Inflation and

cosmological perturbations

David Langlois (APC, Paris)

Outline

1.  Homogeneous inflation

2.  Cosmological perturbations: from quantum fluctuations to observations

3.  Beyond the simplest models

More details can be found in •  Lectures on inflation and cosmological perturbations, arXiv:1001.5259 [astro-ph.CO] •  Inflation, quantum fluctuations and cosmological perturbations (Cargese lectures) arXiv:hep-th/0405053

Standard cosmological model

•  General relativity:

•  FLRW geometry (spatial homogeneity and isotropy)

•  Matter:

•  Friedmann equations

Cosmological evolution

•  Three regimes with different eqs of state 1.  Radiation dominated regime ( ) 2.  Matter dominated regime ( ) 3.  Dark energy dominated regime ( )

•  Evolution of the scale factor (for )

•  The Hot Big Bang model has been very successful but leaves several puzzles unsolved…

q =2

3(1 + w)< 1 (w ⌘ P/⇢ = const)

P = w ⇢

w = 0w = 1/3

w = �1

w 6= �1

with

⇢̇+ 3H(⇢+ P ) = 0 =) ⇢ / a�3(1+w)

Flatness problem

•  Deviation from flatness:

•  Assuming a small curvature term initially

increases with time !

•  Today, . must have been extremely small in the past !

H2 / ⇢ / a�3(1+w) (aH)�2 / a1+3w

w > �1/3

|⌦� 1| . 10�2

Horizon problem

•  Horizon = maximum distance covered by a particle

•  Using

•  If , this is finite, with

For a radial light ray dr =dt

a(t)

aH / a�(1+3w)/2

w > �1/3

�hor

=2

1 + 3w(aH)�1

1�

⇣aia

⌘ 1+3w2

�

�hor

⇠ (aH)�1

hds2 = �dt2 + a2(t)

�dr2 + . . .

� i

comoving space

�hor

(ti; t) =

Z t

ti

dt0

a(t0)=

Z a(t)

ai

d ln a

aH

Horizon problem

In standard cosmology…

How to explain the quasi-isotropy of the CMB ?

Causally disconnected regions ?

Last scattering surface

Big Bang

�hor

(t0

)

�hor

(tls

)⇠ (a0H0)

�1

(als

Hls

)�1⇠

✓a0

als

◆1/2

Inflation

•  A period of acceleration in the early Universe

The horizon can be much bigger than the Hubble radius.

•  Inflation also solves the flatness problem .

•  Inflation also provides an explanation for the origin of the

primordial perturbations, which will give birth to structures in the Universe.

Scalar field inflation

•  How to get inflation ?

•  Cosmological constant : but no end… •  Scalar field

in a homogeneous universe

Slow-roll regime

•  Slow-roll equations

•  Slow-roll parameters

•  Number of e-folds

How much inflation ?

•  Inflation must last long enough to solve the horizon problem.

•  Radiation era: the (comoving)

Hubble radius increases like

•  Slow-roll inflation: decreases like

•  Taking into account the matter dominated era

Example 1

•  Potential of the form

•  Slow-roll:

•  Integration of the slow-roll eqs of motion yields

•  Using

one finds

d ln a

d�=

H

�̇' �3H

2

V 0' � V

m2PV0 = �

�

2m2P

Example 2

•  Exponential potential:

•  Exact solution

•  Slow-roll ( )

Main families of inflation models

•  Huge number of models of inflation •  Three main categories…

•  Large field models

initially. The scalar field rolls down toward the minimun of the potential. Typically

Main families of inflation models

•  Small field models

The initial value of the scalar field is small, typically near a maximum.

•  Hybrid models Requires a second field to end inflation

Hybrid inflation

•  Potential:

•  Effective mass:

•  If then

•  If , end of inflation…

Higgs inflation

•  Non-minimal coupling of the Higgs to gravity:

•  From the Jordan frame to the Einstein frame

with the new field defined by

L = ⇠H†HR

S =

Zd

4x

p�g

M

2P + ⇠h

2

2R� 1

2@µh@

µh� �

4

�h

2 � v2�2�

gµ⌫ ! g̃µ⌫ = ⌦2gµ⌫ ⌦2 ⌘ 1 +⇠h2

M2P

SE =

Zd

4x

p�g̃

M

2P

2R̃� 1

2@µ�@

µ�� V (�)

�

d�

dh=

r⌦2 + 6⇠2h2/M2P

⌦4�

Higgs inflation

•  Potential

•  For

The potential is very flat:

•  Number of e-folds before the end of inflation

•  Observational constraints

V (�) =1

⌦(�)4�

4

�h(�)2 � v2

�2

V (�) ' �M4P

4⇠2

1 + exp

✓� 2�p

6MP

◆�h � MP /

p⇠

✏ ' 4M4P

3⇠2h4, ⌘ ' �4M

2P

3⇠h2

hend ' MP /p

⇠N(h) ' 68

h2 � h2endM2P /⇠

⇠p�' 47⇥ 103