7 January 2008
Taitung Winter School
N formalism for curvature
perturbations from inflation
Yukawa Institute (YITP)
2. Linear perturbation theory
• metric perturbation & time slicing
• N formalism
3. Nonlinear extension on superhorizon scales
• gradient expansion, conservation law
• local Friedmann equation
4. Nonlinear DN formula
• DN for slowroll inflation
• diagrammatic method for DN
Standard (single-field, slowroll) inflation predicts scale-
invariant Gaussian curvature perturbations.
CMB (WMAP) is consistent with the prediction.
Linear perturbation theory seems to be valid.
So, why bother doing more on theoretical models?
Because observational data does not exclude other models.
Tensor perturbations have not been detected yet.
T/S ~ 0.2 - 0.3? or smaller?
Inflation may not be so simple.
multi-field, non-slowroll, extra-dim’s, string theory…
future CMB experiments may detect non-Gaussianity
Y=Ygauss+ fNLY2gauss+ ∙∙∙ ; |fNL| ≳ 5?
Pre-bigbang, ekpyrotic, bouncing,....?
Need to know the dynamics on super-horizon scales
2. Linear perturbation theory
Bardeen ‘80, Mukhanov ‘81, Kodama & MS ‘84, ….
metric on a spatially flat background (g0j=0 for simplicity)
ds2 = 1 2 A dt 2 a2 t (1 2R) ij Hij dx i dx j
S(t+dt) H ij scalar = i j E
S(t) ij = transverse-traceless
xi = const.
• propertime along xi = const.: d = (1 A)dt
• curvature perturbation on S(t): R R= 2 DR
• expansion (Hubble parameter): H = H 1 A t R D E
Choice of time-slicing
• comoving slicing
T i = 0 = t for a scalar field
• uniform density slicing T 00 = t
• uniform Hubble slicing
H = H t
% H A t R D E = 0
geometrical slices 3
• flat slicing
R = 2 D R=0 R = 0
• Newton (shear-free) slicing
scalar 1 ( 3)
t H ij
i j ij D t E = 0 t E = 0 E = 0
comoving = uniform = uniform H on superhorizon scales
N formalism in linear theory MS & Stewart ’96
e-folding number perturbation between S(t) and S(tfin):
N t ;tfin
H d background
1 ( 3) dep only on
= t R D E dt = R D E ini and fin t
3 3 t
S (tfin), R(tfin)
N(t,tfin) N0 (t,tfin)
S (t), R(t)
N=O(k2) if both S(t) and S(tfin) are chosen to be ‘flat’ (R=0).
Choose S(t) = flat (R=0) and S(tfin) = comoving:
N t;tfin = R Ctfin on superhorizon scales
curvature perturbation on comoving slice
(suffix ‘C’ for comoving)
The gauge-invariant variable ‘z’ used in the literature
is related to RC as z = -RC or z = RC on superhorizon scales
By definition, Nt; tfin) is t-independent
Example: slow-roll inflation
• single-field inflation, no extra degree of freedom
RC becomes constant soon after horizon-crossing (t=th):
N th ;tfin = R Ctfin = R Cth
Also N = H(th) tF→C , where tF→C is the time difference
between the comoving and flat slices at t=th.
SC(th) : comoving =0, R=RC
SF(th) : flat
F th tFC , x i = C th F th tFC = 0
R C tfin = N t h ;tfin = F t h dN = Hdt
d / dt
= F t h ··· N formula
d Starobinsky ‘85
Only the knowledge of the background evolution
is necessary to calculate RC(tfin) .
• N for a multi-component scalar:
(for slowroll-type inflation)
R C tfin = N = a F th MS & Stewart ’96
N.B. RC (=z) is no longer constant in time:
RC t = H 2 ··· time varying even on
& superhorizon scales
H 2 th N
2 2 2 2
RC = N F = N a N a
Further extension to non-slowroll case is possible, if
general slow-roll condition is satisfied at horizon-crossing.
Lee, MS, Stewart, Tanaka & Yokoyama ‘05
= O , &= O , 2 & O , ..., = 1
3. Nonlinear extension
• On superhorizon scales, gradient expansion is valid:
Q Q HQ ; H G
Belinski et al. ’70, Tomita ’72, Salopek & Bond ’90, …
This is a consequence of causality:
light cone H-1
• At lowest order, no signal propagates in spatial directions.
Field equations reduce to ODE’s
metric on superhorizon scales
• gradient expansion:
i i , = expansion parameter
ds 2 = N 2dt 2 e 2 ij dx i i dt dx j j dt
det ij = 1,
% i = O
the only non-trivial assumption
contains GW (~ tensor) modes
t , x i = ln a t t , x i ; : curvature perturbation
e.g., choose (t* ,0) = 0
• Energy momentum tensor:
T = u u p g u u ; u T = 0
u p = 0 ; u = 3
assumption: v 0 = O
u – n = O ()
(absence of vorticity mode) u n
• Local Hubble parameter:
H% 1 n = 1 u O 2
n dx = N dt normal to t = const.
At leading order, local Hubble parameter on any slicing is
equivalent to expansion rate of matter flow.
So, hereafter, we redefine H to be H% 1 u
Local Friedmann equation
%2 (t , x i ) = 8 G (t , x i ) O( 2 ) % = t
xi : comoving (Lagrangean) coordinates.
3H p = 0
d = N dt : proper time along matter flow
exactly the same as the background equations.
uniform slice = uniform Hubble slice = comoving slice
as in the case of linear theory
no modifications/backreaction due to super-Hubble
perturbations. cf. Hirata & Seljak ‘05
Noh & Hwang ‘05
4. Nonlinear DN formula
(applicable to each independent matter component)
O = t = t = H N O 2
3 p a
N t 2 , t1 ; x H %N dt = 1 t2 t dt
3 t1 P xi
where xi=const. is a comoving worldline.
This definition applies to any choice of time-slicing.
t2 , x i t1 , x i = DN t2 ,t1 ; x i
DN t2 ,t1 ; x N t2 ,t1 ; x ln
DN - formula Lyth & Wands ‘03, Malik, Lyth & MS ‘04,
Lyth & Rodriguez ‘05, Langlois & Vernizzi ‘05
Let us take slicing such that S(t) is flat at t = t1 [ SF (t1) ]
and uniform density/uniform H/comoving at t = t2 [ SC (t1) ] :
( ‘flat’ slice: S (t) on which = 0 ↔ e = a(t) )
SC(t2) : uniform density (t2)=const.
N (t2,t1 ;xi)
SC(t1) : uniform density (t1)=const.
SF (t1) : flat
N (t 2 , t1 ; x i ) = N 0 (t 2 , t1 ) DN F
N 0 (t2 , t1 ) = ln between SC (t1 ) and SC (t2 )
DN F = t2 , x i t1 , x i = C t2 , x i
suffix C for comoving/uniform /uniform H
where DNF is equal to e-folding number from SF(t1) to SC(t1):
1 SC ( t 2 ) t 1 SC (t2 ) t
DN F = dt dt
3 SF (t1 ) P x i 3 SC (t1 ) P
1 SC (t1 ) t
3 SF (t1 ) P x i
For slow-roll inflation in linear theory, this reduces to
C (t2 ) R C (t2 ) = N t1 ;t2 = H t1 t F C = a F (t1 )
Conserved nonlinear curvature perturbation
Lyth & Wands ’03, Rigopoulos & Shellard ’03, ...
For adiabatic case (P=P() ,or single-field slow-roll inflation),
1 t2 t
N t 2 , t1 ; x =
3 t1 P ( )
1 ( t2 , x i ) d a(t2 )
= = (t2 , x ) (t1 , x ) ln
3 ( t1 , x i ) P ( )
1 (t ,x i ) d
NL ( x i ) (t , x i ) ···slice-independent
3 (t ) P ( )
Lyth, Malik & MS ‘04
non-linear generalization of ‘gauge’-invariant quantity z or Rc
• and can be evaluated on any time slice
• applicable to each decoupled matter component
Example: curvaton model Lyth & Wands ’02
Moroi & Takahashi ‘02
2-field model: inflaton () + curvaton ()
1 2 2 8 GV
V = V ( ) m m2
• During inflation dominates.
• After inflation, begins to dominate (if it does not decay).
= a-4 and a-3, hence / a
• final curvature pert amplitude depends on when decays.
• Before curvaton decay
1 (t , xi ) 1 (t , xi )
z = ln z = ln
3 (t )
3(z ) 4(z )
(t , xi ) (t , xi ) = e e
• On uniform total density slices, = z
3(z z ) 4(z z )
(t , x ) (t , x ) = e
( A PA )z A
nonlinear version of z =
• With sudden decay approx, final curvature pert amp z
is determined by
4(z z )
3(z z )
=1 MS, Valiviita & Wands ‘06
: density fraction of at the moment of its decay
DN for ‘slowroll’ inflation
MS & Tanaka ’98, Lyth & Rodriguez ‘05
• In slowroll inflation, all decaying mode solutions of the
(multi-component) inflaton field die out.
• If the value of determines H uniquely (such as in the
slowroll case) when the scale of our interest leaves the
horizon, N is only a function of , no matter how
complicated the subsequent evolution would be.
• Nonlinear DN for multi-component inflation :
DN = N A A N A
1 n N
= A1 A2 A
n n ! An
where =F (on flat slice) at horizon-crossing.
(F may contain non-gaussianity from subhorizon interactions)
cf. Weinberg ’05, ...
Diagrammatic method for nonlinear D N
Byrnes, Koyama, MS & Wands ‘07
N A1A2 Dn N
z = DN = A A A ;
1 2 n
A1 A2 An
‘basic’ 2-pt function: A ( x ) B ( y) = h AB ( ) G0 ( x y)
field space metric
is assumed to be Gaussian
for non-Gaussian , there will be basic n-pt functions
• connected n-pt function of z:
z ( x )z ( y) c = N A N AG0 ( x y) N AB N AB G0 ( x y)2
x y N ABC N ABC G0 ( x y)3 L
A A 3!
1 A A 1 A A
+ x y + x
y + ···
2! B B BC
z ( x )z ( y)z ( z ) c = N A N AB N BG0 ( x y)G0 ( y z ) perm.
N AB N BC N CAG0 ( x y)G0 ( y z )G0 ( z x )
N N ABC N BC G0 ( x y)2 G0 ( y z ) perm.
A A C
+ perm. + 2! x A C
y z 2! y z
B B B B
+ A z
+ perm. + ···
Superhorizon scale perturbations can never affect local
(horizon-size) dynamics, hence never cause backreaction.
nonlinearity on superhorizon scales are always local.
However, nonlocal nonlinearity (non-Gaussianity) may
appear due to quantum interactions on subhorizon scales.
cf. Weinberg ‘06
There exists a nonlinear generalization of N formula which
is useful in evaluating non-Gaussianity from inflation.
diagrammatic method can by systematically applied.