Non-Gaussian perturbations from mixed inflaton-curvaton scenario

Non-Gaussian perturbations from mixed inflaton-curvaton scenario

José Fonseca - University of PortsmouthBased on a paper with David Wands arxiv:1101.1254/Phys. Rev. D 83, 064025 (2011) and current work

13-Feb at AIMS

Quarta-feira, 15 de Fevereiro de 12

The Plan• Motivation;

• Perturbations from inflation in a nutshell;

• Curvaton Scenario;

• Mixed perturbations;

• Constraints on curvaton dominated theories;

• Including inflation perturbations;

• Summary


This is a pic of the Cosmic Microwave Background, aka CMB!

E. Komatsu et al, Seven-year WMAP Observations: Cosmological Interpretation - arXiv:1001.4538v1

P⇣(k) = �2⇣(k0)

✓k

k0

◆ns(k0)�1

Power Spectrum of primordial density

perturbations

k0 = 0.002Mpc�1

�2�(k0) = (2.430+0.091

�0.091)⇥ 10�9

ns(k0) = 0.968+0.012�0.012


This is a pic of the Cosmic Microwave Background, aka CMB!

E. Komatsu et al, Seven-year WMAP Observations: Cosmological Interpretation - arXiv:1001.4538v1

⇣ = ⇣1 +35fnl(⇣2

1 � h⇣21 i)

Bispectrum from quadratic corrections

�10 < fnl < 74

S i ng l e fie l d i nfla t ion i s perfectly fine but...

P⇣(k) = �2⇣(k0)

✓k

k0

◆ns(k0)�1

Power Spectrum of primordial density

perturbations

k0 = 0.002Mpc�1

�2�(k0) = (2.430+0.091

�0.091)⇥ 10�9

ns(k0) = 0.968+0.012�0.012


Perturbations from inflation in a nutshell


PERTURBATIONS FROM INFLATION IN A NUTSHELLSTANDARD INFLATION = INFLATON + SLOW ROLL

INFLATON

SCALAR FIELD LIVING IN A FRW UNIVERSE THAT DRIVES INFLATION.

3H�̇ ⇥ �V�

H2 ' V (�)

3m2Pl


PERTURBATIONS FROM INFLATION IN A NUTSHELLSTANDARD INFLATION = INFLATON + SLOW ROLL

INFLATON

SCALAR FIELD LIVING IN A FRW UNIVERSE THAT DRIVES INFLATION.

3H�̇ ⇥ �V�

H2 ' V (�)

3m2Pl

SLOW ROLLTHE FIELD HAS AN OVER-DAMPED EVOLUTION, I.E., IT ROLLS DOWN THE POTENTIAL SLOWLY.

THE EXPANSION IS ALMOST EXPONENTIAL.

KINETIC ENERGY DOES NOT VARY WITHIN 1 HUBBLE TIME.

THE POTENTIAL NEEDS TO BE FLAT.

✏� ⌘ 1

2m2

Pl

✓V�

V

◆2

⌧ 1

|⌘��| ⌘��m

2Pl

V��

V

�� ⌧ 1


PERTURBATIONS FROM INFLATION IN A NUTSHELLPOWER SPECTRUM OF PERTURBATIONS FOR A MASSLESS FIELD DURING INFLATION

SPLIT QUANTITIES BETWEEN BACKGROUND AND 1ST ORDER PERTURBATION (GAUSSIAN VACUUM FLUCTUATIONS)

⇥(t, x) = ⇥(t) + �⇥(t, x)

POWER SPECTRUM OF FIELD PERTURBATIONS AT HORIZON EXIT

P�⇥� �⇥

H�2�

⇤2��k=aH�

EQ. OF MOTION FOR THE FIELD PERTURBATIONS (IN FOURIER SPACE) FOR A “MASSLESS” FIELD.

¨�⇥k + 3H ˙�⇥k +

✓k2

a2+m2

�

◆�⇥k = 0


PERTURBATIONS FROM INFLATION IN A NUTSHELLDELTA N FORMALISM AND THE SEPARATE UNIVERSE PICTURE

ba

t1

t2

0h

-1cH

h

sh

Wands et al., astro-ph/0003278v2

THE “SEPARATE UNIVERSE” PICTURE SAYS 2 SUPER-HORIZON REGIONS OF THE UNIVERSE EVOLVE AS IF THEY WERE SEPARATE FRIEDMANN-ROBERTSON-WA L K E R U N I V E R S E S W H I C H A R E L O C A L LY HOMOGENEOUS BUT MAY HAVE DIFFERENT DENSITIES AND PRESSURE.

THE CURVATURE PERTURBATION ZETA IS THEN GIVEN BY THE DIFFERENCE OF THE INTEGRATED EXPANSION FROM A SPATIALY-FLAT HYPERSURFACE TO A UNIFORM-DENSITY HYPERSURFACE .DIFFERENT PATCHES OF THE UNIVERSE WILL HAVE DIFFERENT EXPANSION HISTORIES DUE TO DIFFERENT INITIAL CONDITIONS

⇥ = �N

�N = N 0�⇤⇤ +1

2N 00�⇤2

⇤


PERTURBATIONS FROM INFLATION IN A NUTSHELLOBSERVATIONAL PREDICTIONS FOR SINGLE FIELD

POWER SPECTRUM AND SCALE DEPENDENCE

RUNNING �s ⇥ �24⇥2� + 16⇥�⇤� � 2⌅2�

TILT ns � 1 ⇥ �6�� + 2⇥�

POWER SPECTRUM OF CURVATURE PERTURBATIONS WHICH REMAINS CONSTANT FOR ADIABATIC PERTURBATIONS ON LARGE SCALES

P⇣�⇤ ' 1

2✏⇤

✓H⇤

2⇡mPl

◆2

n⇣ � 1 ⌘ d lnP⇣

d ln k

↵⇣ ⌘ dn⇣

d ln k




RUNNING �s ⇥ �24⇥2� + 16⇥�⇤� � 2⌅2�

TILT ns � 1 ⇥ �6�� + 2⇥�


P⇣�⇤ ' 1

2✏⇤

✓H⇤

2⇡mPl

◆2

n⇣ � 1 ⌘ d lnP⇣

d ln k

↵⇣ ⌘ dn⇣

d ln k

rT ⌘ PG/P⇣

GRAVITATIONAL WAVES

TENSOR-TO-SCALAR RATIO rT =2

m2PlP⇣

✓H⇤2⇡

◆2




RUNNING �s ⇥ �24⇥2� + 16⇥�⇤� � 2⌅2�

TILT ns � 1 ⇥ �6�� + 2⇥�


NON-GAUSSIANITYCONSERVED CURVATURE PERTURBATION REMAINS GAUSSIAN fnl =

5

6(2�⇤ � ⇥��) ⌧ 1

P⇣�⇤ ' 1

2✏⇤

✓H⇤

2⇡mPl

◆2

n⇣ � 1 ⌘ d lnP⇣

d ln k

↵⇣ ⌘ dn⇣

d ln k

rT ⌘ PG/P⇣

GRAVITATIONAL WAVES

TENSOR-TO-SCALAR RATIO rT =2

m2PlP⇣

✓H⇤2⇡

◆2


Curvaton Scenario


CURVATON SCENARIOMAIN PRINCIPLES

IT IS AN INFLATIONARY MODEL. THE INFLATON DRIVES THE ACCELERATED EXPANSION WHILE THE CURVATON PRODUCES THE STRUCTURE IN THE UNIVERSE.

Lyth&Wands: hep-th/0110002Enqvist&Sloth: hep-ph/0109214Moroi&Takahashi: hep-ph/0110096

H⇤ > m� > ��THE CURVATON IS A LIGHT FIELD DURING INFLATION, WEAKLY COUPLED AND LATE DECAYING, I.E., DECAYS INTO RADIATION AFTER INFLATION.


CURVATON SCENARIOMAIN PRINCIPLES

IT IS AN INFLATIONARY MODEL. THE INFLATON DRIVES THE ACCELERATED EXPANSION WHILE THE CURVATON PRODUCES THE STRUCTURE IN THE UNIVERSE.

Lyth&Wands: hep-th/0110002Enqvist&Sloth: hep-ph/0109214Moroi&Takahashi: hep-ph/0110096

H⇤ > m� > ��THE CURVATON IS A LIGHT FIELD DURING INFLATION, WEAKLY COUPLED AND LATE DECAYING, I.E., DECAYS INTO RADIATION AFTER INFLATION.

DURING INFLATIONSUBDOMINANT COMPONENT �� ⌧ 1SINCE IT IS EFFECTIVELLY MASSLESS IT IS IN AN OVER-DAMPED REGIME. THEREFORE OBEYS TO THE SLOW-ROLL CONDITIONS

�� ⌧ 1 , ⇥�� ⌧ 1

AND ACQUIRES A SPECTRUM OF GAUSSIAN FIELD PERTURBATIONS AT HORIZON EXIT P��

✓H⇤2�

◆2


CURVATON SCENARIOMAIN IDEAS

AFTER INFLATION

THE CURVATON IS AN ENTROPY DIRECTION, SO THE CURVATURE PERTURBATION ON UNIFORM DENSITY HYPER-SURFACES IS NO LONGER CONSERVED

⇣̇ 6= 0



AFTER INFLATION


⇣̇ 6= 0

THE FIELD STARTS COHERENT OSCILLATIONS IN THE BOTTOM OF THE POTENTIAL AND BEHAVES LIKE A MATTER FLUID.

H ' m�



AFTER INFLATION


⇣̇ 6= 0

THE FIELD STARTS COHERENT OSCILLATIONS IN THE BOTTOM OF THE POTENTIAL AND BEHAVES LIKE A MATTER FLUID.

H ' m�

DECAYS INTO RADIATION AND TRANSFERS ITS PERTURBATIONS H ' ��


MIXED PERTURBATIONSTRANSFER OF LINEAR PERTURBATIONS

DURING AND AFTER INFLATION, THE CURVATON IS AN ENTROPY PERTURBATION S� ⌘ 3 (⇣� � ⇣�)

ZETA IS THE CURVATURE PERTURBATION ON UNIFORM DENSITY HYPERSURFACES ⇣ = � � H�⇢

⇢̇




ZETA IS THE CURVATURE PERTURBATION ON UNIFORM DENSITY HYPERSURFACES

LOCAL CURVATON ENERGY DENSITY ⇢�

= ⇢̄�

eS� = m2�

�2osc

⇣ = � � H�⇢

⇢̇






= ⇢̄�

eS� = m2�

�2osc

EXPAND LOCAL VALUE OF THE FIELD DURING OSCILLATION IN TERMS OF ITS VEV AND FIELD FLUCTUATIONS DURING INFLATION

�osc

' g + g0��⇤ +1

2g00��2

⇤

�osc

⌘ g(�⇤)G ACCOUNTS FOR NON-LINEAR EVOLUTION OF CHI

⇣ = � � H�⇢

⇢̇






= ⇢̄�

eS� = m2�

�2osc

EXPAND LOCAL VALUE OF THE FIELD DURING OSCILLATION IN TERMS OF ITS VEV AND FIELD FLUCTUATIONS DURING INFLATION

�osc

' g + g0��⇤ +1

2g00��2

⇤

�osc

⌘ g(�⇤)G ACCOUNTS FOR NON-LINEAR EVOLUTION OF CHI

THE ENTROPY PERTURBATION IS S� = SG +

1

4

✓gg00

g02� 1

◆S2G

SG ⌘ 2g0

g��⇤

⇣ = � � H�⇢

⇢̇



THE FINAL CURVATON POWER SPECTRUM COMES FROM THE MODES EXCITED DURING INFLATION AND FROM NON-LINEAR EVOLUTION OF THE FIELD UNTIL DECAY.

PS� = 4

✓g0

g

H⇤2⇡

◆2




PS� = 4

✓g0

g

H⇤2⇡

◆2

AT DECAY ON UNIFORM TOTAL ENERGY DENSITY HYPERSURFACES WE HAVE

⇥(1� ⌦�) e

4(⇣��⇣)+

+⌦�e3(⇣��⇣)

⇤= 1⇢ = ⇢r + ⇢�




PS� = 4

✓g0

g

H⇤2⇡

◆2

AFTER THE DECAY ZETA IS CONSERVED ON SUPER-HORIZON SCALES. WE DEFINE R AS THE TRANSFER EFFICIENCY AT DECAY.

P⇣ = P⇣� +R2

�

9PS�


⇥(1� ⌦�) e

4(⇣��⇣)+

+⌦�e3(⇣��⇣)

⇤= 1⇢ = ⇢r + ⇢�



FOR A SUDDEN DECAY APPROXIMATION R�,dec =3��

4� ��

��dec


PS� = 4

✓g0

g

H⇤2⇡

◆2

AFTER THE DECAY ZETA IS CONSERVED ON SUPER-HORIZON SCALES. WE DEFINE R AS THE TRANSFER EFFICIENCY AT DECAY.

P⇣ = P⇣� +R2

�

9PS�


⇥(1� ⌦�) e

4(⇣��⇣)+

+⌦�e3(⇣��⇣)

⇤= 1⇢ = ⇢r + ⇢�



THE WEIGHT OF THE CURVATON CONTRIBUTION TO THE FINAL POWER SPECTRUM w� ⌘

R2�/9 P⇣�

P⇣




R2�/9 P⇣�

P⇣

CRITICAL EPSILON FOR THE CURVATON. DEFINES THE FRONTIER BETWEEN RELEVANT CONTRIBUTIONS OF THE CURVATON TO THE TOTAL POWER SPECTRUM

✏c ⌘9

2

1

R�P⇣�

✓H⇤2⇡

◆2




R2�/9 P⇣�

P⇣


✏c ⌘9

2

1

R�P⇣�

✓H⇤2⇡

◆2

T H E P O W E R S P E C T R U M O F C U R V AT U R E PERTURBATIONS IN TERMS OF EPSILON CRITICAL P⇣ =

1

2

✓1

✏⇤+

1

✏c

◆✓H⇤

2⇡mPl

◆2




R2�/9 P⇣�

P⇣


✏c ⌘9

2

1

R�P⇣�

✓H⇤2⇡

◆2

T H E P O W E R S P E C T R U M O F C U R V AT U R E PERTURBATIONS IN TERMS OF EPSILON CRITICAL P⇣ =

1

2

✓1

✏⇤+

1

✏c

◆✓H⇤

2⇡mPl

◆2

T H E C U RVAT O N I S T H E M A I N S O U R C E O F PERTURBATIONS IF ✏⇤ � ✏c


MIXED PERTURBATIONSTRANSFER OF LINEAR PERTURBATIONS: OBSERVABLE QUANTITIES

SCALE DEPENDENCE OF POWER SPECTRUM

RUNNING

TILTn⇣ � 1 = w�(nS� � 1) + (1� w�)(n⇣� � 1)

= �2✏⇤ + 2⌘��w� + (1� w�)(�4✏⇤ + 2⌘��)

↵⇣ = w�↵S� + (1� w�)↵⇣� + w� (1� w�)�nS� � n⇣�

�2




RUNNING

TILTn⇣ � 1 = w�(nS� � 1) + (1� w�)(n⇣� � 1)

= �2✏⇤ + 2⌘��w� + (1� w�)(�4✏⇤ + 2⌘��)

↵⇣ = w�↵S� + (1� w�)↵⇣� + w� (1� w�)�nS� � n⇣�

�2

GRAVITATIONAL WAVESTENSOR-TO-SCALAR RATIO rT = 16w�✏c = 16✏⇤(1� w�)




RUNNING

TILTn⇣ � 1 = w�(nS� � 1) + (1� w�)(n⇣� � 1)

= �2✏⇤ + 2⌘��w� + (1� w�)(�4✏⇤ + 2⌘��)

↵⇣ = w�↵S� + (1� w�)↵⇣� + w� (1� w�)�nS� � n⇣�

�2

GRAVITATIONAL WAVESTENSOR-TO-SCALAR RATIO rT = 16w�✏c = 16✏⇤(1� w�)

NON-GAUSSIANITY

fnl =5

6

N��

N2�

w2� fnl =

5

4R�

✓1 +

gg00

g02

◆+

(g/g0)R0� � 2R�

R�

�w2

�FNL

TNL⌧nlf2nl

=36

25w�


CONSTRAINS ON CURVATON DOMINATED THEORIES

Fonseca & Wands arxiv:1101.1254


Solve the Friedmann and the curvaton field evolution equations prior to decay for diferent potential;

Ensure that the curvaton starts subdominant and overdamped;

We match it with fluid description of the curvaton decay studied by Malik et al (2003) and Gupta et al (2004).

In principle there are 4 free parameters:m

��⇤ H⇤

But the COBE normalisation of the power spectrum fixes the Hubble

scale during inflation.

Numerical Studies

✏⇤

In the curvaton limit the observables becomes independent of epsilon.


log10 m/�

log 10

�*/G

eV

-1

1

10

30

100

0.001

0.01

0.1

1

3 4 5 6 7 8 9 10 11 12 13 1414

14.5

15

15.5

16

16.5

17

17.5

18

18.5

Quadratic potential

V (�) =1

2m2�2

fNL

rT

fNL '(

�5/4 for �⇤ � (�/m)1/4mPl

3.9q

�m

m2Pl

�2⇤

for �⇤ ⌧ (�/m)1/4mPl

H⇤ '(

4.7⇥ 10�4�⇤ for �⇤ � (�/m)1/4mPl

1.5⇥ 10�3q

�m

m2Pl

�⇤for �⇤ ⌧ (�/m)1/4mPl

Tensors and non-

linearities can be used

in a complementary

way to constrain the

model parameters.

�⇤ < 5.7⇥ 1017 GeV

�

m< 0.023

✓�⇤mPl

◆2

Curvaton Limit


log10 m/�

log 10

�*/G

eV

-1

1

10

30

100

0.001

0.01

0.1

1

3 4 5 6 7 8 9 10 11 12 13 1414

14.5

15

15.5

16

16.5

17

17.5

18

18.5

Quadratic potential

V (�) =1

2m2�2

fNL

rT

fNL '(

�5/4 for �⇤ � (�/m)1/4mPl

3.9q

�m

m2Pl

�2⇤

for �⇤ ⌧ (�/m)1/4mPl

H⇤ '(

4.7⇥ 10�4�⇤ for �⇤ � (�/m)1/4mPl

1.5⇥ 10�3q

�m

m2Pl

�⇤for �⇤ ⌧ (�/m)1/4mPl

Tensors and non-

linearities can be used

in a complementary

way to constrain the

model parameters.

�⇤ < 5.7⇥ 1017 GeV

�

m< 0.023

✓�⇤mPl

◆2

rg . 0.1

fnl . 100

Curvaton Limit


log10 m/�

log 10

�*/G

eV

0.01

0.1

1

3010

1

3 4 5 6 7 8 9 10 11 12 13 141

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

x 1017

log 10

�*/G

eV

log10 m/�

11030100

0.0010.010.11

3 4 5 6 7 8 9 10 11 12 13 141

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

x 1016

log10 m/�

log 10

�*/G

eV

1

1030100

0.0010.010.11

3 4 5 6 7 8 9 10 11 12 13 141

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

x 1015

Axion potential

V = M4(1� cos(�/f))

fNL rT

f = 1016GeV

f = 1017GeV

f = 1015GeV


log10 m/�

log 10

�*/G

eV

-1000

-100

-10

0

10

1

0.1

3 4 5 6 7 8 9 10 11 12 13 141

1.5

2

2.5

3

3.5

4

4.5

5x 1017

log10 m/�

log 10

�*/G

eV

0.01

1

0.1

0.001

-1000

-100

-10

0

101001000

3 4 5 6 7 8 9 10 11 12 13 141

1.5

2

2.5

3

3.5

4

4.5

5x 1016

log10 m/�

log 10

�*/G

eV

-1000

-100-10

0

101001000

10.0010.010.1

3 4 5 6 7 8 9 10 11 12 13 141

1.5

2

2.5

3

3.5

4

4.5

5x 1015

Hyperbolic-cosine potential

fNL rT

f = 1016GeV

f = 1017GeV

f = 1015GeV

V = M4(cosh(�/f)� 1)


When is the curvaton limit valid?


INCLUDING INFLATION PERTURBATIONS


log10 m/�

log 10

�*/G

eV0.0001

0.01

1

3 4 5 6 7 8 9 10 11 12 13 1414

14.5

15

15.5

16

16.5

17

17.5

18

18.5

✏c

Limits of Epsilon critical

Quadratic Potential

Curvaton limit

Inflaton contributions

✏⇤ � ✏c

✏⇤ . ✏c

We need to fix the first slow-roll parameter to identify each region.

P⇣� =1

2✏c

✓H⇤

2⇡mPl

◆2

✏c =9

8

✓g

g0mPl

◆2 1

R2�


✏⇤ = 0.02

In the curvaton

limit region

rT ' 16✏c↵⇣ ' �2 (n⇣ � 1)2

✏⇤ � ⌘�� ⌘��For and w� ' 1 we expect

log10 m/�

log 10

�*/G

eV

0.001

0.01

0.1

0.24

100

30

10

1

0

−1

7 8 9 10 11 12 13 1414

14.5

15

15.5

16

16.5

17

17.5

fNL rT

✏c = 0.02

✏⇤ = 0.02

from n⇣


log10 m/�

log 10

�*/G

eV

10030

10

1

0 −10.24

0.1

0.01

0.001

6 7 8 9 10 11 12 13 1414

14.5

15

15.5

16

16.5

17

17.5

fNL rT

✏⇤ = 0.1

This case requires fine

tuning of the slow roll

parameters

(✏⇤ � ⌘��) ' 0.02For and w� ' 1 we expect from n⇣✏⇤ ⇠ ⌘��

N o i n f l a t i o n

d o m i n a t e d p o w e r

spectrum allowed

✏c = 0.1


log10 m/�

log 10

�*/G

eV

10030

10

1

0 −10.24

0.1

0.01

0.001

6 7 8 9 10 11 12 13 1414

14.5

15

15.5

16

16.5

17

17.5

✏⇤ = 0.1

log10 m/�

log 10

�*/G

eV

0.001

0.01

0.1

0.24

100

30

10

1

0

−1

7 8 9 10 11 12 13 1414

14.5

15

15.5

16

16.5

17

17.5

✏⇤ = 0.02

log10 m/�

log 10

�*/G

eV-1

1

10

30

100

0.001

0.01

0.1

1

3 4 5 6 7 8 9 10 11 12 13 1414

14.5

15

15.5

16

16.5

17

17.5

18

18.5


Summary• The curvaton is an inflation model to

source structure in the universe and predicts non-Gaussianities;

• The tensor-to-scalar ration and fnl can be used in a complementary way to constrain the curvaton model;

• Studied inflation contributions to the power spectrum and in which regimes are important.


Summary• The curvaton is an inflation model to

source structure in the universe and predicts non-Gaussianities;

• The tensor-to-scalar ration and fnl can be used in a complementary way to constrain the curvaton model;

• Studied inflation contributions to the power spectrum and in which regimes are important.

Thanks!Quarta-feira, 15 de Fevereiro de 12

Date post:	17-Jan-2015
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Non-Gaussian perturbations from mixed inflaton-curvaton scenario

Technology