On the peculiar momentum of baryons after Reionization

The standard cosmological model predicts the statistical properties and evolution of the large scale matter distribution. Initially small inhomogeneities in the matter density field grow under gravity in an expanding universe, giving rise to a cosmic web made of filaments, sheets and superclusters of galaxies that we see today. Large, comoving flows of matter fall onto deep potential wells under gravity, with typical speeds of $\sim 150$ km s^-1 in the standard $\Lambda$CDM cosmological model (Hinshaw et al., 2008). These peculiar velocities ( peculiar with respect to the Hubble flow induced by the universal expansion) cannot be easily measured. Indeed, despite recent claims pending for confirmation (e.g., Kashlinsky et al., 2008; Watkins et al., 2008), there is no conclusive observational evidence for such bulk flows in the present universe.

Nevertheless, the Cosmic Microwave Background (CMB) radiation has provided an accurate measurement of these bulk flows when the universe was 380,000 years old ($z\simeq 1,050$). Before most of the electrons recombined with the protons and the CMB could propagate freely, the electron plasma and the CMB radiation were in thermal equilibrium since Thomson scattering kept both fluids coupled. The free electrons Thomson scattered CMB photons, leaving Doppler imprint according to their relative velocities caused by the oscillation of the plasma-photon fluid in the gravitational potential wells (Sunyaev & Zeldovich, 1970; Hu & Sugiyama, 1995). Since energy and density fluctuations were very small at that epoch, the linear theory is able to describe very accurately the anisotropy pattern that was introduced by Thompson scattering on the CMB, and this has been confirmed by observations (e.g. Mauskopf et al., 2000; Miller et al., 2002; Grainge et al., 2003; Hinshaw et al., 2008).

At redshift $z\sim 10$ the first stars began to ionize hydrogen in the Intergalactic Medium (IGM), and therefore CMB photons interacted again with the free electrons via Thomson scattering. In this scenario, electron peculiar velocities introduced new temperature anisotropies on the CMB. Likewise, due to the anisotropic nature of Thomson scattering, the CMB was also linearly polarized on the large angular scales, as it has also been confirmed by the WMAP experiment, (Hinshaw et al., 2008). These epochs describe the last scenario where all baryons have been observed: at more recent epochs, the amount of baryon matter that we are able to detect is at most of half of what is observed during both recombination and reionization (Fukugita & Peebles, 2004). This constitutes the missing baryon problem.

In this work, and just as in Hernández-Monteagudo & Sunyaev (2008), we investigate bulk flows and the missing baryons utilizing Thomson scatterings between CMB photons and free electrons at low redshift. This mechanism is referred to as the kinetic Sunyaev-Zel’dovich effect (kSZ, Sunyaev & Zeldovich, 1972), if it occurs in galaxy clusters, but it is called the Ostriker Vishniac effect (OV, Ostriker & Vishniac, 1986) in the context of cosmological second order perturbation theory, (for which the electron density and velocity are continuous fields). We shall follow this terminology, although in some occasions we may use either when referring to both effects for simplicity. We shall restrict ourselves to the impact of Thomson scattering on the intensity or temperature anisotropies of the CMB, since the effect on the CMB polarization is already studied in Hernández-Monteagudo & Sunyaev (2008). Unlike in Hernández-Monteagudo et al. (2006), we shall not consider the spectral distortions that the hot fraction of the missing baryons introduce in the CMB black body spectrum. Previous efforts have been made in the computation of the OV effect (Ostriker & Vishniac, 1986; Dodelson & Jubas, 1995; Hu, 2000; Castro, 2003), some of them in the context of the non-Gaussian signal they give rise to. In some recent works, the kSZ in halos has been used as a probe for Dark Energy (Bhattacharya & Kosowsky, 2008), while Moodley et al. (2008) have used the gas in groups to probe the missing baryons, and Lee (2009) studies the prospects of constraining patchy reionization. Here, we present the first full sky projection of the temperature anisotropies induced by peculiar momenta of ionized baryons during and after reionization up to the present: if applied to the smooth electron distribution our calculations provide a full description of the OV effect, while when applied to the halo distribution they describe the contribution of galaxy group/cluster correlation on the kSZ effect. The formalism also permits the computation of the cross correlation between the OV and the kSZ effects. At the same time, our computations provide the linear approximation under which the electron density is approximated by its average value. We can then compare our code with standard linear theory Boltzmann codes such as CMBFAST (Seljak & Zaldarriaga, 1996) or CAMB (Lewis et al., 2000), as well as normalize our predictions to the linear fluctuation amplitudes measured by WMAP. Finally, we analyze our results in the context of future CMB and Large Scale Structure observations, and provide predictions for the detectability of bulk flows and missing baryons in the light of those future data sets. The paper is structured as follows: in Section 2 we describe the line of sight projection of the baryon peculiar momentum, and this is discussed in Section 3 in the context of its application to the halo population. The amplitude of the resulting power spectra are given in Section 4, and their interpretation in terms of bulk flow/missing baryon detection is provided in Section 5. We discuss our main results in Section 6 and conclude in Section 7.

The computation of the OV/kSZ effect requires the projection along the line of sight the product of electron peculiar velocity and density. If we consider the spatial variations of the electron density, then this product becomes a second order quantity in cosmological perturbation theory provided that peculiar velocities are very small ($v/c\ll 1$). I.e., the CMB temperature anisotropies introduced by the peculiar motion of the scatterers reads

with $\tau_{T}$ the Thomson optical depth

$a(\eta)=1/(1+z(\eta))$ the cosmological scale factor, $\sigma_{T}$ the Thomson cross section and $n_{e}(\eta)=\bar{n}_{e}(1+\delta_{e})$ is the electron number density in terms of its average value ($\bar{n}_{e}$) and its spatial fluctuations ($\delta_{e}$). The symbol $\eta$ denotes conformal time or, equivalently, comoving distance. In Hernández-Monteagudo et al. (2006) (hereafter HVJS) a full sky description of the kSZ generated by the low redshift galaxy cluster population was provided. However, as noted later by Gordon et al. (2007), those computations observed only the diagonal part of the Fourier peculiar velocity tensor $\langle v^{i}_{\mbox{\bf{k}}}v^{j\;*}_{\mbox{\bf{q}}}\rangle$ (where k and q are Fourier wave vectors and $i$ and $j$ refer to one of the three spatial components of the peculiar velocity vectors), neglecting all non-diagonal components. In Appendix A, we present a full sky computation considering all terms in this tensor, and find that the complete treatment introduces small changes to the results given in HVJS. The results of the Appendix are summarized here.

In brief, the second order statistic for the electron momentum ($p_{e}\propto n_{e}v_{e}$) is proportional to the quantity $\langle n_{e,1}v_{e,1}n_{e,2}v_{e,2}\rangle$. If $n_{e,i}$ is constant, then it simplifies to ${\bar{n}}_{e}^{2}\langle v_{e,1}v_{e,2}\rangle$, and this is the $vv$ term. Furthermore, it is possible that $n_{e}$ is driven by Poissonian statistics, in which case $\langle n_{e,1}v_{e,1}n_{e,2}v_{e,2}\rangle\propto n_{e}\langle v_{e,1}v_{e,2}\rangle$ (Poisson term). If however $n_{e,i}$ is correlated to the velocity field, then one must use the cumulant expansion theorem, and three new terms arise, of which one is positive, another one is zero and the third one is negative. In all cases, the correlation function $C(\hat{\mbox{\bf{n}}}_{1},\hat{\mbox{\bf{n}}}_{2})$ for each term can be written as Fourier integrals over some wave vector k of functions with an angular dependence of either the type $(\hat{\mbox{\bf{k}}}\cdot\hat{\mbox{\bf{n}}}_{1})(\hat{\mbox{\bf{k}}}\cdot\hat{\mbox{\bf{n}}}_{2})$ or $(\hat{\mbox{\bf{n}}}_{1}\cdot\hat{\mbox{\bf{n}}}_{2})$. Those contributions with $(\hat{\mbox{\bf{k}}}\cdot\hat{\mbox{\bf{n}}}_{1})(\hat{\mbox{\bf{k}}}\cdot\hat{\mbox{\bf{n}}}_{2})$ type dependence will give negligible contribution at the small scales, i.e., the corresponding angular power spectrum multipoles $C_{l}$ will go to zero as the multipole $l$ grows. In this limit, only the contributions with the $(\hat{\mbox{\bf{n}}}_{1}\cdot\hat{\mbox{\bf{n}}}_{2})$ type angular dependence will survive. Let us consider the above more carefully in case by case basis.

Before we show the amplitudes of the OV/kSZ power spectra presented in the previous section, we compare the effect of Thomson scattering as it arises in collapsed halos to the OV generated in the intergalactic medium (IGM). It is relevant to understand under which conditions would collapsed gas in halos contribute more in terms of CMB angular anisotropies than the gas in the IGM. In both scenarios, Thomson scattering introduces anisotropies according to

where we have approximated the Thomson visibility function $\Lambda=\dot{\tau_{T}}\exp{(-\tau_{T})}$ by the opacity, $\Lambda\simeq\dot{\tau_{T}}$, since $\tau_{T}$ is $\sim 0.09$ in the redshift regime of interest (Hinshaw et al., 2008).

We assume that the peculiar velocity of gas with respect to the CMB is very similar for both gas in IGM and halos, the comparison of the OV and kSZ effects must then be focused on the optical depths, i.e., in the line of sight integral of $\dot{\tau_{T}}$ in each case. As shown in HVJS, the effective opacity generated in halos at a given epoch is given by

where $dn/dM$ is the halo mass function (we choose Sheth and Tormen (ST, Sheth & Tormen, 1999), $V(M,z)$ is the proper volume occupied by gas in those halos, and $\dot{\tau}_{T,halo}(M,z)$ is the opacity generated by a halo of mass $M$ at redshift $z$. The effect is in the end sensitive to the total number of electrons in a given halo population, and can be expressed by the average halo optical depth times the volume fraction of halos.

The halo population opacity can be compared directly to the opacity associated to the smooth distribution of all electrons. The top panel of Figure (1) displays the opacity in three scenarios: (i) a smooth distribution of electrons (solid line)¹¹1We assume a sudden reionization model within WMAP V cosmology, for which $\tau_{T}=0.084\pm 0.016$ and $z_{reio}=10.8\pm 1.4$, (ii) electrons within all halos of mass greater than $5\times 10^{13}h^{-1}M_{\odot}$ (dashed line), and (iii) electrons within all halos of mass greater than $5\times 10^{5}h^{-1}M_{\odot}$ (dotted line). The most massive halo population provides a very low opacity due to the small amount of cosmological volume it takes. The latter halo population contains almost all the electrons in the Universe, and therefore its opacity is very close to the one in case (i)²²2According to the halo model, all mass in the Universe must be found in collapsed halos.

We have ignored the effect of the bias until now. Dark matter halos can either be biased or anti-biased tracers of the underlying matter distribution, and we need to include the bias factor when we compute the angular anisotropies. In the bottom panel of Figure (1) the bias factors for the two halo populations considered ($M>5\times 10^{5}h^{-1}M_{\odot}$ and $M>5\times 10^{13}h^{-1}M_{\odot}$) are shown by the dotted thin and dotted thick lines, respectively. These bias factors were computed by using the expressions of Mo & White (1996), and integrating them with the ST mass function:

Although at high redshifts bias factors grow above unity, the volume fraction occupied by the halo population are very low: this is shown by bottom solid lines, (thin line for $M>5\times 10^{5}h^{-1}M_{\odot}$, thick line for $M>5\times 10^{13}h^{-1}M_{\odot}$). The volume fraction is so low that it cannot be compensated by gas overdensity within halos: the product of gas overdensity and the volume fraction is below unity at all redshifts (as shown by the thin ($M>5\times 10^{5}h^{-1}M_{\odot}$) and thick ($M>5\times 10^{13}h^{-1}M_{\odot}$) dashed lines). The thin solid green line shows the level of unity.

According to these results, it seems that the anisotropy due to halos will always be below the level of anisotropy induced by a smooth and continuous distribution of electrons. There are however two caveats: (i) Halos might show bias in their peculiar velocities, although this is unlikely to change the picture by more than a 30%-40% (Peel, 2006), (ii) on the small scales, halos follow Poissonian statistics, and this will add a considerable amount of anisotropy, as we shall see below. These results differ from previous works (e.g., Hu, 2000; Castro, 2003) where the power spectrum for kSZ in non linear structures had been computed. In previous works, the matter power spectrum was simply replaced by the non linear power spectrum, but no changes were introduced in the related opacity. No attention was paid to the Poisson term either, which, in the case of halos, turns out to be dominant. This difference must be motivated by the fact that, they were computing the combined effect of both linear and non-linear density perturbations at high redshift. Here we try to separate the OV contribution from the one generated in halos, and the contribution to the latter arises mostly at low ($z<2$) redshift.

Of all kSZ terms shown in Figure (3), one of the easiest to observe is the Poisson term at $l\sim 2,000-3,000$. Its amplitude is almost as high as the combination of the $vv-dd$ and $vd-vd$ terms from the OV and more importantly, it is associated with massive halos that can be identified at other frequencies. Nevertheless, one must note that measuring the kSZ Poisson term at positions of galaxy clusters does not properly test the theoretical predictions of the model on the bulk flows. This term is sensitive to the average velocity dispersion of those objects (which is likely to be contaminated by non-linear contributions) but tells nothing about the spatial correlations of the peculiar velocity field.

In the context of the standard model, it was shown in HVJS that typical correlation lengths for bulk flows are about 20 – 40 $h^{-1}$Mpc in comoving units. This means that all electrons within a sphere of radius from $20$-$40$ $h^{-1}$Mpc share a similar peculiar velocity. If this is an accurate description of reality, then one can use the kSZ detected in the most massive halos as a template for the peculiar velocity field in their surroundings. This idea was used in Hernández-Monteagudo & Sunyaev (2008) to track the missing baryons by kSZ – E polarization mode cross-correlation, and can also be applied in this context. Let us assume that future high resolution CMB experiments provide accurate measurements of the kSZ in halos more massive than, say, $2\times 10^{14}\;h^{-1}M_{\odot}$, so that a template of the kSZ on the sky (hereafter $\mbox{\bf{M}}_{kSZ}$) can be built. If this template is then cross-correlated with the CMB sky (hereafter $\mbox{\bf{T}}_{CMB}$), then not only does signal come from the known clusters, but also from the surrounding gas. Fundamental sources of noise for this cross-correlation are, to first order, the primordial CMB fluctuations and the instrumental noise (which usually becomes relevant on the scales of the PSF). This cross-correlation can be picked-up via a matched filter (MF) approach,

where the CMB signal is assumed to intrinsically contain some kSZ component proportional to the kSZ template, i.e., $\alpha\mbox{\bf{M}}_{kSZ}$. The covariance matrix C contains the noise sources (namely primordial CMB and instrumental noise). The signal to noise ratio (S/N) of this cross correlation is given by

If the kSZ template is written in multipole space, then the last equation on the S/N can be written for each multipole $l$. In this basis, the covariance matrix C is diagonal for white instrumental noise $N_{l}$, $(\mbox{\bf{C}})_{l,l^{\prime}}=(C_{l}^{CMB}+N_{l})\delta_{l,l^{\prime}}$, and Equation (20) reads

For non-full sky coverage ($f_{sky}<1$), there are not $(2l+1)$ degrees of freedom per multipole $l$, but roughly $(2l+1)f_{sky}$. In this case, the last equation should be multiplied by $f_{sky}^{1/2}$. The term $C_{l,kSZ}^{h,h}$ accounts for the halo-halo kSZ auto-correlation (i.e., the Poisson, $vv$, $vv-dd$ and $vd-vd$ terms), whereas $C_{l,kSZ}^{h,OV}$ expresses the cross-correlation between the kSZ induced in halos and the OV generated by the continuous electron distribution on the large scales (i.e., mostly outside halos). This cross-power spectrum can be easily computed by introducing, in our angular cross-correlation function $C(\hat{\mbox{\bf{n}}}_{1},\hat{\mbox{\bf{n}}}_{2})$, the halo visibility function in the $los$ integrals ${\cal I}_{l}$-s along direction $\hat{\mbox{\bf{n}}}_{1}$, and the OV visibility function along direction $\hat{\mbox{\bf{n}}}_{2}$, (see Appendix). It is worth to stress that in this case there is no contribution from the Poisson term, since this term appears exclusively for the halo auto correlation. If we are searching for the kSZ signal of the gas outside halos, one should mask the known clusters, which is equivalent to dropping the $C_{l,kSZ}^{h,h}$ term in the numerator of Equation (21).

The result of adding all the S/N below a given multipole $l$ is shown in Figure (4). Black lines refer to an ACT-SPT like all-sky CMB experiment with a sensitivity level of $\sim 1\mu K$ per resolution element of $\sim 1$ arcmin. The red color lines (both solid and dashed) display results for (all sky, $f_{sky}=1$) Planck’s HFI 217 GHz channel. If kSZ clusters are not masked, then the cross-correlation test yields a very high S/N, as displayed by the dashed lines. Instead, if known kSZ clusters are masked, then the S/N drops to a level than only an ACT-SPT type CMB experiment would be able to detect it, since it mostly comes from the small angular scales. This is shown by the solid lines, which depict the ideal case of full sky coverage ($f_{sky}=1$). On the contrary, filled triangles and squares refer to an ACT-SPT type experiment covering 2,000 and 10,000 square degrees, respectively. We see that these scenarios lie in the limit of detectability.

The latter case describes the situation in which kSZ measurements in galaxy clusters are used to build the kSZ template and search for the baryons in the IGM. The requirements on the sky coverage, angular resolution and instrumental sensitivity are demanding if a kSZ detection of the missing baryons is to be provided. Since the OV is mostly generated at moderately high redshifts, so if the cluster sample does not extend deep enough, then the S/N drops accordingly (different survey depths are considered in the different panels of Figure (4)). Note that we do not expect to find massive clusters at high (say $z>4$) redshifts.

If kSZ measurement in halos are not available, it has been suggested by Ho, Dedeo & Spergel (2009) that kSZ templates may be built from the large scale matter distribution. Present reconstruction algorithms like e.g., ARGO (Kitaura & Enßlin, 2008) seem to reconstruct peculiar velocities accurately in scales of interest. This velocity field reconstruction could be an additional way to obtain the kSZ templates, and to detect the kSZ in clusters. In this case the S/N is much higher (dashed lines), and there is no need to have deep surveys: most of the S/N of the angular cross-correlation is coming from the Poisson term. This term does not depend critically on depth, but simply relies on the discrete nature of halos. However, since the kSZ template would be a prediction of the model based upon the large scale structure observations, this exercise would still provide a test for the cosmological theory of bulk flows.

Our formalism allows us to study consistently the OV/kSZ temperature anisotropies introduced by both the gas collapsed in halos and the gas present in the IGM, regardless of the redshift range of relevance in each case. Our computations differ from previous ones in mainly two aspects: (i) ours are full sky calculations, where no flat sky/Limber approximation have been invoked and therefore predictions apply to all multipole range, and (ii) the kSZ computations take into account the fact that halos above a given mass threshold host only a fraction of the total mass, and hence the amplitude of the kSZ they give rise to is limited. Indeed, even after accounting for the enhanced clustering of the massive halo population, we find that the kSZ generated by groups and clusters is below the OV anisotropies produced during and shortly after reionization. Only the Poisson term induced by the halo population reaches an amplitude that is comparable to the OV $vv-dd$ term, although it should be possible to partially remove it by masking out the most massive halos. By excising all halos above $2\times 10^{14}h^{-1}M_{\odot}$, the Poisson term should drop by a $\sim$40 % (in the linear units of Figure (3)). Distinguishing these two contributions may be a relevant issue, since the kSZ is mostly generated at low redshifts ($z<3-4$), whereas the OV is one of the very few existing probes of the reionization epoch ($z\in[5,12]$). This gives significance to the cross-over of the total OV power (red filled circles in the left panel of Figure 3) above the intrinsic linear CMB power spectrum at $l\sim 4,000$. We remark that this OV computation is merely second order perturbation theory, and since the linear level of perturbations has been normalized via, e.g. WMAP observations, there are no free parameters for our (second order) OV predictions. Nevertheless, our computations do not observe patchy reionization, that is, the extra amount of anisotropy introduced by a non fully ionized IGM at $z>10$. Zahn et al. (2005) claim that this phase of reionization should introduce fluctuations comparable in amplitude to the OV, and this would provide further relevance to the study of the anisotropies in these angular scales.

This work is also motivated by the search of the missing baryons via their peculiar motions with respect to the CMB. If predictions from linear theory are correct, the baryons should comoving on scales of 20 – 40 $h^{-1}$Mpc, and hence the kSZ pattern generated in clusters and groups of galaxies should be very similar to that generated by the surrounding gas, and can be used to probe the latter. However, it is not clear yet to what extent kSZ measurements in future CMB surveys like ACT or SPT will be contaminated by the intrinsic CMB and point source emission. In the ideal scenario in which errors in the kSZ estimates are unimportant for all clusters more massive than $2\times 10^{14}h^{-1}M_{\odot}$, then it is clear that the signature of missing baryons can be unveiled if the linear theory predictions on bulk flows are correct. One can consider to combine them with peculiar velocity field reconstructions obtained from LSS surveys, in which accuracy would be required at relatively large scales (10–20 $h^{-1}$Mpc). It should be noted that estimates of the kSZ would be needed for each bulk flow, and this can be achieved by combining kSZ/peculiar velocity estimates from different halos belonging to the same filament/supercluster.

Our predictions have larger uncertainties in the kSZ case than in the OV case. We have assumed a Sheth-Tormen mass function for the halo abundance versus mass and redshift. We also model their physical size and their gas profile, and more importantly, we have adopted the Mo & White prescription for their clustering bias. When comparing our results with previous works, we find that our kSZ power spectra are at the level of Schäfer et al. (2006) both in shape and amplitude: when imposing the same mass threshold ($M>5\times 10^{13}h^{-1})M_{\odot}$, our kSZ power spectrum is within a factor of 2, despite our different choice for $\sigma_{8}$, (0.817 versus 0.9). Further, we are introducing an effective bias for the halo velocities of 1.3, while it is not clear to us how close are halo peculiar velocities compared to linear theory expectations in the simulation they used. As Schäfer et al. (2006) pointed out, there may be a significant amount of extra kSZ power at smaller scales due to the presence of sub-structure within the halos, (Springel et al., 2001). This could partially be avoided by smoothing the signal within the halo solid angle, although part of this excess could be associated to a thermal (non-linear) component of halo velocities. The addition of all this extra power could have an impact on the S/N estimations of Figure (4), which increases significantly at high $l$-s. A lower mass for the kSZ estimation would then be required in order to achieve the same S/N limit. However, despite all these sources of uncertainty, we conclude that future kSZ and LSS surveys would shed some light in the problematic of missing baryons and bulk flows in the late universe.

We have revisited the problem of the secondary anisotropies in the CMB introduced by the peculiar motion of baryons during and after reionization. We have addressed this problem in the context of the search for the missing baryons and the measurements of bulk flows at low redshifts. We have paid particular attention in distinguishing the different sources of anisotropy, according to their redshift, in an attempt to isolate the kSZ/OV generated by gas in filaments and superclusters at low redshift from its counterpart arising during reionization. For this purpose, we present the first all sky projection of the peculiar momentum of baryons from reionization to the present epoch. We consider two different scenarios: the OV generated by a smooth distribution of all electrons, and the kSZ generated by only those baryons located in collapsed structures.

The former provides the largest amplitude, and contributes mostly during reionization ($z>6$). Since it corresponds to second order in perturbation theory, WMAP observations at the linear level already provide a normalization. Once those observations fix the set of cosmological parameters, the predictions at second order should also be fixed: the OV should cross over the intrinsic linear CMB temperature angular power spectrum at $l\sim 4,000$, with an amplitude close to $1.5\;\mu K$ (for a sudden reionization scenario at $z_{reio}=10.8$ and $\tau=0.084$). The amplitude of the OV should be above the angular anisotropy level induced by the kSZ in the halo population. The discrete character of halos (and the Poisson statistics associated to it) gives rise to angular fluctuations that constitute the main contaminant (in terms of kSZ/OV effects) to the signal generated during reionization. By masking out all halos more massive than $2\times 10^{14}\;h^{-1}M_{\odot}$, this Poisson term should drop to a level of $\sim 30\%$ of the OV amplitude, leaving room for this window to the reionization epoch.

The terms accounting for the velocity – velocity and velocity – density correlation of the halo population are well below the Poisson-induced anisotropy level, but are relevant since those terms express the correlation of velocities of halos with the velocities of the surrounding gas. We propose using kSZ estimates in halos (to be obtained by ACT or STP-type experiments) in order to build peculiar velocity templates to be cross-correlated to future CMB data. If the gas surrounding halos is co-moving with them as the theory predicts, then a signal should arise from this test even after masking the contribution from the halos on the CMB maps. An ACT or SPT-like CMB experiment covering 10,000 (2,000) square degrees should achieve a S/N of 5 (2.5). The detection of this cross-correlation would provide evidence for the presence of missing baryons in the late universe and confirm the predictions of the theory of cosmological bulk flows. In case there are no quality kSZ estimates at the halo angular positions, then, following Ho, Dedeo & Spergel (2009), it would also be possible to build peculiar velocity templates from future deep LSS surveys. These would provide an estimate of the density field that can be inverted (within our cosmological frame) into a peculiar velocity template. By cross-correlating this template with future CMB maps one should be able to detect the kSZ and bulk flows at high significance, if our understanding of cosmological bulk flows is correct. In summary, cross-correlation studies of future CMB and kSZ/LSS data becoming a promising tool in the characterization of bulk flows and missing baryons in the low redshift universe.

We thank D.N.Spergel for useful discussions. CHM acknowledges fruitful discussions with S.D.White.