2016.04
Minxi He, Junyu Liu, Shiyun Lu, Siyi Zhou, Yi-Fu Cai, Yi Wang, Robert Brandenberger
A characteristic signature of the string gas cosmology is to predict primordial power spectra with a red scalar but a blue
tensor tilt. Nevertheless, this feature can also be realized in the so-called G-inflation model, in which the Horndeski operators have been
introduced, where a blue tilt is realized by softly breaking the null energy condition. In this article we search for potential observational
differences between these two cosmologies by performing detailed perturbation analyses based on the effective field theory approach. Our
results show that, although both two models may produce blue tilted tensor perturbations, they behave differently on three aspects. Firstly,
the string gas cosmology explicitly predicts a specific consistency relation between the index of the scalar modes 𝑛𝑠 and that of tensor
ones 𝑛𝑡 , which can hardly be reproduced by G-inflation. Secondly, the string gas cosmology typically predicts invisible nonlinear
fluctuations while G-inflation gives rise to observationally large non-Gaussianities as its kinetic term becomes important during inflation.
However, after finely tuning the model parameters of G-inflation, there could remain a degeneracy between two models. Nevertheless, this
degeneracy can be broken by the third aspect, that is, the scale dependence of the nonlinearity parameter, which is vanishing for Ginflation but blue tilt for the string gas cosmology. Therefore, we conclude that the string gas cosmology is in principle observationally
distinguishable from the single field inflationary cosmology including G-inflation.
String Gas
Cosmology is an extension of the standard big bang
cosmology, where a gas of closed superstrings are coupled
to the background space-time rather than point particles,
with new symmetries and degrees of freedom. It predicts a
scale-invariant tensor spectrum with a slight blue tilt.
NONGAUSSIANITIES
OF DIFFERENT
ORDERS
𝑓𝑁𝐿(𝐺) ≫ 𝑓𝑁𝐿(𝑆)
We mainly studied the nonGaussianity of G-inflation by
using the framework of effective
field theory of inflation, where
we could conveniently focus on
the perturbations, 𝜋. To produce
blue tilt, we would softly break
the null energy condition but,
because of the presence of
Galileon terms, we could still
avoid ghosts and gradient
instabilities. Under these
conditions, we obtained the 𝑓𝑁𝐿
by calculating the three point
correlation function 𝜋 3 , finding
that the non-Gaussianity here is
of order 1, although there exists
fine-tuning by which the large
non-Gaussianity can be canceled
out. On the other hand, the 𝑓𝑁𝐿
of String Gas Cosmology has
already been found to be
extremely smaller than 1. Thus,
from this point of view, we could
distinguish G-inflation from
String Gas Cosmology by the
different orders of nonGaussianity in certain regime.
VS
G-inflation is a class of inflation
models where the inflation is driven by the Galileon-like
scalar field. They have equations of motion in which the
gravitational fields and scalar fields have no more than
second order derivatives. With slight violation of null
energy condition, G-inflation can have blue tensor spectrum.
THE SCALE
DEPENDENCE OF
THE 𝑓𝑁𝐿 ’S
𝑑
𝑓
𝑑𝑘 𝑁𝐿(𝐺)
= 0,
𝑑
𝑓
𝑑𝑘 𝑁𝐿(𝑆)
≠0
By working out the 𝑓𝑁𝐿 of Ginflation, we also found that
even if in some regime the nonGaussianity of this model
would be invisible as in String
Gas Cosmology, there is one
more property that could help
us to remove the degeneracy
between them. Our calculation
results tell us that the 𝑓𝑁𝐿 of Ginflation is scale independent
while that of String Gas
Cosmology is strongly scale
dependent. Therefore, if
unfortunately the nonGaussianity of G-inflation is
small even vanishing, there is
still another way to distinguish
these two models.
DIFFERENT
CONSISTENCY
RELATIONS
𝑛𝑠 − 1 ≠ −𝑛𝑡 , 𝑛𝑠 − 1 = −𝑛𝑡
String Gas Cosmology has a
very special consistency relation,
𝑛𝑠 = 1 − 𝑛𝑡 . However, in the
description of effective field
theory of inflation with minimally
coupled gravity, 𝑛𝑡 = −2𝜖 and
𝑛𝑠 − 1 = −2𝜖 − 𝜂 + 𝑓 𝜖, 𝜂, …
for a special limit and 𝑛𝑠 − 1 =
− 4𝜖 + 𝑔(𝜖, 𝜂, … ) for other
cases. It is hard to produce 𝑛𝑠 =
1 − 𝑛𝑡 for inflation if we insist
on slow roll expansion and
minimal coupling gravity. Thus,
different consistency relations
provide us with a third way to
distinguish G-inflation and String
Gas Cosmology even though
both of them can produce blue
tilt.