Giuseppe De Risi M. Cavaglià, G.D., M. Gasperini, Phys. Lett. B 610:9-17, 2005 - hep-th/0501251 QG05, Sept. 2005

Giuseppe De RisiGiuseppe De Risi

M. Cavaglià, G.D., M. Gasperini, Phys. Lett. B 610:9-17, 2005 - hep-th/0501251

QG05, Sept. 2005QG05, Sept. 2005

Plan of the talkPlan of the talk

• Perturbations equation on the Kasner brane

• Amplification of the fluctuations

• Conclusions

• Kasner solution on the brane

• The braneworld scenario

The existence of extra dimensions in our universe was put forward in the far 20’s (Kaluza-Klein), and have became a major topic of the modern theoretical physics

In the end of the past century there was a turning point of how we “implement” extra-dimension in the universe (HDD ‘98 and Randall-Sundrum ‘99)

Before RS After RS

The braneworld scenarioThe braneworld scenario

BAAB

y dxdxz

dydxdxeds

2222

RS propose a solution of the form:

This metric is a solution of the 5D Einstein equation only if the bulk cosmological constant and the AdS radius are related

256

The wave equation that is obtained for the 5D transverse traceless perturbation hij(t,x,z) of the RS metric can be expanded on a basis of 4D modes:

m

mm ztvzth )(),(),,( xx

branedT

RgxdydMS L2

22

1 435

43

RS model consists of a flat brane (with tension) embedded in an AdS5 bulk: The starting point is, of course, the Einstein action

The braneworld scenario

0)(

14

15)(

3)(

22

z

zzmz mm

The massive modes contribution results in a correction to the Newtonian potential

2

221

21

1)(

rrmm

MrV

P

The z-dependent part of the equation decouples

The braneworld scenario

It is possible to generalize the RS scenario to have a cosmological evolution (Binetruy, Deffayet, Langlois ’99, Shiromizu, Maeda, Sasaki ’99).The model we analyze consists of an anisotropic brane embedded in AdS

DDp RgxdMS 2

21

)1()(2

1 pXgXXdT

ABBApp

We assume that the brane is rigidly located at the origin: aAX

we impose, for the metric, the ansatz

22222222 )()()( dztbtadtzfds dydx

d “external” dimensions x which should expand

Kasner solutions on the braneKasner solutions on the brane

n “internal” dimensions y which should contract

The brane and extradimensional equations can be decoupled

Evolution on the brane is characterized by a Kasner inflationary regime

0

)(tt

ta

0

)(tt

tbnddndn

)1(1

ndnndd

)1(1

The z-dependent part of the equation gives the AdS warp factor:

1

0

1)(

z

zzf

D

ppz

2)1(

0

Kasner solutions on the brane

And the tension and the bulk cosmological constant are related by the equation: 1

322

pp

M

TD

pp

a(t)

b(t)

To discuss metric perturbations we introduce the expansion:

ABABABABAB hgggg )1(

00 zAaAA hhh

where the perturbation takes values only in the external space and is in the TT gauge

jijij

ij hhg 0 zxthh iijij ,,

Perturbations equation on the Kasner branePerturbations equation on the Kasner brane

Studying the evolution of the cosmological perturbations is one of the most investigated topics in recent braneworld cosmology, because it would (possibly/hopefully) lead to direct comparison with observational data.

An interesting feature in the model under discussion is that, unlike other models (see, for example Langlois, Maartens, Wands ’00, Kobayashi, Kudoh, Tanaka ’03, Easther, Langlois, Maartens, Wands ’03) the massless and massive modes are always decoupled, as in ordinary RS

As in the standard RS case, we expand h in eigenmodes

)(,,,,, zxtvdmzxthdmzxth mi

mi

mi

so that the new variables satisfy

mmmm vmva

vnGdHv 22

2

mmm mpF 2

Perturbations equation on the Kasner brane

This equation can be obtained by the perturbed (to order o(h2)) action:

i

j

j

ii

jj

ii

jj

ipndD

p

hhha

hhhfbaxdM

S ''2

2)2(

8

The equation of motion for the perturbation is:

0)(

'''2

2

j

i

j

ij

ij

ij

i pFhhhta

hnGdHh

So the solution can be normalized in a canonical way, i.e. to the delta function

)()(ˆ)(ˆ ' mmzzdz mm

Substituting this into the action we can define an action for the single massive mode:

22

2

221

2

1)2(

)0(4mmmm

ndd

m

d

m hmha

hhbaxdM

S


Introducing the auxiliary field we obtain the Schrödinger-like equation

mpp

m fM 2ˆ

0ˆ

14

)2()(ˆ

200

2

mm

zz

ppz

zp

m

Volcano-like potential

22222)2( '21

mmmmd

m uamuuuxddS

We get the action

and the canonical equation for the Fourier modes um,k

0,222

,

kmkm uamku

immi

m xhxu ,)(, )0(2

)(22)1(2)1(

m

ndd

mbaM

This action can be put in a canonical form by describing the time evolution by the conformal time )(ta

dt

and introducing an auxiliary field um via the pump field m


The massless mode

The solution of the Schrödinger-like equation we obtain for m = 0, after imposing the normalization condition is

00 2

1Mzp

From this we can get the effective value for the 4D Plank mass

Amplification of the fluctuationsAmplification of the fluctuations

ddP M

p

zM

1

2 01

To study the production of relic gravitons we consider a transition between the inflationary Kasner regime and a final era with a simple Minkowsky metric. Since massless and massive modes do not mix, they can be treat separately

reduces to the RS result for d = 3 and n = 0

21

1

1

0 2)(

dPM

1

constM d

P

2)(

1

0 1

The auxiliary field u0 we have defined has the correct canonical dimension, so that we can normalize its Fourier modes to an initial state of vacuum fluctuations

ke

uik

k 2)(,0

The pump fields before and after the transition (which occurs at the conformal time 1) are

Amplification of the fluctuations

The solution of the (Bessel-like) canonical equation for m = 0 can be expressed in terms of Hankel functions. By imposing continuity at the transition epoch –1 we get the correct solution

kHu k

)2(0,0 4

)(

Curvature scale at the transition

and we can compute the spectral distribution

111k cutoff frequency

1

2

1

1

12,0

20 log)(

kk

kk

MH

hkkdd

Pk

d

1kk

Massive modes

The solution of the SL equation we find for m ≠ 0 is

)()(

)()()()(

2ˆ

02

2

102

2

1

0

2

10

2

10

2

10

2

1

0

mzYmzJ

zzmYmzJzzmJmzYzzm

pp

pppp

m


There is no difference with the standard result (unlike in de Sitter models studied in LMW)

We can then deduce the (differential) coupling parameter Mm that multiplies the massive mode action and, in the light mass regime, the effective measure controlling the light modes contribution to the static two-point function

202

221

8

p

dmmzp

M

dmdmG

Solving the canonical equation for the massive modes can be quite hard, BUT...

11

Hm

let us consider relativistic modes

In this case the amplification equation is exactly the same as in the massless case for each mode

1

0,

H

kmk hdmhDefining we can evaluate the massive contribution to the spectrum

reduces to the RS result for d = 3 and n = 0


The importance of the massive contribution depends on the ratio between the curvature transition scale and the AdS length

1

2

1

1

*

122log)(

1 kk

kk

MH

hkkdd

kd

H

2

00

011*

1

,2

)1(

H

dP

d

zmFdm

MzpMM 0ˆ, 0 mzmF

101 zH 1I PMM *

massive contribution to the spectrum highly suppressed

101 zH 01zHI PMM *possibility that

12

1

1

HM

MM

d

P

strongly enhanced amplification of the massive fluctuations


ConclusionsConclusions

• It is possible to have a cosmological evolution which behaves as the standard flat RS scenario

• Transition from an inflationary phase to a “standard” phase still produces a stochastic background of metric fluctuation

• If the curvature at the transition epoch is low, there is no significative difference with the standard scenario

• If the curvature at the transition epoch is high enough, there can be an important amplification of massive modes that contributes to the spectral amplitude

• Further developments can be analyzed including other fields (i.e. the dilaton) and some “bulk” influence on the brane fluctuations

Documents

Giuseppe De Risi M. Cavaglià, G.D., M. Gasperini, Phys. Lett. B 610:9-17, 2005 - hep-th/0501251 QG05, Sept. 2005