ific, 6 february 2007 julien lesgourgues (lapth, annecy)

IFIC, 6 FebruaryIFIC, 6 February 2007 2007

Julien Lesgourgues (LAPTHJulien Lesgourgues (LAPTH, Annecy, Annecy))

1) « historical arguments »- flatness problem- horizon problem- monopoles & topological defects

2) basic model- slow rolling scalar field

- primordial fluctuations

3) agreement between CMB maps and inflation- coherence- scale invariance - gaussianity- adiabaticity

4) current constraints on inflation, prospects…

1) « historical arguments »- flatness problem- horizon problem- monopoles & topological defects

2) basic model- slow rolling scalar field

- primordial fluctuations

3) agreement between CMB maps and inflation- coherence- scale invariance - gaussianity- adiabaticity

4) current constraints on inflation, prospects…

1979-1982: A.Starobinsky A. Guth

1) « historical arguments » : flatness problem

Definitions :

-scale factor : a(t) ds2 = dt2 - a(t)2 dx2 c=1

-e-fold number : N = ln a e.g. “a stage lasts for N=10 e-folds” a(t) increases by factor e10=22000

Friedmann equation :

or :

matter(nr, r)

spatial curvature

a-2 a-3, a-4


decelerated expansion

H

ln a

ln

matter

radiation

dark energy

?curvature

today


?curvature

Mp4

1062


ln a

ln

matter

radiation

dark energy

today

?

TeV4

1032


curvature

ln a

ln

matter

radiation

dark energy

today

Inflation = stage of accelerated expansion

Friedmann Energy cons. ä(t) > 0 + 3 p < 0

an, -2 < n < 0


ln a

ln

curvature

matter

radiation

dark energy

today

inflation


What is the

minimal duration

of inflation ?



ln a

ln

radiation a -4

today

inflation ~cst

curvature a -2

Ninflation = Npost-inflation

Minimal duration of inflation :


Ninflation Npost-inflation

transition infl. → rad.

minimal Ninflation

(1016 GeV)4

…(1 TeV)4

~67…

~37

1) « historical arguments » : horizon problem

t

x

y


t

x

y

last scattering surface (LSS)

are all LSS points within causal contact ?

photondecoupling


t

x

y

Last scattering surface (LSS)

↓initial singularity

Hubble radius at decoupling: ~1°

photondecoupling


t

x

y

photondecoupling last scattering surface (LSS)

x

inflation

?curvature

1) « historical arguments » : monopoles and other defects

ln a

ln

matter

radiation

darkenergy

today

phase transition

defects

ln a

ln

curvature

matter

radiation

dark energy

inflation

phase transition

1) « historical arguments » : monopoles and other defects

today

2) Basic model : a slow-rolling scalar field


Friedmann + e. c. : ä(t) > 0 + 3 p < 0 nearly homogeneous slow-rolling scalar fields :

= ½ ‘2 + V()

p = ½ ‘2 - V()

|dV/d | < V/mP , |d2V/d2|< V/mP2

V



Friedmann + e. c. : ä(t) > 0 + 3 p < 0 nearly homogeneous slow-rolling scalar field :

= ½ ‘2 + V()

p = ½ ‘2 - V()

|dV/d | < V/mP , |d2V/d2|< V/mP2

V



Friedmann + e. c. : ä(t) > 0 + 3 p < 0 nearly homogeneous slow-rolling scalar field :

= ½ ‘2 + V()

p = ½ ‘2 - V()

|dV/d | < V/mP , |d2V/d2|< V/mP2

V

end of inflation: field oscillates

and decays in particles

which finally thermalize

2) Basic model : primordial cosmological fluctuations

fluctuations today


fluctuations at decoupling


origin of fluctuations ?


decelerated expansion : - causal horizon = Hubble radius ( RH = c/H )

- RH(t) grows faster than a(t)

causal

acausal

timeMATTERDOMINATION

RADIATIONDOMINATION

RHprimodialcosmological perturbations

distance

distance RH


phase transition

no coherent fluctuations

decelerated expansion : - causal horizon = Hubble radius ( RH = c/H )

- RH(t) grows faster than a(t)


RADIATIONDOMINATION

primodialcosmological perturbations


RHdistance

INFLATION

accelerated expansion : - causal horizon » Hubble radius

- RH(t) grows more slowly than a(t)


RADIATIONDOMINATION



RHdistance

INFLATION

1

1- quantum fluctuations of and h grow to macroscopic scales- normalization and evolution imposed by quantum mechanics


RADIATIONDOMINATION



RHdistance

INFLATION

1

2- Hubble crossing, Bogolioubov transformation - “squeezed state” → classical stochastic fluctuations

2


RADIATIONDOMINATION



RHdistance

INFLATION

12

3- perturbation amplitude frozen since - «primordial spectrum» of scalar and tensor perturbations

2

3


RADIATIONDOMINATION



RHdistance

INFLATION

12

3

4- insensitive to microscopical evolution (reheating, phase transition)- primordial spectrum mediated to , b, , CDM

4


RADIATIONDOMINATION



RHdistance

INFLATION

12

3

4

5- acoustic oscillations and decoupling- CMB anisotropies → primordial spectrum inherited from 3

5


RADIATIONDOMINATION


3) Agreement between CMB maps and inflation

inflation predict that perturbations are:

1. coherent

2. nearly gaussian

3. adiabatic*

4. nearly scale invariant*

*for simplest inflationary models


RHdistance

INFLATION

decoupling

time coherence of inflationary fluctuations :




RADIATIONDOMINATION

RHdistance

INFLATION

absence of coherence in the case of topological defects :

decoupling




RADIATIONDOMINATION


1. coherent

2. nearly gaussian

3. adiabatic*




validated (existence of acoustic

peaks)


1. coherent

2. nearly gaussian

3. adiabatic*




validated (statistical analysisof CMB maps)


peaks)


1. coherent

2. nearly gaussian

3. adiabatic*



validated (peak scale)




peaks)

slow rolling

scalar field :

AS

kamplitude V3/2/V’

tilt (1-nS) (V’/V)2 , V’’/V

+ hider order corrections

(tilt running, …)

AT

kamplitude V1/2

tilt nT (V’/V)2

+ higher order corrections

(tilt running, …)

V


scale invariance :


1. coherent

2. nearly gaussian

3. adiabatic*



validated (peak scale)




peaks)

validated (peak amplitudes)

single field

slow-roll inflation :

AS


tilt (1-nS) (V’/V)2 , V’’/V

+ next-order corrections

(running of the tilt, …)

AT

kamplitude V1/2

tilt nT (V’/V)2



V

4) current constraints on inflation

AS


tilt (1-nS) (V’/V)2 , V’’/V



AT

kamplitude V1/2

tilt nT (V’/V)2



overall amplitude

= 0.5x10-5 mp3


AS

kamplitude V3/2/V’ = 0.5x10-5 mp

3

tilt (1-nS) 2.25 (V’/V)2 - V’’/V = 0.5

mp-2



AT

kamplitude V1/2

tilt nT (V’/V)2



overall slope


AS

k

AT

kamplitude V1/2 < (3.7x1016 GeV)2

tilt nT (V’/V)2



amplitude V3/2/V’ = 0.5x10-5 mp3

tilt (1-nS) 2.25 (V’/V)2 - V’’/V = 0.5

mp-2



absence of tensors


AS

k

AT


tilt nT (V’/V)2




tilt (1-nS) 2.25 (V’/V)2 - V’’/V = 0.5

mp-2



absence of tensors


Energy scale of inflation still unknown !!

Self-consistency relation still not checked !!

future CMB experiments (B-polarization) : r ~ 10-2

(factor 50 pour V)

future space-based GW interferometers : r ~ 10-4

(BBO) (factor 5000

pour V)

• measure r, nt : inflationary energy scale + self-consistency r=-8nt

• measure r : inflationary energy scale

• no GW detected : inflation unconstrained

new physics

at 1016 GeV

(extra-D ?)

ordinary QFT

(SUSY, PNGB…)


Energy scale of inflation still unknown !!

Self-consistency relation still not checked !!

AS

k

AT


tilt nT (V’/V)2




tilt (1-nS) 2.25 (V’/V)2 - V’’/V = 0.5

mp-2



?


AS

k

AT


tilt nT (V’/V)2




tilt (1-nS) 2.25 (V’/V)2 - V’’/V = 0.5

mp-2



?


negative running, or no running????



no running (power law spectrum)

negative running (convex spectrum)

WMAP3+SDSS



Theoretical prejudice:

Deep in the slow-roll limit, running ≈ 0

( ns-1 ~ , nrun~ 2 )

Do we expect to be deep in the slow-roll regime?

Question of philosophy and aesthetics…



1) Minimalistic aesthetics:

simple potential (monomial, polynomial, simple function)

slow-roll params () monotonically growing/decreasing

60 e-folds before the end, must be deep in slow-roll

expect running ≈ 0

2) Modesty and pragmatism:

V() may have any shape (many scalars, landscape…)

we can only reconstruct “observable region”

(no assumptions on what’s before/after)

possible large running (and beyond)…


J.L. & W.Valkenburg, in preparation

0

0.2

0.4

0.6

0.8

r

10.9

Small field models

«mP

CONCAVE, V’’>0

CONVEX, V’’<0

n

large field models

~mP

V3/2/V’ ~ 10-5mp if V’ ~ V/ , V~(1016GeV)4

~mP

0

0.2

0.4

0.6

0.8

r

10.9

CONCAVE, V’’>0

CONVEX, V’’<0

=6

=4

=2

monomial potentials V=(...)

=1

n

0

0.2

0.4

0.6

0.8

r

10.9

CONCAVE, V’’>0

CONVEX, V’’<0

=6

=4

=2

new inflationV=V0 [1- (…) +...]

=1


n

0

0.2

0.4

0.6

0.8

r

10.9

CONCAVE, V’’>0

CONVEX, V’’<0

=6

=4

=2

monomial

=1

Loop correction


Hybrid inflation

=1new inflation

V=V0 [1- (…) +...]

n

0

0.2

0.4

0.6

0.8

r

n 10.9

CONCAVE, V’’>0

CONVEX, V’’<0

=4

=2

monomial

=1

Loop correction


new inflationV=V0 [1- (…) +...]

=1

=6

WMAP-3

WMAP-3+SDSS

ific, 6 february 2007 julien lesgourgues (lapth, annecy)

Documents

flatness problem slide

flatness problem ln

flatness problem n inflation

flatness problem curvature

annecy slide

guth slide

ln radiation

inflation transition