Constraints on the missing baryons from the kinetic Sunyaev-Zeldovich effect in Planck data

We estimate the amount of the {\it missing baryons} detected by the \Planck\ measurements of the cosmic microwave background in the direction of Central Galaxies (CGs) identified in the Sloan galaxy survey. The peculiar motion of the gas inside and around the CGs unveils values of the Thomson optical depth $\tau_{\rm T}$ in the range $0.2$--$2\times 10^{-4}$, indicating that the regions probed around CGs contain roughly half of the total amount of baryons in the Universe at the epoch where the CGs are found. If baryons follow dark matter, the measured $\tau_{\rm T}$s are compatible with the detection all the baryons existing inside and around the CGs.


Introduction
The interplay between baryons and dark matter is a key problem in galaxy formation.Understanding the distribution of baryonic and dark matter in galaxies, groups, and clusters of galaxies is an essential step towards the full picture of how these objects form and evolve.It is well known (e.g., Fukugita & Peebles 2004;Cen & Ostriker 2006;Bregman 2007) that only about 10% of all baryons in the Universe reside in the form of stellar mass, while other 90% reside in a diffuse component.Currently there is an ongoing debate (Gupta et al. 2012;Gatto et al. 2013;Planck Collaboration Int. XI 2013;Werk et al. 2014;Le Brun et al. 2015;Miller & Bregman 2015) on whether a significant fraction of the latter component is present in the circumgalactic medium around halos or if instead most of the gas has been expelled or never accreted due to feedback processes.This connects the missing baryon issue with the complex problematic of feedback and galaxy formation.Certainly a more complete census of the baryon distribution in the universe would be of great relevance in this context.
The kinematic Sunyaev-Zeldovich effect (hereafter kSZ Sunyaev & Zeldovich 1972a, 1980) describes the Doppler shift in frequency induced on photons of the Cosmic Microwave Background (hereafter CMB) after they scatter off free electrons.This Thomson scattering induces frequency independent brightness temperature anisotropies in the CMB, which are given by (Sunyaev & Zeldovich 1970) ⋆ Corresponding author: C.Hernández-Monteagudo, chm@cefca.es In this expression, the integral τ T = σ T dl n e is conducted along the line of sight (LOS) given by n and v • n stands for the LOS peculiar velocity of the electrons.We have made the approximation that the typical correlation length of LOS velocities is much larger than the density correlation length, in such a way that the LOS velocity term may be pulled out of the kSZ integral.This is justified by the results of Planck Collaboration Int.XXXVII (2015), who find a typical correlation length of peculiar velocities of 20-40 h −1 Mpc, well above the typical galaxy correlation length (∼ 5 h −1 Mpc).The expression above also shows that the kSZ constitutes an integral over the electron momentum, independently of the temperature, and thus counts all electrons in the bulk flows, regardless they belong to collapsed structures or not.Large scale, bulk matter flows were detected via the kSZ effect firstly by Hand et al. (2012) and more recently by Planck Collaboration Int.XXXVII (2015).We build upon the latter work and extract physical constraints on the amount of baryons contributing to the kSZ signal and the implications of those measurements in the problem of the missing baryons and homogeneity of the Universe.As in Planck Collaboration Int.XXXVII (2015), we use the latest Planck data release (DR2) available at the Planck's Legacy Archive server 1 , the Central Galaxy Catalogue (hereafter CGC) obtained from the Sloan Digital Sky Survey (SDSS/DR7 Abazajian et al. 2009), and a mock catalogue of central galaxies obtained from the Millennium numerical simulation, to which we henceforth refer as the GALAXY mock catalogue.In this work we quote results for the Planck SEVEM CMB clean map, although similar results are obtained for all Measured pairwise peculiar momenta under different aperture sizes for the SEVEM clean map.In both panels the green solid lines provide the expectation from the GALAXY mock catalogue, while the dot-dashed lines correspond to expectations found after averaging over 100 Gaussian simulations.In both cases, we assume τ T = 5 × 10 −5 .the other cleaning algorithms.As in Planck Collaboration Int.XXXVII (2015), our estimates of the kSZ temperature of Eq. 1 are obtained after applying aperture photometry (hereafter AP) in the direction of the CGC members.For any pointing centred upon any given CG and aperture angle θ AP , the average temperature within an annuli of inner and outer radii equal to θ AP and √ 2θ AP is subtracted from the average temperature in a circle of radius equal to θ AP .By subtracting the outer ring we attempt to decrease the amount of large scale, background contamination to the kSZ signal, although we unavoidably remove some fraction of kSZ signal as well.Following Eqs. 4 and 5 in Sect.3.1 of Planck Collaboration Int. XXXVII (2015), we also remove the DC (constant) level in the kSZ temperature estimates of all CGs lying in a similar redshift range.We defer the reader to section 2 of that paper for a more detailed description of the CMB and Sloan data under use, and section 3.1 for a deeper discussion on the measurements of the kSZ temperature estimates.

Fig. 2. Top panel:
Probability distribution for the optical depth τ T inferred from the w δT v rec (r) correlation function (solid lines) and the kSZ peculiar momenta (dashed lines), under different apertures, for the SEVEM map.These estimates of τ T are obtained after fitting the data to the prediction inferred from the GALAXY mock catalogue.Bottom panel: Same as top panel, but using the average of the Gaussian simulations rather than the GALAXY prediction as a reference.The recovered τ T estimates from the two statistics under consideration seem more compatible for the Gaussian simulations than for the GALAXY mock sample.

Methodology and results
The two statistics presented in Planck Collaboration Int.XXXVII (2015) yielding kSZ evidence are the kSZ pairwise peculiar momentum (hereafter p kSZ (r)) and the cross correlation function of the kSZ temperature and the recovered radial peculiar velocity (hereafter where the weights c i, j are given by (Ferreira et al. 1999) Following the convention of Hand et al. (2012), in this equation r i and r j are the vectors pointing to the positions of the i-th and j-th galaxies on the celestial sphere, r i and r j are the comoving distances to those objects, and r i, j = r i − r j refers to the distance vector for this pair of galaxies.The hat symbol (r) denotes a unit vector in the direction of r, and θ is the angle separating ri and r j .Note as well that ri, j refers to the difference vector (r i − r j )/|r i − r j |.On the other hand, the w δT v rec (r) function is computed as the spatial correlation function between the measured kSZ temperature estimates and the linear theory predictions of the radial peculiar velocity of the CGs (v rec ), w δT v rec (r) = i, j (δT i v rec j w i w j )/ i, j w i w j , where as before the i, j sub-indices refer to CGs and w i , w j to their weights.For both statistics, the δT j s correspond to the AP estimates of the kSZ fluctuations of Eq. 1.We refer again to Planck Collaboration Int.XXXVII (2015) for details on how linear theory predictions of CG radial peculiar velocities are obtained.Fig. 1 displays the measured p kSZ (r) and w δT v rec (r) functions under 5, 8, and 12 arcmin aperture radii for the SEVEM clean map of Planck (very similar estimates are found for other clean maps).In order to estimate the amount of free electrons giving rise to these signals, we compare these measurements with predictions obtained from (i) the GALAXY mock catalogue of central galaxies, and (ii) a suite of 100 Gaussian simulations of the matter density contrast field that is then inverted into a peculiar velocity field by means of the linearized continuity equation.The Gaussian simulations of the density contrast are generated from a dark matter linear power spectrum compatible with Planck's cosmology, and have no power at wavemodes k > 0.15 h Mpc −1 since those are regarded as non-linear scales.The first approach provides velocities for the halos hosting the CGs, while the second computes an estimate of the smooth, linear peculiar velocity in a region surrounding each CG.Following the approximation in Eq. 1, green curves in Fig. 1 provide predictions for p kSZ (r) and w δT v rec (r) from the GALAXY mock (solid lines) and the Gaussian simulations (dot-dashed lines), with a choice of τ T = 5 × 10 −5 for display purposes.We find that, while for w δT v rec (r) the two predictions just differ in a ∼ 15 % amplitude factor, for p kSZ (r) they differ both in shape and amplitude.Since the kSZ is built upon all electrons present in the volume sampled by the aperture photometry, and not necessarily bound to the CG halos, we expect the kSZ to lie between the two predictions.
After conducting minimum χ 2 fits of the data to the predictions and assuming Gaussian errors, we obtain the density probabilities for τ T as given in Fig. 2. When fitting to the predictions inferred from the GALAXY catalogue, we obtain τ T estimates from p kSZ (r) measurements falling in the (0.1-1.1) × 10 −4 range, in slight tension (between 1 and 2 σ low) compared to estimates from w δT v rec (r).This apparent mismatch is mitigated and better consistency among τ T estimates is found when fitting to the predictions provided by the Gaussian simulations (see bottom panel of Fig. 2).The p kSZ (r) and w δT v rec (r) measurements should yield similar τ T estimates, even when these measurements are obtained on different CG samples.The p kSZ (r) computation uses the full CG sample, while the w δT v rec (r) is restricted to the CGs fitting inside a 3D grid of 512 h −1 Mpc on a side (Planck Collaboration Int.XXXVII 2015).However, we have checked that both CG samples are equivalent in terms of mass and redshift distribution, and hence Fig. 2 suggests that the motion of electrons in and around CGs is better described by the Gaussian simulations, which we adopt in subsequent analyses.Note that this choice is not crucial for our discussion below.
We next estimate the baryon fraction giving rise to the observed kSZ signal under different apertures.We focus on w δT v rec (r) measurements in this case since they have higher Fig. 3. Top panel: Baryon fraction versus aperture angle.Red circles denote positive estimates, while green squares display the absolute value of negative quantities that are compatible with zero.We find 2-2.5 σ evidence for ∼ 45-55 % of the total amount of baryons.Bottom panel: Recovered profile of gas overdensity from w δT v rec (r) measurements after assuming L corr = 38 h −1 Mpc, (filled circles).In both panels, we compare our data to the prediction for dark matter for different choices of L corr .signal-to-noise ratios.For any line of sight along any CG, all electrons inside a cylinder whose base radius is given by the angular aperture (θ AP ) should contribute to the signal.The LOS depth of this cylinder corresponds to the typical velocity correlation length (L corr ), and is suggested by Fig. 1 to lie in the 20-40 h −1 Mpc range (comoving units).We find however that the baryon fraction should not depend upon L corr for any given aperture: where f vol (z) ≃ n CG (z) π(r(z) θ AP ) 2 L corr is the fraction of the comoving volume sampled by cylinders centred upon CGs, L corr is the LOS depth of the cylinder in comoving units, ne (z) is the average physical electron number density at redshift z (whereas the "0" subscript denotes at present), n CG (z) is the comoving CG number density and r(z) and d Ang (z) are the comoving and angular distances to redshift z, respectively.The brackets ... z denote redshift averages, and thus our previous measurements of τ T correspond to τ T (θ AP ) = τ T (θ AP , z) z .Note that Eq. 4 holds as long as there is no overlap between cylinders along two different LOSs: we have verified that such overlap is negligible for apertures below 13 arcmin, (it affects less than 1 % of pairs falling in the first distance bin at ∼ 9 h −1 Mpc, and this ratio is still smaller for the other distance bins).We remark that this definition of baryon fraction computes the ratio of detected baryons to the total amount of baryons under the effective footprint and redshift range probed by the CGC.The top panel of Fig. 3 shows that the baryon fraction increases with the aperture, amounting to 45-55 % of the total of all baryons for θ AP ∼ 8-10 arcmin.For larger apertures, our measurements become compatible with noise and oscillate around zero.The blue, dot-dashed line provides the prediction for a scenario where the baryons trace perfectly the distribution of dark matter.This prediction follows the approach of Hayashi & White (2008), who compute the dark matter -central halo correlation function ξ h,m (r) which is dependent of both mass and redshift of the halos, and requires some linear bias relation b(M, z) that we approximate by that of Sheth & Tormen (1999).The halo masses are obtained from the stellar masses following the approach in Planck Collaboration Int.XI (2013).This prediction computes, as in Eq. 4, the ratio of the mean dark matter overdensity inside the cosmological volume sampled by the cylinders centred upon the CGs over the inverse of the fraction of the total cosmological volume sampled by the same cylinders.We thus need to integrate δρ m (r) ∝ ξ h,m (r) inside the cylinders, and, as see in Eq. 4, the dependence upon L corr practically vanishes.We can also incorporate the impact of the outer ring of our AP filter in the prediction, after taking into account the redshift and mass distribution of our CG sample for each aperture θ AP .The top panel of Fig. 3 shows that our measurements of the baryon fraction are compatible with this prediction, although slightly falling on the high amplitude side.This means that Planck's measurements are compatible with detecting all missing baryons expected to be found in the outskirts of the CGs.
In the bottom panel of Fig. 3 we address the question whether the amount of baryons found around CGs is close or not to expectations for dark matter.We compare the gas overdensity values inferred from our τ T measurements (filled symbols) and L corr = 38 h −1 Mpc with the predictions for dark matter according to Hayashi & White 2008 (blue lines), after testing different values for L corr ranging from 8 to 38 h −1 Mpc.In this case, the conversion between angular apertures and transversal distances accounts for the redshift distribution of the CG sources.Values for gas overdensity are obtained from τ T (θ AP ) estimates via Our measurements are compatible with the prediction for dark matter regardless the choice of L corr (we only display the case of L corr = 38 h −1 Mpc).While the impact of feedback should be to deplete gas from the inner regions, the data show no strong indication for that.In fact, noise in our measurements and uncertainties in both the predictions for p kSZ (r)/w δT v rec (r) and the stellar mass-halo mass calibration for CGs discourage inferring strong conclusions about feedback.We can only state that our data are compatible with the no-feedback scenario.

Discussion
Observations of metallic lines in the X-ray in the direction of high energy sources provide information about the gas density in the so called circumgalactic medium of the Milky Way (Gupta et al. 2012;Gatto et al. 2013;Werk et al. 2014;Miller & Bregman 2015).While some authors claim to have found evidence for all the baryons to be expected in our halo (Gupta et al. 2012), some other authors seem to find only between 10 to 50 % of the expected amount of baryons (Gatto et al. 2013;Werk et al. 2014;Miller & Bregman 2015).Both the thermal Sunyaev-Zeldovich (hereafter tSZ, Sunyaev & Zeldovich 1972b) and the kSZ effects provide alternative approaches to detect diffuse gas, different to the window available from X-ray observations.The tSZ and kSZ effects provide statistical measurements of gas fraction in the visible halo population and thus do not restrict to the Milky Way host halo.In Planck Collaboration Int.XI (2013) it is claimed that there is no apparent evidence for feedback effects in the tSZ luminosity of CGs, in contradiction with X-ray observations.Very recently, those analyses have been revisited by Le Brun et al. (2015) after including explicitly the impact of feedback in the filters extracting the tSZ signal.They conclude that Planck tSZ observations up to 5 R 500 are actually compatible with AGN-induced feedback effects, and incompatible with the no-feedback hypothesis (or self similarity in the tSZ luminosity -halo mass relation).On the other hand, our kSZ study is blind to any assumed gas profile in CG host halos, and given the compensated structure of the AP filter it provides lower limits to the amount of gas in CGs. Figure 3 shows that our measurements are compatible with having detected all missing baryons in case these follow the dark matter distribution irrespectively of the gas temperature profile.There exist uncertainties in the predictions for the gas peculiar velocities, although these are relatively small if we compare the predictions from the Millennium and the Gaussian simulations in the top panel of Fig. 1.The impact of dust and other contaminants seems to have been characterised and kept under control in Planck Collaboration Int.XXXVII (2015).We thus conclude that the kSZ provides evidence for all the missing baryons around CGs and roughly half the total amount of baryons under the SDSS DR7 angular footprint at redshift z ≃ 0.12.Nevertheless, a more detailed comparison with state-of-the-art hydrodynamical numerical simulations should shed more light on these results.
Our measurement of τ T around the CGs allows us setting constraints on a kSZ dipole, just like in Planck Collaboration Int.XIII (2014): In this equation, δT j correspond to the estimate of the AP kSZ temperature estimate in the direction of the j-th CG, and ( n • nj ) corresponds to the dot product of the unit vectors pointing to the j-th CG and the assumed dipole direction.An estimate of the rms of A dip (denoted by σ A ) is obtained after applying Eq. 6 on the δT j s from the rotated positions.After sweeping n over the entire celestial sphere we conclude that the S/N of the dipole (A dip /σ A ) is always below 1.9.For the SEVEM map and an aperture of 8 arcmin, we find that the amplitude of the dipole from the CGC is below 0.37 µK at 95 % C.L. Taking at face value τ T = 1.4 × 10 −4 for an aperture of 8 arcmin, this results in an upper limit for the velocity of 290 km s −1 at 95 % C.L for a sphere of radius ≈ 350 h −1 Mpc.This limit is clearly inconsistent with the claim of long range flows of Kashlinsky et al. (2008Kashlinsky et al. ( , 2010)), while consistent (and stronger) than the limit of Feindt et al. (2013).Actually, it is also slightly stronger than the limit presented in Planck Collaboration Int.XIII (2014).These analyses provide further evidence for the Copernican principle and the homogeneity of the Universe.

Fig. 1 .
Fig. 1.Top panel: Measured w δT v rec (r) correlation function under different apertures for the SEVEM clean map.Bottom panel:Measured pairwise peculiar momenta under different aperture sizes for the SEVEM clean map.In both panels the green solid lines provide the expectation from the GALAXY mock catalogue, while the dot-dashed lines correspond to expectations found after averaging over 100 Gaussian simulations.In both cases, we assume τ T = 5 × 10 −5 .