6.4 Slow-roll predictions from Ward identities
6.4.1 Two-point function
Power spectrum in the conformal fixed point
The power spectrumhζζiof curvature perturbations can be found from the field the- ory via the stress-energy tensor two-point functionhΘuΘuiu, which in its turn is fully
determined by the two-point function of the dual operator O. As we have seen in chapter 3, in a conformal fixed point, the two-point function for an operatorOwith scaling dimension∆is completely specified by conformal symmetry [83, 84],
hO(x)O(y)i0= 1
|x−y|2∆.
In principle, this completely specifies the holographic power spectrum for the confor- mal fixed point. However, before we can compare with (2.14), we need to perform a Fourier transform. This is necessary as the constraints by conformal symmetry are naturally given in terms of the real space variablesxj, whereasn-point functions in
cosmology are naturally given in terms of the outgoing momenta kj. Any connec-
tion between conformal correlation functions and inflationary correlation functions is therefore necessarily obtained only after a Fourier transform. Although finding the Fourier transform for the two-point function is readily done, in general the Fourier transform leads to a technical obstruction for any quick use of the holographic corre- spondence [149, 150, 255]. As we will see, already for the three-point function this obstruction is difficult to overcome.
The Fourier transform of the two-point function is
hOOi0−−−F.→T. Z d3xd3y|x−y|−2∆ei(k·x+k0·y) = Z d3ueiu·(k+k0) Z d3vv−2∆eiv·(k2+k3−k1) =δ(k+k0)k2∆−3 Z dξ ξ−2∆+2 Z dθsinθe−iξcosθ ∝δ(k+k0)k2∆−3, (6.18)
up to factors of 2 and 2π. SincehOi0 =0, if we takeβ(u) to be an overall constant, the stress-energy tensor correlation function in the conformal fixed point is
hΘkΘk0i0∝δ(k+k0)β2k2∆0−3. Usinghζζi0 ∝ 1
RehΘΘi0 and the expressions (6.6) for andηin the conformal fixed point,
=1 2β
2=0, η=−(∆
the Fourier transform of the two-point function of primary operators agrees with the standard result (2.14), whereβ2describes the singular behavior of the power spectrum as→0 and where the spectral index is given by
ns−1=−2(∆0−3)=2η. (6.20)
Power spectrum in the renormalization group flow
As explained in section 6.2, away from the conformal fixed point, the perturbation by the operatorOwill lead to a renormalization group flow. Holographically this is understood as the deviation from the pure de Sitter phase to a (quasi-de Sitter) inflationary phase and was interpreted by [264] in the light of the known AdS/CFT holographic renormalization methods [139, 141, 142]. Conceptually, it is understood in the Wilsonian sense as a flow between theories, specified by the running of the cou- pling constants. Technically, the renormalization group flow is the result of the need to renormalize the operators Oappearing in (6.16). At the level of the correlation functions, the differential Callan-Symanzik equation dictates the scale dependence of the correlation functions, which was introduced by the inclusion of non-marginal cou- pling constants. For the truly marginal stress-energy tensorΘu, the Callan-Symanzik
equation determines its two-point function via
µ∂ ∂µ+β ∂ ∂u ! hΘuΘui0u=0. (6.21)
Theµ-dependence can be traded for momentum dependence via dimensional analy- sis, hΘuΘui0u=k 3F " k2 µ2,u(µ) # , telling us that µ ∂ ∂µ+k ∂ ∂k ! hΘuΘui0u=3hΘuΘui0u.
Following the literature [72], the Callan-Symanzik equation acting on the renormal- ized operators ˜O=Z(u)O, can be solved by investigating the ansatz
hΘuΘui0u=Z
2(u)β2(u)k3. (6.22) This seems to separate the k- andu-dependence completely, although the two are inherently related through the defining equation of the running couplingu,
β=k∂u
Applying the Callan-Symanzik equation on this ansatz, the wavefunction renormal- izationZis given by ∂Z Z∂u+ ∂β β∂u =0, which is solved by Z(u)=Z0e −Ru 0du 0λ(u0) β(u0). (6.23) Sinceβ=k∂∂uk, we can trade the integration variable fork, introducingµas the only other scale in the problem,
Z(k)=Z0e Rk
µd log(k
0
µ)λ. (6.24)
In this form, it is clear that the wavefunction renormalization just introduces an anomalous dimension to the correlation function. For a constantλ(u) = ∆−3, the two-point function reads
hΘuΘui0u=Z
2 0β
2k3+2(∆−3). (6.25)
Foru → 0, this returns to the earlier found result with an exponent 2∆0−3. For completely arbitraryλ, the result is expressed through the integral in (6.24), which provides a possible method to go beyond the lowest order in slow-roll [265, 288].
As was mentioned in [265], and which deserves renewed emphasis, to connect the conformal correlation function with the inflationary power spectrum, one has to express the two-point function with respect to theaverage Hubble flow. The stan- dard inflationary perturbation theory calculates correlation functions of the quantum fluctuations around the classical inflationary evolution. This evolution is driven by an almost —but not exactly— constant Hubble parameterH(u). Of course, the fact that we have to consider a quasi-de Sitter phase rather than a pure de Sitter evolution is precisely expressed through the slow-roll approximation, something we have already incorporated in the holographic description by studying the renormalization group flow. Still, to correctly identify the fluctuations, we need to express the result with respect to the classical evolution. Since conformal perturbation theory only works around a fixed point of the renormalization group flow, one might wonder how we can express our results with respect to an arbitrary point on the flow, corresponding to the quasi-de Sitter phase. For the two-point function this can be remedied by iso- lating an explicit Hubble parameter dependence, expressed as an integrated effect of the slow-roll parameter (6.5),
H(u)=H0e− 1 2 Ru 0du 0β(u0) =H0e 1 2 Rk µd log(k 0 µ)β2. (6.26)
Using the expression in terms ofk, we find hζkζk0i ∝δ(k+k0)k−3 H2 β2e −Rµkd log(k0 µ)(β2+2λ). (6.27)
The relation (6.6) between the slow-roll parameters and the (derivative of the)βfunc- tion for constantβ(k) andλ(k) then immediately gives
ns−1=−β2−2λ=2η−4, (6.28)
in agreement with (2.15).