1. INDUCED COSMOLOGY ON A CODIMENSION-2 BRANE IN A CONICAL BULK
G. De Risi
Università degli studi di Bari “Aldo Moro”
and
I.N.F.N. Sez. di Bari
arXiv:  [hep-th]

2. PLAN OF THE TALK Introduction to codimension-2 brane
The 6D static solution
Cosmological equations on the brane
Effective 4-D cosmology
Comments and conclusions

3. INTRODUCTION TO CODIMENSION-2 BRANE
Braneworld: Our universe is a “brane” embedded in an higher-dimensional space-time.
Codimension-1 braneworlds:
Tensional brane in an anti-de Sitter spacetime. Gravity dynamically localized on the brane
Randall - Sundrum
Tensionless brane with a 4D curvature term. “Naturally” reproduce late time acceleration
Dvali – Gabadadze - Porrati

4. Codimension-2 braneworlds:
Arkani-Hamed – Dimopoulos - Dvali
Compact but large extra-dimensions. Gauge interactions are confined on the brane, while gravity is free to propagate into the bulk.
The scale of quantum gravity is not fundamental!
It is possible to set the bulk Planck mass at the electroweak scale if the volume of the extradimension is large enough

5. Carrol-Guica
Self-tuning: Vacuum energy on the brane can be “off-loaded” in the bulk, generating a deficit angle in the geometry of the extra dimension.
SUSY breaking in the bulk backreacts by generating an effective cosmological constant on the brane
SLED

6. However…
Static soluton are not stable under phase transitions, since the flux of the magnetic field must be conserved
It is not possible to accommodate on a cod-2 brane any kind of energy-momentum tensor different from the pure tension.
Need to regularize the brane somehow to study cosmolgy on the brane
(maybe this can adress also the phase-transition problem…)

7. THE 6D STATIC SOLUTION
We consider a 5-dimensional brane embedded in a 6-dimensional bulk. The action is
The solution we aim to study consists of two manifolds joined together at the radial position of the brane

8. The metric has a conical singularity at z = m which is just a coordinate singularity if
We can impose the “in” solution to be regular at the horizon. The “out” solution “would” then have a conical singularity governed by a deficit angle b. This fixes a relation between the -in and -out cosmological constants
-in space
-out space
brane

9. COSMOLOGICAL EQUATIONS ON THE BRANE
Study of cosmology would imply solving partial nonlinear differential equations.
Trick (Mirage cosmology): We let the brane move through the bulk. Cosmology is induced by implementing time-dependent Israel junction conditions
Intrinsic e.m. tensor
The equations of motions are
Extrinsic curvature
Intrinsic Einstein tensor
Intrinsic metric

10. For the energy-momentum tensor we choose the form
Extra-dimensional pressure, has the role of an auxilary field
We thus get the e.o.m.

11. EFFECTIVE 4-D COSMOLOGY
We “split” the dynamics into different epochs in which different components of the energy content of the brane dominate.
Late time: dynamics governed by vacuum energy
This is the regime in which the scale factor is large
The e.o.m becomes a set of two algebraic equations in the variables H0, L6 and l (the conservation equation is trivially satisfied).
Expansion rate depends on the brane tension!!! (and there is fine-tuning too…)

12. “Intermediate time”: dynamics govened by ordinary energy density
In this case the e.o.m simplify into two effective Friedman equations
In this case e.o.m simplify into two effective Friedman equations. Of course, the actual 4-D energy density is btained by integrating over the azimuthal dimension

13. We can further distinguish two cases:
1. Sub-crossing regime
In this cas the Hubble radius is much smaller than the crossover scale, so we can drop terms not proportional to rc
The solutions for r, in the interesting case of radiation and matter equation of state, are:
1. Super-crossing regime
In this cas the Hubble radius is much larger than the crossover scale, so we can drop terms proportional to rc

14. The solutions for r, are, in this case
Early time: dynamics influenced by the conical singularity
This is the regime in which the scale factor is small
The e.o.m. become

15. These equation can not be solved analytically, so we must turn to numeric. Notice that, in this case, the 4-D effective energy density is
Plots for the scale factor, the Hubble parameter and the 4-D energy density are
a(t)
H(t)
(4)r(t)

16. COMMENTS AND CONCLUSIONS
By mplementing dynamical junction conditions on a 5-D brane embedded in a 6-D bulk, we could obtain effective 4-D comological equations.
At early times the dynamics is singular, even though the bulk 6-D spacetime is everywhere regular
During radiation and matter dominated epochs, provided the Hubble radius is large enough, the brane evolution mimcs fairly well standard cosmology
At late times, the brane undergoes a de Sitter evolution controlled by an effective cosmological constant which depends on the brane tension
NO SELF TUNING!!!!!

17. Future developments
SUSY extension, considering also the influence of the scalar axion field that keeps the brane wrapped in the azimuthal direction (check SLED hypoteses at nonlinear level)
Step forward in nonlinear analysis: brane movement mdify bulk dynamics (at least to some approximation…)
Very difficult….
Possibly… Study perturbations around this background, to check growth of structures.