$\delta \mathcal {N}$ formalism on the past light-cone

Abstract

We apply the gradient expansion approximation to the light-cone gauge, obtaining a separate universe picture at non-linear order in perturbation theory within this framework. Thereafter, we use it to generalize the $\delta \mathcal {N}$ formalism in terms of light-cone perturbations. As a consistency check, we demonstrate the conservation of the gauge invariant curvature perturbation on uniform density hypersurface $\zeta$ at the completely non-linear level. The approach studied provides a self-consistent framework to connect at non-linear level quantities from the primordial universe, such as $\zeta$, written in terms of the light-cone parameters, to late time observables.

Linear \(\delta \mathcal {N}\) formalism on the light-cone

In this appendix we derive the $\delta \mathcal {N}$ formalism at linear order in perturbation theory using the framework developed in [36, 42] and reviewed in Sect. 4. In this way we explicitly show the consistency of our results.

To begin, we linearize Eq. (97) and obtain

$$\begin{aligned} \delta \mathcal {N}&= \frac{1}{3}\ln \left[ \frac{\left( \Upsilon \sqrt{\gamma }\right) _{UD}}{\left( \Upsilon \sqrt{\gamma }\right) _{UC}}\right] +\mathcal {O}(\epsilon ^{2})\nonumber \\&= \frac{1}{3}\ln \left[ \frac{\left( \bar{\Upsilon }\sqrt{\bar{\gamma }}\right) (1+\delta \Upsilon )(1+2\nu )_{UD}}{\left( \Upsilon \sqrt{\gamma }\right) _{UC}}\right] +\mathcal {O}(\delta ^{2},\epsilon ^{2})\nonumber \\&= \frac{1}{3}\ln \left[ 1+\left( \delta \Upsilon +2\nu \right) _{UD}\right] +\mathcal {O}(\delta ^{2},\epsilon ^{2})\nonumber \,\\&= \frac{1}{3}\left( \delta \Upsilon +2\nu \right) _{UD}+\mathcal {O}(\delta ^{2},\epsilon ^{2}) \end{aligned}$$

(A1)

where we have defined

$$\begin{aligned} \Upsilon =\bar{\Upsilon }(1+\delta \Upsilon ) \end{aligned}$$

(A2)

and we recall that $(\Upsilon \sqrt{\gamma })_{UC}$ is equal to the background value, being $\psi =0$ within the uniform curvature gauge. Also, we have used the metric in Eq. (56) and the scalar/pseudoscalar decomposition of Eq. (58). Since $\delta \Upsilon =N/2$, we then have that

$$\begin{aligned} \delta \mathcal {N}(\lambda _{1,}\lambda _{2,}x^{i})=\frac{1}{6}\left( N+4\nu \right) _{UD} +\mathcal {O}(\delta ^{2},\epsilon ^{2}). \end{aligned}$$

(A3)

From the relation between the light-cone perturbation and the standard ones in Eq. (79), (see also [42]), one gets that

$$\begin{aligned} \psi =-\frac{1}{6}(N+4\nu ). \end{aligned}$$

(A4)

Therefore, Eq. (A3) together with Eq. (A4) and the fact that $\psi _{UD}=\zeta$, gives the well known relation $\delta \mathcal {N}=-\zeta$ [15].

The result obtained in Eq. (A3) proves that the $\delta \mathcal {N}$ formalism on the past light-cone is consistent with the light-cone perturbation theory developed in [36, 42]. Thereby, the $\delta \mathcal {N}$ formalism within the past light-cone framework, at linear order in perturbation theory, could also be obtained by starting from the results presented in Sect. 4. To this aim, one should integrate Eqs. (62) and (63), evaluating them between the uniform curvature and the uniform density slices.