extent of galaxy. The radius is obtained by the correlation of the Luminosity with structural parameter of BCG. For details see Postman1995 () and is the velocity of the center of mass of a SNIa progenitor. In this work we set the dispersion velocity (progenitor velocity) to .

Now by using Eq.(13), we can find the line of sight normalized velocity as

(14) |

where is the difference of the observed distance modulus and the one predicted from background cosmology of CDM in the specific redshift of . The parameter is a unique term which is independent of the local physics, instead it depends on the background cosmological parameters and the redshift of the supernova. Note that is the normalized Hubble parameter to its present value. Now by using Eq.(14) we can calculate the line of sight velocity of a SN, then we can assign this to the bulk velocity of the galaxy cluster which is the host of the SN where superscript represent the host of a SN. Now by this identification we can use Eq.(9) to extract the baryon fraction. This can be done because the RHS of this equation is fixed by two independent observation. As mentioned in introduction the peculiar velocity can be measured by different method like velocity reconstruction with a galaxy fieldHo:2009iw () as well. Accordingly the obtained velocity can be cross-checked with other methods. In the next section we will discuss the observational prospects of finding the missing baryons by the method described in this work.

## Iv Observational Prospects

In this section, we will discuss the observational prospects of the idea proposed in previous sections. In the first subsection, we use the data sample of Union 2.1 in order to represent the logical path of extracting the bulk flow of the clusters from the data with assuming that each SNIa resides in a BCG. We should note that the first subsection is just an example of how this method works and shows the current status of quality of the data. In the second subsection we discuss the error estimation for realistic and optimistic case for finding the baryon fraction and finally in the third subsection we present an estimate for the number of events that we anticipate in future surveys which are suitable for our case of study.

### iv.1 Union 2.1 SNe Ia data sample as preliminary example

In this direction first we assume that all the SNe data from the known catalog are a potentially plausible candidates for our proposal. It means we assume that they are hosted by a central galaxy of a cluster. This is just an assumption to show the details of the proposal. We use the Union 2.1 SNe sample Suzuki:2011hu () to show the procedure. First of all we extract the peculiar velocity with the method which is described in Sec.(III)by assuming that the standard CDM model with best parameters fixed by supernova data Ade:2015xua () describe the cosmological flat background (, ). We obtain the difference of distance modulus versus redshift for the sample of SNe. Note that is the distance modulus from the known background cosmology.

In Fig.(2) we plot versus redshift for SN Ia in redshift range of . As it mentioned, in this redshift range the peculiar velocity correction is the dominant effect in the luminosity change of the SNe Ia Bacon:2014uja (); Habibi:2014cva (), where we neglect the contribution of weak lensing convergence and the Sachs-Wolfe effect. In Fig.(2) the red data points are SNe Ia in the redshift rang of our interest with average error of mag. The data points show the SNe data that are consistent with background CDM predictions in almost . The blue low amplitude dashed curves represent the amount of correction that we expect from the peculiar velocity in linear regime. The linear regime velocity in the line of sight can be obtained via linear matter spectrum as:

(15) |

where is the average linear velocity in the line of sight (assuming an isotropic velocity) in a window function with a comoving radius . The parameter is the growth rate which for standard CDM is equal to (Note that is the normalized Hubble parameter to its present value). The blue curves shows the prediction of standard model in perturbation level. The dotted high amplitude green lines are obtained by assuming a maximum line of sight velocity of . In the next step, we assign the deviation of the distance modulus change to the peculiar velocity of the host galaxies. Reasonably, the next step is to find the line of sight ( ) velocity of each SNe.

In Fig.(3) we plot the reconstructed line of sight velocity in terms of redshift. We use Eq.(14) to extract the line of sight velocity. The error bars are obtained from the propagation of the error in distance modulus and in this step also an error is added due to the velocity of the SNe progenitors. For this task we use a Monte-Carlo method to add a Gaussian error .
The very interesting point to indicate here is that the SNe that seems brighter ( ) moves away from us . This observation is consistent to the argument we made before for peculiar velocity effect in low redshifts.
In Fig.(3), we also obtained the peculiar velocity with an optimistic resolution. The data with smaller blue error-bars come from the assumption that the error-bars of distance modulus in the optimistic case become mag in average ( improvement), which is the characteristic magnitude resolution which can be obtained from LSST survey Abell:2009aa (). Fig.(3) shows that how the accuracy in SNIa luminosity distance measurement are important in this study.

In order to find out about the baryon fraction, now by knowing the kSZ temperature change for each galaxy cluster, we can find the baryon fraction keeping in mind that each SNIa galaxy host is the BCG under investigation. Then
with this assumption we can find the baryon fraction with the relation below:

(16) |

where obtained from Eq.(8) and depends on the background cosmology. We should note that Eq.(16) is the key equation of our proposal. The baryon fraction of cluster can be obtained by its kSZ temperature and the luminosity change of a SN which is hosted by the central galaxy of a cluster. In the next subsection we discuss the error propagation due to other components of Eq. (16), which will introduce the uncertainty in baryon fraction calculation. In Fig. (4), we plot the Fisher forecast of the baryon fraction measurement via SNe Ia in the redshift range of for our configuration. We set the fiducial parameter and plot the prediction of our method for realistic and optimistic cases. In the optimistic case we consider two categories of red-line with the corresponding errors and (blue-line). The optimistic case in this stage is just chosen to show how the baryon fraction can be constrained by future experiments. Note that is the error due to the SNe Ia uncertainties related to intrinsic magnification errors, photometric errors and etc. In the next subsection, we discuss the error estimation and we study in more detail how different aspects of this proposal introduce uncertainties. Also we will show that the improvement in luminosity distance measurement has the main contribution for make this proposal a observationally practical one.

### iv.2 The error estimation

In this section, we study the different error budget which has a role in baryon fraction measurement via the method which is proposed in this work. For simplicity we assume that the background cosmological model is fixed by other observations such as CMB and the errors on density parameters and Hubble constant is much more smaller than the uncertainties in the physics of the cluster, supernova observations and kSZ signal extraction. This means that the uncertainty in baryon fraction calculation can be written as

(17) |

where is the error in kSZ signal from CMB analysis is due to SNe Ia uncertainties which can be due to intrinsic magnification errors, photometric errors, the peculiar velocity of the SNe Ia progenitors and … and is the error induced from the physics of the clusters. We also assume that errors from different contributions are uncorrelated and Gaussian distributed, which is almost a reasonable assumption. In what follows we will numerate the different contributions to the total error introduced in this proposal and then we will address that which errors can be reduced in future experiments.

Kinematic Sunyeav Zeldovich: For kSZ the main problem is that the spectrum has almost a flat power in frequency range, that is why the primordial CMB anisotropies themselves are the most important source of contamination. Accordingly most of kSZ
extraction methods try to separate the cluster signal of kSZ and primordial anisotropies.
Regarding the angular resolution, the galaxy cluster is in order of a few arc-minutes which is related to the angular moment of . This is a good news since the primordial CMB anisotropies are damped for large moments. Accordingly a proper filter can separate the two signals. In a realistic error estimation, we set the relative uncertainty as Swetz:2010fy () and for futuristic (e.g. CMB stage IV experiments) one we set Alonso:2016jpy () (Note that tilde is an indication of relative errors and subscripts ”r” and ”opt” are for realistic and optimistic cases respectively).

Supernovae Type Ia: In the case of the SNe Ia, nowadays with the observation of around supernovae the systematics become the dominant one in error budget in comparison to the statistical errors Howell:2010vd (). This systematics are also survey dependent, accordingly all the supernovae compilation have the problem of calibration matching where photometric offsets is one of the main challenges of using more than one survey. In this direction the future surveys like LSST which has the plan to make a dedicated long run surveys of sky for hunting the SNe Ia, can overcome this problem Abell:2009aa (). Generally the systematics of the SNe Ia intrinsic luminosity measurement can categorized due to the effects such as:

a) calibration: one of the major obstacles in low redshift SNe Ia measurements becomes from the usage of old Landolt photometric system. However for each SN we need a calibration by knowing the filter transmission rate and K-correction. Note that one needs the knowledge of the spectral energy distribution (SED) for K-correction Hsiao:2007pv (). Also we should indicate that large number of low redshift SNe will solve this problem by using a new system of calibration.

b) the ultraviolet treatment of high redshift SNe Ia: one of the main challenges of high redshift SN is the intrinsic scatter in their light curves and their poor calibration Kessler:2009ys (). In the case of our proposal, we can neglect this error as we are dealing with low redshift SNe Ia.

c) reddening due to dust: This is one of the major uncertainties introduced in the modeling of the SNe Ia. It seems that the redder SNe are also dimmer Riess:1996pa (). This means that there is an intrinsic color-luminosity relation in SN data which can become degenerate from the reddening of the host galaxy’s dust. As the color is correlated to the SN’s properties so associating the wrong color to the host of SN can be wrongly assigned to the reddening. The idea to overcome this error is to study the physics of the color-luminosity and reddening simultaneously with independent observations. The infrared observations of the host galaxy can be a solution to decrease the contamination of the dust Shariff:2016cxw (). A final note in this category is that the different SN progenitors can have different circumstellar dust properties.

d) environment dependence and redshift evolution of SNe Ia: It seems that there is an indication that most luminous SNe Ia occurred in late type galaxies Hamuy:1996sq () and sub luminous ones are in old population galaxies Howell:2001hr (). This can be regarded as a strong evidence for redshift and environment dependence of SNe Ia. Now we have an indication that the width and luminosity of SNe Ia light curve is correlated to the mass, star formation rate and the metallicity of the host galaxy Sullivan:2006ah (); Gallagher:2008zi (); Sullivan:2010mg (). As the star formation rate is a redshift dependent quantity accordingly we can anticipate that the SNe Ia absolute magnitude can be a redshift dependent as well. However, now we are entering a large data set era, which by comparing the SNe Ia and their host galaxies properties we can improve the SNe Ia modeling. In other words the large statistics can be used to control the systematics.

e) the physics of progenitor: the last but not the least effect of the error estimation of SNe Ia as a standard candle is the progenitor. There are mainly three scenarios of SNe Type Ia, known as single degenerate (SD) scenario Whelan1973 (), double degenerate (DD) scenario Iben1984 (); Webbink1984 () and sub-Chandra Weesley1986 (). The first scenario assumes that a white dwarf with a companion from main sequence of stars or a red giant is the generator of the nova and in the second one, we assume that the binaries are both white dwarfs. The sub-Chandra model assumes that a layer of helium appears on the surface of a white dwarf below the Chandrasekhar mass until it detonates.
The main goal of future SNe Ia survey like LSST is to observe a large number of SNe Ia (e.g. 50,000 Type Ia per year in LSST survey ), where the good statistics in color and light curves, combined with a small number of sample spectra, any dependence of the supernova standard candle relation on parameters
other than light curve shape and extinction can be extracted. Accordingly these can be minimize the systematics via statistics. The LSST has a plan to reduce the distance indicator error to mag which independently can constrain the dark energy equation of state with statistics better than .

Another important observation which can shed light on the distance measurements in local universe is the GAIA project which is a space observatory of the European Space Agency (ESA) designed for astrometry which can be a great help to calibrate the cepheid variables and make the distance ladder more accurate Barstow:2014dda ().

Galaxy cluster: Finally we come up with the relative error due to the physics of galaxy cluster, where the main contribution is raised form the size of the galaxy and the gas fraction of the cluster. The relative uncertainty is set .

In order to show the effect of different type of the errors on baryon fraction measurement, in Fig.(5) we plot the relative error with respect to the redshift. The redshift range of is divided to four bins with almost equal number of SNe Ia. The mean and the variance of the relative error is plotted (red solid line) for the realistic case that we considered in previous section. Then in order to study the effect of different error budget, we study the case where relative error in kSZ measurement can be improved by a factor of four (by e.g CMB stage IV) and the measurement of the characteristic of the galaxy cluster will become better by a factor of two. As it is shown in Fig.(5) there is much improvement in the relative error. In the next step we plot the relative error estimation by assuming the characteristic resolution of LSST project which can improve the magnitude measurement to (blue dashed data points) also the GAIA project which can improve the current status of the luminosity measurement in local universe by an order of magnitude (pink dotted data points). In order to conclude this subsection, we assert that the main contribution of the error comes from the SNe Ia measurement. The future experiments of SNeIa can improve the distance measurement drastically, which can bring this method in a new and practical state. In the next subsection we will discuss a very important issue of the probability of the detection of the configuration (SNe Ia in a BCG) which we proposed in this work.

### iv.3 The expected number of SNe Ia in a galaxy cluster

In order to finalize our proposal for obtaining the baryon fraction, we should address the important question: ”How many events of SNe Type Ia explosions are happened in the central galaxy of a cluster”. Accordingly in this subsection, we propose two distinct scenarios for the number of expected events for SNe happened in galaxy cluster. First we anticipate that the SNe Ia happens in the central galaxy of a cluster. Then we assume that more than one SNe Ia happens in galaxy cluster. For the first scenario, we formulate this probability due to the simplified relation below where is the number of observed BCG with the desired condition of our proposal.

(18) |

The is a function of the observation time and also it depends on the cosmological volume of the survey and the portion of the sky it spans. All of this information is encapsulated in symbolic parameter of which stands for survey. It is obvious that the observed number of configuration depends on physics of BCG ( indicated by ”bcg” ) and the physics of SNIa (indicated by ”” in eq.(18)). The is the number of BCGs that resides in clusters which has a mass with low bound , where indicate lower limit of a typical BCG which resides in the redshift range of up to . The parameter is the probability that a SNIa occurred in a BCG of a cluster. The number of BCG in galaxy clusters in a cosmological volume, which is limited by the redshift range, that we are interested in is as below

(19) |

where we set as the lower limit of BCG, and is the lower and upper limit of the redshift survey that we are interested and is the fraction of luminous massive galaxies that resides in the center of galaxy cluster.
Due to the fact that BCGs are assembled mainly by major mergers in galaxy formation and evolution process, there is an indication that the most massive galaxies must reside in galaxy clusters, accordingly we set for this study.
In order to obtain the number density of cluster, we use the Press-Schechter approach to find , specifically when we use the fitting function introduced in Jenkins et al Jenkins:2000bv (). Accordingly the number of BCG galaxies become , where is the cosmological volume. Note that the effective volume is for a survey which is in search of galaxies in redshift span of .
Another important parameter is to estimate is the rate of SN Ia in a galaxy with the given mass range. In Graur et al. Graur:2014bua (), there is an extensive study on the rate of Type Ia supernova. Graur et al. used SNe samples to measure mass-normalized SNe rates as a function of stellar mass of the host galaxy and the star formation rate. By assuming the stellar mass of (with the assumption of a mass to light ratio of ) and the star formation rate we will have the marginalized fitting function as . It is worth to mention that the SN rate is proportional to star formation rate and specific start formation rate, accordingly the rate of SN decrease with evolution of BCGs from active to passive ones.
However in low redshift (), the specific star formation rate in BCGs is declining more slowly with time than for field or cluster galaxies, most likely due to the fuel from the cooling of inter-cluster medium McDonald:2015kym (). With all this complicated physics which is governing the SNIa rate relation with the host galaxy, we set the conservative rate of .

A project like Large Synoptic Survey Telescope (LSST) which is designed to operate for 10 years starting from 2019 and will capable of spanning the square degree of the sky (which means ) in 6 optical bandwidth with a limited magnitude to a total point-source depth of Abell:2009aa (), we can estimate the per year with the conservative assumptions we made in this section.
However we should note that one should also take into account that precise spectra for each of the SNe are absolutely necessary: the SNe detection does
not suffice, but a spectroscopic follow-up program should follow. Assuming that we will have a follow up of of the SNe Ia in a very conservative point of view in LSST life time project we can estimate the baryon fraction of more than 30 clusters of galaxies.
As a final word, the other scenario that can be used as a proposal for using the SNe for estimating the baryon fraction is that in a cluster of galaxy one can monitor and measure all SNe in all galaxies within cluster, then on average the peculiar velocities of those SNe would be close to the bulk flow.
In this case, the error on the bulk velocity measurement can be decreased due to the number of SNe Ia found in a group/cluuster of galaxies.
In the next section we will conclude the paper by future prospects.

## V Conclusion and Future Prospects

The distribution of the baryons in the Universe is one of the main questions in cosmology, the big bang nucleosynthesis and cosmic microwave background radiation independently fix the baryon fraction. However the accessability of baryons in late time is a challenging task and it seems that there is a missing baryon problem out there. The galaxy clusters as the main reservoir of baryonic matter which are filled with ionized electrons are suitable environments to study the distribution and physics of baryons in the Universe. One of the promising venues to address the distribution of baryons in the sky is the study of thermal and kinetic Sunyaev-Zeldovich effect which are used as a probe of ionized gas in clusters. In this work we propose a new idea/method to measure the baryon fraction using the kSZ effect and the SNe-Type Ia as the standard candle. For this method to work we assume that a SNIa explodes in a BCG. This is essential in a sense that the peculiar velocity of a BCG in a cluster can be used as an almost fair representative of the cluster’s bulk flow. The BCG resides in the depth of gravitational potential of cluster and its velocity with respect to the center of mass of the system is almost zero.

However in this work we indicate that the main uncertainty of velocity measurement comes from the systematics of SNe Ia including its progenitor velocity.
Accordingly we show that the SNe Ia in low redshift can be used to estimate the peculiar velocity of host galaxy.
In the other hand The kSZ effects on CMB temperature change depends on the baryon fraction and the bulk velocity of galaxy cluster. We assert that the deviation of a standard candle distance modulus from background prediction of the CDM can be related to the peculiar velocity of SNIa host galaxy in lower redshifts, keeping in mind that the host galaxy is chosen to be a BCG. We showed that by knowledge of the peculiar velocity and the temperature change of CMB we can constrain the baryon fraction. We investigate the Fisher forecast for the fiducial value of baryon fraction in the realistic (current) case and also in optimistic state. The analysis are discussed in the second subsection of Sec.(IV).
In the observational prospect part, we also study the possibility of the observation of this effect. We estimate that in a future large scale survey, like LSST which spans half of the sky in years, we can do an optical spectroscopic follow up at least for SNe Ia which explodes in a BCG in a cluster of galaxy.
It is worth to mention that in the case of more statistics in galaxy cluster we can average the peculiar velocity of each host galaxy of SNIa , the average of squared velocity will be a representative of the bulk flow and the dispersion of velocities represent the error bar on bulk velocity.
As a final remark we want to insist that future SNe Ia surveys will decrease the errors on baryon fraction and will bring the introduced method as a practically viable proposal.

###### Acknowledgements.

We would like to thank Farhang Habibi for detailed and insightful comments on the manuscript. Also we would like to thank Saeed Ansari, Alireza Hojati, Nima Khosravi, Sohrab Rahvar and Saeed Tavasoli for useful discussions. SB acknowledges the hospitality of the Abdus Salam International Centre for Theoretical Physics (ICTP) during the final stage of this work.## References

- (1) P. A. R. Ade et al. [Planck Collaboration], “Planck 2015 results. XIII. Cosmological parameters,” arXiv:1502.01589 [astro-ph.CO].
- (2) G. Steigman, “Primordial Nucleosynthesis in the Precision Cosmology Era,” Ann. Rev. Nucl. Part. Sci. 57, 463 (2007) doi:10.1146/annurev.nucl.56.080805.140437 [arXiv:0712.1100 [astro-ph]].
- (3) J. N. Bregman, “The Search for the Missing Baryons at Low Redshift,” Ann. Rev. Astron. Astrophys. 45, 221 (2007) doi:10.1146/annurev.astro.45.051806.110619 [arXiv:0706.1787 [astro-ph]].
- (4) D. Eckert et al., Nature 528, 105 (2015) doi:10.1038/nature16058 [arXiv:1512.00454 [astro-ph.CO]].
- (5) A. de Graaff, Y. C. Cai, C. Heymans and J. A. Peacock, “Missing baryons in the cosmic web revealed by the Sunyaev-Zel’dovich effect,” arXiv:1709.10378 [astro-ph.CO].
- (6) F. Habibi, M. Moniez, R. Ansari and S. Rahvar, “Simulation of Optical Interstellar Scintillation,” Astron. Astrophys. 552, A93 (2013) doi:10.1051/0004-6361/201220172 [arXiv:1301.0514 [astro-ph.IM]].
- (7) S. S. McGaugh, J. M. Schombert, W. J. G. de Blok and M. J. Zagursky, “The Baryon Content of Cosmic Structures,” Astrophys. J. 708, L14 (2010) doi:10.1088/2041-8205/708/1/L14 [arXiv:0911.2700 [astro-ph.CO]].
- (8) Y. B. Zeldovich and R. A. Sunyaev, “The Interaction of Matter and Radiation in a Hot-Model Universe,” Astrophys. Space Sci. 4, 301 (1969).
- (9) R.A. Sunyaev and Y.B. Zeldovich, Comments on Astrophysics and. Space. Physics, 4 ,173 (1972)
- (10) M. Birkinshaw, “The Sunyaev-Zel’dovich effect,” Phys. Rept. 310, 97 (1999) doi:10.1016/S0370-1573(98)00080-5 [astro-ph/9808050].
- (11) R. A. Sunyaev and Y. B. Zeldovich, “The Velocity of clusters of galaxies relative to the microwave background. The Possibility of its measurement,” Mon. Not. Roy. Astron. Soc. 190, 413 (1980).
- (12) S. Das et al., “The Atacama Cosmology Telescope: temperature and gravitational lensing power spectrum measurements from three seasons of data,” JCAP 1404, 014 (2014) doi:10.1088/1475-7516/2014/04/014 [arXiv:1301.1037 [astro-ph.CO]].
- (13) E. M. George et al., “A measurement of secondary cosmic microwave background anisotropies from the 2500-square-degree SPT-SZ survey,” Astrophys. J. 799, no. 2, 177 (2015) doi:10.1088/0004-637X/799/2/177 [arXiv:1408.3161 [astro-ph.CO]].
- (14) T. Plagge et al., “Sunyaev-Zel’dovich Cluster Profiles Measured with the South Pole Telescope,” Astrophys. J. 716, 1118 (2010) doi:10.1088/0004-637X/716/2/1118 [arXiv:0911.2444 [astro-ph.CO]].
- (15) P. A. R. Ade et al. [Planck Collaboration], “Planck Intermediate Results. V. Pressure profiles of galaxy clusters from the Sunyaev-Zeldovich effect,” Astron. Astrophys. 550, A131 (2013) doi:10.1051/0004-6361/201220040 [arXiv:1207.4061 [astro-ph.CO]].
- (16) C. HernÃ¡ndez-Monteagudo, Y. Z. Ma, F. S. Kitaura, W. Wang, R. GÃ©nova-Santos, J. MacÃas-PÃ©rez and D. Herranz, “Evidence of the Missing Baryons from the Kinematic Sunyaev-Zeldovich Effect in Planck Data,” Phys. Rev. Lett. 115, no. 19, 191301 (2015) doi:10.1103/PhysRevLett.115.191301 [arXiv:1504.04011 [astro-ph.CO]].
- (17) E. Schaan et al., “Evidence for the kinematic Sunyaev-Zeľdovich effect with ACTPol and velocity reconstruction from BOSS,” arXiv:1510.06442 [astro-ph.CO].
- (18) M. G. Haehnelt and M. Tegmark, “Using the kinematic Sunyaev-Zeldovich effect to determine the peculiar velocities of clusters of galaxies,” Mon. Not. Roy. Astron. Soc. 279, 545 (1996) doi:10.1093/mnras/279.2.545 [astro-ph/9507077].
- (19) A. Diaferio, R. A. Sunyaev and A. Nusser, “Large scale motions in superclusters: their imprint in the cmb,” Astrophys. J. 533, L71 (2000) doi:10.1086/312627 [astro-ph/9912117].
- (20) S. Bhattacharya and A. Kosowsky, “Cosmological Constraints from Galaxy Cluster Velocity Statistics,” Astrophys. J. 659, L83 (2007) doi:10.1086/517523 [astro-ph/0612555].
- (21) S. Bhattacharya and A. Kosowsky, “Dark Energy Constraints from Galaxy Cluster Peculiar Velocities,” Phys. Rev. D 77, 083004 (2008) doi:10.1103/PhysRevD.77.083004 [arXiv:0712.0034 [astro-ph]].
- (22) F. Bianchini and A. Silvestri, “Kinetic Sunyaev-Zelâdovich effect in modified gravity,” Phys. Rev. D 93, no. 6, 064026 (2016) doi:10.1103/PhysRevD.93.064026 [arXiv:1510.08844 [astro-ph.CO]].
- (23) N. Hand et al., “Evidence of Galaxy Cluster Motions with the Kinematic Sunyaev-Zel’dovich Effect,” Phys. Rev. Lett. 109, 041101 (2012) doi:10.1103/PhysRevLett.109.041101 [arXiv:1203.4219 [astro-ph.CO]].
- (24) D. S. Swetz et al., Astrophys. J. Suppl. 194, 41 (2011) doi:10.1088/0067-0049/194/2/41 [arXiv:1007.0290 [astro-ph.IM]].
- (25) C. P. Ahn et al. [SDSS Collaboration], “The Ninth Data Release of the Sloan Digital Sky Survey: First Spectroscopic Data from the SDSS-III Baryon Oscillation Spectroscopic Survey,” Astrophys. J. Suppl. 203, 21 (2012) doi:10.1088/0067-0049/203/2/21 [arXiv:1207.7137 [astro-ph.IM]].
- (26) P. A. R. Ade et al. [Planck Collaboration], “Planck intermediate results. XXXVII. Evidence of unbound gas from the kinetic Sunyaev-Zeldovich effect,” Astron. Astrophys. 586, A140 (2016) doi:10.1051/0004-6361/201526328 [arXiv:1504.03339 [astro-ph.CO]].
- (27) B. Soergel et al. [DES and SPT Collaborations], “Detection of the kinematic Sunyaev-Zel’dovich effect with DES Year 1 and SPT,” [arXiv:1603.03904 [astro-ph.CO]].
- (28) K. Mody and A. Hajian, “One Thousand and One Clusters: Measuring the Bulk Flow with the Planck ESZ and X-Ray Selected Galaxy Cluster Catalogs,” Astrophys. J. 758, 4 (2012) doi:10.1088/0004-637X/758/1/4 [arXiv:1202.1339 [astro-ph.CO]].
- (29) M. Li, R. E. Angulo, S. D. M. White and J. Jasche, “Matched filter optimization of kSZ measurements with a reconstructed cosmological flow field,” Mon. Not. Roy. Astron. Soc. 443, no. 3, 2311 (2014) doi:10.1093/mnras/stu1224 [arXiv:1404.0007 [astro-ph.CO]].
- (30) S. Appleby, A. Shafieloo and A. Johnson, “Probing bulk flow with nearby SNe Ia data,” Astrophys. J. 801, no. 2, 76 (2015) doi:10.1088/0004-637X/801/2/76 [arXiv:1410.5562 [astro-ph.CO]].
- (31) N. Padmanabhan, X. Xu, D. J. Eisenstein, R. Scalzo, A. J. Cuesta, K. T. Mehta and E. Kazin, Mon. Not. Roy. Astron. Soc. 427, no. 3, 2132 (2012) doi:10.1111/j.1365-2966.2012.21888.x [arXiv:1202.0090 [astro-ph.CO]].
- (32) C. Bonvin, R. Durrer and M. A. Gasparini, Phys. Rev. D 73, 023523 (2006) [Phys. Rev. D 85, 029901 (2012)] doi:10.1103/PhysRevD.85.029901, 10.1103/PhysRevD.73.023523 [astro-ph/0511183].
- (33) D. J. Bacon, S. Andrianomena, C. Clarkson, K. Bolejko and R. Maartens, Mon. Not. Roy. Astron. Soc. 443, no. 3, 1900 (2014) doi:10.1093/mnras/stu1270 [arXiv:1401.3694 [astro-ph.CO]].
- (34) S. Baghram, S. Tavasoli, F. Habibi, R. Mohayaee and J. Silk, “Unraveling the nature of Gravity through our clumpy Universe,” Int. J. Mod. Phys. D 23, no. 12, 1442025 (2014) doi:10.1142/S0218271814420255 [arXiv:1411.7010 [astro-ph.CO]].
- (35) F. Habibi, S. Baghram and S. Tavasoli, “Peculiar velocity measurement in a clumpy universe,” arXiv:1412.8457 [astro-ph.CO].
- (36) M. Boylan-Kolchin, C. P. Ma and E. Quataert, “Red mergers and the assembly of massive elliptical galaxies: the fundamental plane and its projections,” Mon. Not. Roy. Astron. Soc. 369, 1081 (2006) doi:10.1111/j.1365-2966.2006.10379.x [astro-ph/0601400].
- (37) M. Bernardi, J. B. Hyde, R. K. Sheth, C. J. Miller and R. C. Nichol, “The luminosities, sizes and velocity dispersions of Brightest Cluster Galaxies: Implications for formation history,” Astron. J. 133, 1741 (2007) doi:10.1086/511783 [astro-ph/0607117].
- (38) T. R. Lauer, M. Postman, M. A. Strauss, G. J. Graves and N. E. Chisari, “Brightest Cluster Galaxies at the Present Epoch,” Astrophys. J. 797, no. 2, 82 (2014) doi:10.1088/0004-637X/797/2/82 [arXiv:1407.2260 [astro-ph.GA]].
- (39) S. Ho, S. Dedeo and D. Spergel, “Finding the Missing Baryons Using CMB as a Backlight,” arXiv:0903.2845 [astro-ph.CO].
- (40) N. Suzuki et al., “The Hubble Space Telescope Cluster Supernova Survey: V. Improving the Dark Energy Constraints Above z¿1 and Building an Early-Type-Hosted Supernova Sample,” Astrophys. J. 746 (2012) 85 doi:10.1088/0004-637X/746/1/85 [arXiv:1105.3470 [astro-ph.CO]].
- (41) A. Jenkins, C. S. Frenk, S. D. M. White, J. M. Colberg, S. Cole, A. E. Evrard, H. M. P. Couchman and N. Yoshida, “The Mass function of dark matter halos,” Mon. Not. Roy. Astron. Soc. 321, 372 (2001) doi:10.1046/j.1365-8711.2001.04029.x [astro-ph/0005260].
- (42) O. Graur, F. B. Bianco and M. Modjaz, “A unified explanation for the supernova rate-galaxy mass dependence based on supernovae detected in Sloan galaxy spectra,” Mon. Not. Roy. Astron. Soc. 450, no. 1, 905 (2015) doi:10.1093/mnras/stv713 [arXiv:1412.7991 [astro-ph.HE]].
- (43) M. McDonald et al., “Star-Forming Brightest Cluster Galaxies at : A Transitioning Fuel Supply,” Astrophys. J. 817, 86 doi:10.3847/0004-637X/817/2/86 [arXiv:1508.06283 [astro-ph.GA]].
- (44) P. A. Abell et al. [LSST Science and LSST Project Collaborations], “LSST Science Book, Version 2.0,” arXiv:0912.0201 [astro-ph.IM].
- (45) D. Alonso, T. Louis, P. Bull and P. G. Ferreira, “Reconstructing cosmic growth with kinetic Sunyaev-Zeldovich observations in the era of stage IV experiments,” Phys. Rev. D 94, no. 4, 043522 (2016) doi:10.1103/PhysRevD.94.043522 [arXiv:1604.01382 [astro-ph.CO]].
- (46) D. A. Howell, “Type Ia Supernovae as Stellar Endpoints and Cosmological Tools,” Nature Commun. 2, 350 (2011) doi:10.1038/ncomms1344 [arXiv:1011.0441 [astro-ph.CO]].
- (47) P. A. Abell et al. [LSST Science and LSST Project Collaborations], “LSST Science Book, Version 2.0,” arXiv:0912.0201 [astro-ph.IM].
- (48) E. Y. Hsiao et al. [SNLS Collaboration], “K-corrections and spectral templates of Type Ia supernovae,” Astrophys. J. 663, 1187 (2007) doi:10.1086/518232 [astro-ph/0703529 [ASTRO-PH]].
- (49) R. Kessler et al., “First-year Sloan Digital Sky Survey-II (SDSS-II) Supernova Results: Hubble Diagram and Cosmological Parameters,” Astrophys. J. Suppl. 185, 32 (2009) doi:10.1088/0067-0049/185/1/32 [arXiv:0908.4274 [astro-ph.CO]].
- (50) A. G. Riess, W. H. Press and R. P. Kirshner, “A Precise distance indicator: Type Ia supernova multicolor light curve shapes,” Astrophys. J. 473, 88 (1996) doi:10.1086/178129 [astro-ph/9604143].
- (51) H. Shariff, S. Dhawan, X. Jiao, B. Leibundgut, R. Trotta and D. A. van Dyk, “Standardizing Type Ia supernovae optical brightness using near-infrared rebrightening time,” Mon. Not. Roy. Astron. Soc. 463, no. 4, 4311 (2016) doi:10.1093/mnras/stw2278 [arXiv:1605.08064 [astro-ph.CO]].
- (52) M. Hamuy, M. M. Phillips, R. A. Schommer, N. B. Suntzeff, J. Maza and R. Aviles, “The Absolute luminosities of the Calan/Tololo type IA supernovae,” Astron. J. 112, 2391 (1996) doi:10.1086/118190 [astro-ph/9609059].
- (53) D. A. Howell, “The progenitors of subluminous type ia supernovae,” Astrophys. J. 554, L193 (2001) doi:10.1086/321702 [astro-ph/0105246].
- (54) M. Sullivan et al. [SNLS Collaboration], “Rates and properties of type Ia supernovae as a function of mass and star-formation in their host galaxies,” Astrophys. J. 648, 868 (2006) doi:10.1086/506137 [astro-ph/0605455].
- (55) J. S. Gallagher, P. M. Garnavich, N. Caldwell, R. P. Kirshner, S. W. Jha, W. Li, M. Ganeshalingam and A. V. Filippenko, “Supernovae in Early-Type Galaxies: Directly Connecting Age and Metallicity with Type Ia Luminosity,” Astrophys. J. 685, 752 (2008) doi:10.1086/590659 [arXiv:0805.4360 [astro-ph]].
- (56) M. Sullivan et al. [SNLS Collaboration], “The Dependence of Type Ia Supernova Luminosities on their Host Galaxies,” Mon. Not. Roy. Astron. Soc. 406, 782 (2010) doi:10.1111/j.1365-2966.2010.16731.x [arXiv:1003.5119 [astro-ph.CO]].
- (57) J. Whelan, I.J. Iben, Binaries and Supernovae of Type I. Astrophys. J. 186, 10071014 (1973).
- (58) I. Iben and A.V. Tutukov, Supernovae of type I as end products of the evolution of binaries with components of moderate initial mass (M not greater than about 9 solar masses). Astrophys. J. Suppl. 54, 335372 (1984).
- (59) R.F. Webbink, Double white dwarfs as progenitors of R Coronae Borealis stars and Type I supernovae. Astrophys. J. 277, 355360 (1984).
- (60) S.E. Woosley and T.A. Weaver, The physics of supernova explosions. Ann. Rev. Astron. Astrophys. 24, 205253 (1986).
- (61) M. A. Barstow et al., “White paper: Gaia and the end states of stellar evolution,” arXiv:1407.6163 [astro-ph.SR].
- (62) L. D. Shaw, D. H. Rudd and D. Nagai, “Deconstructing the kinetic SZ Power Spectrum,” Astrophys. J. 756, 15 (2012) doi:10.1088/0004-637X/756/1/15 [arXiv:1109.0553 [astro-ph.CO]].
- (63) M. Postman and R.T. Lauer, Brightest cluster galaxies as standard candles, ApJ 440, 28 (1995)