Diapositiva 1

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BRANEWORLD COSMOLOGY WITH A KALB-RAMOND FIELD
Giuseppe De Risi
43rd Rencontres de Moriond La Thuile (Val
d'Aosta, Italy) March 15 - 22, 2008
Phys.Rev.D77044030,2008, arXiv0711.3781 hep-th
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Plan of the talk

• Introduction to the braneworld scenario

• Bouncing cosmology supported by a EM field

• Bouncing cosmology from a KR field

• Comments and conclusions

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The braneworld scenario
In the past decade the idea that our universe is
a brane embedded in a higher dimensional manifold
has become very popular. The Randall-Sundrum (RS)
scenario, in particular, has been very successful
in accommodating all the well-established
theoretical and observational results of standard
cosmology (inflation, perturbations etc.)
RS model consists of a flat brane (with tension)
embedded in an AdS5 bulk.
Newtonian gravity is recovered at low energies
because the e.o.m. of the tensor fluctuation has
a volcano-like potential that bounds the zero
mode.
The massive modes contribution results in a
correction to the Newtonian potential
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Going to the non-linear level, it is possible
(Shiromizu, Maeda, Sasaki) to show that the
Einstein equation are modified
Matter induced from the bulk
High energy correction
Dark radiation from the bulk
The effective 4D theory is still GR with
corrections that become effective at high energies
Cosmology on the brane can be captured in a less
general, though simpler way (Kraus) allow the
brane to move trough a static bulk. In this case
the Israel junction conditions becomes dynamical,
and give the modified Einstein equations.
High energy correction
Dark radiation from the bulk
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Bouncing cosmology on the brane
The standard cosmological model, even in the
inflationary extension, despite its success in
explaining a wide range of cosmological and
astrophysical observation, still suffer from a
major problem the big bang.
The presence of the singularity indicates that we
are extending GR beyond its validity regime
We need a modification that take place at high
energies
Brane cosmology?!
In 2003 Mukherij and Peloso proposed a braneworld
model in which the cosmological evolution on the
brane was non-singular.
In their model, the bulk solution was a
Reissner-Nordström-AdS black hole, i. e. a black
hole with electric charge.
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By letting the brane move one obtain, following
the procedure sketched before, a Friedman-like
equation of the form.
Dark stiff matter
The additional term sourced by the electric
charge is repulsive, and it is dominant at small
scales, i.e. at high energies, so it could
hopefully drive a bounce in the cosmological
evolution.
In fact, it is possible to find exact bouncing
solution for the critical case in which the brane
cosmological constant is zero
closed universe
flat universe
open universe
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However, this proposal was subject to several
criticisms (Kanti and Tamvakis, Hovdebo and Myers)
Perhaps the strongest pathology that has been
pointed out about this kind of model is that the
brane encounter a singularity before undertaking
the bounce.
During its journey trough the bulk, the brane
crosses both horizons of the black hole.
But it is known that the inner horizon of a
Reissner-Nordström black hole is unstable.
In particular, fluctuations which are normalized
it the outer horizon, blow up at the inner one.
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Brane with a Kalb-Ramond field
G.D.R., Phys.Rev.D77044030,2008, arXiv0711.3781
hep-th
Our aim is to study the braneworld setup in
presence of bulk supergravity fields. The action
is
The brane is neutral with respect to U(1) charge
of the KR field
Variation of the action leads to the e.o.m. for
the metric, the dilaton and the Kalb-Ramond field
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The equation for the KR field is solved by the
Ansatz
Duality
This allow us to write the equations in terms of
the dualized fields (Copeland, Lahiri, Wands)
Field source terms are opposite in sign!
We seek for a static solution, with the usual
ansatz for the metric, and assume that the dual
Maxwell field is purely electric.
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We found that the dilaton is constant, therefore
it can be set to zero without any loss of
generality. The solution for the metric and the
Maxwell field are
The term proportional to the U(1) charge is
negative
No repulsive gravity at high energies.
However, the negative sign in the charge term
give rise to a rather exotic possibility set m
lt 0 without having a naked singularity.
In fact, for a negative bulk cosmological
constant we have one horizon located at.
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Let us now consider the cosmological evolution of
the brane. For simplicity, we will consider a
pure tension spatially flat brane. The modified
Friedman equation can be obtained in the usual
way, following the lines depicted before
The universe undergoes a bounce if the scale
factor initially decreasing, reaches a minimum
and start to expand again, i. e. H 0
Here is a picture of the behavior of H as a
function of a for different values of the
parameter
Analytically, one find that the bounce occurs at
the value
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Having found that the universe actually undergoes
a bounce, the crucial requirement is that it
occurs before the brane crosses the horizon. In
fact, possible instabilities can occur only on
the horizon.
The figure represents the behavior of ab as a
function of the brane tension l for different
values of RKR. There is always a range of the
tension in which the bounce occurs outside the
horizon.
It is possible to find an analytical solution of
the allowed values for the brane tension (and
thus for the induced cosmological constant)
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Conclusions

• We study braneworld models with a bulk
  Kalb-Ramond field, and we find that cosmology on
  the brane is free from singularities

• In addition, the cosmogical do not suffer from
  instabilities, such as other models presented in
  the literature

BUT
A crucial assumption was to assume m lt 0, where m
is proportional to the mass of the central body
who sources the black hole

• Difficult interpretation of the Newtonian limit

Not sure that it is an issue
No definite answer even for the standard
Reissner-Nordström black hole (Kodama-Ishibashi)

• Problems related to the overall instability of
  the space-time

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and outlook
Further developments include

• Studying models with matter on the brane

• Presence of enough ordinary radiation could
  spoil the singular-free behavior

• Test the model against astrophysical bounds

• Introduce a DBI coupling between the brane and
  the Kalb-Ramond field

• The dual ansatz could be no longer valid

• Studying 4D scalar and tensor perturbation

• WMAP data favour a scale-invariant spectrum,
  difficult to have with bouncing models.