Backreaction of super-Hubble cosmological perturbations
beyond perturbation theory
Robert Brandenberger,
1,*
Leila L. Graef,
2,†
Giovanni Marozzi,
3,4,‡
and Gian Paolo Vacca
5,§
1
Physics Department, McGill University, Montreal, QC H3A 2T8, Canada
2
Departamento de Fisica Teorica, Universidade do Estado do Rio de Janeiro,
20550-013 Rio de Janeiro, RJ, Brazil
3
Dipartimento di Fisica, Universit`a di Pisa, and INFN, Sezione di Pisa,
Largo B. Pontecorvo 3, I-56127 Pisa, Italy
4
Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud 150, Urca,
CEP 22290-180 Rio de Janeiro, RJ, Brazil
5
INFN—Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
(Received 23 July 2018; published 21 November 2018)
We discuss the effect of super-Hubble cosmological fluctuations on the locally measured Hubble
expansion rate. We consider a large bare cosmological constant in the early Universe in the presence of
scalar field matter (the dominant matter component), which would lead to a scale-invariant primordial
spectrum of cosmological fluctuations. Using the leading-order gradient expansion, we show that the
expansion rate measured by a (secondary) clock field which is not comoving with the dominant matter
component obtains a negative contribution from infrared fluctuations, a contribution whose absolute value
increases in time. This is the same effect that a decreasing cosmological constant would produce. This
supports the conclusion that infrared fluctuations lead to a dynamical relaxation of the cosmological
constant. Our analysis does not make use of any perturbative expansion in the amplitude of the
inhomogeneities.
DOI: 10.1103/PhysRevD.98.103523
I. INTRODUCTION
The cosmological constant problem (see [1] for reviews)
is a key challenge for fundamental physics. Basic argu-
ments imply that the vacuum energy of matter fields should
act as a cosmological constant Λ and cause the Universe to
accelerate. Assuming that an ultraviolet cutoff scale close to
the Planck scale is used, the resulting value for Λ is about
120 orders of magnitude larger than the maximal one
allowed by observations. The discovery of the acceleration
of the Universe [2] has added a new perspective to this
problem. If the observed acceleration is due to a small
cosmological constant, then we do not only have to explain
why the cosmological constant is not of the Planck scale,
but also why it happens to be rearing its head at the present
time in the cosmological history of the Universe. This is the
coincidence problem (see e.g., [3] for reviews of the dark
energy problem).
It has been conjectured for some time that de Sitter space
is unstable because of infrared effects [4,5] (see, however,
[6] for arguments supporting the stability of de Sitter
space), and that hence the bare cosmological constant in
the Lagrangian would be invisible today.
1
Specifically, it
was suggested by Tsamis and Woodard [10] that the
backreaction of super-Hubble scale gravitational waves
could give a negative contribution to the effective cosmo-
logical constant and cause the latter to relax. The problem
was studied in perturbation theory, and it was found that
one needs to go to two-loop order (fourth order in the
amplitude of the gravitational waves) in order to obtain a
nonvanishing effect. The study of the backreaction effect of
long wavelength cosmological perturbations was initiated
in [11] and it was found that at one-loop order (second
order in the amplitude of the perturbations) super-Hubble
cosmological perturbations lead to a negative contribution
to the cosmological constant (see, also, [12 –14]). Based on
this analysis, it was then conjectured in [15] that this
backreaction could lead to a late time scaling solution for
*
rhb@physics.mcgill.ca
†
leilagraef@gmail.com
‡
giovanni.marozzi@unipi.it
§
Gianpaolo.Vacca@bo.infn.it
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP
3
.
1
See also [7–9] for a discussion on the difficulty of obtaining
de Sitter space in string theory.
PHYSICAL REVIEW D 98, 103523 (2018)
2470-0010=2018=98(10)=103523(9) 103523-1 Published by the American Physical Society