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

G. Fanizza  G. Marozzi  and M. Medeiros

###### 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.

##  1 Introduction

Advances in cosmological observations have provided us with high-precision
methods to study the universe [1, 2, 3]. So far, linear cosmological
perturbation theory has been the main tool to describe the early universe,
particularly the primordial seeds that are believed to be produced by quantum
mechanical fluctuations during inflation. These fluctuations grow during the
quasi-exponential expansion epoch and freeze outside the horizon. Later on,
they re-enter the horizon during a power-law expansion epoch, giving rise to
the large-scale structure observed in the universe.

In order to link the gauge invariant quantity that characterizes such
primordial fluctuations with the observations, we need to have a good
understanding of their behaviour outside the horizon. An interesting example
is given by the primordial curvature perturbation on the uniform density
hypersurface $\zeta$, which is expected to be of order $10^{-5}$ at the last-
scattering surface and has been shown to be conserved outside the horizon,
both at the linear [4] and the non-linear level [5]. The first-order treatment
for the primordial fluctuations agrees with the observations of a nearly
Gaussian, scale-invariant power spectrum. Although non-linearities are
expected to be small, they are however unavoidable as a consequence of the
non-linear evolution of the perturbations. Detection of the related non-
Gaussianities can then provide important insights into early universe models,
such as the inflationary ones [6, 7].

The evolution of $\zeta$ is proportional to the non-adiabatic (if any)
contribution in the energy-momentum tensor, as shown in [4, 5]. In the linear
regime, $\zeta$ has been successfully calculated using the $\delta\mathcal{N}$
formalism [8, 9, 10, 11, 12, 13], which has been extended to the exact non-
linear level [5] by applying the first-order gradient expansion directly in
the equations of motion provided by the Arnowitt-Deser-Misner (ADM) formalism
[14]. The first-order gradient expansion, also known as the separate universe
(SU) scheme, describes the universe as a set of FLRW geometries with
independent equations of motion and is a good approximation in the regime of
large comoving wavelengths compared to the horizon [8, 9, 4, 5].

The great advantage of the SU scheme is that the equations of motion within
this approximation have the same form both for the background and perturbed
universe with the exception of the momentum constraint, which vanishes in the
background. As a consequence, one can obtain the non-linear field’s evolution
from the background one by imposing non-linear initial conditions [15].

In [16], this formalism has been generalized to include stochastic effects and
derive, within the framework of the stochastic approach [17] and its relation
with QFT [18, 19], non-perturbative correlation functions for single-field
slow-roll inflation. Further extensions have also studied ultra-slow-roll
inflation [20, 21, 22, 23], allowing for the investigation of primordial
black-hole production [20, 21, 22, 23, 24, 25]. The $\delta\mathcal{N}$
formalism has also been extended to the case when cosmic shear is included to
describe the anisotropic expansion. In such a framework, the evolution of
gravitational waves has been explored both for the case of a Bianchi I
universe [26] and when couplings with external fields are present [27, 28].

A formalism that connects the picture of the primordial universe presented so
far to late-time observables would be greatly welcome, especially if such a
connection can account for non-linearities. Since the Geodesic Light-Cone
(GLC) gauge [29] gives the chance of describing light-like observables in the
late universe exactly, such as the redshift and the distance-redshift relation
[29, 30, 31, 32, 33, 34, 35, 36], the galaxy number count [37, 38], the non-
linear corrections to the CMB spectra [39, 40], and also Ultra-Relativistic
particles [41], this is a natural framework to pursue the aforementioned
program.

Moreover, there has also been recent interest in the GLC gauge application to
the study of backreaction effects from the primordial universe [36, 42, 43,
44]. Although these are very interesting prospects, one still must face the
fact that the evolution of perturbations in the GLC gauge is quite involved
already at linear order (see [44] for an analytical treatment). An alternative
approach to this may be provided by numerical attempts, as done for instance
in [45] for the linearized evolved solution for the gravitational potential on
the past light-cone. In this manuscript, we take a different route by
providing simplified equations of motion using the SU approach on the past
light-cone.

The connection between the primordial origin of inhomogeneities and their
observations has to deal with the fact that the latter are done along our past
light-cone, whereas the primordial universe is usually described using spatial
hypersurfaces. During primordial epochs, these hypersurfaces are naturally
described in terms of uniform field slices. In fact, in a single-field
inflationary scenario, the inflaton is the only clock available. Therefore,
the natural slicing which describes the dynamical space-time evolution is the
one given by uniform inflaton hypersurfaces, which also fixes the time gauge
mode. Another interesting fixing for the time coordinate is the one describing
uniform density slices. This is an interesting fixing because it directly
translates the density perturbations into curvature perturbations, providing
the initial conditions for large-scale structure formation and allowing us to
use the $\delta\mathcal{N}$ formalism to connect current inhomogeneities to
the primordial ones. These two gauge fixings are usually called, respectively,
uniform field gauge (UFG) and uniform density gauge (UDG).

To make contact between these gauges and the GLC one, we recall that, although
the GLC time coordinate is fixed to the time measured by a free-falling
observer, a generalization of this gauge is provided in [44], the so-called
Light-Cone (LC) gauge. In this generalization, the time gauge choice is left
unspecified, allowing us to fix the lapse function for describing the uniform
field and uniform density slicing on the past light-cone. An alternative
approach could be to start from the cosmological perturbation theory on the
past light-cone developed in [36, 42], and then perform the necessary gauge
transformations. A description of the primordial universe in terms of the LC
gauge could be a promising framework to connect non-linearly the late universe
to the primordial one, given in terms of light-cone parameters.

Here, we will take a first step in this direction. In particular, along this
manuscript, we will discuss the gradient expansion as done on the observer’s
past light-cone, which allows us to obtain a SU picture and the
$\delta\mathcal{N}$ formalism in terms of light-cone perturbations. We will
provide this both at the fully non-perturbative level, using the LC gauge
[44], and, as a consistency check, at the linear level using the light-cone
perturbation theory [36, 42]. By considering the LC gauge as a non-linear ADM
decomposition (see [44] for more details), we will show that, unlike previous
literature, where the shift vector was a first-order term in the gradient
expansion [15, 5, 26], in the LC gauge the shift vector has to be taken into
account also for the background. This is an important difference, since in
this case the shift vector corresponds to the direction of propagation of the
photon, and it is used to take into account inhomogeneities along the photon
propagation direction. However, we will neglect spatial derivatives of such
shift vector since they correspond to light-cone distortion effects which are
expected to be negligible on large scales.

After such implementations, we will show that the SU picture can be realized
in the LC gauge (i.e., we will obtain evolution equations with the same form
for both perturbed and background universe). Furthermore, within the gradient
expansion approximation, we will verify at the fully non-linear level that the
curvature perturbation on uniform density slices $\zeta$ is a conserved
quantity (for adiabatic pressure) also when the light-like slicing of
spacetime is used. This is a sanity check that confirms how the SU picture can
be extended also to the case of the light-cone gauge.

In summary, we will present a novel approach for connecting the primordial
universe with observations along the past light-cone, by developing the
$\delta\mathcal{N}$ formalism in the LC gauge. We will verify that the LC
gauge allows for a non-linear description of the primordial universe in terms
of light-cone parameters, and that the SU picture can be realized in this
gauge by neglecting spatial derivatives of the shift vector.

The manuscript is organized as follows. In Sect. 2 we obtain a generic SU
description where we keep inhomogeneities along the geodesics in terms of an
ADM metric. In Sect. 3 we present the set of light-cone gauges used here and,
with a non-linear diffeomorphism, we show how the standard ADM formalism
relates with the LC gauge. Moreover, we provide the LC gauge fixing condition
in terms of the ADM variables up to non-linear order. Thereafter, we discuss
the SU formalism on the past light-cone, which is also presented for the GLC
gauge. Sect. 4 is devoted to the computation of the non-linear number of
e-folds and its relation to spacetime perturbations, which allow us to obtain
the non-linear scale factor in the LC. We then give a proof for the super-
horizon conservation of $\zeta$, at both linear and non-linear order in
perturbation theory, and first order in the gradient expansion, for adiabatic
fluids. Finally, we provide a generalization of the $\delta\mathcal{N}$
formalism in the LC gauge. In Sect. 5, our main conclusions are summarized and
discussed. In Appendix A we provide the linear $\delta\mathcal{N}$ formalism
as a consistency check of the obtained results.

##  2 Separate universe

Let us begin by introducing a systematic approximation scheme, which can be
used when the wavelength of the perturbations is larger than the physical
horizon. This approximation, widely known as the SU approach [8, 9, 4, 5],
consists of employing the already mentioned gradient expansion perturbative
scheme. This is based on the quantity $\epsilon\equiv k/(aH)$, rather than on
the amplitude of the perturbations. This quantity compares the comoving
wavenumber of a given mode $k/a$ with the expansion rate $H$. As an example,
within this approximation scheme, terms with one spatial derivative will be
first order111As an example in a flat space, under a Fourier transformation,
spatial derivatives give rise to terms proportional to $k$. Here we are
considering that for a quantity $Q$,
$\frac{1}{a}\partial_{i}Q\ll\partial_{t}Q\approx HQ$ [10]. in $\epsilon$. The
first order gradient expansion is known as the SU approach, since in this case
the equations of motion for a local patch of the perturbed space-time have the
same form of the FLRW background ones [15]. Thus, in this view the universe
can be described as a collection of FLRW geometries, each one locally
described by a different scale factor.

This approximation can be particularly interesting to study the super-horizon
evolution of the curvature perturbation $\zeta$ with a light-cone foliation of
the spacetime. In fact, the SU approach is used in [4, 5] to show the
conservation of $\zeta$ on super-horizon scales, for adiabatic pressure, at
linear and non-linear order in perturbation theory.

By pplying the gradient expansion to the non-linear ADM formalism [15] one can
provide a SU scheme in the uniform curvature gauge (UCG). It has been shown
that the shift vector has a decaying evolution, and therefore during the
exponential expansion of the universe, it can be considered as a first order
term in the gradient expansion. Also analyzing the consistency between the
Hamiltonian and momentum constraints, it has been shown that, taking into
account also the momentum constraints, the results differ only by a decaying
solution [15]. Moreover, it is shown that the additional information in the
momentum constraints should be $\mathcal{O}(\epsilon^{3})$ in the gradient
expansion. Thereby, for super-horizon perturbations, the SU scheme is a good
approximation.

In this manuscript, we will provide a SU picture for the LC [44] and GLC [29]
gauges. As we will see later, one difference with the previous works is that,
when we consider the LC gauge as an ADM decomposition, the shift vector does
not vanish, not even on the background (see, for instance [44]). In fact, in
the LC gauge, the shift vector describes the direction of observation. On the
other hand, we will neglect the divergence of the shift vector, which
describes the divergence of the direction of observation in the language of
$1+3$ formalism.

###  2.1 ADM formalism

In this section we provide the SU set of equations for generic perturbations.
Firstly, we introduce the ADM splitting and the $1+3$ evolution equations,
then we obtain general conditions which allow a SU evolution of the
perturbations. Thereafter, we show how also the LC gauge satisfies these
conditions. Starting with the ADM metric

$ds_{ADM}^{2}=-\mathcal{M}^{2}dt^{2}+f_{ij}\left(dx^{i}+N^{i}dt\right)\left(dx^{j}+N^{j}dt\right)\,,$
(2.1)

one can prove that, with a suitable choice of the coordinates,
$N^{i}=\mathcal{O}\left(\epsilon\right)$ and therefore
$\partial_{i}N^{i}=\mathcal{O}\left(\epsilon^{2}\right)$. This condition was
assumed in the references [5, 4], and it was proved in [15] considering the
UCG.

Let us now work with the ADM foliation of Eq. (2.1). Thanks to this
description of non-linear general perturbations, made on top of a FLRW
background, we will show how to recover a SU picture even if the shift vector
does not vanish in the background. It rather combines with the time derivative
to provide a derivative along the time-like motion.

The vector $n^{\mu}$ normal to the space-like hypersurfaces $t=const$ is given
by

$n^{\mu}=\frac{\partial^{\mu}t}{\left(-\partial_{\nu}t\partial^{\nu}t\right)^{\frac{1}{2}}}\,,$
(2.2)

which satisfies

$n_{\mu}=-\mathcal{M}\delta_{\mu}^{t}\,,\qquad
n^{\mu}\partial_{\mu}=\frac{1}{\mathcal{M}}\left(\partial_{t}-N^{i}\partial_{i}\right)\,,$
(2.3)

with correspondent induced metric given by

$f_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu}\,.$ (2.4)

This metric can be used to define the following induced quantities

$E\equiv n^{\mu}n^{\nu}T_{\mu\nu}\,,\quad\quad
p^{\mu}\equiv-f^{\mu\nu}n^{\rho}T_{\nu\rho}\,,\quad\quad S^{\mu\nu}\equiv
f^{\mu\rho}f^{\nu\sigma}T_{\rho\sigma}\,,$ (2.5)

where $E$ is the energy, $p^{\mu}$ is the energy flux (or momentum) and
$S^{\mu\nu}$ is stress tensor. Then, the standard energy-momentum tensor can
be written as

$T_{\mu\nu}=\rho n_{\mu}n_{\nu}+p_{\mu}n_{\nu}+p_{\nu}n_{\mu}+S_{\mu\nu}\,,$
(2.6)

where $\rho$ and $p=\frac{1}{3}f^{\mu\nu}S_{\mu\nu}$ are respectively the
energy density and the pressure of the given fluid.

Now we have everything that we need to present the decomposed Einstein
equations. These are developed in full details in [44], where the authors have
specialized to the LC gauge as a $1+1+2$ ADM foliation. As a starting point,
we can extract the energy (time-time) and momentum (time-space) constraints
respectively given by222From now on, for a generic tensor $C_{ij}$, we will
denote its trace with $C\equiv f^{ij}C_{ij}$.

${}^{(3)}R+\Theta_{n}^{2}-K_{ij}K^{ij}=\,2E\,,\qquad\qquad-
D_{j}K_{i}^{j}+D_{i}\Theta_{n}=\,p_{i}\,,$ (2.7)

where we defined the extrinsic curvature as
$K_{\mu\nu}\equiv\nabla_{(\mu}n_{\nu)}$. We can then also define the expansion
rate $\Theta_{n}$ as

$\Theta_{n}\equiv f^{\mu\nu}K_{\mu\nu}\,.$ (2.8)

The evolution of the induced metric $f_{ij}$ and of $K_{ij}$ is then obtained
from the space-space decomposition

$\displaystyle\left(\partial_{t}-\mathcal{L}_{N}\right)f_{ij}$
$\displaystyle=\,2\mathcal{M}K_{ij}\,,$
$\displaystyle\left(\partial_{t}-\mathcal{L}_{N}\right)K_{ij}$
$\displaystyle=\,\mathcal{M}\left[2K_{ik}K_{j}^{k}-K_{ij}\Theta_{n}-^{(3)}R_{ij}+S_{ij}-\frac{1}{2}f_{ij}\left(S-E\right)\right]+D_{i}D_{j}\mathcal{M}\,,$
(2.9)

where $\mathcal{L}_{N}$ is the Lie derivative along the field $N^{i}$.

Finally, the equations for the matter sector $\nabla_{\mu}T^{\mu\nu}=0$ are
given by

$\displaystyle\left(\partial_{t}-\mathcal{L}_{N}\right)E$
$\displaystyle=\,-D_{i}\left(\mathcal{M}p^{i}\right)-\mathcal{M}\left(\Theta_{n}E+K_{ij}S^{ij}\right)\,,$
$\displaystyle\left(\partial_{t}-\mathcal{L}_{N}\right)p_{i}$
$\displaystyle=\,-D_{j}\left(S_{i}^{j}\mathcal{M}\right)-\mathcal{M}\Theta_{n}p_{i}-ED_{i}\mathcal{M}\,.$
(2.10)

Let us now follow the decomposition of [26] by extracting the shape-preserving
volume expansion out of the spatial metric. First, we make a conformal re-
scaling of $f_{ij}$ to

$f_{ij}\equiv e^{2\Xi}\hat{f}_{ij}\,,$ (2.11)

by requiring that the determinant $\hat{f}=\text{det}[\hat{f}_{ij}]=1$. In
this way, we can interpret $e^{\Xi}$ as the local effective scale factor.
Then, we re-scale accordingly also the other quantities

$\displaystyle\hat{A}_{ij}=$
$\displaystyle\,e^{-2\Xi}\left(K_{ij}-\frac{1}{3}f_{ij}\Theta_{n}\right)\,,$
$\,{}^{(3)}\hat{\mathcal{R}}_{ij}=$
$\displaystyle\,e^{-2\Xi}\left({}^{(3)}R_{ij}-\frac{1}{3}\,^{(3)}R\,f_{ij}\right)\,,$
$\displaystyle\hat{\mathcal{S}}_{ij}=$
$\displaystyle\,e^{-2\Xi}\left(S_{ij}-\frac{1}{3}f_{ij}S\right)\,.$ (2.12)

Before applying the decomposition (2.12) to Eqs. (2.9) and (2.10), we define
$\mathcal{M}\frac{d}{d\widetilde{\lambda}}\equiv\left(\partial_{t}-\mathcal{L}_{N}\right)$
and compute the trace of the evolution of $A_{ij}\equiv e^{2\Xi}\hat{A}_{ij}$

$\displaystyle f^{ij}\left(\partial_{t}-\mathcal{L}_{N}\right)A_{ij}=$
$\displaystyle\frac{1}{\mathcal{M}}\left[f^{ij}f_{il}\frac{d}{d\widetilde{\lambda}}\left(f_{jk}A^{lk}\right)+f^{ij}f_{jk}A^{lk}\frac{d}{d\widetilde{\lambda}}f_{il}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\mathcal{M}}\frac{d}{d\widetilde{\lambda}}\left(f_{lk}A^{lk}\right)+f^{ij}f_{jk}A^{lk}2K_{il}$
$\displaystyle=$ $\displaystyle 2A_{ij}A^{ij}=2\hat{A}_{ij}\hat{A}^{ij}\,,$
(2.13)

where we have used the first of Eqs. (2.9) from the first to the second line,
and the fact that $A_{ij}$ is trace-less from the second to the last line.

Now we can apply the decomposition (2.12) to Eqs. (2.9), using also Eqs. (2.7)
and (2.13), to obtain

$\displaystyle\frac{d\Theta_{n}}{d\widetilde{\lambda}}=$
$\displaystyle-\frac{\Theta_{n}^{2}}{3}-\hat{A}_{ij}\hat{A}^{ij}-\frac{1}{2}\left(E+S\right)+\frac{1}{\mathcal{M}}D^{2}\mathcal{M}\,,$
$\displaystyle\frac{d\Xi}{d\widetilde{\lambda}}=$
$\displaystyle\frac{\Theta_{n}}{3}\,,$
$\displaystyle\frac{d\hat{f}_{ij}}{d\widetilde{\lambda}}=$ $\displaystyle
2\hat{A}_{ij}\,.$ (2.14)

Then the evolution of the trace-less quantity $\hat{A}_{ij}$ is given by

$\frac{d\hat{A}_{ij}}{d\widetilde{\lambda}}=-\frac{1}{3}\Theta_{n}\hat{A}_{ij}+2e^{-2\Xi}A_{ik}A_{j}^{k}+\hat{\mathcal{S}}_{ij}-\,^{(3)}\hat{\mathcal{R}}_{ij}+\frac{1}{\mathcal{M}}\left(D_{i}D_{j}-\frac{1}{3}f_{ij}D^{2}\right)\mathcal{M}\,.$
(2.15)

At this point, we will apply our gradient expansion scheme without any gauge
fixing. Hence, considering that
$n^{\mu}=\frac{1}{\mathcal{M}}\left(1,-N^{i}\right)$, for a generic tensor
$l_{ij}$, it holds

$\frac{1}{\mathcal{M}}\left(\partial_{t}-\mathcal{L}_{N}\right)l_{ij}=n^{\mu}\partial_{\mu}l_{ij}+l_{ik}\partial_{j}N^{k}+l_{jk}\partial_{i}N^{k}=\frac{d}{d\lambda}l_{ij}+\mathcal{O}\left(\epsilon^{2}\right)\,,$
(2.16)

where we define $n^{\mu}\partial_{\mu}\equiv\frac{d}{d\lambda}$ and then we
have
$\frac{d}{d\widetilde{\lambda}}=\frac{d}{d\lambda}+\mathcal{O}\left(\epsilon^{2}\right)$.
This corresponds to neglect terms proportional to
$\partial_{i}N^{j}=\mathcal{O}\left(\epsilon^{2}\right)$ in Eq. (2.16), as
done also in [15, 26]. This is our main assumption and comes from the fact
that, due to the spatial derivatives, the momentum constraint of Eq. (2.7) is
a first order relation, i.e. $p_{i}=\mathcal{O}\left(\epsilon\right)$. Hence,
using Eq. (2.5), we get

$\displaystyle p_{i}=-\frac{1}{\mathcal{M}}\left(T_{it}-N^{j}T_{ij}\right)\,.$
(2.17)

Such Eq. (2.17) can be satisfied either by the strong condition

$\displaystyle T_{it}\sim N^{i}=\mathcal{O}\left(\epsilon\right)\,,$ (2.18)

or by the weaker condition that only the combination on its r.h.s. is
$\mathcal{O}(\epsilon)$. Along this paper, we will adopt the stronger
condition to justify our claim that
$\partial_{i}N^{j}=\mathcal{O}(\epsilon^{2})$, in agreement with [15, 5, 4,
26].

Moreover, just as done in some previous works [15, 5, 4], we will also neglect
${}^{(3)}R_{ij}\thicksim
R\thicksim\hat{\mathcal{S}}_{ij}=\mathcal{O}\left(\epsilon^{2}\right)$ since
all of these terms contain double spatial derivatives. For what concerns the
anisotropic stress $\hat{\mathcal{S}}_{ij}$, this is given by combinations of
double spatial derivatives acting on the scalar fields in the matter sector.
The condition $\hat{\mathcal{S}}_{ij}=\mathcal{O}(\epsilon^{2})$ was relaxed
in [26] only for Bianchi geometries and on [27, 28] due to the presence of
gauge fields.

With these considerations, we can decompose again the metric evolution
provided by the first of Eqs. (2.9) thanks Eqs. (2.12). Hence, at first order
in the gradient expansion, we obtain333Note that, although Eqs. (2.19) are
very similar to Eqs. (2.14), after the gradient expansion we have replaced
$\frac{d}{d\tilde{\lambda}}=\frac{d}{d\lambda}+\mathcal{O}(\epsilon^{2})$.

$\frac{d\Xi}{d\lambda}=\frac{\Theta_{n}}{3}\,,\qquad\qquad\frac{d\hat{f}_{ij}}{d\lambda}=2\hat{A}_{ij}\,.$
(2.19)

Moreover, we can prove that $\hat{A}_{ij}$ is at least a second order term in
the gradient expansion at every order in perturbation theory. In fact,
following [15], from Eq. (2.15) we get

$\frac{d}{d\lambda}\hat{A}_{ij}=-\frac{1}{3}\Theta_{n}\hat{A}_{ij}+2\hat{A}_{ik}\hat{A}_{j}^{k}+\mathcal{O}\left(\epsilon^{2}\right)\,,$
(2.20)

and we choose a coordinate system such that $A_{ij}$ vanishes on the
background444This is a quite general condition for isotropic spaces and, as we
will show later, this is also the case for an isotropic LC background. Hence,
at $\mathcal{O}(\delta)$ in perturbation theory, we get that
$\hat{A}_{ik}\hat{A}_{j}^{k}\sim\mathcal{O}(\delta^{2})$ and then Eq. (2.20),
with first of Eqs. (2.19), becomes

$\frac{d}{d\lambda}\hat{A}_{ij}=-\frac{d\Xi}{d\lambda}\hat{A}_{ij}+\mathcal{O}\left(\delta^{2},\epsilon^{2}\right)\,.$
(2.21)

The latter equation is clearly solved by

$\hat{A}_{ij}\propto e^{-\Xi}\,,$ (2.22)

and proves that the $\mathcal{O}(1)$ term in the gradient expansion of
$\hat{A}_{ij}$ decays when $\Xi$ grows in time. Then it can be neglected.
Therefore, we obtain that $\hat{A}_{ij}$ is at least first order in
$\epsilon$. As a consequence, $\hat{A}_{ik}\hat{A}_{j}^{k}$ is not only second
order in $\delta$ expansion but also at least
$\mathcal{O}\left(\epsilon^{2}\right)$. Thanks to this result, this proof can
be repeated iteratively to any order $n$-th in perturbation theory, leading
then to

$\frac{d}{d\lambda}\hat{A}_{ij}=-\frac{d\Xi}{d\lambda}\hat{A}_{ij}+\mathcal{O}\left(\delta^{n+1},\epsilon^{2}\right)\,.$
(2.23)

The solution in Eq. (2.22) solves also Eq. (2.23) and this proves that also
the term $\hat{A}_{ij}$, which is $\mathcal{O}(\epsilon)$, is decaying and can
be neglected. So we have proven our initial claim that $\hat{A}_{ij}$ is at
least of order $\epsilon^{2}$. Hence, considering the evolution of the spatial
metric in Eqs. (2.19), we have that

$\frac{d}{d\lambda}\hat{f}_{ij}=\mathcal{O}\left(\epsilon^{2}\right)\,,$
(2.24)

at any order in perturbation theory.

The above analysis was performed in [15] leading also to
$N^{i}=\mathcal{O}\left(\epsilon\right)$ when $N^{i}$ vanishes at the
background. For the sake of clarity, we underline that in [15] the UCG has
been fixed and then $\hat{f}_{ij}$ corresponds to the tensor modes. This also
shows that the evolution of the tensor modes can be neglected at linear order
in $\epsilon$.

Finally, our complete set of equations to order
$\mathcal{O}\left(\delta^{n+1},\epsilon^{2}\right)$ is given by the energy and
momentum constraints

$E=\frac{\Theta_{n}^{2}}{3}\,,\hskip
85.35826ptp_{i}=\frac{2}{3}D_{i}\Theta_{n}\,,$ (2.25)

with their respective evolution equations given by

$\frac{dE}{d\lambda}=-\Theta_{n}\left(E+\frac{1}{3}S\right)\,,\hskip
56.9055pt\frac{dp_{i}}{d\lambda}=-\frac{1}{3\mathcal{M}}D_{i}\left(\mathcal{M}S\right)-\Theta_{n}p_{i}\,.$
(2.26)

In order to complete our set of SU equations we also need the decomposed
spatial metric evolution given by Eqs. (2.19) and (2.24). Also we have that
the expansion rate evolution is given by

$\displaystyle\frac{d\Theta_{n}}{d\lambda}=$
$\displaystyle-\frac{1}{3}\Theta_{n}^{2}-\frac{1}{2}\left(S+E\right)\,.$
(2.27)

As one can easily see, Eqs. (2.19) and (2.24)-(2.27), valid at first order in
the gradient expansion and to all orders in perturbation theory, exactly
correspond to the homogeneous and isotropic background equations if one
neglects the momentum constraint. These then prove that the condition

$\left(\partial_{t}-\mathcal{L}_{N}\right)f_{ij}=\frac{1}{\mathcal{M}}\frac{d}{d\lambda}f_{ij}+\mathcal{O}\left(\delta^{n+1},\,\epsilon^{2}\right)\,,$
(2.28)

with the fact that $\partial_{i}N^{i},\,\hat{\mathcal{S}}_{ij},\,R_{ij}$ and
$R$ are $\mathcal{O}\left(\delta^{n+1},\,\epsilon^{2}\right)$, reproduces the
SU picture, and matches with previous works [15, 5, 4] if $\lambda=t$ and
$N^{i}\partial_{i}=\mathcal{O}(\epsilon^{2})$.

In the next section, we will specialize this construction to non-linear LC
perturbations on the top of a FLRW background. Thanks to the freedom of the
choice of the lapse function in the LC gauge, we will then provide a general
SU formalism. Furthermore, within the synchronous fixing of the lapse
function, our formalism will be extended also to the GLC gauge.

##  3 Light-Cone gauge

Let us now introduce the LC gauge [44]. This is a generalization of the GLC
gauge [29], where the lapse function is left unfixed and is built as a
foliation of the spacetime thanks to a set of four coordinates adapted to the
observed past light-cone. In particular, the proper time of a generic observer
is described by the coordinate $t$. This corresponds to the proper-time of a
free-falling observer when the GLC fixing of the lapse function occurs. The
coordinates $w$ and $\theta^{a}$ satisfy the same properties as in the GLC
gauge, i.e. $w$ describes the observer’s past light-cone and
$\theta^{a}=const$ describes the light-like geodesics. Given that, the non-
linear line element is [44]

$ds_{LC}^{2}=\Upsilon^{2}dw^{2}-2\mathcal{M}\Upsilon
dwdt+\gamma_{ab}\left(d\theta^{a}-U^{a}dw\right)\left(d\theta^{b}-U^{b}dw\right)\,.$
(3.1)

In this case, the vector $n^{\mu}$ in Eq. (2.2) is given by

$n^{\mu}=\left(\frac{1}{\mathcal{M}},\,\frac{1}{\Upsilon},\,\frac{U^{a}}{\Upsilon}\right)\,.$
(3.2)

The advantage of the metric (3.1) is that it simplifies the description of
light-like signal. For instance, the light-like geodesics are exactly solved
by $k_{\mu}=-\omega\delta_{\mu}^{w}$, where $k_{\mu}$ is the four-momentum of
the photon and $\omega$ is its physical frequency. Moreover, for the GLC gauge
where $\mathcal{M}=1$, also the time-like geodesic is exactly solved by
$u_{\mu}=-\partial_{\mu}t$. In this case $u_{\mu}=n_{\mu}$ is the four-
velocity of the geodesic observer and is perpendicular to the three-
dimensional hypersurfaces of $t=const$. This particular choice simplifies the
description of late-time cosmological observables and allows a completely non-
linear description of such observables as a factorization of the metric
entries [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. As relevant
examples, the expressions for the cosmological redshift and the angular
distance are given, in a exact way and for an arbitrary geometry, directly as
[29, 33]

$1+z=\frac{\left(u_{\mu}k^{\mu}\right)_{s}}{\left(u_{\nu}k^{\nu}\right)_{o}}=\frac{\Upsilon_{o}}{\Upsilon_{s}}\,,\qquad\qquad
d^{2}_{A}=\frac{\sqrt{\gamma}}{\left(\frac{\text{det}\partial_{\tau}\gamma_{ab}}{4\sqrt{\gamma}}\right)_{o}}\,,$
(3.3)

where $z$ is the redshift of the source, the subscript $s$ and $o$ stands for
a quantity evaluated at the source and observer position, and $\gamma$ is the
determinant of $\gamma_{ab}$.

###  3.1 LC gauge shift vector

Here, we will use $N_{LC}^{i}$ (instead of $N^{i}$)555For the LC metric, Latin
indices will always refer to the coordinates $w$ and $\theta^{a}$. to describe
the shift vector of the LC gauge, thus avoiding confusion when we relate
$N_{LC}^{i}$ to the standard $N^{i}$. Hereafter, we will provide a SU picture
allowing for different lapse function fixings within the LC gauge [44]. This
is a crucial step to obtain the $\delta\mathcal{N}$ formalism on the past
light-cone.

In [44] was shown that the non-linear LC gauge can be interpreted as a $1+1+2$
ADM decomposition with coordinates $x^{\mu}=\left(t,w,\theta^{a}\right)$. This
proviso, the shift vector for the first $1+3$ decomposition is then given by

$N_{LC}^{i}=-\mathcal{M}\left(\frac{1}{\Upsilon},\frac{U^{a}}{\Upsilon}\right)\,,\qquad\qquad
N^{LC}_{j}=-\Upsilon\delta_{j}^{w}\,.$ (3.4)

For what concerns the shift vector $N^{i}_{LC}$, this is orthogonal to the
surface at constant $t$ and $w$. Hence, if we recall that the photon four-
momentum in the LC coordinates is
$k^{\mu}=\omega\mathcal{M}^{-1}\Upsilon^{-1}\delta^{\mu}_{t}$, within the 1+3
decomposition we can write the shift vector as

$N^{i}_{LC}=\frac{k^{i}}{\omega}-n^{i}\,.$ (3.5)

Hence, since $n^{\mu}$ is a time-like vector, $N^{i}_{LC}$ can be interpreted
as the space-like component of the propagation direction of an incoming photon
(see [46]). To completely fix the LC gauge, we still have to fix three
conditions. These are given by the following ones

$f_{ww}=\Upsilon^{2}+U^{2}\,,\qquad\qquad f_{wa}=-U_{a}\,,\qquad\qquad
f_{ab}=\gamma_{ab}\,.$ (3.6)

As one can see, $N_{LC}^{w}=\frac{1}{a}$ on the background level, so it does
not vanish when $\epsilon\rightarrow 0$. We then have that
$N_{LC}^{w}=\mathcal{O}\left(1\right)$. With this choice of coordinates,
however, we will show that the condition
$\partial_{i}N_{LC}^{i}=\mathcal{O}\left(\epsilon^{2}\right)$ holds. In order
to do so, we first perform a finite background coordinate transformation on
the metric in Eq. (2.1), from the $x^{i}=\left(r,\theta^{a}\right)$
coordinates to the light-cone ones $y^{i}=\left(w,\theta^{a}\right)$, given by

$dr=dw-\frac{dt}{a}\,,\qquad\qquad d\theta^{a}=d\theta^{a}\,,\qquad\qquad
dt=dt\,,$ (3.7)

in order to relate $N^{i}$ to $N_{LC}^{i}$. A direct computation for the
controvariant components of $N^{i}$ returns that

$N^{r}=\frac{1}{a}-\frac{\mathcal{M}}{\Upsilon}\,,\qquad\qquad
N^{a}=-\frac{U^{a}}{a}\left(1-\frac{1}{a}+\frac{\mathcal{M}}{\Upsilon}\right)\,,$
(3.8)

or, equivalently, in a covariant form

$N_{r}=-\mathcal{M}\Upsilon+\frac{\Upsilon^{2}+U^{2}}{a}\,,$ (3.9)

and

$N_{a}=-\frac{U_{a}}{a}\,.$ (3.10)

Finally, using Eqs. (3.4) and (3.8), the gradient expansion condition of Eq.
(2.18) given by $N^{i}=\mathcal{O}(\epsilon)$ returns
$\partial_{i}N^{i}=\partial_{i}N_{LC}^{i}=\mathcal{O}\left(\epsilon^{2}\right)$.
Within the gradient expansion, we then have

$\displaystyle\partial_{r}N^{r}=$
$\displaystyle-\partial_{w}\left(\frac{\mathcal{M}}{\Upsilon}\right)=\mathcal{O}\left(\epsilon^{2}\right)\,,$
$\displaystyle\partial_{a}N^{a}=$
$\displaystyle-\frac{1}{a}\left(1-N^{r}\right)\partial_{a}U^{a}+U^{a}\partial_{a}N^{r}=\mathcal{O}\left(\epsilon^{2}\right)\,,$
(3.11)

which indeed show that
$\partial_{w}\Upsilon^{-1}\sim\partial_{w}\mathcal{M}\sim\partial_{a}U^{a}=\mathcal{O}\left(\epsilon^{2}\right)$.
This comes from the fact that both $\mathcal{M}$ and $\Upsilon$ have
background counterparts, therefore, both $\partial_{w}\Upsilon^{-1}$ and
$a^{-1}\partial_{w}\mathcal{M}$ are of order $\epsilon^{2}$. The condition
$\partial_{a}U^{a}=\mathcal{O}(\epsilon^{2})$ is obtained by using the fact
that $N^{r}=\mathcal{O}(\epsilon)$ on the second of Eqs. (3.11).

###  3.2 Separate Light-Cones

As it has been shown in Sect. 2.1, one can still obtain a SU picture when the
shift vector combines to form an integral along the geodesics and only its
spatial derivatives are neglected. We will prove that this is the case for
$\partial_{i}N^{i}$ in Eq. (2.18) and for $\partial_{i}N_{LC}^{i}$ in Eq.
(3.11). Additionally, we need the trace-less part of the extrinsic curvature
$\hat{A}_{ij}$ to be negligible in order to obtain the SU scheme for the LC
metric.

The condition for the shift vector is given by

$\partial_{i}N_{LC}^{i}=\partial_{w}\left(\frac{\mathcal{M}}{\Upsilon}\right)+\partial_{a}\left(\frac{\mathcal{M}U^{a}}{\Upsilon}\right)=\mathcal{O}\left(\epsilon^{2}\right)\,,$
(3.12)

which, with Eqs. (2.9), implies

$\frac{d}{d\lambda}f_{ij}=2K_{ij}+\mathcal{O}\left(\epsilon^{2}\right)\,,$
(3.13)

where in $\mathcal{L}_{N}f_{ij}$ we have neglected $\partial_{i}N^{i}$ but not
$N^{i}\partial_{i}f_{jk}$. In fact, the latter combines with
$\partial_{t}f_{ij}$ to reconstruct $\frac{d}{d\lambda}f_{ij}$, following the
general prescription given in Eq. (2.16). One may note the similarity between
Eqs. (3.13) and (2.9). This is because Eq. (2.9) is the non-perturbative
version in the gradient expansion of Eq. (3.13), where we consider instead the
gradient expansion on the parameter
$\frac{d}{d\tilde{\lambda}}=\frac{d}{d\lambda}+\mathcal{O}(\epsilon^{2})$.

Using Eqs. (3.6) at the background level, i.e.

$f_{ww}=a^{2}\,,\qquad\qquad f_{wa}=0\,,\qquad\qquad
f_{ab}=a^{2}r^{2}\bar{q}_{ab}\,,$ (3.14)

one gets that $df_{ij}/d\lambda=2Hf_{ij}$, where $H$ is the background
expansion rate defined as $H\equiv\bar{\Theta}_{n}/3$, $\bar{\Theta}_{n}$ is
the extrinsic curvature on the background and
$\bar{q}_{ab}=\text{diag}\left(1,\sin\theta\right)$. Thus, from Eq. (3.13), we
also see that $\hat{A}_{ij}$ vanishes on the background. Following the same
procedure adopted in Eqs. (2.21) and (2.23), we get that
$\hat{A}_{ij}=\mathcal{O}\left(\epsilon^{2}\right)$ also when the LC gauge is
fixed.

Now, by taking the trace of Eq. (3.13), we obtain

$\displaystyle\Theta_{n}$
$\displaystyle=\,\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}n^{\mu}\right)$
$\displaystyle=\,\frac{1}{\mathcal{M}\Upsilon\sqrt{\gamma}}\frac{d}{d\lambda}\left(\mathcal{M}\Upsilon\sqrt{\gamma}\right)+\partial_{\mu}n^{\mu}$
$\displaystyle=\,\frac{1}{\Upsilon\sqrt{\gamma}}\frac{d}{d\lambda}\left(\Upsilon\sqrt{\gamma}\right)+\frac{1}{\mathcal{M}}\frac{d}{d\lambda}\mathcal{M}+\partial_{\mu}n^{\mu}$
$\displaystyle=\,\frac{1}{\Upsilon\sqrt{\gamma}}\frac{d}{d\lambda}\left(\Upsilon\sqrt{\gamma}\right)+\frac{1}{\mathcal{M}}\left[\partial_{w}\left(\frac{\mathcal{M}}{\Upsilon}\right)+\partial_{a}\left(\frac{\mathcal{M}U^{a}}{\Upsilon}\right)\right]$
$\displaystyle=\,\frac{1}{\Upsilon\sqrt{\gamma}}\frac{d}{d\lambda}\left(\Upsilon\sqrt{\gamma}\right)-\frac{1}{\mathcal{M}}\partial_{i}N_{LC}^{i}\,.$
(3.15)

Thanks to this last equation, within the fixing $\mathcal{M}=1$, we realize
that the difference between $\Theta_{u}\equiv\nabla_{\mu}u^{\mu}$ and
$\Theta_{n}$ is of order $\epsilon^{2}$. We also have from Eq. (2.19), where
we neglect $\partial_{i}N^{i}_{LC}$ in $\Theta_{n}$

$\frac{d\,\Xi}{d\lambda}=\frac{1}{3\Upsilon\sqrt{\gamma}}\frac{d\left(\Upsilon\sqrt{\gamma}\right)}{d\lambda}+\mathcal{O}\left(\epsilon^{2}\right)\,,$
(3.16)

which preserves the background form.

Considering now the homogeneous and isotropic LC background, namely
$\bar{\Upsilon}\sqrt{\bar{\gamma}}=a^{3}r^{2}$, and also identifying
$\bar{E}\equiv\bar{\rho}$ and $\frac{1}{3}\bar{S}\equiv\bar{p}$, Eqs. (2.25)
return

$\displaystyle\frac{\bar{\Theta}_{n}^{2}}{3}=3H^{2}=3\left(\frac{\partial_{t}a}{a}\right)^{2}=\bar{\rho}\,,\hskip
170.71652pt\text{background}$
$\displaystyle\frac{{\Theta_{n}}^{2}}{3}=\frac{1}{3}\left[\frac{1}{(\Upsilon\sqrt{\gamma})}\frac{d\left(\Upsilon\sqrt{\gamma}\right)}{d\lambda}\right]^{2}=\rho+\mathcal{O}\left(\delta^{n},\epsilon^{2}\right)\,.\hskip
56.9055pt\text{non-perturbative}$ (3.17)

Moreover, Eqs. (2.26) and (2.27) give at the background level

$\partial_{t}\bar{\rho}=\,-3H\left(\bar{\rho}+\bar{p}\right)\,,\qquad\qquad\partial_{t}\bar{\Xi}=\,H\,,\qquad\qquad\partial_{t}H=\,-3H^{2}-\frac{1}{2}\left(\bar{\rho}+3\bar{p}\right)\,.$
(3.18)

Here we remark that Eqs. (3.17) mean that the non-linear LC perturbations in a
FLRW universe at first order in the gradient expansion do evolve as a set of
glued background universes with a different set of $\left(a,\,\rho,\,p\right)$
in each patch. Within this picture, then $\Upsilon\sqrt{\gamma}$ can be linked
to the effective local scale factor. To complete the set of equations that
have the same form for the background (last of Eq. (3.18)) and the perturbed
universe, we also have from Eq. (2.27)

$\frac{d}{d\lambda}\left[\frac{1}{(\Upsilon\sqrt{\gamma})}\frac{d(\Upsilon\sqrt{\gamma})}{d\lambda}\right]=-\frac{1}{3}\left[\frac{1}{(\Upsilon\sqrt{\gamma})}\frac{d(\Upsilon\sqrt{\gamma})}{d\lambda}\right]^{2}-\frac{1}{2}\left(3p+\rho\right)+\mathcal{O}\left(\delta^{n},\epsilon^{2}\right)\,.$
(3.19)

Therefore, the SU scheme holds with the universe evolving as a set of
homogeneous and isotropic LC background, where $\lambda$ provides the
evolution of inhomogeneities along $n^{\mu}$. So far we have provided a
consistent SU on the past light-cone in terms of LC gauge entries, which is a
fundamental step to provide the $\delta\mathcal{N}$ formalism on the past
light-cone in the next Sect. 4.

###  3.3 The Geodesic Light-Cone gauge

Let us now apply the SU scheme described in the previous subsection to the
case of the GLC gauge given by Eq. (3.1) with $\mathcal{M}=1$, and show how it
simplifies the evolution of the density perturbations on the past light-cone.
This gauge automatically provides $\nabla_{\mu}u^{\mu}$=$\nabla_{\mu}n^{\mu}$,
then, one can relate the expansion of the 3D-hypersurfaces orthogonal to
$n^{\mu}$ to the matter content described in terms of $u^{\mu}$.

Hence, we recall that the comoving four velocity is given by
$u^{\mu}=\left(1,\Upsilon^{-1},\Upsilon^{-1}U^{a}\right)$ [46, 29], we then
have that the expansion of the 3D-hypersurfaces is

$\Theta_{u}=\nabla_{\mu}u^{\mu}=\frac{\partial_{t}\Upsilon}{\Upsilon}+\frac{\gamma^{ab}\partial_{t}\gamma_{ab}}{2}+\frac{\gamma^{ab}}{2\Upsilon}\partial_{w}\gamma_{ab}+\frac{1}{\Upsilon}\partial_{a}U^{a}+\frac{U^{a}\gamma^{bc}}{2\Upsilon}\partial_{a}\gamma_{bc}\,,$
(3.20)

which can be re-written in a more suitable form as

$\Theta_{u}=\frac{1}{\Upsilon\sqrt{\gamma}}\frac{d\left(\Upsilon\sqrt{\gamma}\right)}{d\lambda}+\partial_{\mu}u^{\mu}\,,$
(3.21)

where $\frac{d}{d\lambda}\equiv u^{\mu}\partial_{\mu}$ accounts for
inhomogeneities along the geodesics and
$\partial_{\mu}u^{\mu}=\partial_{i}N^{i}$ is $\mathcal{O}(\epsilon^{2})$, as
shown in Eqs. (3.11).

An interesting feature of Eq. (3.21) is that the first term contributes both
to the background and to the perturbative level whereas the last term
contributes only to the perturbative level. We thus obtain a separate universe
description using Eq. (3.21) as aimed.

In order to provide the conservation of $\zeta$, when the pressure is
adiabatic, we need to analyze the energy-momentum conservation in the GLC
gauge for the case of a perfect fluid. Starting from

$T_{\mu\nu}=\left(\rho+p\right)u_{\mu}u_{\nu}+g_{\mu\nu}p\,,$ (3.22)

the conservation law along the direction of $u^{\nu}$, i.e.
$u^{\nu}\nabla_{\mu}T_{\,\nu}^{\mu}=0$, exactly returns

$\frac{d\rho}{d\lambda}=-\left(\rho+p\right)\Theta_{u}\,.$ (3.23)

Hence, by using Eq. (3.21), we have

$\displaystyle\frac{d\rho}{d\lambda}=-\left(\rho+p\right)\left[\frac{1}{\Upsilon\sqrt{\gamma}}\frac{d\left(\Upsilon\sqrt{\gamma}\right)}{d\lambda}+\partial_{\mu}u^{\mu}\right]\,.$
(3.24)

Eq. (3.24) is a fully non-linear relation between geometry and matter content.
As an important remark, since Eq. (3.24) is a fully non-linear equation, it
can be seen as a dynamic equation for the exact density perturbations.

According to what outlined so far, within the gradient expansion, where
$\partial_{\mu}u^{\mu}=\partial_{i}N^{i}=\mathcal{O}\left(\epsilon^{2}\right)$,
Eq. (3.24) can then be written as

$\displaystyle\frac{d\rho}{d\lambda}=$
$\displaystyle-\left(\rho+p\right)\frac{1}{\Upsilon\sqrt{\gamma}}\frac{d\left(\Upsilon\sqrt{\gamma}\right)}{d\lambda}+\mathcal{O}(\epsilon^{2})\,.$
(3.25)

In general, pressure and energy density are linked by an equation of state as
$p=q\rho$. The value of $q$ can be time-dependent, accordingly to the specific
era when inhomogeneities are evolving (as happens, for instance, during the
slow-roll inflationary stage). This makes Eq. (3.25) in general quite
complicated to be solved. However, during the late time epochs (e.g.
radiation, matter or cosmological constant dominated universe), $q$ is
constant. In this case, Eqs. (3.25) becomes

$\displaystyle\frac{d\rho}{d\lambda}=$
$\displaystyle-\rho\frac{1+q}{\Upsilon\sqrt{\gamma}}\frac{d\left(\Upsilon\sqrt{\gamma}\right)}{d\lambda}+\mathcal{O}(\epsilon^{2})\,,$
(3.26)

which is exactly solved by

$\rho(\lambda)=A\left(\Upsilon\sqrt{\gamma}\right)^{-(1+q)}(\lambda)+\mathcal{O}(\epsilon^{2})\,,$
(3.27)

where $A$ is a constant. Eq. (3.27) gives the exact link between the geometry
and the energy density in terms of light-cone metric entries. It is also the
starting point to describe non-linear features of inhomogeneities on super-
horizon scales. In fact, in the following we will discuss how
$\Upsilon\sqrt{\gamma}$ relates with the gauge invariant curvature
perturbation $\zeta$ and how both can be computed using the
$\delta\mathcal{N}$ formalism. Furthermore, we will show under which
conditions the quantity $\zeta$ is conserved on super-horizon scales.

##  4 Curvature perturbation evolution

In this section we will study the curvature perturbations along the past
light-cone using the SU scheme developed in the previous section. To this aim,
we will start by considering the linear order in perturbation theory, and by
using the scalar-pseudoscalar decomposition developed in [36, 42]. Hence, we
will first review this perturbation theory, following the approach of [4, 5]
to show the conservation of $\zeta$ along the past light-cone. Thereafter, we
red will generalize this proof to non-linear order in the amplitude of the
perturbations and to first order in the gradient expansion, which allow us to
obtain $\zeta$ at this perturbative level in terms of light-cone
perturbations. Finally, we will generalize the $\delta\mathcal{N}$ formalism
on the past light-cone.

###  4.1 Linear evolution and comparison with previous results

In this section, we want to linearize Eqs. (3.26). To this aim, we first
recall the linear perturbation theory for the GLC coordinates, presented in
[36, 42]. Firstly, we consider general perturbations, i.e. without fixing the
GLC gauge. The metric and its perturbed inverse are then given by

$g_{\mu\nu}=\bar{g}^{\,GLC}_{\mu\nu}+\delta
g_{\mu\nu}=a^{2}\left\\{\begin{pmatrix}0&-a^{-1}&\vec{0}\\\
-a^{-1}&1&\vec{0}\\\
\vec{0}^{\mathbf{T}}&\vec{0}^{\mathbf{T}}&\bar{\gamma}_{ab}\end{pmatrix}+\begin{pmatrix}L&M&V_{b}\\\
M&N&\mathcal{U}_{b}\\\
V_{a}^{\mathbf{T}}&\mathcal{U}_{a}^{\mathbf{T}}&\delta\gamma_{ab}\end{pmatrix}\right\\}\,,$
(4.1)

and

$\delta
g^{\mu\nu}=\begin{pmatrix}-\left(a^{2}L+N+2aM\right)&-a^{-1}\left(a^{2}L+aM\right)&-a\left(aV^{a}+\mathcal{U}^{a}\right)\\\
-a^{-1}\left(a^{2}L+aM\right)&-L&aV^{a}\\\
-a\left(aV^{a}+\mathcal{U}^{a}\right)&aV^{a}&-a^{-2}\delta\gamma^{ab}\end{pmatrix}\,,$
(4.2)

where, following [36, 42], we can decompose $\mathcal{U}_{a},V_{a}$ and
$\delta\gamma_{ab}$ as

$\displaystyle\mathcal{U}_{a}$
$\displaystyle=r^{2}\left(D_{a}u+\tilde{D}_{a}\hat{u}\right)\,,$
$\displaystyle V_{a}$
$\displaystyle=r^{2}\left(D_{a}v+\tilde{D}_{a}\hat{v}\right)\,,$
$\displaystyle\delta\gamma_{ab}$
$\displaystyle=a^{2}r^{2}\left[\left(1+2\nu\right)\bar{q}_{ab}+D_{ab}\mu+\tilde{D}_{ab}\hat{\mu}\right]\,.$
(4.3)

Here, $u,\,v,\,\nu$ and $\mu$ are scalars, and $\hat{u},\,\hat{v}$ and
$\hat{\mu}$ are pseudoscalar degrees of freedom under spatial rotations.
Moreover, the angular derivatives are defined as

$D_{ab}=D_{(a}D_{b)}-\frac{1}{2}q_{ab}D^{2}\,,\qquad\qquad\tilde{D}_{ab}=D_{(a}\tilde{D}_{b)}\,,$
(4.4)

where $\tilde{D}_{a}=\epsilon_{a}^{b}D_{b}$, and $\epsilon_{a}^{b}$ is the
anti-symmetric tensor. We remark that $D_{ab}$ and $\tilde{D}_{ab}$ are trace-
less, so that the trace of $\delta\gamma_{ab}$ is given by the trace of
$q_{ab}=\left(1+2\nu\right)\bar{q}_{ab}$.

At this point, let us also linearize the energy density as

$\rho=\bar{\rho}\left(1+\delta\rho\right)\,.$ (4.5)

In this expansion, however, we keep both $\bar{\rho}$ and $\delta\rho$ as
function of the time-like parameter $\lambda$. By doing this, we then keep
track of the perturbations as projected onto the exact time-like geodesic.
Starting from Eqs. (4.3), we then obtain that

$\sqrt{\gamma}=a^{2}\sqrt{\bar{\gamma}}\,\left(1+2\nu\right)\,.$ (4.6)

At this point, we want to use the metric in Eq. (4.1) to compute the expansion
volume $\Theta_{n}=\nabla_{\mu}n^{\mu}$ of the hypersurfaces orthogonal to $t$
defined in Eq. (2.2). Then, the volume expansion will be given by

$\Theta_{n}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}\,n^{\mu}\right)=\frac{1}{2}g^{\alpha\beta}\frac{d}{d\lambda}g_{\alpha\beta}+\partial_{\mu}n^{\mu}\,,$
(4.7)

where $n^{\mu}\partial_{\mu}=\frac{d}{d\lambda}$. Using this last equation,
and the fact that $\Theta_{n}=\Theta_{u}+\mathcal{O}(\epsilon^{2})$, as shown
in Eq. (3.15), we then have

$\Theta_{u}=3H\left[1+\frac{1}{2}\left(a^{2}L+N+2aM\right)\right]+\frac{1}{2}\frac{d}{d\lambda}\left(N+4\nu\right)+\mathcal{O}(\epsilon^{2})\,,$
(4.8)

where we are neglecting the terms

$\left(\partial_{w}+\frac{2}{r}\right)\left(N+aM\right)=\mathcal{O}\left(\epsilon^{2}\right)\,,\qquad\qquad
D_{a}\left(\mathcal{U}^{a}+aV^{a}\right)=\mathcal{O}\left(\epsilon^{2}\right)\,,$
(4.9)

since they are proportional to $\nabla_{i}\mathcal{B}^{i}$ in standard
perturbation theory (see [36, 42]). The same happens in the standard approach,
see [4, 5]. The reason why $\nabla_{i}\mathcal{B}^{i}$ can be neglect is that,
in the language of the gradient expansion applied to the ADM metric, this term
is the divergence of the shift vector, hence second order in the gradient
expansion. As a side remark, we underline that Eq. (4.7) is given in terms of
the cosmic time by

$\Theta_{n}=3H-3\partial_{t}\psi+\mathcal{O}(\epsilon^{2})\,,$ (4.10)

which is in agreement with [4].

Let us now consider the energy-momentum tensor conservation projected onto
$u^{\mu}$. From Eq. (3.23), we have that

$\displaystyle\frac{d\rho}{d\lambda}+\left(\rho+p\right)\Theta_{u}=0\,.$
(4.11)

Thanks to Eqs. (4.5) and (4.8), and by using the equation of state $p=q\rho$,
Eq. (4.11) gives

$\displaystyle\frac{d\bar{\rho}}{d\lambda}+3H\bar{\rho}(1+q)=0\,,$
$\displaystyle\frac{d\left(\bar{\rho}\delta\rho\right)}{d\lambda}+\bar{\rho}\left(1+q\right)\left\\{3H\left[\delta\rho+\frac{1}{2}\left(a^{2}L+N+2aM\right)\right]+\frac{1}{2}\frac{d}{d\lambda}\left(N+4\nu\right)\right\\}+\mathcal{O}(\epsilon^{2})=0\,,$
(4.12)

for the background and perturbed quantities respectively. Let us now fix the
GLC gauge, which satisfies the following conditions [36]

$\displaystyle a^{2}L+N+2aM$ $\displaystyle=0\,,$
$\displaystyle\partial_{w}\left(N+aM\right)$
$\displaystyle=\frac{1}{2}\partial_{w}N=\mathcal{O}(\epsilon^{2})\,,$
$\displaystyle\partial_{a}\left(\mathcal{U}^{a}+aV^{a}\right)$
$\displaystyle=\partial_{a}\mathcal{U}^{a}=\mathcal{O}\left(\epsilon^{2}\right)\,.$
(4.13)

Then, by inserting first of Eqs. (4.12) into the second one, by using the GLC
gauge conditions given in Eqs. (4.13), and, finally, by integrating over the
affine parameter $\lambda$, we get

$\delta\rho(\lambda)=-\frac{1+q}{2}\left(N+4\nu\right)\,.$ (4.14)

It is worth to stress that we would obtain the same result by linearizing Eqs.
(3.27). Therefore, this show the self-consistency of our gradient expansion
method.

At first order in the gradient expansion, for a generic adiabatic equation of
state, i.e. $p=p(\rho)$, and by fixing the GLC gauge, we obtain, from Eqs.
(4.8), (4.11) and (4.13), that

$\frac{1}{\rho+p(\rho)}\frac{d\rho}{d\lambda}=-3H-\frac{1}{2}\frac{d\left(N+4\nu\right)}{d\lambda}\,.$
(4.15)

This equation can be integrated between two different hypersurfaces marked by
the values $\lambda_{1}$ and $\lambda_{2}$, and gives

$\frac{1}{6}\left(N+4\nu\right)|_{\lambda_{1}}^{\lambda_{2}}+\bar{\mathcal{N}}(\lambda_{2},\lambda_{1})=-\frac{1}{3}\int_{\rho(\lambda_{1},x^{i})}^{\rho(\lambda_{2},x^{i})}\frac{d\rho}{\rho+p(\rho)}\,,$
(4.16)

where the dependence on $x^{i}$ in the extremes of integration denotes that we
are considering inhomogeneous hyper-surfaces. At the same time, we have
defined $\bar{\mathcal{N}}$ as

$\displaystyle\bar{\mathcal{N}}\left(\lambda_{2},\,\lambda_{1}\right)\equiv\int^{\lambda_{2}}_{\lambda_{1}}Hd\lambda$
(4.17)

Moreover, as done for standard perturbations in [5], we want to extract the
background contribution of Eq. (4.16). Therefore, we start by doing this in
the r.h.s. of Eq. (4.16), which can then be written as

$\int_{\rho(\lambda_{1},x^{i})}^{\rho(\lambda_{2},x^{i})}\frac{d\rho}{\rho+p(\rho)}=\int_{\bar{\rho}(\lambda_{2})}^{\rho(\lambda_{2},x^{i})}\frac{d\rho}{\rho+p(\rho)}-\int_{\bar{\rho}(\lambda_{1})}^{\rho(\lambda_{1},x^{i})}\frac{d\rho}{\rho+p(\rho)}+\int_{\bar{\rho}(\lambda_{1})}^{\bar{\rho}(\lambda_{2})}\frac{d\rho}{\rho+p(\rho)}\,,$
(4.18)

where $\bar{\rho}$ determines the background value of $\rho$. The last term is
the background value given by $\bar{\mathcal{N}}$ (see Eq. (4.17)). Therefore,
we have

$\frac{1}{6}\left(N+4\nu\right)\left(\lambda_{1}\right)+\frac{1}{3}\int_{\bar{\rho}(\lambda_{1})}^{\rho(\lambda_{1},x^{i})}\frac{d\rho}{\rho+p(\rho)}=\frac{1}{6}\left(N+4\nu\right)\left(\lambda_{2}\right)+\frac{1}{3}\int_{\bar{\rho}(\lambda_{2})}^{\rho(\lambda_{2},x^{i})}\frac{d\rho}{\rho+p(\rho)}\,,$
(4.19)

then, the quantity

$\tilde{\zeta}=-\frac{1}{6}\left(N+4\nu\right)-\frac{1}{3}\int_{\bar{\rho}(\lambda)}^{\rho(\lambda,x^{i})}\frac{d\rho}{\rho+p(\rho)}\,,$
(4.20)

is conserved. One interesting aspect of Eq. (4.20) is that the geometrical
terms $N$ and $\nu$, present on the conservation of $\tilde{\zeta}$, are
precisely the same terms that contribute to the linearized angular distance-
redshift relation for the linearized GLC gauge [36].

Let us now link our results to the standard perturbation theory. We follow the
notation of [36, 42], where the standard metric is given by

$\displaystyle ds^{2}$
$\displaystyle=\,a^{2}\left[-\left(1+2\phi\right)d\eta^{2}-2\mathcal{B}_{i}dx^{i}d\eta+\left(\bar{\gamma}_{ij}+\mathcal{C}_{ij}\right)dx^{i}dx^{j}\right]\,,$
(4.21)

and the SVT decomposition given by

$\displaystyle\mathcal{B}_{i}$ $\displaystyle=B_{i}+\partial_{i}B\,,$
$\displaystyle C_{ij}$
$\displaystyle=\,-2\bar{\gamma}_{ij}\psi+2D_{ij}E+2\nabla_{(i}F_{j)}+2h_{ij}\,,$
(4.22)

where $B_{i}$ and $F_{i}$ are divergenceless vectors and $h_{ij}$ is a trace-
less and divergenceless tensor. We also have

$\displaystyle
D_{ij}E=\left(\nabla_{(i}\nabla_{j)}-\bar{\gamma}_{ij}\frac{\Delta_{3}}{3}\right)E\,.$
(4.23)

In this case, the trace of $g_{ij}$ is proportional to $-\psi$, and then by
using the relation between standard and GLC perturbations given by [36, 42]

$\psi=-\frac{1}{6}\left(N+4\nu\right)\,,$ (4.24)

we get that

$\tilde{\zeta}=\psi-\frac{1}{3}\int_{\bar{\rho}(\lambda)}^{\rho(\lambda,x^{i})}\frac{d\rho}{\rho+p(\rho)}\,,$
(4.25)

or, equivalently, with the use of Eq. (4.5),

$\tilde{\zeta}=\psi-\frac{1}{3}\int_{\bar{\rho}(\lambda)}^{\rho(1+\delta\rho)}\frac{d\rho}{\rho+p(\rho)}\approx\psi-\frac{1}{3}\frac{\bar{\rho}\delta\rho}{\rho+p(\rho)}\,,$
(4.26)

where we have expanded at linear order in the density perturbations in the
last equality. Since we are working at first order in the gradient expansion,
where the spatial gauge modes occur at the next-to-leading order, and the time
gauge mode is fixed, the quantity $\tilde{\zeta}$ given in Eq. (4.20) is gauge
invariant. Hence, within this approximation scheme, we may identify it with
the curvature perturbation $\zeta$. For the complete expression of $\zeta$ to
order $\mathcal{O}(\delta,\epsilon^{n})$, see Eqs. (2.27) of [42], where we
also provide its gauge invariance proof in terms of light-cone perturbations.

###  4.2 Non-linear $\zeta$

As we have seen, our non-linear SU approach in the GLC gauge allowed us to
obtain Eq. (3.25) by neglecting the last term in Eq. (3.24), which we have
shown to correspond to the terms neglected in the standard perturbation theory
[4] at linear order. Now, with the aim of going beyond this result, we leave
the lapse function $\mathcal{M}$ unspecified and use the SU approach on the
light-cone to prove the non-linear conservation of the curvature perturbation
in terms of LC parameters. Just as done in the previous subsection, we start
from Eq. (3.23) which gives

$\displaystyle\Theta_{u}=-\frac{1}{\left(\rho+p\right)}\frac{d\rho}{d\lambda}\,.$
(4.27)

Moreover, we consider Eq. (3.15) and the approximation exploited after Eq.
(4.7), namely $\Theta_{n}\equiv\nabla_{\mu}n^{\mu}\simeq\Theta_{u}$. We then
get that

$\displaystyle\frac{1}{\Upsilon\sqrt{\gamma}}\frac{d\left(\Upsilon\sqrt{\gamma}\right)}{d\lambda}=-\frac{1}{\left(\rho+p\right)}\frac{d\rho}{d\lambda}\,.$
(4.28)

We now integrate this equation along $\lambda$ and consider that the pressure
is adiabatic. In this way, one obtains that

$\displaystyle\ln\left[\frac{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{2}}}{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{1}}}\right]=-\int_{\rho\left(\lambda_{1},x^{i}\right)}^{\rho\left(\lambda_{2},x^{i}\right)}\frac{d\rho}{\rho+p\left(\rho\right)}\,.$
(4.29)

At this point, the r.h.s. can be manipulated in the same spirit of what done
in Eq. (4.18) (see also [4]). By doing so, we get that

$\displaystyle\int_{\rho(\lambda_{1},x^{i})}^{\rho(\lambda_{2},x^{i})}\frac{d\rho}{\rho+p(\rho)}$
$\displaystyle=\int_{\bar{\rho}(\lambda_{2})}^{\rho(\lambda_{2},x^{i})}\frac{d\rho}{\rho+p(\rho)}-\int_{\bar{\rho}(\lambda_{1})}^{\rho(\lambda_{1},x^{i})}\frac{d\rho}{\rho+p(\rho)}+\int_{\bar{\rho}(\lambda_{1})}^{\bar{\rho}(\lambda_{2})}\frac{d\rho}{\rho+p(\rho)}$
$\displaystyle=\int_{\bar{\rho}(\lambda_{2})}^{\rho(\lambda_{2},x^{i})}\frac{d\rho}{\rho+p(\rho)}-\int_{\bar{\rho}(\lambda_{1})}^{\rho(\lambda_{1},x^{i})}\frac{d\rho}{\rho+p(\rho)}+\ln\left(\frac{\bar{\Upsilon}\sqrt{\bar{\gamma}}|_{\lambda_{2}}}{\bar{\Upsilon}\sqrt{\bar{\gamma}}|_{\lambda_{1}}}\right)\,,$
(4.30)

where, from the first to the second line, we have used Eq. (4.29) at the
background level. Now, thanks to Eq. (4.30), we can rewrite Eq. (4.29) as

$\displaystyle\ln\left(\frac{\Upsilon\sqrt{{\gamma}}}{\bar{\Upsilon}\sqrt{\bar{\gamma}}}\right)_{\lambda_{2}}+\int_{\bar{\rho}(\lambda_{2})}^{\rho(\lambda_{2},x^{i})}\frac{d\rho}{\rho+p(\rho)}=\ln\left(\frac{\Upsilon\sqrt{\gamma}}{\bar{\Upsilon}\sqrt{\bar{\gamma}}}\right)_{\lambda_{1}}+\int_{\bar{\rho}(\lambda_{1})}^{\rho(\lambda_{1},x^{i})}\frac{d\rho}{\rho+p(\rho)}\,.$
(4.31)

This shows that there is a conserved quantity at first order in the gradient
expansion. This quantity corresponds to the non-linear curvature perturbation
$\zeta$

$\displaystyle\zeta=\frac{1}{3}\ln\left(\frac{\Upsilon\sqrt{{\gamma}}}{\bar{\Upsilon}\sqrt{\bar{\gamma}}}\right)+\frac{1}{3}\int_{\bar{\rho}(t)}^{\rho(t,x^{i})}\frac{d\rho}{\rho+p(\rho)}+\mathcal{O}\left(\epsilon^{2}\right)\,.$
(4.32)

which then generalizes the linear result in Eq. (4.20).

Thus, the curvature perturbation defined in Eq. (4.32) generalizes at the non-
linear level, but at the first order in the gradient expansion, the expression
of the gauge invariant curvature perturbation in the LC gauge formalism given
in [42]. In fact, the first term corresponds to the curvature perturbations
whilst the second term corresponds to density perturbations in the same spirit
as done in Eq. (4.25). Finally, we remark that we have obtained Eq. (4.32)
without specifying the lapse function $\mathcal{M}$, therefore we still have
the freedom to fix the time gauge mode, as we will see better later.

###  4.3 $\delta\mathcal{N}$ formalism on the LC gauge

Let us begin by writing explicitly the exact expression for the expansion rate
defined by the normal vector $n^{\mu}$ given by Eq. (3.2). This we will be
then applied to the evaluation of the number of e-folds using the SU picture
of the LC gauge. Such approach will allow us to obtain a generalization of the
non-linear $\delta\mathcal{N}$ formalism in terms of LC gauge metric entries.
The expansion rate in the LC gauge is given by

$\Theta_{n}=\nabla_{\mu}n^{\mu}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(n^{\mu}\sqrt{-g}\right)=\frac{1}{\mathcal{M}\Upsilon\sqrt{\gamma}}\frac{d}{d\lambda}\left(\mathcal{M}\Upsilon\sqrt{\gamma}\right)+\partial_{\mu}n^{\mu}\,,$
(4.33)

where, from Eqs. (3.2) and (3.12), we have that

$\partial_{\mu}n^{\mu}=-\frac{\partial_{t}\mathcal{M}}{\mathcal{M}^{2}}-\partial_{i}\left(\frac{N^{i}_{LC}}{\mathcal{M}}\right)=-\frac{\partial_{t}\mathcal{M}}{\mathcal{M}^{2}}+\mathcal{O}\left(\epsilon^{2}\right)\,.$
(4.34)

Now, using Eqs. (4.33) and (4.34), we obtain

$\Theta_{n}=\frac{1}{\Upsilon\sqrt{\gamma}}\frac{d}{d\lambda}\left(\Upsilon\sqrt{\gamma}\right)+\mathcal{O}\left(\epsilon^{2}\right)\,.$
(4.35)

An interesting thing about this result is that it is invariant in form on the
past light-cone, i.e. all the dependence on the lapse function is hidden in
$n^{\mu}\partial_{\mu}\equiv\frac{d}{d\lambda}$. Let us now integrate Eq.
(4.35) to compute the non-linear number of e-folds at first order in the
gradient expansion in terms of light-cone entries. We can then easily obtain
the following result

$\mathcal{N}\left(\lambda_{f},\lambda_{i}\right)\equiv\frac{1}{3}\int_{\lambda_{i}}^{\lambda_{f}}\Theta_{n}d\lambda^{\prime}=\frac{1}{3}\ln\left[\frac{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{f}}}{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{i}}}\right]+\mathcal{O}\left(\epsilon^{2}\right)\,.$
(4.36)

Let us note that $\mathcal{N}(\lambda_{f},\,\lambda_{i})$ from Eq. (4.36) is a
biscalar, i.e. depends on the gauge fixing both at the initial and final
slicing. One possible fixing of the lapse function is given by the uniform
curvature light-cone (UCLC) gauge: in this case the effective local scale
factor is given by its background value666We will be using the subscript UC to
describe the UCLC gauge fixing and the subscript UD to describe the UDLC gauge
fixing.

$\displaystyle\left(\Upsilon\sqrt{\gamma}\right)_{UC}=\bar{\Upsilon}\sqrt{\bar{\gamma}}\,.$
(4.37)

Therefore, if we fix both initial and final slices on the UCLC gauge, the
number of e-folds will be given by its background value, as

$\mathcal{N}_{UC}\left(\lambda_{f},\,\lambda_{i}\right)=\bar{\mathcal{N}}\left(\lambda_{f},\,\lambda_{i}\right)\,.$
(4.38)

Let us also introduce the uniform density light-cone (UDLC) gauge defined by

$\displaystyle\rho_{UD}\left(\lambda,\,x^{i}\right)=\bar{\rho}\left(\lambda\right)\,.$
(4.39)

Then, within the UDLC gauge, we have

$\zeta=\,\frac{1}{3}\ln\left[\frac{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{f\,UD}}}{\left(\bar{\Upsilon}\sqrt{\bar{\gamma}}\right)_{\lambda_{i}}}\right]=\,\frac{1}{3}\ln\left[\frac{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{f\,UD}}}{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{i\,UC}}}\right]\,$
(4.40)

where from the first to the second equality we have used Eq. (4.37). We can
now compare Eqs. (4.36) and (4.40), and obtain that $\zeta$ can be related to
the number of e-folds in the following way

$\displaystyle\mathcal{N}\left(\lambda_{f\,UD},\,\lambda_{i\,UC}\right)=$
$\displaystyle\frac{1}{3}\ln\left[\frac{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{f\,UD}}}{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{i\,UC}}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{3}\ln\left[\frac{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{f\,UD}}}{\left(\bar{\Upsilon}\sqrt{\bar{\gamma}}\right)_{\lambda_{f}}}\frac{\left(\bar{\Upsilon}\sqrt{\bar{\gamma}}\right)_{\lambda_{i}}}{\left(\Upsilon\sqrt{\gamma}\right)_{\lambda_{i\,UC}}}\right]+\bar{\mathcal{N}}\left(\lambda_{f},\lambda_{i}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{3}\ln\left(\frac{\Upsilon\sqrt{\gamma}}{\bar{\Upsilon}\sqrt{\bar{\gamma}}}\right)_{\lambda_{f\,UD}}+\bar{\mathcal{N}}\left(\lambda_{f},\lambda_{i}\right)$
$\displaystyle=$
$\displaystyle-\zeta(\lambda_{f})+\bar{\mathcal{N}}\left(\lambda_{f},\lambda_{i}\right)\,.$
(4.41)

Hence, if we define

$\delta\mathcal{N}\equiv\,\mathcal{N}\left(\lambda_{f\,UD},\,\lambda_{i\,UC}\right)-\mathcal{N}\left(\lambda_{f\,UC},\,\lambda_{i\,UC}\right)\,,$
(4.42)

we straightforwardly get, from (4.38) and (4.41), that
$\delta\mathcal{N}=-\zeta$, which is valid at any order in perturbation
theory, in agreement with [15]. As a remark, we underline that, while Eq.
(4.36) depends directly on the initial and final slice, Eq. (4.42) depends
only on the difference of the perturbations of the e-fold number in the UDLC
and UCLC on the final slices. Now we will give an example of how this last
result can be used to evaluate the power spectrum of $\zeta$ in terms of the
LC metric entries. Following the procedure used in [15], we fix the UCLC gauge
and adopt the SU approximation. In this way, the density perturbations can be
written in terms of the background density with perturbed initial conditions
as follows

$\displaystyle\rho_{UC}(\mathcal{N}_{UC},\,\mathbf{x})=\bar{\rho}(\mathcal{\bar{N}},\,\varphi_{*}^{A}(x))\,,$
(4.43)

where we use Eq. (4.38). Also, $\varphi_{*}^{A}$ is the field content of the
underlying inflationary model evaluated just after the horizon exit. The index
$A=1,...,d$ refers to the possibility that inflation could happen with $d$
scalar fields. Instead, by considering the UDLC gauge, we would have

$\displaystyle\rho_{UD}(\mathcal{N}_{UD},\,\mathbf{x})=\bar{\rho}(\mathcal{N}_{UD})\,.$
(4.44)

Since $\rho$ is a scalar, we can then write

$\displaystyle\rho^{\prime}(\mathcal{N}^{\prime},\,x^{\prime})=\rho(\mathcal{N},\,x)\,,$
(4.45)

which holds between two generic sets of coordinates $\mathcal{N},\,x$ and
$\mathcal{N}^{\prime},\,x^{\prime}$ i.e. the value of a scalar function in a
given physical point does not depend on the choice of coordinates.

Using Eqs. (4.43) and (4.44) on Eq. (4.45), we get

$\displaystyle\bar{\rho}(\bar{\mathcal{N}},\,\varphi_{*}^{A}(x))=\bar{\rho}(\mathcal{N}_{UD})\,,$
(4.46)

where the $x^{\prime}=x$ from Eq. (4.45) corresponds to the choice of the
spatial threading to fix the LC gauge. Also, we choose
$\mathcal{N}^{\prime}=\mathcal{N}_{UC}$, and $\mathcal{N}=\mathcal{N}_{UD}$.
Then, Eq. (4.46) can be inverted as

$\displaystyle\mathcal{N}_{UD}=\bar{\mathcal{N}}(\bar{\rho},\,\varphi_{*}^{A}(\mathbf{x}))\,.$
(4.47)

Hence, by expanding the fields in Eq. (4.47) at linear order as
$\varphi^{A}_{*}=\bar{\varphi}^{A}_{*}+\delta\varphi^{A}_{*}$, using Eq.
(4.42), we obtain

$\displaystyle\frac{1}{3}\ln\left[\frac{\left(\Upsilon\sqrt{\gamma}\right)_{UD}}{\left(\Upsilon\sqrt{\gamma}\right)_{UC}}\right]=$
$\displaystyle\mathcal{N}(\bar{\rho},\,\varphi_{*}^{A}(\mathbf{x}))|_{UD}-\bar{\mathcal{N}}(\bar{\rho})|_{UC}$
$\displaystyle=$
$\displaystyle\delta\varphi_{*}^{A}\partial_{A}\bar{\mathcal{N}}+\frac{1}{2}\delta\varphi_{*}^{A}\delta\varphi_{*}^{B}\partial_{A}\partial_{B}\bar{\mathcal{N}}+...\,.$
(4.48)

Therefore, given an inflationary model one may link the value of
$\varphi_{*}^{A}$ to the left hand side of Eq. (4.48).

Altogether, Eq. (4.48) can be a starting point for the evaluation of $f_{NL}$
in terms of light-cone perturbations (see, for example, [13] for an evaluation
of $f_{NL}$ using the $\delta\mathcal{N}$ formalism). Therefore, the results
presented in Eqs. (4.42) and (4.48) constitute a further step to obtain non-
Gaussian predictions on the past light-cone, from the primordial universe and
directly in terms of the metric entries.

##  5 Conclusions

In this manuscript we have developed a separate universe (SU) description in
the non-linear LC gauge. We provide the non-linear conditions to fix the LC
and the GLC gauges in terms of standard coordinates on the ADM formalism. The
main difference with the previous works [5, 15, 26], where the SU was
considered, is that for the LC and GLC gauges we cannot neglect the shift
vector, since this contains information about inhomogeneities along the world-
line.

As an application of our results, and a consistency check, we repeated the
procedures of [4] and [5] to prove the super-horizon conservation of the
comoving curvature perturbation $\zeta$, when a light-like foliation of
spacetime is taken. This conservation has been achieved by neglecting the non-
adiabatic pressure within the SU scheme.

We then generalize the $\delta\mathcal{N}$ formalism [10, 8, 9, 5, 15, 11, 12,
13], in terms of the combination of the LC metric entries
$\Upsilon\sqrt{\gamma}$ within the uniform density light-cone gauge which is
one of our most important results. Let us remark that the gradient expansion
employed here simplifies the expression of the expansion rate at the non-
linear level (see Eq. (4.33)). This could help in simplifying also the
perturbative expressions (see, for instance, the one presented in Eq. (6.11)
of [43]).

The $\delta\mathcal{N}$ formalism provides a procedure to investigate the
evolution of the perturbations for different inflationary models. The
extension of the $\delta\mathcal{N}$ formalism on the past light-cone, as
developed in this manuscript, allows the evaluation of such dynamical
evolution directly over the past light-cone. This moves us one step forward to
the evaluation of non-linear effects (such as backreaction effects and non-
Gaussianities) since the primordial universe until the late-time one along
such past light-cone. Indeed, as a future step, we aim to investigate
primordial backreaction effects on different expansion rates using the above-
mentioned formalism and well-posed averaging procedures on the past light-cone
[29, 35].

Finally, for what regards the possible non-Gaussianities associated to any
inflationary model, the $\delta\mathcal{N}$ formalism is a very useful tool.
In fact, as shown in [13], this formalism provides very simple expressions for
$f_{NL}$ in terms of $\mathcal{N}$. In other words, the overall goal of the
research program where this work is nested is to obtain a formalism to compute
the curvature perturbations at horizon re-entry expressed in terms of light-
cone entries. We remark this point since the subsequent evolution of this
metric entries could then be compared to late-time expression of cosmological
observables as, for instance, the ones presented in Eq. (3.3). This would
provide a self-consistent framework entirely given on the light-cone to
disentangle the primordial non-Gaussianities from the ones naturally emerging
during the non-linear late-time dynamics [47, 48, 49].

## Acknowledgement

GM and MM are supported in part by INFN under the program TAsP (Theoretical
Astroparticle Physics). GF acknowledges support by the FCT under the program
“Stimulus” with the grant no. CEECIND/04399/2017/CP1387/CT0026 and through the
research project with ref. number PTDC/FIS-AST/0054/2021. GF is also member of
the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale
di Alta Matematica (INdAM).

##  Appendix A 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. (4.42) and obtain

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

where we have defined

$\Upsilon=\bar{\Upsilon}(1+\delta\Upsilon)$ (A.2)

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. (4.1) and the scalar/pseudoscalar decomposition of Eq.
(4.3). Since $\delta\Upsilon=N/2$, we then have that

$\delta\mathcal{N}(\lambda_{1,}\lambda_{2,}x^{i})=\frac{1}{6}\left(N+4\nu\right)_{UD}+\mathcal{O}(\delta^{2},\epsilon^{2})\,.$
(A.3)

From the relation between the light-cone perturbation and the standard ones in
Eq. (4.24), (see also [42]), one gets that

$\displaystyle\psi=-\frac{1}{6}(N+4\nu)\,.$ (A.4)

Therefore, Eq. (A.3) together with Eq. (A.4) and the fact that
$\psi_{UD}=\zeta$, gives the well known relation $\delta\mathcal{N}=-\zeta$
[15].

The result obtained in Eq. (A.3) 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. (4.7) and (4.8), evaluating them between the uniform
curvature and the uniform density slices.

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