TTI, Vallico Sotto, Tuscany - 09/02/10 P Pmdt26/tti_talks/deBB_10/peter_tti...Vallico Sotto - 2nd September 2010 TTI, Vallico Sotto, Tuscany - 09/02/10 P P Institut d’Astrophysique

Vallico Sotto - 2nd September 2010

TTI, Vallico Sotto, Tuscany - 09/02/10

P PInstitut d’Astrophysique de Paris

GRεCO

Cosmological problems and a possible non-inflationary solution in the framework of de Broglie-Bohm quantum cosmology

Vallico Sotto - 2nd September 2010 2

Basic cosmology: the standard model and its difficulties

Homogeneous & Isotropic metric (FLRW):

Conformal time Conformally flat space

LastScatteringsurface

ds2 = !dt2 + a2(t)!ij

!

xk"

dxidxj

d" =dt

a(t)="

ds2 = a2(")!

!d" + !ijdxidxj"

S =

# #!g

$

!R

6#2P

+ p

%

d4x

p = $%

nT = nS ! 1 =12$

1 + 3$

T

S$ 4 % 10!2

&

nS ! 1

1


ds2 = !dt2 + a2(t)!ij

!

xk"

dxidxj

d" =dt

a(t)="

ds2 = a2(")!

!d"2 + !ijdxidxj"

S =

# #!g

$

!R

6#2P

+ p

%

d4x

p = $%

nT = nS ! 1 =12$

1 + 3$

T

S$ 4 % 10!2

&

nS ! 1

1


ds2 = !dt2 + a2 (t)

!

dr2

1 !Kr2+ r2

"

d!2 + sin2 !d"2#

$

,

d# =dt

a(t)="

ds2 = a2(#)"

!d#2 + $ijdxidxj#

S =

% #!g

&

!R

6%2P

+ p

'

d4x

p = &'

nT = nS ! 1 =12&

1 + 3&

T

S$ 4 % 10!2

(

nS ! 1

1

Matter component: perfect fluid:


Tµ! = pgµ! + (! + p)uµu!

ds2 = !dt2 + a2 (t)

!

dr2

1 !Kr2+ r2

"

d"2 + sin2 "d#2#

$

,

d$ =dt

a(t)="

ds2 = a2($)"

!d$2 + %ijdxidxj#

S =

% #!g

&

!R

6&2P

+ p

'

d4x

p = '!

nT = nS ! 1 =12'

1 + 3'

1


p = !"

Tµ! = pgµ! + (" + p)uµu!

ds2 = !dt2 + a2 (t)

!

dr2

1 !Kr2+ r2

"

d#2 + sin2 #d$2#

$

,

d% =dt

a(t)="

ds2 = a2(%)"

!d%2 + &ijdxidxj#

S =

% #!g

&

!R

6'2P

+ p

'

d4x

p = !"

1

dustLastScatteringsurface

! =1

3

p = !"

Tµ! = pgµ! + (" + p)uµu!

ds2 = !dt2 + a2 (t)

!

dr2

1 !Kr2+ r2

"

d#2 + sin2 #d$2#

$

,

d% =dt

a(t)="

ds2 = a2(%)"

!d%2 + &ijdxidxj#

S =

% #!g

&

!R

6'2P

+ p

'

d4x

1

radiation


H2 +Ka2

=1

3(8!GN" + !)

# =1

3

p = #"

Tµ! = pgµ! + (" + p)uµu!

ds2 = !dt2 + a2 (t)

!

dr2

1 !Kr2+ r2

"

d$2 + sin2 $d%2#

$

,

d& =dt

a(t)="

ds2 = a2(&)"

!d&2 + 'ijdxidxj#

1

+ cosmological constant = Einstein equation:


a

a=

1

3[! ! 4!GN (" + p)]

H2 +Ka2

=1

3(8!GN" + !)

# =1

3

p = #"

Tµ! = pgµ! + (" + p)uµu!

ds2 = !dt2 + a2 (t)

!

dr2

1 !Kr2+ r2

"

d$2 + sin2 $d%2#

$

,

d& =dt

a(t)="

1


! = 0

p = !"

Tµ! = pgµ! + (" + p)uµu!

ds2 = !dt2 + a2 (t)

!

dr2

1 !Kr2+ r2

"

d#2 + sin2 #d$2#

$

,

d% =dt

a(t)="

ds2 = a2(%)"

!d%2 + &ijdxidxj#

S =

% #!g

&

!R

6'2P

+ p

'

d4x

p = !"

1


Problems with standard model:

Singularity

Horizon

Flatness

Homogeneity

Perturbations

Dark matter

Dark energy / cosmological constant

Baryogenesis

...

Topological defects (monopoles)

1

also available fromOXFORD UNIVERSITY PRESS

Cosmology Steven Weinberg

Particle Astrophysics: Second EditionDonald H. Perkins

Cosmic Anger—Abdus Salam: The First Muslim Nobel Scientist Gordon Fraser

Revolutionaries of the Cosmos—The Astro-Physicists Ian S. Glass

Cover image: ‘Le vent’ by Madeleine Attal (1997).

This book provides an extensive survey of all the physics necessary to understand the current developments in the field of fundamental cosmology, as well as an overview ofthe observational data and methods. It will help students to get into research by providingdefinitions and main techniques and ideas discussed today. The book is divided into threeparts. Part 1 summarises the fundamentals in theoretical physics needed in cosmology(general relativity, field theory, particle physics). Part 2 describes the standard model ofcosmology and includes cosmological solutions of Einstein equations, the hot big bangmodel, cosmological perturbation theory, cosmic microwave background anisotropies,lensing and evidence for dark matter, and inflation. Part 3 describes extensions of thismodel and opens up current research in the field: scalar-tensor theories, supersymmetry,the cosmological constant problem and acceleration of the universe, topology of the universe, grand unification and baryogenesis, topological defects and phase transitions,string inspired cosmology including branes and the latest developments. The book provides details of all derivations and leads the student up to the level of research articles.

Patrick Peter and Jean-Philippe Uzan are at the Institut d'Astrophysique de Paris, France.

‘Fills a niche that other recent cosmology texts leave open, namely self-contained derivations in cosmology that span both fundamental issues and applications to the realuniverse that are of great interest to observers.’ Joseph Silk, University of Oxford

‘A remarkable book. Written with great authority and enthusiasm, it gives a comprehensiveview of primordial cosmology today. The style is accessible to a novice for a good introduction, as well as being technically precise enough to be useful for specialists.’

Ted Jacobson, University of Maryland

Primordial Cosmology

Patrick PeterJean-Philippe Uzan

oxford graduate texts

9 780199 209910

2

ogt

PrimordialCosm

ologyPeter&

Uzan

peter&uzan_ppc.qxd 27/2/09 09:11 Page 1



a(t) ! 0

a

a=

1

3[! " 4!GN (" + p)]

H2 +Ka2

=1

3(8!GN" + !)

# =1

3

p = #"

Tµ! = pgµ! + (" + p)uµu!

ds2 = "dt2 + a2 (t)

!

dr2

1 "Kr2+ r2

"

d$2 + sin2 $d%2#

$

,

1

Problems:

Singularity

Horizon

Flatness

Homogeneity

Perturbations

Dark matter


Baryogenesis

...



Problems:

Singularity

Horizon

Flatness

Homogeneity

Perturbations

Dark matter


Baryogenesis

...


O

A”

A

A’

B

B’ B”

Space

Tim

e

Last scattering surface

1°

Decouplingzdec

Horizonat decoupling

O

Infinite redshift


Problems:

Singularity

Horizon

Flatness

Homogeneity

Perturbations

Dark matter


Baryogenesis

...


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

6


! !!tot

!crit

a

a=

1

3[" " 4"GN (! + p)]

H2 +Ka2

=1

3(8"GN! + ")

# =1

3

p = #!

Tµ! = pgµ! + (! + p)uµu!

ds2 = "dt2 + a2 (t)

!

dr2

1 "Kr2+ r2

"

d$2 + sin2 $d%2#

$

,

1


! !!tot

!crit

! = 1 =" K = 0

a

a=

1

3[" # 4"GN (! + p)]

H2 +Ka2

=1

3(8"GN! + ")

# =1

3

p = #!

Tµ! = pgµ! + (! + p)uµu!

ds2 = #dt2 + a2 (t)

!

dr2

1 #Kr2+ r2

"

d$2 + sin2 $d%2#

$

,

1

Flat space

Unstable!


!crit !3H2

8"GN

! !!tot

!crit

! = 1 =" K = 0

a

a=

1

3[" # 4"GN (! + p)]

H2 +Ka2

=1

3(8"GN! + ")

# =1

3

p = #!

Tµ! = pgµ! + (! + p)uµu!

1


Problems:

Singularity

Horizon

Flatness

Homogeneity

Perturbations

Dark matter


Baryogenesis

...

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

6


Accepted solution = INFLATION


a > 0 & a > 0

!crit !3H2

8"GN

! !!tot

!crit

! = 1 =" K = 0

a

a=

1

3[" # 4"GN (! + p)]

H2 +Ka2

=1

3(8"GN! + ")

# =1

3

p = #!

1


Problems:

Singularity

Horizon

Flatness

Homogeneity

Perturbations

Dark matter


Baryogenesis

...




Problems:

Singularity

Horizon

Flatness

Homogeneity

Perturbations

Dark matter


Baryogenesis

...



(Linde’s book)


Problems of Inflation 3

t

xtR

t0

ti

tf (k)

ti (k)

k

H!1

Fig. 1. Space-time diagram (sketch) showing the evolution of scales in inflationarycosmology. The vertical axis is time, and the period of inflation lasts between ti andtR, and is followed by the radiation-dominated phase of standard big bang cosmol-ogy. During exponential inflation, the Hubble radius H

!1 is constant in physicalspatial coordinates (the horizontal axis), whereas it increases linearly in time aftertR. The physical length corresponding to a fixed comoving length scale labelled byits wavenumber k increases exponentially during inflation but increases less fast thanthe Hubble radius (namely as t

1/2), after inflation.

From R. Brandenberger, in M. Lemoine, J. Martin & P. P. (Eds.), “Inflationary cosmology”,Lect. Notes Phys. 738 (Springer, Berlin, 2007).

Scalar field origin?

Trans-Planckian

Hierarchy (amplitude)

Singularity

Validity of classical GR?

V (!)

!!4! 10!12

1

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

1

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

1

!t; !(t) = !0a(t)

a0" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

a(#)

!

"

"+g

"+d

#

$ = Tij(k)

!

"

"!g

"!

d

#

$

vk %e!ic

Sk!

%

2cSk

1


Inflation:

solves cosmological puzzlesuses GR + scalar fields [(semi-)classical]can be implemented in high energy theoriesmakes falsifiable predictions ...... consistent with all known observationsstring based ideas (brane inflation, ...)

11


Alternative model???

Inflation:


11



purely classical theory

Inflation:


11




Inflation:


Quantum gravity / cosmology

11




singularity, initial conditions & homogeneity

Inflation:



11




singularity, initial conditions & homogeneity

Inflation:



bounces (always in WdW - recall N. Pinto-Neto’s talk)

11


!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

#

1

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

a(#)

1


Ekpyrotic scenario:

3

inspiration in the extra dimensional scenarios, a la Ran-dall – Sundrum [4], and can be motivated by compact-ifying the action of 11 dimensional supergravity on anS1/Z2 orbifold, compactified on a Calabi–Yau three-fold.This results in an e!ectively five dimensional action read-ing

S5 !!

M5

d5x"

"g5

#

R(5)

"1

2(!")2 "

3

2

e2!F2

5 !

$

, (1)

where # is the scalar modulus, and F the field strength ofa four-form gauge field. Two four–dimensional boundarybranes (orbifold fixed planes), one of which to be lateridentified with our universe, are separated by a finite gap.Both are BPS states [13], i.e., they can be described atlow energy by an e!ective N = 1 supersymmetric model,so that their curvature vanishes. This is how the flatnessproblem is addressed in the ekpyrotic model.

Rorb

Bulk

K=0

Visible

Hidden

4 D

bra

ne,

Qua

si B

PS

FIG. 1: Schematic representation of the old ekpyrotic modelas a bulk – boundary branes in an e!ective five dimensionaltheory. Our Universe is to be identified with the visible brane,and a bulk brane is spontaneously nucleated near the hiddenbrane, moving towards our universe to produce the Big-Bangsingularity and primordial perturbations. In the new ekpy-rotic scenario, the bulk brane is absent and it is the hiddenbrane that collides with the visible one, generating the hotBig Bang singularity.

In the “old” scenario [8], the five dimensional bulk isalso assumed to contain various fields not described here,whose excitations can lead to the spontaneous nucleationof yet another, much lighter, freely moving, brane. Inthe so-called “new” scenario [9], and its cyclic exten-sion [23], it is the hidden boundary brane that is ableto move in the bulk. In both cases, this extra brane, ifassumed BPS (as demanded by minimization of the ac-tion) is flat, parallel to the boundary branes and initially

at rest. Non perturbative e!ects yield an interaction po-tential between the visible and the bulk brane. The dis-tance of the former to the latter can be regarded as ascalar field living on the four dimensional visible bound-ary brane whose e!ective action is thus that of four di-mensional GR together with a scalar field " evolving inan exponential potential, namely

S4 =

!

M4

d4x"

"g4

#

R(4)

2$"

1

2(!#)2 " V (#)

$

, (2)

with

V (") = "Vi exp

#

"4#

%&

mPl

(" " "i)

$

, (3)

where & is a constant and $ = 8%G = 8%/m2Pl

. Apartfrom the sign, the potential is the one that leads to thewell known power-law inflation model if the value of &lies in a given range [24].

The interaction between the two branes results in one(bulk or hidden) brane moving towards the other (vis-ible) boundary until they collide. This impact time isthen identified with the Big-Bang of standard cosmol-ogy. Slightly before that time, the exponential potentialabruptly goes to zero so the boundary brane is led to asingular transition at which the kinetic energy of the bulkbrane is converted into radiation. The result is, from thispoint on, exactly similar to standard big bang cosmology,with the di!erence that the flatness problem is claimedto be solved by saying our Universe originated as a BPSstate (see however [23]).

FIG. 2: Scale factor in the new ekpyrotic scenario. TheUniverse starts its evolution with a slow contraction phasea ! ("!)1+! with " = "0.9 on the figure. The bounce itselfis explicitly associated with a singularity which is approachedby the scalar field kinetic term domination phase, and theexpansion then connects to the standard Big-Bang radiationdominated phase.

3


S5 !!

M5

d5x"

"g5

#

R(5)

"1

2(!")2 "

3

2

e2!F2

5 !

$

, (1)


Rorb

Bulk

K=0Visible

Hidden

4 D

bra

ne,

Qua

si B

PS




S4 =

!

M4

d4x"

"g4

#

R(4)

2$"

1

2(!#)2 " V (#)

$

, (2)

with

V (") = "Vi exp

#

"4#

%&

mPl

(" " "i)

$

, (3)





3


S5 !!

M5

d5x"

"g5

#

R(5)

"1

2(!")2 "

3

2

e2!F2

5 !

$

, (1)


Rorb

Bulk

K=0

Visible

Hidden

4 D

bra

ne,

Qua

si B

PS




S4 =

!

M4

d4x"

"g4

#

R(4)

2$"

1

2(!#)2 " V (#)

$

, (2)

with

V (") = "Vi exp

#

"4#

%&

mPl

(" " "i)

$

, (3)





3


S5 !!

M5

d5x"

"g5

#

R(5)

"1

2(!")2 "

3

2

e2!F2

5 !

$

, (1)


Rorb

Bulk

K=0

Visible

Hidden

4 D

bra

ne,

Qua

si B

PS




S4 =

!

M4

d4x"

"g4

#

R(4)

2$"

1

2(!#)2 " V (#)

$

, (2)

with

V (") = "Vi exp

#

"4#

%&

mPl

(" " "i)

$

, (3)





3


S5 !!

M5

d5x"

"g5

#

R(5)

"1

2(!")2 "

3

2

e2!F2

5 !

$

, (1)


Rorb

Bulk

K=0

Visible

Hidden

4 D

bra

ne,

Qua

si B

PS




S4 =

!

M4

d4x"

"g4

#

R(4)

2$"

1

2(!#)2 " V (#)

$

, (2)

with

V (") = "Vi exp

#

"4#

%&

mPl

(" " "i)

$

, (3)






Standard Failures and some (bouncing) solutions

Singularity

Horizon

Flatness

Homogeneity

PerturbationsOthers



Singularity

Horizon

Flatness

Homogeneity

PerturbationsOthers

Merely a non issue in the bounce case!



Singularity

Horizon

Flatness

Homogeneity

PerturbationsOthers


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

6

can be made divergent easily if

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

6



Singularity

Horizon

Flatness

Homogeneity

PerturbationsOthers


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

6

accelerated expansion (inflation) or decelerated contraction (bounce)


a < 0 & a < 0

!crit !3H2

8"GN

! !!tot

!crit

! = 1 =" K = 0

a

a=

1

3[" # 4"GN (! + p)]

H2 +Ka2

=1

3(8"GN! + ")

# =1

3

p = #!

1

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

6


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

6



Singularity

Horizon

Flatness

Homogeneity

PerturbationsOthers


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

6



a < 0 & a < 0

!crit !3H2

8"GN

! !!tot

!crit

! = 1 =" K = 0

a

a=

1

3[" # 4"GN (! + p)]

H2 +Ka2

=1

3(8"GN! + ")

# =1

3

p = #!

1

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

6


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

6

Large & flat Universe + low initial density + diffusion

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

tdi!usion

tHubble%

#

R1/3H

#

1 +#

AR2H

$

="

6

enough time to dissipate any wavelength

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

tdi!usion

tHubble%

#

R1/3H

#

1 +#

AR2H

$

="

6

vacuum state!

tdissipation

!0

I = R !!

3 (4Rµ!Rµ! ! R2)

="dV

d"= I

S =1

16#GN

"

d4x#!g

#

R +N

$

i=1

"iI(i) ! V (!)

%

s $ n#

max(s) =

Aa sin&

s#

1 + x2'

+ Ba cos&

s#

1 + x2'

sin s

1



Singularity

Horizon

Flatness

Homogeneity

PerturbationsOthers


see coming slides

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

6



a < 0 & a < 0

!crit !3H2

8"GN

! !!tot

!crit

! = 1 =" K = 0

a

a=

1

3[" # 4"GN (! + p)]

H2 +Ka2

=1

3(8"GN! + ")

# =1

3

p = #!

1

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

6


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

6


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

tdi!usion

tHubble%

#

R1/3H

#

1 +#

AR2H

$

="

6


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

tdi!usion

tHubble%

#

R1/3H

#

1 +#

AR2H

$

="

6

vacuum state!

tdissipation

!0

I = R !!

3 (4Rµ!Rµ! ! R2)

="dV

d"= I

S =1

16#GN

"

d4x#!g

#

R +N

$

i=1

"iI(i) ! V (!)

%

s $ n#

max(s) =

Aa sin&

s#

1 + x2'

+ Ba cos&

s#

1 + x2'

sin s

1



Singularity

Horizon

Flatness

Homogeneity

PerturbationsOthers


see coming slides

dark matter/energy, baryogenesis, ...

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

6



a < 0 & a < 0

!crit !3H2

8"GN

! !!tot

!crit

! = 1 =" K = 0

a

a=

1

3[" # 4"GN (! + p)]

H2 +Ka2

=1

3(8"GN! + ")

# =1

3

p = #!

1

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

6


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" nS #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

6


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

tdi!usion

tHubble%

#

R1/3H

#

1 +#

AR2H

$

="

6


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

tdi!usion

tHubble%

#

R1/3H

#

1 +#

AR2H

$

="

6

vacuum state!

tdissipation

!0

I = R !!

3 (4Rµ!Rµ! ! R2)

="dV

d"= I

S =1

16#GN

"

d4x#!g

#

R +N

$

i=1

"iI(i) ! V (!)

%

s $ n#

max(s) =

Aa sin&

s#

1 + x2'

+ Ba cos&

s#

1 + x2'

sin s

1


Initial conditionsfixed in the contracting era

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

#

1

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

a(#)

1

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

a(#)

!

"

"+g

"+d

#

$ = Tij(k)

!

"

"!g

"!

d

#

$

1

a = a0

!

1 +

"

T

T0

#2$

13(1!!)

K = +1

a(T )

Q(T )

H = H(0) + H(2) + · · ·

! = !(0) (a, T )!(2) [v, T ; a (T )]

"# = !3!2

Pl

2

%

" + p

#a

d

d$

&v

a

'

ds2 = a2($)(

(1 + 2#) d$2 ! [(1 ! 2#) %ij + hij ] dxidxj)

d$ = a3!!1dT

i&!(2)

&$=

*

d3x

"

!1

2

'2

'v2+

#

2v,iv

,i !a""

a

#

!(2)

v""k +

"

c2Sk2 !

a""

a

#

vk = 0

4

Perturbations:


Initial conditionsfixed in the contracting era

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

#

1

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

a(#)

1

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

a(#)

!

"

"+g

"+d

#

$ = Tij(k)

!

"

"!g

"!

d

#

$

1

???????

a = a0

!

1 +

"

T

T0

#2$

13(1!!)

K = +1

a(T )

Q(T )

H = H(0) + H(2) + · · ·

! = !(0) (a, T )!(2) [v, T ; a (T )]

"# = !3!2

Pl

2

%

" + p

#a

d

d$

&v

a

'

ds2 = a2($)(

(1 + 2#) d$2 ! [(1 ! 2#) %ij + hij ] dxidxj)

d$ = a3!!1dT

i&!(2)

&$=

*

d3x

"

!1

2

'2

'v2+

#

2v,iv

,i !a""

a

#

!(2)

v""k +

"

c2Sk2 !

a""

a

#

vk = 0

4

Perturbations:


100 1000

k or n

10-6

10-4

10-2

100

102

k3|!

BS

T|2

-1.5 -1 -0.5 0 0.5 1 1.5

t

0

50

100

150

200

Vu(t)

-1 0 1-15

-10

-5

0

5

10

15

!BST(t)

16

a = a0

!

1 +

"

T

T0

#2$

13(1!!)

K = +1

a(T )

Q(T )

H = H(0) + H(2) + · · ·

! = !(0) (a, T )!(2) [v, T ; a (T )]

"# = !3!2

Pl

2

%

" + p

#a

d

d$

&v

a

'

ds2 = a2($)(

(1 + 2#) d$2 ! [(1 ! 2#) %ij + hij ] dxidxj)

d$ = a3!!1dT

i&!(2)

&$=

*

d3x

"

!1

2

'2

'v2+

#

2v,iv

,i !a""

a

#

!(2)

v""k +

"

c2Sk2 !

a""

a

#

vk = 0

4

Perturbations:

S =

!

d4x!"g

"

R

6!2Pl

"1

2"µ#"µ# " V (#)

#

ds2 = dt2 " a2(t)

$

dr2

1 "Kr2+ r2d!2

%

H2 =1

3

$

1

2#2 + V

%

"Ka2

" =3Hu

2a2$$ #

1

a

&

%!

%! + p!

$

1 "3K

%!a2

%

7

S =

!

d4x!"g

"

R

6!2Pl

"1

2"µ#"µ# " V (#)

#

ds2 = dt2 " a2(t)

$

dr2

1 "Kr2+ r2d!2

%

H2 =1

3

$

1

2#2 + V

%

"Ka2

" =3Hu

2a2$$ #

1

a

&

%!

%! + p!

$

1 "3K

%!a2

%

u!! +

"

k2 "$!!

$" 3K

'

1 " c2S

(

#

u = 0

P" = AknS"1 cos

$

&kph

k#+ '

%

L = p(X, #)

X #1

2gµ$"µ#"$#

T µ$ = (% + p)uµu$ " pgµ$

% # 2X"p

"X" p

uµ #"µ#!2X

8

! > !1

3

S =

!

d4x"!g

"

R

6"2Pl

!1

2#µ$#µ$ ! V ($)

#

ds2 = dt2 ! a2(t)

$

dr2

1 !Kr2+ r2d!2

%

H2 =1

3

$

1

2$2 + V

%

!Ka2

" =3Hu

2a2%% #

1

a

&

&!

&! + p!

$

1 !3K

&!a2

%

u!! +

"

k2 !%!!

%! 3K

'

1 ! c2S

(

#

u = 0

P" = AknS"1 cos2$

!kph

k#+ '

%

L = p(X, $)

X #1

2gµ$#µ$#$$

T µ$ = (& + p)uµu$ ! pgµ$

& # 2X#p

#X! p

8

Bunch-Davies vacuum initial conditions: quantized perturbations over a classical background!!!


T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

d

dt|! ' 1| = '2

a

a3

tdi!usion

tHubble%

#

R1/3H

#

1 +#

AR2H

$

="

$ > '1

3

6

17

A specific model: 4D Quantum cosmology

Perfect fluid: bounce

no horizon problem if

Results:


H =

!

!p2

a

4a!Ka +

pT

a3!

"

N

a3!

H! = 0

i!!

!T=

1

4a3(!!1)/2 !

!a

#

a(3!!1)/2 !

!a

$

! + Ka!

K = 0 =" " #2a3(1!!)/2

3(1 ! #)=" i

!!

!T=

1

4

!2!

!"2

" > 0

!!!

!"= !

!!

!"

! =

%

eiET $(E)%E(T )dE

$ e!(ET0)2

! =

&

8T0

& (T 20 + T 2)

2

'14

exp

!

!T0"

2

T 20 + T 2

"

e!iS(",T )

S =T"2

T 20 + T 2

+1

2arctan

T0

T!

&

4

4

Quantum cosmology+ canonical transformation+ rescaling (volume …)+ units

= a simple Hamiltonian:

constraint

i∂Ψ∂T

=14a3(ω−1)/2 ∂

∂a

a(3ω−1)/2 ∂

∂a

Ψ +KaΨ

K = 0 =⇒ χ ≡ 2a2(1−ω/23(1− ω)

=⇒ i∂Ψ∂T

=14

∂2Ψ∂χ2

χ > 0

Ψ∂Ψ∂χ

= Ψ∂Ψ∂χ

3

Wheeler-De Witt

∃ x(t)

md2

x(t)dt2

= −∇ (V + Q)

∃t0; ρ (x, t0) = |Ψ (x, t0)|2

Q −→ 0

ds2 = N

2(τ)dτ − a2(τ)γijdx

idxj

p = p0

ϕ + θs

N(1 + ω)

1+ωω

(ϕ, θ, s) =

T = −pse−s/s0p−(1+ω)ϕ s0ρ

−ω0

H =−p

2a

4a−Ka +

pT

a3ω

N

a3ω

HΨ = 0

2

+ Technical trick:

18


a = a, H

a = a0

!

1 +

"

T

T0

#2$

13(1!!)

K = +1

a(T )

Q(T )

H = H(0) + H(2) + · · ·

! = !(0) (a, T )!(2) [v, T ; a (T )]

"# = !3!2

Pl

2

%

" + p

#a

d

d$

&v

a

'

ds2 = a2($)(

(1 + 2#) d$2 ! [(1 ! 2#) %ij + hij ] dxidxj)

d$ = a3!!1dT

i&!(2)

&$=

*

d3x

"

!1

2

'2

'v2+

#

2v,iv

,i !a""

a

#

!(2)

5

19

WKB exact superposition:

Gaussian wave packet

phase

Bohmian trajectory

i∂Ψ∂T

=14a3(ω−1)/2 ∂

∂a

a(3ω−1)/2 ∂

∂a

Ψ +KaΨ

K = 0 =⇒ χ ≡ 2a2(1−ω/23(1− ω)

=⇒ i∂Ψ∂T

=14

∂2Ψ∂χ2

χ > 0

Ψ∂Ψ∂χ

= Ψ∂Ψ∂χ

Ψ =

eiET ρ(E)ψE(T )dE

3

i∂Ψ∂T

=14a3(ω−1)/2 ∂

∂a

a(3ω−1)/2 ∂

∂a

Ψ +KaΨ

K = 0 =⇒ χ ≡ 2a2(1−ω/23(1− ω)

=⇒ i∂Ψ∂T

=14

∂2Ψ∂χ2

χ > 0

Ψ∂Ψ∂χ

= Ψ∂Ψ∂χ

Ψ =

eiET ρ(E)ψE(T )dE

∝ e−(ET0)2

3

i∂Ψ∂T

=14a3(ω−1)/2 ∂

∂a

a(3ω−1)/2 ∂

∂a

Ψ +KaΨ

K = 0 =⇒ χ ≡ 2a2(1−ω/23(1− ω)

=⇒ i∂Ψ∂T

=14

∂2Ψ∂χ2

χ > 0

Ψ∂Ψ∂χ

= Ψ∂Ψ∂χ

Ψ =

eiET ρ(E)ψE(T )dE

∝ e−(ET0)2

Ψ =

8T0

π (T 20 + T 2)2

14

exp− T0χ2

T 20 + T 2

e−iS(χ,T )

3

i∂Ψ∂T

=14a3(ω−1)/2 ∂

∂a

a(3ω−1)/2 ∂

∂a

Ψ +KaΨ

K = 0 =⇒ χ ≡ 2a2(1−ω/23(1− ω)

=⇒ i∂Ψ∂T

=14

∂2Ψ∂χ2

χ > 0

Ψ∂Ψ∂χ

= Ψ∂Ψ∂χ

Ψ =

eiET ρ(E)ψE(T )dE

∝ e−(ET0)2

Ψ =

8T0

π (T 20 + T 2)2

14

exp− T0χ2

T 20 + T 2

e−iS(χ,T )

S =Tχ2

T 20 + T 2

+12

arctanT0

T− π

4

3


quantum potential

i∂Ψ∂T

=14a3(ω−1)/2 ∂

∂a

a(3ω−1)/2 ∂

∂a

Ψ +KaΨ

K = 0 =⇒ χ ≡ 2a2(1−ω

/23(1− ω)

=⇒ i∂Ψ∂T

=14

∂2Ψ

∂χ2

χ > 0

Ψ∂Ψ∂χ

= Ψ∂Ψ∂χ

Ψ =

eiETρ(E)ψE(T )dE

∝ e−(ET0)2

Ψ =

8T0

π (T 20 + T 2)2

14

exp− T0χ

2

T20 + T 2

e−iS(χ,T )

S =Tχ

2

T20 + T 2

+12

arctanT0

T− π

4

a = a,H

a = a0

1 +

T

T0

2 1

3(1−ω)

K = 0

3

i∂Ψ∂T

=14a3(ω−1)/2 ∂

∂a

a(3ω−1)/2 ∂

∂a

Ψ +KaΨ

K = 0 =⇒ χ ≡ 2a2(1−ω

/23(1− ω)

=⇒ i∂Ψ∂T

=14

∂2Ψ

∂χ2

χ > 0

Ψ∂Ψ∂χ

= Ψ∂Ψ∂χ

Ψ =

eiETρ(E)ψE(T )dE

∝ e−(ET0)2

Ψ =

8T0

π (T 20 + T 2)2

14

exp− T0χ

2

T20 + T 2

e−iS(χ,T )

S =Tχ

2

T20 + T 2

+12

arctanT0

T− π

4

a = a,H

a = a0

1 +

T

T0

2 1

3(1−ω)

K = −1

3

i∂Ψ∂T

=14a3(ω−1)/2 ∂

∂a

a(3ω−1)/2 ∂

∂a

Ψ +KaΨ

K = 0 =⇒ χ ≡ 2a2(1−ω

/23(1− ω)

=⇒ i∂Ψ∂T

=14

∂2Ψ

∂χ2

χ > 0

Ψ∂Ψ∂χ

= Ψ∂Ψ∂χ

Ψ =

eiETρ(E)ψE(T )dE

∝ e−(ET0)2

Ψ =

8T0

π (T 20 + T 2)2

14

exp− T0χ

2

T20 + T 2

e−iS(χ,T )

S =Tχ

2

T20 + T 2

+12

arctanT0

T− π

4

a = a,H

a = a0

1 +

T

T0

2 1

3(1−ω)

K = +1

3

a(T )

Q(T )

4

a(T )

Q(T )

4

Trajectory


Usual treatment of the perturbations?

conformal time

Einstein-Hilbert action up to 2nd order

Wave function? No question about it ...

Bardeen (Newton) gravitational potential

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

4

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

ds2 = a

2(η)(1 + 2Φ) dη

2 − [(1− 2Φ) γij + hij ] dxidx

j

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

4

SE!H =!

d4x"

R(0) + !(2)R#

" =1

3

tdissipation

#0

I = R !$

3 (4Rµ!Rµ! ! R2)

="dV

d$= I

S =1

16%GN

!

d4x#!g

%

R +N

&

i=1

$iI(i) ! V (!)

'

1

Mukhanov-Sasaki variable

Simple scalar field with varying mass in Minkowski space!!! z = z[a(!)]

!

d4x "(2)L !!

d4x

"

(#v)2 +z!!

zv2

#

$ =1

3

tdissipation

!0

I = R "$

3 (4Rµ!Rµ! " R2)

=#dV

d%= I

1

d! = a(t)!1dt

z = z[a(!)]

!

d4x "(2)L !!

d4x

"

(#v)2 +z""

zv2

#

$ =1

3

tdissipation

!0

I = R "$

3 (4Rµ!Rµ! " R2)

1

d! = a(t)!1dt

z = z[a(!)]

!

d4x "(2)L =1

2

! !#d3x d!

"

($!v)2 " #ij$iv$jv +z""

zv2

#

% =1

3

tdissipation

!0

I = R "$

3 (4Rµ"Rµ" " R2)

1



conformal time




a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

4

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

ds2 = a

2(η)(1 + 2Φ) dη


j

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

4

SE!H =!

d4x"

R(0) + !(2)R#

" =1

3

tdissipation

#0

I = R !$

3 (4Rµ!Rµ! ! R2)

="dV

d$= I

S =1

16%GN

!

d4x#!g

%

R +N

&

i=1

$iI(i) ! V (!)

'

1



!

d4x "(2)L !!

d4x

"

(#v)2 +z!!

zv2

#

$ =1

3

tdissipation

!0

I = R "$

3 (4Rµ!Rµ! " R2)

=#dV

d%= I

1

Classical

d! = a(t)!1dt

z = z[a(!)]

!

d4x "(2)L !!

d4x

"

(#v)2 +z""

zv2

#

$ =1

3

tdissipation

!0

I = R "$

3 (4Rµ!Rµ! " R2)

1

d! = a(t)!1dt

z = z[a(!)]

!

d4x "(2)L =1

2

! !#d3x d!

"

($!v)2 " #ij$iv$jv +z""

zv2

#

% =1

3

tdissipation

!0

I = R "$

3 (4Rµ"Rµ" " R2)

1



conformal time




a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

4

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

ds2 = a

2(η)(1 + 2Φ) dη


j

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

4

SE!H =!

d4x"

R(0) + !(2)R#

" =1

3

tdissipation

#0

I = R !$

3 (4Rµ!Rµ! ! R2)

="dV

d$= I

S =1

16%GN

!

d4x#!g

%

R +N

&

i=1

$iI(i) ! V (!)

'

1



!

d4x "(2)L !!

d4x

"

(#v)2 +z!!

zv2

#

$ =1

3

tdissipation

!0

I = R "$

3 (4Rµ!Rµ! " R2)

=#dV

d%= I

1

Classical Quantum

d! = a(t)!1dt

z = z[a(!)]

!

d4x "(2)L !!

d4x

"

(#v)2 +z""

zv2

#

$ =1

3

tdissipation

!0

I = R "$

3 (4Rµ!Rµ! " R2)

1

d! = a(t)!1dt

z = z[a(!)]

!

d4x "(2)L =1

2

! !#d3x d!

"

($!v)2 " #ij$iv$jv +z""

zv2

#

% =1

3

tdissipation

!0

I = R "$

3 (4Rµ"Rµ" " R2)

1


Our treatment of the perturbations? Self-consistent ...

Hamiltonian up to 2nd order

factorization of the wave function

comes from 0th order

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

4

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

4

conformal time

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

4

K = +1

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

ds2 = a

2(η)(1 + 2Φ) dη


j

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

4


a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

ds2 = a

2(η)(1 + 2Φ) dη


j

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

4


Need dBB!!!

22

Our treatment of the perturbations? Self-consistent ...

Hamiltonian up to 2nd order

factorization of the wave function

comes from 0th order

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

4

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

4

conformal time

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

4

K = +1

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

ds2 = a

2(η)(1 + 2Φ) dη


j

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

4


a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

ds2 = a

2(η)(1 + 2Φ) dη


j

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

4


+ canonical transformations:

Fourier mode

Bunch-Davies vacuum initial conditions

+ evolution (matchings and/or numerics)

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

4

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

4

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

4

!t; !(t) = !0a0

a(t)" !

Pl

V (")

!"4" 10!12

!t(±"); a(t) # 0

Einf $ 10!3MPl

S3 > S2 > S1

a(#)

!

"

"+g

"+d

#

$ = Tij(k)

!

"

"!g

"!

d

#

$

vk %e!ic

Sk!

%

2cSk

1


100 1000

k or n

10-6

10-4

10-2

100

102

k3|!

BS

T|2

24

4

-1!108

-8!107

-6!107

-4!107

-2!107 0

2!107

4!107

6!107

8!107

1!108

x

102

104

106

108

1010

S(x

)

|"e S(x)|

|#m S(x)||S(x)|

~k=10

-3

-1!108

-5!107 0

5!107

1!108

x

100

102

104

106

108

v(x

)

|"e v(x)|

|#m v(x)||v(x)|

~k=10

-3

FIG. 1: Time evolution of the scalar mode function for the equation of state ! = 0.1 in the one-fluid model of the bounce. Theleft panel shows the full time evolution which was computed, i.e., the function S(x), while the right panel shows v(x) itself,both plots having k = 10!3. For x < 0, there are oscillations only in the real and imaginary parts of the mode, so the amplitudeshown is a non oscillating function of time. It however acquires an oscillating piece after the bounce has taken place.

!(a, T ) = a(1!3!)/2!

!!(0)(a, T )!

!

2. Its continuity equation

coming from Eq. (17) reads

"!

"T!

"

"a

"

a(3!!1)

2

"S

"a!

#

= 0, (21)

which implies in the Bohm interpretation that

a = !a(3!!1)

2

"S

"a, (22)

in accordance with the classical relations a = a, H =!a(3!!1)Pa/2 and Pa = "S/"a.

Inserting the phase of (20) into Eq. (22), we obtain theBohmian quantum trajectory for the scale factor:

a(T ) = a0

$

1 +

%

T

T0

&2'

13(1!!)

. (23)

Note that this solution has no singularities and tends tothe classical solution when T " ±#. Remember that weare in the gauge N = a3!, and T is related to conformaltime through

NdT = ad# =$ d# = [a(T )]3!!1 dT. (24)

The solution (23) can be obtained for other initial wavefunctions (see Ref. [8]).

The Bohmian quantum trajectory a(T ) can be usedin Eq. (16). Indeed, since one can view a(T ) as a func-tion of T , it is possible to implement the time dependentcanonical transformation generated by

U = exp

"

i

%(

d3xav2

2a

&#

% (25)

% exp

)

i

"(

d3x

%

v$ + $v

2

&

ln

%

1

a

&#*

. (26)

As a(T ) is a given quantum trajectory coming fromEq. (17), Eq. (25) must be viewed as the generator ofa time dependent canonical transformation to Eq. (17).It yields, in terms of conformal time, the equation for!(2)[v, a(#), #]

i"!(2)

"#=

(

d3x

%

!1

2

%2

%v2+

&

2v,iv

,i !a""

2av2

&

!(2).

(27)

This is the most simple form of the Schrodinger equa-tion which governs scalar perturbations in a quantumminisuperspace model with fluid matter source. The cor-responding time evolution equation for the operator v inthe Heisenberg picture is given by

v"" ! &v,i,i !

a""

av = 0, (28)

where a prime means derivative with respect to confor-mal time. In terms of the normal modes vk, the aboveequation reads

v""k +

%

&k2 !a""

a

&

vk = 0. (29)

These equations have the same form as the equations forscalar perturbations obtained in Ref. [1]. This is quitenatural since for a single fluid with constant equation ofstate &, the pump function z""/z obtained in Ref. [1] isexactly equal to a""/a obtained here. The di"erence isthat the function a(#) is no longer a classical solutionof the background equations but a quantum Bohmiantrajectory of the quantized background, which may leadto di"erent power spectra.

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

s(x) = a12 (1−3ω) v(x)√

T0

4

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

s(x) = a12 (1−3ω) v(x)√

T0

x ≡ T

T0

4

4

-1!108

-8!107

-6!107

-4!107

-2!107 0

2!107

4!107

6!107

8!107

1!108

x

102

104

106

108

1010

S(x

)

|"e S(x)|

|#m S(x)||S(x)|

~k=10

-3

-1!108

-5!107 0

5!107

1!108

x

100

102

104

106

108

v(x

)

|"e v(x)|

|#m v(x)||v(x)|

~k=10

-3


!(a, T ) = a(1!3!)/2!

!!(0)(a, T )!

!



"!

"T!

"

"a

"

a(3!!1)

2

"S

"a!

#

= 0, (21)


a = !a(3!!1)

2

"S

"a, (22)



a(T ) = a0

$

1 +

%

T

T0

&2'

13(1!!)

. (23)


NdT = ad# =$ d# = [a(T )]3!!1 dT. (24)



U = exp

"

i

%(

d3xav2

2a

&#

% (25)

% exp

)

i

"(

d3x

%

v$ + $v

2

&

ln

%

1

a

&#*

. (26)


i"!(2)

"#=

(

d3x

%

!1

2

%2

%v2+

&

2v,iv

,i !a""

2av2

&

!(2).

(27)


v"" ! &v,i,i !

a""

av = 0, (28)


v""k +

%

&k2 !a""

a

&

vk = 0. (29)



100 1000

k or n

10-6

10-4

10-2

100

102

k3|!

BS

T|2

24

4

-1!108

-8!107

-6!107

-4!107

-2!107 0

2!107

4!107

6!107

8!107

1!108

x

102

104

106

108

1010

S(x

)

|"e S(x)|

|#m S(x)||S(x)|

~k=10

-3

-1!108

-5!107 0

5!107

1!108

x

100

102

104

106

108

v(x

)

|"e v(x)|

|#m v(x)||v(x)|

~k=10

-3


!(a, T ) = a(1!3!)/2!

!!(0)(a, T )!

!



"!

"T!

"

"a

"

a(3!!1)

2

"S

"a!

#

= 0, (21)


a = !a(3!!1)

2

"S

"a, (22)



a(T ) = a0

$

1 +

%

T

T0

&2'

13(1!!)

. (23)


NdT = ad# =$ d# = [a(T )]3!!1 dT. (24)



U = exp

"

i

%(

d3xav2

2a

&#

% (25)

% exp

)

i

"(

d3x

%

v$ + $v

2

&

ln

%

1

a

&#*

. (26)


i"!(2)

"#=

(

d3x

%

!1

2

%2

%v2+

&

2v,iv

,i !a""

2av2

&

!(2).

(27)


v"" ! &v,i,i !

a""

av = 0, (28)


v""k +

%

&k2 !a""

a

&

vk = 0. (29)


a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

s(x) = a12 (1−3ω) v(x)√

T0

4

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

s(x) = a12 (1−3ω) v(x)√

T0

x ≡ T

T0

4

4

-1!108

-8!107

-6!107

-4!107

-2!107 0

2!107

4!107

6!107

8!107

1!108

x

102

104

106

108

1010

S(x

)

|"e S(x)|

|#m S(x)||S(x)|

~k=10

-3

-1!108

-5!107 0

5!107

1!108

x

100

102

104

106

108

v(x

)

|"e v(x)|

|#m v(x)||v(x)|

~k=10

-3


!(a, T ) = a(1!3!)/2!

!!(0)(a, T )!

!



"!

"T!

"

"a

"

a(3!!1)

2

"S

"a!

#

= 0, (21)


a = !a(3!!1)

2

"S

"a, (22)



a(T ) = a0

$

1 +

%

T

T0

&2'

13(1!!)

. (23)


NdT = ad# =$ d# = [a(T )]3!!1 dT. (24)



U = exp

"

i

%(

d3xav2

2a

&#

% (25)

% exp

)

i

"(

d3x

%

v$ + $v

2

&

ln

%

1

a

&#*

. (26)


i"!(2)

"#=

(

d3x

%

!1

2

%2

%v2+

&

2v,iv

,i !a""

2av2

&

!(2).

(27)


v"" ! &v,i,i !

a""

av = 0, (28)


v""k +

%

&k2 !a""

a

&

vk = 0. (29)


Best fit to the data!!!


10 20 50 100 200 500 1000

1000

2000

3000

4000

5000

6000

l

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!l"1"Cl

2#

Data!

105

104

1000

100

10

1

104 1000

0.001 0.01 0.1 1 10

100 10 1

Wavelength (h–1 Mpc)

Wave number k (h / Mpc)

Pow

er s

pect

rum

tod

ay P

(k)

(h–1

Mpc

)3

CMBSDSS Galaxies Cluster abundanceGravitational lensingLyman – ! forest

No obvious oscillations ...


1 10 100 1000l

02

00

04

00

06

00

0

l(l

+1

)Cl /2!!!"!#"$%

WMAP1WMAP3

#=103 k

*=10

-2 Mpc

-1


! ! 1

! =1

3

tdissipation

"0

I = R "!

3 (4Rµ!Rµ! " R2)

=#dV

d#= I

S =1

16$GN

"

d4x$"g

#

R +N

$

i=1

#iI(i) " V (!)

%

1

27

spectrum

id. grav. waves:

same dynamics + initial conditions same spectrum

CMB normalisation

bounce curvature

x ≡ T

T0

PΦ ∝ k3 |Φk|2 ∝ A2SknS−1

5

x ≡ T

T0


µ +

k2 − a

a

µ = 0

µ ≡ h

a

Ph ∝ k3 |hk|2 ∝ A2TknT

µini ∝exp(−ikη)√

kη

nT = nS − 1 =12ω

1 + 3ω

A2S

= 2.08× 10−10

T0a3ω0 1500Pl

5

x ≡ T

T0


µ +

k2 − a

a

µ = 0

µ ≡ h

a



kη

nT = nS − 1 =12ω

1 + 3ω

A2S

= 2.08× 10−10

T0a3ω0 1500Pl

5

x ≡ T

T0


µ +

k2 − a

a

µ = 0

µ ≡ h

a



kη

nT = nS − 1 =12ω

1 + 3ω

A2S

= 2.08× 10−10

T0a3ω0 1500Pl

5

x ≡ T

T0


µ +

k2 − a

a

µ = 0

µ ≡ h

a



kη

nT = nS − 1 =12ω

1 + 3ω

A2S

= 2.08× 10−10

T0a3ω0 1500Pl

5

x ≡ T

T0


µ +

k2 − a

a

µ = 0

µ ≡ h

a



kη

nT = nS − 1 =12ω

1 + 3ω

A2S

= 2.08× 10−10

T0a3ω0 1500Pl

5

x ≡ T

T0


µ +

k2 − a

a

µ = 0

µ ≡ h

a



kη

nT = nS − 1 =12ω

1 + 3ω

A2S

= 2.08× 10−10

T0a3ω0 1500Pl

5

x ≡ T

T0


µ +

k2 − a

a

µ = 0

µ ≡ h

a



kη

nT = nS − 1 =12ω

1 + 3ω

A2S

= 2.08× 10−10

T0a3ω0 1500Pl

5

a(T )

Q(T )

H = H(0) + H(2) + · · ·

Ψ = Ψ(0) (a, T ) Ψ(2)) [v, T ; a (T )]

∆Φ = −3

2Pl

2

ρ + p

ωa

ddη

v

a

ds2 = a

2(η)(1 + 2Φ) dη


j

dη = a3ω−1dT

i∂Ψ(2)

∂η=

d3

x

−1

2δ2

δv2+

ω

2v,iv

,i − a

a

Ψ(2)

vk +

c2Sk

2 − a

a

vk = 0

c2S

=√

ω = 0

vk ∝exp(−icSkη

2cSk

4

scale invariance + amplification! ! 1

! =1

3

tdissipation

"0

I = R "!

3 (4Rµ!Rµ! " R2)

=#dV

d#= I

S =1

16$GN

"

d4x$"g

#

R +N

$

i=1

#iI(i) " V (!)

%

1

2 stages are necessary


7

-100 -75 -50 -25 0 25 50 75 100~kx

10-20

10-15

10-10

10-5

100

105

Sp

ectr

a

P!

(x)/ ~k

nS-1

, ~k=10

-7

Ph(x)/

~k

nT,

~k=10

-7

P!

(x)/ ~k

nS-1

, ~k=10

-5

Ph(x)/

~k

nT,

~k=10

-5

"=0.01

FIG. 2: Rescaled power spectra for scalar and tensor pertur-bations as functions of time for ! = 0.01 and two di!erentvalues of k. It is clear from the figure that not only bothspectra reach a constant mode, but also that this mode doesbehave as indicated in Eqs. (46) and (50). It is purely inci-dental that the actual constant value of both modes are veryclose for that particular value of !. The constant values ob-tained in this figure are the one used to derive the spectrumbelow. In this figure and the following, the value of nS usedto rescale " is the one derived in Eq. (46), thus proving thevalidity of the analytic calculation.

Defining

L0 = T0a3!0 , (53)

the curvature scale at the bounce (the characteristicbounce length-scale), namely, L0 ! 1/

"R0 where R0 is

the scalar curvature at the bounce, from which one canwrite

k =k

a0L0 #

L0

!bouncephys

, (54)

one obtains the scalar perturbation density spectrum asa function of time through

P! =2(" + 1)

#2"

" (1 $ "))2k|f(x)|2

!

$Pl

L0

"2

, (55)

where

f(x) ##

1 + x2$! 1+3!

3(1!!)dS

dx$ x

#

1 + x2$! 4

3(1!!) S,

while the tensor power spectrum is

Ph =2k3

#2

|v|2

1 + x2

!

$Pl

L0

"2

, (56)

in which the rescaled function v, defined through [11]

v #a

12 (1!3!)µ

$Pl

"T0

,

10-6

10-5

10-4

10-3

10-2

10-1

100

101

~k

10-4

10-2

100

102

P!

( ~ k)/

~ kn

S-1

, P

h(

~ k)/

~ kn

T a

nd

T

/S

P!

( ~k)/

~k

nS-1

Ph(

~k)/

~k

nT

T/S

FIG. 3: Rescaled power spectra for ! = 8!10!4, correspond-ing to the conservative maximum bound on the deviation froma scale invariant spectrum nS = 1.01, as function of k. Thescalar spectrum P! is the full line, while the dashed line isthe gravitational wave spectrum Ph. Also shown is the ratioT/S (dotted); in this case, the T/S " 5.2 ! 10!3, i.e. almosttwo orders of magnitude below the current limit. This casehas a typical bounce length-scale of L0 # 1.47 ! 103"Pl . Theamplitude of the modes is obtained as the constant part ofFig. 2.

satisfies the same dynamical equation (51) with cs % 1,with µ subject to initial condition

µini =

%

3

k$Ple

!ik" . (57)

From the above defined spectra, one reads the ampli-tudes

A2S#

4

25P# (58)

and

A2T#

1

100Ph, (59)

where we assume the classical relation between ! and thecurvature perturbation % through

% =5 + 3"

3(1 + ")! % P# =

&

5 + 3"

3(1 + ")

'2

P!, (60)

to obtain the observed spectrum. Since both spectra areidentical power laws, and indeed almost scale invariantpower laws, the tensor-to-scalar (T/S) ratio, defined bythe CMB multipoles C$ at $ = 10 as

T

S#

C(T)10

C(S)10

= F(", · · ·)A2

T

A2S

, (61)

can easily be computed (see, e.g., [16] and referencestherein). In Eq. (61), the function F depends entirely


WMAP constraint

predictions

spectrum slightly blue

power-law + concordance

x ≡ T

T0


µ +

k2 − a

a

µ = 0

µ ≡ h

a



kη

nT = nS − 1 =12ω

1 + 3ω

A2S

= 2.08× 10−10

T0a3ω0 1500Pl

nS = 0.96± 0.02 =⇒ nS ∼< 8× 10−4

0.62

5

T

S=

C(T)10

C(S)10

= F (Ω, · · ·)A2

T

A2S

∝√

w

T

S 4× 10−2

nS − 1

6

T

S=

C(T)10

C(S)10

= F (Ω, · · ·)A2

T

A2S

∝√

w

T

S 4× 10−2

nS − 1

6

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

6


WMAP constraint

predictions

spectrum slightly blue

power-law + concordance

x ≡ T

T0


µ +

k2 − a

a

µ = 0

µ ≡ h

a



kη

nT = nS − 1 =12ω

1 + 3ω

A2S

= 2.08× 10−10

T0a3ω0 1500Pl

nS = 0.96± 0.02 =⇒ nS ∼< 8× 10−4

0.62

5

T

S=

C(T)10

C(S)10

= F (Ω, · · ·)A2

T

A2S

∝√

w

T

S 4× 10−2

nS − 1

6

T

S=

C(T)10

C(S)10

= F (Ω, · · ·)A2

T

A2S

∝√

w

T

S 4× 10−2

nS − 1

6

T0a3!0 ! 1500!Pl

nS = 0.96 ± 0.02 =" w #< 8 $ 10!4

! 0.62

T

S=

C(T)10

C(S)10

= F (!, · · ·)A2

T

A2S

%&

w

T

S! 4 $ 10!2

!

nS ' 1

dH ( a(t)

" t

ti

d"

a(")

ti ) '*

6


Numerical aside (Mike’s talk monday):


Numerical aside (Mike’s talk monday):

Scaling


monopoles = ???

Dark energy ...

dBB Cosmology without inflation?

New solutions to old puzzles

No singularity

G.R. ...

(oscillations, T/S ...)New predictions

Other models (many fluids, scalar fields, ...)

Possible and testable!

Model dependence

Future

Non gaussianities

Relaxation?Polarizations

Documents

TTI, Vallico Sotto, Tuscany - 09/02/10 P Pmdt26/tti_talks/deBB_10/peter_tti...Vallico Sotto - 2nd September 2010 TTI, Vallico Sotto, Tuscany - 09/02/10 P P Institut d’Astrophysique