Page 1

arXiv:0908.0192v2 [astro-ph.CO] 14 May 2010

From CFT Spectra to CMB Multipoles in

Quantum Gravity Cosmology

Ken-ji Hamada1, Shinichi Horata2and Tetsuyuki Yukawa3

1Institute of Particle and Nuclear Studies, KEK, Tsukuba 305-0801, Japan

1Department of Particle and Nuclear Physics, The Graduate University for

Advanced Studies (Sokendai), Tsukuba 305-0801, Japan

2,3Hayama Center for Advanced Studies, The Graduate University for

Advanced Studies (Sokendai), Hayama 240-0193, Japan

Abstract

We study the inflation process of universe based on the renormal-

izable quantum gravity formulated as a conformal field theory (CFT).

We show that the power-law CFT spectrum approaches to that of

the Harrison-Zel’dovich-Peebles type as the amplitude of gravitational

potential gradually reduces during the inflation. The non-Gaussanity

parameter is preserved within order of unity due to the diffeomor-

phism invariance.Sharp fall-off of the angular power spectrum of

cosmic microwave background (CMB) at large scale is understood as

a consequence of the existence of dynamical scale of the quantum

gravity ΛQG(≃ 1017GeV). The angular power spectra are computed

and compared with the WMAP5 and ACBAR data with a quality of

χ2/dof ≃ 1.1.

Page 2

1Introduction

Recent observations of anisotropies in the cosmic microwave background

(CMB) by various groups such as the cosmic background explore (COBE)

[1], the Wilkinson microwave anisotropy probe (WMAP) [2, 3], and the ar-

cminute cosmology bolometer array receiver (ACBAR) [4] have provided a

refined picture of the history of universe after the big bang. Cosmologi-

cal parameters are determined with high accuracy based on the cosmolog-

ical perturbation theory [5, 6, 7, 8], assuming only the primordial spec-

trum close to that of the Harrison-Zel’dovich-Peebles [9, 10, 11]. We be-

lieve that one of the important problems remained in the study of inflation

[12, 13, 14, 15, 16, 17, 18, 19, 20] is to clarify dynamics producing such a scale-

invariant spectrum from the fundamental theory rather than introducing an

artificial field by hands just for the phenomenological purpose.

As the fundamental theory, we will employ the renormalizable quantum

gravity formulated based on the conformal field theory (CFT) in four dimen-

sions [21, 22, 23, 24]. It predicts that quantum fluctuations of the conformal

mode in gravitational fields become so large at very high energies beyond

the Planck scale, and a conformally invariant space-time is realized as a

consequence of background metric independence. It then produces a power-

low spectrum and a non-Gaussian fluctuation distribution for the theoretical

generation of CMB spectrum.

Evolution of the early universe can be regarded as a violating process

of conformal invariance [25, 23, 26, 27]. The conformal symmetry starts

to be broken at the Planck scale, and the space-time dynamics shifts to

the inflationary epoch with the expansion time constant about the Planck

mass mpl(= 1/√G). The conformal invariance is completely broken at the

dynamical scale of quantum gravity ΛQG, which is expected to be 1017GeV.

At this energy scale, the inflation terminates, and the universe turns to the

classical Friedmann universe.

The purpose of this paper is to clarify how the Harrison-Zel’dovich-

Peebles spectrum is prepared for the initial condition of the cosmological

1

Page 3

perturbation equation for computing the CMB angular power spectra. The

time evolution of gravitational fluctuations in the inflationary background

has been studied within the linear approximation [26]. It was shown that

during the inflation the amplitude of scalar fluctuation decreases to the size

which solves the flatness problem.Combining this result with smallness

of the non-Gaussianity, we will show that the ‘almost’ Harrison-Zel’dovich-

Peebles spectrum (a constant spectrum with rapid fall-off at small momenta)

emerges after the inflation.

The tensor mode which measures a degree of deviation from the confor-

mal invariance is expected to be small initially because of the asymptotically

free behavior of this mode, while its amplitude is preserved during the infla-

tion. Thus, the tensor mode also gives a significant contribution at the later

stage in the primordial spectra for the computation of the CMB multipole

distribution.

The correlation length of quantum gravity is given by the order of ξΛ=

1/ΛQG, and it brings the absence of correlations of two points separated larger

than ξΛinitially prepared before the universe starts inflation. This explains

the sharp fall off of the angular power spectra at low multipoles [25].

2 Quantum Gravity Cosmology

The renormalizable quantum gravity formulated as a perturbed theory

from CFT is defined by the dimensionless action [21, 22, 23, 24],

I =

?

d4x√−g

?

−1

t2C2

µνλσ− bG4+1

¯ h

?

1

16πGR − Λ + LM

??

,(1)

where Cµνλσ is the Weyl tensor, G4is the Euler density, and t is a dimen-

sionless coupling constant which measures a degree of deviation from CFT.

The cosmological constant is denoted by Λ, whose effect can be neglected in

the early universe. LMrepresents the Lagrangean for conformally invariant

matter fields, and ¯ h is the Planck constant which is taken to be unity.

2

Page 4

The metric field is decomposed to the conformal mode φ and the traceless

tensor mode hµνas

gµν= e2φ¯ gµν,¯ gµν= ηµν+ hµν, (2)

with tr(h) = 0. Signature of the flat background metric ηµνis (−1,1,1,1),

which defines the conformal time and the comoving frame with the coordi-

nates xµ= (η,xi) and ∂µ= (∂η,∂i).

The traceless tensor mode, which is governed by the Weyl action, is

handled perturbatively in terms of the coupling t. The renormalized cou-

pling constant tr indicates the asymptotic freedom with the beta function

βt= −β0t3

mode, and also implies the existence of a dynamical energy scale ΛQG. The

running coupling constant is then written as 1/¯t2

physical momentum p.

A recent significant progress in quantization techniques is that the con-

formal mode has been managed non-perturbatively, as in the case of two-

dimensional quantum gravity [29, 30, 31].

morphism invariant measure in terms of the practical measures defined on

the flat background metric ηµν, the partition function is expressed as Z =

?[dφdh···]ηexp(iS + iI). The induced action S is the Jacobian needed to

recover the diffeomorphism invariance. At the lowest order of the coupling

tr, it gives the kinetic term of the conformal mode, called the Riegert action

[32, 33, 34, 35],

S = −b1

8π2

where¯∆4= (∂λ∂λ)2+o(h) is a conformally invariant fourth-order differential

operator for a scalar field variable. This action is a four-dimensional counter

part of the Liouville-Polyakov action in two dimensions. The coefficient has

been computed to be b1= (NX+ 11NW/2 + 62NA)/360 + 769/180, where

NX, NW, and NA are numbers of scalar fields, Weyl fermions, and gauge

fields, respectively, and the last term is a quantum loop correction from

r(β0> 0) [28, 21]. It justifies the perturbative treatment of this

r(p) = β0log(p2/Λ2

QG) for a

When we rewrite the diffeo-

?

d4xφ¯∆4φ, (3)

3

Page 5

gravitational fields. Since b1for various GUT models is given about 10, we

consider this case in the following.

There are various proporsals in the past how to treat the problem of

ghosts. For example, Tomboulis [36] proposed that ghosts might be removed

in the IR region based on the idea of Lee and Wick using the resummed

propagator in asymptotically free theories. The idea of asymptotic safety by

Weinberg [37] is defined as a cutoff model expanded in derivatives about the

ghost-free Einstein theory assuming the existence of a non-trivial UV fixed

point where the cutoff is taken to be infinity. Recently, Horava [38] proposed

a higher-derivative gravity model by making ghosts non-dynamical at the

cost of diffeomorphism invariance in the UV limit.

Our proposal [24] is that the problem of ghosts should be reconsidered

under the light of CFT described by the combined system of the Riegert and

the Weyl actions which appears in the UV limit. Since the conformal symme-

try realized as a part of diffeomorphism invariance mixes positive-metric and

negative-metric modes of the gravitational field, we cannot consider these

modes separately and thus the field acts as a whole in physical quantities.

At present we do not have a complete proof on the unitarity problem yet,

but there is no unphysical behavior at least within discussions given in this

paper.

The quantum gravity cosmology [25, 23, 26, 27, 39] is now defined by the

effective action described by S+I together with higher trcorrections in which

the two mass scales are ordered as mpl≫ ΛQG. When the universe is at the

Planck scale, the coupling constant is negligibly small due to the asymptotic

freedom, and the equation of motion for the homogeneous component of

conformal mode,ˆφ, is obtained to be

b1∂4

ηˆφ − 3πm2

ple2ˆφ(∂2

ηˆφ + ∂ηˆφ∂ηˆφ) = 0.(4)

We introduce the proper time τ through dτ = adη with the scale factor a =

eˆφ. In terms of H = ˙ a/a, where dot represents the proper time derivative, the

stable solution is written as H = HDwhich corresponds to the inflationary

4