Three-Point Correlation Function

The real and imaginary parts of the initial conditions for this case are then

Σ^IJ∗Re = −G^IJH

We follow a similar procedure to consider the momentum-momentum correlation

hP^I(k1, τ1)P^J(k2, τ2)i = (2π)³δ(k1+ k2)Π^IJ The initial conditions for Σ^ab_Re where also given by Dias, Frazer and Seery [118].

4.3. Three-Point Correlation Function

We now move on to calculating the three-point function using the in-in formalism we outlined in Eqn. (4.14). After expanding the exponents to first order we find that the non-vanishing terms that remain are given by,

hδX^aδX^bδX^ci^? =−i Z τinit

−∞

dτhδX_?^aδX_?^bδX_?^c,H^efgδX^eδX^fδX^gi?, (4.25)

whereH^efg is the interaction part of the Hamiltonian extracted from the cubic part of the action in Eqn. (3.25).

The Hamiltonian contains the kernel tensors aIJ K, bIJ K and cIJ K and is defined as,

where the indices are organized so that a block of field labels are followed by a block of momentum labels and are contracted over the internal legs only. The bold font

4.3: Three-Point Correlation Function 70

on the indices indicates that the usual summation over phase space indices is ac-companied by an integration over Fourier space. This order is sufficient as all higher order terms are suppressed by factors of Q ≈ H/M^pl. We can rewrite Eqn. (4.25) by including the integration over Fourier space and removing bold indices,

hδX^aδX^bδX^ci? =−i Z τinit

−∞

dτHef g

Z Πid³ki

(2π)⁹ (2π)³δ(Σik_i)hδX?^aδX_?^bδX_?^cδX^eδX^fδX^gi . (4.27) By using Wick’s theorem the 6-point correlation function above can be broken into six permutations of two-point correlation functions,

hδX^aδX^bδX^ci^? =− i Z τinit

−∞

dτH^{ef g}

Z Πid³ki

(2π)⁹ (2π)³δ(Σiki)hδX?^aδX^eihδX?^bδX^fihδX?^cδX^gi + cyclic.

(4.28) Each permutation represents a different way of contracting a pair of internal and external legs of the Feynman diagram. For convenience we can write the three-point function as the tensor

hδX^aδX^bδX^ci^? = (2π)³δ(Σik_i)B_?^abc(k1, k2, k3) . (4.29) We can rewrite the three-point function Babc in terms of permutations of the two-point function Σab,

B_?^abc=−6i Z τinit

−∞

dτH^efgΣ^ae(τ?, τ )Σ^bf(τ?, τ )Σ^cg(τ?, τ ) + c.c. (4.30)

If we then substitute Eqn. (4.26) into Eqn. (4.30) we get,

B_?^abc=− 6i Z τ?

−∞

dτa⁴

2 AefgΣ^aeΣ^bfΣ^cg

− 1 3a

B¯e(fg)Σ^a¯eΣ^bfΣ^cg+ B(e|f |g)¯ Σ^aeΣ^b¯fΣ^cg+ B(ef )¯gΣ^aeΣ^bfΣ^c¯g

− 1 3a²

C_{(¯e¯f )g}Σ^a¯eΣ^b¯fΣ^cg+ C(¯e|f |¯g)Σ^a¯eΣ^bfΣ^c¯g+ C_{e(¯f ¯g)}Σ^aeΣ^bfΣ^c¯g ,

(4.31)

where the tensors Σ^abhave a dependence on two times (i.e. Σ^ab(τ?, τ )), τ representing the internal legs and τ? representing the external legs and the bars over the indices label P^I components. From our calculations of the two-point functions in Sec. (4.2)

4.3: Three-Point Correlation Function 71

We can now begin to explicitly calculate the three-point function by substituting Eqns. (4.32) into Eqn. (4.31). As we only need to integrate over the internal legs, the external components of Eqn. (4.32) can be brought outside. The time dependence of the bIJ K and cIJ K tensors which appear in the interaction Hamiltonian is slow-roll suppressed and their time dependence can be neglected. On the other hand, the aIJ K tensor contains fast changing terms proportional to (k/a)² ≈ (kτ)² which grow exponentially into the past and whose time dependence must be included. This splitting of the aIJ K into the fast and slow parts is best illustrated when we convert Eqn. (3.32) to conformal time τ =−1/aH

aIJ K = φ˙IGJ K

2H³M_pl²

(kJ· kK)

τ² + aIJ K(slow). (4.33)

It is also assumed that H and Γ^ab which appear in the expression for Σ(τ1, τ2) are also sufficiently slowly varying that their time dependence can be neglected. The integral is dominated by its upper limit, and these assumptions mean that when evaluating it one takes Γ^IJ → G^IJ(τ?) and H → H(τ?). The assumptions need only be true for a short period around the time the initial conditions are fixed. In the resulting expressions for the initial conditions for Babc, we keep both the terms which grow fastest as τ → −∞ as well as the sub-leading terms.

To illustrate how this is evaluated in practice, let us consider this explicitly for the case of a field-field-field correlation.

• a,b,c → Field-Field-Field

Substituting in the expression for the two-point function we obtain

B_∗^abc=− iH⁶

4.3: Three-Point Correlation Function 72

where we assume that H and Π^IJ are sufficiently slowly varying to be taken as constants and that we can take Π^IJ → G^IJ.

In order to perform the integration we need to know the time dependence of the tensors. As discussed earlier the aIJ K tensor contains fast and slow varying parts.

The part containing terms quadratic in τ vary quickly and so are included in the integral separately (the first term in Eq. (4.34)), the remaining parts we label a^{IJ K}_s and we assume can be considered constant in time. The next step is to perform the integration, recalling that the result is dominated by the upper limit (because the integral is highly oscillatory into the past). Keeping the leading and sub-leading terms in τ , and writing in terms of a and H, the final result is

4.3: Three-Point Correlation Function 73 We reiterate that the calculation for the three-point function in our set up may be perform very early when the modes of interest are deep within the horizon. Using the in–in formalism we have reviewed in this chapter we can calculate the result in single-field inflation. We quote the result from Maldacena [106, 119] for the three-point function of ζ (which similarly obtained by the gauge relations in Sec. (2.5)),

hζ(k¹)ζ(k2)ζ(k3)∼ (2π)³δ(Σiki) 1 time when modes k1 and k2 cross the horizons and t3 is the time when the mode k3. Using the definition of the reduced bispectrum in Eqn. (2.147) we arrive at the famed consistency relation,

fNL = 5

12(1− n^s) , (4.40)

in the squeezed limit. This result is calculated when the mode of interest is integrated over from deep inside the horizon to the end of inflation, incorporating the sub-and super-horizon evolution of the mode. In contray to this, the transport method uses the in–in formalism only for its initial condition and incorporates the sub- and super horizon evolution in the transport equations. With these sets of equations as our initial conditions and the equation from Ch. (3) we have all the ingredients necessary our transport approach in the next chapter.

5. Evaluating Statistics from