1. Outline Motivation Stochastic Warm Inflation model Results and conclusions
Power spectrum for inflation models with thermal
and quantum noises
Leandro A. da Silva, Rudnei O. Ramos
Rio de Janeiro State University
Department of Theoretical Physics
XXXIII Encontro Nacional de F´ısica de Part´ıculas e Campos
28/08/2012

2. Outline Motivation Stochastic Warm Inflation model Results and conclusions
1 Motivation
2 Stochastic Warm Inflation model
3 Results and conclusions

3. Outline Motivation Stochastic Warm Inflation model Results and conclusions
Contributions to the Power Spectrum
Great achievement of Inflationary models: fluctuations generated
during an early phase of inflation provide a source of nearly scale
invariant density perturbations, what nicely agrees with
observations.
Cold inflation: quantum fluctuations of the inflaton field make
the sole contribution
Warm inflation: the dominant contribution comes from
thermal fluctuations
⇓
These models look like extreme cases... Questions:
What about the intermediate regime?
How can we explicitly account for both quantum and thermal
contributions under the same framework?
In what conditions do thermal fluctuations overcome the
quantum ones?

4. Outline Motivation Stochastic Warm Inflation model Results and conclusions
The stochastic approach to warm inflation
Possible answer: stochastic approach to warm inflation
Extension of the original stochastic inflation model
(Starobinsky ∼ 1987)
Explicitly accounts for both quantum and thermal fluctuations
in a very transparent way
Recovers cold inflation and warm inflation in appropriate limits
Start point:
ds2
= dt2
− e2Ht
d2
x
∂2
∂t2
+ (3H + Υ)
∂
∂t
−
1
a2
2
Φ +
∂Veff,r(Φ)
∂Φ
= ξT ,
Effective equation of motion ⇒ Langevin-like equation with local
dissipation (Υ) and white thermal noise (ξT ).
(Berera, A., Moss, I. G., Ramos, R. O. (2009).Reports on Progress in
Physics, 72(2))
ξT (x, t)ξT (x , t ) = 2ΥTa−3
δ(x − x )δ(t − t ) ,

5. Outline Motivation Stochastic Warm Inflation model Results and conclusions
The stochastic approach to warm inflation
Introducing the field split
Φ(x, t) = ϕ(t) + δϕ(x, t) + φq(x, t) ,
where
ϕ(t) =
1
Ω Ω
d3
xΦ(x, t) ,
and
φq(x, t) =
d3k
(2π)3/2
W(k, t) φk(t)e−ik·x
ˆak + φ∗
k(t)eik·x
ˆa†
k
We get:
∂2
ϕ
∂t2
+ [3H + Υ(ϕ)]
∂ϕ
∂t
+ V,ϕ(ϕ) = 0 ,
∂2
∂t2
+ [3H + Υ(ϕ)]
∂
∂t
−
1
a2
2
+ Υ,ϕ(ϕ) ˙ϕ + V,ϕϕ(ϕ) δϕ = ˜ξq + ξT ,

6. Outline Motivation Stochastic Warm Inflation model Results and conclusions
The stochastic approach to warm inflation
˜ξq = −
∂2
∂t2
+ [3H + Υ(ϕ)]
∂
∂t
−
1
a2
2
+ Υ,ϕ(ϕ) ˙ϕ + V,ϕϕ(ϕ) φq ,
˜ξq → quantum noise term; ˜ξq(x, t), ˜ξq(x , t ) = 0 → Classical
behaviour
In terms of z =
k
aH
coordinate, the fluctuation EoM becomes:
δϕ (k, z) −
1
z
(3Q + 2)δϕ (k, z) + 1 + 3
η − βQ/(1 + Q)
z2
δϕ(k, z) =
1
H2z2
ξT (k, z) + ˜ξq(k, z) .
β and η → Slow-roll parameters, Q =
Υ
3H

7. Outline Motivation Stochastic Warm Inflation model Results and conclusions
The stochastic approach to warm inflation
From the solution of EoM, we write the inflaton power spectrum:
Pδϕ =
k3
2π2
d3
k
(2π)3
δϕ(k, z)δϕ(k , z) = P
(th)
δϕ (z) + P
(qu)
δϕ (z) .
P
(th)
δϕ (z) =
ΥT
π2
∞
z
dz z 2−4ν
G(z, z )2
, P
(qu)
δϕ = [2n(k) + 1]
k3
2π2
|Fk(z)|2
Fk(z) =
π3/2
zν
H
4k3/2
∞
z
dz (z )3/2−ν
[Jα(z)Yα(z ) − Jα(z )Yα(z)]
×
βQ
1 + Q
W,z (z )H(1)
µ (z ) + z W,z (z )H
(1)
µ−1(z )
−
3Q
z
βQ
1 + Q
+ η W(z )H(1)
µ (z ) − 3QW(z )H
(1)
µ−1(z )
−
π3/2
zν
H
4k3/2
∞
z
dz (z )5/2−ν
[Jα(z)Yα−1(z ) − Jα−1(z )Yα(z)] W,z (z )H(1)
µ (z )

8. Outline Motivation Stochastic Warm Inflation model Results and conclusions
The stochastic approach to warm inflation
Which filter function? Step filter case: W(k, t) ≡ θ(k − aH)
Pδϕ(z) ≈
HT
4π2
3Q
2
√
π
22α
z2ν−2α Γ (α)
2
Γ (ν − 1) Γ (α − ν + 3/2)
Γ ν − 1
2 Γ (α + ν − 1/2)
+
H
T
2
eH/T − 1
+ 1 z2η
,
where
ν = 3(1 + Q)/2 , α = ν2 +
3βQ
1 + Q
− 3η
Restrict model parameters → Curvature perturbations, spectral index and
tensor to scalar ratio:
∆2
R =
H2
˙φ2
Pδϕ , ns = 1 +
d ln ∆2
R
d ln k
, r =
2 H2
(1 + Q)2Pδϕ
WMAP 7 years: ns = 0.967 ± 0.014 (at 68% CL), r < 0.24
(WMAP+BAO+SN) Martin, J., Ringeval, C. (2010). Physical Review D,
82(2), 1-17.

9. Outline Motivation Stochastic Warm Inflation model Results and conclusions
Results and conclusions
V (φ) =
10−14
m4
P
p
φ
mP
p
Figure: Full line: p = 2, dashed line: p = 4 and the dotted line: p = 6.

10. Outline Motivation Stochastic Warm Inflation model Results and conclusions
Results and conclusions
Figure: (a) p = 2, (b) p = 4, (c) p = 6

11. Outline Motivation Stochastic Warm Inflation model Results and conclusions
The stochastic approach to warm inflation
Recovering Cold Inflation and Warm Inflation:
Q = 0 limit:
ns = 1 + 2η − 6 .
Q 1 limit:
ns = 1 + 2η − 6 + (8 − 2η)Q + O(Q2
) .
Large Q, T limit:
ns = 1 +
1
Q
−
9
4
−
3
4
β +
3
2
η + O(Q−3/2
) + O(1/T) ,

12. Outline Motivation Stochastic Warm Inflation model Results and conclusions
Results and conclusions
Stochastic warm inflation is able to describe in a natural way
both cold and thermal fluctuations.
It recovers Cold Inflation in Q = 0 limit and Warm Inflation in
Q 1 limit.
Thermal fluctuation and dissipation make V ∼ φp (p > 3)
again compatible with observations.
Perspectives: check the results using a more “physical” filter,
like a gaussian one (work in progress)
Impose others observational constraints, like running ns (work
in progress)