Relativity & Gravitation
100 yrs after Einstein in Prague
26 June 2012
Misao Sasaki
Yukawa Institute for Theoretical Physics
Kyoto University
2
0. Horizon & flatness problem
ds 2  dt 2  a2 (t )d (23)
4 G

(   3P )a  0 for P  
3
3
if a  t n , n  1 gravity=attractive
ds  a ( )(d  d )
2
2
2
2
( 3)
& 
a&
dt
d 
: conformal time
a(t )
t
dt
 finite

a(t )
t0  0
conformal time is
bounded from below
E
• horizon problem
particle horizon
3
• solution to the horizon problem
4 G
& 
a&
(   3P )a  0
3
for a sufficient lapse of time in the early universe
t
dt 
  0  
t0  0 
a(t )
t0
or large enough to
cover the present
horizon size
NB: horizon problem≠
homogeneity & isotropy problem
4
• flatness problem (= entropy problem)
8 G
K
H 
  2 ;    K  
3
a
|K |
4
if   a ,  ?
in the early universe.
2
a
conversely if  | K |/ a 2 at an epoch in the early universe,
the universe must have either collapsed (if K  0)
or become completely empty (if K  0) by now.
2
alternatively, the problem is the existence of huge
entropy within the curvature radius of the universe
3
3
 a 
 a0 
3
S T 
 T0 
 T03 H 03  1087


 |K |
 |K |




3
(# of states = exp[S])
solution to horizon & flatness problems
spatially homogeneous scalar field:
1 &2
1 &2
    V ( ), P    V ( )
2
2
  3P  2 &2  V ( )  0 if &2  V ( )


   P  V ( ) if &2 = V ( )
potential dominated
V ~ cosmological const./vacuum energy
K
  const .
decreases rapidly
2
a
8 G
2
H 
  const . inflation
3
“vacuum energy” converted to radiation
after sufficient lapse of time
solves horizon & flatness problems simultaneously
5
6
1．Slow-roll inflation and vacuum fluctuations
• single-field slow-roll inflation
Linde ’82, ...
metric： ds 2  dt 2  a 2 (t ) ij dx i dx j
V()
field eq.：
& 3H& V ( )=0
&
V ( )
&
 
3H
8 G  1 &2
 a&

2
 a   H  3  2   V ( )
 


2

may be realized for various potentials
3 &2

&2
H&
3

 2 2

= 1 ∙∙∙ slow variation of H
1 &2
H
 V 2 V
2
a ~ e Ht
inflation!
7
comoving scale vs Hubble horizon radius
k: comoving wavenumber
log L
k
H
a
k
H
a
a
L
k
k
H
a
superhorizon
L  H 1
subhorizon
k
H
a
subhorizon
t=tend
inflation
k
H
a
log a(t)=N
hot bigbang
8
e-folding number: N
# of e-folds from = (t)
until the end of inflation
a(tend )
 exp[ N (t  tend )] 
a(t )
N  N ( )  
tend
t ( )
redshift
Hdt ~ ln 1  z( )
log L
N=N()
L=H-1 ~ t
L=H-1~ const
log a(t)
t=t()
t=tend
 determines comoving scale k
9
Curvature (scalar-type) Perturbation
 intuitive (non-rigorous) derivation
• inflaton fluctuation (vacuum fluctuations=Gaussian)
v
k
2
 k
2
1
k
 iwkt
, k ~
e
; wk  ? H
a
2wk
rapid expansion renders oscillations frozen at k/a < H
(fluctuations become “classical” on superhorizon scales)
k ~
H
2k 3
;
k
= H 
a
2
k2
H


2


 k / a~ H
• curvature perturbation on comoving slices
H
R c   & ∙∙∙ conserved on superhorizon scale,

for purely adiabatic pertns.
evaluated on ‘flat’ slice (vol of 3-metric unperturbed)
10
Curvature perturbation spectrum
2
H 
PR (k )  
~ almost scale-invariant
& k
2


 H
2
• spectrum
(rigorous/1st
• N - formula
N ( )  
tend
t ( )
a
principle derivation by Mukhanov ’85, MS ’86)
Starobinsky (’85)
Hdt  
end

H
d
&

 H 
 N 
 N ( )  
 
   & 
R
   k  H  
 k H
a
2
a
a
2
 H2 
 N 
2
PR (k )  


k
  
& k
2



 H 
c
k
H
a
;
k2
k3
2
H
 2 k  

2
2



2
N
A
geometrical justification  N   A  MS & Stewart (’96)
A 
non-linear extension
Lyth, Marik & MS (’04)
11
Tensor Perturbation
 i hijTT   ij hijTT  0 : transverse-traceless
Starobinsky (’79)
• canonically normalized tensor field
2
1  ij 
4
S ~  d x g 
  ggg
2  t 
1
M P TT
TT
ij 
hij 
hij ;
2
32 G
ij (k;t ) 


 ,
1
MP 
8 G
ak Pij (k )k (t )  h.c.
k (t ) : same as massless scalar
• tensor spectrum


hijTT
v
k ,
2
4
 2
MP


v
ij k ,
2
 2 4
k2
M P2
8H 2

( 2 M P )2
Tensor-to-scalar ratio
N
H2
• scalar spectrum:
Ps  k  
• tensor spectrum:
8 H2
Pg (k )  2
M P  2 2
 2 
2
N
2
2
 G ab ( )
N N
 a  b
 k ns  1
k
ng
2
&

2
&

2H&
• tensor spectral index: ng   2  2 2 
H
MP H
M P2 & N
g
dN
H 
  a a N
dt
r
Pg
Ps
 8 ng

1
M N
2
P
2

2
Pg
8 Ps
··· valid for all slow-roll models
with canonical kinetic term
12
Comparison with observation
Standard (single-field, slowroll) inflation predicts scaleinvariant Gaussian curvature perturbations.
WMAP 7yr
CMB (WMAP) is consistent with the prediction.
Linear perturbation theory seems to be valid.
13
14
CMB constraints on inflation
Komatsu et al. ‘10
scalar spectral index: ns = 0.95 ~ 0.98
tensor-to-scalar ratio: r < 0.15
15
However,….
Inflation may be non-standard
multi-field, non-slowroll, DBI, extra-dim’s, …
PLANCK, … may detect non-Gaussianity
(comoving) curvature perturbation:
3
2
 5?
R C  R gauss  f NL R gauss
 L ; f NL ~
5
B-mode (tensor) may or may not be detected.
2 
2
10-10 M Planck
?
energy scale of inflation H 
modified (quantum) gravity? NG signature?
Quantifying NL/NG effects is important
16
2. Origin of non-Gaussianity
• self-interactions of inflaton/non-trivial “vacuum”
quantum physics, subhorizon scale during inflation
• multi-field
classical physics, nonlinear coupling to gravity
superhorizon scale during and after inflation
• nonlinearity in gravity
classical general relativistic effect,
subhorizon scale after inflation
17
Origin of NG and cosmic scales
log L
k: comoving wavenumber
a
L
k
classical/local
effect
classical
gravity
L  H 1
quantum effect
t=tend
inflation
log a(t)
hot bigbang
18
Origin１：self-interaction/non-trivial vacuum
Non-Gaussianity generated on subhorizon scales
(quantum field theoretical)
• conventional self-interaction by potential is ineffective
Maldacena (’03)
ex. chaotic inflation
1 2 2
V  m  ∙∙∙ free field!
2
(grav. interaction is Planck-suppressed)
~ O(1/ M Pl2 )
V   4   ~ 1015
extremely small!
• need unconventional self-interaction
→ non-canonical kinetic term can generate large NG
19
1a. Non-canonical kinetic term: DBI inflation
Silverstein & Tong (2004)
kinetic term：
1 1
2
&

f

K ~ f ( ) 1  f ( )
1
~ (Lorenz factor)-1
perturbation expansion
1 3

2
&
   2   X ; X  f  


K  K 0  1K   2 K   3 K  
∝
∝
=
0
3
 3+2
 ~ 0   202  
large NG for large 
20
Bi-spectrum (3pt function) in DBI inflation
Alishahiha et al. (’04)
R C ( p1 )R C ( p2 )R C ( p3 )
~  ( p j ) f NL ( p1 , p2 , p3 )  R C ( p1 )R C ( p2 )  cyclic 
j
f NL ~ 
fNLlarge for equilateral
configuration
v
v
v
p1 ~ p2 ~ p3
2
v
p1
v
p2
v
p3
v v v
 p1  p2  p3  0 
equil
f NL  f NL
WMAP 7yr：
equil
241  f NL
 266 (95% CL)
21
1b. Non-trivial vacuum
• de Sitter spacetime = maximally symmetric SO(3,1)
(same degrees of sym as Poincare (Minkowski) sym)
gravitational interaction (G-int) is negligible in vacuum
(except for graviton/tensor-mode loops)
• slow-roll inflation : dS symmetry is slightly broken
G-int induces NG but suppressed by    H&/ M 2
Pl
But large NG is possible if the initial state (or state at
horizon crossing) does NOT respect dS symmetry
(eg, initial state ≠ Bunch-Davies vacuum)
various types of NG :
scale-dependent, oscillating, featured, folded ...
Chen et al. (’08), Flauger et al. (’10), ...
23
Origin 2：superhorizon generation
• NG may appear if T mn depends nonlinearly on  ,
even if  itself is Gaussian.
This effect is small in single-field slow-roll model
(⇔ linear approximation is valid to high accuracy)
Salopek & Bond (’90)
• For multi-field models, contribution to T mn from each field
can be highly nonlinear.
tot
NG is always of local type:
local
f NL ( p1 , p2 , p3 )  f NL
 const.
WMAP 7yr：
A = tot 
local
10  f NL
 74 (95% CL)
N formalism for this type of NG
x
24
Origin 3：nonlinearity in gravity
ex. post-Newtonian metric in asymptotically flat space




ds 2   1  2  2 2   dt 2  1  2  2 2   dr 2  
Newton
potential
NL (post-Newton) terms
(in both local and nonlocal forms)
• important when scales have re-entered Hubble horizon
distinguishable from NL matter dynamics?
• effect on CMB bispectrum may not be negligible
f NL ~ O(5) ?
Pitrou et al. (2010)
(for both squeezed and equilateral types)
25
3. N formalism
What is N?
• N is the perturbation in # of e-folds counted
backward in time from a fixed final time tf
therefore it is nonlocal in time by definition
• tf should be chosen such that the evolution of the
universe has become unique by that time.
=adiabatic limit
• N is equal to conserved NL comoving curvature
perturbation on superhorizon scales at t>tf
• N is valid independent of gravity theory
26
3 types of N
2
1
t  tf
end of/after
inflation
originally adiabatic
entropy/isocurvature → adiabatic
27
Nonlinear N - formula
Choose flat slice at t = t1 [ SF (t1) ] and
comoving (=uniform density) at t = t2 [ SC (t1) ] :
( ‘flat’ slice: S(t) on which det  (3)  a3 (t )e 3 R (t , x )  a3 (t ) )
SC(t2) : comoving
 (t2)=const.
SF(t2) : flat
SF (t1) : flat

N  t 2 , t1   N 0  t 2 , t1    N F t 2 , t1 ; x i
R(t2)=0
R(t1)=0

 N F  t 2 , t1 ; x i   R C  t 2 , x i 
28
• Nonlinear N for multi-component inflation :
 N  N  A   A   N  A 
1
n N
A1
A2



An
A1
A2
n
!






n
 A
n
where  =F is fluctuation on initial flat slice at or after
horizon-crossing.
F may contain non-Gaussianity from
subhorizon (quantum) interactions
eg, in DBI inflation
4．NG generation on superhorizon scales
two efficient mechanisms to convert
isocurvature to curvature perturbations:
curvaton-type & multi-brid type
• curvaton-type
Lyth & Wands (’01), Moroi & Takahashi (‘01),...
curv() << tot during inflation
highly nonlinear dep of curv on  possible
curv() can dominate after inflation
curvature perturbation can be highly NG
29
30
• multi-brid inflation
MS (’08), Naruko & MS (’08),...

sudden change/transition in the trajectory
1 2
 N   a N    ab N  a b L
2
curvature of this surface
determines sign of fNL
a
tensor-scalar ratio r may be large in multi-brid models,
while it is always small in curvaton-type if NG is large.
31
5. Summary
• inflation explains observed structure of the universe
flatness: W0=1 to good accuracy
curvature perturbation spectrum
almost scale-invariant
almost Gaussian
• inflation also predicts scale-invariant tensor spectrum
will be detected soon if tensor-scalar ratio r>0.1
any new/additional features?
32
non-Gaussianities
• 3 origins of NG in curvature perturbation
1. subhorizon ∙∙∙ quantum origin
2. superhorizon ∙∙∙ classical (local) origin
NG from
inflation
3. NL gravity ∙∙∙ late time classical dynamics
• DBI-type model: origin 1.
equil
f NL
may be large
need to be quantified
• non BD vacuum: origin 1.
any type of f NL may be large
• multi-field model: origin 2.
local
f NL
may be large:
In curvaton-type models r≪1.
Multi-brid model may give r~0.1.
33
Identifying properties of non-Gaussianity
is extremely important for understanding
physics of the early universe
not only bispectrum(3-pt function) but also
trispectrum or higher order n-pt functions
may become important.
Confirmation of primordial NG?
PLANCK (February 2013?) ...