Relativity & Gravitation 100 yrs after Einstein in Prague 26 June 2012. Inflation and Birth of. Cosmological Perturbations. Misao Sasaki. Yukawa Institute for Theoretical Physics Kyoto University. 0. Horizon & flatness problem. horizon problem. gravity=attractive. conformal time is
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100 yrs after Einstein in Prague
26 June 2012
Inflation and Birth of
Yukawa Institute for Theoretical Physics
conformal time is
bounded from below
for a sufficient lapse of time in the early universe
or large enough to
cover the present
NB: horizon problem≠
homogeneity & isotropy problem
alternatively, the problem is the existence of huge
entropy within the curvature radius of the universe
(# of states = exp[S])
spatially homogeneous scalar field:
V ~ cosmological const./vacuum energy
“vacuum energy” converted to radiation
after sufficient lapse of time
solves horizon & flatness problems simultaneously
Linde ’82, ...
may be realized for various potentials
∙∙∙ slow variation of H
k: comoving wavenumber
fdetermines comoving scale ke-folding number: N
# of e-folds from f=f(t)
until the end of inflation
L=H-1 ~ t
rapid expansion renders oscillations frozen at k/a < H
(fluctuations become “classical” on superhorizon scales)
∙∙∙ conserved on superhorizon scale,
for purely adiabatic pertns.
evaluated on ‘flat’ slice (vol of 3-metric unperturbed)
~ almost scale-invariant
(rigorous/1st principle derivation by Mukhanov ’85, MS ’86)
MS & Stewart (’96)
Lyth, Marik & MS (’04)
··· valid for all slow-roll models
with canonical kinetic term
Komatsu et al. ‘10
Inflation may be non-standard
multi-field, non-slowroll, DBI, extra-dim’s, …
PLANCK, …may detect non-Gaussianity
(comoving) curvature perturbation:
B-mode (tensor)may or may not be detected.
energy scale of inflation
modified (quantum) gravity? NG signature?
Quantifying NL/NG effects is important
quantum physics, subhorizon scaleduring inflation
classical physics, nonlinear coupling to gravity
superhorizon scaleduring and after inflation
classical general relativistic effect,
subhorizon scaleafter inflation
k: comoving wavenumber
Non-Gaussianity generated on subhorizon scales
(quantum field theoretical)
ex. chaotic inflation
∙∙∙ free field!
(grav. interaction is Planck-suppressed)
g 31a. Non-canonical kinetic term: DBI inflation
Silverstein & Tong (2004)
~ (Lorenz factor)-1
large NG for large g
fNLlarge for equilateral configurationBi-spectrum (3pt function) in DBI inflation
Alishahiha et al. (’04)
(same degrees of sym as Poincare (Minkowski) sym)
gravitational interaction (G-int) is negligible in vacuum
(except for graviton/tensor-mode loops)
G-int induces NG but suppressed by
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), ...
rtotOrigin 2：superhorizon generation
This effect is smallinsingle-field slow-roll model
(⇔ linear approximation is valid to high accuracy)
Salopek & Bond (’90)
NG is always of local type:
dN formalism for this type of NG
ex. post-Newtonian metric in asymptotically flat space
NL (post-Newton) terms
(in both local and nonlocal forms)
distinguishable from NL matter dynamics?
Pitrou et al. (2010)
(for both squeezed and equilateral types)
What is dN?
therefore it is nonlocal in time by definition
f13 types of dN
entropy/isocurvature → adiabatic
Chooseflat slice at t = t1 [ SF (t1) ] and
comoving (=uniform density) at t = t2 [ SC (t1) ] :
NonlineardN - formula
( ‘flat’ slice: S(t) on which )
SC(t2) : comoving
SF(t2) : flat
SF (t1) : flat
wheredf =dfFis fluctuation on initial flat slice at or after horizon-crossing.
dfF may contain non-Gaussianity from
subhorizon (quantum) interactions
eg, in DBI inflation
two efficient mechanisms to convert
isocurvature to curvature perturbations:
curvaton-type & multi-brid type
Lyth & Wands (’01), Moroi & Takahashi (‘01),...
rcurv(f) << rtot during inflation
highly nonlinear dep of drcurv on df possible
rcurv(f) can dominate after inflation
curvature perturbation can be highly NG
MS (’08), Naruko & MS (’08),...
sudden change/transition in the trajectory
curvature of this surface
determines sign of fNL
tensor-scalar ratio r may be large in multi-brid models,
while it is always small in curvaton-type if NG is large.
flatness:W0=1 to good accuracy
curvature perturbation spectrum
will be detected soon if tensor-scalar ratio r>0.1
any new/additional features?
1. subhorizon ∙∙∙ quantum origin
2. superhorizon ∙∙∙ classical (local) origin
3. NL gravity ∙∙∙ late time classical dynamics
need to be quantified
may be large
any type ofmay be large
may be large:
In curvaton-type modelsr≪1.
Multi-brid model may giver~0.1.
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?) ...