Galaxy Clustering: An EFT Approachresearch.ipmu.jp/seminar/sysimg/seminar/2062.pdf ·...

x = xfl(⌧)

q = xfl(0)

x = xfl(⌧)

q = xfl(0)

Galaxy Clustering:An EFT Approach

Fabian SchmidtMPA

with Alex Barreira, Elisa Chisari, Vincent Desjacques, Cora Dvorkin,

Donghui Jeong, Mehrdad Mirbabayi, Zvonimir Vlah, Matias Zaldarriaga

April 2018

Motivation

• The clustering of galaxies (large-scale structure, LSS) is historically one of the key probes of cosmology

• From ~1998 until recently, most spectacular results came from “cleaner” probes - Supernovae and the cosmic microwave background (CMB)

• Now, again, in a new golden age of LSS with plenty of experiments under way: BOSS, DES, DESI, PFS, SphereX, Euclid, WFIRST, ...

Peebles; Efstathiou+ ’90 predicted a positive cosmological constant Λ from LSS observations

Motivation• Using large-scale structure, we can learn about

• Inflation (or, which mechanism generated seeds of structure ?)

• Dark Energy and Gravity (is General Relativity correct ?)

• Dark Matter (does it cluster as expected ?)

• the formation history of galaxies, clusters, and the IGM

• Uniquely broad set of science opportunities!

• As first demonstrated by 2dF and SDSS

Motivation

• Inflation: reconstruct the properties of the initial conditions, and look for gravitational waves

• Dark Energy and Gravity: the growth of structure depends sensitively on the expansion history of the Universe, and the nature of gravity

• Dark Matter: how “cold” is cold dark matter ? What is the sum of neutrino masses ?

D00 + aHD

0 = 4⇡G ⇢DGrowth equation:

Cold Dark Matter cosmology in a nutshell

Millennium simulation / MPA

• Assume scale-invariant, adiabatic, approx. Gaussian initial conditions

• Large-scale fluctuations are small (still linear today)

• Structure forms hierarchically from small to large scales

• Perturbative expansion in fluctuations on large scales

Cold Dark Matter cosmology in a nutshell

• Assume scale-invariant, adiabatic, approx. Gaussian initial conditions

• Large-scale fluctuations are small (still linear today)

• Structure forms hierarchically from small to large scales

• Perturbative expansion in fluctuations on large scales

Millennium simulation / MPA

Theory of Large-Scale Structure

�(k, t) = ⇢m(k, t)/⇢m � 1

aH

k

linear nonlinear

large scales

small scales

1

1

* for CDM component

horizon scales

subhorizon (“Newtonian”)

scales


�(k, t) = ⇢m(k, t)/⇢m � 1

aH

k

linear nonlinear

large scales

small scales

1

1

Fully GR

* for CDM component

horizon scales


scales


�(k, t) = ⇢m(k, t)/⇢m � 1

aH

k

linear nonlinear

large scales

small scales

1

1

Fully GR

* for CDM component

horizon scales


scales

Typical perturbations in (observable) universe


• Well-established tools:

• linear Boltzmann

• N-body* methods

�(k, t) = ⇢m(k, t)/⇢m � 1

aH

k

linear nonlinear

large scales

small scales

1

1

Fully GR

Boltzmann

N-body*

Typical perturbations in (observable) universe

* and hydrodynamics


• Foundation: separation between nonlinear scale and horizon

• Linear theory: Fourier modes evolve independently; solved problem

• However, bulk of information in LSS is on nonlinear scales (Nmodes ~ kmax3)

kNL ' 0.1hMpc�1 � aH

What we observe, is this, however…

Hubble UDF

What we observe, is this, however…

Hubble UDFHubble UDFGil-Marin et al, 2016

• We cannot yet simulate the formation of galaxies* fully realistically

• Need to abstract from the incomplete understanding on small scales

• Only hope for rigorous results is on scales k < kNL

• Goal: describe galaxy clustering up to a given scale and accuracy using a finite number of free bias parameters :

�g(x) =X

O

bO O(x)

bO

Theory of galaxy clustering

(at fixed time)

* Of course, everything in following will apply to any tracer of LSS.

Large-Scale Galaxy BiasVincent Desjacquesa,b, Donghui Jeongc, Fabian Schmidtd

aPhysics department, Technion, 3200003 Haifa, IsraelbDepartement de Physique Theorique and Center for Astroparticle Physics, Universite de Geneve, 24 quai Ernest Ansermet,

CH-1221 Geneve 4, SwitzerlandcDepartment of Astronomy and Astrophysics, and Institute for Gravitation and the Cosmos, The Pennsylvania State

University, University Park, PA 16802, USAd Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching, Germany

Abstract

This review presents a comprehensive overview of galaxy bias, that is, the statistical relation betweenthe distribution of galaxies and matter. We focus on large scales where cosmic density fields are quasi-linear. On these scales, the clustering of galaxies can be described by a perturbative bias expansion, and thecomplicated physics of galaxy formation is absorbed by a finite set of coe�cients of the expansion, called biasparameters. The review begins with a pedagogical proof of this very important result, which forms the basisof the rigorous perturbative description of galaxy clustering, under the assumptions of General Relativityand Gaussian, adiabatic initial conditions. Key components of the bias expansion are all leading localgravitational observables, which includes the matter density but also tidal fields and their time derivatives.We hence expand the definition of local bias to encompass all these contributions. This derivation is followedby a presentation of the peak-background split in its general form, which elucidates the physical meaning ofthe bias parameters, and a detailed description of the connection between bias parameters and galaxy (orhalo) statistics. We then review the excursion set formalism and peak theory which provide predictions forthe values of the bias parameters. In the remainder of the review, we consider the generalizations of galaxybias required in the presence of various types of cosmological physics that go beyond pressureless matterwith adiabatic, Gaussian initial conditions: primordial non-Gaussianity, massive neutrinos, baryon-CDMisocurvature perturbations, dark energy, and modified gravity. Finally, we discuss how the description ofgalaxy bias in the galaxies’ rest frame is related to observed clustering statistics measured from the observedangular positions and redshifts in actual galaxy catalogs.

Keywords: cosmology, large-scale structure, galaxy surveys, galaxy bias, dark matter,primordial non-gaussianity

Email addresses: [email protected] (Vincent Desjacques), [email protected] (Donghui Jeong),[email protected] (Fabian Schmidt)

Preprint submitted to Physics Reports November 30, 2016

arX

iv:1

611.

0978

7v1

[astr

o-ph

.CO

] 29

Nov

201

6

Large-Scale

Galaxy

Bias

Vin

centD

esjacques

a,b,

Don

ghuiJeon

gc,

Fab

ianSch

mid

td

aPhysics

depa

rtmen

t,Tech

nion,3200003Haifa

,Isra

elbD

epartem

entdePhysiqu

eTheo

riqueandCen

terforAstro

particle

Physics,

Universite

deGen

eve,24qu

aiErn

estAnserm

et,CH-1221Gen

eve4,Switzerla

nd

cDepa

rtmen

tofAstro

nomyandAstro

physics,

andIn

stitute

forGravita

tionandtheCosm

os,

ThePen

nsylva

nia

State

University,

University

Park,

PA

16802,USA

dMax-P

lanck-In

stitutfurAstro

physik,

Karl-S

chwarzsch

ild-Stra

ße1,85748Garch

ing,

Germ

any

Abstra

ct

This

reviewpresents

acom

preh

ensive

overviewof

galaxybias,

that

is,th

estatistical

relationbetw

eenth

edistrib

ution

ofgalaxies

and

matter.

We

focus

onlarge

scalesw

here

cosmic

den

sityfield

sare

quasi-

linear.

On

these

scales,th

eclu

stering

ofgalaxies

canbe

describ

edby

apertu

rbative

bias

expan

sion,an

dth

ecom

plicated

physics

ofgalaxy

formation

isab

sorbed

bya

finite

setof

coe�cients

ofth

eexp

ansion

,called

biasparam

eters.T

he

reviewbegin

sw

itha

ped

agogicalproof

ofth

isvery

important

result,

which

forms

the

basis

ofth

erigorou

spertu

rbative

descrip

tionof

galaxyclu

stering,

under

the

assum

ption

sof

Gen

eralR

elativityan

dG

aussian

,ad

iabatic

initial

condition

s.K

eycom

pon

entsof

the

bias

expan

sionare

alllead

ing

localgravitation

alob

servables,

which

inclu

des

the

matter

den

sitybut

alsotid

alfield

san

dth

eirtim

ederivatives.

We

hen

ceexp

and

the

defi

nition

oflocal

biasto

encom

pass

allth

esecontrib

ution

s.T

his

derivation

isfollow

edby

apresentation

ofth

epeak-b

ackground

split

inits

general

form,w

hich

elucid

atesth

ephysical

mean

ing

ofth

ebias

param

eters,an

da

detailed

descrip

tionof

the

connection

betw

eenbias

param

etersan

dgalaxy

(orhalo)

statistics.W

eth

enreview

the

excursion

setform

alisman

dpeak

theory

which

provid

epred

ictions

forth

evalu

esof

the

bias

param

eters.In

the

remain

der

ofth

ereview

,w

econ

sider

the

generalization

sof

galaxybias

required

inth

epresen

ceof

various

types

ofcosm

ologicalphysics

that

gobeyon

dpressu

relessm

atterw

ithad

iabatic,

Gau

ssianin

itialcon

dition

s:prim

ordial

non

-Gau

ssianity,

massive

neu

trinos,

baryon

-CD

Misocu

rvature

pertu

rbation

s,dark

energy,

and

mod

ified

gravity.Fin

ally,w

ediscu

sshow

the

descrip

tionof

galaxybias

inth

egalaxies’

restfram

eis

relatedto

observed

clusterin

gstatistics

measu

redfrom

the

observed

angu

larposition

san

dred

shifts

inactu

algalaxy

catalogs.

Keyw

ords:cosm

ology,large-scale

structu

re,galaxy

surveys,

galaxybias,

dark

matter,

prim

ordial

non

-gaussian

ity

Email

addresses:

[email protected]

(Vincen

tDesja

cques),

[email protected]

(Donghui

Jeo

ng),

[email protected](F

abianSch

midt)

Prep

rintsu

bmitted

toPhysics

Repo

rtsNovem

ber30,2016

arXiv:1611.09787v1 [astro-ph.CO] 29 Nov 2016

All of what I will talk about, and much more, can be found in:

Physics Reports, in press

Large-Scale Galaxy BiasVincent Desjacquesa,b, Donghui Jeongc, Fabian Schmidtd

aPhysics department, Technion, 3200003 Haifa, IsraelbDepartement de Physique Theorique and Center for Astroparticle Physics, Universite de Geneve, 24 quai Ernest Ansermet,

CH-1221 Geneve 4, SwitzerlandcDepartment of Astronomy and Astrophysics, and Institute for Gravitation and the Cosmos, The Pennsylvania State

University, University Park, PA 16802, USAd Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching, Germany

Abstract

This review presents a comprehensive overview of galaxy bias, that is, the statistical relation betweenthe distribution of galaxies and matter. We focus on large scales where cosmic density fields are quasi-linear. On these scales, the clustering of galaxies can be described by a perturbative bias expansion, and thecomplicated physics of galaxy formation is absorbed by a finite set of coe�cients of the expansion, called biasparameters. The review begins with a pedagogical proof of this very important result, which forms the basisof the rigorous perturbative description of galaxy clustering, under the assumptions of General Relativityand Gaussian, adiabatic initial conditions. Key components of the bias expansion are all leading localgravitational observables, which includes the matter density but also tidal fields and their time derivatives.We hence expand the definition of local bias to encompass all these contributions. This derivation is followedby a presentation of the peak-background split in its general form, which elucidates the physical meaning ofthe bias parameters, and a detailed description of the connection between bias parameters and galaxy (orhalo) statistics. We then review the excursion set formalism and peak theory which provide predictions forthe values of the bias parameters. In the remainder of the review, we consider the generalizations of galaxybias required in the presence of various types of cosmological physics that go beyond pressureless matterwith adiabatic, Gaussian initial conditions: primordial non-Gaussianity, massive neutrinos, baryon-CDMisocurvature perturbations, dark energy, and modified gravity. Finally, we discuss how the description ofgalaxy bias in the galaxies’ rest frame is related to observed clustering statistics measured from the observedangular positions and redshifts in actual galaxy catalogs.

Keywords: cosmology, large-scale structure, galaxy surveys, galaxy bias, dark matter,primordial non-gaussianity

Email addresses: [email protected] (Vincent Desjacques), [email protected] (Donghui Jeong),[email protected] (Fabian Schmidt)

Preprint submitted to Physics Reports November 30, 2016

arX

iv:1

611.

0978

7v1

[astr

o-ph

.CO

] 29

Nov

201

6

Large-Scale

Galaxy

Bias

Vin

centD

esjacques

a,b,

Don

ghuiJeon

gc,

Fab

ianSch

mid

td

aPhysics

depa

rtmen

t,Tech

nion,3200003Haifa

,Isra

elbD

epartem

entdePhysiqu

eTheo

riqueandCen

terforAstro

particle

Physics,

Universite

deGen

eve,24qu

aiErn

estAnserm

et,CH-1221Gen

eve4,Switzerla

nd

cDepa

rtmen

tofAstro

nomyandAstro

physics,

andIn

stitute

forGravita

tionandtheCosm

os,

ThePen

nsylva

nia

State

University,

University

Park,

PA

16802,USA

dMax-P

lanck-In

stitutfurAstro

physik,

Karl-S

chwarzsch

ild-Stra

ße1,85748Garch

ing,

Germ

any

Abstra

ct

This

reviewpresents

acom

preh

ensive

overviewof

galaxybias,

that

is,th

estatistical

relationbetw

eenth

edistrib

ution

ofgalaxies

and

matter.

We

focus

onlarge

scalesw

here

cosmic

den

sityfield

sare

quasi-

linear.

On

these

scales,th

eclu

stering

ofgalaxies

canbe

describ

edby

apertu

rbative

bias

expan

sion,an

dth

ecom

plicated

physics

ofgalaxy

formation

isab

sorbed

bya

finite

setof

coe�cients

ofth

eexp

ansion

,called

biasparam

eters.T

he

reviewbegin

sw

itha

ped

agogicalproof

ofth

isvery

important

result,

which

forms

the

basis

ofth

erigorou

spertu

rbative

descrip

tionof

galaxyclu

stering,

under

the

assum

ption

sof

Gen

eralR

elativityan

dG

aussian

,ad

iabatic

initial

condition

s.K

eycom

pon

entsof

the

bias

expan

sionare

alllead

ing

localgravitation

alob

servables,

which

inclu

des

the

matter

den

sitybut

alsotid

alfield

san

dth

eirtim

ederivatives.

We

hen

ceexp

and

the

defi

nition

oflocal

biasto

encom

pass

allth

esecontrib

ution

s.T

his

derivation

isfollow

edby

apresentation

ofth

epeak-b

ackground

split

inits

general

form,w

hich

elucid

atesth

ephysical

mean

ing

ofth

ebias

param

eters,an

da

detailed

descrip

tionof

the

connection

betw

eenbias

param

etersan

dgalaxy

(orhalo)

statistics.W

eth

enreview

the

excursion

setform

alisman

dpeak

theory

which

provid

epred

ictions

forth

evalu

esof

the

bias

param

eters.In

the

remain

der

ofth

ereview

,w

econ

sider

the

generalization

sof

galaxybias

required

inth

epresen

ceof

various

types

ofcosm

ologicalphysics

that

gobeyon

dpressu

relessm

atterw

ithad

iabatic,

Gau

ssianin

itialcon

dition

s:prim

ordial

non

-Gau

ssianity,

massive

neu

trinos,

baryon

-CD

Misocu

rvature

pertu

rbation

s,dark

energy,

and

mod

ified

gravity.Fin

ally,w

ediscu

sshow

the

descrip

tionof

galaxybias

inth

egalaxies’

restfram

eis

relatedto

observed

clusterin

gstatistics

measu

redfrom

the

observed

angu

larposition

san

dred

shifts

inactu

algalaxy

catalogs.

Keyw

ords:cosm

ology,large-scale

structu

re,galaxy

surveys,

galaxybias,

dark

matter,

prim

ordial

non

-gaussian

ity

Email

addresses:

[email protected]

(Vincen

tDesja

cques),

[email protected]

(Donghui

Jeo

ng),

[email protected](F

abianSch

midt)

Prep

rintsu

bmitted

toPhysics

Repo

rtsNovem

ber30,2016


All of what I will talk about, and much more, can be found in:

EFT approach in LSS

• Effective field theory: write down all terms (in Lagrangian or equations of motion) that are consistent with symmetries

• Gravity: general covariance

• Galaxy density: 0-component of 4-vector (momentum density)

• Order contributions by perturbative order, and number of spatial derivatives

EFT approach in LSS• LSS is non-relativistic: velocities v << c

• Only relevant metric component is time-time component: gravitational potential Φ

• Relevant remaining gauge symmetries:

Time rescaling

Time-dependent Lorentz boost(“generalized Galilei transformation”

Rotations

⌧ ! ⌧ + c(⌧) , � ! �+ C(⌧)

xi ! xi + ⇠i(⌧) , � ! �+Ai(⌧)xi

vi ! vi + ⇠i0(⌧)

xi ! Rijx

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EFT approach in LSS• LSS is non-relativistic: velocities v << c

• Only relevant metric component is time-time component: gravitational potential Φ

• Relevant remaining gauge symmetries:

Time rescaling

Time-dependent Lorentz boost(“generalized Galilei transformation”)

Rotations

⌧ ! ⌧ + c(⌧) , � ! �+ C(⌧)

xi ! xi + ⇠i(⌧) , � ! �+Ai(⌧)xi

vi ! vi + ⇠i0(⌧)

xi ! Rijx


EFT bias expansion• What can (and thus has to) appear?

• Stress-energy (matter):

• But not velocity (forbidden by gauge symmetry)

• Time derivatives have to be convective:

• Gravity (potential):

• But not Φ or r�<latexit sha1_base64="oJ/F6EoSjnCBCR1qKwtHH+IANdg=">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</latexit><latexit sha1_base64="oJ/F6EoSjnCBCR1qKwtHH+IANdg=">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</latexit><latexit sha1_base64="oJ/F6EoSjnCBCR1qKwtHH+IANdg=">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</latexit><latexit sha1_base64="oJ/F6EoSjnCBCR1qKwtHH+IANdg=">AAAInXichZVbb9s2FMfV7lI3u7TdHvdQYUGAPaiGlDlrHoahaNMlD3bjDHFTLHINiqItwRSpUJQTg9C32Ov2vfZtdnRLGHLDBFimzu/Pc3g5h4xymhbS9/9+8PCTTz/7/NHg8c4XX3719ZOnz755X/BSYDLDnHLxIUIFoSkjM5lKSj7kgqAsouQiWr+p+cWGiCLl7FxuczLP0IqlyxQjCabfQ4YiisJpki6e7vpDv3lcuxF0jV2ne6aLZ4+qMOa4zAiTmKKiuAz8XM4VEjLFlFQ7YVmQHOE1WpFLaDKUkWKumiFX7h5YYnfJBfyYdBur3kOhrCi2WQTKDMmkMFlt9KLs3/BlKZeHc5WyvJSE4TbWsqSu5G69BG6cCoIl3UIDYZHCcF2cIIGwhIXa2XNLlkrwGJNlmGQ5VqGXfFQvgir0VCgyd5LjqupxUTKNTxYq5DGXd7zrr3VtpMAZucY8yxCLVQj2pPqPOLouvRX+n0MY1z2P+sh04c8gooitmg0TREe/ABI90oFIeASMQ17VadcYqkVmqE6zqn6RFbJRdIsiE41vUTiGNI6R5ZbdKlTmm/M5jTQc2Xis4S5ALdpzxyinCBOX50QgWSdjvYEUrKotko/71Z2tuOqto86a6lL1Yr+CvHNTVtce6Z3vtNIJCCeLUJIbqaYUopsrfwpjrNMaI6pOzSnMNDgz4RGhGj4y8bkGz014psEzK6E0ODEhPtHoiUmPNXhcWWk21fDUxmMNj+u12nM3UL0cKtVQbqrLYK7CKFO7Vj1sSNUQYoF1C9YWyFuQW+CqBVcWEC0QFihbUFrgugXXFrhpwY0Fti3Y9ifMhiVIqrB5bRSr7uyg4zSuz1De52UPl4yqZZeA77pFXQlC1kYyVU1CSbMEC7AX6SpD1mbFQOKux32CgCCaJzbh9VFSF6QRheR1nA0S0EgpZ80om3uiPZvh5Gm+1G8kvoCPSokVnCv+8CfP9/zK0PAtoq9peafaP/CGo8AbHh4a0l+5IIU8huVgnXgY/OgND0Ye/BvaI0LyaSly2vsdjsAteH15YMwmqmM3+VmvujUoO2MFiS19P9FGDRd2YF7PduP9/jDwh8HZaPfV6+7qHjjfOd87PziB89J55Zw4U2fmYIc5fzh/On8Nng/eDsaDd6304YOuz7fOvWdw8Q+6yCXn</latexit>

r2� , (@i@j�)2 ,

D

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D

D⌧= @⌧ + vi@i

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� , �2 , r2� , ✓ = @ivi ,

D�

D⌧, · · ·

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⌧ ! ⌧ + c(⌧) , � ! �+ C(⌧)

xi ! xi + ⇠i(⌧) , � ! �+Ai(⌧)xi

vi ! vi + ⇠i0(⌧)

xi ! Rijx


EFT bias expansion• What can (and thus has to) appear?

• Stress-energy (matter):

• But not velocity (forbidden by gauge symmetry)

• Time derivatives have to be convective:

• Gravity (potential):

• But not Φ or r�<latexit sha1_base64="oJ/F6EoSjnCBCR1qKwtHH+IANdg=">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</latexit><latexit sha1_base64="oJ/F6EoSjnCBCR1qKwtHH+IANdg=">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</latexit><latexit sha1_base64="oJ/F6EoSjnCBCR1qKwtHH+IANdg=">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</latexit><latexit sha1_base64="oJ/F6EoSjnCBCR1qKwtHH+IANdg=">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</latexit>

r2� , (@i@j�)2 ,

D

D⌧r2� , · · ·

<latexit 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D

D⌧= @⌧ + vi@i

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� , �2 , r2� , ✓ = @ivi ,

D�

D⌧, · · ·

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⌧ ! ⌧ + c(⌧) , � ! �+ C(⌧)

xi ! xi + ⇠i(⌧) , � ! �+Ai(⌧)xi

vi ! vi + ⇠i0(⌧)

xi ! Rijx


� ⌘ ⇢m � ⇢m⇢m

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EFT bias expansion

• We are not done yet however… Two issues:

• Many terms are redundant, as they are related through the equations of motion for matter and gravity

• Cumbersome, but no problem - can eliminate redundant terms order by order in perturbations

• So far, we have written the EFT as local in time and space

• Only makes sense if spatial and time derivatives are suppressed

• True for spatial derivatives, but not for time derivatives! Galaxies form over many Hubble times (as does matter field)

• Theory is “nonlocal in time”

Galaxy formation

• Consider coarse-grained (large scale) view of region that forms a galaxy at conformal time τ

• Formation happens over long time scale, but small spatial scale R*

• For halos, expect R⇤ . RL

Galaxy formation• Consider large-scale perturbations

• Galaxy density then becomes a local function in space*

• Using equations of motion, we can eliminate dependence on matter density and velocity

• We are left with nonlinear, nonlocal-in-time functional of tidal tensor:ng(x, ⌧) = Fg [@i@j�(xfl(⌧

0), ⌧ 0)]

* higher spatial derivatives are suppressed by (λ/R*)2 -> later

x = xfl(⌧)

xfl(⌧0)

Mirbabayi, FS, Zaldarriaga ’14

• Consider operator (field) O(x,t) that is constructed from *

• For simplicity, consider linear dependence of galaxy on O

• Linear functional in time:

• In perturbation theory, we know the time evolution of all these operators, e.g.

• At n-th order, there can be at most n different time dependences, and hence <= n independent terms!

• Equivalently: arbitrarily high time derivatives can be written in terms of <= n terms

Non-locality in time2.3 Systematics of the Bias Expansion

We now describe how to systematically carry out the bias expansion up to higher orders, starting

from Eq. (2.5). We will restrict ourselves to lowest order in (spatial) derivatives, which yields

the leading operators on large scales. Let us begin by assuming Gaussian initial conditions. As

discussed above, Eq. (2.5) still involves a functional dependence on the long-wavelength modes

along the past fluid trajectory. Consider a general operator O constructed out of the field3

⇧`

ij⌘ ⇧ij . At linear order, the dependence of ng(x, ⌧) on O can formally be written as

ng(x, ⌧) =

Z⌧

⌧in

d⌧ 0 fO(⌧, ⌧0)O(xfl(⌧

0), ⌧ 0) (2.28)

=

Z⌧

⌧in

d⌧ 0 fO(⌧, ⌧0)

�O(x, ⌧) +

Z⌧

⌧in

d⌧ 0 fO(⌧, ⌧0)(⌧ 0 � ⌧)

�D

D⌧O(x, ⌧) + · · · ,

where D/D⌧ is a convective time derivative. In Eulerian coordinates, D/D⌧ is given by

D

D⌧=

@

@⌧+ ui

@

@xi, (2.29)

where ui is the peculiar velocity. The expansion in (2.28) shows that we have to allow for convec-

tive time derivatives such as D(⇧ij)/D⌧ , in the basis of operators. Including time derivatives of

arbitrary order then provides a complete basis of operators. Note however that the higher-order

terms in the expansion (2.28) are not suppressed, since both galaxies and matter fields evolve

over the Hubble time scale. Fortunately, it is possible to reorder the terms in (2.28) so that only

a finite number need to be kept at a given order in perturbation theory [15].

To do this, we do not work with the convective time derivatives of operators directly, but

instead take special linear combinations which are chosen in such a way that the contributions

from lower-order operators cancel. Let O[n] denote an operator that starts are n-th order in

perturbation theory, while O(n) stands for the n-th order contribution of O. Consider for example

⇧ij at linear order in perturbation theory, i.e. ⇧(1)ij

. We will also define ⇧[1]ij

⌘ ⇧ij , as it will turn

out to be the lowest-order operator in a hierarchy. Taking the convective derivative of ⇧(1)ij

(at

this order, this is simply equivalent to the ordinary time derivative) with respect to the logarithm

of the growth factor D(⌧), we have

D

D lnD⇧(1)

ij= (Hf)�1 D

D⌧⇧(1)

ij= ⇧(1)

ij, (2.30)

where f ⌘ d lnD/d ln a is the logarithmic growth rate. Hence, the operator

⇧[2]ij

⌘

✓D

D lnD� 1

◆⇧[1]

ij, (2.31)

involves the first time derivative of ⇧ij , but starts at second order in perturbation theory. This

can be generalized to a recursive definition at n-th order [15],

⇧[n]⌘

1

(n� 1)!

(Hf)�1 D

D⌧⇧[n�1]

� (n� 1)⇧[n�1]

�. (2.32)

3To avoid clutter in the expressions, we will drop the labels ` on the long-wavelength fields from now on.

11

� = D(⌧)�(1) +D2(⌧)�(2) + · · ·

Senatore ’14; Mirbabayi, FS, Zaldarriaga ’14

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* From Poisson equation, , so this includes “local bias” terms δn

• Consider operator (field) O(x,t) that is constructed from *

• For simplicity, consider linear dependence of galaxy on O

• Linear functional in time:

• In perturbation theory, we know the time evolution of all these operators, e.g.

• At n-th order, there can be at most n different time dependences, and hence <= n independent terms!

• Equivalently: arbitrarily high time derivatives can be written in terms of <= n terms

Non-locality in time2.3 Systematics of the Bias Expansion

We now describe how to systematically carry out the bias expansion up to higher orders, starting

from Eq. (2.5). We will restrict ourselves to lowest order in (spatial) derivatives, which yields

the leading operators on large scales. Let us begin by assuming Gaussian initial conditions. As

discussed above, Eq. (2.5) still involves a functional dependence on the long-wavelength modes

along the past fluid trajectory. Consider a general operator O constructed out of the field3

⇧`

ij⌘ ⇧ij . At linear order, the dependence of ng(x, ⌧) on O can formally be written as

ng(x, ⌧) =

Z⌧

⌧in

d⌧ 0 fO(⌧, ⌧0)O(xfl(⌧

0), ⌧ 0) (2.28)

=

Z⌧

⌧in

d⌧ 0 fO(⌧, ⌧0)

�O(x, ⌧) +

Z⌧

⌧in

d⌧ 0 fO(⌧, ⌧0)(⌧ 0 � ⌧)

�D

D⌧O(x, ⌧) + · · · ,

where D/D⌧ is a convective time derivative. In Eulerian coordinates, D/D⌧ is given by

D

D⌧=

@

@⌧+ ui

@

@xi, (2.29)

where ui is the peculiar velocity. The expansion in (2.28) shows that we have to allow for convec-

tive time derivatives such as D(⇧ij)/D⌧ , in the basis of operators. Including time derivatives of

arbitrary order then provides a complete basis of operators. Note however that the higher-order

terms in the expansion (2.28) are not suppressed, since both galaxies and matter fields evolve

over the Hubble time scale. Fortunately, it is possible to reorder the terms in (2.28) so that only

a finite number need to be kept at a given order in perturbation theory [15].

To do this, we do not work with the convective time derivatives of operators directly, but

instead take special linear combinations which are chosen in such a way that the contributions

from lower-order operators cancel. Let O[n] denote an operator that starts are n-th order in

perturbation theory, while O(n) stands for the n-th order contribution of O. Consider for example

⇧ij at linear order in perturbation theory, i.e. ⇧(1)ij

. We will also define ⇧[1]ij

⌘ ⇧ij , as it will turn

out to be the lowest-order operator in a hierarchy. Taking the convective derivative of ⇧(1)ij

(at

this order, this is simply equivalent to the ordinary time derivative) with respect to the logarithm

of the growth factor D(⌧), we have

D

D lnD⇧(1)

ij= (Hf)�1 D

D⌧⇧(1)

ij= ⇧(1)

ij, (2.30)

where f ⌘ d lnD/d ln a is the logarithmic growth rate. Hence, the operator

⇧[2]ij

⌘

✓D

D lnD� 1

◆⇧[1]

ij, (2.31)

involves the first time derivative of ⇧ij , but starts at second order in perturbation theory. This

can be generalized to a recursive definition at n-th order [15],

⇧[n]⌘

1

(n� 1)!

(Hf)�1 D

D⌧⇧[n�1]

� (n� 1)⇧[n�1]

�. (2.32)

3To avoid clutter in the expressions, we will drop the labels ` on the long-wavelength fields from now on.

11

Senatore ’14; Mirbabayi, FS, Zaldarriaga ’14

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* From Poisson equation, , so this includes “local bias” terms δm

O(xfl, ⌧) = Dm(⌧)O(m)(xfl, ⌧0) +D

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Complete bias expansion


By reordering, we obtain a compact list of terms up to any order, e.g. up to fourth: due to short-scale modes whose statistics are uncorrelated over large distances. Such stochasticity

is described by introducing a set of random variables ✏i(x) which are uncorrelated with the matter

variables. They are thus completely described by their moments h(✏i)n(✏j)m · · ·i (n+m > 1, with

h✏ii = 0 since any expectation value can be absorbed into the mean galaxy density). Let us

restrict to Gaussian initial conditions for the moment. We can demand that these moments only

depend on the statistics of the initial small-scale fluctuations '(ks), |ks| & ⇤. The influence

of these small-scale initial conditions on the late-time galaxy density will then depend on the

long-wavelength observables through the gravitational evolution of the initial conditions. Thus,

we need to allow for stochastic terms in combination with each of the operators in the basis

discussed in Sec. 2.3. Counting the stochastic fields as linear perturbations, we have to add four

stochastic fields ✏i up to cubic order, namely

1st ✏1 (2.36)

2nd ✏2Tr[⇧ij ]

3rd ✏3Tr[(⇧ij)2] , ✏4 (Tr[⇧ij ])

2 ,

where ⇧ij = ⇧[1]ij, as defined in (2.2). Let us note that, in principle, one could also have stochastic

terms of the form ✏ij⇧ij . However, in position space, correlation functions of ✏ij are proportional

to (products of) Kronecker delta tensors and Dirac delta functions. For this reason, the e↵ects of

these terms on the statistics of galaxies are indistinguishable from those written in (2.36). Hence,

the basis (2.36) fully captures the e↵ects of stochastic noise terms.

Let us now consider the non-Gaussian case, and study under what conditions PNG induces

additional stochastic terms. By assumption, stochastic variables ✏i only depend on the statistics

of the small-scale initial perturbations. As long as the coupling between long and short modes

is completely captured by the relation (2.15), all e↵ects of this coupling are accounted for in our

non-Gaussian basis (2.34). In this case, Eq. (2.36) only needs to be augmented by terms of the

same type multiplied by ,

1st � (2.37)

2nd ✏

3rd ✏ � Tr[⇧ij ] .

As we show in App. B.2, this holds whenever the initial conditions are derived from a single

random degree of freedom, corresponding to a single set of random phases. This is the case for

the ansatz in (2.11).

Consider the correlation of the amplitude of small-scale initial perturbations over large dis-

tances. This can be quantified by defining the small-scale potential perturbations 's(x) through

a high-pass filter Ws. Writing 's(k) ⌘ Ws(k)'(k) in Fourier space, where Ws(k) ! 0 for k ⌧ ⇤,

we obtain the following two-point function of ('s)2(k):

h('s)2(k) ('s)

2(k0)i0 =4Y

i=1

Z

ki

�D(k � k12) �D(k0� k34)⇥ h's(k1)'s(k2)'s(k3)'s(k4)i . (2.38)

13

…

Small-scale modes lead to stochastic contributions:

⇧[1]ij = @i@j�(x, ⌧)

⇧[n]ij / D

D⌧⇧[n�1]

ij

where

starts at n-th order in pert. theory

Up to fourth order, we therefore have, in exact formal analogy to Eq. (2.61),

1st tr[⇧[1]] (2.64)

2nd tr[(⇧[1])2] , (tr[⇧[1]])2

3rd tr[(⇧[1])3] , tr[(⇧[1])2] tr[⇧[1]] , (tr[⇧[1]])3 , tr[⇧[1]⇧[2]]

4th tr[(⇧[1])4] , tr[(⇧[1])3] tr[⇧[1]] ,⇣tr[(⇧[1])2]

⌘2

, (tr[⇧[1]])4 ,

tr[⇧[1]] tr[⇧[1]⇧[2]] , tr[⇧[1]⇧[1]⇧[2]] , tr[⇧[1]⇧[3]] , tr[⇧[2]⇧[2]] .

This basis o↵ers the advantage of having a close connection to the standard Eulerian bias expansion. Forexample, the coe�cient of the term (tr[⇧[1]])n is precisely b�n = bn/n!, since tr[⇧[1](x, ⌧)] = �(x, ⌧) at allorders. The term tr[(⇧[1])2] = (Kij)2 + �2/3 on the other hand contains the tidal field squared. Of course,as in the Lagrangian case, explicit time derivatives appear in the bias expansion at third order through the

operator tr[⇧[1]⇧[2]], which again is directly related to the operator O(3)

td(see p. 31 and Appendix C). As

mentioned after Eq. (2.61), gravitational and velocity shear are no longer su�cient in the bias expansion

starting at fourth order, as evidenced by the appearance of ⇧[3]

ij .

We now discuss the key approximation made in constructing the convenient bases Eq. (2.61) andEq. (2.64), namely that all operator contributions at a given perturbative order n have the same timedependence, [D(⌧)]n. This is only strictly true in an EdS Universe, while in ⇤CDM and quintessence cos-mologies new time dependences appear at each new order. For example, second-order operators can havea time dependence given by [D(⌧)]2 or by D2(⌧), where D2(⌧) /

RD2d ln D is the second-order growth

factor. This means that the operators in the bases described above are in general not su�cient anymore.However, we show in Appendix B.6 that the first instance of a new term in the bias expansion appears onlyat fourth order. Specifically, the tr[⇧[1]⇧[3]] term in Eq. (2.64) splits into two terms which, however, haveto have very similar bias coe�cients if the nonlinear growth factors approximately obey Dn(⌧) ' [D(⌧)]n;the departures from this relation are at the percent level for a standard ⇤CDM cosmology. This meansthat the additional operators added to complete the operator bases described here will be (i) fourth andhigher order; (ii) suppressed by a numerical prefactor . 0.1 relative to the terms included in Eq. (2.64).They will thus be irrelevant in most practical applications. Note that, since we only work to third order inperturbation theory there, all results given in Sec. 4 hold in ⇤CDM and quintessence cosmologies.

2.6 Higher-derivative bias

In the treatment of bias so far, we have approximated the formation of halos and galaxies as perfectlylocal in a spatial sense. After reordering the time derivatives along the fluid trajectory, we have written thebias expansion for �g(x, ⌧) in terms of operators evaluated at the same location: O(x, ⌧) in the Eulerianbasis, or OL(q, ⌧) in the Lagrangian basis. However, we know that the formation of halos and galaxiesinvolves the collapse of matter from a finite region in space, and thus, the local bias expansion derived abovecannot be completely correct on all scales. In this section, we study the limitation of the spatially-localapproximation and derive the set of additional operators to include in the expansion Eq. (2.56). We referto these operators as higher-derivative operators. They naturally arise in peak theory or the excursion-setapproach [13, 150, 151] (see Sec. 5 and Sec. 6 for a detailed discussion).

In order to incorporate the deviation from perfect locality of galaxy formation, we should replace thelocal operators O(x, ⌧) appearing in Eq. (2.56) with functionals [152, 153]. For example, the linear-orderoperator in the Eulerian basis, O = �, now becomes

b�(⌧)�(x, ⌧) !Z

d3y F�(y, ⌧)�(x + y, ⌧) , (2.65)

where F�(y, ⌧) is a kernel that is in general time dependent. Here, we have used the homogeneity of theUniverse, or the absence of preferred locations, which dictates that F� is independent of x. We can now

38



By reordering, we obtain a compact list of terms up to any order, e.g. up to fourth: due to short-scale modes whose statistics are uncorrelated over large distances. Such stochasticity

is described by introducing a set of random variables ✏i(x) which are uncorrelated with the matter

variables. They are thus completely described by their moments h(✏i)n(✏j)m · · ·i (n+m > 1, with

h✏ii = 0 since any expectation value can be absorbed into the mean galaxy density). Let us

restrict to Gaussian initial conditions for the moment. We can demand that these moments only

depend on the statistics of the initial small-scale fluctuations '(ks), |ks| & ⇤. The influence

of these small-scale initial conditions on the late-time galaxy density will then depend on the

long-wavelength observables through the gravitational evolution of the initial conditions. Thus,

we need to allow for stochastic terms in combination with each of the operators in the basis

discussed in Sec. 2.3. Counting the stochastic fields as linear perturbations, we have to add four

stochastic fields ✏i up to cubic order, namely

1st ✏1 (2.36)

2nd ✏2Tr[⇧ij ]

3rd ✏3Tr[(⇧ij)2] , ✏4 (Tr[⇧ij ])

2 ,

where ⇧ij = ⇧[1]ij, as defined in (2.2). Let us note that, in principle, one could also have stochastic

terms of the form ✏ij⇧ij . However, in position space, correlation functions of ✏ij are proportional

to (products of) Kronecker delta tensors and Dirac delta functions. For this reason, the e↵ects of

these terms on the statistics of galaxies are indistinguishable from those written in (2.36). Hence,

the basis (2.36) fully captures the e↵ects of stochastic noise terms.

Let us now consider the non-Gaussian case, and study under what conditions PNG induces

additional stochastic terms. By assumption, stochastic variables ✏i only depend on the statistics

of the small-scale initial perturbations. As long as the coupling between long and short modes

is completely captured by the relation (2.15), all e↵ects of this coupling are accounted for in our

non-Gaussian basis (2.34). In this case, Eq. (2.36) only needs to be augmented by terms of the

same type multiplied by ,

1st � (2.37)

2nd ✏

3rd ✏ � Tr[⇧ij ] .

As we show in App. B.2, this holds whenever the initial conditions are derived from a single

random degree of freedom, corresponding to a single set of random phases. This is the case for

the ansatz in (2.11).

Consider the correlation of the amplitude of small-scale initial perturbations over large dis-

tances. This can be quantified by defining the small-scale potential perturbations 's(x) through

a high-pass filter Ws. Writing 's(k) ⌘ Ws(k)'(k) in Fourier space, where Ws(k) ! 0 for k ⌧ ⇤,

we obtain the following two-point function of ('s)2(k):

h('s)2(k) ('s)

2(k0)i0 =4Y

i=1

Z

ki

�D(k � k12) �D(k0� k34)⇥ h's(k1)'s(k2)'s(k3)'s(k4)i . (2.38)

13

…

Small-scale modes lead to stochastic contributions:

Time derivatives <~>nonlocal terms, e.g.

⇧[1]ij = @i@j�(x, ⌧)

⇧[n]ij / D

D⌧⇧[n�1]

ij

where

starts at n-th order in pert. theory

Up to fourth order, we therefore have, in exact formal analogy to Eq. (2.61),

1st tr[⇧[1]] (2.64)

2nd tr[(⇧[1])2] , (tr[⇧[1]])2

3rd tr[(⇧[1])3] , tr[(⇧[1])2] tr[⇧[1]] , (tr[⇧[1]])3 , tr[⇧[1]⇧[2]]

4th tr[(⇧[1])4] , tr[(⇧[1])3] tr[⇧[1]] ,⇣tr[(⇧[1])2]

⌘2

, (tr[⇧[1]])4 ,

tr[⇧[1]] tr[⇧[1]⇧[2]] , tr[⇧[1]⇧[1]⇧[2]] , tr[⇧[1]⇧[3]] , tr[⇧[2]⇧[2]] .

This basis o↵ers the advantage of having a close connection to the standard Eulerian bias expansion. Forexample, the coe�cient of the term (tr[⇧[1]])n is precisely b�n = bn/n!, since tr[⇧[1](x, ⌧)] = �(x, ⌧) at allorders. The term tr[(⇧[1])2] = (Kij)2 + �2/3 on the other hand contains the tidal field squared. Of course,as in the Lagrangian case, explicit time derivatives appear in the bias expansion at third order through the

operator tr[⇧[1]⇧[2]], which again is directly related to the operator O(3)

td(see p. 31 and Appendix C). As

mentioned after Eq. (2.61), gravitational and velocity shear are no longer su�cient in the bias expansion

starting at fourth order, as evidenced by the appearance of ⇧[3]

ij .

We now discuss the key approximation made in constructing the convenient bases Eq. (2.61) andEq. (2.64), namely that all operator contributions at a given perturbative order n have the same timedependence, [D(⌧)]n. This is only strictly true in an EdS Universe, while in ⇤CDM and quintessence cos-mologies new time dependences appear at each new order. For example, second-order operators can havea time dependence given by [D(⌧)]2 or by D2(⌧), where D2(⌧) /

RD2d ln D is the second-order growth

factor. This means that the operators in the bases described above are in general not su�cient anymore.However, we show in Appendix B.6 that the first instance of a new term in the bias expansion appears onlyat fourth order. Specifically, the tr[⇧[1]⇧[3]] term in Eq. (2.64) splits into two terms which, however, haveto have very similar bias coe�cients if the nonlinear growth factors approximately obey Dn(⌧) ' [D(⌧)]n;the departures from this relation are at the percent level for a standard ⇤CDM cosmology. This meansthat the additional operators added to complete the operator bases described here will be (i) fourth andhigher order; (ii) suppressed by a numerical prefactor . 0.1 relative to the terms included in Eq. (2.64).They will thus be irrelevant in most practical applications. Note that, since we only work to third order inperturbation theory there, all results given in Sec. 4 hold in ⇤CDM and quintessence cosmologies.

2.6 Higher-derivative bias

In the treatment of bias so far, we have approximated the formation of halos and galaxies as perfectlylocal in a spatial sense. After reordering the time derivatives along the fluid trajectory, we have written thebias expansion for �g(x, ⌧) in terms of operators evaluated at the same location: O(x, ⌧) in the Eulerianbasis, or OL(q, ⌧) in the Lagrangian basis. However, we know that the formation of halos and galaxiesinvolves the collapse of matter from a finite region in space, and thus, the local bias expansion derived abovecannot be completely correct on all scales. In this section, we study the limitation of the spatially-localapproximation and derive the set of additional operators to include in the expansion Eq. (2.56). We referto these operators as higher-derivative operators. They naturally arise in peak theory or the excursion-setapproach [13, 150, 151] (see Sec. 5 and Sec. 6 for a detailed discussion).

In order to incorporate the deviation from perfect locality of galaxy formation, we should replace thelocal operators O(x, ⌧) appearing in Eq. (2.56) with functionals [152, 153]. For example, the linear-orderoperator in the Eulerian basis, O = �, now becomes

b�(⌧)�(x, ⌧) !Z

d3y F�(y, ⌧)�(x + y, ⌧) , (2.65)

where F�(y, ⌧) is a kernel that is in general time dependent. Here, we have used the homogeneity of theUniverse, or the absence of preferred locations, which dictates that F� is independent of x. We can now

38

@i@jr2

�2<latexit 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• Some virtues of this expansion:

• Linearly independent at each order

• Complete in the EFT sense: closed under renormalization (proof see MSZ)

• Equivalent expansions in Eulerian and Lagrangian space

• Bias parameters can be mapped from one frame to another unambiguously


Spatial nonlocality and scale-dependent bias

• Beyond large-scale limit: need to expand spatial nonlocality of galaxy formation

• Higher derivative biases are suppressed with scale R*

• E.g.,

• This also allows for baryonic physics, which has to come with additional derivatives

• Example: pressure perturbations

• Pressure force:

• At higher order in derivatives, time evolution no longer determined by gravity alone

R2⇤r2�, R2

⇤(r�)2, R2⇤r2(sij)

2, · · ·�g(k, ⌧) =�b1 + br2�k

2R2⇤��(k, ⌧)

�p = c2s�⇢<latexit 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F = r�p / r�<latexit 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Velocity bias• Galaxy velocities are important probe of cosmology - but how

related to matter velocity?

• Recall that bias expansion for galaxy density cannot include

• The same is true for any observable - in particular also the relative velocity between matter and galaxies

• Hence, relative velocity can be written as

• Necessarily higher derivative ~ R*2 ! Cf. pressure forces

• Also small-scale stochastic velocities, with power spectrum ~ k4, which captures virial motions

• Summary: Galaxy velocities are unbiased on large scales.

vig � vi = @in� , (@i@j�)

2 , · · ·o

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Transformation to redshift space

• Observed galaxy statistics are obtained from rest-frame statistics via coordinate transformation to redshift space

• By combining three ingredients, we can obtain consistent theoretical description for observed galaxy statistics (n-point functions in redshift space):

• Perturbation theory for matter

• Bias expansion

• Velocity bias expansion

xobs = x+v · naH

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• So far, assumed Gaussian, adiabatic initial conditions

• If these assumptions are violated at most weakly (as indicated by CMB), can perturbatively include these:

• Primordial non-Gaussianity

• Relative density/velocity perturbations between CDM and baryons (from pre-recombination)

• In each case, obtain well-defined finite set of additional terms in bias expansion

Impact of initial conditions

Application: galaxy power spectrum

• Assume we can measure rest-frame galaxy density

• That is, neglect redshift-space distortions and other projection effects

• Leading-order galaxy power spectrum at fixed time:

• Valid on very large scales

• 2 free parameters

di�cult, since measurements of higher-order statistics become necessary, for instance the trispectrum inthe case of cubic-order bias parameters. The implementation and the required computational resources forhigher-order statistics become increasingly demanding.

Since this is a substantial subsection, we provide a brief outline here. We begin with the leading two-and three-point functions in Eulerian space, both in the Fourier- and real-space representations (Sec. 4.1.1).We then briefly discuss the corresponding results in Lagrangian space (Sec. 4.1.2), which are relevant forestimating bias parameters from halos identified in N-body simulations. Sec. 4.1.3 then provides a quantita-tive, albeit simplified and idealized, forecast of the ability of current and future galaxy surveys to measurethe bias parameters and amplitude of the matter power spectrum using the results of Sec. 4.1.1. Next, wederive the next-to-leading correction to the galaxy two-point function (1-loop power spectrum) in Sec. 4.1.4,illustrating how the predictions of Sec. 4.1.1 can be taken to higher order and what scalings the higher-orderterms obey.

4.1.1 Two- and three-point functions at leading orderWe begin with the leading-order (LO), or tree-level, predictions for the power spectrum and bispectrum of

halos, that is, the two- and three-point correlation functions in Fourier space. The leading-order calculationof the halo power spectrum and bispectrum requires, respectively, linear- and second-order perturbationtheory (see Appendix B). These leading-order predictions are accurate on su�ciently large scales, roughlyat the level of 10% for k . 0.03 h Mpc�1 in Fourier space at z = 0 (a more precise calculation is the subjectof Sec. 4.1.4); the range increases at higher redshifts [100]. We will present the corresponding real-spaceresults, the correlation functions, at the end of this section.

The halo auto-power spectrum and halo-matter cross-power spectrum are given by

P lohh(k) ⌘ h�h(k)�h(k0)i0lo = b2

1PL(k) + P {0}

"

P lohm(k) ⌘ h�h(k)�m(k0)i0lo = b1PL(k) , (4.2)

where, here and throughout, a prime on an expectation value denotes that the momentum-conserving Diracdelta, (2⇡)3�D(k+k

0) in case of Eq. (4.2), is to be dropped (see Tab. 2). As mentioned in the introduction,we drop the time argument throughout this section for clarity. Again, we would obtain the same relationfor galaxies if we were able to measure their proper rest-frame density at the true physical position, that

is, without redshift-space distortions and other projection e↵ects. P {0}

" = limk!0h"(k)"(k0)i0 is the scale-independent large-scale stochastic contribution [see Eq. (2.83) in Sec. 2.8]. Note that this is a renormalizedstochastic term which absorbs scale-independent terms from higher loop integrals (see Sec. 4.1.4). We will

discuss P {0}

" in more detail in Sec. 4.5.3. The next-to-leading-order corrections to Phh(k) as well as Phm(k)from nonlinear evolution of both matter and bias, and from higher-derivative biases, will be described inSec. 4.1.4.

Since the halo stochasticity contributes to the halo auto-power spectrum Phh(k) but not to the halo-matter cross-power spectrum Phm(k), the latter o↵ers the simplest and cleanest measurement of the linearbias parameter b1 for halos (see e.g. [228, 229, 125]). This technique can also be applied to galaxies,by measuring the matter distribution through weak gravitational lensing, specifically, the cross-correlation(“galaxy-galaxy lensing”) of the projected galaxy density with the tangential shear measured from sourcegalaxies at higher redshifts [93, 230, 231, 232, 233, 234] (see [235] for a recent review). Briefly, for lens galaxiesat a known comoving distance �L and source galaxies following a normalized redshift distribution p(z), thestacked tangential shear around galaxies in angular multipole space corresponds to a projection of the real-space galaxy-matter power spectrum, Pgm = b1Pmm at leading order, given in the Limber approximation[236] by

Cg�(l) =3

2⌦m0H

2

0

Zdz p(z)

�(z) � �L

�(z)

�1 + z(�L)

�LPgm

k =

pl(l + 1)

�L, z(�L)

!. (4.3)

By itself, this observable su↵ers from a degeneracy between b1 and the matter power spectrum normalization.This degeneracy can be broken by including the projected auto-correlation of galaxies Cgg(l), and/or thecosmic shear power spectrum C��(l), as recently applied in [237, 31].

74

Pgg(k)<latexit sha1_base64="Whwo2OoRMsKKkXPb5q8OgBWyb5k=">AAAInHichZXdbts2FMfVdlu97KvdLgcMwoIAHeAaUuasuSiGoguWALMXZ4iTDpFrUBQtC6ZIhaKcGASfYrfbg+1tdvThhCE3TIBl6vz+POeQPCTjgmalDIK/Hz1+8sGHHz3tfbzzyaefff7Fs+dfXpS8EphMMadcvItRSWjGyFRmkpJ3hSAojym5jFc/1fxyTUSZcXYuNwWZ5Shl2SLDSILp98lcpal+sfpu/mw3GATN47uNsGvset0zmT9/qqOE4yonTGKKyvIqDAo5U0jIDFOid6KqJAXCK5SSK2gylJNyppqMtb8HlsRfcAE/Jv3GavZQKC/LTR6DMkdyWdqsNvbj/N/wVSUXhzOVsaKShOE21qKivuR+PQN+kgmCJd1AA2GRQbo+XiKBsIR52tnzK5ZJ8JiQRbTMC6yi/vK9ehnqqK8ikfvjAmu9xWXFDD6eq4gnXN7zrr/RtZECZ+QG8zxHLFER2Jf6P+KYuuxO+H8OIa8HHs3MTOFrEFHE0mbBBDHRj4DEFplALHkMjENZ1VXXGPQ8t1Snua5fJEUuiu9QbKPRHYpGUMUJctyyO4XKA3s8p7GBYxePDNwFqEV7/ggVFGHi84IIJOtirBeQglVFDMUUvd/X97byemsddtbMlKqX+xrqzs9YvfXI1vlOKx2DcDyPJLmVakIhuj3zp5BjXdYYUXVqD2FqwKkNjwg18JGNzw14bsMzA545BWXAsQ3xiUFPbHpswGPtlNnEwBMXjww8qudqz1/D7uWwUy3lWl+FMxXFudp19sOa6IYQB6xasHJA0YLCAdctuHaAaIFwQNWCygE3LbhxwG0Lbh2wacFme8Ks2RJJFTWvtWL63g46TpP6DOXbutzCBaNq0RXgr92kpoKQlVVMuikoaW/BEuxllubIWawESNL1eEgQEESLpUt4fZTUG9KKQoo6zhoJaGSUsybL5p5oz2Y4eZov9RtJLuFDK5HCuRIMfugH/UBbGr5B9C2t7lX7B/3BMOwPDg8t6c9ckFIew3SwTjwIv+8PDoZ9+Le0R4QUk0oUdOt3MAS34PXVgTWauI7d1Gc9605SbsUKkjj67UAbNVzYoX09u42L/UEYDMKz4e6bt93V3fO+9r71Xnih98p74514E2/qYS/3/vD+9P7qfdM76v3SG7fSx4+6Pl95D57exT88LiU3</latexit><latexit sha1_base64="Whwo2OoRMsKKkXPb5q8OgBWyb5k=">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</latexit><latexit sha1_base64="Whwo2OoRMsKKkXPb5q8OgBWyb5k=">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</latexit><latexit sha1_base64="Whwo2OoRMsKKkXPb5q8OgBWyb5k=">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</latexit>


4.1.4 Next-to-leading-order corrections to the two-point functionsSo far, we have derived the leading contributions to the two- and three-point functions of galaxies

and halos on large scales. In order to illustrate how higher-order corrections to the above results can bederived, we also present the next-to-leading-order (NLO, or 1-loop) correction to the two-point function.We only discuss real-space predictions without any projection e↵ects. Hence our expressions mostly applyto halos, but the structure of the perturbative expansion remains the same even when including projectione↵ects. Deriving the nonlinear correction to the two-point functions requires a third-order calculation inperturbation theory, since, for any operator O, contributions of the type hO(1)O0(3)i contribute at the sameorder as hO(2)O0(2)i [260, 126, 195]. Thus, the bias expansion in Eq. (4.1) is necessary and su�cient toderive this correction. Writing

Phm(k) = P lohm(k) + P nlo

hm (k) + · · ·Phh(k) = P lo

hh(k) + P nlohh (k) + · · · , (4.21)

and following the notation of [195], the NLO contributions to the halo-matter and halo-halo power spectrumare respectively given by13

P nlohm (k) = b1

⇥P nlomm(k) � 2C2

s,e↵k2PL(k)⇤+ P nlo

hm (k)

P nlohm (k) ⌘ b�2I [�(2),�2](k) + bK2I [�(2),K2

](k) +

✓bK2 +

2

5btd

◆fnlo(k)PL(k)

� br2�k2PL(k) + k2P {2}

""m (4.22)

P nlohh (k) = (b1)

2⇥P nlomm(k) � 2C2

s,e↵k2PL(k)⇤+ 2b1P

nlohm (k) +

X

O,O02{�2,K2}

bObO0I [O,O0](k) + k2P {2}

" ,

where P nlomm(k) is the NLO correction to the matter power spectrum [Eq. (B.18)], C2

s,e↵ ⌘ (2⇡)c2

s,e↵/k2nl is

the scaled e↵ective sound speed of the matter fluid [82, 83] [cf. Eq. (B.28)] and knl is defined in Eq. (4.25),while "m is the e↵ective stochastic contribution to the matter density (Appendix B.3). Further,

fnlo(k) = 4

Z

p

[p · (k � p)]2

p2|k � p|2 � 1

�F2(k, �p)PL(p)

I [O,O0](k) = 2

" Z

pSO(k � p,p)SO0(k � p,p)PL(p)PL(|k � p|)

�Z

pSO(�p,p)SO0(�p,p)PL(p)PL(p)

#, (4.23)

where SO(k1,k2) =

8><

>:

F2(k1,k2), O = �(2)

1, O = �2

(k1 · k2)2 � 1/3, O = K2

. (4.24)

We see that, in addition to14 b�2 = b2/2 and bK2 , which also enter the leading-order halo three-point

function, one new local bias term appears in the NLO halo power spectra, namely O(3)

tddefined in Eq. (2.50),

with associated bias parameter btd. Thus, only one of four cubic-order local bias parameters contributesto the next-to-leading-order halo power spectra. We have also included the leading higher-derivative biasbr2� / R2

⇤in Eq. (4.22), where R⇤ is the nonlocality scale of the halos or galaxies considered.

Correspondingly, we have also included the scale-dependent stochastic contributions to P nlohm and P nlo

hh (k).

The latter is expanded following Eq. (2.88), and is expected to scale as |P {2}

" | ⇠ R2⇤P {0}

" [125, 175, 176, 177].

13We include the relevant stochastic terms which were not considered in [195].14We use the bias parameter b�2 here to make the notation in Eqs. (4.22)–(4.24) more compact.

82




hm (k) + · · ·Phh(k) = P lo

hh(k) + P nlohh (k) + · · · , (4.21)


P nlohm (k) = b1



hm (k)

P nlohm (k) ⌘ b�2I [�(2),�2](k) + bK2I [�(2),K2

](k) +

✓bK2 +

2

5btd

◆fnlo(k)PL(k)

� br2�k2PL(k) + k2P {2}

""m (4.22)

P nlohh (k) = (b1)



nlohm (k) +

X

O,O02{�2,K2}

bObO0I [O,O0](k) + k2P {2}

" ,


s,e↵ ⌘ (2⇡)c2

s,e↵/k2nl is


fnlo(k) = 4

Z

p

[p · (k � p)]2

p2|k � p|2 � 1

�F2(k, �p)PL(p)

I [O,O0](k) = 2

" Z


�Z


#, (4.23)

where SO(k1,k2) =

8><

>:

F2(k1,k2), O = �(2)

1, O = �2

(k1 · k2)2 � 1/3, O = K2

. (4.24)







hh (k).


" | ⇠ R2⇤P {0}

" [125, 175, 176, 177].


82

with

• Next-to-leading order (NLO): involve 2 additional quadratic, 1 cubic, and 2 higher-derivative parameters:






hm (k) + · · ·Phh(k) = P lo

hh(k) + P nlohh (k) + · · · , (4.21)


P nlohm (k) = b1



hm (k)

P nlohm (k) ⌘ b�2I [�(2),�2](k) + bK2I [�(2),K2

](k) +

✓bK2 +

2

5btd

◆fnlo(k)PL(k)

� br2�k2PL(k) + k2P {2}

""m (4.22)

P nlohh (k) = (b1)



nlohm (k) +

X

O,O02{�2,K2}

bObO0I [O,O0](k) + k2P {2}

" ,


s,e↵ ⌘ (2⇡)c2

s,e↵/k2nl is


fnlo(k) = 4

Z

p

[p · (k � p)]2

p2|k � p|2 � 1

�F2(k, �p)PL(p)

I [O,O0](k) = 2

" Z


�Z


#, (4.23)

where SO(k1,k2) =

8><

>:

F2(k1,k2), O = �(2)

1, O = �2

(k1 · k2)2 � 1/3, O = K2

. (4.24)







hh (k).


" | ⇠ R2⇤P {0}

" [125, 175, 176, 177].


82




hm (k) + · · ·Phh(k) = P lo

hh(k) + P nlohh (k) + · · · , (4.21)


P nlohm (k) = b1



hm (k)

P nlohm (k) ⌘ b�2I [�(2),�2](k) + bK2I [�(2),K2

](k) +

✓bK2 +

2

5btd

◆fnlo(k)PL(k)

� br2�k2PL(k) + k2P {2}

""m (4.22)

P nlohh (k) = (b1)



nlohm (k) +

X

O,O02{�2,K2}

bObO0I [O,O0](k) + k2P {2}

" ,


s,e↵ ⌘ (2⇡)c2

s,e↵/k2nl is


fnlo(k) = 4

Z

p

[p · (k � p)]2

p2|k � p|2 � 1

�F2(k, �p)PL(p)

I [O,O0](k) = 2

" Z


�Z


#, (4.23)

where SO(k1,k2) =

8><

>:

F2(k1,k2), O = �(2)

1, O = �2

(k1 · k2)2 � 1/3, O = K2

. (4.24)







hh (k).


" | ⇠ R2⇤P {0}

" [125, 175, 176, 177].


82

with

⌘ b�2I

b

) + bK2I

) + k2P






hm (k) + · · ·Phh(k) = P lo

hh(k) + P nlohh (k) + · · · , (4.21)


P nlohm (k) = b1



hm (k)

P nlohm (k) ⌘ b�2I [�(2),�2](k) + bK2I [�(2),K2

](k) +

✓bK2 +

2

5btd

◆fnlo(k)PL(k)

� br2�k2PL(k) + k2P {2}

""m (4.22)

P nlohh (k) = (b1)



nlohm (k) +

X

O,O02{�2,K2}

bObO0I [O,O0](k) + k2P {2}

" ,


s,e↵ ⌘ (2⇡)c2

s,e↵/k2nl is


fnlo(k) = 4

Z

p

[p · (k � p)]2

p2|k � p|2 � 1

�F2(k, �p)PL(p)

I [O,O0](k) = 2

" Z


�Z


#, (4.23)

where SO(k1,k2) =

8><

>:

F2(k1,k2), O = �(2)

1, O = �2

(k1 · k2)2 � 1/3, O = K2

. (4.24)







hh (k).


" | ⇠ R2⇤P {0}

" [125, 175, 176, 177].


82




hm (k) + · · ·Phh(k) = P lo

hh(k) + P nlohh (k) + · · · , (4.21)


P nlohm (k) = b1



hm (k)

P nlohm (k) ⌘ b�2I [�(2),�2](k) + bK2I [�(2),K2

](k) +

✓bK2 +

2

5btd

◆fnlo(k)PL(k)

� br2�k2PL(k) + k2P {2}

""m (4.22)

P nlohh (k) = (b1)



nlohm (k) +

X

O,O02{�2,K2}

bObO0I [O,O0](k) + k2P {2}

" ,


s,e↵ ⌘ (2⇡)c2

s,e↵/k2nl is


fnlo(k) = 4

Z

p

[p · (k � p)]2

p2|k � p|2 � 1

�F2(k, �p)PL(p)

I [O,O0](k) = 2

" Z


�Z


#, (4.23)

where SO(k1,k2) =

8><

>:

F2(k1,k2), O = �(2)

1, O = �2

(k1 · k2)2 � 1/3, O = K2

. (4.24)







hh (k).


" | ⇠ R2⇤P {0}

" [125, 175, 176, 177].


82

with

⌘ b�2I

b

) + bK2I

) + k2P

2

5btd

◆






hm (k) + · · ·Phh(k) = P lo

hh(k) + P nlohh (k) + · · · , (4.21)


P nlohm (k) = b1



hm (k)

P nlohm (k) ⌘ b�2I [�(2),�2](k) + bK2I [�(2),K2

](k) +

✓bK2 +

2

5btd

◆fnlo(k)PL(k)

� br2�k2PL(k) + k2P {2}

""m (4.22)

P nlohh (k) = (b1)



nlohm (k) +

X

O,O02{�2,K2}

bObO0I [O,O0](k) + k2P {2}

" ,


s,e↵ ⌘ (2⇡)c2

s,e↵/k2nl is


fnlo(k) = 4

Z

p

[p · (k � p)]2

p2|k � p|2 � 1

�F2(k, �p)PL(p)

I [O,O0](k) = 2

" Z


�Z


#, (4.23)

where SO(k1,k2) =

8><

>:

F2(k1,k2), O = �(2)

1, O = �2

(k1 · k2)2 � 1/3, O = K2

. (4.24)







hh (k).


" | ⇠ R2⇤P {0}

" [125, 175, 176, 177].


82




hm (k) + · · ·Phh(k) = P lo

hh(k) + P nlohh (k) + · · · , (4.21)


P nlohm (k) = b1



hm (k)

P nlohm (k) ⌘ b�2I [�(2),�2](k) + bK2I [�(2),K2

](k) +

✓bK2 +

2

5btd

◆fnlo(k)PL(k)

� br2�k2PL(k) + k2P {2}

""m (4.22)

P nlohh (k) = (b1)



nlohm (k) +

X

O,O02{�2,K2}

bObO0I [O,O0](k) + k2P {2}

" ,


s,e↵ ⌘ (2⇡)c2

s,e↵/k2nl is


fnlo(k) = 4

Z

p

[p · (k � p)]2

p2|k � p|2 � 1

�F2(k, �p)PL(p)

I [O,O0](k) = 2

" Z


�Z


#, (4.23)

where SO(k1,k2) =

8><

>:

F2(k1,k2), O = �(2)

1, O = �2

(k1 · k2)2 � 1/3, O = K2

. (4.24)







hh (k).


" | ⇠ R2⇤P {0}

" [125, 175, 176, 177].


82

with

⌘ I⌘ b�2I ) + bK2I

) + k2P

2

5btd

◆

� br2�k

b1)2⇥P nlo

b

2P {2}

"P"P ,

• Quadratic and cubic terms scale like

• Controlled by shape of P(k) and nonlinear scale

• Higher-derivative contributions scale as

• Obviously, NLO corrections become important toward smaller scales (higher k)

• Importantly: Two independent expansion parameters!


disentangle the various higher-order bias parameters in practice; note that there is only a limited rangein wavenumbers that can be used for the parameter estimation, due to the presence of higher loop andderivative corrections (see below). Nevertheless, the leading-order bispectrum can be used to determine b2

and bK2 , leaving only btd, br2�, and P {2}

""m to be constrained from the NLO correction to the halo-mattercross-power spectrum.

In order to gain a more detailed understanding of the magnitude of the corrections in Eq. (4.22), let usapproximate the matter power spectrum by a power law,

PL(k) ⇡ 2⇡2

k3nl

✓k

knl

◆n

, (4.25)

where knl is the nonlinear scale at which the dimensionless matter power spectrum �2(k) = k3PL(k)/(2⇡2)becomes unity. This yields, for example,

I [�2,�2]

PL(k)= 2

✓k

knl

◆3+n Z1

�1

dµ

2

Z1

0

x2dxh⇣

xp

1 + x2 � 2xµ⌘n

� x2ni

. (4.26)

While other NLO loop-integral terms have di↵erent angular integrands, the scaling / (k/knl)3+n is commonto all (see also Fig. 12). Note that, depending on the value of n, the integral over x might need to beregularized in the UV (ultraviolet, small-scale, or large-x, limit of the integral), while the integral is safe fromdivergence in the IR (infrared, large-scale, or small-x, limit of the integral), because of the term subtractedin Eq. (4.23); in any case, this does not a↵ect the scaling with k/knl. This scaling allows us to estimate theimportance of higher-order terms. For example, 2-loop corrections correspondingly scale as (k/knl)2(3+n)

for a scale-free power spectrum [216]. For our reference ⇤CDM cosmology, we have approximately15 knl(z =0) = 0.25 h Mpc�1 and n = d ln PL/d ln k|knl = �1.7, so that the one-loop terms scale approximately as(k/knl)1.3. Of course, this is only a rough approximation as PL(k) cannot be approximated as a powerlaw over the entire relevant range of scales. In particular, since n becomes positive for k . 0.02 h Mpc�1,the NLO terms eventually scale as k2 for su�ciently small values of k. Nevertheless, such estimates areimportant as they allow us to marginalize over higher-loop corrections and rigorously take into account theuncertainty in the prediction of Eq. (4.22) [261].

The higher-derivative term / br2� obeys a scaling with k (/ k2) that is in general di↵erent from thatof the NLO corrections (/ k3+n). Further, the former involves an additional scale, R⇤. Thus, we have twoindependent expansion parameters,

✏loop ⌘✓

k

knl

◆3+n

⇡✓

k

0.25 h Mpc�1

◆1.3

, and ✏deriv. ⌘ k2R2

⇤. (4.27)

Thus, depending on the halo or galaxy sample, the leading higher-derivative term could be negligible com-pared to the NLO corrections on the scales of interest, e.g. 0.01 . k[ h Mpc�1] . 0.2, or could be significantlylarger. If ✏deriv. is comparable to ✏loop on the scales considered, then both NLO and leading higher-derivativecorrections should be included. This is what we have assumed in Eq. (4.22). More generally, when going tohigher orders, one would then include terms that involve the same powers of ✏loop and ✏deriv.. For example,at 2-loop order, these are the terms of order ✏2

loop, ✏loop✏deriv., and ✏2

deriv.. On the other hand, if the two

expansion parameters are substantially di↵erent, then it is necessary to retain terms that are higher orderin the larger parameter. For example, if ✏deriv. � ✏loop, one should allow for additional higher-derivativeterms, which leads to contributions / {k4R4

⇤, k6R6

⇤, · · · } PL(k) in Eq. (4.22) [126, 262, 263]. The cuto↵ of

the perturbative approach then is at k ⇡ 1/R⇤. All of this applies analogously to the bispectrum and highern-point functions.

Finally, the higher-derivative stochastic contributions, which scale as k2 (as opposed to k2PL(k) asthe higher-derivative bias contribution), are higher order in terms of their k scaling, but the amplitude

15This was obtained by fitting a power law to PL(k) over the range k 2 [0.1, 0.25]hMpc�1.

84

disentangle the various higher-order bias parameters in practice; note that there is only a limited rangein wavenumbers that can be used for the parameter estimation, due to the presence of higher loop andderivative corrections (see below). Nevertheless, the leading-order bispectrum can be used to determine b2

and bK2 , leaving only btd, br2�, and P {2}

""m to be constrained from the NLO correction to the halo-mattercross-power spectrum.

In order to gain a more detailed understanding of the magnitude of the corrections in Eq. (4.22), let usapproximate the matter power spectrum by a power law,

PL(k) ⇡ 2⇡2

k3nl

✓k

knl

◆n

, (4.25)

where knl is the nonlinear scale at which the dimensionless matter power spectrum �2(k) = k3PL(k)/(2⇡2)becomes unity. This yields, for example,

I [�2,�2]

PL(k)= 2

✓k

knl

◆3+n Z1

�1

dµ

2

Z1

0

x2dxh⇣

xp

1 + x2 � 2xµ⌘n

� x2ni

. (4.26)

While other NLO loop-integral terms have di↵erent angular integrands, the scaling / (k/knl)3+n is commonto all (see also Fig. 12). Note that, depending on the value of n, the integral over x might need to beregularized in the UV (ultraviolet, small-scale, or large-x, limit of the integral), while the integral is safe fromdivergence in the IR (infrared, large-scale, or small-x, limit of the integral), because of the term subtractedin Eq. (4.23); in any case, this does not a↵ect the scaling with k/knl. This scaling allows us to estimate theimportance of higher-order terms. For example, 2-loop corrections correspondingly scale as (k/knl)2(3+n)

for a scale-free power spectrum [216]. For our reference ⇤CDM cosmology, we have approximately15 knl(z =0) = 0.25 h Mpc�1 and n = d ln PL/d ln k|knl = �1.7, so that the one-loop terms scale approximately as(k/knl)1.3. Of course, this is only a rough approximation as PL(k) cannot be approximated as a powerlaw over the entire relevant range of scales. In particular, since n becomes positive for k . 0.02 h Mpc�1,the NLO terms eventually scale as k2 for su�ciently small values of k. Nevertheless, such estimates areimportant as they allow us to marginalize over higher-loop corrections and rigorously take into account theuncertainty in the prediction of Eq. (4.22) [261].

The higher-derivative term / br2� obeys a scaling with k (/ k2) that is in general di↵erent from thatof the NLO corrections (/ k3+n). Further, the former involves an additional scale, R⇤. Thus, we have twoindependent expansion parameters,

✏loop ⌘✓

k

knl

◆3+n

⇡✓

k

0.25 h Mpc�1

◆1.3

, and ✏deriv. ⌘ k2R2

⇤. (4.27)

Thus, depending on the halo or galaxy sample, the leading higher-derivative term could be negligible com-pared to the NLO corrections on the scales of interest, e.g. 0.01 . k[ h Mpc�1] . 0.2, or could be significantlylarger. If ✏deriv. is comparable to ✏loop on the scales considered, then both NLO and leading higher-derivativecorrections should be included. This is what we have assumed in Eq. (4.22). More generally, when going tohigher orders, one would then include terms that involve the same powers of ✏loop and ✏deriv.. For example,at 2-loop order, these are the terms of order ✏2

loop, ✏loop✏deriv., and ✏2

deriv.. On the other hand, if the two

expansion parameters are substantially di↵erent, then it is necessary to retain terms that are higher orderin the larger parameter. For example, if ✏deriv. � ✏loop, one should allow for additional higher-derivativeterms, which leads to contributions / {k4R4

⇤, k6R6

⇤, · · · } PL(k) in Eq. (4.22) [126, 262, 263]. The cuto↵ of

the perturbative approach then is at k ⇡ 1/R⇤. All of this applies analogously to the bispectrum and highern-point functions.

Finally, the higher-derivative stochastic contributions, which scale as k2 (as opposed to k2PL(k) asthe higher-derivative bias contribution), are higher order in terms of their k scaling, but the amplitude

15This was obtained by fitting a power law to PL(k) over the range k 2 [0.1, 0.25]hMpc�1.

84

10�2 10�1

wavenumber k [h/Mpc]

103

104

105

P(k

)[M

pc3

/h3]

Pmm(k)

Phm(k)

Phh(k)

10�1

wavenumber k [h/Mpc]

�0.5

0.0

0.5

1.0

1.5

�P

(k)/

PL(k

)

PNLOmm (k)/PL(k)

PNLOhm (k)/(b1PL(k))

br2�

b2

bK2

btd

Figure 12: Left panel: illustration of halo auto- (red, top line) and cross-power spectra (green, middle line), and the matterpower spectrum (blue, bottom line) at z = 0. The solid lines show the total LO plus NLO result, while the dashed curvesshow the LO (linear) prediction only. The bias parameters used here are b1 = 1.50, b2 = �0.69, and bK2 = �0.14, as inTab. 6, while br2� = R2

⇤ with R⇤ = 2.61h�1 Mpc. btd = 23/42(b1 � 1) is taken from the Lagrangian LIMD prediction

(Sec. 2.4). The stochastic amplitudes are taken from the Poisson expectation, P{0}" = 1/nh and P

{2}" = �R2

⇤/nh, with

nh = 1.41 · 10�4(h�1 Mpc)�3. We have set P{2}""m = 0 in P nlo

hm (k). Right panel: fractional size of the NLO contributions tothe matter and halo-matter cross-power spectrum at z = 0. The red dashed line shows the result for Phm(k) for the fiducialbias parameters given above. The di↵erent shaded areas around P nlo

hm show the e↵ect of rescaling the various bias parametersby a factor in the range [0.5, 2]. Clearly, the contributions from di↵erent bias parameters exhibit similar dependencies on k,and are in general di�cult to disentangle using only the power spectrum. The perturbative description is expected to fail fork & 0.25hMpc�1, where P nlo

mm(k) becomes as large as the LO prediction PL(k).

We will return to this in Sec. 4.5.3. It is often assumed that there is no stochastic contribution to thehalo-matter cross-power spectrum. However, this is only true at lowest order. The nonlinear small-scalemodes of the density field are responsible for both the halo stochasticity " and the stochastic contributionto the matter density field "m, which, as discussed in Appendix B.3, is due to the e↵ective pressure of thenonlinear matter fluctuations and scales as k2 in the low-k limit. Hence, one has to allow for a correlation

between the two stochastic fields, leading to the term k2P {2}

""m in P nlohm , which is comparable to the other

NLO contributions. Note that it could be either positive or negative.The magnitude and scale dependence of the NLO corrections to the halo and matter power spectra is

shown in Fig. 12. As expected, we see that the corrections become increasingly important towards smallerscales (higher k). We see a particularly steep suppression of Phh(k), which, for our fiducial parameters, is

dominated by the higher-derivative stochastic contribution k2P {2}

" . The right panel of Fig. 12 shows thefractional size of the NLO correction to Pmm(k) and Phm(k). Depending on the value of the various biasand stochastic parameters, the NLO correction could be either positive or negative (shaded regions), andcancellations between the di↵erent NLO contributions can occur. In any case, as soon as the fractionalsize of the NLO correction approaches order unity, we expect that higher-order loop contributions which wehave not included become comparable to P nlo

hm (k) as well, and hence the perturbative expansion ceases toconverge.

The NLO halo-matter power spectrum adds five additional free parameters to the ones present at leading

order (b1, P {0}

" ). These can, in principle, be disentangled due to the di↵erent scale dependence of each term.However, as illustrated in Fig. 12, these scale dependences are su�ciently similar that it is di�cult to

83

• Many contributions have very similar shape

• If only interested in power spectrum, can significantly reduce number of free parameters

Illustration of NLO contributions to galaxy power spectrum

Further applications• Galaxy density and velocity are not the only

application of EFT approach/general bias expansion:

1. Galaxy shapes = intrinsic alignments: symmetric, trace-free 2-tensor

2. Complete description of counterterms in EFT of matter

3. Small scale power spectrum = “responses”: symmetric 2-/4-/6-/… tensor

• Describes nonlinear matter n-point functions in squeezed limit (bispectrum, covariance, …)

Wagner, FS et al 2015Barreira & FS 2017a,b

FS, Chisari, Dvorkin 2015Vlah, Chisari, FS, in prep.

Zaldarriaga & Mirbabayi, 2015

Summary• LSS contains a wealth of information on dark energy, growth of structure, and

the early Universe

• To use this, we need to understand nonlinear (and nonlocal) relation between initial conditions and observed galaxies

• We now have a complete framework for galaxy biasing (on perturbative scales)

• Also for galaxy velocities

• Leads to well-defined prediction for all n-point functions of galaxies

• “Just compute”

• Lots of free parameters, but many of them quasi-degenerate

• Next challenges:

• How much information in nonlinear galaxy clustering, given many free parameters?

• How best to extract it?

Large-Scale

Galaxy

Bias

Vin

centD

esjacques

a,b,

Don

ghuiJeon

gc,

Fab

ianSch

mid

td

aPhysics

depa

rtmen

t,Tech

nion,3200003Haifa

,Isra

elbD

epartem

entdePhysiqu

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riqueandCen

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particle

Physics,

Universite

deGen

eve,24qu

aiErn

estAnserm

et,CH-1221Gen

eve4,Switzerla

nd

cDepa

rtmen

tofAstro

nomyandAstro

physics,

andIn

stitute

forGravita

tionandtheCosm

os,

ThePen

nsylva

nia

State

University,

University

Park,

PA

16802,USA

dMax-P

lanck-In

stitutfurAstro

physik,

Karl-S

chwarzsch

ild-Stra

ße1,85748Garch

ing,

Germ

any

Abstra

ct

This

reviewpresents

acom

preh

ensive

overviewof

galaxybias,

that

is,th

estatistical

relationbetw

eenth

edistrib

ution

ofgalaxies

and

matter.

We

focus

onlarge

scalesw

here

cosmic

den

sityfield

sare

quasi-

linear.

On

these

scales,th

eclu

stering

ofgalaxies

canbe

describ

edby

apertu

rbative

bias

expan

sion,an

dth

ecom

plicated

physics

ofgalaxy

formation

isab

sorbed

bya

finite

setof

coe�cients

ofth

eexp

ansion

,called

biasparam

eters.T

he

reviewbegin

sw

itha

ped

agogicalproof

ofth

isvery

important

result,

which

forms

the

basis

ofth

erigorou

spertu

rbative

descrip

tionof

galaxyclu

stering,

under

the

assum

ption

sof

Gen

eralR

elativityan

dG

aussian

,ad

iabatic

initial

condition

s.K

eycom

pon

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sionare

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hen

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and

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esecontrib

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edby

apresentation

ofth

epeak-b

ackground

split

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general

form,w

hich

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ephysical

mean

ing

ofth

ebias

param

eters,an

da

detailed

descrip

tionof

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connection

betw

eenbias

param

etersan

dgalaxy

(orhalo)

statistics.W

eth

enreview

the

excursion

setform

alisman

dpeak

theory

which

provid

epred

ictions

forth

evalu

esof

the

bias

param

eters.In

the

remain

der

ofth

ereview

,w

econ

sider

the

generalization

sof

galaxybias

required

inth

epresen

ceof

various

types

ofcosm

ologicalphysics

that

gobeyon

dpressu

relessm

atterw

ithad

iabatic,

Gau

ssianin

itialcon

dition

s:prim

ordial

non

-Gau

ssianity,

massive

neu

trinos,

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-CD

Misocu

rvature

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s,dark

energy,

and

mod

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gravity.Fin

ally,w

ediscu

sshow

the

descrip

tionof

galaxybias

inth

egalaxies’

restfram

eis

relatedto

observed

clusterin

gstatistics

measu

redfrom

the

observed

angu

larposition

san

dred

shifts

inactu

algalaxy

catalogs.

Keyw

ords:cosm

ology,large-scale

structu

re,galaxy

surveys,

galaxybias,

dark

matter,

prim

ordial

non

-gaussian

ity

Email

addresses:

[email protected]

(Vincen

tDesja

cques),

[email protected]

(Donghui

Jeo

ng),

[email protected](F

abianSch

midt)

Prep

rintsu

bmitted

toPhysics

Repo

rtsNovem

ber30,2016


• From the review…

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