Dark Energy/Cosmological Constant in Models with Warped Extra

Dark Energy/Cosmological Constant in Models with Warped Extra Dimensions

Ishwaree Neupane

COSPA09 -- 20 Nov 2009 -- Melbourne University

Observations fit 4d Einstein GR with a Cosmological Constant term (dark energy density) and dark matter: How to get this picture from a compactified 10d string theory or supergravity models?

ISH NEUPANE 18ISH NEUPANE 18--20 Nov 2009, Cospa09, Melbourne 20 Nov 2009, Cospa09, Melbourne 22

• String theory - the best (only?) theory of quantum gravity

• Warped Extra dimensions - a generic prediction of string theory

• Implications:

– Particle physics (very model dependent)

– Cosmology (has universal aspects through extra-

dimensional gravity)

- Testing String theory on CMB /Cosmological

Observations (inflation, dark energy, dark matter….)

General Motivation:


Warped Compactification

Basic Ideas

Extra Dimensions Higher dim spacetime: “bulk” Localization of matter to a subspace “3-brane”

(in contrast with Kaluza-Klein approach)

Motivations Particle Physics: Mass Hierarchy (why ? )

Brane Gravity: A New Compactification Scheme Brane inflation / braneworld cosmology

ISH NEUPANE 18ISH NEUPANE 18--20 Nov 2009, Cospa09, Melbourne 20 Nov 2009, Cospa09, Melbourne

44

2

5

6;y

yb e -

Z2-symmetry

5 4

5 452

5

12

2 MS d x g R d x q

RandallRandall--SundrumSundrum single single branebrane model model

22 2 25 5for 6yds dy e dx dx

•• Warp factor Warp factor AA((yy))==ee--||yy|/|/ decreases exponentially decreases exponentially as |as |yy||→→inftyinfty

•• ExtraExtra dimdim isiseffectivelyeffectivelycompactcompactwithwith sizesize ~~

gMust be for a realistic

cosmology



RS type (effectively) 5d-regions can arise in warped compactifications of type IIB string theory

The KS geometry in D=10 dimensions

has some common features with warped

Randall-Sundrum 5D braneworld model

1221,16 )( SSSRTRY

77

Inflation in String TheoryInflation in String TheoryInflation in String TheoryInflation in String Theory

A typical potential obtained by string compactificationA typical potential obtained by string compactification

The potential with respect to Φ and Ψ is very steep: these fields rapidly run down, and the potential energy V vanishes. One needs to stabilize these fields.

Φ and Ψ are canonically normalized fields corresponding to the dilaton field and to the volume of the compactified

space; is the field driving inflation

Dilaton stabilization: Polchinski et al 2001Kachru et al 2003 Burgess et al 2003

Polchinski et al 2001Kachru et al 2003 Burgess et al 2003

)(V~),,(V 62 e


100 150 200 250 300 350 400

-2

-1.5

-1

-0.5

0.5V

100 150 200 250 300 350 400

0.2

0.4

0.6

0.8

1

1.2

V

ΛΛ or dark energy in string theoryor dark energy in string theoryΛΛ or dark energy in string theoryor dark energy in string theory

AdSAdS minimumminimum dSdS minimumminimum

KKLT (Kachru, Kallosh, Linde,Trivedi ) 2003KKLT (Kachru, Kallosh, Linde,Trivedi ) 2003

1) Start with a theory with runaway potential (AdS minimum)

2) Bend this potential invoking some (non-perturbative) effects

3) Uplift the minimum to the state with positive vacuum energy by adding a positive energy of an anti-D3 brane in warped Calabi-Yau space


This proposal suffers from fine-tuning issues associated with the necessary flatness of the potential and or the level of fine tuning required for Λto be the present-day gravitational vacuum (dark energy) density

99

String Theory and the Dark Universe

• Dark energy / cosmological constant Λ :• Quintessence – likely to be linked to moduli fields, e.g., size and shape of compact space

Λ – could depend on fundamental (UV) structure of theory and statistics of string theory vacua: internal space geometry, branes, external fluxes, etc


Λ>0

Λ<0Λ=0

1010

The runaway involves one or more scalars, depending on the model considered

)(V

Is there a room for a ``Cosmological Constant-like term”

and/or “Quintessence” in String or Supergravity theory?


What are the obstacles in finding explicit (inflationary) de Sitter

solutions of 10D supergravity equations?

1111

The celebrated answer: Brane world “no-go” theorem!

“No-go” theorem forbids cosmic acceleration in cosmological solution arising from compactification of pure SUGRA where the internal space is time-dependent, non-singular compact manifold without boundary.

- Gibbon (1984)- Maldacena-Nunez (2001)Why?

Acceleration requires violation of 4D strong energy condition

If extra dimensions are warped and static, then in a compactified theory

0Tor 0R AB(4)00 BA

0R (D)00

0R (4)00


Provided that SEC holds for D=10 or 11D SUGRA

1212

To see this one considers a FRW metric

The time-time component of 4D Ricci tensor gives

Acceleration requires and hence

The 4D Einstein field equations imply that

An accelerated expansion is possible in a universe governed by Einstein's gravity only if the matter in it violates SEC.

Why should the SEC be violated?

2

22

2

2222

4 1)( dr

kr

drtadtds

a

aR

3)4(00

0/ aa0)4(

00 R

pTgTR ijij 300

)4(00


0/ aa

1313

Consider a (4+m)-dimesional metric ansatz

This gives

A celebrated version of “no-go” theoreom

)(42)(24

1)4(00

(D)00 ),(R yA

yyA eeRyx

nmmn

yA dydyygxdse )()(ds 24

)(22D

0Rifonly 0R )4(00

)(00 D

)(42)(00

)(2)4(00

)(2

4

1 yADyAyA eReRe

If the last term above vanishes, then



Any time that one does not understand something, one can point to details

that do not work.

This has been more or less the case with braneworld models and string theory

compactifications with warped extra dimensions

It is always important to identity what iswrong qualitatively and give the best

clue to possible future progress/success.

1. Allow internal space to be time-dependent, analogue of time-dependent scalar fields – Lukas et al ‘00

2. Drop condition that internal space is flat or positive, it may be negatively curved (hyperbolic)

Townsend-Wohlfarth, N. Ohta, IPN et al 2003

A compactified theory on hyperbolic spaces leads to cosmologies with transient accelerating phase.

SUGRA solutions describing accelerating SUGRA solutions describing accelerating cosmologies from twisted spaces: cosmologies from twisted spaces: IPN&WiltshireIPN&Wiltshire: : hephep--th/0502003 (PLB), hepth/0502003 (PLB), hep--th/0504135 (PRD), hepth/0504135 (PRD), hep--th/0609086 th/0609086

(PRL)(PRL)

ISH NEUPANE March 2009, Seoul Nat'l University ISH NEUPANE March 2009, Seoul Nat'l University

The limitation with warped models studied previously have arisen from an over simplification of 10d metric ansatz.

IPN: arXiv:0901.2568 [hep-th]IPN: arXiv:0903.4190 [hep-th]IPN:arxiv:0911.xxxx

These papers give a few explicit examples of non-singular warped compactifications on de Sitter space dS4.

The explicit de Sitter solutions given in these papers serve as explicit models of accelerating universe (attributed to dark energy or cosmological constant-like term).


Recent Finding

1717

)()(ds 26

)(24

)(2210 ydsexdse yAyA

Start with

221

2

26

5dy

)(ds

X

nmmn

dsy

dydyyg

1,15 or TS

21

16

)1(20

yR

23

2224 dx)()( tadtxds


A general 6d metric

)()(ds 26

)(24

)(2210 ydsexdse yAyA

)dy()(ds 221

2226 5X

nmmn dsydydyyg

The metric ansatz

2

22)2(

32

)2(3

L

ye A

8

)2( 2

1

Hteta )( 22

1

LH

explicitly solves the 10d vacuum Einstein equations when


The drawback of this solution is that the 6d metric and hence the warp factor is singular at y=0

1919

It’s more convenient to write our solution in z-coordinate:

)(

2

9

))((

2412

322

222

23

22

||210

0

iic

czkk

eer

dzkr

dxtadt

eeds

2

2

27

2

||54~

00 463||4

463w

6

kczk

kc er

eerk

V

6

w6

8102

)2( VM

M Pl

The warped volumeof 6d space


To get a finite 6d warped volume one introduces a Dp-brane (imposing Z2 symmetry) at z=0


Why did the previous authors -- including Gibbons, Maldacena-Nunez, Giddings et al and many others -- not realise (or somehow rule out) these explicit de Sitter solutions of 10D Einstein equations or background supergravity equations?


The two major assumptions that went into the earlier discussions of braneworld no-go theorems are

!)2(

,

6

w6

8102

)32(6

6w6

constVM

M

constegydV

Pl

A

0)(2

ynA

y e for any n and A(y)

1.

2. These constraints are ‘strict’ which are generally not satisfied by cosmological warped solutions, especially, in the presence of some localised (brane) sources at y=0 where A(y)=const

Can the natural “constants” change in time and or as one moves away from the D3-brane?

)()(ds 26

)(23

22)(2210 ydsexdtadte yAyA

))(cosh)y(ysinh(

)(ds2

02

1022

26

5X

nmmn

dsyy

dydyyg

An alternative to a true cosmological “constant” ?

32

)2(3)cosh(ln

2

2)(

22

0

yyyA 8

)2( 2

1

Hteta )(2

2

H

This solves the 10D Einstein equations when



AAyy

exRyxR y

A22

022

)2(

00)4(

00)10( )2(2

)(sinh)(ˆ),(

)()coth(

8

)2(33)(ˆ

02

2

00)4( yyyxR

Here it is possible to get acceleration without violating the 10D strong energy condition

0ˆ00 R

000)10( R

222

2222

224

)2/(2222210

5coth

8

)2(tanh

)2(3

32

32

cosh)2(3 0

Xdsydyyds

yAds e

yyV ~coshln)/(~ 66


dyyyVM

RgxdVM

RgxdMds

)(coshe

)2ˆ(ˆe

......~

)2/(16

65

810

4A-

4)4(44

66

810

4A-

)10(10108

10210

0

0

2

26

Requiring that and

forces us to make the choice

TeVM 10

21204 10~ PlM

1360 A

Dimensional Reduction

It may not be natural to just set to zero!4


This is true also in lower dimensions: Take D=5

)(2

)2(2 3

435

555

35 Tgxd

MRgxd

MS b

222222

2

222)(22

5 )sin(1

)( dzddrkr

drtadteds zA

/

2

2/

22

1)( tt e

keta ||2

5||2

2

24

24)(

zz eezA

)(254

3A(z)4

435 2ˆeˆ2

zAeff eLRdzgxd

MS

)(24

2416

24

1601

12||2

5||2

||25

||2

22||25

||2

225

2

2

zee

ee

eeL zz

zz

zz

Summary Warped models give a new picture of the universe

as well as new opportunities for both particle physics and cosmology (inflation, dark energy, dark matter, …)

Cosmic acceleration (attributed dark energy) is intrinsically an extra-dimensional phenomenon in many compactifications of supergravity models.

String theory can, of course, accommodate some null energy condition violating objects -- negative tension branes and O-planes -- but they may not be required just to get an accelerating universe from higher dimensional Einstein's theory


Dark Energy/Cosmological Constant in Models with Warped Extra

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