Slides from the talk - CP3-Origins

Testing the Initial Conditions of the Universe
with the Large-Scale Structure
University of Southern Denmark, Odense - September 24th, 2012
Emiliano SEFUSATTI - ICTP
in collaboration with Martin Crocce, Vincent Desjacques (ArXiv:1003.0007, ArXiv: 1111.6966)
+ Dani Figueroa, Toni Riotto & Filippo Vernizzi (ArXiv: 1205.2015)
+ Xingang Chen, James Fergusson & Paul Shellard (ArXiv: 1204.6318)

Outline
• Inflation, Primordial non-Gaussianity & the CMB
• (Non-primordial) non-Gaussianity in the Large-Scale Structure
• Primordial non-Gaussianity in the matter distribution
• Primordial non-Gaussianity in the galaxy distribution

Part 1
The “shapes” of Inflation & the CMB
Text
!"#$"%&'(%")* +, )-).)
Polarization * +, )-).)
!"#$"%&'(%")* +, )-)/...)
012&%34&516)* +, )-)/...)

Primordial non-Gaussianity
The Gaussianity of primordial fluctuations is one of the basic
predictions of the simplest model of inflation
confirmed (so far) at 0.1% level by CMB observations
Therefore ...
the possible detection of a large non-Gaussianity component in
the intial conditions would, by itself, rule-out canonical,
single-field, slow-roll inflation, providing a tool to
discriminate between models of inflation

Non-Gaussian Initial Conditions
Gaussian initial conditions:
• their statistical properties are completely specified by the two-point
correlation function or the power spectrum of the curvature
perturbations:
• All higher-order correlation functions are vanishing
Bispectrum:
Φ k1 Φ k2 = δ D ( k 1 + k 2 )P Φ (k 1 )
Φ k1 Φ k2 Φ k3 = δ D ( k 1 + k 2 + k 3 ) B Φ (k 1 ,k 2 ,k 3 )=0
Trispectrum:
Φ k1 Φ k2 Φ k3 Φ k4 = δ D ( k 1 + ... + k 4 ) T Φ ( k 1 , k 2 , k 3 , k 4 )=0
Non-Gaussian initial conditions are
characterized by an infinite set of functions:
B Φ (k 1 ,k 2 ,k 3 ) = 0
T Φ ( k 1 , k 2 , k 3 , k 4 ) = 0
, etc ...

Models of primordial non-Gaussianity
Local non-Gaussianity:
A quadratic correction to the initial curvature perturbations, Φ:
Φ(x) =φ(x)+f NL φ 2 (x)
leading to the initial, curvature bispectrum
B Φ (k 1 ,k 2 ,k 3 )=2f NL P φ (k 1 )P φ (k 2 )+2perm.
with specific properties:
• large values for squeezed triangular configurations
• well represents theoretical models where perturbations are
generated outside the horizon: curvaton, multiple-field
inflation, inhomogeneous preheating, ....
Equilateral non-Gaussianity:
k 1
k 2
B Φ (k 1 ,k 2 ,k 3 )=6f NL −P φ (k 1 )P φ (k 2 )+perm. − 2P 2/3
φ (k 1)P 2/3
φ (k 2)P 2/3
φ (k 3)+...
k 3
• large values for equilateral triangular configurations
• well approximates the bispectrum predicted by DBI inflation
or higher derivative models
Orthogonal non-Gaussianity ... and many more
k 1 k 2
k 3

The slang of non-Gaussianity
Most inflationary models predict a scale-invariant curvature bispectrum
B Φ (k, k, k) ∼ P 2 Φ(k) ∼ 1 k 6
What distinguish them is the shape
“shape” = the dependence of the curvature bispectrum predicted
by a given model of inflation on the shape of the triangular
configuration k1, k2, k3
B Φ (k 1 ,k 2 ,k 3 )=f NL
1
k 2 1 k2 2 k2 3
F
r 2 = k 2
k 1
,r 3 = k 3
k 1

Models of primordial non-Gaussianity
id est,
Multiple fields
courtesy of G. DʼAmico

The CMB Bispectrum
The Bispectrum of the CMB is the most
direct probe of the initial bispectrum
B l1 l 2 l 3
∼
B Φ (k 1 ,k 2 ,k 3 )∆ l1 (k 1 )∆ l2 (k 2 )∆ l3 (k 3 )j l1 (k 1 r)j l2 (k 2 r)j l3 (k 3 r)
CMB angular
bispectrum
curvature
bispectrum
transfer
functions
• The CMB provides a snapshot of density perturbations at early times
• Power spectrum measurements (Clʼs) are matched by linear predictions (so far)
• The CMB bispectrum is an optimal estimator for the fNL parameter

The CMB Bispectrum
The Bispectrum of the CMB is the most
direct probe of the initial bispectrum
B l1 l 2 l 3
∼
B Φ (k 1 ,k 2 ,k 3 )∆ l1 (k 1 )∆ l2 (k 2 )∆ l3 (k 3 )j l1 (k 1 r)j l2 (k 2 r)j l3 (k 3 r)
CMB angular
bispectrum
curvature
bispectrum
transfer
functions
• The CMB provides a snapshot of density perturbations at early times
• Power spectrum measurements (Clʼs) are matched by linear predictions (so far)
• The CMB bispectrum is an optimal estimator for the fNL parameter
No evidence (yet)!
(despite 2-sigma “hints” in
previous data releases ...)
WMAP 7 years:
Local -10 < fNL < 74
Equilateral -214 < fNL < 266
Orthogonal -410 < fNL < 6
@ 95% CL Komatsu et al. (2011)
The CMB bispectrum is equally sensitive to any model of non-Gaussianity

The CMB Bispectrum
The Bispectrum of the CMB is the most
direct probe of the initial bispectrum
B l1 l 2 l 3
∼
B Φ (k 1 ,k 2 ,k 3 )∆ l1 (k 1 )∆ l2 (k 2 )∆ l3 (k 3 )j l1 (k 1 r)j l2 (k 2 r)j l3 (k 3 r)
CMB angular
bispectrum
curvature
bispectrum
transfer
functions
• The CMB provides a snapshot of density perturbations at early times
• Power spectrum measurements (Clʼs) are matched by linear predictions (so far)
• The CMB bispectrum is an optimal estimator for the fNL parameter
No evidence (yet)!
(despite 2-sigma “hints” in
previous data releases ...)
Planck will improve WMAP
results by a factor of ~ 4
ΔfNL
local
ΔfNL
equilateral
WMAP 21 140
Planck ~ 5 ~ 30
The CMB bispectrum is equally sensitive to any model of non-Gaussianity

Part 1I
(Non-primordial) non-Gaussianity in the Large-Scale Structure

Non-Gaussianity in the Large-Scale Structure
250
250
200
200
150
150
100
100
50
50
0
0
0 50 100 150 200 250
Mock galaxy distribution
0 50 100 150 200 250
Rayleigh-Lévy flight
ES & Scoccimarro (2005)

Non-Gaussianity in the Large-Scale Structure
250
250
200
150
100
200
150
100
Cosmological parameters and
constraints on the initial
conditions are typically obtained
from power spectrum
measurements
50
50
0
0
0 50 100 150 200 250
0 50 100 150 200 250
-Gaussianity in the Large-Scale Structure
the power spectrum
cannot distinguish the
two distributions!
edshift surveys, information on cosmological parameters and the initial
ditions is typically obtained from measurements of the power spectrum
δ k δ k ≡δ D ( k + k )P(k) where δ k FT ↔ δ(x) ≡
ρ(x) − ¯ρ
¯ρ
250
ES & Scoccimarro (2005)

Non-Gaussianity in the Large-Scale Structure
250
250
200
150
100
200
150
100
Cosmological parameters and
constraints on the initial
conditions are typically obtained
from power spectrum
measurements
-Gaussianity in the Large-Scale Structure
50
50
t their non-Gaussian properties, quantifiedbytheirhigher-ordercorrelation
0
ctions are clearly different:
0 50 100 150 200 250
0
0 50 100 150 200 250
Bispectrum:
δ k1 δ k2 δ k3 ≡δ D ( k 123 )B(k 1 , k 2 , k 3 )
cture
k3
k1
✁
❍ ❍
✁☛ ✟ ✟✟✯ k2
dbytheirhigher-ordercorrelation
Trispectrum:
rispectrum:
250
δ k1 δ k2 δ k3 δ k4 c ≡ δ D ( k 1234 )T ( k 1 , k 2 , k 3 , k 4 )
200
✁
✁☛
150
k4
❍ ❍
✲ ✻
k1
k2
k3
trispectrum:
δ k1 δ k2 δ k3 δ k4 c ≡ δ D ( k 1234 )T ( k 1 , k 2 , k 3 , k 4 )
⇒
k4
k1
✁
❍ ❍
✁☛ ✲ ✻ k3
k2
the power spectrum
cannot distinguish the
two distributions!
but their non-Gaussian
properties are very
different!
100
ES & Scoccimarro (2005)
50

Non-Gaussianity from Gravitational Instability
A fundamental quantity: the matter overdensity
δ(x) ≡
ρ(x) − ¯ρ
¯ρ
≥−1
At early times (z ~ 1000)fluctuations
are small (σδ≪1):
⟨δδ⟩≠0,
⟨δδδ⟩≃0
⟨δδδδ⟩≃0
At late times fluctuations grow, σδ >1
Non-Gaussianity is induced by
gravitational evolution
⟨δδ⟩≠0,
⟨δδδ⟩≠0
⟨δδδδ⟩≠0

Non-Gaussianity from Gravitational Instability
At large scales fluctuations are small, σδ≪1, even at low redshift
we can study their evolution in terms of Perturbation Theory
Equations of motion for
matter density and velocity:
δ, v
• Continuity eq.
• Euler eq.
• Poisson eq.
∂δ
+ ∇ · [(1 + δ)v] =0
∂τ
∂v
∂τ + Hv +(v · ∇)v = −∇φ
∇ 2 φ = 3 2 H2 Ω m δ

Non-Gaussianity from Gravitational Instability
At large scales fluctuations are small, σδ≪1, even at low redshift
we can study their evolution in terms of Perturbation Theory
Equations of motion for
matter density and velocity:
δ, v
• Continuity eq.
• Euler eq.
• Poisson eq.
∂δ
+ ∇ · [(1 + δ)v] =0
∂τ
∂v
∂τ + Hv +(v · ∇)v = −∇φ
∇ 2 φ = 3 2 H2 Ω m δ
Perturbative solution for the
matter density, in Fourier space
δ k = δ (1)
k
+ δ (2)
k
+ ...
Linear solution
δ (2)
k
=
d 3 qF 2 ( k − q, q) δ (1)
δ (1)
k−q q
Quadratic nonlinear correction

Non-Gaussianity from Gravitational Instability
At large scales fluctuations are small, σδ≪1, even at low redshift
we can study their evolution in terms of Perturbation Theory
Equations of motion for
matter density and velocity:
δ, v
Perturbative solution for the
matter density, in Fourier space
• Continuity eq.
• Euler eq.
• Poisson eq.
δ k = δ (1)
k
+ δ (2)
k
+ ...
Linear solution
∂δ
+ ∇ · [(1 + δ)v] =0
∂τ
∂v
∂τ + Hv +(v · ∇)v = −∇φ
∇ 2 φ = 3 2 H2 Ω m δ
δ (2)
k
=
Initial conditions δ (1)
k1
δ (1)
k2
= δ D ( k 1 + k 2 )P 0 (k 1 )
B0 and T0 vanish for
Gaussian initial conditions!
d 3 qF 2 ( k − q, q) δ (1)
δ (1)
k−q q
Quadratic nonlinear correction
δ (1)
k1
δ (1)
k2
δ (1)
k3
= δ D ( k 1 + k 2 + k 3 ) B 0 (k 1 ,k 2 ,k 3 )
δ (1)
k1
δ (1)
k2
δ (1)
k3
δ (1)
k4
= δ D ( k 1 + ... + k 4 ) T 0 ( k 1 , k 2 , k 3 , k 4 )

Non-Gaussianity from Gravitational Instability
At large scales fluctuations are small, σδ≪1, even at low redshift
we can study their evolution in terms of Perturbation Theory
Equations of motion for
matter density and velocity:
δ, v
Perturbative solution for the
matter density, in Fourier space
δ k = δ (1)
k
+ δ (2)
k
+ ...
Linear solution
δ (2)
k
=
Initial conditions δ (1)
k1
δ (1)
k2
= δ D ( k 1 + k 2 )P 0 (k 1 )
B0 and T0 vanish for
Gaussian initial conditions!
• Continuity eq.
• Euler eq.
• Poisson eq.
∂δ
+ ∇ · [(1 + δ)v] =0
∂τ
∂v
∂τ + Hv +(v · ∇)v = −∇φ
∇ 2 φ = 3 2 H2 Ω m δ
d 3 qF 2 ( k − q, q) δ (1)
δ (1)
k−q q
Quadratic nonlinear correction
δ (1)
k1
δ (1)
k2
δ (1)
k3
= δ D ( k 1 + k 2 + k 3 ) B 0 (k 1 ,k 2 ,k 3 )
δ (1)
k1
δ (1)
k2
δ (1)
k3
δ (1)
k4
= δ D ( k 1 + ... + k 4 ) T 0 ( k 1 , k 2 , k 3 , k 4 )
Perturbative solution for the
matter 3-point function
δδδ = δ (1) δ (1) δ (1) + δ (1) δ (1) δ (2) + ...
loop corrections
= B0 = 0 for Gaussian
initial conditions
non-zero bispectrum
induced by gravity!

The Matter Bispectrum induced by Gravity
ateral configurations of the matter bispectrum
B G = BG
tree [P 0 ]+B loop
G [P 0]
and the linear and 1-loop predictions in PT a
BG
tree
(k 1 ,k 2 ,k 3 )=2F 2 ( k 1 , k 2 ) P 0 (k 1 ) P 0 (k 2 )+2perm.
The bispectrum induced by gravity has a well defined
dependence on scale and on the shape
Bk, k, k
110 4
5000
1000
500
matter bispectrum
equilateral configurations
f NL 0
z 0.5
1-loop
Bk, k, k
The equilateral configurations
of the matter bispectrum:
B(k, k, k) vs. k
Numerical 1000 simulations and PT
predictions
100
100
50
Tree-level
10
10 4 k h Mpc 1
E.S., M. Crocce, & V. Desjacques (2010)
F. Bernardeau, M. Crocce, E.S. (2010)
0.01 0.02 0.05 0.10 0.20
k h Mpc 1
0.01 0.02 0.05

The Matter Bispectrum induced by Gravity
The bispectrum represents the probability for three particles
(or galaxies) to form a triangle of a given size and shape
Plot of the reduced bispectrum Q(k1, k2, k3)
with fixed k1 and k2 as a function of the
angle between the two wavenumbers
2.5
Q(k 1 ,k 2 ,k 3 )=
B(k 1 ,k 2 ,k 3 )
P (k 1 )P (k 2 )+P (k 1 )P (k 3 )+P (k 2 )P (k 3 )
Qk1,k2,Θ
2.0
1.5
Nbody
treelevel
1loop
k 1 0.094 h 1 Mpc
k 2 2k 1
1.0
0.5
matter bispectrum
f NL 0, z 0.5
0.0 0.2 0.4 0.6 0.8 1.0
Θ
k 1
k 2
k 3 Θ
ΘΠ

Non-Gaussianity from Galaxy Bias (more problems?)
Additional non-Gaussianity in the galaxy distribution
is induced by nonlinear galaxy bias
The relation between the
observed galaxy
overdensity and the matter
density is nonlinear
δ g (x) ≡ n g(x) − ¯n g
¯n g
= f [δ(x)]
local bias
At large scales, we expand it in
a Taylor series
δ g (x) =b 1 δ(x)+ 1 2 b 2 δ 2 (x)+...
Linear bias
Quadratic bias correction
Perturbative solution for the
galaxy 3-point function
δ g δ g δ g = b 3 1 δδδ + b 2 1 b 2 δδδ 2 + ...
matter bispectrum
bispectrum induced
by nonlinear bias
B g (k 1 ,k 2 ,k 3 )=b 3 1 B(k 1 ,k 2 ,k 3 )+b 2 1 b 2 P (k 1 ) P (k 2 )+2perm. + ...
The component induced by bias has a different
dependence on the shape of the triangle

From the CMB to the Large-Scale Structure
The CMB provides a snapshot of the Universe at
very high redshift and its power spectrum
(the Cl’s) is matched by linear predictions
The observable Universe to the
CMB last scattering surface
The Large-Scale Structure is the result
of a highly nonlinear evolution, with
different sources of non-Gaussianity
How can we distinguish and detect
the primordial component?
Why bother at all?!
The Sloan Digital Sky Survey
(white dots are galaxies)

From the CMB to the Large-Scale Structure
The CMB provides a snapshot of the Universe at
very high redshift and its power spectrum
(the Cl’s) is matched by linear predictions
The observable Universe to the
CMB last scattering surface
The Large-Scale Structure is the result
of a highly nonlinear evolution, with
different sources of non-Gaussianity
How can we distinguish and detect
the primordial component?
Why bother at all?!
Two reasons:
The Sloan Digital Sky Survey
(white dots are galaxies)
1. The CMB is 2D field, the LSS is 3D
2. Galaxy bias reserved a surprise for us ...

Part III
Matter
400 particles. Since we are interested mainly in the masses
and positions of cluster-sized halos, and not their internal
structure, we have not used high force resolution: we
employ a Plummer softening length l of 0.2 times the
mean interparticle spacing. We have checked that using
higher force resolution (l half as large) does not appreciably
change the mass function. All simulations were performed
at the Sunnyvale cluster at CITA; depending upon
We c
z ¼ 1, 0
[66], w
tions, th
extensiv
are plot
fNL = - 5000
fNL = - 500
fNL = 0
Havin
like to c
used fo
above,
Gaussia
Schecht
function
order o
interest
to the
adopted
fNL = + 500
( )
fNL = + 5000
Dalal et al. (2008)
FIG. 1 (color online). Slice through simulation outputs at z ¼
0 generated with the same Fourier phases but with f ¼

Matter Power Spectrum
In Perturbation Theory ...
P = P 0 + P loop
G
[P 0]+P loop
NG [P 0,B 0 ]
matter power spectrum
Linear power
spectrum
Gravity-induced
contributions
(depending on P0 alone)
Additional gravity-induced contributions
present only for NG initial conditions (B0)

Matter Power Spectrum
In Perturbation Theory ...
P = P 0 + P loop
G
[P 0]+P loop
NG [P 0,B 0 ]
matter power spectrum
Linear power
spectrum
Gravity-induced
contributions
(depending on P0 alone)
Additional gravity-induced contributions
present only for NG initial conditions (B0)
Few percent effect at small scales
for allowed values of fNL
Ratio of the non-Gaussian
to the Gaussian power
spectrum for fNL = ±100
(local) at z =1
Smith, Desjacques & Marian (2010)

Matter Power Spectrum
In Perturbation Theory ...
P = P 0 + P loop
G
[P 0]+P loop
NG [P 0,B 0 ]
Linear power
spectrum
Gravity-induced
contributions
(depending on P0 alone)
Additional gravity-induced contributions
present only for NG initial conditions (B0)
Few percent effect at small scales
. for Fedeli allowed and L. Moscardini values of fNL
Weak Lensing
Ratio of the non-Gaussian
to the Gaussian power
spectrum for fNL = ±100
(local) at z =1
Smith, Desjacques & Marian (2010)
Fedeli & Moscardini (2010)
Giannantonio et al. (2011)
From EUCLID we expect:
ΔfNL ~ 20 (local)
ΔfNL ~ 3 (orthogonal)
ΔfNL ~ 10 (equilateral)
(with Planck priors on
cosmology)

Matter Power Spectrum & Bispectrum
In Perturbation Theory ...
P = P 0 + P loop
G
[P 0]+P loop
NG [P 0,B 0 ]
matter power spectrum
Linear power
spectrum
Gravity-induced
contributions
(depending on P0 alone)
Additional gravity-induced contributions
present only for NG initial conditions (B0)
B = B 0 + B tree
G
[P 0 ]+B loop
G
[P 0]+B loop
NG [P 0,B 0 ]
& bispectrum
Primordial
component

The matter bispectrum and PNG: large scales
At large scales I can approximate the matter bispectrum with the tree-level expression on PT:
B(k 1 ,k 2 ,k 3 ) B 0 + B tree
G [P 0 ]
Primordial
component
Gravity-induced
component
Bk,k,k
5000
2000
1000
500
200
Gravity
Initial, f NL 100
Gravity Initial
0.01 0.02 0.05 0.1
k h Mpc 1
Equilateral configurations
of the matter bispectrum
B 0 (k, k, k) k→0
(k, k, k) ∼
B tree
G
f NL
D(z)k 2
The primordial component has a
different dependence on scale
than the gravity-induced one!
It is relevant at large scales
and early times
This is true for almost all models (local,
equilateral, orthogonal ...)

The matter bispectrum and PNG: large scales
At large scales I can approximate the matter bispectrum with the tree-level expression on PT:
B(k 1 ,k 2 ,k 3 ) B 0 + B tree
G [P 0 ]
Primordial
component
Gravity-induced
component
1.4
1.2
k 1 =0.01 h Mpc −1 , k 2 =1.5 k 1
Reduced bispectrum as a function of
the angle between two wavenumbers
Qk 1 ,k 2 ,Θ
1.0
0.8
0.6
0.4
0.2
Gravity
Initial, f NL 100
Gravity Initial
Primordial component for
local non-Gaussianity:
large contribution in the
squeezed limit!
The primordial component has
a different dependence on the
shape of the triangular
configurations
0.0 0.2 0.4 0.6 0.8 1.0
ΘΠ
and it is specific to the non-Gaussian
model (local, equilateral, orthogonal ...)

The matter bispectrum and PNG: large scales
Qk 1 ,k 2 ,Θ
2.5
2.0
1.5
1.0
Local NG, 10 f NL 74
Current CMB constraints for different
models of non-Gaussianity as uncertainties
on the generic configurations of the matter
bispectrum, B B 0 + BG tree [P 0 ]
0.5
The matter bispectrum can distinguish
different non-Gaussian models
0.0 0.2 0.4 0.6 0.8 1.0
ΘΠ
2.5
Equilateral NG, 214 f NL 266
2.5
Orthogonal NG, 410 f NL 6
2.0
2.0
Qk 1 ,k 2 ,Θ
1.5
1.0
Qk 1 ,k 2 ,Θ
1.5
1.0
0.5
0.5
0.0 0.2 0.4 0.6 0.8 1.0
ΘΠ
0.0 0.2 0.4 0.6 0.8 1.0
ΘΠ

The matter bispectrum and PNG: large scales
In Perturbation Theory ...
P = P 0 + P loop
G
[P 0]+P loop
NG [P 0,B 0 ]
matter power spectrum
Linear power
spectrum
Gravity-induced
contributions
(depending on P0 alone)
Additional gravity-induced contributions
present only for NG initial conditions (B0)
B = B 0 + B tree
G
[P 0 ]+B loop
G
[P 0]+B loop
NG [P 0,B 0 ]
& bispectrum
Primordial
component
If B0 was the only effect of NG initial conditions on the LSS
then future, large volume surveys (~100 Gpc 3 ) could
provide:
ΔfNL local < 5 and ΔfNL eq < 10
Scoccimarro, ES & Zaldarriaga (2004), ES & Komatsu (2007)

The matter bispectrum and PNG: small scales
In Perturbation Theory ...
P = P 0 + P loop
G
[P 0]+P loop
NG [P 0,B 0 ]
matter power spectrum
Linear power
spectrum
Gravity-induced
contributions
(depending on P0 alone)
Additional gravity-induced contributions
present only for NG initial conditions (B0)
B = B 0 + B tree
G
[P 0 ]+B loop
G
[P 0]+B loop
NG [P 0,B 0 ]
& bispectrum
Primordial
component
Nonlinear corrections are also
affected by the initial conditions!

B G B G, tree, nw
BNG
BNG BG
BG
1.6
1.4
The matter bispectrum and PNG:
1.4
small scales
1.4
1.2
1.2
B = 1.3 B 0 + BG
tree
1.0
BNG BG
0.8 1.2 Primordial Gravity-induced Additional gravity-induced 0.8 1.2 contributions
component contributions present for NG initial conditions (B0)
0.6
0.05 0.10 0.15 0.20 0.25 0.30
1.1
1.1
0.05 0.10 0.15 0.20 0.25 0.30
k h Mpc 1
k h Mpc 1
Squeezed 1.0 configurations B(Δk, k, k)
as a function of k with Δk = 0.01 h/Mpc
1.4
1000
1.3500
1.2200
100
1.1
50
treelevel
oneloop
SC01
ES (2009)
ES, Crocce & Desjacques (2010)
1.0
5000
Ratio B f NL 100 B f NL 0
[P 0 ]+B loop
G
1.0
0.05Squeezed 0.10 0.15configurations 0.20 0.25 0.30 B(∆k, k, k) vs. 0.05 k, 0.10 non-Gaussian ES 0.15 (2009) 0.20 0.25 initial 0.30c
k h Mpc 1
Ratio Difference B f NL B f100 NL 100 B f NL B f NL 00
z 0z 1
0.010.05 0.02 0.10 0.15 0.05 0.200.10 0.25 0.20.30
k hk Mpc h Mpc 1
1
Nbody
treelevel
oneloop
[P 0]+B loop
B G B G, tree, nw
z 1
1.0
1.3
NG [P 0,B 0 ]
BNG BG
BNG BG
200
1.3
BNG BG
1.4
1.2
1.1
1.0
100
50
20
10
Ratio B f NL 100 B f NL 0
k h Mpc 1
z 2
ES, Crocce & Desjacques (2010)
Ratio Difference B f NL B f100 NL 100 B f NL B f NL
00
z z1
2
0.010.05 0.02 0.10 0.15 0.05 0.200.10 0.25 0.200.30
k hk Mpc h Mpc 1
1

0.8
0.6
0.01 0.1 1 10
k h Mpc 1
The matter bispectrum and PNG: even smaller scales
N
Beyond PT: The Halo Model
1000
Difference B f NL 100 B f NL 0
BNG BG
100
10
There is a significant effect of NG
initial conditions of about 5-15% on all
triangles, at small scales and at late
times for fNL = 100
Weak lensing?
NbodyHM
1
1.40.01 0.1 Nbody 1 HM
1.2
1.0
k h Mpc 1
10
0.8
0.6
0.01 0.1 1 10
k h Mpc 1
1.3
Ratio B f NL 100 B f NL 0
BNG BG
1.2
1.1
Figueroa, ES, Riotto & Vernizzi (2012)
Squeezed configurations B(Δk, k, k)
as a function of k with Δk = 0.01 h/Mpc
1.0
0.01 0.1 1 10
k h Mpc 1

Part IV
Galaxies
IMPRINTS OF PRIMORDIAL NON-GAUSSIANITIES ON ...
fNL = - 5000
fNL = - 500
fNL = 0
fNL = + 500
fNL = + 5000
Dalal et al. (2008)
FIG. 7 (color online). Cross-power spectra for various f NL .
The upper panel displays P h ðkÞ, measured in our simulations at
z ¼ 1 for halos of mass 1:6 10 13 M

Galaxy bias and the galaxy power spectrum
Dalal et al. (2008):
The bias of galaxies receives a significant scale-dependent
correction for NG initial conditions of the local type
P g (k) =[b 1 + ∆b 1 (f NL ,k)] 2 P (k)
“Gaussian” Scale-dependent correction
IMPRINTS OF biasPRIMORDIAL due NON-GAUSSIANITIES to local non-GaussianityON ...
Large effect on large scales!
Dalal et al. (2008)
PHYSICAL REVIEW
using the halo auto spectra to compute
results as the cross spectra; i.e. we
stochasticity. Examples of the variou
resulting bias factors are plotted in F
As can be seen, we numerically co
∆b predicted 1,NG (fscale NL ,k) dependence. ∼
f NLBecau
statistics of rare objects, D(z) the errors k 2 on
simulations plotted in Fig. 8 are lar
tempt to improve the statistics on the
bining the bias measurements from
Figure 8 plots the average ratio betwe
in our simulations and our analytic
using c ¼ 1:686 as predicted from t
model [78]. In computing the average
we used a uniform weighting across
Measurements of the power spectrum of
dark matter halos in N-body simulation
with local NG initial conditions

Galaxy bias and the galaxy power spectrum
Dalal et al. (2008):
The bias of galaxies receives a significant scale-dependent
correction for NG initial conditions of the local type
P g (k) =[b 1 + ∆b 1 (f NL ,k)] 2 P (k)
“Gaussian”
bias
Scale-dependent correction
due to local non-Gaussianity
QSOs+LRGs: -31 < fNL < 70 (95% CL)
[Slosar et al. (2008)]
AGNs+QSOs+LRGs: 8 < fNL < 88 (95% CL)
[Xia et al. (2011)]
Limits from LSS are
already competitive
with the CMB!
(at least for the local model ...)
high-redshift sources: quasars & AGNs
CMB limits (95% CL): -10 < fNL < 74
[Komatsu et al. (2009)]
From EUCLID we expect:
ΔfNL ~ 5
from the 3D power spectrum alone
[e.g. Giannantonio et al. (2011)]

shear
by
→
n-
ad
we
0).
er-
L)
ng
nd
nt
llias
=
et
ses
by
w-
Galaxy bias and the galaxy power spectrum
Dalal et al. (2008):
The bias of galaxies receives a significant scale-dependent
correction for NG initial conditions of the local type
FIG. 4: The scale-dependent part of bias b − b ∞ divided by
b ∞ − 1, for f NL =100andz =0. Thescale-independent
part of bias, b ∞ =1.5 and3forsolidanddottedcurves.The
dashed line is Dalal et al.’s derivation [1].
P g (k) =[b 1 + ∆b 1 (f NL ,k)] 2 P (k)
“Gaussian”
bias
Afshordi & Tolley (2008)
Scale-dependent correction
due to local non-Gaussianity
FIG. 5: This figure illustrates the contrast between the gaus-
Introduction
PNG and the Matter Bispectrum
PNG and the Galaxy Bispectrum
Future Perspectives
A local model for primordial non-Gau
Non-Gaussian Local non-Gaussianity
correction to the curvatur
introduces a correlation between h
large-scale Φ(x) fluctuations =φ(x)+f and NL
loc.
the φ 2 (x) −
small-scales responsible for the
[Salopek & Bond (19
formation (collapse) of dark matter
halos (and therefore, galaxies)
⇒ B Φ (k 1 , k 2 , k 3 )=2fNL loc. [P φ (k 1 )P
k 1
k 2
k 3
Large val
Represen
are gener
Curvato
Multipl
Inhomo

What about other models?
approach [40]
∆b I = b NG
1 − b G 1 = − ∂ ln RNG (M)
, (6.5)
∂δ c
The scale-dependence of bias can be different
for other models, or not be there at all ...
R NG (M) being the ratio of the non-Gaussian to the Gaussian mass function. Here,
ver, we treat ∆b I as a free parameter and compare it later on with the prediction
ed from the mass functions. We choose q as the second free parameter. All other 1
tities in Eq. (6.4) are we derive from the theory and are kept fixed.
In Fig. 8, we show as an example the effect of local non-Gaussianity on the halo bias
alos of mass 1.2 − 2.4 × 10 14 M ⊙ /h at z = 0. Note that we plot ∆b(k)+0.1. As ∆b I is
tive, this addition of 0.1 is needed to still make use of the logarithmic scale.
The different line types visualize the effect of the different terms in Eq. (6.4). The 0.1
s lines show the best fit to the data (using all modes up to k max =0.1Mpc/h) and
des all terms given above. The short-dashed lines neglects ∆b I appearing inside the
re brackets in Eq. (6.4). The inclusion of this term makes the non-Gaussian bias nonr
in f NL [55], since ∆b I depends on f NL . The dot-dashed line neglects ∆b I completely.
scale-dependent bias shift becomes important on smaller scales (k >0.02), for which
cale-dependent part becomes small.
!b + 0.1
10
1
0.1
1
k
Orthogonal orthogonal NG
Gaussian bias. This has the advantage that by doing this we do not need to include β(k
the modelling of ∆b(k) (seeEq.(2.1) and Eq. (2.4)).
!b + 0.1
0.4
f NL = -1000
1.2x10 14 < M
f NL = -250
halo < 2.4x10 14
3x10 13 < M halo < 6x10
Figure 8. Local non-Gaussian bias of halos with mass 1.2 × 10 14 Mpc/h < 13
M < 2.4 × 10 14 M
at z = 0. The difference of the bias measured from the non-Gaussian simulations and Gau
simulations is depicted 0.2 by the data points (for computation of the errorbars, see App. B). The
lines show the best fit using the model given in Eq. 6.4. The dot-dashed lines show the m
predictions when the scale-independent bias shift, ∆b I , is neglected. The short-dashed lines ne
the term which is non-linear 0 in f NL (see text for details). Thick lines and red symbols correspon
f NL = 250, while thin lines and blue symbols show the results for f NL = 60. Note that we act
show ∆b +0.1 to allow for a logarithmic scale.
!b
10
0.01
0.003 0.01 0.1
-0.2
-0.4
1
k 2
local
Local NG
k [h/Mpc]
f NL = 250
f NL = 60
As the non-Gaussian and the Gaussian simulation share the same realization of
initial Gaussian field, ∆b(k) is almost free of sample variance and, in addition, has sm
shot noise than b NG (k) and b G (k) individually. We estimate the error on ∆b(k) directly f
const.
the distribution of ∆b(k) in each k bin (for details, see App. B).
Following Eq. 2.1, wemodel∆b(k) by equilateral
Wagner & Verde (2010)
Equilateral NG
0.01
0.003 0.01 0.1
k [h/Mpc]
where b G 1
∆b(k) =∆b I + f NL b
G
1 + ∆b I − 1 qδ c F M (k)
D(z) M M (k) , (
-0.6
0.003 0.01 0.1
k [h/Mpc]
is the linear halo bias obtained from the Gaussian simulation on large scales

What about other models?
approach [40]
∆b I = b NG
1 − b G 1 = − ∂ ln RNG (M)
, (6.5)
∂δ c
The scale-dependence of bias can be different
R NG (M) being the ratio of the non-Gaussian to the Gaussian mass function. Here,
ver, for we other treat ∆b I
models, as a free parameter or not and be compare there it at later all on ... with the prediction
ed from the mass functions. We choose q as the second free parameter. All other 1
tities in Eq. (6.4) are we derive from the theory and are kept fixed.
In Fig. 8, we show as an example the effect of local non-Gaussianity on the halo bias
alos of mass 1.2 − 2.4 × 10 14 M ⊙ /h at z = 0. Note that we plot ∆b(k)+0.1. As ∆b I is
tive, this addition of 0.1 is needed to still make use of the logarithmic scale.
The different line types visualize the effect of the different terms in Eq. (6.4). The 0.1
s lines show the best fit to the data (using all modes up to k max =0.1Mpc/h) and
des all terms given above. The short-dashed lines neglects ∆b I appearing inside the
re brackets in Eq. (6.4). The inclusion of this term makes the non-Gaussian bias nonr
inNB: f NL [55], A since careful ∆b I depends definition f NL . The dot-dashed of the line neglects ∆b I completely.
scale-dependent bias shift becomes important on smaller scales (k >0.02), for which 0.01
cale-dependent part becomes small.
templates might be in order ...
!b + 0.1
10
1
0.1
1
k
?
Orthogonal orthogonal NG
Gaussian bias. This has the advantage that by doing this we do not need to include β(k
the modelling of ∆b(k) (seeEq.(2.1) and Eq. (2.4)).
!b + 0.1
Text
0.4
f NL = -1000
1.2x10 14 < M
f NL = -250
halo < 2.4x10 14
3x10 13 < M halo < 6x10
Figure 8. Local non-Gaussian bias of halos with mass 1.2 × 10 14 Mpc/h < 13
M < 2.4 × 10 14 M
at z = 0. The difference of the bias measured from the non-Gaussian simulations and Gau
simulations is depicted 0.2 by the data points (for computation of the errorbars, see App. B). The
lines show the best fit using the model given in Eq. 6.4. The dot-dashed lines show the m
predictions when the scale-independent bias shift, ∆b I , is neglected. The short-dashed lines ne
the term which is non-linear 0 in f NL (see text for details). Thick lines and red symbols correspon
f NL = 250, while thin lines and blue symbols show the results for f NL = 60. Note that we act
show ∆b +0.1 to allow for a logarithmic scale.
!b
10
-0.2
-0.4
1
k 2
local
Local NG
0.003 0.01 0.1
k [h/Mpc]
f NL = 250
f NL = 60
As the non-Gaussian and the Gaussian simulation share the same realization of
initial Gaussian field, ∆b(k) is almost free of sample variance and, in addition, has sm
shot noise than b NG (k) and b G (k) individually. We estimate the error on ∆b(k) directly f
const.
the distribution of ∆b(k) in each k bin (for details, see App. B).
Following Eq. 2.1, wemodel∆b(k) by equilateral
Wagner & Verde (2010)
Equilateral NG
0.01
0.003 0.01 0.1
k [h/Mpc]
where b G 1
∆b(k) =∆b I + f NL b
G
1 + ∆b I − 1 qδ c F M (k)
D(z) M M (k) , (
-0.6
0.003 0.01 0.1
k [h/Mpc]
is the linear halo bias obtained from the Gaussian simulation on large scales

The interesting case of Quasi-Single Field Inflation
k −3/2
3 ln k 3
,
k 1
QSF inflation predicts a family of models
parametrized by
α = 1 2 − ν ,
m 2
H 2 ≈ 9 (1)
4 ,
9 ν ≡ 4 − m2
H 2 . “isocurvaton” fields, i.e. field (2)
non-Gaussianity with intermediate and “shapes” is present between eventhe
if the non-Gaussianity is perfectly scaleheequilateral
inflaton requires and local m 2 ≥ 0, so α ≥−1. As m 2 /H 2 > 9/4, the isocurvatons
significant effects on density perturbations. So we are mainly interested in
a momentum dependence lies between that of the equilateral shape (α = 1)
large non-Gaussianity or its direct multifield generalization, and that of the
ultifield models with light isocurvatonssqueezed-limit
m H. (See [9–11] for reviews.)
n the isocurvaton masses may be model-dependent in more general situations,
dependence in the squeezed limit is a robust evidence for the existence of such
atively as follows [6, 7]. The fluctuations of the massive scalars decay after
y decay immediately after the horizon-exit, and for lighter scalars they decay
responsible for the large non-Gaussianities, are therefore generated between
n scales. The former
ν = 0.2
is responsible for the equilateral-like ν = 1 shapes, and the
cy check, if we look at the special limit of massless scalars, the superhorizon
over the characteristic local shape in the squeezed limit. This momentum
titatively as follows [8], at least for α close to −1 . Ignoring the physics within
ed limit of the three-point function can be regarded as the modulation of the
ngth modes from a long-wavelength mode. After horizon exit, we know that
ays as ∼ a −1−α • Inflationary phenomenology.
as a function of the scale factor a. So the amplitude of the
1+α
m is the mass of the
orthogonal to the inflaton
trajectory in field-space
Figure 1. Thisfigureillustratesamodelofquasi-singlefieldinflation in ter
The θ direction is the inflationary direction, with a slow-roll potential. The
isocurvature direction, which typically has mass of order H.
PHYSICAL REVIEW D 81, 063511 (2010)
larger than O(H), quasi-single field inflation makes the same predic
inflation. However, once large couplings exist and the mass are of orde
these massive isocurvatons can have important effects on density pert
In this paper, we shall study a simple model of quasi-single fiel
model, the coupling between the inflaton and the massive isocurvat
turning trajectory. The tangential direction of this turning trajector
direction, while the orthogonal direction is lifted by a mass oforderH
The motivations for investigating quasi-single field inflation are
• UV completion and fine-tuning in inflation models. To satisfy t
tion, fine-tunings or symmetries should generally be evoked. Atl
the case for models that have reasonable UV completion in string
ity [3, 4]. For slow-roll inflation, this means that, in the inflati
light fields will typically acquire mass of order the Hubble para
heavy to be the inflaton candidates. On the other hand, in a UV
tiple light fields arise naturally. Taking these facts into considera
of inflation emerges: There is one inflation direction with mass m
directions in the field space with m ∼ H. In contrast to the sl
higher order terms in the potential such as V ′′′ ∼ H and V ′′′′ ∼
these non-flat directions. To have more than one flat direction n
The above picture for inflation suggests the quasi-single field in
When Chen the & inflaton Wang trajectory (2010) tu
isocurvature perturbation is converted to curvature perturbation
ing the effect of such a conversion on density perturbations is in

The interesting case of Quasi-Single Field Inflation
11
The scale-dependent
bias correction
b bG
b bG
b bG
10
1
QSF, Ν 1.5 10 1
QSF, Ν 1
1
1
k 2 k 3/2
0.1
0.1
0.01
0.01
0.001
0.001
0.001 0.01 0.1
0.001 0.01 0.1
k h Mpc 1
k h Mpc 1
10
QSF, Ν 0.5 10
QSF, Ν 0
1
1
1
1 √
k
k
0.1
0.1
0.01
0.01
0.001
0.001
0.001 0.01 0.1
0.001 0.01 0.1
k h Mpc 1
k h Mpc 1
10
Local NG 10 Equilateral NG
1
z 1
1
k 2 M 10 12.5 h 1 M
b 1,G 1.2
0.1
b bG
b bG
b bG
0.1
0.01
M 10 13.5 h 1 M
b 1,G 2.0
∼ const
0.001
0.001
ES, Chen, Fergusson & Shellard (2012)
0.001 0.01 0.1
k h Mpc 1
0.001 0.01 0.1
k h Mpc 1

The interesting
Νcase 1.5
of Quasi-Single Field Inflation
Ν 1.5
50 100
Given a positive
detection of fNL,
how well can we
constrain
the parameter ν,
150 200
in future galaxy surveys?
f NL
f NL 50
0.2
0.0
1.4
1.2
f NL 100
0 50 100 150 200
f NL
0.2
0.0
1.4
1.2
0
f N
Ν
1.0
1.0
Fisher matrix analysis of
galaxy power spectrum
measurements at k < kmax(z)
kmax(0) = 0.075 h Mpc -1
Ν
1.4
1.2
1.0
0.8
f NL 0.6 50
Ν 1
0.4
0.2
50 100 150 200
f NL
0.0
Ν
0 50 Ν 1.5 100 150 200
ES, Chen, Fergusson & Shellard (2012)
1.4 1.4
0.8
0.6
0.4
0.2
0.0
f NL 0.0
0 50 100 150 200
f NL
LSS, V1
CMB
LSS, V1 CMB
f NL 50
Ν
f NL 100
Ν 1
Ν
0.8
0.6
0.4
0.2
0.0
1.4 1.4
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
f N
Ν
0 50 100 150 2
f NL
f NL 10
Ν 1.

What about other models?
approach [40]
∆b I = b NG
1 − b G 1 = − ∂ ln RNG (M)
, (6.5)
∂δ c
The scale-dependence of bias can be different
R NG (M) being the ratio of the non-Gaussian to the Gaussian mass function. Here,
ver, for we other treat ∆b I
models, as a free parameter or not and be compare there it at later all on ... with the prediction
ed from the mass functions. We choose q as the second free parameter. All other 1
tities in Eq. (6.4) are we derive from the theory and are kept fixed.
In Fig. 8, we show as an example the effect of local non-Gaussianity on the halo bias
alos of mass 1.2 − 2.4 × 10 14 M ⊙ /h at z = 0. Note that we plot ∆b(k)+0.1. As ∆b I is
tive, this addition of 0.1 is needed to still make use of the logarithmic scale.
ThePower different linespectrum
types visualize the effect of the different terms in Eq. (6.4). The 0.1
s lines
measurements
show the best fit to the data
are
(using
not
all modes
equally
up to k max =0.1Mpc/h) and
des all terms given above. The short-dashed lines neglects ∆b I appearing inside the
re brackets sensitive in Eq. (6.4). to The all inclusion NG ofmodels!
this term makes the non-Gaussian bias nonr
in f NL [55], since ∆b I depends on f NL . The dot-dashed line neglects ∆b I completely.
scale-dependent bias shift becomes important on smaller scales (k >0.02), for which 0.01
cale-dependent part becomes small.
!b + 0.1
10
1
0.1
1
k
Quasi-Single Field NG
orthogonal
ν = 0.5
0.01
0.003 0.01 0.1
k [h/Mpc]
Gaussian bias. This has the advantage that by doing this we do not need to include β(k
the modelling of ∆b(k) (seeEq.(2.1) and Eq. (2.4)).
!b + 0.1
Text
0.4
f NL = -1000
1.2x10 14 < M
f NL = -250
halo < 2.4x10 14
3x10 13 < M halo < 6x10
Figure 8. Local non-Gaussian bias of halos with mass 1.2 × 10 14 Mpc/h < 13
M < 2.4 × 10 14 M
at z = 0. The difference of the bias measured from the non-Gaussian simulations and Gau
simulations is depicted 0.2 by the data points (for computation of the errorbars, see App. B). The
lines show the best fit using the model given in Eq. 6.4. The dot-dashed lines show the m
predictions when the scale-independent bias shift, ∆b I , is neglected. The short-dashed lines ne
the term which is non-linear 0 in f NL (see text for details). Thick lines and red symbols correspon
f NL = 250, while thin lines and blue symbols show the results for f NL = 60. Note that we act
show ∆b +0.1 to allow for a logarithmic scale.
!b
10
-0.2
local
Local NG
0.003 0.01 0.1
k [h/Mpc]
-0.6
0.003 0.01 0.1
k [h/Mpc]
f NL = 250
f NL = 60
As the non-Gaussian and the Gaussian simulation share the same realization of
initial Gaussian field, ∆b(k)
const.
is almost free of sample variance and, in addition, has sm
shot noise than b NG (k) and b G (k) individually. We estimate the error on ∆b(k) directly f
-0.4
the distribution of ∆b(k) in each k bin (for details, see App. B).
Following Eq. 2.1, wemodel∆b(k) byEquilateral equilateral
NG
where b G 1
1
k 2
Wagner & Verde (2010)
∆b(k) =∆b I + f NL b
G
1 + ∆b I − 1 qδ c F M (k)
D(z) M M (k) , (
is the linear halo bias obtained from the Gaussian simulation on large scales

The Galaxy Bispectrum: NG and nonlinear bias
The galaxy bispectrum at large scales
B g (k 1 ,k 2 ,k 3 )=b 3 1 B(k 1 ,k 2 ,k 3 )+b 2 1 b 2 P (k 1 ) P (k 2 )+2perm. + ...

The Galaxy Bispectrum: NG and nonlinear bias
The galaxy bispectrum at large scales
B g (k 1 ,k 2 ,k 3 )=b 3 1 B(k 1 ,k 2 ,k 3 )+b 2 1 b 2 P (k 1 ) P (k 2 )+2perm. + ...
B = B 0 + B tree
G
Primordial component
(large scales)
[P 0 ]+B loop
G
[P 0]+B loop
NG [P 0,B 0 ]

The Galaxy Bispectrum: NG and nonlinear bias
The galaxy bispectrum at large scales
B g (k 1 ,k 2 ,k 3 )=b 3 1 B(k 1 ,k 2 ,k 3 )+b 2 1 b 2 P (k 1 ) P (k 2 )+2perm. + ...
B = B 0 + B tree
G
P = P 0 + P loop
G
[P 0]+P loop
NG [P 0,B 0 ]
[P 0 ]+B loop
G
[P 0]+B loop
NG [P 0,B 0 ]
Primordial component
(large scales)
Effect on nonlinear
evolution (small scales)

The Galaxy Bispectrum: NG and nonlinear bias
The galaxy bispectrum at large scales
b 1,G + ∆b 1,NG (f NL ,k) b 2,G + ∆b 2,NG (f NL , k 1 , k 2 )
Scale-dependent
bias corrections
B g (k 1 ,k 2 ,k 3 )=b 3 1 B(k 1 ,k 2 ,k 3 )+b 2 1 b 2 P (k 1 ) P (k 2 )+2perm. + ...
B = B 0 + B tree
G
P = P 0 + P loop
G
[P 0]+P loop
NG [P 0,B 0 ]
[P 0 ]+B loop
G
[P 0]+B loop
NG [P 0,B 0 ]
Primordial component
(large scales)
∆b 1,NG (f NL , k)=∆b 1,si (f NL )+∆b 1,sd (f NL ,b 1,G , k)
∆b 2,NG (f NL , k 1 , k 2 )=∆b 2,si (f NL )+∆b 2,sd (f NL ,b 1,G ,b 2,G , k 1 , k 2 )
Effect on nonlinear
evolution (small scales)
ES (2009)
Giannantonio & Porciani (2010)
Baldauf, Seljak & Senatore (2010)

The Galaxy Bispectrum: NG and nonlinear bias
The galaxy bispectrum at large scales
b 1,G + ∆b 1,NG (f NL ,k) b 2,G + ∆b 2,NG (f NL , k 1 , k 2 )
Scale-dependent
bias corrections
B g (k 1 ,k 2 ,k 3 )=b 3 1 B(k 1 ,k 2 ,k 3 )+b 2 1 b 2 P (k 1 ) P (k 2 )+2perm. + ...
B = B 0 + B tree
G
P = P 0 + P loop
G
[P 0]+P loop
NG [P 0,B 0 ]
[P 0 ]+B loop
G
[P 0]+B loop
NG [P 0,B 0 ]
Primordial component
(large scales)
∆b 1,NG (f NL , k)=∆b 1,si (f NL )+∆b 1,sd (f NL ,b 1,G , k)
∆b 2,NG (f NL , k 1 , k 2 )=∆b 2,si (f NL )+∆b 2,sd (f NL ,b 1,G ,b 2,G , k 1 , k 2 )
Effect on nonlinear
evolution (small scales)
ES (2009)
Giannantonio & Porciani (2010)
Baldauf, Seljak & Senatore (2010)
• We test this model in N-body simulations
with local NG initial conditions
• We fit all triangular configurations
up to k = 0.07 h/Mpc
for b1,G, b2,G, Δb1,G and Δb2,G
δ h δ h δ h = δ D ( k 123 ) B h
P h
→ b 1,G , ∆b 1,si
B h,G → b 2,G
∆B h,NG
→ ∆ b 2,si

B
B NG
Sims Sims Theory Theory
B NG B G
1.10
4000
1.05
2000
1.00
0
B h, NG
b 10
b 10
c 01 k
b 20
b 20
c 11 k
b 20 c 01
B
∆ 2
The 20 000 Halo Bispectrum:
φ (k) ≡ k3 P φ (k)/(2π 2 )=A
theory
φ (k/k
vs.
0 ) ns−1 . The non-Gaussianity is of the local fo
0
0 simulations
standard (CMB) convention in which Φ(x) is primordial, and not extrapolated to
Ω
5000
m =0.279, Ω b =0.0462, n s =0.96, and a normalization of the Gaussian curvatu
Halo bispectrum:
B NG B NG Bat G the pivot point k 2000 B NG B NG B 0 =0.02Mpc −1 , close to the best-fitting G values inferred from C
1.5 3
a density
2
B h (k 1 ,k 2 ,k 3 )=b 3 fluctuations amplitude σ3
8 0.81 when the initial conditions are Gauss
each 1(f of which NL ,k) has B(kf NL 1 ,k 2 =0, ,k 3 ) ±100, 2 were run with the N-body code gadget [8]
1.0 1
field φ is employed in each set of 1runs so as to minimize the sampling variance.
0.5 0
generated +b 1 (f NL at redshift ,k 1 )b 1 (fz NL ,k 2 )b 2 (f 0 NL ,k 1 ,k 2 ) P (k 1 ) P (k 2 )+cyc.
1
i = 99 using the Zel’dovich approximation [9].
1
0.2 0.4
We study 0.6
the 0.8
distribution of FoF halos 0.2
in two0.4 mass bins.
0.6The low-mass
0.8
bin is d
B(k1, k2, θ) as a function of 1.6 θ × with 10
ΘΠ
k1 13 = h −1 0.05 Mh/Mpc, k2 =0.07 h/Mpc
⊙ while the high-mass bin is given by M>1.6 × 10
ΘΠ
13 h −1 M ⊙ . W
z =0.5 (precisely z =0.509).
Sims Theory
B NG B G
1.10
Ratio B NG B G
1.05
1.00
2
O f NL
0.95
2000
3
2
1
0
1000
1
500
0
500
1000
1.10
Sims Theory
Ratio B NG B G
0.2 0.4 0.6 0.8
B NG B NG B G
0.2 0.4 0.6 0.8
ΘΠ
B NG B NG 2 B G 2
0.2 0.4 0.6 0.8
ΘΠ
0.2
Ratio B NG
0.4
B G 0.6 0.8
ES, Crocce & Desjacques (2011)
ΘΠ
2
O f NL
0.95
1000
500
0
500
1000
ΘΠ
B NG B NG 2 B G 2
ES, Crocce & Desjacques (2011)
k 1 k 3 Θ
Θ
0.2 0.4 0.6 0.8
ΘΠ
k 2

B
B NG
2
O f NL
20500
000
∆ 2
The Halo Bispectrum:
φ (k) ≡ k3 P φ (k)/(2π 2 )=A
theory
φ (k/k
vs.
0 ) ns−1 . The non-Gaussianity is of the local fo
simulations
10 000 0
standard (CMB) convention in which Φ(x) is primordial, and not extrapolated to
0
Ω m =0.279, Ω b =0.0462, n s =0.96, and a normalization of the Gaussian curvatu
5000
Halo bispectrum:
B NG B NG at B G the pivot point k 0 =0.02Mpc −1 , close B NG to the B NG best-fitting B G values inferred from C
1.5 3 0.2 0.4
a density
0.6 0.8
3
B
2 h (k 1 ,k 2 ,k 3 )=b 3 fluctuations amplitude σ 8 0.81 when the initial conditions are Gauss
each 1(f of which NL ,k) has B(kf NL 1 ,k 2 =0, ,k 3 ) ±100, 2 were run with the N-body code gadget [8]
1.0 1
field φ is employed in each set of 1runs so as to minimize the sampling variance.
0.5 0
generated +b 1 (f NL at redshift ,k 1 )b 1 (fz NL ,k 2 )b 2 (f 0
i = 99 using the NL Zel’dovich ,k 1 ,k 2 ) P approximation (k 1 ) P (k 2 )+cyc. [9].
1
1
We study the distribution of FoF halos in two mass bins. The low-mass bin is d
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
B(k1, k2, θ) as a function of 1.6 θ with × 10k1 13 = h −1 0.07 Mh/Mpc, k2 =0.08 h/Mpc
⊙ while the high-mass bin is given by M>1.6 × 10 13 h −1 M
ΘΠ
ΘΠ
⊙ . W
z =0.5 (precisely z =0.509).
Sims Theory
B NG B G
1.15
10 000
1.10
1.05
5000
1.00
0.95
0
0.90
3
2
1
10000
500 1
0
500
1000
1.25
Sims Theory
1.20
ΘΠ
B
Ratio B NG
1.25
B
B G Ratio B NG B G
h, NG b 20
1.20
b 10
b 20
b 1.15
10
c 11 k
c 01 k
b 20 c 01
1.10
1.05
1.00
B NG B NG B G
0.95
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
B NG B NG 2 B G 2
0.2 0.4 0.6 0.8
ΘΠ
0.2 Ratio B NG 0.4 B G 0.6 0.8
ΘΠ
ES, Crocce & Desjacques (2011)
2
O f NL
Sims Theory
B NG B G
1000
500
0
500
1000
ΘΠ
B NG B NG 2 B G 2
ES, Crocce & Desjacques (2011)
k 1 k 3 Θ
0.2 0.4 0.6 0.8
Θ
ΘΠ
k 2

Halo Power Spectrum vs. Halo Bispectrum
10
Low mass halos:
8.8 10 12 Mh 1 M
1.6 10 13
z 0.5
f NL 100
10
High mass halos:
M 1.6 10 13 h 1 M
z 0.5
f NL 100
S N kmax
S N kmax
1
1
P m
P h
B m , matter
B h , halos
P m
P h
B m , matter
B h , halos
0.05 0.10 0.15 0.20 0.25
k max h Mpc 1
Cumulative signal-to-noise for the effect of NG initial conditions
on matter and galaxy correlators (P & B)
Sum of all configurations up to kmax
S
N
2
P
=
k
max
k
(P NG − P G ) 2
∆P 2 S
N
2
B
=
k
max
(B NG − B G ) 2
∆B 2
k 1 ,k 2 ,k 3
0.05 0.10 0.15 0.20 0.25
k max h Mpc 1
The cumulative NG
effect is comparable
at mildly nonlinear
scales

A Fisher matrix analysis for Galaxy correlators
The uncertainty on fNL (local)
from Power Spectrum &
Bispectrum (& both)
100
50
V 10 h 3 Gpc 3 , z 1
k min 0.009 h Mpc 1
b 1 2, b 2 0.8
fNL
20
marginalized (b1, b2)
10
5
P
B
PB
0.02 0.04 0.06 0.08 0.10
k max h Mpc 1
FIG. 15: One-σ uncertainty on the f NL parameter, mar

Conclusions
• The CMB is currently the best test of the Gaussianity of the Initial Conditions:
Planck will soon improve WMAP constraints by a factor of 4
• The impact of NG on nonlinear evolution of structure is significant,
particularly in terms of the matter bispectrum: can this be detected in
weak lensing surveys?
• Due to a scale-dependent correction to linear bias, galaxy power spectrum
measurements are currently providing constraints on fNL comparable to
WMAP’s, but for local NG only!? Quasi-Single Field inflation is the
perfect case study of initial bispectrum in the squeezed limit
• We do have a good understanding of the multiple effects of PNG on the
galaxy bispectrum at large scales (with room for improvement!)
• A complete analysis of the large-scale structure (e.g. galaxy power
spectrum and bispectrum) can do better than power spectrum alone:
smaller uncertainties on NG parameters for virtually any model of non-
Gaussianity