Phenomenological Implications of an Alternative

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Transcript Phenomenological Implications of an Alternative

Inhomogeneities in Loop Cosmology
Mikhail Kagan
Institute for Gravitational Physics and Geometry,
Pennsylvania State University
in collaboration with
M. Bojowald, P. Singh
(IGPG, Penn State)
H.H.Hernandez, A. Skirzewski
(Max-Planck-Institut für Gravitationsphysik,
Albert-Einstein-Institut, Potsdam, Germany)
Tuesday, July 21, 2015
1
Outline
1. Motivation
2. Classical description
3. Canonical formulation
a) Quantization
b) Correction functions
c) Effective Equations
4. Implications
5. Summary
2
Motivation.
Test robustness of results of homogeneous and isotropic
Loop Quantum Cosmology.
Evolution of inhomogeneities is expected to explain
cosmological structure formation and lead observable
results.
3
Lagrangean Formulation. Background metric.
Action
S  Sgr [ gab ]  Sm[ ]
Matter
Gravity
S gr [ gab ]   21  d 4 x  g R
Sm [ ]    d 4 x  g 12 g abab  V ( ) 
g  det(gab ),   8G
2
ds  a ()(d  d r )
2
Friedman equation
Klein-Gordon equation
Raychaudhuri equation
2
2
   2

  a 2V ( ) 
3 2

  2 H  a 2V, ( )  0
H 
2

H    2  a 2V ( ) 
3
4
Lagrangean Formulation. Perturbations.
2
ds  a ()((1  2 )d  (1  2 )d r )
2
Einstein Equations
2
2
2  3H  3H 2  

2
  3H  2 H   H 2 
 a   H   

2
a 2T00

2
a 2Taa
a 2Ta0
Klein-Gordon Equation
  2H  2  a2V, ( )  2a2V, ( )  4  0
5
Canonical Formulation. Basic variables.
Poisson brackets
Matter
Scalar field
Field momentum
     ( x)
     ( y )
1
 ( x), ( y ) 
V0
 ( x), ( y)  ( x, y)
Gravity
(densitized) Triad
Ashtekar connection
Spin connection
Extrinsic curvature
average quantities
Eia  p( x ) ia
Aai  ai ( x )  Kai ( x )
1  p( x )
ai   aij j
2
p
Kai  k  k ( x)  ai
X :
1
3
X
(
x
)
d
x
V0 
A ( x), E ( y)    ( x, y)
i
a
b
j
b
a
k ( x), p( y)

i
j

3V0
k ( x ), p( y )   ( x, y )
3
  Immirzi parameter
6
Canonical Formulation. Constraints.
Hamiltonian
Diffeomorphism
(vector)
Gravity
HG [ N ] 
1
N
3

d
x
 ijk Fabi E aj Ekb  21   2 Kai Kbj Ei[ a E bj] 

2
| det E |
DG [ N a ] 
1
d
 
3
xN a Fabi Eib
Matter
a b
1 2

1 Ei E j  a  b

H [ N ]   d xN 

 | det E |V ( ) 
| det E |
 2 | det E | 2

3
Total
H [ N ]  HG [ N ]  H [ N ]
D [ N a ]   d 3 xN a a
D[ N a ]  DG [ N a ]  D [ N a ]
7
Canonical Formulation. Classical EoM.
Constraint equations
H [ N ]
0
 (N )
H [ N ]
0
N
BG Friedmann
D[ N a ]
0
a
 (N )
Pert S-T Einstein
Pert Friedmann
Dynamical equations
k  {k , H }

 p  { p, H }
k  {k , H }

p  {p, H }
  { , H }

  { , H }
  { , H }

  { , H }
BG Raychaudhuri
Pert Raychaudhuri
BG K-G
Pert K-G
with identification
 
p
2p
N p
N 
p
2p
p  a2
8
Canonical Formulation. Constraints.
Hamiltonian
Diffeomorphism
(vector)
Gravity
HG [ N ] 
1
N
3

d
x
 ijk Fabi E aj Ekb  21   2 Kai Kbj Ei[ a E bj] 

2
| det E |
DG [ N a ] 
1
d
 
3
xN a Fabi Eib
Matter
a b
1 2

1 Ei E j  a  b

H [ N ]   d xN 

 | det E |V ( ) 
| det E |
 2 | det E | 2

3
Total
H [ N ]  HG [ N ]  H [ N ]
D [ N a ]   d 3 xN a a
D[ N a ]  DG [ N a ]  D [ N a ]
9
Quantization. Correction functions.
Sources of corrections:
inverse powers of triad
Modified constraints:
1
Na
3
i
2  j k  K j K k E [ a E b ]
b



d
x




b
ijk a
b
a b
a b
i
j
2 
| det E |
a b
2


E

s
i E j  a  b
3
D
H [ N ]   d xN 

 | det E |V ( ) 
| det E |
 2 | det E | 2

HG [ N ] 
Typical behavior
of
correction functions:
D
10
Quantization. Effective EoM.
Pert Friedmann


ab 2  3 H  3 H 2 1  a'p     pT00
a
a
a 
2

Pert S-T Einstein

 2a'p  

    pTa0
 a   H 1 
a 
2



Pert Raychaudhuri
Pert K_G
 2a'p 
 a'p  1 2
  2 H  1 
    ab b  1  4a'pb 
  3H 1 
a
3
a



 3
2

2


5
a
'p
2a''p
2a'p  a 
2 
 
 H  1

 
pTaa
a
a

 a   2


  2 H 1 


D'p 
D'p
  Ds 2  D pV,  21 
D pV,
D
D 


2


2




D'p
D'p
D'p
D''p



  4 H
   0
 4  1 

 
D
2D 
D

 D  

classically
0,
1
11
Implications. Newton’s potential.
Pert Friedmann
Pert S-T Einstein
Corrected Poisson Equation
2  k2 

2ab


3
p   H   P 
a


Length Scale
_
1
1 a3b
1
2 _ n
as
a
(p)~1+c(l


,
P/p) , (c, n>0)
2
2
2
k H 3a'p
H
_
_
so |a'p|=n(a -1)~(lP2/p)n
Green’s Function
 (r) 

2r
Within one Hubble Radius
_

~ k  a'p
N H
exp  i kr    N  
classically
0,
1
12
Implications. Power spectrum.
BG, Pert Raychaudhuri
BG, Pert Friedmann
(P = w
  31  w 1H  w 2 2 3 H 2  0
_
where 3  2ap2/a < 0
Large-scale Fourier Modes
  1  v  /  3 / 2  0
Two Classical Modes
decaying ( < 0) const (_=0)
 ( )    , with   

v
1  1  43 / v
2

With Quantum Corrections
decaying ( < 0)
growing (_ 3/n  0)
_ mode describes measure of inhomogeneity)
classically
0,
1
13
Summary.
1. Formalism for canonical treatment of inhomogeneities.
2. Now correction functions depend on p(x).
3. Effective equations for cosmological perturbations.
4. Quantum corrections arise on large scales:
a) Newton’s potential is modified by a factor smaller than one,
which can be interpreted as small repulsive quantum contribution.
b) Cosmological modes evolve differently,
resulting in non-conservation of curvature perturbations.
5. Results can be generalized to describe vector & tensor modes.
14