# Gauge invariance on the light-cone: curvature perturbations and radiative
degrees of freedom

G. Fanizza  G. Marozzi  and M. Medeiros

###### Abstract

We derive the expressions on the observed light-cone for some relevant
cosmological gauge invariant variables, such as the Mukhanov-Sasaki variable
and $E$\- and $B$\- modes of the tensor perturbations. Since the structure of
the light-cone does not reflect in a direct way the FLRW symmetries, we
develop a formalism which is coordinate independent and classifies the
perturbations according to their helicities. Even though we work with linear
perturbations, our formalism can be readily extended to non-linear theory and
put the basis to study the evolution of cosmological perturbations, since the
early- until the the late-time Universe, directly along the observed light-
cone.

##  1 Introduction

The Geodesic Light-Cone (GLC) gauge [1] has shown several advantages in many
aspects of the late-time theoretical cosmology. Few examples are the exact
computation of light-like cosmological observables, such as the redshift [1,
2], the luminosity distance [3, 4, 5, 6, 7], the Jacobi map [6, 8], the galaxy
number count [9, 10] and the evaluation of well-posed averages of these
observables [1, 11]. Furthermore, the GLC gauge has been successfully applied
also to the study of non-linear corrections to CMB spectra [12, 13, 14] and to
investigate the behavior of ultra-relativistic particles in the inhomogeneous
Universe [15, 16].

These advantages have their root in the fact that the GLC gauge is fixed in
such a way that the light-like geodesics are exactly solved. Notwithstanding
this remarkable feature, a detailed study of the cosmological dynamics in the
GLC coordinates [17] has shown how this is quite involved already at linear
order, at least for an analytic treatment.

Because of these facts, a sort of dualism seems to emerge between the standard
perturbative description of cosmological inhomogeneities and the fully non-
linear light-cone one. The former admits manageable (and explicitly solvable)
equation of motions but rather complicated expressions for light-like
cosmological observables, whereas the latter has simple non-linear forms for
the cosmological observables (and their averages) but a quite cumbersome
evolution of the dynamics.

A possible way out of this dichotomy may be provided by numerical attempts. As
a relevant example, in a recent work [18], the linearized equation of motion
on the light-cone for the gravitational potential has been solved. This open
an unprecedented opportunity to study the evolution of perturbations along the
observed past light-cone. However, for this program to be successful, a
systematic formulation of perturbation theory along the light-cone is needed,
in order to properly identify the relevant (combinations of the) degrees of
freedom for the quantities of cosmological interest.

In [19], we have built a cosmological perturbation theory directly on the
light-cone and provided a general road map for the computation of gauge
invariant observables. This has been applied to the study of the distance-
redshift relation. Here we aim to extend our approach to describe other gauge
invariant degrees of freedom of cosmological interest, such as the curvature
perturbation and the tensor ones. As we are going to show, a key ingredient
for the evaluation of the latter will be a decomposition of the perturbations
based on their behavior as Scalar/PseudoScalar (SPS) under rotation on the
sphere rather than the usual Scalar-Vector-Tensor (SVT) decomposition. Thanks
to this, we will be able to provide the explicit expression for the $E$\- and
$B$\- mode of the tensor perturbations, which correspond to the physical
radiative degrees of freedom of the theory.

Finally, we acknowledge that in [20] the study of the gauge invariant
curvature perturbation in the GLC gauge has also been applied to obtain
expressions for the expansion rate in the early universe in order to
investigate the backreaction effects along the past light-cone. In this
regard, what this work adds to this computation is the explicit gauge
invariant expression for this quantity, whereas in [20] the linearized GLC has
been adopted. As a sanity check, we prove that our generalized result agrees
with [20].

The manuscript is organized as follows. In Sect. 2, we first recall the
standard perturbation theory and set the notation. Afterwards, we report some
of the main results of [19] and provide the connection between the standard
SVT perturbations and the SPS ones. In Sect. 3, we introduce a manner to
decompose the perturbations based on a set of projections which takes into
account the helicity of the perturbations, is coordinate independent and
extendable to higher perturbative orders. Also in this section, we apply this
formalism to both standard perturbation theory and the light-cone one. Hence,
in Sect. 4 we extract the $E$\- and $B$\- modes for the gauge invariant tensor
perturbations as function of the light-cone SPS perturbations. Finally, the
main conclusions are discussed in Sect. 5. Furthermore, in Appendix A we
explicitly check that the relations between SVT and SPS satisfy the expected
gauge transformations. Appendix B is devoted to show the agreement with the
results and the formalism of [20], whereas Appendix C reports some technical
aspects concerning the spin-raising and spin-lowering operators needed along
the paper. Appendix D contains the technical details for the evaluation of the
$E$\- and $B$\- modes of the gauge invariant tensor perturbations.

##  2 A conventional approach

In this section, we will make contact between the standard perturbation theory
and the light-cone one. We will briefly review the standard approach to linear
perturbations and the main gauge invariant variables of cosmological interest.
Then, we will introduce a suitable set of perturbations for the light-cone
structure. Finally, after having developed the appropriate dictionary between
the two sets, we will derive gauge invariant quantities such as tensor
perturbations and Mukhanov-Sasaki variable in terms of the light-cone
perturbations. This work is a natural continuation of the formalism already
developed in [19, 17].

###  2.1 Standard Perturbation Theory

Here, we briefly report the well-known results of linear perturbations of
Friedmann-Lemaître-Robertson-Walker (FLRW) metric in order to set the notation
that will be followed in the rest of the paper. By defining with $g_{\mu\nu}$
the inhomogeneous metric tensor and with $\delta g_{\mu\nu}$ the set of
perturbations on top of the FLRW metric $\bar{g}_{\mu\nu}$, we have111Greek
indices run from $0$ to $4$, Latin ones $i,j,k,\dots$ range from $1$ to $3$
and Latin ones $a,b,c,\dots$ label angular coordinates.

$g_{\mu\nu}\equiv\bar{g}_{\mu\nu}+\delta
g_{\mu\nu}=a^{2}\left[\begin{pmatrix}-1&0\\\
0&\bar{\gamma}_{ij}\end{pmatrix}+\begin{pmatrix}-2\phi&-\mathcal{B}_{i}\\\
-\mathcal{B}_{i}&\mathcal{C}_{ij}\end{pmatrix}\right]\,.$ (2.1)

In spherical coordinates, the background spatial metric is
$\bar{\gamma}_{ij}=\text{diag}\left(1,\bar{\gamma}_{ab}\right)$, the metric on
the 2-sphere is
$\bar{\gamma}_{ab}=r^{2}\text{diag}\left(1,\sin^{2}\theta\right)$ and $r$ is
the space-like radial coordinate. Perturbations are typically decomposed
according to the irreducible representations of the background symmetry groups
$SO\left(3\right)$ as

$\displaystyle\mathcal{B}_{i}$ $\displaystyle=\partial_{i}B+B_{i}\,,$
$\displaystyle\mathcal{C}_{ij}$
$\displaystyle=-2\,\psi\,\gamma_{ij}+2\,D_{ij}E+2\,\nabla_{(i}F_{j)}+2\,h_{ij}\,,$
(2.2)

where the covariant derivative $\nabla_{i}$ is defined as
$\nabla_{i}S^{j}=\partial_{i}S^{j}+\Gamma_{il}^{j}S^{l}$ with
$\Gamma_{il}^{j}$ the Christoffel connection w.r.t. $\bar{\gamma}_{ij}$,
$D_{ij}\equiv\nabla_{(i}\nabla_{j)}-\frac{1}{3}\bar{\gamma}_{ij}\Delta_{3}$
and $\Delta_{3}\equiv\bar{\gamma}^{ij}\nabla_{i}\nabla_{j}$. Eq. (2.2) is
known as Scalar-Vector-Tensor (SVT) decomposition, since $B,\psi,E$ and $\phi$
are scalars, $B_{i}$ and $F_{i}$ are divergence-less vectors
($\nabla^{i}B_{i}=\nabla^{i}F_{i}=0$), and $h_{ij}$ is a trace-less and
divergence-less tensor ($\bar{\gamma}^{ij}h_{ij}=0$ and $\nabla^{i}h_{ij}=0$)
when the metric is transformed according to the above-mentioned symmetry
group.

Eq. (2.2) still admits a gauge freedom in the definition of the perturbations
$\delta g_{\mu\nu}$. This is fixed by exploring the transformation property
under diffeomorphism: given a coordinate transformation
$y^{\mu}\rightarrow\widetilde{y^{\mu}}=y^{\mu}+\epsilon^{\mu}$, perturbations
$\delta g_{\mu\nu}$ change as

$\widetilde{\delta g_{\mu\nu}}(y^{\mu})=\delta
g_{\mu\nu}(y^{\mu})-\mathcal{L}_{\mathbf{\epsilon}}g_{\mu\nu}(y^{\mu})\,,$
(2.3)

where
$\mathcal{L}_{\mathbf{\epsilon}}(g_{\mu\nu})\equiv\hat{\nabla}_{\mu}\epsilon_{\nu}+\hat{\nabla}_{\nu}\epsilon_{\mu}$
is the Lie derivative of $g_{\mu\nu}$ w.r.t. the field $\epsilon^{\mu}$ and
$\hat{\nabla}_{\mu}$ is the covariant derivative w.r.t. the 4-dimensional
background metric $\bar{g}_{\mu\nu}$. The SVT decomposition can be applied to
the space-like part of $\epsilon^{\mu}$ as well. This simply returns
$\epsilon^{i}=e^{i}+\partial^{i}\epsilon$. We are then left with two scalars
gauge fields $\epsilon^{\eta}$ and $\epsilon$ and a divergence-less vector
gauge field $e^{i}$. The straightforward combination of Eqs. (2.2) and (2.3)
leads to the well-known gauge transformations for scalars

$\displaystyle\widetilde{\phi}=$
$\displaystyle\,\phi-\partial_{\eta}\epsilon^{\eta}+\mathcal{H}\epsilon^{\eta}\,,$
$\displaystyle\widetilde{E}=$ $\displaystyle\,E-\epsilon\,,$
$\displaystyle\widetilde{\psi}=$
$\displaystyle\,\psi+\mathcal{H}\epsilon^{\eta}+\frac{1}{3}\Delta_{3}\epsilon\,,$
$\displaystyle\widetilde{B}=$
$\displaystyle\,B-\epsilon^{\eta}+\partial_{\eta}\epsilon\,,$ (2.4)

and vectors

$\widetilde{B}_{i}=B_{i}+\partial_{\eta}\left(\frac{e_{i}}{a^{2}}\right)\,,\qquad\qquad\widetilde{F}_{i}=F_{i}-\frac{e_{i}}{a^{2}}\,,$
(2.5)

whereas tensor perturbations are gauge invariant $\widetilde{h}_{ij}=h_{ij}$,
since there is not any linear tensor gauge mode in $\epsilon^{\mu}$.

Gauge dependence affects also the matter sector. Indeed, given a scalar $f$,
its gauge transformation is given by

$\widetilde{f}(y^{\mu})=f(y^{\mu})-\mathcal{L}_{\epsilon}f(y^{\mu})\,.$ (2.6)

By splitting the general scalar field $f$ in a time-dependent background
$\bar{f}$ and its linear fluctuations $\delta f$, we will then obtain the
following gauge transformation

$\widetilde{\delta f}=\delta f-\epsilon^{\eta}\partial_{\eta}\bar{f}\,.$ (2.7)

Of particular interest is the case of a single field slow-roll inflationary
model, where we have a perfect fluid, with energy-density $\rho$ and pressure
$p=w\rho$, and a scalar field $\varphi$. In this case we have

$\displaystyle\widetilde{\delta\rho}=$
$\displaystyle\delta\rho-\epsilon^{\eta}\partial_{\eta}\bar{\rho}\,,$
$\displaystyle\widetilde{\delta\varphi}=$
$\displaystyle\delta\varphi-\epsilon^{\eta}\partial_{\eta}\bar{\varphi}\,,$
(2.8)

where $\delta\rho$ and $\delta\varphi$ are the linear fluctuations of
$\bar{\rho}$ and $\bar{\varphi}$ respectively.

We can then get rid of the gauge dependence by building linear combinations of
perturbations. Concerning the scalar sector, few relevant examples are given
by the following gauge invariant variables. We can work exclusively with the
metric perturbations, where particularly important variables are the so-called
Bardeen potentials

$\displaystyle\Psi=$
$\displaystyle\,\psi+\mathcal{H}\left(B+\partial_{\eta}E\right)+\frac{1}{3}\Delta_{3}E\,,$
$\displaystyle\Phi=$
$\displaystyle\,\phi-\partial_{\eta}\left(B+\partial_{\eta}E\right)-\mathcal{H}\left(B+\partial_{\eta}E\right)\,.$
(2.9)

Other realizations of gauge invariant quantities relate matter to geometry and
are provided by

$\displaystyle\zeta$
$\displaystyle=\psi+\frac{1}{3}\Delta_{3}E-\frac{1}{3}\frac{\delta\rho}{\rho+p}\,,$
$\displaystyle\mathcal{R}$
$\displaystyle=\psi+\frac{1}{3}\Delta_{3}E+\left(\frac{\mathcal{H}}{\partial_{\eta}\bar{\varphi}}\right)\delta\varphi\,,$
(2.10)

which are both gauge invariant expressions related to the curvature
perturbations. Indeed, we recognize the combination
$\psi+\frac{1}{3}\Delta_{3}E$ as the gauge dependent curvature perturbation
which, as well-known, transforms with a temporal gauge mode. The different
gauge fixings for this gauge mode in the curvature perturbation then play an
important role in the current cosmological paradigm. Indeed $\zeta$ represents
the gauge invariant quantity whose value equals the spatial curvature
perturbation as evaluated on constant uniform energy density hypersurfaces. In
the same manner, $\mathcal{R}$ represents the spatial curvature perturbation
evaluated on hypersurfaces where the scalar field $\varphi$ is uniform. This
last one defines the physical curvature perturbations in the primordial
Universe which has to be connected with late-time inhomogeneities, since the
inflaton is the only clock available for single field models. Such variable is
then directly connected with the so-called Mukhanov-Sasaki variable [21, 22,
23] given by

$Q=\frac{\partial_{\eta}\bar{\varphi}}{\mathcal{H}}\mathcal{R}\,.$ (2.11)

For a later use, let us see the way to extract the SVT parameters from Eqs.
(2.2). Starting from Eqs. (2.2), we can first extract the scalar perturbation
$B$ by applying $\nabla^{i}$ to $\mathcal{B}_{i}$. We then have that

$\Delta_{3}B=\,\nabla^{i}\mathcal{B}_{i}\qquad\text{and}\qquad
B_{i}=\,\mathcal{B}_{i}-\partial_{i}\Delta^{-1}_{3}\left(\nabla^{j}\mathcal{B}_{j}\right)\,.$
(2.12)

In a similar manner, we can express the SVT perturbations in
$\mathcal{C}_{ij}$ from (2.2). By projecting $\mathcal{C}_{ij}$ onto
$\bar{\gamma}^{ij}$ and $D^{ij}$, we get the scalars

$\psi=-\frac{1}{6}\bar{\gamma}^{ij}\mathcal{C}_{ij}\qquad\text{and}\qquad\left(\Delta_{3}\right)^{2}E=\frac{3}{4}D^{ij}\mathcal{C}_{ij}\,.$
(2.13)

Hence, by using Eqs. (2.13), we can extract $F_{i}$ from
$\nabla^{j}\mathcal{C}_{ij}$, and obtain

$\Delta_{3}F_{i}=\nabla^{j}\mathcal{C}_{ij}+6\nabla_{i}\psi-4\nabla_{i}\left(\psi+\frac{1}{3}\Delta_{3}E\right)\,,$
(2.14)

where we split the contribution of $\psi$ in order to reconstruct the gauge
dependent curvature in the last term. Finally, thanks to Eqs. (2.13) and
(2.14), we have that the gauge invariant tensor perturbations are

$h_{ij}=\frac{1}{2}\mathcal{C}_{ij}-\frac{3\nabla_{i}\nabla_{j}}{\Delta_{3}}\psi+\left[\gamma_{ij}+\frac{\nabla_{i}\nabla_{j}}{\Delta_{3}}\right]\left(\psi+\frac{1}{3}\Delta_{3}E\right)-\frac{1}{\Delta_{3}}\left[\nabla^{l}\nabla_{(i}\mathcal{C}_{j)l}\right]\,.$
(2.15)

We remark that $h_{ij}$ in Eq. (2.15) is defined up to a free-function $f$
such that $\Delta_{3}f=0$. This follows from the fact that we needed to invert
the Laplacian in Eqs. (2.13) and (2.14).

The SVT decomposition here provided simply makes use of the irreducible
representations in terms of the $SO(3)$ group. This is a symmetry of the
background, once a foliation of the manifold in terms of constant-time
hypersurfaces is adopted, where also the operators $\bar{\gamma}_{ij}$,
$D_{ij}$, $\nabla_{i}$ are defined. However, no assumptions about a
perturbative expansion for $\delta g_{\mu\nu}$ has been required in order to
derive Eqs. (2.12)-(2.15). Therefore, all these equations can be interpreted
as a non-linear decomposition.

###  2.2 Light-Cone Perturbations

Here we will discuss a set of perturbations better adapted to the structure of
the light-cone. We will follow the formalism already developed in [19].

To this end, we start from the Geodesic Light-Cone (GLC) gauge. This is a
fully non-linear metric expressed in terms of the light-cone coordinates
$x^{\mu}=\left(\tau,\,w,\,\theta^{a}\right)$, where $\tau$ is the proper time
of a geodetic observer with four-velocity
$u_{\mu}=-\partial_{\mu}\tau=-\delta_{\mu}^{\tau}$, $w$ is a null coordinate
which describes the observer’s past light-cone and $\theta^{a}$ label the
incoming directions of light-like signals in the observer’s frame. This
coordinates system is built in a manner that both time-like geodesics and
light-like geodesics are exactly solved respectively by
$u_{\mu}=-\delta_{\mu}^{\tau}$ and $k_{\mu}=\partial_{\mu}w=\delta_{\mu}^{w}$.
This is possible thanks to the fixing of the gauge freedom allowed by general
relativity, rendering then the fully inhomogeneous metric tensor as [1]

$\displaystyle g^{GLC}_{\mu\nu}=\begin{pmatrix}0&-\Upsilon&\vec{0}\\\
-\Upsilon&\Upsilon^{2}+U^{2}&-U_{a}\\\
\vec{0}^{\mathbf{T}}&-U_{a}^{\mathbf{T}}&\gamma_{ab}\end{pmatrix}\,.$ (2.16)

As shown in [19], the linearization of (2.16) corresponds to the following
gauge fixing in standard perturbation theory in Eq. (2.1)

$\phi=0\qquad,\qquad\mathcal{B}_{r}=-\frac{C_{rr}}{2}\qquad,\qquad\mathcal{B}_{a}=-\mathcal{C}_{ar}\,,$
(2.17)

and this gauge fixing has been dubbed as Observational Synchronous Gauge
(OSG).

Now, in order to apply the SPS decomposition and its classification for the
gauge modes, we start from the homogeneous FLRW-like background metric (2.16).
This is given by the condition $\Upsilon=a$, where $a(\tau)$ is the scale
factor and $U_{a}=0$. The latter just follows from the isotropy of the
background, where there is no misalignment of the light-geodesics between two
light-cones in the FLRW geometry (differently from anisotropic geometries, as
shown in [24]). Additionally, we take

$d\tau=ad\eta\qquad\text{and}\qquad dw=dr+d\eta$ (2.18)

to link $\tau$ and $w$ to $\eta$ and $r$, whereas the angles remain unaltered,
i.e. $d\theta^{a}_{FLRW}=d\theta^{a}_{GLC}$222Thanks to this equality, in the
following we will omit the label FLRW or GLC when referring to the angular
part of the metric.. In this way, the desired framework for the light-cone
perturbations is [19]

$f_{\mu\nu}=\bar{f}_{\mu\nu}+\delta f_{\mu\nu}=\begin{pmatrix}0&-a&\vec{0}\\\
-a&a^{2}&\vec{0}\\\
\vec{0}^{\mathbf{T}}&\vec{0}^{\mathbf{T}}&a^{2}\bar{\gamma}_{ab}\end{pmatrix}+a^{2}\begin{pmatrix}L&M&V_{a}\\\
M&N&U_{a}\\\
V^{\mathbf{T}}_{a}&U^{\mathbf{T}}_{a}&\delta\gamma_{ab}\end{pmatrix}\,.$
(2.19)

Given that $U_{a}$ in Eq. (2.19) is a purely linear quantity, since now on, it
will be meant as a perturbation, unless differently specified.

The angular perturbations in $\delta f_{\mu\nu}$ may be further decomposed in
their irreducible form as

$\displaystyle V_{a}=$
$\displaystyle\,r^{2}\left(D_{a}v+\widetilde{D}_{a}\hat{v}\right)\,,$
$\displaystyle U_{a}=$
$\displaystyle\,r^{2}\left(D_{a}u+\widetilde{D}_{a}\hat{u}\right)\,,$
$\displaystyle\delta\gamma_{ab}=$
$\displaystyle\,2r^{2}\left(\bar{q}_{ab}\nu+D_{ab}\mu+\widetilde{D}_{ab}\hat{\mu}\right)\,,$
(2.20)

where $D_{a}$ is the covariant derivative w.r.t. the metric on the sphere
$\bar{q}_{ab}=\text{diag}\left(1,\sin^{2}\theta^{1}\right)$,
$\widetilde{D}_{a}=\epsilon_{a}^{\,b}D_{b}$, where $\epsilon_{ab}$ is the
Levi-Civita symbol, $D_{ab}\equiv D_{(a}D_{b)}-\frac{1}{2}\bar{q}_{ab}D^{2}$
and $\widetilde{D}_{ab}=D_{(a}\widetilde{D}_{b)}$.

Perturbations are classified as scalars $v,\,u,\,\mu$ and $\nu$, and pseudo-
scalar $\hat{v},\,\hat{u}$ and $\hat{\mu}$ under a rotation on the 2-D sphere
where $\tau$ and $w$ are constants. We underline that this perturbative scheme
has been built on a background where the light-cone structure is encoded.
Indeed, as proven in [17], the exact GLC gauge is equivalent to a 1+1+2 ADM-
like foliation of the Lorentzian manifold, where the 1+1 subspace contains a
time-like and a light-like hypersurface. On the other hand, the remaining 2-D
space-like hypersurface can be interpreted as a set of spheres at constant
time on the same light-cone. In our case, this structure precisely reflects
into the background metric $\bar{f}_{\mu\nu}$.

Here we anticipate that the trace-less perturbations of the 2-D sphere $\mu$
and $\hat{\mu}$ describe purely spin-2 gauge invariant perturbations in the
helicity basis. In fact, as we will also show along this work, and previously
claimed in [25, 17, 19], the scalar/pseudo-scalar decomposition presented in
(2.20) is most suitable to describe $E$ and $B$ modes polarization of the
perturbations in terms of the helicity decomposition (see also [26, 27, 28]
and the references therein, and [8] for applications to the GLC gauge).

###  2.3 SVT-GLC relation

In this subsection, we will make contact between the standard SVT
decomposition and the linear light-cone perturbations just presented. To this
end, we first report the relations between $\delta f_{\mu\nu}$ and $\delta
g_{\mu\nu}$ already found in Eqs. (3.7) of [19]. Indeed, thanks to the
coordinate transformation (2.18), we have

$\displaystyle\phi=$ $\displaystyle-\frac{1}{2}\left(a^{2}L+N+2aM\right)\,,$
$\displaystyle\mathcal{B}_{r}$ $\displaystyle=-\left(N+aM\right)\,,$
$\displaystyle\mathcal{B}_{a}$ $\displaystyle=-\left(U_{a}+aV_{a}\right)\,,$
$\displaystyle\mathcal{C}_{rr}$ $\displaystyle=N\,,$
$\displaystyle\mathcal{C}_{ra}$ $\displaystyle=U_{a}\,,$
$\displaystyle\mathcal{C}_{ab}$ $\displaystyle=\delta\gamma_{ab}\,.$ (2.21)

Now we can use the results from Sect. 2.2 and relate the SVT decomposition to
the GLC perturbations.

This can be done by extracting the SVT parameters directly from operators
appearing within the GLC d.o.f., once the decomposition in Eqs. (2.2) is
adopted. Hence, by inserting Eqs. (2.21) into Eqs. (2.12)-(2.15), we have that
the scalars are

$\displaystyle\psi=$ $\displaystyle-\frac{1}{6}\left(N+4\nu\right)\,,$
$\displaystyle\Delta_{3}B=$
$\displaystyle-\left(\partial_{w}+\frac{2}{r}\right)\left(N+aM\right)-D^{2}\left(u+av\right)\,,$
$\displaystyle\left(\Delta_{3}\right)^{2}E=$
$\displaystyle\,\frac{1}{2}\left(\partial_{w}^{2}+\frac{5}{r}\partial_{w}+\frac{3}{r^{2}}\right)\left(N-2\nu\right)-\frac{1}{4\,r^{2}}D^{2}\left(N+4\nu\right)$
$\displaystyle+\frac{1}{6\,r^{2}}\left(\partial_{w}+\frac{1}{r}\right)D^{2}u+\frac{3}{2\,r^{2}}D^{2}\nu+\frac{3}{4\,r^{2}}\left(D^{2}\right)^{2}\mu+\frac{3}{2r^{2}}D^{2}\mu\,,$
(2.22)

where we made use of the following useful relations

$\displaystyle\nabla^{i}\mathcal{C}_{ir}=$
$\displaystyle\left(\partial_{w}+\frac{2}{r}\right)N+D^{2}u-\frac{4}{r}\nu\,,$
$\displaystyle\nabla^{i}\mathcal{C}_{ia}=$
$\displaystyle\left(\partial_{w}+\frac{2}{r}\right)\left[r^{2}\left(D_{a}u+\tilde{D}_{a}\hat{u}\right)\right]+2D_{a}\nu\,+D_{a}\left(D^{2}\mu+2\mu\right)+\tilde{D}_{a}\left(D^{2}\hat{\mu}+2\hat{\mu}\right)\,,$
$\displaystyle\nabla^{i}\nabla^{j}\mathcal{C}_{ij}=$
$\displaystyle\,\partial_{w}^{2}N+\frac{4}{r}\left(\partial_{w}+\frac{1}{2r}\right)N+\frac{2}{r^{2}}\left(\partial_{w}+\frac{1}{r}\right)\left(r^{2}D^{2}u\right)\,$
$\displaystyle-\frac{4}{r}\left(\partial_{w}+\frac{1}{r}\right)\nu+\frac{2}{r^{2}}D^{2}\nu\
+\frac{1}{r^{2}}\left(D^{2}+2\right)D^{2}\mu\,.$ (2.23)

In the same way, we find that the vector field $B_{i}$ is

$\displaystyle\Delta_{3}B_{r}=$
$\displaystyle-\Delta_{3}\left(N+aM\right)+\partial_{w}\left[\left(\partial_{w}+\frac{2}{r}\right)\left(N+aM\right)+r^{2}D^{2}\left(u+av\right)\right]\,,$
$\displaystyle\Delta_{3}B_{a}=$
$\displaystyle\,\Delta_{3}\left[r^{2}D_{a}\left(u+av\right)+r^{2}\tilde{D}_{a}\left(\hat{u}+a\hat{v}\right)\right]\,$
$\displaystyle+\partial_{a}\left[\left(\partial_{w}+\frac{2}{r}\right)\left(N+aM\right)+r^{2}D^{2}\left(u+av\right)\right]\,.$
(2.24)

Finally, by inserting Eqs. (2.22) and (2.23) into Eq. (2.14), we also express
the components of the vector field $F_{i}$ in terms of GLC perturbations as

$\displaystyle\Delta_{3}F_{r}=$
$\displaystyle-\partial_{w}\left(N+4\nu\right)+\left(\partial_{w}+\frac{2}{r}\right)N+D^{2}u-\frac{4}{r}\nu$
$\displaystyle-\frac{4\partial_{w}}{\Delta_{3}}\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.\,\,$
$\displaystyle+\left.\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]\,,$
$\displaystyle\Delta_{3}F_{a}=$
$\displaystyle-\partial_{a}\left(N+4\nu\right)+\left(\partial_{w}+\frac{2}{r}\right)\left[r^{2}\left(D_{a}u+\tilde{D}_{a}\hat{u}\right)\right]+2D_{a}\nu\,$
$\displaystyle+D_{a}\left(D^{2}\mu+2\mu\right)+\tilde{D}_{a}\left(D^{2}\hat{\mu}+2\hat{\mu}\right)\,$
$\displaystyle-\frac{4\partial_{a}}{\Delta_{3}}\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.\,,\,$
$\displaystyle+\left.\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]\,.$
(2.25)

What emerges from Eqs. (2.22)-(2.25) is that the standard scalar and vector
perturbations look quite involved once expressed in terms of the light-cone
perturbations, even though their gauge transformation is consistent with the
ones expected in the SVT approach (see Appendix A for the explicit proves).
Before discussing how this procedure is applied to the tensor perturbations
and to other gauge invariant variables of cosmological interest, such as the
curvature perturbations and the Mukhanov-Sasaki variable, we report the gauge
dependent curvature in terms of GLC perturbations. This will be useful for
later evaluations and is given by

$\displaystyle\psi+\frac{1}{3}\Delta_{3}E=$
$\displaystyle\frac{1}{\Delta_{3}}\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.$
$\displaystyle\left.+\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]\,,$
(2.26)

where we have explicitly used also the SPS decomposition introduced in Eq.
(2.20). As a consistency check, we notice that Eq. (2.26) does not depend on
pseudo-scalar perturbations. The physical reason for that stands in the fact
that l.h.s. of Eq. (2.26) is entirely sourced by scalar perturbations in the
SVT decomposition and this means that no magnetic modes can be present on the
r.h.s. as well. Since in the SPS decomposition only the pseudo-scalars can
source the magnetic modes (see [17] for a detailed discussion in this regard),
they cannot appear on the r.h.s. We will discuss in details this relation in
Sect. 2.4 for the science case study of the tensor perturbations where both
electric and magnetic modes are present.

###  2.4 Gauge Invariant Quantities

With the formalism introduced above, we can straightforwardly focus on some
relevant gauge invariant quantities within our light-cone perturbation theory.
By using Eq. (2.26), we have that the quantities defined in Eqs. (2.10) are

$\displaystyle\zeta=$
$\displaystyle-\frac{1}{3}\frac{\delta\rho}{\rho+p}+\frac{1}{\Delta_{3}}\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.\,$
$\displaystyle+\left.\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]\,,$
$\displaystyle\mathcal{R}=$ $\displaystyle
H\frac{\delta\varphi}{\partial_{\tau}\varphi}+\frac{1}{\Delta_{3}}\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.\,$
$\displaystyle+\left.\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]\,,$
(2.27)

and we recall that $Q\equiv\partial_{\tau}\delta\varphi/H$ and
$H\equiv\partial_{\tau}a/a$. We notice that all the quantities in Eqs. (2.27)
do not depend on perturbations which are null in the linearized GLC gauge,
namely $V_{a}=L=0$ and $N+2aM=0$ (see Eq. (3.1) of [19]). This means that a
direct computation of $\zeta$, $\mathcal{R}$ and $Q$ within the linearized GLC
gauge is equivalent in form to the actual gauge invariant expressions for
these quantities. Moreover, as a side fact, we checked that our expression for
$\mathcal{R}$ agrees with the result of [20], where the linearized GLC gauge
has been fixed from the beginning. This agreement is shown in Appendix B.

Finally, we compute the tensor components as presented in Eq. (2.15) by using
Eqs. (2.22), (2.23) and (2.26). We then get the result for the radial
projection

$\displaystyle h_{rr}=$
$\displaystyle\,\frac{1}{2}N+\frac{1}{2\Delta_{3}}\partial_{w}^{2}\left(N+4\nu\right)-\frac{1}{\Delta_{3}}\partial_{w}\left[\left(\partial_{w}+\frac{2}{r}\right)N+D^{2}u-\frac{4}{r}\nu\right]$
$\displaystyle+\frac{1}{\Delta_{3}}\left(1+\frac{\partial_{w}^{2}}{\Delta_{3}}\right)\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.\,$
$\displaystyle+\left.\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]\,,$
(2.28)

for the radial-angular component

$\displaystyle h_{ar}=$
$\displaystyle\,\frac{r^{2}}{2}\left(D_{a}u+\tilde{D}_{a}\hat{u}\right)+\frac{1}{2\Delta_{3}}\left(\partial_{w}-\frac{1}{r}\right)D_{a}\left(N+4\nu\right)-\frac{D_{a}}{2\Delta_{3}}\left[\left(\partial_{w}+\frac{2}{r}\right)N+D^{2}u-\frac{4}{r}\nu\right]\,$
$\displaystyle-\frac{1}{2\Delta_{3}}\left(\partial_{w}-\frac{2}{r}\right)\left[\left(\partial_{w}+\frac{2}{r}\right)\left(r^{2}D_{a}u+r^{2}\tilde{D}_{a}\hat{u}\right)+2D_{a}\nu\right.\,$
$\displaystyle\left.+D_{a}\left(D^{2}\mu+2\mu\right)+\tilde{D}_{a}\left(D^{2}\hat{\mu}+2\hat{\mu}\right)\right]$
$\displaystyle+\frac{1}{\left(\Delta_{3}\right)^{2}}\left[\left(\partial_{w}-\frac{1}{r}\right)D_{a}\right]\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.\,$
$\displaystyle+\left.\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]\,,$
(2.29)

and for the pure angular part

$\displaystyle h_{ab}=$
$\displaystyle\,r^{2}\left(\bar{q}_{ab}\nu+D_{ab}\mu+\tilde{D}_{ab}\hat{\mu}\right)+\frac{1}{\Delta_{3}}\left(D_{a}D_{b}+\frac{\gamma_{ab}}{r}\partial_{w}\right)\left[\frac{1}{2}\left(N+4\nu\right)\right]\,$
$\displaystyle-\frac{1}{\Delta_{3}}\left\\{\frac{\gamma_{ab}}{r}\left[\left(\partial_{w}+\frac{2}{r}\right)N+D^{2}u\right]\right\\}-\frac{1}{\Delta_{3}}\left\\{\left(\partial_{w}+\frac{2}{r}\right)\left(r^{2}D_{a}D_{b}u+r^{2}\tilde{D}_{ab}\hat{u}\right)\,\right.$
$\displaystyle+\left.D_{a}D_{b}\left(2\nu+D^{2}\mu+2\mu\right)+\tilde{D}_{ab}\left(D^{2}\hat{\mu}+2\hat{\mu}\right)\right\\}$
$\displaystyle+\frac{1}{\Delta_{3}}\left[\gamma_{ab}+\frac{\left(D_{a}D_{b}+\frac{\gamma_{ab}}{r}\partial_{w}\right)}{\Delta_{3}}\right]\left\\{\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu\right.\,$
$\displaystyle+\left.\frac{1}{2r^{2}}D^{2}\mu+\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right\\}\,,$
(2.30)

which are the first order gauge invariant tensor perturbation within our new
perturbation theory. In Appendix A, we explicitly checked the expected gauge
transformation of the SVT perturbations in terms of the light-cone
perturbations. As expected, tensor perturbations are gauge invariant.

In Appendix B, we have also checked that our results for the tensor
perturbations agrees with the ones which can be evaluated by following the
procedure outlined in [20], where the authors provide a general formula to
evaluate the tensor perturbations, even though they do not provide their
explicit evaluations.

At first look, the expressions for the light-cone perturbations look a way
messier than the ones typically used in the SVT decomposition for the 1+3
spacetime foliation. Despite this complication, the advantage in using the
light-cone perturbations stands in the fact that our expressions are ready to
be decomposed in terms of electric and magnetic modes. We partially discussed
this feature after Eq. (2.26). To show the actual advantage of this
decomposition, we will introduce in the next Section the helicity basis. Even
though tensor perturbations contain both scalar and pseudo-scalar
contributions, let us note that the pure radial projection is exclusively
sourced by scalar perturbations. This is not surprising, since $h_{rr}$ is
just the helicity-0 mode of the tensor perturbations, hence it cannot contain
any pseudo-scalar term.

##  3 A coordinate independent Scalar-PseudoScalar decomposition

The results obtained in the previous sections show that a link between the
light-cone and the SVT perturbations does not necessarily bring to manageable
expressions. It is worth then to provide a way to classify perturbations in
the SPS manner which is coordinate independent, in order to better understand
the link between different perturbative schemes. To this end, we first
consider an unspecified metric $g_{\mu\nu}$ which can be linearized as follows

$\displaystyle g_{\mu\nu}\approx\bar{g}_{\mu\nu}+\delta g_{\mu\nu}\,,$ (3.1)

where $\bar{g}_{\mu\nu}$ is a generic background and $\delta g_{\mu\nu}$ are a
set of metric perturbations added on top of the former. With respect to the
background metric, we then define the following set of vectors

$\displaystyle\bar{u}^{\mu}\bar{u}_{\mu}=-1\qquad,\qquad\bar{n}^{\mu}\bar{n}_{\mu}=1\,,\qquad,\qquad\bar{n}^{\mu}u_{\mu}=0\,.$
(3.2)

Here $\bar{u}^{\mu}$ plays the role of the time-like flow for the
$\bar{g}_{\mu\nu}$, whereas $\bar{n}^{\mu}$ is the ”radial” space-like
direction within the spanned 3-D hyper-surface orthogonal to $\bar{u}^{\mu}$.
Since this hyper-surface is 3-D, it admits three independent directions to
form an orthonormal space-like basis. Indeed, besides $\bar{n}^{\mu}$, we can
also introduce $\bar{e}^{\mu}_{A}$, where $A=1,2$ and

$\bar{e}^{\mu}_{A}\bar{e}_{\mu
B}=\delta_{AB}\qquad\text{and}\qquad\bar{n}^{\mu}\bar{e}_{\mu A}=0\,.$ (3.3)

We then identify $\\{\bar{n}^{\mu},\bar{e}^{\mu}_{A}\\}$ as the orthonormal
space-like triad and $\bar{e}^{\mu}_{A}$ as the unit-vectors orthogonal to the
”radial” direction $\bar{n}^{\mu}$. The vectors $\bar{e}^{\mu}_{A}$ can be
rewritten in the so-called helicity basis, namely a complex linear combination
of their elements given by

$\bar{e}^{\mu}_{\pm}\equiv\frac{1}{\sqrt{2}}\left(\bar{e}^{\mu}_{1}\pm
i\bar{e}^{\mu}_{2}\right)\,,$ (3.4)

where it follows that $\bar{e}^{\mu}_{\pm}\bar{e}_{\mu\pm}=0$ and
$\bar{e}^{\mu}_{\pm}\bar{e}_{\mu\mp}=1$. The advantage in working with Eq.
(3.4) stands in the fact that $\bar{e}^{\mu}_{\pm}$ have well-defined
transformation rules under 2-D rotations in the Euclidean plane spanned by
$\bar{e}^{\mu}_{A}$ itself. Indeed, such a rotation is given by

$R^{A}_{B}=\cos\beta\,\delta^{A}_{B}+i\sin\beta\,\left(\sigma_{2}\right)^{A}_{B}\,,$
(3.5)

where $\sigma_{2}$ is the antisymmetric Pauli matrix and $\beta$ is the
rotation angle. Hence, for the rotated basis
$\widetilde{\bar{e}^{\mu}_{A}}\equiv R^{B}_{A}\bar{e}^{\mu}_{B}$, the helicity
basis given by
$\widetilde{\bar{e}^{\mu}_{\pm}}\equiv\frac{1}{\sqrt{2}}\left(\widetilde{\bar{e}^{\mu}_{1}}\pm
i\widetilde{\bar{e}^{\mu}_{2}}\right)$ transforms as

$\widetilde{\bar{e}^{\mu}_{\pm}}=e^{\pm i\beta}\,\bar{e}^{\mu}_{\pm}\,.$ (3.6)

In this way, the perturbations $\delta g_{\mu\nu}$ can be classified according
to their transformation properties under such a rotation. Explicitly, we have
4 helicity-0 projections, i.e.

$\mathcal{S}\equiv\bar{u}^{\mu}\bar{u}^{\nu}\delta
g_{\mu\nu}\quad,\quad\mathcal{T}_{\rVert}\equiv\bar{n}^{\mu}\bar{n}^{\nu}\delta
g_{\mu\nu}\quad,\quad\mathcal{V}_{\rVert}\equiv\bar{n}^{\mu}\bar{u}^{\nu}\delta
g_{\mu\nu}\quad,\quad\mathcal{T}\equiv\bar{e}^{\mu}_{+}\bar{e}^{\nu}_{-}\delta
g_{\mu\nu}\,,$ (3.7)

two helicity-1 projections

$\mathcal{V}_{\pm}\equiv\bar{u}^{\mu}\bar{e}^{\nu}_{\pm}\delta
g_{\mu\nu}\quad,\quad\mathcal{T}_{\rVert\pm}\equiv\bar{n}^{\mu}\bar{e}^{\nu}_{\pm}\delta
g_{\mu\nu}\,,$ (3.8)

and a single set of helicity-2 perturbations, namely

$\mathcal{T}_{\pm\pm}\equiv\bar{e}^{\mu}_{\pm}\bar{e}^{\nu}_{\pm}\delta
g_{\mu\nu}\,.$ (3.9)

This procedure to classify linear perturbations has the manifest advantage
that is independent of the (possible) symmetries present in the Killing
vectors of $\bar{g}_{\mu\nu}$. This is not the case for the standard
cosmological perturbation theory, where the SVT representation with respect to
the rotational symmetry of the background is largely used. Moreover, also the
gauge properties of our projections can be treated without an explicit notion
of the adopted background. Indeed, under a linear diffeomorphism
$x^{\mu}\rightarrow\tilde{x}^{\mu}=x^{\mu}+\epsilon^{\mu}$, the metric
transforms as

$\widetilde{\delta g}_{\mu\nu}(x^{\alpha})=\delta
g_{\mu\nu}(x^{\alpha})-\bar{\nabla}_{\mu}\epsilon_{\nu}(x^{\alpha})-\bar{\nabla}_{\nu}\epsilon_{\mu}(x^{\alpha})\,,$
(3.10)

where $\bar{\nabla}_{\mu}$ is the covariant derivative with respect to
$\bar{g}_{\mu\nu}$. In this way, we can project $\epsilon^{\mu}$ as well onto
the background vectors and get

$T\equiv\bar{u}^{\mu}\epsilon_{\mu}\qquad,\qquad
R\equiv\bar{n}^{\mu}\epsilon_{\mu}\qquad,\qquad
X_{\pm}\equiv\bar{e}^{\mu}_{\pm}\epsilon_{\mu}\,.$ (3.11)

Hence, it follows that helicity-0 perturbations (3.7) transform as

$\displaystyle\widetilde{\mathcal{S}}=$
$\displaystyle\,\mathcal{S}-2\,\bar{u}^{\mu}\bar{\nabla}_{\mu}T+2\left(\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)\epsilon_{\nu}\,,$
$\displaystyle\widetilde{\mathcal{T}}_{\rVert}=$
$\displaystyle\,\mathcal{T}_{\rVert}-2\,\bar{n}^{\mu}\bar{\nabla}_{\mu}R+2\left(\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)\epsilon_{\nu}\,,$
$\displaystyle\widetilde{\mathcal{V}}_{\rVert}=$
$\displaystyle\,\mathcal{V}_{\rVert}-\bar{n}^{\mu}\bar{\nabla}_{\mu}T-\bar{u}^{\mu}\bar{\nabla}_{\mu}R+\left(\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)\epsilon_{\nu}+\left(\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)\epsilon_{\nu}\,,$
$\displaystyle\widetilde{\mathcal{T}}=$
$\displaystyle\,\mathcal{T}-\bar{e}^{\mu}_{+}\bar{\nabla}_{\mu}X_{-}-\bar{e}^{\mu}_{-}\bar{\nabla}_{\mu}X_{+}+\left(\bar{e}^{\mu}_{+}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{-}\right)\epsilon_{\nu}+\left(\bar{e}^{\mu}_{-}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{+}\right)\epsilon_{\nu}\,,$
(3.12)

helicity-1 projections in (3.8) change according to

$\displaystyle\widetilde{\mathcal{V}}_{\pm}=$
$\displaystyle\,\mathcal{V}_{\pm}-\bar{u}^{\mu}\bar{\nabla}_{\mu}X_{\pm}-\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}T+\left(\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)\epsilon_{\nu}+\left(\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)\epsilon_{\nu}\,,$
$\displaystyle\widetilde{\mathcal{T}}_{\rVert\pm}=$
$\displaystyle\,\mathcal{T}_{\rVert\pm}-\bar{n}^{\mu}\bar{\nabla}_{\mu}X_{\pm}-\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}R+\left(\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)\epsilon_{\nu}+\left(\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)\epsilon_{\nu}\,,$
(3.13)

and, finally, helicity-2 perturbations (3.9) transform as

$\widetilde{\mathcal{T}}_{\pm\pm}=\,\mathcal{T}_{\pm\pm}-2\,\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}X_{\pm}+2\left(\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)\epsilon_{\nu}\,.$
(3.14)

At this point, we first remark that the projected gauge modes (3.11) are
scalars under background coordinate transformations, such that all the
covariant derivatives $\bar{\nabla}_{\mu}$ acting on $T,\,R,\,X_{\pm}$ in
(3.12)-(3.14) can be immediately written as ordinary partial derivatives.
Moreover, $T$ and $R$ can be respectively interpreted as ”time” and ”radial”
gauge modes, although the explicit form of time-like and space-like coordinate
for $\bar{g}_{\mu\nu}$ has not been provided. The same happens for $X_{\pm}$,
which can be read as the clockwise and anti-clockwise gauge modes on the 2-D
space-like screen where the angular coordinates are typically defined.

To conclude this section, we do a further step in the gauge transformations
(3.12)-(3.14). Indeed, it is still a bit unsatisfactory to have those gauge
transformation explicitly in terms of $\epsilon_{\mu}$ rather than its
projections $T,\,R,\,X_{\pm}$. To this end, we then write explicitly each
vector in the form $h^{\mu}\bar{\nabla}_{\mu}f^{\nu}$ in terms of its
projections onto $\\{\bar{u}_{\nu},\bar{n}_{\nu},\bar{e}_{\nu\pm}\\}$, namely

$h^{\mu}\bar{\nabla}_{\mu}f^{\nu}=\left(\bar{u}_{\rho}h^{\mu}\bar{\nabla}_{\mu}f^{\rho}\right)\bar{u}^{\nu}+\left(\bar{n}_{\rho}h^{\mu}\bar{\nabla}_{\mu}f^{\rho}\right)\bar{n}^{\nu}+\left(\bar{e}_{\rho\mp}h^{\mu}\bar{\nabla}_{\mu}f^{\rho}\right)\bar{e}^{\nu}_{\pm}\,,$
(3.15)

where the last component projects $\mp$ helicity onto the $\pm$ vector because
of the properties $\bar{e}^{\mu}_{\pm}\bar{e}_{\mu\mp}=1$ and
$\bar{e}^{\mu}_{\pm}\bar{e}_{\mu\pm}=0$. In this way, the gauge transformation
can be written as

$\displaystyle\widetilde{\mathcal{S}}=$
$\displaystyle\,\mathcal{S}-2\,\bar{u}^{\mu}\bar{\nabla}_{\mu}T+2\left(\bar{u}_{\nu}\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)T+2\left(\bar{n}_{\nu}\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)R+2\left(\bar{e}_{\nu\mp}\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)X_{\pm}\,,$
$\displaystyle\widetilde{\mathcal{T}}_{\rVert}=$
$\displaystyle\,\mathcal{T}_{\rVert}-2\,\bar{n}^{\mu}\bar{\nabla}_{\mu}R+2\left(\bar{u}_{\nu}\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)T+2\left(\bar{n}_{\nu}\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)R+2\left(\bar{e}_{\nu\mp}\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)X_{\pm}\,,$
$\displaystyle\widetilde{\mathcal{V}}_{\rVert}=$
$\displaystyle\,\mathcal{V}_{\rVert}-\bar{n}^{\mu}\bar{\nabla}_{\mu}T-\bar{u}^{\mu}\bar{\nabla}_{\mu}R$
$\displaystyle+\left(\bar{u}_{\nu}\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)T+\left(\bar{n}_{\nu}\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)R+\left(\bar{e}_{\nu\mp}\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)X_{\pm}$
$\displaystyle+\left(\bar{u}_{\nu}\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)T+\left(\bar{n}_{\nu}\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)R+\left(\bar{e}_{\nu\mp}\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)X_{\pm}\,,$
$\displaystyle\widetilde{\mathcal{T}}=$
$\displaystyle\,\mathcal{T}-\bar{e}^{\mu}_{+}\bar{\nabla}_{\mu}X_{-}-\bar{e}^{\mu}_{-}\bar{\nabla}_{\mu}X_{+}$
$\displaystyle+\left(\bar{u}_{\nu}\bar{e}^{\mu}_{+}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{-}\right)T+\left(\bar{n}_{\nu}\bar{e}^{\mu}_{+}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{-}\right)R+\left(\bar{e}_{\nu\mp}\bar{e}^{\mu}_{+}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{-}\right)X_{\pm}$
$\displaystyle+\left(\bar{u}_{\nu}\bar{e}^{\mu}_{-}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{+}\right)T+\left(\bar{n}_{\nu}\bar{e}^{\mu}_{-}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{+}\right)R+\left(\bar{e}_{\nu\mp}\bar{e}^{\mu}_{-}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{+}\right)X_{\pm}\,,$
(3.16)

for the helicity-0 components, as

$\displaystyle\widetilde{\mathcal{V}}_{\pm}=$
$\displaystyle\,\mathcal{V}_{\pm}-\bar{u}^{\mu}\bar{\nabla}_{\mu}X_{\pm}-\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}T$
$\displaystyle+\left(\bar{u}_{\nu}\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)T+\left(\bar{n}_{\nu}\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)R+\left(\bar{e}_{\nu\mp}\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{u}^{\nu}\right)X_{\pm}$
$\displaystyle+\left(\bar{u}_{\nu}\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)T+\left(\bar{n}_{\nu}\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)R+\left(\bar{e}_{\nu\mp}\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)X_{\pm}\,,$
$\displaystyle\widetilde{\mathcal{T}}_{\rVert\pm}=$
$\displaystyle\,\mathcal{T}_{\rVert\pm}-\bar{n}^{\mu}\bar{\nabla}_{\mu}X_{\pm}-\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}R$
$\displaystyle+\left(\bar{u}_{\nu}\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)T+\left(\bar{n}_{\nu}\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)R+\left(\bar{e}_{\nu\mp}\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{n}^{\nu}\right)X_{\pm}$
$\displaystyle+\left(\bar{u}_{\nu}\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)T+\left(\bar{n}_{\nu}\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)R+\left(\bar{e}_{\nu\mp}\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)X_{\pm}\,,$
(3.17)

for the helicity-1 projections and finally as

$\widetilde{\mathcal{T}}_{\pm\pm}=\,\mathcal{T}_{\pm\pm}-2\,\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}X_{\pm}+2\left(\bar{u}_{\nu}\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)T+2\left(\bar{n}_{\nu}\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)R+2\left(\bar{e}_{\nu\mp}\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}\right)X_{\pm}\,,$
(3.18)

for the helicity-2 ones. Although Eqs. (3.16)-(3.18) might look more
complicated than Eqs. (3.12)-(3.14), they have the advantage that all the
gauge modes are explicitly written in terms of their scalar projections
$T,\,R$ and $X_{\pm}$. This renders their manipulation quite easier when the
background (and its symmetries, if any!) are left unspecified.

The fact that the projections (3.7)-(3.9) and their gauge transformations
(3.16)-(3.18) are readily given only in terms of scalar quantities means that
they can be evaluated for different backgrounds and directly compared between
each other.

To conclude this section, we underline that for the gauge transformations
(3.16)-(3.18) no notion about the transport of the basis
$\\{\bar{u}^{\mu},\bar{n}^{\nu},\bar{e}^{\mu}_{\pm}\\}$ has been adopted.
Particular choices can heavily simplify the overall form of the gauge
transformations. This will be manifest in the subsequent sections.

###  3.1 Geodesic Light-Cone background

Here we want to apply the general formalism developed in the previous section
to the perturbation theory on the light-cone already introduced in [19]. We
can now profit of the properties of the GLC gauge to find the suited
background basis for the helicity projections presented in this section. Since
$\bar{f}_{\mu\nu}$ shares the same features of $g^{GLC}_{\mu\nu}$, we start
from the background vectors

$\bar{u}_{\mu}=-\delta^{\tau}_{\mu}\qquad\text{and}\qquad\bar{k}_{\mu}=\delta^{w}_{\mu}\,,$
(3.19)

and build $\bar{n}_{\nu}$ as

$\bar{n}_{\nu}=\bar{u}_{\mu}+\left(\bar{u}^{\alpha}\bar{k}_{\alpha}\right)^{-1}\bar{k}_{\mu}=-\delta^{\tau}_{\mu}+a\,\delta^{w}_{\mu}\,.$
(3.20)

Indeed, it is straightforward to check that $\bar{u}^{\mu}\bar{u}_{\mu}=-1$
and $\bar{n}^{\mu}\bar{n}_{\mu}=1$. For what concerns the basis
$\bar{e}^{\nu}_{A}$, the requirement that
$\bar{u}_{\nu}\bar{e}^{\nu}_{A}=\bar{n}_{\nu}\bar{e}^{\nu}_{A}=0$ returns that
they can only admit non-null components in their angular part. We fix the
latter as follows

$\bar{e}_{\nu 1}=ar\,\delta^{\theta^{1}}_{\nu}\qquad,\qquad\bar{e}_{\nu
2}=ar\sin\theta^{1}\,\delta^{\theta^{2}}_{\nu}\,.$ (3.21)

We remark that this choice is not only consistent with the requirements (3.3)
but also provides that

$\Pi^{\mu}_{\nu}\bar{k}^{\alpha}\bar{\nabla}_{\alpha}\bar{e}^{\nu}_{A}=0\,,$
(3.22)

where
$\Pi_{\mu\nu}\equiv\bar{f}_{\mu\nu}+\bar{u}_{\mu}\bar{u}_{\nu}-\bar{n}_{\mu}\bar{n}_{\nu}$
is the 2-D sub-manifold orthogonal to $\bar{u}_{\mu}$ and $\bar{n}_{\mu}$. In
other words, $\bar{e}^{\mu}_{A}$ is chosen to be the background Sachs basis
[6] and a notion of transport for it is provided.

Thanks to Eqs. (3.21), it is easy now to obtain also the circular basis (3.4)
as

$\bar{e}_{\nu\pm}=\frac{ar}{\sqrt{2}}\,\left(\delta^{\theta^{1}}_{\nu}\pm
i\sin\theta^{1}\,\delta^{\theta^{2}}_{\nu}\right)\,.$ (3.23)

With Eqs. (3.19), (3.20) and (3.23), the SPS perturbations (3.7)-(3.9) in the
light-cone framework are readily written as four helicity-0 perturbations

$\mathcal{S}=a^{2}L+2aM+N\quad,\quad\mathcal{T}_{\rVert}=N\quad,\quad\mathcal{V}_{\rVert}=N+aM\quad,\quad\mathcal{T}=2\,\nu\,,$
(3.24)

two helicity-1 projections

$\mathcal{V}_{\pm}=r\,\partial_{\pm}\left[\left(u+av\right)\mp
i\left(\hat{u}+a\hat{v}\right)\right]\quad,\quad\mathcal{T}_{\rVert\pm}=r\,\partial_{\pm}\left(u\mp
i\hat{u}\right)\,,$ (3.25)

and a single set of helicity-2 perturbations, namely

$\mathcal{T}_{\pm\pm}=\,\partial^{2}_{\pm}\left(\mu\mp i\hat{\mu}\right)\,.$
(3.26)

Moreover, the background basis
$\\{\bar{u}_{\mu},\bar{n}_{\nu},\bar{e}_{\nu\pm}\\}$ explicitly given by Eqs.
(3.19), (3.20) and (3.23) also provides the needed transport properties of the
basis along the manifold. These are resumed as follows

$\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\nu}=\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\nu}=\bar{u}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}=\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{e}^{\nu}_{\pm}=0\,,$
(3.27)

where the transport of $\bar{u}^{\mu}$ is a direct consequence of the fact
that $\bar{u}^{\mu}$ follows a time-like geodesic flow and the transport of
$\bar{e}^{\mu}_{\pm}$ along $\bar{u}^{\nu}$ and $\bar{n}^{\nu}$ follows from
Eq. (3.22). In addition to them, we also have that

$\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{n}^{\rho}=H\,\bar{u}^{\rho}\qquad,\qquad\bar{n}^{\mu}\bar{\nabla}_{\mu}\bar{u}^{\rho}=H\,\bar{n}^{\rho}\,,$
(3.28)

and

$\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{u}^{\rho}=H\bar{e}^{\rho}_{\pm}\qquad,\qquad\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{n}^{\rho}=\frac{1}{ar}\,\bar{e}^{\rho}_{\pm}\,.$
(3.29)

Finally, for the circular basis, we have

$\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{e}^{\rho}_{\mp}=H\bar{u}^{\rho}-\frac{\bar{n}^{\rho}}{ar}-\frac{\cot\theta}{\sqrt{2}\,ar}\bar{e}^{\rho}_{\mp}\qquad\text{and}\qquad\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}\bar{e}^{\rho}_{\pm}=\frac{\cot\theta}{\sqrt{2}\,ar}\bar{e}^{\rho}_{\pm}\,.$
(3.30)

The knowledge of transport for the background basis then immediately returns
(and simplifies) the gauge transformations of the helicity projections
reported in (3.12)-(3.14), namely

$\displaystyle\widetilde{\mathcal{S}}=$
$\displaystyle\,\mathcal{S}-2\,\bar{u}^{\mu}\bar{\nabla}_{\mu}T\,,$
$\displaystyle\widetilde{\mathcal{T}}_{\rVert}=$
$\displaystyle\,\mathcal{T}_{\rVert}-2\,\bar{n}^{\mu}\bar{\nabla}_{\mu}R-2\,HT\,,$
$\displaystyle\widetilde{\mathcal{V}}_{\rVert}=$
$\displaystyle\,\mathcal{V}_{\rVert}-\bar{n}^{\mu}\bar{\nabla}_{\mu}T-\bar{u}^{\mu}\bar{\nabla}_{\mu}R+HR\,,$
$\displaystyle\widetilde{\mathcal{T}}=$
$\displaystyle\,\mathcal{T}-\bar{e}^{\mu}_{+}\bar{\nabla}_{\mu}X_{-}-\bar{e}^{\mu}_{-}\bar{\nabla}_{\mu}X_{+}-2\,HT-\frac{2}{ar}R-\frac{\cot\theta}{\sqrt{2}\,ar}X_{-}-\frac{\cot\theta}{\sqrt{2}\,ar}X_{+}$
$\displaystyle=$
$\displaystyle\,\mathcal{T}+\frac{1}{ar}\left(\partial_{+}X_{-}+\partial_{-}X_{+}\right)-2\,HT-\frac{2}{ar}R\,,$
(3.31)

for the helicity-0 perturbations, then

$\displaystyle\widetilde{\mathcal{V}}_{\pm}=$
$\displaystyle\,\mathcal{V}_{\pm}-\bar{u}^{\mu}\bar{\nabla}_{\mu}X_{\pm}-\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}T+HX_{\pm}=\,\mathcal{V}_{\pm}-\bar{u}^{\mu}\bar{\nabla}_{\mu}X_{\pm}-\frac{1}{ar}\partial_{\pm}T+HX_{\pm}\,,$
$\displaystyle\widetilde{\mathcal{T}}_{\rVert\pm}=$
$\displaystyle\,\mathcal{T}_{\rVert\pm}-\bar{n}^{\mu}\bar{\nabla}_{\mu}X_{\pm}-\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}R+\frac{1}{ar}X_{\pm}=\,\mathcal{T}_{\rVert\pm}-\bar{n}^{\mu}\bar{\nabla}_{\mu}X_{\pm}-\frac{1}{ar}\partial_{\pm}R+\frac{1}{ar}X_{\pm}\,,$
(3.32)

for the helicity-1 projections and finally

$\widetilde{\mathcal{T}}_{\pm\pm}=\,\mathcal{T}_{\pm\pm}-2\,\left(\bar{e}^{\mu}_{\pm}\bar{\nabla}_{\mu}X_{\pm}-\frac{\cot\theta}{\sqrt{2}\,ar}X_{\pm}\right)=\,\mathcal{T}_{\pm\pm}-\frac{2}{ar}\partial_{\pm}X_{\pm}\,,$
(3.33)

for the helicity-2 modes. We notice few points at this stage:

  * •

in the last of Eqs (3.31) and (3.33), the emergent structure from general
gauge transformations automatically reproduces the derivatives
$\partial_{\pm}$, which are related to the spin-lowering and raising operators
${/\partial}$ and $\overline{{/\partial}}$, discussed in Appendix C;

  * •

moreover, all the projections correctly transform according to their helicity
properties. This is evident from the fact that $T$ and $R$ and helicity-0
gauge modes. Moreover, $X_{\pm}$ have helicity $\pm 1$ and the action of
$\partial_{\pm}$ is such that it increases/decreases by one unity the helicity
of the field they act on;

  * •

gauge transformations in Eqs. (3.31)-(3.33) have the advantage to make
manifest the physical interpretations of the gauge modes themselves, since $T$
can be interpreted as the time-like mode, $R$ is the radial mode and $X_{\pm}$
correspond to the angular gauge freedom. Clearly, it is possible to express
them according to the infinitesimal shift $\xi^{\mu}$ of the light-cone
coordinates
$\left(\tau,w,\theta^{a}\right)\rightarrow\left(\tau,w,\theta^{a}\right)+\xi^{\mu}$,
as done in [19], having then

$T=-\xi^{0}\quad,\quad R=-\xi^{0}+a\xi^{w}\quad,\quad
X_{\pm}=\bar{e}_{a\pm}\xi^{a}\,.$ (3.34)

It is evident, however, how this would mix the different gauge modes.
Moreover, the expressions in terms of $T,\,R,\,X_{\pm}$ have the advantage to
deal with scalar quantities. This is not the case if one works with
$\xi^{\mu}$ instead.

###  3.2 Friedmann-Lemaître-Robertson-Walker background

We now want to apply our formalism for the projected inhomogeneities to the
standard linear perturbation theory built on the FLRW background. In fact, our
final goal is to connect the light-cone perturbations in the helicity basis to
the standard perturbations in the SVT decomposition. To this end, we will
profit of the invariance for finite background coordinate transformation of
our perturbative scheme. In fact, by using the background coordinate
transformation in Eqs. (2.18), we get that for the FLRW backgound, our basis
is

$\bar{u}_{\mu}=-a\,\delta^{\eta}_{\mu}\qquad,\qquad\bar{n}_{\mu}=a\,\delta_{\mu}^{r}\,,\qquad,\qquad\bar{e}_{\nu
1}=ar\,\delta^{\theta^{1}}_{\nu}\qquad,\qquad\bar{e}_{\nu
2}=ar\sin\theta^{1}\,\delta^{\theta^{2}}_{\nu}\,.$ (3.35)

Moreover, the transport equations are exactly the same as Eqs. (3.27)-(3.30),
with the substitution $H=\mathcal{H}/a$ and $r=w-\eta$. In this way, we can
immediately write the helicity-0 projections as

$\mathcal{S}=-2\phi\quad,\quad\mathcal{T}_{\rVert}=\mathcal{C}_{rr}\quad,\quad\mathcal{V}_{\rVert}=-\mathcal{B}_{r}\quad,\quad\mathcal{T}=\frac{\bar{\gamma}^{ab}\mathcal{C}_{ab}}{2}\,.$
(3.36)

The two helicity-1 projections become

$\mathcal{V}_{\pm}=-a\,\bar{e}^{a}_{\pm}\mathcal{B}_{a}\quad,\quad\mathcal{T}_{\rVert\pm}=a\,\bar{e}^{a}_{\pm}\mathcal{C}_{ra}\,.$
(3.37)

Finally the helicity-2 perturbations is given by

$\mathcal{T}_{\pm\pm}=a^{2}\,\bar{e}^{a}_{\pm}\bar{e}^{b}_{\pm}\mathcal{C}_{ab}\,.$
(3.38)

The knowledge of transport for the background basis then immediately returns
(and simplify) the gauge transformations of the helicity projections. These
are formally equivalent to (3.16)-(3.18), once it is taken into account that
$H=\mathcal{H}/a$ and $r=w-\eta$ and now $T,\,R$ and $X_{\pm}$ are given by

$T=-a\,\epsilon^{\eta}\quad,\quad R=a\,\epsilon^{r}\quad,\quad
X_{\pm}=\bar{e}_{a\pm}\epsilon^{a}\,.$ (3.39)

##  4 E and B modes for tensor perturbations

The results achieved in the previous sections allow to relate directly the
projected perturbations as evaluated w.r.t. different backgrounds. Indeed,
since the helicity perturbations are independent of the chosen background, we
can straightforwardly equal Eqs. (3.36)-(3.38) with (3.24)-(3.26). We then
obtain for the helicity-0 projections

$\phi=-\frac{1}{2}\left(a^{2}L+2aM+N\right)\quad,\quad\mathcal{C}_{rr}=N\quad,\quad\mathcal{B}_{r}=-N-aM\quad,\quad\bar{\gamma}^{ab}\mathcal{C}_{ab}=4\nu\,,$
(4.1)

for the helicity-1

$\displaystyle\mathcal{B}_{\pm}\equiv$
$\displaystyle\,\bar{e}^{a}_{\pm}\mathcal{B}_{a}=-\frac{r}{a}\,\partial_{\pm}\left[\left(u+av\right)\mp
i\left(\hat{u}+a\hat{v}\right)\right]\,,$
$\displaystyle\mathcal{C}_{r\pm}\equiv$
$\displaystyle\,\bar{e}^{a}_{\pm}\mathcal{C}_{ra}=\frac{r}{a}\,\partial_{\pm}\left(u\mp
i\hat{u}\right)\,,$ (4.2)

and finally for the helicity-2

$\mathcal{C}_{\pm\pm}\equiv\bar{e}^{a}_{\pm}\bar{e}^{b}_{\pm}\mathcal{C}_{ab}=\frac{2}{a^{2}}\,\partial^{2}_{\pm}\left(\mu\mp
i\hat{\mu}\right)\,.$ (4.3)

As a check, we notice that the results of Eqs. (4.1) are in agreement with the
ones provided in (2.21).

A remarkable feature is that Eqs. (4.1)-(4.3) are given by construction in
terms of the spin raising and lowering operators. This is a clear advantage if
one is interested in studying the $E$\- and $B$\- modes of the perturbations.
Indeed, for a generic tensor field of rank $s$ $T_{{a_{1}}...{a_{s}}}$, we
recall that the projections onto $\bar{e}^{a}_{\pm}$ can be written as

$T_{\pm}\equiv\bar{e}^{a_{1}}_{\pm}...\bar{e}^{a_{s}}_{\pm}T_{{a_{1}}...{a_{s}}}\,,$
(4.4)

and then one can extract its $E$\- and $B$\- modes thanks to the action of the
spin raising and lowering operators, respectively ${/\partial}$ and
$\overline{{/\partial}}$, as

$\displaystyle
T^{\mathbf{E}}\equiv\frac{\overline{{/\partial}}^{\,s}T_{+}+{/\partial}^{\,s}T_{-}}{2}\qquad\text{and}\qquad
T^{\mathbf{B}}\equiv-i\frac{\overline{{/\partial}}^{\,s}T_{+}-{/\partial}^{\,s}T_{-}}{2}\,.$
(4.5)

We redirect to Appendix C for details and useful relations about
$\overline{{/\partial}}$ and ${/\partial}$.

Let us notice that the projections (4.4) have helicity $s$. Hence, for the
helicity-1 and 2 projections, we can directly obtain the $E$\- and $B$\- modes
of the perturbations (4.2) and (4.3) as

$\displaystyle\mathcal{B}^{\mathbf{E}}\equiv$
$\displaystyle\,\frac{\overline{{/\partial}}\mathcal{B}_{+}+{/\partial}\mathcal{B}_{-}}{2}=\frac{r}{a\sqrt{2}}D^{2}\left(u+av\right)\quad,\quad\mathcal{B}^{\mathbf{B}}\equiv\,-i\frac{\overline{{/\partial}}\mathcal{B}_{+}-{/\partial}\mathcal{B}_{-}}{2}=\frac{r}{a\sqrt{2}}D^{2}\left(\hat{u}+a\hat{v}\right)\,,$
$\displaystyle\mathcal{C}_{r}^{\mathbf{E}}\equiv$
$\displaystyle\,\frac{\overline{{/\partial}}\mathcal{C}_{r+}+{/\partial}\mathcal{C}_{r-}}{2}=-\frac{r}{a\sqrt{2}}D^{2}u\quad,\quad\mathcal{C}_{r}^{\mathbf{B}}\equiv-i\frac{\overline{{/\partial}}\mathcal{C}_{r+}-{/\partial}\mathcal{C}_{r-}}{2}=-\frac{r}{a\sqrt{2}}D^{2}\hat{u}\,,$
$\displaystyle\mathcal{C}^{\mathbf{E}}\equiv$
$\displaystyle\,\frac{\overline{{/\partial}}^{2}\mathcal{C}_{++}+{/\partial}^{2}\mathcal{C}_{--}}{2}=\,\frac{\left(D^{2}+2\right)D^{2}\mu}{a^{2}}\quad,\quad\mathcal{C}^{\mathbf{B}}\equiv-i\frac{\overline{{/\partial}}^{2}\mathcal{C}_{++}-{/\partial}^{2}\mathcal{C}_{--}}{2}=\frac{\left(D^{2}+2\right)D^{2}\hat{\mu}}{a^{2}}\,.$

In Eqs. (LABEL:eq:E-B-modes) we have widely made use of the properties
reported in Appendix C and used the relations
$\partial_{+}\equiv-{/\partial}/\sqrt{2}$ and
$\partial_{-}\equiv-\overline{{/\partial}}/\sqrt{2}$.

We notice that an important connection emerges when we relate standard
perturbations (on top of a 1+3 background) with the light-cone ones (on top of
a 1+1+2 background). Indeed, as can be seen in Table 1, an interesting block
correspondence between the two sets of perturbations arises. In particular,
among the four helicity-0 projections, the trace of the angular part is
equivalent to $\nu$ and decouples from the other projections. Hence, the pure
time-time standard perturbation $\phi$ and the radial projections
$\mathcal{B}_{r}$ and $\mathcal{C}_{rr}$ are entirely sourced to $N$, $M$ and
$L$.

|  | $\nu$ | $N$ | $M$ | $L$ | $u$ | $v$ | $\hat{u}$ | $\hat{v}$ | $\mu$ | $\hat{\mu}$
---|---|---|---|---|---|---|---|---|---|---|---
Helicity 0 | $\mathcal{C}$ | ✓ |  |  |  |  |  |  |  |  |
3-6[1pt/1pt] | $\mathcal{C}_{rr}$ |  | ✓ |  |  |  |  |  |  |  |
| $\mathcal{B}_{r}$ |  | ✓ | ✓ |  |  |  |  |  |  |
| $\phi$ |  | ✓ | ✓ | ✓ |  |  |  |  |  |
Helicity 1 | $\mathcal{C}_{r}^{\mathbf{E}}$ |  |  |  |  | ✓ |  |  |  |  |
$\mathcal{B}^{\mathbf{E}}$ |  |  |  |  | ✓ | ✓ |  |  |  |
7-10[1pt/1pt] | $\mathcal{C}_{r}^{\mathbf{B}}$ |  |  |  |  |  |  | ✓ |  |  |
| $\mathcal{B}^{\mathbf{B}}$ |  |  |  |  |  |  | ✓ | ✓ |  |
Helicity 2 | $\mathcal{C}^{\mathbf{E}}$ |  |  |  |  |  |  |  |  | ✓ |
11-12[1pt/1pt] | $\mathcal{C}^{\mathbf{B}}$ |  |  |  |  |  |  |  |  |  | ✓
Table 1: Block scheme for the correspondence between the 1+3 perturbations and
the 1+1+2 ones, according to the explicit relations (4.1)-(4.3).

Instead, for what regards the helicity-1 standard perturbations, i.e.
$\mathcal{B}_{a}$ and $\mathcal{C}_{ra}$, their $E$\- and $B$\- modes are
respectively sourced by $u$, $v$ and $\hat{u}$, $\hat{v}$, whereas $\mu$ and
$\hat{\mu}$ are in a one-to-one correspondence with the $E$\- and $B$\- modes
of the helicity-2 perturbations.

We remark that, within the GLC gauge fixing, where $v=\hat{v}=L=0$ and
$N=-2aM$, Eqs. (4.1)-(4.3) become

$\displaystyle\phi=$ $\displaystyle
0\qquad,\qquad\mathcal{C}_{rr}=-2\,\mathcal{B}_{r}=N\qquad,\qquad\mathcal{T}=2\nu\,,$
$\displaystyle\mathcal{B}^{\mathbf{E}}=$
$\displaystyle-\mathcal{C}_{r}^{\mathbf{E}}=\frac{r}{a\sqrt{2}}D^{2}u\qquad,\qquad\mathcal{B}^{\mathbf{B}}=-\mathcal{C}_{r}^{\mathbf{B}}=\frac{r}{a\sqrt{2}}D^{2}\hat{u}\,,$
$\displaystyle\mathcal{C}^{\mathbf{E}}=$
$\displaystyle\,\frac{1}{a^{2}}\left[\left(D^{2}\right)^{2}\mu+2D^{2}\mu\right]\qquad,\qquad\mathcal{C}^{\mathbf{B}}=\frac{1}{a^{2}}\left[\left(D^{2}\right)^{2}\hat{\mu}+2D^{2}\hat{\mu}\right]\,.$
(4.7)

Although the resume in Table 1 shows a clear correspondence between the SPS
perturbations in 1+3 and 1+1+2 frameworks, we have to do a further step to
show how the expressions for $E$\- and $B$\- modes of tensor perturbations
look like in the light-cone setup. Therefore, we conclude this section by
discussing these explicit expressions as functions of the SPS light-cone
perturbations, while the full derivation is reported in Appendix D.

Let us start from Eq. (2.2) and project the tensor perturbations on the
helicity basis $\bar{e}^{i}_{\pm}$

$h_{\pm\pm}\equiv\bar{e}_{\pm}^{i}\bar{e}_{\pm}^{j}h_{ij}=\frac{\mathcal{C}_{\pm\pm}}{2}-\bar{e}_{\pm}^{i}\bar{e}_{\pm}^{j}\nabla_{i}S_{j}=\frac{\mathcal{C}_{\pm\pm}}{2}-\frac{1}{ar}\partial_{\pm}S_{\pm}\,,$
(4.8)

where we have defined $S_{i}\equiv\partial_{i}E+F_{i}$ and used the results of
Appendix C. Using Eq. (4.3) we can write

$h_{\pm\pm}=\frac{1}{a^{2}}\partial_{\pm}^{2}\left(\mu\mp
i\hat{\mu}\right)-\frac{1}{ar}\partial_{\pm}S{}_{\pm}\,.$ (4.9)

Hence, thanks to Eqs. (4.5) the $E$\- and $B$\- modes are given by

$h^{\mathbf{E}}=\left(D^{2}+2\right)\left(\frac{D^{2}\mu}{2a^{2}}+\frac{1}{ar\sqrt{2}}S^{\mathbf{E}}\right)\qquad\text{and}\qquad
h^{\mathbf{B}}=\left(D^{2}+2\right)\left(\frac{D^{2}\hat{\mu}}{2a^{2}}+\frac{1}{ar\sqrt{2}}S^{\mathbf{B}}\right)\,.$
(4.10)

As one can check from Eqs. (2.4) and (2.5), $S_{i}$ transforms under a gauge
redefinition of the fields as $\tilde{S}_{i}=S_{i}-\epsilon_{i}$ and then,
thanks to Eq. (3.39), we have that $\tilde{S}_{\pm}=\mathcal{S}_{\pm}-X_{\pm}$
or, equivalently, in terms of the $E/B$ modes,

$\tilde{S}^{\mathbf{E}}=S^{\mathbf{E}}-X^{\mathbf{E}}\qquad\text{and}\qquad\tilde{S}^{\mathbf{B}}=S^{\mathbf{B}}-X^{\mathbf{B}}\,,$
(4.11)

where $X^{\mathbf{E}/\mathbf{B}}$ are given in the SPS decomposition by

$X^{\mathbf{E}}=-\frac{r}{a\sqrt{2}}D^{2}\chi\qquad\text{and}\qquad
X^{\mathbf{B}}=-\frac{r}{a\sqrt{2}}D^{2}\hat{\chi}\,.$ (4.12)

By recalling that $\mu$ and $\hat{\mu}$ transform under a gauge redefinition
of the fields as $\tilde{\mu}=\mu-\chi$ and
$\tilde{\hat{\mu}}=\hat{\mu}-\hat{\chi}$, Eqs. (4.12) explicitly prove that
Eqs. (4.10) are gauge invariant. Finally, we have that the $E$\- modes of the
tensor perturbations are given by

$\displaystyle h^{\mathbf{E}}=$
$\displaystyle\frac{\left(D^{2}+2\right)}{2a^{2}}\left\\{D^{2}\mu-\frac{1}{r\Delta_{3}}\left[\left(\partial_{w}+\frac{3}{r}\right)\left(rD^{2}u\right)+\frac{\left(D^{2}+2\right)}{r}D^{2}\mu+\frac{2D^{2}}{r}\nu\right]\right\\}$
$\displaystyle+\frac{\left(D^{2}+2\right)}{2a^{2}r}\frac{1}{\Delta_{3}}\left\\{\frac{D^{2}}{r}\left\\{\frac{1}{3}\left(-\frac{2}{r}\partial_{w}+\Delta_{3}\right)\left\\{\frac{1}{\Delta^{2}_{3}}\left[\frac{1}{2}\left(\partial_{w}^{2}+\frac{5}{r}\partial_{w}+\frac{3}{r^{2}}\right)\left(N-2\nu\right)\right.\right.\right.\right.$
$\displaystyle\left.\left.-\frac{1}{4\,r^{2}}D^{2}\left(N+4\nu\right)+\frac{1}{6\,r^{2}}\left(\partial_{w}+\frac{1}{r}\right)D^{2}u+\frac{3}{2\,r^{2}}D^{2}\nu+\frac{3}{4r^{2}}\left(D^{2}+2\right)D^{2}\mu\right]\right\\}$
$\displaystyle\left.\left.+\frac{2}{r}\frac{1}{\Delta_{3}}\left[\left(\frac{1}{3}\partial_{w}+\frac{1}{r}\right)\left(2N-4\nu\right)+D^{2}u\right]-\frac{1}{3}\left(N+4\nu\right)\right\\}\right\\}\,,$
(4.13)

whereas the $B$\- modes are given by

$\displaystyle h^{\mathbf{B}}=$
$\displaystyle\frac{\left(D^{2}+2\right)}{2a^{2}}\left\\{D^{2}\hat{\mu}-\frac{1}{r\Delta_{3}}\left[\left(\partial_{w}+\frac{3}{r}\right)\left(rD^{2}\hat{u}\right)+\frac{\left(D^{2}+2\right)}{r}D^{2}\hat{\mu}\right]\right\\}\,,$
(4.14)

Eqs. (4.13) and (4.14) provide the expressions of the two independent
helicity-2 degrees of freedom in terms of the light-cone perturbations. Even
though this result has been achieved without fixing any gauge, their values
are given only in terms of the GLC gauge perturbations. This supports the
claim that the non-linear GLC line element is a natural choice for the study
of cosmological observables. As expected, we have that the electric modes are
only sourced by scalar perturbations in the SPS scheme, whereas the magnetic
modes are sourced by the pseudoscalar fields. Therefore, the importance of
Eqs. (4.13) and (4.14) stands in the fact that they explicitly provide the SPS
fields combination to be identified with the radiative gravitational degrees
of freedom.

Now, we can then use Eqs. (3.36)-(3.38) and (LABEL:eq:E-B-modes) to write the
$E/B$ modes of gravitational waves in terms of the coordinate independent
decomposition introduced in Sect. 3. To this aim, we split

$h^{\mathbf{E/B}}\equiv h^{\mathbf{E/B}}_{2}+h^{\mathbf{E}}_{01}\,,$ (4.15)

where

$\displaystyle h^{\mathbf{E/B}}_{2}\equiv$
$\displaystyle\frac{1}{2a^{2}}\left\\{\mathcal{T}^{\mathbf{E/B}}-\frac{\left(D^{2}+2\right)}{ar\Delta_{3}}\left[\left(\partial_{\rVert}+\frac{3}{r}\right)\mathcal{T}_{\rVert}^{\mathbf{E/B}}-\frac{1}{r\sqrt{2}}\mathcal{T}^{\mathbf{E/B}}\right]\right\\}\,,$
$\displaystyle h^{\mathbf{E}}_{01}\equiv$
$\displaystyle\frac{1}{2a^{2}}\left\\{\frac{\left(D^{2}+2\right)}{ar\Delta_{3}}\left[\frac{1}{r\sqrt{2}}D^{2}\mathcal{T}+\frac{D^{2}}{ar\sqrt{2}}\left[2\left(\psi+\frac{1}{3}\Delta_{3}E\right)-\Delta_{3}E+\frac{2}{r}\left(\partial_{r}E+F_{r}\right)\right]\right]\right\\}\,.$

The rationale behind this splitting stands in the fact that
$h^{\mathbf{E/B}}_{2}$ is only sourced by $\mu,\,u$ and $\hat{\mu},\,\hat{u}$
respectively for the electric and magnetic modes. This then reflects the
symmetry in the SPS decomposition of $U_{a}$ and $\gamma_{ab}$. We remark that
Eq. (LABEL:eq:411) is valid for both $E$\- and $B$\- modes but the second line
is present only for the $E$\- modes of the tensor perturbations. Furthermore,
Eq. (LABEL:eq:411) allows us to interpret $h^{\mathbf{E}/\mathbf{B}}_{2}$ in a
coordinate independent way as the actual content related to the gravitational
wave propagation along the observed light-cone.

###  4.1 Link with the initial conditions

The results presented in the previous sections tell us what are the relevant
combinations to be studied along the past light-cone. However, a neat
connection with the primordial initial conditions is yet to be provided. In
fact, usually initial conditions are given on spatial hypersurfaces at the
horizon re-entering redshift $z_{*}$, whereas we have discussed how to study
the radiative degrees of freedom along the light-cone.

In order to provide the initial conditions to be evolved, we refer to the
results of [8]. In particular, the link between the $E/B$ modes found in Sect.
4 and the spectra for primordial tensor perturbations in terms of their linear
polarization $h_{\pm 1}$ is given, according to Eq. (B.14) of [8] as333To
directly apply the results of [8] in this paper, the following dictionary is
intended: $h^{\mathbf{E}/\mathbf{B}}=\sum_{\ell,m}\mathcal{T}^{E/B}_{\ell m}$

$\displaystyle h^{\mathbf{E}}(z_{*},\bf{n})=$
$\displaystyle-4\pi\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\,\left(-i\right)^{\ell}\left[\frac{\ell(\ell-1)}{2}\frac{j_{\ell}(kr_{*})}{(kr_{*})^{2}}-j_{\ell}(kr_{*})+\frac{j_{\ell-1}(kr_{*})}{kr_{*}}\right]$
$\displaystyle=$ $\displaystyle-2\pi\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\sum_{\ell=0}^{\infty}\,\mathcal{K}^{\mathbf{E}}_{\ell}(kr_{*})\left[\sum_{m=-\ell}^{\ell}\left(\,{}_{-2}Y^{*}_{\ell
m}(\hat{{\bf k}};{\bf E})+\,_{2}Y^{*}_{\ell m}(\hat{{\bf k}};{\bf
E})\right)\,h_{+}(z_{*},{\bf k})\right.$
$\displaystyle\left.-\sum_{m=-\ell}^{\ell}\left(\,{}_{-2}Y^{*}_{\ell
m}(\hat{{\bf k}};{\bf E})-\,_{2}Y^{*}_{\ell m}(\hat{{\bf k}};{\bf
E})\right)\,h_{\times}(z_{*},{\bf k})\right]\,,$ $\displaystyle
h^{\mathbf{B}}(z_{*},\bf{n})=$ $\displaystyle-4\pi\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\,\left(-i\right)^{\ell}\left[(\ell+2)\frac{j_{\ell}(kr_{*})}{kr_{*}}-j_{\ell+1}(kr_{*})\right]$
$\displaystyle\times\sum_{p=\pm 1}p\,_{-2p}Y^{*}_{\ell m}(\hat{{\bf k}};{\bf
E})\,h_{p}(z_{*},{\bf k})$ $\displaystyle=$
$\displaystyle-2\pi\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\sum_{\ell=0}^{\infty}\,\mathcal{K}^{\mathbf{B}}_{\ell}(kr_{*})\left[\sum_{m=-\ell}^{\ell}\left(\,{}_{-2}Y^{*}_{\ell
m}(\hat{{\bf k}};{\bf E})-\,_{2}Y^{*}_{\ell m}(\hat{{\bf k}};{\bf
E})\right)\,h_{+}(z_{*},{\bf k})\right.$
$\displaystyle\left.-i\sum_{m=-\ell}^{\ell}\left(\,{}_{-2}Y^{*}_{\ell
m}(\hat{{\bf k}};{\bf E})+\,_{2}Y^{*}_{\ell m}(\hat{{\bf k}};{\bf
E})\right)\,h_{\times}(z_{*},{\bf k})\right]\,,$ (4.17)

where $j_{\ell}$’s are the spherical Bessel functions of $\ell$-th order,
$\,{}_{s}Y_{\ell m}(\hat{\bf k};\bf{E})$ are the spin-weighted spherical
harmonics (see Appendix C for details) and $\bf{k}$ is the momentum in Fourier
space of the given wave. Moreover, in the second equalities of Eqs. (4.17) we
have defined

$\displaystyle\mathcal{K}^{\mathbf{E}}_{\ell}(kr_{*})\equiv$
$\displaystyle\left(-i\right)^{\ell}\left[\frac{\ell(\ell-1)}{2}\frac{j_{\ell}(kr_{*})}{(kr_{*})^{2}}-j_{\ell}(kr_{*})+\frac{j_{\ell-1}(kr_{*})}{kr_{*}}\right]\,,$
$\displaystyle\mathcal{K}^{\mathbf{B}}_{\ell}(kr_{*})\equiv$
$\displaystyle\left(-i\right)^{\ell}\left[(\ell+2)\frac{j_{\ell}(kr_{*})}{kr_{*}}-j_{\ell+1}(kr_{*})\right]\,,$
(4.18)

and introduced the basis for the longitudinal and circular polarizations
$h_{+}$ and $h_{\times}$ given by $h_{\pm 1}\equiv\frac{1}{2}\left(h_{+}\mp
ih_{\times}\right)$.

Eqs. (4.17) then connect the $E/B$ modes of the tensor perturbations to the
primordial gravitational waves evolved until the re-entering epoch and the
latter provides the initial conditions for the evolution of the observed $E/B$
modes of the tensor perturbations until to the late-time Universe.

##  5 Summary and Conclusions

In this work, we have developed a formalism to classify a set of perturbations
on a Lorentzian manifold according to their helicity rather than to their
transformation properties under $SO(3)$. We have then applied this new
formalism to a cosmological perturbation theory well-suited to the structure
of the light-cone previously proposed in [19] (see also [17]).

We have first considered the entire set of linear perturbations built on top
of the so-called Geodesic Light-Cone (GLC) background metric and used the
gauge transformation properties for this set of variables to find what is the
expression for relevant gauge invariant cosmological variables, such as the
Mukhanov-Sasaki variable and the $E$\- and $B$\- modes of the linear tensor
perturbations. Some previous attempts in this regards have been already
proposed by other authors (see [20]). As we have proven in the Appendix B, our
results agree with the ones obtained in [20]. However, thanks to what we
called the Scalar-PseudoScalar (SPS) decomposition, we have been able to move
further and decouple the electric degrees of freedom from the magnetic ones.

Our results have been achieved thanks to the SPS decomposition and to the
formalism of the spin-raising and spin-lowering operators. Indeed, the
advantages in using this become manifest in order to decouple the tensor
degrees of freedom from the vector and scalar ones, when mixed on the light-
cone structure. This was a not trivial milestone to be achieved for two
reasons. First of all, it provides a clear answer to the claim done in [17]
about the physical spin-2 radiative degrees of freedom evolving along the
light-cone. In fact, Eqs. (4.13) and (4.14) precisely give what is the
combination of the SPS perturbations leading to the electric and magnetic
modes of the tensor perturbations, rather than just referring to their parity
transformations. Secondly, this SPS decomposition of cosmological
perturbations can be readily extended to the study of higher-order
perturbations and then compared to other recent coordinate independent
formulation of non-linear cosmological observables [29].

These two features are of particular importance when one is interested to
evolve the perturbations directly along the observed light-cone rather than
through constant-time hypersurfaces. In this framework, beyond the few
analytical attempts to study the light-cone dynamics presented in [17, 30],
this work furnishes a scheme to evolve the entangled structure of the light-
cone dynamics via numerical attempts, as already investigated in [18]. With
this in mind, having a neat, systematic classification for the relevant light-
cone degrees of freedom is crucial to evolve the right variables and the
results presented in this work are in line with this program. For this program
to be successful, we have also discussed how to connect the light-cone
variables to the primordial spectra for the case of tensor perturbations.

Finally, we remark that the capability to define our perturbation theory
directly on the past light-cone is also relevant to connect the evolved non-
linear inhomogeneities we see today to the primordial fluctuations when the
perturbations are still linear. Hence, combined with the capability to
evaluate the exact light-cone averages in a covariant way within the GLC gauge
[11], the results obtained provide a step towards the evaluation of
backreaction since the primordial until the late-time Universe.

## Acknowledgements

GF is thankful to the Pisa section of the INFN for the hospitality provided
during the development and finalisation of this project. 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). GM and MM are supported in part by INFN under the program TAsP
(Theoretical Astroparticle Physics).

##  Appendix A Gauge transformations

In this appendix, we evaluate the gauge transformations of the SVT
perturbations once expressed in terms of the SPS decomposition. We will then
show that these transformations agree with what is expected in the standard
perturbation theory and will then provide a sanity check of our evaluations.

We start by recalling the gauge transformations for the light-cone
perturbations in Eq. (2.19). Following [19], we define the gauge field for the
light-cone perturbation with
$\xi^{\mu}\equiv\left(\xi^{0},\xi^{w},\hat{\xi}^{a}\right)$ and get

$\displaystyle\tilde{L}$
$\displaystyle=L+\frac{2}{a}\partial_{\tau}\xi^{w}\,,$
$\displaystyle\tilde{M}$
$\displaystyle=M+\partial_{\tau}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)+\frac{1}{a}\partial_{w}\xi^{w}+2\frac{H}{a}\xi^{0}\,,$
$\displaystyle\tilde{N}$
$\displaystyle=N-2H\xi^{0}+2\partial_{w}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)\,,$
$\displaystyle\tilde{V}_{a}$
$\displaystyle=V_{a}+\frac{1}{a}\partial_{a}\xi^{w}-\bar{\gamma}_{ab}\partial_{\tau}\hat{\xi}^{b}\,,$
$\displaystyle\tilde{U}_{a}$
$\displaystyle=U_{a}+\partial_{a}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-\bar{\gamma}_{ab}\partial_{w}\hat{\xi}^{b}\,,$
$\displaystyle\tilde{\delta\gamma}_{ab}$
$\displaystyle=\delta\gamma_{ab}-2\bar{\gamma}_{ab}H\xi^{0}+\frac{2\bar{\gamma}_{ab}}{r}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-\left(\bar{\gamma}_{ac}D_{b}+\bar{\gamma}_{bc}D_{a}\right)\hat{\xi}^{c}\,.$
(A.1)

The latter can be further manipulated in order to give the gauge
transformations for the SPS degrees of freedom. Following again [19], we
decompose also $\hat{\xi}^{a}$ in terms of scalar and pseudo-scalar modes
$\chi$ and $\hat{\chi}$, namely

$\hat{\xi}^{a}\equiv q^{ab}\left(D_{a}\chi+\tilde{D}_{a}\hat{\chi}\right)\,,$
(A.2)

leading then to

$\displaystyle\tilde{\nu}=$
$\displaystyle\nu-\frac{1}{2}D^{2}\chi-\xi^{0}\left(H-\frac{1}{ar}\right)-\frac{\xi^{w}}{r}\,,$
$\displaystyle\tilde{v}=$ $\displaystyle
v+\frac{1}{ar^{2}}\xi^{w}-\partial_{\tau}\chi\,,$
$\displaystyle\tilde{\hat{v}}=$
$\displaystyle\hat{v}-\partial_{\tau}\hat{\chi}\,,$ $\displaystyle\tilde{u}=$
$\displaystyle
u+\frac{1}{r^{2}}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-\partial_{w}\chi\,,$
$\displaystyle\tilde{\hat{u}}=$
$\displaystyle\hat{u}-\partial_{w}\hat{\chi}\,,$ $\displaystyle\tilde{\mu}=$
$\displaystyle\mu-\chi\,,$ $\displaystyle\tilde{\hat{\mu}}=$
$\displaystyle\hat{\mu}-\hat{\chi}\,.$ (A.3)

The relation between the SVT gauge field $\epsilon^{\mu}$ used in Eq. (2.3)
and the SPS one here introduced is then given by the finite background
coordinate transformation (2.18), namely

$\xi^{\mu}=\frac{\partial x^{\mu}}{\partial y^{\nu}}\epsilon^{\nu}\,,$ (A.4)

leading to

$\xi^{0}=a\epsilon^{\eta}\qquad,\qquad\xi^{w}=\epsilon^{\eta}+\epsilon^{r}\qquad,\qquad\hat{\xi}^{a}=\epsilon^{a}\,.$
(A.5)

Moreover, for later uses, we also recall that Eqs. (2.18) gives the relations
between the derivatives

$\partial_{\eta}=a\partial_{\tau}+\partial_{w}\qquad,\qquad\partial_{r}=\partial_{w}\qquad,\qquad\partial_{a}=\partial_{a^{GLC}}\,.$
(A.6)

These relations will be useful later to verify the correct gauge
transformations of the SVT perturbations when expressed as a linear
combination of SPS fields. Other useful relations that we provide are the
following ones

$\displaystyle\Delta_{3}\epsilon=$
$\displaystyle\nabla_{i}\epsilon^{i}=\left(\partial_{r}+\frac{2}{r}\right)\epsilon^{r}+D_{a}\epsilon^{a}=\left(\partial_{w}+\frac{2}{r}\right)\left(\xi^{w}-\frac{\xi^{0}}{a}\right)+D^{2}\chi\,,$
$\displaystyle\Delta_{3}\epsilon^{r}=$
$\displaystyle\left(\partial_{r}^{2}+\frac{2}{r}\partial_{r}-\frac{2}{r^{2}}+\frac{1}{r^{2}}D^{2}\right)\epsilon^{r}-\frac{2}{r}D_{a}\epsilon^{a}\,,$
$\displaystyle\Delta_{3}\epsilon^{a}=$
$\displaystyle\left(\partial_{r}^{2}+\frac{4}{r}\partial_{r}+\frac{1}{r^{2}}+\frac{1}{r^{2}}D^{2}\right)\epsilon^{a}+\frac{2}{r^{3}}D^{a}\epsilon^{r}\,.$
(A.7)

Now we can use the set of gauge transformations of the general GLC
perturbations given in Eqs. (A.1) and (A.3) to check the consistency of the
SVT perturbations in terms of GLC perturbations. Starting with $\phi$ given in
Eq. (2.21), we use Eqs. (A.5) and (A.6) to get

$\displaystyle\widetilde{\phi}=$
$\displaystyle-\frac{1}{2}\left(a^{2}\widetilde{L}+\widetilde{N}+2a\widetilde{M}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(a^{2}L+N+2aM\right)-\left(\partial_{\tau}+\frac{1}{a}\partial_{w}\right)\xi^{0}=\phi-\frac{1}{a}\partial_{\eta}\left(a\epsilon^{\eta}\right)\,,$
(A.8)

which is in agreement with the expected gauge transformation for $\phi$ in
Eqs. (2.4).With the use of the same equations, also the gauge transformation
of $B$ can be checked. We get (2.22)

$\displaystyle\Delta_{3}\widetilde{B}=$
$\displaystyle-\left(\partial_{w}+\frac{2}{r}\right)\left(\widetilde{N}+a\widetilde{M}\right)-D^{2}\left(\widetilde{u}+a\widetilde{v}\right)\,$
$\displaystyle=$
$\displaystyle-\left(\partial_{w}+\frac{2}{r}\right)\left[\left(N+aM\right)+\left(a\partial_{\tau}+\partial_{w}\right)\left(\frac{\xi^{0}}{a}-\xi^{w}\right)+\partial_{w}\left(\frac{\xi^{0}}{a}\right)\right]\,$
$\displaystyle-D^{2}\left[u+av+\frac{\xi^{0}}{ar^{2}}-\left(a\partial_{\tau}+\partial_{w}\right)\chi\right]\,$
$\displaystyle=$
$\displaystyle-\left(\partial_{w}+\frac{2}{r}\right)\left(N+aM\right)-D^{2}\left(u+av\right)\,$
$\displaystyle+\left(a\partial_{\tau}+\partial_{w}\right)\left[-\left(\partial_{w}+\frac{2}{r}\right)\left(\frac{\xi^{0}}{a}-\xi^{w}\right)+D^{2}\chi\right]$
$\displaystyle-\left(\partial_{w}^{2}+\frac{2}{r}\partial_{w}+r^{-2}D^{2}\right)\left(\frac{\xi^{0}}{a}\right)=\Delta_{3}\left(B+\partial_{\eta}\epsilon-\epsilon^{\eta}\right)\,,$
(A.9)

where on the last line we use again (A.5) and (A.6). Hence, the gauge
transformation of $B$ is the one expected. On the other hand, $\psi$, as
written in (2.22), transforms as

$\displaystyle\widetilde{\psi}=-\frac{1}{6}\left(\widetilde{N}+4\widetilde{\nu}\right)=$
$\displaystyle-\frac{1}{6}\left(N+4\nu\right)+H\xi^{0}-\frac{1}{3}\left[\left(\partial_{r}+\frac{2}{r}\right)\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-D_{a}\hat{\xi}^{a}\right]$
$\displaystyle=$
$\displaystyle\psi+\mathcal{H}\epsilon^{\eta}+\frac{1}{3}\Delta_{3}\epsilon\,.$
(A.10)

We have now all the ingredients to check directly the gauge transformation for
gauge dependent curvature perturbation (2.26). We preliminary derive the
following gauge transformations

$\displaystyle\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2\,r^{2}}D^{2}\right)\widetilde{N}=$
$\displaystyle\,\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2\,r^{2}}D^{2}\right)\left[N-2H\xi^{0}+2\partial_{w}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)\right]\,$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2\,r^{2}}D^{2}\right)\left(N-2H\xi^{0}\right)\,$
$\displaystyle+\,\left(\frac{1}{r}\partial^{2}_{w}+\frac{1}{r^{2}}\partial_{w}-\frac{1}{2\,r^{2}}D^{2}\partial_{w}\right)\left(\frac{\xi^{0}}{a}-\xi^{w}\right)\,,$
$\displaystyle-\left(\partial^{2}_{w}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2\,r^{2}}D^{2}\right)\widetilde{\nu}=$
$\displaystyle\,-\left(\partial^{2}_{w}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2\,r^{2}}D^{2}\right)\left[{\nu}-H\xi^{0}\right.\,$
$\displaystyle+\left.\frac{1}{r}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-\frac{1}{2}D^{2}\chi\right]\,,$
$\displaystyle=$
$\displaystyle\,-\left(\partial^{2}_{w}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2\,r^{2}}D^{2}\right)\left({\nu}-H\xi^{0}-\frac{1}{2}D^{2}\chi\right)\,$
$\displaystyle-\frac{1}{r}\left(\partial^{2}_{w}+\frac{1}{r}\partial_{w}+\frac{1}{2\,r^{2}}D^{2}\right)\left(\frac{\xi^{0}}{a}-\xi^{w}\right)\,,$
$\displaystyle\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}\widetilde{u}=$
$\displaystyle\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u+\frac{1}{2\,r^{2}}D^{2}\left[\left(\partial_{w}+\frac{1}{r}\right)\left(\frac{\xi^{0}}{a}-\xi^{w}\right)\right]\,$
$\displaystyle-\frac{1}{2}\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}\right)D^{2}\chi\,,$
$\displaystyle\frac{1}{4\,r^{2}}D^{2}\left(2\widetilde{\mu}+D^{2}\widetilde{\mu}\right)=$
$\displaystyle\frac{1}{4\,r^{2}}D^{2}\left(2\mu+D^{2}\mu-2\chi-D^{2}\chi\right)\,.$
(A.11)

Hence, using Eqs. (A.11) in the gauge transformation of $\psi$ given in Eq.
(2.26), we get

$\displaystyle\widetilde{\psi}+\frac{1}{3}\Delta_{3}\widetilde{E}=$
$\displaystyle\frac{1}{\Delta_{3}}\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)\widetilde{N}+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\widetilde{\mu}+\frac{1}{2r^{2}}D^{2}\widetilde{\mu}\right.$
$\displaystyle\left.+\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}\widetilde{u}-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\widetilde{\nu}\right]\,$
$\displaystyle=$
$\displaystyle\frac{1}{\Delta_{3}}\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.$
$\displaystyle\left.+\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]+\left(H\xi^{0}\right)\,$
$\displaystyle=$
$\displaystyle\psi+\frac{1}{3}\Delta_{3}E+\mathcal{H}\epsilon^{\eta}\,,$
(A.12)

where, once again, we have used Eqs. (A.5) and (A.6). Eqs. (A.12) and (2.8)
show the gauge invariance of $\zeta,\mathcal{R}$ and $Q$, defined in Eq.
(2.10) and calculated in terms of GLC perturbations in Eqs. (2.27). Indeed, to
this aim it is enough to recall that also from (A.10) and (A.12) we have that
the perturbations of an homogeneous and isotropic fluid $\bar{\rho}$ or a
scalar field $\bar{\phi}$, transform accordingly as
$\widetilde{\delta\phi}=\delta\phi-\xi^{0}\partial_{\tau}\bar{\phi}$ and
$\widetilde{\delta\rho}=\delta\rho-\xi^{0}\partial_{\tau}\bar{\rho}$.

With the complete set of gauge transformations for $\psi$ and $E$, we can also
check the transformation of $F_{i}$. To this end, we consider Eq. (2.14) and
then use the gauge transformations (A.10) and (A.12). With the use of the
gauge transformations for $\nabla^{i}C_{ir}$, i.e.

$\displaystyle\nabla^{i}\widetilde{C}_{ir}=$
$\displaystyle\left(\partial_{r}+\frac{2}{r}\right)\widetilde{N}+r^{-2}D^{2}\widetilde{u}-\frac{4}{r}\widetilde{\nu}$
$\displaystyle=$
$\displaystyle\left(\partial_{r}+\frac{2}{r}\right)\left[N-2H\xi^{0}+2\partial_{w}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)\right]$
$\displaystyle+r^{-2}D^{2}u+r^{-2}D^{2}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-r^{-2}D^{2}\partial_{w}\chi\,$
$\displaystyle-\frac{4}{r}\nu+\frac{2}{r}\chi+\frac{4}{r}H\xi^{0}-\frac{4}{r^{2}}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)\,$
$\displaystyle=$
$\displaystyle\left(\partial_{r}+\frac{2}{r}\right)N+r^{-2}D^{2}u-\frac{4}{r}\nu$
$\displaystyle-2\partial_{w}\left(H\xi^{0}\right)+\partial_{w}\left[\left(\partial_{w}+\frac{2}{r}\right)\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-r^{-2}D^{2}\chi\right]\,$
$\displaystyle+\left(\partial^{2}_{w}+\frac{2}{r}\partial_{w}-\frac{2}{r^{2}}+r^{-2}D^{2}\right)\left(\frac{\xi^{0}}{a}-\xi^{w}\right)+\frac{2}{r}D^{2}\chi\,$
$\displaystyle=$
$\displaystyle\left(\partial_{r}+\frac{2}{r}\right)N+r^{-2}D^{2}u-\frac{4}{r}\nu\,$
$\displaystyle-2\partial_{w}\left(H\xi^{0}\right)+\Delta_{3}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-\partial_{w}\left(\Delta_{3}\epsilon\right)\,$
(A.13)

where on the last equality we used Eq. (A.7) with
$D_{a}\epsilon^{a}=D^{2}\chi$, we get

$\displaystyle\Delta_{3}\widetilde{F}_{r}=$
$\displaystyle\nabla^{i}\widetilde{C}_{ir}+6\nabla_{i}\widetilde{\psi}-4\nabla_{i}\left(\widetilde{\psi}+\frac{1}{3}\Delta_{3}\widetilde{E}\right)\,$
$\displaystyle=$
$\displaystyle\left(\partial_{r}+\frac{2}{r}\right)N+r^{-2}D^{2}u-\frac{4}{r}\nu\,$
$\displaystyle-2\partial_{w}\left(H\xi^{0}\right)+\Delta_{3}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-\partial_{w}\left(\Delta_{3}\epsilon\right)\,$
$\displaystyle-\partial_{w}\left(N+4\nu\right)+6\partial_{w}\left(H\xi^{0}\right)+2\partial_{w}\left(\Delta_{3}\epsilon\right)\,$
$\displaystyle-4\partial_{w}\frac{1}{\Delta_{3}}\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.$
$\displaystyle\left.+\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]-4\partial_{w}\left(H\xi^{0}\right)\,,$
(A.14)

which is precisely what expected since

$\displaystyle\Delta_{3}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-\partial_{w}\left[\left(\partial_{w}+\frac{2}{r}\right)\left(\frac{\xi^{0}}{a}-\xi^{w}\right)+r^{-2}D^{2}\chi\right]=-\Delta_{3}\left(e^{r}\right)$
(A.15)

with the SVT decomposition for the gauge mode
$\epsilon_{r}=\partial_{r}\epsilon+e_{r}$.

With the same procedure, the gauge transformation of
$\nabla^{i}\tilde{C}_{ia}$ will provide the gauge transformation of
$\Delta_{3}\tilde{F}_{a}$. The former is given by

$\displaystyle\nabla^{i}\tilde{C}_{ia}=$
$\displaystyle\left(\partial_{w}+\frac{2}{r}\right)\left[r^{2}\left(D_{a}\widetilde{u}+\tilde{D}_{a}\widetilde{\hat{u}}\right)\right]+2D_{a}\widetilde{\nu}\,$
$\displaystyle+D_{a}\left(D^{2}\widetilde{\mu}+2\widetilde{\mu}\right)+\tilde{D}_{a}\left(D^{2}\widetilde{\hat{\mu}}+\widetilde{\hat{\mu}}\right)\,$
$\displaystyle=$
$\displaystyle\left(\partial_{w}+\frac{2}{r}\right)\left\\{\left[r^{2}\left(D_{a}u+\tilde{D}_{a}\hat{u}\right)\right]+D_{a}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-r^{2}\partial_{w}\left[D_{a}\chi+\tilde{D}_{a}\hat{\chi}\right]\right\\}\,$
$\displaystyle+2D_{a}\nu-
D_{a}\left(D^{2}\chi\right)-2D_{a}\left(H\xi^{0}\right)+\frac{2}{r}D_{a}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)\,$
$\displaystyle+D_{a}\left[\left(D^{2}\mu+2\mu\right)-\left(D^{2}\chi+2\chi\right)\right]+\tilde{D}_{a}\left[\left(D^{2}\hat{\mu}+2\hat{\mu}\right)-\left(D^{2}\hat{\chi}+2\hat{\chi}\right)\right]\,$
$\displaystyle=$
$\displaystyle\left(\partial_{w}+\frac{2}{r}\right)\left[r^{2}\left(D_{a}u+\tilde{D}_{a}\hat{u}\right)\right]+2D_{a}\nu+D_{a}\left(D^{2}\mu+2\mu\right)\,$
$\displaystyle+\tilde{D}_{a}\left(D^{2}\hat{\mu}+2\hat{\mu}\right)-2D_{a}\left(H\xi^{0}\right)-D_{a}\left(\Delta_{3}\epsilon\right)\,$
$\displaystyle-r^{2}\left(\partial_{w}^{2}+\frac{4}{r}\partial_{w}+\frac{1}{r^{2}}+r^{-2}D^{2}\right)\left(D_{a}\chi+\tilde{D}_{a}\hat{\chi}\right)+\frac{2}{r}D_{a}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)\,,$
(A.16)

where we used the relations

$\displaystyle D_{a}\left(D^{2}\chi+2\chi\right)$
$\displaystyle=D^{2}\left(D_{a}\chi\right)+D_{a}\chi\,,$
$\displaystyle\tilde{D}_{a}\left(D^{2}\hat{\chi}+2\chi\right)$
$\displaystyle=D^{2}\left(\tilde{D}_{a}\hat{\chi}\right)+\tilde{D}_{a}\chi\,,$
(A.17)

and, thanks to Eq. (A.7), last line of Eq. (A.16) becomes

$r^{2}\left(\partial_{w}^{2}+\frac{4}{r}\partial_{w}+\frac{1}{r^{2}}+r^{-2}D^{2}\right)\left(D_{a}\chi+\tilde{D}_{a}\hat{\chi}\right)+\frac{2}{r}D_{a}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)=\Delta_{3}\left(a^{-2}\bar{\gamma}_{ab}\hat{\xi}^{b}\right)\,.$
(A.18)

We then have that $F_{a}$ transforms as

$\displaystyle\Delta_{3}\tilde{F}_{a}=$
$\displaystyle\nabla^{i}\widetilde{C}_{ia}+6\nabla_{a}\widetilde{\psi}-4\nabla_{a}\left(\widetilde{\psi}+\frac{1}{3}\Delta_{3}\widetilde{E}\right)\,$
$\displaystyle=$
$\displaystyle\nabla^{i}\widetilde{C}_{ia}+6\nabla_{a}\left(N+4\nu\right)-4\nabla_{a}\left(\widetilde{\psi}+\frac{1}{3}\Delta_{3}\widetilde{E}\right)\,$
$\displaystyle=$
$\displaystyle\nabla^{i}C_{ia}-2D_{a}\left(H\xi^{0}\right)-D_{a}\left(\Delta_{3}\epsilon\right)-\Delta_{3}\left(a^{-2}\bar{\gamma}_{ab}\hat{\xi}^{b}\right)\,$
$\displaystyle+6\nabla_{a}\left(N+4\nu\right)+6D_{a}\left(H\xi^{0}\right)+2D_{a}\left(\Delta_{3}\epsilon\right)\,$
$\displaystyle-4\frac{\nabla_{a}}{\Delta_{3}}\left[\frac{1}{2}\left(\frac{1}{r}\partial_{w}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\frac{1}{4r^{2}}\left(D^{2}\right)^{2}\mu+\frac{1}{2r^{2}}D^{2}\mu\right.$
$\displaystyle\left.+\frac{1}{2}\left(\partial_{w}+\frac{3}{r}\right)D^{2}u-\left(\partial_{w}^{2}+\frac{3}{r}\partial_{w}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\right]-4D_{a}\left(H\xi^{0}\right)$
$\displaystyle=$
$\displaystyle\Delta_{3}F_{a}+D_{a}\left(\Delta_{3}\epsilon\right)-\Delta_{3}\left(a^{-2}\bar{\gamma}_{ab}\hat{\xi}^{b}\right)\,,$
(A.19)

where the gauge modes in the last line of Eq. (A.19) can be rewritten as

$D_{a}\left(\Delta_{3}\epsilon\right)-\Delta_{3}\left(a^{-2}\bar{\gamma}_{ab}\hat{\xi}^{b}\right)=-\Delta_{3}\left(a^{-2}\epsilon_{b}-D_{a}\epsilon\right)=\Delta_{3}\left(a^{-2}e_{b}\right)\,.$
(A.20)

We remark that Eqs. (A.14) and (A.19) can be written in the following compact
form

$\widetilde{F}_{i}^{GLC}=F_{i}^{GLC}-a^{-2}\left[\bar{f}_{i\mu}\xi^{\mu}-\partial_{i}\xi\right]\,,$
(A.21)

where the superscript (GLC) is just to indicate the expression for $F_{i}$ in
terms of GLC perturbations.

What we have shown is that GLC perturbations correctly reproduce all the gauge
transformations for the scalar and vector perturbations in the SVT
decomposition. We conclude this appendix by showing also the gauge invariance
of the tensor perturbations in the same framework. In fact, thanks to the
expected transformations for $\psi$ in Eq. (A.10), for $E$ in Eq. (A.12) and
for $F_{i}$ in Eq. (A.21) we have that

$\displaystyle\tilde{h}_{rr}=$
$\displaystyle\frac{N}{2}-H\xi^{0}+\partial_{w}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)$
$\displaystyle+\left(\psi+\frac{1}{3}\Delta_{3}E\right)+H\xi^{0}-\nabla_{r}\left(F_{r}+\partial_{r}E\right)-\left(\frac{\xi^{0}}{a}-\xi^{w}\right)=h_{rr}\,,$
$\displaystyle\tilde{h}_{ra}=$
$\displaystyle\frac{U_{a}}{2}-\frac{\bar{\gamma}_{ab}}{2}\partial_{w}\hat{\xi}^{b}+\frac{1}{2}\partial_{a}\left(\frac{\xi^{0}}{a}\right)$
$\displaystyle-\frac{1}{2}\nabla_{a}\left(F_{r}+\partial_{r}E\right)+\frac{1}{2}\partial_{a}\left(\frac{\xi^{0}}{a}\right)-\frac{1}{2}\nabla_{r}F_{a}+\frac{\bar{\gamma}_{ab}}{2}\partial_{w}\hat{\xi}^{b}=h_{ra}\,,$
$\displaystyle\tilde{h}_{ab}=$
$\displaystyle\frac{\delta\gamma_{ab}}{2}-\bar{\gamma}_{ab}H\xi^{0}+\frac{\bar{\gamma}_{ab}}{r}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)-\frac{1}{a^{2}}D_{(a}\xi_{b)}$
$\displaystyle+\bar{\gamma}_{ab}\left(\psi+\frac{1}{3}\Delta_{3}E\right)+\bar{\gamma}_{ab}H\xi^{0}$
$\displaystyle-\nabla_{(a}F_{b)}+\nabla_{(a}\partial_{b)}E-\frac{\bar{\gamma}_{ab}}{r}\left(\frac{\xi^{0}}{a}-\xi^{w}\right)+\frac{1}{a^{2}}D_{(a}\xi_{b)}=h_{ab}\,,$
(A.22)

where the gauge transformation for $N$, $U_{a}$ and $\delta\gamma_{ab}$ are
given in Eqs. (A.1), the gauge dependent curvature transforms according to Eq.
(A.12) and for the vector components satisfy Eqs. (A.14) and (A.19).

##  Appendix B Comparison with Fröb and Lima

In this appendix we compare our results with the ones obtained in [20]. We
will discuss two aspects: we first compare our gauge invariant curvature
perturbation (2.27) with the one provided in [20] where the linear GLC gauge
is fixed. Afterwards, we will show that our results for the gauge invariant
tensor perturbations (2.28)-(2.30) are in agreement with the ones we can
calculate with the procedure outlined in [20].

###  B.1 Gauge invariant curvature perturbation

As we have already commented in the main text, although Eq. (2.27) is valid in
any gauge, its value coincide with the one when GLC gauge is fixed. The latter
is the one provided in Eq. (5.33) of [20]. Since now on, we will refer to
their result as $\mathcal{R}^{FL}$. In order to properly set the comparison,
we preliminary point out that the definition $\mathcal{R}^{FL}$ adopted in
[20] coincides with $2\mathcal{R}$. We then have that

$\displaystyle\Delta_{3}\mathcal{R}^{FL}=$
$\displaystyle\frac{2H}{\partial_{\tau}\phi}\Delta_{3}\delta\varphi+\frac{2}{ar}\left(\partial_{r}+\frac{1}{r}\right)\Upsilon^{(1)}-\frac{1}{ar^{2}}D^{2}\Upsilon^{(1)}-\frac{1}{a^{2}r^{2}}\left(\partial_{r}+\frac{1}{r}\right)D_{a}U_{(1)}^{a}$
$\displaystyle+\frac{1}{2a^{2}r^{4}}\left(D_{a}D_{b}\gamma_{(1)}^{ab}-D^{2}\gamma_{(1)}\right)-\frac{1}{2a^{2}r^{2}}\left(\partial_{r}^{2}-\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\right)\gamma_{(1)}\,,$
(B.1)

where we have already identified the inflaton perturbations $\delta\varphi$
with their $\Phi^{(1)}$, since what they call $\tau^{(1)}$ is null within the
GLC gauge fixing. Moreover, the subscript and superscript $(1)$ indicate
linear order expansion in Eq. (2.16). According to SPS decomposition, then, we
have the following dictionary

$\displaystyle\Upsilon^{(1)}=$ $\displaystyle\frac{a}{2}N\,,$ $\displaystyle
U^{(1)}_{a}=$
$\displaystyle-a^{2}r^{2}\left(D_{a}u+\tilde{D}_{a}\hat{u}\right)\,,$
$\displaystyle\gamma_{(1)}^{ab}=$ $\displaystyle
2\,a^{2}r^{2}\left(\bar{q}^{ab}\nu+D^{ab}\mu+\tilde{D}^{ab}\hat{\mu}\right)\,.$
(B.2)

Furthermore, still following the definition given in [20], we have that

$\gamma_{(1)}\equiv\bar{q}^{ab}\gamma_{ab}^{(1)}=4a^{2}r^{2}\nu\,.$ (B.3)

Now we can compare term by term the two results, by using (B.2) and (B.3). We
get

$\displaystyle\frac{2}{ar}\left(\partial_{r}+\frac{1}{r}\right)\Upsilon^{(1)}-\frac{1}{ar^{2}}D^{2}\Upsilon^{(1)}$
$\displaystyle=\left(\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N\,,$
(B.4)
$\displaystyle\frac{1}{a^{2}r^{2}}\left(\partial_{r}+\frac{1}{r}\right)D_{a}U_{(1)}^{a}$
$\displaystyle=\left(\frac{1}{r}\partial_{r}+\frac{3}{r}\right)D^{2}u\,,$
$\displaystyle\frac{1}{2a^{2}r^{4}}\left(D_{a}D_{b}\gamma_{(1)}^{ab}-D^{2}\gamma_{(1)}\right)$
$\displaystyle=-\frac{1}{r^{2}}\left[D^{2}\nu+D^{2}\mu+\frac{1}{2}\left(D^{2}\right)^{2}\mu\right]\,,$
$\displaystyle\frac{1}{2a^{2}r^{2}}\left(\partial_{r}^{2}-\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\right)\gamma_{(1)}$
$\displaystyle=2\left(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\frac{1}{r^{2}}\right)\nu\,.$

Hence, when expressed in terms of the SPS perturbations, the result found in
[20] becomes

$\displaystyle\Delta_{3}\mathcal{R}^{FL}=$
$\displaystyle\frac{2H}{\partial_{\tau}\bar{\varphi}}\Delta_{3}\delta\varphi+\left(\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N+\left(\frac{1}{r}\partial_{r}+\frac{3}{r}\right)D^{2}u$
$\displaystyle-2\left(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\frac{1}{r^{2}}+\frac{1}{2\,r^{2}}D^{2}\right)\nu+\frac{1}{r^{2}}D^{2}\mu+\frac{1}{2\,r^{2}}\left(D^{2}\right)^{2}\mu\,,$
(B.5)

which precisely returns $\Delta_{3}\mathcal{R}^{FL}=2\Delta_{3}\mathcal{R}$.

###  B.2 Tensor perturbations

As a further comparison between our results and [20], here we discuss how to
compare our results in Eqs. (2.28)-(2.30) for the tensor perturbations
$h_{ij}$ given in [20]. To this end, since in [20] the tensor perturbations
have not been explicitly expressed in terms of the light-cone perturbations,
we first recall their approach to evaluate $h_{ij}$.

Let us start from their relation

$2\left(\Delta_{3}\right)^{2}h_{ij}=\left(\Pi_{i}^{l}\Pi_{j}^{k}-\frac{1}{2}\Pi_{ij}\Pi^{lk}\right)\mathcal{C}_{lk}\,,$
(B.6)

where $\Pi_{ij}$, defined in Eq. $\left(5.27\right)$ of [20], translates to
our notation as

$\Pi_{ij}=\left(\bar{\gamma}_{ij}\Delta_{3}-\nabla_{i}\nabla_{j}\right)\,.$
(B.7)

and $\mathcal{C}_{lk}$ is given by Eq. (2.2).

Since, for a generic scalar function $f$, we have that
$\Pi^{ij}\nabla_{i}\nabla_{j}f=0$ and
$\Pi^{ij}\bar{\gamma}_{ij}f=2\Delta_{3}f$, we have that

$\Pi^{ij}\mathcal{C}_{ij}=-4\Delta_{3}\left(\psi+\frac{1}{3}\Delta_{3}E\right)\,.$
(B.8)

To proceed on the evaluation of $h_{ij}$, we then explicitly write the first
term in the r.h.s. of Eq. (B.6) as

$\Pi_{i}^{l}\Pi_{j}^{k}\mathcal{C}_{lk}=\left(\Delta_{3}\right)^{2}\mathcal{C}_{ij}-2\Delta_{3}\left[\nabla_{(i}\nabla^{k}\mathcal{C}_{j)k}\right]-\nabla_{i}\nabla_{j}\left[\left(\Pi^{lk}-\Delta_{3}\bar{\gamma}^{lk}\right)\mathcal{C}_{lk}\right]\,,$
(B.9)

and then obtain

$\displaystyle 2\left(\Delta_{3}\right)^{2}h_{ij}=$
$\displaystyle\left(\Delta_{3}\right)^{2}\mathcal{C}_{ij}-2\Delta_{3}\left[\nabla_{(i}\nabla^{k}\mathcal{C}_{j)k}\right]$
$\displaystyle-\nabla_{i}\nabla_{j}\left[\left(\frac{1}{2}\Pi^{lk}-\Delta_{3}\bar{\gamma}^{lk}\right)\mathcal{C}_{lk}\right]-\frac{1}{2}\Delta_{3}\left[\bar{\gamma}_{ij}\Pi^{lk}\mathcal{C}_{lk}\right]\,.$
(B.10)

Hence, in terms of the SPS decomposition, we get

$\displaystyle\bar{\gamma}^{ij}\mathcal{C}_{ij}=$ $\displaystyle N+4\nu\,,$
$\displaystyle\Pi^{ij}\mathcal{C}_{ij}=$
$\displaystyle-2\left(\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N-2\left(\partial_{r}+\frac{3}{r}\right)D^{2}u$
$\displaystyle-\frac{1}{r^{2}}\left[\left(D^{2}\right)^{2}+2D^{2}\right]\mu+4\left(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\frac{1}{r^{2}}+\frac{1}{2}r^{-2}D^{2}\right)\nu\,.$
(B.11)

Finally, thanks to Eqs. (B.10), (B.11) and (2.23), we compute all the
components for the tensor perturbations according to Eq. (B.6). For $h_{rr}$,
we get

$\displaystyle\left(\Delta_{3}\right)^{2}h_{rr}=$
$\displaystyle\left(\Delta_{3}\right)^{2}\frac{N}{2}+\frac{\Delta_{3}}{2}\partial_{w}^{2}\left(N+4\nu\right)-\Delta_{3}\left\\{\partial_{w}\left[\left(\partial_{w}+\frac{2}{r}\right)N+D^{2}u-\frac{4}{r}\nu\right]\right\\}$
$\displaystyle-\frac{1}{4}\left(\Delta_{3}-\partial_{w}^{2}\right)\left[-2\left(\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N-2\left(\partial_{r}+\frac{3}{r}\right)D^{2}u-\frac{2}{r^{2}}D^{2}\mu\right.$
$\displaystyle\left.-\frac{1}{r^{2}}\left(D^{2}\right)^{2}\mu+4\left(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\,\right]\,.$
(B.12)

Analogously, $h_{ra}$ is

$\displaystyle\left(\Delta_{3}\right)^{2}h_{ra}=$
$\displaystyle\frac{1}{2}\left(\Delta_{3}\right)^{2}\left[r^{2}\left(D_{a}u+\tilde{D}_{a}\hat{u}\right)\right]+\frac{\Delta_{3}}{2}\left[D_{a}\left(\partial_{r}-\frac{1}{r}\right)\left(N+4\nu\right)\right]$
$\displaystyle-\left[\frac{\Delta_{3}}{2}\left(\partial_{r}-\frac{2}{r}\right)\right]\left\\{\left(\partial_{r}+\frac{2}{r}\right)\left[r^{2}\left(D_{a}u+\tilde{D}_{a}\hat{u}\right)\right]+2D_{a}\nu\right.$
$\displaystyle\left.+D_{a}\left(D^{2}\mu+2\mu\right)+\tilde{D}_{a}\left(D^{2}\hat{\mu}+2\hat{\mu}\right)\right\\}$
$\displaystyle-\frac{\Delta_{3}}{2}\left\\{D_{a}\left[\left(\partial_{r}+\frac{2}{r}\right)N+D^{2}u-\frac{4}{r}\nu\right]\right\\}$
$\displaystyle+\frac{1}{4}D_{a}\left(\partial_{r}-\frac{2}{r}\right)\left\\{-2\left(\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N-2\left(\partial_{r}+\frac{3}{r}\right)D^{2}u-\frac{2}{r^{2}}D^{2}\mu\right.$
$\displaystyle\left.-\frac{1}{r^{2}}\left(D^{2}\right)^{2}\mu+4\left(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\,\right\\}\,.$
(B.13)

Finally, we have that $h_{ab}$ can be expressed as

$\displaystyle\left(\Delta_{3}\right)^{2}h_{ab}=$
$\displaystyle\left(\Delta_{3}\right)^{2}\left[r^{2}\left(\bar{q}_{ab}\nu+D_{ab}\mu+\tilde{D}_{ab}\hat{\mu}\right)\right]+\frac{\Delta_{3}}{2}\left[\left(D_{a}D_{b}+\frac{\bar{\gamma}_{ab}}{r}\partial_{w}\right)\left(N+4\nu\right)\right]$
$\displaystyle-\Delta_{3}\left\\{\frac{\bar{\gamma}_{ab}}{r}\left[\left(\partial_{w}+\frac{2}{r}\right)N+D^{2}u-\frac{4}{r}\nu\right]+\left(\partial_{w}+\frac{2}{r}\right)\left[r^{2}\left(D_{a}D_{b}u+\tilde{D}_{ab}\hat{u}\right)\right]\right.$
$\displaystyle\left.+D_{a}D_{b}\left(2\nu+D^{2}\mu+2\mu\right)+\tilde{D}_{ab}\left(D^{2}\hat{\mu}+2\hat{\mu}\right)\right\\}$
$\displaystyle+\frac{1}{4}\left[\bar{\gamma}_{ab}+\left(D_{a}D_{b}+\frac{\bar{\gamma}_{ab}}{r}\partial_{w}\right)\right]\left\\{-2\left(\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}-\frac{1}{2r^{2}}D^{2}\right)N-\frac{2}{r^{2}}D^{2}\mu\right.$
$\displaystyle\left.-\frac{1}{r^{2}}\left(D^{2}\right)^{2}\mu-2\left(\partial_{r}+\frac{3}{r}\right)D^{2}u+4\left(\partial_{r}^{2}+\frac{3}{r}\partial_{r}+\frac{1}{r^{2}}+\frac{1}{2r^{2}}D^{2}\right)\nu\,\right\\}\,.$
(B.14)

We then have that Eqs. (B.12)-(B.14) respectively agree with Eqs.
(2.28)-(2.30), proving then the equivalence between our approach and [20].

##  Appendix C Spin raising and lowering operators

In order for the paper to be as self-contained as possible, we provide here
some useful relations of spin-weighted spherical harmonics and the spin
raising and lowering operators which are extensively used in this work. This
appendix is not meant to give an exhaustive discussion of the subject. It
rather provides the reader with some basic useful formulas needed to obtain
our results. Anyone interested in a more detailed discussion about the topic
can refer to [28, 26, 27, 31].

Let us start by considering a function $f(\bf n)$ with spin $s$. We define the
spin raising and lowering operators respectively as

$\displaystyle{/\partial}f({\bf n})\equiv$
$\displaystyle-\sin^{s}\theta\left[\partial_{\theta}+\frac{i}{\sin\theta}\partial_{\phi}\right]\left[\sin^{-s}\theta\,f({\bf
n})\right]=-\left(\partial_{\theta}+\frac{i}{\sin\theta}\partial_{\phi}\right)f({\bf
n})+s\cot\theta f({\bf n})$ $\displaystyle\overline{{/\partial}}f({\bf
n})\equiv$
$\displaystyle-\sin^{-s}\theta\left[\partial_{\theta}-\frac{i}{\sin\theta}\partial_{\phi}\right]\left[\sin^{s}\theta\,f({\bf
n})\right]=-\left(\partial_{\theta}-\frac{i}{\sin\theta}\partial_{\phi}\right)f({\bf
n})-s\cot\theta f({\bf n})\,.$

Here $(\theta,\phi)$ are the usual polar angles of the direction n. The effect
of the operator ${/\partial}$ ($\overline{{/\partial}}$) on the function $f$
is to raise (lower) its spin, such that ${/\partial}f$
($\overline{{/\partial}}f$) has spin $s+1$ ($s-1$). For a spin zero function
$\overline{{/\partial}}f=0$. In this way, the spin-weighted spherical
harmonics ${}_{s}Y_{\ell m}$ are obtained from the spin zero spherical
harmonics $Y_{\ell m}$ by acting with ${/\partial}$ and
$\overline{{/\partial}}$ respectively

$\,{}_{s}Y_{\ell m}({\bf n};{\bf
E})=\sqrt{\frac{\left(\ell-|s|\right)!}{\left(\ell+|s|\right)!}}\begin{cases}{/\partial}^{s}Y_{\ell
m}({\bf n};{\bf E})\qquad&,\qquad s\geq 0\\\
\left(-1\right)^{s}\overline{{/\partial}}^{|s|}Y_{\ell m}({\bf n};{\bf
E})\qquad&,\qquad s<0\,.\end{cases}$ (C.2)

The prefactor is needed to ensure that they remain normalized, i.e.

$\int d\Omega_{\bf n}\,{}_{s}Y_{\ell m}({\bf n};{\bf
E})\,_{s}Y^{*}_{\ell^{\prime}m^{\prime}}({\bf n};{\bf
E})=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,.$ (C.3)

Our notation is such that the pair (n;E) indicates that the angles of the
direction ${\bf n}$ are taken with respect to the reference frame where ${\bf
E}$ is the $z$-direction. Hence, the angular derivatives in the definition of
the spin operators and the dependence on ${\bf n}$ in Eq.
(LABEL:eq:spin_operators) are referring to the same basis.

With reference to the basis $\bar{e}^{i}_{\pm}$ in Eq. (3.35), we define
$s^{a}_{\pm}\equiv ar\,\bar{e}^{a}_{\pm}$ and then write Eqs.
(LABEL:eq:spin_operators) as

$\displaystyle{/\partial}f({\bf n})=$
$\displaystyle\left(-\sqrt{2}\,s^{a}_{+}\partial_{a}+s\cot\theta\right)f({\bf
n})=-\sqrt{2}\,\partial_{+}f({\bf n})\,,$
$\displaystyle\overline{{/\partial}}f({\bf n})=$
$\displaystyle\left(-\sqrt{2}\,s^{a}_{-}\partial_{a}-s\cot\theta\right)f({\bf
n})=-\sqrt{2}\,\partial_{-}f({\bf n})\,,$ (C.4)

where $\partial_{\pm}\equiv s^{a}_{\pm}\partial_{a}\mp
s\frac{\cot\theta}{\sqrt{2}}$ and

$s^{a}_{\pm}\nabla_{a}s^{c}_{\pm}=\frac{\cot\theta}{\sqrt{2}}\,s^{c}_{\pm}\,.$
(C.5)

For a generic field $f(\bf{n})$ with spin $s$, the following relations hold

$\displaystyle\overline{{/\partial}}{/\partial}f({\bf n})=$ $\displaystyle
D^{2}f({\bf n})+s\left(1-s\cot^{2}\theta\right)f({\bf
n})+2\,i\,s\frac{\cot\theta}{\sin\theta}\partial_{\phi}f({\bf n})\,,$
$\displaystyle\left[\overline{{/\partial}},{/\partial}\right]f({\bf n})=$
$\displaystyle\,2sf({\bf n})\,,$
$\displaystyle\left\\{\overline{{/\partial}},{/\partial}\right\\}f({\bf n})=$
$\displaystyle\,2\left(D^{2}f({\bf n})-s^{2}\cot^{2}\theta\,f({\bf
n})+2\,i\,s\frac{\cot\theta}{\sin\theta}\partial_{\phi}f({\bf n})\right)\,,$
$\displaystyle\overline{{/\partial}}^{2}{/\partial}^{2}f({\bf n})=$
$\displaystyle\,\overline{{/\partial}}{/\partial}\overline{{/\partial}}{/\partial}f({\bf
n})+2\left(s+1\right)\overline{{/\partial}}{/\partial}f({\bf n})\,,$
$\displaystyle\left[{/\partial},\left\\{\overline{{/\partial}},{/\partial}\right\\}\right]f({\bf
n})=$ $\displaystyle-2\left(2s+1\right){/\partial}f({\bf n})\,,$
$\displaystyle\left[\overline{{/\partial}},\left\\{\overline{{/\partial}},{/\partial}\right\\}\right]f({\bf
n})=$ $\displaystyle 2\left(2s-1\right)\overline{{/\partial}}f({\bf n})\,.$
(C.6)

Since they have a particular importance for the results obtained in the main
text, we report some useful applications of Eqs. (C.6) for $s=0,\pm 1,\pm 2$.
For a spin-0 field $A(\bf{n})$, we get

$\displaystyle s^{a}_{+}s^{b}_{+}D_{ab}A({\bf n})=$
$\displaystyle\frac{1}{2}{/\partial}^{2}A({\bf n})\qquad,\qquad
s^{a}_{-}s^{b}_{-}D_{ab}A({\bf n})=\frac{1}{2}\overline{{/\partial}}^{2}A({\bf
n})\,,$ $\displaystyle s^{a}_{+}s^{b}_{+}\widetilde{D}_{ab}A({\bf n})=$
$\displaystyle-\frac{i}{2}{/\partial}^{2}A({\bf n})\qquad,\qquad
s^{a}_{-}s^{b}_{-}\widetilde{D}_{ab}A({\bf
n})=\frac{i}{2}\overline{{/\partial}}^{2}A({\bf n})\,,$
$\displaystyle\overline{{/\partial}}^{2}{/\partial}^{2}A({\bf n})=$
$\displaystyle\,\left(D^{2}\right)^{2}A({\bf n})+2D^{2}A({\bf n})$ (C.7)

where we have used $s^{b}_{\pm}\epsilon^{a}_{b}=\mp is^{a}_{\pm}$. For a
spin-1 vector field $S_{i}({\bf n})$, we have that

$s^{a}_{\pm}D^{2}S_{a}({\bf
n})=\frac{1}{2}\left\\{\overline{{/\partial}},{/\partial}\right\\}S_{\pm}({\bf
n})\,,$ (C.8)

and for a spin-2 tensor field $S_{ij}({\bf n})$, we also have

$s^{a}_{\pm}s^{b}_{\pm}D^{2}S_{ab}({\bf
n})=\frac{1}{2}\left\\{\overline{{/\partial}},{/\partial}\right\\}S_{\pm\pm}({\bf
n})\,.$ (C.9)

##  Appendix D Derivations of the E and B modes for tensor perturbations

In this appendix we report the derivation of the $E$\- and $B$\- modes of the
helicity-2 radiative degrees of freedom, reported in the main text in Eqs.
(4.13) and (4.14). To this aim, let us first consider the field $S_{i}$ needed
to evaluate Eq. (4.8)

$\Delta_{3}S_{i}=\Delta_{3}\left(F_{i}+\partial_{i}E\right)=\nabla^{j}\mathcal{C}_{ji}+\nabla_{i}\left[2\left(\psi+\frac{1}{3}\Delta_{3}\right)-\Delta_{3}E\right]\,.$
(D.1)

Hence, in order to compute the $E$\- and $B$\- modes of $S_{i}$, we consider
the first term on the r.h.s. of Eq. (D.1) given by

$\nabla^{j}\mathcal{C}_{ji}=a^{2}\left(\bar{n}^{l}\bar{n}^{k}+2\,\bar{e}^{(l}_{+}\bar{e}^{k)}_{-}\right)\nabla_{l}\mathcal{C}_{ki}\,,$
(D.2)

where we have rewritten the background $3$-D metric as
$\bar{g}^{ij}=\bar{n}^{i}\bar{n}^{j}+2\,\bar{e}^{(i}_{+}\bar{e}^{j)}_{-}$. In
this way, the three terms on the r.h.s. of Eq. (D.2) projected onto
$\bar{e}^{i}_{\pm}$ are given by

$\displaystyle
a^{2}\bar{e}_{\pm}^{i}\bar{n}^{l}\bar{n}^{k}\nabla_{l}\mathcal{C}_{ki}=$
$\displaystyle\,a\partial_{\rVert}\mathcal{C}_{\rVert\pm}=\frac{1}{a}\partial_{\rVert}\mathcal{T}_{\rVert\pm}\,,$
$\displaystyle
a^{2}\bar{e}_{\pm}^{i}\bar{e}_{\pm}^{l}\bar{e}_{\mp}^{k}\nabla_{l}\mathcal{C}_{ki}=$
$\displaystyle\,\frac{a}{r}\left(\partial_{\pm}\mathcal{C}_{\pm\mp}+\mathcal{C}_{\rVert\pm}\right)=\frac{1}{ar}\mathcal{T}_{\rVert\pm}+\frac{1}{ar}\partial_{\pm}\mathcal{T}\,,$
$\displaystyle
a^{2}\bar{e}_{\pm}^{i}\bar{e}_{\mp}^{l}\bar{e}_{\pm}^{k}\nabla_{l}C_{ki}=$
$\displaystyle\,\frac{a}{r}\left(\partial_{\mp}\mathcal{C}_{\pm\pm}+2\,\mathcal{C}_{\pm\rVert}\right)=\frac{2}{ar}\mathcal{T}_{\rVert\pm}+\frac{1}{ar}\partial_{\mp}\mathcal{T}_{\pm\pm}\,,$
(D.3)

where we have defined $\partial_{\rVert}\equiv\bar{n}^{i}\partial_{i}$.

Using Eqs. (D.3) and applying Eqs. (4.4) and (4.5), we then obtain

$\displaystyle\left(\bar{e}_{\pm}^{i}\nabla^{j}\mathcal{C}_{ij}\right)^{\mathbf{E}}=$
$\displaystyle\frac{1}{a}\left(\partial_{r}+\frac{3}{r}\right)\mathcal{T}_{\rVert}^{\mathbf{E}}-\frac{1}{ar\sqrt{2}}\mathcal{T}^{\mathbf{E}}-\frac{D^{2}}{ar\sqrt{2}}\mathcal{T}\,,$
$\displaystyle\left(\bar{e}^{i}_{\pm}\nabla^{j}\mathcal{C}_{ji}\right)^{\mathbf{B}}=$
$\displaystyle\frac{1}{a}\left(\partial_{\rVert}+\frac{3}{r}\right)\mathcal{T}_{\rVert}^{\mathbf{B}}-\frac{1}{ar\sqrt{2}}\mathcal{T}^{\mathbf{B}}\,,$
(D.4)

where all the $\mathcal{T}$ quantities are given in Sect. 3. At this point, we
obtain that the $E/B$ modes of $S_{i}$ are given by

$\displaystyle\left(\bar{e}_{\pm}^{i}\Delta_{3}S_{i}\right)^{\mathbf{E}}=$
$\displaystyle\,\Delta_{3}S^{\mathbf{E}}-\frac{2}{ar^{2}\sqrt{2}}D^{2}\left(\partial_{r}E+F_{r}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{a\sqrt{2}}\left(\partial_{w}+\frac{3}{r}\right)\left(rD^{2}u\right)-\frac{1}{ar\sqrt{2}}\left(D^{2}+2\right)D^{2}\mu-\frac{2D^{2}}{ar\sqrt{2}}\nu$
$\displaystyle-\frac{D^{2}}{ar\sqrt{2}}\left[2\left(\psi+\frac{1}{3}\Delta_{3}E\right)-\Delta_{3}E\right]\,,$
$\displaystyle\left(\bar{e}_{\pm}^{i}\Delta_{3}S_{i}\right)^{\mathbf{B}}=$
$\displaystyle-\frac{1}{a\sqrt{2}}\left(\partial_{w}+\frac{3}{r}\right)\left(rD^{2}\hat{u}\right)-\frac{1}{ar\sqrt{2}}\left(D^{2}+2\right)D^{2}\hat{\mu}$
(D.5)

where we have used
$\left(\bar{e}_{\pm}^{i}\Delta_{3}S_{i}\right)^{\mathbf{E}}=\Delta_{3}S^{\mathbf{E}}-\frac{2}{r^{2}\sqrt{2}}D^{2}S_{\rVert}$,
with $S_{\rVert}\equiv\bar{n}^{i}S_{i}$, while $E$, $F_{r}$ and the gauge
dependent curvature perturbation $\psi+\frac{1}{3}\Delta_{3}E$ are given in
Eqs. (2.22), (2.25) and (2.26). Finally, the final expressions of the $E/B$
modes of the radiative degrees of freedom in terms of the SPS perturbations
are given by

$\displaystyle h^{\mathbf{E}}=$
$\displaystyle\frac{\left(D^{2}+2\right)}{2a^{2}}\left\\{D^{2}\mu-\frac{1}{r\Delta_{3}}\left[\left(\partial_{w}+\frac{3}{r}\right)\left(rD^{2}u\right)+\frac{\left(D^{2}+2\right)}{r}D^{2}\mu+\frac{2D^{2}}{r}\nu\right]\right\\}$
$\displaystyle+\frac{\left(D^{2}+2\right)}{2a^{2}r}\frac{1}{\Delta_{3}}\left\\{\frac{D^{2}}{r}\left\\{\frac{1}{3}\left(-\frac{2}{r}\partial_{w}+\Delta_{3}\right)\left\\{\frac{1}{\Delta^{2}_{3}}\left[\frac{1}{2}\left(\partial_{w}^{2}+\frac{5}{r}\partial_{w}+\frac{3}{r^{2}}\right)\left(N-2\nu\right)\right.\right.\right.\right.$
$\displaystyle\left.\left.-\frac{1}{4\,r^{2}}D^{2}\left(N+4\nu\right)+\frac{1}{6\,r^{2}}\left(\partial_{w}+\frac{1}{r}\right)D^{2}u+\frac{3}{2\,r^{2}}D^{2}\nu+\frac{3}{4r^{2}}\left(D^{2}+2\right)D^{2}\mu\right]\right\\}$
$\displaystyle\left.\left.+\frac{2}{r}\frac{1}{\Delta_{3}}\left[\left(\frac{1}{3}\partial_{w}+\frac{1}{r}\right)\left(2N-4\nu\right)+D^{2}u\right]-\frac{1}{3}\left(N+4\nu\right)\right\\}\right\\}\,,$
(D.6)

and

$\displaystyle h^{\mathbf{B}}=$
$\displaystyle\frac{\left(D^{2}+2\right)}{2a^{2}}\left\\{D^{2}\hat{\mu}-\frac{1}{r\Delta_{3}}\left[\left(\partial_{w}+\frac{3}{r}\right)\left(rD^{2}\hat{u}\right)+\frac{\left(D^{2}+2\right)}{r}D^{2}\hat{\mu}\right]\right\\}\,.$
(D.7)

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