giuseppe de risi m. cavaglià, g.d., m. gasperini, phys. lett. b 610:9-17, 2005 - hep-th/0501251...
TRANSCRIPT
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
Perturbations equation on the Kasner brane
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
Perturbations equation on the Kasner brane
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
Amplification of the fluctuations
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
Amplification of the fluctuations
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
Amplification of the 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