The statistical properties of the primordial fluctuations

Transcription

1 The tatitical propertie of the primordial fluctuation Lecturer: Prof. Paolo Creminelli Trancriber: Alexander Chen July 5, 0 Content Lecture Lecture 4 3 Lecture 3 6

2 Primordial Fluctuation Lecture Lecture Looking at the WMAP, we believe that the aniotropie that we ee i due to the primordial fluctuation. We want to talk about how to calculate them. The quantity we are intereted in i ζ, which i the metric fluctuation in the patial part. We write the patial metric a a (t)e ζ(x,t) δ ij dx i dx j () We are intereted in, and can oberve the -point function ζ k ζ k, or even the 3-point or 4-point function. At the moment we don t oberve anything that i non-gauian, which mean only -point function i nonzero. But future obervation may lead to non-gauianitie, which are predicted by theorie. We will do ingle field inflation, but do it in an unconventional way. We call it effective theory of inflation. The original paper i Cheung et al, The idea i to apply effective field theory to inflation. To decribe a ytem, we have to identify the degree of freedom, the ymmetrie of the ytem, and the dynamic i governed by the lowet dimenional operator. Let apply the ame logic to inflation. During the period of inflation the pacetime i cloe to de- Sitter. Becaue the period of inflation i finite, there i a previleged licing which i decided by the final product of inflation. Thi previleged licing i the flat licing becaue the reulting pacetime ha negligible curvature. Now with thi the time diffeomorphim ymmetry i broken t t + ξ 0, but the pace diffeomorphim ymmetry i preerved. So when we conider the action it hould be invariant under patial diffeomorphim. We can write down the action S = d 4 x [ ] M p g R c(t)g00 + Λ(t) + M (t) a (g 00 + ) + 6 H 3(t)(g 00 + ) +... () The fact that g 00 appear i due to that g 00 i invariant under patial reparametrization. From the above action we can extract the energy momentum tenor a T µν = g δs δg µν (3) For low-roll inflation, the pacetime i de-sitter for firt approximation d = H η ( dη + dx ) (4) To firt approximation the time-dependent component of the action evolve lowly. The above pacetime ha another iometry which i dilation (x, η) λ(x, η). Thi mean that ζ k λ 3 ζ λk i a ymmetry of the ytem. The correlation function mut then go like ζ k ζ k = (π) 3 δ(k + k ) k 3 (5) It i very important that ζ i a contant which i independent of time, o that it i conerved from the end of inflation to the time of recombination. Becaue after the inflation the variation of ζ i on the cale larger than the Hubble radiu, the time dependence can be ignored to firt approximation becaue the time dependence i aborbed in the cale factor a(t).

3 Primordial Fluctuation Lecture Let come back to effective theory. Conider the implet model of inflation S = d 4 x [ g ] ( ϕ) V (ϕ) d 4 x [ g ϕ 0 (t) ] g00 V (ϕ 0 (t)) (6) Thi i made poible by chooing the gauge that we dicued above. In general we conider model with action in the form of a polynomial P (X, ϕ) where X = /( ϕ), then in thi gauge we will have S d 4 x gp ( ) ϕ 0 g00, ϕ 0 (t) For ome reaon it i more convenient to change gauge. Breaking a gauge ymmetry i equivalent to introducing a new degree of freedom. The above action break time diffeomorphim ymmetry, and we want to change into another gauge where thi i not the cae. Conider a U() gauge field with an action 4 F µν F µν m A µ A µ (8) Now the theory i not gauge invariant, but we are not afraid of that, and we can introduce a new field π which tranform a π π Λ. Then the action become 4 F µν F µν m (A µ µ π) (9) Then the action i gauge invariant. We will do exactly the ame thing here, and introduce a new field π(x, t) which tranform a π(x) π(x) ξ 0 (x) (0) Then we will change anything involving g 00 into B(t)g 00 B(t + π(x, t)) (7) (t + π(x, t)) (t + π(x, t)) x µ x ν g µν () The original action will now be turned into d 4 x [ M ( ) ] p g R M p H π ( π) + M 4 ( π + π 3 π ( π) a ) 4 3 M 3 4 π 3 The above formula only contain the term involving π. Thi will be the tarting point. () 3

4 Primordial Fluctuation Lecture Lecture We tart with the effective action S = d 4 x [ M ( p g R M p Ḣ π ( iπ) ) ( a + M 4 π + π 3 π ( iπ) ) ] a 4 3 M 3 4 π (3) where the calar field π can be thought of a a Goldtone boon field ϕ 0 (t + π(x, t)) for the broken time diffeomorphim ymmetry. Let try to ee what thi action implie. We define the peed of ound in the following way c = M 4/M p Ḣ. Then we can write the ame action in term of thi M p Ḣ ( c π c ( π) ) ( a + Mp Ḣ ) ( ) c π 3 π ( π) a 4 3 M 3 4 π 3 (4) We can calculate the -point function of π uing thi action π k π k = (π) 3 δ(k + k ) H k 3 c M p Ḣ c 3 (5) Thi i for the calar field π, but we are intereted in the calar perturbation ζ. Thi can be recovered uing ζ = Hπ. Becaue π i like change in time, we can jut do a change in time δt = π, and the metric i perturbed, and to linear order the above relation i recovered. The -point function i then ζ k ζ k = (π) 3 δ(k + k ) c H 4M p ε k 3 (6) The tilt in the power pectrum i defined by k 3+(n ). The way to calculate it i d n = d log k log H4 = 4 Ḣ Ḣ c H Ḧ ḢH ċ c H c k/a H (7) By cale invariance and becaue we have N e e-fold after inflation, thi tilt will be around /N e, but the exact number depend on model. Let now look at correction to thi picture. The ratio of the cubic lagrangian and the quadratic lagrangian will be a meaure of how good it i to neglect higher order term. Thi ratio i L 3 = (M p Ḣ/c ) π L Mp Ḣ c ζ (8) Thi i a ratio meauring how good the Gauian approximation i, or the interaction term with the free field theory. Thi number multiplied by 0 5 will be what people call f NL. Of coure experiment ha not detected large non-gauianitie, and they have et bound on thi quantity, and f NL of about 00 i what we have today. If we tart with a potential term controlled by ε, then the magnitude of non-gauianity will be NG 3 ε ζ, f NL ε (9) 4

5 Primordial Fluctuation Lecture Thi will be even maller than the 0 3 that our experiment bound i. We can even include quartic term, and the quartic non-gauianity can be calculated a NG 4 ζ ε 3 (0) So thi i even more uppreed. Let u now try to calculate the 3-point function of the calar perturbation. Let review how to do the calculation firt. In ordinary quantum field theory we know the way to calculate the expectation value of a tring of time ordered product of field operator Ω T ϕ(x)ϕ(y) Ω = lim 0 Ũ(T, x 0)ϕ I (x)ũ(x 0, y 0 )ϕ I (y)ũ(y 0, T ) 0 () T + ( iε) In the end we have lim 0 T ϕ(x)ϕ(y)e i T T H I(t)dt 0 T + ( iε) However in de-sitter pace our vacuum can only be defined in the far pat, but not in the future. So at both ide we evolute to the far pat in order to hit the Minkowki vacuum. () 5

6 Primordial Fluctuation Lecture 3 3 Lecture 3 We will calculate the correlation function. Remember that lat time we defined the way of evaluating the correlation function. We will recale the field a Mp Ḣ π c = π (3) c And the interaction term in the hamiltonian will be accordingly modified. To firt order, the 3-point function will jut be the commutator of the 3 π field with the interaction Hamiltonian π k (η = 0)π k π k3 = i 0 dη [ π k (0)π k (0)π k3 (0), H int (η ) ] (4) The olution of the claical equation of motion i π k (η) = π cl k (η)a k + πcl k (η) a k, where π cl k (η) = H c 3 k 3 ( ikc η) e ickη, πk cl H (η) = c 3 k 3 c k ηe ikcη (5) The interaction Hamiltonian in conformal time will be H int = A d 3 x Hη π 3 (6) In the end of the calculation the reult will be π k π k π k3 = (π) 3 δ(k + k + k 3 )6 The integration can eaily be evaluated by 0 iε H 6 i c3 k 3 i dη η e iktcη = 0 ( i)a dη Hη c6 kk k 3η 3 e iktcη + c.c. (7) i (k t c ) 3 (8) Now let look at the reult. If we write ζ = Hπ then the above reult tranlate into ζ k ζ k ζ k3 = (π) 3 δ(k + k + k 3 )P ζ c M p M Ḣ k k k 3 (k + k + k 3 ) 3 (9) where P ζ = H 4 /(4Mp Ḣ c ). The quantity c M3 4/(M p Ḣ ) will be called f NL. At the end of inflation we have reheating, which mean that the inflaton ocillate around the minimum of the inflaton potential and the energy change into other thing. At the beginning we are in matter dominance when Γ H end. Then after Γ H, we enter the period of radiation dominance becaue of decay. We can check that a(t f ) a(de) H /6 a(de) a(t i ) decay Γ /6 (30) There i a conitency relation for 3-point function, and it very generic for ingle-field model. We look at the queezed mode where a k i much maller than the other. Conider the function ζ(x )ζ(x )ζ(x 3 ), then we have ζ(x )ζ(x )ζ(x 3 ) = ζ(x ) ζ(x )ζ(x 3 ) ζ (3) 6

7 Primordial Fluctuation Lecture 3 where ζ(x )ζ(x 3 ) ζ = F ( ) e ζ x x 3 (3) We can expand thi around ζ. After calculation, the bottom line i that lim ζ(k )ζ(k )ζ(k 3 ) = (π) 3 δ(k + k + k 3 )P (k )P (k ) d log k3 P (k ) + O k 0 d log k ( ) k k (33) where k i the wave vector of the hort wavelength wave. Thi ay that baically there i no correlation between the long and hort mode. So if we ee otherwie in the ky, then there mut be more than one field. 7