ENFPC 2010

Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm Inflation
Stochastic Approach to Warm Inflation
Leandro Alexandre da Silva
1
Rio de Janeiro State University
Department of Theoretical Physics
XXXI Encontro Nacional de F´ısica de Part´ıculas e Campos
01/09/2010
1
in collaboration with R.O. Ramos

1 Stochastic Approach to Inflation
2 Warm Inflation
3 Stochastic approach to Warm Inflation

Stochastic Approach to Inflation
When?
Stochastic Inflation: ∼ 1987
Why?
Exponentially rapid expansion → “freeze” of inflaton quantum
fluctuations on super-horizon scales
⇓
Inflaton fluctuations behave effectivelly as classical fluctuation
modes with random amplitudes
⇓
Emulates the growth of vacuum fluctuations by an effective
stochastic noise field which drives the dynamics of the
volume-smoothed inflaton → effective dynamics for
coarse-grained field φ

How?
Usual approach → decomposition of φ in a classical,
coarse-grained component and in a quantum fluctuation part:
Φ(x, t) → φ(x, t) + q(x, t) .
φ → coarse-grained scalar field averaged over approximatelly
all de Sitter horizon size 1/χ
q(x, t) → summarizes high frequency (k kh ≈ χ) quantum
fluctuations.
q(x, t) aproximated as a free, massless scalar field.

de Sitter metric:
ds2
= dxµ
dxν
gµν = −dt2
+ e2χt
dx2
,
Lagrangian density:
L =
1
2
√
−g [gµν
∂µΦ∂νφ − 2V (Φ)]
Equation of motion(EoM):
−3χ
∂
∂t
−
∂2
∂t2
+ e−2χt 2
Φ(x, t) −
∂V (Φ)
∂Φ
= 0

Making the field decomposition (slow-roll: ¨φ(x, t) ≈ 0):
3χ
∂
∂t
− e−2χt 2
[φ(x, t) + q(x, t)] +
∂V (φ)
∂φ
= 0
3χ
∂
∂t
− e−2χt 2
φ(x, t) +
∂V (φ)
∂φ
= 3χη(x, t) ,
Noise term:
η(x, t) ≡ −
∂
∂t
+
e−2χt
3χ
2
q(x, t)
Fourier mode expansion in de Sitter background:
q(x, t) ≡ d3
kWχ(k) σk
(t)e−ik·x
âk
+ σ∗
k
(t)eik·x
â†
k
.
Wχ(k) → filter or window function. Sharp momentum cutoff
implementation: Wχ(k) ≡ θ(k − χeχt).

σk
(t) ≡
1
2k(2π)3
χτ − i
χ
k
e−ikτ
.
Commutator:
[η(x, τ), η(y, τ)] = 0
⇓
Classical behaviour of quantum noise!

σk
(t) ≡
1
2k(2π)3
χτ − i
χ
k
e−ikτ
.
Commutator:
[η(x, τ), η(y, τ)] = 0
⇓
Classical behaviour of quantum noise!
Propagator:
0 | η(x, t)η(y, t ) | 0 =
χ3
4π2
δ(t − t )
sin τ | x − y |
τ | x − y |
.

Warm Inflation
Same basic ideas of cold inflation
Inflaton interacts with other fields → radiation production
A reheating process is no more necessary
Smooth transition to the radiation dominated regime
Thermal origin for the density perturbations
A. Berera and L. Z. Fang,Phys. Rev. Lett. 74, 1912 (1995):
Inflaton dynamics → Langevin-like equation

Microscopic motivation to Warm Inflation
Example:
S[φ, χ, σ] = d4
x
1
2
(∂µφ)2
−
1
2
m2
φφ2
−
λ
4!
φ4
+
1
2
(∂µχ)2
−
1
2
m2
χχ2
+
1
2
(∂µσ)2
−
1
2
m2
σσ2
−
g2
2
φ2
χ2
− f χσ2
.
φ → classical field in which dynamics we are interested in
χ → intermediate field that couples to σ and φ
σ → Thermally equilibrated field at temperature T

Microscopic motivation to Warm Inflation
Detailed calculation: Berera and Ramos, PRD63, 103509
(2001),Gleiser and Ramos, PRD50, 2441 (1994):
General Effective Equation of Motion (Homogeneous
approximation)
d2φ(t)
dt2
= −
dVeff(φ)
dφ
− φn
(t)
t
−∞
dt φn
(t ) ˙φ(t )Kχ(t − t )
+ φn
(t) ξ (t) ,
n = 0: additive noise→ φχ2 interaction
n = 1: multiplicative noise → φ2χ2 interaction

Markovian Approximation
Non-Markovian Equation of Motion:
∂2
t + m2
φ +
λ
3!
φ(t)2
φ(t) + φn
(t)
t
t0
dt K(t − t )φn
(t ) ˙φ(t )
= φn
(t)ξ(t) .
Markovian Approximation:
φn
(t)
t
t0
dt K(t − t )φn
(t ) ˙φ(t ) φ2n
(t) ˙φ(t)
t
t0→−∞
dt K(t − t )
→ Υ φ2n
(t) ˙φ(t) .
Markovian Equation of Motion:
¨φ + Υ φ2n ˙φ + m2
φφ +
λ
6
φ3
= φn
(t) ξ(t)

Figure: (a) mχ = 50H(0), (b) mχ = 150H(0) and (c) mχ = 250H(0)

More details about Markovian dynamics reliability:
R. L. S. Farias, R. O. Ramos and L. A. da Silva, Nonlinear
eﬀects in the dynamics governed by non-Markovian stochastic
Langevin-like equations, JPCS, in press
R. L. S. Farias, R. O. Ramos and L. A. da Silva, Numerical
Solutions for non-Markovian Stochastic Equations of Motion
Comp. Phys. Comm. 180, 574 (2009).
R. L. S. Farias, L. A. da Silva and R. O. Ramos,
Non-Markovian stochastic Langevin equations: Markovian and
non-Markovian dynamics, Phys. Rev. E 80, 031143 (2009)
R. L. S. Farias, R. O. Ramos and L. A. da Silva,Langevin
Simulations with Colored Noise and Non-Markovian
Dissipation Brazilian Journal of Physics, vol. 38 , no. 3B,
(2008)

Considering the additive case:
¨φ + [3H + Υ] ˙φ + V (φ) = ξ ,
¨a = −
8π
3m2
pl
ρr + ˙φ2
− V (φ) a ,
˙ρφ = −3
˙a
a
˙φ2
− Υ ˙φ2
+ ν ˙φ , ˙ρr = −4
˙a
a
ρr + Υ ˙φ2
− ξ ˙φ .

Stochastic approach to Warm Inflation
Back to stochastic approach: implement Warm Inflation
Modification of EoM:
∂φ(x, t)
∂t
=
1
3χ + Υ
e−2χt 2
φ(x, t) −
∂V (φ)
∂φ
+η(x, t)+ξ(x, t) .

Back to stochastic approach: implement Warm Inﬂation
Modiﬁcation of EoM:
∂φ(x, t)
∂t
=
1
3χ + Υ
e−2χt 2
φ(x, t) −
∂V (φ)
∂φ
+η(x, t)+ξ(x, t) .
Global analysis:
∂φ(t)
∂t
= −
1
3χ + Υ
∂V (φ)
∂φ
+ η(t) + ξ(t) ,
η(t)η(t ) =
χ3
4π2
δ(t − t )
ξ(t)ξ(t ) = βδ(t − t ) .

Considering a general SDE of the form
˙φ = h(φ, t) + g(φ, t)η(t) + f (φ, t)ξ(t) ,
we obtain a Fokker-Planck equation for η(t)ξ(t ) = 0 of the form
∂P(φ, t)
∂t
= −
∂
∂φ
D(1)
+
∂2
∂φ2
D(2)
P(φ, t) .
where the Kramers-Moyal coeﬃcients are:
D(1)
(φ, t) = h(φ, t) +
α
2
∂g(φ, t)
∂φ
g(φ, t) +
β
2
∂f (φ, t)
∂φ
f (φ, t)
D(2)
(φ, t) =
α
2
g(φ, t)2
+
β
2
f (φ, t)2
.
(1)

In our particular case:
∂P(φ, t)
∂t
= −
∂
∂φ
−
1
3χ + Υ
∂V (φ)
∂φ
+
∂2
∂φ2
χ3
4π2
+
β
2
P(φ, t) .
A trick to obtain φn(t) :
∂
∂t
φn
(t) ≡
∞
−∞
dφφn ∂
∂t
P(φ, t) .
∂
∂t
φn
(t) =
χ3
8π2
+
β
2
n(n−1) φn−2
(t) −
n
3χ + Υ
φn−1
(t)
∂V (φ)
∂t
.
φ2
(t) =
3χ + Υ
2m2
φ
χ3
4π2
+ β 1 − exp −
2m2
φ
3χ + Υ
t .

The Fokker-Planck operator
LFP ≡ −
∂
∂φ
−
1
3χ + Υ
∂V (φ)
∂φ
+
∂2
∂φ2
χ3
4π2
+
β
2
,
is so that LFP = L†
FP.
⇒ Transformation of variables:
t → (3χ + Υ)t
φ → (3χ + Υ)
χ3
4π2
+
β
ψ

The Fokker-Planck becomes:
∂P(ψ, t )
∂t
=
2
∂2P(ψ, t )
∂ψ2
+
∂
∂ψ
∂ ˜V (ψ)
∂ψ
P(ψ, t ) ,
where
˜V (ψ) ≡ (3χ + Υ)
χ3
4π2
+
β
−1
V φ = (3χ + Υ)
χ3
4π2
+
β
ψ
One more transformation
P(ψ, t ) = exp
1 ˜V (0) − ˜V (ψ) F(ψ, t ) ,
and then the Fokker-Planck equation becomes...

...a Schr¨odinger-like equation with imaginary time:
−
2
∂2
∂ψ2
+ U(ψ) F(ψ, t ) = −
∂F(ψ, t )
∂t
, (2)
with
U(ψ) ≡
1
2

 ∂ ˜V (ψ)
∂ψ
2
−
∂2 ˜V (ψ)
∂ψ2

 . (3)
Analogy with quantum mechanics:
ψ | α, t =
λ
ψ | λ λ | α, t → F(ψ, t ) =
λ
cλe− i
Eλt
ϕλ(ψ) ,
with
Hϕλ(ψ) = Eλϕλ(ψ) , H ≡ −
2
∂2
∂ψ2
+ U(ψ) , t →
i
t

Then using standard operator techniques and recovering the
original variables φ and t, we get:
P(φ, t) =
λ
cλe−m2
φσ−1
λt
×
Λ
1
2
π
1
4
√
2λλ!
m2
φ
(λ
2 − 3
4 )
m2
φΛ−1
φ −
∂
∂φ
λ
exp
m2
φΛ−1
φ2
,
And the propagator K(φ2, t2; φ1, t1):
K(φ2, t2; φ1, t1) = 1 − e−2m2
φσ−1
(t2−t1)
− 1
2
×
exp



−
m2
φΛ−1
φ2
2 + φ2
1 − 2φ2φ1e−m2
φσ−1
(t2−t1)
1 − e−2m2
φσ−1(t2−t1)



with Λ ≡ (3χ + Υ) χ3
4π2 + β
and σ ≡ 3χ + Υ

Discussions and perspectives
Planck energy scale at V (φ = φmax ) → breakdown of
semiclassical picture of spacetime
⇓
P(φmax ) = 0
⇓
Bounded from above eigenvalues
H eigenvalues ∈ N → LFP eingenvalues ∈ Z−
⇓
Highest eigenvalue (˜λmax ) of volume-weighted FP equation
(LFP → LFP + 3H) can be negative or positive: warm inﬂation
→ ˜λmax > 0 → number of inﬂating domains increases without
limit. (to appear in JCAP)

Discussions and perspectives
Work out the local (φ(x, t)) analysis
Study the pathway to classicalization: when the thermal
fluctuations overcome the quantum ones?
Better quantify eternal inflation based on Warm Inflation

Thanks for your attention!!!

ENFPC 2010

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