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
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.
Recent observations of anisotropies in the cosmic microwave background
(CMB) by various groups such as the cosmic background explore (COBE)
, the Wilkinson microwave anisotropy probe (WMAP) [2, 3], and the ar-
cminute cosmology bolometer array receiver (ACBAR)  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
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 . 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
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 .
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],
16πGR − Λ + LM
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.
where | ∂2= ∂i∂iandˆφ is an inflationary solution of homogeneous equation of
motion (6). The second equation is of a second order obtained by factoring
out the operator | ∂2.
The dynamical factors B0(τ) in both equations and the inverse of¯t2
in the second equation, which come from the Riegert and the Weyl actions
respectively, show that the conformal gravity dynamics disappears at the
transition, and thus the Einstein gravity dominates in the equation of motion.
The second equation then plays an important role in connecting between
the inflation and the Einstein phases, namely initially in the limit¯tr → 0
the conformal mode dominates such that Φ = Ψ, while at the transition
point where the coupling diverges, the configuration with Φ = −Ψ should be
 C. Bennett et al., Astrophys. J. 464 (1996) L1.
 D. Spergel et al., Astrophys. J. Suppl. 148 (2003) 175.
 E. Komatsu et al., Astrophys. J. Suppl. 180 (2009) 330.
 C. Reichardt et. al, Astrophys. J. 694 (2009) 1200.
 J. Bardeen, Phys. Rev. D22 (1980) 1882.
 H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78 (1984) 1.
 W. Hu and N. Sugiyama, Astrophys. J. 444 (1995) 489; Phys. Rev. D51
 A. Liddle and D. Lyth, Cosmological Inflation and Large-Scale Structure,
(Cambridge Univ. Press, 2000).
 E. Harrison, Phys. Rev. D1 (1970) 2726.
 Ya. B. Zel’dovich, Mon. Not. R. Astron. Soc. 160 (1972) P1.
 P. Peebles and J. Yu, Astrophys. J. 162 (1970) 815.
 A. Guth, Phys. Rev. D23 (1981) 347.
 K. Sato, Mon. Not. R. Astron. Soc. 195 (1981) 467.
 A. Linde, Phys. Lett. B108 (1982) 389.
 A. Albrecht and P. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220.
 A. Starobinsky, Phys. Lett. B91 (1980) 99.
 V. Mukhanov and G. Chibisov, Pis’ma Zh. Eksp. Theor. Fiz. 33 (1981)
549 [JETP Lett. 33 (1981) 532].
 A. Starobinsky, Pis’ma Zh. Eksp. Theor. Fiz. 34 (1981) 460 [JETP Lett.
34 (1981) 438].
 A. Vilenkin, Phys. Rev. D32 (1985) 2511.
 S. Hawking, T. Hertog and H. Reall, Phys. Rev. D63 (2001) 083504.
 K. Hamada, Prog. Theor. Phys. 108 (2002) 399.
 K. Hamada, Found. Phys. 39 (2009) 1356.
 K. Hamada, S. Horata and T. Yukawa, Focus on Quantum Gravity Re-
search, (Nova Science Publisher, NY, 2006), Chap. 1.
 K. Hamada, Int. J. Mod. Phys. A24 (2009) 3073.
 K. Hamada and T. Yukawa, Mod. Phys. Lett. A20 (2005) 509.
 K. Hamada, S. Horata and T. Yukawa, Phys. Rev. D74 (2006) 123502.
 K. Hamada, S. Horata, N. Sugiyama and T. Yukawa, Prog. Theor. Phys.
119 (2008) 253.
 E. Fradkin and A. Tseytlin, Nucl. Phys. B201 (1982) 469.
 A. Polyakov, Phys. Lett. 103B (1981) 207; Mod. Phys. Lett. A2 (1987)
 V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A3
 J. Distler and H. Kawai, Nucl. Phys. B321 (1989) 509; F. David, Mod.
Phys. Lett. A3 (1988) 1651.
 R. Riegert, Phys. Lett. 134B (1984) 56.
 I. Antoniadis and E. Mottola, Phys. Rev. D45 (1992) 2013.
 I. Antoniadis, P. Mazur and E. Mottola, Nucl. Phys. B388 (1992) 627.
 K. Hamada and F. Sugino, Nucl. Phys. B553 (1999) 283.
 E. Tomboulis, Phys. Lett. 70B (1977) 361.
 S. Weinberg, in Understanding the Fundamental Constituents of Matter,
ed. A. Zichichi (Plenum Press, NY, 1977).
 P. Horava, Phys. Rev. D79 (2009) 084008.
 K. Hamada, A. Minamizaki and A. Sugamoto, Mod. Phys. Lett. A23
 E. Komatsu and D. Spergel, Phys. Rev. D63 (2001) 063002.
 I. Antoniadis, P. Mazur and E. Mottola, Phys. Rev. Lett 79 (1997) 14.
 T. Pyne and S. Carroll, Phys. Rev. D53 (1996) 2920.
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