Randall-Sundrum vs. holographic cosmology

Neven BilićRuđer Bošković InstituteZagreb

VCES15,Vienna, November 2015

Basic idea

Braneworld cosmology is based on the scenario in which matter is confined on a brane moving in the higher dimensional bulk with only gravity allowed to propagate in the bulk.We will consider two types of braneworlds in a 5-dim asymptotically Anti de Sitter space (AdS5)

1. In a holographic braneworld universe a 3-brane is located at the boundary of the asymptotic AdS5. The cosmology is governed by matter on the brane in addition to the boundary CFT

2. In the second Randall-Sundrum (RS II) model a 3-brane is located at a finite distance from the boundary of AdS5. The model was proposed as an alternative to compactification of extra dimensions.

.

time

Conformal boundary at z=0

space

z

xRSII brane at z=zbr

extra dimension

Foliation of the bulk:

There exists a map between these two substantially different scenarios

This talk is based on:

N.B., arXiv:1511.07323

and related works

E. Kiritsis, JCAP 0510 (2005) [hep-th/0504219]. P. Brax and R. Peschanski, Acta Phys. Polon. B 41 (2010) [arXiv:1006.3054].

Outline

1. Randall–Sundrum model - basics

2. Connection with AdS/CFT

3. Holographic cosmology

4. Holographic map

5. Effective energy density

6. Conclusins & Outlook

RS model is a 4+1-dim. universe with AdS5 geometry

containing two 3-branes with opposite brane tensions separated in the 5th dimension.

1. Randall-Sundrum model

bulk actionbulk GH br1 br2S S S S S

Gibbons-Hawking term

brane action4 4

br mattS d x h d x h

L

;c d

ab a b d cK h h n

K – trace of the extrinsic curvature tensor

(5)5

bulk 55

1det

8 2

RS d x G

G

5GH

5

1det

8S d x hK

G

AdS bulk is a space-time with negative cosmological constant:

22 2(5) 2

( )a babds G dx dx g dx dx dz

z

Various coordinate representations:

Fefferman-Graham coordinates

2 2 / 2(5)

yds e g dx dx dy

Gaussian normal coordinates

ℓ - curvature radius of AdS5

22 2(5) 2

( )ds G dx dx g dx dx dzz

/yz e

2 2 2(5)

yds e g dx dx dy

(5)2

6

Schwarzschild coordinates (static, spherically symmetric)

22 2 2 2(5) ( )

( )

drds f r dt r d

f r

2 2

2 2( )

rf r

r

22 2 2sin

d d d

5 bh2

8

3

G M

0 open flat1 open hyperbolic

1 closed spherical

Relation with z 2 2 2 2

2 2 2

4

2 16

r z

z

1/ 0.1mmnV Long et al, Nature 421 (2003).

RSII brane-world does not rely on compactification to localize gravity at the brane, but on the curvature of the bulk (“warped compactification”). The negative cosmological constant Λ(5) acts to “squeeze” the gravitational field closer to the brane. One can see this in Gaussian normal coordinates on the brane at y = 0, for which the AdS5 metric takes the form

RS II was proposed as an alternative to compactification of extra dimensions. If extra dimensions were large that would yield unobserved modification of Newton’s gravitational law. Eperimental bound on the volume of n extra dimensions

2 2 / 2(5)

yds e dx dx dy

warp factor

Second Randall-Sundrum model (RS II)

L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999)

“Sidedness”

In the original RSII model one assumes the Z2 symmetry

so the region is identied with with the observer brane at the fixed point z = zbr. The braneworld is sitting between two patches of AdS5, one on either side, and is therefore dubbed “two-sided”. In contrast, in the “one-sided” RSII model the region is simply cut off.

br0 z z

brz z z

brz z

br0 z z

1-sided and 2-sided versions are equivalent from the point of view of an observer at the brane. However, in the 1-sided RSII model, by shifting the boundary in the bulk from z = 0 to z = zbr , the model is conjectured to be dual to a cutoff CFT coupled to gravity, with z = zbr providing the cutoff. This connection involves a single CFT at the boundary of a single patch of AdS5. In the 2-sided RSII model one would instead require two copies of the CFT, one for each of the AdS5 patches.

M. J. Duff and J. T. Liu, Class. Quant. Grav. 18 (2001); Phys. Rev. Lett. 85, (2000)

or br bry y y y

x5x y

0y y l5( )d x

y

0

In RSII observers reside on the positive tension brane at y=0 and the negative tension brane is pushed off to infinity in the fifth dimension.

One usually imposes the RS fine tuning condition

0 25 N

3 3

8 8G G

5x y

The Planck mass scale is determined by the curvature of the five-dimensional space-time

2 /

N 5 50

1

2ye dy

G G G

1 one-sided

2 two-sided

which eliminates the 4-dim cosmological constant.

y y

Conformal boundary

RSII Cosmology – Dynamical Brane

Cosmology on the brane is obtained by allowing the brane to move in the bulk. Equivalently, the brane is kept fixed at y=0 while making the metric in the bulk time dependent.

0y

Consider a time dependent brane hypersurface defined by( ) 0r a t

in AdS-Schwarzschild background. The induced line element on the brane is

The junction conditions on the brane with matter

2 2 2 2 2ind ( ) ( ) kds n t dt a t d

2 2 22

2 2

( )( ) ; ( )

( )ta a

n f a f af a a

yield

Hubble expansion rate on the brane

The Friedmann equation on the brane is modified

22 2N N

4

8 4

3 3

G G

a

H

dark radiationdue to a black hole in the bulk – should not exceed 10% of the total radiation content in the epoch of BB nucleosynthesis

Quadratic deviation from the standard FRW.Decays rapidly as in the radiation epoch

8~ a

RSII cosmology is thus subject to astrophysical tests

2 22

2

2 2 2t

Ha

a

a n a

H

AdS/CFT correspondence is a holographic duality between gravity in d+1-dim space-time and quantum CFT on the d-dim boundary. Original formulation stems from string theory:

Conformal Boundary at z=0

AdS bulktime

Equivalence of 3+1-dimN =4 Supersymmetric YM Theory and string theory in AdS5S5

Examples of CFT:quantum electrodynamics, Yang Mills gauge theory, massless scalar field theory,massless spin ½ field theory

J. Maldacena, Adv. Theor. Math. Phys. 2 (1998)

2. Connection with AdS/CFT

In the RSII model by introducing the boundary in AdS5 at z = zbr instead of z = 0, the model is conjectured to be dual to a cutoff CFT coupled to gravity, with z = zbr providing the IR cutoff (corresponding to the UV catoff of the boundary CFT)The on-shell bulk action is IR divergent because physical distances diverge at z=0

A 4-dim asymptotically AdS metric near z=0 can be expanded as (0) 2 (2) 4 (4)g g z g z g

de Haro, Solodukhin, Skenderis, Comm. Math. Phys. 217 (2001)

22 2(5) 2

( )ds g dx dx dzz

Explicit expressions for , in terms of arbitrary may be found in

(2 ) , 2,4ng n (0)g

We regularize the action by placing the RSII brane near the AdS boundary, i.e., at z = εℓ, ε<<1, so that the induced metric is

(0) 2 2 (2)2

1( )h g g

The bulk splits in two regions: 0≤ z ≤εℓ, and εℓ ≤ z ≤∞. We can either discard the region 0≤ z ≤εℓ (one-sided regularization, ) or invoke the Z2 symmetry and

identify two regions (two-sided regularization, ). The regularized bulk action is

2 1

(5)reg 5bulk 0 5 GH

5

2 det8 2z

RS S d x G S

G

We obtain the renormalized boundary action by adding counterterms and taking the limit ε→0

ren (0)0 0 1 2 3

0[ ] lim( [ ] [ ] [ ] [ ])S g S G S h S h S h

The necessary counterterms are

Hawking, Hertog and Reall, Phys. Rev. D 62 (2000), hep-th/0003052

Now we demand that the variation with respect to hμν of the total RSII action (the regularized on shell bulk action together with the brane action) vanishes, i.e.,

regbulk br( [ ] [ ]) 0S h S h

Which may be expressed asmatter on the brane

cosmological constant

Einstein Hilbert term

AdS/CFT prescription

The variation of the action yields Einstein’s equations on theboundary

(0) CFT mattN

18

2R Rg G T T

de Haro, Solodukhin, Skenderis, Comm. Math. Phys. 217 (2001)

This equation (for γ=1) was derived in a different way in withde Haro, Solodukhin, Skenderis, Class. Quant. Grav. 18 (2001)

Conformal anomaly

3

CFT 2GB

5128T G C

G

AdS/CFT prescription yields the trace of the boundary stress tensor TCFT

compared with the general result from field theory

2 212

3C R R R R R

2GB 4G R R R R R

Gauss-Bonnet invariant

Weyl tensor squared

The two results agree if we ignore the last term and identify 3

5128b c

G

CFT 2GB 'T bG cC b R

Generally b ≠ c because

s f v2

(11 / 2) 62

360(4 )

n n nb

but in the N = 4 U(N) super YM theory b=c with 2 2 2

s f v6 , 4 ,n N n N n N

s f v2

3 12

120(4 )

n n nc

The conformal anomaly is correctly reproduced if we identify

3 2

5

2N

G

s f v2

3 12

120(4 )

n n nc

2 22 2 2 2 2 2 2(5) 2 2

( ) ( , ) ( , ) kds g dx dx dz n z d a z d dzz z

we start from AdS-Sch static coordinates and make the coordinate transformation . The line element will take a general form

with boundary conditions at z=0:

0( ,0) 1, ( ,0) ( )n a a

3. Holographic cosmology

Assuming the induced metric at the boundary to be of FRW type

2 (0) 2 2 2(0) 0 ( ) kds g dx dx d a d

( , ), ( , )t t z r r z

Solving Einstein’s equations in the bulk one finds

22 2 42 2 0

0 40

11 ,

4 4

z za a

a

H

0

,a

na

2 20 0 2

0

Ha

Hwhere 0

00

aH

a Hubble rate at

the boundary

Comparing the exact solution with the expansion

(0) 2 (2) 4 (4)g g z g z g

we can extract and . Then, using the de Haro et al expression for TCFT we obtain

(2)g(4)g

P.S. Apostolopoulos, G. Siopsis and N. Tetradis, Phys. Rev. Lett. 102, (2009)arXiv:0809.3505

CFT CFT (0)1

4T t T g

The second term corresponds to the conformal anomaly

The first term is a traceless tensor with non-zero components3

4 2000 0 04

5 0 0

3 43

64ii

at t

G a a

H H

3CFT 20

05 0

3

16

aT

G a

H

Hence, apart from the conformal anomaly, the CFT dual to the time dependent asymptotically AdS5 bulk metric is a conformal fluid with the equation of state where ,

CFT CFT 3p CFT 00t

CFTiip t

From the previously derived Einstein equations at the boundary we obtain the holographic Friedmann equations

22 4N0 0 0 4

0

8 4

3 4

G

a

H H

24 20 N0 0 0 0

0

41 ( 3 )

2 3

a Gp

a

H H

where matt matt0 00 0, i

iT p T

dark radiation

quadratic deviation

quadratic deviation

The second Friedmann equation may be derived from energy-momentum conservation

4. Holographic map

22 2 2 2 2 2(5) 2

( , ) ( , ) kds n z d a z d dzz

The time dependent bulk spacetime with metric

may be regarded as a z-foliation of the bulk with FRW cosmology on each z-slice. In particular:at z=zbr: RSII cosmology, at z=0: holographic cosmology.

A map between z-cosmology and z=0-cosmology can be constructed using 22 2 4

2 2 00 4

0

11 ,

4 4

z za a

a

H0

,a

na

and the inverse relation

2 2 42 2 20 4

1 12 2

a z za z

a

HHE 1 one-sided

1 two-sidedE

and a mapping between the Hubble rates

0 0a aH H

From one finds a simple relationship

122 2 42 2 0

0 40

11 ,

4 4

z z

a

HH H

and its inverse1

2 2 42 2 2 20 4

2 1 12

z zz

a

HH H HE

0n a a

22

2 2 2

a

a n a

H

22 00 2 2

0 0

a

a a

H

In these equations we have defined

0 4

40

1

2a

one sided one sided

physical region (ρ0 ≥ 0)

4 4 2 2 4 40 0 02 2 1 2 2 1a a H

The regime of large H0 violates the weak energy condition

Hubble rate at as a function of the Hubble rate at2z 2 22 H 2 22 H

2 202 H2 2

02 H

0z

Holographic map

2 2 2 20 0 kds d a d 2 2 2 2

0 02

1kds d a d

n

z z

2 2 2 2

kds d a d

0z

holographiccosmology

RSIIcosmology

brz z 2 2 2 2 2

kds n d a d

6. Effective energy density

Holographic scenario: Primary cosmology is on the AdS boundary at z = 0. Observers on the RSII brane on an arbitrary z-slice experience an emergent cosmology which is a reflection of the boundary cosmology.

We shall assume the presence of matter on the primary brane only and no matter in the bulk

RSII scenario: Primary cosmology is on the RSII brane at z = zbr The cosmology on the z = 0 brane emerges as a reflection of the RSII cosmology.

We analyze two cosmologic scenarios:

Holographic scenario

Suppose the cosmology on the z=0 brane is known, i.e., the density ρ0, pressure p0, and scale a0 are known. If there is no matter in the bulk the induced cosmology on an arbitrary z-slice is completely determined. For simplicity we take z=ℓ .Then, assuming the modified Friedmann equations hold on the RSII brane, the effective energy density is given by

0 2N

3

8 G

1e

Effective density ρ at z/ℓ = 0.5 (red), 1 (black), 2 (blue line) for as a function of ρ0

0 0

0 0

0 0 1e 1e

4 40 0.5a In both plots

Low density regime (relevant for the one sided version only)4 4

0 0 0, 1a

For the energy density at quadratic order is1e

Hence, at linear order the effective fluid on the RSII brane satisfies the same equation ofstate as the fluid on the holographic brane. The dark radiation contribution is the same on both branes whereas the cosmological constant term may be different on the two branes depending on the brane tensions. We recover the standard cosmology on both branes by choosing ℓ such that σ0 becomes sufficiently large to suppress the quadratic and higher terms

and pressure at linear order is

For we find a substantial difference even at linear order.

• The effective density ρ on the RSII brane differs from ρ0 on the holographic brane by a multiplicative constant.

• The effective cosmological constant does not vanish and is equal to

• For zbr/ℓ =1 the effective density ρ diverges in the limit ρ0→0

Cosmological constant term

1e

RSII scenario

In the RSII scenario the primary braneworld is the RSII brane at z = zbr . Observers at the boundary brane at z = 0 experience the emergent cosmology. For simplicity we take σ=σ0 and z=ℓ .Then, assuming the modified Friedmann equations hold on the holographic, the effective energy density is given by

1 one-sided

1 two-sidedE0 0

2 4 40 0

4 ( 1 )

( 1 ) a

E E

E

Thus, the two-sided model with positive energy density and positive μ maps into a holographic cosmology with negative effective energy density ρ0. For μ = 0 the density ρ0 Diverges as 1/ ρ. The one-sided model maps into two branches: ε = -1 branch identical with the two-sided map and

the ε = +1 branch with smooth positive function ρ0= ρ0(ρ).

Conclusions and outlook

We have explicitly constructed the holographic mapping between two cosmological braneworlds: holographic and RSII .

The cosmologies are governed by the corresponding modified Friedman equations.

There is a clear distinction between 1-sided and 2-sided holographic map with respective 1-sided and 2-sided versions of RSII model.

In the 2-sided map the low-density regime on the two-sided RSII brane corresponds to the high negative energy density on the holographic brane

The low density regime can be made simultaneous only in the one-sided RSII

Speculations

It is conceivable that we live in a braneworld with emergent cosmology. That is, dark energy and dark matter could be emergent phenomena induced by what happens on the primary braneworld.For example, suppose our universe is a one-sided RSII braneworld the cosmology of which is emergent in parallelwith the primary holographic cosmology. If ρ0 describes matter with the equation of state satisfying 3p0 + ρ0 >0, as for, e.g., CDM, we will have an asymptotically de Sitter universe on the RSII brane. If we choose ℓ so that Λ fits the observed value, the quadratic term will be comparable with the linear term today but will strongly dominate in the past and hence will spoil the standard cosmology. However, the standard ΛCDM cosmology could be recovered by including a negative cosmological constant term in ρ0 and fine tune it to cancel Λ up to a small phenomenologically acceptable contribution.

Thank you

a) The horizon temperature is proportional to the physical temperature of the expanding conformal fluid.

b) There exist a maximal z equal to the critical proper time , where corresponds to the critical temperature of some phase transition (e.g, the chiral phase transition in hadronic physics)

c) The induced metric on the zc -slice corresponds to the effective acoustic metric in the regime in which the perturbations (e.g, massless pions) propagate.

Analog geometry connection with AdS/CFT: AdS/QGP

cz cc c1 /T

N.B., S. Domazet, D. Tolic, PLB 743 (2015)

cz