A path-integral approach to CMB

Transcription

1 A path-integal appoach to CMB Based on PHR and L. R. Abamo, CMB and Random Flights: tempeatue and polaization in position space, JCAP6(3)43 Paulo Henique Reimbeg IF-USP II JBPCosmo, 4/3, Peda Azul, ES

2 CMB: basics µ (z) e µ (z) x z µ (z) µ(z) e visibility function z coupled intemediate phase (ecombination) fee popagation g = a (η)[ ( + Φ)d η + ( Ψ) (3) g]

3 Integal vesion of Boltzmann s equations Tempeatue Θ(x o, η o, ô) = Θ l (k, η o )= ηo d 3 k (π) 3/ eik x o 4π lm i l Θ l (k, η o )Y lm (ˆk)Y lm (ô) { [ [ [ dη e {µ µ(η) (η) θ SW (k, η) j l (k η ) kv b (k, η) j l (k η ) + [ Θ (k, η) ] [ 3 6α (k, η) } l (k η )+ ] ] +(Ψ + Φ )(k, η) j l (k η ) j l(k η ) [ j ] [ ] ] } Polaization } ] Recombination visibility function Q + iu 4I (x o, η o, ô) = d 3 k (π) 3/ eik x o 4π lm i l α l (k, η o )Y lm (ˆk) Y lm (ô) α l (k, η o )= 3 (l + )! (l )! ηo dη µ (η)e µ(η) [ Θ (k, η) ] jl (k η ) 6α (k, η) (k η ) o^

4 Polaization in position space - uncoupled tempeatue { { [ { [ ] θ () Θ(η o, ô) = lm ] } } lm (η o)y lm (ô) S lm (k, η) := d ˆk e µ(η) ηo lm =il dη θ () { µ (η) [ S lm (k, η) =( i) l dk (π) / k S lm (k, η) j l (k η o ) Θ SW (k, η) V b (k, η) η dx (π) / X S lm (X, η) j l (kx) ] +(Ψ + Φ )(k, η) } Y lm(ˆk) (3.8) θ () lm = ηo dη S lm (X = η, η)

5 Polaization in position space - uncoupled tempeatue } Q + iu 4I (η o, ô) = lm π lm (η o ) Y lm (ô) π () lm (η o)= 3 π π () lm (η o)= 3 π π (3) lm (η o)= 3 π ] (l + )! (l )! π (n) lm (η o)= 3 π ηn (l + )! (l )! (l + )! (l )! ηo ηo dη g(η ) (l + )! (l )! dη ηo dη g(η ) [ η ηo π lm (η o )= dη g(η ) η dη g(η ) η dη g(η ) η dη g(η ) dx X S lm (X, η) dη η dη 3 g(η 3 ) n= η3 dη dη (n )! 9 (n ) π (n) lm (η o) dx X S lm (X, η) η [ dxx S lm (X, η) 9 dx X S lm (X, η) 9 dη... dη n T{g(η )... g(η n )} dk k j l (kx) j l(k η o ) (k η o ) j (k η ) ] dkk j l (kx) j l(k η o ) j (k η ) (k η o ) (k η ) j (k η ) dk k j l (kx) j l(k η o ) j (k η ) j (k η ) (k η o ) (k η ) (k η ) j (k η 3 ) dk k j l (kx) j l(k η o ) j (k η ) (k η o ) (k η )... j (k η n ) (k η n ) }{{ j (k η n ) } (n ) times

6 Random Flights H. Spohn, Commun. Math. Phys., 6, 978

7 Random Flights: position of the poblem 3 3 n A walke moves with constant speed in a D- dimensional and pefoms n steps of lengths,,..., n. Afte each step the walke changes isotopically its diection of motion. What is the pobability that afte n steps, the walke will be at a distance fom the oigin? { 3 n n n p n (;,..., n D) := d { [Γ(D/)] n dk d ( ) D/ k J D/(k) n i= } J D/ (k i ) (k i ) D/ ( ) [ ( )] ( π ) (n+)/ n m [ ( )] Γ m + 3 n p n(; m+,..., n m +3) = dk k j m (k) n q= j m (k q ) (k q ) m j m(k n )

8 Extended Random Flights X R

9 Extended andom flights 3 3 n 3 n n n H lm (R, X,,..., n ):= dk k j l(kr) (kr) j l(kx) m n q= ( ) j m (k q ) (k q ) m j m(k n ) j l (k ) j l (k )= ( )m + ( d k m ) m ( ) P m l (cos α) sin m α j m (k) Gegenbaue addition theoem X H lm (R, X,,..., n ) R+X ( π ) (n+)/ = ( )m [ ( )] Γ m + 3 n d R X ( ) J m RX P m R m l (cos α) sin m α n m m+ n p n (;,..., n m +3) n R

10 Recoveing the hieachy Θ(η o, ô) = lm θ () lm (η o)y lm (ô) θ () lm = ηo dη S lm (X = η, η) π lm = 3 (l + )! ηo dη g(η ) 4π (l )! ( ( ) ) 9n π (n+)/ ηn [ ( Γ 7 )] n dη ( ) Q + iu 4I (n )! n= (η o, ô) = lm η dx X S lm (X, η) π lm (η o ) Y lm (ô) ( ) dη... dη n T{g(η )... g(η n )} [ ( )] d(cos α) ( X ηn 3 ) P l (cos α) sin α p n (; η,..., η n 7)

11 Intepetation π lm = 3 (l + )! 4π (l )! ηo ) (n+)/ dη g(η ) ( 9n π ηn [ ( Γ 7 )] n dη ( X ηn d(cos α) 3 n= (n )! η dx X S lm (X, η) dη... dη n T{g(η )... g(η n )} ) P l (cos α) sin α p n (; η,..., η n 7). 3 n n i. e., Aveage ove all possible shapes of the polygon with given sides; x Compute the souce tem contibutions mediated by the weight given the the aveage shape of the polygon. We should emak that the integation ove x no longe extends to infinity: x is limited by the sum of intemediate steps! Aveage the lengths of the intemediate steps weighted by the visibility functions; Sum fo all possible numbe of intemediate steps; 7= + (. + ); Uppe limit in the integation in X assues that souces ae computed in the suppot of the obseves past light cone.

12 Diagams Solid lines mean polaization, and dashed lines mean tempeatue; The vetical lines to the ight of the diagams epesent the obsevable being calculated (time uns upwad); The inteception of two lines detemine what souces ae being consideed fo a given obsevable. π lm (η o )= Note: Diagams with moe legs coespond to highe powes of the visibility function.

13 Convegence ) ( ) ( )( ) ( ) We shall conside the special case when It can be shown (Watson, A Teatise on the Theoy of Bessel Functions) that: and then, fo / finite: p n (;,..., 7) η = η =... = η n =: ( 7 n lim n p n(;,..., 7) = δ(). ) 7/ e 7 n n in the limit. In this case we show that: S lm = lim ηo η dxx S lm (x, η) n 3 ( x d(cos α) ( ) = π 3 ( η ) S lm( η, η).. We conclude, then, that high ode tems (coesponding to lage numbe of intemediate scatteings) ae suppessed in the seies expansion, as one would expect fom Boltzmann s H-theoem. 3 ) P l (cos α)(sin α) p n (;,..., 7)

14 Recoupling the tempeatue [ ] - θ () lm = + { [ Complicato: dk k j l (kx) [ ] +3 (k η o ) j l (k η o ) j (k η ) H l n }{{}, H (l+) ( θ () lm = θ (n) lm = n q= θ (n) lmq ( η, η,..., η n,x) H (l+q) ( η,x, η,..., η n )

15 Polaization at second ode π () lm = + + π (n) n lm = q= π (n) lmq ( η, η,..., η n,x)) H (l+q) ( η,x, η,..., η n )

16 Geneal ule fo diagams diag. segment numeical facto Bessel function Contibution = ηo ηn dη g(η ) dη (n )! η { [ ] dk k j l (kx) [ a Diagam ] j (k η n ) η stan dx X S lm (X, η) dη... dη n T{g(η )... g(η n )} 3 (l + )! π (l )! j l (k ) (k ) [ ] +3 (k ) j l (k ) j (k ) (k ) = 3 (l + )! ηo η η η3 dη g(η ) dη g(η ) dη 3 g(η 3 ) dη π (l )! j (k ) (k ) { dx X S lm (X, η) dk k j l (kx) j l(k η ) (k η ) [ ] [ ] } +3 (k η ) j (k η ) +3 (k η ) j (k η ) j (k η 3 ). [ ] +3 (k ) j (k ) 9 j (k ) (k )

17 Conclusions The Boltzmann s equations fo CMB can be descibed in position space, whee the causal stuctue becomes evident; In position space CMB tempeatue fluctuations and polaization can be given in tems of a seies expansion in tems of the numbe of scatteings duing ecombination; Contibutions fom high ode tems ae suppessed. (Figues