Features in the primordial spectra

Jinn-Ouk Gong

APCTP, Pohang 790-784, Korea

WHEPP XVIIIT-Kanpur, Kanpur, India

8th December, 2015

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Outline

1 Introduction

2 Origins of featuresModelling the sourcesA case of effective single field inflation

3 Features in the correlation functionsGeneral slow-roll spectraCorrelation of correlation functions

4 Inferred features from CMB

5 Conclusions

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Why inflation?

Hot big bang

Horizon problem

Flatness problem

Monopole problem

Initial perturbations

Inflation

Single causal patch

Locally flat

Diluted away

Quantum fluctuations

1 Initial conditions for hot big bang

2 A certain amount of expansion is required:

Number of e-folds : N = log

(ae

ai

)& 50−60 is necessary

3 Consistent with most recent observations, but...

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Why bothering about extra structure?

Observations seem to prefer SR inflation (with large fieldexcursion?)

L eff[φ] = L0[φ]︸ ︷︷ ︸slow-roll

+ ∑n

cnOn[φ]

Λn−4︸ ︷︷ ︸EFT allows these

Smooth structure over a few

Constructing EFT typically introduces sub-Planckian structure thatgenerally prevents long enough SR inflation without anything else

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Why bothering about extra structure?

Observations seem to prefer SR inflation (with large fieldexcursion?)

Leff[φ] = L0[φ]︸ ︷︷ ︸slow-roll

+ ∑n

cnOn[φ]

Λn−4︸ ︷︷ ︸EFT allows these

Smooth structure over a few

Structure beyond a cutoff

Constructing EFT typically introduces sub-Planckian structure thatgenerally prevents long enough SR inflation without anything else

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Why features in the primordial spectrum?

Intervening structure gives deviations from otherwise smoothevolution during inflationDeviations permeate throughout perturbation: features!Tantalizing observational hints: have we already seen?

2 10 500

1000

2000

3000

4000

5000

6000

D `[µ

K2 ]

90◦ 18◦

500 1000 1500 2000 2500

Multipole moment, `

1◦ 0.2◦ 0.1◦ 0.07◦Angular scale

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Why correlated features in the primordial spectra?

Effects of deviations permeate the mode function solution

They manifest themselves in the coupling of EFT expansion

All correlation functions are correlated

Q1: How are features possibly generated?

Q2: How do we compute the features in the spectra?

Q3: How are the spectra correlated quantitatively?

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Where could the devations occur from?

Quadratic action for the curvature perturbation R

S2 =∫

d4xa3m2Plε

[R2

c2s− (∇R)2

a2

]Deviations may occur in

1 background expansion (ε) and/or

2 speed of sound (cs)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Phenomenological modelling

Change in the parameter(s) can be modelled appropriately

Potential with features typically leads to violent change in ε

e.g. Starobinsky model V (φ) = V0

{1− [

A+θ(φ)∆A]φ

}(Starobinsky 1992)

step potential V (φ) = V0(φ)

[1+ c tanh

(φ−φ0

d

)](Adams, Cresswell & Easther 2001, Stewart 2002...)

Sound of speed can be modelled similarly

e.g. Gaussian profile c−2s −1 ∼ exp

[− (N −N0)2

2∆N2

]cosh profile c−2

s −1 ∼ cosh−2(

N −N0

∆N

)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Theoretical foundations: effective single field inflation

Multi-field inflation with hierarchies

Different from single field case only if the trajectory is curved

Decomposition of perturbation: along and pert to trajc.f. Gordon et al. 2000

!0a (t)

! aa (t, x) = !0a (t +! )+ Na (t +! )F

aN

background trajectory !0

a t +! (t, x)( )

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Effective single field theory: recipe

1 Write the action in terms of R (along traj) and F (off traj)

2 Integrate out F : eSeff[R] = ∫[DF ]eS[R,F ][

= equiv to plugging linear sol:(−2+M2

eff

)F =−2θ

(φ0/H

)R

]3 Effective single field action Seff[R]

4 ,

Effects of heavy physics in “speed of sound”

c−2s ≡ 1+ 4θ2

M2eff

(θ : angular velocity of traj

)

Single field theory with non-trivial c2s : footprint of heavy physics

(Achucarro et al. 2012a)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Digression: “effective field theory”

Truncation in 2/M2eff when feeding back the solution of F :

F = (−2+M2eff

)−1 −2θφ0

HR = 1

M2eff

(1+ 2

M2eff

+·· ·)−2θφ0

HR

(b)

φ1

φ2 η⊥

t

φ1

φ2 η⊥

t

φa0(t)

φa0(t)

κ

(a)

∆φ

(b)

φ1

φ2 η⊥

t

φ1

φ2 η⊥

t

φa0(t)

φa0(t)

κ

(a)

∆φ

φ1

φ2

φa0(t)

η⊥

tT⊥∆φ ∼ κ∆h

Valid for “adiabatic traj”: |2F |¿ M2eff |F |

(∣∣∣θ/θ∣∣∣¿ Meff

)→

Creation of heavy quanta suppressed (Achucarro et al. 2012b)

EFT remains valid even for c−2s = 1+ 4θ2

M2eff

À 1 (strong turn)

As Meff → H derivative terms become important (JG, Seo & Sypsas 2014)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

General slow-roll approximation

Curvature pert Rk(τ) = de Sitter piece + departure from dSTreating all corrections equally: no hierarchies bet SR para(Stewart 2002, Choe, JG and Stewart 2004)

Mode equation: z2 ≡ 2a2m2Plε, y ≡p

2kzRk, dx =≡−kcsdt/a, f ≡ 2πxz/k

d2y

dx2 +(1− 2

x2

)y︸ ︷︷ ︸

de Sitter solution

= 1

x2

f ′′−3f ′

f︸ ︷︷ ︸≡g(logx)

y

︸ ︷︷ ︸departure from dS

(f ′ ≡ df

d logx

)→ y0(x) =

(1+ i

x

)eix

Green’s function solution (JG & Stewart 2001)

y(x) =y0(x)+ i

2

∫ ∞

x

du

u2 g(logu)[y∗0 (u)y0(x)−y∗0 (x)y0(u)

]︸ ︷︷ ︸

≡L(x,u)

y(u)

=y0(x)+L(x,u)y0(u)+L(x,u)L(u,v)y0(v)+·· ·Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Spectra with GSR functions

Spectra can be written in terms of

source g(logτ) and window functions (power spectrum) and

source gB(logτ), window functions and power spectrum(bispectrum) (Adshead, Hu, Dvorkin & Peiris 2011, JG, Schalm & Shiu 2014)

PR (k) = limx→0

∣∣∣∣ xy

f

∣∣∣∣2= 1

f 2?

{1+ 2

3

f ′?f?

+ 2

3

∫ ∞0

dτ

τ[W (kτ)−θ(τ?−τ)]g(logτ)

}

BR (k1,k2,k3) = (2π)4

4

√PR (k1)

k21

√PR (k2)

k22

√PR (k3)

k23

∫ ∞0

dτ

τgB(logτ)

×{(

dτ−3f ′f

)WB + 1

3dτ (XB +XB3)

∫ ∞0

dτ

τg(log τ)X(k3τ)+2 perm

− 1

3dτWB3

∫ ∞τ

dτ

τg(log τ)W (k3τ)− 1

3dτXB3

∫ τ

0

dτ

τg(log τ)X(k3τ)+2 perm

− 1

2dτ (XB +XB3)

∫ ∞τ

dτ

τg(log τ)

(1

k3τ+ 1

k33 τ

3

)+2 perm

}

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Important notes regarding the GSR spectra

Window functions W (kτ), X(kτ), WB(k1,k2,k3;τ) etc areconstructed from the 0th order solution y0, e.g.

W (x) =−3

xℜ[y0(x)]ℑ[y0(x)] = 3sin(2x)

2x3− 3cos(2x)

x2− 3sin(2x)

2x

PR can be inverted to write f and g as (Joy, Stewart, JG & Lee 2005)

log

(1

f 2

)=

∫ ∞

0

dk

km(kτ)logPR

f ′f= 1

2

∫ ∞

0

dk

km(kτ)(nR −1) where m(x) = 2

π

[1

x− cos(2x)

x− sin(2x)

]g =−1

2

∫ ∞

0

dk

km(kτ)

[αR +3(nR −1)

]Or we can split S2 into free + deviations that source features

S2 = S2,free +S2,int → e.g.∆PR

PR(k) = k

∫ 0

−∞dτ

(1− 1

c2s

)sin(2kτ)

Another inversion method between PR and its source(Achucarro, JG, Palma & Patil 2013)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

View from effective field theory

In the low-energy regime with π=−R/H (Cheung et al. 2008)

Sπ =∫

d4xp−g

{m2

Pl

2R−m2

PlH

[π2 − (∇π)2

a2

]→ usual SR

+2M42

[π2 + π3 − π (∇π)2

a2

]− 4

3M4

3 π3 +·· ·

}→ departure

M4n is the expansion parameters of EFT and in principle

independent from each other

M42 is common to S2 and S3: explicit correlation between

power- and bispectra (Achucarro et al. 2013, JG, Schalm & Shiu 2014)

M43 manifests itself as an independent bispectrum source gB

(JG, Schalm & Shiu 2014)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

General structure of correlation

With changes in ε and/or cs being most important, gB

(equivalently M43 ) can be specified i.t.o. power spectrum

source (Palma 2015)

BR is completely fixed by PR , nR and αR (Achucarro et al. 2013)

Directly translated into templates as

fNL(k1,k2,k3) = 10

3

k1k2k3

k31 +k3

2 +k33

(k1k2k3)2BR

(2π)4P 2R

= c∆0 (k) logPR + c∆1 (k) (nR −1)+ c∆2 (k)αR

Applicable to any configuration by projection

Standard consistency relation in the squeezed limit

fNL −→k3→0

5

12(1−nR)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

A glance at the correlations

Comparison with direct calculation versus GSR reconstruction forsimple analytic models (JG, Schalm & Shiu 2014)

0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

DP

R�P

R

0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

-5

0

5

f NL

0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

-0.5

0.0

0.5

k1�k*

Df N

L

0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

-0.02

-0.01

0.00

0.01

0.02

DP

R�P

R0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

-4

-2

0

2

4

f NL

0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

-0.05

0.00

0.05

k1�k*

Df N

L

Starobinsky model (left) and top-hat speed of sound (right)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Searching for correlated features

There have been considerable efforts to construct PR from theCMB data during last decades (Hunt & Sarkar, Hazra, Shafieloo & Souradeep, Hamann,

Peiris & Verde, Bucher...) and tantalizing evidences from a number ofmultipole locations (`∼ 20...)

No direct reconstruction of the bispectrum from the CMB data(or is there any?)

Very few studies on the combined searches for 2- and 3-pointfunctions

Recent report from the Cambridge group (Fergusson, Gruetjen, Shellard &

Liguori 2015) reported no correlated features for 2- and 3-pointfunctions, under the assumption of featureless CMB

Joint constraints should place stronger constraints!

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Inferred deviations from CMB with Planck 2013

(Hazra, Shafieloo & Souradeep 2014)

0

1000

2000

3000

4000

5000

6000

ℓ(ℓ+

1) C ℓ

TT/2π

[µ

K2]

Planck data

Power-law best fit

Uniformly smoothed reconstructed spectrum

Error weighted reconstructed spectrum

-200

-100

0

100

200

2 10 20

∆[ℓ

(ℓ+

1) C ℓ

TT/2π

] [µ

K2]

ℓ50 500 1000 1500 2000 2500

There are 2σ deviations from perfect featureless power spectrum

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Inferred deviations from CMB with Planck 2013

(Hazra, Shafieloo & Souradeep 2014)

2e-09

3e-09

1e-09 1e-05 0.0001 0.001 0.01 0.1

PS(k

)

k in Mpc-1

Power law best fit

Uniformly smoothed PPS

Error weighted reconstructed spectrum

2e-09

2.5e-09

0.01 0.05 0.1 0.15

There are 2σ deviations from perfect featureless power spectrum

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Inferred deviations from CMB with Planck 2013

(Hazra, Shafieloo & Souradeep 2014)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.02 0.04 0.06 0.08 0.1 0.12 0.14

∆P

S(k

)/P

S(k

)

k in Mpc-1

α+a+b+1+2 [i=10, ∆=0.02]

There are 2σ deviations from perfect featureless power spectrum

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Where to look for in the equilateral limit

10-3 10-2

k (Mpc−1 )

−6

−4

−2

0

2

4

6

f nl (e

quila

tera

l)

Accuracy of GSR is slightly off by O (1)%, but the positions and(qualitative) amplitudes of oscillations are faithfully captured,

butwithin 95% CL featureless bispectrum is consistent with data, withoscillations being likely to represent noise. (Appleby, JG, Hazra, Shafieloo & Sypsas)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Where to look for in the equilateral limit

10-3 10-2 10-1

k (Mpc−1 )

200

150

100

50

0

50

100

150

200

fnl(k

) (eq

uila

tera

l)

0.010 0.012 0.014 0.0166040200

204060

0.020 0.022 0.024201510

505

101520

Accuracy of GSR is slightly off by O (1)%, but the positions and(qualitative) amplitudes of oscillations are faithfully captured, butwithin 95% CL featureless bispectrum is consistent with data, withoscillations being likely to represent noise. (Appleby, JG, Hazra, Shafieloo & Sypsas)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Where to look for in the squeezed limit

10-3 10-2

k (Mpc−1 )

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

f nl (s

quee

zed)

For the squeezed limit the GSR approximation is surprisinglyperfect,

and at k ∼ 0.1 Mpc−1 and especially at k ∼ 0.013 Mpc−1

there are marginal 2σ features. (Appleby, JG, Hazra, Shafieloo & Sypsas)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Where to look for in the squeezed limit

10-3 10-2 10-1

k (Mpc−1 )

8

6

4

2

0

2

4

6

8

fnl(k

) (sq

ueez

ed)

0.010 0.012 0.014 0.016432101234

0.09 0.10 0.110.200.150.100.050.000.050.100.150.20

For the squeezed limit the GSR approximation is surprisinglyperfect, and at k ∼ 0.1 Mpc−1 and especially at k ∼ 0.013 Mpc−1

there are marginal 2σ features. (Appleby, JG, Hazra, Shafieloo & Sypsas)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

What about general configurations?

For k1 = 0.1 Mpc−1, plotting fNL as a function of k2/k1 and k3/k1

0.0 0.2 0.4 0.6 0.8 1.0

k3/k1

0.5

0.6

0.7

0.8

0.9

1.0

k2/k

1

0.00

1.25

2.50

3.75

5.00

6.25

7.50

8.75

10.00

0.0 0.2 0.4 0.6 0.8 1.0

k3/k1

0.5

0.6

0.7

0.8

0.9

1.0

k2/k

1

−8

−7

−6

−5

−4

−3

−2

−1

0

Regions with f +2σNL − f fid

NL < 0 (black strips, left) and f −2σNL − f fid

NL > 0(white strips, right) are most distinctive in the squeezed limit.(Appleby, JG, Hazra, Shafieloo & Sypsas)

Features in the primordial spectra Jinn-Ouk Gong

Introduction Origins of features Features in the correlation functions Inferred features from CMB Conclusions

Conclusions

Features are most likely to arise from changes in ε and/or cs

These changes can be described byinflaton potential with steps/kink... (e.g. Starobinsky model)integrating out heavy d.o.f. (effective single field inflation)

GSR provides a conveninent analytic approach

Correlation functins are correlatedSource function of the bispectrum gB is formally independentSpecified by PR and its first 2 derivatives2σ deviations of fNL from 0 can be inferredPolarization spectra provide cleaner signals (under progress)

Features manifest themselves in all correlation functions ina specific correlated manner, which can serve as a usefuland maybe unique way for detecting higher order functions

Features in the primordial spectra Jinn-Ouk Gong