Transcript IR back reaction during inflation

Takahiro Tanaka (YITP, Kyoto univ.)
in collaboration with Yuko Urakawa （Barcelona univ.)
arXiv:1208.XXXX
PTP125:1067 arXiv:1009.2947,
Phys.Rev.D82:121301 arXiv:1007.0468
PTP122: 779 arXiv:0902.3209
PTP122:1207 arXiv:0904.4415
Various IR issues
IR divergence coming from k-integral
Secular growth in time ∝(Ht)n
Adiabatic perturbation,
which can be locally absorbed by the choice of time slicing.
Isocurvature perturbation
≈ field theory on a fixed curved background
Tensor perturbation
Background trajectory
in field space
isocurvature
perturbation
adiabatic
perturbation
IR problem for isocurvature perturbation
f : a minimally coupled scalar field with a small mass (m2≪H2) in dS.
f
2 reg

aH
0
2
H
 k 
3
d k 3 

k  aH 
2m2
3H 2
H4
 2
m
summing up only long wavelength modes beyond the Horizon scale
De Sitter inv. vac. state does not exist in the massless limit.
Allen & Folacci(1987)
Kirsten & Garriga(1993)
distribution
m2⇒0
Large vacuum fluctuation
potential
If the field fluctuation is too large, it is
easy to imagine that a naïve
perturbative analysis will break down
once interaction is introduced.
Let’s consider local average of f :
aH
f   d k fk e
3
ikx
0
(Starobinsky & Yokoyama (1994))
More and more short wavelength modes
participate in f as time goes on.
Equation of motion for f :
df
V f  f


2
dN
3H
H
Newly participating modes
act as random fluctuation
in slow roll approximation
fkfk  H 2 k 3
f N  f N   H 4 N  N 
In the case of massless lf4 : f
2
→
H2
l
Namely, in the end, thermal
equilibrium is realized : V ≈ T 4

Distant universe is quite different from ours.
Our observable
universe


Each small region in the above picture
gives one representation of many parallel universes.
However: wave function of the universe
= “a superposition of all the possible parallel universes”
must be so to keep translational invariance of the wave fn. of the universe

Question is “simple expectation values are really observables for us?”
“Are simple expectation values
really observables for us?”
Answer will be No!
 f 
Before
 f 
Correlated
After
Un-correlated
Cosmic expansion
Various interactions
|a＞ |b＞|c＞
Decoherence
f
f
Superposition of wave packets
   
  
Coarse graining
Unseen d.o.f.
  a  b  c   a  b  c  
Statistical ensemble
  a a  b b  c c 
Our classical observation
picks up one of the
decohered wave packets.
How can we evaluate the actual observables?
Setup: 4D Einstein gravity + minimally coupled scalar field
Broadening of averaged field can be absorbed by the
proper choice of time coordinate.
Factor coming from this loop:
  y   y    d 3k Pk   log aH / k min 
1 k3
curvature perturbation in
co-moving gauge.
scale invariant spectrum
- no typical mass scale
 ij  e2 2 exphij
f  0
Transverse
traceless
Yuko Urakawa and T.T., PTP122: 779 arXiv:0902.3209

In conventional cosmological perturbation theory,
gauge is not completely fixed.
Time slicing can be uniquely specified: f =0
but spatial coordinates are not.
h 0h
j
j
Residual gauge:
j
i,j
 g hij  i, j   j ,i

To solve the equation for  i, by
imposing boundary condition at
infinity, we need information about
un-observable region.
OK!
Elliptic-type differential
equation for  i.
 i  
Not unique locally!
observable
region
time
direction



The local spatial average of  can be set to 0 identically
by an appropriate gauge choice. Time-dependent scale
transformation.
Even if we choose such a local gauge, the evolution
equation for  stays hyperbolic. So, the interaction
vertices are localized inside the past light cone.
Therefore, IR divergence does not appear as long as we
compute  in this local gauge. But here we assumed
that the initial quantum state is free from IR divergence.

Local gauge conditions.
  
i
But unsatisfactory?
The results depend
on the choice of
boundary conditions.
No influence from outside
Translation
Complete gauge fixing
invariance is lost.
Imposing boundary
conditions on the boundary
of the observable region
☺

Genuine coordinate-independent quantities.
Correlation functions for 3-d scalar curvature on f =constant slice.
R(x1) R(x2)
Coordinates do not have gauge invariant meaning.
(Giddings & Sloth 1005.1056)
Use of geodesic coordinates:
(Byrnes et al. 1005.33307)
x
x(XA, l=1) =XA +  xA
XA
Specify the position by solving geodesic eq. D2 xi dl2  0
x origin
i
 Xi
with initial condition Dx dl
l 0
gR(X )
A
:= R(x(XA, l=1)) = R(XA) + xA R(XA) + …
gR(X1) gR(X2) should be truly coordinate independent.
In f =0 gauge, EOM is very simple

2
t

Only relevant terms in
the IR limit were kept.
 3   2  t  e2     0
Non-linearity is concentrated on this term.
Formal solution in IR limit can be obtained as
   I  2 I 1e2   I  
with
g
g
-1
d2
 2   2 log H
d
2
2 







3





e

being the formal inverse of
t
2
t




R  4e2    I   I 21e2    x   x  I  




R x1  R x2  ∋  I2  21e 2    x   x  I  x1   21e 2    x   x  I  x1 
g
IR divergent factor


IR regularity may require 21e2     x    I  0
IR regularity may require
2
1 2 
e

   x    I  0
However, -1 should be defined for each Fourier component.
 f t , x    d k e
1
3
ik  x
~
 f k t 
-1
k
for arbitrary function f (t,x)
with k  t2  3   2  t  e2  k 2
Then, 21e2   I   x   I  0 is impossible,
Because for I ≡  d 3k (eikx vk(t) ak + h.c.),
1e2   I  eik x ak while  x   I  ik  x eik x ak
Instead, one can impose
2
1 2 
e



   x     I   d 3 k ak Dk e ikx vk t   h.c.
d
k 3 / 2eif k 
with Dk  ,k 3 / 2e if k 
,
d log k
which reduces to conditions on the mode functions.
 2k 2k1e2  vk  Dk vk
・extension to the higher order:




2
1


1  2 
3
2 ikx
2

e


2



d
k
a
D
e vk t   h.c.


x


x


I
k
k



2


With this choice, IR divergence disappears.
g
R X 1  R X 2 
g
4 
 
2
I
  d log k 
IR divergent factor
2
log k
k
7
2 ik ( X1  X 2 )
vk e
total derivative

In addition to considering gR, we need additional conditions
 2k 2k1e2  vk  Dk vk
and its higher order extension.
What is the physical meaning of these conditions?
~~
s
Background gauge: ~
x  e x  x    x
ds2  dt2  e2  dx 2
d~
s 2  dt2  e2   2 s d~
x2



~
~
~
H  H0    Hint  
H  H0   Hint   s
•Quadratic part in ~ and s is identical to s = 0 case.
•Interaction Hamiltonian is obtained just by replacing
~
the argument  with   s.
Therefore, one can use
~
1) common mode functions for I and I
I ≡ 
d 3k
(eikx v
k(t) ak
+ h.c.)
2) common iteration scheme.
   I    I 
~
~
~
I ≡  d 3k (eikx vk(t) a~k + h.c.)
~ 
   I    I  s
We may require
~~~ ~~
~~ ~
0   x1   x2   xn  0  0   x1   x2   xn  0
 2k 2k1e2  vk  Dk vk
the previous condition compatible with Fourier
decomposition
Retarded integral with (h0)=I(h0) guarantees the commutation relation of 
Dkvk(h0)=0 : incompatible with the normalization condition.
It looks quite non-trivial to find consistent IR regular states.
However, the Euclidean vacuum state (h0 →±i ∞ ) (should)
satisfies this condition. (Proof will be given in our new paper
(still incomplete!))
We obtained the conditions for the absence of IR
divergences.
“Wave function must be homogeneous in the
direction of background scale transformation”
Euclidean vacuum and its excited states (should)
satisfy the IR regular condition.
It requires further investigation whether there are
other (non-trivial and natural) quantum states
compatible with the IR regularity.