Non-Gaussian perturbations from mixed inflaton-curvaton scenario

Non-Gaussian perturbations from mixed
inflaton-curvaton scenario

José Fonseca - University of Portsmouth
Based 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!
Power Spectrum of
primordial density
perturbations
✓ ◆ns (k0 ) 1
k
P⇣ (k) = ⇣ (k0 )
2
k0
1
k0 = 0.002Mpc
2 +0.091 9
(k0 ) = (2.430 0.091 ) ⇥ 10
+0.012
ns (k0 ) = 0.968 0.012

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


This is a pic of the Cosmic Microwave Background, aka CMB!
Power Spectrum of
Bispectrum from quadratic
primordial density
corrections
perturbations
✓ ◆ns (k0 ) 1
k 3
P⇣ (k) = ⇣ (k0 )
2
⇣ = ⇣1 + fnl (⇣1
2 2
h⇣1 i)
k0 5
1
k0 = 0.002Mpc
2 +0.091 9
(k0 ) = (2.430 0.091 ) ⇥ 10 10 < fnl < 74
+0.012
ns (k0 ) = 0.968 0.012

E. Komatsu et al, Seven-year
WMAP Observations:
Single field inflation is
Cosmological Interpretation
perfectly fine but...
- arXiv:1001.4538v1


Perturbations from
inﬂation in a nutshell


INFLATON
3H ˙ ⇥ V
SCALAR FIELD LIVING IN A FRW UNIVERSE THAT
DRIVES INFLATION. 2 V( )
H '
3m2 l
P

PERTURBATIONS FROM INFLATION IN A NUTSHELL
STANDARD INFLATION = INFLATON + SLOW ROLL

INFLATON
3H ˙ ⇥ V
SCALAR FIELD LIVING IN A FRW UNIVERSE THAT
DRIVES INFLATION. 2 V( )
H '
3m2 l
P

SLOW ROLL
✓ ◆2
THE FIELD HAS AN OVER-DAMPED EVOLUTION, I.E., IT
1 2 V
✏ ⌘ mP l ⌧1
ROLLS DOWN THE POTENTIAL SLOWLY. 2 V
THE EXPANSION IS ALMOST EXPONENTIAL.

KINETIC ENERGY DOES NOT VARY WITHIN 1 HUBBLE 2 V
TIME.
|⌘ |⌘ mP l ⌧1
V

THE POTENTIAL NEEDS TO BE FLAT.

STANDARD INFLATION = INFLATON + SLOW ROLL

SPLIT QUANTITIES BETWEEN BACKGROUND AND 1ST
O R D E R P E R T U R B AT I O N ( G A U S S I A N VA C U U M ⇥(t, x) = ⇥(t) + ⇥(t, x)
FLUCTUATIONS)

✓ ◆
EQ. OF MOTION FOR THE FIELD PERTURBATIONS (IN ¨ ˙
⇥k + 3H ⇥k +
k2
+ m2 ⇥k = 0
FOURIER SPACE) FOR A “MASSLESS” FIELD. a2

⇥ ⇤2
POWER SPECTRUM OF FIELD PERTURBATIONS AT H
P ⇥
HORIZON EXIT 2 k=aH

POWER SPECTRUM OF PERTURBATIONS FOR A MASSLESS FIELD DURING INFLATION

0

THE “SEPARATE UNIVERSE” PICTURE SAYS 2 SUPER-
HORIZON REGIONS OF THE UNIVERSE EVOLVE AS IF t2
THEY WERE SEPARATE FRIEDMANN-ROBERTSON-
W A 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
-1
cH s

HOMOGENEOUS BUT MAY HAVE DIFFERENT DENSITIES
AND PRESSURE. t1

a b
Wands et al., astro-ph/0003278v2

THE CURVATURE PERTURBATION ZETA IS THEN GIVEN
BY THE DIFFERENCE OF THE INTEGRATED EXPANSION ⇥ = N
FROM A SPATIALY-FLAT HYPERSURFACE TO A
UNIFORM-DENSITY HYPERSURFACE .
1 00 2
DIFFERENT PATCHES OF THE UNIVERSE WILL HAVE N = 0
N ⇤⇤ + N ⇤⇤
DIFFERENT EXPANSION HISTORIES DUE TO DIFFERENT 2
INITIAL CONDITIONS

DELTA N FORMALISM AND THE SEPARATE UNIVERSE PICTURE

POWER SPECTRUM AND SCALE DEPENDENCE
✓ ◆2
POWER SPECTRUM OF CURVATURE PERTURBATIONS
1 H⇤
P⇣ ⇤
'
W H I C H R E M A I N S C O N S TA N T F O R A D I A B AT I C 2✏⇤ 2⇡mP l
PERTURBATIONS ON LARGE SCALES
d ln P⇣
TILT n⇣ 1⌘
d ln k
ns 1⇥ 6 + 2⇥
dn⇣
RUNNING ↵⇣ ⌘ s ⇥ 24⇥2 + 16⇥ ⇤ 2⌅ 2
d ln k

OBSERVATIONAL PREDICTIONS FOR SINGLE FIELD

✓ ◆2
1 H⇤
P⇣ ⇤
'
d ln P⇣
TILT n⇣ 1⌘
d ln k
ns 1⇥ 6 + 2⇥
dn⇣
RUNNING ↵⇣ ⌘ s ⇥ 24⇥2 + 16⇥ ⇤ 2⌅ 2
d ln k

GRAVITATIONAL WAVES
✓ ◆2
2 H⇤
TENSOR-TO-SCALAR RATIO rT ⌘ PG /P⇣ rT =
m2 l P⇣
P 2⇡


✓ ◆2
1 H⇤
P⇣ ⇤
'
d ln P⇣
TILT n⇣ 1⌘
d ln k
ns 1⇥ 6 + 2⇥
dn⇣
RUNNING ↵⇣ ⌘ s ⇥ 24⇥2 + 16⇥ ⇤ 2⌅ 2
d ln k

GRAVITATIONAL WAVES
✓ ◆2
2 H⇤
TENSOR-TO-SCALAR RATIO rT ⌘ PG /P⇣ rT =
m2 l P⇣
P 2⇡
NON-GAUSSIANITY
CONSERVED CURVATURE PERTURBATION REMAINS 5
GAUSSIAN
fnl = (2 ⇤ ⇥ )⌧1
6


Curvaton Scenario


IT IS AN INFLATIONARY MODEL.
THE INFLATON DRIVES THE ACCELERATED EXPANSION Lyth&Wands: hep-th/0110002
Enqvist&Sloth: hep-ph/0109214
WHILE THE CURVATON PRODUCES THE STRUCTURE IN Moroi&Takahashi: hep-ph/0110096
THE UNIVERSE.

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

CURVATON SCENARIO
MAIN PRINCIPLES

IT IS AN INFLATIONARY MODEL.
THE INFLATON DRIVES THE ACCELERATED EXPANSION Lyth&Wands: hep-th/0110002
Enqvist&Sloth: hep-ph/0109214
WHILE THE CURVATON PRODUCES THE STRUCTURE IN Moroi&Takahashi: hep-ph/0110096
THE UNIVERSE.

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

DURING INFLATION
SUBDOMINANT COMPONENT ⌧1
SINCE IT IS EFFECTIVELLY MASSLESS IT IS IN AN
OVER-DAMPED REGIME. THEREFORE OBEYS TO THE ⌧1 , ⇥ ⌧1
SLOW-ROLL CONDITIONS
✓ ◆2
AND ACQUIRES A SPECTRUM OF GAUSSIAN FIELD H⇤
PERTURBATIONS AT HORIZON EXIT P
2

CURVATON SCENARIO
MAIN PRINCIPLES

AFTER INFLATION

THE CURVATON IS AN ENTROPY DIRECTION, SO THE
CURVATURE PERTURBATION ON UNIFORM DENSITY ˙ 6= 0
⇣
HYPER-SURFACES IS NO LONGER CONSERVED

CURVATON SCENARIO
MAIN IDEAS

AFTER INFLATION

⇣

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

CURVATON SCENARIO
MAIN IDEAS

AFTER INFLATION

⇣

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

DECAYS INTO RADIATION AND TRANSFERS ITS
PERTURBATIONS
H'

CURVATON SCENARIO
MAIN IDEAS

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

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

MIXED PERTURBATIONS
TRANSFER OF LINEAR PERTURBATIONS

S ⌘ 3 (⇣ ⇣ )

⇣=
˙

LOCAL CURVATON ENERGY DENSITY ⇢ = ⇢ eS = m2
¯ 2
osc

MIXED PERTURBATIONS

S ⌘ 3 (⇣ ⇣ )

⇣=
˙

¯ 2
osc

EXPAND LOCAL VALUE OF THE FIELD DURING 1 00
0 2
OSCILLATION IN TERMS OF ITS VEV AND FIELD osc 'g+g ⇤+ g ⇤
FLUCTUATIONS DURING INFLATION 2

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

MIXED PERTURBATIONS

S ⌘ 3 (⇣ ⇣ )

⇣=
˙

¯ 2
osc

EXPAND LOCAL VALUE OF THE FIELD DURING 1 00
0 2
OSCILLATION IN TERMS OF ITS VEV AND FIELD osc 'g+g ⇤+ g ⇤
FLUCTUATIONS DURING INFLATION 2

G ACCOUNTS FOR NON-LINEAR EVOLUTION OF CHI osc ⌘ g( ⇤)
✓ ◆
1 gg 00 2
S = SG + 1 SG
4 g 02
THE ENTROPY PERTURBATION IS
g0
SG ⌘ 2 ⇤
g

MIXED PERTURBATIONS

THE FINAL CURVATON POWER SPECTRUM COMES ✓ 0
◆2
FROM THE MODES EXCITED DURING INFLATION AND g H⇤
FROM NON-LINEAR EVOLUTION OF THE FIELD UNTIL PS = 4
DECAY.
g 2⇡

MIXED PERTURBATIONS

◆2
DECAY.
g 2⇡

⇥
AT DECAY ON UNIFORM TOTAL ENERGY DENSITY
(1 ⌦ ) e4(⇣ ⇣)
+
HYPERSURFACES WE HAVE ⇢ = ⇢r + ⇢ ⇤
+⌦ e3(⇣ ⇣)
=1

MIXED PERTURBATIONS

◆2
DECAY.
g 2⇡

⇥
(1 ⌦ ) e4(⇣ ⇣)
+
+⌦ e3(⇣ ⇣)
=1

AFTER THE DECAY ZETA IS CONSERVED ON SUPER- R2
HORIZON SCALES. WE DEFINE R AS THE TRANSFER P⇣ = P⇣ + PS
EFFICIENCY AT DECAY. 9

MIXED PERTURBATIONS

◆2
DECAY.
g 2⇡

⇥
(1 ⌦ ) e4(⇣ ⇣)
+
+⌦ e3(⇣ ⇣)
=1

AFTER THE DECAY ZETA IS CONSERVED ON SUPER- R2
HORIZON SCALES. WE DEFINE R AS THE TRANSFER P⇣ = P⇣ + PS
EFFICIENCY AT DECAY. 9
3
FOR A SUDDEN DECAY APPROXIMATION R ,dec =
4 dec

MIXED PERTURBATIONS

2
THE WEIGHT OF THE CURVATON CONTRIBUTION TO R /9 P⇣
THE FINAL POWER SPECTRUM
w ⌘
P⇣

MIXED PERTURBATIONS

2
w ⌘
P⇣
✓ ◆2
CRITICAL EPSILON FOR THE CURVATON. DEFINES THE 9 1 H⇤
FRONTIER BETWEEN RELEVANT CONTRIBUTIONS OF ✏c ⌘
2 R P⇣ 2⇡
THE CURVATON TO THE TOTAL POWER SPECTRUM

MIXED PERTURBATIONS

2
w ⌘
P⇣
✓ ◆2
2 R P⇣ 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 1 1 1 H⇤
P⇣ = +
PERTURBATIONS IN TERMS OF EPSILON CRITICAL 2 ✏⇤ ✏c 2⇡mP l

MIXED PERTURBATIONS

2
w ⌘
P⇣
✓ ◆2
2 R P⇣ 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 1 1 1 H⇤
P⇣ = +
PERTURBATIONS IN TERMS OF EPSILON CRITICAL 2 ✏⇤ ✏c 2⇡mP l

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

MIXED PERTURBATIONS

SCALE DEPENDENCE OF POWER
SPECTRUM
n⇣ 1 = w (nS 1) + (1 w )(n⇣ 1)
TILT = 2✏⇤ + 2⌘ w + (1 w )( 4✏⇤ + 2⌘ )
2
RUNNING ↵⇣ = w ↵S + (1 w )↵⇣ + w (1 w ) nS n⇣

MIXED PERTURBATIONS
TRANSFER OF LINEAR PERTURBATIONS: OBSERVABLE QUANTITIES

SPECTRUM
n⇣ 1 = w (nS 1) + (1 w )(n⇣ 1)
TILT = 2✏⇤ + 2⌘ w + (1 w )( 4✏⇤ + 2⌘ )
2

GRAVITATIONAL WAVES
TENSOR-TO-SCALAR RATIO rT = 16w ✏c = 16✏⇤ (1 w )

MIXED PERTURBATIONS

SPECTRUM
n⇣ 1 = w (nS 1) + (1 w )(n⇣ 1)
TILT = 2✏⇤ + 2⌘ w + (1 w )( 4✏⇤ + 2⌘ )
2

GRAVITATIONAL WAVES
TENSOR-TO-SCALAR RATIO rT = 16w ✏c = 16✏⇤ (1 w )

NON-GAUSSIANITY

5N ✓ ◆
5 gg 00 (g/g 0 )R0 2R
FNL fnl = w2 fnl = 1 + 02 + w2
6 N2 4R g R

⌧nl 36
TNL 2 = 25w
fnl

MIXED PERTURBATIONS

CONSTRAINS ON
CURVATON
DOMINATED THEORIES
Fonseca & Wands
arxiv:1101.1254


Numerical Studies

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).
m
In principle there are 4 free parameters: ⇤ ✏⇤ H⇤
But the COBE normalisation of the power spectrum fixes the Hubble
scale during inflation.
In the curvaton limit the observables becomes independent of epsilon.


Curvaton Limit 18.5

18 -1

Quadratic potential 17.5
1
1
rT
10
1 2 2 17 30 0.1

V( )= m

/GeV
100
2 16.5 0.01

fN L

*
Tensors and non-

10
16 0.001

log
linearities can be used 15.5

in a complementary 15

way to constrain the 14.5

model parameters. 14
3 4 5 6 7 8
log
9
m/
10 11 12 13 14
10
( 4
4.7 ⇥ 10 q⇤ for ⇤ ( /m)1/4 mP l
H⇤ ' 3 m2 l 1/4 < 5.7 ⇥ 1017 GeV
1.5 ⇥ 10 P
for ⇤ ⌧ ( /m) mP l ⇤
(
m ⇤ ✓ ◆2
5/4 for ( /m)1/4 mP l ⇤
q ⇤ < 0.023
fN L ' m2 l 1/4
m mP l
3.9 m P2 for ⇤ ⌧ ( /m) mP l
⇤


rg . 0.1
Curvaton Limit 18.5

18 -1

Quadratic potential 17.5
1
1
rT
10
1 2 2 17 30 0.1

V( )= m

/GeV
100
2 16.5 0.01

fN L

*
Tensors and non-

10
16 0.001

log
linearities can be used 15.5

in a complementary 15

way to constrain the 14.5

model parameters. 14
3 4 5 6 7 8 9 10 11 12 13 14
log
10
m/ fnl . 100
( 4
4.7 ⇥ 10 q⇤ for ⇤ ( /m)1/4 mP l
H⇤ ' 3 m2 l 1/4 < 5.7 ⇥ 1017 GeV
1.5 ⇥ 10 P
for ⇤ ⌧ ( /m) mP l ⇤
(
m ⇤ ✓ ◆2
5/4 for ( /m)1/4 mP l ⇤
q ⇤ < 0.023
fN L ' m2 l 1/4
m mP l
3.9 m P2 for ⇤ ⌧ ( /m) mP l
⇤


15
x 10

3 100
30 10
2.8

2.6 0.01 0.001
0.1
1
2.4

Axion potential

/GeV
2.2
1

*
2

log10
4
V = M (1 cos( /f )) 1.8

1.6

fN L rT 1.4

1.2 f = 1015 GeV
1
3 4 5 6 7 8 9 10 11 12 13 14
log m/
10
17 16
x 10 x 10

3 0.01 3

2.8 2.8
f = 1016 GeV
2.6 2.6 0.01
1 0.1 0.001
0.1
2.4 2.4
/GeV

2.2
/GeV 2.2
*

*
2 2
10

log10
log

1.8 1.8

1
1.6 1.6

1
1.4 1.4
100 30 10 1
10

1.2
30
f = 1017 GeV 1.2

1 1
3 4 5 6 7 8 9 10 11 12 13 14 3 4 5 6 7 8 9 10 11 12 13 14
log m/ log m/
10 10


15
x 10
5

4.5
f = 1015 GeV
4

0.01 0.001
1 0.1

Hyperbolic-cosine potential
3.5

/GeV
*
3

10
-1000

log
4
V = M (cosh( /f ) 1) 2.5
-100
-10

2
0

fN L rT 1.5 1000 100 10

1
3 4 5 6 7 8 9 10 11 12 13 14
log m/
10
17 16
x 10 x 10
5 5
-1000
16 1

4.5 4.5
f = 10 GeV
-100
0.1
4 4

-1000
3.5 3.5
/GeV

/GeV
*

*
3 1 3 -100
-10
10

10
log

log

0.01
2.5 2.5
-10

2 2
0
0
1.5 0.1 1.5
17 10 0.001
10
f = 10 GeV 1000 100

1 1
3 4 5 6 7 8 9 10 11 12 13 14 3 4 5 6 7 8 9 10 11 12 13 14
log10 m/ log10 m/


When is the curvaton limit valid?


INCLUDING INFLATION
PERTURBATIONS


Limits of Epsilon critical

✓ ◆2 Quadratic Potential
1 H⇤
✏c
18.5
P⇣ =
2✏c 2⇡mP l 18

1
✓ ◆2 17.5
9 g 1
✏c = 17
8 g 0 mP l R2

/GeV
0.01
16.5

*
10
16
Curvaton limit
log
0.0001
15.5

✏⇤ ✏c 15

Inflaton contributions 14.5

✏⇤ . ✏c 14
3 4 5 6 7 8
log
10
9
m/
10 11 12 13 14

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


For ✏⇤ ⌘ ⌘ and w ' 1 we expect ✏⇤ = 0.02 from n⇣

17.5

0.24

17
✏c = 0.02
0.1

−1
16.5
0
0.01 In the curvaton
/GeV

1
16

limit region
*

0.001
10

15.5 10
rT ' 16✏c
log

30 2
15
✏⇤ = 0.02 ↵⇣ ' 2 (n⇣ 1)
100
14.5
fN L rT
14
7 8 9 10 11 12 13 14
log m/
10


For ✏⇤ ⇠ ⌘ and w ' 1 we expect (✏⇤ ⌘ ) ' 0.02 from n⇣
17.5

✏c = 0.1
17
This case requires fine
0.24
−1
0
0.1 tuning of the slow roll
16.5

0.01 parameters
/GeV

16
*

0.001
No inflation
10

15.5
log

1
dominated power
15

✏⇤ = 0.1 10
spectrum allowed
14.5 30

fN L rT 100

14
6 7 8 9 10 11 12 13 14
log m/
10


18.5 17.5

18 -1
17 0.24
−1
1 0
1
17.5 0.1

10 16.5
17 30 0.1
0.01
/GeV

/GeV
100
0.01 16
16.5
*

*
0.001
10

10
16 0.001
15.5
log

log
15.5
1
15

15
✏⇤ = 0.1 10
30
14.5
14.5 100

14 14
3 4 5 6 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 13 14
log m/ log m/
10 10
17.5

0.24

17
0.1

−1
16.5
0
0.01
/GeV

1
16
*

0.001
10

15.5 10
log

30
15
✏⇤ = 0.02
100
14.5

14
7 8 9 10 11 12 13 14
log10 m/


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!

Non-Gaussian perturbations from mixed inflaton-curvaton scenario

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