Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Holographic Slow-Roll Ination
Adam Bzowski
Work in collaboration with
Kostas Skenderis and Paul McFadden.
University of Sussex, 15.10.2012.
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Rotate the Universe
Euclidean AdS
ds 2 = dr 2 + e 2αr d~x 2 .
de Sitter solution
ds 2 = −dt 2 +e 2Ht d~x 2 .
Cosmological observables ↔ correlation functions in CFT
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Overview
1
Standard ination and cosmology.
2
AdS/CFT and the holographic cosmology.
Two models:
1 Slow-roll ination as a deformation of a dual CFT.
2 Holographic cosmology based on the super-renormalizable dual QFT.
3
4
Conclusions.
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Cosmic Microwave Background
Non-geometric ination
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Primordial perturbations
Perturbations δφ of the inaton (matter) ⇒
perturbations in the stress-energy tensor δ Tµν ⇒
perturbations of the metric δ gµν via Einstein equations.
In the gauge where δφ = 0,
ds 2 = −dt 2 + a(t )2 δij + e 2ζ [e γ ]ij ,
where
ζ is the scalar perturbation,
γij is the transverse-traceless perturbation.
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Observables
We use a notation
(s )
γ (s ) = ¯ij γ ij ,
(s )
where ij is a helicity projector, s = ±1, i.e. it spans a two-dimensional
space of transverse-traceless, symmetric tensors
(s )
(s )
ij = ji ,
p i (ijs ) = 0,
(s )
ii = 0.
Standard observables are:
scalar and tensor power spectra
hhζ(p )ζ(−p )ii,
hhγ (s ) (p )γ (s ) (−p )ii,
bispectra or non-gaussianities
hhζ(p1 )ζ(p2 )ζ(p3 )ii,
( s1 )
hhγ (p )γ (s2 ) (p )ζ(p )ii,
1
2
3
higher-point correlations functions.
hhγ (s1 ) (p1 )ζ(p2 )ζ(p3 )ii
hhγ (s1 ) (p1 )γ (s2 ) (p2 )γ (s3 ) (p3 )ii,
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Power spectra
One usually denes
∆2S (p ) =
p3
hhζ(p )ζ(−p )ii,
(2π)2
∆2T (p ) =
p3
(2π)2
hhγ (s ) (p )γ (s ) (−p )ii.
If ∆S ,T does not depend on any dimensionful parameters dierent
than the momentum the spectrum is scale-invariant.
If correlation functions are those of free elds the spectrum is
gaussian.
Non-gaussianities are related to interactions in the underlying
fundamental theory.
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
What can be measured?
Experimental parametrisation:
∆2S (p ) = ∆2S (p0 ) ·
p nS −1
,
p0
∆2T (p ) = ∆2T (p0 ) ·
p nT −3
.
p0
Current data:
∆2S (p0 ) = (2.445 ± 0.096) · 10−9 ,
ns − 1 = 0.040 ± 0.013,
at p0 = 0.002Mpc−1 .
1
Scalar power spectrum has small amplitude and is almost scale
invariant.
2
In principle non-gaussianities were measured, but uncertainties
usually exceed the measured value.
3
Not sucient data for the remaining characteristics.
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Cosmology/domain-wall duality
Non-geometric ination
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
More explicitly
More formally, cosmology/domain-wall duality changes the following signs
ds 2 = ∓dz 2 + a2 (z )d~x 2 ,
Φ = Φ(z ),
as well as
S=
∓1
2κ2
Z
d4 x |g | −R + (∂Φ)2 + 2κ2 V (Φ) .
p
Therefore, cosmology/domain-wall duality leads to
κ̄2 = −κ2 ,
p̄ = −ip ,
[Maldacena (2005)], [Skenderis, Townsend (2006, 2007)], [McFadden,
Skenderis (2010)]
Introduction
Cosmology
Holographic cosmology
Gauge/Gravity duality
Holographic slow-roll ination
Non-geometric ination
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
To be precise
For every eld Φ in AdSd +1 there exists an operator OΦ in CFT such that
Z
ZSUGRA [φ(0) ] = exp −
φ(0) OΦ
.
∂ AdS
CFT
For example
scalar eld Φ in AdS of mass m ⇔ conformal operator OΦ of
dimension ∆ given by m2 = ∆(d − ∆).
metric g µν in AdS ⇔ energy-momentum tensor T µν .
Similarly, there exists a gauge/gravity duality between some gauge theory
without gravity and a gravity theory with the metric asymptotically power
law
ds 2 = −dt 2 + t 2α d~x 2 ,
[Kanitscheider, Taylor, Skenderis (2008)]
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Analytic continuation
Euclidean AdS
de Sitter
CFT side
AdS side
κ̄2
= −κ2 ,
p̄ = −ip .
N̄ = −iN ,
p̄ = −ip .
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Cosmology/CFT correspondence
Two-point functions
hhζ(p )ζ(−p )ii =
where
−1
,
8 Im B (p̄ )
hhγ (s ) (p )γ (s ) (−p )ii =
0
−δ ss
,
Im A(p̄ )
0
hhT ij (p̄ )T kl (p̄ )ii = A(p̄ )Πijkl + B (p̄ )π ij π kl .
In [McFadden, Skenderis (2011)] all three-point function has been worked
out, for example
1
1
× Q3
Im [hhT (p̄1 )T (p̄2 )T (p̄3 )ii
256
Im
j =1 B (q̄j )

3
X
δ
T
+3
hhT (p̄j )T (−p̄j )ii − 2 δ ij hh ij (p̄1 , p̄2 )T (p̄3 )iig =δ + 2 perm.  .
δg
j =1
hhζ(p1 )ζ(p2 )ζ(p3 )ii = −
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
The models
The basic questions that can be asked are:
1
2
Given a cosmology, what is the dual QFT?
Given a dual QFT, what is the corresponding cosmology?
In our papers [AB, McFadden, Skenderis (2011)], [AB, McFadden,
Skenderis (2012) to appear] we have been considering two models:
1
The cosmology is given by an inaton with a fake superpotential.
What is the dual QFT?
2
A dual QFT is super-renormalizable, dened by perturbations around
free theories. What is the cosmology?
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Holographic slow-roll ination
The inaton potential is given by a fake superpotential [Townsend
(1984)], [Bond, Salopek (1991)], . . .
Z
√
1
S = 2 d4 x −g R − (∂Φ)2 − 2κ2 V (Φ)
2κ
with
2 − 3 W 2,
−2κ2 V = W,Φ
2
where λ 1 and c ∼ 1.
λ
c
W = − 2 − Φ2 − Φ 3 ,
2
3
Equations of motion can be solved exactly and the background geometry
turns out to be asymptotically de Sitter.
Φ(t ) =
3λ
c
·
1
,
1 + e λt
λ2
λt − 3c 2
a(t ) = 1 + e
λ3
λ2 e λt
.
exp t 1 + 2 + 2
3c
3c (1 + e λt )2
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Cosmology to follow
For this model
∗
=
η∗
=
2λ4
p 2λ
+ O (λ7 )
c 2 (1 + p λ )4
−λ +
2λ
+ O (λ4 )
1 + pλ
This is a `hilltop' inationary
model.
In this model ∗ ∼ η∗4 , which
diers from a textbook slow-roll
ination, where ∗ ∼ η∗ .
In particular,
hhζ(p )ζ(−p )ii =
H∗2
4∗
(1 + 2b η∗ + O (λ2 )).
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
The dual description
With the potential
λ
c
W = − 2 − Φ2 − Φ 3 ,
2 − 3 W 2,
−2κ2 V = W,Φ
2
2
3
the mass of the inaton Φ is m2 = λ(3 − λ).
In the dual description it corresponds to the operator OUV of dimension
∆UV = 3 − λ turned on in the UV CFT,
Z
2λ
S = SUV CFT +
d3 x OUV .
c
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
The dual description in the IR
The inationary phase is
described by the inverse RG
ow from the IR to the UV
CFT.
In the IR the deformation is
given by an irrelevant conformal
primary operator OIR of
dimension
∆IR = 3 + λ + O (λ4 ) and
Z
2λ
S = SIR CFT −
d3 x OIR .
c
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
What we want to achieve
We want to show that the cosmological results can be obtained via
the cosmology/CFT duality.
In other words, the slow-roll ination can be understood as a simple
deformation of the dual CFT.
Since λ 1, we can use the perturbation theory around the CFTs.
CFT does not have to be weakly coupled!
Since the gravity side is weakly coupled, the agreement on the both
sides of the correspondence leads to a non-trivial check on
AdS/CFT.
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Sketch of the calculations in the CFT
We need to calculate the correlation functions
hhTT ii,
hhT (s1 ) TT ii,
hhT (s ) T (s ) ii,
hhT (s1 ) T (s2 ) T ii,
0
hhTTT ii
hhT (s1 ) T (s2 ) T (s3 ) ii
in the deformed CFT, i.e. in the QFT with the action
Z
2λ
S = SCFT +
d3 x O,
c
where O is a scalar conformal primary of dimension ∆ = 3 − λ.
Since
hT (x )isrc = (∆ − d )φ0 (x )hO(x )isrc ,
all correlation functions involving trace of T are related to the correlation
functions involving O.
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Sketch of the calculations in the CFT, continued
c2
1
,
8λ4 Im hhO(p̄ )O(−p̄ )ii
c hhO(p̄1 )T (s2 ) (p̄2 )T (s3 ) (p̄3 )ii + semi-local terms
(s2 ) (s3 )
hhζγ γ ii =
2λ2 ImhhO(p̄1 )O(−p̄1 )ii · Im A(p̄2 ) Im A(p̄3 )
hhζ(p )ζ(−p )ii = −
and then the perturbative expansion, e.g.
hO(x )O(y )i =
=
hO(x )O(y )e −
2λ R O
c
iCFT
hO(x )O(y )iCFT −
2λ
c
Z
d3 u hO(x )O(y )O(u )iCFT + . . .
Two challenges:
1
2
To nd out the general expressions for three-point functions such as
hhO(p1 )T (s2 ) (p2 )T (s3 ) (p3 )ii in momentum space.
Use the perturbative expansion in the leading order in λ (more
complex than it looks).
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Three-point functions in CFT
The form of all three-point functions in CFTs involving a scalar
conformal primary and the stress-energy tensor are given in position
space by Osborn and Petkos.
We have worked out a complete set of three-point functions of a
scalar primary operator, conserved spin-1 current and stress-energy
tensor. We chose a dierent approach via the direct solution of
conformal Ward identities. This is a topic for another presentation...
We use the method to evaluate all necessary three-point functions of
stress-energy tensor and the operator of dimension 3 − λ to leading
order in λ.
Now we want to evaluate
hO(x )O(y )i =
=
hO(x )O(y )e −
2λ R O
c
iCFT
hO(x )O(y )iCFT −
in the leading order in λ.
2λ
c
Z
d3 u hO(x )O(y )O(u )iCFT + . . .
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Limits of n-point functions
Problem:
hhO(p1 )O(p2 )O(p3 )ii =
|p1 |3 + |p2 |3 + |p3 |3
+ O (λ0 ).
λ
One can show that the leading λ behaviour of
Z
In = d3 u1 . . . d3 un hO(x )O(y )O(u1 ) . . . O(un )i
in the momentum space is
1
In (p ) ∼ n p 3−(n+2)λ .
λ
In the perturbative expansion
Z
2λ
hO(x )O(y )iCFT −
d3 u hO(x )O(y )O(u )iCFT + . . .
c
all terms contribute!
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Summation of the series
One can show that the leading λ behaviour of the n-point function is
lim
hhO(p )O(−p )O(p1 ) . . . O(pn )ii
p1 ,p2 ,...,pn →0
(−1)n (n + 3)! p 3−(n+2)λ
1
·
+O
.
=
2 · 3n+1
λn
λ n −1
The behaviour is universal assuming λ has `nothing to do' with other
operators in the theory.
The summation is possible and we nd the perfect agreement with
cosmological two and three-point functions.
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
The model with non-geometric ination
This model is analysed in [AB, McFadden, Skenderis (2011)]. We start
with the dual QFT
Z
1
1
1
1
3
S = 2
d x Tr (FijI )2 + (D φJ )2 + (D χK )2 + ψ̄ L D/ ψ L
2
2
2
gYM
+ interactions] ,
where we have
NA SU (N )-gauge elds AaI
i ,
Nφ minimal real scalars φJ ,
N conformal real scalars χK ,
χ
Nψ fermions ψαL ,
with all elds transforming in the adjoint representation of SU (N ).
The QFT is assumed to be weakly coupled, therefore
The perturbative analysis on the QFT side is possible.
We should expect a strongly coupled gravity on the cosmology side.
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Predictions
Predictions dier from the conventional ination but are compatible with
the current data. The scalar power spectrum is
hhζζii ∼
1
hhTT ii
∼
1
2 ln(p /p ) + O (g 4 ) ,
N 2 (Nφ + NA )p 3 1 + Cge
0
e
2 = g 2 N̄ /p̄ .
where the dimensionless eective coupling is ge
YM
1
2
3
4
The amplitude is small ⇒ N ∼ 104 large N expansion is valid.
2 ∼ 10−2 at p = 0.002Mpc−1 QFT
Almost scale invariance ⇒ ge
0
is weakly coupled.
The dual cosmology is strongly coupled and stringy corrections are
important. The ination phase is non-geometric.
The geometry is asymptotically the power-law ination
ds 2 ∼ −dt 2 + t 2n d~x 2 for t → ∞ and some n.
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Running of the spectral index
Holographic models predict [McFadden, Skenderis (2010)], [Dias (2011)]
d ns
4 ),
= −(ns − 1) + O (ge
d log p
while conventional slow-roll ination predicts that αs /(ns − 1) is very
small (of rst order in slow-roll).
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
ΛCDM vs. our model
ΛCDM model, our holographic model.
The dedicated analysis was carried out in [Easther, Flauger,
McFadden, Skenderis (2011)]. Related work: [Dias (2011)].
Other characteristics such as non-gaussianities were calculated as
well.
Conclusions
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Conclusions
1
Ination is holographic: standard observables such as power spectra
and non-gaussianities can expressed and calculated by means of the
correlation functions of the dual QFT.
2
3
The QFT dual to slow-roll ination is a deformation of a CFT.
Slow-roll results are essentially xed by conformal invariance.
4
There are new holographic models based on perturbative QFT that
describe the ination starting in a non-geometric, strongly coupled
phase.
5
A class of models based on super-renormalizable QFT was t to data
and shown to provide a competitive model to ΛCDM. Data from the
Planck satellite should permit a denitive test of this scenario.
Introduction
Cosmology
Holographic cosmology
Holographic slow-roll ination
Non-geometric ination
Conclusions
Thank you for your attention.
arXiv:
[hep-th/0907.5542]
[hep-th/1001.2007]
[hep-th/1010.0244]
[hep-th/1011.0452]
[hep-th/1104.2040]
[hep-th/1104.3894]
[hep-th/1112.1967]