Inflation with Weakly Broken Galileon Symmetry

Transcript

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[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1505.00007],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1506.06750],
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
[arXiv:1511.01817].
Outline
The standard Big Bang model and inflation
The Galileon symmetry
The notion of Weakly Broken Galileon (WBG) symmetry
The Effective Field Theory of inflation
Inflation with WBG symmetry
Conclusions
The Big Bang model. In the last decades, experiments have provided
the following picture of the Universe:
• it is expanding and, in particular, accelerating;
• it appears very homogeneous and isotropic;
• it is extremely flat;
• primordial perturbations, imprinted on the CMB (δT/ T ∼ 10−5 ), are
almost
◦ scale-invariant (ns = 0.968 ± 0.006),
◦ adiabatic (|αnon-adi | ≲ 10−2 ),
equil
◦ Gaussian (ƒNL
= −4 ± 43);
• no tensor modes (r < 0.07, 95% CL).
6000
5000
D`T T [µK2 ]
Tensor to scalar ratio r
0.15
0.30
Planck 2013
Planck TT+lowP
Planck TT,TE,EE+lowP
3000
2000
1000
0
600
∆D`T T
0.00
Luca Santoni – SNS and INFN, Pisa
4000
60
30
0
-30
-60
300
0
-300
-600
0.94
0.96
0.98
Primordial tilt (ns )
1.00
May 17-20, 2016
2
10
30
500
1000
1500
2000
2500
`
3 / 16
Inflation consists in an accelerated
expansion at early times,
⃗2 ,
ds2 = −dt 2 + (t)2 d
̈ > 0,
and provides an explanation of the
observed features and of the origin
of the density perturbations, as
seeds for the large-scale structure
formation in the Universe.
Many models of inflation. In terms of a single scalar field:
◦ L = (∂φ)2 − V(φ), with a flat V(φ) (slow-roll inflation);
◦ L = P (∂φ)2 , φ (DBI, k-inflation);
◦ …
◦ …
May 17-20, 2016
4 / 16
The Galileon symmetry
What about higher derivative operators? Higher derivative operators
are not easy to deal with. Usually, either
• they are irrelevant (for instance, in the effective field theory of
inflation† , which can parametrize all single-field models in a
single framework, they are generically expected to be negligible
with respect to lower derivative operators)
or
• lead to instabilities,
unless some symmetry protects the theory.
This is the case of the Galileon symmetry:
φ → φ + c + bμ μ .
Irrespective of inflation, this is of some theoretical interest. Before
considering its consequences in cosmology, let’s study its main
properties.
† See
next slides.
May 17-20, 2016
5 / 16
Galileons in flat space-time
[Nicolis, Rattazzi, Trincherini ’09]
What are the lowest order operators in a 4-d flat space-time invariant
under φ → φ + c + bμ μ ? Apart from constant terms and total
derivatives, the kinetic operator
L2 = (∂φ)2
is clearly invariant. Moreover, one finds for instance
L3 = (∂φ)2 □φ .
A generic Galileon theory
1
(∂φ)2 □φ
L = − (∂φ)2 + c3
+ ...
2
Λ33
satisfies some remarkable properties:
• the coefficient c3 is not renormalized (non-renormalization
theorem) [Luty, Porrati, Rattazzi ’03];
• second order equations of motion.
May 17-20, 2016
6 / 16
Weakly Broken Galileon symmetry: a simple
example in flat space-time
It turns out to be interesting to break slightly† the Galileon symmetry.
Let’s introduce a small breaking (Λ2 ≫ Λ3 ) of the form
1
(∂φ)2 □φ (∂φ)4
L′ = − (∂φ)2 +
+
2
Λ42
Λ33
• We expect symmetry breaking operators of the form (∂φ)2n to
be generated by quantum corrections.
• All the symmetry breaking operators are generated at a scale
which is parametrically higher than Λ2 .
⇒ The operator (∂φ)4 gets only small corrections through loop
effects.
This is the remnant of the non-renormalization theorem.
† This is not a mere exercise, but it is necessary once gravity is turned on. Indeed,
the Galileon symmetry can only be defined in a flat space-time, being explicitly broken
by gravity.
May 17-20, 2016
7 / 16
Coupling to gravity
As anticipated, gravity explicitly breaks the Galileon symmetry:

1
(∂φ)2 □φ
p
2
L = −g − (∂φ) +
,
□ = ∇μ ∇μ .
3
2
Λ3
• As in the previous example, the symmetry breaking operators
(∂φ)2n are quantum mechanically generated.
• In particular, the smallest scale by which the operators (∂φ)2n
are suppressed is (MPl Λ33 )1/ 4 .
Therefore, we define a WBG theory to be of the form

1
(∂φ)2 □φ (∂φ)4
p
WBG
2
L
= −g − (∂φ) +
+
,
2
Λ42
Λ33
where Λ2 ≡ (MPl Λ33 )1/ 4 .
Non-renormalization theorem. Loop corrections are suppressed by
Λ3
powers of M
.
Pl
May 17-20, 2016
8 / 16
The most general WBG theory
Introducing the other Galileon interactions, a further generalization (in
Horndeski class), that does not spoil the properties, is still possible:
LWBG
= Λ42 G2 (X)
2
LWBG
=
3
LWBG
=
4
LWBG
=
5
Λ42
Λ33
G3 (X)□φ
Λ82
Λ42
Λ3
Λ63
G (X)R + 2
6 4
G4X (X) [] 2 − [2 ]
Λ82
Λ42
Λ3
3Λ93
G (X)Gμν ∇μ ∇ν φ −
9 5
where X ≡ −
(∂φ)2
,
Λ4
2
G5X [] 3 − 3[][2 ] + 2[3 ]
[2 ] = ∇μ ∇ν φ∇ν ∇μ φ, ...
Remark. The coefficients of the polynomials G (X) are not
Λ3
renormalized up to some powers of M
(WBG symmetry).
Pl
[Pirtskhalava, L.S., Trincherini, Vernizzi ’15]
May 17-20, 2016
9 / 16
WBG theories in Cosmology: de Sitter solutions
Let’s study the consequences of the (approximate) Galileon
symmetry for the cosmological scalar fields:
φ → φ + c + bμ μ .
An inflationary theory with WBG symmetry can be written as
∫
S=
p
d4  −g

1
2
M2Pl R − V(φ) +
5
∑

LWBG
.

=2
From the equations of motion one infers that:
• if V ≡ 0, exact linear solution φ ∼ t: the acceleration is driven by
the derivative operators;
• if V ̸= 0, slight deviation from de Sitter (|Ḣ| ≪ H2 ): the evolution
is of slow-roll type.
Remark. In both cases quantum corrections are fully under control.
May 17-20, 2016
10 / 16
[Creminelli, Luty, Nicolis, Senatore ’06], [Cheung, Creminelli, Fitzpatrick, Kaplan, Senatore, ’08]
Parametrizing all single-field models, irrespective of the underlying
microscopic theory, the effective theory of inflation turns out to be a
useful framework to study cosmological perturbations.
General idea. The inflationary phase has to be connected to a
standard decelerated evolution, inducing a privileged slicing. In ADM
variables:
ds2 = −N2 dt 2 + γj (N dt + d )(Nj dt + dj ) .
⃗ ) = 0, we write down the most general
Fixing the gauge δφ(t, 
Lagrangian compatible with the residual symmetries,
⃗ ):
 →  + ξ (t, 
∫
M2 Ḣ
p
4
S = d  −g LEH − Pl2 − M2Pl (3H2 + Ḣ)
N
+ M41 δN2 − M̂31 δKδN + . . .
May 17-20, 2016
11 / 16
∫
S=

p
d4  −g Lbck + M41 δN2 − M̂31 δKδN + . . . .
M1
MPl2H
• How can a theory with equally
dominant δN2 and δKδN be
consistent, without exiting the
regime of validity of the
effective theory?
3
<
• One expects that physical
observables are
predominantly determined by
the operators with the least
number of derivatives.
May 17-20, 2016
O(1)
O(ε)
4
O(ε)
O(1)
M1
MPl2 H2
12 / 16
The EFT of inflation with WBG symmetry
[Pirtskhalava, L.S., Trincherini ’15]
∫
S=

p
d4  −g Lbck + M41 δN2 − M̂31 δKδN + . . . .
M41 = Λ42 (2X 2 G2XX + . . .)
3
<
• In the case of theories with
WBG symmetry one can
explore a wider region beyond
M̂31 H ≪ M41 . Indeed,
M̂31 H = Λ42 (−2XZG3X + . . .)
M1
MPl2H
O(1)
WBG
|M41 | ∼ |M̂31 H| ≲ M2Pl H2 .
• δKδN is relevant for
non-Gaussianity.
O(ε)
4
O(ε)
• Various regimes show up.
WBG
May 17-20, 2016
O(1)
M1
MPl2 H2
13 / 16
Experimental constraints on single-field inflation
A qualitative sketch of the experimental constraints on the effective
parameter space and examples of fundamental theories:
∫

p
S = d4  −g Lbck + M41 δN2 − M̂31 δKδN + . . .
<
3
• The bound on r alone
puts strong and robust
constraints on the
parameter space of the
effective theory.
• Further constraints are
imposed by ƒNL .
M1
MPl2H
O(1)
WBG
r > 0.1
r < 0.1
O(ε)
WBG
Slow-rollO(ε)
May 17-20, 2016
DBI
O(1)
4
M1
MPl2 H2
14 / 16
Conclusions
• The consequences of an approximate φ → φ + c + bμ μ
symmetry have been studied.
• We have established the notion of weakly broken Galileon
invariance, introducing a particular non-minimal coupling to
gravity and a higher scale Λ2 ≡ (MPl Λ33 )1/ 4 .
• The resulting effective theory is characterized by the technically
natural hierarchy Λ2 ≫ Λ3 , allowing to retain the quantum
non-renormalization properties of the Galileon for a broad range
of physical backgrounds.
• The de Sitter solutions that these theories possess are
insensitive to loop corrections.
• The accelerate expansion can be sustained by the potential or
the derivative operators, leading to potentially and kinetically
driven phases.
• On slowly-rolling backgrounds large non-Gaussian signals are in
principle allowed and detectable, well within the validity of the
effective theory.
May 17-20, 2016
15 / 16
Thank you
May 17-20, 2016
16 / 16
Backup slides
May 17-20, 2016
1 / 31
Galileon transformations
General idea. In presence of gravity we want to study the properties
of the following (necessarily approximate) symmetry of some scalar
fields:
φ → φ + c + bμ μ .
Why approximate? Couplings to gravity break the Galileon symmetry
explicitly.
Purpose. Studying theories which retain as much as possible the
Galileon symmetry ⇒ Weakly Broken Galileon invariance.
May 17-20, 2016
2 / 31
Weakly Broken Galileon symmetry: a simple
example
1
1
L = − (∂φ)2 + 3 (∂φ)2 □φ
2
Λ3
• Exactly invariant under φ → φ + c + bμ μ .
• Λ3 is the cutoff.
Let’s introduce a small breaking (Λ2 ≫ Λ3 ):
1
1
1
L′ = − (∂φ)2 + 3 (∂φ)2 □φ + 4 (∂φ)4
2
Λ2
Λ3
• We expect symmetry breaking operators of the form (∂φ)2n to
be generated by quantum corrections.
• All the symmetry breaking operators are generated at a scale
which is parametrically higher than Λ2 .
⇒ The operator (∂φ)4 gets only small corrections through loop
effects.
May 17-20, 2016
3 / 31
Weakly Broken Galileon symmetry
What happens if we introduce gravity?
Let’s generalize the simple example
1
1
1
L′ = − (∂φ)2 + 3 (∂φ)2 □φ + 4 (∂φ)4
2
Λ2
Λ3
following this guide principle:
• preserve the quantum properties (small quantum corrections).
We will find a class of theories with Weakly Broken Galileon (WBG)
invariance. As a consequence, they will have second order equations
of motion.
Foretaste. Applied to inflation, this will provide a radiatively stable
cosmological evolution with interesting phenomenological properties.
May 17-20, 2016
4 / 31
Coupling to gravity
How can we couple our scalar theory to gravity? There are many
ways to couple the theory to gravity...
This necessarily breaks Galileon invariance.
Let’s concentrate on quantum generated operators of the form
(∂φ)2n .
We will see that coupling to gravity defines the sense in which the
breaking can be considered weak.
May 17-20, 2016
5 / 31
Coupling to gravity: minimal coupling
The minimally coupled Galileon theory is obtained by replacing
∂μ → ∇μ .
Operators of the form
(∂φ)2n
MkPl Λ4n−k−4
3
,
k, n > 0,
are generated.
One can show that the smallest scale by which such operators are
suppressed is (MPl Λ53 )1/ 6 .
Can one do better? Can one enhance such a scale?
May 17-20, 2016
6 / 31
Coupling to gravity: non-minimal coupling
Let’s consider the following non-minimally coupled Galileon theory:
Lnm
=
3
Lnm
4
Lnm
5
p
−g(∂φ)2 [] ,
= −g (∂φ)4 R − 4(∂φ)2 [] 2 − [2 ] ,
2
p
= −g (∂φ)4 Gμν ∇μ ∇ν φ + (∂φ)2 [] 3 − 3[][ 2 ] + 2[3 ] .
3
p
One can prove that in this case only operators of the form
(∂φ)2n
MnPl Λ3n−4
3
are generated.
Result. The smallest scale by which the operators (∂φ)2n are
suppressed is Λ2 ≡ (MPl Λ33 )1/ 4 ⇒ WBG symmetry.
Remark. The theory turns out to have 2nd order e.o.m. also for the
metric: Covariant Galileon. [Deffayet, Esposito-Farese, Vikman ’09]
May 17-20, 2016
7 / 31
WBG symmetry in presence of a scalar potential
Let’s introduce a flat (ϵv , ηv ≪ 1) potential V(φ):
∫
p
d4  −g LWBG − V(φ) ,
V(φ) ∼ φm .
hn
Warning. Do quantum corrections to the vertices V(φ) Mcn spoil the
Pl
notion of WBG symmetry of the theory? No. Indeed, for instance,

Λ3 n
• only internal graviton lines: ∼ M
;
Pl
p Λ 3 2
V ′ Λ2
• one internal scalar field: ∼ VM 3 ∼ ϵv M
;
Pl
Pl
• …
Result. These loop corrections are consistent with the notion of WBG
symmetry.
May 17-20, 2016
8 / 31
The inflationary epoch
Big Bang puzzles:
• horizon problem;
• flatness;
• generation of density
perturbations as seeds for
Universe large-scale structure
formation.
What we observe:
(almost)
• scale-invariant (|ns − 1| ∼ 10−2 ),
• adiabatic (|αnon-adi | ≲ 10−2 ),
equil
• Gaussian (ƒNL
= −4 ± 43)
spectrum for density perturbations;
• no tensor modes (r < 10−1 ).
Inflation, as a possible solution,
with SEC violation:
⃗2 ,
ds2 = −dt 2 + (t)2 d
̈ > 0,
with approximate shift symmetry,
φ → φ + c, of the scalar field
driving the background evolution,
with degree of breaking ϵ ≡ − HḢ2 .
May 17-20, 2016
9 / 31
The inflationary epoch
Contact with observations
r < 0.07 (95% CL) ;
• amplitude of scalar modes, As :
δT/ T ∼ 10−5 ;
• scalar spectral index, ns :
ns = 0.968 ± 0.006 ;
equil
ƒNL
∝
0.96
0.98
Primordial tilt (ns )
1.00
5000
B(k,k,k)
:
Ps (k)2
= −4 ± 43 .
0.94
6000
4000
3000
2000
1000
0
600
∆D`T T
• non-Gaussianity,
equil
ƒNL
Tensor to scalar ratio r
0.15
0.30
Pγ (k)
:
Ps (k)
0.00
• tensor-to-scalar ratio, r ∝
D`T T [µK2 ]
Experimental constraints
(Planck2015+BICEP2+Keck Array):
Planck 2013
Planck TT+lowP
Planck TT,TE,EE+lowP
60
30
0
-30
-60
300
0
-300
-600
2
10
30
500
1000
1500
2000
2500
`
May 17-20, 2016
10 / 31
The inflationary phase
A simple action for the inflaton:
∫
1 2
1
p
4
2
S = d  −g
M R − (∂φ) − V(φ) .
2 Pl
2
Inflaton fluctuations:
⃗ ) = φ0 (t) + δφ(t, 
⃗) .
φ(t, 
Equations of motions:
φ̈0 + 3Hφ̇0 + V ′ (φ0 ) = 0 ,
H(t) ≡
̇(t)
(t)
.
In order to have a sufficiently long inflationary expansion, the inflaton
field φ is assumed to slowly roll down a very flat potential:
|Ḣ| ≪ H2 ,
MPl |V ′ | ≪ |V| .
May 17-20, 2016
11 / 31
Contact with observations: scalar modes
The inflaton fluctuations δφ are responsible for the CMB anisotropies.
The distribution function of the scalar fluctuations is parametrized in
terms of
• power spectrum:
〈δφk⃗ δφk⃗′ 〉 = (2π)3 Ps (k)δ(k⃗ + k⃗′ ), Ps (k)k 3 ≡ As k ns −1 ;
6000
D`T T [µK2 ]
5000
4000
3000
2000
1000
∆D`T T
0
600
60
30
0
-30
-60
300
0
-300
-600
2
10
30
500
1000
1500
2000
2500
`
May 17-20, 2016
12 / 31
Contact with observations: scalar modes
• bispectrum:
〈δφk⃗1 δφk⃗2 δφk⃗3 〉 = (2π)3 B(k1 , k2 , k3 )δ(k⃗1 + k⃗2 + k⃗1 ) .
The bispectrum is related to the field’s self-interactions. A
possible detection could discriminate among different models.
Simple parametrization:
B(k1 , k2 , k3 ) ∝ ƒNL Ps (k1 )Ps (k2 )
+ Ps (k1 )Ps (k3 ) + Ps (k2 )Ps (k3 ) .
ƒNL is highly suppressed in slow-roll models: the extreme
flatness of the potential gives rise to small non-Gaussianity.
What about derivative interactions?
May 17-20, 2016
13 / 31
Observable non-Gaussianity
In a generic low-energy EFT

1
(∂φ)4
p
2
+ ... ,
L = −g − (∂φ) − V(φ) +
2
Λ4
what is the impact of derivative contributions at the level of
non-Gaussianity? One finds [Creminelli ’03]
ƒNL ∼
φ̇20
Λ4
.
Any amount of ƒNL ≳ 1 would be out of the regime of validity of the
effective theory.
It could not be trusted, unless the infinite series of derivative
operators can be re-summed because of some symmetry protecting
the theory against large quantum corrections, e.g. in DBI inflation
equil
which predicts ƒNL ∼ c12 , c2s ≪ 1.
s
May 17-20, 2016
14 / 31
Observable non-Gaussianity
Message. Theories with weakly broken Galileon symmetry admit
sub-luminal scalar perturbations within a well defined low-energy EFT,
resulting in possibly large non-Gaussian deviations, ƒNL ≫ 1.
May 17-20, 2016
15 / 31
Contact with observations: tensor modes
Fluctuations of the metric tensor, gμν = ḡμν + γμν , give rise to
′
′
〈γs⃗ γs⃗′ 〉 = (2π)3 Pγ (k)δss δ(k⃗ + k⃗′ ) ,
k
k
Pγ (k) =
2H2
M2Pl k 3
.
• The tensor modes are much more model independent and
robust. [Creminelli, Gleyzes, Noreña, Vernizzi ’14]
Conversely, scalar modes can be adjusted in many ways (shape
of the potential, many scalars, speed of sound cs , …).
• The amplitude fixes the energy scale of inflation.
May 17-20, 2016
16 / 31
One more brief comment…
Because of the breaking of time diffeomorphisms, the unitary gauge
action describes 3 d.o.f. : the 2 graviton helicities + 1 scalar mode.
2
2
(In analogy, the same happens for a spin-1 massive particle: L = − 41 Fμν
− m2 A2μ )
This mode can be made explicit by performing a broken gauge
⃗ ) (Stückelberg trick).
transformation, t → t + π(t, 
(For a non-Abelian gauge group: Aμ → UAμ U† − gi U∂μ U† , U ≡ eiπ T  )
In some cases the physics of the Goldstone decouples from the two
graviton helicities at short distances, when the mixing with gravity can
be neglected (decoupling limit).
(In analogy with the equivalence theorem for the longitudinal components of the massive gauge
boson: m ≪ E ≪ 4πm/ g)
May 17-20, 2016
17 / 31
Inflation with WBG symmetry
The Friedmann equations:
3M2Pl H2 = V + Λ42 X
2M2Pl Ḣ = −
1
−
2
G2
X
+ 2G2X − 6ZG3X + . . . ,
Λ42 X [1 + 2G2X − 3ZG3X + . . .]
1 + 2G4 − 4XG4X − 2ZXG5X
X=
φ̇20
.
Λ42
≲ 1,
Z≡
Hφ̇0
≲ 1.
Λ33
Depending on the values of X and Z, two phenomenologically distinct
regimes:
Background quantum stability:
• kinetically driven phase (X ∼ Z ∼ 1) ⇒ mixing with gravity is
order-one important at all scales (i.e. decoupling limit does not
apply);
p
• potentially driven phase (X ∼ ϵ, Z ∼ 1) with
M2Pl H2 ∼ V ≫ (∂φ)2 ⇒ decoupling limit applies at the Hubble
scale and possible large non-Gaussianity.
May 17-20, 2016
18 / 31
[Creminelli, Luty, Nicolis, Senatore ’06], [Cheung, Creminelli, Fitzpatrick, Kaplan, Senatore, ’08]
The inflationary phase has to be connected to a standard decelerated
evolution, inducing a privileged slicing. In ADM variables:
ds2 = −N2 dt 2 + γj (N dt + d )(Nj dt + dj ) .
⃗ ) = 0, the most general perturbative
In unitary gauge, i.e. δφ(t, 
⃗ ), is
Lagrangian, invariant under  →  + ξ (t, 
∫
S=
4
p
d  γN
M2Pl
M2 Ḣ
R + Kμν K μν − K 2 − Pl2 − M2Pl (3H2 + Ḣ)
N
(3)
2
+ M41 δN2 + M42 δN3 + . . .
− M̂31 δKδN + M̂32 δKδN2 + . . .
−
M̄21

2
δK − δKμν δK
May 17-20, 2016
μν
−
(3)

RδN + . . .
19 / 31
∫
S=

p
d4  −g Lbck + M41 δN2 − M̂31 δKδN + . . . .
3
<
In other words:
• One expects to have control
only over the region
M̂31 H ≪ M41 .
• Are we able to explore a wider
region of the parameter
space? WBG symmetry
allows this…
M1
MPl2H
O(1)
O(ε)
4
O(ε)
May 17-20, 2016
O(1)
M1
MPl2 H2
20 / 31
The EFT of inflation with WBG symmetry
LWBG

M41 Λ−4
2
M̂31 HΛ−4
2
...
=2
2X 2 G2XX
×
...
=3
−3XZ (G3X + 2XG3XX )
...
=4
12XZ 2 (3G4XX + ...)
G5X
XZ 3 3
+ ...
X
−2XZG3X
G4X
8XZ 2
+ ...
X
G5X
2XZ 3 3
+ ...
X
=5
...
...
• As anticipated, a radiatively stable hierarchy follows:
M41 ∼ M̂31 H.
• kinetically driven phase (X ∼ Z ∼ 1):
M41 ∼ M2Pl H2 ,
• potentially driven phase (X ∼
M41
∼
ϵM2Pl H2 ,
p
M̂31 ∼ M2Pl H,
…
ϵ, Z ∼ 1):
M̂31 ∼ ϵM2Pl H,
May 17-20, 2016
…
21 / 31
A more quantitative analysis
[Maldacena ’05], [Chen, Huang, Kachru, Shiu ’08], [Pirtskhalava, L.S., Trincherini ’15]
Let’s consider the effective theory
∫

p
S = d4  γN Lbck + M41 δN2 + M42 δN3 − M̂31 δKδN + M̂32 δKδN2 .
We want to explore configurations in which the decoupling limit does
not apply and the full computation is required.
• Defining the gauge γj = (t)2 e2ζ(t,⃗ ) δj ,
• solving linearly the Hamiltonian constraints,
• expanding up to the 3rd order
yields the action for the scalar mode

∫
⃗ ζ)2
1
(∇
4
3
2
2
(3)
d   ζ̇ − cs
S=
+L
.
2
2
May 17-20, 2016
22 / 31
Constraints on single-field inflation
In terms of the dimensionless parameters
α≡
M̂31
M2Pl H
,
β≡
M41
M2Pl H2
,
γ≡
M̂32
M2Pl H
δ≡
,
M42
M2Pl H2
,
the speed of sound is
c2s
=
(ϵ + α)(1 − α) +
α̇
H
ϵ + β − 3α(2 − α)
.
Different choices for the values of the parameters can yield a small
speed of sound.
May 17-20, 2016
23 / 31
Constraints on single-field inflation: summary
Parameters:
ϵ = 10−2 ,
γ≃δ≃1
0.05
0.000
0.04
- 0.002
2
M1 /(2MPl
H)
r=0.1
0.02
⋀3
⋀3
2
M1 /(2MPl
H)
0.03
0.01
r=0.01
- 0.004
- 0.006
- 0.008
0.00
- 0.01
- 0.010
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
2 2
M14 /(MPl
H )
2 2
M14 /(MPl
H )
May 17-20, 2016
24 / 31
Constraints on single-field inflation: summary
ϵ = 10−3 ,
Parameters:
γ≃δ≃1
0.010
0.008
⋀3
2
M1 /(2MPl
H)
Remarks.
• DBI allowed.
• WBG inflation is allowed as
well with the (stable) tuning
α ≲ 10−2 , while β, γ, δ ≲ 1.
• In particular, slow-roll-WBG
inflation is allowed.
r=0.01
0.006
0.004
0.002
0.000
- 0.002
0.0
0.2
0.4
0.6
0.8
1.0
2 2
M14 /(MPl
H )
May 17-20, 2016
25 / 31
Inflationary models
α≡
M̂3
1
M2
H
Pl
,
β≡
M4
1
M2
H
Pl
,
2
γ≡
M̂3
2
M2
H
Pl
,
δ≡
M4
2
M2
H2
Pl
.
Inflationary models
Parameter hierarchy
c2s
Non-Gaussianity amplitude
Canonical slow-roll
ϵ ≪ 1; α, β, γ, δ = 0
1
ƒNL ∼ ϵ
DBI-like theories
α, γ = 0; ϵ ≪ β ≪ δ
ϵ
β
Galileon inflation
ϵ ≪ α, β, γ, δ
α(1−α)
β−3α(2−α)
Kinetically driven WBG
ϵ ≪ α, β, γ, δ ∼ 1
α(1−α)
β−3α(2−α)
ƒNL ∼
1 †
c6
s
Potentially driven WBG
ϵ ∼ α, β, γ, δ
ϵ+α
ϵ+β−6α
ƒNL ∼
1 ‡
c4
s
ƒNL ∼
ƒNL ∼
1
c2
s
1
c2
s
(or ∼
1
)
c4
s
†
Such a steep growth is ruled out by experimental constraints.
Slight deformation of slow-roll inflation, but with possible large
non-Gaussianity.
‡
May 17-20, 2016
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de Sitter space and symmetries
Inflation takes place in (an approximate) de Sitter space,
ds2 =
1
H2 η2
⃗ 2 = −dt 2 + e2Ht d
⃗2 ,
−dη2 + d
whose isometry group is SO(4,1). Spatial translations and rotations
are exact symmetries because of homogeneity and isotropy of the
background evolution. The other de Sitter isometries are
⃗ → λ
⃗,
• dilation: η → λη, 
⃗ ),  →  + b (−η2 + 
⃗ 2 ) − 2 (b⃗ · 
⃗ ).
• η → η − 2η(b⃗ · 
In general, these are not symmetries of the background. However,
invariance under dilation, which guarantees a scale-invariant power
spectrum and constraints the 2-point function to be
F(kη)
δφk⃗ δφk⃗′ = (2π)3 δ(k⃗ + k⃗′ )
,
k3
can be recovered by an additional φ → φ + c invariance.
May 17-20, 2016
[Creminelli ’12]
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Potentially driven WBG inflation
The de Sitter phase is sustained by the potential, that satisfies the
conditions
′′ 2 ′ 2
MPl
V V
2
,
ϵv ≡
,
|ηv | ≡ MPl 2
V
V as in standard slow-roll inflation. The equations of motion are
3Hφ̇0 F(X, Z) ≃ −V ′ (φ0 ) ,
X=
φ̇20
Λ42
∼
p
ϵ,
Z ∼ 1.
Moreover, c2s ∝ ϵ + α: then, α ≃ −ϵ ⇒ strongly sub-luminal scalar
perturbations. The WBG symmetry guarantees that this is respected
by loop corrections.
L(3) ⊃ (∂ζ)2 ∂2 ζ
⇒
May 17-20, 2016
equil
ƒNL
∼
1
c4s
.
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Summary of Part I
• The consequences of a theory with an approximate Galileon
symmetry (φ → φ + c + bμ μ ) in presence of gravity have been
studied.
• We have introduced the notion of WBG invariance, which
characterizes the unique class of couplings of such a theory to
gravity that maximally retain the defining symmetry: it is the
curved-space remnant of the Galileon’s non-renormalization
properties.
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Conclusions (1)
• The consequences of an approximate φ → φ + bμ μ symmetry
have been studied.
• Among all possible couplings to gravity, the “more invariant” one
under Galileon transformations has been chosen: this has
provided the notion of weakly broken Galileon invariance.
• The resulting action defines a sub-class of Horndeski theories
(with second order equations of motion), that one with fine
quantum properties.
• The resulting effective theory is characterized by two scales (with
a technically natural hierarchy Λ2 ≫ Λ3 ), allowing to retain the
quantum non-renormalization properties of the Galileon for a
broad range of physical backgrounds.
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Conclusions (2)
• The de Sitter solutions that these theories possess are
insensitive to loop corrections.
• The accelerate expansion can be sustained by the potential or
the derivative operators, leading to potentially and kinetically
driven phases.
• On slowly-rolling backgrounds large non-Gaussian signals are in
principle allowed and detectable, well within the validity of the
effective theory.
• The theories characterized by WBG symmetry can also be
applied in the context of late-time cosmic acceleration.
May 17-20, 2016
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