Measuring the baryon fraction in cluster of galaxies with Kinematic Sunyaev Zeldovich and a Standard Candle
aa r X i v : . [ a s t r o - ph . C O ] F e b Measuring the baryon fraction in cluster of galaxies withKinematic Sunyaev Zeldovich and a Standard Candle
Shant Baghram ∗ Department of Physics, Sharif University of Technology, P. O. Box 11155-9161, Tehran, Iran
We propose a new method to use the kinematic Sunyaev-Zeldovich for measuring the baryon fraction incluster of galaxies. In this proposal we need a configuration in which a supernova Type Ia resides in a brightestcluster galaxy of intermediate redshift clusters. We show this supernova Type Ia can be used to measure thebulk velocity of a galaxy cluster. We assert that the redshift range of . < z < . is suitable for thisproposal. The main contribution to the deviation of standard candles distance modulus from cosmologicalbackground prediction in this redshift range comes from peculiar velocity of the host galaxy and gravitationallensing. In this work we argue that by the knowledge of the bulk flow of the galaxy cluster and the cosmicmicrowave background photons temperature change due to kinematic Sunyaev-Zeldovich, we can constrain thebaryon fraction of galaxy cluster. The probability of this configuration for clusters is obtained. We estimatein a conservative parameter estimation the large synoptic survey telescope can find spectroscopically followed ∼ galaxy clusters with a bright cluster galaxy which hosts a type Ia Supernova each year. Finally, weshow the improving of the distance modulus measurement is the key improvement in future surveys which willbe 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 [1] and Big Bang Nucleosynthesis (BBN) to a fraction of f b = ρ b /ρ m ≃ . [2]. However this ratio is smallerby a factor of 2 or 3 in galaxies and cluster of galaxies. This is known as the missing baryon problem [3]. There is an idea thatthe missing baryons are in intergalactic medium in diffuse warm-hot plasma which is hard to detect in X-ray. Also there are hintsthat baryons are not only in the virialized gravitationally bound objects but also they reside in voids and filaments [4, 5]. Worthto mention that 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 [6]. The search for the missing baryons is one of the ambitious quests ofcosmology, which will shed light on the process of galaxy formation and evolution. One of the promising cosmological probesto address this problem is the study of the galaxy clusters as the largest gravitationally bound systems which represent the laststep of the hierarchical structure formation. It seems that the galaxy clusters suffer less from the missing baryon problem thanthe other structures and they can be considered as prominent candidates to check the universality of the baryon fraction ratio inthe structures[7]. We can learn about the physics of baryons in galaxy clusters by studying the interaction of the galaxy clusterwith CMB photons. Although less than one percent of the CMB photons passed through the galaxy clusters but the physicsof the interaction is known and under control. The inverse Compton scattering of CMB photons by hot intra-cluster gas ofelectrons change the intensity of the observed CMB. This effect is known as Sunyaev-Zeldovich (SZ) effect [8–10]. The bulkmotion of the galaxy cluster also introduce a Doppler shift effect on the CMB photons known as kinetic Sunyaev Zeldovich(kSZ) effect [11]. The kSZ is a physical process of electron-photon scattering which is color blind (i.e. keeps the CMB spectrumalmost unchanged), while the thermal Sunyaev-Zeldovich (tSZ) is a process that changes the CMB spectrum. We should notethat for a typical cluster of galaxies the thermal velocities are higher than the bulk velocity and accordingly kSZ amplitude isan order of magnitude smaller than the tSZ. Thermal SZ is observed via CMB temperature [12, 13] and also from individualcluster studies[14, 15]. The kSZ is more challenging to be detected because of its smaller amplitude and also the fact that itdoes not change the spectrum of CMB. Despite to its observational challenges, kSZ is a very valuable quantity to be measuredin clusters where it can be used to address some astrophysical questions like missing baryon problem [16, 17] and also it canbe used to address cosmological questions (e.g. kSZ can be used as a method to measure the growth of structures to constraindark energy and modified gravity theories [18–21]). It is worth to mention that there is a frequency band in CMB observations ν ∼ GHz where the tSZ effect on the intensity of CMB photons is almost zero. The first detection of kSZ is reported byHand et al. (2012) [22] used the correlation of the Atacama Cosmology Telescope (ACT) data [23] with the pair-wise velocityof the Baryon Oscillation Spectroscopic Survey (BOSS) spectroscopic catalogue [24]. The kSZ signal is also detected in Planckand Sloan data[25], South Pole Telescope and Dark Energy Survey cross correlation as well [26]. ∗ Electronic address: baghram-AT-sharif.edu
We should note that there are attempts to detect the kSZ signal from individual clusters (for example see [27–30]). The casestudy of the kSZ signal in individual cluster of galaxies is an important attempt to understand the thermodynamic, merger andevolution of each cluster.In this work we propose a novel idea to measure the baryon fraction in cluster of galaxies by using the kSZ effect. Thetemperature change due to the kSZ is proportional to the baryon fraction and the bulk velocity. In the case if we can findout the bulk velocity of the galaxy cluster then we can pin down the baryon fraction more accurately by adding the CMB-kSZinformation. The main idea of this work is centered on the individual cluster kSZ signal and bulk velocity measurement. Thebulk velocity is usually measured by using the X-ray catalog of clusters as a complimentary probe to kSZ [34] or it is obtainedvia reconstructed matter density field [35] or by using the distance indicators[36]. In this direction we suggest to use standardcandles such as Supernova (SNe) type Ia to measure the bulk velocity. Traditionally SNe Ia used as a probe of cosmic bulk flow[37]. In this work we assume that the SN resides in the brightest cluster galaxy (BCG), accordingly this can be used as a probeof bulk velocity of cluster. Finally adding up the two independent observations of kSZ and SNe Ia, one can use it as a probe ofbaryon fraction in cluster. We also discuss the observational prospects of this proposal and the probability that this configurationcan 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 ofthe 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 ∆ T ¯ T | kSZ (ˆ n ) = − σ T c Z dχ z e − τ ( χ ) n e ( χ ˆ n, χ ) ~v e . ˆ n, (1)where ¯ T is the mean CMB temperature, χ = χ ( z ) is the comoving distance which is a function of redshift, σ T is Thomsonscattering cross section, τ is the optical depth, n e is the physical number density of free electrons, ~v e is the peculiar velocity offree electrons and ˆ n is the direction defined from observer to source. The optical depth is defined as τ = R dln e σ T , where l isthe physical length of the ionized patch in the sky. Note that the optical depth is very small and we set e − τ ≃ hereafter. Alsonote that the kSZ has almost frequency independent amplitude, for more details see Sec. (IV).The kSZ signal is an order of magnitude lower than the tSZ and mainly is extracted statistically from the cross correlationwith the late time tracers of gravitational potential. In this framework the leading science is done by Atacama CosmologyTelescope(ACT), which detect the kSZ effect by correlating the signal with reconstructed velocity field obtained for the galaxiesin LSS surveys such as Baryon Acoustic Oscillation Sky Survey (BOSS)[17]. However there are attempts to obtain the kSZsignal from an individual cluster (e.g. SZ measurement of MACS J0717.5+3745 galaxy cluster at redshift z = 0 . using thedata obtained with the NIKA camera at the IRAM 30m telescope [31] and SZ measurement of RX J1347.5-1145 galaxy clusterat redshift z = 0 . with data collected at 147, 213, 281, and 337 GHz using the Multiwavelength Submillimeter InductanceCamera (MUSIC) and at 140 GHz with Bolocam [30]). These type of studies are essential to understand the physics of baryonsin a galaxy cluster. For example a kSZ signal plus the knowledge on properties of galaxy cluster can be used to obtain thepeculiar velocity of 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 anindependent 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 perturbationcan be used as a probe of peculiar velocity due to the Euler equation [38]. The other method is to use the standard candles likeSNe 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. In order toproceed we write number density of electrons n e as n e ≃ ∆¯ n e , (2)where ¯ n e 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 ¯ n e = Υ ρ g ( z ) µ e m p , (3)where ρ g is the mean gas density in redshift z, m p is the proton mass and µ e is the effective number of electrons per nucleon.Accordingly µ e m p is the mean mass per electron. Υ is the ionization fraction which is defined as Υ = [1 − Y p (1 − N He / / (1 − Y p / , where Y p is the primordial abundance of Helium and N He is the number of ionized electrons corresponding to Heliumatoms [82] Now we can relate the gas density to the baryon fraction of universe and matter density of universe as below ρ g = 3 H πG f g f b Ω m (1 + z ) , (4)where f g is the gas fraction of baryons in a cluster and f b is our crucial parameter of the study, known as baryon fractionparameter. Ω m is the matter density parameter and H is the Hubble parameter both defined in present time. Now by assumingthat the evolution of all above parameters inside a cluster is negligible and position independent, the kSZ effect will become ∆ T ¯ T | ikSZ ≃ − σ T ℓ i (∆¯ n e ) × ~β i . ˆ n, (5)where ℓ i is the size of the cluster labeled by superscript i and ~β ≡ ~v/c . Now by using Eq.(3) and Eq.(4), it is straightforward toshow that the kSZ signal amplitude can be written as ∆ T ¯ T | ikSZ ≃ −C i f b × ( ~β i . ˆ n ) , (6)where C i is a specific parameter for each cluster, which depends on the mean redshift, the physics of intra-cluster medium andthe line of sight length of the cluster as C i ( z i ) = 3 H πG σ T µ e m p (∆ f g Υ ℓ i )Ω m (1 + z i ) . (7)Now C i can be written in the terms of cluster’s gas fraction, ionization factor and line of sight length C i ( z i ) ≃ × − × Ω m h (1 + z i ) f g Υ ℓ i M pc . (8)Finally we can define a parameter x i = −C i f b for each cluster which is related to kSZ signal and the bulk flow of the cluster as x i ≃ ( ∆ T ¯ T | ikSZ ) / ( ~β i . ˆ n ) , (9) where the approximation works for Ω m = 0 . , h = 0 . , ℓ i ≃ M pc , Υ ≃ , f g ≤ and f b ≤ . . Accordingly by theknowledge of kSZ signal and the peculiar velocity 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
The classical methods for peculiar velocity measurements from galaxy surveys needs an independent distance indicator tosubtract the Hubble flow from redshift position ( v p ≃ cz − Hr phy , where v p is the peculiar velocity, z is the redshift of thegalaxy/cluster and r phy is the physical distance.) Many distance indicators are used to measure the peculiar velocity such asTully-Fisher relation [39], fundamental plane [40] and the SNe Ia as well. The Tully-Fisher and fundamental plane measurementsintroduce a − error in peculiar velocity measurement [41, 42]. Although there is an assertion by Springob et al.[43]that the fundamental plane analysis results to relative error in peculiar velocity less than at all redshifts. However there isan approximation in [43], that the ratio of the apparent and physical size of a galaxy in redshift z is proportional to ( z − z p ) /z (where z p is the redshift assigned to peculiar velocity). This assumption means that in small redshifts the physical distanceto the galaxy is equal to cz/H . The SNe Ia as a distance indicator is also introduce an error in velocity measurement duethe fact that the peculiar velocity of the host galaxy change the magnitude of the SN Ia [44]. The other method that can beused for peculiar velocity measurements is via reconstruction of density field and peculiar velocity[45]. We argue that in futureexperiments the method we propose in this work will be comparable with classic methods of peculiar velocity measurement. Inthis section we will discuss how standard candles can be used to determine the peculiar velocity of the structures without the useof standard relation (i.e. v p ≃ cz − Hr phy ). The idea is straightforward, the standard candles are used to establish a luminositydistance-redshift relation for a given background cosmology model. However the deviation from the homogenous backgroundwill change this relation. One of the important modifications is due to the peculiar velocity of the host galaxy of a SN, whichaffects both the luminosity and redshift of the standard candle. Accordingly, we can use the deviation of the luminosity distanceof a SN type Ia as a probe of peculiar velocity.We assume a perturbed FRW universe with a Newtonian comoving gauge metric ds = a ( η ) (cid:2) − (1 + 2Ψ( x, t )) dη + (1 − x, t )) δ ij dx i dx j (cid:3) , (10)where η is the conformal time, Ψ and Φ are the scalar perturbations of metric. If we assume that the General Relativity (GR) isthe correct classical theory of gravity and the universe is filled with components that have no anisotropic pressure, then we get Φ = Ψ .In a perturbed universe, the luminosity distance of a standard candle is modulated due to propagation of light in perturbed FRWuniverse (Sachs Wolfe effect and gravitational lensing). The luminosity distance is also corrected due to the peculiar velocity ofthe source and observer. These corrections can be formulated as[46–48] d L ( z s , ˆ n ) = (1 + z s ) χ ( z s ) [1 − κ v − κ g − κ SW − κ ISW ] , (11)where d L ( z s , ˆ n ) is the luminosity distance of a supernova in observed redshift of the source z s and direction ˆ n (Note that ˆ n isunit vector in the direction of observer toward source). χ ( z s ) is the comoving distance of the source in FRW universe. Theparameter κ v is the correction due to peculiar velocity of source. The κ g , κ SW and κ ISW are the lensing convergence, Sachs-Wolfe and Integrated Sachs-Wolfe correction terms. (These terms are defined and studied extensively in [47]). In intermediateredshift . < z < . which is the redshift range which we are interested in this work the peculiar velocity and gravitationallensing has comparable effect, accordingly [44] d L ( z s , ˆ n ) ≃ ¯ d L ( z s )[1 − κ v − κ g ] , (12)where ¯ d L ( z s ) is the background luminosity distance and κ v is luminosity correction due to peculiar velocity [46] defined as κ v = − (cid:18) − z s χ ( z s ) c − H ( z s ) (cid:19) ( ~β s . ˆ n ) , (13)where H is the Hubble parameter and ~β s = ~v p /c where ( v p is the peculiar velocity). In low and intermediate redshifts ( z < . ),the term in parentheses is negative, accordingly the objects moving toward us ( ~β s . ˆ n < ) introduce a κ v < where if wereplace 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( ~β s . ˆ n > ), that introduces a κ v > and accordingly the source become brighter. These chain of conclusions are changed inhigher redshifts. The lensing correction term is defined as κ g = Z χ ( z s )0 dχ ( χ ( z s ) − χ ) χχ ( z s ) ∇ ⊥ Φ ≃
32 Ω m H Z χ ( z s )0 dχ ( χ ( z s ) − χ ) χχ ( z s ) (1 + z ) δ m , (14)where δ m is the total matter density contrast. Note that the second term is an approximation as we change the 2D Laplacian witha 3D.The main idea of this work is that we can use SNe for finding the peculiar velocity (bulk flow) of the galaxy cluster. Beforeproceed further, we should numerate the velocity contributions to our specific case of study. The velocities which can be assignedto 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 thegravitational potential of the cluster and c) The peculiar velocity due to the bulk flow of the galaxy cluster. In this work we aremerely interested in bulk flow, accordingly in order to extract this term, which is the source of the kSZ effect, we assume, thatthe host galaxy of the supernova is the one which is located in the gravitational/ optical center of the galaxy cluster (bright clustergalaxy (BCG)). The assumption that the SN host galaxy is in the center of the cluster is needed, as we want to assign the bulkvelocity of the cluster to the peculiar velocity of the BCG. This assumption comes from the idea that the central bright galaxiesreside in the minimum of the gravitational potential well of the cluster and the peculiar velocity with respect to the center of themass 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 ofthe cluster. This configuration is schematically shown in Fig.(1). However we should keep in mind that the effect of SNe Ia’sprogenitor’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 progenitorvelocity introduce a Gaussian error to the velocity estimation related to the dynamical mass of BCG. The contribution of thiserror can be calculated by setting the velocity of the center of mass of a SNIa progenitor equal to the dispersion velocity inside atypical BCG using the fundamental plane relation of elliptical galaxies [49, 50], for a specific example, we express the dispersionvelocity in terms of magnitude [51] as log ( σ pro / kms − ) = − (0 . ± . M m / . − . ± . where M m is the FIG. 1: A schematic configuration of a galaxy cluster is plotted which has a kSZ signal and its bright cluster galaxy hosts a supernova typeIa. The dotted red arrow indicate the line of sight direction and the blue dashed arrow is the direction of the bulk flow. The solid black arrowsshows the peculiar velocities of cluster members. Orange dashed dotted arrows shows the CMB photons which will interact with ionizedplasma of galaxy cluster and the red star represent a SNIa exploded in the BCG. metric magnitude [83] and σ pro is the velocity of the center of mass of a SNIa progenitor. In this work we set the dispersionvelocity (progenitor velocity) to σ pro ∼ km/s ± .Now by using Eqs.(12 and 13), we can find the line of sight normalized velocity as ( ~β s . ˆ n ) = (1 − ∆ µ/ − ¯ κ g ) / ˜ κ v , (15)where ∆ µ is the difference of the observed distance modulus and the one predicted from background cosmology of Λ CDM inthe specific redshift of z s . The parameter ˜ κ v = (1 + z s ) cH − /χ ( z s ) E ( z s ) − is a unique term which is independent of thelocal physics, instead it depends on the background cosmological parameters and the redshift of the supernova. Note that E ( z s ) is the normalized Hubble parameter to its present value. Note that ¯ κ g is the correction term due to gravitational lensing term,and the bar indicate that this term can be obtained by using the matter power spectrum of density perturbations. ¯ κ g = 94 Ω m H Z χ ( z s )0 dχ (cid:20) ( χ ( z s ) − χ ) χχ ( z s ) (1 + z ) (cid:21) Z ∞ kdk π P m ( k, z ) , (16)where P m ( k, z ) is the matter power spectrum in redshift z . Now by using Eq.(15) we can calculate the line of sight velocityof a SN, then we can assign this to the bulk velocity of the galaxy cluster which is the host of the SN ~β s . ˆ n ≡ ( ~β i . ˆ n ) wheresuperscript i represent the host of a SN. Now we can use Eq.(9) to extract the baryon fraction. This can be done becausethe RHS of this equation is fixed by two independent observations. As mentioned in introduction and the beginning of thissection the peculiar velocity can be measured by different method like velocity reconstruction with a galaxy field[52] or usingdistance indicators, which can be cross-checked with other methods. Now that we obtained the peculiar velocity via SNe Iamethod, we should mention the peculiar velocity of the the host introduce an error on its measurement which is proportionalto δv p /v p ∝ H ( z s )¯ r phy κ v / ( cz s − H ¯ r phy ) , which this new source of uncertainty is not considered in classical methods ofpeculiar velocity measurement by distance indicators. In the next section we will discuss the observational prospects of findingthe 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 a logical path of extracting the bulk flow of the clusters from the datawith assuming that each SNIa resides in a BCG. We should note that the first subsection is just an example of how this methodworks and shows the current status of quality of the data. In the second subsection we discuss the error estimation for realisticand optimistic case for finding the baryon fraction and finally in the third subsection we present an estimate for the number ofevents that we anticipate in future surveys which are suitable for our case of study. -1.5-1-0.5 0 0.5 1 1.5 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 ∆ µ z Union 2.1 data Λ CDM
FIG. 2: The difference of the SNe Ia distance modules with respect to the Λ CDM prediction ( ∆ µ ) is plotted versus redshift for Union 2.1 inthe redshift range of (0 . , . with σ error bar. The black solid zero line shows the baseline of Λ CDM for comparison. -0.04-0.02 0 0.02 0.04 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 ∆ µ z Λ CDMpeculiar velocity correctionMax V los =1200/ 3 km/s κ g FIG. 3: The theoretical difference of the SNe Ia distance modules with respect to the Λ CDM prediction ( ∆ µ ) in the perturbation level is plottedversus redshift. The dashed blue curves are obtained from linear theory prediction for peculiar velocity corrections and the green long dashedlines are for velocity corrections with larger amplitude, assuming a maximum velocity of km/s for a typical cluster. The red dotted linecorrespond to the correction to the magnitude due gravitational lensing convergence. A. 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 ourproposal. We assume that they are hosted by a central galaxy of a cluster. This is just an assumption to show the details ofthe proposal. In this section, we use the Union 2.1 SNe sample [53] to show the procedure. First of all we extract the peculiarvelocity with the method which is described in Sec.(III) by assuming that the standard Λ CDM model with best parameters fixedby supernova data [1] describe the cosmological flat background ( Ω m = 0 . , h = 0 . ). We obtain the difference of distancemodulus ∆ µ = µ obs − ¯ µ versus redshift for the sample of SNe. Note that ¯ µ is the distance modulus from the known backgroundcosmology.In Fig.(3) we plot ∆ µ versus redshift for SN Ia in redshift range of (0 . , . . In this redshift range the host galaxy clusterof the SN can be observed in arc-minutes. This resolution is needed for kSZ measurements. The clusters in lower redshiftare observed in lower moments (large angular resolution), where the primordial CMB dominate the kSZ signal. On the otherhand in higher redshifts the peculiar velocity corrections to SNe data become subdominant plus the fact that the errors in SNmagnitude become larger. In Fig.(3) the red data points are SNe Ia in the redshift rang of our interest with average error of ∼ .
304 + ± . mag. In Fig.(IV A), we plot the corrections we anticipate from peculiar velocity in linear theory (blue longdashed line) and to the typical peculiar velocity of v p ∼ / √ km/s . The red line is the contribution due to gravitational -10000-5000 0 5000 10000 15000 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 v l o s [ k m / s] z line of sight velocity LSST like SNe Ia Max V los =1200 / 3 km/s FIG. 4: The line of sight velocity of SNe is obtained from Eq.(15) is plotted versus redshift for the SNe of the Union sample. The horizontaldashed green lines shows the typical maximum velocity of galaxy clusters v p ≃ ± km/s . This plot is obtained for realistic and optimisticerrors on SNe luminosity distance measurement. The uncertainty due to the progenitors are the same in considered cases. Note that the SNeIa have less than σ tension with the prediction of distance modulus change via v p ≃ ± km/s peculiar velocity lensing which is considered as a source of correction to the distance modulus. The convergence corrections is obtained fromEq.16. From the plot it is obvious that the lensing correction increase with the redshift while the peculiar velocity correctionsdecrease. We choose the redshift range in a way that we can observe the cluster of galaxies with CMB experiments and extracttheir kSZ effect and also be in a plausible range where the corrections to distance modulus due to peculiar velocity are calculable.The blue low amplitude dashed curves represent the amount of correction that we expect from the peculiar velocity in linearregime. The linear regime velocity in the line of sight can be obtained via linear matter spectrum P ( k ) as: h v p i ( z ) ≃ H f π Z P ( k ) W ( kR ( z )) dk, (17)where h v p i ( z ) is the average linear velocity in the line of sight (assuming an isotropic velocity) in a window function with acomoving radius R . The parameter f is the growth rate which for standard Λ CDM is equal to f ≃ [(Ω m (1 + z ) ) /E ( z )] . (Note that E ( z ) is the normalized Hubble parameter to its present value). The blue curves shows the prediction of standardmodel in perturbation level. The dotted high amplitude green lines are obtained by assuming a maximum line of sight velocityof ∼ / √ km/s ∼ km/s . In the next step, we assign the deviation of the distance modulus change to the peculiarvelocity of the host galaxies.In Fig.(4) we plot the reconstructed line of sight velocity in terms of redshift. We use Eq.(15) to extract the line of sightvelocity. The error bars are obtained from the propagation of the error in distance modulus and gravitational lensing correctionsand 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 toadd a Gaussian error σ pro ∼ km/s ± . The very interesting point to indicate here is that the SNe that seems brighter( ∆ µ < ) moves away from us ~β. ˆ n > . This observation is consistent to the argument we made before for peculiar velocityeffect in low and intermediate redshifts. In Fig.(4), we also obtained the peculiar velocity with an optimistic resolution. The datawith 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 LSSTsurvey [65]. Fig.(4) shows that how the accuracy in SNIa luminosity distance measurement are important in this study. Weshould also note that the SNe Ia line of sight velocity is valuable for each cluster and the binning of data like the case of bulkvelocity measurement is not useful.The baryon fraction can be obtained by the knowledge of kSZ signal for each galaxy cluster and the assumption that the SNIahost is the BCG as below f ib ≃ (cid:20) ( ∆ T ¯ T | ikSZ ) / (cid:16) ∆ µ/ − − ¯ κ g (cid:17)(cid:21) ˜ κ v C i , (18)where C i obtained from Eq.(8) and ˜ κ v depends on the background cosmology. We should note that Eq.(18) is the key equationof our proposal. The baryon fraction of cluster can be obtained by its kSZ temperature and the luminosity change of a SN whichis hosted by the central galaxy of a cluster. In the next subsection we discuss the error propagation due to other components L ( f b ) f b realisticCMB IVLSST SNe + CMB IV FIG. 5: The constraints on fiducial parameter of baryon fraction due to realistic (current) errors on physics of cluster, kSZ and SNe Ialuminosity distance measurement and the optimistic errors for CMB stage IV (blue dashed line) and CMB stage IV and LSST like SNe Ia data(green dotted line) of Eq. (18), which will introduce the uncertainty in baryon fraction calculation. In Fig. (5), we plot the Fisher forecast of thebaryon fraction measurement via SNe Ia in the redshift range of (0 . , . for our configuration. We set the fiducial parameter f b = 0 . and plot the σ prediction of our method for realistic and optimistic cases. In the optimistic case we consider twocategories of a) The CMB stage IV case with S/N ∼ for kSZ measurement. (dashed blue line) and b) The CMB IV + LSSTlike SNe (dotted green line). The optimistic cases in this stage is just chosen to show how the baryon fraction can be constrainedby future experiments. In the next subsection, we discuss the error estimation and we study in more detail how different aspectsof this proposal introduce uncertainties. Also we will show that the improvement in luminosity distance measurement has themain contribution for make this proposal a observationally practical one. B. The error estimation
In this section, we study the different error budget which has a role in baryon fraction measurement via the method which isproposed in this work. For simplicity we assume that the background cosmological model is fixed by other observations suchas CMB and the errors on density parameters and Hubble constant is much more smaller than the uncertainties in the physics ofthe cluster, supernova observations and kSZ signal extraction. This means that the uncertainty in baryon fraction calculation canbe written as σ f b = ( ∂f b ∂ (∆ T /T | kSZ ) ) σ ksz + ( ∂f b ∂ ∆ µ ) σ sn + ( ∂f b ∂C i ) σ c + ( ∂f b ∂ ¯ κ g ) σ κ g , (19)where σ ksz is the error in kSZ signal from CMB analysis σ sn is due to SNe Ia uncertainties which can be due to intrinsicmagnification errors, photometric errors, the peculiar velocity of the SNe Ia progenitors and ... σ c is the error induced fromthe physics of the clusters and σ κ g is the error due to gravitational lensing effect. We also assume that errors from differentcontributions are uncorrelated and Gaussian distributed, which is almost a reasonable assumption. In what follows we willnumerate the different contributions to the total error introduced in this proposal and then we will address that which errors canbe reduced in future experiments. Kinematic Sunyeav Zeldovich : For kSZ the main problem is that the spectrum has almost a flat power in frequencyrange, that is why the primordial CMB anisotropies themselves are the most important source of contamination. Accordinglymost of kSZ extraction methods try to separate the cluster signal of kSZ and primordial anisotropies. In the case if the galaxycluster can be observed in redshifts high enough to be studied in arc-minute resolution then we can use the kSZ measurements.The arc-minute resolution corresponds to angular moment of ℓ ∼ , the CMB primordial signal become less important dueto Silk damping and accordingly the kSZ signal can be extracted from primordial one. The other source of contamination is thetSZ effect, which is an order of magnitude higher than the kSZ in ordinary clusters. An important step toward separation of kSZfrom tSZ is multi-frequency observations of individual clusters. The total intensity change ∆ I ν due to a galaxy cluster observedin frequency ν with respect to CMB I is given as ∆ I ν I = f ( ν, T e ) y tsz + g ( ν, T e , v p ) y ksz , (20)where T e is the temperature of intra cluster medium (ICM) , v p is the line of sight velocity. y tsz and y ksz are the amplitude oftSZ and kSZ signal and f ( ν, T e ) and g ( ν, T e , v p ) are the characteristic frequency dependent function of tSZ and kSZ as [10] f ( x, T e ) = x e x ( e x − ( x coth( x − δ tsz ( x, T e )) , (21)where x = hν/k B T CMB and δ tsz ( x, T e ) is the relativistic correction to the characteristic function[60]. g ( x, v p , T e ) = x e x ( e x − (1 + δ ksz ( x, v p , T e )) , (22)where δ ksz ( x, v p , T e ) is the relativistic correction to the characteristic function[61]. Now if we have signal in two frequencybands (like The New IRAM KID Array (NIKA) camera at the IRAM 30m telescope at ν = 150 and ν = 260 GHz [62] wecan extract the amplitude of kSZ amplitude as below y ksz = f ( x , T e )∆ I ν − f ( x , T e )∆ I ν I f ( x , T e ) g ( x , v p , T e ) − I f ( x , T e ) g ( x , v p , T e ) (23)where x i = hν i /k B T CMB , i = 1 , . This procedure is done for observing the cluster MACS J0717.5+3745 with NIKA whichis composed of four distinguishable sub-clusters. The detection of kSZ signal in one of sub-clusters is obtained by . σ [31].Also the technique of component separation is used in a recent work to extract the kSZ signal to study the thermodynamicof a cluster [32]. One of the main future project which will use the technique of multi-wavelength detection of SZ effectis the 6-meter CCAT-prime telescope[33]. Note that in distinguishing procedure, the X-ray observation of the clusters canbe considered essential and also complimentary in order to study the effective temperature of ICM [31]. There is othermulti-wavelength studies of the MACS J0717.5+3745 such as the study of the SZ with MUSTANG and Bolocam [63] Exceptthe primordial CMB anisotropy and tSZ, there are other sources of noise such as diffuse foreground emissions which can bedivided to atmospheric, galactic and extragalactic sources. For comparison on the level of error dependence, we set the relativeuncertainty as σ ksz / (∆ T /T | kSZ ) ≃ (i.e. S/N ∼ ) for a realistic case [23, 58] and for futuristic (e.g. CMB stage IVexperiments) one we set ˜ σ ksz ( opt ) ≃ . σ ksz ( r ) (i.e. S/N ∼ ) [58]. (Note that tilde is an indication of relative errors andsubscripts ”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 systemat-ics become the dominant one in error budget in comparison to the statistical errors [64]. This systematics are also surveydependent, accordingly all the supernovae compilation have the problem of calibration matching where photometric offsets isone of the main challenges of using more than one survey. In this direction the future surveys like LSST which has the plan tomake a dedicated long run surveys of sky for hunting the SNe Ia, can overcome this problem [65]. Generally the systematics ofthe 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 Landoltphotometric system. However for each SN we need a calibration by knowing the filter transmission rate and K-correction. Notethat one needs the knowledge of the spectral energy distribution (SED) for K-correction [66]. Also we should indicate that largenumber 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 z > . SN is the intrinsic scatterin their light curves and their poor calibration [67].c) reddening due to dust : This is one of the major uncertainties introduced in the modeling of the SNe Ia. It seems that theredder SNe are also dimmer [68]. This means that there is an intrinsic color-luminosity relation in SN data which can becomedegenerate from the reddening of the host galaxy’s dust. As the color is correlated to the SN’s properties so associating thewrong color to the host of SN can be wrongly assigned to the reddening. The idea to overcome this error is to study the physicsof the color-luminosity and reddening simultaneously with independent observations. The infrared observations of the hostgalaxy can be a solution to decrease the contamination of the dust [69]. A final note in this category is that the different SNprogenitors 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 Iaoccurred in late type galaxies [70] and sub luminous ones are in old population galaxies [71]. This can be regarded as a strongevidence for redshift and environment dependence of SNe Ia. Now we have an indication that the width and luminosity of SNe0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 σ f b / f b z realistic CMB IV LSST SNe + CMB IV FIG. 6: The relative error of the baryon fraction is plotted for different redshift bins for the realistic case (red solid line). and the optimisticerrors for CMB stage IV (blue dashed line) and CMB stage IV and LSST like SNe Ia data (green dotted line)
Ia light curve is correlated to the mass, star formation rate and the metallicity of the host galaxy [72–74]. As the star formationrate is a redshift dependent quantity accordingly we can anticipate that the SNe Ia absolute magnitude can be a redshiftdependent as well. However, now we are entering a large data set era, which by comparing the SNe Ia and their host galaxiesproperties 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 theprogenitor. There are mainly three scenarios of SNe Type Ia, known as single degenerate (SD) scenario [75], double degenerate(DD) scenario [76, 77] and sub-Chandra [78]. The first scenario assumes that a white dwarf with a companion from mainsequence of stars or a red giant is the generator of the nova and in the second one, we assume that the binaries are both whitedwarfs. The sub-Chandra model assumes that a layer of helium appears on the surface of a white dwarf below the Chandrasekharmass until it detonates. The main goal of future SNe Ia survey like LSST is to observe a large number of SNe Ia (e.g. 50,000Type Ia per year in LSST survey ), where the good statistics in color and light curves, combined with a small number of samplespectra, any dependence of the supernova standard candle relation on parameters other than light curve shape and extinctioncan be extracted. Accordingly these can be minimize the systematics via statistics. The LSST has a plan to reduce the distanceindicator 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 whichis a space observatory of the European Space Agency (ESA) designed for astrometry which can be a great help to calibrate thecepheid variables and make the distance ladder more accurate [79]. Galaxy cluster : The main contribution to the relative error of the physics of galaxy cluster is raised form the size of thegalaxy and the gas fraction of the cluster. The relative uncertainty is set ˜ σ c ≃ . gravitational lensing : Finally we come up with the relative error of the gravitational lensing. This error can be esti-mated by the theoretical convergence introduce in Eq.(16) using the standard Λ CDM cosmology with cosmological parameter Ω m = 0 . , H = 0 . and σ . . In order to show the effect of different type of the errors on baryon fraction measurement,in Fig.(6) we plot the relative error with respect to the redshift. The redshift range of (0 . , . is divided to four bins withalmost equal number of SNe Ia. The mean and the σ variance of the relative error is plotted (red solid line) for the realisticcase that we considered in previous section. Then in order to study the effect of different error budget, we study the casewhere relative error in kSZ measurement can be improved by a factor of ten (by e.g CMB stage IV experiments) (blue dashedline). In the next step we plot the relative error estimation by assuming the characteristic resolution of LSST project which canimprove the magnitude measurement to . mag . As it is shown in Fig.(6) there is much improvement in the relative error dueto luminosity. In order to conclude this subsection, we assert that the main contribution of the error comes from the SNe Iameasurement. The future experiments of SNeIa can improve the distance measurement drastically, which can bring this methodin a new and practical state. In the next subsection we will discuss a very important issue of the probability of the detection ofthe configuration (SNe Ia in a BCG) which we proposed in this work.1 C. 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 manyevents of SNe Type Ia explosions are happened in the central galaxy of a cluster”. Accordingly in this subsection, we proposetwo distinct scenarios for the number of expected events for SNe happened in galaxy cluster. First we anticipate that the SNe Iaexplode in the central galaxy of a cluster. Then we assume that more than one SNe Ia happens in galaxy cluster. For the firstscenario, we formulate this probability due to the simplified relation below where N snobs is the number of observed BCG with thedesired condition of our proposal. N snobs ( T obs ; s ; bcg ; ⊘ ) = f sky × N bcg × P ⊘ × T obs , (24)where N snobs = N snobs ( T obs ; s ; bcg ; ⊘ ) is a function of the observation time T obs and also it depends on the cosmological volumeof the survey and the portion of the sky f sky it spans. All of this information is encapsulated in symbolic parameter of s whichstands for survey. It is obvious that the observed number of configuration depends on physics of BCG ( indicated by ”bcg” ) andthe physics of SNIa (indicated by ” ⊘ ” in eq.(24)). The N bcg is the number of BCGs that resides in clusters which has a darkmatter host of mass M with lower bound M l < M bcg , where M l indicate lower limit of a typical BCG which resides in theredshift range of z i up to z f . The parameter P ⊘ is the probability that a SNIa occurred in a BCG of a cluster in a year. Thenumber of BCG in galaxy clusters in a cosmological volume, which is limited by the redshift range, that we are interested in isas below N bcg ( M l , z i , z f ) = f bcg Z M l dM Z z f z i dz dVdz dn ( M, z ) dM , (25)where we set M l = 10 . M ⊙ as the lower limit of dark matter halo, the host of the BCG, z i = 0 . and z f = 0 . is thelower and upper limit of the redshift survey that we are interested and f bcg is the fraction of luminous massive galaxies thatresides in a center of galaxy cluster. Due to the fact that BCGs are assembled mainly by major mergers in galaxy formationand evolution process, there is an indication that the most massive galaxies must reside in galaxy clusters, accordingly we set f bcg ∼ for this study. In order to obtain the number density of cluster, we use the Press-Schechter approach to find dn/dM ,specifically when we use the fitting function introduced in Jenkins et al [54]. Accordingly the number of BCG galaxies become N bcg ≃ × × f bcg × ( V c / Gpc ) , where V c is the cosmological volume. Note that the effective volume is V c ≈ Gpc for a survey which is in search of galaxies in redshift span of . < z < . . Another important parameter is P ⊘ to estimate isthe rate of SN Ia in a galaxy with the given mass range. In Graur et al. [55], there is an extensive study on the rate of Type Iasupernova. Graur et al. used SNe samples to measure mass-normalized SNe rates as a function of stellar mass of the host galaxyand the star formation rate. By assuming the stellar mass of . < M ∗ < (with the assumption of a mass to light ratioof ∼ ) and the star formation rate we will have the marginalized fitting function as P ⊘ ≃ . × ( M ∗ / M ⊙ ) /year . Itis worth to mention that the SN rate is proportional to star formation rate and specific start formation rate, accordingly the rateof SN decrease with evolution of BCGs from active to passive ones. To finalize this part, by considering all the complicatedphysics which is governing the SNIa rate relation with the host galaxy, we set very conservative rate of P ⊘ ∼ . /year .A project like Large Synoptic Survey Telescope (LSST) which is designed to operate for 10 years starting from 2019 and willcapable of spanning the , square degree of the sky (which means f sky ∼ . ) in 6 optical bandwidth with a limitedmagnitude to a total point-source depth of r ∼ . [65], we can estimate the N snobs ∼ per year with the conservativeassumptions we made in this section. However we should note that one should also take into account that precise spectra foreach of the SNe are absolutely necessary. This means that the SNe detection does not suffice, but a spectroscopic follow-upprogram should follow. Assuming that we will have a follow up of of the SNe Ia in a very conservative point of view inLSST life time project we can estimate the baryon fraction of ∼ clusters of galaxies. As a final word, the other scenariothat can be used as a proposal for using the SNe for estimating the baryon fraction is that in a cluster of galaxy one can monitorand measure all SNe in all galaxies within cluster, then on average the peculiar velocities of those SNe would be close to thebulk flow. In this case, the error on the bulk velocity measurement can be decreased due to the number of SNe Ia found in agroup/cluster 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 andcosmic microwave background radiation independently fix the baryon fraction. However the accessability of baryons in late timeis a challenging task and it seems that there is a missing baryon problem out there. The galaxy clusters as the main reservoir ofbaryonic matter which are filled with ionized electrons are suitable environments to study the distribution and physics of baryons2in the Universe. One of the promising venues to address the distribution of baryons in the sky is the study of thermal and kineticSunyaev-Zeldovich effect which are used as a probe of ionized gas in clusters. In this work we propose a new idea/methodto measure the baryon fraction using the kSZ effect and the SNe-Type Ia as the standard candle. For this method to work weassume 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 usedas an almost fair representative of the cluster’s bulk flow. The BCG resides in the depth of gravitational potential of cluster andits 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 Iaincluding its progenitor velocity. Accordingly we show that the SNe Ia in low redshift can be used to estimate the peculiarvelocity of host galaxy. In the other hand The kSZ signal 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 Λ CDM can be related tothe peculiar velocity of SNIa host galaxy in lower redshifts, keeping in mind that the host galaxy is chosen to be a BCG. Weshowed that by knowledge of the peculiar velocity and the temperature change of CMB we can constrain the baryon fraction. Weinvestigate 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 possibilityof 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 for almost ∼ SNe Ia which explodes in a BCG in a cluster of galaxy.Also we should note the CMB stage IV experiments bring the number of individual cluster which can be examined for kSZsignal will grow gradually. It is worth to mention that in the case of more statistics in galaxy cluster we can average the peculiarvelocity of each host galaxy of SNIa , the average of squared velocity will be a representative of the bulk flow and the dispersionof velocities represent the error bar on bulk velocity. As a final remark we want to insist that future SNe Ia surveys will decreasethe errors on baryon fraction and will bring the introduced method as a practically viable proposal.
Acknowledgments
We would like to thank Farhang Habibi for detailed and insightful comments on the manuscript. Also we would like to thankSaeed Ansari, Alireza Hojati, Nima Khosravi, Sohrab Rahvar, Sina Taamoli and Saeed Tavasoli for useful discussions. We thankthe anonymous referee for his/her thorough review and highly appreciate the comments and suggestions, which significantlycontributed to improving the quality of the manuscript. SB acknowledges the hospitality of the Abdus Salam InternationalCentre for Theoretical Physics (ICTP) during the final stage of this work. [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. , 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. 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