Effective Field Theory for Inflation
The effective theory approach to calculations of higher order
correlators of curvature perturbations
Matteo Fasiello
based on work with N.Bartolo, S.Matarrese, A.Riotto.
JCAP 1012 026 (2010) &JCAP1009,035 (2010)&JCAP 1008, 008 (2010)
+ work in progress..
Cosmo 2011, Porto
Motivations
The power spectrum from the simplest models of standard single-field slow-roll
inflation already nicely accounts for CMB and LSS observations;
higher order because
I A detailed analysis of the amplitude and shape of higher order correlators
(e.g. bispectrum,trispectrum) for each inflationary mechanism is
instrumental in discriminating among all the allowed scenarios.
I In studying interactions one is probing the dynamics of inflation.
I
The timing is quite good.
New experiments provide a better sensitivity to deviations from Gaussian
statistics: Planck et al. present us with the opportunity to test this zoo of
possibilities.
The effective approach
I The effective theory approach offers several advantages:
• A unifying perspective on inflation in that the effective Lagrangian
comprises many known inflationary models (standard, Ghost, DBI, etc)
• Symmetries and properties of the action emerge that would not
otherwise be manifest
• Considerable calculational advantages in the decoupling regime.
Ref: [Cheung et al. JHEP 0803 (2008) 014]
The Lagrangian of the theory
The background solution is φ0 (t) . It identifies a privileged spacetime slicing 1 .
Perturbations around the background:
φ(t, ~x ) = φ0 (t) + δφ(t, ~x )
Choose the priviledged slicing to coincide with surfaces of constant t, it
amounts to:
δφ(t, ~x ) = 0
unitary/comoving gauge.
The scalar fluctuation is eaten by the metric, i.e. no explicit scalar fluctuations.
Lagrangian only space diffs invariant now.
Ref: [Cheung et al.]
Once the most general Lagrangian around a given FRW background is written,
use the Stuckelberg trick to restore full rep. inv. and make the scalar explicit.
1
The geometry of this slicing is described by the extrinsic curvature of surfaces at constant time we will
employ later on.
Decoupling
For sufficiently high energy, Emix , the dynamics of the scalar π decouples.
If H > Emix , ⇒ the dynamics of the scalar is well described by:
Z
√ 2
(∂i π)2
S2π = d 4 x −g MPl
Ḣ(∂µ π)2 + 2M2 (t)4 π̇ 2 − M̄1 (t)3 H 3π̇ 2 −
2 a2
2
2
2
2
2
2
M̄2 (t) (∂i π)(∂j π)
M̄3 (t) (∂i π)(∂j π) −
−
(1)
2
a4
2
a4
At the quadratic level we can see:
• M̄1 = M̄2 = M̄3 = 0 = M2 → standard single-field slow-roll inflation with
unitary sound speed
• M2 6= 0 opens up the gate for cs < 1 models which are often associated to
large higher order correlators (DBI inflation&Co)
• Ḣ → 0, M̄1 = 0 gives back Ghost inflation
These correspondences continue at third and fourth order.
e.o.m.
In Fourier space with u(τ, ~k) = a(τ )π(τ, ~k) and τ conformal time, the e.o.m.
for the scalar π is:
00
uk −
2
uk + α0 k 2 uk + β0 k 4 τ 2 uk = 0
τ2
(2)
see also [Senatore, Phys.Rev. D71 (2005) ]
α0 =
2
−MPl
Ḣ − M̄13 H/2
2
−MPl
Ḣ + 2M24 − 3M̄13 H
β0 =
(M̄22 + M̄32 )H 2
2
−MPl
Ḣ + 2M24 − 3M̄13 H
(3)
Note that:
• in general single-field slow-roll α0 = cs2 ;
• ghost inflation corresponds to α0 = 0;
β0 = 0
β0 6= 0
• for α0 6= 0; β0 6= 0 we are in a more general playground, the solution to the
e.o.m. would interpolate between known and possibly new inflationary models.
We solve this e.o.m. for the first time.
Solution
√ 2 2 √
1
iα0
α0
−i H e 2 i β0 k τ Γ 45 − √
HypG(− 41 − i √
, − 21 , −i β0 k 2 τ 2 )
4 β0
4 β0
πk (τ ) = q
√
α0
√ )
2(MP2 H 2 + 2M24 − 3M̄13 H) k 3/2 (α0 + β0 )3/4 Γ( 45 +
4α0 −4i
β0
It gives back Ghost Inflation (α0 = 0) and P(X , φ) (β0 → 0).
From here, using ζ = −Hπ, the power spectrum:
√
(α0 + β0 )−3/2 H 4
Pζ =
2
16π(MPl
H 2 + 2M24 − 3M̄13 H) |Γ( 45 +
α0
4α0 −4i
√ )|2
,
(4)
β0
Here, schematically, its tilt is:
ns − 1 = Θ (1) × + Θ1 (1) × 1 + Θs (1) × s + Θ0 (1) × 0 + Θη (1) × η − Γ̇/(HΓ) (
In the paper we also give the explicit expression for the running.
One can put mild bounds on the new, generalized slow-roll parameters from the
value of the power spectrum itself and from the requirement that they are not
so large as to produce too large a value for fNL .
Higher order correlators
(2)
Following the same algorythm that gave us Sπ one obtains:
N.B. α0 1 and β0 1 put mild bounds on M2 , M̄1 , M̄2 , M̄3 ,.
Other parameters first appearing at higher orders bounded by loop corrections
to the spectrum and Mn M, with M mass of the underlying theory.
Perspective
• We are going to look at higher order correlators of all non trivial interaction
terms. All the terms are driven by the Mn coefficients and at this stage we do
not worry about their including higher derivative operators which would spoil
the e.o.m. in the sense of Ostrogradski
(more rigorous treatment work in progress with X. Chen et al).
• A larger value of some of the the Mn ’s might be more natural for example,
but in the spirit of the effective theory we consider all the interactions which
would provide a distinctive signature in the form of higher correlators.
The specific weight of each coefficient is known once one specifies the model,
imposes symmetries etc...
Bispectrum
hζ~k1 ζ~k2 ζ~k3 i ≡ (2π)3 δ (3) (k~1 + k~2 + k~3 ) × fNL × F (k1 , k2 , k3 )
fNL is the amplitude
F (k1 , k2 , k3 ) accounts for the shape of the 3-point function.
All results obtained using the IN-IN formalism.
fNL
We calculate it for a grid of values of the α0 , β0 parameters:
Benchmarks
α0
β0
1
10−2
0
2
0
0.5 · 10−4
3
0.5 · 10−2
0.25 · 10−4
4
2 · 10−7
5 · 10−5
5
10−4
0
6
10−6
0
(1)K-inflation wavefunction (2)Ghost (3) Intermediate A (4) Int. B
(5)K-inflation wf with very small generalized speed of sound (6) even smaller
Equilateral shape as a consistency check
• M24 π̇(∇π)2 , typical interaction term, its wf is known to peak on the equilateral
configuration k1 = k2 = k3 . F (1, x2 , x3 )x22 x32 is plotted (x3 = k3 /k1 , x2 = k2 /k1 )
Figure: P(X , φ) ↑ w.f.
Ghost % w.f. No news, just obtaining what we should
Figure: interpolating solution ↑, still equilateral shape
News!
• M̄62 π̇(∂ij π)2 /a4 , extr. curvature-generated term we consider for the first time.
benchmarks
M̄6
fNL
1
2
104 γ7
4 · 103 γ7
3
5 · 103 γ7
4
4 · 103 γ7
5
6
108 γ7
1012 γ7
2
γ7 = (M̄62 H 2 )/(MPl
H 2 + 2M24 − 3M̄13 H).
Figure: P(X , φ) ↑
Interpolating solution ↑
Here a shape function of a single, independent interaction operator peaks on a flat
(k1 = 2k2 = 2k3 ) configuration. See also [Senatore et al. JCAP 1001 (2010) 028 ]
where this shape was first obtained by considering two operators and [Creminelli et al.
JCAP 1102 (2011) 006] where this shape was obtained in the context of galileon
models.
More news!
• M̄8 ∂i 2 π (∂jk π)2 /a6 , another new extr. curvature-generated term. Flat shape.
• M̄9 ∂ij π∂jk π∂ki π/a6 , extr. curvature term. A slightly different but still flat shape.
M̄6
In the papers we also look at the running of fNL
and find that it can be dominated by
generalized slow roll parameters other than , η, s.
Four-point function contributions: diagrams
Figure: Scalar exchange diagram -. Contact interaction diagram ↑
• s.e.: third-order interaction terms integrated twice over time
• c.i.: fourth-order interactions integrated once
Symmetries
Quite a number of interactions ⇒ ordering principle/selection criterium would
be useful.
Symmetries come handy!
Symmetries are also useful if one is looking for small bispectrum & large
trispectrum examples.
S1 :
π → −π;
S2 :
π → −π and t → −t.
(6)
Table 1
Coefficients
S1
S2
Coefficients
S1
S2
M2
X
X
M̄10
X
X
M3
X
X
M̄11
X
X
M4
X
X
M̄12
X
X
M̄1
X
X
M̄13
X
X
M̄2
X
X
M̄14
X
X
M̄3
X
X
M̄15
X
X
M̄4
X
X
N̄1
X
X
M̄5
X
X
N̄2
X
X
M̄6
X
X
N̄3
X
X
M̄7
X
X
N̄4
X
X
M̄8
X
X
N̄5
X
X
The Coefficients marked with “ X” in correspondence of a given symmetry S are
S-invariant, those marked with “X” violate the S symmetry.
Will be studying the interactions driven by the parameters in red.
M̄9
X
X
/
Six variables ⇒ need to pick a specific configuration
I equilateral: k1 = k2 = k3 = k4 , plot k12 ≡ |k~1 + k~2 | and k14 ≡ |k~1 + k~4 |
I folded: k12 → 0,
k1 = k2 ,
k3 = k4 , plot k14 /k1 and k4 /k1
I specialized planar limit: k1 = k3 = k14 ,
k12 = f (k1 , k2 , k4 ), plot k2 /k1
and k4 /k1
I near the double squeezed limit:
k3 = k4 = k12
k2 = g (k1 , k3 , k4 , k14 , k12 )
k12 → 0, plot k14 and k4
Trispectrum shape functions
1
hζ~k1 ζ~k2 ζ~k3 ζ~k4 i ∝ (2π)9 Pζ3 δ (3) (k~1 + k~2 + k~3 + k~4 )Π4i=1 3 T (k1 , k2 , k3 , k4 , k12 , k14 )
ki
ζ = −Hπ
Plotted quantities
I Equilateral
−→
T
I Folded
−→
T
I Specialized planar limit
−→
T
I Near double squeezed limit
−→
T
Π4i=1 k1i
Shapes literature
Equilateral conf. for P(X , φ) models, [Chen et al. JCAP 0908 (2009) 008 ]
Our equilateral findings
Shape Literature
Double squeezed limit for P(X , φ) models, [Chen et al. ’09]
Our double-squeezed findings
Compare with P(X , φ) models: here the double squeezed limit cannot remove degeneracies.
One cannot tell if an interaction is third or fourth order by looking at the plot in d.s. configuration
Summary and Future Work
Message
Extrinsic curvature-generated terms should be taken into account because:
I Significantly enlarge the useful parameter space spanned by the effective
theory
I Can produce large NG and present interesting distinctive features in the
form of their shape functions
Future Work
I Non Bunch-Davies vacua
I Relax the shift symmetry requirement
I Specially for the trispectrum, focus on concrete and well grounded models
spanned by the effective theory.
Explicit expressions II
6M02
Θ0 = q
√
1
2M0 2 + 2 (2H 2 MP2 + M1 4 ) H 2 M 2 +2M
P
0 =
Ṁ0
;
HM0
2 ≡
1 =
Ṁ1
;
HM1
Γ =
η
s
Ṁ2
= −+
.
HM2
2
2(cs2 − 1)
M04 ≡ M̄02 H 2 ;
M14 ≡ M̄13 H .
.
2
4 +3M 4
1
Γ̇
.
HΓ
(7)
(8)
(9)
WMAP 7 bounds on fNL , (95% CL)
local
−10 ≤ fNL
≤ 74 ;
equilateral
−214 ≤ fNL
≤ 266 ;
orthogonal
−410 ≤ fNL
≤6.
(10)
How to get to the π Lagrangian in detail
Whenever one breaks time reparametrization invariance there appears
automatically a preferred slicing of spacetime described by a function t̃(x).
This function is such that if, for example, the breaking is realized by a scalar
function of time φ0 (t) then whenever t̃ is constant so is φ0 (t).
Unitary gauge consists in requiring that t̃(x) coincides with t so as to make the
additional scalar degree of freedom t̃ not to appear explicitly in the Lagrangian.
1
(11)
Unperturbed space diffs invariant Lagrangian
In Creminelli et al. ’07 it is shown that the most general unperturbed
Lagrangian in unitary gauge has the form:
Z
√
S = d 4 x −g F (Rµνρσ , g 00 , Kµν , ∇µ , t)
(12)
with the requirement that all the remaining free indices in F are upper
’0’. Here Kµν is the extrinsic curvature of surfaces at constant time.
Take the unit vector perpendicular to the surface of constant t̃,
∂µ t̃
nµ = p
µν
−g ∂µ t̃∂ν t̃
and consider the induced metric on the aforementioned surfaces,
hµν = gµν + nµ nν . Kµν is then given by
Kµν = hµσ ∇σ nν
Expanding around FRW
The most generic theory (with broken time diffs) around a FRW
background:
Z
S =
4
d x
p
−g
h1
2
2
2
MPl R + MPl Ḣg
−
M̄1 (t)3
2
00
(g
2
2
− MPl (3H + Ḣ) +
00
µ
+ 1)δK µ −
1
2!
M̄2 (t)2
2
4
M2 (t) (g
µ
2
00
2
+ 1) +
δK µ −
M̄3 (t)2
2
1
3!
4
M3 (t) (g
00
3
+ 1)
i
µ
ν
δK ν δK µ + ... .
(13)
The dots stand for terms starting at third and fourth order in
perturbations.
This is all good but one eventually wants to see explicitly the scalar
degree of freedom rather than working with the metric; something which
is readily obtained using the Stueckelberg trick.
Making the scalar d.o.f. explicit again
Consider a simplified version of the action written above:
Z
√
S=
d 4 x −g A(t) + B(t)g 00 (x) .
Upon t → e
t = t + ξ 0 (x), ~x → ~e
x = ~x ; one gets:
Z
d 4x
S=
p
∂e
x
∂x 0 ∂x 0 µν
e (e
−e
g (e
x (x)) A(t) + B(t) µ
g
x (x)) .
ν
∂x
∂e
x ∂e
x
which can be written as:
Z
S =
4
d e
x
q
"
0
0
−e
g (e
x ) A(e
t − ξ (x(e
x ))) + B(e
t − ξ (x(e
x )))
∂(e
t − ξ0 (x(e
x ))) ∂(e
t − ξ0 (x(e
x ))) µν
e
g
(e
x)
∂e
xµ
∂e
xν
#
Now, promote ξ 0 (x) to a field, ξ 0 (x) → −π(x), whose transformation under time
reparametrization reads:
π(x) → π
e(e
x (x)) = π(x) − ξ 0 (x);
It’s easy to verify that the action below is now invariant under full diffs:
Z
S =
4
d x
q
∂(t + π(x)) ∂(t + π(x)) µν
g
(x) .
−g (x) A(t + π(x)) + B(t + π(x))
∂x µ
∂x ν
Pay back time I
The scalar degree of freedom is now explicit in the time dependence of the coefficients
and the metric transformation.
This choice pays off in that it is expected, from standard gauge theory, that the
dynamics of the scalar π decouples from the one of the metric above some energy
range. Consider one second order sample term in the action:
S =
Z p
M2 (t)4
00 2 (1 + g ) →
−g ... +
2!
Z
=
Z
=
"
4
d x
p
−g ... +
"
4
d x
p
−g ... +
Z p
∂(t + π(x)) ∂(t + π(x)) µν
M2 (t + π)4
2
(1 +
g
(x))
−g ... +
2!
∂x µ
∂x ν
#
2
M2 (t + π)4 2 00
0i
ij
(1 + π̇) g + 2(1 + π̇)g ∂i π + g ∂i π∂j π + 1
,
2!
#
M2 (t + π)4 0i
0i
00
2 00
(1 + π̇) (g(0) + g(1) + ..) + 2(1 + π̇)∂i π(g(0) + g(1) ..) + ..
2!
Second order fluctuations: the π-terms contain always one or two more derivatives
than the g -ones. For sufficiently high energies, the derivatives (think in Fourier space)
will provide the π flucuations with a much bigger weight.
Pay back time II
If one sets the working regime (H, in our cosmological scenario) above the so called
mixing energy, which can be read off from the canonically normalized coefficients of
the kinetic quadratic terms in the action, calculations are greatly simplified. One now
is concerned only with Sπ . At second order:
Z
S2 =
√ 2
Ḣ(∂µ π)2 + 2M2 (t)4 π̇ 2 − M̄1 (t)3 H
d 4 x −g MPl
−
3π̇ 2 −
(∂i π)2
2 a2
2
2
2
2
M̄3 (t)2 (∂i π)(∂j π) M̄2 (t)2 (∂i π)(∂j π)
−
2
a4
2
a4
(14)
Already at the quadratic level we see different inflationary mechanisms captured:
• M̄1 = M̄2 = M̄3 = 0 = M2 → standard single-field slow-roll inflation with unitary
sound speed
• M2 6= 0 opens up the gate for cs < 1 models which are often associated to large
higher correlators (DBI inflation&Co)
• Ḣ → 0, M̄1 = 0 gives back Ghost inflation
These clear-cut correspondences continue at third and fourth order.
Schwinger-Keldysh (IN-IN) formalism
Main formula:
D Rt
Rt
E
0
0
0
0 hΩ|Θ(t)|Ωi = 0 T̄ e i 0 HI (t )dt ΘI (t)T e −i 0 HI (t )dt 0
Θ(t) = field operator
|Ωi = vacuum of the interaction theory
T , T̄ =time-ordering, anti-time-ordering operators
HI = interaction Hamiltonian
• All the fields in the interaction picture so free-field operator
expansion.
• Just like in usual QFT, only here correlations between
observables are at the same spacetime points and there’s the
anti-time order operator as well.
Fourth order, Trispectrum
Running
Want to consider the running of the fNL contribution generated by the
M̄6 -driven interaction term: this coefficient regulates operators that produce
interesting bispectrum and trispectrum shape-functions.
nNG ≡
M̄6
M̄
d ln |fNL
(k)|
1 d fNL6
'
= Θ(, η, s, 1 , 0 ) + 26
d ln k
HfNL dt
(15)
M̄˙ 6
H M̄6
There is a window for the parameter 6 to be larger than the usual slow roll
M̄6
parameters without having a small M̄6 spoil the importance of fNL
6 =
M̄62 H 2
√
4
M2 (α0 + β0 )
> 1;
M̄62 H 2
√
> 1.
+ β 0 )2
M34 (α0
(16)
and without having a large M̄˙ 6 give too large a correction to the power
spectrum.
√
√
(α0 + β0 )M34
(α0 + β0 )2 M44
M24
>
;
>
;
> 6 . (17)
6
6
M̄62 H 2
M̄62 H 2
M̄62 H 2