δ N formalsm for curvaure perurbaons from nflaon Msao Sasak Yukawa Insue (YITP) Kyoo Unversy
. Inroducon. Lnear perurbaon heory merc perurbaon & me slcng δn N formalsm 3. Nonlnear exenson on superhorzon scales graden expanson, conservaon law local Fredmann equaon 4. Nonlnear ΔN N formula ΔN N for slowroll nflaon dagrammac mehod for ΔN IR dvergence ssue 5. Summary
. Inroducon Sandard (sngle-feld, slowroll) nflaon predcs scale- nvaran Gaussan curvaure perurbaons. CMB (WMAP) s conssen wh he predcon. Lnear perurbaon heory seems o be vald.
So, why boher dong more research on nflaon? Because observaonal daa does no exclude oher models. Tensor perurbaons have no been deeced ye. T/S ~ 0. - 0.3? or smaller? In fac, nflaon may no be so smple. mul-feld, non-slowroll, exra-dm dm s, srng heory PLANCK, CMBpol, may deec non-gaussany Ψ=Ψ gauss gauss + f NL NL Ψ gauss gauss + ; f NL 5? Nonlnear backreacon on superhorzon scales? Re-consder he dynamcs on super-horzon scales
. Lnear perurbaon heory Σ(+d) Σ() = + + + R + ( A) () j ds d a ( ) δj Hj dx dx x = cons. dτ properme along x = cons.: dτ = ( + A) d curvaure perurbaon on Σ(): Ρ expanson (Hubble parameer): Bardeen 80, Mukhanov 8, Kodama & MS 84,. merc on a spaally fla background ( H ) ( H ) scalar ensor 4 = ΔR a (3) (3) expanson (Hubble parameer): H% H ( A) j j (g 0j =0 for smplcy) R = = j E ransverse-raceless (3) = + R + Δ E 3
Choce of me-slcng comovng slcng maer-based slces unform densy slcng μ comovng slcng T = φ φ( ) unform densy slcng T 0 ρ = ρ( ) ( = for a scalar ) 0 feld 0 unform Hubble slcng geomercal slces fla slcng (3) Newon (shear-free) slcng (3) H% = H( ) HA+ R + Δ E 3 = (3) 4 R = ΔR=0 R = 0 a scalar ( 3) H j 0 0 raceless j δj Δ E = E = 3 comovng = unform ρ = unform H on superhorzon scales E 0 = 0
δn N formalsm n lnear heory MS & Sewar 96 e-foldng number perurbaon beween Σ() ) and Σ( fn ): ( ) ( ) ; fn Hd % Hd fn fn δ N τ τ background (3) E d R = 3 R R ( ) ( ) ( ) fn O ε fn = + Δ + Σ 0 ( fn ) Σ 0 () δn(, fn ) x =cons. N 0 (, fn ) Σ ( fn ), Ρ( fn ) Σ (), Ρ() δn=0 f boh Σ() ) and Σ( fn ) are chosen o be fla (Ρ=0).
Choose Σ() ) = fla (Ρ=0)( and Σ( fn ) = comovng: Σ( fn ), Ρ C ( fn ) δ N ( ; ) = R ( ) fn C fn x =cons. Σ(), Ρ()=0 curvaure perurbaon on comovng slce (suffx C for comovng) The gauge-nvaran varable ζ used n he leraure s relaed o Ρ C as ζ = -Ρ C or ζ = Ρ C By defnon, δn(; fn ) s -ndependen
Example: slow-roll nflaon sngle-feld nflaon, no exra degree of freedom Ρ C becomes consan soon afer horzon-crossng (=( h ): log L ( ; ) R ( ) R ( ) δ N = = h fn C fn C h Ρ C = cons. L=H - = h nflaon = fn log a
Also δn = H( h ) δ F C, where δ F C s he me dfference beween he comovng and fla slces a = h. δ F C Σ C ( h ) : comovng Σ F ( h ) : fla (, x ) ( ) φ δ φ + = ( ) F h F C C h δφ=0, Ρ=Ρ C δφ & φ δ Ρ=0, δφ=δφ F F + h F C = 0 R H dφ / d ( ) = δn( ; ) = δφ ( ) C fn h fn F h = dn F dφ δφ ( ) h δn formula dn = Hd Sarobnsky 85 Only he knowledge of he background evoluon s necessary o calculae Ρ C ( fn ).
for a mul-componen scalar: δ N for a (for slowroll nflaon) N a R = δn = δφ a φ ( ) ( ) C fn F h a MS & Sewar 96 N.B. Ρ C (=ζ) ) s no longer consan n me: & F C ( ) H φ R = δφ me varyng even on & φ superhorzon scales R ( ) = N δφ = N C fn F Furher exenson o non-slowroll case s possble, f general slow-roll condon s sasfed a horzon-crossng. Lee, MS, Sewar, Tanaka & Yokoyama 05 & φ O( ), O( ), O( ),..., H = φ& φ ξ ξ ξ ξ H = & & H & = = φ φ ( h ) ( π ) H a N N a φ
3. Nonlnear exenson On superhorzon scales, graden expanson s vald: x Q = Q : HQ; H : Gρ Belnsk e al. 70, Toma 7, Salopek & Bond 90, Ths s a consequence of causaly: L»H - lgh cone H - A lowes order, no sgnal propagaes n spaal drecons. Feld equaons reduce o ODE s
merc on superhorzon scales graden expanson: ε, ε = expanson parameer merc: α j de %, ( )( j j + β + β ) ds = N d + e % γ dx d dx d ( ) γ j = β = O ε he only non-rval assumpon conans GW (~ ensor) modes (, ) ( ) ( x = ln a +, x ); : curvaure perurbaon α ψ ψ fducal `background e.g., choose ψ ( *,0) = 0
Energy momenum ensor: ( ) μν μ ν μν μ ν μν T = ρu u + p g + u u ; u T = d μ α ρ+ μu + = = + dτ N u assumpon: v = O 0 ( ε ) u u μ n μ = O(ε) (absence of vorcy mode) Local Hubble parameer: μ μ H% μn = μu + O 3 3 n μ dx μ μ μ ( ) 0 ; ( ρ p u 3 O ) μ ε = Nd ( ε ) ν 0 normal o = cons. A leadng order, local Hubble parameer on any slcng s equvalen o expanson rae of maer flow. ~ So, hereafer, we redefne H o be H% u μ μ 3 u μ n μ =cons.
Local Fredmann equaon 8 G H% (, x ) π = ρ (, x ) + O( ε ) 3 d 3H( p) 0 d ρ + % τ ρ + = x : comovng (Lagrangean) coordnaes. dτ = d : proper me along maer flow exacly he same as he background equaons. separae unverse unform ρ slce = unform Hubble slce = comovng slce as n he case of lnear heory no modfcaons/backreacon due o super-hubble perurbaons. cf. Hraa & Seljak 05 Noh & Hwang 05
4. Nonlnear ΔN formula energy conservaon: (applcable o each ndependen maer componen) ρ ( ) a ( O ε = ) α = & + + ψ = H% O ε 3 + a N + ( ρ p) e-foldng number: ( ), ; % N N x H d = d 3 ρ + where x =cons. s a comovng worldlne. Ths defnon apples o any choce of me-slcng. ψ where (, x ) ψ (, x ) =ΔN (, ; x ) (, ; ) ( x N, ; x ) ΔN ln ρ P x a ( ) a ( )
ΔN - formula Lyh & Wands 03, Malk, Lyh & MS 04, Lyh & Rodrguez 05, Langlos & Vernzz 05 Le us ake slcng such ha Σ() ) s fla a = [ Σ F ( ) ] and unform densy/unform H/comovng a = [ Σ C ( ) ] : ( fla slce: Σ() ) on whch ψ = 0 e α = a() ) Σ C ( ) : unform densy ρ ( )=cons. N(, ;x ) ΔN F Σ C ( ) : unform densy Σ F ( ) : fla N (, ; x) = N0(, ) +ΔN F a ( ) N (, ) = ln beween Σ ( ) and Σ 0 C C a ( ) ρ ( )=cons. ψ ( )=0 ( )
Then (, ) (, ) ( x, x ) Δ N = ψ ψ = ψ F x C suffx C for comovng/unform ρ/unform H where ΔΝ s equal o e-foldng number from Σ F F ( ) o Σ C ( ): Σ ( ) ρ Σ ( ) ρ Δ = + 3 ρ+ P 3 ρ+ P C C N F d d ΣF( ) Σ ( ) C = 3 Σ Σ C F x ( ) ( ) ρ + ρ P x d For slow-roll nflaon n lnear heory, hs reduces o N ( ) ( ) a ψ C( ) R C( ) = δn ; = H δf C = δφ ( ) a F a φ
ΔN N for slowroll nflaon In slowroll nflaon, all decayng mode soluons of he (mul-componen) nflaon feld φ de ou. Nonlnear ΔN for mul-componen nflaon : A A A Δ N = N φ + δφ N φ = n ( ) ( ) n N n! L A A An φ φ φ MS & Tanaka 98, Lyh & Rodrguez 05 If φ s slow rollng when he scale of our neres leaves he horzon, N s only a funcon of φ (ndep of dφ/d,, apar from rval dep. on me fn from whch N s measured), no maer how complcaed he subsequen evoluon would be. A δφ δφ A L δφ where δφ =δφδφ F (on fla slce) a horzon-crossng. (δφδφ F may conan non-gaussany from subhorzon neracons) cf. Maldacena 03, Wenberg 05,... A n
Dagrammac mehod for nonlnear Δ N n DN ζ =Δ = AA L A n A A An N δφ δφ L δφ ; N AA L A n! A φ A φ An L φ n N n basc -p funcon: A B AB δφ ( x) δφ ( y) = h ( φ) G0( x y) feld space merc δφ s assumed o be Gaussan for non-gaussan δφ,, here wll be basc n-p funcons conneced n-p funcon of ζ: Byrnes, Koyama, MS & Wands 07 -p funcon A AB ζ( x) ζ ( y) = N A N G0( x y) + N ABN G0( x y) c! x y N N G x y A A 3! A A A A + x y + x y! B B 3! BC BC ABC 3 + ABC 0( ) +L +
3-p funcon A B ζ( x) ζ( y) ζ ( z) = N N N G ( x y) G ( y z) + perm. x A A y B + c AB 0 0 AB CA + N N N G ( x y) G ( y z) G ( z x) BC 0 0 0 A BC + N N ABC N G0( x y) G0( y z) + perm.! +L x A C + perm. +! x A C z! y z B B B x A A + perm. +! y z B C B C
IR dvergence problem Loop dagrams lke x A C AB CA N N A C BC N G0( x y) G0( y zg ) 0( z x) y z B B n he m-p funcon gve rse o IR dvergence n he (m-)( )-specrum f P(k)~k n 4 wh n. eg,, he above dagram gves Boubekeur & Lyh 05 Pk (, k, k) : δ ( k+ k + k) d p P( p) P( k+ p ) P( k p ) 3 3 3 3 cuoff-dependen! Is hs IR cuoff physcally observable? (real space 3-p 3 fcn s perfecly regular f G 0 (x)) s regular.)
8. Summary Superhorzon scale perurbaons can never affec local (horzon-sze) dynamcs,, hence never cause backreacon. nonlneary on superhorzon scales are always local. However, nonlocal nonlneary (non-gaussany) may appear due o quanum neracons on subhorzon scales. cf. Wenberg 06 There exss a nonlnear generalzaon of δ N formula whch s useful n evaluang non-gaussany from nflaon. dagrammac mehod can by sysemacally appled. IR dvergence from loop dagrams needs furher consderaon.