Consistency relations for large-scale structures: Applications for the integrated Sachs-Wolfe effect and the kinematic Sunyaev-Zeldovich effect
aa r X i v : . [ a s t r o - ph . C O ] J a n Astronomy & Astrophysicsmanuscript no. ISW_AA_Aug7 c (cid:13)
ESO 2018October 24, 2018
Consistency relations for large-scale structures: Applications forthe integrated Sachs-Wolfe effect and the kinematicSunyaev-Zeldovich effect
Luca Alberto Rizzo , David F. Mota , and Patrick Valageas Institut de Physique Théorique, CEA, IPhT, F-91191 Gif-sur-Yvette, Cédex, France Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, NorwayOctober 24, 2018
ABSTRACT
Consistency relations of large-scale structures provide exact nonperturbative results for cross-correlations of cosmic fields in thesqueezed limit. They only depend on the equivalence principle and the assumption of Gaussian initial conditions, and remain nonzeroat equal times for cross-correlations 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 Sachs-Wolfe e ff ect or the kinematic Sunyaev-Zeldovich e ff ect. In the squeezed limit, this allows us to express the three-point cross-correlations, or bispectra, of two galaxy or matterdensity fields, or weak lensing convergence fields, with the secondary Cosmic Microwave Background (CMB) distortion in terms ofproducts of a linear and a nonlinear power spectrum. In particular, we find that cross-correlations with the integrated Sachs-Wolfee ff ect 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 large-scale structures. Key words.
Cosmology – large-scale structure of the Universe
1. Introduction
Measuring statistical properties of cosmological structures isnot only an e ffi cient tool to describe and understand the maincomponents of our Universe, but also it is a powerful probe ofpossible new physics beyond the standard Λ -Cold Dark Matter( Λ CDM) concordance model. However, on large scales, cosmo-logical structures are described by perturbative methods, whilesmaller scales are described by phenomenological models orstudied with numerical simulations. It is therefore di ffi cult to ob-tain accurate predictions on the full range of scales probed bygalaxy and lensing surveys. Furthermore, if we consider galaxydensity fields, theoretical predictions remain sensitive to thegalaxy bias, which involves phenomenological modeling of starformation, even if we use cosmological numerical simulations.As a consequence, exact analytical results that go beyond low-order perturbation theory and also apply to biased tracers arevery rare.Recently, some exact results have been obtained(Kehagias & Riotto 2013; Peloso & Pietroni 2013;Creminelli et al. 2013; Kehagias et al. 2014a; Peloso & Pietroni2014; 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 large-scalewave numbers and n small-scale wave numbers, to the n -pointsmall-scale density correlation. These relations, obtained atthe leading order over the large-scale wave numbers, arisefrom the equivalence principle (EP) and the assumption ofGaussian initial conditions. The equivalence principle ensuresthat small-scale structures respond to a large-scale perturbationby a uniform displacement, while primordial Gaussianityprovides a simple relation between correlation and response functions (see Valageas et al. (2016) for the additional termsassociated with non-Gaussian initial conditions). Therefore,such relations express a kinematic e ff ect that vanishes forequal-times statistics, as a uniform displacement has no impacton the statistical properties of the density field observed at agiven time.In practice, it is, however, di ffi cult to measure di ff erent-timesdensity correlations and it would therefore be useful to obtainrelations that remain nonzero at equal times. One possibility toovercome such a problem is to go to higher orders and takeinto account tidal e ff ects, which at leading order are given bythe response of small-scale structures to a change in the back-ground density. Such an approach, however, introduces some ad-ditional approximations (Valageas 2014a; Kehagias et al. 2014b;Nishimichi & Valageas 2014).Fortunately, it was recently noticed that by cross-correlatingdensity fields with velocity or momentum fields, or with the timederivative of the density field, one obtains consistency relationsthat do not vanish at equal times (Rizzo et al. 2016). Indeed, thekinematic e ff ect modifies the amplitude of the large-scale veloc-ity and momentum fields, while the time derivative of the densityfield is obviously sensitive to di ff erent-times e ff ects.In this paper, we investigate the observational applicabilityof these new relations. We consider the lowest-order relations,which relate three-point cross-correlations or bispectra in thesqueezed limit to products of a linear and a nonlinear powerspectrum. To involve the non-vanishing consistency relations,we study two observable quantities, the secondary anisotropy ∆ ISW of the cosmic microwave background (CMB) radiation dueto the integrated Sachs-Wolfe e ff ect (ISW), and the secondaryanisotropy ∆ kSZ due to the kinematic Sunyaev-Zeldovich (kSZ)e ff ect. The first process, associated with the motion of CMB pho- Article number, page 1 of 10 & Aproofs: manuscript no. ISW_AA_Aug7 tons through time-dependent gravitational potentials, depends onthe time derivative of the matter density field. The second pro-cess, associated with the scattering of CMB photons by free elec-trons, depends on the free electrons velocity field. We investigatethe cross correlations of these two secondary anisotropies withboth galaxy density fields and the cosmic weak lensing conver-gence.This paper is organized as follows. In Section 2 we recall theconsistency relations of large-scale structures that apply to den-sity, momentum, and momentum-divergence (i.e., time deriva-tive of the density) fields. We describe the various observationalprobes that we consider in this paper in Section 3. We study theISW e ff ect in Section 4 and the kSZ e ff ect in Section 5. We con-clude in Section 6.
2. Consistency relations for large-scale structures
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; Valageas2014b; Horn et al. 2014, 2015), it is possible to obtain exact re-lations between density correlations of di ff erent orders in thesqueezed limit, where some of the wavenumbers are in the linearregime and far below the other modes that may be strongly non-linear. These “kinematic consistency relations”, obtained at theleading order over the large-scale wavenumbers, arise from theequivalence principle and the assumption of Gaussian primordialperturbations. They express the fact that at leading order wherea large-scale perturbation corresponds to a linear gravitationalpotential (hence a constant Newtonian force) over the extent ofa small-size structure, the latter falls without distortions in thislarge-scale potential.Then, in the squeezed limit k →
0, the correlation betweenone large-scale density mode ˜ δ ( k ) and n small-scale densitymodes ˜ δ ( k j ) can be expressed in terms of the n -point small-scalecorrelation, as h ˜ δ ( k , η ) n Y j = ˜ δ ( k j , η j ) i ′ k → = − P L ( k , η ) h n Y j = ˜ δ ( k j , η j ) i ′ × n X i = D ( η i ) D ( η ) k i · k k , (1)where the tilde denotes the Fourier transform of the fields, η isthe conformal time, D ( η ) is the linear growth factor, the primein h . . . i ′ denotes that we factored out the Dirac factor, h . . . i = h . . . i ′ δ D ( P k j ), and P L ( k ) is the linear matter power spectrum. Itis worth stressing that these relations are valid even in the non-linear regime and for biased galaxy fields ˜ δ g ( k j ). The right-handside gives the squeezed limit of the (1 + n ) correlation at the lead-ing order, which scales as 1 / k . It vanishes at this order at equaltimes, because of the constraint associated with the Dirac factor δ D ( P k j ).The geometrical factors ( k i · k ) vanish if k i ⊥ k . Indeed,the large-scale mode induces a uniform displacement along thedirection of k . This has no e ff ect on small-scale plane waves ofwavenumbers k i with k i ⊥ k , as they remain identical after sucha displacement. Therefore, the terms in the right-hand side ofEq.(1) must vanish in such orthogonal configurations, as we cancheck from the explicit expression. The simplest relation that one can obtain from Eq.(1) is forthe bispectrum with n = h ˜ δ ( k , η )˜ δ g ( k , η )˜ δ g ( k , η ) i ′ k → = − P L ( k , η ) k · k k ×h ˜ δ g ( k , η )˜ δ g ( k , η ) i ′ D ( η ) − D ( η ) D ( η ) , (2)where we used that k = − k − k → − k . For generality, weconsidered here the small-scale fields ˜ δ g ( k ) and ˜ δ g ( k ) to beassociated with biased tracers such as galaxies. The tracers as-sociated with k and k can be di ff erent and have di ff erent bias.At equal times the right-hand side of Eq.(2) vanishes, as recalledabove. The density consistency relations (1) express the uniform mo-tion of small-scale structures by large-scale modes. This simplekinematic e ff ect vanishes for equal-time correlations of the den-sity field, precisely because there are no distortions, while thereis a nonzero e ff ect at di ff erent times because of the motion ofthe small-scale structure between di ff erent times. However, aspointed out in Rizzo et al. (2016), it is possible to obtain non-trivial equal-times results by considering velocity or momentumfields, which are not only displaced but also see their amplitudea ff ected by the large-scale mode. Let us consider the momentum p defined by p = (1 + δ ) v , (3)where v is the peculiar velocity. Then, in the squeezed limit k →
0, the correlation between one large-scale density mode ˜ δ ( k ), n small-scale density modes ˜ δ ( k j ), and m small-scale momentummodes ˜ p ( k j ) can be expressed in terms of ( n + m ) small-scalecorrelations, as h ˜ δ ( k , η ) n Y j = ˜ δ ( k j , η j ) n + m Y j = n + ˜ p ( k j , η j ) i ′ k → = − P L ( k , η ) × (cid:26) h n Y j = ˜ δ ( k j , η j ) n + m Y j = n + ˜ p ( k j , η j ) i ′ n + m X i = D ( η i ) D ( η ) k i · k k + n + m X i = n + ( dD / dn )( η i ) D ( η ) h n Y j = ˜ δ ( k j , η j ) i − Y j = n + ˜ p ( k j , η j ) × i k k [ δ D ( k i ) + ˜ δ ( k i , η i )] ! n + m Y j = i + ˜ p ( k j , η j ) i ′ (cid:27) . (4)These relations are again valid in the nonlinear regime and forbiased galaxy fields ˜ δ g ( k j ) and ˜ p g ( k j ). As for the density con-sistency relation (1), the first term vanishes at this order at equaltimes. The second term, however, which arises from the ˜ p fieldsonly, remains nonzero. This is due to the fact that ˜ p involvesthe velocity, the amplitude of which is a ff ected by the motioninduced by the large-scale mode.The simplest relation associated with Eq.(4) is the bispec-trum among two density-contrast fields and one momentumfield, h ˜ δ ( k , η )˜ δ g ( k , η ) ˜ p g ( k , η ) i ′ k → = − P L ( k , η ) × (cid:18) k · k k h ˜ δ g ( k , η ) ˜ p g ( k , η ) i ′ D ( η ) − D ( η ) D ( η ) + i k k h ˜ δ g ( k , η )˜ δ g ( k , η ) i ′ D ( η ) dDd η ( η ) (cid:19) . (5) Article number, page 2 of 10uca Alberto Rizzo et al.: Consistency relations for the ISW and kSZ e ff ects For generality, we considered here the small-scale fields ˜ δ g ( k )and ˜ p g ( k ) to be associated with biased tracers such as galaxies,and the tracers associated with k and k can again be di ff erentand have di ff erent bias. At equal times, Eq.(5) reads as h ˜ δ ( k )˜ δ g ( k ) ˜ p g ( k ) i ′ k → = − i k k d ln Dd η P L ( k ) P g ( k ) , (6)where P g ( k ) is the galaxy nonlinear power spectrum and weomitted the common time dependence. This result does not van-ish thanks to the term generated by ˜ p in the consistency relation(5). In addition to the momentum field p , we can consider its diver-gence λ , defined by λ ≡ ∇ · [(1 + δ ) v ] = − ∂δ∂η . (7)The second equality expresses the continuity equation, that is,the conservation of matter. In the squeezed limit we obtain fromEq.(4) (Rizzo et al. 2016) h ˜ δ ( k , η ) n Y j = ˜ δ ( k j , η j ) n + m Y j = n + ˜ λ ( k j , η j ) i ′ k → = − P L ( k , η ) × (cid:26) h n Y j = ˜ δ ( k j , η j ) n + m Y j = n + ˜ λ ( k j , η j ) i ′ n + m X i = D ( η i ) D ( η ) k i · k k − n + m X i = n + h ˜ δ ( k i , η i ) n Y j = ˜ δ ( k j , η j ) n + m Y j = n + j , i ˜ λ ( k j , η j ) i ′ × ( dD / d η )( η i ) D ( η ) k i · k k (cid:27) . (8)These relations can actually be obtained by taking derivativeswith respect to the times η j of the density consistency relations(1), using the second equality (7). As for the momentum consis-tency relations (4), these relations remain valid in the nonlinearregime and for biased small-scale fields ˜ δ g ( k j ) and ˜ λ g ( k j ). Thesecond term in Eq.(8), which arises from the ˜ λ fields only, re-mains nonzero at equal times. This is due to the fact that λ in-volves the velocity or the time-derivative of the density, whichprobes the evolution between (infinitesimally close) di ff erenttimes.The simplest relation associated with Eq.(8) is the bispec-trum among two density-contrast fields and one momentum-divergence field, h ˜ δ ( k , η )˜ δ g ( k , η ) ˜ λ g ( k , η ) i ′ k → = − P L ( k , η ) k · k k × (cid:18) h ˜ δ g ( k , η ) ˜ λ g ( k , η ) i ′ D ( η ) − D ( η ) D ( η ) + h ˜ δ g ( k , η )˜ δ g ( k , η ) i ′ D ( η ) dDd η ( η ) (cid:19) . (9)At equal times, Eq.(9) reads as h ˜ δ ( k )˜ δ g ( k ) ˜ λ g ( k ) i ′ k → = − k · k k d ln Dd η P L ( k ) P g ( k ) . (10)
3. Observable quantities
To test cosmological scenarios with the consistency relations oflarge-scale structures we need to relate them to observable quan-tities. We describe in this section the observational probes thatwe consider in this paper. We use the galaxy numbers counts orthe weak lensing convergence to probe the density field. To ap-ply the momentum consistency relations (6) and (10), we use theISW e ff ect to probe the momentum divergence λ (more preciselythe time derivative of the gravitational potential and matter den-sity) and the kSZ e ff ect to probe the momentum p . δ g From galaxy surveys we can typically measure the galaxy den-sity contrast within a redshift bin, smoothed with a finite-sizewindow on the sky, δ s g ( θ ) = Z d θ ′ W Θ ( | θ ′ − θ | ) Z d η I g ( η ) δ g [ r , r θ ′ ; η ] , (11)where W Θ ( | θ ′ − θ | ) is a 2D symmetric window function centeredon the direction θ on the sky, of characteristic angular radius Θ , I g ( η ) is the radial weight along the line of sight associated witha normalized galaxy selection function n g ( z ), I g ( η ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dzd η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n g ( z ) , (12) r = η − η is the radial comoving coordinate along the line ofsight, and η is the conformal time today. Here and in the fol-lowing we use the flat sky approximation, and θ is the 2D vectorthat describes the direction on the sky of a given line of sight. Thesuperscript “s” in δ s g denotes that we smooth the galaxy densitycontrast with the finite-size window W Θ . Expanding in Fourierspace, we can write the galaxy density contrast as δ s g ( θ ) = Z d θ ′ W Θ ( | θ ′ − θ | ) Z d η I g ( η ) × Z d k e i k k r + i k ⊥ · r θ ′ ˜ δ g ( k , η ) , (13)where k k and k ⊥ are respectively the parallel and the perpendicu-lar components of the 3D wavenumber k = ( k k , k ⊥ ) (with respectto the reference direction θ =
0, and we work in the small-anglelimit θ ≪ W Θ as˜ W Θ ( | ℓ | ) = Z d θ e − i ℓ · θ W Θ ( | θ | ) , (14)we obtain δ s g ( θ ) = Z d η I g ( η ) Z d k ˜ W Θ ( k ⊥ r ) e i k k r + i k ⊥ · r θ ˜ δ g ( k , η ) . (15) κ From weak lensing surveys we can measure the weak lensingconvergence, given in the Born approximation by κ s ( θ ) = Z d θ ′ W Θ ( | θ ′ − θ | ) Z d η r g ( r ) ∇ Ψ + Φ r , r θ ′ ; η ] , (16)where Ψ and Φ are the Newtonian gauge gravitational potentialsand the kernel g ( r ) that defines the radial depth of the survey is g ( r ) = Z ∞ r dr s dz s dr s n g ( z s ) r s − rr s , (17) Article number, page 3 of 10 & Aproofs: manuscript no. ISW_AA_Aug7 where n g ( z s ) is the redshift distribution of the source galaxies.Assuming no anisotropic stress, that is, Φ = Ψ , and using thePoisson equation, ∇ Ψ = π G N ¯ ρ δ/ a , (18)where G N is the Newton constant, ¯ ρ is the mean matter densityof the Universe today, and a is the scale factor, we obtain κ s ( θ ) = Z d η I κ ( η ) Z d k ˜ W Θ ( k ⊥ r ) e i k k r + i k ⊥ · r θ ˜ δ ( k , η ) , (19)with I κ ( η ) = π G N ¯ ρ r g ( r ) a . (20) ∆ ISW
From Eq.(7) λ can be obtained from the momentum divergenceor from the time derivative of the density contrast. These quanti-ties are not as directly measured from galaxy surveys as den-sity contrasts. However, we can relate the time derivative ofthe density contrast to the ISW e ff ect, which involves the timederivative of the gravitational potential. Indeed, the secondarycosmic microwave background temperature anisotropy due tothe integrated Sachs-Wolfe e ff ect along the direction θ reads as(Garriga et al. 2004) ∆ ISW ( θ ) = Z d η e − τ ( η ) ∂ Ψ ∂η + ∂ Φ ∂η ! [ r , r θ ; η ] = Z d η e − τ ( η ) ∂ Ψ ∂η [ r , r θ ; η ] , (21)where τ ( η ) is the optical depth, which takes into account the pos-sibility of late reionization, and in the second line we assumedno anisotropic stress, that is, Φ = Ψ . We can relate ∆ ISW to λ through the Poisson equation (18), which reads in Fourier spaceas − k ˜ Ψ = π G N ¯ ρ ˜ δ/ a . (22)This gives ∂ ˜ Ψ ∂η = π G N ¯ ρ k a ( ˜ λ + H ˜ δ ) , (23)where H = d ln a / d η is the conformal expansion rate. Integrat-ing the ISW e ff ect δ ISW over some finite-size window on the sky,we obtain, as in Eq.(15), ∆ s ISW ( θ ) = Z d η I ISW ( η ) Z d k ˜ W Θ ( k ⊥ r ) e i k k r + i k ⊥ · r θ × ˜ λ + H ˜ δ k , (24)with I ISW ( η ) = π G N ¯ ρ e − τ a . (25) ∆ kSZ Thomson scattering of CMB photons o ff moving free electronsin the hot galactic or cluster gas generates secondary anisotropies(Sunyaev & Zeldovich 1980; Gruzinov & Hu 1998; Knox et al.1998). The temperature perturbation, ∆ kSZ = δ T / T , due to thiskinematic Sunyaev Zeldovich (kSZ) e ff ect, is ∆ kSZ ( θ ) = − Z d l · v e σ T n e e − τ = Z d η I kSZ ( η ) n ( θ ) · p e , (26)where τ is again the optical depth, σ T the Thomson cross sec-tion, l the radial coordinate along the line of sight, n e the numberdensity of free electrons, v e their peculiar velocity, and n ( θ ) theradial unit vector pointing to the line of sight. We also definedthe kSZ kernel by I kSZ ( η ) = − σ T ¯ n e ae − τ , (27)and the free electrons momentum p e as n e v e = ¯ n e (1 + δ e ) v e = ¯ n e p e . (28)Because of the projection n · p e along the line of sight, some caremust be taken when we smooth ∆ kSZ ( θ ) over some finite-size an-gular window W Θ ( | θ ′ − θ | ). Indeed, because the di ff erent lines ofsight θ ′ in the conical window are not perfectly parallel, if wedefine the longitudinal and transverse momentum componentsby the projection with respect to the mean line of sight n ( θ ) ofthe circular window, for example, p e k = n ( θ ) · p e , the projection n ( θ ′ ) · p e receives contributions from both p e k and p e ⊥ . In thelimit of small angles we could a priori neglect the contributionassociated with p e ⊥ , which is multiplied by an angular factor andvanishes for a zero-size window. However, for small but finiteangles, we need to keep this contribution because fluctuationsalong the lines of sight are damped by the radial integrations andvanish in the Limber approximation, which damps the contribu-tion associated with p e k .For small angles we write at linear order n ( θ ) = ( θ x , θ y , θ =
0. Then, the integration overthe angular window gives for the smoothed kSZ e ff ect ∆ s kSZ ( θ ) = Z d η I kSZ ( η ) Z d k e i k · n r (cid:20) ˜ p e k ˜ W Θ ( k ⊥ r ) − i k ⊥ · ˜ p e ⊥ k ⊥ ˜ W ′ Θ ( k ⊥ r ) (cid:21) . (29)Here we expressed the result in terms of the longitudinal andtransverse components of the wave numbers and momenta withrespect 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 , n ⊥ x = (1 , , − θ x ) and n ⊥ y = (0 , , − θ y ), and we write for instance k = k ⊥ x n ⊥ x + k ⊥ y n ⊥ y + k k n .We denote ˜ W ′ Θ ( ℓ ) = d ˜ W Θ / d ℓ . The last term in Eq.(29) is due tothe finite size Θ of the smoothing window, which makes the linesof sight within the conical beam not strictly parallel. It vanishesfor an infinitesimal window, where W Θ ( θ ) = δ D ( θ ) and ˜ W Θ = W ′ Θ =
0. We find in Section 5.1 that this contribution is typicallynegligible in the regime where the consistency relations apply, asthe width of the small-scale windows is much smaller than theangular size associated with the long mode.
As we explained above, in order to take advantage of the con-sistency relations we use the ISW or kSZ e ff ects because they Article number, page 4 of 10uca Alberto Rizzo et al.: Consistency relations for the ISW and kSZ e ff ects involve the time-derivative of the density field or the gas ve-locity. The reader may then note that redshift-space distortions(RSD) also involve velocities, but previous works that studiedthe galaxy density field in redshift space (Creminelli et al. 2014;Kehagias et al. 2014a) found that there is no equal-time e ff ect,as in the real-space case. Indeed, in both real space and redshiftspace, the long mode only generates a uniform change of coordi-nate (in the redshift-space case, this shift involves the radial ve-locity). Then, there is no e ff ect at equal times because such uni-form shifts do not produce distortions and observable signatures.In contrast, in our case there is a nonzero equal-time e ff ect be-cause the e ff ect of the long mode cannot be absorbed by a simplechange of coordinates. Indeed, the kSZ e ff ect, associated withthe scattering of CMB photons by free electons in hot ionizedgas (e.g., in X-ray clusters), actually probes the velocity di ff er-ence between the rest-frame of the CMB and the hot gas. Thus,the CMB last-scattering surface provides a reference frame andthe long mode generates a velocity di ff erence with respect to thatframe that cannot be described as a change of coordinate. Thisexplains why the kSZ e ff ect makes the long-mode velocity shiftobservable, without conflicting with the equivalence principle.There is also a nonzero e ff ect for the ISW case, because the lat-ter involves the time derivative of the density field, so that anequal-time statistics actually probes di ff erent-times properties ofthe density field (e.g., if we write the time derivative as an in-finitesimal finite di ff erence).If we cross-correlate real-space and redshift-space quantities,there will also remain a nonzero e ff ect at equal times, becausethe long mode generates di ff erent shifts for the real-space andredshift-space fields. Thus, we can consider the e ff ect of a longmode on small-scale correlations of the weak lensing conver-gence κ with redshift-space galaxy density contrasts δ s g . How-ever, weak lensing observables have broad kernels along the lineof sight, so that a small di ff erential shift along the radial directionis suppressed. In contrast, in the kSZ case the e ff ect is directlydue to the change of velocity by the long mode, and not by theindirect impact of the change of the radial redshift coordinate.Another observable e ff ect of the long mode was pointed outin Baldauf et al. (2015). These authors noticed that a long modeof wave length 2 π/ k of the same order as the baryon acoustic os-cillation (BAO) scale, x BAO ∼ h − Mpc, gives a di ff erent shiftto galaxies separated by this distance. This produces a spreadof the BAO peak, after we average over the long mode. Thereason why this e ff ect is observable is that the correlation func-tion shows a narrow peak at the BAO scale, with a width of or-der ∆ x BAO ∼ h − Mpc. This narrow feature provides a probeof the small displacement of galaxies by the long mode, whichwould otherwise be negligible if the galaxy correlation were aslow power law. As noticed above, the absence of such a narrowfeature suppresses the signal associated with cross-correlationsamong weak-lensing (real-space) quantities and redshift-spacequantities, because of the radial broadening of the weak-lensingprobes.This BAO probe is actually a second-order e ff ect, in the senseof the consistency relations. Indeed, the usual consistency rela-tions are obtained in the large-scale limit k →
0, where the longmode generates a uniform displacement of the small-scale struc-tures. In contrast, the spread of the BAO peak relies on the di ff er-ential displacement between galaxies separated by x BAO . In theTaylor expansion of the displacement with respect to the posi-tions of the small-scale structures, beyond the lowest-order con-stant term one takes into account the linear term over x , whichscales as kx . This is why this e ff ect requires that k be finite andnot too small, of order k ∼ π/ x BAO .
4. Consistency relation for the ISW temperatureanisotropy
In this section we consider cross correlations with the ISW ef-fect. This allows us to apply the consistency relation (9), whichinvolves the momentum divergence λ and remains nonzero atequal times. To take advantage of the consistency relation (9), we must con-sider three-point correlations ξ (in configuration space) withone observable that involves the momentum divergence λ . Here,using the expression (24), we study the cross-correlation be-tween two galaxy density contrasts and one ISW temperatureanisotropy, ξ ( δ s g , δ s g , ∆ s ISW ) = h δ s g ( θ ) δ s g ( θ ) ∆ s ISW ( θ ) i . (30)The subscripts g , g , and ISW denote the three lines of sight as-sociated with the three probes. Moreover, the subscripts g and g recall that the two galaxy populations associated with δ s g and δ s g can be di ff erent and have di ff erent bias. As we recalled in Sec-tion 2, the consistency relations rely on the undistorted motionof small-scale structures by large-scale modes. This correspondsto the squeezed limit k → k ≪ k L , k ≪ k j , (31)where k L is the wavenumber associated with the transition be-tween the linear and nonlinear regimes. The first condition en-sures that ˜ δ ( k ) is in the linear regime, while the second condi-tion ensures the hierarchy between the large-scale mode and thesmall-scale modes. In configuration space, these conditions cor-respond to Θ ≫ Θ L , Θ ≫ Θ j , | θ − θ j | ≫ | θ − θ | . (32)The first condition ensures that δ s g ( θ ) is in the linear regime,whereas the next two conditions ensure the hierarchy of scales.The expressions (15) and (24) give ξ = Z d η d η d η I g ( η ) I g ( η ) I ISW ( η ) × Z d k d k d k ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) × e i( k k r + k k r + k k r + k ⊥ · r θ + k ⊥ · r θ + k ⊥ · r θ ) ×h ˜ δ g ( k , η )˜ δ g ( k , η ) ˜ λ ( k , η ) + H ˜ δ ( k , η ) k i . (33)The configuration-space conditions (32) ensure that we satisfythe Fourier-space conditions (31) and that we can apply the con-sistency relations (2) and (9). This gives ξ = − Z d η d η d η b g ( η ) I g ( η ) I g ( η ) I ISW ( η ) × Z d k d k d k ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) × e i( k k r + k k r + k k r + k ⊥ · r θ + k ⊥ · r θ + k ⊥ · r θ ) × P L ( k , η ) k · k k δ D ( k + k + k ) × h ˜ δ g ˜ λ + H ˜ δ k i ′ D ( η ) − D ( η ) D ( η ) + h ˜ δ g ˜ δ k i ′ D ( η ) dDd η ( η ) ! . (34) Article number, page 5 of 10 & Aproofs: manuscript no. ISW_AA_Aug7
Here we assumed that on large scales the galaxy bias is lin-ear, k → δ g ( k ) = b g ( η )˜ δ ( k ) + ˜ ǫ ( k ) , (35)where ˜ ǫ is a stochastic component that represents shot noise andthe e ff ect of small-scale (e.g., baryonic) physics on galaxy for-mation. From the decomposition (35), it is uncorrelated with thelarge-scale density field (Hamaus et al. 2010), h ˜ δ ( k )˜ ǫ ( k ) i = h ˜ ǫ ˜ δ g ( ˜ λ + H ˜ δ ) i . In-deed, the small-scale local processes within the region θ shouldbe very weakly correlated with the density fields in the dis-tant regions θ and θ , which at leading order are only sensi-tive to the total mass within the large-scale region θ . Therefore, h ˜ ǫ ˜ δ g ( ˜ λ + H ˜ δ ) i should exhibit a fast decay at low k , whereasthe term in Eq.(34) associated with the consistency relation onlydecays as P L ( k ) / k ∼ k n s − with n s ≃ .
96. In Eq.(34), we alsoassumed 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 scaledependence [by keeping the factor b g ( k , η ) in the integral over k ]. The small-scale two-point correlations h · i ′ are dominatedby contributions at almost equal times, η ≃ η , as di ff erent red-shifts would correspond to points that are separated by severalHubble radii along the lines of sight and density correlations arenegligible beyond Hubble scales. Therefore, ξ is dominated bythe second term that does not vanish at equal times. The integralsalong the lines of sight suppress the contributions from longitu-dinal wavelengths below the Hubble radius c / H , while the angu-lar windows only suppress the wavelengths below the transverseradii c Θ / H . Then, for small angular windows, Θ ≪
1, we canuse Limber’s approximation, k k ≪ k ⊥ hence k ≃ k ⊥ . Integratingover k k through the Dirac factor δ D ( k k + k k + k k ), and next over k k and k k , we obtain the Dirac factors (2 π ) δ D ( r − r ) δ D ( r − r ).This allows us to integrate over η and η and we obtain ξ = − (2 π ) Z d η b g ( η ) I g ( η ) I g ( η ) I ISW ( η ) d ln Dd η × Z d k ⊥ d k ⊥ d k ⊥ δ D ( k ⊥ + k ⊥ + k ⊥ ) ˜ W Θ ( k ⊥ r ) × ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) e i r ( k ⊥ · θ + k ⊥ · θ + k ⊥ · θ ) × P L ( k ⊥ , η ) k ⊥ · k ⊥ k ⊥ k ⊥ P g , m ( k ⊥ , η ) , (36)where P g , m is the galaxy-matter power spectrum. The integra-tion over k ⊥ gives ξ = − (2 π ) Z d η b g I g I g I ISW d ln Dd η Z d k ⊥ d k ⊥ ˜ W Θ ( k ⊥ r ) × ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) P L ( k ⊥ , η ) P g , m ( k ⊥ , η ) × e i r [ k ⊥ · ( θ − θ ) + k ⊥ · ( θ − θ )] k ⊥ · k ⊥ k ⊥ k ⊥ , (37)and the integration over the angles of k ⊥ and k ⊥ gives ξ = ( θ − θ ) · ( θ − θ ) | θ − θ || θ − θ | (2 π ) Z d η b g I g I g I ISW d ln Dd η × Z ∞ dk ⊥ dk ⊥ ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) × P L ( k ⊥ , η ) P g , m ( k ⊥ , η ) J ( k ⊥ r | θ − θ | ) × J ( k ⊥ r | θ − θ | ) , (38)where J is the first-order Bessel function of the first kind. As the expression (38) arises from the kinematic consistencyrelations, it expresses the response of the small-scale two-pointcorrelation h δ s g ( θ ) ∆ s ISW ( θ ) i to a change of the initial conditionassociated with the large-scale mode δ s g ( θ ). The kinematic e ff ectgiven at the leading order by Eq.(38) is due to the uniform mo-tion of the small-scale structures by the large-scale mode. Thisexplains why the result (38) vanishes in the two following cases:1. ( θ − θ ) ⊥ ( θ − θ ). There is a nonzero response of h δ λ i if there is a linear dependence on δ ( θ ) of h δ λ i , so that itsfirst derivative is nonzero. A positive (negative) δ ( θ ) leads toa uniform motion at θ towards (away from) θ , along the di-rection ( θ − θ ). From the point of view of θ and θ , thereis a reflection symmetry with respect to the axis ( θ − θ ).For instance, if δ > θ typically decreases in the mean with the radius | θ − θ | , andfor ∆ θ ⊥ ( θ − θ ) the points θ ± = θ ± ∆ θ are at the samedistance from θ and have the same density contrast δ in themean, with typically δ < δ as | θ ± − θ | > | θ − θ | . There-fore, the large-scale flow along ( θ − θ ) leads to a positive λ = − ∆ δ / ∆ η independently of whether the matter movestowards or away from θ (here we took a finite deviation ∆ θ ).This means that the dependence of h δ λ i on δ ( θ ) is quadratic(it does not depend on the sign of δ ( θ )) and the first-order re-sponse function vanishes. Then, the leading-order contribu-tion to ξ vanishes. (For infinitesimal deviation ∆ θ we have λ = − ∂δ /∂η =
0; by this symmetry, in the mean δ is anextremum of the density contrast along the orthogonal direc-tion to ( θ − θ )).2. θ = θ . This is a particular case of the previous configura-tion. Again, by symmetry from the viewpoint of δ , the twopoints δ ( θ +∆ θ ) and δ ( θ − ∆ θ ) are equivalent and the meanresponse associated with the kinematic e ff ect vanishes.This also explains why Eq.(38) changes sign with ( θ − θ )and ( θ − θ ). Let us consider for simplicity the case where thethree points are aligned and δ ( θ ) >
0, so that the large-scaleflow points towards θ . We also take δ >
0, so that in the meanthe density is peaked at θ and decreases outwards. Let us take θ close to θ , on the decreasing radial slope, and on the otherside of θ than θ . Then, the large-scale flow moves matter at θ towards θ , so that the density at θ at a slightly later time comesfrom more outward regions (with respect to the peak at θ ) witha lower density. This means that λ = − ∂δ /∂η is positive sothat ξ >
0. This agrees with Eq.(38), as ( θ − θ ) · ( θ − θ ) > J >
0. If we flip θ to the otherside of θ , we find on the contrary that the large-scale flow bringshigher-density regions to θ , so that we have the change of signs λ < ξ <
0. The same arguments explain the change ofsign with ( θ − θ ). In fact, it is the relative direction between ( θ − θ ) and ( θ − θ ) that matters, measured by the scalar product ( θ − θ ) · ( θ − θ ). This geometrical dependence of the leading-ordercontribution to ξ could provide a simple test of the consistencyrelation, without even computing the explicit expression in theright-hand side of Eq.(38). The three-point correlation ξ in Eq.(38) cannot be written as aproduct of two-point correlations because there is only one in-tegral along the line of sight that is left. However, if the linearpower spectrum P L ( k , z ) is already known, we may write ξ in Article number, page 6 of 10uca Alberto Rizzo et al.: Consistency relations for the ISW and kSZ e ff ects terms of some two-point correlation ξ . For instance, the small-scale cross-correlation between one galaxy density contrast andone weak lensing convergence, ξ ( δ s g , κ s ) = h δ s g ( θ ) κ s ( θ ) i (39)reads as ξ = (2 π ) Z d η I g I κ Z ∞ dk ⊥ k ⊥ ˜ F Θ ( k ⊥ r ) × ˜ F Θ ( k ⊥ r ) J ( k ⊥ r | θ − θ | ) P g , m ( k ⊥ ) , (40)where we again used Limber’s approximation. Here we denotedthe angular smoothing windows by ˜ F to distinguish ξ from ξ .Then, we can write ξ = ( θ − θ ) · ( θ − θ ) | θ − θ || θ − θ | ξ , (41)if the angular windows of the two-point correlation are chosensuch that˜ F Θ ( k ⊥ r ) ˜ F Θ ( k ⊥ ) = (2 π ) I g I ISW I κ b g d ln Dd η × Z ∞ dk ⊥ ˜ W Θ ( k ⊥ r ) J ( k ⊥ r | θ − θ | ) P L ( k ⊥ , η ) ! × ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) J ( k ⊥ r | θ − θ | ) k ⊥ J ( k ⊥ r | θ − θ | ) . (42)This implies that the angular windows ˜ F Θ and ˜ F Θ of the two-point correlation ξ 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 prac-tical to first measure the power spectra P L and P g , m indepen-dently, at the redshifts needed for the integral along the line ofsight (38), and next compare the measure of ξ with the expres-sion (38) computed with these power spectra. From Eq.(38) we can directly obtain the lensing-lensing-ISWthree-point correlation, ξ ( κ s , κ s , ∆ s ISW ) = h κ s ( θ ) κ s ( θ ) ∆ s ISW ( θ ) i , (43)by replacing the galaxy kernels b g I g and I g by the lensing con-vergence kernels I κ and I κ , ξ = ( θ − θ ) · ( θ − θ ) | θ − θ || θ − θ | (2 π ) Z d η I κ I κ I ISW d ln Dd η × Z ∞ dk ⊥ dk ⊥ ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) × P L ( k ⊥ , η ) P ( k ⊥ , η ) J ( k ⊥ r | θ − θ | ) × J ( k ⊥ r | θ − θ | ) . (44)As compared with Eq.(38), the advantage of the cross-correlation with the weak lensing convergence κ is that Eq.(44)involves the matter power spectrum P ( k ⊥ ) instead of the morecomplicated galaxy-matter cross power spectrum P g , m ( k ⊥ ). In the previous Section 4.1, we considered the three-pointgalaxy-galaxy-ISW correlation (30), to take advantage of themomentum dependence of the ISW e ff ect (or more preciselyits dependence on the time derivative of the density field),which gives rise to consistency relations that do not vanish atequal times. The reader may wonder whether we could also usethe galaxy-ISW-ISW correlation for the same purpose. FromEq.(23), this three-point correlation involves h ˜ δ ( ˜ λ + ˜ δ )( ˜ λ + ˜ δ ) i ′ , instead of h ˜ δ ˜ δ ( ˜ λ + ˜ δ ) i ′ in Eq.(33), where we use compactnotations. Thus, we obtain the combination h δ ∆ ISW ∆ ISW i ∝ h ˜ δ ˜ λ ˜ λ i ′ + H h h ˜ δ ˜ λ ˜ δ i ′ + h ˜ δ ˜ δ ˜ λ i ′ i + H h ˜ δ ˜ δ ˜ δ i ′ . (45)On the other hand, at equal times the consistency relation (8)writes as h ˜ δ ( k ) n Y j = ˜ δ ( k j ) n + m Y j = n + ˜ λ ( k j ) i ′ k → = P L ( k ) D ′ D n + m X i = n + k · k i k × h ˜ δ ( k i ) n Y j = ˜ δ ( k j ) n + m Y j = n + j , i ˜ λ ( k j ) i ′ , (46)where we only keep the contributions of order 1 / k and the sec-ond line in Eq.(8) cancels out. The first contribution to the three-point correlation (45) reads as h ˜ δ ˜ λ ˜ λ i ′ = P L ( k ) D ′ D " k · k k h ˜ δ ˜ λ i ′ + k · k k h ˜ δ ˜ λ i ′ = P L ( k ) D ′ D k · k k h h ˜ δ ( k ) ˜ λ ( − k ) i ′ − h ˜ δ ( − k ) ˜ λ ( k ) i ′ i = . (47)Here again, we only consider the leading contribution of order1 / k and we use k = − k in the limit k →
0. The term inthe bracket in the second line vanishes because the cross-powerspectrum h ˜ δ ( k ) ˜ λ ( − k ) i ′ = P δ,λ ( k ) only depends on | k | , because ofstatistical isotropy. The second contribution to Eq.(45) reads as h ˜ δ ˜ λ ˜ δ i ′ + h ˜ δ ˜ δ ˜ λ i ′ = P L ( k ) D ′ D " k · k k h ˜ δ ˜ δ i ′ + k · k k h ˜ δ ˜ δ i ′ = . (48)The third contribution h ˜ δ ˜ δ ˜ δ i ′ vanishes as usual at equal times,as it only involves the density field. Thus, we find that theleading-order contribution to the galaxy-ISW-ISW three-pointcorrelation vanishes, in contrast with the galaxy-galaxy-ISWthree-point correlation studied in section 4.1. This is why we fo-cus on the three-point correlations (30) and (43), with only oneISW field.This cancellation can be understood from symmetry. Letus consider the maximal case where the points { θ , θ , θ } arealigned. There is a nonzero consistency relation if the depen-dence of h λ λ i ′ to δ ( θ ) contains a linear term. In the long-modelimit, this means that h λ λ i ′ changes sign with the sign of thelarge-scale velocity flow. However, by symmetry h λ λ i ′ doesnot select a left or right direction along the line ( θ , θ ), so thatit cannot depend on the sign of the large-scale velocity flow, noron the sign of δ ( θ ). In contrast, in the case of the three-pointcorrelation (30), with only one ISW observable, the consistency Article number, page 7 of 10 & Aproofs: manuscript no. ISW_AA_Aug7 relation relies on the dependence of h δ λ i ′ on the large-scalemode δ (see the discussion after Eq.(38)). Then, it is clear thatthe nonsymmetrical quantity h δ λ i ′ defines a direction along theaxis ( θ , θ ), and a linear dependence on δ ( θ ) and on the sign ofthe large-scale velocity is expected.
5. Consistency relation for the kSZ effect
In this section we consider cross correlations with the kSZ ef-fect. This allows us to apply the consistency relation (5), whichinvolves the momentum p and remains nonzero at equal times. In a fashion similar to the galaxy-galaxy-ISW correlation studiedin Section 4.1, we consider the three-point correlation betweentwo galaxy density contrasts and one kSZ CMB anisotropy, ξ ( δ s g , δ s g , ∆ s kSZ ) = h δ s g ( θ ) δ s g ( θ ) ∆ s kSZ ( θ ) i , (49)in the squeezed limit given by the conditions (31) in Fourierspace and (32) in configuration space. The expressions (15) and(29) give ξ = ξ k + ξ ⊥ , (50)with ξ k = Z d η d η d η I g ( η ) I g ( η ) I kSZ ( η ) Z d k d k d k × e i( k · n r + k · n r + k · n r ) ˜ W Θ ( k ( n ) ⊥ r ) ˜ W Θ ( k ( n )1 ⊥ r ) × ˜ W Θ ( k ( n )2 ⊥ r ) h ˜ δ g ( k , η )˜ δ g ( k , η ) ˜ p ( n ) e k )( k , η ) i (51)and ξ ⊥ = − i Z d η d η d η I g ( η ) I g ( η ) I kSZ ( η ) Z d k d k d k × e i( k · n r + k · n r + k · n r ) ˜ W Θ ( k ( n ) ⊥ r ) ˜ W Θ ( k ( n )1 ⊥ r ) × ˜ W ′ Θ ( k ( n )2 ⊥ r ) h ˜ δ g ( k , η )˜ δ g ( k , η ) k ( n )2 ⊥ · ˜ p ( n ) e ⊥ k ( n )2 ⊥ ( k , η ) i , (52)where we split the longitudinal and transverse contributionsto Eq.(29). Here { n , n , n } are the radial unit vectors thatpoint to the centers { θ , θ , θ } of the three circular windows,and { ( k ( n ) k , k ( n ) ⊥ ) , ( k ( n )1 k , k ( n )1 ⊥ ) , ( k ( n )2 k , k ( n )2 ⊥ ) } are the longitudinal andtransverse wave numbers with respect to the associated centrallines of sight [e.g., k ( n ) k = n · k ].The computation of the transverse contribution (52) is sim-ilar to the computation of the ISW three-point correlation (34),using again Limber’s approximation. At lowest order we obtain ξ ⊥ = ( θ − θ ) · ( θ − θ ) | θ − θ || θ − θ | (2 π ) Z d η b g I g I g I kSZ d ln Dd η × Z ∞ dk ⊥ dk ⊥ k ⊥ ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W ′ Θ ( k ⊥ r ) × P L ( k ⊥ , η ) P g , e ( k ⊥ , η ) J ( k ⊥ r | θ − θ | ) × J ( k ⊥ r | θ − θ | ) , (53)where P g , e is the galaxy-free electrons cross power spectrum. The computation of the longitudinal contribution (51) re-quires slightly more care. Applying the consistency relation (5)gives ξ k = − Z d η d η d η b g ( η ) I g ( η ) I g ( η ) I kSZ ( η ) × Z d k d k d k ˜ W Θ ( k ( n ) ⊥ r ) ˜ W Θ ( k ( n )1 ⊥ r ) ˜ W Θ ( k ( n )2 ⊥ r ) × e i( k · n r + k · n r + k · n r ) D ( η ) P L ( k ) dDd η ( η ) × i n · k k h ˜ δ g ˜ δ e i ′ δ D ( k + k + k ) , (54)where we only kept the contribution that does not vanish at equaltimes, as it dominates the integrals along the lines of sight, andwe used P L ( k , η ) = D ( η ) P L ( k ). If we approximate the threelines of sight as parallel, we can write n · k = k k , where the lon-gitudinal and transverse directions coincide for the three lines ofsight. Then, Limber’s approximation, which corresponds to thelimit where the radial integrations have a constant weight on theinfinite real axis, gives a Dirac term δ D ( k k ) and ξ k = k k . H / c while the angular window gives k ⊥ . H / ( c Θ ) sothat k k ≪ k ⊥ ). Taking into account the small angles between thedi ff erent lines of sight, as for the derivation of Eq.(29), the inte-gration over k through the Dirac factor gives at leading order inthe angles ξ k = − Z d η d η d η b g ( η ) I g ( η ) D ( η ) I g ( η ) I kSZ ( η ) dDd η ( η ) × Z dk k d k ⊥ dk k d k ⊥ ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) × e i[ k k ( r − r ) + k ⊥ · ( θ − θ ) r + k k ( r − r ) + k ⊥ · ( θ − θ ) r ] × P L ( k ⊥ ) P g , e ( k ⊥ ; η , η )i k k + k ⊥ · ( θ − θ ) k ⊥ . (55)We used Limber’s approximation to write for instance P L ( k ) ≃ P L ( k ⊥ ), but we kept the factor k k in the last term, as the trans-verse factor k ⊥ · ( θ − θ ), due to the small angle between the linesof sight n and n , is suppressed by the small angle | θ − θ | . Weagain split ξ k over two contributions, ξ k = ξ k k + ξ ⊥ k , associatedwith the factors k k and k ⊥ · ( θ − θ ) of the last term. Let us firstconsider the contribution ξ k k . Writing i k k e i k k ( r − r ) = ∂∂ r e i k k ( r − r ) ,we integrate by parts over η . For simplicity we assume that thegalaxy selection function I g vanishes at z = I g ( η ) = , (56)so that the boundary term at z = k k and k k give a factor (2 π ) δ D ( r − r ) δ D ( r − r ), andwe can integrate over η and η . Finally, the integration over theangles of the transverse wavenumbers yields ξ k k = − (2 π ) Z d η dd η h b g I g D i I g I kSZ dDd η × Z ∞ dk ⊥ dk ⊥ ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) × k ⊥ k ⊥ P L ( k ⊥ ) P g , e ( k ⊥ , η ) J ( k ⊥ r | θ − θ | ) × J ( k ⊥ r | θ − θ | ) , (57)where J is the zeroth-order Bessel function of the first kind.For the transverse contribution ξ ⊥ k we can proceed in the same Article number, page 8 of 10uca Alberto Rizzo et al.: Consistency relations for the ISW and kSZ e ff ects fashion, without integration by parts over η . This gives ξ ⊥ k = − (2 π ) Z d η b g I g I g I kSZ D dDd η × Z ∞ dk ⊥ dk ⊥ ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) × k ⊥ P L ( k ⊥ ) P g , e ( k ⊥ , η ) | θ − θ | J ( k ⊥ r | θ − θ | ) × J ( k ⊥ r | θ − θ | ) . (58)It is useful to estimate the orders of magnitude of the threecontributions ξ ⊥ , ξ k k , and ξ ⊥ k . Using ˜ W ′ Θ ( ℓ ) ∼ Θ ˜ W Θ ( ℓ ), and con-sidering the case where we only have two angular scales for theangles (32), Θ ∼ Θ ∼ | θ − θ | , Θ ∼ | θ − θ | ≃ | θ − θ | , Θ ≪ Θ , (59)the transverse wavenumbers are of order k ⊥ ∼ / r Θ and k i ⊥ ∼ / r Θ i . This gives ξ ⊥ ∼ b g I g I g I kSZ D Θ k ⊥ k ⊥ P L ( k ⊥ ) P g , e ( k ⊥ ) , (60) ξ k k ∼ b g I g I g I kSZ D η k ⊥ P L ( k ⊥ ) P g , e ( k ⊥ ) , (61)and ξ ⊥ k ∼ b g I g I g I kSZ D k ⊥ k ⊥ | θ − θ | P L ( k ⊥ ) P g , e ( k ⊥ ) , (62)hence ξ ⊥ ξ k k ∼ Θ k ⊥ η ∼ Θ Θ ≪ , ξ ⊥ k ξ k k ∼ | θ − θ | k ⊥ η ∼ . (63)Thus, we find that the contribution ξ ⊥ associated with the sec-ond term in Eq.(29), which is due to the angle between the linesof sight within the small conical beam of angle Θ , is negligi-ble as compared with the contribution ξ k associated with thefirst term in Eq.(29), which is the zeroth-order term. However,the two components ξ k k and ξ ⊥ k are of the same order. The firstone, ξ k k , is the zeroth-order contribution when the lines of sight n and n are taken to be parallel, whereas the second one, ξ ⊥ k ,is the first-order contribution over this small angle, measured by | θ − θ | (which is, however, much larger than the width Θ thatgives rise to ξ ⊥ ). This first-order contribution can be of the sameorder as the zeroth-order contribution because the latter is sup-pressed by the radial integration along the line of sight, whichdamps longitudinal modes, k k ≪ k ⊥ .In contrast with Eq.(38), the kSZ three-point correlation,given by the sum of Eqs.(53), (57), and (58), does not van-ish for orthogonal directions between the small-scale separation( θ − θ ) and the large-scale separation ( θ − θ ). Indeed, the lead-ing order contribution in the squeezed limit to the response of h δ p i to a large-scale perturbation δ factors out as h δ δ i v δ ,where we only take into account the contribution that does notvanish at equal times (and we discard the finite-size smoothinge ff ects). The intrinsic small-scale correlation h δ δ i does not de-pend on the large-scale mode δ , whereas v δ is the almost uni-form velocity due to the large-scale mode, which only dependson the direction to δ ( θ ) and is independent of the orientation ofthe small-scale mode ( θ − θ ).Because the measurement of the kSZ e ff ect only probesthe radial velocity of the free electrons gas along the line ofsight, which is generated by density fluctuations almost paral-lel to the line of sight over which we integrate and which are damped by this radial integration, the result (57) is suppressedas compared with the ISW result (38) by the radial derivative d ln( b g I g D ) / d η ∼ / r . Also, the contribution (57), associatedwith transverse fluctuations that are almost orthogonal to the sec-ond line of sight, is suppressed as compared with the ISW result(38) by the small angle | θ − θ | between the two lines of sight.One drawback of the kSZ consistency relation, (53) and (57)-(58), is that it is not easy to independently measure the galaxy-free electrons power spectrum P g , e , which is needed if we wishto test this relation. Alternatively, Eqs.(57)-(58) may be used asa test of models for the free electrons distribution and the crosspower spectrum P g , e . Again, from Eqs.(53) and (57)-(58) we can directly obtain thelensing-lensing-kSZ three-point correlation, ξ ( κ s , κ s , ∆ s kSZ ) = h κ s ( θ ) κ s ( θ ) ∆ s kSZ ( θ ) i , (64)by replacing the galaxy kernels b g I g and I g by the lensing con-vergence kernels I κ and I κ . This gives ξ = ξ ⊥ + ξ k k + ξ ⊥ k with ξ ⊥ = ( θ − θ ) · ( θ − θ ) | θ − θ || θ − θ | (2 π ) Z d η I κ I κ I kSZ d ln Dd η × Z ∞ dk ⊥ dk ⊥ k ⊥ ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W ′ Θ ( k ⊥ r ) × P L ( k ⊥ , η ) P m , e ( k ⊥ , η ) J ( k ⊥ r | θ − θ | ) × J ( k ⊥ r | θ − θ | ) , (65) ξ k k = − (2 π ) Z d η dd η [ I κ D ] I κ I kSZ dDd η Z ∞ dk ⊥ dk ⊥ × ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) k ⊥ k ⊥ P L ( k ⊥ ) × P m , e ( k ⊥ , η ) J ( k ⊥ r | θ − θ | ) J ( k ⊥ r | θ − θ | ) , (66)and ξ ⊥ k = − (2 π ) Z d η I κ I κ I kSZ D dDd η Z ∞ dk ⊥ dk ⊥ × ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) ˜ W Θ ( k ⊥ r ) k ⊥ P L ( k ⊥ ) × P m , e ( k ⊥ , η ) | θ − θ | J ( k ⊥ r | θ − θ | ) J ( k ⊥ r | θ − θ | ) . (67)This now involves the matter-free electrons cross power spec-trum P m , e .The application of the relations above is, unfortunately, anontrivial task in terms of observations: to test those relationsone would require the mixed galaxy (matter) - free electronspower spectrum. One possibility would be to do a stacking anal-ysis of several X-ray observations of the hot ionized gas bymeasuring the bremsstrahlung e ff ect. For instance, one could in-fer n e n p T − / , by making some reasonable assumptions aboutthe plasma state, as performed in Fraser-McKelvie et al. (2011),with the aim of measuring n e in filaments. We would of courseneed to cover a large range of scales. For kpc scales, inside galax-ies and in the intergalactic medium, one could use for instancesilicon emission line ratios (Kwitter & Henry 1998; Henry et al.1996). For Mpc scales, or clusters, one may use the Sunyaev-Zeldovich (SZ) e ff ect (Rossetti et al. 2016). Nevertheless, allthese proposed approaches are quite speculative at this stage. Article number, page 9 of 10 & Aproofs: manuscript no. ISW_AA_Aug7
As for the ISW e ff ect, we investigate whether the galaxy-kSZ-kSZ correlation provides a good probe of the consistency re-lations. For the same symmetry reasons as in Section 4.4, wefind that the leading-order contribution to this three-point corre-lation vanishes. Let us briefly sketch how this cancellation ap-pears. First, from the hierarchy (63) we neglect the contributionassociated with the second term in Eq.(29), that is, the widths ofthe small-scale windows are small and we can approximate eachconical beam as a cylinder (flat-sky limit). Then, we only havethe component ξ k k similar to Eq.(51), which gives in compactnotations h δ ∆ kSZ ∆ kSZ i ∝ h ˜ δ ( k )[ n · ˜ p e ( k )][ n · ˜ p e ( k )] i ′ . (68)The consistency relation (4) gives at equal times h δ ∆ kSZ ∆ kSZ i ∝ n · k k h ˜ δ e ( k )[ n · ˜ p e ( k )] i ′ + n · k k h [ n · ˜ p e ( k )]˜ δ e ( k ) i ′ . (69)In the regime (59), we can take n ≃ n , hence h δ ∆ kSZ ∆ kSZ i ∝ n · k k n · h h ˜ δ e ( k ) ˜ p e ( − k ) i ′ + h ˜ p e ( k )˜ δ e ( − k ) i ′ i = . (70)Here we used the fact that the density-momentum crosspower spectrum obeys the symmetry h ˜ δ e ( k ) ˜ p e ( − k ) i ′ = −h ˜ δ e ( − k ) ˜ p e ( k ) i ′ , associated with a change of sign of the coor-dinate axis.This cancellation can again be understood in configura-tion space. At leading order in the squeezed limit, the linearchange of h p k ( θ ) p k ( θ ) i ′ due to a large-scale perturbation δ ( θ ) is( h δ p k i ′ + h p k δ i ′ ) v δ k , where v δ is the large-scale velocity gener-ated by the large-scale mode (the second-order term h + δ δ i v δ k does not contribute to the response function and the consistencyrelation). By symmetry the sum in the parenthesis vanishes.Therefore, in this paper we focus on the three-point correlations(49) and (64), with only one kSZ field.
6. Conclusions
In this paper, we have shown how to relate the large-scaleconsistency relations with observational probes. Assuming thestandard cosmological model (more specifically, the equiva-lence principle and Gaussian initial conditions), nonzero equal-times consistency relations involve the cross-correlations be-tween galaxy or matter density fields with the velocity, momen-tum, or time-derivative density fields. We have shown that theserelations can be related to actual measurements by consideringthe ISW and kSZ e ff ects, which indeed involve the time deriva-tive of the matter density field and the free electrons momen-tum field. We focused on the lowest-order relations, which applyto three-point correlation functions or bispectra, because higher-order correlations are increasingly di ffi cult to measure.The most practical relation obtained in this paper is prob-ably the one associated with the ISW e ff ect, more particularlyits cross-correlation with two cosmic weak-lensing convergencestatistics. Indeed, it allows one to write this three-point correla-tion function in terms of two matter density field power spectra(linear and nonlinear), which can be directly measured (e.g., bytwo-point weak lensing statistics). Moreover, the result, which is the leading-order contribution in the squeezed limit, showsa specific angular dependence as a function of the relative an-gular positions of the three smoothed observed statistics. Then,both the angular dependence and the quantitative prediction pro-vide a test of the consistency relation, that is, of the equivalenceprinciple and of primordial Gaussianity. If we consider insteadthe cross-correlation of the ISW e ff ect with two galaxy densityfields, we obtain a similar relation but it now involves the mixedgalaxy-matter density power spectrum P g, m and the large-scalegalaxy bias b g . These two quantities can again be measured (e.g.,by two-point galaxy-weak lensing statistics) and provide anothertest of the consistency relation.The relations obtained with the kSZ e ff ect are more intricate.They do not show a simple angular dependence, which wouldprovide a simple signature, and they involve the galaxy-free elec-trons or matter-free electrons power spectra. These power spec-tra are more di ffi cult to measure. One can estimate the free elec-tron density in specific regions, such as filaments or clusters,through X-ray or SZ observations, or around typical structuresby stacking analysis of clusters. This could provide an estimateof the free electrons cross power spectra and a check of the con-sistency relations. Although we can expect significant error bars,it would be interesting to check that the results remain consistentwith the theoretical predictions. A violation of these consistencyrelations would signal either a modification of gravity on cos-mological scales or non-Gaussian initial conditions. We leave tofuture works the derivation of the deviations associated with var-ious nonstandard scenarios. Acknowledgements.
This work is supported in part by the French Agence Na-tionale de la Recherche under Grant ANR-12-BS05-0002. DFM thanks the sup-port of the Research Council of Norway.
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