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:
ISH NEUPANE 18ISH NEUPANE 18--20 Nov 2009, Cospa09, Melbourne 20 Nov 2009, Cospa09, Melbourne 33
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
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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
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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
ISH NEUPANE 18ISH NEUPANE 18--20 Nov 2009, Cospa09, Melbourne 20 Nov 2009, Cospa09, Melbourne
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
ISH NEUPANE 18ISH NEUPANE 18--20 Nov 2009, Cospa09, Melbourne 20 Nov 2009, Cospa09, Melbourne
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?
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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
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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
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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
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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).
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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
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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
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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
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To get a finite 6d warped volume one introduces a Dp-brane (imposing Z2 symmetry) at z=0
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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?
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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
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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
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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
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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
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