arXiv:1506.02671v3 [astro-ph.CO] 24 Mar 2016

Prepared for submission to JCAP
Imprint of inﬂation on galaxy shape correlations
Fabian Schmidt,a Nora Elisa Chisari,b,c Cora Dvorkind,†
aMax-Planck-Institute for Astrophysics, D-85748 Garching, Germany bDepartment of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA cDepartment of Physics, University of Oxford, Oxford, OX1 3RH, United Kingdom dInstitute for Theory and Computation, Harvard University, 60 Garden St., Cambridge, MA 02138 †Hubble Fellow
Abstract. We show that intrinsic (not lensing-induced) correlations between galaxy shapes
oﬀer a new probe of primordial non-Gaussianity and inﬂationary physics which is complementary to galaxy number counts. Speciﬁcally, intrinsic alignment correlations are sensitive to an anisotropic squeezed limit bispectrum of the primordial perturbations. Such a feature arises in solid inﬂation, as well as more broadly in the presence of light higher spin ﬁelds during inﬂation (as pointed out recently by Arkani-Hamed and Maldacena). We present a derivation of the all-sky two-point correlations of intrinsic shapes and number counts in the presence of non-Gaussianity with general angular dependence, and show that a quadrupolar (spin-2) anisotropy leads to the analog in galaxy shapes of the well-known scale-dependent bias induced in number counts by isotropic (spin-0) non-Gaussianity. Moreover, in the presence of non-zero anisotropic non-Gaussianity, the quadrupole of galaxy shapes becomes sensitive to far superhorizon modes. These eﬀects come about because long-wavelength modes induce a local anisotropy in the initial power spectrum, with which galaxies will correlate. We forecast that future imaging surveys could provide constraints on the amplitude of anisotropic nonGaussianity that are comparable to those from the Cosmic Microwave Background (CMB). These are complementary as they probe diﬀerent physical scales. The constraints, however, depend on the sensitivity of galaxy shapes to the initial conditions which we only roughly estimate from observed tidal alignments.

Contents
1 Introduction
2 Large-scale non-Gaussianity
3 Intrinsic alignments and non-Gaussianity
4 Observed shape and number count correlations 4.1 Projection 4.2 Intrinsic shape correlations 4.3 Signatures of non-Gaussianity 4.4 Lensing shear
5 Modeling the red galaxy population and alignments
6 Fisher matrix analysis
7 Constraining non-Gaussianity with clustering and shapes
8 Discussion and conclusions
A Derivation of Eq. (3.11)
B Renormalized shape biases B.1 Non-Gaussian initial conditions
C Non-Gaussian contributions to weak lensing shear

1
3
4
10 10 12 14 15
16
18
19
21
23
25 27
29

1 Introduction
The intrinsic alignments of red galaxies are thought to have their origin in the interaction of galaxy shapes with the tidal ﬁeld of the large-scale structure [1]. These alignments remain a relatively unexplored cosmological probe, complementary to weak gravitational lensing and the statistics of the galaxy distribution. Upcoming imaging surveys, such as Euclid1 and the Large Synoptic Survey Telescope (LSST2), will measure the shapes of billions of galaxies over half of the sky, providing a unique opportunity to exploit alignments as a cosmological probe.
Alignments can contribute to the correlations between shapes and positions of galaxies with a strength larger than weak lensing at low redshifts, and sub-dominant at larger redshifts [2, 3]. Much of the eﬀort in the study of alignments has been focused on measuring, modeling and removing intrinsic alignment contamination to weak gravitational lensing observables [4– 7]. The tidal alignment model [1] has been shown to reproduce the scale-dependence of the observed alignment signal for red galaxies in the linear regime [8, 9] with an amplitude that depends on luminosity [10], while blue galaxies have no measured alignments and only upper limits for their intrinsic shape correlations are available [11, 12]. In this work, we use “red”
1http://sci.esa.int/euclid/ 2http://www.lsst.org/lsst/

–1–

as a synonym of early-type and “blue”, of late-type galaxies. In the low-redshift Universe, a very small fraction of early-types have blue colors [13], but this assumption has not been tested at the high redshifts to be probed by upcoming imaging surveys. On the other hand, recent results from numerical hydrodynamical simulations suggest that indeed color is well correlated with galaxy dynamics as tracers of the intrinsic alignment signal [14]. Alignments are typically regarded as a contaminant to ‘cosmic shear’ (the two-point correlation of lensinginduced ellipticities), and they can contribute signiﬁcantly to the correlation of galaxy shapes with the weak lensing of the Cosmic Microwave Background (CMB) [15, 16].
The exploration of alignments as a cosmological probe has only started recently. The wealth of shape information expected from upcoming imaging surveys, and the availability of a model that reproduces the observed alignment correlations of red galaxies, has triggered the exploration of the use of intrinsic alignments for cosmology. For example, the correlation between galaxy shapes and CMB B-mode polarization induced by primordial gravitational waves could be detected from future surveys, albeit with limitations in conﬁrming its primordial origin [17]. While the CMB B-mode polarization is a purely linear eﬀect induced by the gravitational redshift of tensor metric perturbations, the latter also induce an eﬀective tidal ﬁeld which modiﬁes the local growth of structure and acts to align galaxies [18, 19]. Another example of the use of intrinsic alignments for cosmology is the possible detection of baryon acoustic oscillations in the cross-correlation of galaxy positions and shapes from future surveys [20]. While intrinsic alignments are weaker in amplitude than galaxy clustering, they do have some advantages such as the absence of redshift-space distortions (RSD) [21].
In this work, we show that intrinsic alignments can probe inﬂationary models where the squeezed-limit bispectrum of the primordial curvature perturbations, usually denoted R or ζ, is anisotropic. Phrased in terms of the primordial potential perturbation φ during matter domination, which is related to the curvature perturbation by R = (5/3)φ, the squeezed-limit bispectrum can be expressed as

Bφ(k1, k2, k3 = kL) =

AℓPℓ(kˆL · kˆS)

ℓ=0,2,...

kL kS

∆
Pφ(kL)Pφ(kS ) 1 + O

kL2 kS2

, (1.1)

where k3 = kL ≪ k1, k2 while kS = k1 − kL/2, Pℓ are the Legendre polynomials, and Aℓ are dimensionless amplitudes. Note that the angular dependence has to be even; see Section 2. In general, there can be several contributions with diﬀerent power indices ∆.
In this paper, we will focus on the “local” scaling with ∆ = 0. However, our results are
easily generalized to other scalings. In Eq. (1.1), the coeﬃcient A0 of the isotropic term is related to the usual local non-Gaussianity parameter fNloLc via A0 = 4fNloLc. We are interested in constraining the parameter that governs the leading anisotropic contribution, namely the
quadrupolar dependence of the bispectrum, A2. As we will show, A2 does not lead to a signiﬁcant large-scale eﬀect in galaxy clustering, speciﬁcally the two-point function. It does
however lead to the analog of the well-known scale-dependent bias for galaxy counts, which scales as fNloLc(k/aH)−2, in the two-point statistics of galaxy shapes, adding a contribution ∝ A2(k/aH)−2. Combining galaxy clustering with shapes, we can then constrain both isotropic and anisotropic non-Gaussianity simultaneously.
Thus, alignments oﬀer a golden opportunity to explore a new dimension in the parameter
space of inﬂationary models. Anisotropic non-Gaussianity can arise in several early-Universe
scenarios. In solid inﬂation [22], inﬂation is driven by the exponential stretching of an
unusual solid. A feature of solid inﬂation is that it produces large anisotropic non-Gaussianity

–2–

while fNloLc is small (A2 ≫ A0) [22]. Both isotropic and anisotropic non-Gaussianity are produced by primordial curvature perturbations generated by large-scale magnetic ﬁelds [23, 24]. Anisotropic non-Gaussianity can also be produced in inﬂationary models with a generalized bispectrum from excited Bunch-Davies vacuum [25]. Recently, Arkani-Hamed and Maldacena [26] pointed out (see also [27–29]) that the squeezed-limit bispectrum of primordial perturbations oﬀers a clean probe of the spectrum of particles with masses of order the Hubble scale and smaller during inﬂation (see [30] for a corresponding study of the tensor-scalar-scalar three-point function). The index ∆ is related to the mass of the particle, with ∆ → 0 for degrees of freedom much lighter than the Hubble scale. The order of the angular dependence L encodes the spin of the particles, as the leading contribution of a spin-s particle yields a contribution of order L = s in Eq. (1.1). Thus, our results show that the combination of galaxy clustering and shapes can probe the presence of scalar (spin 0) and tensor (spin 2) degrees of freedom with m H during inﬂation.
Of course, this type of primordial non-Gaussianity can also be constrained using the CMB, which yields the currently tightest constraints of A2 = 13 ± 93 (1σ) [31]. We stress, however, that galaxy clustering and shapes probe the three-point function in Eq. (1.1) for small-scale modes that are on signiﬁcantly smaller scales than the CMB. Given that the amplitude of non-Gaussianity can well be scale-dependent, the constraints we forecast are to be seen as complementary to the CMB.
Our work is organized as follows. Section 2 presents the non-Gaussian matter density ﬁeld and our conventions for parametrizing anisotropic non-Gaussianity through the squeezed-limit bispectrum. The impact of tidal ﬁelds and primordial non-Gaussianity on the shapes and clustering of galaxies is described in Section 3. In Section 4, we derive full sky observables of intrinsic alignments. Section 5 summarizes the adopted model for the red galaxy population, based on [3], while we outline the Fisher forecast in Section 6. In Section 7, we present a forecast of the detectability of anisotropic non-Gaussianity in an LSST-like weak lensing survey. Finally, we discuss our results in Section 8.
Throughout this paper, we assume the following Planck [32] ﬁducial ﬂat ΛCDM cosmology: Ωbh2 = 0.022, ΩCDMh2 = 0.12, h = 0.67, ΩK = 0, As = 2.2 × 10−9, ns = 0.9645, kp = 0.05 Mpc−1 and we deﬁne Ωm = Ωb + ΩCDM.

2 Large-scale non-Gaussianity

When computing two-point correlations of large-scale structure in the large-scale limit, the

leading relevant non-Gaussian statistic is the squeezed-limit potential bispectrum Bφ(k1, k2, kL), where kL ≪ k1, k2. While bispectra are in general non-separable functions of the momenta, they can usually be well approximated by a sum of separable terms [33]. For simplicity, we

will only consider one of those terms, which in the squeezed limit can be written in general

as

Bφ(kL, kS, kS ) = f (kL, kS )Pφ(kL)Pφ(kS ) ,

(2.1)

with “beyond-squeezed limit” terms suppressed by (kL/kS)2. These are negligible when considering large-scale correlations, and we will not consider them here. If the bispectrum is scale-invariant, which we will assume in the following, then f (kL, kS ) = f (kˆL · kˆS, kL/kS). Further, one can parametrize the dependence on kL/kS as a power law (kL/kS)∆, so that f (kL, kS) → f (kˆL · kˆS) (kL/kS)∆. In the following sections, we will restrict to bispectra of the local-type for simplicity (∆ = 0). The generalization beyond the simplifying assumptions

made here is straightforward and will be pointed out below where relevant.

–3–

Finally we can decompose the angular dependence f (µ) into multipole moments, where µ = kˆL · kˆS:

f (µ) =

AℓPℓ(µ) .

(2.2)

ℓ=0, 2, 4,···

It will become clear below why we neglect odd multipole moments. This then leads to Eq. (1.1). Standard local non-Gaussianity corresponds to L = 0 and A0 = 4fNloLc. We are interested in the two lowest order contributions: a monopole A0, and a quadrupole A2, which
corresponds to the anisotropic shape generated, for example, by solid inﬂation [34] or spin-2 particles [26]. Thus, fNloLc = 0 or “no local non-Gaussianity” will correspond to the statement that A0 = 0 in Eq. (2.2), while A2 = 0 corresponds to a vanishing quadrupole.
To interpret the eﬀect of the bispectrum deﬁned in Eq. (2.1), consider a single long-
wavelength plane wave potential perturbation φ(kL). According to Eq. (2.1), the local small-
scale power spectrum at position x is given by



Pφ (kS, x|φ(kL)) = 1 +

AℓPℓ(µ)

ℓ=0, 2, 4,···

kL kS

∆

φ(kL

)eikL

x


Pφ

(kS

)

,

(2.3)

where P (kS) is the Gaussian power spectrum. This can be easily veriﬁed by multiplying by φ(k′L), integrating over d3x, and taking the expectation value. Since φ is a real ﬁeld, Pφ(−kS) = Pφ(kS ) has to hold, and Eq. (2.3) implies that only even multipoles can contribute
to the leading squeezed-limit bispectrum Eq. (2.1).

3 Intrinsic alignments and non-Gaussianity

In this section, we derive the linear relation between intrinsic galaxy shapes and the underlying matter distribution in the presence of primordial non-Gaussianity, including the eﬀect of the anisotropic bispectrum. Intrinsic alignments can be treated in exactly the same way as galaxy biasing, the only diﬀerence is that projected shapes are a spin-2 quantity on the sky, as opposed to the scalar galaxy density. Thus, our discussion parallels that of [35–38] who studied the impact of primordial non-Gaussianity on galaxy clustering. We describe galaxy shape statistics in two stages. First, we provide a description of the statistics of the physical, intrinsic three-dimensional shapes of galaxies, denoted as gij. We will assume that gij is related to the trace-free part of the second moment tensor Iij of, for example, the intrinsic emissivity of a given galaxy:

gij = [Tr Ikl]−1

Iij

−

1 3

δij

Tr

Ikl

,

(3.1)

so that gij is dimensionless. Second, we project the shapes onto the sky plane, thus providing the connection to the observed two-dimensional shapes, and add the contribution due to gravitational lensing. Since the processes that determine galaxy shapes are intrinsically three-dimensional, we argue that a physical treatment of alignments should proceed in this way, rather than starting with an expansion on the projected galaxy shapes.
The precise location and shape of a given galaxy is determined by the nonlinear distribution of matter within some ﬁnite spatial region around the galaxy (along its past trajectory) described by a typical scale R∗. For example, if the dominant mechanism for determining position and shape is gravitational collapse, R∗ will be of the order of the Lagrangian radius of the parent halo of the galaxy. Since we are interested in the large-scale statistics of

–4–

galaxies, let us coarse-grain the galaxy distribution on some scale much larger than R∗ (but

smaller than the scale of correlations we are interested in). Then, the only properties of the

matter distribution that are relevant to galaxy formation are the matter density δ = ρ/ρ¯ − 1

and tidal ﬁeld Kij, coarse-grained over a scale R∗ (e.g., [39, 40]).3 We deﬁne the tidal ﬁeld

via

Kij

=

1 4πG ρ¯a2

∂i∂j

−

1 3

δij

∇2

Φ = Dij δ,

(3.2)

where Φ is the gravitational potential and

Dij

≡

∂i∂j ∇2

−

1 3

δij

.

(3.3)

The quantity Kij is trace-free, so that the six degrees of freedom contained in {δ, Kij} fully describe the second derivative tensor ∂i∂jΦ. Due to the equivalence principle, an observable such as galaxy shapes cannot depend on ∂iΦ or Φ itself.
Thus, on large scales, r ≫ R∗, the statistics of galaxy shapes and number, quantiﬁed by the fractional overdensity δn, can be eﬀectively described through local functions of the coarse-grained δ and Kij:

δn(x, τ ) = F [δ(x, τ ), Kkl(x, τ ), τ ] gij(x, τ ) = Gij[δ(x, τ ), Kkl(x, τ ), τ ] ,

(3.4)

where τ denotes conformal time. The leading correction to Eq. (3.4) due to the non-locality of galaxy formation is given by terms of order R∗2∇2δ and R∗2∇2Kkl, which we will neglect throughout this work. In Fourier space, they lead to a scale-dependent bias ∝ k2R∗2 which becomes relevant on small scales.
In Eq. (3.4), we have ignored the small-scale perturbations which are also present in the real universe. This is ﬁne in the case of Gaussian initial conditions, since the small-scale perturbations are not correlated with the long-wavelength perturbations [apart from terms that already appear in Eq. (3.4)]. As we will see, this will not be the case for non-Gaussian initial conditions.
A Taylor expansion of Eq. (3.4) then leads to the usual local bias expansion for the galaxy number density perturbation δn, i.e.

δn(x,

τ)

=

bn1 (τ )

δ(x,

τ)

+

1 2

bn2 (τ

)

δ2(x,

τ

)

+

1 2

bnt

(τ

)

(Kij

)2(x,

τ

)

+

·

·

·

.

(3.5)

The analogous expansion for gij, up to quadratic order, leads to

gij (x,

τ

)

=

bI1(τ )

Kij (x,

τ)

+

1 2

bI2

(τ

)

Kij (x,

τ )δ(x,

τ)

+

1 2

bIt

(τ

)

Kik

K

k j

−

1 3

δij

(Klm

)2

(x, τ )

+··· .

(3.6)

Note that we have allowed for all possible terms consistent with the symmetries of δn and gij, and in the absence of preferred directions in the galaxy frame. The coeﬃcient bni corresponds to the i-th order response of the mean number density of galaxies to a change in
3More accurately, the relevant properties are the coarse-grained density and tidal ﬁeld along the entire past trajectory of the ﬂuid elements. However, this is not relevant at the order up to which we work here. See [41, 42] for a discussion in the context of galaxy bias.

–5–

the background density, i.e. bni = (ρ¯i/n¯g)∂in¯g/∂ρ¯i. Similarly, bI1 corresponds to the linear response of the mean shape of galaxies to an external tidal ﬁeld Kij. Due to a lack of preferred directions in the frame of the galaxy, this response must have the simple form given in Eq. (3.6). That is, bI1 is simply a number rather than a tensor. Similarly, bI2 corresponds to the response of galaxy shapes to an external tidal ﬁeld and a simultaneous change in
the background density. Such a term is expected to be present intrinsically, however it is also induced by the weighting by the galaxy number density [43]. Finally, bIt quantiﬁes the quadratic tidal alignment. We emphasise that Eqs. (3.5)–(3.6) are eﬀective relations describ-
ing galaxy statistics on large scales rather than deterministic relations valid at a given scale.
We have also neglected stochastic terms in the bias relations, which have to be included in
general. The cubic and higher order terms neglected in Eqs. (3.5)–(3.6), while important on
small scales, are not relevant for the discussion here as they do not lead to scale-dependent
signatures of primordial non-Gaussianity on large scales.
Let us now consider the cross-correlation of galaxy shapes with the matter density ﬁeld
on large scales. Eq. (3.6) yields

δ(x)gij (y)

= bI1 δ(x)Kij (y)

+

1 2

bI2

δ(x)δ(y)Kij (y)

+

1 2

bIt

(τ

)

δ(x)

Kik

K

k j

−

1 3

δij

(Klm

)2

(y)

+··· ,

(3.7)

where the ellipsis stands for higher order correlators which are not relevant on the large scales we are interested in. The ﬁrst term in Eq. (3.7) is the leading term in the case of Gaussian initial conditions. It is given, in terms of the matter correlation function ξ(r), by

δ(x)Kij (y) = Dijξ(|x − y|) .

(3.8)

The second and third terms in Eq. (3.7) involving three-point functions are exclusively in-

duced by non-linear evolution in the case of Gaussian initial conditions, in which case they

are only relevant on small scales. Hence, the large-scale limit of the matter-shape cross-

correlation becomes

δ(x)gij (y) Gauss=ian IC bI1Dijξ(|x − y|) ,

(3.9)

or equivalently in Fourier space,

δ(k)gij (k′) = bI1

kikj k2

−

1 3

δij

Pm(k) (2π)3δD(k + k′) ,

(3.10)

where Pm(k) is the matter power spectrum. This implies that bI1 is the analog of the usual linear bias for galaxy density correlations. Eq. (3.10) corresponds to the prediction of the
so-called linear alignment model [1]. This is generically the leading contribution to galaxy shape correlations on large scales, unless bI1 happens to be very small for a given galaxy sample.
We now consider the case of non-Gaussian initial conditions of the local type, quantiﬁed
by Eq. (1.1). The three-point terms in Eq. (3.7) are sensitive to the bispectrum in the
squeezed limit, since two ﬁelds are evaluated at the same point. In the following, for simplicity of the presentation we will only focus on the ﬁrst term ∝ bI2; the term ∝ bIt behaves in essentially the same way (see App. A), so including it here would add nothing new. As
shown in App. A, for a squeezed-limit bispectrum of the form Eq. (1.1) we have

δ(x)δ(y)Kij (y)

=

2 5

A2Dij

ξδφ

(|x

−

y|)

δ2

,

(3.11)

–6–

where ξδφ(r) is the linear matter-primordial potential cross-correlation. Equivalently, in Fourier space,

δ(k)[δ Kij](k′)

=

2 5

A2

kikj k2

−

1 3

δij

M−1(k)Pm(k) δ2 (2π)3δD(k + k′) ,

(3.12)

where Pm(k) is the matter power spectrum in synchronous-comoving gauge, and

M(k, z)

=

2 3

k2T (k)D(z) ΩmH02

(3.13)

is the linear relation between primordial potential and density, δ(k, z) = M(k, z)φ(k), with T (k) denoting the transfer function and D(z) denoting the growth factor normalized to D(z) = 1/(1 + z) during matter domination. Thus, there is a non-Gaussian contribution to the large-scale shape correlation if the squeezed-limit bispectrum is anisotropic, A2 = 0.
However, Eqs. (3.11)–(3.12) involve the variance of the matter density ﬁeld δ2 , which is formally divergent if we let the smoothing scale go to zero, and hence needs to be regularized by introducing a counter-term. Equivalently, it corresponds to an unphysical dependence of our observable on the arbitrary smoothing scale. We see that the counter-term we need to add to the bias expansion of galaxy shapes, Eq. (3.6), is Dijφ. This term is not due to gravitational evolution, since it involves the potential itself and does not obey the equivalence principle, but is due to the initial conditions. Note that in Eq. (3.4) we have implicitly assumed that the statistics of the initial small-scale modes which inﬂuence local galaxy formation are the same everywhere. The squeezed-limit bispectrum of Eq. (1.1), however, precisely describes a modulation of the power spectrum of small-scale density ﬂuctuations by long-wavelength potential perturbations. This exactly parallels the case of galaxy density correlations in the case of local-type primordial non-Gaussianity [35]. The key diﬀerence for the generalized squeezed limit of Eq. (1.1) is that now the local power spectrum becomes anisotropic. Thus, in the case of primordial non-Gaussianity with A2 = 0 we have to explicitly take into account the dependence of galaxy shapes on the amplitude and anisotropy of small-scale ﬂuctuations.
Consider the following transformation of the initial matter power spectrum,

Pmin,iα(k) = 1 + 2αlmkˆlkˆm Pmin,iiso(k)

(3.14)

where αlm is trace-free and Pmin,iiso is the ﬁducial isotropic power spectrum. We can deﬁne the mean response of galaxy shapes to such an anisotropic initial power spectrum,

bING ≡

∂ gij α ∂αij

,
α=0

(3.15)

where no summation is to be taken (the absence of other preferred directions requires bING to be a number rather than a tensor). As we show in App. B, allowing for this dependence of
galaxy shapes on the anisotropy of the initial power spectrum in Eq. (3.4) provides the correct counter-term 3bINGA2Dijφ to cancel the divergence appearing in Eqs. (3.11)–(3.12) (the same holds for the divergence appearing in the other three-point term ∝ bIt ). This is not surprising, since according to Eq. (2.3) (with ∆ = 0) a long-wavelength potential perturbation yields a
modulation of the type Eq. (3.14) with

2αlm = A2

kˆLl kˆLm

−

1 3

δlm

φ(kL)eikLx = A2DlmφL(x) .

(3.16)

–7–

The end result is that the large-scale matter-shape cross-correlation becomes

δ(x)gij (y) = bI1Dij ξ(|x − y|) + 3bINGA2Dijξδφ(|x − y|) , and equivalently in Fourier space

(3.17)

δ(k′)gij (k) =

kikj k2

−

1 3

δij

bI1 + 3bINGA2 M−1(k) Pm(k) (2π)3δD(k + k′) .

(3.18)

This result is very similar to the well-known scale-dependent bias induced by local primordial non-Gaussianity in the two-point function of the galaxy density. Speciﬁcally, the galaxy density-matter cross-correlation is given by [37]

δ(k′)δn(k) =

bn1

+

1 2

bnNGA0

M−1(k)

Pm(k) (2π)3δD(k + k′) ,

(3.19)

where bnNG = n¯−g 1∂n¯g/∂ ln σ8 is deﬁned as the response of the galaxy abundance to a change in the (isotropic) initial power spectrum amplitude quantiﬁed by σ8. For dark matter halos and assuming a universal mass function, we can relate this to the linear density bias via bnNG = (bn1 − 1)δc.
Two facts are noteworthy about Eqs. (3.18)–(3.19). First, isotropic local type non-
Gaussianity (A0 = 0; A2 = 0) does not generate any large-scale correlations of galaxy shapes at linear order in the non-Gaussianity. The reason is that, due to symmetry, an isotropic
rescaling of the small-scale ﬂuctuations cannot aﬀect galaxy shapes. Second, there is no
impact of anisotropic non-Gaussianity (A0 = 0; A2 = 0) on large-scale galaxy density correlations. The number density of tracers depends on the amplitude of small scale ﬂuctua-
tions. It can be shown (App. A) that this is not modiﬁed by anisotropic non-Gaussianity.
Thus, at the level of two-point functions, only galaxy shapes are aﬀected by anisotropic
non-Gaussianity, while the same is true for the galaxy density and isotropic non-Gaussianity.
By comparing shape correlations with density correlations we can thus disentangle isotropic
from anisotropic shapes of the local type.
Finally, it is straightforward to generalize Eq. (3.19) to diﬀerent squeezed-limit power
scalings ∆. Speciﬁcally,

δ(k′)gij (k) =

kikj k2

−

1 3

δij

bI1 + 3bI01,∆A2

k kp

∆
M−1(k)

Pm(k) (2π)3δD(k + k′),

(3.20)

where kp is a pivot scale and we have indicated that now bI01,∆ depends on ∆ (and kp). For ∆ ≈ 2, the non-Gaussian contribution is only weakly scale-dependent and thus becomes more
diﬃcult to constrain.
To summarize, the large-scale correlation between matter and galaxy shapes has a Gaussian piece ∝ bI1, where bI1 represent the linear response of galaxy shapes to a long-wavelength tidal ﬁeld; and a non-Gaussian piece, which is proportional to the quadrupolar part of the squeezed-limit bispectrum. The coeﬃcient of the non-Gaussian term, bING, quantiﬁes the linear response of galaxy shapes to an anisotropic power spectrum of initial ﬂuctuations
[Eq. (3.14)]. Our derivations in this section represent the generalization to a spin-2 ﬁeld of
the well-known non-Gaussian halo bias [35, 37, 44].
While it is clear that the scale-dependent bias of galaxy shapes has to be present in general, it is diﬃcult to predict a precise number of bING for dark matter halos, let alone

–8–

0.05 0.00 −0.05

z=0.0,A2 = −10 z=0.0,A2 = 10 z=0.5,A2 = −10 z=0.5,A2 = 10 z=1.0,A2 = −10 z=1.0,A2 = 10

alignment bias

−0.10

−0.1150−3

k (h/Mpc)

10−2

Figure 1. The tidal bias induced by anisotropic non-Gaussianity, Eq. (3.21), for bI1 = −0.1ΩmD(0)/D(z), bING = −0.1Ωm and A2 = −10 (see Section 5). The increasing thickness of the curves represents diﬀerent redshifts from z = 0 to z = 1.5. At k > 10−2 h Mpc−1, the eﬀect of non-Gaussianity is negligible and all curves asymptote to −bI1. Anisotropic non-Gaussianity changes
the tidal bias on large scales and the eﬀect is larger at high redshift at a given scale.

galaxies. If it is very small, then this eﬀect becomes less interesting observationally. The value of bING could be measured by running N-body simulations with anisotropic initial power spectrum and measuring the mean shapes of halos along the anisotropy axes. This is beyond
the scope of this paper. In the context of the excursion set [45] however, one would expect that bING is comparable to the linear tidal alignment coeﬃcient bI1. In both cases the collapse threshold ν = δc/σ becomes direction-dependent, i.e. diﬀerent along diﬀerent axes. In the tidal case, this is due to the faster infall along the steeper potential gradient (e.g., [46]), while
an anisotropic power spectrum of small-scale ﬂuctuations will lead to a similar anisotropic ν. For our forecasts, we will thus assume that bING ∼ bI1. This should only be understood as an estimate at the order-of-magnitude level however. That is, while we expect the strongest
signature of anisotropic primordial non-Gaussianity to appear for those galaxies which show
the strongest linear tidal alignment (i.e. massive elliptical galaxies), the amplitude of the
two coeﬃcients could certainly diﬀer by a factor of several.
Figure 1 shows the total eﬀective bias of intrinsic alignments,

bI1 + 3bINGA2M−1(k, z) ,

(3.21)

i.e. the term in curly brackets multiplying Pm(k) in Eq. (3.18), as a function of scale. In the Gaussian case, it is a constant, while for A2 = 0 it is strongly scale-dependent. Here, we assume a ﬁducial value of bI1 = −0.1ΩmD(0)/D(z) (consistent with LRG measurements at low redshift [8]), and bING = −0.1Ωm (see Section 5). The impact of non-Gaussianity increases with redshift. Depending on the sign of A2, the anisotropic tidal bias can have the same sign as the isotropic linear alignment model (if A2 > 0) or the opposite sign (if A2 < 0). In the latter case, the eﬀect of the anisotropic tidal bias is to generate tangential alignments
on large scales. This transition scale can be estimated by simply setting Eq. (3.21) to zero, and it is k ≃ 1.5 × 10−3 h/Mpc for A2 = −10. The impact of this bias on angular shape correlations will be shown in the next section.

–9–

4 Observed shape and number count correlations

In this section we derive the implications of the scale-dependent bias in galaxy alignments for observed angular correlations of galaxy shapes and number counts. We will also include the contributions from gravitational lensing to shapes, which are comparable to or larger than alignments at cosmological redshifts. However, we will neglect the impact of primordial nonGaussianity on the lensing correlations, since they are highly suppressed by the projection along the lightcone [47]. This is discussed in detail in App. C.

4.1 Projection
In imaging surveys, one does not have access to the 3D shape of galaxies which we have written in Eq. (3.17). Instead, one only measures (in the absence of lensing) the intrinsic shape projected onto the sky:

γI,ij ≡

Pi

kPj l

−

1 2

Pij

P

kl

gkl ,

(4.1)

where

Pij = δij − nˆinˆj

(4.2)

is the projection operator and nˆ is the unit vector along the line of sight. Note that PijPjk = Pi k, and any term ∝ δkl in gkl drops out in the projection Eq. (4.1). In Eq. (4.1) we have introduced the notation γI,ij for the intrinsic shape, as we will also encounter the lensing contribution γG,ij below. The observed shape γij is the sum of both contributions.
Further, in imaging surveys galaxies follow a redshift distribution, dN/dz. Thus, fol-
lowing Eq. (3.17), the intrinsic alignment contribution to observed shapes of galaxies in the
direction nˆ are related to the density ﬁeld and gravitational potential through

γI,ij(nˆ) =

dz

dN dz

bI1Di⊥j δm(χ(z)nˆ; τ (z)) + 3bINGA2Di⊥j φ(χ(z)nˆ; τ (z))

,

(4.3)

where χ(z) is the comoving distance out to redshift z, τ (z) is the conformal time at that

redshift, and we have deﬁned the projection onto the sky of the derivative operator Dij

through

Di⊥j =

Pi kPj l

−

1 2

Pij

P

kl

Dkl .

(4.4)

Given the discussion of the renormalization approach in Section 3, Eq. (4.3) is not to be seen
as a truly local relation, but merely as formal relation in order to derive angular correlations.
Eq. (3.18) shows that the eﬀects of primordial non-Gaussianity on alignment correlations
are relevant on large scales. For this reason, we calculate proper full-sky angular correlations.
We begin by reviewing the formalism employed in the following to obtain full-sky correlations eﬃciently. γij(nˆ) is a traceless 2-tensor on the sky, with two degrees of freedom. Following the literature developed for CMB polarization [48–50], we decompose γij into components that transform with a spin s = ±2. Consider an orthonormal coordinate system (e1, e2, nˆ). If we rotate the coordinate sys√tem around nˆ by an angle ψ, so that ei → e′i, the linear combinations m± ≡ (e1 ∓ i e2)/ 2 transform as

m± → m′± = e±iψm±.

(4.5)

– 10 –

A general function f (nˆ) is spin-s if it transforms under the same transformation as

f (nˆ) → f (nˆ)′ = eisψf (nˆ).

(4.6)

An ordinary scalar function on the sphere is clearly spin-0, while the unit vectors m± deﬁned above are spin±1 ﬁelds. This decomposition is particularly useful for deriving multipole coeﬃcients and angular power spectra. For a summary of useful results, see App. A of [51]. We also deﬁne

k± ≡ mi∓ki E± ≡ mi∓mj∓Eij

(4.7)

for any 3-vector ki and 3-tensor Eij. We then decompose γij (this equally applies to the individual I and G contributions) as

γij = 2γ mi+mj+ + −2γ mi−mj− ±2γ = mi∓mj∓γij ,

(4.8)

where ±2γ are spin ±2 functions on the sphere (in analogy to Stokes parameters Q ± iU ). Consider the contribution of a single Fourier mode k of the density and potential, which we choose along the z axis, to ±2γI (nˆ):

±2γI (nˆ, kˆz) =

dz

dN dz

k± k

2
bI1δm(k; τ (z)) + 3bINGA2φ(k; τ (z)) eik·nˆχ(z)

=

−

1 2

(1

−

µ2

)

dz

dN dz

bI1δm(k; τ (z)) + 3bINGA2φ(k; τ (z)) eixµ ,

(4.9)

where µ is the cosine between kˆ and nˆ, and we have deﬁned x ≡ kχ(z). We have also used

that Pijmi±mj± trivially under

= 0. Since ±2γ are rotations. Instead,

not scalar quantities on the sky, they do not transform we apply spin-raising ð2 and lowering ð¯2 operators to

obtain a scalar on the sky (see App. A of [51] for details). Speciﬁcally, for a spin-2 quantity

such as +2γ, the spin-lowering operator is given by [see Eq. (A8) of [51] for m = 0]

ð¯2

+2γI

=

∂2 ∂µ2

(1 − µ2)+2γI (nˆ, ˆz)

=

1 2

dz

dN dz

bI1δm(k; τ (z)) + 3bINGA2φ(k; τ (z))

(1 + ∂x2)2

x2eixµ

.

(4.10)

In the last line, we have used

∂2 ∂µ2

(1 − µ2)2eixµ

=

∂2 ∂µ2

(1 + ∂x2)2eixµ

= −(1 + ∂x2)2

x2eixµ

.

(4.11)

This derivative operator is real, which reﬂects the fact that the linear alignments from scalar
perturbations considered here do not produce B-mode shape correlations even in the presence
of anisotropic non-Gaussianity. B-modes will be generated on small scales by the nonlinear alignment terms written in Eq. (3.6), however. Applying the spin-raising operator ð2 to −2γI yields exactly the same result. Crucially, ð¯2 +2γI as well as ð2−2γI transform as scalars on the sky, which allows us to evaluate their two-point function (angular power spectrum) in a

– 11 –

straightforward way. The angular power spectrum of ð¯2 +2γI is then directly proportional to that of γI , which is our ﬁnal goal (see Section 4.2).
We emphasize that Eq. (4.10) is the exact, proper relativistic expression of intrinsic shapes on linear scales. This is because, as shown in [40], the tidal ﬁeld deﬁned in conformalNewtonian gauge corresponds to the physical, locally observable tidal ﬁeld at linear order. The lensing contribution discussed in Section 4.4 below then corresponds to the mapping of the intrinsic locally observable shape to our observations from Earth. Note also that redshift-space distortions do not aﬀect alignments at linear order, since at that order they only modify the physical volume which does not aﬀect shapes (see also [21]).
Correspondingly, we can also write down the observed galaxy overdensity at linear order:

δn(nˆ, kˆz) =

dz

dN dz

bn1 δm(k;

τ (z))

+

1 2

bnNGA0φ(k;

τ (z))

eixµ .

(4.12)

While the expression for the intrinsic shapes Eq. (4.10) is exact in the linear regime, we have neglected several linear-order contributions in the galaxy overdensity:

• Redshift-space distortions: We will consider photometric galaxy samples throughout, for which RSD are a subdominant eﬀect. Further, at l = 2 it is inconsistent to only include the linear RSD term without further relativistic corrections (see below).

• Magniﬁcation bias: This contribution, while straightforward to include, is typically subdominant and furthermore aﬀected negligibly by primordial non-Gaussianity as shown in [47] (see also App. C).

• Relativistic corrections: There are corrections to the galaxy clustering kernels (e.g., [52] and App. B in [53]) that scale as (aH/k) and (aH/k)2. The latter terms are of the
same form as local-type non-Gaussianity with A0 1 for typical sample parameters.

These simpliﬁcations are justiﬁable, since the focus of this paper is shape correlations rather than clustering, and since these approximations are not expected to have a signiﬁcant impact on the forecasted constraints for A2.

4.2 Intrinsic shape correlations
We can now apply the results of App. A1 of [51] to obtain the angular power spectra of galaxy overdensity and shapes, and their cross-spectrum. In the following, we will consider three diﬀerent redshift distributions of galaxies in order to keep the treatment general: dNG/dz denotes the redshift distribution of all galaxies with measured shapes; dNred/dz denotes the subset of these galaxies that are early-type and known to show linear alignment of the type considered here (a linear alignment has so far not been detected for late-type galaxies). Finally, dNn/dz denotes the galaxy sample used in the measurement of the clustering. For our main results, we will assume that Nn, Nred and NG are the same population of red galaxies. We summarize our model for this population in Section 5.
Eq. (4.10) is a special case, r = 0, of Eq. (A19) in [51], and Eq. (A24) there (for |s| = 2 and NP = 1, where NP denotes the number of polarization states) immediately yields the

– 12 –

10−7

10−7

10−8

10−8

CI (l) CI (l)

10−9

10−9

10−10 10−112

46

10 15
l

(a) A2 = 10.

A2 = 0 A2 = 10
30 50

10−10 10−112

46

10 15
l

(b) A2 = −10.

A2 = 0 A2 = −10
30 50

Figure 2. The angular auto-power spectrum of intrinsic shapes (without accounting for weak gravitational lensing) for our ﬁducial choice of bING and A2 = +10 (left) and for A2 = −10 (right). The Gaussian case is shown in solid black for three redshifts, z = {0.5, 1, 1.5} from top to bottom. We
show the case with anisotropic non-Gaussianity at the same redshifts (dashed).

auto-power spectrum of the intrinsic component of the shapes:

CI (l)

=

2 π

(l (l

− +

2)! 2)!

k2dk Pm(k)|FlI (k)|2

FlI (k) =

dz

dNred dz

bI1 + 3bINGA2M−1(k, z)

D(z) D(0)

(1 + ∂x2)2 x2jl(x) x=kχ(z)

=

(l (l

+ −

2)! 2)!

dz

dNred dz

bI1 + 3bINGA2M−1(k, z)

D(z) D(0)

jl(x) x2

.
x=kχ(z)

(4.13) (4.14)

Here, Pm(k) is the matter power spectrum at z = 0, and M(k, z) is deﬁned in Eq. (3.13). jl denotes the spherical Bessel function of order l. Note that no B-modes are generated by linear scalar contributions. Also, by symmetry the lowest non-zero multipole of shape correlations is the quadrupole l = 2.
Similarly, we can write down the cross-correlation between galaxy overdensity and shape E-modes (again, only the intrinsic part). This is the multipole space version of the densitytangential shape correlation function in real space, and reads

CnI (l)

=

2 π

(l − 2)! (l + 2)!

k2dk Pm(k)FlI (k)Fln(k)

Fln(k) =

dz

dNn dz

D(z) D(0)

bn1

+

1 2

bnNGA0M−1

(k,

z)

jl(x) x=kχ(z) .

The clustering auto-power spectrum, on the other hand, is simply given by

Cnn(l)

=

2 π

k2dk Pm(k)|Fln(k)|2 .

Eqs. (4.13)–(4.17) are valid on the full sky.

(4.15) (4.16)
(4.17)

– 13 –

4.3 Signatures of non-Gaussianity
We now illustrate the eﬀect of isotropic and anisotropic non-Gaussianity on the angular power spectra of galaxy counts and the intrinsic contribution to shapes (the lensing contribution will be considered in the next section). Depending on the sign of A0 and A2, we obtain an enhancement or suppression of the angular correlations at low multipoles (as expected from Figure 1). In this section, we will assume the same values for the bias parameters as for Figure 1, i.e. bI1 = −0.1ΩmD(0)/D(z) and bING = −0.1Ωm.
By deﬁnition, the intrinsic shape auto-power spectrum (Figure 2) is positive regardless of anisotropic non-Gaussianity. This power spectrum decreases with redshift, as expected from Eq. (4.20), from z = 0.5 (top) to z = 1.5 (bottom). Anisotropic non-Gaussianity changes the power spectrum for small multipoles (l ≤ 20) and asymptotes to the Gaussian case on small angular scales. The fractional eﬀect increases towards higher redshifts.
An interesting and novel eﬀect of primordial non-Gaussianity appears in the quadrupole of shape correlations for A2 = 0: consider the term ∝ (A2)2 in CI (l). Focusing just on the k-dependence at an eﬀective source redshift zs, this contribution to CI (l) is proportional to

CI (l) ∝

k2dk Pφ(k)(bINGA2)2

jl(x) x2

.
x=kχ(zs)

(4.18)

Now consider the quadrupole l = 2. We see that the factor in brackets asymptotes to a constant for kχ(zs) ≪ 1, and this term becomes, for k ≪ 1/χ(zs),

CI (l) kχ∝≪1 (bINGA2)2 k2dk Pφ(k) .

(4.19)

This quantity diverges logarithmically in the IR for a near scale-invariant spectrum of primor-
dial ﬂuctuations. Thus, the quadrupole of galaxy shapes depends on the lowest wavenumber
kmin at which non-zero anisotropic primordial non-Gaussianity is present, no matter how small. In principle, a measurement of CI (l = 2), together with the higher l which constrain A2, would thus allow us to measure this low k cut-oﬀ, which is presumably related to the beginning of inﬂation (kmin ∼ 1/τinf where τinf is the conformal time at the beginning of inﬂation). Quantitatively, the contribution of such super-survey modes is roughly of order (A2)2As ln[τinf/χ(zs)], where we have neglected the contribution from the small tilt ns − 1. Given the smallness of the amplitude of ﬂuctuations As ∼ 10−10, this is probably only observable for an extremely long period of inﬂation with correspondingly large τinf (or, alternatively, if the ﬂuctuations near the begining of inﬂation happened to be much larger
than the extrapolation of the near scale-invariant spectrum). For deﬁniteness, we choose kmin = 2 × 10−8 h Mpc−1 for our numerical results; for A2 = 10, CI (l = 2) decreases by roughly a factor of 3 when increasing kmin to 10−5 h Mpc−1.
It might be surprising that modes that are far outside our current horizon aﬀect observ-
ables. Physically, this is because for A2 = 0, the matter power spectrum in our observable universe is anisotropic and has a spatially constant quadrupole with an expectation value proportional to (A2)2 k2dk Pφ(k). Galaxy shapes respond to this anisotropy, as quantiﬁed by bING, and the spatially constant power spectrum quadrupole will lead to a corresponding quadrupole of galaxy shapes. Related eﬀects of superhorizon modes have been found in
[54, 55]. Note that a similar eﬀect also exists for isotropic non-Gaussianity A0 = 0, where the monopole of the galaxy overdensity similarly receives a contribution ∝ (A0)2 k2dk Pφ(k). However, this eﬀect is unobservable, since the amplitude of scalar ﬂuctuations themselves

– 14 –

100 10−1

A0 = 0, A2 = 10 A0 = 10, A2 = 0 A0 = 10, A2 = 10

0.2 0.0 −0.2

∆CnI /CnI ∆CnI /CnI

10−2 −0.4

−0.6

A0 = 0, A2 = −10 A0 = 10, A2 = 0

10−32

46

10

20

50

A0 = 10, A2 = −10

2

46

10

20

50

ll

(a) Positive A2 combined with A0 = {0, 10} and com- (b) Negative A2 combined with A0 = {0, 10} and

pared to the Gaussian case.

compared to the Gaussian case.

Figure 3. Fractional change in the angular cross-spectrum of galaxy positions and intrinsic shapes at z = 1 for diﬀerent levels of non-Gaussianity and our ﬁducial choice of bING. We assume bn1 = 2 and bnNG = (bn1 − 1)δc. The left panel shows results for A2 = +10, while the right panel shows A2 = −10.
In addition, we show results for isotropic non-Gaussianity, A2 = 0, A0 = +10 in both panels.

receives the same correction, which is thus absorbed in the normalization As. In this regard, anisotropic PNG and galaxy shapes in principle oﬀer qualitatively new information over the well-known isotropic case.
When the cross-spectrum of galaxy positions and intrinsic shear is considered, both A2 and A0 have an impact on the large-scale correlations. Figure 3 shows the eﬀect of the diﬀerent types of non-Gaussianity on this cross-spectrum and assuming bn1 = 2. For A2 < 0 (right panel), the correlation between galaxy positions and shapes changes sign on large scales, corresponding to a transition from radial to tangential alignment. That is, intrinsic shapes in this case have the same sign as the lensing signal, which is also tangential behind overdensities. If A2 > 0, non-Gaussianity strengthens the eﬀect of the tidal ﬁeld, enhancing the radial alignment of red galaxies. In this case, there is no sign change. We emphasize that both A2 and A0 are theoretically allowed to have either positive or negative signs.

4.4 Lensing shear
Correlations in galaxy shapes are not only present intrinsically through tidal alignments, but are also induced by gravitational lensing (in particular, the shear; see [56] for a review). In general, these two eﬀects are diﬃcult to separate, making it necessary to model them jointly [4]. We thus include the eﬀect of gravitational lensing in the shape correlations in our quantitative forecasts of Section 7. At linear order, the alignment (I) and lensing (G) contributions are simply additive, so that the full shape auto-power spectrum becomes

Cγγ (l)

=

2 (l − 2)! π (l + 2)!

dk k2 Pm(k)|FlI (k) + FlG(k)|2 ,

(4.20)

where the lensing shear kernel for a distribution of source redshifts dNG/dz is given by (e.g., App. F of [51])

FlG(k)

=

1 (l + 2)! 2 (l − 2)!

χmax 0

dχ

k2

DΦ(k,

z(χ))

jl(x) x2

χ

χmax dχ˜H(χ˜) dNG (χ˜ − χ) , (4.21) χ dz χ˜

– 15 –

and DΦ is given by

DΦ(k, z)

=

3H02Ωm,0 k2

(1

+ z)D(z) D(0)

.

(4.22)

The expression in Eq. (4.20) includes the intrinsic alignment auto-power spectrum, crosscorrelations between weak lensing shear and intrinsic shapes, and cosmic shear. The correlation between intrinsic shapes and alignments occur when a galaxy in the background is being lensed by the same matter density ﬁeld that tends to align the second galaxy [2, 57]. Similarly, the density-observed shape cross-correlation becomes

Cnγ (l)

=

2 π

(l − 2)! (l + 2)!

k2dk Pm(k) FlI (k) + FlG(k) Fln(k) .

(4.23)

As this is an important point, we reiterate that the result in Eq. (4.20), with Eq. (4.14) and Eq. (4.21), is exact on linear scales, i.e. it does not receive further relativistic corrections.4 For the intrinsic contribution, we have argued this in Section 4.1. For the lensing contribution,
Ref. [51] showed that the seemingly “Newtonian” expression for the shear in conformal-
Newtonian gauge [Eq. (4.21)] is the proper relativistic expression for the observable shear
at linear order. Note that this does not hold for the convergence κ, where the observable
magniﬁcation in fact receives several relativistic corrections. As discussed above, Eq. (4.23) assumes several simpliﬁcations in the number count kernel Fln(k).
For the numerical implementation, we use the Limber approximation for l ≥ 60 for
all angular power spectra, following [58]. At these multipoles, the Limber approximation
provides an estimation of the angular power spectra with suﬃcient accuracy for our purposes,
and yields better numerical convergence. We have veriﬁed that using the full integrals up to
l = 400 does not change our results. For the quadrupole (l = 2), we use a value of kmin = 2 × 10−8 h Mpc−1 as minimum wavenumber, which matters for the shape auto-correlation as
discussed above. For all other multipoles the value of kmin is chosen small enough to ensure a converged result.

5 Modeling the red galaxy population and alignments

Red galaxies have been observed to be subject to intrinsic alignments with a scale dependence that is well approximated by the linear alignment model (Eq. 3.10) on scales above ∼ 10 Mpc/h [3, 8–10, 57]. On the other hand, late-type galaxies do not exhibit detectable alignment on large scales [11, 12]. Thus, in order to estimate the intrinsic alignment power spectrum as a function of redshift, it is necessary to model the red galaxy fraction.
The fraction of red sources,

fred(z)

=

dNred/dz dNG/dz

,

(5.1)

depends on the intrinsic properties of red galaxies (their spectral energy distribution and luminosity function) and on the properties of the survey (ﬁlter, magnitude limit, source redshift distrubution). We use the model constructed by Ref. [3] and applied by Ref. [59] to obtain the fraction of red galaxies and their mean i-band luminosity as a function of redshift for an LSST-like survey. The source redshift distribution expected for LSST, which

4This assumes that certain potential gauge artefacts have been removed from A2 [40].

– 16 –

0.35 0.30

Red fraction, ilim = 25.3 dNred/dz

0.25

Red fraction

0.20

0.15

0.10

0.05

0.000.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Redshift

Figure 4. The expected fraction of red galaxies observed by LSST as a function of redshift for a limiting magnitude of ilim = 25.3 (AB). The gray dashed curve shows dNred/dz with an arbitrary normalization.

extends from z = 0.1 to z = 3, is given in [60]; there are 26 galaxies per arcmin2 and we assume an i-band limiting magnitude of 25.3 (AB), consistent with the so called ‘gold sample’ of galaxies with signal-to-noise ratio > 20 and measured shears in LSST [61]. We assume a red galaxy luminosity function as observed by Ref. [62] from a combination of galaxy surveys between 0.2 < z < 1.2. In our model, the paucity of red galaxies at z > 1.5 implies that intrinsic alignments have no contribution to the total shear above those redshifts. The redshift distribution of red galaxies in LSST using this model is shown in Figure 4. Throughout, we assume a survey with an area coverage of 18, 000 square degrees, since the eﬀect of masking is accounted for in the eﬀective number density of galaxies. When projected over the range z = [0, 1.5], LSST should observe 3 red galaxies per arcmin2. Notice that this is optimistic because we are not modeling any additional selection cuts on galaxy shapes that could potentially aﬀect the red population. In our model, blue galaxies are assumed to have no alignments.
We can relate the linear shape bias bI1 to the convention commonly used in papers reporting on measurements of intrinsic alignment as follows:

bI1

=

−AI (Lr)ΩmC1ρcrit

D(0) D(z)

,

(5.2)

where Ωm is the matter density parameter today and C1ρcrit = 0.0134 is ﬁxed by convention to the SuperCOSMOS intrinsic alignment amplitude [63]. Later on, we will treat C1ρcrit as a free parameter when making predictions of non-Gaussianity constraints from future surveys.
AI is a luminosity-dependent amplitude measured by Ref. [3] from a sample of Luminous Red Galaxies (LRGs) in the Sloan Digital Sky Survey [64],

AI (Lr) = (5.76+−00..6602)

Lr L0

1.13+−00..225
,

(5.3)

where Lr is the average r-band luminosity at a given redshift and L0 is a pivot luminosity corresponding to an absolute magnitude of M0(z) = −22 − 1.2z − 5 log10 h. We emphasize

– 17 –

that ongoing imaging surveys, such as the Kilo-Degree Survey5, the Dark Energy Survey6,
the PAU survey7 and Hyper-Suprime Cam8 will be able to test the assumptions on the
dependence of bI1 with redshift and luminosity we have made here. Unlike in the case of the scale-dependent bias in galaxy number counts, bnNG ≈ (bn1 −1)δc,
for which a fairly accurate estimate is known, no such estimate exists for the response of
shapes to the power spectrum anisotropy quantiﬁed by bING. We will thus assume that bING is comparable to bI1 at any given redshift, speciﬁcally

bING

=

˜bING

×

bI1(z)

D(z) D(0)

(5.4)

where we assume throughout ˜bING = 1 but keep it explicit in order to illustrate the residual modeling uncertainty.
Note that the eﬀect of the luminosity dependence of the intrinsic alignment bias was
not taken into account in Section 4.3 above, but will be included in the quantitative forecast.
Its most important consequence is the existence of a Malmquist-like bias for the population
of intrinsically aligned galaxies in a magnitude-limited survey. As a result, the red galaxy population at higher redshifts has a larger bI1, which results in a larger contribution from anisotropic non-Gaussianity compared to the case presented in Section 4.3, where bI1 was set to a constant.

6 Fisher matrix analysis

In order to estimate the ability of future surveys to set joint constraints on isotropic and
anisotropic non-Gaussianity, we perform a Fisher matrix analysis over the parameter set {bn1 , A0, A2, C1ρcrit, σ8}. The remainder of the cosmological parameters are ﬁxed to their values given at the end of Section 1. Further, we include a prior in σ8 = 0.831 ± 0.013 from Planck [32, Table 3, column 4].
The predictions for the angular auto- and cross-power spectra were given in Section 4.
All auto-power spectra also contain noise terms, which we assume to be white,

Nγ

=

σγ2 n¯G

and

Nn

=

1 n¯n

,

(6.1)

where σγ2 is the dispersion of the intrinsic shape including measurement noise per component, and n¯G is the projected surface density of galaxies with shapes per steradian. Similarly, n¯n is the surface density of galaxies used to compute the clustering signal.
Since all perturbations are Gaussian at the order we work in, the covariance between
two power spectra is given by

Cov[Cαβ (l),

Cγδ (l′)]

=

δKll′

(2l

1 + 1)fsky

[Cαγ (l)Cβδ(l)

+

Cαδ(l)Cβγ (l)]

where the indices {α, β, γ, δ} run over {n, G}, i.e. number densities and shapes. The Fisher matrix is given by [65, 66]

(6.2)

5http://kids.strw.leidenuniv.nl 6http://www.darkenergysurvey.org 7“Physics of the Accelerating Universe”, http://www.pausurvey.org/home-PAU.html 8http://www.naoj.org/Projects/HSC/

– 18 –

Fµν =

l

∂D(l) ∂pµ

Cov−1

(l)

∂D(l) ∂pν

(6.3)

where Cov is the covariance matrix of the data vector D = {Cnn, Cnγ, Cγγ} and ∂µ indicates

the partial derivative with respect to the parameters of interest. We perform these derivatives

analytically on the expressions for Cαβ(l), and evaluate the results numerically. In practice,

A2 cannot the power

bspeeccotnrastraarieneedﬀeicntdiveepleyndcoenntsltyraoinf i˜bnINgG.thWe hpirloedwuectas˜bsINuGmAe2˜.bINTG h=e

1, this means that 1σ uncertainty in

each parameter, after marginalizing over all other parameters, are given by σµ,full−margin =

(F −1)µµ. We also consider forecasted errors after marginalizing only over {bn1 , C1ρcrit, σ8},

which allows us to determine whether degeneracies between A0 and A2 exist. If A0 and A2

correspond to coordinates 1 and 2 of the Fisher matrix, we obtain the partially-marginalized

bounds from deﬁning the following sub-matrix,

(G−1)i,j = [F −1]i=[1,2],j=[1,2]

(6.4)

and then the partially-marginalized uncertainties are given by

σi,part−margin = Gi−i1/2.

(6.5)

7 Constraining non-Gaussianity with clustering and shapes

In this section, we forecast the ability of galaxy imaging surveys to detect anisotropic non-
Gaussianity using galaxy shapes, in particular those of red galaxies. We consider an LSST-like survey with 26 galaxies per arcmin2 [60] and we model the red galaxy population as described
in Section 5. We assume that the dispersion in the intrinsic ellipticities per component is
σγ = 0.3 [67]. We do not account for dispersion due to photometric redshift scatter in the red galaxy population in this section, since we are integrating over the full redshift range of the
red galaxy population. Figure 5 shows the position-shape and shape-shape angular power
spectra and their expected uncertainties as a function of multipole number for the cases with ˜bINGA2 = 0 and ˜bINGA2 = 100. Clearly, the shape correlations are dominated by the lensing contribution, and the fractional eﬀect of anisotropic non-Gaussianity is signiﬁcantly smaller
than in the intrinsic-only correlations shown in Figures 2 and 3.
We combine the clustering, galaxy-lensing and cosmic shear information for the red galaxy population to obtain constraints on {C1ρcrit, bn1 , A0, (˜bINGA2), σ8}. Notice that we assume that the luminosity and redshift dependence of the intrinsic alignment model is
ﬁxed by current observational data, but we allow the overall amplitude of alignments to be
constrained through changing the normalization given by C1ρcrit (no priors are considered). Analogously, we assume a ﬁducial constant, redshift-independent bias for red galaxies, bn1 = 2, that is allowed to vary in the Fisher analysis. We obtain the following fully marginalized 1σ
bounds on the uncertainties in the parameter set from the multipole range 2 ≤ l ≤ 600:

∆(C1ρcrit) = 8.6 × 10−4, ∆bn1 = 2.6 × 10−2,
∆A0 = 44, and ∆(˜bINGA2) = 150.

(7.1)

– 19 –

Cnγ Cγγ

7 ×10−7 6 5 4 3 2 1 0 −1 −2100

A2 = 0 A2 = 100

101
l
(a) Cnγ (l)

102

1.4 ×10−7 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.2 −0.4100

101
l
(b) Cγγ (l)

102

Figure 5. Position-shape correlations and shape-shape correlations of red galaxies expected to be observed in LSST between 0.1 < z < 1.5. The case of Gaussian initial conditons is shown in solid black, and the case with A2 = 100 is shown as dashed gray lines. The error bars show the expected uncertainties for a Gaussian ﬁducial cosmology.

Our forecast for the uncertainty in A2 (assuming ˜bING = 1) is somewhat worse than the current CMB constraints, ∆AC2 MB ∼ 90 [31, 68, at 68% conﬁdence level]. We emphasize however that galaxy alignments probe diﬀerent scales than the CMB, and thus these constraints should
be seen as complementary to the CMB in view of a possible scale-dependence of the non-
Gaussianity. This applies in particular to the small-scale modes, where the modes relevant for galaxy formation are kS 1 h Mpc−1, while the CMB probes modes with kS 0.1 h Mpc−1.
Note further that the constraints depend on the ﬁducial value of A2 chosen, where our choice of ﬁducial A2 = 0 is conservative. This is because the terms ∝ (A2)2 do not contribute to the constraints in this case. In particular, this applies to the quadrupole of
the shape auto-power spectrum, which diverges as kmin → 0 as discussed in Section 4.3, and is the most prominent non-Gaussian feature in Figure 5. Thus, the true log-likelihood is
steeper than the quadratic approximation around A2 = 0 suggests, so that even in case of a non-detection the actual constraints obtained for a survey with these speciﬁcations could be
tighter. Further, in case of a detection of non-zero A2 with shape correlations, the amplitude of Cγγ(l = 2) could be used to constrain kmin via its logarithmic dependence on it.
For the well-known isotropic non-Gaussianity on the other hand, our results indicate an uncertainty of ∆fNloLc = 11. This latter constraint is signiﬁcantly worse than what is expected for Stage IV spectroscopic surveys, since we consider only a single redshift bin which loses
information on all line of sight modes. We will return to this issue at the end of this section. Figure 6 shows likelihood contours in the A0 − ˜bINGA2 plane, obtained by marginalizing
over C1ρcrit, σ8 and bn1 . They correspond to {1, 2, 3}σ conﬁdence intervals. We ﬁnd that including all auto- and cross-correlations of number counts and shapes almost entirely removes
any degeneracy between the two non-Gaussianity parameters.
Given our linear treatment of the matter power spectrum, we consider the eﬀect of
restricting the forecasts to lower multipoles (note however that nonlinear corrections to the
lensing contribution are straightforward to incorporate). Figure 7 shows the forecasted un-
certainty in the anisotropic non-Gaussianity parameter as the maximum l is increased. This
ﬁgure shows that although the eﬀect of non-Gaussianity is most signiﬁcant on large scales, the small scales still contain signiﬁcant information in particular on bn1 and C1ρcrit which are

– 20 –

300

200

100

b˜INGA2

0

−100

−200

−30−0100

−50

0

50 100

A0

Figure 6. {1, 2, 3}σ uncertainties in A0 and ˜bINGA2 (thick to thin black, blue and red solid lines, respectively) marginalizing over C1ρcrit, σ8 and bn1 . These constraints are obtained from considering
the combination of clustering, position-shape correlations and shape-shape correlations and using the
redshift distribution of red galaxies expected to be observed in LSST between 0 < z < 1.5.

helpful in improving the constraints on the non-Gaussianity parameters. While extending the modeling of the power spectra to nonlinear scales is beyond the scope of this work, there is thus potential for better constraints on the non-Gaussianity parameters from l > 600. However, the eﬀects of baryonic physics on the nonlinear lensing and alignment power spectrum on those scales are still a matter of debate (see, for example, [69]). On the other hand, removing scales below l < 10 while keeping lmax = 600 ﬁxed increases the uncertainty in the marginalized ∆(˜bINGA2) by a factor of 1.6. Not surprisingly, the very low multipoles are important for constraining primordial non-Gaussianity, requiring careful control of the systematics in imaging surveys.
In our constraints we have included the full redshift distribution of galaxies, thus suppressing any information contained in modes along the line of sight. This is the reason why our forecasted constraint on fNloLc is roughly an order of magnitude worse than what is obtained from spectroscopic surveys of similar volume. Given that, unlike the lensing ﬁeld, the tidal alignment is an intrinsically three-dimensional eﬀect, we thus expect a potential signiﬁcant improvement over the constraints quoted here by using a full tomographic analysis rather than a single redshift bin, which would restore information on the line-of-sight modes. This has been found to hold for the scale-dependent bias in the galaxy density induced by fNloLc [70, 71], and for IA in the case of Gaussian initial conditions [7, 72]. Further, tomography should help break any degeneracy between the alignment strength and cosmological parameters such as σ8, which we have not considered here. In fact, a much smaller spectroscopic sample of red galaxies with well-measured shapes could deliver interesting constraints as well. We defer the forecast for the full tomographic and spectroscopic cases to future work.
8 Discussion and conclusions
We have shown that the intrinsic alignment of red (elliptical) galaxies will become an interesting probe of inﬂationary physics in the future. Through their sensitivity to a quadrupolar

– 21 –

400

350

∆(b˜I01A2)

300

250

200

150 101

102
Maximum l

Figure 7. Bound on ∆(˜bINGA2) after marginalizing over all other parameters and as a function of the maximum multipole considered (holding the minimum l ﬁxed at 2).

anisotropic mode coupling, galaxy shapes expand the parameter space of inﬂationary models that can be probed using large-scale structure observables. The anisotropic squeezed-limit bispectrum translates into a local anisotropy in the matter power spectrum, which imprints itself in the shapes of halos and galaxies forming at that location. This non-Gaussian bias produces a tendency towards radial or tangential alignments, depending on the sign of the non-Gaussianity, which increases rapidly towards large scales. This is the analog in galaxy shapes of the familiar scale-dependent bias in galaxy clustering induced by isotropic primordial non-Gaussianity.
As a speciﬁc example, models such as solid inﬂation, the realization of inﬂation through the symmetries of a solid [22], could be probed through this method. More generally, as was recently pointed out by Ref. [26], the angular dependence of the bispectrum encodes information about the spin of light particles during inﬂation. Therefore, constraints on the quadrupolar part of the bispectrum, analyzed in this work, provide information about tensor degrees of freedom during inﬂation. Moreover, we have shown that should anisotropic non-Gaussianity be detected, the quadrupole of galaxy shapes can be used to constrain the maximum scale at which this mode coupling persists, which is presumably related to the beginning of or some other special event during inﬂation. This is a unique possibility of probing physics outside of our horizon, which is not possible with isotropic non-Gaussianity.
We have modeled the red fraction of galaxies in the LSST survey following the approach suggested by [3], and conservatively assumed that late-type (blue) galaxies do not align, which is consistent with currently available constraints [11, 12]. We have shown that the constraints on A2 for future surveys are comparable to, though slightly worse than, current CMB constraints. Speciﬁcally, we ﬁnd ∆A2 = 150(˜bING)−1 for LSST, while current constraints on A2 from Planck require −80 < A2 < 110 at the 68% conﬁdence level. Note that this constraint was derived for a ﬁducial A2 = 0, which neglects any terms scaling as (A2)2. To achieve these constraints, imaging surveys will need to access scales below l < 10, otherwise the constraint worsens by a factor of 1.6. However, a three-dimensional reconstruction of the auto and cross-spectra of positions and shapes could potentially yield signiﬁcantly tighter

– 22 –

constraints on both A2 and A0, as the alignment process is intrinsically three-dimensional, in contrast to lensing. Just as importantly, the constraints on A2 from galaxy shapes are to be considered complementary to those from the CMB, as they come from diﬀerent physical scales (k 1 h Mpc−1 vs k 0.1h Mpc−1).
One key caveat to these forecasts is the unknown value of the parameter bING, which quantiﬁes the response of galaxy shapes to an anisotropic initial power spectrum. In this work, we have assumed that bING is of the same order as the tidal alignment strength bI1 which has been measured observationally. Ideally, bING should be estimated from hydrodynamic simulations of galaxy formation, by running a simulation with an anisotropic initial power spectrum and measuring the response of halos and galaxies to this preferred direction (also as a function of luminosity or mass). For the purpose of this work, we chose to simply phrase our constraints in terms of the combination ˜bINGA2, where ˜bING ∼ 1 if our assumption is valid.
Acknowledgments
We are thankful to Michael Strauss, Rachel Mandelbaum and Simon White for useful comments and discussions. CD was supported by NASA through Hubble Fellowship grant HSTHF2-51340.001 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. This work has made use of the Python package mpmath [73].

A Derivation of Eq. (3.11)

The left-hand side of Eq. (3.11) is given in terms of the potential three-point function by

δ(x)δ(y)Kij (y) =

d3k (2π)3

eik·rM(k)

d3k1 (2π)3

d3k2 (2π)3

k1ik1j k12

−

1 3

δij

× M(k1)M(k2) φkφk1 φk2 ,

where r = x − y, and M is deﬁned in Eq. (3.13). Using the deﬁnition of the bispectrum,

φkφk1 φk2 = (2π)3δD(k + k1 + k2)Bφ(k, k1, k2),

(A.1)

we have

δ(x)δ(y)Kij (y) =

d3k (2π)3

eik·rM(k)

d3k1 (2π)3

k1ik1j k12

−

1 3

δij

× M(k1)M(|k + k1|)Bφ(k, k1, |k + k1|) .

(A.2)

We now expand the integrand in powers of q = k/k1. Further, we deﬁne µ = kˆ · kˆ1. As shown in App. C of [35], but generalized to the anisotropic case Eq. (1.1), the bispectrum is
given by

Bφ(k, k1, |k + k1|) = AℓPℓ(µ)Pφ(k)Pφ(k1) 2 + (2qµ + [1 − 2µ2]q2)nφ + O(q3),
ℓ

where nφ = ns − 4 and we have assumed a pure power-law Pφ(k) for simplicity. Further,

M(|k + k1|) = M(k1) 1 + qµ nM(k1) + O(q2) ;

nM(k1)

≡

∂

ln M(k1) ∂ ln k1

.

(A.3)

– 23 –

Inserting these expressions into Eq. (A.2), we obtain

δ(x)δ(y)Kij (y) =

×

1
dµ
−1

2π dφ 0 2π

d3k (2π)3

eik·r M(k)Pφ (k)

k12dk1 (2π)2

M2(k1)Pφ(k1)

k1ik1j k12

−

1 3

δij

AℓPℓ(µ) 2 + 2qµ {nφ + nM(k1)} + O(q2)
ℓ

.

In the following we will neglect any O(q2) corrections; in terms of observed correlations, these would lead to a small scale-independent correction which is degenerate with the Gaussian part. We now choose k to lie along the z axis. We can then write

k1ik1j k12

 cos2 φ(1 − µ2) =  cos φ sin φ(1 − µ2)
 cos φ µ 1 − µ2

cos φ sin φ(1 − µ2) sin2 φ(1 − µ2)
sin φ µ 1 − µ2

cos φ µ 1 − µ2 
sin φ µ 1 − µ2  
µ2

.

(A.4)

Performing the φ integral only leaves the diagonal terms (as expected by symmetry):

2π 0

dφ k1ik1j 2π k12

=

1 2

 1 − µ2 0
0

0 1 − µ2
0

0
0 2µ2

.

(A.5)

In order to obtain an expression for general directions of k, we notice that by symmetry, the ﬁnal expression can only involve the tensors kˆikˆj and δij. Straightforward algebra then yields

2π dφ 0 2π

k1ik1j k12

−

1 3

δij

= P2(µ)

kikj k2

−

1 3

δij

.

(A.6)

We thus have

δ(x)δ(y)Kij (y) =

d3k (2π)3

eik·rM(k)Pφ

(k)

kikj k2

−

1 3

δij

k12dk1 (2π)2

M2(k1)Pφ(k1

)

1
× dµ P2(µ) AℓPℓ(µ) 2 + 2qµ {nφ + nM(k1)} + O(q2)
−1 ℓ

=

2 5

A2

d3k (2π)3

eik·r M(k)Pφ (k)

kikj k2

−

1 3

δij

k12dk1 2π2

M2(k1)Pφ(k1

)

.

In the last line, we have used the orthonormality of the Legendre polynomials. Note that since L is even, the qµ term does not contribute and corrections are order q2. The k1 integral now simply gives the (formally divergent) variance of the density ﬁeld δ2 . By deﬁning

ξφδ(r) =

d3k (2π)3

eik·rM(k)Pφ

(k)

(A.7)

we then obtain

δ(x)δ(y)Kij (y)

=

2 5

A2Dij

ξφδ(r)

δ2

.

(A.8)

This is Eq. (3.11). It is easy to see that the corresponding result for the quadratic tidal term

simply becomes

δ(x)

Kik

K

k j

−

1 3

δij

(Klm)2

(y)

=

2 15

A2Dij

ξφδ

(r)

δ2

.

(A.9)

– 24 –

This is because the analog of Eq. (A.2) for this term diﬀers by factors constructed out of k1i (k + k1)j/(k1|k + k1|), which at leading order are simply k1ik1j /k12.
Also, let us show that anisotropic non-Gaussianity does not lead to a scale-dependent
bias in the number density of tracers. The tracer density locally depends on the variance of the density ﬁeld smoothed on some scale, as well as the square of the tidal tensor Ki2j [Eq. (3.2)]. Let us calculate these local quantities given the power spectrum Eq. (2.3):

δ2 x = (Kij )2 x =

d3kS (2π)3
d3kS (2π)3

1 + f (kˆL · kˆS)φ(kL)eikLx P (kS)

kˆSi kˆSj

−

1 3

δij

2

1 + f (kˆL · kˆS)φ(kL)eikLx

P (kS ) .

(A.10)

Aligning kL with the z axis, we see that both terms involve the integral
1
dµ f (µ) = 2A0 ,
−1

(A.11)

i.e. these quantities pick out the monopole of f . Thus, non-Gaussianity of the quadrupolar type does not aﬀect the local averages Eq. (A.10) at leading order in the non-Gaussianity.

B Renormalized shape biases

In this section, we provide a rigorous derivation of the reasoning outlined in Section 3. For
this purpose, we introduce a ﬁctitious coarse-graining scale RL which we assume to be much smaller than the scale on which we measure correlations. We denote coarse-grained ﬁelds by
a subscript L. Since we will be dealing exclusively with galaxy shapes and not with number
counts, we will drop the superscript I for the remainder of the appendix.
Let us consider the case of Gaussian initial conditions ﬁrst. Since RL ≫ R∗, the shape of galaxies at position x becomes a local function of the coarse-grained density δL and tidal ﬁeld KLij. In addition, the shapes are in general a functional of the small-scale ﬂuctuations within a region of size ∼ R∗ around x. Hence,

gij (x) = FiLj (δL(x); KLkl(x); δs) ,

(B.1)

where δs = δ − δL are the small-scale ﬂuctuations. The superscript L denotes the fact that FiLj depends on RL. We can now perform a formal Taylor expansion of FiLj in δL and KLij, taking into account that FiLj has the same symmetries as gij. The ﬁrst few terms of the expansion are

gij (x)

=

c1 KLij (x)

+

1 2

c2δL

(x)KLij

(x)

+

1 2

ct

KLik KLkj

−

1 3

δij

(KLlm)2

+ O(δL3 , ∇2δL) .

(B.2)

The coeﬃcients ci depend on RL and on the small-scale modes δs. In the Gaussian case, the small-scale modes are uncorrelated with the long-wavelength perturbations, so that they
merely lead to stochastic terms which do not correlate over large scales. Then, the ci can be regarded as eﬀective constants. We have not written the most general leading order term here, which would be cijklKLkl, and instead reduced cijkl to a scalar c1. The reason is that

– 25 –

in order to construct anything more general we would need some preferred direction which is absent in 3D space.9
The expectation value of gij has to vanish by symmetry. Note that this is achieved at second order without having to add any counter term in Eq. (B.2). Now consider the
transformation

KLij(x, τ ) → KLij(x, τ ) + D(τ )βij; δL(x, τ ) → δL(x, τ ) ,

(B.3)

where D(τ ) is the matter growth factor, βij is constant and βi i = 0. This corresponds to adding a uniform tidal ﬁeld to the Newtonian potential,

ΦN (x)

→

ΦN (x)

+

3 4

Ωm0H02(1

+

z)D(z)βij xixj

.

Under this transformation, the expectation value of g changes to

(B.4)

gij (x, τ ) β = c1(τ )βij + O(δL3 , ∇2δL) .

(B.5)

In the following, we will assume we are evaluating the correlations at some ﬁxed time t, and drop this argument for clarity. We now introduce the renormalized bias parameters through

b1 ≡

∂ gij β ∂βij

β=0 = c1 + O([δL, KLij]2) .

(B.6)

Note that the deﬁnition of b1 is explicitly independent of RL. The RL-dependent quadratic corrections in Eq. (B.6) come from cubic order terms in Eq. (B.2), hence we have not written
them here. Similarly, the renormalized quadratic tidal bias is deﬁned via

bt ≡

∂2 gij β ∂(βij )2

β=0 = ct + O([δL, KLij ]2) ,

(B.7)

where now the quadratic corrections come from fourth-order terms in Eq. (B.2). In order to derive the renormalized bias corresponding to c2, we generalize Eq. (B.3) to

KLij(x, τ ) → KLij(x, τ ) + D(τ ) βij; δL(x, τ ) → δL(x, τ ) + D(τ ) c ,

(B.8)

where the transformation of δL is the same as the one used to deﬁne renormalized local bias parameters in [35]. Speciﬁcally, it corresponds to adding a uniform matter density component

cρ¯. We then deﬁne

b2

≡

∂2 gij β,c ∂βij ∂c

= c2 + O([δL,
β=0,c=0

KLij]2) .

(B.9)

Again, b2 is independent of RL by construction. We now turn to correlations of gij on large scales. For simplicity, let us consider the
cross-correlation with matter here, i.e. δ(x)gij (y) . After renormalizing the cn into bn we obtain

δ(x)gij (y)

= b1 δ(x)KLij (y)

+

1 2

b2

δ(x)δL(y)KLij (y)

+

1 2

bt

δ(x)

KLik KLkj

−

1 3

δij

(KLlm

)2

(y)

+··· ,

(B.10)

9In reality, the line of sight nˆ provides such a preferred direction. Thus, in principle a linear term like nˆinˆjnˆknˆltkLl could be present due to redshift space eﬀects. However such a term will not be relevant for photometric galaxy catalogues.

– 26 –

where the ellipsis stands for higher order no-zero-lag correlators, i.e. correlators that do not
contain any zero-lag terms such as δL2 , KijKij when expanded into connected correlators. If |x − y| ≫ RL (and the correlation function does not have sharp features), then

δ(x)KLij (y) = δ(x)Kij (y) = Dijξ(|x − y|)

(B.11)

where ξ(r) is the matter correlation function. This renormalized bias quantiﬁes the linear response of galaxy shapes to a long-wavelength tidal ﬁeld as described above. For Gaussian initial conditions, the three-point function in Eq. (B.10) is only due to non-linear evolution and relevant on smaller scales.

B.1 Non-Gaussian initial conditions
We now turn to the case of non-Gaussian initial conditions, quantiﬁed by Eq. (1.1). Note that the three-point term in Eq. (B.10) is sensitive to the bispectrum in the squeezed limit if |x − y| ≫ RL, the coarse-graining scale adopted. As shown in App. A, for a squeezed-limit bispectrum of the form Eq. (1.1) we have

δ(x)δL(y)KLij (y)

=

2 5

A2Dij

ξδφ(|x

−

y|)σL2

,

(B.12)

where ξδφ(r) is the matter-potential cross-correlation and σL2 is the variance of the coarsegrained density ﬁeld, σL2 ≡ δL2 . A similar result (with prefactor 2/15 instead of 2/5) holds for the term ∝ bt. The fact that this term scales as σL2 , i.e. that it strongly depends on the arbitrary coarse-graining scale indicates that the bias expansion Eq. (B.2) is not suﬃcient in
the presence of primordial non-Gaussianity. This exactly parallels the case of galaxy density correlations, where the corresponding three-point term also scales as σL2 [35].
In order to remedy this unsatisfying situation, we explicitly include the dependence of
gij on the small scale ﬂuctuations indicated in Eq. (B.1). Of course, this dependence has to be consistent with the symmetries of gij. Physically, we expect that the lowest order dependence on δs in Eq. (B.1) will be through a quadrupolar asymmetry in the statistics of the local small-scale density ﬁeld; an isotropic change cannot produce an eﬀect at linear
order because we are expanding a spin-2 function. Note that it is suﬃcient to phrase the
dependence on the small-scale modes in terms of δs, since the small-scale tidal ﬁeld is related to δs through the Poisson equation. Let us deﬁne the parameter ysij as the anisotropy of the local small-scale correlation function within a region of size RL:

yskl(x)

=

1 σy2

d3r WL(|r|)δs(x)Drklδs(x + r)

σy2 =

d3k1 (2π)3

W˜ L(k1

)W˜ s2(k1

)Pm

(k1

)

,

(B.13)

where WL is an arbitrary isotropic ﬁlter function on the scale RL (e.g., Gaussian or tophat), and W˜ L is its Fourier transform. Drij denotes the derivative operator Eq. (3.3) acting with respect to r. While this deﬁnition is convenient, other deﬁnitions are certainly possible and do not change the results. Note that since by deﬁnition δs = δ − δL, the statistics of δs also depend on RL. We now let gij depend locally on yskl:

gij (x) = FiLj (δL(x); KLkl(x); yskl(x)) .

(B.14)

– 27 –

We expand Fij to linear order in yskl, which adds a term cNGysij (x)

(B.15)

to the expansion on the r.h.s. of Eq. (B.2). At linear order in the non-Gaussianity parameters Aℓ, we do not need to consider higher order terms involving ysij. The same symmetry argument as for KLij dictates a simple scalar coeﬃcient cNG. Correspondingly, we have another contribution to the matter-shape correlation:

δ(x)gij (y)

= b1 δ(x)KLij (y)

+

1 2

b2

δ(x)δL(y)KLij (y)

+

1 2

bt

δ(x)

KLik KLkj

−

1 3

δij

(KLlm

)2

(y)

+ cNG δ(x)ysij (y) + · · · . (B.16)

The last term can be calculated along the same lines as the calculation in App. A. The Fourier transform of Eq. (B.13) is given by

ysij (k)

=

1 σy2

d3k1 (2π)3

k1i k1j k12

−

1 3

δij

W˜ L(k1)δs(k1)δs(k − k1)

(B.17)

We then obtain

δ(x)ysij (y)

=

1 σy2

d3k (2π)3

eik·rM(k)

d3k1 (2π)3

W˜ L(k1)

k1i k1j

−

1 3

δij

k12

× Ms(k1)Ms(|k + k1|)Bφ(k, k1, |k + k1|)

(B.18)

where Ms(k) = M(k)W˜ s(k). The derivation now exactly parallels that of Eq. (A.8), where the k1 integral now yields

k12dk1 2π2

W˜ L(k1)M2s(k1)Pφ(k1)

=

σy2

,

(B.19)

cancelling the prefactor. We obtain

δ(y)ysij (x)

=

2 5

A2Dij

ξφδ(r)

.

(B.20)

There are thus three non-Gaussian terms that are proportional to A2Dijξδφ, two of which are strongly dependent on RL. Intuition tells us that these contributions should combine to an RL-independent term involving a renormalized bias parameter bNG. As before, bNG should be given as the response of the mean shape of galaxies to a speciﬁc (RL-independent) transformation of the density ﬁeld. Consider the transformation

δ(x) → δα(x) = 1 + αlmDlm δ(x)

(B.21)

or equivalently in Fourier space

δα(k) = 1 + αlm

klkm k2

−

1 3

δlm

δ(k) .

(B.22)

– 28 –

This corresponds to an anisotropic, scale-independent rescaling of the density ﬁeld. In the
following we will assume without loss of generality that αlm is trace-free. The expectation value of the shape after this transformation is

where

gij

α

=

1 2

b2

δLKLij

α+

1 2

bt

KLik KLkj

−

1 3

δij

(KLlm

)2

+ cNG ysij α .
α

δLKLij α = 3 =

KLik KLkj

−

1 3

δij

(KLlm)2

α

d3k (2π)3

kikj k2

−

1 3

δij

1

+

αlm

klkm k2

W˜ L2(k)P (k)

ysij

α

=

1 σy2

d3k (2π)3

kikj k2

−

1 3

δij

1

+

αlm

klkm k2

W˜ L(k)W˜ s2(k)P (k) .

After some algebra, this yields

(B.23) (B.24)

gij α =

1 15

b2σL2

+

1 45

btσL2

+

2 15

cNG

αij .

(B.25)

The linear response of the mean shape of galaxies, which shall deﬁne our bias bNG, is thus

given by

bNG ≡

∂ gij α ∂αij

=
α=0

1 15

b2σL2

+

1 45

btσL2

+

2 15

cNG

.

(B.26)

Solving this for cNG yields

cNG

=

15 2

bNG

−

1 2

b2σL2

−

1 6

bt

σL2

,

(B.27)

which we insert into Eq. (B.16) to obtain

δ(x)gij (y) = b1Dij ξδδ(|x − y|) + 3bNGA2Dij ξδφ(|x − y|) .

(B.28)

Our intuition is thus conﬁrmed: by adding the bias bNG, which quantiﬁes the linear response
of galaxy shapes to an anisotropic power spectrum of initial ﬂuctuations, we have absorbed the unphysical dependence on σL2 present in the local quadratic bias expression.

C Non-Gaussian contributions to weak lensing shear

In Section 4 and following, we have neglected any modiﬁcation of the lensing contribution to galaxy shapes due to primordial non-Gaussianity. We now show that there is a modiﬁcation, but it is negligible compared to the scale-dependent signature in the intrinsic alignments. The observed shear is not exactly given by its linear expression γ introduced in Section 4, but rather there are nonlinear corrections,

γobs(θ) = γ(θ) + cγκ(θ)γ(θ) + · · · ,

(C.1)

where cγ is a constant of order unity. The nonlinear corrections are due to reduced shear [56] as well as lensing bias corrections induced by source density weighting [74]. This leads to a
three-point contribution to shear two-point functions of the form

γobs(θ)γobs(θ′) = γ(θ)γ(θ′) + 2cγ κ(θ)γ(θ)γ(θ′) + · · · .

(C.2)

– 29 –

Primordial non-Gaussianity in general contributes to this three-point function. This was studied in detail by Ref. [47], who showed that

Cγobs (l) = Cγ(l) + 2cγ

d2l1 (2π)2

cos

2φl1 Bκ(l1, |l

−

l1|, l) .

(C.3)

Here, Bκ is the convergence bispectrum, and we have aligned l1 so that φl = 0. Further, we have used that in Fourier space γ(l) = γ1(l) + iγ2(l) = exp(2iφl)κ(l). Assuming the Limber approximation, the convergence bispectrum is related to the matter bispectrum by

Bκ(l1, l2, l3) =

3 2

ΩmH02

3

χs 0

dχ χ

WL3 χ3

(χs, χ) a3(χ)

Bm

l1 χ

,

l2 χ

,

l3 χ

;

χ

,

(C.4)

where WL(χs, χ) ≡ χ/χs(χs−χ) denotes the lensing kernel. Ref. [47] only considered isotropic

local non-Gaussianity, however their results are easily generalized to the anisotropic case by

making use of the squeezed limit. While in the isotropic case, the low-l contribution to

Eq. (C.3) is canceled by the cos 2φl1 modulation, this factor will lead to a non-zero contribution in the anisotropic case, which will be of the same order as the isotropic contribution to

Cκobs(l) discussed in [47]. For our purposes, an order of magnitude estimate suﬃces. By using the squeezed limit

for the bispectrum and approximating the projection kernels, we can derive in analogy with

Sec. II in [47]

Cγobs (l) − 1 ∼ 10−5 A2

Cγ (l)

100

l 100

−2
.

(C.5)

Thus, at the values for A2 relevant for detectable alignment signatures, the fractional modiﬁcation to the shear is much less than one even at the lowest multipoles.

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