Towards detection of relativistic effects in galaxy number counts using kSZ Tomography

TL;DR

This paper explores how future high-resolution CMB and galaxy surveys can detect relativistic effects in galaxy counts through kSZ tomography, revealing biases in primordial non-Gaussianity estimates and improving bias parameter constraints.

Contribution

It demonstrates the potential of using correlations between the remote dipole field and galaxy counts to detect relativistic corrections and improve cosmological parameter estimation.

Findings

Neglecting relativistic corrections biases primordial non-Gaussianity measurements.

Correlations can detect relativistic effects in galaxy counts.

Remote dipole field helps constrain bias parameters and reduce alignment bias.

Abstract

High-resolution, low-noise observations of the cosmic microwave background (CMB) planned for the near-future will enable new cosmological probes based on re-scattered CMB photons -- the secondary CMB. At the same time, enormous galaxy surveys will map out huge volumes of the observable Universe. Using the technique of kinetic Sunyaev Zel'dovich (kSZ) tomography these new probes can be combined to reconstruct the remote dipole field, the CMB dipole as observed from different vantage points in our Universe. The volume accessible to future galaxy surveys is large enough that general relativistic corrections to the observed distribution of galaxies must be taken into account. These corrections are interesting probes of gravity in their own right, but can also obscure potential signatures of primordial non-Gaussianity. In this paper, we demonstrate that correlations between the reconstructed remote dipole field and the observed galaxy number counts can in principle be used to detect general relativistic corrections. We show that neglecting general relativistic corrections leads to an $\mathcal{O}(1)$ bias on the inferred amplitude of primordial non-Gaussianity, $f_{\rm NL}$. In addition, we demonstrate that the reconstructed remote dipole field can provide useful constraining power on various bias parameters appearing in the galaxy number counts, and can significantly mitigate the effects of alignment bias.

Tables

Table 1. Table 1: Cosmological and relativistic parameters, bias functions, and their fiducial values. The biases b v , b G , b A , f evo , s subscript 𝑏 𝑣 subscript 𝑏 𝐺 subscript 𝑏 𝐴 subscript 𝑓 evo 𝑠 b_{v},\,b_{G},\,b_{A},\,f_{\rm evo},\,s , that we refer to throughout are, respectively, the optical depth bias, the galaxy bias, the alignment bias, the evolution bias, and the magnification bias all defined explicitly in Appendix  A and  B . The values of bias functions indicated with a † † \dagger vary with redshift, the modeling of which is described in Appendix  A .

Parameter	${ϵ}_{\mathrm{GR}}^{\mathrm{kSZ}}$	${ϵ}_{\mathrm{GR}}^N$	$f_{NL}$	${10}^{10}{A}_s$	$n_s$	${\Omega }_b$	${\Omega }_c$	$h$	$\tau$	$b_v^z$	$b_G^z$	$b_A^z$	$f_{\mathrm{evo}}^z$	$s^z$
Fiducial value	1	1	0	2.2	0.96	0.0528	0.2647	0.675	0.06	1	$†$	0	$†$	$†$

Table 2. Table 2: Forecasted uncertainties on parameters of interest, and bias on f NL subscript 𝑓 NL f_{\rm NL} due to neglecting relativistic effects from number counts and kSZ measurements. Forecasts are shown with and without alignment bias, and as additional observables and priors are added.

	N	N, no $b_A$	N + CMB	N + CMB + kSZ	N + CMB + kSZ + $f_{\mathrm{evo}}$ prior
$\sigma \left(f_{\mathrm{NL}}\right)$	16	14	9.0	1.3	1.0
$b\left(f_{\mathrm{NL}}\right)$	-9.3	-8.3	-2.2	-2.0	-3.3
$\sigma \left({ϵ}_{\mathrm{GR}}^N\right)$	5.1	4.1	3.2	0.58	0.09
$\sigma \left({ϵ}_{\mathrm{GR}}^{\mathrm{kSZ}}\right)$	–	–	–	0.61	0.60

Equations

$$C_{\ell }^{XY}=4\pi \int \frac{dk}{k}\mathcal{P}(k){\Delta }_{\ell }^X(k){\Delta }_{\ell }^{Y\ast }(k),$$

$$\mathcal{P}(k)=A_s{\left(\frac{k}{k_0}\right)}^{n_s-1}.$$

$${\Delta }_{\ell }^{\mathrm{N}}(k)=$$

$${\Delta }_{\ell }^{\mathrm{kSZ}}(k)=$$

$$F_{\alpha \beta }=\sum _{\ell =1}^{{\ell }_{\max }}\frac{2\ell +1}{2}\mathrm{Tr}\left[\left({\partial }_{\alpha }{\mathcal{C}}_{\ell }\right){\mathcal{C}}_{\ell }^{-1}\left({\partial }_{\beta }{\mathcal{C}}_{\ell }\right){\mathcal{C}}_{\ell }^{-1}\right]+F_{\alpha \beta }^{\mathrm{CMB},\mathrm{high}-\mathrm{\ell }},$$

$$\mathcal{C}_{\ell}=\left(\begin{array}[]{ccc}C_{\ell}^{{\rm N,N}}&C_{\ell}^{{\rm N,CMB}}&C_{\ell}^{{\rm N,kSZ}}\\ C_{\ell}^{{\rm CMB,N}}&C_{\ell}^{{\rm CMB,CMB}}&C_{\ell}^{{\rm CMB,kSZ}}\\ C_{\ell}^{{\rm kSZ,N}}&C_{\ell}^{{\rm kSZ,CMB}}&C_{\ell}^{{\rm kSZ,kSZ}}\\ \end{array}\right)+N_{\ell}\,.$$

$$\Delta \alpha ={\left(F^{-1}\right)}^{\alpha \beta }{v}_{\beta },$$

$$v_{\beta }=\sum _{\ell =1}^{{\ell }_{\max }}\frac{2\ell +1}{2}\mathrm{Tr}\left[\left({\partial }_{\alpha }{\mathcal{C}}_{\ell }\right){\mathcal{C}}_{\ell }^{-1}\Delta {\mathcal{C}}_{\ell }{\mathcal{C}}_{\ell }^{-1}\right],$$

$${\Delta }_{\ell }^{\mathrm{D},i}(k)$$

$${\Delta }_{\ell }^{\mathrm{RSD},i}(k)$$

$${\Delta }_{\ell }^{\mathrm{lens},i}(k)$$

$${\Delta }_{\ell }^{D1,i}(k)$$

$${\Delta }_{\ell }^{D2,i}(k)$$

$${\Delta }_{\ell }^{P1,i}(k)$$

$${\Delta }_{\ell }^{P2,i}(k)$$

$${\Delta }_{\ell }^{P3,i}(k)$$

$${\Delta }_{\ell }^{P4,i}(k)$$

$${\Delta }_{\ell }^{P5,i}(k)$$

$$W_i(z)\equiv \left|\frac{d\chi }{dz}\right|{\tilde{W}}_i(\chi )$$

$$b_{\mathrm{NG}}(k,z)=3f_{\mathrm{NL}}(b_{\mathrm{G}}(k,z)-1){\Omega }_m{H}_0^2{\delta }_c/(k^{2}T(k)S_{\psi }),$$

$$s(\chi )$$

$$f_{\mathrm{evo}}(\chi )$$

$$n_s\equiv \frac{\#\mathrm{galaxies}}{dVd\ln L},$$

$$\mathcal{N}(\chi ,>\ln L_{\mathrm{cut}})\equiv {\int }_{\ln L_{\mathrm{cut}}}^{\infty }d\ln L^{\prime }{n}_s(\chi ,\ln L^{\prime }).$$

$$n_s(M)dM$$

$$M_{\ast }(z)$$

$${\varphi }_{\ast }(z)$$

$$M$$

$${\Delta }_{\ell }^{\mathrm{N},\mathrm{GR},i}={\Delta }_{\ell }^{D1,i}+{\Delta }_{\ell }^{D2,i}+{\Delta }_{\ell }^{P1,i}+{\Delta }_{\ell }^{P2,i}+{\Delta }_{\ell }^{P3,i}+{\Delta }_{\ell }^{P4,i}+{\Delta }_{\ell }^{P5,i}.$$

$${\Delta }_{\ell }^{A,i}(k)=\frac{b_v}{2\ell +1}\int d\chi W_i(\chi )[\ell j_{\ell -1}(k\chi )-(\ell +1)j_{\ell +1}(k\chi )]{\mathcal{K}}^A(\chi )$$

$${\mathcal{K}}^{\mathrm{LD}}(\chi )$$

$${\mathcal{K}}^{\mathrm{RD}}(\chi )$$

$${\mathcal{K}}^{\mathrm{SW}}(\chi )$$

$${\mathcal{K}}^{\mathrm{ISW}}(\chi )$$

$${\Delta }_{\ell }^{\mathrm{kSZ},\mathrm{GR},i}={\Delta }_{\ell }^{\mathrm{RD},i}+{\Delta }_{\ell }^{\mathrm{SW},i}+{\Delta }_{\ell }^{\mathrm{ISW},i}.$$

Full text

I Introduction

Galaxy surveys and cosmic microwave background (CMB) measurements will provide us with exceptionally accurate and precise measurements of our Universe over the coming decade. Galaxy surveys such as LSST 0912.0201 will produce massive redshift catalogues on the volume frontier, mapping out structures on ultra-large scales. Several exciting opportunities present themselves in the era of large-volume surveys, including the potential to measure subtle general relativistic effects in the observed clustering of galaxies Sasaki:1987ad ; Pyne:2003bn ; Hui:2005nm ; Bonvin:2005ps ; Barausse:2005nf ; Yoo:2008tj ; 2009PhRvD..80h3514Y ; Yoo:2010ni ; 1105.5280 ; 1105.5292 ; 1107.5427 and an opportunity to detect primordial non-Gaussianity through its scale-dependent effect on galaxy bias Dalal:2007cu . CMB experiments such as Simons Observatory Ade:2018sbj and CMB-S4 1610.02743 will produce measurements of the CMB on the sensitivity frontier, mapping out small-scale CMB temperature and polarization anisotropies near the nano-Kelvin level. At this sensitivity, it will be possible to accurately measure secondary CMB anisotropies such as the kinetic Sunyaev Zel’dovich (kSZ) effect SZ80 , temperature anisotropies induced by the scattering of CMB photons from free electrons in bulk motion after reionization.

At first sight, these developments might appear only vaguely related. However, because the large scale structure (LSS) is responsible for the secondary CMB anisotropies, there is a strong correlation between the small-angular scale CMB and the distribution of structure probed by e.g. galaxy surveys. In the case of the kSZ effect, the statistical anisotropies in such correlations encode information about the structure of the Universe on the largest scales. This information can be extracted using a technique known as kSZ tomography Ho09 ; Shao11b ; Zhang11b ; Zhang01 ; Munshi:2015anr ; Schaan15 ; Ferraro:2016ymw ; Hill:2016dta ; Zhang10d ; Zhang:2015uta ; Terrana2016 ; Deutsch:2017ybc ; Smith:2018bpn ; Munchmeyer:2018eey ; Sehgal:2019nmk , described in more detail below. In this paper, we demonstrate that a comparison of the large-scale distribution of galaxies to the large-scale inhomogeneities reconstructed using kSZ tomography can yield valuable new information about general relativistic contributions to the observed distribution of galaxies, with consequences for the measurement of primordial non-Gaussianity and various astrophysical bias parameters.

Considering the ways in which relativistic effects contribute to observables dates back to the birth of general relativity. In the present context, such effects arise from a precise treatment of photon geodesics in an inhomogeneous Universe. On large scales, cosmological perturbation theory is formulated within general relativity (GR), so there are no additional “dynamical” effects to consider. The earliest perturbative calculations that carefully considered all relativistic contributions to observables for scalar perturbations are perhaps for fluctuations in the luminosity distance-redshift relation Sasaki:1987ad ; Pyne:2003bn ; Hui:2005nm ; Bonvin:2005ps . Considering these effects will be important for future supernova surveys to constrain the properties of dark energy Hui:2005nm . Subsequently, the importance of relativistic effects has also been considered beyond leading order Barausse:2005nf ; 1205.5221 ; 1207.1286 ; 1207.2109 ; 1401.7973 ; 1402.1933 ; 1405.7860 ; 1406.1135 ; 1406.4140 ; 1812.04336 , and in an exact, numerical setting Giblin:2016mjp ; Giblin:2017ezj .

The galaxy number density as a function of redshift and angle on the sky–what we actually observe in a galaxy redshift survey–is also subject to relativistic corrections Yoo:2008tj ; 2009PhRvD..80h3514Y ; Yoo:2010ni ; 1105.5280 ; 1105.5292 ; 1107.5427 , similar in nature to effects commonly considered for other observables such as the CMB. In addition to the projected galaxy number density, redshift-space distortions (RSD; anisotropies in the mapping from real-space to redshift-space induced by peculiar velocities), magnification from lensing, and subdominant “general relativistic” (GR) effects also contribute to the observed number counts at linear order.111Rigorous treatments of relativistic effects have been extended to second order in Bertacca:2014dra ; Bertacca:2014wga . These GR effects include additional Doppler (magnification) terms and potential (Sachs-Wolfe, integrated Sachs-Wolfe, time delay) terms, analogous to the CMB. GR effects on number counts become important on scales approaching the cosmological horizon, and therefore are only accessible to surveys which have very large volume. Using a single tracer, it is unlikely that GR effects can be detected by any near-term survey 1505.07596 ; 1710.02477 . However, multi-tracer techniques can be used to pull them out Alonso:2015sfa . Should these effects be detectable, they could serve as a probe of modifications to GR on ultra-large scales Baker:2015bva ; Renk:2016olm ; Duniya:2019mpr .

A prime science target of future galaxy surveys, for example LSST 0912.0201 and SPHEREX Dore:2014cca , is primordial non-Gaussianity (PNG); see Alvarez:2014vva for an overview. Local-type PNG induces a scale-dependent galaxy bias Dalal:2007cu , proportional to the parameter $f_{\rm NL}$, leading to a measurable enhancement ($f_{\rm NL}>0$) or suppression ($f_{\rm NL}<0$) of galaxy clustering on the largest scales. An important milestone is constraining PNG at the level of $\sigma(f_{\rm NL})<1$. This is because a generic prediction of a large class of multi-field inflationary models is $f_{\rm NL}\sim\mathcal{O}(1)$, making a constraint at this level a potentially powerful discriminator between single-field and multi-field models of inflation Bartolo:2004if ; Alvarez:2014vva . Both PNG and GR effects modify the galaxy-galaxy power spectrum on large physical/angular scales, making it important to incorporate both into forecasts for the constraining power of future experiments. In particular, it is known that neglecting GR effects can lead to an $\mathcal{O}(1)$ bias on measurements of $f_{\rm NL}$ 1710.02477 . A proper understanding of GR effects is therefore essential for properly interpreting future surveys, and the implication of their results for inflationary cosmology.

Measurements of PNG Seljak:2008xr and GR effects Alonso:2015sfa can benefit greatly from incorporating cross-correlations of galaxy surveys with other tracers of the underlying dark matter distribution. In this case, a mode-by-mode comparison between the galaxy number density and dark matter density can be used to measure (scale-dependent) bias or to identify extra contributions to the observed galaxy number density from GR effects. Because such a comparison depends only on the properties of our observed realization, there is in principle no sample variance Seljak:2008xr . Taking advantage this “sample variance cancellation”, the ability to measure PNG or GR effects is limited only by the fidelity of the reconstruction of the dark matter density field (which depends on how correlated the tracer is with the distribution of dark matter) and shot noise on the galaxy survey.

Recently, kSZ tomography was introduced as a new and powerful tool for constraining PNG Munchmeyer:2018eey . The kSZ effect induces CMB temperature anisotropies due to the scattering of CMB photons from free electrons in the post-reionization Universe. The kSZ effect comprises the dominant blackbody contribution to the observed CMB temperature on small angular scales ($\ell\gtrsim 4000$). A number of detections of the kSZ effect have been made with existing datasets Hand12 ; Schaan15 ; PlanckkszI ; DeBernardis2016pdv ; Soergel2016mce ; Sugiyama2017uvr ; Li2017uin ; Hill2016dta ; PlanckkszII , and future experiments promise to obtain very high significance measurements Ade:2018sbj ; 1610.02743 . The amplitude of the kSZ temperature anisotropy from locations along our past light cone is proportional to the value of the remote dipole field, the projected temperature dipole at different locations in the observable Universe.

The three-dimensional remote dipole field can be reconstructed from statistical anisotropies in the cross-correlation between the CMB temperature on small angular scales and a galaxy survey, a technique called kSZ tomography Terrana2016 ; Deutsch:2017ybc ; Smith:2018bpn . The dominant contribution to the remote dipole field is the local peculiar velocity, which can be related to the dark matter density through linear theory. Correlating the reconstructed dipole field with the galaxy survey can therefore be used to isolate the scale-dependent galaxy bias due to PNG; the high fidelity of the reconstruction possible with future surveys allows one to take strong advantage of sample variance cancellation. Ref. Munchmeyer:2018eey forecasted that it will in principle be possible to constrain PNG at the level of $\sigma(f_{\rm NL})<1$ with the next generation of CMB instruments and galaxy surveys using kSZ tomography.

The present paper makes a number of important contributions to this previous analysis. First, we extend the analysis of Ref. Munchmeyer:2018eey , which utilized a simplified geometry, to the light cone. Our analysis includes all contributions to the remote dipole field beyond the local peculiar velocity (Doppler, Sachs-Wolfe, and integrated Sachs-Wolfe). We include RSD, lensing, and GR effects in the galaxy number density. We leave as free parameters in our model the redshift-dependent galaxy bias, evolution bias, magnification bias, and a multiplicative bias on the reconstructed remote dipole field that describes the optical depth degeneracy (see e.g. Hall2014 ; Battaglia:2016xbi ) in measurements of the kSZ effect (see Ref. Smith:2018bpn for an argument that this is sufficient). We also include an additional bias in the galaxy survey from intrinsic alignments due to large-scale tidal fields 0903.4929 . From here on we refer to such a bias as the alignment bias, defined explicitly in 0903.4929 and Appendix A. Finally, we include information from the primary CMB temperature and polarization in our constraints.

The goals of the present paper are to answer the questions:

• •

Is it possible to isolate GR effects in galaxy surveys using kSZ tomography?

• •

To what extent does neglecting GR effects bias the measurement of $f_{\rm NL}$ using kSZ tomography?

• •

Is sample variance cancellation between the remote dipole field and the galaxy survey useful for measuring various bias parameters?

• •

Does incorporating information from the primary CMB temperature and polarization help?

In summary, we find that kSZ tomography is a useful tool for isolating GR effects and improving the measurement of a variety of bias parameters when information from the primary CMB is incorporated. We further demonstrate that neglecting GR effects leads to an $\mathcal{O}(1)$ bias on measurements of $f_{\rm NL}$ from kSZ tomography.

The paper is structured as follows. In Sec. II, we outline the parameters of the Fisher forecast. In Sec. III, we discuss the results of our forecast, and in Sec. IV we conclude. A number of appendices are included to summarize the observables which go into our forecast.

II Forecasting method

In order to address the questions posed in the introduction, we forecast the constraining power of future CMB measurements and galaxy surveys. Performing this forecast will require two main ingredients: one or more observable quantities, and any noise associated with measuring these observables. The observables we consider are angular power spectra,

[TABLE]

where $X$ and $Y$ are one of: the perturbations in number counts of galaxies, the primary CMB temperature and polarization perturbations, or the remote dipole field reconstructed using kSZ tomography. Here, $\mathcal{P}(k)$ is the dimensionless primordial power spectrum,

[TABLE]

The noise associated with each of these observables will be shot noise in the case of galaxy number counts, instrument noise in the case of the primary CMB, and reconstruction noise in the case of the remote dipole field. The angular power spectra themselves are computed at linear order for the different tracers we consider. We explicitly state the form of the contributions to the transfer functions for number counts and the remote dipole field in Appendices A and B. The noise models we use are partly described below, and further details appear in these appendices, along with details on various bias functions that the transfer functions depend on.

As we are interested in exploring the importance of relativistic, non-Newtonian corrections, we parametrize the amplitude of these effects following 1505.07596 . We describe the amplitude of the relativistic corrections by defining the parameters $\epsilon^{\rm N}_{\rm GR}$ and $\epsilon^{\rm kSZ}_{\rm GR}$, defined at the level of the transfer functions as

[TABLE]

Here, the relativistic terms for number counts ($\Delta^{\rm N,GR}_{\ell}$) include all effects except standard intrinsic density fluctuations ($\Delta^{\rm D}_{\ell}$), RSD ($\Delta^{\rm RSD}_{\ell}$), and lensing terms ($\Delta^{\rm L}_{\ell}$), all of which are defined in Appendix A. While lensing itself is a general relativistic effect that must be taken into account, it has been considered separately in previous literature, and we follow this convention. The relativistic contributions to the remote dipole field ($\Delta^{\rm kSZ,GR}_{\ell}$) we take to include all contributions except the local (Newtonian) peculiar velocity Doppler term ($\Delta^{\rm LD}_{\ell}$). These contributions are also “primordial” in the sense that they represent the Sachs-Wolfe, Integrated Sachs-Wolfe, and primordial Doppler components.

We compute the galaxy number counts spectrum and reconstruct the remote dipole field in a set of redshift bins between $z\in[0.1,3]$.222kSZ tomography can in principle be performed at higher redshifts, however the number of observed galaxies beyond $z\sim 3$ in our fiducial survey is quite small, leading to a large reconstruction noise on the remote dipole field. We divide this range into 30 bins equally spaced in comoving distance, and integrate the transfer functions over the redshift range within each bin weighted by tophat window functions (eg. $W_{i}(z)$ in Eq. 19 is a tophat). The width of these bins roughly corresponds to photometric redshift uncertainties, with $\sigma_{z}\sim 3/30\sim 0.1$ in our binning scheme, although a more optimistic forecast could include additional bins as $\sigma_{z}$ is projected to be of order 0.05 for an LSST-like survey, and as low as $\sigma_{z}\sim 0.02$ for a “red” galaxy population 0912.0201 . However, generally do not find that changing the number of bins strongly affects our constraints, as we illustrate further below.

The final parameters we consider include standard cosmological ones, $f_{NL}$, bias parameters in each redshift bin, and the lightcone-projection/GR/primordial correction coefficients. In Table 1 we list each of the parameters we constrain, along with the fiducial values we use in our forecast.

We compute the Fisher matrix,

[TABLE]

for parameters $\alpha$ and $\beta$, and using a covariance matrix that includes all of the tracers,

[TABLE]

The $C_{\ell}^{XY}$ functions are the power spectra as defined in Eq. (1) and $N_{\ell}$ noise sources associated with these measurements, described in more detail below. These spectra are computed in each redshift bin for the remote dipole field and number counts; cross-spectra between all bins are accounted for, although this is not made explicit in Eq. (6). The CMB spectra are computed for both temperature and $E$-mode polarization. Derivatives are computed numerically with respect to the parameters $\{\Omega_{b},\Omega_{c},h,\tau\}$ using a second-order accurate upwind finite difference stencil. The remaining derivatives are computed analytically, explicitly commuting derivatives through any integrals over the redshift bins.

Due to the contributions from relativistic effects and non-Gaussianities manifesting on large scales, we consider auto- and cross-spectra of all tracers included in the forecast at low $\ell<60$. We also account for the high-$\ell$ CMB temperature and $E$-mode constraints on cosmological parameters separately, using lensed CMB power spectra generated using CAMB. The Fisher matrix from this is included as the $F^{\rm CMB,\,high-\ell}_{\alpha\beta}$ term in Eq. (5). In producing the high-$\ell$ CMB constraints, we have assumed a maximum available $\ell$ of 4000 in both $T$ and $E$. The high-$\ell$ constraint is also only used to constrain the standard cosmological parameters $\{A_{s},n_{s},\Omega_{b},\Omega_{c},h,\tau\}$, and so elements are considered zero when an index corresponds to the remaining parameters and bias functions.

In order to study constraints from specific tracers or combinations of tracers, we can selectively exclude tracers from the covariance matrix $C_{\ell}$ by removing the row and column associated with a particular tracer, eg. removing the last column and row in order to neglect the contribution from the remote dipole field. When excluding the CMB, we also exclude the high-$\ell$ constraint (the $F^{\rm CMB,\,high-\ell}_{\alpha\beta}$ term). In this way, we can examine how constraints from number counts alone improve as additional information is added from the primary CMB, and subsequently the remote dipole field.

The noise spectra $N_{\ell}$ are computed consistently for each observable we consider. For the primary CMB, we assume a CMB instrument noise of $1\mu$k-arcmin for both temperature and polarization measurements (although we vary this later), and a 1 arcminute beam in each. Galaxy shot noise is computed from the luminosity function and limiting magnitude of the survey, as described in Appendix A. This is equivalent to the model adopted in Ref. 1505.07596 . The reconstruction noise for the remote dipole field is then computed following Deutsch:2017ybc , using the galaxy number counts and CMB power spectra and noise described previously. We further assume that the electron distribution follows the dark matter distribution for our fiducial model. The uncertainty in this assumption is folded into the optical depth bias on the remote dipole, which we marginalize over in our analysis. For the reconstruction noise we assume a maximum available $\ell$ of $\ell_{\rm max}=9000$, which largely saturates the signal-to-noise of the relevant modes at the assumed CMB noise levels. We also assume that foregrounds and systematics can be mitigated and we do not consider these here. This may have an effect on the realistically accessible $\ell_{\rm max}$, and we comment on the implications for our constraints below.

We lastly seek to evaluate the bias in parameters, $\Delta\alpha$, due to neglecting various terms that arise from relativistic considerations. Following 1710.02477 , we have

[TABLE]

where $F^{-1}$ is the inverse fisher matrix, and

[TABLE]

where $\Delta\mathcal{C}_{\ell}=\mathcal{C}_{\ell}^{\rm true}-C_{\ell}^{\rm fiducial}$. The quantity $\Delta\alpha$ then describes the extent to which the measurement of a parameter $\alpha$ is biased when assuming a fiducial spectrum $C_{\ell}^{{\rm fiducial}}$ instead of using the true spectrum $C_{\ell}^{{\rm true}}$. The fiducial spectrum we use here is the one where the general relativistic contributions are neglected in both the number counts and kSZ measurements, ie. the transfer functions are evaluated with $\epsilon_{{\rm GR}}^{{\rm kSZ}}=0$ and $\epsilon_{{\rm GR}}^{{\rm N}}=0$.

III Results

We now examine the impact of the relativistic effects on future observations of galaxy number counts, CMB temperature and polarization, and the reconstruction of the remote dipole field. We wish to highlight the importance of alignment bias in the context of number counts alone. We explicitly focus on this as alignment bias is often not taken into account in forecasts, yet has the potential to interfere with measurements of the growth function and galaxy bias, thereby degrading constraints on cosmological parameters. We then incorporate both primary CMB measurements, and the remote dipole field into the forecast, and explore how constraints change as each new observable is included. We will demonstrate that the degradation on parameter constraints incurred by including the alignment bias is no longer an issue once CMB and kSZ tomography are included. In particular constraints on $f_{\rm evo}$ are dramatically improved when kSZ tomography is used (for $z\lesssim 2.5$).

We begin by examining constraints on the various bias functions in Fig. 1. These can be seen to drastically deteriorate as alignment bias is taken into account (compare the gray-dashed curves with the others), especially galaxy bias with which the alignment bias is degenerate. The bias functions subsequently recover as information from the CMB is included in the constraint. Notably, beyond the primary CMB and number counts alone, kSZ tomography can offer a substantial improvement in constraints on these biases, from a factor of a few, up to an order of magnitude in the case of evolution bias (for $z\lesssim 2.5$). No priors on any parameters are included in the constraints in this figure, and cosmological parameters including $f_{\rm NL}$ and the $\epsilon_{\rm GR}$ parameters are marginalized over.

The fiducial values of the galaxy and magnification bias as described in Appendix A are of order unity, and thus should be detected at high significance. The evolution bias is somewhat smaller, so detection prospects are perhaps marginal even when including information from kSZ tomography in the constraint. The evolution bias itself is degenerate with the amplitude of relativistic effects, and to a smaller extent $f_{\rm NL}$. An improved constraint on this bias can therefore help to reduce forecasted uncertainties, especially in $\epsilon_{\rm GR}^{\rm N}$. The amplitude of the alignment bias, on the other hand, is expected to be at most a few percent 0903.4929 ; while also marginal, we find that kSZ tomography may offer a way to detect alignment bias. The bias due to optical depth degeneracy, $b_{v}$, is only constrained when including information from the remote dipole field. The sub-percent constraint on $b_{v}$ is highly significant relative to the fiducial value of unity of this bias, and is comparable to constraints forecasted using other methods 1901.02418 .

We next examine how detectable relativistic contributions to observables are, looking at $\epsilon_{\rm GR}^{\rm N}$ and $\epsilon_{\rm GR}^{\rm kSZ}$, along with how an inferred bias for $f_{\rm NL}$ may be incurred when neglecting relativistic effects. The constraints and bias values are listed in Table 2. These have been marginalized over other cosmological parameters noted in Table (1), as well as all bias functions, including the bias due to the kSZ optical depth degeneracy where relevant. In both the case where we include and neglect alignment bias, relativistic effects are not detectable with number counts alone, and the bias due to neglecting these effects is unimportant. As constraints from the primary CMB are added, which primarily serve to pin down standard cosmological parameters, constraints on relativistic effects recover from alignment bias. As the reconstructed kSZ remote dipole field is included in the analysis, the constraints improve substantially, to the point where both a bias on $f_{\rm NL}$ and relativistic effects are detectable at moderate significance.

Following 1505.07596 , we also consider the impact of including a Gaussian prior on the evolution bias of $\Delta f_{\rm evo}=1$ in all redshift bins. The constraints on relativistic effects we find are then comparable to the multiple tracers considered in Alonso:2015sfa , however the prior we use is significantly less strict. We additionally consider what happens when we do not simultaneously constrain the $\epsilon_{\rm GR}$ parameters and $f_{\rm NL}$: this improves constraints to $\sigma(f_{\rm NL})\sim 0.9$ both with and without the evolution bias prior.

Using all tracers considered–number counts including alignment bias and the $f_{\rm evo}$ prior, the primary CMB, and the remote dipole field–we lastly explore the constraints as a function of instrumental parameters. We show these in Figure 2, as the CMB experiment noise, number of redshift bins (ie. photometric redshift error), and limiting magnitude are varied about the fiducial values considered above. The constraints presented in these figures include the evolution bias prior.

Generally, the constraints improve as one might expect. Increasing CMB instrument noise results in higher reconstruction noise on the remote dipole field, and constraints on the parameters we consider all become mildly worse. Decreasing the number of redshift bins (e.g. increasing the photometric redshift error) similarly reduces the information available, resulting again in moderately worse constraints. Varying the limiting magnitude of the galaxy survey provides a more complicated picture: constraints on $f_{\rm NL}$ and $\epsilon_{\rm GR}^{\rm kSZ}$ improve, while constraints on $\epsilon_{\rm GR}^{\rm N}$ actually worsen. This is because the fiducial bias model changes in such a way that the signal of relativistic effects (as well as lensing, which we do not consider) all decrease relative to intrinsic density perturbations and RSD effects. The “noise” due to cosmic variance from these dominant effects is therefore increased, resulting in a worse overall constraint on relativistic effects.

IV Conclusions and discussion

In this paper, we have illustrated the potential importance of the remote dipole field reconstructed using kSZ tomography for the detectability of relativistic effects in galaxy number counts, and the need to account for relativistic effects to obtain an unbiased measurement of primordial non-Gaussianity using kSZ tomography. We also highlighted the improvement on the measurement of various (redshift-dependent) bias parameters from the correlation between the remote dipole field and number counts. The forecasted constraints on most bias parameters can improve by a factor of a few, and constraints on the evolution bias can improve by more than an order of magnitude. This is significant, as evolution bias is strongly degenerate with both relativistic effects and scale-dependent bias from primordial non-Gaussianity. Even with this improvement, evolution bias is poorly constrained; we demonstrate that a weak prior can substantially improve the detectability of relativistic effects. Importantly, our analysis shows that kSZ tomography can significantly mitigate the effects of alignment bias, which can seriously degrade constraints on primordial non-Gaussianity using number counts alone.

Our forecasts have included a number of optimistic assumptions. In particular, we have presented the constraints in the limit where we have data on the full sky and foregrounds and systematics are negligible. Partial-sky data will weaken measurements on the largest scales, which could significantly degrade constraints on relativistic effects and primordial non-Gaussianity. Foregrounds and systematics on small angular scales have the potential to degrade the reconstruction of the remote dipole field. To estimate the effect this might have, we repeated our forecast using an $\ell_{\rm max}$ of 5000 in the reconstruction noise. This yields only a roughly 25 % increase in our forecasted uncertainties.

Several additional assumptions made in this work are perhaps pessimistic. For example, the galaxy number counts we compute contain very few observable galaxies beyond redshift $z\sim 2$, while calculations of the galaxy number counts using different assumptions can yield a higher number of galaxies at comparable redshifts (see e.g. Smith:2018bpn ; Schmittfull:2017ffw for similar studies). The magnitude limits we assume are also conservative, and information from fainter galaxies up to a magnitude of $r\sim 27.5$ may be accessible, albeit with larger uncertainties in photometric redshifts. We have also not split our analysis into separate galaxy populations (red, blue) nor included additional tracers, such as intensity mapping, that have been shown to improve constraints 1505.07596 ; 1710.02477 ; Alonso:2015sfa .

While changing our assumptions about the fiducial CMB and galaxy surveys has the potential to nudge our forecasted uncertainties up or down, this work demonstrates that kSZ tomography promises to be an important and useful tool for isolating both relativistic effects and new physics. We have further highlighted a significant bias on measurements of primordial non-Gaussianity incurred when relativistic effects are neglected, and illustrated a significant improvement on various bias functions that can be obtained using kSZ tomography. This paper strengthens the science case for performing kSZ tomography using future observations.

V Acknowledgments

We would like to thank Juan Cayuso, Neal Dalal, and Moritz Munchmeyer for helpful discussions. This research was enabled in part by support provided by the Shared Hierarchical Academic Research Computing Network (SHARCNET:www.sharcnet.ca), Compute Canada (www.computecanada.ca), and the Kenyon College Department of Physics. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. MCJ and DC were supported by the National Science and Engineering Research Council through a Discovery grant. JBM acknowledges support as a CITA National Fellow.

Appendix A Number counts transfer functions

Corrections to relativistic perturbations arise when considering how to map from the observed $\left(N(z)-\bar{N}(z)\right)/\bar{N}(z)$ (or the unperturbed $N(z)$) to the theoretical $\left(N(\bar{z})-\bar{N}(\bar{z})\right)/\bar{N}(\bar{z})$. Here we describe the standard linear theory equations used to compute the perturbations in number counts, and the adjustments we make to these equations in order to account for non-Gaussianities and alignment bias.

We define all number counts transfer functions used below. These transfer functions are the same as can be found in literature 1105.5292 ; 1105.5280 ; Alonso:2015sfa ; 1505.07596 , with two amendments: two additional bias functions are included, one for alignment bias, $b_{\rm A}$ 0903.4929 , and one for primordial non-Gaussianities, $b_{\rm NG}$ Dalal:2007cu .

[TABLE]

where $\chi$ is comoving distance and the redshift-space window function

[TABLE]

is a tophat function in redshift, i.e., nonzero and constant in the i-th redshift bin, and is normalized so that its integral over redshift is unity. The transfer functions for determining various fields from primordial perturbations are noted by $S_{\delta_{M},{\rm syn}}$ for the synchronous gauge matter density, $S_{\phi}$ and $S_{\psi}$ for the Newtonian potentials, and $S_{v}$ for the scalar velocity field as defined in Terrana2016 , which is related to the divergence of the velocity field $\theta$ as $S_{\theta}=k^{2}S_{v}$. The bias due to non-Gaussianities (NG) is described by

[TABLE]

with $\delta_{c}=1.686$ the linearized collapse threshold.

The remaining bias functions introduced in Eqs. (9)–(18) are defined with respect to the background source population. Specifically we can take Eq. (9) as the definition of the galaxy bias ($b_{G}$), and use the parameterization (for a full galaxy sample) based on the simulations of Weinberg2002 and quoted in the LSST science book 0912.0201 , of $b_{G}=0.95/D(z)\simeq 0.95+0.67z$. The magnification bias ($s$) and evolution bias ($f_{\rm evo}$) are defined as

[TABLE]

Here, the quantity $n_{s}$ is the luminosity function: the galaxy number counts density per luminosity on a spatial hypersurface, ie. not projected onto a lightcone,

[TABLE]

and $\mathcal{N}$ is the integrated number density above a threshold luminosity, derived from the luminosity function,

[TABLE]

We model the evolution and magnification biases following the approach outlined in Appendix B.4 of 1505.07596 . Explicitly, we assume a Schechter luminosity function of the form

[TABLE]

To model the full sample of galaxies we use the $r^{\prime}$-band luminosity function found by Gabasch2005 to approximate the $r$ band luminosity function,

[TABLE]

The parameters of this model have the explicit values: $\alpha=-1.33$, $g=-1.25$, $M_{0}=-21.49$, $\phi_{0}=2.59$, $\phi_{1}=-0.136$, and $\phi_{2}=-0.081$.

We then relate the absolute magnitude in Eq. (25) to an apparent magnitude, $m$, via

[TABLE]

where the $d_{L}$ is the luminosity distance and $k$ is a $k$-correction due to the corresponding galaxy’s spectral energy distribution redshifted to $z$. We use the extrapolated values of $k(z)$ found in 1505.07596 (with $k(z)\propto z$). The quantities, $s$ and $f_{\rm evo}$ (and their redshift dependence) will thus depend on the magnitude limit of the survey in question (we need to integrate Eq. (24) to $M_{\rm cut}$). Here we use the limit $m_{\rm cut}=r=25.3$ as a representative choice for an LSST type survey 0912.0201 , although we also explore $m_{\rm cut}=r=\,26.3,\,27.3$.

Lastly, and explicitly, we consider the general relativistic terms to include

[TABLE]

The remaining lensing, RSD, and intrinsic perturbation terms we do not consider as part of the relativistic effects. Together, both the relativistic and non-relativistic contributions comprise the first-order gauge-independent observable angular power spectrum.

Appendix B Remote dipole (kSZ) transfer functions

The contributions to the remote dipole field transfer function are given as

[TABLE]

where $b_{v}$ is an overall bias in the amplitude of the reconstructed remote dipole field that arises due to uncertainty in the electron density field, the “optical depth bias”, and the kernels $\mathcal{K}^{A}$ are given by one of

[TABLE]

for the various terms $A$ (local Doppler LD, remote Doppler RD, Sachs-Wolfe SW, and integrated Sachs-Wolfe ISW) that contribute to the remote dipole field, and for $\Delta\chi\equiv\chi-\chi_{\rm CMB}$. The fiducial optical depth bias value is chosen to be unity, $b_{v}=1$, and is marginalized over independently in each redshift bin considered. Explicitly, we consider the general relativistic (or primary CMB) terms to include

[TABLE]

The non-relativistic (non-primordial) remaining term is the local Doppler contribution, attributable to only the Newtonian-gauge peculiar velocity of the remote electron. As noted in Terrana2016 , the Newtonian velocity contribution alone is not a (linear) gauge-invariant quantity, and the observable should include the additional relativistic contributions.