Just enough inﬂation: power spectrum modiﬁcations on large...

Francisco Gil Pedro !

!!

in collaboration with: Michele Cicoli, Bhaskar Dutta,

Sean Downes, and Alexander Westphal

Just enough inflation: power spectrum modifications

on large scales[1407.1048]

Outline:

Observational hints The standard inflationary picture The spectrum of slow-roll inflation Pre-inflation and the power spectrum Degeneracy in power loss Summary

Francisco Gil Pedro, Trieste, 8 July 2014

Observational hints


Observational hints


large r increases: -significance of power loss

-level of power loss

Tensor modes contribute to TT spectrum on large

scales

Cosmic varianceWe only have one sky

Measurements on largest scales are statistically limited[Planck XV]

Cosmic variance

[1309.4060]

CV can be decreased using LSS data

CV is the simplest explanation for low-

anomaly`


The standard pictureln aH

Ne

radiation era:

ln aH � �Ne

matter era:

ln aH � �12Ne

60 e-folds of slow-roll:

ln aH � +Ne

tet60Francisco Gil Pedro, Trieste, 8 July 2014


Ne

radiation era:

ln aH � �Ne

matter era:

ln aH � �12Ne


ln aH � +Ne

tet60

decays



Ne

radiation era:

ln aH � �Ne

matter era:

ln aH � �12Ne


ln aH � +Ne

tet60

decays freezes



Ne

radiation era:

ln aH � �Ne

matter era:

ln aH � �12Ne


ln aH � +Ne

tet60

decays freezes decays


The Mukhanov-Sasaki Equation

u00 +

✓k2 � z00

z

◆u = 0

z ⌘ ap2✏Hu ⌘ ⇣ zcurvature perturbation: ⇣ with

Useful to use efolds as ‘time’ coordinate

aH ⇠ e⇠NAssume background of the form:



u00 +

✓k2 � z00

z

◆u = 0

z ⌘ ap2✏Hu ⌘ ⇣ zcurvature perturbation: ⇣ with



⇠ = 1 +O (✏H , ⌘H)

inflation:



u00 +

✓k2 � z00

z

◆u = 0

z ⌘ ap2✏H

u↵↵ + ⇠ u↵ +

(✓k

aH

◆2

� (1 + ⇠)

)u = 0

u ⌘ ⇣ zcurvature perturbation: ⇣ with



⇠ = 1 +O (✏H , ⌘H)

inflation:


The Mukhanov-Sasaki EquationLhor

⌘ H�1

k < a H $ �phys > H�1

k > a H $ �phys < H�1 inside horizon, decays

outside horizon, freezes

⇣

⇣

Behaviour of curvature pert. depends on during inflation:


The Mukhanov-Sasaki EquationLhor

⌘ H�1

k < a H $ �phys > H�1

k > a H $ �phys < H�1 inside horizon, decays

outside horizon, freezes

⇣

⇣

Behaviour of curvature pert. depends on

u = C(1) 1p⇠aH

H(1)⌫

✓k

⇠aH

◆+ C(2) 1p

⇠aHH(2)

⌫

✓k

⇠aH

◆

set by ics

during inflation:


The spectrum of slow-roll inflation



deep inside horizon modes do not feel curvature



deep inside horizon modes do not feel curvature

⇠ = 1 +O (✏H , ⌘H)

flat space mode functions



ns = 0.9603± 0.0073

k⇤ = 0.002 Mpc�1

ln(1010As) = 3.089+0.024�0.027

Scale invariance is excluded by more than 5�

Planck XXII:

Pk = As

✓k

k⇤

◆ns�1


Pre-inflation and the power spectrum

?? What if inflation was short ??




theoretically well motivated:

Freivogel et al. 2007 McAllister et al. 2013




??

Modifies large scale part of spectrum

Can this leave an imprint on the power spectrum?

theoretically well motivated:

Freivogel et al. 2007 McAllister et al. 2013


ln aH

Ne

slow-roll inflationpre-inflation

?



ln aH

Ne


? k � a0H0



ln aH

Ne


? k � a0H0

k ⇠ a0H0



What pre-inflation?• Fast-roll

• Super inflation

• Climbing scalars

• Radiation domination

• Matter domination

• Curvature domination

⇠ = 2

Contaldi et al. 2003

Liu et al. 2013

Sagnotti et al. 2012/14

Nicholson&Contaldi 2007

Kinney&Powell 2008

⇠ = 0 Linde et al. 1998,…,2011


What pre-inflation?• Fast-roll

• Super inflation

• Climbing scalars

• Radiation domination

• Matter domination

• Curvature domination

⇠ = 2

Contaldi et al. 2003

Liu et al. 2013

Sagnotti et al. 2012/14

Nicholson&Contaldi 2007

Kinney&Powell 2008

What similarities and what differences?

⇠ = 0 Linde et al. 1998,…,2011


! = �1 + 2 ⇠

3

What pre-inflation?

⇠ > 1

0 < ⇠ < 1IIIlnaH

Ne

II

IV

�1/2 < ⇠ < 0

w < �1

�1 < w < �1/3

�1/3 < w < 0

w > 0

! ⇡ �1⇠ ⇡ 1

I

⇠ < �1/2 aH ⇠ e⇠N


What pre-inflation?

⇠ > 1

lnaH

Ne

IV

w < �1

w > 0

! ⇡ �1⇠ ⇡ 1

I

⇠ < �1/2

! = �1 + 2 ⇠

3

aH ⇠ e⇠N


Type I backgroundsln

aH

Ne

⇠ ⇠ 1

⌫ ⇠ 3/2⇠ < �1/2

⌫ < 3/2

I k � aH|0

k ⌧ aH|0

k ⇠ aH|0

Large scale spectrum from superhorizon modesDecelerated expansion: decreasesH


Type I backgroundsln

aH

Ne

⇠ ⇠ 1

⌫ ⇠ 3/2⇠ < �1/2

⌫ < 3/2

I k � aH|0

k ⌧ aH|0

k ⇠ aH|0

Large scale spectrum from superhorizon modeslarge scales set by pre-inflationary

vacuum

Decelerated expansion: decreasesH


Type I backgrounds

! Pk ⇠H2

inf

✏H

lnaH

Ne

⇠ ⇠ 1

⌫ ⇠ 3/2⇠ < �1/2

⌫ < 3/2

I k � aH|0

k ⌧ aH|0

k ⇠ aH|0


vacuum



Type I backgrounds

! Pk ⇠H2

inf

✏H

lnaH

Ne

⇠ ⇠ 1

⌫ ⇠ 3/2⇠ < �1/2

⌫ < 3/2

I k � aH|0

k ⌧ aH|0

k ⇠ aH|0


vacuum

frozen: decaying: with

⇠ < �2�2 < ⇠ < 0

ns > 1



Type I backgrounds

! Pk ⇠H2

inf

✏H

lnaH

Ne

⇠ ⇠ 1

⌫ ⇠ 3/2⇠ < �1/2

⌫ < 3/2

I k � aH|0

k ⌧ aH|0

k ⇠ aH|0


vacuum


⇠ < �2�2 < ⇠ < 0

ns > 1 frozen



Type I backgrounds

! Pk ⇠H2

inf

✏H

lnaH

Ne

⇠ ⇠ 1

⌫ ⇠ 3/2⇠ < �1/2

⌫ < 3/2

I k � aH|0

k ⌧ aH|0

k ⇠ aH|0


vacuum


⇠ < �2�2 < ⇠ < 0

ns > 1 frozen


! Pk ⇠ H2 ⇥✓

k

aH

◆ns�1

deep IR behaviour determined by :ns


Type I backgrounds

! Pk ⇠H2

inf

✏H

lnaH

Ne

⇠ ⇠ 1

⌫ ⇠ 3/2⇠ < �1/2

⌫ < 3/2

I k � aH|0

k ⌧ aH|0

k ⇠ aH|0


vacuum


⇠ < �2�2 < ⇠ < 0

ns > 1 frozen


! Pk ⇠ H2 ⇥✓

k

aH

◆ns�1

deep IR behaviour determined by :ns

IR power suppression

⌧1z }| {


Different peak amplitudeDifferent low-k fall-off

Same broad features

red green blue ⇠ = �1

⇠ = �2⇠ = �1/2 matter

fast-roll radiation

background:

0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

kphysêHinf

P k

Type I backgrounds


We see what we put in: choice of ICs determines power

on large scales

Type IV backgrounds

⇠ > 11

2< ⌫ <

3

2

lnaH

Ne

IV

⇠ ⇠ 1

⌫ ⇠ 3/2 k � aH|0

k ⌧ aH|0

k ⇠ aH|0

Large scale spectrum from pre-inf. subhorizon modesAccelerated expansion: is growingH H / e(⇠�1)Ne


Type IV backgrounds

! Pk ⇠H2

inf

✏H

⇠ > 11

2< ⌫ <

3

2

lnaH

Ne

IV

⇠ ⇠ 1

⌫ ⇠ 3/2 k � aH|0

k ⌧ aH|0

k ⇠ aH|0



Type IV backgrounds

! Pk ⇠H2

inf

✏H

⇠ > 11

2< ⌫ <

3

2

lnaH

Ne

IV

⇠ ⇠ 1

⌫ ⇠ 3/2 k � aH|0

k ⌧ aH|0

k ⇠ aH|0


frozen with ns > 1


Type IV backgrounds

! Pk ⇠H2

inf

✏H

⇠ > 11

2< ⌫ <

3

2

lnaH

Ne

IV

⇠ ⇠ 1

⌫ ⇠ 3/2 k � aH|0

k ⌧ aH|0

k ⇠ aH|0


frozenfrozen with ns > 1


Type IV backgrounds

! Pk ⇠H2

inf

✏H

⇠ > 11

2< ⌫ <

3

2

lnaH

Ne

IV

⇠ ⇠ 1

⌫ ⇠ 3/2 k � aH|0

k ⌧ aH|0

k ⇠ aH|0


! Pk ⇠ H2 ⇥✓

k

aH

◆ns�1



Type IV backgrounds

! Pk ⇠H2

inf

✏H

⇠ > 11

2< ⌫ <

3

2

lnaH

Ne

IV

⇠ ⇠ 1

⌫ ⇠ 3/2 k � aH|0

k ⌧ aH|0

k ⇠ aH|0


! Pk ⇠ H2 ⇥✓

k

aH

◆ns�1


Low power on large scales

|{z}⌧H2

inf|{z}⌧1


0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

kphysêHinf

P k

⇠ = 3/2⇠ = 2⇠ = 5/2

red green blue

!super-inflation

background:

Type IV backgrounds


0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

kphysêHinf

P k


Same broad features

⇠ = 3/2⇠ = 2⇠ = 5/2

red green blue

!super-inflation

background:

Type IV backgrounds


0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

kphysêHinf

P k


Same broad features

⇠ = 3/2⇠ = 2⇠ = 5/2

red green blue

!super-inflation

background:

Type IV backgrounds


More robust than Type I: Insensitive to choice of ICs

Pre-inflation and power loss

Degeneracy: one-parameter family of primordial spectraln

aH

Ne0.001 0.01 0.1 1 10 100 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

kphysêHinf

P k


Pre-inflation and power loss

Interesting hints:[1311.1599] & [1402.1418]

claim these spectra are better fits than simpler monotonic

parametrization 0.001 0.01 0.1 1 10 100 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

kphysêHdS

P Q

Degeneracy: one-parameter family of primordial spectraln

aH

Ne0.001 0.01 0.1 1 10 100 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

kphysêHinf

P k


TT Power spectrum

5 10 50 100 500 1000

1000

2000

3000

4000

5000

{

{H{+

1L

2pC {@mK

2 D

10-5 10-4 0.001 0.01 0.10

5. ¥ 10-10

1. ¥ 10-9

1.5 ¥ 10-9

2. ¥ 10-9

2.5 ¥ 10-9

3. ¥ 10-9

k in MPc-1

PHkL

�

�(�+

1)2�

C�[µ

K2]

PQ(k)

k in MPc�1


Summary

Persistent hints from COBE / WMAP / PLANCK !Short inflation modifies large scale/low- power spectrum

!Different ways to reduce power

!Degeneracy/universality in power loss !Better understanding requires fit to the data

!We might be seeing pre-inflationary phase

`


Thank you


[1407.1048]

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