Imprints of local lightcone projection effects on the galaxy bispectrum. II

TL;DR

This paper details the correction of the galaxy bispectrum for local lightcone projection effects, including Doppler, potential, lensing, magnification bias, and evolution bias, showing significant deviations from Newtonian predictions at certain scales.

Contribution

It provides a comprehensive second-order correction to the galaxy bispectrum accounting for all local lightcone projection effects and biases, extending previous work.

Findings

Correction to Newtonian bispectrum is ~30% at z~1 on equality scales.

Includes second-order effects of magnification and evolution biases.

Generalizes previous results to more accurately model observational data.

Abstract

General relativistic imprints on the galaxy bispectrum arise from observational (or projection) effects. The lightcone projection effects include local contributions from Doppler and gravitational potential terms, as well as lensing and other integrated contributions. We recently presented for the first time, the correction to the galaxy bispectrum from all local lightcone projection effects up to second order in perturbations. Here we provide the details underlying this correction, together with further results and illustrations. For moderately squeezed shapes, the correction to the Newtonian prediction is ~30% on equality scales at z ~ 1. We generalise our recent results to include the contribution, up to second order, of magnification bias (which affects some of the local terms) and evolution bias.

Figures

Tables

Table 1. Table 1: Percentage GR corrections at equality, as defined in ( 71 ), for the bispectra in Fig.  1 .

	${\alpha }^{\left(s\right)}\times {10}^2$ for $z=1,\mathrm{\hspace{0.33em}1.5}$	Triangle shape
$\Delta B^{\parallel }$	$32.8,\mathrm{\hspace{0.33em}32.2}$	squeezed
	$.17,\mathrm{\hspace{0.33em}.096}$	equilateral
$\Delta B^{\perp }$	$54.1,\mathrm{\hspace{0.33em}69.6}$	squeezed
	$.17,\mathrm{\hspace{0.33em}.096}$	equilateral
$\Delta B^0$	$33.5,\mathrm{\hspace{0.33em}69.4}$	squeezed
	$.15,\mathrm{\hspace{0.33em}.14}$	equilateral

Table 2. Table 2: Fourier transform kernel and coefficient of each term of ( 30 ) and ( 31 ), ordered according to their k 𝑘 k -dependence. N denotes a Newtonian term ( k 0 superscript 𝑘 0 k^{0} ), Γ 1 subscript Γ 1 \Gamma_{1} is for k − 4 superscript 𝑘 4 k^{-4} , Γ 2 subscript Γ 2 \Gamma_{2} is for k − 3 superscript 𝑘 3 k^{-3} , Γ 3 subscript Γ 3 \Gamma_{3} to Γ 8 subscript Γ 8 \Gamma_{8} are for k − 2 superscript 𝑘 2 k^{-2} and Γ 9 subscript Γ 9 \Gamma_{9} to Γ 14 subscript Γ 14 \Gamma_{14} are for k − 1 superscript 𝑘 1 k^{-1} . For convenience, the superscript (1) is dropped from first-order variables δ ( 1 ) , v ( 1 ) , Φ ( 1 ) superscript 𝛿 1 superscript 𝑣 1 superscript Φ 1 \delta^{(1)},v^{(1)},\Phi^{(1)} .

Term	$\Gamma$	Fourier kernel $ℱ$	Coefficient
${\delta }^{\left(2\right)}$	N	$F_2({𝒌}_1,{𝒌}_2)$	$b_1$
${\partial }_{\parallel }^2{v}^{\left(2\right)}$	N	$f^2\mathscr{H}{\mu }_3^2{G}_2({𝒌}_1,{𝒌}_2)$	$-1/\mathscr{H}$
$\delta {\partial }_{\parallel }^{2}v$	N	$-f\mathscr{H}\left({\mu }_1^2+{\mu }_2^2\right)/2$	$-2b_1/\mathscr{H}$
${\partial }_{\parallel }v{\partial }_{\parallel }\delta$	N	$-f\mathscr{H}{\mu }_1{\mu }_2\left(k_1^2+k_2^2\right)/\left(2k_1{k}_2\right)$	$-2b_1/\mathscr{H}$
${\partial }_{\parallel }v{\partial }_{\parallel }^{3}v$	N	$f^2{\mathscr{H}}^2\left({\mu }_1{\mu }_2^3{k}_2^2+{\mu }_2{\mu }_1^3{k}_1^2\right)/\left(k_1{k}_2\right)$	$2/{\mathscr{H}}^2$
${\left[{\partial }_{\parallel }^{2}v\right]}^2$	N	$f^2{\mathscr{H}}^2{\mu }_1^2{\mu }_2^2$	$2/{\mathscr{H}}^2$
${\left[\Phi \right]}^2$	${\Gamma }_1$	$9{\Omega }_m^2{\mathscr{H}}^4/\left(4k_1^2{k}_2^2\right)$	$𝒜$
$\Phi v$	${\Gamma }_1$	$-3{\Omega }_m{\mathscr{H}}^{3}f/\left(2k_1^2{k}_2^2\right)$	$𝒞$
${\nabla }^{-2}\left(v{\nabla }^2{v}^{\prime }-v^{\prime }{\nabla }^{2}v-6{\partial }_i\Phi {\partial }^{i}v-6\Phi {\nabla }^{2}v\right)$	${\Gamma }_1$	$9{\Omega }_m{\mathscr{H}}^{3}f/\left(2k_1^2{k}_2^2\right)$	$\left(3-b_e\right)\mathscr{H}$
$v{v}^{\prime }$	${\Gamma }_1$	$f{\mathscr{H}}^3\left(3{\Omega }_m-2f\right)/\left(2k_1^2{k}_2^2\right)$	$\left(b_e-3\right)\mathscr{H}$
${\left[v\right]}^2$	${\Gamma }_1$	$f^2{\mathscr{H}}^2/\left(k_1^2{k}_2^2\right)$	${\left(b_e-3\right)}^2{\mathscr{H}}^2+b_e^{\prime }\mathscr{H}+\left(b_e-3\right){\mathscr{H}}^{\prime }$
$v{\partial }_{\parallel }v$	${\Gamma }_2$	$i{f}^2{\mathscr{H}}^2\left({\mu }_1{k}_1+{\mu }_2{k}_2\right)/\left(2k_1^2{k}_2^2\right)$	$ℬ$
$\Phi {\partial }_{\parallel }v$	${\Gamma }_2$	$-3if{\Omega }_m{\mathscr{H}}^3\left({\mu }_1{k}_1+{\mu }_2{k}_2\right)/\left(4k_1^2{k}_2^2\right)$	$𝒟$
$\Phi {\partial }_{\parallel }\Phi$	${\Gamma }_2$	$9i{\Omega }_m^2{\mathscr{H}}^4\left({\mu }_1{k}_1+{\mu }_2{k}_2\right)/\left(8k_1^2{k}_2^2\right)$	$2\left(f-2+2𝒬\right)/\mathscr{H}$
${\Psi }^{\left(2\right)}={\Phi }^{\left(2\right)}$	${\Gamma }_3$	$-3{\Omega }_m{\mathscr{H}}^2{F}_2({𝒌}_1,{𝒌}_2)/\left(2k_3^2\right)$	$4𝒬-1-b_e+R$
${\Phi }^{\left(2\right)\prime }$	${\Gamma }_3$	$-3{\Omega }_m{\mathscr{H}}^3\left(2f-1\right)F_2({𝒌}_1,{𝒌}_2)/\left(2k_3^2\right)$	$1/\mathscr{H}$
$v^{\left(2\right)}$	${\Gamma }_4$	$f\mathscr{H}{G}_2({𝒌}_1,{𝒌}_2)/k_3^2$	$\left(3-b_e\right)\mathscr{H}$
${\left[{\partial }_{\parallel }v\right]}^2$	${\Gamma }_5$	$-f^2{\mathscr{H}}^2{\mu }_1{\mu }_2/\left(k_1{k}_2\right)$	$ℰ$
${\partial }_{\parallel }v{\partial }_{\parallel }\Phi$	${\Gamma }_5$	$3f{\Omega }_m{\mathscr{H}}^3{\mu }_1{\mu }_2/\left(2k_1{k}_2\right)$	$2\left(2-f-2𝒬\right)/\mathscr{H}$
${\partial }_{i}v{\partial }^{i}v$	${\Gamma }_6$	$-f^2{\mathscr{H}}^2{𝒌}_1\cdot {𝒌}_2/\left(k_1^2{k}_2^2\right)$	$b_e-1-2𝒬-R$
${\partial }_{i}v{\partial }^i\Phi$	${\Gamma }_6$	$3f{\Omega }_m{\mathscr{H}}^3{𝒌}_1\cdot {𝒌}_2/\left(2k_1^2{k}_2^2\right)$	$2/\mathscr{H}$

$\Phi \delta$	${\Gamma }_7$	$-3{\Omega }_m{\mathscr{H}}^2\left(k_1^2+k_2^2\right)/\left(4k_1^2{k}_2^2\right)$	$2b_1\left(f-2-b_e+4𝒬+R\right)-S$
$\Phi {\delta }^{\prime }$	${\Gamma }_7$	$-3f{\Omega }_m{\mathscr{H}}^3\left(k_1^2+k_2^2\right)/\left(4k_1^2{k}_2^2\right)$	$-2b_1/\mathscr{H}$
$v\delta$	${\Gamma }_7$	$f\mathscr{H}\left(k_1^2+k_2^2\right)/\left(2k_1^2{k}_2^2\right)$	$b_1^{\prime }+2b_1\left(3-b_e\right)\mathscr{H}$
$v{\delta }^{\prime }$	${\Gamma }_7$	$f^2{\mathscr{H}}^2\left(k_1^2+k_2^2\right)/\left(2k_1^2{k}_2^2\right)$	$-2b_1$
$\Phi {\partial }_{\parallel }^{2}v$	${\Gamma }_8$	$3f{\Omega }_m{\mathscr{H}}^3\left({\mu }_1^2{k}_1^2+{\mu }_2^2{k}_2^2\right)/\left(4k_1^2{k}_2^2\right)$	$2\left(1-2f+2b_e-6𝒬-2R-{\mathscr{H}}^{\prime }/{\mathscr{H}}^2\right)/\mathscr{H}$
$\Phi {\partial }_{\parallel }^2\Phi$	${\Gamma }_8$	$-9{\Omega }_m^2{\mathscr{H}}^4\left({\mu }_1^2{k}_1^2+{\mu }_2^2{k}_2^2\right)/\left(4k_1^2{k}_2^2\right)$	$-2/{\mathscr{H}}^2$
$v{\partial }_{\parallel }^{2}v$	${\Gamma }_8$	$-f^2{\mathscr{H}}^3\left({\mu }_1^2{k}_1^2+{\mu }_2^2{k}_2^2\right)/\left(2k_1^2{k}_2^2\right)$	$2\left(b_e-3\right)/\mathscr{H}$
$\Phi {\partial }_{\parallel }\delta$	${\Gamma }_9$	$-3i{\Omega }_m{\mathscr{H}}^2\left({\mu }_1{k}_1^3+{\mu }_2{k}_2^3\right)/\left(4k_1^2{k}_2^2\right)$	$2b_1/\mathscr{H}$
${\partial }_{i}v{\partial }_{\parallel }{\partial }^{i}v$	${\Gamma }_{10}$	$-i{f}^2{\mathscr{H}}^2{𝒌}_1\cdot {𝒌}_2\left({\mu }_1{k}_1+{\mu }_2{k}_2\right)/\left(2k_1^2{k}_2^2\right)$	$-4/\mathscr{H}$
${\delta }^{\prime }{\partial }_{\parallel }v$	${\Gamma }_{11}$	$i{f}^2{\mathscr{H}}^2\left({\mu }_1{k}_2+{\mu }_2{k}_1\right)/\left(2k_1{k}_2\right)$	$2b_1/\mathscr{H}$
$\delta {\partial }_{\parallel }v$	${\Gamma }_{11}$	$if\mathscr{H}\left({\mu }_1{k}_2+{\mu }_2{k}_1\right)/\left(2k_1{k}_2\right)$	$2b_1\left(b_e-2𝒬-R\right)+S$
$\Phi {\partial }_{\parallel }^{3}v$	${\Gamma }_{12}$	$3if{\Omega }_m{\mathscr{H}}^3\left({\mu }_1^3{k}_1^3+{\mu }_2^3{k}_2^3\right)/\left(4k_1^2{k}_2^2\right)$	$-2/{\mathscr{H}}^2$
${\partial }_{\parallel }v{\partial }_{\parallel }^{2}v$	${\Gamma }_{13}$	$-i{f}^2{\mathscr{H}}^2\left({\mu }_1{\mu }_2^2{k}_2+{\mu }_2{\mu }_1^2{k}_1\right)/\left(2k_1{k}_2\right)$	$2\left(3-2b_e+4𝒬+2R+{\mathscr{H}}^{\prime }/{\mathscr{H}}^2\right)/\mathscr{H}$
${\partial }_{\parallel }v{\partial }_{\parallel }^2\Phi$	${\Gamma }_{13}$	$3if{\Omega }_m{\mathscr{H}}^3\left({\mu }_1{\mu }_2^2{k}_2+{\mu }_2{\mu }_1^2{k}_1\right)/\left(4k_1{k}_2\right)$	$2/{\mathscr{H}}^2$
${\partial }_{\parallel }{v}^{\left(2\right)}$	${\Gamma }_{14}$	$if\mathscr{H}{\mu }_3{G}_2({𝒌}_1,{𝒌}_2)/k_3$	$b_e-2Q-R$

Equations

$$\text{mbox}dN(z,\mathbf{n},>\ln L)=\mathcal{N}(z,\mathbf{n},>\ln L)D_A^2(z,\mathbf{n})k_{\mu }{u}^{\mu }\frac{\text{mbox}d\lambda }{\text{mbox}dz}\text{mbox}dz\text{mbox}d{\Omega }_o.$$

$$\mathcal{N}(z,\mathbf{n},>\ln L)={\int }_{\ln L}^{\infty }\text{mbox}d\ln \tilde{L}{n}_g(z,\mathbf{n},\ln \tilde{L}).$$

$$\frac{\text{mbox}dN(z,\mathbf{n},>\ln L)}{\text{mbox}dz\text{mbox}d{\Omega }_o}$$

$${\Delta }_g(z,\mathbf{n})$$

$${\Delta }_g^{(r)}={\Delta }_{g\mathrm{N}}^{(r)}+{\Delta }_{g\mathrm{GR}}^{(r)},r=1,2.$$

$$\displaystyle\mathcal{Q}(a,\bar{L})=-\frac{\partial\ln\big{[}a^{3}\bar{\mathcal{N}}(a,{>}\bar{L})\big{]}}{\partial\ln\bar{L}}.$$

$$b_{e}(a,\bar{L})=\frac{\partial\ln\big{[}a^{3}\bar{\mathcal{N}}(a,{>}\bar{L})\big{]}}{\partial\ln a}.$$

$${\partial }_{\Vert }=n^i{\partial }_i,{\partial }_{\perp i}={\partial }_i-n_i{\partial }_{\Vert },$$

$$\frac{\text{mbox}d}{\text{mbox}d\chi }=-\frac{\text{mbox}d}{\text{mbox}d\eta }=-{\partial }_{\eta }+{\partial }_{\parallel },$$

$${\nabla }_{\perp }^2={\nabla }^2-{\partial }_{\parallel }^2-\frac{2}{\chi }{\partial }_{\parallel }.$$

$$a^{-2}\text{mbox}d{s}^2$$

$$v^i$$

$${\nabla }^2{\Phi }^{(1)}=\frac{3}{2}{\Omega }_m{\mathcal{H}}^2{\delta }_{m\mathrm{C}}^{(1)},{\delta }_{m\mathrm{C}}^{(1)\prime }=-{\nabla }^2{v}^{(1)},$$

$${\Phi }^{(1)}=-\frac{3}{2}{\Omega }_m\frac{{\mathcal{H}}^2}{k^2}{\delta }_{m\mathrm{C}}^{(1)}$$

$$\mathcal{H}{v}^{(1)}=f\frac{{\mathcal{H}}^2}{k^2}{\delta }_{m\mathrm{C}}^{(1)}$$

$${\delta }_{g\mathrm{C}}^{(1)}(a,\mathbf{x},<\ln L)=b_1(a,\ln \bar{L}){\delta }_{m\mathrm{C}}^{(1)}(a,\mathbf{x}).$$

$${\delta }_g^{(1)}={\delta }_{g\mathrm{C}}^{(1)}+\left(3-b_e\right)\mathcal{H}{v}^{(1)}=b_1{\delta }_{m\mathrm{C}}^{(1)}+\left(3-b_e\right)\mathcal{H}{v}^{(1)}.$$

$${\delta }_{m\mathrm{C}}^{(1)}={\delta }_{m\mathrm{T}}^{(1)},{\delta }_{g\mathrm{C}}^{(1)}={\delta }_{g\mathrm{T}}^{(1)}=b_1{\delta }_{m\mathrm{T}}^{(1)}.$$

$$\delta_{g{\rm C}}=b_{1}\delta_{m{\rm C}}+{1\over 2}b_{2}\big{(}\delta_{m{\rm C}}\big{)}^{2}+\cdots,$$

$$\delta_{g{\rm C}}^{{{({2})}}}=b_{1}\delta_{m{\rm C}}^{{{({2})}}}+b_{2}\big{[}\delta_{m{\rm C}}^{{{({1})}}}\big{]}^{2}.$$

$$\delta_{m{\rm T}}^{(2)}=\delta_{m{\rm C}}^{(2)}+{2}\big{[}\partial_{i}\delta_{m{\rm C}}^{(1)}\big{]}\nabla^{-2}\partial^{i}\delta_{m{\rm C}}^{(1)},$$

$$\delta_{g{\rm T}}^{(2)}={\delta_{g{\rm C}}^{(2)}+2\big{[}\partial_{i}\delta_{g{\rm C}}^{(1)}\big{]}\nabla^{-2}\partial^{i}\delta_{mC}^{(1)}}.$$

$${\delta }_{g\mathrm{T}}^{(2)}$$

$$\delta_{g{\rm T}}^{{{({2})}}}=b_{1}\delta_{m{\rm T}}^{{{({2})}}}+b_{2}\big{[}\delta_{m{\rm T}}^{{{({1})}}}\big{]}^{2}.$$

$${\delta }_g^{(2)}$$

$${\Delta }_{g\mathrm{N}}^{(1)}$$

$${\Delta }_{g\mathrm{GR}}^{(1)}$$

$${\Delta }_g^{(2)}$$

$${\Delta }_{g\mathrm{N}}^{(2)}$$

$${\Delta }_{g\mathrm{GR}}^{(2)}$$

$$\mathcal{A}$$

Full text

1. I Introduction

2. II Galaxy number counts in general relativity

3. II.1 Local model of galaxy bias on ultra-large scales

4. II.2 Observed galaxy number counts in Poisson gauge

5. III Galaxy number overdensity in Fourier space and the bispectrum

6. IV Numerical Results

7. V Conclusion

8. A Second-order gauge transformation of number density contrast

9. B Expansion of perturbed variables in Fourier space

10. C The coefficients in the GR kernel ${{\cal K}^{(2)}_{\rm GR}}$

I Introduction

The galaxy power spectrum has been central to the cosmological constraints extracted from galaxy surveys up to now. For an accurate comparison of observations to theory, observational projection effects on the galaxy power spectrum must be taken into account. The main projection effect comes from redshift-space distortions (RSD) Jackson:2008yv ; Sargent:1977ApJ…212L…3S ; Kaiser:1987qv , which must be included in the analysis of the power spectrum. But it is not only accuracy that is gained – there is additional information to be extracted from the RSD themselves.

In addition to RSD, the galaxy power spectrum is also affected by lensing magnification Moessner:1997vm ; Hui:2007cu ; Hui:2007tm . In the analysis of current 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 the galaxy power spectrum for an accurate theoretical analysis – and, as with RSD, the lensing itself will deliver additional information Alonso:2015uua ; Montanari:2015rga ; Dizgah:2016bgm .

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 power spectrum on ultra-large scales ($H_{0}\lesssim k\lesssim k_{\rm eq}$) Yoo:2010ni ; Challinor:2011bk ; Bonvin:2011bg . These include Doppler, Sachs-Wolfe, integrated Sachs-Wolfe and time-delay terms. As in the case of RSD and lensing, these terms need to be incorporated for accuracy, and they also contain extra information.

The ultra-large scale GR corrections have a qualitatively similar effect on the galaxy power spectrum to primordial non-Gaussianity (PNG), which also modifies the power spectrum on ultra-large scales via scale-dependent galaxy bias. The GR corrections must therefore be taken into account when super-equality scales are probed to measure or constrain the PNG parameter ${f}_{\text{NL}}$ Bruni:2011ta ; Baldauf:2011bh ; Jeong:2011as ; Camera:2014bwa ; Camera:2014sba .

The galaxy bispectrum can provide additional information, partly independent of the power spectrum Sefusatti:2006pa ; Sefusatti:2007ih . The effects on the bispectrum from RSD have been computed in Verde:1998zr ; Scoccimarro:1999ed and from lensing in Schmidt:2008mb . Recently, the galaxy bispectrum has been used to detect the RSD and baryon acoustic oscillation (BAO) features in the BOSS survey, and to give independent measurements of growth rates and distances Gil-Marin:2016wya ; Slepian:2016kfz .

As in the case of the galaxy power spectrum, we need to take account of the observational lightcone effects in the galaxy bispectrum which distort the information on the underlying dark matter distribution, but which also provide new information. These projection effects are the same as for the power spectrum – with one major difference: for the 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, out to higher redshifts and across larger sky areas. Recent forecasts, using a Newtonian model with RSD but no GR projection effects, indicate that the bispectrum can considerably enhance the constraining power of future surveys Tellarini:2016sgp – especially for probing the initial conditions of the Universe via PNG. In order to fully exploit the improved precision from upcoming surveys, we need theoretical accuracy that matches and moves beyond observational precision. One important part of this theoretical requirement is to include all the GR projection effects in modelling the galaxy bispectrum.

The GR lightcone effects on the galaxy angular bispectrum from lensing convergence were computed on intermediate scales in DiDio:2015bua , neglecting the other, ultra-large scale, GR corrections to the galaxy overdensity. Another partial result was given in Kehagias:2015tda , using a separate-universe approximation to compute the galaxy angular bispectrum with all GR lightcone effects in the squeezed limit.

We recently provided a further partial result, valid for all triangle shapes, by computing all the local GR projection corrections to the galaxy bispectrum, including all second order terms and couplings Umeh:2016nuh . Crucial to our result is the expression for the observed galaxy number counts on the past lightcone, up to second order. This is given in the most general case by Bertacca:2014hwa (see also Bertacca:2014dra ; Bertacca:2014wga ; Yoo:2014sfa ; DiDio:2014lka ). Our work is complementary to the subsequent work by DiDio:2016gpd , who include lensing and terms of order $(\mathcal{H}/k)[\delta^{(1)}]^{2}$, but neglect all other GR effects on ultra-large scales.

Here we provide details of the derivation of the results given in Umeh:2016nuh , with additional illustrations, and we generalise some of those results. In particular, we include the magnification bias (which also contributes to local terms in the number counts) and the evolution bias. In Umeh:2016nuh , 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 Sefusatti:2006pa ; Sefusatti:2007ih (schematically, $\delta_{g}\sim b_{1}\delta_{m}+b_{2}\delta_{m}^{2}/2$). However, we take care to ensure that the definition of bias is gauge-independent and applies on ultra-large scales.

• •

For simplicity, we use standard Newtonian results to evaluate the second-order velocity potential $v^{(2)}$ and metric potentials $\Phi^{(2)},\Psi^{(2)}$, which contribute to the projection effects.

• •

We neglect the second-order effect of the radiation era on initial conditions for sub-equality modes Tram:2016cpy .

• •

We compute the galaxy bispectrum at fixed redshift and in Fourier space, and we use the plane-parallel approximation. Consequently, the following are not included in our approach: wide-angle correlations, radial correlations, lensing and other integrated contributions.

At second order in GR, scalar perturbations generate secondary vector and tensor modes Mollerach:1997up ; Matarrese:1997ay . These modes also enter the projection effects in the observed galaxy number density contrast at second order Bertacca:2014hwa ; Bertacca:2014dra ; Bertacca:2014wga ; Yoo:2014sfa ; DiDio:2014lka . As shown by Lu:2008ju ; Bruni:2013mua for vector modes and Ananda:2006af ; Baumann:2007zm ; Jeong:2012nu for tensor modes, the power in the secondary vector and tensor modes is much smaller 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 Ade:2015xua ; in particular, $h=H_{0}/(100\,{\rm km\,s}^{-1}{\rm Mpc}^{-1})=0.678$ and $\Omega_{m0}=1-\Omega_{\Lambda 0}=0.308$.

II Galaxy number counts in general relativity

The observer looks down the past lightcone and counts $\mbox{d}N$ galaxies, above a threshold luminosity $L$, within a redshift interval $\mbox{d}z$ about the observed redshift $z$, and within a solid angle element $\mbox{d}\Omega_{o}$ about the observed direction $\bm{n}$, where Challinor:2011bk ; Jeong:2011as ; Bertacca:2014hwa ; Alonso:2015uua

[TABLE]

Here $D_{A}$ is the angular diameter distance, $u^{\mu}$ is the 4-velocity of the source, $k^{\mu}=\mbox{d}x^{\mu}/\mbox{d}\lambda$ is the geodesic photon 4-momentum, and $\mathcal{N}$ is the flux-limited number density of sources:

[TABLE]

In the integrand, $n_{g}$ is the proper number density of sources, and only sources with luminosity above the detection threshold are counted by the observer.

The fractional perturbation $\Delta_{g}$ of the observed number counts is defined by

[TABLE]

where $\mathcal{H}(\eta)=a^{\prime}(\eta)/a(\eta)$ is the conformal Hubble rate, the comoving line-of-sight distance is given by $\mbox{d}\chi=\mbox{d}z/[(1+z)\mathcal{H}(z)]$, and $\bar{\mathcal{N}}$ is the background magnitude-limited number density. Henceforth, we suppress the dependence of $\Delta_{g}$ on ${\ln L}$ to reduce clutter. We expand ${\Delta_{g}}$ up to second order in perturbation theory:

[TABLE]

where we subtract off the average of ${\Delta_{g}^{{{({2})}}}}$ in order to ensure that $\langle{\Delta_{g}}\rangle=0$. For later convenience, we split the observed number density contrast into Newtonian and GR parts:

[TABLE]

We only consider the bispectrum at fixed redshift, so that all correlations are in the same redshift bin. There are integrated GR contributions to $\Delta_{g}^{(1)}$, 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 integrated contributions, and we neglect all such terms. Specifically, we neglect the integrated contributions in Bertacca:2014hwa , which gives the fully general $\Delta_{g}^{(1)}$ and $\Delta_{g}^{(2)}$ in Poisson gauge.111We also neglect all terms at the observer, which do not contribute to the bispectrum. A complete treatment would include the integrated terms, with all cross-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 integrated term, but also includes local (non-integrated) terms Bonvin:2008ni . This means that the magnification bias will still enter the bispectrum, even if we neglect all integrated terms. The magnification bias is given by the logarithmic slope of the background number density at the threshold luminosity:

[TABLE]

We have used the comoving number density in the definition above since it arises also in the definition of the evolution bias:

[TABLE]

This quantity describes the deviation of the background number density of sources from the idealised case of $a^{3}\bar{\mathcal{N}}=\bar{\mathcal{N}}_{0}$.

Radial and transverse derivatives are defined as

[TABLE]

the derivative down rays of the past lightcone is

[TABLE]

and the screen space projected Laplacian is

[TABLE]

Since $\Delta_{g}$ is defined as an observable, it is gauge-independent and we can use any gauge to compute it. In a given gauge, it will be of the form $\Delta_{g}=\delta_{g}+$ terms that describe projection effects in that gauge, where $\delta_{g}=\delta{\cal N}/\bar{\cal N}=\delta_{g}^{{{({1})}}}+{1\over 2}\delta_{g}^{{{({2})}}}$ is the galaxy number density contrast in the chosen gauge. We choose the Poisson gauge since it is convenient for splitting into Newtonian and GR parts. Neglecting the vector and tensor modes, the metric and the peculiar velocity of galaxies (equal to the dark matter velocity on the scales of interest) are given by

[TABLE]

The observed comoving coordinates Bertacca:2014dra of a galaxy are $\bm{x}=\chi(z)\bm{n}=[\eta_{0}-\eta(z)]\bm{n}$. We have assumed that anisotropic stress vanishes at first order, which implies $\Psi^{{{({1})}}}=\Phi^{{{({1})}}}$ in GR.

We will also use the comoving-synchronous (C) overdensities of matter and galaxy counts $\delta_{m{\rm C}},\delta_{g{\rm C}}$. The first-order Poisson and continuity equations are then

[TABLE]

which lead to

[TABLE]

II.1 Local model of galaxy bias on ultra-large scales

We start by considering the Poisson-gauge number density contrast $\delta_{g}^{{{({1})}}}$ at linear order, which is related to the dark matter density contrast $\delta^{{{({1})}}}_{m}$ via the galaxy bias. We need to ensure that the definition of scale-independent galaxy bias is gauge-independent and valid on ultra-large scales. As explained in detail in Challinor:2011bk ; Bruni:2011ta ; Jeong:2011as , the physical definition of scale-independent bias is in the matter rest-frame, which coincides with the galaxy rest-frame (on large scales there is 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 $L$):

[TABLE]

The Poisson-gauge number density contrast is related to the C-gauge one by Challinor:2011bk

[TABLE]

The velocity potential term in (17) ensures gauge-independence of the bias model on ultra-large scales. This term is the GR part of $\delta_{g}^{{{({1})}}}$, 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 Bertacca:2015mca ; Villa:2015ppa . There is no unique Eulerian frame in GR, but a 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 Bertacca:2015mca :

[TABLE]

The last equality is the definition of the Eulerian bias parameter at first order. This means that $b_{1}$ in (16) is the Eulerian bias parameter.

We extend (16) to higher order with the simplest possible model of scale-independent bias. This model assumes that 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 are scale-independent in the galaxy rest-frame, i.e. in C gauge. Expanding in powers of the mass density contrast, we have

[TABLE]

where $b_{I}=b_{I}(a,\ln L)$. At first order, this recovers (16). At second order we have:222For convenience, we have omitted the term -b_{2}{\big{\langle}\big{[}\delta_{m{\rm C}}^{{{({1})}}}\big{]}^{2}\big{\rangle}} on the right of (20).

[TABLE]

The relation between C- and T-gauge matter overdensities at second order is Bertacca:2015mca ; Villa:2015ppa

[TABLE]

where $-2\nabla^{-2}\partial^{i}\delta_{mC}^{(1)}$ is a gauge generator. Since the C$\,\to\,$T gauge transformation is purely spatial, (21) also applies to the galaxy counts:

[TABLE]

From (20)–(22), using (18), we find that

[TABLE]

which implies

[TABLE]

Therefore local-in-mass-density and scale-independent bias in C and T gauge are equivalent up to second order, with the 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

[TABLE]

By (16) and (24), this leads to the final expression for the Poisson-gauge galaxy density contrast in the simplest local bias model:

[TABLE]

The velocity and metric potential terms ensure gauge-independence on ultra-large scales. Equation (26) is the second-order generalisation of (17).

II.2 Observed galaxy number counts in Poisson gauge

At first order, we replace $\delta_{g}^{(1)}$ using the bias relations (16)–(18), and then split $\Delta_{g}^{{{({1})}}}$ into Newtonian and GR parts:

[TABLE]

The Newtonian part = T-gauge density contrast + Kaiser RSD, and the GR part = Doppler + potential + velocity potential. The velocity potential arises from the term in (17), which may be expressed in terms of the metric potential via (14) and (15). The Doppler term in (28) is the one proportional to the line-of-sight velocity $\partial_{\|}v^{{{({1})}}}$.

At second order, we use the gauge-independent bias model (26) to replace the Poisson-gauge $\delta_{g}^{{{({2})}}}$ term in $\Delta_{g}^{{{({2})}}}$. The remaining terms in $\Delta_{g}^{{{({2})}}}$ are second-order generalisations of RSD, Doppler and potential terms, together with quadratic 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 $\Delta_{g}^{(2)}$, including evolution bias and magnification bias, as well as all integrated effects, is given in Bertacca:2014hwa (including recent corrections Bertacca:2017 ). We include in this general expression our gauge-independent model of the galaxy bias at second order, (26), and we neglect the terms with integrated contributions. The result is

[TABLE]

The Newtonian part of (29) is formed from the density contrast and Kaiser RSD terms and their couplings:333Note that the GR correction to $\delta_{m{\rm T}}^{(2)}$ does not enter the bias term $b_{1}\delta_{m{\rm T}}^{(2)}$, as explained in Dai:2015jaa ; dePutter:2015vga ; Bartolo:2015qva . There is a GR correction to $v^{(2)}$, which we neglect here.

[TABLE]

The remaining terms form the GR correction:

[TABLE]

The background coefficients in the last line are

[TABLE]

In deriving (30)–(LABEL:bce) from (29), we used the following:

(a) eliminate $\mbox{d}/\mbox{d}\chi$ using (9), and $\partial_{\perp i}$ using (8);

(b) show, using the commutator relation \big{[}\partial_{\perp i},\partial_{\|}\big{]}=\chi^{-1}\partial_{\perp i}, that

[TABLE]

(c) express $\delta_{g}^{{{({1})}}}$ in terms of $\delta_{m{\rm T}}^{{{({1})}}}$ and $v^{(1)}$, using (17) and (18);

(d) rewrite the term from the perturbation of the magnification bias, using (16)–(18), as

[TABLE]

where the second equality uses (6), (7) and $\partial/\partial\ln a=\mathcal{H}^{-1}\partial/\partial\eta$.

In summary: we have used the general formula for $\Delta_{g}^{(2)}$ in Poisson gauge, given in Bertacca:2014hwa , neglecting the terms with line-of-sight integrals, to derive (30)–(LABEL:bce). In these equations we have broken down the highly complex formula in Bertacca:2014hwa into simple parts, facilitating analytical and then numerical analysis. Our new contribution is to determine the Poisson-gauge $\delta_{g}^{(2)}$ via a simple local-in-mass-density model of bias (26), that is gauge independent and valid on ultra-large scales.444Three groups have computed $\Delta_{g}^{(2)}$ – in Bertacca:2014hwa ; Bertacca:2014dra ; Bertacca:2014wga , Yoo:2014sfa and DiDio:2014lka . All have used different formalisms. The collective task of cross-checking these independent results has been initiated but is not complete, even in the simplest case with no integrated contributions and $b_{e}=0={\cal Q}$.

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 variables depend on $\bm{n}$ and can be computed in Fourier space at fixed $\eta(z)$. With $\bm{n}$ and $z$ fixed, we transform $\bm{x}=[\eta_{0}-\eta(z)]\bm{n}+\bm{x}_{0}\to\bm{k}$, which is equivalent to transforming over all observer positions $\bm{x}_{0}$. Our Fourier convention is

[TABLE]

where we suppress the redshift dependence. The transform of a product $h({\bm{x}})=g({\bm{x}})f({\bm{x}})$ leads to a convolution in Fourier space

[TABLE]

For notational convenience we write the T-gauge matter density contrast as

[TABLE]

from now on.

At second order, the matter density contrast and the velocity and metric potentials are given in a Newtonian approximation by Bernardeau:2001qr :

[TABLE]

The kernels for the dark matter and peculiar velocity perturbations in a matter-dominated model are

[TABLE]

The corrections to these kernels from the presence of $\Lambda$ are small Tram:2016cpy , and we neglect them. Within the same approximation, we have $\delta^{(2)}\propto D^{2}\delta^{(2)}_{0}$, so that $\delta^{(2)\prime}=2f\mathcal{H}\delta^{(2)}$. Then it follows from (44) that

[TABLE]

We write $\Delta_{g}^{(1,2)}$ in terms of kernels:

[TABLE]

and we split the kernels into Newtonian and GR parts, $\mathcal{K}^{(1,2)}=\mathcal{K}^{(1,2)}_{\rm N}+\mathcal{K}^{(1,2)}_{\rm GR}$. In (49), we subtracted off the ensemble average of $\Delta_{g}$:

[TABLE]

in order to ensure that $\langle\Delta_{g}\rangle=0$. Here $P(k)\equiv P_{\delta^{(1)}}(k)$ is the linear matter power spectrum.

By (27) and (28), the linear order kernel is given by

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are redshift dependent:

[TABLE]

At second order, the Newtonian part of the kernel is

[TABLE]

where $\mu_{i}=\hat{\bm{k}}_{i}\cdot{\bm{n}}$. The second line in (54) is the nonlinear Kaiser RSD contribution Verde:1998zr ; Scoccimarro:1999ed .

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:

[TABLE]

where the $\Gamma_{I}(z)$ are given in Appendix C.

We have ordered the $\Gamma_{I}$ according to the powers of $\mathcal{H}/k$, starting with the ${\cal O}(\mathcal{H}^{4}/k^{4})$ term and ending with the ${\cal O}(\mathcal{H}/k)$ terms. This is our key result – transforming the highly complicated second-order GR projection corrections given by (31) into a manageable Fourier-space kernel (55). In the special case $b_{e}=0={\cal Q}$, (31) reduces to the form given in Umeh:2016nuh . When $b_{e},{\cal Q}$ are nonzero, the $\Gamma_{I}$ become much more complicated.

In Fourier space, the observed galaxy bispectrum $B_{g}$ at fixed redshift is given by

[TABLE]

At second order, the only combinations of terms that contribute at tree-level are

[TABLE]

where the factors of $2$ arise from the factor $1/2$ in the perturbative expansion of $\Delta_{g}$. In the second equality, we have further separated the bispectrum into purely Newtonian and purely GR parts (first line), and cross-correlations between Newtonian and GR terms (following lines). The cross-correlation terms become important on smaller scales than the pure GR term.

The full expression for the galaxy bispectrum in terms of kernels follows from (58) as:

[TABLE]

The bispectrum in the Newtonian approximation is

[TABLE]

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 $B_{g}=B_{g}^{\rm{R}}+{\rm i}\,B_{g}^{\rm{I}}$ and compute the absolute value of the galaxy bispectrum, given by $|B_{g}|^{2}=(B_{g}^{\rm R})^{2}+({B_{g}^{\rm{I}}})^{2}$.

There are four different angles implicit in (59):

• three $\theta_{i}$ between the observer line of sight and the mode vectors (with cosines $\mu_{i}=\cos\theta_{i}=\hat{\bm{k}}_{i}\cdot{\bm{n}}$)
• • one of the angles $\theta_{ij}$ between $\bm{k}_{i}$ and $\bm{k}_{j}$ (with cosines ${\mu_{ij}=\cos\theta_{ij}=\hat{\bm{k}}_{i}\cdot\hat{\bm{k}}_{j}}$).

Two of the $\mu_{i}$ are independent, since $\mu_{1}k_{1}+\mu_{2}k_{2}+\mu_{3}k_{3}=0$, where $k_{3}=|\bm{k}_{1}+\bm{k}_{2}|$. Two of the $\mu_{ij}$ can be determined by the third via trigonometric identities. Finally, one of the two remaining $\mu_{i}$ may be expressed in terms of the other one and the choice of independent $\mu_{ij}$, using the trigonometric addition formula. If we choose $\mu_{1}$ and $\mu_{12}$, then

[TABLE]

where $\mu_{12}$ can be determined from the $k_{i}$. Here $\phi$ is the azimuthal angle, characterizing the orientation of the triangle in Fourier space, and the $\pm$ arises due to invariance under reflection of ${\bm{n}}$ about $\hat{\bm{k}}_{2}$ in their plane.

Implementing these conditions, the galaxy bispectrum is a function of $\mu_{1}$ and $\phi$, together with the magnitudes of the three mode vectors. The dependence of $B_{g}$ on $\mu_{1}$ and $\phi$ may be expanded in spherical harmonics:

[TABLE]

where the multipoles of $B_{g}$ are given by

[TABLE]

This can be compared to the Legendre multipole expansion of the galaxy power spectrum

[TABLE]

Note that we can also expand the bispectrum in Associated Legendre polynomials and still recover the multipoles as given in (63).

Typically, only the $m=0$ multipoles of $B_{g}$ are considered, and we will do this, so that $B_{g}=B_{g}({k}_{1},{k}_{2},{k}_{3},\mu_{1})$. In fact, this does not lose much information Gagrani:2016rfy . For the monopole, we use the shorthand $B^{0}_{g}\equiv B^{00}_{g}$.

IV Numerical Results

In order to illustrate quantitatively the imprint of GR effects on the galaxy bispectrum, we specialise to an isosceles configuration, with

[TABLE]

We evaluate the following cases:

[TABLE]

For redshifts and astrophysical parameters, we choose:

[TABLE]

where the galaxy bias parameters are similar to Pollack:2013alj .

In each case, we compare the Newtonian prediction (60) for the galaxy bispectrum, to the GR prediction (59). We consider the galaxy bispectrum $B_{g}$ as a function of triangle size for two isosceles shapes. We fix $\mu_{12}=\cos\theta_{12}$ and vary $k$, for two special cases:

[TABLE]

Figure 1 shows the radial, transverse and monopole parts of $B_{g}$, together with the percentage correction relative to the Newtonian case without the GR projection effects, on scales $0.01\leq k\leq 0.1$, which includes BAO scales. In all cases, as expected, the GR corrections become increasingly important on larger scales. The squeezed configuration has a larger correction than the equilateral. For the monopole, the GR correction at equality scales reaches ${\cal O}(30-{70\%})$ at $z\sim 1-1.5$, and then grows larger. Note that when the short modes are equality scale, the long mode is still within the Hubble horizon:

[TABLE]

On the largest scales, our results need to be corrected for wide-angle correlations that are absent in the plane-parallel approximation.

It is interesting to identify the various contributions to the galaxy bispectrum monopole in Fig. 1. We do this in two ways, as illustrated in Fig. 2, for the moderately squeezed (left) and equilateral (right) shapes, at $z=1.5$:

• •

In the top panel, we show the contributions from the various 3-point correlations \big{\langle}\Delta_{g}(\bm{k}_{1})\Delta_{g}(\bm{k}_{2})\Delta_{g}(\bm{k}_{3})\big{\rangle}, as given in (58).

The pure Newtonian correlation gives the standard curve (dashed, black). The 5 solid curves are the correlations with GR corrections: 1 pure GR correlation (red), which dominates on horizon scales, and 4 correlations between GR and Newtonian. It can be seen that 3 of the mixed correlation terms (blue, green, magenta) dominate the GR correction on subhorizon scales.

For the squeezed case, the dominant correlation is \big{\langle}\Delta^{{{({1})}}}_{g\rm{N}}(\bm{k}_{1})\Delta^{{{({1})}}}_{g\rm{GR}}(\bm{k}_{2})\Delta^{{{({2})}}}_{g\rm{GR}}(\bm{k}_{3})\big{\rangle} (blue). If we omitted the second-order GR projection effects, we would miss this dominant GR contribution to the squeezed galaxy bispectrum.

Note that the correlation with only one GR first-order projection term, i.e., \big{\langle}\Delta^{{{({1})}}}_{g\rm{N}}(\bm{k}_{1})\Delta^{{{({1})}}}_{g\rm{GR}}(\bm{k}_{2})\Delta^{{{({2})}}}_{g\rm{N}}(\bm{k}_{3})\big{\rangle} (magenta), has a constant contribution on super-equality scales.

• •

In the bottom panel, we show the contributions from the first-order GR kernel ${\cal K}^{(1)}_{\rm GR}$ , (51), on its own (red), and then together with the terms in the second-order GR kernel ${\cal K}^{(2)}_{\rm GR}$, (55), split into powers of $k^{-1}$.

The first-order GR correction (red) clearly under-estimates the full GR correction, especially in the squeezed case.

Amongst the second-order GR corrections in the squeezed case, the $k^{-1}$ term (blue) dominates on ultra-large scales until close to the comoving horizon, $k=\mathcal{H}$, when the $k^{-n}$, $n=2,3,4$ terms (green, magenta, orange) dominate.

On scales around equality, we can find a power-law fit for the fractional GR corrections to the Newtonian prediction:

[TABLE]

We find that $n=2$ is a good fit for all $s$ and redshift, and for squeezed and equilateral cases. This shows that the dominant GR corrections add up to behave as ${\cal O}(\mathcal{H}^{2}/k^{2})$ around equality scales. The amplitude on equality scales, $\alpha^{(s)}$, varies weakly with $s$ and $z$, but is significantly smaller for equilateral shapes – see Table 1.

V Conclusion

We considered the local relativistic projection effects on the galaxy bispectrum, up to second order, providing the details behind the results presented in Umeh:2016nuh , and generalizing those results to include evolution bias and magnification bias. We transformed the local GR contribution into Fourier space, to form the kernel $\mathcal{K}^{{{({2})}}}_{\mathrm{GR}}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3})$ given by (55), with further details presented in Appendix B, and the $\Gamma_{I}$ 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 from GR effects to the bispectrum.

We incorporated a careful treatment of galaxy bias on ultra-large scales, which is essential in order to avoid spurious gauge effects. We assumed a simple local-in-mass-density model of nonlinear bias that neglects tidal effects, leading to 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 squeezed triangles in the radial, transverse and monopole parts of the bispectrum. On equality scales at $z\sim 1-1.5$ they alter the bispectrum monopole in the moderately squeezed case by $\sim 30-70\%$. On ultra-large scales, the bispectrum is dominated by the local GR terms.

The contributions to the total GR correction of the monopole are shown in Fig. 2. The top panel presents the contributions from the various 3-point correlations given in (58). In the squeezed case, the dominant correlation is

[TABLE]

If we included only the first-order GR projection effects in our analysis, we would miss this dominant GR contribution to the squeezed galaxy bispectrum. The bottom panel breaks down the terms in the second-order GR kernel ${\cal K}^{(2)}_{\rm GR}$ according to powers of $k^{-1}$. For the squeezed case, the $k^{-1}$ term dominates on ultra-large scales until close to the comoving horizon, $k=\mathcal{H}$.

Our main aim was to highlight the importance of the effects from observations, properly analysed in GR, and to this 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 $v^{(2)}$, $\Phi^{(2)}$ and $\Psi^{(2)}$ 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 Tram:2016cpy . The last requires one to use the 3-point correlation function, for example through a spherical harmonic decomposition.555After our paper was completed, Bertacca:2017dzm presented a formalism for analysing the 3-point correlation function with all GR effects included, but without computation of the effects.

[TABLE]

**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 anonymous referee for very useful comments. All authors are funded in part by the NRF (South Africa). OU, SJ and RM are also supported by the South African SKA Project. RM and CC are also supported by the UK STFC, Grants ST/N000668/1 (RM) and ST/P000592/1 (CC).

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 Bertacca:2014dra :

[TABLE]

Here $\xi^{(1)}$ is a gauge generator, and the residual C-gauge freedom is fixed by imposing $\xi^{{{({1})}}\prime}=2v^{{{({1})}}}$ Bertacca:2014dra .

It follows from the identity

[TABLE]

that the last line of (72) reduces to $-\left(b_{e}-3\right)\mathcal{H}\,\partial_{i}\xi^{{{({1})}}}\partial^{i}v^{{{({1})}}}/2$, which cancels the first term on the third line. Thus (72) may be simplified to

[TABLE]

By the continuity equation, given in (13), the gauge fixing condition $\xi^{{{({1})}}\prime}=2v^{{{({1})}}}$ implies that

[TABLE]

Using this, the relation (22) between C- and T-gauge number density contrasts becomes

[TABLE]

Then it follows from (74) and (76) that (72) can be rewritten as the second-order map from the Poisson-gauge $\delta_{g}$ to the T-gauge $\delta_{g{\rm T}}$:

[TABLE]

This is (25).

Appendix B Expansion of perturbed variables in Fourier space

We express all variables in terms of the T-gauge matter density contrast, $\delta({\bm{k}})$. For the gravitational and velocity potentials, (14), (15) and (18) give

[TABLE]

The growth rate and growth suppression factor in $\Lambda$CDM obey

[TABLE]

The galaxy number density contrast in Fourier space is expanded using (16), (17):

[TABLE]

The evolution of the velocity potential follows from the Euler equation as

[TABLE]

The time derivative of the galaxy number density contrast follows from (80) and (81) as

[TABLE]

At second order, a typical term such as $v^{{{({1})}}}({\bm{x}})\delta_{g}^{{{({1})}}}({\bm{x}})$ can be expressed as:

[TABLE]

where we used (39) and the definition of the Dirac delta function in three dimensions. Then we express the perturbative variables in terms of $\delta^{(1)}$, using (78) and (80):

[TABLE]

This leads to

[TABLE]

where the kernel is

[TABLE]

Table 2 gives the Fourier kernels for all second-order terms in $\Delta_{g}^{(2)}$.

where ${\cal A},{\cal B},{\cal C},{\cal D},{\cal E}$ are given by (32)–(LABEL:bce), and

[TABLE]

Note that the kernels for quadratic terms in Table 2 can be obtained from an algorithm. Consider a term such as

[TABLE]

where ${\rm D}=\partial_{i}$ or $\partial_{\|}$, and $X,Y=\delta,v$ or $\Phi$. The corresponding term in the kernel is formed as follows:

[TABLE]

Appendix C The coefficients in the GR kernel ${{\cal K}^{(2)}_{\rm GR}}$

The coefficients $\Gamma_{I}(z)$ in (55) follow from (30)–(LABEL:bce), using Table 2.

Evolution bias $b_{e}$ and magnification bias ${\cal Q}$ make the $\Gamma_{I}$ much more complicated than for the case $b_{e}=0={\cal Q}$, which is considered in Umeh:2016nuh . (Note that when ${\cal Q}=0$, all the terms with $\partial/\partial\ln\bar{L}$ vanish.)

[TABLE]

[TABLE]