squeezing the cmb bispectrumbenasque.org/2012cosmology/talks_contr/232_vernizzi.pdf · squeezing...

With L. Boubekeur, P. Creminelli, G. D’Amico, J. Noreña,(arXiv:0806.1016, arXiv:0906.0980)

and P. Creminelli, C. Pitrou (arXiv:1109.1822)

Filippo Vernizzi - IPhT, CEA Saclay

Benasque - 23 August 2012

Squeezing the CMB bispectrum

Related talks: Antony Lewis, Guido Pettinari, Takahiro Tanaka, ...

Credits: ESA (animation by C. Carreau)

�∆T

T(n)

∆T

T(n�)� Cl ≡

�

m

�almal�m��δll�δmm�⇒If CMB is Gaussian it is fully characterized by the angular 2-point function or power spectrum (~103 numbers):

∆T

T(n) =

�

lm

almYlm(n)Spherical harmonic expansion of the temperature fluctuations:

⇒�∆T

T(n1)

∆T

T(n2)

∆T

T(n3)�

A CMB map contains ~106 pixels. Information could be hidden in higher-order statistics. Bispectrum: angular 3-point function

Credit : ESA/D. Ducros

WMAP

Planck

�al1m1al2m2al3m3� = Gm1m2m3l1l2l3

bl1l2l3

Curvature fluctuations of a constant inflaton field hypersurface:

Standard picture

gij = a2(t)e2ζδij

φ = φ(t)

Each curvature/inflaton Fourier mode behaves as an harmonic oscillator with time dependent mass

m =k

a(t)

S =1

2

�dtd

3xa

3 φ2

H2

�ζ2 − 1

a2(∂ζ)2

�

initial potential final potentialH ≡ a

a

S =

�d4x

√−g

�M2

p

2R− 1

2(∂φ)2 − V (φ)

�


Standard picture

gij = a2(t)e2ζδij

φ = φ(t)S =

1

2

�dtd

3xa

3 φ2

H2

�ζ2 − 1

a2(∂ζ)2

�

S =

�d4x

√−g

�M2

p

2R− 1

2(∂φ)2 − V (φ)

�

Each mode exits the Hubble radius, , and get frozen out the horizon with an almost scale-invariant spectrum:

These modes re-enter the horizon and we observe them today...

k/a(t) � H

�ζ�kζ�k�� = (2π)3δ(�k + �k�)Pζ(k)

ns − 1 � −0.04Pζ(k) =1

2k3H

4

φ2

��k=aH

=Aζ

k3

�k

k∗

�ns−1


Standard picture

gij = a2(t)e2ζδij

φ = φ(t)S =

1

2

�dtd

3xa

3 φ2

H2

�ζ2 − 1

a2(∂ζ)2

�

S =

�d4x

√−g

�M2

p

2R− 1

2(∂φ)2 − V (φ)

�

Each mode exits the Hubble radius, , and get frozen out the horizon with an almost scale-invariant spectrum:

k/a(t) � H

Inflation Standard evolution ln a

ln (aH)−1

k−1

k

aH� 1

k

aH� 1

Recombination

Beyond GaussianityAre there correlations between these modes?

S =

�d4xa

3 φ2

2H2

�ζ2 − 1

a2(∂ζ)2 +

2

H

∂i

∂2ζ∂iζ

∂2

a2ζ + . . .

�Gravity induces interactions which are suppressed by slow-roll:

�ζ�k1ζ�k2

ζ�k3� = (2π)3δ(�k1 + �k2 + �k3)Fζ(k1, k2, k3)

Present bounds consistent with this small signal. Any modification from this standard picture enhances non-Gaussianity:➡ Modified inflaton Lagrangian: higher-derivative terms, DBI, ghost inflation, etc...

➡ Multi-field inflation; curvaton or varying decay rate reheating

➡ Alternative to inflation

�ζζζ��ζζ� 3

2

∼ 0.01× 10−5!!⇒

potentially contains a wealth of information about the source of perturbations

�k1

�k2

�k3

Fζ(k1, k2, k3) ∼ O(�, η)P (k1)P (k2) + . . .

Fζ(k1, k2, k3)

Squeezed limitkL ≡ k1 � kS ≡ k2 ∼ k3

Maldacena ’02

�k1�k2

�k3

Maldacena’s consistency relation - for any single-field mode, in the squeezed limit:

�ζ�kLζ�kS

ζ−�kS� = −(ns − 1)P (kL)P (kS)


Maldacena ’02

�k1�k2

�k3

�ζ�kLζ�kS

ζ−�kS� = −(ns − 1)P (kL)P (kS)

The effect of the long mode translates into a rescaling of the momenta:

⇒ k = ke−ζLgijdxidxj = a2(t)e2ζL(�x)d�x2 = a2(t)d�x2

kS(xA) = kSe−ζL(xA) kS(xB) = kSe

−ζL(xB)

Flat spectrum ns-1=0

H−1

�ζ�kLζ�kS

ζ−�kS� = �ζ�kL

�ζ�kSζ−�kS

�ζL� ≈ �ζ�kLP (kS)� = −d ln(k3SPkS )

d ln kSP (kS)P (kL)



Maldacena ’02

�k1�k2

�k3

�ζ�kLζ�kS

ζ−�kS� = −(ns − 1)P (kL)P (kS)




−ζL(xB)

Red spectrum ns-1<0

H−1

�ζ�kLζ�kS




d ln kSP (kS)P (kL)



Maldacena ’02

�k1�k2

�k3

�ζ�kLζ�kS

ζ−�kS� = −(ns − 1)P (kL)P (kS)




−ζL(xB)

Blue spectrum ns-1>0

H−1

�ζ�kLζ�kS




d ln kSP (kS)P (kL)

All single-field models predict negligible NG in the squeezed limit: a detection of local NG rules out all single-field models!


CMB non-GaussianityEven in absence of primordial non-Gaussianity, , the CMB is non-Gaussian! �ζ�k1

ζ�k2ζ�k3

� = 0

2nd-order effects induce NG:

• at recombination: 2nd-order perturbations in the fluid + GR nonlinearities;• late time: ISW-lensing (extensively studied in the literature).

D[δ(1)] = 0

D[δ(2)] = S[δ(1)2]δ = δ(1) + δ(2) ⇒

All these effects are order ~few: important to interpret Planck data!

⇒ fNL ∼ �δ(2)δ(1)δ(1)��δ(1)δ(1)�2

∼ few

⇒Nonlinear relation

ζ δT

T


ln (aH)−1

k−1

k

aH� 1

k

aH� 1

Recombination

http://www2.iap.fr/users/pitrou/cmbquick.htm

Recent progressComplete Boltzmann calculation at 2nd-order is extremely challenging. Complicated non-linearities in baryon-photon fluid & from GR.

➡ Small-scale growth of nonlinear perturbations in the dark matter fluid:

δT

T=

δTrec

Trec+ Φ

➡ Perturbed recombination, free electron density (100<l<3000):

Bernardeau, Pitrou, Uzan ’08; Bartolo, Riotto ’08

δne

ne≈ ne

neδt ≈ 5

δnb

nb

Senatore, Tassev, Zaldarriaga ’09; Khatri, Wandelt ’09

• Sub-Hubble scales at recombination:

• All scales, numerical Boltzmann code:

➡ 2nd-order perts from product of 1st-order (not gauge invariant!):

➡ Full Boltzmann code of Cyril Pitrou: f localNL ≈ 5

f localNL ≈ 1

Nitta et al, ’09

Pitrou, Uzan, Bernardeau, ’08, ’09, ’10; Beneke, Fidler, ’10

⇒ f equilNL ∼ O(10)

• Super-Hubble scales ⇒ 2nd-order GR-only problem:

Bartolo, Matarrese, Riotto ’04;Boubekeur et al, ’09

f localNL = −(

1

6+ cos 2θ) (cos θ = llong · lshort)

⇒ f localNL ∼ −3.5/− 1



Particular squeezed limit

at recombinationH−1

One of the angles must subtend a scale longer than Hubble radius at recombination (but smaller than Hubble radius today):


ln (aH)−1

k−1

k

aH� 1

k

aH� 1

Recombination

101 102 1030

1000

2000

3000

4000

5000

6000

7000

!/3ISW

Total

Doppler

!!

4

l

l(l+

1)C

l/2!

µK

2

l2Cl � const

at recombinationH−1

Cl �AT

l2

�l

l∗

�ns−1

Sachs-Wolfe effect:

δT

T=

1

3Φ(�xrec) = −1

5ζ(�xrec) ⇒

l(l+1)C

l/2π

(µK

2)

l

A. Lewis ‘12

One of the angles must subtend a scale longer than Hubble radius at recombination (but smaller than Hubble radius today):

l � 60

Bl1l2l3 , l1 � l2 � l3 & l1 � 200For the bispectrum:

Particular squeezed limit

friend 1

�kL

ClS

Physical argument

LSS

Creminelli, Zaldarriaga, ’04

friend 2

Two friends at different positions receive photons a bit after recombination at the same physical temperature or Hubble time. The long mode is out of their horizon: they will see the same CMB anisotropies.

long wavelength mode

friend 1

�kL

ClSLSS

friend 2

me now

Rescaling of spatial coords ⇒ rescaling of angles:

The long mode is inside my horizon. Have to take into account of the modulation due to it. This induces a bispectrum.

ClS → ClS +ΘLd

dΘLClS Θ ≡ ∆T/T

ClS → ClS −ΘL(5n ·∇nClS )

BlLlSlS = �ΘLClS � = 5ClL

l2S

d(l2SClS )

d ln lSLong mode changes the local average temperature: BlLlSlS = 2ClLClS

(ΘL = −1

5ζ)

long wavelength mode

f locNL = −1

6Lensing close to last scattering displaces the 2-p function

2nd-order evolution as a coord change Maldacena ’02; Weinberg ’03; Fitzpatrick et al. ’09

Locally, possible to rewrite a perturbed FRW metric as an unperturbed one by reabsorbing the long mode with a coordinate transformation. Ex, in matter dominance:

ds2 = a2(η)�−(1 + 2ΦL)dη

2 + (1− 2ΦL)dx2�

ds2 = a2(η)�−dη2 + dx2

�

η = η(1 + ΦL/3)

xi = xi(1− 5ΦL/3)

FRWH

−1

(ζ = −5

3ΦL)

⇒

Conversely, start from a perturbed metric at 1st-order and “generate” 2nd-order couplings between short and long modes by the inverse coordinate transformation:

ds2 = a2(η)�−(1 + 2ΦS)dη

2 + (1− 2ΨS)dx2�

ds2 = a2(η)�−e2Φdη2 + e2Ψdx2

�⇒

Φ = ΦS + ΦL +1

3ΦL

∂ΦS

∂ ln η− 5

3ΦLx

i ∂ΦS

∂xi

2nd-order evolution as a coord change

0.1 0.2 0.5 1.0 2.0 5.0�4

�3

�2

�1

0

1

Η�Ηeq

�

kL�0.01�Ηeq, kS�10�Ηeq, cos Θ�0.5

eq. �49�Einstein's eqs.

0.1 0.2 0.5 1.0 2.0 5.0�600

�400

�200

0

200

400

600

Η�Ηeq

∆ rad

kL�0.01�Ηeq, kS�10�Ηeq, cos Θ�0.5

eq. �50�Einstein's eqs.

Solution of Einstein’s eqs. at 2nd order

Coordinate transformation:Φ(2)

|�kS+�kL|

δ(2)r,|�kS+�kL|

ΦkS → ΦkS − 5

3ΦkL

�f(η)

∂ΦkS

∂ ln η− ∂ΦkS

∂ ln k

�

δkS → δkS − 5

3ΦkL

�f(η)

∂δkS

∂ ln η− ∂δkS

∂ ln k

�

Example: radiation-to-matter transition:

f(η) = −20 + 15αη + 3α2η2

5(2 + αη)

α = (√2− 1)/ηeq

Coordinate transformation on the CMBApply this coordinate transformation to the observed CMB: Θobs(n) =

Tobs(n)− �Tobs��Tobs�

In pure matter dominance and instantaneous recombination

Θobs = [Θ+ Φ− n · �v](ηrec, �xrec)

Coordinate transformation on the CMBApply this coordinate transformation to the observed CMB: Θobs(n) =

Tobs(n)− �Tobs��Tobs�

Θobs = Θobs,S +Θobs,L +Θobs,L

�1 +

∂

∂ ln ηrec− 5n ·∇n

�Θobs,S

Θobs,S = [ΘS + ΦS − n · �vS ](ηrec, �xrec) Θobs,L =1

3ΦL(ηrec, �xrec)

Time derivative is geometrically suppressed as

Holds also when including radiation/matter transition (early Sachs-Wolfe) and finite recombination.

Check: a mode out of Hubble radius today is unobservable. Cancels out from this expression.

∼ ηrecηobs

BlLlSlS = ClLClS

�2 + 5

d ln(l2SClS )

d ln lS

�Bispectrum:

Extension of the Maldacena relation. Cf: BkLkSkS = −PkLPkS

d ln(k3SPkS )

d ln kS

Boltzmann code: CMBquick

Bartolo, Matarrese, Riotto ’06; Bernardeau, Pitrou, Uzan ’08; Pitrou ’08; Bartolo, Riotto ’08; Khatri, Wandelt ’08; Senatore, Tassev, Zaldarriaga ’09; Nitta et al. ’09, Beneke and Fidler ’10

There have been several contributions to development of Boltzmann numerical code at 2nd order.

One of the most complete code is Pitrou’s CMBquick (http://www2.iap.fr/users/pitrou/).

This relation can be used as consistency check of Boltzmann codes based on a physical limit

CMBquick

http://www2.iap.fr/users/pitrou/


Boltzmann code: CMBquick

Bartolo, Matarrese, Riotto ’06; Bernardeau, Pitrou, Uzan ’08; Pitrou ’08; Bartolo, Riotto ’08; Khatri, Wandelt ’08; Senatore, Tassev, Zaldarriaga ’09; Nitta et al. ’09, Beneke and Fidler ’10

There have been several contributions to development of Boltzmann numerical code at 2nd order.

One of the most complete code is Pitrou’s CMBquick (http://www2.iap.fr/users/pitrou/).

This relation can be used as consistency check of Boltzmann codes based on a physical limit

50 100 200 500 1000 2000�50

�40

�30

�20

�10

0

10

20

l

B 20,l,l

l1�20, l2�l3�l

eq. �42�CMBquick

eq. abovesqueezed

50 100 200 500 1000 2000�50

�40

�30

�20

�10

0

10

20

l

B l�10,l,

ll1�l�10, l2�l3�l

squeezed

The check is nontrivial! Even though analytically the squeezed limit is easy, in the code all 2nd-order effects must conspire to reproduce the simple analytical formula.

BlLlSlS = ClLClS

�2 + 5

d ln(l2SClS )

d ln lS

�



Final result

BlLlSlS = ClLClS

�2 + 5

d ln(l2SClS )

d ln lS

�+ 6ClLClS

�2 cos 2θ − (1 + cos 2θ)

d ln(l2SClS )

d ln lS

�

Lensing due to correlation between temperature at recombination and transverse displacement:

Θlensedobs = Θobs,S(ηrec, �x∗ + δ�x⊥) = Θobs,S(η∗, �x∗) + δ�x⊥ · �∇Θobs,S(η∗, �x∗)

Coordinate and average temperature redefinition Lensing

(cos θ = lL · lS)

Boubekeur et al. ’09

δ�x⊥ = −2

� η0

ηrec

(1− ηrecη

)�∇nΦ(�x)dη

Final result

BlLlSlS = ClLClS

�2 + 5

d ln(l2SClS )

d ln lS

�+ 6ClLClS

�2 cos 2θ − (1 + cos 2θ)

d ln(l2SClS )

d ln lS

�Coordinate and average temperature redefinition Lensing

BlLlSlS = ClLClS (1 + 6 cos 2θ)

�2− d ln(l2SClS )

d ln lS

�

The bispectrum vanishes for a white noise spectrum ClS ∝ const

We expect this for anisotropic lensing: shear just remaps the points preserving the area, so white noise remains the same.

We do not have a simple argument for the cancellation of the isotropic part. However, the above equation is correct only at lowest order in the scale invariance of the long mode.

Zaldarriaga, ’99

Partial cancellation between (isotropic) lensing convergence and space redefinition. Overdense regions (negative potential) give positive convergence, moving the spectrum towards larger angles, while coordinate redefinition shrink it.

Combining the two contributions:

Contamination

5 10 50 100 500 1000�1.2

�1.0

�0.8

�0.6

�0.4

�0.2

0.0

0.2

l

f NLloc�cos2Θ

�0�

We can define the contamination to as:f locNL

f locNL =

��d2l2d2l3(2π)2

[Bloc(l1, l2, l3)]2

6Cl1Cl2Cl3

�−1

·�

d2l2d2l3(2π)2

Bloc(l1, l2, l3)BX(l1, l2, l3)

6Cl1Cl2Cl3

f locNL = −0.39 , lmax = 2000Integrating only in the squeezed limit (100<l<lmax):

Negligible contamination to Planck.

f localNL = −(

1

6+ cos 2θ)

�1− 1

2

d ln(l2SClS )

d ln lS

�

In the limit of scale invariance: NG computed in Boubekeur et al, ’09

See also Bartolo, Riotto ’11

Observability

100 1000500200 2000300 3000150 15007000.001

0.005

0.010

0.050

0.100

0.500

1.000

lmax

S�N

eq. �44�local signal fNL�1

�S

N

�2

=1

π

�d2l2d2l3(2π)2

[B(l1, l2, l3)]2

6Cl1Cl2Cl3

Can we observe this signal? Signal-to-noise ratio is:

∝ l2max

∝ lmax

We are integrating only in the squeezed limit and we do not include polarization. Boltzmann code would give a better estimate. Possibly measurable effect.



d ln lS

�

Conclusion

Valid for adiabatic (single clock) perturbations. Already takes into account NG from single-field models. It is a consistency relation on the observable (CMB temperature) in the squeezed limit.

In the squeezed limit (one mode longer than horizon at recombination), it is possible to compute the CMB bispectrum exactly.



d ln lS

�

➡ Improve numerical reliability of the code and compute contamination (G. Pettinari’s talk).

• Planck will not be biased by 2nd-order effects at recombination (detectability?). Late time effects important (A. Lewis’ talk).

f localNL ≈ 5 f local

NL ≈ 3The value of the contamination by Pitrou may be overestimated:

• Is it compatible with these results? Yes if large signal comes from lL>100.• Is it compatible with Senatore & Zaldarriaga’s results fNL ~ -3.5?

• Test Boltzmann codes at 2nd order. Reasonable agreement with Pitrou’s CMBquick code. Code reliable in the regime that we studied.

squeezing the cmb bispectrumbenasque.org/2012cosmology/talks_contr/232_vernizzi.pdf · squeezing...

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