Imprints of local lightcone projection effects on the galaxy bispectrum. II
IImprints of local lightcone projection effects on the galaxy bispectrum. II
Sheean Jolicoeur a , Obinna Umeh a , Roy Maartens a,b and Chris Clarkson a,c,d a Department of Physics & Astronomy, University of the Western Cape, Cape Town 7535, South Africa b Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX, United Kingdom c School of Physics & Astronomy, Queen Mary University of London, London E1 4NS, United Kingdom d Department of Mathematics & Applied Mathematics, University of Cape Town, Cape Town 7701, South Africa (Dated: October 2, 2017)
General relativistic imprints on the galaxy bispectrum arise from observa-tional (or projection) effects. The lightcone projection effects include localcontributions from Doppler and gravitational potential terms, as well as lens-ing and other integrated contributions. We recently presented for the firsttime, the correction to the galaxy bispectrum from all local lightcone projec-tion effects up to second order in perturbations. Here we provide the detailsunderlying this correction, together with further results and illustrations. Formoderately squeezed shapes, the correction to the Newtonian prediction is ∼
30% on equality scales at z ∼
1. We generalise our recent results to includethe contribution, up to second order, of magnification bias (which affects someof the local terms) and evolution bias.
CONTENTS
I. Introduction 2II. Galaxy number counts in general relativity 4II.1. Local model of galaxy bias on ultra-large scales 5II.2. Observed galaxy number counts in Poisson gauge 7III. Galaxy number overdensity in Fourier space and the bispectrum 10IV. Numerical Results 13V. Conclusion 17A. Second-order gauge transformation of number density contrast 18B. Expansion of perturbed variables in Fourier space 19C. The coefficients in the GR kernel K (2)GR a r X i v : . [ a s t r o - ph . C O ] S e p I. INTRODUCTION
The galaxy power spectrum has been central to the cosmological constraints extracted from galaxy surveys upto now. For an accurate comparison of observations to theory, observational projection effects on the galaxy powerspectrum must be taken into account. The main projection effect comes from redshift-space distortions (RSD) [1–3],which must be included in the analysis of the power spectrum. But it is not only accuracy that is gained – there isadditional information to be extracted from the RSD themselves.In addition to RSD, the galaxy power spectrum is also affected by lensing magnification [4–6]. In the analysis ofcurrent surveys, the lensing contribution to galaxy number counts is typically not included in the power spectrum.For future surveys, which will probe higher redshifts, this lensing projection effect will need to be included in thegalaxy power spectrum for an accurate theoretical analysis – and, as with RSD, the lensing itself will deliver additionalinformation [7–9].Lensing convergence contributes a general relativistic (GR) projection effect, which is a correction to the Newtonian(overdensity + RSD) galaxy power spectrum. There are further GR projection effects which modify the galaxy powerspectrum on ultra-large scales ( H (cid:46) k (cid:46) k eq ) [10–12]. These include Doppler, Sachs-Wolfe, integrated Sachs-Wolfeand time-delay terms. As in the case of RSD and lensing, these terms need to be incorporated for accuracy, and theyalso contain extra information.The ultra-large scale GR corrections have a qualitatively similar effect on the galaxy power spectrum to primordialnon-Gaussianity (PNG), which also modifies the power spectrum on ultra-large scales via scale-dependent galaxybias. The GR corrections must therefore be taken into account when super-equality scales are probed to measure orconstrain the PNG parameter f NL [13–17].The galaxy bispectrum can provide additional information, partly independent of the power spectrum [18, 19]. Theeffects on the bispectrum from RSD have been computed in [20, 21] and from lensing in [22]. Recently, the galaxybispectrum has been used to detect the RSD and baryon acoustic oscillation (BAO) features in the BOSS survey, andto give independent measurements of growth rates and distances [23, 24].As in the case of the galaxy power spectrum, we need to take account of the observational lightcone effects in thegalaxy bispectrum which distort the information on the underlying dark matter distribution, but which also providenew information. These projection effects are the same as for the power spectrum – with one major difference: forthe bispectrum, we require the projection effects up to at least second order in perturbations.Next-generation galaxy surveys will enable increasingly accurate measurements of the galaxy bispectrum, outto higher redshifts and across larger sky areas. Recent forecasts, using a Newtonian model with RSD but no GRprojection effects, indicate that the bispectrum can considerably enhance the constraining power of future surveys [25]– especially for probing the initial conditions of the Universe via PNG. In order to fully exploit the improved precisionfrom upcoming surveys, we need theoretical accuracy that matches and moves beyond observational precision. Oneimportant part of this theoretical requirement is to include all the GR projection effects in modelling the galaxybispectrum.The GR lightcone effects on the galaxy angular bispectrum from lensing convergence were computed on intermediatescales in [26], neglecting the other, ultra-large scale, GR corrections to the galaxy overdensity. Another partial resultwas given in [27], using a separate-universe approximation to compute the galaxy angular bispectrum with all GRlightcone effects in the squeezed limit.We recently provided a further partial result, valid for all triangle shapes, by computing all the local GR projectioncorrections to the galaxy bispectrum, including all second order terms and couplings [28]. Crucial to our result isthe expression for the observed galaxy number counts on the past lightcone, up to second order. This is given in themost general case by [29] (see also [30–33]). Our work is complementary to the subsequent work by [34], who includelensing and terms of order ( H /k )[ δ (1) ] , but neglect all other GR effects on ultra-large scales.Here we provide details of the derivation of the results given in [28], with additional illustrations, and we generalisesome of those results. In particular, we include the magnification bias (which also contributes to local terms in thenumber counts) and the evolution bias. In [28], both of these were set to zero.We focus on large enough scales that perturbation theory is accurate, and we make the following assumptions: • A Gaussian primordial curvature perturbation. • A simple local-in-mass-density model of galaxy bias, as in [18, 19] (schematically, δ g ∼ b δ m + b δ m / • For simplicity, we use standard Newtonian results to evaluate the second-order velocity potential v (2) and metricpotentials Φ (2) , Ψ (2) , which contribute to the projection effects. • We neglect the second-order effect of the radiation era on initial conditions for sub-equality modes [35]. • We compute the galaxy bispectrum at fixed redshift and in Fourier space, and we use the plane-parallel ap-proximation. Consequently, the following are not included in our approach: wide-angle correlations, radialcorrelations, lensing and other integrated contributions.At second order in GR, scalar perturbations generate secondary vector and tensor modes [36, 37]. These modesalso enter the projection effects in the observed galaxy number density contrast at second order [29–33]. As shown by[38, 39] for vector modes and [40–42] for tensor modes, the power in the secondary vector and tensor modes is muchsmaller than the scalar power at second order, so we neglect the vector and tensor contributions.We adopt a standard concordance model, with parameters given by the latest Planck best-fit values [43]; in partic-ular, h = H / (100 km s − Mpc − ) = 0 .
678 and Ω m = 1 − Ω Λ0 = 0 . II. GALAXY NUMBER COUNTS IN GENERAL RELATIVITY
The observer looks down the past lightcone and counts d N galaxies, above a threshold luminosity L , within aredshift interval d z about the observed redshift z , and within a solid angle element dΩ o about the observed direction n , where [7, 11, 15, 29] d N ( z, n , > ln L ) = N ( z, n , > ln L ) D A ( z, n ) k µ u µ d λ d z d z dΩ o . (1)Here D A is the angular diameter distance, u µ is the 4-velocity of the source, k µ = d x µ / d λ is the geodesic photon4-momentum, and N is the flux-limited number density of sources: N ( z, n , > ln L ) = (cid:90) ∞ ln L d ln ˜ L n g ( z, n , ln ˜ L ) . (2)In the integrand, n g is the proper number density of sources, and only sources with luminosity above the detectionthreshold are counted by the observer.The fractional perturbation ∆ g of the observed number counts is defined byd N ( z, n , > ln L )d z dΩ o = χ ( z )(1 + z ) H ( z ) ¯ N ( z, > ln L ) (cid:2) g ( z, n , > ln L ) (cid:3) , (3)where H ( η ) = a (cid:48) ( η ) /a ( η ) is the conformal Hubble rate, the comoving line-of-sight distance is given by d χ = d z/ [(1 + z ) H ( z )], and ¯ N is the background magnitude-limited number density. Henceforth, we suppress the dependence of ∆ g on ln L to reduce clutter. We expand ∆ g up to second order in perturbation theory:∆ g ( z, n ) = ∆ (1) g ( z, n ) + 12 (cid:104) ∆ (2) g ( z, n ) − (cid:10) ∆ (2) g ( z, n ) (cid:11)(cid:105) , (4)where we subtract off the average of ∆ (2) g in order to ensure that (cid:104) ∆ g (cid:105) = 0. For later convenience, we split theobserved number density contrast into Newtonian and GR parts:∆ ( r ) g = ∆ ( r ) g N + ∆ ( r ) g GR , r = 1 , . (5)We only consider the bispectrum at fixed redshift, so that all correlations are in the same redshift bin. There areintegrated GR contributions to ∆ (1) g , from weak lensing convergence and also from integrated Sachs-Wolfe and time-delay terms, and we neglect these terms. At second order, there are many more terms with line-of-sight integratedcontributions, and we neglect all such terms. Specifically, we neglect the integrated contributions in [29], which givesthe fully general ∆ (1) g and ∆ (2) g in Poisson gauge. A complete treatment would include the integrated terms, with allcross-bin correlations. This far more complicated analysis is left for future work.An important point to note is that the GR weak lensing convergence consists not only of the standard integratedterm, but also includes local (non-integrated) terms [44]. This means that the magnification bias will still enter thebispectrum, even if we neglect all integrated terms. The magnification bias is given by the logarithmic slope of thebackground number density at the threshold luminosity: Q ( a, ¯ L ) = − ∂ ln (cid:2) a ¯ N ( a, > ¯ L ) (cid:3) ∂ ln ¯ L . (6)We have used the comoving number density in the definition above since it arises also in the definition of the evolutionbias: b e ( a, ¯ L ) = ∂ ln (cid:2) a ¯ N ( a, > ¯ L ) (cid:3) ∂ ln a . (7)This quantity describes the deviation of the background number density of sources from the idealised case of a ¯ N = ¯ N .Radial and transverse derivatives are defined as ∂ (cid:107) = n i ∂ i , ∂ ⊥ i = ∂ i − n i ∂ (cid:107) , (8) We also neglect all terms at the observer, which do not contribute to the bispectrum. the derivative down rays of the past lightcone isdd χ = − dd η = − ∂ η + ∂ (cid:107) , (9)and the screen space projected Laplacian is ∇ ⊥ = ∇ − ∂ (cid:107) − χ ∂ (cid:107) . (10)Since ∆ g is defined as an observable, it is gauge-independent and we can use any gauge to compute it. In a givengauge, it will be of the form ∆ g = δ g + terms that describe projection effects in that gauge, where δ g = δ N / ¯ N = δ (1) g + δ (2) g is the galaxy number density contrast in the chosen gauge. We choose the Poisson gauge since it isconvenient for splitting into Newtonian and GR parts. Neglecting the vector and tensor modes, the metric and thepeculiar velocity of galaxies (equal to the dark matter velocity on the scales of interest) are given by a − d s = − (cid:104) (1) + Φ (2) (cid:105) d η + (cid:104) − (1) − Ψ (2) (cid:105) d x , (11) v i = ∂ i (cid:104) v (1) + 12 v (2) (cid:105) . (12)The observed comoving coordinates [30] of a galaxy are x = χ ( z ) n = [ η − η ( z )] n . We have assumed that anisotropicstress vanishes at first order, which implies Ψ (1) = Φ (1) in GR.We will also use the comoving-synchronous (C) overdensities of matter and galaxy counts δ m C , δ g C . The first-orderPoisson and continuity equations are then ∇ Φ (1) = 32 Ω m H δ (1) m C , δ (1) (cid:48) m C = −∇ v (1) , (13)which lead to Φ (1) = −
32 Ω m H k δ (1) m C where Φ (1) ( a, k ) = D ( a ) a Φ (1) (1 , k ) , (14) H v (1) = f H k δ (1) m C where f = d ln D d ln a and δ (1) m C ( a, k ) = D ( a ) δ (1) m C (1 , k ) . (15) II.1. Local model of galaxy bias on ultra-large scales
We start by considering the Poisson-gauge number density contrast δ (1) g at linear order, which is related to the darkmatter density contrast δ (1) m via the galaxy bias. We need to ensure that the definition of scale-independent galaxybias is gauge-independent and valid on ultra-large scales. As explained in detail in [11, 13, 15], the physical definitionof scale-independent bias is in the matter rest-frame, which coincides with the galaxy rest-frame (on large scales thereis no velocity bias). The matter rest-frame corresponds to the C gauge, so that the correct definition at first order is(restoring the dependence on L ): δ (1) g C ( a, x , < ln L ) = b ( a, ln ¯ L ) δ (1) m C ( a, x ) . (16)The Poisson-gauge number density contrast is related to the C-gauge one by [11] δ (1) g = δ (1) g C + (3 − b e ) H v (1) = b δ (1) m C + (3 − b e ) H v (1) . (17)The velocity potential term in (17) ensures gauge-independence of the bias model on ultra-large scales. This term isthe GR part of δ (1) g , since it is suppressed on small scales but grows on ultra-large scales, as shown by (15).In GR, the Lagrangian frame corresponds to the C gauge [45, 46]. There is no unique Eulerian frame in GR, buta convenient choice is the total-matter (T) gauge. This is related to the C gauge by a purely spatial transformation,so that at first order, the matter and galaxy overdensities are the same [45]: δ (1) m C = δ (1) m T , δ (1) g C = δ (1) g T = b δ (1) m T . (18)The last equality is the definition of the Eulerian bias parameter at first order. This means that b in (16) is theEulerian bias parameter.We extend (16) to higher order with the simplest possible model of scale-independent bias. This model assumesthat galaxy number density contrast is a local function of only the matter density contrast – the so-called local-in-mass-density model. For a physical definition valid on ultra-large scales, we require that the bias coefficients arescale-independent in the galaxy rest-frame, i.e. in C gauge. Expanding in powers of the mass density contrast, wehave δ g C = b δ m C + 12 b (cid:0) δ m C (cid:1) + · · · , (19)where b I = b I ( a, ln L ). At first order, this recovers (16). At second order we have: δ (2) g C = b δ (2) m C + b (cid:2) δ (1) m C (cid:3) . (20)The relation between C- and T-gauge matter overdensities at second order is [45, 46] δ (2) m T = δ (2) m C + 2 (cid:2) ∂ i δ (1) m C (cid:3) ∇ − ∂ i δ (1) m C , (21)where − ∇ − ∂ i δ (1) mC is a gauge generator. Since the C → T gauge transformation is purely spatial, (21) also appliesto the galaxy counts: δ (2) g T = δ (2) g C + 2 (cid:2) ∂ i δ (1) g C (cid:3) ∇ − ∂ i δ (1) mC . (22)From (20)–(22), using (18), we find that δ (2) g T = b δ (2) m C + b (cid:2) δ (1) m C (cid:3) + 2 b (cid:2) ∂ i δ (1) m C (cid:3) ∇ − ∂ i δ (1) mC = b (cid:104) δ (2) m C + 2 (cid:2) ∂ i δ (1) m T (cid:3) ∇ − ∂ i δ (1) mC (cid:105) + b (cid:2) δ (1) m C (cid:3) (23)which implies δ (2) g T = b δ (2) m T + b (cid:2) δ (1) m T (cid:3) . (24)Therefore local-in-mass-density and scale-independent bias in C and T gauge are equivalent up to second order, withthe same Eulerian bias coefficients.We will use the T gauge, since the relation to the Poisson gauge overdensity is simpler for T gauge than C gauge.In Appendix A, we show that δ (2) g = δ (2) g T + (3 − b e ) H v (2) + 2(3 − b e ) H v (1) δ (1) g T − v (1) δ (1) (cid:48) g T + (cid:104) ( b e − H (cid:48) + b (cid:48) e H + ( b e − H (cid:105)(cid:2) v (1) (cid:3) + ( b e − H v (1) v (1) (cid:48) − ( b e − H∇ − (cid:20) v (1) ∇ v (1) (cid:48) − v (1) (cid:48) ∇ v (1) − ∂ i Φ (1) ∂ i v (1) − (1) ∇ v (1) (cid:21) . (25)By (16) and (24), this leads to the final expression for the Poisson-gauge galaxy density contrast in the simplest localbias model: δ (2) g = b δ (2) m T + b (cid:2) δ (1) m T (cid:3) + (cid:104) ( b e − H + b (cid:48) e H + ( b e − H (cid:48) (cid:105)(cid:2) v (1) (cid:3) + ( b e − H v (1) v (1) (cid:48) + 2 b (3 − b e ) H v (1) δ (1) m T − v (1) (cid:104) b δ (1) (cid:48) m T + b (cid:48) δ (1) m T (cid:105) + (3 − b e ) H∇ − (cid:20) v (1) ∇ v (1) (cid:48) − v (1) (cid:48) ∇ v (1) − ∂ i Φ (1) ∂ i v (1) − (1) ∇ v (1) (cid:21) . (26)The velocity and metric potential terms ensure gauge-independence on ultra-large scales. Equation (26) is the second-order generalisation of (17). For convenience, we have omitted the term − b (cid:10)(cid:2) δ (1) m C (cid:3) (cid:11) on the right of (20). II.2. Observed galaxy number counts in Poisson gauge
At first order, we replace δ (1) g using the bias relations (16)–(18), and then split ∆ (1) g into Newtonian and GR parts:∆ (1) g N = b δ (1) m T − H ∂ (cid:107) v (1) , (27)∆ (1) g GR = (cid:20) b e − Q + 2 ( Q − χ H − H (cid:48) H (cid:21) (cid:104) ∂ (cid:107) v (1) − Φ (1) (cid:105) + (2 Q −
1) Φ (1) + 1 H Φ (1) (cid:48) + (3 − b e ) H v (1) . (28)The Newtonian part = T-gauge density contrast + Kaiser RSD, and the GR part = Doppler + potential + velocitypotential. The velocity potential arises from the term in (17), which may be expressed in terms of the metric potentialvia (14) and (15). The Doppler term in (28) is the one proportional to the line-of-sight velocity ∂ (cid:107) v (1) .At second order, we use the gauge-independent bias model (26) to replace the Poisson-gauge δ (2) g term in ∆ (2) g .The remaining terms in ∆ (2) g are second-order generalisations of RSD, Doppler and potential terms, together withquadratic couplings amongst all the first-order terms. The quadratic terms encode an interaction between two effects;in Fourier space, they correspond to mode coupling.The general equation for ∆ (2) g , including evolution bias and magnification bias, as well as all integrated effects, isgiven in [29] (including recent corrections [47]). We include in this general expression our gauge-independent modelof the galaxy bias at second order, (26), and we neglect the terms with integrated contributions. The result is∆ (2) g = b δ (2) m T + b (cid:2) δ (1) m T (cid:3) + (cid:104) ( b e − H + b (cid:48) e H + ( b e − H (cid:48) (cid:105)(cid:2) v (1) (cid:3) + ( b e − H v (1) v (1) (cid:48) + 2 b (3 − b e ) H v (1) δ (1) m T − v (1) (cid:104) b δ (1) (cid:48) m T + b (cid:48) δ (1) m T (cid:105) + (3 − b e ) H∇ − (cid:20) v (1) ∇ v (1) (cid:48) − v (1) (cid:48) ∇ v (1) − ∂ i Φ (1) ∂ i v (1) − (1) ∇ v (1) (cid:21) − H ∂ (cid:107) v (2) + (3 − b e ) H v (2) + (cid:20) b e − Q − − Q ) χ H − H (cid:48) H (cid:21) (cid:104) ∂ (cid:107) v (2) − Φ (2) (cid:105) + 2( Q − (2) + Φ (2) + 1 H Ψ (2) (cid:48) + (cid:20) b e − Q − H (cid:48) H − (1 − Q ) 2 χ H (cid:21) (cid:20) (cid:2) Φ (1) (cid:3) − (cid:2) ∂ (cid:107) v (1) (cid:3) + ∂ ⊥ i v (1) ∂ i ⊥ v (1) − ∂ (cid:107) v (1) Φ (1) − H (cid:16) Φ (1) − ∂ (cid:107) v (1) (cid:17) (cid:16) Φ (1) (cid:48) − ∂ (cid:107) v (1) (cid:17) (cid:21) + 2 (2 Q −
1) Φ (1) δ (1) g − H δ (1) g ∂ (cid:107) v (1) + 2 H δ (1) g Φ (1) (cid:48) + (cid:18) Q − Q − ∂ Q ∂ ln ¯ L (cid:19) (cid:2) Φ (1) (cid:3) + 2 H (cid:18) Q + H (cid:48) H (cid:19) Φ (1) Φ (1) (cid:48) − H (cid:18) Q + H (cid:48) H (cid:19) Φ (1) ∂ (cid:107) v (1) + 2 H (cid:2) Φ (1) (cid:48) (cid:3) + 2 H (cid:2) ∂ (cid:107) v (1) (cid:3) + 2 H ∂ (cid:107) v (1) ∂ (cid:107) Φ (1) + 4 H ∂ (cid:107) v (1) ∂ (cid:107) Φ (1) − H Φ (1) ∂ (cid:107) v (1) − H Φ (1) ∂ (cid:107) Φ (1) + 2 H Φ (1) dΦ (1) (cid:48) d χ − H ∂ (cid:107) v (1) dΦ (1) (cid:48) d χ + 2 H (cid:18) H (cid:48) H (cid:19) ∂ (cid:107) v (1) ∂ (cid:107) v (1) − H Φ (1) ∂ (cid:107) Φ (1) + 2 H (cid:18) − H (cid:48) H (cid:19) ∂ (cid:107) v (1) Φ (1) (cid:48) − H ∂ (cid:107) v (1) Φ (1) (cid:48) + 2 H ∂ ⊥ i v (1) ∂ i ⊥ Φ (1) − H ∂ ⊥ i v (1) ∂ i ⊥ ∂ (cid:107) v (1) + (cid:18) χ H − (cid:19) ∂ ⊥ i v (1) ∂ i ⊥ v (1) + 2 H ∂ (cid:107) v (1) ∂ (cid:107) v (1) + (cid:40)(cid:20) b e Q − b e − Q − Q + 8 ∂ Q ∂ ln ¯ L + 4 ∂ Q ∂ ln ¯ a +2 H (cid:48) H (1 − Q ) + 4 χ H (cid:18) Q − Q − ∂ Q ∂ ln ¯ L (cid:19) (cid:21) Φ (1) + 2 (cid:20) b e − Q − H (cid:48) H − χ H (1 − Q ) (cid:21) δ (1) g − H d δ (1) g d χ + 2 H (cid:20) Q − b e + H (cid:48) H + 2 χ H (1 − Q ) (cid:21) ∂ (cid:107) v (1) + 2 H (cid:20) b e − − χ H (1 − Q ) − H (cid:48) H (cid:21) Φ (1) (cid:48) − H Q ∂ (cid:107) Φ (cid:41) (cid:104) ∂ (cid:107) v (1) − Φ (1) (cid:105) + (cid:40) b e − b e + ∂b e ∂ ln ¯ a + 6 Q − Q b e + 4 Q − ∂ Q ∂ ln ¯ L − ∂ Q ∂ ln ¯ a + 6 χ H (cid:48) H (1 − Q ) + (1 − b e + 4 Q ) H (cid:48) H − H (cid:48)(cid:48) H + 3 H (cid:48) H + 2 χ H (cid:18) − Q + 2 Q − ∂ Q ∂ ln ¯ L (cid:19) + 2 χ H (cid:20) − b e − Q +2 b e Q − Q + 4 ∂ Q ∂ ln ¯ L + 2 ∂ Q ∂ ln ¯ a (cid:21)(cid:41) (cid:104) ∂ (cid:107) v (1) − Φ (1) (cid:105) + 4 (cid:20)(cid:18) − χ H (cid:19) ∂ (cid:107) v (1) − (cid:18) − χ H (cid:19) Φ (1) (cid:21) ∂δ (1) g ∂ ln ¯ L . (29)The Newtonian part of (29) is formed from the density contrast and Kaiser RSD terms and their couplings: ∆ (2) g N = b δ (2) m T + b (cid:2) δ (1) m T (cid:3) − H ∂ (cid:107) v (2) − b H (cid:20) δ (1) mT ∂ (cid:107) v (1) + ∂ (cid:107) v (1) ∂ (cid:107) δ (1) mT (cid:21) + 2 H (cid:20)(cid:2) ∂ (cid:107) v (1) (cid:3) + ∂ (cid:107) v (1) ∂ (cid:107) v (1) (cid:21) . (30)The remaining terms form the GR correction:∆ (2) g GR = H (3 − b e ) v (2) + (cid:104) (9 − b e + b e ) H + b (cid:48) e H + ( b e − H (cid:48) (cid:105)(cid:2) v (1) (cid:3) + ( b e − H v (1) v (1) (cid:48) − ( b e − H∇ − (cid:20) v (1) ∇ v (1) (cid:48) − v (1) (cid:48) ∇ v (1) − ∂ i Φ (1) ∂ i v (1) − (1) ∇ v (1) (cid:21) + 2(3 − b e ) b H v (1) δ (1) m T − v (1) (cid:16) b (cid:48) δ (1) m T + b δ (1) (cid:48) m T (cid:17) + (cid:20) b e − Q − − Q ) χ H − H (cid:48) H (cid:21) ∂ (cid:107) v (2) + (cid:20) − b e + 2 Q + 2(1 − Q ) χ H + H (cid:48) H (cid:21) Φ (2) − − Q )Ψ (2) + 1 H Ψ (2) (cid:48) + 2 H (cid:20) b δ (1) (cid:48) m T ∂ (cid:107) v (1) + ( f − Q )Φ (1) ∂ (cid:107) Φ (1) + (2 − f − Q ) ∂ (cid:107) v (1) ∂ (cid:107) Φ (1) − b Φ (1) δ (1) (cid:48) m T + b Φ (1) ∂ (cid:107) δ (1) m T − ∂ i v (1) ∂ (cid:107) ∂ i v (1) + ∂ i v (1) ∂ i Φ (1) (cid:21) + 2 H (cid:20) ∂ (cid:107) v (1) ∂ (cid:107) Φ (1) − Φ (1) ∂ (cid:107) Φ (1) − Φ (1) ∂ (cid:107) v (1) (cid:21) − − b e ) v (1) ∂ (cid:107) v (1) + 2 (cid:20) b (cid:18) b e − Q − − Q ) χ H − H (cid:48) H (cid:19) + b (cid:48) H + 2 (cid:18) − χ H (cid:19) ∂b ∂ ln ¯ L (cid:21) δ (1) m T ∂ (cid:107) v (1) + 2 H (cid:20) − b e + 4 Q + 4(1 − Q ) H χ + 3 H (cid:48) H (cid:21) ∂ (cid:107) v (1) ∂ (cid:107) v (1) + 2 (cid:20) b (cid:18) f − − b e + 4 Q + 2(1 − Q ) χ H + H (cid:48) H (cid:19) − b (cid:48) H− (cid:18) − χ H (cid:19) ∂b ∂ ln ¯ L (cid:21) Φ (1) δ (1) m T + (cid:20) b e − − Q − − Q ) χ H − H (cid:48) H (cid:21) ∂ i v (1) ∂ i v (1) + 2 H (cid:20) − f + 2 b e − Q− − Q ) χ H − H (cid:48) H (cid:21) Φ (1) ∂ (cid:107) v (1) + A (cid:2) Φ (1) (cid:3) + B v (1) ∂ (cid:107) v (1) + C Φ (1) v (1) + D Φ (1) ∂ (cid:107) v (1) + E (cid:2) ∂ (cid:107) v (1) (cid:3) . (31)The background coefficients in the last line are A = − f (cid:18) − b e + 4 Q + 4(1 − Q ) χ H + 2 H (cid:48) H (cid:19) − f (cid:48) H + b e + 6 b e − b e Q + 4 Q + 16 Q − ∂ Q ∂ ln ¯ L − Q (cid:48) H + b (cid:48) e H + 2 χ H (cid:18) − Q + 2 Q − ∂ Q ∂ ln ¯ L (cid:19) − χ H (cid:20) b e − b e Q − Q + 8 Q − H (cid:48) H (1 − Q ) − ∂ Q ∂ ln ¯ L − Q (cid:48) H (cid:21) + H (cid:48) H (cid:18) − − b e + 8 Q + 3 H (cid:48) H (cid:19) − H (cid:48)(cid:48) H , (32) B = 2 H (cid:20) − b e + 2 b e (1 − Q ) χ H − b e + 2 b e Q − Q − b (cid:48) e H − − Q ) χ H + 2 (cid:18) − χ H (cid:19) Q (cid:48) H (cid:21) , (33) C = 2 H (cid:20) − f (3 − b e ) − b e − b e (1 − Q ) χ H + b (cid:48) e H + b e − b e Q + 12 Q + 6(1 − Q ) χ H − (cid:18) − χ H (cid:19) Q (cid:48) H (cid:21) , (34) D = 4 + 2 f (cid:20) − f + 2 b e − Q − − Q ) χ H − H (cid:48) H (cid:21) + 2 f (cid:48) H − b e − b e + 12 b e Q − Q − Q + 16 ∂ Q ∂ ln ¯ L +12 Q (cid:48) H − b (cid:48) e H − χ H (cid:18) − Q + 2 Q − ∂ Q ∂ ln ¯ L (cid:19) − χ H (cid:20) − − b e + 2 b e Q + Q − Q + 3 H (cid:48) H (1 − Q )+6 ∂ Q ∂ ln ¯ L + 2 Q (cid:48) H (cid:21) + 2 H (cid:48) H (cid:18) b e − Q − H (cid:48) H (cid:19) + 2 H (cid:48)(cid:48) H , (35) E = − − b e + b e − b e Q + 6 Q + 4 Q − ∂ Q ∂ ln ¯ L − Q (cid:48) H + b (cid:48) e H + 2 χ H (cid:18) − Q + 2 Q − ∂ Q ∂ ln ¯ L (cid:19) + 2 χ H (cid:20) − b e + 2 b e Q − Q − Q + 3 H (cid:48) H (1 − Q ) + 4 ∂ Q ∂ ln ¯ L + 2 Q (cid:48) H (cid:21) + H (cid:48) H (cid:18) − b e + 4 Q + 3 H (cid:48) H (cid:19) − H (cid:48)(cid:48) H . (36) Note that the GR correction to δ (2) m T does not enter the bias term b δ (2) m T , as explained in [48–50]. There is a GR correction to v (2) ,which we neglect here. In deriving (30)–(36) from (29), we used the following:(a) eliminate d / d χ using (9), and ∂ ⊥ i using (8);(b) show, using the commutator relation (cid:2) ∂ ⊥ i , ∂ (cid:107) (cid:3) = χ − ∂ ⊥ i , that ∂ ⊥ i v (1) ∂ i ⊥ ∂ (cid:107) v (1) = ∂ i v (1) ∂ (cid:107) ∂ i v (1) − ∂ (cid:107) v (1) ∂ (cid:107) v (1) + 1 χ (cid:104) ∂ i v (1) ∂ i v (1) − (cid:2) ∂ (cid:107) v (1) (cid:3) (cid:105) ; (37)(c) express δ (1) g in terms of δ (1) m T and v (1) , using (17) and (18);(d) rewrite the term from the perturbation of the magnification bias, using (16)–(18), as ∂δ (1) g ∂ ln ¯ L = ∂b ∂ ln ¯ L δ (1) m T − ∂b e ∂ ln ¯ L H v (1) = ∂b ∂ ln ¯ L δ (1) m T + Q (cid:48) v (1) , (38)where the second equality uses (6), (7) and ∂/∂ ln a = H − ∂/∂η .In summary: we have used the general formula for ∆ (2) g in Poisson gauge, given in [29], neglecting the terms withline-of-sight integrals, to derive (30)–(36). In these equations we have broken down the highly complex formula in[29] into simple parts, facilitating analytical and then numerical analysis. Our new contribution is to determine thePoisson-gauge δ (2) g via a simple local-in-mass-density model of bias (26), that is gauge independent and valid onultra-large scales. Three groups have computed ∆ (2) g – in [29–31], [32] and [33]. All have used different formalisms. The collective task of cross-checkingthese independent results has been initiated but is not complete, even in the simplest case with no integrated contributions and b e = 0 = Q . III. GALAXY NUMBER OVERDENSITY IN FOURIER SPACE AND THE BISPECTRUM
We will only consider correlations at the same observed redshift. At fixed redshift z , the perturbative variablesdepend on n and can be computed in Fourier space at fixed η ( z ). With n and z fixed, we transform x = [ η − η ( z )] n + x → k , which is equivalent to transforming over all observer positions x . Our Fourier convention is f ( x ) = (cid:90) d k (2 π ) e i k · x f ( k ) , f ( k ) = (cid:90) d x e − i k · x f ( x ) = (cid:90) d k (cid:48) (2 π ) (2 π ) δ D ( k − k (cid:48) ) f ( k (cid:48) ) , (39)where we suppress the redshift dependence. The transform of a product h ( x ) = g ( x ) f ( x ) leads to a convolution inFourier space h ( k ) = (cid:90) d k (2 π ) d k (2 π ) f ( k ) g ( k )(2 π ) δ D ( k + k − k ) . (40)For notational convenience we write the T-gauge matter density contrast as δ m T ≡ δ = δ (1) + 12 δ (2) , (41)from now on.At second order, the matter density contrast and the velocity and metric potentials are given in a Newtonianapproximation by [51]: δ (2) ( k ) = (cid:90) d k (2 π ) d k (2 π ) δ (1) ( k ) δ (1) ( k ) F ( k , k )(2 π ) δ D ( k + k − k ) , (42) v (2) ( k ) = f H k (cid:90) d k (2 π ) d k (2 π ) δ (1) ( k ) δ (1) ( k ) G ( k , k )(2 π ) δ D ( k + k − k ) , (43)Φ (2) ( k ) = Ψ (2) ( k ) = −
32 Ω m H k δ (2) ( k ) . (44)The kernels for the dark matter and peculiar velocity perturbations in a matter-dominated model are F ( k , k ) = 107 + k · k k k (cid:18) k k + k k (cid:19) + 47 (cid:18) k · k k k (cid:19) , (45) G ( k , k ) = 67 + k · k k k (cid:18) k k + k k (cid:19) + 87 (cid:18) k · k k k (cid:19) . (46)The corrections to these kernels from the presence of Λ are small [35], and we neglect them. Within the sameapproximation, we have δ (2) ∝ D δ (2)0 , so that δ (2) (cid:48) = 2 f H δ (2) . Then it follows from (44) thatΦ (2) (cid:48) = (2 f − H Φ (2) . (47)We write ∆ (1 , g in terms of kernels:∆ (1) g ( k ) = (cid:90) d k (2 π ) K (1) ( k ) δ (1) ( k )(2 π ) δ D ( k − k ) , (48)∆ (2) g ( k ) = (cid:90) d k (2 π ) d k (2 π ) K (2) ( k , k , k ) δ (1) ( k ) δ (1) ( k )(2 π ) δ D ( k + k − k ) − δ ( D ) ( k ) (cid:10) ∆ (2) g (cid:11) , (49)and we split the kernels into Newtonian and GR parts, K (1 , = K (1 , + K (1 , . In (49), we subtracted off the ensembleaverage of ∆ g : (cid:10) ∆ (2) g (cid:11) = (cid:90) d k (2 π ) P ( k ) K (2) ( k , − k , , (50)in order to ensure that (cid:104) ∆ g (cid:105) = 0. Here P ( k ) ≡ P δ (1) ( k ) is the linear matter power spectrum.1By (27) and (28), the linear order kernel is given by K (1)N ( k ) = b + f µ , K (1)GR ( k ) = i µk γ + γ k , µ = ˆ k · n , (51)where γ and γ are redshift dependent: γ H = f (cid:20) b e − Q − − Q ) χ H − H (cid:48) H (cid:21) , (52) γ H = f (3 − b e ) + 32 Ω m (cid:20) b e − f − Q − − Q ) χ H − H (cid:48) H (cid:21) . (53)At second order, the Newtonian part of the kernel is K (2)N ( k , k , k ) = b F ( k , k ) + b + f G ( k , k ) µ + f µ µ k k (cid:0) µ k + µ k (cid:1) + b fk k (cid:104)(cid:0) µ + µ (cid:1) k k + µ µ (cid:0) k + k (cid:1)(cid:105) , (54)where µ i = ˆ k i · n . The second line in (54) is the nonlinear Kaiser RSD contribution [20, 21].The GR part follows from (31), after transformation to Fourier space. The details, with all the necessary transforms,are given in Appendix B, and they lead to the GR kernel: K (2)GR ( k , k , k ) = 1 k k (cid:26) Γ + i ( µ k + µ k ) Γ + k k k (cid:104) F ( k , k ) Γ + G ( k , k ) Γ (cid:105) + ( µ µ k k ) Γ + ( k · k ) Γ + (cid:0) k + k (cid:1) Γ + (cid:0) µ k + µ k (cid:1) Γ + i (cid:20) (cid:0) µ k + µ k (cid:1) Γ + ( µ k + µ k ) ( k · k ) Γ + k k ( µ k + µ k ) Γ + (cid:0) µ k + µ k (cid:1) Γ + µ µ k k ( µ k + µ k ) Γ + µ k k k G ( k , k ) Γ (cid:21)(cid:27) , (55)where the Γ I ( z ) are given in Appendix C.We have ordered the Γ I according to the powers of H /k , starting with the O ( H /k ) term and ending with the O ( H /k ) terms. This is our key result – transforming the highly complicated second-order GR projection correctionsgiven by (31) into a manageable Fourier-space kernel (55). In the special case b e = 0 = Q , (31) reduces to the formgiven in [28]. When b e , Q are nonzero, the Γ I become much more complicated.In Fourier space, the observed galaxy bispectrum B g at fixed redshift is given by (cid:10) ∆ g ( k )∆ g ( k )∆ g ( k ) (cid:11) = (2 π ) B g ( k , k , k ) δ D ( k + k + k ) . (56)At second order, the only combinations of terms that contribute at tree-level are2 (cid:10) ∆ g ( k )∆ g ( k )∆ g ( k ) (cid:11) = (cid:10) ∆ (1) g ( k )∆ (1) g ( k )∆ (2) g ( k ) (cid:11) + 2 cyc. perm. (57)= (cid:10) ∆ (1) g N ( k )∆ (1) g N ( k )∆ (2) g N ( k ) (cid:11) + (cid:10) ∆ (1) g GR ( k )∆ (1) g GR ( k )∆ (2) g GR ( k ) (cid:11) + (cid:10) ∆ (1) g N ( k )∆ (1) g N ( k )∆ (2) g GR ( k ) (cid:11) + (cid:10) ∆ (1) g GR ( k )∆ (1) g GR ( k )∆ (2) g N ( k ) (cid:11) + 2 (cid:104)(cid:10) ∆ (1) g N ( k )∆ (1) g GR ( k )∆ (2) g N ( k ) (cid:11) + (cid:10) ∆ (1) g N ( k )∆ (1) g GR ( k )∆ (2) g GR ( k ) (cid:11)(cid:105) + 2 cyc. perm. , (58)where the factors of 2 arise from the factor 1 / g . In the second equality, wehave further separated the bispectrum into purely Newtonian and purely GR parts (first line), and cross-correlationsbetween Newtonian and GR terms (following lines). The cross-correlation terms become important on smaller scalesthan the pure GR term.The full expression for the galaxy bispectrum in terms of kernels follows from (58) as: B g ( k , k , k ) = (cid:20) K (1)N ( k ) K (1)N ( k ) K (2)N ( k , k , k ) + K (1)GR ( k ) K (1)GR ( k ) K (2)GR ( k , k , k )+ K (1)N ( k ) K (1)N ( k ) K (2)GR ( k , k , k ) + K (1)GR ( k ) K (1)GR ( k ) K (2)N ( k , k , k )+ 2 K (1)N ( k ) K (1)GR ( k ) (cid:110) K (2)N ( k , k , k ) + K (2)GR ( k , k , k ) (cid:111) (cid:21) P ( k ) P ( k ) + 2 cyc. perm. (59)2The bispectrum in the Newtonian approximation is B g N ( k , k , k ) = K (1)N ( k ) K (1)N ( k ) K (2)N ( k , k , k ) P ( k ) P ( k ) + 2 cyc. perm. (60)All other terms in (59) are GR corrections, i.e., they vanish if the GR projection effects are neglected.Calculation of the galaxy bispectrum including all the GR terms leads to a complex-valued function. We split (59)into real and imaginary parts B g = B R g + i B I g and compute the absolute value of the galaxy bispectrum, given by | B g | = ( B R g ) + ( B I g ) .There are four different angles implicit in (59):three θ i between the observer line of sight and the mode vectors (with cosines µ i = cos θ i = ˆ k i · n )+ one of the angles θ ij between k i and k j (with cosines µ ij = cos θ ij = ˆ k i · ˆ k j ).Two of the µ i are independent, since µ k + µ k + µ k = 0, where k = | k + k | . Two of the µ ij can be determinedby the third via trigonometric identities. Finally, one of the two remaining µ i may be expressed in terms of the otherone and the choice of independent µ ij , using the trigonometric addition formula. If we choose µ and µ , then µ = µ µ ± (cid:113) − µ (cid:113) − µ cos φ , (61)where µ can be determined from the k i . Here φ is the azimuthal angle, characterizing the orientation of the trianglein Fourier space, and the ± arises due to invariance under reflection of n about ˆ k in their plane.Implementing these conditions, the galaxy bispectrum is a function of µ and φ , together with the magnitudes ofthe three mode vectors. The dependence of B g on µ and φ may be expanded in spherical harmonics: B g ( k , k , k , µ , φ ) = (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) B (cid:96)mg ( k , k , k ) Y (cid:96)m ( µ , φ ) , (62)where the multipoles of B g are given by B (cid:96)mg ( k , k , k ) = (2 (cid:96) + 1)4 π (cid:90) π d φ (cid:90) − d µ B g ( k , k , k , µ , φ ) Y ∗ (cid:96)m ( µ , φ ) . (63)This can be compared to the Legendre multipole expansion of the galaxy power spectrum P g ( k, µ ) = (cid:96) max (cid:88) (cid:96) =0 P (cid:96)g ( k ) L (cid:96) ( µ ) with P (cid:96)g ( k ) = (2 (cid:96) + 1)2 (cid:90) − d µ P g ( k, µ ) L (cid:96) ( µ ) . (64)Note that we can also expand the bispectrum in Associated Legendre polynomials and still recover the multipoles asgiven in (63).Typically, only the m = 0 multipoles of B g are considered, and we will do this, so that B g = B g ( k , k , k , µ ). Infact, this does not lose much information [52]. For the monopole, we use the shorthand B g ≡ B g .3 IV. NUMERICAL RESULTS
In order to illustrate quantitatively the imprint of GR effects on the galaxy bispectrum, we specialise to an isoscelesconfiguration, with k = k ≡ k, k = k (cid:112) µ ) . (65)We evaluate the following cases:radial: µ = 1 → B (cid:107) g , transverse: µ = 0 → B ⊥ g , monopole: (cid:90) d µ → B g . (66)For redshifts and astrophysical parameters, we choose: z = 1 . , . , b ( z ) = √ z, b ( z ) = − . √ z, b e = 0 = Q , (67)where the galaxy bias parameters are similar to [53].In each case, we compare the Newtonian prediction (60) for the galaxy bispectrum, to the GR prediction (59). Weconsider the galaxy bispectrum B g as a function of triangle size for two isosceles shapes. We fix µ = cos θ andvary k , for two special cases:equilateral: µ = − , moderately squeezed: µ = − . ⇒ k ≈ k . (68)Figure 1 shows the radial, transverse and monopole parts of B g , together with the percentage correction relativeto the Newtonian case without the GR projection effects, on scales 0 . ≤ k ≤ .
1, which includes BAO scales. In allcases, as expected, the GR corrections become increasingly important on larger scales. The squeezed configuration hasa larger correction than the equilateral. For the monopole, the GR correction at equality scales reaches O (30 − z ∼ − .
5, and then grows larger. Note that when the short modes are equality scale, the long mode is still withinthe Hubble horizon: k ∼ k eq ⇒ k ∼ k eq ∼ H . (69)On the largest scales, our results need to be corrected for wide-angle correlations that are absent in the plane-parallelapproximation.It is interesting to identify the various contributions to the galaxy bispectrum monopole in Fig. 1. We do this intwo ways, as illustrated in Fig. 2, for the moderately squeezed (left) and equilateral (right) shapes, at z = 1 . • In the top panel, we show the contributions from the various 3-point correlations (cid:10) ∆ g ( k )∆ g ( k )∆ g ( k ) (cid:11) , asgiven in (58).The pure Newtonian correlation gives the standard curve (dashed, black). The 5 solid curves are the correlationswith GR corrections: 1 pure GR correlation (red), which dominates on horizon scales, and 4 correlations betweenGR and Newtonian. It can be seen that 3 of the mixed correlation terms (blue, green, magenta) dominate theGR correction on subhorizon scales.For the squeezed case, the dominant correlation is (cid:10) ∆ (1) g N ( k )∆ (1) g GR ( k )∆ (2) g GR ( k ) (cid:11) (blue). If we omitted thesecond-order GR projection effects, we would miss this dominant GR contribution to the squeezed galaxybispectrum.Note that the correlation with only one GR first-order projection term, i.e., (cid:10) ∆ (1) g N ( k )∆ (1) g GR ( k )∆ (2) g N ( k ) (cid:11) (ma-genta), has a constant contribution on super-equality scales. • In the bottom panel, we show the contributions from the first-order GR kernel K (1)GR , (51), on its own (red),and then together with the terms in the second-order GR kernel K (2)GR , (55), split into powers of k − .The first-order GR correction (red) clearly under-estimates the full GR correction, especially in the squeezedcase.Amongst the second-order GR corrections in the squeezed case, the k − term (blue) dominates on ultra-largescales until close to the comoving horizon, k = H , when the k − n , n = 2 , , − − − − k / Mpc − B k g / M p c k = H ( z = 1 .
0) sq ., z = 1 . ., z = 1 . ., z = 1 . ., z = 1 . − − k / Mpc − − − − (cid:0) | B k g − B k g N | / B k g N (cid:1) × % − − − − k / Mpc − B ⊥ g / M p c k = H ( z = 1 .
0) sq ., z = 1 . ., z = 1 . ., z = 1 . ., z = 1 . − − k / Mpc − − − − (cid:0) | B ⊥ g − B ⊥ g N | / B ⊥ g N (cid:1) × % − − − − k / Mpc − B g / M p c k = H ( z = 1 .
0) sq ., z = 1 . ., z = 1 . ., z = 1 . ., z = 1 . − − k / Mpc − − − − (cid:0) | B g − B g N | / B g N (cid:1) × % FIG. 1.
Left:
Galaxy bispectrum for moderately squeezed ( k ≈ k/
16, solid) and equilateral ( k = k , dashed) shapes,at z = 1 . , .
5. From top to bottom: radial, transverse and monopole parts.
Right:
Percentage difference relative to theNewtonian approximation for 0 . ≤ k ≤ .
1, which includes BAO scales. − − − − k / Mpc − B g / M p c k = H ( z = 1 . h ∆ (1)N ∆ (1)N ∆ (2)N ih ∆ (1)GR ∆ (1)GR ∆ (2)GR i h ∆ (1)N ∆ (1)N ∆ (2)GR ih ∆ (1)GR ∆ (1)GR ∆ (2)N i h ∆ (1)N ∆ (1)GR ∆ (2)N ih ∆ (1)N ∆ (1)GR ∆ (2)GR i − − − − k / Mpc − B g / M p c k = H ( z = 1 . h ∆ (1)N ∆ (1)N ∆ (2)N ih ∆ (1)GR ∆ (1)GR ∆ (2)GR i h ∆ (1)N ∆ (1)N ∆ (2)GR ih ∆ (1)GR ∆ (1)GR ∆ (2)N i h ∆ (1)N ∆ (1)GR ∆ (2)N ih ∆ (1)N ∆ (1)GR ∆ (2)GR i − − − − k / Mpc − B g / M p c k = H ( z = 1 . N O (1) GR O (2) k − O (2) k − O (2) k − O (2) k − − − − − k / Mpc − B g / M p c k = H ( z = 1 . N O (1) GR O (2) k − O (2) k − O (2) k − O (2) k − FIG. 2. Contributions to the galaxy bispectrum monopole for the moderately squeezed ( left ) and equilateral ( right ) shapes ofFig. 1, at z = 1 . Top:
The different 3-point correlations that contribute to the galaxy bispectrum – purely Newtonian, purely GR and mixedcorrelations – as given in (58).
Bottom:
The different contributions to the galaxy bispectrum from the first order GR kernel (51) on its own, and then togetherwith the terms in the second order GR kernel (55), split into powers of k − . B ( s ) g = B ( s ) g N (cid:104) B ( s ) (cid:105) s = radial, transverse, monopole , (70)∆ B ( s ) = α ( s ) (cid:18) kk eq (cid:19) − n .
007 Mpc − (cid:46) k (cid:46) .
07 Mpc − . (71)We find that n = 2 is a good fit for all s and redshift, and for squeezed and equilateral cases. This shows that thedominant GR corrections add up to behave as O ( H /k ) around equality scales. The amplitude on equality scales, α ( s ) , varies weakly with s and z , but is significantly smaller for equilateral shapes – see Table I. α ( s ) × for z = 1 , . B (cid:107) . , . . , .
096 equilateral∆ B ⊥ . , . . , .
096 equilateral∆ B . , . . , .
14 equilateralTABLE I. Percentage GR corrections at equality, as defined in (71), for the bispectra in Fig. 1. V. CONCLUSION
We considered the local relativistic projection effects on the galaxy bispectrum, up to second order, providing thedetails behind the results presented in [28], and generalizing those results to include evolution bias and magnificationbias. We transformed the local GR contribution into Fourier space, to form the kernel K (2)GR ( k , k , k ) given by (55),with further details presented in Appendix B, and the Γ I coefficients given in Appendix C. Once we have this kernel,computing the bispectrum is a relatively straightforward procedure, which allows us to analyse the contribution fromGR effects to the bispectrum.We incorporated a careful treatment of galaxy bias on ultra-large scales, which is essential in order to avoid spuriousgauge effects. We assumed a simple local-in-mass-density model of nonlinear bias that neglects tidal effects, leadingto the relativistic bias relation (26) for the Poisson-gauge galaxy number density contrast.The GR effects can be significant, as illustrated in Fig. 1 and Table I, for equilateral and moderately squeezedtriangles in the radial, transverse and monopole parts of the bispectrum. On equality scales at z ∼ − . ∼ − (cid:10) ∆ (1) g N ( k )∆ (1) g GR ( k )∆ (2) g GR ( k ) (cid:11) . If we included only the first-order GR projection effects in our analysis, we would miss this dominant GR contributionto the squeezed galaxy bispectrum. The bottom panel breaks down the terms in the second-order GR kernel K (2)GR according to powers of k − . For the squeezed case, the k − term dominates on ultra-large scales until close to thecomoving horizon, k = H .Our main aim was to highlight the importance of the effects from observations, properly analysed in GR, and tothis end, we treated the simplest case, taking the first steps towards a complete analysis. We have not included: • primordial non-Gaussianity; • tidal stress in the galaxy bias; • GR corrections to the v (2) , Φ (2) and Ψ (2) terms that contribute to the projection effects; • the second-order effect of the radiation era on initial conditions for sub-equality modes; • integrated contributions to the projection effects, wide-angle correlations and radial (cross-bin) correlations.The first three effects can be incorporated within our Fourier-space analysis using the plane-parallel approximation.The fourth requires numerical integration with a second-order Boltzmann code [35]. The last requires one to use the3-point correlation function, for example through a spherical harmonic decomposition. Acknowledgments:
We are especially grateful to Kazuya Koyama for very helpful comments. We thank Tobias Baldauf, Daniele Bertacca,Ruth Durrer, Sabino Matarrese and David Wands for useful discussions and comments. We also thank an anonymousreferee for very useful comments. All authors are funded in part by the NRF (South Africa). OU, SJ and RMare also supported by the South African SKA Project. RM and CC are also supported by the UK STFC, GrantsST/N000668/1 (RM) and ST/P000592/1 (CC). After our paper was completed, [54] presented a formalism for analysing the 3-point correlation function with all GR effects included,but without computation of the effects. Appendix A: Second-order gauge transformation of number density contrast
At second order, the number density contrasts in Poisson and C gauges are related by a generalisation of (17),which is given in [30]: δ (2) g = δ (2) g C + (3 − b e ) H v (2) + (cid:104) ( b e − H (cid:48) + b (cid:48) e H + ( b e − H (cid:105)(cid:2) v (1) (cid:3) + ( b e − H v (1) v (1) (cid:48) − ( b e − H∇ − (cid:20) v (1) ∇ v (1) (cid:48) − v (1) (cid:48) ∇ v (1) − ∂ i Φ (1) ∂ i v (1) − (1) ∇ v (1) (cid:21) + 2(3 − b e ) H v (1) δ (1) g C − v (1) δ (1) (cid:48) g C − ∂ i ξ (1) (cid:104) (3 − b e ) H ∂ i v (1) + 2 ∂ i δ (1) g C (cid:105) −
12 ( b e − H∇ − (cid:20) ∂ i ξ (1) ∂ i ∇ v (1) + ∂ i v (1) ∂ i ∇ ξ (1) + 2 ∂ i ∂ j ξ (1) ∂ i ∂ j v (1) (cid:21) . (A1)Here ξ (1) is a gauge generator, and the residual C-gauge freedom is fixed by imposing ξ (1) (cid:48) = 2 v (1) [30].It follows from the identity ∇ (cid:104) ∂ i ξ (1) · ∂ i v (1) (cid:105) = ∂ i v (1) · ∇ (cid:2) ∂ i ξ (1) (cid:3) + ∂ i ξ (1) · ∇ (cid:2) ∂ i v (1) (cid:3) + 2 ∂ j ∂ i ξ (1) · ∂ j ∂ i v (1) , (A2)that the last line of (A1) reduces to − ( b e − H ∂ i ξ (1) ∂ i v (1) /
2, which cancels the first term on the third line. Thus(A1) may be simplified to δ (2) g = δ (2) g C + (3 − b e ) H v (2) + (cid:104) ( b e − H (cid:48) + b (cid:48) e H + ( b e − H (cid:105)(cid:2) v (1) (cid:3) + ( b e − H v (1) v (1) (cid:48) − ( b e − H∇ − (cid:20) v (1) ∇ v (1) (cid:48) − v (1) (cid:48) ∇ v (1) − ∂ i Φ (1) ∂ i v (1) − (1) ∇ v (1) (cid:21) + 2(3 − b e ) H v (1) δ (1) g C − v (1) δ (1) (cid:48) g C − (cid:2) ∂ i δ (1) gC (cid:3) ∂ i ξ (1) . (A3)By the continuity equation, given in (13), the gauge fixing condition ξ (1) (cid:48) = 2 v (1) implies that ∂ i ξ (1) = − ∇ − ∂ i δ (1) m C . (A4)Using this, the relation (22) between C- and T-gauge number density contrasts becomes δ (2) g C − (cid:2) ∂ i δ (1) gC (cid:3) ∂ i ξ (1) = δ (2) g T . (A5)Then it follows from (A3) and (A5) that (A1) can be rewritten as the second-order map from the Poisson-gauge δ g to the T-gauge δ g T : δ (2) g = δ (2) g T + (3 − b e ) H v (2) + (cid:104) ( b e − H (cid:48) + b (cid:48) e H + ( b e − H (cid:105)(cid:2) v (1) (cid:3) + ( b e − H v (1) v (1) (cid:48) − ( b e − H∇ − (cid:20) v (1) ∇ v (1) (cid:48) − v (1) (cid:48) ∇ v (1) − ∂ i Φ (1) ∂ i v (1) − (1) ∇ v (1) (cid:21) + 2(3 − b e ) H v (1) δ (1) g T − v (1) δ (1) (cid:48) g T . (A6)This is (25).9 Appendix B: Expansion of perturbed variables in Fourier space
We express all variables in terms of the T-gauge matter density contrast, δ ( k ). For the gravitational and velocitypotentials, (14), (15) and (18) give H v (1) ( k ) = f H k δ (1) ( k ) , Φ (1) ( k ) = −
32 Ω m H k δ (1) ( k ) . (B1)The growth rate and growth suppression factor in ΛCDM obey f (cid:48) H = 12 (cid:0) m − (cid:1) f − f + 32 Ω m , H g (cid:48) g = f − . (B2)The galaxy number density contrast in Fourier space is expanded using (16), (17): δ (1) g = b δ (1) + (3 − b e ) H v (1) . (B3)The evolution of the velocity potential follows from the Euler equation as v (1) (cid:48) = −H v (1) − Φ (1) . (B4)The time derivative of the galaxy number density contrast follows from (B3) and (B4) as δ (1) (cid:48) g = (cid:0) b (cid:48) + b f H (cid:1) δ (1) + (cid:2) (3 − b e ) (cid:0) H (cid:48) − H (cid:1) − b (cid:48) e H (cid:3) v (1) − (3 − b e ) H Φ (1) . (B5)At second order, a typical term such as v (1) ( x ) δ (1) g ( x ) can be expressed as: v (1) ( x ) δ (1) g ( x ) = (cid:90) d k (2 π ) e i k · x (cid:104) v (1) δ (1) g (cid:105) ( k ) , (B6) (cid:104) v (1) δ (1) g (cid:105) ( k ) = (cid:90) d x e − i k · x v (1) ( x ) δ (1) g ( x )= 12 (cid:90) d x (cid:90) d k (2 π ) d k (2 π ) (cid:104) v (1) ( k ) δ (1) g ( k ) + v (1) ( k ) δ (1) g ( k ) (cid:105) e − i k · x e i k · x e i k · x = 12 (cid:90) d k (2 π ) d k (2 π ) (cid:104) v (1) ( k ) δ (1) g ( k ) + v (1) ( k ) δ (1) g ( k ) (cid:105) (2 π ) δ D ( k + k − k ) , (B7)where we used (39) and the definition of the Dirac delta function in three dimensions. Then we express the perturbativevariables in terms of δ (1) , using (B1) and (B3): v (1) ( k ) δ (1) g ( k ) + v (1) ( k ) δ (1) g ( k ) = (cid:20) b f H (cid:18) k + 1 k (cid:19) + 2 f (3 − b e ) H k k (cid:21) δ (1) ( k ) δ (1) ( k ) . (B8)This leads to (cid:104) v (1) δ (1) g (cid:105) ( k ) = (cid:90) d k (2 π ) d k (2 π ) F (cid:104) v (1) δ (1) g (cid:105) δ (1) ( k ) δ (1) ( k )(2 π ) δ D ( k + k − k ) , (B9)where the kernel is F (cid:104) v (1) ( x ) δ (1) g ( x ) (cid:105) = f H (cid:2) b (cid:0) k + k (cid:1) + 2 (3 − b e ) f H (cid:3) k k . (B10)Table II gives the Fourier kernels for all second-order terms in ∆ (2) g .0 TABLE II. Fourier transform kernel and coefficient of each term of (30) and (31), ordered according to their k -dependence. Ndenotes a Newtonian term ( k ), Γ is for k − , Γ is for k − , Γ to Γ are for k − and Γ to Γ are for k − . For convenience,the superscript (1) is dropped from first-order variables δ (1) , v (1) , Φ (1) .Term Γ Fourier kernel F Coefficient δ (2) N F ( k , k ) b ∂ (cid:107) v (2) N f H µ G ( k , k ) − / H δ∂ (cid:107) v N − f H (cid:0) µ + µ (cid:1) / − b / H ∂ (cid:107) v∂ (cid:107) δ N − f H µ µ (cid:0) k + k (cid:1) / (cid:0) k k (cid:1) − b / H ∂ (cid:107) v∂ (cid:107) v N f H (cid:0) µ µ k + µ µ k (cid:1) / (cid:0) k k (cid:1) / H (cid:2) ∂ (cid:107) v (cid:3) N f H µ µ / H (cid:2) Φ (cid:3) Γ m H / (cid:0) k k (cid:1) A Φ v Γ − m H f/ (cid:0) k k (cid:1) C∇ − ( v ∇ v (cid:48) − v (cid:48) ∇ v − ∂ i Φ ∂ i v − ∇ v ) Γ m H f/ (cid:0) k k (cid:1) (3 − b e ) H vv (cid:48) Γ f H (cid:0) m − f (cid:1) / (cid:0) k k (cid:1) ( b e − H (cid:2) v (cid:3) Γ f H / (cid:0) k k (cid:1) ( b e − H + b (cid:48) e H + ( b e − H (cid:48) v∂ (cid:107) v Γ i f H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) B Φ ∂ (cid:107) v Γ − f Ω m H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) D Φ ∂ (cid:107) Φ Γ
9i Ω m H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) f − Q ) / H Ψ (2) = Φ (2) Γ − m H F ( k , k ) / (cid:0) k (cid:1) Q − − b e + R Φ (2) (cid:48) Γ − m H (2 f − F ( k , k ) / (cid:0) k (cid:1) / H v (2) Γ f H G ( k , k ) /k (3 − b e ) H (cid:2) ∂ (cid:107) v (cid:3) Γ − f H µ µ / (cid:0) k k (cid:1) E ∂ (cid:107) v∂ (cid:107) Φ Γ f Ω m H µ µ / (cid:0) k k (cid:1) − f − Q ) / H ∂ i v ∂ i v Γ − f H k · k / (cid:0) k k (cid:1) b e − − Q − R∂ i v∂ i Φ Γ f Ω m H k · k / (cid:0) k k (cid:1) / H Φ δ Γ − m H (cid:0) k + k (cid:1) / (cid:0) k k (cid:1) b (cid:0) f − − b e + 4 Q + R (cid:1) − S Φ δ (cid:48) Γ − f Ω m H (cid:0) k + k (cid:1) / (cid:0) k k (cid:1) − b / H vδ Γ f H (cid:0) k + k (cid:1) / (cid:0) k k (cid:1) b (cid:48) + 2 b (3 − b e ) H vδ (cid:48) Γ f H (cid:0) k + k (cid:1) / (cid:0) k k (cid:1) − b Φ ∂ (cid:107) v Γ f Ω m H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) (cid:0) − f + 2 b e − Q − R − H (cid:48) / H (cid:1) / H Φ ∂ (cid:107) Φ Γ − m H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) − / H v∂ (cid:107) v Γ − f H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) b e − / H Φ ∂ (cid:107) δ Γ −
3i Ω m H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) b / H ∂ i v∂ (cid:107) ∂ i v Γ − i f H k · k (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) − / H δ (cid:48) ∂ (cid:107) v Γ i f H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) b / H δ∂ (cid:107) v Γ i f H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) b (cid:0) b e − Q − R (cid:1) + S Φ ∂ (cid:107) v Γ f Ω m H (cid:0) µ k + µ k (cid:1) / (cid:0) k k (cid:1) − / H ∂ (cid:107) v∂ (cid:107) v Γ − i f H (cid:0) µ µ k + µ µ k (cid:1) / (cid:0) k k (cid:1) (cid:0) − b e + 4 Q + 2 R + H (cid:48) / H (cid:1) / H ∂ (cid:107) v∂ (cid:107) Φ Γ f Ω m H (cid:0) µ µ k + µ µ k (cid:1) / (cid:0) k k (cid:1) / H ∂ (cid:107) v (2) Γ i f H µ G ( k , k ) /k b e − Q − R where A , B , C , D , E are given by (32)–(36), and R ≡ − Q ) χ H + H (cid:48) H , S ≡ (cid:18) − χ H (cid:19) ∂b ∂ ln ¯ L . (B11)Note that the kernels for quadratic terms in Table II can be obtained from an algorithm. Consider a term such asD n X D m Y, (B12)where D = ∂ i or ∂ (cid:107) , and X, Y = δ, v or Φ. The corresponding term in the kernel is formed as follows: (cid:110) (cid:0) i k (cid:1) n (cid:0) i k (cid:1) m for D = ∂ (cid:107) OR 12 (cid:0) i k · i k (cid:1) n for D = ∂ i , m = n × (cid:2) k − if X is v or Φ (cid:3) × (cid:2) k − if Y is v or Φ (cid:3) × (cid:2) a factor of µ for each ∂ (cid:107) acting on X (cid:3) × (cid:2) a factor of µ for each ∂ (cid:107) acting on Y (cid:3) × (cid:2) a factor of f H for each v (cid:3) × (cid:2) a factor of −
32 Ω m H for each Φ (cid:3)(cid:111) + (cid:110) ↔ (cid:111) (B13)2 Appendix C: The coefficients in the GR kernel K (2)GR The coefficients Γ I ( z ) in (55) follow from (30)–(36), using Table II.Evolution bias b e and magnification bias Q make the Γ I much more complicated than for the case b e = 0 = Q ,which is considered in [28]. (Note that when Q = 0, all the terms with ∂/∂ ln ¯ L vanish.)Γ H = 94 Ω m (cid:34) − f (cid:18) − b e + 4 Q + 4(1 − Q ) χ H + 2 H (cid:48) H (cid:19) − f (cid:48) H + b e + 6 b e − b e Q + 4 Q + 16 Q − ∂ Q ∂ ln ¯ L − Q (cid:48) H + b (cid:48) e H + 2 χ H (cid:18) − Q + 2 Q − ∂ Q ∂ ln ¯ L (cid:19) − χ H (cid:18) b e − b e Q − Q + 8 Q − H (cid:48) H (1 − Q ) − ∂ Q ∂ ln ¯ L − Q (cid:48) H (cid:19) + H (cid:48) H (cid:18) − − b e + 8 Q + 3 H (cid:48) H (cid:19) − H (cid:48)(cid:48) H (cid:35) + 3Ω m f (cid:34) − f (3 − b e ) + b e (cid:18) − Q ) χ H (cid:19) − b (cid:48) e H − b e + 4 b e Q − Q − − Q ) χ H + 2 (cid:18) − χ H (cid:19) Q (cid:48) H (cid:35) + f (cid:34) − b e + b e + b (cid:48) e H + ( b e − H (cid:48) H (cid:35) (C1)Γ H = 94 Ω m ( f − Q ) + 32 Ω m f (cid:34) − − f (cid:18) − f + 2 b e − Q − − Q ) χ H − H (cid:48) H (cid:19) − f (cid:48) H + 3 b e + b e − b e Q + 4 Q + 8 Q − ∂ Q ∂ ln ¯ L − Q (cid:48) H + b (cid:48) e H + 2 χ H (cid:18) − Q + 2 Q − ∂ Q ∂ ln ¯ L (cid:19) + 2 χ H (cid:18) − − b e + 2 b e Q + Q − Q + 3 H (cid:48) H (1 − Q ) + 6 ∂ Q ∂ ln ¯ L + 2 Q (cid:48) H (cid:19) − H (cid:48) H (cid:18) b e − Q − H (cid:48) H (cid:19) − H (cid:48)(cid:48) H (cid:35) + f (cid:34) − b e (cid:18) − Q ) χ H (cid:19) − b e + 2 b e Q − Q − b (cid:48) e H − − Q ) χ H + 2 (cid:18) − χ H (cid:19) Q (cid:48) H (cid:35) (C2)Γ H = 32 Ω m (cid:34) − f + b e − Q − − Q ) χ H − H (cid:48) H (cid:35) (C3)Γ H = f (3 − b e ) (C4)Γ H = 3Ω m f (2 − f − Q ) + f (cid:34) b e − b e + 4 b e Q − Q − Q + 4 ∂ Q ∂ ln ¯ L + 4 Q (cid:48) H − b (cid:48) e H− χ H (cid:18) − Q + 2 Q − ∂ Q ∂ ln ¯ L (cid:19) − χ H (cid:18) − b e + 2 b e Q − Q − Q + 3 H (cid:48) H (1 − Q ) + 4 ∂ Q ∂ ln ¯ L + 2 Q (cid:48) H (cid:19) − H (cid:48) H (cid:18) − b e + 4 Q + 3 H (cid:48) H (cid:19) + H (cid:48)(cid:48) H (cid:35) (C5)Γ H = 3Ω m f − f (cid:34) − b e − Q − Q ) χ H − H (cid:48) H (cid:35) (C6)Γ H = 32 Ω m (cid:34) b (cid:18) b e − Q − − Q ) χ H − H (cid:48) H (cid:19) + b (cid:48) H + 2 (cid:18) − χ H (cid:19) ∂b ∂ ln ¯ L (cid:35) − f (cid:34) b ( f − b e ) + b (cid:48) H (cid:35) (C7)Γ H = 94 Ω m + 32 Ω m f (cid:34) − f + 2 b e − Q − − Q ) χ H − H (cid:48) H (cid:35) + f (3 − b e ) (C8)3Γ H = −
32 Ω m b (C9)Γ H = 2 f (C10)Γ H = f (cid:34) b (cid:18) f + b e − Q − − Q ) χ H − H (cid:48) H (cid:19) + b (cid:48) H + 2 (cid:18) − χ H (cid:19) ∂b ∂ ln ¯ L (cid:35) (C11)Γ H = −
32 Ω m f (C12)Γ H = 32 Ω m f − f (cid:34) − b e + 4 Q + 4(1 − Q ) χ H + 3 H (cid:48) H (cid:35) (C13)Γ H = f (cid:34) b e − Q − − Q ) χ H − H (cid:48) H (cid:35) (C14)4 [1] J. C. Jackson, Fingers of God: A critique of Rees’ theory of primoridal gravitational radiation , Mon. Not. Roy. Astron.Soc. (1972) 1P–5P, [ arXiv:0810.3908 ].[2] W. L. W. Sargent and E. L. Turner,
A statistical method for determining the cosmological density parameter from theredshifts of a complete sample of galaxies , Astrophys. J. (Feb., 1977) L3–L7.[3] N. Kaiser,
Clustering in real space and in redshift space , Mon. Not. Roy. Astron. Soc. (1987) 1–27.[4] R. Moessner and B. Jain,
Angular cross-correlation of galaxies: a probe of gravitational lensing by large scale structure , Mon. Not. Roy. Astron. Soc. (1998) 18, [ astro-ph/9709159 ].[5] L. Hui, E. Gaztanaga, and M. LoVerde,
Anisotropic Magnification Distortion of the 3D Galaxy Correlation. 1. RealSpace , Phys. Rev.
D76 (2007) 103502, [ arXiv:0706.1071 ].[6] L. Hui, E. Gaztanaga, and M. LoVerde,
Anisotropic Magnification Distortion of the 3D Galaxy Correlation: II. Fourierand Redshift Space , Phys. Rev.
D77 (2008) 063526, [ arXiv:0710.4191 ].[7] D. Alonso, P. Bull, P. G. Ferreira, R. Maartens, and M. G. Santos,
Ultra-large scale cosmology with next-generationexperiments , Astrophys. J. (2015) 145, [ arXiv:1505.07596 ].[8] F. Montanari and R. Durrer,
Measuring the lensing potential with tomographic galaxy number counts , JCAP (2015),no. 10 070, [ arXiv:1506.01369 ].[9] A. M. Dizgah and R. Durrer,
Lensing corrections to the E g ( z ) statistics from large scale structure , JCAP (2016)035, [ arXiv:1604.08914 ].[10] J. Yoo,
General Relativistic Description of the Observed Galaxy Power Spectrum: Do We Understand What WeMeasure? , Phys. Rev.
D82 (2010) 083508, [ arXiv:1009.3021 ].[11] A. Challinor and A. Lewis,
The linear power spectrum of observed source number counts , Phys. Rev.
D84 (2011) 043516,[ arXiv:1105.5292 ].[12] C. Bonvin and R. Durrer,
What galaxy surveys really measure , Phys. Rev.
D84 (2011) 063505, [ arXiv:1105.5280 ].[13] M. Bruni, R. Crittenden, K. Koyama, R. Maartens, C. Pitrou, and D. Wands,
Disentangling non-Gaussianity, bias andGR effects in the galaxy distribution , Phys. Rev.
D85 (2012) 041301, [ arXiv:1106.3999 ].[14] T. Baldauf, U. Seljak, L. Senatore, and M. Zaldarriaga,
Galaxy Bias and non-Linear Structure Formation in GeneralRelativity , JCAP (2011) 031, [ arXiv:1106.5507 ].[15] D. Jeong, F. Schmidt, and C. M. Hirata,
Large-scale clustering of galaxies in general relativity , Phys. Rev.
D85 (2012)023504, [ arXiv:1107.5427 ].[16] S. Camera, M. G. Santos, and R. Maartens,
Probing primordial non-Gaussianity with SKA galaxy redshift surveys: afully relativistic analysis , Mon. Not. Roy. Astron. Soc. (2015), no. 2 1035–1043, [ arXiv:1409.8286 ].[17] S. Camera, R. Maartens, and M. G. Santos,
Einstein’s legacy in galaxy surveys , Mon. Not. Roy. Astron. Soc. (2015),no. 1 L80–L84, [ arXiv:1412.4781 ].[18] E. Sefusatti, M. Crocce, S. Pueblas, and R. Scoccimarro,
Cosmology and the Bispectrum , Phys. Rev.
D74 (2006) 023522,[ astro-ph/0604505 ].[19] E. Sefusatti and E. Komatsu,
The bispectrum of galaxies from high-redshift galaxy surveys: Primordial non-Gaussianityand non-linear galaxy bias , Phys. Rev.
D76 (2007) 083004, [ arXiv:0705.0343 ].[20] L. Verde, A. F. Heavens, S. Matarrese, and L. Moscardini,
Large scale bias in the universe. 2. Redshift space bispectrum , Mon. Not. Roy. Astron. Soc. (1998) 747–756, [ astro-ph/9806028 ].[21] R. Scoccimarro, H. Couchman, and J. A. Frieman,
The Bispectrum as a signature of gravitational instability inredshift-space , Astrophys. J. (1999) 531–540, [ astro-ph/9808305 ].[22] F. Schmidt, A. Vallinotto, E. Sefusatti, and S. Dodelson,
Weak Lensing Effects on the Galaxy Three-Point CorrelationFunction , Phys. Rev.
D78 (2008) 043513, [ arXiv:0804.0373 ].[23] H. Gil-Mar´ın, W. J. Percival, L. Verde, J. R. Brownstein, C.-H. Chuang, F.-S. Kitaura, S. A. Rodrguez-Torres, and M. D.Olmstead,
The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: RSD measurement fromthe power spectrum and bispectrum of the DR12 BOSS galaxies , Mon. Not. Roy. Astron. Soc. (2017), no. 21757–1788, [ arXiv:1606.00439 ].[24] Z. Slepian et al.,
Detection of Baryon Acoustic Oscillation Features in the Large-Scale 3-Point Correlation Function ofSDSS BOSS DR12 CMASS Galaxies , arXiv:1607.06097 .[25] M. Tellarini, A. J. Ross, G. Tasinato, and D. Wands, Galaxy bispectrum, primordial non-Gaussianity and redshift spacedistortions , JCAP (2016), no. 06 014, [ arXiv:1603.06814 ].[26] E. Di Dio, R. Durrer, G. Marozzi, and F. Montanari,
The bispectrum of relativistic galaxy number counts , JCAP (2016) 016, [ arXiv:1510.04202 ].[27] A. Kehagias, A. M. Dizgah, J. Norea, H. Perrier, and A. Riotto,
A Consistency Relation for the Observed GalaxyBispectrum and the Local non-Gaussianity from Relativistic Corrections , JCAP (2015), no. 08 018,[ arXiv:1503.04467 ].[28] O. Umeh, S. Jolicoeur, R. Maartens, and C. Clarkson,
A general relativistic signature in the galaxy bispectrum: the localeffects of observing on the lightcone , JCAP (2017) 003, [ arXiv:1610.03351 ].[29] D. Bertacca,
Observed galaxy number counts on the light cone up to second order: III. Magnification bias , Class. Quant.Grav. (2015), no. 19 195011, [ arXiv:1409.2024 ].[30] D. Bertacca, R. Maartens, and C. Clarkson, Observed galaxy number counts on the lightcone up to second order: I. Mainresult , JCAP (2014), no. 09 037, [ arXiv:1405.4403 ]. [31] D. Bertacca, R. Maartens, and C. Clarkson, Observed galaxy number counts on the lightcone up to second order: II.Derivation , JCAP (2014), no. 11 013, [ arXiv:1406.0319 ].[32] J. Yoo and M. Zaldarriaga,
Beyond the Linear-Order Relativistic Effect in Galaxy Clustering: Second-OrderGauge-Invariant Formalism , Phys. Rev.
D90 (2014), no. 2 023513, [ arXiv:1406.4140 ].[33] E. Di Dio, R. Durrer, G. Marozzi, and F. Montanari,
Galaxy number counts to second order and their bispectrum , JCAP (2014) 017, [ arXiv:1407.0376 ]. [Erratum: JCAP1506,no.06,E01(2015)].[34] E. Di Dio, H. Perrier, R. Durrer, G. Marozzi, A. M. Dizgah, J. Norea, and A. Riotto,
Non-Gaussianities due toRelativistic Corrections to the Observed Galaxy Bispectrum , JCAP (2017), no. 03 006, [ arXiv:1611.03720 ].[35] T. Tram, C. Fidler, R. Crittenden, K. Koyama, G. W. Pettinari, and D. Wands,
The Intrinsic Matter Bispectrum in Λ CDM , JCAP (2016), no. 05 058, [ arXiv:1602.05933 ].[36] S. Mollerach and S. Matarrese,
Cosmic microwave background anisotropies from second order gravitational perturbations , Phys. Rev.
D56 (1997) 4494–4502, [ astro-ph/9702234 ].[37] S. Matarrese, S. Mollerach, and M. Bruni,
Second order perturbations of the Einstein-de Sitter universe , Phys. Rev.
D58 (1998) 043504, [ astro-ph/9707278 ].[38] T. H.-C. Lu, K. Ananda, C. Clarkson, and R. Maartens,
The cosmological background of vector modes , JCAP (2009) 023, [ arXiv:0812.1349 ].[39] M. Bruni, D. B. Thomas, and D. Wands,
Computing General Relativistic effects from Newtonian N-body simulations:Frame dragging in the post-Friedmann approach , Phys. Rev.
D89 (2014), no. 4 044010, [ arXiv:1306.1562 ].[40] K. N. Ananda, C. Clarkson, and D. Wands,
The Cosmological gravitational wave background from primordial densityperturbations , Phys. Rev.
D75 (2007) 123518, [ gr-qc/0612013 ].[41] D. Baumann, P. J. Steinhardt, K. Takahashi, and K. Ichiki,
Gravitational Wave Spectrum Induced by Primordial ScalarPerturbations , Phys. Rev.
D76 (2007) 084019, [ hep-th/0703290 ].[42] D. Jeong and F. Schmidt,
Large-Scale Structure with Gravitational Waves I: Galaxy Clustering , Phys. Rev.
D86 (2012)083512, [ arXiv:1205.1512 ].[43]
Planck
Collaboration, P. A. R. Ade et al.,
Planck 2015 results. XIII. Cosmological parameters , Astron. Astrophys. (2016) A13, [ arXiv:1502.01589 ].[44] C. Bonvin,
Effect of Peculiar Motion in Weak Lensing , Phys. Rev.
D78 (2008) 123530, [ arXiv:0810.0180 ].[45] D. Bertacca, N. Bartolo, M. Bruni, K. Koyama, R. Maartens, S. Matarrese, M. Sasaki, and D. Wands,
Galaxy bias andgauges at second order in General Relativity , Class. Quant. Grav. (2015), no. 17 175019, [ arXiv:1501.03163 ].[46] E. Villa and C. Rampf, Relativistic perturbations in Λ CDM: Eulerian & Lagrangian approaches , JCAP (2016),no. 01 030, [ arXiv:1505.04782 ].[47] D. Bertacca,
Private communication (Erratum for [29] to appear) , .[48] L. Dai, E. Pajer, and F. Schmidt,
On Separate Universes , JCAP (2015), no. 10 059, [ arXiv:1504.00351 ].[49] R. de Putter, O. Dor´e, and D. Green,
Is There Scale-Dependent Bias in Single-Field Inflation? , JCAP (2015),no. 10 024, [ arXiv:1504.05935 ].[50] N. Bartolo, D. Bertacca, M. Bruni, K. Koyama, R. Maartens, S. Matarrese, M. Sasaki, L. Verde, and D. Wands,
Arelativistic signature in large-scale structure , Phys. Dark Univ. (2016) 30–34, [ arXiv:1506.00915 ].[51] F. Bernardeau, S. Colombi, E. Gaztanaga, and R. Scoccimarro, Large scale structure of the universe and cosmologicalperturbation theory , Phys. Rept. (2002) 1–248, [ astro-ph/0112551 ].[52] P. Gagrani and L. Samushia,
Information Content of the Angular Multipoles of Redshift-Space Galaxy Bispectrum , arXiv:1610.03488 .[53] J. E. Pollack, R. E. Smith, and C. Porciani, A new method to measure galaxy bias , Mon. Not. Roy. Astron. Soc. (2014) 555, [ arXiv:1309.0504 ].[54] D. Bertacca, A. Raccanelli, N. Bartolo, M. Liguori, S. Matarrese, and L. Verde,
Relativistic wide-angle galaxy bispectrumon the light-cone , arXiv:1705.09306arXiv:1705.09306