The Effective Field Theory of Inflation

TL;DR: The effective field theory of inflation as mentioned in this paper is the most general theory describing the fluctuations around a quasi de Sitter background, in the case of single field models, in which the scalar mode can be eaten by the metric by going to unitary gauge.

Abstract: We study the effective field theory of inflation, i.e. the most general theory describing the fluctuations around a quasi de Sitter background, in the case of single field models. The scalar mode can be eaten by the metric by going to unitary gauge. In this gauge, the most general theory is built with the lowest dimension operators invariant under spatial diffeomorphisms, like g^{00} and K_{mu nu}, the extrinsic curvature of constant time surfaces. This approach allows us to characterize all the possible high energy corrections to simple slow-roll inflation, whose sizes are constrained by experiments. Also, it describes in a common language all single field models, including those with a small speed of sound and Ghost Inflation, and it makes explicit the implications of having a quasi de Sitter background. The non-linear realization of time diffeomorphisms forces correlation among different observables, like a reduced speed of sound and an enhanced level of non-Gaussianity.

Summary

1. Introduction

• The effective field theory approach, i.e. the description of a system through the lowest dimension operators compatible with the underlying symmetries, has been very fruitful in many areas, from particle physics to condensed matter.
• Describing the theory of perturbations during inflation directly around the time evolving vacuum where time diffs are non-linearly realized.the authors.
• The first two operators after the Einstein-Hilbert term are fixed by the requirement of having a given unperturbed solution H(t), while all the others are free and parametrize all the possible different theories of perturbations with the same background solution.
• It is well known that the physics of the longitudinal components of massive gauge bosons can be studied, at sufficiently high energy, concentrating on the scalar Goldstone mode (equivalence theorem).
• However the equivalence among different Lagrangians becomes more complicated when the authors consider more general terms.

2. Construction of the action in unitary gauge

• Inflation is a period of accelerated cosmic expansion with an approximately constant Hubble parameter.
• Besides the usual terms with the Riemann tensor, which are invariant under all diffs, many extra terms are now allowed, because of the reduced symmetry of the system.
• Notice that only the first three terms in the action above contain linear perturbations around the chosen FRW solution, all the others are explicitly quadratic or higher.
• 3Notice that this symmetry has nothing to do with the breaking of time diffeomorphisms.
• Clearly this cannot be true exactly as all the other terms will be generated by loop corrections: they encode all the possible effects of high energy physics on this simple slow-roll model of inflation.

4.1 Slow-roll inflation and high energy corrections

• As discussed in the last section, predictions at the scale H can be made neglecting the mixing with gravity and concentrating on the Goldstone Lagrangian (3.16).
• As additional terms are generated by graviton loops, they may be very small.
• One example of a possible experimental limit on higher dimension operators is the consistency relation for the gravitational wave tilt.

4.2 Small speed of sound and large non-Gaussianities

• The Goldstone action (3.16) shows that the spatial kinetic term (∂iπ) 2 is completely fixed by the background evolution to be M2PlḢ(∂iπ) 2.
• This relationship was stressed in the explicit calculation of the 3-point function in [21].
• Therefore, predictions which depend on very large values of ζ may lie out of the regime of validity of the effective field theory.
• The explicit calculation [21] gives the contribution of the operator π̇(∇π)2 to the experimentally constrained parameter f equil.

4.2.1 Cutoff and naturalness

• Therefore the self-interactions among the Goldstones will become strongly coupled at a certain energy scale, which sets the cutoff of their theory.
• The calculation is straightforward, the only complication coming from the non-relativistic dispersion relation.
• As expected the theory becomes more and more strongly coupled for small cs, so that the cutoff scale decreases.
• Notice that the scale Λ indicates the maximum energy at which their theory is weakly coupled and make sense; below this scale new physics must come into the game.

4.3 De-Sitter limit and the Ghost condensate

• This spatial kinetic term will make the Goldstone propagate even in the limit cs → 0.
• Looking back at the procedure (3.13) to reintroduce the Goldstone π, the authors realize that this symmetry forbids any dependence on π without derivatives.
• This is a quite remarkable feature of this example, as usually the mixing with gravity is related to the background stress energy tensor and therefore to the curvature of spacetime: the more a system curves space, the more it mixes with gravity.
• Notice however that one can not take the limit.
• It is straightforward to see that the operator M42 π̇(∇π)2 coming from the unitary gauge operator M42 (g 00 + 1)2 has dimension 1/4 and it is the least irrelevant operator.

4.3.1 De-Sitter limit without the Ghost condensate

• This would be a total time derivative without the time dependence of the scale factor a(t) and of the metric determinant.
• This implies that the mixing does not become less and less relevant going to high energy.
• Not surprisingly the estimates above are the same as the ones the authors obtained in the Ghost Condensate case eqs (4.32) and (4.33).
• One could imagine that both these spatial kinetic terms are suppressed for some reason and the leading operators come at higher order.

5. Conclusions

• Given the ongoing experimental effort to test inflation and the proliferation of different models, it is quite important to characterize the most general theory of inflation.
• In this approach the role of symmetries is made much more transparent.
• From this point of view, their approach makes clearer the relationship among inflation, theories of modification of gravity and theories which violate the Null Energy Condition (equivalent to Ḣ > 0 in the cosmological context) like the bouncing models [1, 29, 30].
• All the possible deviations from a vanilla slow-roll scenario are systematically encoded in the size of higher order operators, similarly to what happens in the study of the Standard Model of particle physics.
• Moreover all single field models are unified in a common framework and this allows us to draw general conclusions which are independent of the specific realization, as done in [12, 31] for example.

Full text

Journal of High Energy Physics
The effective field theory of inflation
To cite this article: Clifford Cheung et al JHEP03(2008)014
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This content was downloaded from IP address 131.215.225.161 on 19/06/2019 at 20:24

JHEP03(2008)014
Published by Institute of Physics Publishing for SISSA
Received: January 7, 2008
Accepted: February 22, 2008
Published: March 6, 2008
The effective field theory of inflation
Clifford Cheung, A. Liam F itzpatrick, Jared Kaplan and Leonardo Senatore
Jefferson Physical Laboratory, Harvard University,
Cambridge, MA 02138, U.S.A.
E-mail: cwcheung@fas.harvard.edu, fitzpatr@fas.harvard.edu,
kaplan@physics.harvard.edu, senatore@physics.harvard.edu
Paolo Creminelli
Abdus S alam International Center for Theoretical Physics,
Strada Costiera 11, 34014 Trieste, Italy
E-mail: creminel@ictp.it
Abstract: We study the effective field theory of inflation, i.e. the most general theory
describing the fluctuations around a quasi de Sitter background, in the case of single field
models. Th e scalar mode can be eaten by the metric by going to unitary gauge. In this
gauge, the most general theory is built with the lowest dimension operators invariant under
spatial diffeomorphisms , like g
00
and K
µν
, the extrinsic curvature of constant time sur faces.
This approach allows us to characterize all the possible high energy corrections to simple
slow-roll inflation, whose sizes are constrained by experiments. Also, it describes in a com-
mon language all single field models, including those with a small speed of sound and Ghost
Inflation, and it makes explicit the implications of having a quasi de Sitter background.
The non-linear realization of time diffeomorphism s forces correlation among different ob-
servables, like a reduced speed of sound and an enhanced level of non-Gaussianity.
Keywords: Spontaneous Symmetry Breaking, Space-Time Symmetries, Gauge
Symmetry, Cosmology of Theories beyond the SM.

JHEP03(2008)014
Contents
1. Introduction 1
2. Construction of the action in unitary gauge 4
3. Action for the Goldstone boson 7
4. The various limits of single field inflation 10
4.1 Slow-roll inflation and high energy corrections 10
4.2 Small speed of sound and large non-Gaussianities 13
4.2.1 Cutoff and naturalness 16
4.3 De-Sitter limit and the Ghost condensate 17
4.3.1 De-Sitter limit without the Ghost condensate 20
5. Conclusions 21
A. The most general Lagrangian in unitary gauge 22
B. Expanding around a given FRW solution 24
1. Introduction
The effective field theory approach, i.e. the description of a system through the lowest
dimension operators compatible with the underlying symmetries, has been very fruitful
in many areas, from particle physics to condensed matter. The purpose of this paper
is to apply this methodology to describe the theory of fluctuations aroun d an inflating
cosmological background.
The usual way to study a single field inflationary model is to start from a Lagrangian
for a scalar field φ and solve the equation of motion for φ together with the Friedmann
equations for the FRW metric. We are interested in an inflating solution, i.e. an accelerated
expansion with a slowly varying Hub ble parameter, with the scalar following an homoge-
neous time-dependent solution φ
0
(t). At this point one studies perturbations around this
background solution to work out the predictions for the various cosmological observables.
The theory of perturbations around the time evolving solution is quite different from
the theory of φ we started with: while φ is a scalar under all diffeomorphisms (diffs), the
perturbation δφ is a scalar only under spatial diffs while it transforms non-linearly with
respect to time diffs:
t → t + ξ
0
(t, ~x) δφ → δφ +
˙
φ
0
(t)ξ
0
. (1.1)
– 1 –

JHEP03(2008)014
In particular one can choose a gauge φ(t, ~x) = φ
0
(t) where there are no in flaton pertur-
bations, but all degrees of freedom are in the metric. The s calar variable δφ has been
eaten by the graviton, which has now three degrees of f reedom: the scalar mode and the
two tensor helicities. This phenomenon is analogous to what happens in a spontaneously
broken gauge theory. A Goldstone mode, which transf orms non-linearly under the gauge
symmetry, can be eaten by the gauge boson (unitary gauge) to give a massive spin 1 par-
ticle. The non-linear sigma model of the Goldstone can be embedded and UV completed
into a linear representation of the gauge symm etry like in the Higgs sector of the Standard
Model. This is analogous to the standard formulation of inflation, where we start from a
Lagrangian for φ with a linear representation of diffs. In this paper we want to stress the
alternative point of view, describing the th eory of perturbations durin g inflation directly
around the time evolving vacuu m where time diffs are non-linearly realized. This formalism
has been firstly introduced, for a generic FRW backgroun d, in [1] to study the possibility
of violating the Null Energy Condition; here we will extend this formalism focusing on an
inflationary solution.
We will show that in unitary gauge the most generic Lagrangian with broken time
diffeomorphisms (but unbroken spatial diffs) describing perturbations around a flat FRW
with Hubble rate H(t) is given by
S =
Z
d
4
x
√
−g

1
2
M
2
Pl
R + M
2
Pl
˙
Hg
00
− M
2
Pl

3H
2
+
˙
H

+
M
2
(t)
4
2!
(g
00
+ 1)
2
(1.2)
+
M
3
(t)
4
3!
(g
00
+ 1)
3
+ . . . −
¯
M
2
(t)
2
2
δK
µ
µ
2
+ . . .

.
The first two operators after the Einstein-Hilbert term are fixed by the requirement of
having a given un perturbed solution H(t), while all the others are free and parametrize all
the possible different theories of perturbations with the same background solution. As time
diffs are broken one is allowed to wr ite any term that respects spatial diffs, including for ex-
ample g
00
and the extrinsic curvature K
µ
ν
of the surfaces at constant time. The coefficients
of the operators will be in general time dependent. The reader may be worried by the use
of a Lagrangian that is not invariant under diffeomorphisms. But clearly diff. invariance
can be restored as in a standard gauge theory. On e performs a time-diffeomorphism with
parameter ξ
0
(t, ~x) and promotes the p arameter to a field π(t, ~x) which shifts under time
diffs: π(t, ~x) → π(t, ~x) − ξ
0
(t, ~x). The scalar π is the Goldstone mode w hich non linearly
realizes the time diffs and it describes the scalar perturbations around the FRW solution.
It is well known that the physics of the longitudinal components of massive gauge
bosons can be s tudied, at sufficiently high energy, concentrating on the scalar Goldstone
mode (equivalence theorem). The same is true in our case: for su fficiently high energy
the mixing with gravity is irrelevant and we can concentrate on the Goldstone mode. In
this regime the physics is very transparent and most of the information about cosmo-
logical perturbations can be obtained. Performing the broken diff transformation on the
– 2 –

JHEP03(2008)014
Lagrangian (1.2) and concentrating on the Goldstone mode π one gets
S
π
=
Z
d
4
x
√
−g

M
2
Pl
˙
H (∂
µ
π)
2
+ 2M
4
2

˙π
2
+ ˙π
3
− ˙π
1
a
2
(∂
i
π)
2

−
4
3
M
4
3
˙π
3
−
¯
M
2
2
1
a
4
(∂
2
i
π)
2
+ . . .

. (1.3)
Every invariant operator in unitary gauge is promoted to a (non-linear) operator for
the Goldstone: the non-linear realization of diff invariance forces the relation among various
terms.
Let us briefly point out w hat are th e advantages of this appr oach before moving to a
systematic construction of the theory.
• Starting from a “vanilla” scenario of inflation with a scalar field with minimal kinetic
term and slow-roll potential, we have parameterized our ignorance about all the
possible h igh energy effects in terms of the leading invariant operators. Experiments
will put bounds on the various operators, for example with measurements of the
non-Gaussianity of perturbations and studying the deviation from the consistency
relation for the gravitational wave tilt. In some sense this is similar to what one
does in particle physics, where one puts constraints on the size of the operators
that describe deviations from the Standard Model and thus encode th e effect of new
physics.
• It is explicit what is forced by the symmetries and by the requirement of an inflating
background and what is free. For example eq. (1.3) shows that the spatial kinetic term
(∇π)
2
is proportional to
˙
H, while the time kinetic term ˙π
2
is free. Another example
is the u nitary gauge operator (g
00
+ 1)
2
. Once written in terms of the Goldstone π,
this gives a quadratic term ˙π
2
, which reduces the speed of sound of π excitations,
and a cubic term ˙π(∇π)
2
, which increases the interaction among modes, i.e. th e non-
Gaussianity. Therefore, barring cancellations with other operators, a reduced speed
of s ound is r elated by symmetry to an enhanced non-Gaussianity. Notice moreover
that the coefficient of this operator is constrained to be positive, to avoid propagation
of π excitations out of the lightcone.
• One knows all the possible operators. For example, at the leading order in derivatives,
the interaction among three π modes can be changed by (g
00
+ 1)
2
and (g
00
+ 1)
3
.
This will correspond to two different shapes of the 3-point function which can be in
principle experimentally distinguished to fix the size of each operator.
• All the possible single field models are now unified. For example there has been
interest in models with a modified Lagrangian L((∂φ)
2
, φ), like DBI inflation [2 – 6]
which have rather peculiar predictions. In our language th ese correspond to the case
in which the operators (g
00
+ 1)
n
are large. An other interesting limit is when
˙
H → 0;
in this case the leading spatial kinetic term is coming from the operator proportional
to
¯
M
2
and it is of the form (∇
2
π)
2
. This limit describes Ghost Inflation [7].
– 3 –