1. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Non-isentropic stochastic inflation, single field
potentials and Planck data
Leandro A. da Silva, Rudnei O. Ramos1
XXXIV Encontro Nacional de F´
ısica de Part´
ıculas e Campos
27/08/2013
1
Universidade do Estado do Rio de Janeiro
Final remarks

2. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Two main questions:
How the introduction of dissipative and temperature effects
impacts the compatibility between theoretical predictions and
observational data?
Can temperature and dissipation stop eternal self-reproduction
of the universe?

3. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Warm inflation:
Same basic ideas of standard inflation.
Inflaton interacts with its environment → radiation production
during inflation.
No reheating mechanism is necessary.
Smooth transition to radiation domination era.

4. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Warm inflation
Microscopic motivation:
L[φ, χ, σ] = L[φ] + L[χ] + L[σ] + Lint [φ, χ] + Lint [χ, σ]
Procedure: functional integration over χ e σ.
Non-equilibrium dynamics → Real time formalism
Markovian approximation (local dissipation) → system
characteristic time scale
relaxation time scale
Effective equation of motion:
∂
1
∂2
+ (3H + Υ) − 2
2
∂t
∂t a
2
Φ+
∂Veff (Φ)
= ξT
∂Φ
Final remarks

5. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Warm inflation: basic equations
1
∂2
∂
+ (3H + Υ) − 2
∂t2
∂t a
2
Φ+
∂V (Φ)
= ξT
∂Φ
ξT (x, t)ξT (x , t ) = 2ΥT a−3 δ(x − x )δ(t − t )
a=−
¨
8π
˙
ρr + Φ2 − V (Φ) a
3m2
pl
a˙
˙
˙
˙
ρΦ = −3 Φ2 − ΥΦ2 + ξT Φ ,
˙
a
=
m2
pl
16π
< 1 + Q,
V
V
2
,
η=
η <1+Q
m2
pl
8π
e
a
˙
˙
˙
ρr = −4 ρr + ΥΦ2 − ξT Φ
˙
a
V
V
,
β=
β <1+Q,
m2
pl
8π
ΥV
ΥV
Q≡
Υ
3H
Final remarks

6. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Contributions to the power spectrum
Important characteristic of inflation: Natural mechanism to generation of
nearly scale invariant density perturbations.
Cold inflation: quantum fluctuations contributions
Warm inflation: thermal fluctuations contributions

7. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Contributions to the power spectrum
Important characteristic of inflation: Natural mechanism to generation of
nearly scale invariant density perturbations.
Cold inflation: quantum fluctuations contributions
Warm inflation: thermal fluctuations contributions
⇓
extreme cases...
Non-isentropic stochastic inflation:
Quantum and thermal fluctuations taken in account explicitly
and in a transparent way.
Recovers standard results both from cold and warm inflation.

8. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky I: perturbative approach
Central idea:
Φ(x, t) → Φ> (x, t) + Φ< (x, t)
Φ> (x, t) → ϕ(t) + δϕ(x, t)
Mode separation implemented through an “Window
function”: W (k, t) ≡ θ(k − aH)
Goal: Effective dynamics for δϕ.
Φ< (x, t) ≡ φq (x, t) =
d3 k
W (k, t) φk (t)e−ik·x ak + φ∗ (t)eik·x a†
ˆ
ˆk
k
3/2
(2π)
√
H π
(1)
φk (τ ) =
(|τ |)3/2 Hµ (k|τ |) ,
2
where µ =
9/4 − 3η.

9. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky I: perturbative approach
∂ϕ
∂2ϕ
+ [3H + Υ(ϕ)]
+ V,ϕ (ϕ) = 0 ,
2
∂t
∂t
1 2
∂2
∂
˜
−
+ [3H + Υ(ϕ)]
+ Υ,ϕ (ϕ)ϕ + V,ϕϕ (ϕ) δϕ = ξq + ξT ,
˙
∂t2
∂t a2
˜
ξq = −
∂2
1
∂
− 2
+ [3H + Υ(ϕ)]
2
∂t
∂t a
2
+ Υ,ϕ (ϕ)ϕ + V,ϕϕ (ϕ) φq ,
˙
˜
ξq → generalized quantum noise term
˜
˜
ξq (x, t), ξq (x , t ) = 0 → classical behavior preserved
Equation of motion as a function of z = k/(aH):
1
η − βQ/(1 + Q)
δϕ (k, z) − (3Q + 2)δϕ (k, z) + 1 + 3
δϕ(k, z) =
z
z2
1
˜
ξT (k, z) + ξq (k, z) .
H 2z2

10. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky I: perturbative approach
Using the EoM solution, we define the inflaton power spectrum:
Pδϕ =
k3
2π 2
d3 k
(th)
(qu)
δϕ(k, z)δϕ(k , z) = Pδϕ (z) + Pδϕ (z)
(2π)3
2
3Q 2α 2ν−2α Γ (α) Γ (ν − 1) Γ (α − ν + 3/2)
√ 2 z
1
2 π
Γ ν − 2 Γ (α + ν − 1/2)
≈
HT
4π 2
+
H
coth
T
zH 2η
z
2T
,

11. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky I: perturbative approach
Using the EoM solution, we define the inflaton power spectrum:
Pδϕ =
k3
2π 2
d3 k
(th)
(qu)
δϕ(k, z)δϕ(k , z) = Pδϕ (z) + Pδϕ (z)
(2π)3
2
3Q 2α 2ν−2α Γ (α) Γ (ν − 1) Γ (α − ν + 3/2)
√ 2 z
1
2 π
Γ ν − 2 Γ (α + ν − 1/2)
≈
HT
4π 2
+
H
coth
T
zH 2η
z
2T
,
As expected, nearly scale invariant.
Alternative derivation of the enhancement term (Mohanty et al,
Phys. Rev. Lett. 97, 251301 (2006))

12. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky I: perturbative approach
Using the EoM solution, we define the inflaton power spectrum:
Pδϕ =
k3
2π 2
d3 k
(th)
(qu)
δϕ(k, z)δϕ(k , z) = Pδϕ (z) + Pδϕ (z)
(2π)3
2
3Q 2α 2ν−2α Γ (α) Γ (ν − 1) Γ (α − ν + 3/2)
√ 2 z
1
2 π
Γ ν − 2 Γ (α + ν − 1/2)
≈
HT
4π 2
+
H
coth
T
zH 2η
z
2T
,
Recovers all results of cold and warm inflation:
Q
1 and T
H ⇒ Pδϕ ∝ HT (Berera and Fang, Phys.
Rev. Lett. 74 (1995))
√
Q
1 and T
H ⇒ Pδϕ ∝ T HΥ (Hall, Moss and Berera,
Phys. Rev. D 69, 083525 (2004) )
Q
1 and T
H ⇒ Pδϕ ∝ H 2 (cold inflation)

13. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky I: perturbative approach
V (φ) =
4
λMpl
p
φ
Mpl
p
,
Υ(φ, T ) = Cφ
φ2a T c
,
m2b
X
c + 2a − 2b = 1
Figure: blue lines, Υ(φ), red Υ = cte. Dashed lines p = 2, full lines
p = 4, dotted lines p = 6

14. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Cosmological parameters:
Curvature perturbations:
H2
P = ∆2 (k0 )
R
˙ δϕ
φ2
8 H2
∆2 = 2
h
Mpl 4π 2
∆2 =
R
ns −1
k
k0
Spectral index (and running ns ):
ns − 1 =
d ln ∆2
R
d ln k
ns ≡
dns
d ln k
Tensor-to-scalar ratio:
r≡
∆2
4 H2
h
2 = (1 + Q)2 π 2 P
∆R
δϕ
Final remarks

15. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Cosmological parameters:
Some interesting limits:
spectral index:
Q → 0 and T → 0
ns = 1 + 2η − 6ε
r ≈ 16
1 and T
H ⇒
1
9
9
3
− ε − β + η + O(1/Q3/2 ) + O(1/(Q3/2 T 2 ))
ns = 1 +
Q
4
4
2
16 H
r≈ √
3πT Q5/2
(Hall, Moss and Berera, Phys. Rev. D 69, 083525 (2004) )
Q
Q
1 and T
H
ns = 1 + 2η − 6ε + (8ε − 2η)Q + O(Q2 )

16. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Cosmological parameters: WMAP-9yr (arXiv:1212.5226)
Figure: Green, eCMB, red, eCMB+BAO+H0 .Light colors → 95% CL,
dark colors → 68% CL.

17. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Cosmological parameters: Planck (arXiv:1303.5082)
Final remarks

18. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Cosmological parameters: WMAP-9yr (arXiv:1212.5226)

19. Outline
Warm inflation
Starobinsky I
Results: V ∝ φ2 68%CL
Observations
Starobinsky II
Final remarks

20. Outline
Warm inflation
Starobinsky I
Results: V ∝ φ4 95%CL
Observations
Starobinsky II
Final remarks

21. Outline
Warm inflation
Starobinsky I
Results: V ∝ φ6 95%CL
Observations
Starobinsky II
Final remarks

22. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Extending Starobinsky II: large scales
Similar prescription: Φ(x, t) = ϕ(x, t) + φq (x, t)
Large scales (
¨
H −1 ): ≈ homogeneous dynamics, Φ ≈ 0
Resulting equation of motion:
ϕ=−
˙
V,ϕ (ϕ)
H 3/2
+
3H(1 + Q)
2π
1+
2
eH/T
Two-point function:
ζ(t)ζ(t ) = δ(t − t ) .
−1
ζ(t)
Final remarks

23. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Extending Starobinsky II: large scales
Associated Fokker-Planck equation:
∂
∂
1 ∂2
P (ϕ, t) = −
D(1) P (ϕ, t) +
D(2) P (ϕ, t)
∂t
∂ϕ
2 ∂ϕ2
≡ LF P P (ϕ, t)
Drift and difusion coefficients:
V,ϕ (ϕ)
≡ −f (ϕ) ,
3H(1 + Q)
H3
2
= 2 1 + H/T
.
4π
e
−1
D(1) = −
D(2)
Final remarks

24. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky II: large scales
Eternal inflation:
Inflation doesn’t end globally
Qualitative condition to self-reproduction regime of H-regions:
f (ϕ)
H(ϕ)
D(2)
H(ϕ)

25. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky II: large scales
Global picture: physical probability distribution function, PV :
∂
∂
∂PV
=
−D(1) (ϕ)PV +
D(2) (ϕ)PV
∂t
∂ϕ
∂ϕ
PV (ϕ, t) ≡
PV (ϕ, t)
exp 3 dtH
+3 [H(ϕ) − H ] PV

26. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky II: large scales
Global picture: physical probability distribution function, PV :
∂
∂
∂PV
=
−D(1) (ϕ)PV +
D(2) (ϕ)PV
∂t
∂ϕ
∂ϕ
PV (ϕ, t) ≡
+3 [H(ϕ) − H ] PV
PV (ϕ, t)
exp 3 dtH
Dimensionless version:


∂
∂ 
x2n−1
PV (x, t ) = −
PV (x, t )
∂t
∂x xn 1 + γ
3xn
2
3n
∂
λ x
2
3 n
+
1 + xn /T
PV (x, t ) +
[x − xn ] PV (x, t ) ,
∂x2 12n2 8π 2
2n
e
−1
V (ϕ) =
4
λMp
2n
ϕ
Mp
2n

27. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky II: large scales
Transforming to a Schr¨dinger-like equation:
o
dx
σ→
D(2) (x)
1
∂
(2) (x) ∂σ
D
∂
→
∂x
Pn (x) →
1
1
exp
(2) (x)−3/4
2
D
dx
D(1) (x)
ψn (σ) ,
D(2) (x)
∂2
ψn (σ) − VS (σ)ψn (σ) = Λn ψn (σ) ,
∂σ 2
where
(2)
(2)
(2)
(1)
3 (D,x )2 D,xx D,x D(1) D,x
(D(1) )2
VS (σ) =
−
−
+
+
.
16 D(2)
4
2
2D(2)
4D(2)

28. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky II: large scales
Schr¨dinger-like equation ⇒ Sturm-Liouville problem:
o
L≡
d
d
p(x)
y + q(x)y = −λw(x)y
dx
dx
ca y(a) + da y (a) = 0
cb y(b) + db y (b) = 0 .
cn φn (x)
y(x) =
n
Lφn = −λn w(x)φn
λ0 < λ1 < λ2 . . .
−py
λ0 = miny(x)
dy
dx
b
b
a
+
a
b
a
dx p
dy
dx
dx y 2 w(x)
2
− qy 2
(Rayleigh quotient)

29. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Extending Starobinsky II: large scales
Eternal inflation case:
Cn ψn (σ)eΛn t
Ψ(σ, t) =
n
dσ
Λ0 = −minψ(σ)
dψn
dσ
2
2
+ VS (σ)ψn
2
dσ ψn
Λ0 > 0 ⇔ there is some interval σ1 < σ < σ2 so that VS < 0

30. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Extending Starobinsky II: large scales
Ex.: φ4 potential.
Cold limit:
n=2
VS
=−
3 −2 5
σ − + σ 2 − 3π
16
2
3 −1
σ
λ
Final remarks

31. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Extending Starobinsky II: large scales
High dissipation and temperature limit:
n=2
VS
=
Then, if λT <
288π 2
γ(γ+2)
72π 2
288π 2 − γλT (2 + γ) σ −2
(γλT )2
eternal self-reproduction is suppressed!
Final remarks

32. Outline
Warm inflation
Starobinsky I
Observations
Starobinsky II
Final remarks
Answers to the proposed questions:
Non-isentropic stochastic polynomial chaotic inflation model is
viable in face of the most recent results for the cosmological
parameters. (more details and results: Rudnei O. Ramos and L. A.
da Silva JCAP03(2013)032)
Temperature and dissipative effects can suppress the eternal
self-reproduction of H-regions.