M. J. Guzman- Brane Cosmology and Randall-Sundrum Model

8/3/2019 M. J. Guzman- Brane Cosmology and Randall-Sundrum Model

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BRANE COSMOLOGY and Randall-Sundrum

model

M. J. Guzman

June 16, 2009

M. J. Guzman BRANE COSMOLOGY and Randall-Sundrum model
http://find/http://goback/


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Standard Model of Cosmology

CMB and large-scale structure observations provide us ahigh-precision estimation of the cosmological parameters:tot

1, m

0.27



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1, m

0.27 we need

0.73



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1, m

0.27 we need

0.73

But, we have Flatness problem



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1, m

0.27 we need

0.73

But, we have Flatness problem Small-scale inhomogeneity



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1, m

0.27 we need

0.73

But, we have Flatness problem Small-scale inhomogeneity Large-scale smoothness



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1, m

0.27 we need

0.73

But, we have Flatness problem Small-scale inhomogeneity Large-scale smoothness Cosmological constant



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1, m

0.27 we need

0.73

But, we have Flatness problem Small-scale inhomogeneity Large-scale smoothness Cosmological constant Magnetic monopoles



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One of the solutions



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One of the solutionsINFLATION



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One of the solutionsINFLATION

When vaccum energy was the dominant component of the energydensity of the universe,a(t) grew exponentially.

In a time interval t 1032[s]a(t) grows by a factor of

e100 3 1043 !



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Brane Cosmology

One alternative cosmological model is Brane Cosmology.

Embedding our typical 4D space-time in a higher 5D space, andproposing that matter is confined to the brane, yields a model ofuniverse consistent with requirements of General Relativity.Furthermore, it is required a fine-tunning relation betweenparameters in the brane and the bulk.



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Randall-Sundrum model

Properties of the

BULK: 5 : bulk cosmological constant 5 : 5D gravitational coupling constant (5 = 8G

(5))



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Properties of the


(5))

BRANE: : brane tension (a constant) The brane is located at y = 0.



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Properties of the


(5))

BRANE: : brane tension (a constant) The brane is located at y = 0.

5D spacetime

x0 : time coordinate xi : space coordinates, i = 1, 2, 3 y : additional dimension such that y = y (Z2 symmetry)



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Action

From an Action, we can derive equations of motion.Einstein-Hilbert Action

SH =

g Rd4x

S =1

16GSH + SM

Using principle of least action, it is obtained the Einstein Field

Equations,R 1

2Rg = 8GT



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In the Randall-Sundrum model, total action is a sum of thecontribution of the bulk (5D space-time) and the brane (4Dspace-time).

S = SEH + Sbrane

SEH =

d5x

g(5)

R

225+ 5

Sbrane =

d4xg(4)()



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Solution of RS model

But here things are more complicated, it is not enough to calculateS = 0...The brane induces a discontinuity in the bulk, which is solved byimposing junction conditions.

G K Kwith G the Einstein tensor,

G = R1

2 Rg

K is the extrinsic curvature 4-tensor of the brane.



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Using these, and proposing the previous ansatz for the metric,

ds2 = e2K(y)dxdx + dy2

yields two equations for the parameters of the bulk and the brane,

6K2 = 2553K = 25(y)

solving both, we obtain K(y) =

256 5

y , and

6K(y)|y=0 = 25



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Together, the two equations imply

5 =25

62

For static solutions to exist, a fine-tunning must exist between thebrane tension and the bulk cosmological constant.Setting 5 1, and writing the bulk metric of the form

ds2 = a2b2(dt2 dy2) a2ijdxidxj

Leads us to

+ 3a

a( + p) = 0



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It is definedd = abdt, aH = da/d, a = e(t), aH = da/dPlaying with the equations, we obtain:

H2

=

2

36 +

5

6 +

a4

with an integration constant. Splitting energy-density andpressure: = M + , p = pM + we obtain Friedmanns equation

H2 =8G

3M

1 +

M2

+

43

+

a4



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Physical considerations

Identifying8G

3=

18

43 =

2

36 +56

and comparing with the fine-tunning relation 5 =

25

6 2, we see

that 4 = 0. If this is not satisfied, 4 = 0.From Friedmanns equation, ifM

, H

M, instead of

H M. It is obtained an expansion rate larger than in thestandard model.



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Physical considerations

Another models, more complicated, include a scalar field confined to the brane, obtaining inflation in the brane.An interesting possibility, given by more sophisticated models, is avariation in the traditional potencial energy between two masses(confined on the brane):

V(r) =G(5)m1m2

r

1 +

l2

r2+ O(r3)

with l related to the parameters of the bulk:

l =6255



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REFERENCES

D. Langlois, Prog.Theor.Phys.Suppl.148:181-212,2003

Bruck et al 2003, arXiv:hep-th/0303095v1

Papantonopoulos, Lect.Notes Phys. 592 (2002) 458 Randall L, Sundrum R. 1999 Phys.Rev.Lett. 83 3370 Randall L, Sundrum R. 1999 Phys.Rev.Lett. 83 4690

Israel W 1966 Nuovo Cim B44S10 1; Erratum: ibid Nuovo

Cim B48 463

Battye, Carter, Phys.Lett. B, Volume 509, Issue 3-4, p.331-336

Kolb and Turner, The Early Universe Sean Carroll, Spacetime and Geometry. Weinberg, Gravitation and Cosmology. Peacock, Cosmological Physics.

Misner, Thorne and Wheeler, Gravitation.



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Thanks :)


M. J. Guzman- Brane Cosmology and Randall-Sundrum Model

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