Consistency relations for largescale structures: Applications for the integrated SachsWolfe effect and the kinematic SunyaevZeldovich effect
^{1} Institut de Physique Théorique, CEA, IPhT, 91191 GifsurYvette Cedex, France
email: patrick.valageas@cea.fr
^{2} Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway
Received: 15 January 2017
Accepted: 29 July 2017
Consistency relations of largescale structures provide exact nonperturbative results for crosscorrelations of cosmic fields in the squeezed limit. They only depend on the equivalence principle and the assumption of Gaussian initial conditions, and remain nonzero at equal times for crosscorrelations of density fields with velocity or momentum fields, or with the time derivative of density fields. We show how to apply these relations to observational probes that involve the integrated SachsWolfe effect or the kinematic SunyaevZeldovich effect. In the squeezed limit, this allows us to express the threepoint crosscorrelations, or bispectra, of two galaxy or matter density fields, or weak lensing convergence fields, with the secondary cosmic microwave background distortion in terms of products of a linear and a nonlinear power spectrum. In particular, we find that crosscorrelations with the integrated SachsWolfe effect show a specific angular dependence. These results could be used to test the equivalence principle and the primordial Gaussianity, or to check the modeling of largescale structures.
Key words: largescale structure of Universe
© ESO, 2017
1. Introduction
Measuring statistical properties of cosmological structures is not only an efficient tool to describe and understand the main components of our Universe, but also it is a powerful probe of possible new physics beyond the standard Λcold dark matter (ΛCDM) concordance model. However, on large scales, cosmological structures are described by perturbative methods, while smaller scales are described by phenomenological models or studied with numerical simulations. It is therefore difficult to obtain accurate predictions on the full range of scales probed by galaxy and lensing surveys. Furthermore, if we consider galaxy density fields, theoretical predictions remain sensitive to the galaxy bias, which involves phenomenological modeling of star formation, even if we use cosmological numerical simulations. As a consequence, exact analytical results that go beyond loworder perturbation theory and also apply to biased tracers are very rare.
Recently, some exact results have been obtained (Kehagias & Riotto 2013; Peloso & Pietroni 2013; Creminelli et al. 2013; Kehagias et al. 2014a; Peloso & Pietroni 2014; Creminelli et al. 2014; Valageas 2014b; Horn et al. 2014, 2015) in the form of “kinematic consistency relations”. They relate the (ℓ + n)density correlation, with ℓ largescale wave numbers and n smallscale wave numbers, to the npoint smallscale density correlation. These relations, obtained at the leading order over the largescale wave numbers, arise from the equivalence principle (EP) and the assumption of Gaussian initial conditions. The equivalence principle ensures that smallscale structures respond to a largescale perturbation by a uniform displacement, while primordial Gaussianity provides a simple relation between correlation and response functions (see Valageas et al. 2017, for the additional terms associated with nonGaussian initial conditions). Therefore, such relations express a kinematic effect that vanishes for equaltimes statistics, as a uniform displacement has no impact on the statistical properties of the density field observed at a given time.
In practice, it is, however, difficult to measure differenttimes density correlations and it would therefore be useful to obtain relations that remain nonzero at equal times. One possibility to overcome such a problem is to go to higher orders and take into account tidal effects, which at leading order are given by the response of smallscale structures to a change in the background density. Such an approach, however, introduces some additional approximations (Valageas 2014a; Kehagias et al. 2014b; Nishimichi & Valageas 2014).
Fortunately, it was recently noticed that by crosscorrelating density fields with velocity or momentum fields, or with the time derivative of the density field, one obtains consistency relations that do not vanish at equal times (Rizzo et al. 2016). Indeed, the kinematic effect modifies the amplitude of the largescale velocity and momentum fields, while the time derivative of the density field is obviously sensitive to differenttimes effects.
In this paper, we investigate the observational applicability of these new relations. We consider the lowestorder relations, which relate threepoint crosscorrelations or bispectra in the squeezed limit to products of a linear and a nonlinear power spectrum. To involve the nonvanishing consistency relations, we study two observable quantities, the secondary anisotropy Δ_{ISW} of the cosmic microwave background (CMB) radiation due to the integrated SachsWolfe effect (ISW), and the secondary anisotropy Δ_{kSZ} due to the kinematic SunyaevZeldovich (kSZ) effect. The first process, associated with the motion of CMB photons through timedependent gravitational potentials, depends on the time derivative of the matter density field. The second process, associated with the scattering of CMB photons by free electrons, depends on the free electrons velocity field. We investigate the cross correlations of these two secondary anisotropies with both galaxy density fields and the cosmic weak lensing convergence.
This paper is organized as follows. In Sect. 2 we recall the consistency relations of largescale structures that apply to density, momentum, and momentumdivergence (i.e., time derivative of the density) fields. We describe the various observational probes that we consider in this paper in Sect. 3. We study the ISW effect in Sect. 4 and the kSZ effect in Sect. 5. We conclude in Sect. 6.
2. Consistency relations for largescale structures
2.1. Consistency relations for density correlations
As described in recent works (Kehagias & Riotto 2013; Peloso & Pietroni 2013; Creminelli et al. 2013; Kehagias et al. 2014a; Peloso & Pietroni 2014; Creminelli et al. 2014; Valageas 2014b; Horn et al. 2014, 2015), it is possible to obtain exact relations between density correlations of different orders in the squeezed limit, where some of the wavenumbers are in the linear regime and far below the other modes that may be strongly nonlinear. These “kinematic consistency relations”, obtained at the leading order over the largescale wavenumbers, arise from the equivalence principle and the assumption of Gaussian primordial perturbations. They express the fact that at leading order where a largescale perturbation corresponds to a linear gravitational potential (hence a constant Newtonian force) over the extent of a smallsize structure, the latter falls without distortions in this largescale potential.
Then, in the squeezed limit k → 0, the correlation between one largescale density mode and n smallscale density modes can be expressed in terms of the npoint smallscale correlation, as (1)where the tilde denotes the Fourier transform of the fields, η is the conformal time, D(η) is the linear growth factor, the prime in ⟨ ... ⟩ ′ denotes that we factored out the Dirac factor, ⟨ ... ⟩ = ⟨ ... ⟩ ′δ_{D}( ∑ k_{j}), and P_{L}(k) is the linear matter power spectrum. It is worth stressing that these relations are valid even in the nonlinear regime and for biased galaxy fields . The righthand side gives the squeezed limit of the (1 + n) correlation at the leading order, which scales as 1 /k. It vanishes at this order at equal times, because of the constraint associated with the Dirac factor δ_{D}( ∑ k_{j}).
The geometrical factors (k_{i}·k) vanish if k_{i} ⊥ k. Indeed, the largescale mode induces a uniform displacement along the direction of k. This has no effect on smallscale plane waves of wavenumbers k_{i} with k_{i} ⊥ k, as they remain identical after such a displacement. Therefore, the terms in the righthand side of Eq. (1) must vanish in such orthogonal configurations, as we can check from the explicit expression.
The simplest relation that one can obtain from Eq. (1)is for the bispectrum with n = 2,
(2)where we used that k_{2} = − k_{1} − k → − k_{1}. For generality, we considered here the smallscale fields and to be associated with biased tracers such as galaxies. The tracers associated with k_{1} and k_{2} can be different and have different bias. At equal times the righthand side of Eq. (2) vanishes, as recalled above.
2.2. Consistency relations for momentum correlations
The density consistency relations (1) express the uniform motion of smallscale structures by largescale modes. This simple kinematic effect vanishes for equaltime correlations of the density field, precisely because there are no distortions, while there is a nonzero effect at different times because of the motion of the smallscale structure between different times. However, as pointed out in Rizzo et al. (2016), it is possible to obtain nontrivial equaltimes results by considering velocity or momentum fields, which are not only displaced but also see their amplitude affected by the largescale mode. Let us consider the momentum p defined by (3)where v is the peculiar velocity. Then, in the squeezed limit k → 0, the correlation between one largescale density mode , n smallscale density modes , and m smallscale momentum modes can be expressed in terms of (n + m) smallscale correlations, as (4)These relations are again valid in the nonlinear regime and for biased galaxy fields and . As for the density consistency relation (1), the first term vanishes at this order at equal times. The second term, however, which arises from the fields only, remains nonzero. This is due to the fact that involves the velocity, the amplitude of which is affected by the motion induced by the largescale mode.
The simplest relation associated with Eq. (4)is the bispectrum among two densitycontrast fields and one momentum field, (5)For generality, we considered here the smallscale fields and to be associated with biased tracers such as galaxies, and the tracers associated with k_{1} and k_{2} can again be different and have different bias. At equal times, Eq. (5)reads as (6)where P_{g}(k) is the galaxy nonlinear power spectrum and we omitted the common time dependence. This result does not vanish thanks to the term generated by in the consistency relation (5).
2.3. Consistency relations for momentumdivergence correlations
In addition to the momentum field p, we can consider its divergence λ, defined by (7)The second equality expresses the continuity equation, that is, the conservation of matter. In the squeezed limit we obtain from Eq. (4) (Rizzo et al. 2016) (8)These relations can actually be obtained by taking derivatives with respect to the times η_{j} of the density consistency relations (1), using the second equality (7). As for the momentum consistency relations (4), these relations remain valid in the nonlinear regime and for biased smallscale fields and . The second term in Eq. (8), which arises from the fields only, remains nonzero at equal times. This is due to the fact that λ involves the velocity or the timederivative of the density, which probes the evolution between (infinitesimally close) different times.
The simplest relation associated with Eq. (8)is the bispectrum among two densitycontrast fields and one momentumdivergence field, (9)At equal times, Eq. (9)reads as (10)
3. Observable quantities
To test cosmological scenarios with the consistency relations of largescale structures we need to relate them to observable quantities. We describe in this section the observational probes that we consider in this paper. We use the galaxy numbers counts or the weak lensing convergence to probe the density field. To apply the momentum consistency relations (6) and (10), we use the ISW effect to probe the momentum divergence λ (more precisely the time derivative of the gravitational potential and matter density) and the kSZ effect to probe the momentum p.
3.1. Galaxy number density contrast δ_{g}
From galaxy surveys we can typically measure the galaxy density contrast within a redshift bin, smoothed with a finitesize window on the sky, (11)where W_{Θ}(  θ′ − θ ) is a 2D symmetric window function centered on the direction θ on the sky, of characteristic angular radius Θ, I_{g}(η) is the radial weight along the line of sight associated with a normalized galaxy selection function n_{g}(z), (12)r = η_{0} − η is the radial comoving coordinate along the line of sight, and η_{0} is the conformal time today. Here and in the following we use the flat sky approximation, and θ is the 2D vector that describes the direction on the sky of a given line of sight. The superscript “s” in denotes that we smooth the galaxy density contrast with the finitesize window W_{Θ}. Expanding in Fourier space, we can write the galaxy density contrast as (13)where k_{∥} and k_{⊥} are respectively the parallel and the perpendicular components of the 3D wavenumber k = (k_{∥},k_{⊥}) (with respect to the reference direction θ = 0, and we work in the smallangle limit θ ≪ 1). Defining the 2D Fourier transform of the window W_{Θ} as (14)we obtain (15)
3.2. Weak lensing convergence κ
From weak lensing surveys we can measure the weak lensing convergence, given in the Born approximation by (16)where Ψ and Φ are the Newtonian gauge gravitational potentials and the kernel g(r) that defines the radial depth of the survey is (17)where n_{g}(z_{s}) is the redshift distribution of the source galaxies. Assuming no anisotropic stress, that is, Φ = Ψ, and using the Poisson equation, (18)where is the Newton constant, is the mean matter density of the Universe today, and a is the scale factor, we obtain (19)with (20)
3.3. ISW secondary anisotropy Δ_{ISW}
From Eq. (7) λ can be obtained from the momentum divergence or from the time derivative of the density contrast. These quantities are not as directly measured from galaxy surveys as density contrasts. However, we can relate the time derivative of the density contrast to the ISW effect, which involves the time derivative of the gravitational potential. Indeed, the secondary CMB temperature anisotropy due to the integrated SachsWolfe effect along the direction θ reads as (Garriga et al. 2004) (21)where τ(η) is the optical depth, which takes into account the possibility of late reionization, and in the second line we assumed no anisotropic stress, that is, Φ = Ψ. We can relate Δ_{ISW} to λ through the Poisson equation (18), which reads in Fourier space as (22)This gives (23)where ℋ = dlna/ dη is the conformal expansion rate. Integrating the ISW effect δ_{ISW} over some finitesize window on the sky, we obtain, as in Eq. (15), (24)with (25)
3.4. Kinematic SZ secondary anisotropy Δ_{kSZ}
Thomson scattering of CMB photons off moving free electrons in the hot galactic or cluster gas generates secondary anisotropies (Sunyaev & Zeldovich 1980; Gruzinov & Hu 1998; Knox et al. 1998). The temperature perturbation, Δ_{kSZ} = δT/T, due to this kinematic SunyaevZeldovich (kSZ) effect, is (26)where τ is again the optical depth, σ_{T} the Thomson cross section, l the radial coordinate along the line of sight, n_{e} the number density of free electrons, v_{e} their peculiar velocity, and n(θ) the radial unit vector pointing to the line of sight. We also defined the kSZ kernel by (27)and the free electrons momentum p_{e} as (28)Because of the projection n·p_{e} along the line of sight, some care must be taken when we smooth Δ_{kSZ}(θ) over some finitesize angular window W_{Θ}(  θ′ − θ ). Indeed, because the different lines of sight θ′ in the conical window are not perfectly parallel, if we define the longitudinal and transverse momentum components by the projection with respect to the mean line of sight n(θ) of the circular window, for example, p_{e ∥} = n(θ)·p_{e}, the projection n(θ′)·p_{e} receives contributions from both p_{e ∥} and p_{e ⊥}. In the limit of small angles we could a priori neglect the contribution associated with p_{e ⊥}, which is multiplied by an angular factor and vanishes for a zerosize window. However, for small but finite angles, we need to keep this contribution because fluctuations along the lines of sight are damped by the radial integrations and vanish in the Limber approximation, which damps the contribution associated with p_{e ∥}.
For small angles we write at linear order n(θ) = (θ_{x},θ_{y},1), close to a reference direction θ = 0. Then, the integration over the angular window gives for the smoothed kSZ effect (29)Here we expressed the result in terms of the longitudinal and transverse components of the wave numbers and momenta with respect to the mean line of sight n(θ) of the circular window W_{Θ}. Thus, whereas the radial unit vector is n(θ) = (θ_{x},θ_{y},1), we can define the transverse unit vectors as n_{⊥ x} = (1,0, − θ_{x}) and n_{⊥ y} = (0,1, − θ_{y}), and we write for instance k = k_{⊥ x}n_{⊥ x} + k_{⊥ y}n_{⊥ y} + k_{∥}n. We denote . The last term in Eq. (29) is due to the finite size Θ of the smoothing window, which makes the lines of sight within the conical beam not strictly parallel. It vanishes for an infinitesimal window, where W_{Θ}(θ) = δ_{D}(θ) and , . We find in Sect. 5.1 that this contribution is typically negligible in the regime where the consistency relations apply, as the width of the smallscale windows is much smaller than the angular size associated with the long mode.
3.5. Comparison with some other probes
As we explained above, in order to take advantage of the consistency relations we use the ISW or kSZ effects because they involve the timederivative of the density field or the gas velocity. The reader may then note that redshiftspace distortions (RSD) also involve velocities, but previous works that studied the galaxy density field in redshift space (Creminelli et al. 2014; Kehagias et al. 2014a) found that there is no equaltime effect, as in the realspace case. Indeed, in both real space and redshift space, the long mode only generates a uniform change of coordinate (in the redshiftspace case, this shift involves the radial velocity). Then, there is no effect at equal times because such uniform shifts do not produce distortions and observable signatures. In contrast, in our case there is a nonzero equaltime effect because the effect of the long mode cannot be absorbed by a simple change of coordinates. Indeed, the kSZ effect, associated with the scattering of CMB photons by free electrons in hot ionized gas (e.g., in Xray clusters), actually probes the velocity difference between the restframe of the CMB and the hot gas. Thus, the CMB lastscattering surface provides a reference frame and the long mode generates a velocity difference with respect to that frame that cannot be described as a change of coordinate. This explains why the kSZ effect makes the longmode velocity shift observable, without conflicting with the equivalence principle. There is also a nonzero effect for the ISW case, because the latter involves the time derivative of the density field, so that an equaltime statistics actually probes differenttimes properties of the density field (e.g., if we write the time derivative as an infinitesimal finite difference).
If we crosscorrelate realspace and redshiftspace quantities, there will also remain a nonzero effect at equal times, because the long mode generates different shifts for the realspace and redshiftspace fields. Thus, we can consider the effect of a long mode on smallscale correlations of the weak lensing convergence κ with redshiftspace galaxy density contrasts . However, weak lensing observables have broad kernels along the line of sight, so that a small differential shift along the radial direction is suppressed. In contrast, in the kSZ case the effect is directly due to the change of velocity by the long mode, and not by the indirect impact of the change of the radial redshift coordinate.
Another observable effect of the long mode was pointed out in Baldauf et al. (2015). These authors noticed that a long mode of wave length 2π/k of the same order as the baryon acoustic oscillation (BAO) scale, x_{BAO} ~ 110h^{1} Mpc, gives a different shift to galaxies separated by this distance. This produces a spread of the BAO peak, after we average over the long mode. The reason why this effect is observable is that the correlation function shows a narrow peak at the BAO scale, with a width of order Δx_{BAO} ~ 20h^{1} Mpc. This narrow feature provides a probe of the small displacement of galaxies by the long mode, which would otherwise be negligible if the galaxy correlation were a slow power law. As noticed above, the absence of such a narrow feature suppresses the signal associated with crosscorrelations among weaklensing (realspace) quantities and redshiftspace quantities, because of the radial broadening of the weaklensing probes.
This BAO probe is actually a secondorder effect, in the sense of the consistency relations. Indeed, the usual consistency relations are obtained in the largescale limit k → 0, where the long mode generates a uniform displacement of the smallscale structures. In contrast, the spread of the BAO peak relies on the differential displacement between galaxies separated by x_{BAO}. In the Taylor expansion of the displacement with respect to the positions of the smallscale structures, beyond the lowestorder constant term one takes into account the linear term over x, which scales as kx. This is why this effect requires that k be finite and not too small, of order k ~ 2π/x_{BAO}.
4. Consistency relation for the ISW temperature anisotropy
In this section we consider cross correlations with the ISW effect. This allows us to apply the consistency relation (9), which involves the momentum divergence λ and remains nonzero at equal times.
4.1. GalaxygalaxyISW correlation
To take advantage of the consistency relation (9), we must consider threepoint correlations ξ_{3} (in configuration space) with one observable that involves the momentum divergence λ. Here, using the expression (24), we study the crosscorrelation between two galaxy density contrasts and one ISW temperature anisotropy, (30)The subscripts g, g_{1}, and ISW_{2} denote the three lines of sight associated with the three probes. Moreover, the subscripts g and g_{1} recall that the two galaxy populations associated with and can be different and have different bias. As we recalled in Sect. 2, the consistency relations rely on the undistorted motion of smallscale structures by largescale modes. This corresponds to the squeezed limit k → 0 in the Fourierspace Eqs. (1) and (8), which writes more precisely as (31)where k_{L} is the wavenumber associated with the transition between the linear and nonlinear regimes. The first condition ensures that is in the linear regime, while the second condition ensures the hierarchy between the largescale mode and the smallscale modes. In configuration space, these conditions correspond to (32)The first condition ensures that is in the linear regime, whereas the next two conditions ensure the hierarchy of scales.
The expressions (15) and (24) give (33)The configurationspace conditions (32) ensure that we satisfy the Fourierspace conditions (31) and that we can apply the consistency relations (2) and (9). This gives (34)Here we assumed that on large scales the galaxy bias is linear, (35)where is a stochastic component that represents shot noise and the effect of smallscale (e.g., baryonic) physics on galaxy formation. From the decomposition (35), it is uncorrelated with the largescale density field (Hamaus et al. 2010), . Then, in Eq. (34) we neglected the term . Indeed, the smallscale local processes within the region θ should be very weakly correlated with the density fields in the distant regions θ_{1} and θ_{2}, which at leading order are only sensitive to the total mass within the largescale region θ. Therefore, should exhibit a fast decay at low k, whereas the term in Eq. (34) associated with the consistency relation only decays as P_{L}(k) /k ~ k^{ns − 1} with n_{s} ≃ 0.96. In Eq. (34), we also assumed that the galaxy bias b_{g} goes to a constant at large scales, which is usually the case, but we could take into account a scale dependence [by keeping the factor b_{g}(k,η) in the integral over k].
The smallscale twopoint correlations ⟨ 1·2 ⟩ ′ are dominated by contributions at almost equal times, η_{1} ≃ η_{2}, as different redshifts would correspond to points that are separated by several Hubble radii along the lines of sight and density correlations are negligible beyond Hubble scales. Therefore, ξ_{3} is dominated by the second term that does not vanish at equal times. The integrals along the lines of sight suppress the contributions from longitudinal wavelengths below the Hubble radius c/H, while the angular windows only suppress the wavelengths below the transverse radii cΘ /H. Then, for small angular windows, Θ ≪ 1, we can use Limber’s approximation, k_{∥} ≪ k_{⊥} hence k ≃ k_{⊥}. Integrating over k_{∥} through the Dirac factor δ_{D}(k_{∥} + k_{1 ∥} + k_{2 ∥}), and next over k_{1 ∥} and k_{2 ∥}, we obtain the Dirac factors (2π)^{2}δ_{D}(r_{1} − r)δ_{D}(r_{2} − r). This allows us to integrate over η_{1} and η_{2} and we obtain (36)where P_{g1,m} is the galaxymatter power spectrum. The integration over k_{2 ⊥} gives (37)and the integration over the angles of k_{⊥} and k_{1 ⊥} gives (38)where J_{1} is the firstorder Bessel function of the first kind.
As the expression (38) arises from the kinematic consistency relations, it expresses the response of the smallscale twopoint correlation to a change of the initial condition associated with the largescale mode . The kinematic effect given at the leading order by Eq. (38) is due to the uniform motion of the smallscale structures by the largescale mode. This explains why the result (38) vanishes in the two following cases:

1.
(θ − θ_{2}) ⊥ (θ_{1} − θ_{2}). There is a nonzero response of ⟨ δ_{1}λ_{2} ⟩ if there is a linear dependence on δ(θ) of ⟨ δ_{1}λ_{2} ⟩, so that its first derivative is nonzero. A positive (negative) δ(θ) leads to a uniform motion at θ_{2} towards (away from) θ, along the direction (θ − θ_{2}). From the point of view of θ_{1} and θ_{2}, there is a reflection symmetry with respect to the axis (θ_{1} − θ_{2}). For instance, if δ_{1}> 0 the density contrast at a position θ_{3} typically decreases in the mean with the radius  θ_{3} − θ_{1} , and for Δθ_{2} ⊥ (θ_{1} − θ_{2}) the points are at the same distance from θ_{1} and have the same density contrast δ_{3} in the mean, with typically δ_{3}<δ_{2} as . Therefore, the largescale flow along (θ − θ_{2}) leads to a positive λ_{2} = − Δδ_{2}/ Δη_{2} independently of whether the matter moves towards or away from θ (here we took a finite deviation Δθ_{2}). This means that the dependence of ⟨ δ_{1}λ_{2} ⟩ on δ(θ) is quadratic (it does not depend on the sign of δ(θ)) and the firstorder response function vanishes. Then, the leadingorder contribution to ξ_{3} vanishes. (For infinitesimal deviation Δθ_{2} we have λ_{2} = − ∂δ_{2}/∂η_{2} = 0; by this symmetry, in the mean δ_{2} is an extremum of the density contrast along the orthogonal direction to (θ_{1} − θ_{2}).)

2.
θ_{1} = θ_{2}. This is a particular case of the previous configuration. Again, by symmetry from the viewpoint of δ_{1}, the two points δ(θ_{2} + Δθ_{2}) and δ(θ_{2} − Δθ_{2}) are equivalent and the mean response associated with the kinematic effect vanishes.
This also explains why Eq. (38) changes sign with (θ_{1} − θ_{2}) and (θ − θ_{2}). Let us consider for simplicity the case where the three points are aligned and δ(θ) > 0, so that the largescale flow points towards θ. We also take δ_{1}> 0, so that in the mean the density is peaked at θ_{1} and decreases outwards. Let us take θ_{2} close to θ_{1}, on the decreasing radial slope, and on the other side of θ_{1} than θ. Then, the largescale flow moves matter at θ_{2} towards θ_{1}, so that the density at θ_{2} at a slightly later time comes from more outward regions (with respect to the peak at θ_{1}) with a lower density. This means that λ_{2} = − ∂δ_{2}/∂η_{2} is positive so that ξ_{3}> 0. This agrees with Eq. (38), as (θ − θ_{2})·(θ_{1} − θ_{2}) > 0 in this geometry, and we assume the integrals over wavenumbers are dominated by the peaks of J_{1}> 0. If we flip θ_{2} to the other side of θ_{1}, we find on the contrary that the largescale flow brings higherdensity regions to θ_{2}, so that we have the change of signs λ_{2}< 0 and ξ_{3}< 0. The same arguments explain the change of sign with (θ − θ_{2}). In fact, it is the relative direction between (θ − θ_{2}) and (θ_{1} − θ_{2}) that matters, measured by the scalar product (θ − θ_{2})·(θ_{1} − θ_{2}). This geometrical dependence of the leadingorder contribution to ξ_{3} could provide a simple test of the consistency relation, without even computing the explicit expression in the righthand side of Eq. (38).
4.2. Threepoint correlation in terms of a twopoint correlation
The threepoint correlation ξ_{3} in Eq. (38) cannot be written as a product of twopoint correlations because there is only one integral along the line of sight that is left. However, if the linear power spectrum P_{L}(k,z) is already known, we may write ξ_{3} in terms of some twopoint correlation ξ_{2}. For instance, the smallscale crosscorrelation between one galaxy density contrast and one weak lensing convergence, (39)reads as (40)where we again used Limber’s approximation. Here we denoted the angular smoothing windows by to distinguish ξ_{2} from ξ_{3}. Then, we can write (41)if the angular windows of the twopoint correlation are chosen such that (42)This implies that the angular windows and of the twopoint correlation ξ_{2} have an explicit redshift dependence.
In practice, the expression (42) may not be very convenient. Then, to use the consistency relation (38) it may be more practical to first measure the power spectra P_{L} and P_{g1,m} independently, at the redshifts needed for the integral along the line of sight (38), and next compare the measure of ξ_{3} with the expression (38) computed with these power spectra.
4.3. LensinglensingISW correlation
From Eq. (38) we can directly obtain the lensinglensingISW threepoint correlation, (43)by replacing the galaxy kernels b_{g}I_{g} and I_{g1} by the lensing convergence kernels I_{κ} and I_{κ1}, (44)As compared with Eq. (38), the advantage of the crosscorrelation with the weak lensing convergence κ is that Eq. (44) involves the matter power spectrum P(k_{1 ⊥}) instead of the more complicated galaxymatter cross power spectrum P_{g1,m}(k_{1 ⊥}).
4.4. Vanishing contribution to the galaxyISWISW correlation
In the previous section (Sect. 4.1), we considered the threepoint galaxygalaxyISW correlation (30), to take advantage of the momentum dependence of the ISW effect (or more precisely its dependence on the time derivative of the density field), which gives rise to consistency relations that do not vanish at equal times. The reader may wonder whether we could also use the galaxyISWISW correlation for the same purpose. From Eq. (23), this threepoint correlation involves , instead of in Eq. (33), where we use compact notations. Thus, we obtain the combination (45)On the other hand, at equal times the consistency relation (8) writes as (46)where we only keep the contributions of order 1 /k and the second line in Eq. (8) cancels out. The first contribution to the threepoint correlation (45) reads as (47)Here again, we only consider the leading contribution of order 1 /k and we use k_{2} = − k_{1} in the limit k → 0. The term in the bracket in the second line vanishes because the crosspower spectrum only depends on  k , because of statistical isotropy. The second contribution to Eq. (45) reads as (48)The third contribution vanishes as usual at equal times, as it only involves the density field. Thus, we find that the leadingorder contribution to the galaxyISWISW threepoint correlation vanishes, in contrast with the galaxygalaxyISW threepoint correlation studied in section 4.1. This is why we focus on the threepoint correlations (30) and (43), with only one ISW field.
This cancellation can be understood from symmetry. Let us consider the maximal case where the points { θ,θ_{1},θ_{2} } are aligned. There is a nonzero consistency relation if the dependence of ⟨ λ_{1}λ_{2} ⟩ ′ to δ(θ) contains a linear term. In the longmode limit, this means that ⟨ λ_{1}λ_{2} ⟩ ′ changes sign with the sign of the largescale velocity flow. However, by symmetry ⟨ λ_{1}λ_{2} ⟩ ′ does not select a left or right direction along the line (θ_{1},θ_{2}), so that it cannot depend on the sign of the largescale velocity flow, nor on the sign of δ(θ). In contrast, in the case of the threepoint correlation (30), with only one ISW observable, the consistency relation relies on the dependence of ⟨ δ_{1}λ_{2} ⟩ ′ on the largescale mode δ (see the discussion after Eq. (38)). Then, it is clear that the nonsymmetrical quantity ⟨ δ_{1}λ_{2} ⟩ ′ defines a direction along the axis (θ_{1},θ_{2}), and a linear dependence on δ(θ) and on the sign of the largescale velocity is expected.
5. Consistency relation for the kSZ effect
In this section we consider cross correlations with the kSZ effect. This allows us to apply the consistency relation (5), which involves the momentum p and remains nonzero at equal times.
5.1. GalaxygalaxykSZ correlation
In a fashion similar to the galaxygalaxyISW correlation studied in Sect. 4.1, we consider the threepoint correlation between two galaxy density contrasts and one kSZ CMB anisotropy, (49)in the squeezed limit given by the conditions (31) in Fourier space and (32) in configuration space. The expressions (15) and (29) give (50)with (51)and (52)where we split the longitudinal and transverse contributions to Eq. (29). Here { n,n_{1},n_{2} } are the radial unit vectors that point to the centers { θ,θ_{1},θ_{2} } of the three circular windows, and are the longitudinal and transverse wave numbers with respect to the associated central lines of sight [e.g., ].
The computation of the transverse contribution (52) is similar to the computation of the ISW threepoint correlation (34), using again Limber’s approximation. At lowest order we obtain (53)where P_{g1,e} is the galaxyfree electrons cross power spectrum.
The computation of the longitudinal contribution (51) requires slightly more care. Applying the consistency relation (5) gives (54)where we only kept the contribution that does not vanish at equal times, as it dominates the integrals along the lines of sight, and we used P_{L}(k,η) = D(η)^{2}P_{L0}(k). If we approximate the three lines of sight as parallel, we can write n_{2}·k = k_{∥}, where the longitudinal and transverse directions coincide for the three lines of sight. Then, Limber’s approximation, which corresponds to the limit where the radial integrations have a constant weight on the infinite real axis, gives a Dirac term δ_{D}(k_{∥}) and ξ_{3 ∥} = 0 (more precisely, as we recalled above Eq. (36), the radial integration gives k_{∥} ≲ H/c while the angular window gives k_{⊥} ≲ H/ (cΘ) so that k_{∥} ≪ k_{⊥}). Taking into account the small angles between the different lines of sight, as for the derivation of Eq. (29), the integration over k_{2} through the Dirac factor gives at leading order in the angles (55)We used Limber’s approximation to write for instance P_{L0}(k) ≃ P_{L0}(k_{⊥}), but we kept the factor k_{∥} in the last term, as the transverse factor k_{⊥}·(θ_{2} − θ), due to the small angle between the lines of sight n and n_{2}, is suppressed by the small angle  θ_{2} − θ . We again split ξ_{3 ∥} over two contributions, , associated with the factors k_{∥} and k_{⊥}·(θ_{2} − θ) of the last term. Let us first consider the contribution . Writing , we integrate by parts over η. For simplicity we assume that the galaxy selection function I_{g} vanishes at z = 0, (56)so that the boundary term at z = 0 vanishes. Then, the integrations over k_{∥} and k_{1 ∥} give a factor (2π)^{2}δ_{D}(r − r_{2})δ_{D}(r_{1} − r_{2}), and we can integrate over η and η_{1}. Finally, the integration over the angles of the transverse wavenumbers yields (57)where J_{0} is the zerothorder Bessel function of the first kind. For the transverse contribution we can proceed in the same fashion, without integration by parts over η. This gives (58)It is useful to estimate the orders of magnitude of the three contributions , and . Using , and considering the case where we only have two angular scales for the angles (32), (59)the transverse wavenumbers are of order k_{⊥} ~ 1 /rΘ and k_{i ⊥} ~ 1 /rΘ_{i}. This gives (60)(61)and (62)hence (63)Thus, we find that the contribution ξ_{3 ⊥} associated with the second term in Eq. (29), which is due to the angle between the lines of sight within the small conical beam of angle Θ_{2}, is negligible as compared with the contribution ξ_{3 ∥} associated with the first term in Eq. (29), which is the zerothorder term. However, the two components and are of the same order. The first one, , is the zerothorder contribution when the lines of sight n and n_{2} are taken to be parallel, whereas the second one, , is the firstorder contribution over this small angle, measured by  θ − θ_{2}  (which is, however, much larger than the width Θ_{2} that gives rise to ξ_{3 ⊥}). This firstorder contribution can be of the same order as the zerothorder contribution because the latter is suppressed by the radial integration along the line of sight, which damps longitudinal modes, k_{∥} ≪ k_{⊥}.
In contrast with Eq. (38), the kSZ threepoint correlation, given by the sum of Eqs. (53), (57), and (58), does not vanish for orthogonal directions between the smallscale separation (θ_{1} − θ_{2}) and the largescale separation (θ − θ_{2}). Indeed, the leading order contribution in the squeezed limit to the response of ⟨ δ_{1}p_{2} ⟩ to a largescale perturbation δ factors out as ⟨ δ_{1}δ_{2} ⟩ v_{δ}, where we only take into account the contribution that does not vanish at equal times (and we discard the finitesize smoothing effects). The intrinsic smallscale correlation ⟨ δ_{1}δ_{2} ⟩ does not depend on the largescale mode δ, whereas v_{δ} is the almost uniform velocity due to the largescale mode, which only depends on the direction to δ(θ) and is independent of the orientation of the smallscale mode (θ_{1} − θ_{2}).
Because the measurement of the kSZ effect only probes the radial velocity of the free electrons gas along the line of sight, which is generated by density fluctuations almost parallel to the line of sight over which we integrate and which are damped by this radial integration, the result (57) is suppressed as compared with the ISW result (38) by the radial derivative dln(b_{g}I_{g}D) / dη ~ 1 /r. Also, the contribution (57), associated with transverse fluctuations that are almost orthogonal to the second line of sight, is suppressed as compared with the ISW result (38) by the small angle  θ − θ_{2}  between the two lines of sight.
One drawback of the kSZ consistency relation, (53), (57), and (58), is that it is not easy to independently measure the galaxyfree electrons power spectrum P_{g1,e}, which is needed if we wish to test this relation. Alternatively, Eqs. (57) and (58) may be used as a test of models for the free electrons distribution and the cross power spectrum P_{g1,e}.
5.2. LensinglensingkSZ correlation
Again, from Eqs. (53), (57), and (58) we can directly obtain the lensinglensingkSZ threepoint correlation, (64)by replacing the galaxy kernels b_{g}I_{g} and I_{g1} by the lensing convergence kernels I_{κ} and I_{κ1}. This gives with (65)(66)and (67)This now involves the matterfree electrons cross power spectrum P_{m,e}.
The application of the relations above is, unfortunately, a nontrivial task in terms of observations: to test those relations one would require the mixed galaxy (matter) – free electrons power spectrum. One possibility would be to do a stacking analysis of several Xray observations of the hot ionized gas by measuring the bremsstrahlung effect. For instance, one could infer n_{e}n_{p}T^{− 1 / 2}, by making some reasonable assumptions about the plasma state, as performed in FraserMcKelvie et al. (2011), with the aim of measuring n_{e} in filaments. We would of course need to cover a large range of scales. For kpc scales, inside galaxies and in the intergalactic medium, one could use for instance silicon emission line ratios (Kwitter & Henry 1998; Henry et al. 1996). For Mpc scales, or clusters, one may use the SunyaevZeldovich (SZ) effect (Rossetti et al. 2016). Nevertheless, all these proposed approaches are quite speculative at this stage.
5.3. Suppressed contribution to the galaxykSZkSZ correlation
As for the ISW effect, we investigate whether the galaxykSZkSZ correlation provides a good probe of the consistency relations. For the same symmetry reasons as in Sect. 4.4, we find that the leadingorder contribution to this threepoint correlation vanishes. Let us briefly sketch how this cancellation appears. First, from the hierarchy (63) we neglect the contribution associated with the second term in Eq. (29), that is, the widths of the smallscale windows are small and we can approximate each conical beam as a cylinder (flatsky limit). Then, we only have the component ξ_{3 ∥ ∥} similar to Eq. (51), which gives in compact notations (68)The consistency relation (4) gives at equal times (69)In the regime (59), we can take n_{1} ≃ n_{2}, hence (70)here we used the fact that the densitymomentum cross power spectrum obeys the symmetry , associated with a change of sign of the coordinate axis.
This cancellation can again be understood in configuration space. At leading order in the squeezed limit, the linear change of ⟨ p_{∥}(θ_{1})p_{∥}(θ_{2}) ⟩ ′ due to a largescale perturbation δ(θ) is (⟨ δ_{1}p_{∥ 2} ⟩ ′ + ⟨ p_{∥ 1}δ_{2} ⟩ ′)v_{δ ∥}, where v_{δ} is the largescale velocity generated by the largescale mode (the secondorder term does not contribute to the response function and the consistency relation). By symmetry the sum in the parenthesis vanishes. Therefore, in this paper we focus on the threepoint correlations (49) and (64), with only one kSZ field.
6. Conclusions
In this paper, we have shown how to relate the largescale consistency relations with observational probes. Assuming the standard cosmological model (more specifically, the equivalence principle and Gaussian initial conditions), nonzero equaltimes consistency relations involve the crosscorrelations between galaxy or matter density fields with the velocity, momentum, or timederivative density fields. We have shown that these relations can be related to actual measurements by considering the ISW and kSZ effects, which indeed involve the time derivative of the matter density field and the free electrons momentum field. We focused on the lowestorder relations, which apply to threepoint correlation functions or bispectra, because higherorder correlations are increasingly difficult to measure.
The most practical relation obtained in this paper is probably the one associated with the ISW effect, more particularly its crosscorrelation with two cosmic weaklensing convergence statistics. Indeed, it allows one to write this threepoint
correlation function in terms of two matter density field power spectra (linear and nonlinear), which can be directly measured (e.g., by twopoint weak lensing statistics). Moreover, the result, which is the leadingorder contribution in the squeezed limit, shows a specific angular dependence as a function of the relative angular positions of the three smoothed observed statistics. Then, both the angular dependence and the quantitative prediction provide a test of the consistency relation, that is, of the equivalence principle and of primordial Gaussianity. If we consider instead the crosscorrelation of the ISW effect with two galaxy density fields, we obtain a similar relation but it now involves the mixed galaxymatter density power spectrum P_{g,m} and the largescale galaxy bias b_{g}. These two quantities can again be measured (e.g., by twopoint galaxyweak lensing statistics) and provide another test of the consistency relation.
The relations obtained with the kSZ effect are more intricate. They do not show a simple angular dependence, which would provide a simple signature, and they involve the galaxyfree electrons or matterfree electrons power spectra. These power spectra are more difficult to measure. One can estimate the free electron density in specific regions, such as filaments or clusters, through Xray or SZ observations, or around typical structures by stacking analysis of clusters. This could provide an estimate of the free electrons cross power spectra and a check of the consistency relations. Although we can expect significant error bars, it would be interesting to check that the results remain consistent with the theoretical predictions. A violation of these consistency relations would signal either a modification of gravity on cosmological scales or nonGaussian initial conditions. We leave to future works the derivation of the deviations associated with various nonstandard scenarios.
Acknowledgments
This work is supported in part by the French Agence Nationale de la Recherche under Grant ANR12BS050002. D.F.M. thanks the support of the Research Council of Norway.
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