Correlated primordial spectra in effective theory of inflation

0.48 MB·

by  Jinn-Ouk Gong

APCTP Pre2017-001
Correlated primordial spectra
in effective theory of inflation
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2 Jinn-Ouk Gong(a,b) and Masahide Yamaguchi(c)

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A a Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea
2 b Department of Physics, POSTECH, Pohang 37673, Korea
c Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
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Abstract
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r We derive a direct correlation between the power spectrum and bispectrum of the
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s primordial curvature perturbation in terms of the Goldstone mode based on the effective
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[ field theory approach to inflation. We show examples of correlated bispectra for the

parametrized feature models presented by the Planck collaboration. We also discuss
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v the consistency relation and the validity of our explicit correlation between the power
5 spectrum and bispectrum.
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1 Introduction
The high energy scale during inflation, presumably well beyond the reach of the current and
future particle accelerator experiments, calls for an effective theory description of inflation
[1, 2]. This is because by construction the effective field theory approach is systematic through
which we can account for our ignorance. A key observation in writing the effective field theory
of single-field inflation#1 is to note that in the time-dependent background the time transla-
tional symmetry is broken, while spatial diffeomorphism is preserved [1]. The couplings that
determine the expansion of the effective theory of the Goldstone mode π, which realizes the
time diffeomorphism, are represented by a set of mass scales M4. In the so-called decoupling
n
regime the Goldstone π could decouple from the metric fluctuations and the effective action of
π is dramatically simplified. Especially, the first expansion parameter M4 is manifest in both
2
quadratic and cubic order of π: see (5).
The observation that the coefficient M4 is common to the quadratic and cubic action of π
2
indicates that, to leading order in the decoupling limit, the corresponding correlation functions
– the power spectrum and bispectrum – are explicitly correlated. It means that ideally, given
an explicit analytic form of the power spectrum theoretically, we can find unambiguously the
corresponding bispectrum. Or, at the very least observationally, it remains tantalizing because
of the existence of outliers in the power spectrum of the temperature fluctuations of the cosmic
microwave background [4]. The explicit correlation would make possible joint analysis using
the two- and three-point correlation functions [5], which can place much stronger constraints
on cosmological parameters. It can also open a compelling way of searching for new physics
beyond the paradigm of standard slow-roll inflation, since any deviations would strongly signal
the typical mass scale associated with new physics [6].
In this article, we derive a direct and explicit relation between the power spectrum and
bispectrum of the primordial curvature perturbation using the Goldstone mode π. Such a
correlation was first explicitly studied in the top-down approach [7] and expanded into more
general context in [8], in which heavy degrees of freedom are integrated out to lead to an
effectivesinglefielddescriptionofinflation[9](seealso[10]). Toleadingorderoftheheavymass
scale, the speed of sound c uniquely characterizes the effects of the heavy degrees of freedom
s
[9], i.e. the coefficients of the effective theory. Our approach here is conversely bottom-up,
complementary to the previous studies as we will see in the main text.
The article is organized as follows. In the next section, after briefly reviewing the effective
fieldtheoryofinflation, wederivethesimpleexpressionofthecorrectiontothepowerspectrum.
By inverting it we can write the unknown, model-dependent effective theory parameter in terms
of the power spectrum which can be constrained observationally. In Section 3, we derive a
direct and explicit relation between the corrections of the power spectrum and bispectrum. In
Section 4 we discuss the consistency relation of the squeezed bispectrum [11] and the validity
#1 Extensions to the multi-field case are possible under certain constraints [3].
1
of the correlation we derive. The final section is devoted to summary and conclusions.
2 Effective theory and correction to power spectrum
In this section, after briefly reviewing the effective field theory of inflation, we give the for-
mula of the correction to the power spectrum due to the deviation from usual slow-roll phase
parametrized by the expansion coefficient of the effective theory.
2.1 Brief review of effective field theory of inflation
We begin with a brief review of the effective field theory of inflation [1]. In unitary gauge,
the information on the primordial curvature perturbation is encoded in geometrical quantities
respecting the time-dependent spatial diffeomorphism symmetry. Then, the action for the
primordial curvature perturbation is written in general as
(cid:90) √
S = d4x −gF(gµν,g ,K ,R ,∇ ,t), (1)
µν µν µνρσ µ
where K is the extrinsic curvature with respect to t = constant hypersurface. Since the zeroth
µν
and first order terms are determined by the background quantities, the action can be expanded
as
(cid:90) √ (cid:26)1 (cid:104) (cid:105)
S = d4x −g m2 R+m2 H˙ (t)g00 −m2 3H2(t)+H˙ (t)
2 Pl Pl Pl
(cid:27)
(cid:0) (cid:1)
+F δg00,δK ,δR ;g ,gµν,∇ ,t , (2)
µν µνρσ µν µ
where F represents second and higher order perturbation terms and is given by
1 1
F = M4(t)(cid:0)δg00(cid:1)2 + M4(t)(cid:0)δg00(cid:1)3 +···
2 2 3! 3
1 1 1
− M¯3(t)δg00δK − M¯3(t)KδK − M¯3(t)Kµ δKν −··· , (3)
2 1 2 2 2 3 ν µ
with K ≡ Kµ . It is noticed that time diffeomorphism invariance is broken in this action. But,
µ
it can be recovered by the introduction of the Stu¨ckelberg field π(x), which corresponds to the
Nambu-Goldstonebosonandtransformsunderthecoordinatetransformationst → (cid:101)t = t+ξ0(x)
and x → x as
(cid:101)
π(x) −→ π(x(x)) = π(x)−ξ0(x). (4)
(cid:101) (cid:101)
In the decoupling regime H (cid:38) M2/m , the action reduces to
2 Pl
(cid:90) √ (cid:26)m2 (cid:20) (∇π)2(cid:21) (cid:20) (∇π)2(cid:21) 4 (cid:27)
S = d4x −g PlR−m2 H˙ π˙2 − +2M4 π˙2 +π˙3 −π˙ − M4π˙3 +··· ,
π 2 Pl a2 2 a2 3 3
(5)
2
where the dots represent the higher derivative terms. The sound velocity c is related to M as
s 2
2M4
c−2 = 1− 2 . (6)
s m2 H˙
Pl
In this article, we further set M (t) = 0 because M4 ∼ (1−c−2)M4 on general arguments [12].
3 3 s 2
π and R are related to linear order by π = −R/H, so in the regime (5) is valid we can to first
˙
approximation consider π˙ ≈ −R/H.
2.2 Corrections to the power spectrum
We first concentrate on the quadratic part and evaluate the correction to power spectrum
originating from the term with M4. Since the standard slow-roll terms multiplied by H˙ in
2
(5) are dominant as various observations indicate, we treat the quadratic contribution of M4
2
as perturbation. In terms of the speed of sound (6), we assume that for a limited duration
c deviates from unity, with the deviation being not too far away from unity. Neglecting the
s
√
metric perturbation as we consider the decoupling regime so that −g = a3 simply, from (5)
the quadratic part other than the usual slow-roll, which we may call second order interaction,
is
(cid:90)
S = d4xa32M4(t)π˙2. (7)
2,int 2
The interaction Hamiltonian is then#2
(cid:90) (cid:90)
H = d3xa3(−2)c2M4(t)π˙2 ≈ d3xa3(−2)M4(t)π˙2, (8)
int s 2 2
where we have used the assumption that c is not too far away from unity. This interaction
s
Hamiltonian can be expressed in terms of the Fourier mode as
(cid:90) d3q d3q
H = −2aM4 1 2δ(3)(−q )π(cid:48) π(cid:48) , (9)
int 2 (2π)3 12 q1 q2
where q ≡ q +q +···+q , a prime represents a derivative with respect to the conformal
12···n 1 2 n
time dτ = dt/a, and
(cid:90) d3q
π(τ,x) = eiq·xπ (τ). (10)
(2π)3 q
Now we can compute the corrections using the standard in-in formalism. We can straightly
obtain
2π2
∆(cid:104)π π (τ)(cid:105) ≡ (2π)3δ(3)(k ) ∆P
k1 k2 12 k3 π
1
(cid:90) τ→0
= i adτ(cid:48)(cid:104)0|[H (τ(cid:48)),π π (τ)]|0(cid:105)
int k1 k2
τ0→−∞
(cid:20) (cid:90) 0 (cid:21)
(cid:0) (cid:1)
= (2π)3δ(3)(k )2(cid:60) 2iπ∗ π∗ (0) dτ −2a2M4 π(cid:48) π(cid:48) (τ) , (11)
12 (cid:98)k1(cid:98)k2 2 (cid:98)k1(cid:98)k2
−∞
#2One should be careful when the interaction Lagrangian includes derivative terms. Conjugate momentum
must be defined by use of the full Lagrangian rather than the free part.
3
where we have expanded the free field π using the creation and annihilation operators as
k
π = a π +a† π∗ with (cid:2)a ,a†(cid:3) = (2π)3δ(3)(k−q), (12)
k k(cid:98)k −k(cid:98)k k q
and π (τ) is the mode function solution given by
(cid:98)k
R(cid:98) −i
π (τ) = − k = √ (1+ikτ)e−ikτ . (13)
(cid:98)k
H 4(cid:15)k3m
Pl
Thus, we immediately find the correction to the power spectrum as
∆P ∆P k (cid:90) 0
π(k) ≈ R(k) ≈ dτ (cid:0)−2M4(cid:1)sin(2kτ), (14)
P P m2 (cid:15)H2 2
π R Pl −∞
where
P 1
R
P = = (15)
π H2 8π2m2 (cid:15)
Pl
is the featureless flat spectrum.
2.3 Inverting the power spectrum
For future convenience, let us return to (11) and write it in an alternative form. The real part
is obtained by adding the complex conjugate:
2π2 (cid:20) (cid:90) 0 (cid:21)
(cid:0) (cid:1)
∆P = 2(cid:60) 2iπ∗ π∗ (0) dτ −2a2M4 π(cid:48) π(cid:48) (τ)
k3 π (cid:98)k1(cid:98)k2 2 (cid:98)k1(cid:98)k2
1 −∞
(cid:90) 0
(cid:0) (cid:1)
= 2iπ∗ π∗ (0) dτ −2a2M4 π(cid:48) π(cid:48) (τ)+c.c.. (16)
(cid:98)k1(cid:98)k2 2 (cid:98)k1(cid:98)k2
−∞
By noting from (13) that π (−τ) = −π∗(τ) and π(cid:48)(−τ) = π∗(τ), and by oddly extending M4
(cid:98)k (cid:98)k (cid:98)k (cid:98)k 2
to define M(cid:102)4 as
2
(cid:26)
M4(τ) if τ < 0
M(cid:102)4(τ) ≡ 2 , (17)
2 −M4(−τ) if τ > 0
2
(16) can be written as#3
2π2 (cid:90) ∞ (cid:16) (cid:17)
∆P = 2iπ∗ π∗ (0) dτ −2a2M(cid:102)4 π(cid:48) π(cid:48) (τ)
k3 π (cid:98)k1(cid:98)k2 2 (cid:98)k1(cid:98)k2
1 −∞
1 1 (cid:90) ∞ −2M(cid:102)4
= 2π2P dτ 2 ie−ik12τ . (18)
πk k 2 (cid:15)m2 H2
1 2 −∞ Pl
#3 Notice that we are at this stage not directly computing the propagator by adopting the iε prescription
of the contour, which remains unchanged though. Our goal is to invert (14) by incorporating mathematical
manipulations in such a way that the model-dependent parameter M4 is given in terms of ∆P which can be
2 π
observationally constrained.
4
Since we have defined M(cid:102)4 oddly, only the odd part of e−ik12τ survives and finally we have,
2
setting k = k = k,
1 2
∆P k (cid:90) ∞ −2M(cid:102)4
π = dτ 2 sin(2kτ). (19)
P 2 (cid:15)m2 H2
π −∞ Pl
From (19) we can write the coefficient M(cid:102)4, which is essentially M4 in the effective action
2 2
(cid:0) (cid:1)
(5), in terms of ∆P /P as follows. From sin(2kτ) = e2ikτ −e−2ikτ /(2i), we can multiply
π π
e2ikτ(cid:48) to both sides of (19) and integrate over k to obtain
(cid:90) ∞ 2i ∆P 1 (cid:90) ∞ (cid:16) (cid:17)(cid:90) ∞ (cid:104) (cid:105)
dke2ikτ(cid:48) (cid:15)m2 H2 R(k) = dτ −2M(cid:102)4 dk e2ik(τ+τ(cid:48)) −e−2ik(τ−τ(cid:48)) = 2πM(cid:102)4(τ(cid:48)).
k Pl P 2 2 2
−∞ R −∞ −∞
(20)
Thus,
2(cid:15)m2 H2 (cid:90) ∞ dk∆P
2M(cid:102)4(τ) = i Pl R(k)e2ikτ . (21)
2 π k P
−∞ R
This is the inverse formula, in which M4 can be expressed in terms of the correction to power
2
spectrum.
3 Correlation between power spectrum and bispectrum
In this section, we first give the formula of bispectrum coming from the cubic action (5),
and then derive the explicit relation between the correction to the power spectrum and the
bispectrum.
3.1 Bispectrum
As advertised before, we only consider the cubic order action with the coefficient M4:
2
(cid:90) √ (cid:20) (∇π)2(cid:21)
S = d4x −g2M4 π˙3 −π˙ . (22)
3 2 a2
We can follow the same steps as before: the interaction Hamiltonian is
(cid:90) (cid:20) (∇π)2(cid:21)
H = − d3xa3 ·2M4 π˙3 −π˙
int 2 a2
(cid:90) d3q d3q d3q (cid:104) q ·q (cid:105)
= −2a3M4 1 2 3δ(3)(−q ) π˙ π˙ π˙ + 1 2π π π˙ +2 perm . (23)
2 (2π)3·2 123 q1 q2 q3 3a2 q1 q2 q3
Then, the bispectrum of π becomes
(cid:104)π π π (τ)(cid:105) ≡ (2π)3δ(3)(k )B (k ,k ,k )
k1 k2 k3 123 π 1 2 3
(cid:26) (cid:90) 0
(cid:0) (cid:1)
= (2π)3δ(3)(k ) iπ∗ π∗ π∗ (0) dτ −2aM4
123 (cid:98)k1(cid:98)k2(cid:98)k3 2
−∞
(cid:27)
(cid:104) (cid:105)
× 6π(cid:48) π(cid:48) π(cid:48) (τ)+2(k ·k )π π π(cid:48) (τ)+2 perm +c.c. .
(cid:98)q1(cid:98)q2(cid:98)q3 1 2 (cid:98)q1(cid:98)q2(cid:98)q3
(24)
5
Again, we can find that by extending M4 oddly the complex conjugate includes the integral
2
from 0 to ∞, so
(cid:90) ∞ (cid:16) (cid:17)
(cid:2) (cid:3)
B (k ,k ,k ) = iπ∗ π∗ π∗ (0) dτ −2aM(cid:102)4 6π(cid:48) π(cid:48) π(cid:48) (τ)+2(k ·k )π π π(cid:48) (τ)+2 perm .
π 1 2 3 (cid:98)k1(cid:98)k2(cid:98)k3 2 (cid:98)q1(cid:98)q2(cid:98)q3 1 2 (cid:98)q1(cid:98)q2(cid:98)q3
−∞
(25)
3.2 Bispectrum in terms of the power spectrum
In this subsection, we can use (21) and write the bispectrum (25) purely in terms of the
power spectrum and its derivatives. Let us first consider the first term of (25). We can
straightforwardly write, with K ≡ k ,
123
(cid:90) ∞ (cid:16) (cid:17) H 3H (cid:90) ∞ dk∆P (cid:90) ∞
iπ∗ π∗ π∗ (0) dτ −2aM(cid:102)4 6π(cid:48) π(cid:48) π(cid:48) (τ) = (2π2P )2 R(k) dττ2ei(2k−K)τ
(cid:98)k1(cid:98)k2(cid:98)k3 2 (cid:98)q1(cid:98)q2(cid:98)q3 π π k k k k P
−∞ 1 2 3 −∞ R −∞
3 1 (cid:90) ∞ dk∆P d2 (cid:18) K(cid:19)
= (2π2P )2 H R(k) δ k −
π 4 k k k k P dk2 2
1 2 3 −∞ R
= (2π2P )23H 1 d2 (cid:20)1∆PR(k)(cid:21)(cid:12)(cid:12)(cid:12) , (26)
π 4 k k k dk2 k P (cid:12)
1 2 3 R k=K/2
where for the second equality we have replaced τ2 in the time integral with two derivatives with
respect to k, and for the last equality we have iteratively integrated by parts.
To proceed further, with P(total) = P +∆P , from
R R R
∆P
logP(total) ≈ logP + R , (27)
R R P
R
with P being flat, we can find the spectral index and the running respectively as#4
R
dlogP(total) d (cid:18)∆P (cid:19)
n −1 ≡ R = k R , (28)
R
dlogk dk P
R
dn d2 (cid:18)∆P (cid:19) d (cid:18)∆P (cid:19)
α ≡ R = k2 R +k R . (29)
R dlogk dk2 P dk P
R R
Thus (26) can be now written as
(cid:90) ∞ (cid:16) (cid:17)
iπ∗ π∗ π∗ (0) dτ −2aM(cid:102)4 6π(cid:48) π(cid:48) π(cid:48) (τ)
(cid:98)k1(cid:98)k2(cid:98)k3 2 (cid:98)q1(cid:98)q2(cid:98)q3
−∞
(cid:20) (cid:21)(cid:12)
= (2π)4P23H 1 1 α −3(n −1)+2∆PR (cid:12)(cid:12) . (30)
π2 k k k K3 R R P (cid:12)
1 2 3 R k=K/2
#4In case one takes into account the slight tilt of P , the spectral index and the running given here represent
R
onlytheeffectof∆P /P . Sinceweassumedthatforalimiteddurationc deviatesfromunity,wecanseparate
R R s
the correction part from the standard slow-roll part, for both of which, the spectral index and the running can
be defined, respectively.
6
We can proceed in a similar manner for the second term of (25) and find
(cid:90) ∞ (cid:16) (cid:17)
iπ∗ π∗ π∗ (0) dτ −2aM(cid:102)4 2(k ·k )π π π(cid:48) (τ)
(cid:98)k1(cid:98)k2(cid:98)k3 2 1 2 (cid:98)q1(cid:98)q2(cid:98)q3
−∞
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)(cid:12)
= (2π)4P2 1 −k1 ·k2 1+ k12 + 2k1k2 ∆PR + −k12 − 3k1k2 (n −1)+ k1k2α (cid:12)(cid:12) .
πH (k k )3k K K2 P K K2 R K2 R (cid:12)
1 2 3 R k=K/2
(31)
Thus, the bispectrum can be expressed in terms of the correction to power spectrum, its
first and second derivatives as
B (k ,k ,k )
π 1 2 3
(cid:20) (cid:21)(cid:12)
= (2π)4P2 H A(k ,k ,k )∆PR +B(k ,k ,k )(n −1)+C(k ,k ,k )α (cid:12)(cid:12) ,
π(k k k )3 1 2 3 P 1 2 3 R 1 2 3 R (cid:12)
1 2 3 R k=K/2
(32)
where the functions of momenta A, B and C are given by, respectively,
1 (cid:88) 1 (cid:88) 1 (cid:88)
A(k ,k ,k ) = − k2k3 +2 k2k2 − k3, (33)
1 2 3 K2 i j K i j 4 i
i(cid:54)=j i>j i
1 (cid:88) 1 (cid:88) 1 (cid:88) 1
B(k ,k ,k ) = 2 k2k3 −3 k2k2 + k k2 − k k k , (34)
1 2 3 K2 i j K i j 4 i j 4 1 2 3
i(cid:54)=j i>j i(cid:54)=j
1 (cid:88) 1 (cid:88) 1
C(k ,k ,k ) = − k2k3 + k2k2 − k k k . (35)
1 2 3 K2 i j K i j 4 1 2 3
i(cid:54)=j i>j
This expression is one of the main results in this article.
In Figure 1, we show a few examples using the following parametrized feature models [13]:
a localized oscillatory burst due to e.g. step in the inflaton potential, logarithmic and linear
oscillations and cutoff models given by
 (cid:18) (cid:19) (cid:18) (cid:19)
k k/k
 AsW0 k D x s (step)
 s d
 (cid:20) (cid:18) (cid:19) (cid:21)
 k

∆P  Alogcos ωloglog k +ϕlog (logarithmic oscillations)
P R = (cid:18) k (cid:19)nlin (cid:18)log k (cid:19) , (36)
R  A cos ω +ϕ (linear oscillations)
 lin k link lin
 (cid:18) πlink (cid:19) lin
 log |C −D |2 (cutoff model)
 16k c c
c
7
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Figure 1: (Upper panels) power spectrum and (lower panels) the corresponding bispectrum for
various feature models (36) discussed in [13]. For simplicity, we have set the amplitudes of the
features as A = A = A = 1 and phases as ϕ = ϕ = 0. For the step model, we have
s log lin log in
also set the damping scale x = 1. Meanwhile, following [13] we have set log ω = 1.25,
d 10 log
log ω = 1.02 and n = 0.66. We show the bispectrum projected onto the equilateral (red
10 lin lin
dashed), folded (blue dotted) and squeezed (black solid) configurations.
8
where the functions that appear in these parametrized feature models are
1 (cid:104) (cid:105)
(cid:0) (cid:1) (cid:0) (cid:1)
W (x) = 18x−6x3 cos(2x)+ 15x2 −9 sin(2x) , (37)
0 2x4
x
D(x) = , (38)
sinhx
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
−ik k 1 k
C = exp H(2) − +i H(2) , (39)
c k 0 2k k/k 1 2k
c c c c
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
ik k 1 k
D = exp H(2) − −i H(2) , (40)
c k 0 2k k/k 1 2k
c c c c
with H(2) being the Hankel function of the second kind. As we can see, each power spectrum
n
gives distinctively different patterns of the corresponding bispectrum in various configurations.
4 Squeezed bispectrum and consistency relation
We can note that (32) vanishes in the squeezed limit, say, k ≈ k and k → 0.#5 This seems
1 2 3
to contradict the consistency relation between the power spectrum and the squeezed limit of
the bispectrum [11],
B (k ,k ,k ) −→ (1−n )P (k )P (k ), (41)
R 1 2 3 R R 1 R 3
k3→0
because as (14) shows the power spectrum is well away from featureless flat one, so the corre-
sponding spectral index is non-trivial. Indeed, in [7], the consistency relation is recovered for
features caused by non-trivial speed of sound.
Let us first return to the quadratic action for the curvature perturbation. Including the
speed of sound, it is written as
(cid:34) (cid:35)
(cid:90) R˙2 (∇R)2
S = d4xa3m2 (cid:15) − , (42)
2 Pl c2 a2
s
so there are two possible sources of departure from the usual canonical slow-roll [15]: (cid:15) and c .
s
Let us consider these two cases separately. Our goal here is to see the form of the corrections to
the power spectrum for each case. But this seems unclear, since the form of the interaction part
of the quadratic action – just R˙2 for c , and R˙2 and (∇R)2 for (cid:15) – is different. Thus naively
s
thinking the resulting correction terms would be of different structure. We first assume that
c solely supplies the deviations from the standard slow-roll in such a way that for a limited
s
duration c deviates from unity, with the deviation being not too far away from unity. We may
s
then write, with the canonical slow-roll part being the leading, free part,
(cid:90) (cid:20) (∇R)2(cid:21) (cid:90) (cid:18) 1 (cid:19)
S = d4xa3m2 (cid:15) R˙2 − + d4xa3m2 (cid:15) −1 . (43)
2 Pl a2 Pl c2
s
(cid:124) (cid:123)(cid:122) (cid:125)
≡S2,int
#5It was recently claimed that for local observers, the squeezed limit vanishes in single-field inflation [14].
But in this article we do not take such effects into account and hence the consistency relation should hold if we
would calculate it adequately.
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