Large-scale structure tests of inflation Marilena LoVerde - - PowerPoint PPT Presentation

Large-scale structure tests of inflation

Marilena LoVerde (Institute for Advanced Study)

Inflation as the origin of structure

stars, galaxies, clusters of galaxies

gravitational collapse

small matter & energy fluctuations

quantum fluctuations

inflation

δφinflaton Φcurvature δTCMB δρmatter δngalaxies

Inflation as the origin of structure

stars, galaxies, clusters of galaxies

gravitational collapse

small matter & energy fluctuations

quantum fluctuations

inflation

δφinflaton Φcurvature δTCMB δρmatter δngalaxies statistics of these probe of this era

Inflation as the origin of structure

stars, galaxies, clusters of galaxies

gravitational collapse

small matter & energy fluctuations

quantum fluctuations

inflation

δφinflaton Φcurvature δTCMB δρmatter δngalaxies statistics of these probe of this era

powerful probe

Inflation as the origin of structure

stars, galaxies, clusters of galaxies

gravitational collapse

small matter & energy fluctuations

quantum fluctuations

inflation

δφinflaton Φcurvature δTCMB δρmatter δngalaxies statistics of these probe of this era

So long as we can accurately model this relationship

Community Goal: Determine ``initial’’ conditions for density perturbations (learn about inflation) Intermediate Goal: Identify & model signatures of inflation (e.g. non- Gaussianity) that may persist in large- scale structure This talk: Analytic and N-body comparison

• f large-scale structure with 3 simple

examples of non-Gaussian initial conditions

Lots of data!

Lots of data!

Planck

South Pole Telescope

Atacama Cosmology Telescope

Planck

South Pole Telescope

Atacama Cosmology Telescope

Sloan Digital Sky Survey

Dark Energy Survey

Hobby-Eberly Telescope Dark Energy EXperiment

Subaru Hyper Suprime Cam

Lots of data!

What can we learn about the statistics

• f the initial (post-inflationary)

conditions?

<Φ(k)Φ(k’)> ≡ PΦ(k) δ(k+k’)

Power spectum:

k

variance of fluctuations on scale k

~

If the initial conditions were Gaussian, we’ d be done

<Φ(k)Φ(k’)> ≡ PΦ(k) δ(k+k’)

Power spectum:

k

variance of fluctuations on scale k

~

If the initial conditions were Gaussian, we’ d be done

<Φ(k)Φ(k’)> ≡ PΦ(k) δ(k+k’)

Power spectum:

k

variance of fluctuations on scale k

~

but they don’ t have to be perfectly Gaussian!

What do non-Gaussian initial conditions look like?

“fNL” “gNL” “τNL”

probability

Φ value

skewness ~ fNL kurtosis ~ fNL2

Φ value

probability

skewness ~ 0 kurtosis ~ gNL

probability

Φ value

skewness ~ fNL kurtosis ~ τNL

τNL= fNL2(1 + Pφφ/Pσσ )

(Φ=primordial gravitational potential)

fNL = fNL(1 + Pφφ/Pσσ )2

~

and Pφσ = 0

. . . . . .

Φ(x) ∼ δσ(x)+ fNL δσ(x)2 Φ(x) ∼ δσ(x) + gNL δσ(x)3 + . . . Φ(x) ∼ δφ(x)+ δσ(x) + fNLδσ(x)2 + . . . ~

More generally, non-trivial multi-point correlation functions (or polyspectra) are introduced

〈Φ(k)Φ(k’)Φ(k’’)〉= 2fNL (PΦ(k) PΦ(k’) + . . .) (2π)3 δ(k+k’+k’’)

largest in the “squeezed” limit

k k’ k’’

Bispectrum:

function of triangle

k’ k’’ k

} }

(Φ=primordial gravitational potential)

(Φ=primordial gravitational potential)

〈Φ(k)Φ(k’)Φ(k’’)〉= 2fNL (PΦ(k) PΦ(k’) + . . .) (2π)3 δ(k+k’+k’’)

largest in the “squeezed” limit

k k’ k’’

Bispectrum: Trispectrum:

〈Φ(k)Φ(k’)Φ(k’’)Φ(k’’’)〉= gNL (PΦ(k) PΦ(k’) PΦ(k’’) + . . .) (2π)3 δ(k+k’+k’’+k’’’)

c

k k’ k’’ k’’’

peaks in the limit

+ 2 τNL (PΦ(k) PΦ(k’) PΦ(|k+k’’|) + . . .) (2π)3 δ(k+k’+k’’+k’’’)

k k’ k’’ k’’’

peaks in the squeezed limits

function of triangle

k k’ k’’’ k’’

function of a quadrilateral

k’ k’’ k

} } }

(Φ=primordial gravitational potential)

〈Φ(k)Φ(k’)Φ(k’’)〉= 2fNL (PΦ(k) PΦ(k’) + . . .) (2π)3 δ(k+k’+k’’)

largest in the “squeezed” limit

k k’ k’’

Bispectrum: Trispectrum:

〈Φ(k)Φ(k’)Φ(k’’)Φ(k’’’)〉= gNL (PΦ(k) PΦ(k’) PΦ(k’’) + . . .) (2π)3 δ(k+k’+k’’+k’’’)

c

k k’ k’’ k’’’

peaks in the limit

+ 2 τNL (PΦ(k) PΦ(k’) PΦ(|k+k’’|) + . . .) (2π)3 δ(k+k’+k’’+k’’’)

k k’ k’’ k’’’

peaks in the squeezed limits

function of triangle

k k’ k’’’ k’’

function of a quadrilateral

k’ k’’ k

} } }

so gNL and τNL different ``shape” trispectra

Helpful to consider how polyspectra couple different physical scales scales

Φ ∼ δσ+ fNL δσ2

“fNL”

〈Φshort2〉= 〈σG,short2〉(1 + 4 fNL σG,long(x))

small-scale power depends on large-scale fluctuations!

(Φ=primordial gravitational potential)

Slosar, Hirata, Seljak, Ho, Padmanabhan 2008

Φ ∼ δσ+ fNL δσ2

“fNL”

Φ ∼ δσ + gNLδσ3 + . . .

“gNL”

〈Φshort2〉= 〈σG,short2〉(1 + 4 fNL σG,long(x))

〈Φshort3〉= 18 gNL〈σG,short2〉σG,long(x) ≡ fNLeff (x)

2 (Φ=primordial gravitational potential)

〈σG,short2〉

2

small-scale power depends on large-scale fluctuations! small-scale skewness depends on large-scale fluctuations!

Φ ∼ δσ+ fNL δσ2

“fNL”

Φ ∼ δσ + gNLδσ3 + . . .

“gNL” “τNL”

Φ ∼ δφ+ δσ + fNLδσ2 + . . . ~

〈Φshort2〉= 〈σG,short2〉(1 + 4 fNL σG,long(x))

〈Φshort3〉= 18 gNL〈σG,short2〉σG,long(x) ≡ fNLeff (x)

2

〈Φs2〉= 〈ΦG,short2〉(1 + 4 fNLσG,long(x))

Φ = φ+σ

~

(Φ=primordial gravitational potential)

〈σG,short2〉

2

“fNL” “gNL” “τNL”

Φ(x)=σG(x)+fNL σG(x)2 Φ(x)=σG(x)+gNL σG(x)3 Φ(x)=φG(x)+σG(x) +fNL (1+ξ2) σG(x)2 τNL= fNL2(1+ξ2)

〈Φ3〉 ≈ 〈Φ2〉 6fNL

2

〈Φ3〉 = 0 〈Φ3〉 ≈ 〈Φ2〉 6fNL

2

〈Φ4〉≈48 τNL〈Φ2〉

〈Φ4〉≈48 fNL2〈Φ2〉

〈Φ4〉≈24 gNL〈Φ2〉

ξ2 =Pφφ/Pσσ 〈Φs2 〉= 〈Φs2 〉(1+4fNLΦl) 〈Φs2〉=〈Φs2〉(1+6gNLΦl2 )〈Φs2 〉= 〈Φs2 〉(1+4(1+ ξ2)fNLσl) definition skewness kurtosis

short-long scale coupling

〈Φs3〉= 18 gNL〈Φs2〉Φl

• 10 < fNL < 74
• 12.34×105 < gNL < 15.58×105

• 6000 < τNL < 33000

These are meant to be cartoon examples, but these types of initial condition can arise from real models

For instance

φ

inflaton potential ∼ V(φ,σ) δφ

H2 = 8πG/3 V(φ,σ) total energy dominated by inflaton: perturbations from inflaton Gaussian

σ

curvaton δσ

curvature perturbations from curvaton can be non-Gaussian

Φ ∼ δσ + δσ2 Φ ∼ δσ + δσ3 + . . . Φ ∼ δφ+ δσ + δσ2 + . . .

Linde and Mukhanov 1997; Lyth and Wands 2002

“fNL” “gNL” “τNL”

But “local” models (i.e. ΦNG(x)=F(σG(x))) of non-Gaussianity are just some examples Can also get non-Gaussianity from self interactions of inflaton (e.g. k-inflation, DBI, . . . ) that has a very different ``shape’’ But single-field models do not generate such extreme couplings of perturbations on short and long length scales

In fact:

Acquaviva, Bartolo, Matarrese, Riotto 2003; Maldacena 2003; Creminelli & Zaldarriaga 2004 (see also Tanaka, Urakawa 2011)

where ns = dlnPΦ(k)/dlnk + 4 ≈ 1

single-field inflation predicts the so called “consistency relation” so fNL few rules it out

~ >

〈Φ(k)Φ(k’)Φ(k’’-->0)〉≈ (ns-1)(2π)3 δ(k+k’) PΦ(k) PΦ(k’’)

}

≈fNL

Note: also have,

single-field consistency relation

fNL ∂ln k3PΦ ∂ln k (ns-1) ≈

also applies to gNL and τNL

= gNL ∂ln k6BΦ ∂ln k ≈ = nNG τNL ≈ (ns-1)2

e.g. Chen, Huang, Shiu 2008; Leblond & Pajer 2011 Suyama & Yamaguchi 2008; Sugiyama, Komatsu, Futamase 2011; Smith, ML, Zaldarriaga 2011

τNL > fNL2 ~

(see also Tanaka, Urakawa 2011)

in terms of physical observables these are strictly zero

Aside on τNL > fNL2 ~ allowing two fields to generate perturbations gave τNL > fNL2, perhaps this inequality tells us something important about inflation?

Aside on τNL > fNL2 ~ fNL ~ <ΦL PS>/PL τNL ~ <PL PS>/PL define:

〔

ΦLΦL* PsΦL* ΦLPs* PSPs*

〔

Smith, ML, Zaldarriaga 2011

Aside on τNL > fNL2 ~ fNL ~ <ΦL PS>/PL τNL ~ <PL PS>/PL define:

〔

ΦLΦL* PsΦL* ΦLPs* PSPs*

〔 〔

PL fNLPL fNLPL τNLPL

〔

~

pos.-def. so τNL - fNL2 > 0

• doesn’

t have to do with inflation at all

Smith, ML, Zaldarriaga 2011

How does large-scale structure see primordial non-Gaussianity?

φ σ

inflaton curvaton

V(φ,σ)

?

Signatures in LSS I: more/fewer massive halos

δρ/ρ

δc

dark matter halos form in peaks of the density field non-Gaussianity changes the number density of peaks

Gaussian positive skewness no skewness, positive kurtosis Lucchin & Matarrese 1988; Chiu, Ostriker, Strauss 1998; Robinson, Gawiser, Silk 2000

Signatures in LSS I: more/fewer massive halos

Lucchin & Matarrese 1988; Chiu, Ostriker, Strauss 1998; Robinson, Gawiser, Silk 2000

δρ/ρ

δc

non-Gaussianity changes the number density of peaks

Gaussian positive skewness no skewness, positive kurtosis

number of peaks ⇔ number of dark matter halos

~

dark matter halos form in peaks of the density field

Signatures in LSS II: scale-dependent halo bias

a dark matter halo forms when δρ/ρ is larger than the collapse threshold δρ/ρ

δc

Signatures in LSS II: scale-dependent halo bias

a dark matter halo forms when δρ/ρ is larger than the collapse threshold δρ/ρ

δc δc-δl

δρ/ρ which is easier to reach on top of a long wavelength density perturbation

Signatures in LSS II: scale-dependent halo bias

a dark matter halo forms when δρ/ρ is larger than the collapse threshold δρ/ρ

δc δc-δl

δρ/ρ which is easier to reach on top of a long wavelength density perturbation so the number of halos fluctuates depending on δl

δn = δl . . . ∂n ∂δ

Signatures in LSS II: scale-dependent halo bias

a dark matter halo forms when δρ/ρ is larger than the collapse threshold δρ/ρ

δc δc-δl

δρ/ρ which is easier to reach on top of a long wavelength density perturbation so the number of halos fluctuates depending on δl

δn = δl . . . ∂n ∂δ

“halo bias”

Signatures in LSS II: scale-dependent halo bias

the number of halos fluctuates depending on δl BUT with fNL, the small-scale power fluctuates also depending on Φl

Matarrese & Verde 2008; Slosar, Hirata, Seljak, Ho, Padmanabhan 2008; Afshordi & Tolley 2008; McDonald 2008 Dalal, Doré, Huterer, Shirokov 2007

δc-δl

δρ/ρ

Signatures in LSS II: scale-dependent halo bias

the number of halos fluctuates depending on δl

Matarrese & Verde 2008; Slosar, Hirata, Seljak, Ho, Padmanabhan 2008; Afshordi & Tolley 2008; McDonald 2008 Dalal, Doré, Huterer, Shirokov 2007

δc-δl

δρ/ρ BUT with fNL, the small-scale power fluctuates also depending on Φl

= δl + 4fNL Φl. . . ∂n ∂Ps ∂n ∂δ δn

Signatures in LSS II: scale-dependent halo bias

the number of halos fluctuates depending on δl

Matarrese & Verde 2008; Slosar, Hirata, Seljak, Ho, Padmanabhan 2008; Afshordi & Tolley 2008; McDonald 2008 Dalal, Doré, Huterer, Shirokov 2007

δc-δl

δρ/ρ

∇2Φl∼ 4πG δl

Poisson’ s BUT with fNL, the small-scale power fluctuates also depending on Φl

= δl + 4fNL Φl. . . ∂n ∂Ps ∂n ∂δ δn ∂n ∂Ps ∂n ∂δ

( )

4fNL k2 + δl

~

δn

Signatures in LSS II: scale-dependent halo bias

the number of halos fluctuates depending on δl

Matarrese & Verde 2008; Slosar, Hirata, Seljak, Ho, Padmanabhan 2008; Afshordi & Tolley 2008; McDonald 2008 Dalal, Doré, Huterer, Shirokov 2007

δc-δl

δρ/ρ

∇2Φl∼ 4πG δl

Poisson’ s

this 1/k2 scaling is hard to generate with local (post-inflationary) processes

BUT with fNL, the small-scale power fluctuates also depending on Φl

= δl + 4fNL Φl. . . ∂n ∂Ps ∂n ∂δ δn ∂n ∂Ps ∂n ∂δ

( )

4fNL k2 + δl

~

δn

Signatures in LSS II: scale-dependent halo bias

the number of halos fluctuates depending on δl

Matarrese & Verde 2008; Slosar, Hirata, Seljak, Ho, Padmanabhan 2008; Afshordi & Tolley 2008; McDonald 2008 Dalal, Doré, Huterer, Shirokov 2007

δc-δl

δρ/ρ

∇2Φl∼ 4πG δl

Poisson’ s

this 1/k2 scaling is hard to generate with local (post-inflationary) processes powerful test!

BUT with fNL, the small-scale power fluctuates also depending on Φl

= δl + 4fNL Φl. . . ∂n ∂Ps ∂n ∂δ δn ∂n ∂Ps ∂n ∂δ

( )

4fNL k2 + δl

~

δn

Signatures in LSS II: scale-dependent halo bias

a dark matter halo forms when δρ/ρ is larger than the collapse threshold

δc-δl

δρ/ρ

Desjacques & Seljak 2009; Smith, Ferraro, ML 2011

∇2Φl∼ 4πG δl ≈ ( + 18gNL /k2 ) δl(k) . . .

bias depends on Fourier scale k

δn = δl + 18gNL Φl. . .

so the number of halos fluctuates depending on δl and Φ

∂n ∂δ ∂n ∂S3

with gNL non-Gaussianity, the small-scale skewness fluctuates with Φl

∂n ∂δ ∂n ∂S3

Signatures in LSS II: scale-dependent halo bias

e.g. Creminell, D’Amico, Musso, Noreña 2011

Summary:

Φ(x)=ΦG(x)+ fNL (ΦG(x)2-<ΦG2>) + gNL(ΦG(x)3-ΦG<ΦG2>)

local non-Gaussianity bfNL,gNL (k) ∼ b + fNL,gNL x constant k2 scale dependent halo bias impossible to generate with single field inflation!

Signatures in LSS II: scale-dependent halo bias

precise values of fNL, gNL will require care -- but seeing 1/k2 is the most exciting part

e.g. Creminell, D’Amico, Musso, Noreña 2011

Summary:

Φ(x)=ΦG(x)+ fNL (ΦG(x)2-<ΦG2>) + gNL(ΦG(x)3-ΦG<ΦG2>)

local non-Gaussianity bfNL,gNL (k) ∼ b + fNL,gNL x constant k2 scale dependent halo bias impossible to generate with single field inflation!

• bservational systematics hard!

e.g. Scoccimarro et al 2012

Can we model the non-Gaussian effects on halos reliably?

φ σ

inflaton curvaton

V(φ,σ)

?

complicated, non-linear gravitational evolution

Can we model the non-Gaussian effects on halos reliably?

φ σ

inflaton curvaton

V(φ,σ)

?

complicated, non-linear gravitational evolution

N-body simulations modeling gravitational evolution and halo formation with fNL, gNL, and τNL initial conditions

Signatures in LSS I: more/fewer massive halos

N-body simulations with fNL, gNL, and τNL

fNL, τNL = 2fNL2

ML & Smith 2010

kurtosis can have important effects on the mass function!

gNL curves are simple models of halo abundance (Press-Schechter + spherical collapse + different approximations for for P.D.F . of δM)

fNL, τNL = fNL2

Signatures in LSS I: more/fewer massive halos

N-body simulations with fNL, gNL, and τNL

fNL, τNL = 2fNL2

ML & Smith 2010

gNL fNL, τNL = fNL2

the new ``log-Edgeworth’’ mass reliably captures NG effects for fNL, gNL, and τNL types of non-Gaussianity

the old mass function is ok for fNL

Signatures in LSS I: more/fewer massive halos

The log-Edgeworth is a good fit for fNL, gNL, and τNL , even at high masses and redshifts!

but cosmology with clusters is hard try to find my mass to 5%!

Signatures in LSS I: more/fewer massive halos

more to explore: halo finders, mass-observable relation (these issues apply to using clusters for dark energy also)

• poss. advantage is insensitivity to “shape” of NG

(see also Wagner, Verde, Boubekeur 2010)

nNG(M) <δM2> ,<δM3>, <δM4>c

Don’ t need to know B(k1,k2,k3), T(k1,k2,k3,k4); “local”, “equilateral” info integrated out

variance, skewness, kurtosis of density field smoothed on scale M

Recall, Poisson’ s: ∇2 Φl ∼ 4πG δl or k2 Φl(k) ∼ 4πGδl(k)

Signatures in LSS II: scale-dependent halo bias

so on large scales

Dalal, Doré, Huterer, Shirokov 2007

≈ ( + 4fNL /k2 ) δl(k) . . . ∂n ∂Ps ∂n ∂δ Pnδ(k) ≈ ( + 4fNL /k2 ) Pδδ(k) . . . ∂n ∂Ps ∂n ∂δ

OR

= δl + 4fNL Φl. . . ∂n ∂Ps ∂n ∂δ δn

Recall, Poisson’ s: ∇2 Φl ∼ 4πG δl or k2 Φl(k) ∼ 4πGδl(k)

Signatures in LSS II: scale-dependent halo bias

so on large scales

Dalal, Doré, Huterer, Shirokov 2007

≈ ( + 4fNL /k2 ) δl(k) . . . ∂n ∂Ps ∂n ∂δ <δn δ> = Pnδ(k) ≈ ( + 4fNL /k2 ) Pδδ(k) . . . ∂n ∂Ps ∂n ∂δ

OR

= δl + 4fNL Φl. . . ∂n ∂Ps ∂n ∂δ δn

given a Gaussian mass function, we can calculate these derivatives and predict the coefficients

Signatures in LSS II: scale-dependent halo bias

so on large scales

δn ≈ ( + 4fNL /k2 ) δl(k) . . . ∂n ∂Ps ∂n ∂δ

AND IT WORKS!

Slosar, Hirata, Seljak, Ho, Padmanabhan 2008 example data Dalal, Doré, Huterer, Shirokov 2007 Pillepich, Porciani, Hahn 2008; Desjacques, Seljak, Iliev 2008; Grossi et al 2009

k (h/Mpc) halo bias ∼Pnδ(k)/Pδδ (k) ∼ halo power spectrum

fNL = +500 fNL = -500

• 29 < fNL < 69 !!

Wagner & Verde 2011

Recall, Poisson’ s: ∇2 Φl ∼ 4πG δl or k2 Φl(k) ∼ 4πGδl(k)

Scoccimarro et al 2012

Signatures in LSS II: scale-dependent halo bias

And for gNL:

≈ ( + 18gNL /k2 ) δl(k) . . . ∂n ∂S3 ∂n ∂δ

halo mass (Msun/h) k (h/Mpc)

halo bias ∼Pnδ(k)/Pδδ (k)

(see also Desjacques and Seljak 2010; Desjacques, Jeong, Schmidt 2011)

gNL bias coefficient from sims

3dlnn/dfNL measured from sims

gNL bias coefficient× M-1/2

Smith, Ferraro, ML 2011

= ( + 3gNL /k2 ) δl(k) . . . ∂n ∂fNL ∂n ∂δ δn

IT WORKS AGAIN!

bias coefficient for gNL in terms of mass contrast w/fNL where coefficient in terms of bias

Signatures in LSS II: scale-dependent halo bias

bgNL(k)= b + ∂lnn(M) ∂fNL 3gNL k2 bfNL(k)= b + 2 δc fNL (b-1) k2

bias coefficient for gNL in terms of mass contrast w/fNL where coefficient in terms of bias

Signatures in LSS II: scale-dependent halo bias

bgNL(k)= b + ∂lnn(M) ∂fNL 3gNL k2 bfNL(k)= b + 2 δc fNL (b-1) k2

we have a fit for gNL in terms of bias:

bgNL(k) ∼ b +gNL non-linear function(b) k2

form will depend on selection of population in M, z

Expectations?

Dark Enegy Survey

fNL ~ 10, dlnfNL/dk ~ 0.5 fNL ~ 3-5 , dlnfNL/dk ~ 0.2 fNL ~ 6, dlnfNL/dk ~ 0.8

2 years? Euclid 1 year Planck

Sloan Digital Sky Survey now!

fNLeq ~ 20 fNLorth fNL ~ 20

10 years Large Synoptic Survey Telescope

fNL ~ 1

SUMIRE

5 years?

Summary

Non-Gaussian initial conditions significantly change the abundance and clustering of dark matter halos We have an analytic description for the halo mass function that compares well to N-body for fNL, gNL and τNL -- perhaps it works for more general forms of NG? Analytic descriptions of halo bias agree well with sims, for fNL and gNL too! Large-scale structure is a promising probe of the early universe

Signatures in LSS III: stochastic halo bias

fNL , τNL non-Gaussianity gives both scale dependent bias and makes halo # fluctuations stochastic w.r.t. dark matter fluctuations τNL dependent stochasticty is present in N- body sims! But the predicted amplitude is not as accurate as fNL and gNL biases . . .

Tseliakhovich, Hirata, Slosar 2010 Smith & ML 2010

halos

dark matter halos dark matter

just σ

• grav. potential
• tot. grav. potential ∼ φ+σ

non-stochastic stochastic

Signatures in LSS III: stochastic halo bias

Pnδ (k)∼ ( + 4fNL /k2 ) Pδδ Pnn (k)∼ ( + 4fNL /k2 )2Pδδ + (4fNL )2 ξ2 Pδδ

N.B. the bias factor in Pnδ is unchanged from fNL-only model

ξ2 = Pφφ(k)/Pσσ(k) τNL = (1+ ξ2) fNL2

stochasticity, r ≡ Pnn/Pδδ-Pnδ2/Pδδ2 105 k3 r

∂n ∂Ps ∂n ∂Ps ∂n ∂δ ∂n ∂δ ∂n ∂Ps

stochasticity ≈ τNL-fNL2

Signatures in LSS III: stochastic halo bias

N.B. the bias factor in Pnδ is unchanged from fNL-only model

models with ξ ≠ 0 indeed stochastic

stochasticity, r ≡ Pnn/Pδδ-Pnδ2/Pδδ2 105 k3 r

Pnδ (k)∼ ( + 4fNL /k2 ) Pδδ Pnn (k)∼ ( + 4fNL /k2 )2Pδδ + (4fNL )2 ξ2 Pδδ

stochasticity ≈ τNL-fNL2

ξ2 = Pφφ(k)/Pσσ(k) τNL = (1+ ξ2) fNL2

∂n ∂Ps ∂n ∂Ps ∂n ∂δ ∂n ∂δ ∂n ∂Ps

Signatures in LSS III: stochastic halo bias does stochasticity agree with predictions?

excess stochasticity above Gaussian

Pnn (k)∼ ( + 4fNL /k2 )2Pδδ + (4fNL )2 ξ2 Pδδ ∂n ∂Ps ∂n ∂δ ∂n ∂Ps

Signatures in LSS III: stochastic halo bias does stochasticity agree with predictions?

excess stochasticity above Gaussian

um, shape looks good but not amplitude tends to look better at low masses, low fNL

Pnn (k)∼ ( + 4fNL /k2 )2Pδδ + (4fNL )2 ξ2 Pδδ ∂n ∂Ps ∂n ∂δ ∂n ∂Ps