Constraints on primordial non-Gaussianity from halo bias measured through CMB lensing cross-correlations Fiona McCarthy1 2 3Mathew S. Madhavacheril4 2and Abhishek S. Maniyar5

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Constraints on primordial non-Gaussianity from halo bias measured through CMB

lensing cross-correlations

Fiona McCarthy,1, 2, 3, ∗Mathew S. Madhavacheril,4, 2 and Abhishek S. Maniyar5

1Center for Computational Astrophysics, Flatiron Institute, New York, NY, USA 10010

2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

3Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

4Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA

5Center for Cosmology and Particle Physics, Department of Physics,

New York University, New York, NY 10003, USA

(Dated: October 4, 2022)

Local non-Gaussianities in the initial conditions of the Universe, parameterized by fNL, induce

a scale-dependence in the large-scale bias of halos in the late Universe. This eﬀect is a promising

path to constrain multi-ﬁeld inﬂation theories that predict non-zero fNL. While most existing

constraints from the halo bias involve auto-correlations of the galaxy distribution, cross-correlations

with probes of the matter density provide an alternative channel with fewer systematics. We present

the strongest large-scale structure constraint on local primordial non-Gaussianity that utilizes cross-

correlations alone. We use the cosmic infrared background (CIB) consisting of dusty galaxies as a

halo tracer and cosmic microwave background (CMB) lensing as a probe of the underlying matter

distribution, both from Planck data. Milky Way dust is one of the key challenges in using the

large-scale modes of the CIB. Importantly, the cross-correlation of the CIB with CMB lensing is far

less aﬀected by Galactic dust compared to the auto-spectrum of the CIB, since the latter picks up

an additive bias from Galactic dust. We ﬁnd no evidence for primordial non-Gaussianity and obtain

−87 < fNL <19, with a Gaussian σ(fNL)≈41, assuming universality of the halo mass function.

We ﬁnd that future CMB lensing data from Simons Observatory and CMB-S4 could achieve σ(fNL )

of 23 and 20 respectively. The constraining power of such an analysis is limited by current Galactic

dust cleaning techniques which introduce a multiplicative bias on very large scales, requiring us

to choose a minimum multipole of `= 70. If this challenge is overcome with improved analysis

techniques or external data, constraints as tight as σ(fNL) = 4 can be achieved through the cross-

correlation technique. More optimistically, constraints better than σ(fNL) = 2 could be achieved if

the CIB auto-spectrum is dust-free down to the largest scales.

I. INTRODUCTION

The search for non-Gaussianities in the initial condi-

tions of the Universe (“primordial non-Gaussianities”, or

PNG) is a key goal of the cosmology community. Of

particular interest is primordial non-Gaussianity of the

local type, parameterized by flocal

NL , with flocal

NL = 0 in-

dicating exact Gaussianity. Multi-ﬁeld inﬂation mod-

els predict flocal

NL of O(1) (e.g. [1]), and so a detection

of flocal

NL will be key for discriminating between inﬂa-

tion models. To date, all measurements are consistent

with Gaussian initial conditions, with the strongest con-

straint coming from measurements of the early-universe

bispectrum (or three-point function) through the cosmic

microwave background (CMB) as measured by Planck:

flocal

NL =−0.9±5.1 [2]. This constraint is not expected to

improve enough to probe multi-ﬁeld inﬂation with future

measurements of the primary CMB ﬂuctuations (e.g. up

to σ(flocal

NL ) = 2 with the Simons Observatory[3]).

The late-universe large-scale structure (LSS) bis-

pectrum is perhaps the next obvious probe of non-

Gaussianities; although, as gravitational evolution in-

∗Electronic address: fmccarthy@ﬂatironinstitute.org

duces non-Gaussianities in an initially non-Gaussian

ﬁeld, these must ﬁrst be disentangled before constrain-

ing the primordial Universe from a measurement of the

bispectrum of LSS [4–7]. However, there exists a well-

known signature of flocal

NL (henceforth fNL) in the two-

point power spectrum of biased objects such as halos. In

particular, non-zero fNL induces a scale-dependence in

the bias of these objects with respect to dark matter, a

signal that is strongest on the largest scales [8]:

∆b∼fNL

k2(bG−1),(1)

where bGis the Gaussian bias (which is scale-independent

on large scales), and ∆bis the change in bias induced

by fNL. Constraints from the bias of quasars in the

SDSS/BOSS surveys [9–14] have used this signal to con-

strain fNL, with the strongest ﬁnding fNL =−12 ±

21 [14], and recent combined constraints from the BOSS

bispectrum and power spectrum in fact get most of their

constraining power on fNL from the eﬀect on the power

spectrum [15, 16]. Forecasts have indicated that future

LSS surveys such as Rubin Observatory’s Legacy Sur-

vey of Space and Time (LSST) [17] and SPHEREx [18],

a high-number density galaxy clustering survey, will be

able to reach the σ(fNL)∼1 regime if systematics are

well-controlled.

Many of the aforementioned LSS constraints on fNL

arXiv:2210.01049v1 [astro-ph.CO] 3 Oct 2022

involve multiple powers of the halo overdensity ﬁeld (two

in the power spectrum, three in the bispectrum). On

the other hand, constraints from cross-correlations with

probes of the unbiased matter distribution—like those

of [11, 12]—oﬀer advantages: (1) an analysis involving a

cross-correlation of one power of the halo overdensity typ-

ically does not suﬀer from additive systematics in mea-

surements of the LSS survey (e.g. selection eﬀects and

Milky Way dust); and (2) a joint analysis of all cross- and

auto-spectra can signiﬁcantly improve the bias measure-

ment through sample variance cancellation [19]. Such

measurements have been proposed using unbiased trac-

ers of mass such as CMB lensing convergence maps [20]

or velocity such as the kinetic Sunyaev–Zel’dovich (kSZ)

eﬀect [21]; [11, 12] use the integrated Sachs Wolfe (ISW)

eﬀect as well as CMB lensing.

In this work, we present the strongest constraint

on fNL through cross-correlation alone, the previous

strongest being fNL = 46 ±68 from the cross correla-

tion of the ISW eﬀect and galaxies [11]. We use (1) the

cosmic infrared background (CIB) as our halo tracer and

(2) weak lensing of the CMB as our probe of the unbiased

matter distribution.

1. The CIB is sourced by the thermal radiation of dust

grains in distant galaxies; these dust grains absorb

ultraviolet (UV) starlight, which heats them up and

is re-emitted in the infrared (IR). The star forma-

tion rate (SFR) of our Universe peaked at around

z∼2 [22], and the CIB is thus sourced from galax-

ies at around this redshift and higher, although it is

a diﬀuse ﬁeld with contributions from all redshifts

up to reionization at z∼7. The CIB anisotropies

that we measure trace the clustering of these ob-

jects [23]. For this reason, it might be considered a

promising candidate for constraining fNL: the fNL

signal increases with bias, and galaxies at high red-

shift such as those sourcing the CIB are more highly

biased than galaxies at lower redshifts. As well as

this, it is highly correlated with the CMB lensing

convergence ﬁeld κ, giving a potential opportunity

to improve the fNL measurement by using a simul-

taneous measurement of κand the CIB intensity to

exploit sample variance cancellation.

2. The CMB lensing convergence ﬁeld is a map of all

the matter between us and the surface of last scat-

tering, projected along the line of sight [24]. As

the CMB has been traveling through the Universe,

it has interacted gravitationally with this matter

in a well-understood way [25]. The result is that

the CMB we see has been weakly lensed, an eﬀect

which can be detected statistically, and has been

done with high statistical signiﬁcance by the Planck

satellite [26–29] and high-resolution ground-based

CMB experiments such as the Atacama Cosmol-

ogy Telescope (ACT) (e.g. [30–32]) and the South

Pole Telescope (SPT) (e.g. [33–37]).

Previous work has shown that the information

contained in the auto-power spectrum of the CIB

anisotropies could in principle yield a measurement with

σ(fNL)<1 [38]. However, as indicated earlier, there are

signiﬁcant diﬃculties associated with using auto-spectra

for fNL measurements, and this is especially true for the

CIB. The signal of interest is mostly sourced at large

scales, where it is diﬃcult to separate the cosmologi-

cal CIB signal from the emission from dust in our own

Milky Way galaxy. The Galactic dust signal is also scale-

dependent with signiﬁcant power on large scales; even in

maps post-processed through component separation or

foreground cleaning techniques, any spurious dust power

will bias the inference of fNL. For this reason, we do

not use the large-scale CIB auto-power spectrum1in

this work and instead focus on constraining fNL from its

cross-power spectrum with the CMB lensing convergence

ﬁeld Cνκ

`alone, as this statistic does not suﬀer from the

same additive dust bias. There is however a multiplica-

tive bias associated with the dust cleaning procedure that

prevents us from accessing all scales [39]; this is discussed

later in this work.

The paper is organized as follows. In Section II we dis-

cuss the relevant theory, including the scale-dependence

induced in the bias by fNL, and the formalism we use

to model the CIB and the the CIB-CMB-lensing cross-

correlation. In Section III we discuss the data products

used in our analysis and in Section IV we present our

pipeline for the extraction of fNL. We present our re-

sults in Section V. In Section VI we forecast constraints

from future CMB lensing experiments. We conclude in

Section VII.

Throughout, we use the cosmology of [40]: {H0=

67.11 km/s/Mpc,Ωch2= 0.1209,Ωbh2= 0.022068, As=

2.2×10−9, ns= 0.9624}where H0is the Hubble constant

today, Ωch2is the physical cold dark matter density to-

day, Ωbh2is the physical baryon density today, Asis the

amplitude of scalar ﬂuctuations, and nsis the spectral

index (with a pivot scale of 0.05 Mpc−1). All matter

power spectra and transfer functions are calculated with

the Einstein-Boltzmann code CAMB2[41].

II. THEORY

In Part II A of this Section we discuss the induction of

scale-dependence in halo bias from fNL. In Part II B we

present the theory model we use to model the CIB and

the CIB-CMB lensing cross power spectrum.

1We use “CIB auto-power spectrum” to refer to the cross-power

spectra between the diﬀerent frequency channels at which the

CIB is measured. As described later, we do include small-scale

CIB auto-spectra to help constrain the CIB model itself.

2https://camb.info

A. fNL from scale-dependent bias

fNL parameterizes primordial non-Gaussianity of the

local type as follows:

Φ(x) = φ(x) + fNL φ2(x)−φ2 (2)

where Φ(x) is the Newtonian potential at xand φ(x)

is an underlying Gaussian ﬁeld. On sub-horizon scales,

Φ is related to the overdensity δthrough the Poisson

equation.

While the overdensity ﬁeld δis continuous, in several

situations the peaks of δare the objects of interest. This

is because gravitational collapse happened only where δ

was higher than a critical value δc, and so these regions

(with δ > δc) are those in which large scale structure

formed. These peaks of δare biased with respect to δ:

δh=bhδ, (3)

where δhis the overdensity of the peaks (the “halo over-

density”), and bhis their bias (the “halo bias”). This

leads to them following a diﬀerent power spectrum to

that of the underlying dark matter:

Phh(k) = b2

hPmm(k) (4)

where Phh(k) is the halo power spectrum and Pmm(k) is

the matter power spectrum. For Gaussian initial condi-

tions, bhis scale independent on large scales—i.e., it does

not depend on k. However, non-Gaussianity of the form

of Equation (2) serves to induce a scale dependence [8]:

bNG

h=bG

h+fNL

3ΩmH2

k2T(k)D(z)δc(bG

h−1) (5)

where Ωmis the mean density of matter today; H0is

the Hubble constant; T(k) and D(z) are the transfer

and growth functions of the density ﬁeld, respectively,

with T(k) normalized to 1 at low kand D(z) normalized

such that D(z) = 1

1+zduring matter domination; and

δc= 1.686 is the critical overdensity above which objects

undergo gravitational collapse. bG

hrefers to the Gaussian

bias, i.e. the bias in the absence of fNL.

B. The CIB-CMB lensing cross correlation

1. The CIB

The CIB is sourced by thermal emission of dust in

star-forming galaxies. As the physics of star-formation is

not well understood, we lack a ﬁrst-principles model for

the CIB. Instead several parametric models of various

physical motivation have been proposed (see, e.g. [42–

45]).

The CIB intensity at frequency ν Iνis given by

Iν(ˆ

n) = Zχre

dχa(χ)jν(χ, ˆ

n),(6)

where jνis the comoving CIB emissivity density, a(χ) is

the scale factor, and the integral over comoving distance

χis done out to reionization at χre.jν(χ, ˆ

n) can be

separated into its mean value and ﬂuctuations:

jν(χ, ˆ

n) = ¯

jν(χ)1 + δjν(χ, ˆ

jν(χ).(7)

CIB models generally include a model for the mean emis-

sivity ¯

jνas well as a prescription for the clustering of the

ﬂuctuations, in particular the three-dimensional emissiv-

ity power spectrum Pνν0

jj (k, z, z0), which is deﬁned as fol-

lows:

δjν(k, z)δjν0(k0, z0)

jν(z)¯

jν0(z0)≡(2π)3Pνν0

jj (k, z, z0)δ3(k−k0).

(8)

The angular CIB power spectrum can then be integrated

directly according to

Cνν0

`=2

πZdχdχ0Zk2dk (9)

a(χ)a(χ0)¯

jν(χ)¯

jν0(χ0)Pνν0

jj (k, z, z0)j`(kχ)j`(kχ0)

where j`(x) are the spherical Bessel functions of degree

`. As jν(χ) has support on a very wide range of χ, in

most cases the Limber approximation [46] is valid and we

can simplify Equation (9) to reduce to the more standard

expression:

Cνν0

`=Zdχ

χ2a2(χ)¯

jν(χ)¯

jν0(χ)Pνν0

jj k=`

χ, z(10)

where Pjj (k, z)≡Pjj (k, z, z) is the equal-time emissivity

power spectrum. However, at the lowest values of `(`<

∼

40), we should integrate the full expression (9).

In this work, we use the linear CIB model of [47] to

model the CIB. In this model, the mean CIB emissivity

is related directly to the mean star formation rate density

(SFRD) with the Kennicutt relation [48]:

jν(z) = ρSF R(z)(1 + z)Sν,eﬀ (z)χ2

K(11)

where Kis the Kennicutt constant K= 1.7×

10−10Myr−1L−1

and Sν,eﬀ (z) is the mean eﬀective

spectral energy distribution (SED), calculated using the

method of [49] using SEDs calibrated with Herschel

data [50, 51]3. The SFRD is parameterized according

ρSF R(z) = α(1 + z)β

1 + 1+z

γδ(12)

3These are available at this URL [45]

Parameter Value

α0.007

ρSF R(z)β3.590

Evolution γ2.453

δ6.578

CIB b00.83

bias b10.742

evolution b20.318

TABLE I: The ﬁducial values for the parameters of the CIB

model, from [47].

with α, β, γ, δ free parameters of the model. As this is a

linear model, the CIB ﬂuctuations can be parameterized

directly by deﬁning the CIB bias bCIB(z):

Pνν0

jj lin(k, z, z0) = bCIB(z)bCIB(z0)Plin

mm(k, z, z0) (13)

where Plin

mm(k, z, z0) is the linear matter power spectrum;

bCIB(z) is parameterized as

bCIB(z) = b0+b1z+b2z2(14)

with b0,b1,b2free parameters of the model (note that

Pνν0

jj as deﬁned in Equation (8) is thus independent of ν

and ν0, with frequency-dependence of Cνν0

`coming from

the SFRD alone). We expect this linear model to be

suﬃcient since we restrict our analysis to relatively large

scales (`≤610).

The parameters {α, β, γ, δ, b0, b1, b2}were ﬁt to the

Planck CIB auto and CIB-lensing power spectra at

ν={217,353,545,857}GHz in Ref. [47]; their values are

given in Table I. In our analysis, we marginalize over all

of these parameters, with a prior of b0= 0.83 ±0.11. We

note that we do not vary any cosmological parameters,

since these are very well determined by primary CMB

measurements.

There is also a small contribution to the CIB power

from the small-scale regime (1-halo term) and the shot

noise (as the CIB is intrinsically sourced by discrete ob-

jects), which is constant in `. We include these contri-

butions to the power by using the prescription presented

in [45, 47]4. However, in practice we will marginalize over

the values of the shot noise, which we expect to allow for

model uncertainty in both the shot noise and the 1-halo

term which are very degenerate on the linear scales we

use, as the 1-halo term is only very mildly scale depen-

dent in this regime.

Thus, in total, the full model for the CIB power is

Cνν0

`=Cνν0

`linear +Cνν0

`one−halo +Sνν0(15)

4Again, see this URL for the pre-computed 1-halo term.

where Cνν0

`linear is the linear term that we model by cal-

culating Equation(10) using Pνν0

jj lin and ¯

jνas described

above; Cνν0

`one−halo is the (almost-constant) one-halo

contribution, which we pre-compute; and Sνν0is the con-

stant shot-noise (over which we will marginalize in our

analysis).

2. CMB lensing

Gravitational lensing induces a speciﬁc form of statisti-

cal anisotropy in the CMB allowing the use of quadratic

estimators to reconstruct the line-of-sight gravitational

potential φ[52] integrated all the way to the surface of

last scattering. The contribution to the lensing potential

peaks at redshifts around z∼2. As the CIB is sourced

mostly at the same redshifts where the CMB lensing ef-

ﬁciency peaks, the two ﬁelds are expected to be highly

correlated with each other; indeed, their correlation has

been detected by Planck [53], SPT[54] and ACT [30, 55].

Going forward, we may interchangeably refer to both the

lensing potential φand the lensing convergence ﬁeld κ

(proportional to the projected matter density), which are

straightforwardly related through ∇2φ=−2κ.

The CMB lensing potential φis given by

φ(ˆ

n) = −2ZχS

dχχS−χ

χSχΦ(χ, ˆ

n) (16)

where χSis the comoving distance to the surface of last

scattering, where the CMB was released, and Φ(χ, ˆ

n) is

the Newtonian potential. Φ can be related directly to

the matter overdensity δon sub-horizon scales with the

Poisson equation

∇2Φ = −3

2H0

c2ΩmH0

aδ. (17)

As a result of this, in harmonic space the lensing potential

is related to the lensing convergence κby

φ`=2

`(`+ 1)κ`,(18)

where

κ=ZχS

dχW κ(χ)δ(χ, ˆ

n),(19)

with the lensing convergence kernel Wκ(χ) given by

Wκ(χ) = 3

2H0

c2Ωm

aχ1−χ

χS.(20)

The angular power spectrum of the CMB lensing conver-

gence ﬁeld is

Cκκ

`=2

πZdχdχ0Zk2dk (21)

Wκ(χ)Wκ(χ0)Pmm(k, z, z0)j`(kχ)j`(kχ0),

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Constraintsonprimordialnon-GaussianityfromhalobiasmeasuredthroughCMBlensingcross-correlationsFionaMcCarthy,1,2,3,MathewS.Madhavacheril,4,2andAbhishekS.Maniyar51CenterforComputationalAstrophysics,FlatironInstitute,NewYork,NY,USA100102PerimeterInstituteforTheoreticalPhysics,Waterloo,OntarioN2L2Y5,Can...

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Constraints on primordial non-Gaussianity from halo bias measured through CMB lensing cross-correlations Fiona McCarthy1 2 3Mathew S. Madhavacheril4 2and Abhishek S. Maniyar5

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