Ambiguities in second-order cosmological perturbations for non-canonical scalar fields

Abstract

Over the last few years, it was realised that non-canonical scalar fields can lead to the accelerated expansion in the early universe. The primordial spectrum in these scenarios not only shows near scale-invariance consistent with CMB observations,but also large primordial non-Gaussianity. Second-order perturbation theory is the primary theoretical tool to investigate such non-Gaussianity. However, it is still uncertain which quantities are gauge-invariant at second-order and their physical understanding therefore remains unclear. As an attempt to understand second order quantities, we consider a general non-canonical scalar field, minimally coupled to gravity, on the unperturbed FRW background where metric fluctuations are neglected a priori. In this simplified set-up, we show that there arise ambiguities in the expressions of physically relevant quantities, such as the effective speeds of the perturbations. Further, the stress tensor and energy density display a potential instability which is not present at linear order. Comment: 3 Pages, To appear in the Proceedings of the 12th Marcel Grossman Meeting

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Ambiguities in second-order cosmological perturbations for
non-canonical scalar fields
Corrado Appignani(1), Roberto Casadio(1), S. Shankaranarayanan(2,3)∗
(1)Dipartimento di Fisica, Universit‘a di Bologna and I.N.F.N., Sezione di Bologna, via
Irnerio 46, 40126 Bologna, Italy
(2)Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, UK
(3)School of Physics, Indian Institute of Science Education and Research-Trivandrum,
CET campus, Thiruvananthapuram 695 016, India
Over the last few years, it was realised that non-canonical scalar fields can lead to the
accelerated expansion in the early universe. The primordial spectrum in these scenarios
not only shows near scale-invariance consistent with CMB observations, but also large
primordial non-Gaussianity. Second-order perturbation theory is the primary theoretical
tool to investigate such non-Gaussianity. However, it is still uncertain which quantities
are gauge-invariant at second-order and their physical understanding therefore remains
unclear. As an attempt to understand second order quantities, we consider a general non-
canonical scalar field, minimally coupled to gravity, on the unperturbed FRW background
where metric fluctuations are neglected a priori. In this simplified set-up, we show that
there arise ambiguities in the expressions of physically relevant quantities, such as the
effective speeds of the perturbations. Further, the stress tensor and energy density display
a potential instability which is not present at linear order.
Keywords: Inflation, non-canonical scalar field, higher order cosmological perturbation
1. Introduction and motivation
Predictions of inflation seem to be in excellent agreement with the CMB data.1
However, it is still unclear what is the nature of the field which drives inflation.
Historically, a canonical scalar field has been the preferred candidate for inflaton,
but in recent years, also a non-canonical scalar field, dubbed as k-inflaton, was
considered as serious alternative mechanisms to drive inflation.2
Both scenarios lead to nearly scale invariant power-spectra with negligible run-
ning and hence can not be distinguished (or ruled out) from the current CMB
data.1The future missions, including PLANCK,3hold promise in ruling our either
of these two scenarios by looking at the non-Gaussianity of the primordial spectra,
but quantifying non-Gaussianity requires one to go beyond linear order.4–6
There are four different approaches in the literature to study cosmological per-
turbations: 1) solving Einstein’s equations order-by-order;72) the covariant ap-
proach based on a general frame vector uα;83) the Arnowitt-Deser-Misner (ADM)
approach based on the normal frame vector nα;94) the reduced action approach.10
In the case of linear perturbations, it was shown that all of these four approaches
lead to identical equations of motion. However, to our knowledge, a complete anal-
ysis has not been done in the literature for higher-order perturbations.
In this talk, to illustrate the problems that may occur at higher-order, and not
to get bogged-down with the gauge issues, we consider a simple situation: we freeze
∗speaker; E-mail: shanki@iisertvm.ac.in

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all metric perturbations and focus on the perturbations of a minimally-coupled,
generalised scalar field φ, whose Lagrangian density is given by2
L = P(X,φ) ,where2X = ∇αφ∇αφ .(1)
More precisely, we will only consider linear perturbations of the scalar field,
φ(t,x) = φ0(t) + δ(1)φ(t,x) ,(2)
about the 4-dimensional FRW background, while expanding all the dependent
quantities, like X and stress tensor, up to second order, and highlight the main
differences in these approaches. For this purpose, it is convenient to compare the
ratio
c2
coefficients of δ˙φ2
in the components of the stress tensor and related quantities. Since c2
ally the square of a speed, we will refer to this ratio as the “speed of propagation”.
s=coefficients of (δφ,i/a)2
,(3)
sis dimension-
2. Key results
Below, we will provide the key results and, for details, we refer the reader to Ref.11
Perturbed tensor and ADM approach: For an arbitrary scalar field Lagrangian,
δ
most notably for the canonical scalar field, the effective speed of propagation of
δ
(2)T00may represent an unstable perturbation; only under very special conditions,
(2)T00(and δ
(2)Tii) are the same as that of the standard definition12
c2
δφ=
P
(0)
X
P
(0)
X+ P
(0)
XX˙φ2
0
(4)
Using (3), we can define speeds related with the propagation of density perturba-
tions in the background frame from δ
(2)T00and δ
(2)Tii, respectively,
c2
0=
P
(0)
X− P
XX˙φ2
(0)
XX˙φ2
0
XXX˙φ4
P
(0)
X+ 4P
(0)
0+ P
(0)
0
,c2
?= c2
δφ.(5)
Note that these velocities become imaginary indicating that the perturbations
may be unstable. The nature of this instability is easily understood in analogy
with classical mechanics: when c2
δφis negative, the system resembles an inverted
harmonic oscillator and, no matter how small δφ, it will rapidly run away from the
background solution φ0and from the perturbative regime.
Covariant approach: In the fluid frame, the energy density exhibits the same kind
of instability as the perturbed stress-tensor δ
In particular, the perturbations turn out to be unstable because of the negative sign
of the spatial momentum contribution. Only under special conditions, the effective
speed of propagation of the energy density and pressure perturbations are equal.
Using (3), the speed of propagation for density and pressure perturbations in the
fluid frame are
(2)T00, also for the canonical scalar field.
c2
ρ= −
P
(0)
X+ P
XX˙φ2
(0)
XX˙φ2
0
XXX˙φ4
P
(0)
X+ 4P
(0)
0+ P
(0)
0
,c2
p= c2
?= c2
δφ,(6)

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which differ from Eq. (5).
Symmetry reduced approach: The basic idea here is to perturb the action about
the FRW background, up to second (or higher) order, and reduce it so that the
perturbations are described in terms of a single gauge-invariant variable. The second
order canonical Hamiltonian is identical to the stress-tensor for the canonical scalar
field, but differs for general non-canonical fields. This implies that δ
canonical Hamiltonian δ
(2)T00and the
(2)H may become unstable under different conditions.
3. Discussion
The first question is why the canonical Hamiltonian, perturbed stress tensor and
super-Hamiltonian coincide for a canonical scalar field, but not for general scalar
field Lagrangians. To go about answering this question, it is necessary to look at
the four approaches we have employed from a different perspective. In the first
two approaches – perturbed stress tensor and covariant approach – we perturb
the general expression for the scalar field stress tensor and obtain its second order
contribution δ
reduced action – we expand the action to second order in the perturbation and
obtain the super-(canonical) Hamiltonian of the corresponding perturbed action.
While the super-Hamiltonian δ
δ
Under what condition the super-Hamiltonian δ
δ
time-variation of background quantities (like P
the canonical scalar field, these functions are indeed constant and the perturbed
quantities therefore coincide.) Although such an approximation may be valid for
specific non-canonical fields, they fail for some of the known fields like Tachyon,
DBI.11
R. C. is supported by the INFN grant BO11 and S. S. was supported by the
Marie Curie Incoming International Grant IIF-2006-039205.
(2)T00. In the last two approaches – ADM formulation and symmetry-
(2)H is identical to δ
(2)T00, the canonical Hamiltonian
(2)H differs.
(2)H, the canonical Hamiltonian
(2)H for non-canonical scalar fields are identical? They are all identical provided the
(0)
X
and P
(0)
XX) can be neglected. (For
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