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
About Yukawa Institute for Theoretical Physics
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BIEP,
Post docs (JSPS, YITP)
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
§Non-Gaussianity
 CMB non-Gaussianity might be measurable!
Non-linear dynamics
 Once we take into account interaction, we may
feel it unjustified to neglect loop corrections,

Factor coming from this loop:
  y   y    d 3k Pk   log aH / k min 
 1 k 3 for scale invariant spectrum
Prediction looks depending on the IR cutoff scale introduced
by hand!
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
 : a minimally coupled scalar field with a small mass (m2≪H2) in dS.

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  :
aH
   d k k e
3
ikx
0
(Starobinsky & Yokoyama (1994))
More and more short wavelength modes
participate in  as time goes on.
Equation of motion for  :
d
V   f


2
dN
3H
H
in slow roll approximation
Newly participating modes
act as random fluctuation
kk  H 2 k 3
f N  f N   H 4 N  N 
In the case of massless l4 : 
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!
 
Before
 
Correlated
After
Un-correlated
Cosmic expansion
Various interactions
|a＞ |b＞|c＞
Decoherence


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?
(Urakawa & Tanaka PTP122:1207)

Discussing quantum decoherence is annoying.



Which d.o.f. to coarse-grain?
What is the criterion of classicality?
To avoid subtle issues about decoherence,
we propose to introduce a “projection operator”.
 
Picking up one history is difficult.
Instead, we throw away the other histories
presumably uncorrelated with ours.
over-estimate of fluctuations
We compute
PO

P
 2 

with P  exp  
2 
 2 
Projection acts only on the external lines.
How the contribution from the IR modes at k ≈ kmin is suppressed?
x
y
x'
  d 4 y G x, y G x' , y G  y, y 
One of Green fns. is retarded, GR.
<int(y)int(y)>
∵ Expansion in terms of interaction-picture fields:
 x   int x    d 4 yl4 point  y GR x, y int  y 3  
x’
GR

<int(y)int(y)>
GR
x
y
<int(x’)int(y)>
Integration over the vertex y is restricted to the region within the past light-cone.
time
Window fn.
For each hy, IR fluctuation of int (y) is
suppressed since  hfin is restricted.
OK to any order of loop expansion!
~ secular growth in time
Window fn.
Past light cone during inflation
shrinks down to horizon size.
1
2
2
2
dsde


d
h

dr

Sitter
2
 Hh 

time
h ≈ Dh D r :past light cone
Dr
1
Rlight cone 

 Hh
H

However, for hy→  ∞, the suppression due to constraint on  gets weaker.
Then, G( y , y) ≈ <int(y)int(y)> becomes large.

y

  d 4 x GR x, y G x' , y G  y, y 

GR (x , y) →constant for hy→ ∞
hy-integral looks divergent, but
homogeneous part of  is constrained by the projection.
xGR(x , y ) → 0 faster than GR(x , y ) for hy→ ∞
looks OK, at least, at one-loop level !
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
  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:  =0
but spatial coordinates are not.
h 0h
j
j
j
i,j
Residual gauge:
 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.
D 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.
D  
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  =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. D 2 x i 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  =0 gauge, EOM is very simple

2
t

Only relevant terms in
the IR limit were kept.
 3   2   t  e 2   D   0
Non-linearity is concentrated on this term.
Formal solution in IR limit can be obtained as
   I  2 I 1e 2  D I  
with
g
g
-1
d2
 2   2 log H
d
2
2 







3





e
D
being the formal inverse of
t
2
t




R  4e 2  D  I   I 21e 2  D  x   x  I  




R x1  R x2  ∋  I2 D 21e 2  D  x   x  I  x1  D 21e 2  D  x   x  I  x1 
g
IR divergent factor


IR regularity may require 21e 2  D   x    I  0
IR regularity may require
2
1 2 
e

D   x    I  0
However, -1 should be defined for each Fourier component.
1
3
ik  x
-1 ~
 f t , x    d k e k f k t  for arbitrary function f (t,x)
with k   t2  3   2   t  e 2  k 2
Then, 21e2  D I   x   I  0 is impossible,
Because for I ≡  d 3k (eikx vk(t) ak + h.c.),
1e 2  D I  eik  x ak while  x    I  ik  x e ik  x ak
Instead, one can impose
2
1 2 
e



D   x     I   d 3k ak Dk eikx vk t   h.c.
d
k 3 / 2 ei k 
with Dk  ,k 3 / 2e i 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
D

2



d
k
a
D
e vk t   h.c.


x


x


I
k
k



2

With this choice, IR divergence disappears.
g
R X1  R X 2 
g
4 
 
2
I
  d log k 
IR divergent factor
2
logk
k
7
2 ik ( X  X )
1
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    x

x e x
ds 2  dt 2  e 2  dx 2
d~
s 2  dt 2  e 2   2 s d~
x2

H  H 0    H int  


~
~
~
H  H 0   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!))
In the squeezed limit, 3pt fn is given by power and index of spectrum .
 k  k  k   k1  k 2  k3 ns  1Pk Pk for k1 << k2, k3
1
2
3
1
2
Super-horizon long wavelength mode k1 should be irrelevant for the short
wavelength modes k2, k3 .
The only possible effect of k1 mode is to
modify the proper wave numbers corresponding to k2, k3 .
Let’s consider  in geodesic normal coordinates X ~ ek1x
g
 k   d Xe
3
a
  ka
ik a  X g
 X    e
3 k1 / 2
3
d xe
ik a e
 k1
x
  x    
k1
3

  k1  k a   ka   ka   , neglecting tensor modes
2

<k1 gk2 gk3> ≈ 0
 k  k  k   Pk  logk   logk  3  k  k
1
2
3
1
2
3
2
3
∵ gk2 and gk3 are not correlated with the long-wavelength mode k1.
This derivation already indicates that the leading term in the squeezed limit given by the
consistency relation vanishes once we consider “genuine coordinate independent quantities”.
If we consider a state:
<k1 gk2 gk3> ≈ 0
IR divergences!
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.
<k1 gk2 gk3> ≈ 0
In the squeezed limit, 3pt fn vanishes,
(1) for k1 << (aL)-1 << k2, k3
or
(2) for (aL)-1 << k1 << k2, k3
?
size of our observable universe
In the case (1), Fourier mode with such small k1 cannot be resolved!
But approximate expression for geodesic normal coordinates
X ~ ek1 x is valid only for the case (1).
For extension to the case (2), we have to solve the geodesic equation:


d 2 xi
dx j dx k
i
i
2 i
ik  X







i
2
(
k

X
)
X

X
k

k
e
jk
1
1
1
dl2
dl dl
Although it’s too technical to explain it here,
even if we include the above corrections to
the relation between, x and X,
<k1 gk2 gk3> ≈ 0 holds in the squeezed limit.

2-point function of the usual curvature perturbation
is divergent even at the tree level.
 (X1) (X2)2 I (X1) I(X2)2 d(logk)k3uk(X1) u*k (X2) c.c.
where
 I  u k ak + u*k a†k
uk k3/2(1-ik/aH)eik/aH for Bunch Davies vacuum
d(logk)k3uk(X1) u*k (X2) d(logk) Logarithmically divergent!


Of course, artificial IR cutoff removes IR divergence
d(log k)k3uk(X1) u*k (X2) d(log k)Pk but very artificial!
Why there remains IR divergence even in BD vacuum?
 is not gauge invariant, but gR(X) ≈R(X) is.
gR(X1) gR(X2)2 ≈ d(logk)k3Duk(X1) Du*k (X2)  d(log k) k4
Local gauge-invariant quantities do not diverge for the
Bunch-Davies vacuum state.

◎flat gauge  synchronous gauge
No interaction term in the evolution
equation at O(0) in flat gauge.
◎R(XA) ~ e-2 D
◎R  gR
gR(X1) gR(X2)4 gR3(X1) gR1(X2)  gR2(X1) gR2(X2)  gR1(X1) gR3(X2)
I 2 d(logk)k3D(D2uk(X1))D(u*k (X2)) + 2D(Duk(X1))D(Du*k
(X2))
∝
+D(uk(X1))D(D2u*k (X2)) + c.c.
where
pieces)
I  uk +ak(manifestly
+ u*k a†k & finite
D : loga  (x  )

IR divergence from I 2, in general.
However, the integral vanishes for the Bunch-Davies vacuum state.
∵ uk k3/2(1-ik/aH)eik/aH
Duk k3/2 logk (k3/2uk)
2
gR(X1) gR(X2)4 I 2d(logk) logk
D(k3/2uk(X1))D(k3/2u*k (X2))  + c.c.

To remove IR divergence, the positive frequency function corresponding
to the vacuum state is required to satisfy Duk k3/2 logk (k3/2uk) .
(YU and TT, in preparation)
g


H 
 2 H
i
R2   I D 1  2  loga 
 2  x i   I

H H
 H 

At the lowest order in , Duk  (log a xii ) uk k3/2 logk (k3/2uk) was requested.
Some extension of this relation to O( ) is necessary:
Natural extension is


H 
 2 H
i
3 / 2
3/ 2


1





x

u

k

k
uk

i k
log k
2  log a
2

H H
 H 

should have the same coefficient
uk , uk    k  k
uk  eikx
2
H
eikx f k / aH 
3/ 2 
k 
  2 H  
k2
 2 2  3
 2   f  0 consistent with the

a H
above requirement !!
 H H  
uk 
EOM for f : 
2
  log
 aH  3 logaH

Notice that
 2

 
H 

R X 1  R X 2      d log k   logk  
 2 2  logk 

H 
 H


3/ 2
 Dk uk  X 1 Dk 3 / 2uk  X 2 
IR divergence can be removed by an appropriate choice of the initial
vacuum even if we consider the next leading order of slow roll.
g
g
4 
2
I

