# Anisotropic separate universe and Weinberg’s adiabatic mode

Takahiro Tanaka  Yuko Urakawa

###### Abstract

In the separate universe approach, an inhomogeneous universe is rephrased as a
set of glued numerous homogeneous local patches. This is the essence of the
gradient expansion and the $\delta N$ formalism, which have been widely used
in solving a long wavelength evolution of the universe. In this paper, we show
that the separate universe approach can be generically used, as long as a
theory under consideration is local and preserves the spatial diffeomorphism
invariance. Focusing on these two conditions, we also clarify the condition
for the existence of the so-called Weinberg’s adiabatic mode. Remarkably, the
separate universe approach and the $\delta N$ formalism turn out to be
applicable also to models with shear on large scales and also to modified
theories of gravity, accepting violation of four-dimensional diffeomorphism
invariance. The generalized $\delta N$ formalism enables us to calculate all
the large scale fluctuations, including gravitational waves. We also argue
several implications on anisotropic inflation and ultra slow-roll inflation.

##  1 Introduction and summary

The crucial window to probe the physics of cosmic inflation is observing the
primordial fluctuations which were generated during inflation and subsequently
stretched far beyond the Hubble horizon scales. In order to connect the
theoretical prediction of inflation to various observations, we need to solve
the evolution, when the scale of interest is much larger than the accessible
scale by any causal propagation. The hierarchical difference between these two
scales provides a new expansion parameter $\epsilon$, enabling a completely
different expansion scheme, called the gradient expansion [1, 2, 3], from the
standard cosmological perturbation theory.

At the leading order of the gradient expansion, two different local regions in
the universe which are separated more than the scale of the causal
communication evolve mutually independently. This is the basic assumption in
the separate universe approach [1, 4, 5], which identifies the inhomogeneous
universe with glued numerous local regions which evolve independently. With
this identification, the time evolution of the inhomogeneous universe can be
computed merely by solving the time evolution of the background homogeneous
and isotropic universe with different initial conditions without solving the
partial differential equations. This largely facilitates the computation of
the non-linear evolution at large scales, which is necessary to evaluate the
primordial non-Gaussianity. Based on the gradient expansion, the $\delta N$
formalism [6, 7, 8, 5, 9] was developed to calculate the primordial spectrum
of the adiabatic curvature perturbation $\zeta$, including its non-Gaussian
spectrums. In Refs. [10, 11, 12, 13, 14], it was later extended to the next to
leading order of the gradient expansion.

Observing the primordial non-Gaussianity through the cosmic microwave
background (CMB) and the large scale structure (LSS) can uncover a detailed
property of the inflaton. Furthermore, these observations can also explore a
possible imprint of spectator fields, which are not the driving force of
inflation but still contribute to the primordial perturbations. Apart from
spectator scalar fields, which has attracted vast attention also as a source
of the primordial non-Gaussianity or the isocurvature perturbation, non-zero
spin fields might be excited during inflation. The generation of a vector
field perturbation through the coupling with the inflaton $\phi$, e.g.,
$I(\phi)F_{\mu\nu}F^{\mu\nu}$ [15, 16, 17] or non-minimal coupling with
gravity [18], has been discussed as a mechanism of the primordial
magnetogenesis. Here, $F_{\mu\nu}$ denotes the field strength of the
$U(1)_{Y}$ gauge field. A significant growth of the vector field results in a
large backreaction on the dynamics of the inflaton and the geometry of the
universe. While a too large backreaction can spoil inflation [19], a mild
backreaction can lead to a sustainable anisotropic inflation with shear on
large scales [20, 21, 22, 23], providing a counterexample of cosmic no-hair
conjecture [24]. Recently, Gong el al. investigated the effective field theory
prescription [25] of the anisotropic inflation [26]. In Refs. [27, 28], the
possibility that the generated vector field on large scales serves as a vector
oscillating dark matter was explored, while keeping the backreaction
negligible. In Ref. [29], the model in Refs. [15, 16, 17] with the kinetic
coupling $I(\phi)F_{\mu\nu}F^{\mu\nu}$ was generalized to discuss the
generation of general integer spin fields through the kinetic coupling with
the inflaton as a framework of cosmological collider physics [30, 31, 32]. The
imprints of these non-zero spin fields encoded in the primordial curvature
perturbation might be detectable from measurements of CMB [33, 34, 35, 36] and
LSS [37, 38, 39, 40, 41, 42].

The original $\delta N$ formalism [6, 7, 8, 5, 9] only applies to a model
composed of scalar fields as matter contents. Although the $\delta N$
formalism has been used more broadly [43, 44], the range of the validity
remains unclear. Having considered the recent increasing interest in models
with non-zero (integer) spin fields, in this paper, we upgrade the gradient
expansion and the $\delta N$ formalism, enabling an application to such
models. When the contribution of the non-zero spin fields to the geometry is
not negligible, the conventional argument which verifies the separate universe
picture (by ensuring the approximate validity of the momentum constraints) [9,
45, 46] cannot apply.

In this paper, we will show that the separate universe approach can be
justified, as long as the two fundamental conditions are satisfied, including
a model with non-zero spin fields. This enables a generalization of the
$\delta N$ formalism, which we dub the generalized $\delta N$ formalism (the
g$\delta N$ formalism). Another assumption which is usually employed in the
gradient expansion and the $\delta N$ formalism, leaving aside a few
exceptions [47, 48], is the spacetime diffeomorphism (Diff) invariance. It
will turn out that our new formalism can apply even when the spacetime Diff is
broken down to the spatial Diff.

The final time of the $\delta N$ formalism, at which we evaluate the curvature
perturbation $\zeta$, is usually set after the background trajectory in the
field space converges. This is because $\zeta$ is conserved in time after the
convergence of the trajectories [4, 1, 8, 5], when $\zeta$ has the constant
solution in the large scale limit, called the Weinberg’s adiabatic mode (WAM)
[49]. In this paper, we seek for a general condition that ensures the
existence of the WAM. To make a systematic argument possible, going beyond
case studies, we distinguish the existence of the WAM from the conservation of
the adiabatic curvature perturbation $\zeta$. When $\zeta$ is conserved the
WAM should exist, while even if the WAM exists, $\zeta$ is not always
conserved, since it is not necessarily the dominant solution.

Figure 1: This is the flowchart of the main claim. The first arrow leads to
the validity of the separate universe evolution, $(\star)$ (see Sec. 2.1) and
the second one leads to the approximate existence of the Weinberg’s adiabatic
mode (WAM). The models with a cross are excluded and the one with a check is
allowed at each step. We put $\ast$ on the spatially non-flat FLRW, since
$(\star)$ still holds, when the curvature scale is much larger than the size
of the causal patch(see Sec. 2.2).

The main claim of this paper is summarized in Fig. 1. In Sec. 2, we show that
the locality of the theory and the spatial Diff invariance ensure the separate
universe picture, on which the gradient expansion is based [1]. When these two
fundamental conditions are valid, we can capture the large scale evolution of
the universe just by solving the field equations in the homogeneous limit,
which is nothing more than solving the background evolution. This enables us
to apply the $\delta N$ formalism also to models with anisotropies on large
scales [20, 21, 22, 23] and also to models in which the spacetime Diff is
broken down to the spatial Diff. [50]. Furthermore, in the generalized
formalism, not only the curvature perturbation, we can compute all the
quantities which do not vanish in the large scale limit, including
gravitational waves. In Sec. 3, we show that when one more condition, which we
call the $[$approx${\cal G}_{\epsilon}$$]~{}$condition, is also satisfied, the
WAM exists as an approximate large scale solution. In Sec. 4, we apply the
generalized $\delta N$ formalism to a model of anisotropic inflation. In Ref.
[20], it was shown that shear on large scales can survive without being
diluted by the cosmic expansion, but the amplitude is bounded by the slow-roll
parameter $\varepsilon$. We generalize their result, removing the slow-roll
approximation. We also make a brief comment on an inflection type inflation
model, which has attracted attention also as a model which can predict
primordial black holes. In this paper, we only consider a classical theory. In
Sec. 5, we argue possible implications of this paper on quantized system.
Table 1 summarizes the symbols introduced in this paper.

##  2 Gradient expansion and separate universe

In this paper, we express the $(d+1)$-dimensional line element as

$\displaystyle ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt)\,,$
(2.1)

with $i,j=1,\cdots,d$. We express the spatial metric as

$\displaystyle g_{ij}=e^{2\psi}\,\gamma_{ij}\,,$ (2.2)

where $\gamma_{ij}$ satisfies $\det[\gamma]=1$. Using $\psi$, the determinant
of $g_{ij}$ is given by $g=e^{2d\psi}$.

###  2.1 Smoothing and separate universe evolution

The gradient expansion [1, 2] is an expansion scheme with respect to the
spatial gradient at each given time, which provides a useful tool to address
the long wavelength evolution of the fluctuations in a cosmological setup (see
also Ref. [3]). The gradient expansion starts with smoothing out small scale
fluctuations [1]. We express a set of the coarse grained fields which are
smoothed out at a physical scale $\lambda_{\rm s}(t)$ as $\\{\varphi^{a}\\}$.
We require that the smoothing scale $\lambda_{s}$ should be equal to or larger
than the scale of the causal propagation during the time scale of an expanding
universe111In this paper, as a typical time scale of the causal propagation,
we consider the time scale of the cosmic expansion, $1/K$. When we only
consider a much shorter time scale than $1/K$ with $\delta t\ll 1/K$,
obviously the size of the causal patch becomes smaller. Then, $\lambda_{s}$
can be chosen as $c_{\rm max}/K>\lambda_{\rm s}>c_{\rm max}\delta t$. and much
smaller than the scale under consideration, $\lambda$, satisfying [4]

$\displaystyle\lambda\gg\lambda_{\rm s}\geq\frac{c_{\rm max}}{K}\,.$ (2.3)

Here, $K$ denotes the trace of the extrinsic curvature of the time constant
hypersurface, which corresponds to ($d$ times) the Hubble parameter for the
FLRW spacetime (the definition will be given in Sec. 3.3) and $c_{\rm max}$
denotes the maximum speed of propagation in the system under consideration,
i.e., $c_{\rm max}\equiv{\rm max}\,\\{c_{a}\\}$, where $c_{a}$ denotes the
propagation speed of $\varphi^{a}$. The fields $\\{\varphi^{a}\\}$ include
both the metric and matter fields.

Symbol |  Definition
---|---
$(\star)$ | Separate universe evolution, defined in Sec. 2.1
${\cal C}$ | Set of constraints
${\cal G}$ | Set of gauge constraints, which serve generators of gauge symmetries
${\cal G}_{\epsilon}$ | Set of gauge constraints in which all terms vanish in the limit $\epsilon\to 0$
${\cal X}$ | Set of gauge conditions
${\cal X}_{\epsilon}$ | Set of gauge conditions that vanish in the limit $\epsilon\to 0$
${\cal H}$ | Hamiltonian constraint
${\cal H}_{i}$ | Momentum constraints
${\cal H}_{\rm U(1)}$ | Gauge constraint of U(1) gauge symmetry, Eq. (2.26)
${\cal E}_{\rm all}$ | All field equations after solving non-gauge constraints ${\cal C}\cap{\cal G}^{c}$
${\cal E}_{\rm ev}$ | Set of all evolution equations
${\cal E}$ | Sum of ${\cal E}_{\rm ev}$ and ${\cal G}\cap{\cal G}_{\epsilon}^{c}$, i.e., ${\cal E}_{\rm ev}\cup({\cal G}\cap{\cal G}_{\epsilon}^{c})$
$\\{\varphi^{a}\\}$ | All fields after solving ${\cal C}\cap{\cal G}^{c}$ and imposing ${\cal X}$ which are not ${\cal X}_{\epsilon}$
$\\{\varphi^{a^{\prime}}\\}^{\prime}$ | Set of remaining fields after solving ${\cal C}\cap{\cal G}^{c}_{\epsilon}$ for $\\{\varphi^{a}\\}$
$\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$ | Set of all physical degrees of freedom
Table 1: This table summarizes the symbols introduced in this paper. Here $c$ denotes a complementary set. The detailed definition for ${\cal G}_{\epsilon}$ is given in Sec. 2.4.1 and the ones for $\\{\varphi^{a}\\}$, $\\{\varphi^{a^{\prime}}\\}^{\prime}$, and $\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$ are given in Sec. 2.4.2. By definition, ${\cal C}\supseteq{\cal G}\supseteq{\cal G}_{\epsilon}$, ${\cal E}_{\rm all}\supseteq{\cal E}\supseteq{\cal E}_{\rm ev}$, and $\\{\varphi^{a}\\}\supseteq\\{\varphi^{a^{\prime}}\\}^{\prime}\supseteq\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$. We can reduce $\\{\varphi^{a^{\prime}}\\}^{\prime}$ to $\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$ by solving ${\cal G}_{\epsilon}$ and removing residual gauge degrees of freedom (if any). Symbol |  Examples
---|---
${\cal G}$ |  ${\cal H}$ for a $(d+1)$-dim Diff theory, ${\cal H}_{i}$, ${\cal H}_{\rm U(1)}$
${\cal G}_{\epsilon}$ |  ${\cal H}_{i}$, ${\cal H}_{\rm U(1)}$ in the absence of charged fields
${\cal G}\cap{\cal G}^{c}_{\epsilon}$ |  ${\cal H}$ for a $(d+1)$-dim Diff theory, ${\cal H}_{\rm U(1)}$ in the presence of charged fields
${\cal X}_{\epsilon}$ | Transverse condition $\partial_{i}{\gamma^{i}}_{j}=0$, Coulomb gauge condition $\partial_{i}A^{i}=0$
${\cal X}\cap{\cal X}^{c}_{\epsilon}$ |  $N_{i}=0$, $A_{0}=0$
${\cal E}_{\rm ev}$ |  $(i,\,j)$ component of Einstein equations, Klein-Gordon equation
Table 2: This table summarizes examples for sets of the equations listed in
Table 1.

As a consequence of the smoothing, operating the spatial gradient gives rise
to the suppression by a small quantity, which is usually characterized by the
spatial variation of $\varphi^{a}$ within each causally connected patch with
the size $c_{\rm max}/(e^{\psi}K)$,

$\displaystyle\epsilon\equiv{\rm
max}\left\\{\frac{1}{|\varphi^{a}|}\left|\frac{c_{a}}{e^{\psi}K}\partial_{i}\varphi^{a}\right|\right\\}\,.$
(2.4)

The value of $\epsilon$ depends on the smoothing scheme such as the sharpness
of the window function which smooths out the small scale fluctuation. Without
going into the detail of the smoothing scheme, we simply assume that the
window function is an analytic function and $\epsilon$ takes a value of
$(0\leq)\,\epsilon\ll 1$. Then, $|c_{a}\partial_{i}\varphi^{a}/(e^{\psi}K)|$
is approximately bounded by $\epsilon|\varphi^{a}|$, allowing us to expand the
field equations in terms of the small parameter $\epsilon$, which is the
gradient expansion. In the standard cosmological perturbation theory, the
fields $\varphi^{a}$ are divided into the background configuration and
perturbations. The field equations are solved order by order after expanding
them with respect to the perturbations. The $\epsilon$ expansion, which is the
spatial gradient expansion on each time slice, provides a completely different
expansion scheme from the standard cosmological perturbation theory. The
validity of the gradient expansion does not require that the deviation from
the background should be perturbatively small.

Since the field values in each causal patch are almost homogeneous, we can
identify an inhomogeneous universe with glued numerous nearly homogeneous
patches. The basic assumption of the separate universe approach is the
validity of the following identity [1, 5],

$\displaystyle[{\rm Separate~{}universe~{}evolution}~{}(\star)]:$
$\displaystyle\quad\,\,(\mbox{Solving the dynamics of an inhomogeneous
universe})$ $\displaystyle\quad\,\,\,=(\mbox{Solving the dynamics of glued
nearly homogeneous patches independently})\,$

which enables us to avoid solving a set of the partial differential equations
for the coarse-grained fields, $\\{\varphi^{a}(t,\,\hbox{\boldmath{$x$}})\\}$.
When $(\star)$ holds, the time evolution of the inhomogeneous universe is
determined solely by solving a set of the corresponding ordinary differential
equations, where the spatial gradient terms are simply dropped, for a
corresponding initial condition. We express a set of solutions which satisfy
the field equations in this limit as $\\{\bar{\varphi}^{a}(t)\\}$. In order to
emphasize that $(\star)$ is the identification of the evolution in two
pictures for a common initial condition, we call it separate universe
evolution instead of separate universe approach, which has been widely used.

Since $\\{\varphi^{a}(t,\,\hbox{\boldmath{$x$}})\\}$ is inhomogeneous, these
fields in different patches, in general, take different values. This is
incorporated by assigning different initial conditions, correspondingly, to
these fields in different causal patches as

$\displaystyle\lim_{\epsilon\to
0}\varphi^{a}(t,\,\hbox{\boldmath{$x$}})=\bar{\varphi}^{a}(t;\\{\bar{\varphi}^{a^{\prime}}(t_{*})=\varphi^{a^{\prime}}(t_{*},\,\hbox{\boldmath{$x$}})\\}^{\prime})\,,$
(2.5)

where $t_{*}$ denotes the initial time. Here,
$\\{\varphi^{a^{\prime}}(t_{*},\,\hbox{\boldmath{$x$}})\\}^{\prime}$ denotes a
set of metric and matter fields by imposing gauge conditions and also by
solving constraint equations, which will be specified in Sec. 2.4.2. For
clarity, we use the index with a dash to denote the reduced set of the fields.
The first line of $(\star)$ corresponds to the left hand side of Eq. (2.5) and
the second line corresponds to the right hand side. Equation (2.5) equates the
leading terms of the solution expanded with respect to the small parameter of
the gradient expansion $\epsilon$ (the left hand side) with the solution of
the equations in the limit $\epsilon\to 0$ (the right hand side).

As is clear from the definition, (2.4), the expansion in terms of $\epsilon$
is the expansion with respect to the spatial gradient. Lyth, Malik and Sasaki
introduced a small parameter for their gradient exapsion, $\epsilon_{\rm
LMS}$, as the ratio between the Fourier mode $k$ which corresponds to the
scale of interest and the scale of the cosmic expansion, i.e., $\epsilon_{\rm
LMS}\equiv c_{\rm max}k/(e^{\psi}K)$ [5]. Their expansion scheme with respect
to $\epsilon_{\rm LMS}$ is slightly different from our expansion in terms of
$\epsilon$. When the coarse-grained field $\varphi^{a}$ is expressed by the
Fourier modes, our scheme includes all the modes with
$k/e^{\psi}<1/\lambda_{\rm s}\,(\leq K/c_{\rm max})$. By contrast,
$\epsilon_{\rm LMS}$ focuses on a single Fourier mode under consideration.
When we consider non-linear perturbation, different Fourier modes need to be
considered altogether. Therefore, in this paper, we use $\epsilon$ as an
expansion parameter, which takes into account all the Fourier modes below
$e^{\psi}/\lambda_{\rm s}$. When the Fourier mode expansion is possible, the
condition $\epsilon\ll 1$ implies that all the Fourier modes included in the
smeared fields $\varphi^{a}$ should satisfy $k/e^{\psi}\ll K/c_{\rm max}$,
ensuring also $\epsilon_{\rm LMS}\ll 1$.

###  2.2 Homogeneity and smoothing

If non-vanishing gradient of some field is necessary to describe the
configuration, the smoothing condition $\epsilon\ll 1$ is violated. As an
example, let us consider the case where the coarse-grained spatial metric in
each causal patch, $\bar{g}_{ij}$, is approximately given by the Friedmann-
Lemaître-Robertson-Walker (FLRW) metric with a non-zero spatial curvature as

$\displaystyle g_{ij}(t,\,\hbox{\boldmath{$x$}})\simeq
a^{2}(t)\left[\delta_{ij}+\frac{{\cal K}x_{i}x_{j}}{1-{\cal
K}\delta_{kl}x^{k}x^{l}}\right]\,,$ (2.6)

where ${\cal K}=\pm 1$ and $a(t)$ denotes the scale factor. The condition,
$\epsilon\ll 1$, requires that the curvature radius $a/\sqrt{|{\cal K}|}$
should be much larger than the size of the causal patch $c_{\rm max}/K$,
indicating that the spatial curvature within each patch is effectively zero.
This requirement is safely satisfied in computing the superhorizon evolution
of the wavelengths below the present horizon scale, because the spatial
curvature of the present universe $|\Omega_{K}|$ is bounded by ${\cal
O}(10^{-3})$ [51]. Meanwhile, the coarse-grained spatial metric needs to be
almost flat but it is not necessarily isotropic, including Bianchi type I
spacetime (recall that the spatial curvature is not accepted).

Another example is the solid inflation, which was proposed in Ref. [52]. The
matter content in the solid inflation consists of $d$ scalar fields
$\phi^{I}(x)$ with $I=1,\,2,\,\cdots,d$ that preserve the internal shift and
rotational symmetries. The solid action which is compatible with the required
symmetry is given by

$\displaystyle S=\int d^{4}xF(X,\,Y,\,Z)+\cdots\,,$ (2.7)

where $X$, $Y$, and $Z$ are given by $X=[B]$, $Y=[B^{2}]/[B]^{2}$, and
$Z=[B^{3}]/[B]^{3}$ with
$B^{IJ}\equiv\partial_{\mu}\phi^{I}\partial^{\mu}\phi^{J}$. Here, the dots
denote the trace over the matrix indices $I,\,J,\,\cdots$. In Eq. (2.7),
higher derivative terms, which do not contribute to low-energy physics, are
abbreviated. The background configuration of the solid is space dependent,
$\phi^{I}=x^{I}$, while the translation and rotation symmetries are preserved
if simultaneous change of the internal field space coordinates is performed.
In this model with the spatial configuration, a term with
$\partial_{i}\phi^{I}$ can contribute as ${\cal O}(\epsilon^{0})$ without
being suppressed by small quantities. A Lagrangian density which includes
$B^{IJ}$ in denominator can yield a negative power of $\epsilon$, when
$B^{IJ}$ is dominated by the spatial gradient term. Meanwhile, as long as
$B^{IJ}$ is dominated by the time derivative term as a consequence of the
smoothing, the same theory does not give rise to any negative powers of
$\epsilon$.

Since we postulate the smoothing which ensures that operating the spatial
gradient always yields the suppression with a positive power of the small
parameter $\epsilon$, the following discussions on the IR physics do not apply
to the cases mentioned above. In fact, it is known that the WAM does not exist
both for the perturbed non-flat FLRW spacetime [53, 54] and solid inflation
[52] (see also [55]).

###  2.3 Basic conditions

The gradient expansion developed in Refs. [1, 2] assumes general relativity
where any acausal propagation of information is prohibited. The causality
ensures that two different patches which are separated much more than $c_{\rm
max}/K$ evolve almost independently, allowing only the tiny mutual influence
among them, which is mediated by the spatial gradient terms suppressed by
$\epsilon\ll 1$. This effect can be taken into account order by order through
the perturbative expansion with respect to $\epsilon$. In particular, in the
limit $\epsilon\to 0$, the evolution of each causal patch can be determined
completely independently of other patches for scalar field systems [1, 9].

Having considered that the inflationary universe provides a natural laboratory
to explore new physics at an extremely high energy, it is crucially important
to establish a systematic tool to compute the primordial perturbations for a
general model of inflation. For this purpose, in this paper, we derive the
general condition that ensures the validity of the separate universe
evolution, $(\star)$, and the $\delta N$ formalism. It turns out that
$(\star)$ can be verified generically, including extended models of gravity
and also non-zero spin fields as matter contents.

####  2.3.1 Spatial Diff invariance

First, we require

  * •

$[$sDiff$]~{}$: The theory which describes the coarse-grained fields
$\\{\varphi^{a}(t,\,\hbox{\boldmath{$x$}})\\}$ should remain invariant under
the $d$-dim spatial diffeomorphism (Diff),

$\displaystyle x^{i}\to\tilde{x}^{i}(t,\,x^{i}).$ (2.8)

The $d$-dim spatial Diff invariance is a sub group of $(d+1)$-dim Diff
invariance. Variations of the spatial Diff invariant theories are summarized,
e.g., in Ref. [56]. The foliation preserving Diff invariance is satisfied in
the Horava-Lifshitz (HL) gravity [50], which is known to be power-counting
renormalizable. Lately, in Refs. [57, 58], the perturbative renormalizability
was shown for the projectable subclass of the HL gravity. The violation of the
$(d+1)$-dim Diff invariance leads to the appearance of an additional scalar
degree of freedom in the gravity sector, dubbed Khronon.

One may want to consider a theory whose action does not include the lapse
function. Nevertheless, since the action for most theories, including a
$(d+1)$-dim Diff invariant theory and the HL gravity, is naturally written
down using the lapse function, in what follows we include $N$, when we write
down an explicit example of the action. Let us, however, emphasize that the
Hamiltonian constraint, which can be derived by the functional derivative of
the action with respect to $N$, does not play any important role in the
following discussion, enabling an extension to a theory without containing the
lapse function straightforwardly.

In a $(d+1)$-dim Diff invariant theory, using the energy conservation, which
is ensured by the time Diff, we can show the existence of the constant
solution for the curvature perturbation in the uniform density slicing [4, 5].
It is intuitively natural that the time Diff implies the existence of a
conserved quantity. However, this argument cannot apply to a theory where the
$(d+1)$-dim Diff invariance is violated. Meanwhile, it is known that the
curvature perturbation is also conserved at large scales, e.g., in non-
projectable version of the HL gravity [59, 60] (see also Ref. [61]). We show
that the constant solution also exists, even when the $(d+1)$-dim Diff
invariance is broken down to the $d$-dim Diff invariance.

####  2.3.2 Locality condition

As emphasized above, in general relativity, the causality is crucial to verify
the separate universe approach and the gradient expansion. In the absence of
the $(d+1)$-dim Diff invariance, there is no clear bound on the propagation
speed from the causality. Therefore, instead, as the second condition we
require

  * •

$[$locality$]~{}$: After solving all the constraint equations except for gauge
constraints ${\cal G}$, the effective dynamics of the coarse-grained fields is
described by the Lagrangian density ${\cal L}(t,x^{i})$ given locally by a
function of fields at $x^{\mu}=(t,x^{i})$. Taking variation of the
corresponding action gives local field equations for the coarse-grained
fields.

We call a constraint which generates a gauge symmetry a gauge constraint,
expressing it as ${\cal G}$. Gauge constraints can be derived by taking
derivative with respect to Lagrange multipliers associated with the gauge
symmetry. The gauge constraint, which is first class, can transform into a
second class constraint by imposing a gauge condition. The variation of the
action for ${\cal L}(t,x^{i})$ gives field equations for the set of the fields
$\\{\varphi^{a}\\}$ which remain after all the non-gauge constraints are
solved and the gauge conditions are employed. Since the locality is imposed
for the coarse-grained fields, our $[$locality$]~{}$condition does not exclude
a theory in which non-locality appears only in the dynamics of the fine-
grained fields but not in that of the coarse-grained fields (a recent attempt
to construct such theories can be found e.g., in Ref. [62]). We implicitly
assume that all the evolution equations do not vanish in the limit
$\epsilon\to 0$ and can be expanded in powers of spatial gradient.

Eliminating a Lagrange multiplier can transform a local Lagrangian density to
a non-local one, especially when the Lagrange multiplier is accompanied by a
spatial gradient. For example, let us consider a system with a single
dynamical scalar field $\phi$ and a Lagrange multiplier $\chi$ whose
Lagrangian density is given by

$\displaystyle{\cal
L}(x)=\cdots+g^{ij}\partial_{i}\chi\partial_{j}\chi/2+\chi\,g^{ij}\partial_{i}\phi\partial_{j}\phi+\cdots\,,$
(2.9)

where we have written only the terms which include $\chi$. Taking the
functional derivative of the action with respect to $\chi$, we obtain the
constraint equation as

$\displaystyle\frac{1}{N\sqrt{g}}\partial_{i}(g^{ij}N\sqrt{g}\,\partial_{j}\chi(x))=g^{ij}\partial_{i}\phi\partial_{j}\phi\,.$
(2.10)

Once we eliminate $\chi$ by solving this constraint equation, ${\cal L}$
becomes non-local because of the appearance of the inverse Laplacian. Such
models do not satisfy the $[$locality$]~{}$condition, even though the original
Lagrangian does not contain non-local operators manifestly.

The presence of non-local terms can cause a serious problem for the validity
of the separate universe approach, since two patches whose physical distance
is much larger than $c_{\rm max}/K$ can interact through non-local terms.
Especially, if the Lagrangian density includes terms with the inverse
Laplacian $\partial^{-2}$, a term with the spatial derivative operator is no
longer necessarily suppressed by $\epsilon$, since it may be compensated by
$\partial^{-2}$. The $[$locality$]~{}$condition requires that after
eliminating the Lagrange multipliers by solving all the constraints, ${\cal
C}$, except for the gauge constraints ${\cal G}$, the action becomes local
with an appropriate choice of the gauge fixing conditions. Solving a gauge
constraint such as the momentum constraints can also give rise to non-local
terms in the action. Nevertheless, as will be discussed in the next
subsection, unlike non-gauge constraints, the existence of gauge constraints
does not disturb the validity of the separate universe evolution.

The Hamiltonian constraint ${\cal H}$ for a $(d+1)$-dim Diff invariant theory
and momentum constraints ${\cal H}_{i}$ are one of the gauge constraints
${\cal G}$. When the $(d+1)$-dim Diff is broken down to the foliation
preserving Diff, e.g., in the HL gravity, ${\cal H}$ is not necessarily ${\cal
G}$. For the projectable version of the HL gravity, since the lapse function
is not allowed to depend on the spatial coordinates, ${\cal H}$ only gives a
global constraint, which is integrated over a time constant hypersurface.
Therefore, precisely speaking the projectable version does not satisfy the
$[$locality$]~{}$condition. Instead, one may want to consider a theory where
the time reparametrization invariance is also explicitly broken, e.g., by
simply setting $N$ to 1 in the projectable version. In this case, since the
(global) Hamiltonian constraint does not exist, the $[$locality$]~{}$condition
can be fulfilled. Meanwhile, for the non-projectable version, since $N$ also
depends on the spatial coordinates, we obtain ${\cal H}$ at each spacetime
point. Since the time coordinate transformation is limited to
$t\to\tilde{t}(t)$, unlike the global Hamiltonian constraint, the Hamiltonian
constraint for each patch is not ${\cal G}$. As long as the latter can be
solved without giving rise to non-local contributions, the
$[$locality$]~{}$condition is satisfied. This is the situation discussed in
Refs. [59, 60].

Strictly speaking, to figure out when the $[$locality$]~{}$condition holds, we
need to understand how the short modes below the smoothing scale,
$\lambda_{\rm s}$, affect the dynamics of $\\{\varphi^{a}\\}$. As is widely
known in the context of stochastic inflation [63, 64, 65, 66] (see e.g., Refs.
[67, 68, 69] for a recent progress), the effective action obtained by
integrating out the degrees of freedom of short modes is, in general, given by
a non-local effective Lagrangian density, which does not satisfy the
$[$locality$]~{}$condition defined above. Therefore, in our forthcoming paper
[70], we will relax the $[$locality$]~{}$condition. The refined
$[$locality$]~{}$condition restricts the smoothing scheme and also the quantum
system, particularly the quantum state of the short modes to be traced out. In
Ref. [70], we will show that under the refined $[$locality$]~{}$condition, the
discussion proceeds almost in the same way as in a classical theory. In this
paper, focusing on a classical theory, we simply require the
$[$locality$]~{}$condition defined above.

###  2.4 Validity of separate universe evolution

In this subsection, we show that the separate universe evolution $(\star)$
indeed holds under the $[$locality$]~{}$and $[$sDiff$]~{}$conditions.

####  2.4.1 Gauge constraints

When all the terms in a gauge constraint are suppressed by the spatial
gradient, solving the constraint gives rise to non-local terms in the action
(recall the discussion in the previous subsection). In what follows, we
express a set of gauge constraints as ${\cal G}_{\epsilon}$, when all the
terms therein are suppressed by at least one spatial gradient, i.e.,

${\rm a~{}constraint~{}}\chi\in{\cal G}_{\epsilon}\qquad\quad{\rm
if~{}}\chi\in{\cal G}{\rm~{}and~{}}\chi={\cal O}(\epsilon)\,.$

For example, the momentum constraints ${\cal H}_{i}=0$, which are obtained by
taking the variation of the action with respect to $N_{i}$, is one of ${\cal
G}_{\epsilon}$. We can confirm that all the terms in ${\cal H}_{i}$ are indeed
accompanied by at least one spatial gradient as follows. Using ${\cal H}_{i}$,
which is the generator of the spatial Diff, the change of a field
$\varphi^{a}$ under a spatial coordinate transformation (2.13) is given by

$\displaystyle\left\\{\varphi^{a}(x),\int d^{d}y\sqrt{g(y)}\xi^{i}{\cal
H}_{i}(y)\right\\}={{\cal L}\raise 1.5pt\hbox{-}\,}_{\xi}\varphi^{a}(x)\,.$
(2.11)

Here, $\\{\cdot,\,\cdot\\}$ denotes the Poisson brackets and ${{\cal L}\raise
1.5pt\hbox{-}\,}_{\xi}$ denotes the Lie derivative. The generator, which
satisfies Eq. (2.11), is expressed as

$\displaystyle\int d^{d}x\sqrt{g(x)}\xi^{i}{\cal
H}_{i}(x)=\sum_{a}{}^{\prime}\int d^{d}x\,\pi_{a}(x){{\cal L}\raise
1.5pt\hbox{-}\,}_{\xi}\varphi^{a}(x)\,,$ (2.12)

where $\sum{}^{\prime}$ sums over all the fields in $\\{\varphi^{a}\\}$ other
than $N$ and $N^{i}$. Tensor indices of $\\{\varphi^{a}\\}$ are abbreviated,
and $\pi_{a}(x)\equiv\partial(N\sqrt{g}{\cal L})/\partial\dot{\varphi}^{a}$ is
the conjugate momentum of $\varphi^{a}$.

One can easily understand that ${\cal H}_{i}$ in Eq. (2.12) is equivalent to
the usual momentum constraints defined by $\tilde{\cal H}_{i}\equiv
N\partial{\cal L}/\partial N^{i}$, when we use the equations of motion $\delta
S/\delta\varphi^{a}=0$. The variation of the action caused by the spatial Diff
transformation,

$\displaystyle x^{i}\to e^{\xi^{\mu}\partial_{\mu}}x^{i}=x^{i}+\xi^{i}+\cdots$
(2.13)

with $\xi^{\mu}=(0,\,\xi^{i}(t,\,\hbox{\boldmath{$x$}}))$, which should
vanish, can be calculated as

$\displaystyle 0=\delta_{\xi}S=$ $\displaystyle\,\int
d^{d+1}x\left\\{\xi^{i}\partial_{t}\left(\sqrt{g}{\cal\tilde{H}}_{i}\right)+\sum_{a}{}^{\prime}\frac{\delta
S}{\delta\varphi^{a}}\delta_{\xi}\varphi^{a}+{\cal O}(\epsilon^{2})\right\\}$
(2.14) $\displaystyle+\left[-\int d^{d}x\sqrt{g(x)}\xi^{i}\tilde{\cal
H}_{i}(x)+\sum_{a}{}^{\prime}\int d^{d}x\,\pi_{a}(x){{\cal L}\raise
1.5pt\hbox{-}\,}_{\xi}\varphi^{a}(x)\ \right]_{t_{i}}^{t_{f}}\,,$ (2.15)

where we used $\delta_{\xi}N=0$ and
$\delta_{\xi}N^{i}=-\partial_{t}\xi^{i}+{\cal O}(\epsilon)$. Here
$\delta_{\xi}$ denotes the variation under the spatial Diff transformation
(2.13). The variation of $\varphi^{a}$ is given by
$\delta_{\xi}\varphi^{a}={{\cal L}\raise 1.5pt\hbox{-}\,}_{\xi}\varphi^{a}$.
Using the equations of motion $\delta S/\delta\varphi^{a}=0$, we find
$\tilde{\cal H}_{i}={\cal H}_{i}$ and

$\displaystyle\frac{\partial(\sqrt{g}{\cal H}_{i})}{\partial t}=0\,,$ (2.16)

at the leading order of $\epsilon$.

Performing the integration by parts in the left hand side of (2.12), we can
derive the explicit form of ${\cal H}_{i}$. Since all the terms in ${{\cal
L}\raise 1.5pt\hbox{-}\,}_{\xi}\varphi^{a}(x)$ are accompanied with at least
one spatial derivative, so are the terms in ${\cal H}_{i}$, i.e., ${\cal
H}_{i}\in{\cal G}_{\epsilon}$. Using

$\displaystyle{{\cal L}\raise
1.5pt\hbox{-}\,}_{\xi}g_{ij}=-\xi^{k}\partial_{k}g_{ij}-\partial_{i}\xi^{k}g_{kj}-\partial_{j}\xi^{k}g_{ik}\,,$
(2.17)

and performing the integration by parts, the contribution of the variation of
the spatial metric $\delta_{\xi}g_{ij}$ in the right hand side of Eq. (2.12)
can be further rewritten into a more widely used form as

$\displaystyle\int d^{d}x\,\pi^{ij}(x){{\cal L}\raise
1.5pt\hbox{-}\,}_{\xi}g_{ij}(x)=2\int
d^{d}x\,\sqrt{g(x)}\,\xi^{i}\nabla_{j}\\!\left(\frac{{\pi^{j}}_{i}(x)}{\sqrt{g(x)}}\right)\,,$
(2.18)

where $\pi^{ij}$ is the conjugate momentum of $g_{ij}$, defined by

$\displaystyle\pi^{ij}\equiv\frac{\partial(N\\!\sqrt{g}{\cal
L})}{\partial\dot{g}_{ij}}\,,$ (2.19)

and ${\pi^{j}}_{i}\equiv g_{ik}\pi^{jk}$. Here, $\nabla_{i}$ is the covariant
derivative for the spatial metric $g_{ij}$. The conjugate momentum divided by
$\\!\sqrt{g}$, which transforms as a rank $(1,\,1)$ tensor, satisfies
$\sqrt{g}\,\nabla_{j}({\pi^{j}}_{i}/\sqrt{g})=\partial_{j}{\pi^{j}}_{i}-\pi^{jk}g_{jk,i}$.
Unlike the momentum constraints, the Hamiltonian constraint is not ${\cal
G}_{\epsilon}$.

Another example of ${\cal G}_{\epsilon}$ is the gauge constraint for a U(1)
gauge field. Let us consider a matter Lagrangian density ${\cal L}_{\rm
matter}$, given by

$\displaystyle{\cal L}_{\rm matter}={\cal
F}(\phi^{I},\,X^{IJ},\,Y,\,\tilde{Y})\,,$ (2.20)

where $\phi^{I}$ with $I=1,\,2,\,\cdots$ denotes neutral or charged scalar
fields with $X^{IJ}$, $Y$ and $\tilde{Y}$ being

$\displaystyle
X^{IJ}\equiv-\frac{1}{4}(D^{(I)}_{\mu}\phi^{I}D^{(J)\mu}\phi^{J*}+{\rm
c.c.})\,,$ (2.21)

and

$\displaystyle
Y\equiv-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,\qquad\tilde{Y}\equiv-\frac{1}{4}\tilde{F}_{\mu\nu}F^{\mu\nu}\,.$
(2.22)

Here $F_{\mu\nu}$ and $\tilde{F}_{\mu\nu}$ are the field strength and its dual
for a U(1) gauge field $A_{\mu}$, respectively, and $D^{(I)}_{\mu}$ is given
by

$\displaystyle D^{(I)}_{\mu}\equiv\partial_{\mu}-ie^{(I)}A_{\mu}\,,$ (2.23)

where $e^{(I)}$ is the U(1)-charge of $\phi^{I}$. For a neutral scalar field,
the charge $e^{(I)}$ should be set to 0. The Lagrangian density, (2.20),
includes the typical Lagrangian for the axion and also the Euler-Heisenberg
Lagrangian as a special case.

In what follows (unless stated), we impose the gauge condition

$\displaystyle N_{i}=0\,,$ (2.24)

which is local in the sense that it does not vanish in the limit $\epsilon\to
0$. As is known, $N_{i}=0$ does not completely remove the degrees of freedom
for the spatial small gauge transformations. In fact, at linear perturbation,
$N_{i}$ transforms under the spatial coordinate transformation,
$x^{i}\to\tilde{x}^{i}=x^{i}+\xi^{i}$ as

$\displaystyle\tilde{N}_{i}=N_{i}-\dot{\xi}^{j}g_{ji}\,.$ (2.25)

This indicates that we can still perform a time-independent spatial coordinate
transformation, maintaining the gauge condition $N_{i}=0$. We will discuss an
additional gauge condition to remove this residual gauge degree of freedom at
the end of Sec. 2.4.2.

Taking derivative with respect to $A^{0}$, we obtain the gauge constraint,
which corresponds to the Gauss law, as

$\displaystyle{\cal H}_{\rm U(1)}\equiv\partial_{i}\pi^{i}_{A}+J^{0}=0\,,$
(2.26)

where $\pi_{A}^{i}$ is the conjugate momentum given by

$\displaystyle\pi_{A}^{i}\equiv\frac{\partial(N\sqrt{g}{\cal L}_{\rm
matter})}{\partial\dot{A}_{i}}=\sqrt{g}{\cal
F}_{Y}g^{ij}\frac{\dot{A}_{j}}{N}+{\cal O}(\epsilon)\,,$ (2.27)

with ${\cal F}_{Y}\equiv\partial{\cal F}/\partial Y$, and $J^{0}$ is the 0-th
component of the current, which corresponds to the electric charge density
(multiplied by $\sqrt{g}$) and is given by

$\displaystyle J^{0}=\frac{\partial(N\sqrt{g}{\cal L}_{\rm matter})}{\partial
A_{0}}=\frac{\sqrt{g}{\cal
F}_{IJ}}{4N}\left\\{ie^{(I)}\phi^{I}(D_{0}^{(J)}\phi^{J})^{*}-iD_{0}\phi^{I}e^{(J)}\phi^{J*}+{\rm
c.c.}\right\\}\,,$ (2.28)

with ${\cal F}_{IJ}\equiv\partial{\cal F}/\partial X^{IJ}$. In the presence of
the charged scalar fields, since the current $J^{0}$ includes terms which are
not suppressed by $\epsilon$, the gauge constraint does not become ${\cal
G}_{\epsilon}$. To satisfy consistently the gauge constraint (2.26), which we
denote as ${\cal H}_{\rm U(1)}$, $J^{0}={\cal O}(\epsilon)$ should hold,
requiring the fall-off of the current $J^{0}$ in the large scale limit. In the
temporal gauge with $A_{0}=0$, the gauge constraint reads222As an example, let
us consider the case when only one of the scalar fields $I=\bar{I}$ is charged
and the other fields are neutral. Then, expressing
$\phi_{\bar{I}}=\bar{\phi}e^{i\theta}$, we can rewrite Eq. (2.29) as
$\sqrt{g}{\cal F}_{\bar{I}\bar{I}}\bar{\phi}^{2}\frac{\dot{\theta}}{N}={\cal
O}(\epsilon)\,,$ which requires the absence of the azimuth motion in the large
scale limit. This is consistent with the known background evolution (see,
e.g., Ref. [71] and references therein). The constant phase $\theta$ is
canceled out from all the field equations at the leading order of the gradient
expansion.

$\displaystyle\frac{\sqrt{g}{\cal
F}_{IJ}}{N}\left\\{ie^{(I)}(\phi^{I}\dot{\phi}^{J*}-{\rm
c.c.})+(I\leftrightarrow J)\right\\}={\cal O}(\epsilon)\,.$ (2.29)

This is nothing but the charge neutrality on large scales. In the absence of
charged scalar fields, since $J^{0}$ vanishes, the gauge constraint ${\cal
H}_{\rm U(1)}$ becomes ${\cal G}_{\epsilon}$.

Similarly to the gauge constraints, the gauge conditions also can vanish in
the limit $\epsilon\to 0$. We express the whole set of gauge conditions as
${\cal X}$ and those which vanish in the limit $\epsilon\to 0$ as ${\cal
X}_{\epsilon}$. A typical example of ${\cal X}_{\epsilon}$ is the transverse
condition, $\partial_{i}\gamma^{ij}=0$. Meanwhile, our gauge condition
$N_{i}=0$ is not ${\cal X}_{\epsilon}$. When we remove gauge degrees of
freedom by imposing ${\cal X}_{\epsilon}$, the $[$locality$]~{}$condition can
fail to hold due to non-locality caused by employing ${\cal X}_{\epsilon}$.

####  2.4.2 Proof of separate universe evolution

Let us classify all the field equations which remain after solving the non-
gauge constraints, ${\cal E}_{\rm all}$, into the evolution equations ${\cal
E}_{\rm ev}$ and the gauge constraints ${\cal G}$, i.e., ${\cal E}_{\rm
all}={\cal E}_{\rm ev}\cup{\cal G}$. For our convenience, we introduce ${\cal
E}\equiv{\cal E}_{\rm ev}\cup({\cal G}\cap{\cal G}_{\epsilon}^{c})$, where the
gauge constraints whose leading terms are suppressed by $\epsilon$, ${\cal
G}_{\epsilon}$, are excluded. Here, the superscript $c$ denotes a
complementary set. The $[$locality$]~{}$condition ensures that solving the
non-gauge constraints ${\cal C}\cap{\cal G}^{c}$ does not give rise to non-
local terms in the action. In addition, solving the gauge constraints whose
leading terms are not suppressed by $\epsilon$, i.e., ${\cal G}\cap{\cal
G}^{c}_{\epsilon}$ does not yield non-local contributions, either. Therefore,
the solution of ${\cal E}$ remains local, verifying the basic condition of the
separate universe $(\star)$ and allowing us to systematically solve equations
${\cal E}$ expanded by $\epsilon$. As discussed in Sec. 2.4.1, we solve ${\cal
E}$, choosing the gauge conditions which are not ${\cal X}_{\epsilon}$.

We express the set of the fields which remain after solving ${\cal C}\cap{\cal
G}^{c}$ and employing the gauge conditions which are not ${\cal X}_{\epsilon}$
as $\\{\varphi^{a}\\}$. This set of the fields, $\\{\varphi^{a}\\}$, includes
redundant variables, since the constraints ${\cal G}$ are not yet solved. In
addition, the gauge conditions ${\cal X}\cap{\cal X}^{c}_{\epsilon}$ may still
leave residual gauge degrees of freedom. Distinguishing it from
$\\{\varphi^{a}\\}$, we denote the variables which are necessary and
sufficient to specify a physical initial configuration by $\\{\varphi^{a_{\rm
phys}}\\}_{\rm phys}$ with the indices $a_{\rm phys}$.

Obviously, the solution space of ${\cal E}$ becomes broader than the physical
solution space of the system, since ${\cal G}_{\epsilon}$ is not yet solved.
To address this aspect, let us emphasize the following two properties of
${\cal G}_{\epsilon}$. First, solving only ${\cal E}$ without ${\cal
G}_{\epsilon}$ completely determines the time evolution of $\\{\varphi^{a}\\}$
for a given initial condition. Second, when ${\cal G}_{\epsilon}$ hold at a
certain time, say at the initial time, they continue to hold at any time. For
example, under the gauge transformation,

$A_{\mu}\to A_{\mu}+\partial_{\mu}\theta\,,$ (2.30)

and spatial Diff transformation (2.13), the variation of the action whose
Lagrangian density is given by Eq. (2.20) can be evaluated as

$\displaystyle 0$
$\displaystyle\,=\left(\delta_{\theta}+\delta_{\xi}\right)S_{\rm matter}$
(2.31) $\displaystyle\,=\int
d^{d+1}x\,\left[-\theta\partial_{t}\left(\sqrt{g}{\cal H}_{\rm
U(1)}\right)+\xi^{i}\partial_{t}\left(\sqrt{g}{\cal H}_{\rm
U(1)}A_{i}\right)+\xi^{i}\partial_{t}\left(\sqrt{g}{\cal H}_{i}\right)+{\cal
O}(\epsilon^{2})\right]\,,$ (2.32)

where we have used $\delta_{\xi}A_{0}=-\dot{\xi}^{i}A_{i}+{\cal O}(\epsilon)$,
and we have abbreviated the boundary terms and the terms which vanish by using
the evolution equations ${\cal E}_{\rm ev}$. From this expression, we find

$\displaystyle{\partial\over\partial t}\left(\sqrt{g}\,{\cal H}_{\rm
U(1)}A_{i}+\sqrt{g}{\cal H}_{i}\right)=0\,,$ (2.33)
$\displaystyle{\partial\sqrt{g}\,{\cal H}_{\rm U(1)}\over\partial t}=0\,,$
(2.34)

holds when all the evolution equations ${\cal E}_{\rm ev}$ are satisfied, at
the leading order of the expansion with respect to $\epsilon$. In particular,
to show the conservation of U(1) gauge constraint (2.34), we only need to use
the evolution equations for the matter fields that have non-zero U(1) charges,
for which $\delta_{\theta}$ does not vanish. When ${\cal H}_{\rm U(1)}$ holds
at an initial time, it continues to hold at an arbitrary time, i.e., being
conserved in time. Meanwhile, only the linear combination of ${\cal H}_{\rm
U(1)}$ and ${\cal H}_{i}$ is conserved. This was not explicit in Eq. (2.16),
since all the matter field equations, including ${\cal H}_{\rm U(1)}$, was
used to derive it. For $\dot{A}_{i}\neq 0$, only when both ${\cal H}_{i}$ and
${\cal H}_{\rm U(1)}$ hold at an initial time, ${\cal H}_{i}$ continues to
hold at an arbitrary time.

Therefore, we can reduce the solution space of $\\{\varphi^{a}\\}$ to the one
of $\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$ merely by choosing a proper
initial condition that satisfies ${\cal G}$ (and also by removing residual
gauge degrees of freedom). The right hand side of Eq. (2.5) follows a local
time evolution determined by solving the equations ${\cal E}$ in the limit
$\epsilon\to 0$ at each Hubble patch for a given initial condition
$\\{\varphi_{*}^{a^{\prime}}\\}^{\prime}$. We can specify the initial
condition of the separate universe by using
$\\{\varphi^{a^{\prime}}\\}^{\prime}$ which consist of the remaining fields of
$\\{\varphi^{a}\\}$ after solving the gauge constraints which do not vanish in
the limit $\epsilon\to 0$, i.e., ${\cal G}\cap{\cal G}^{c}_{\epsilon}$ such as
the Hamiltonian constraint for a $(d+1)$-dim Diff invariant theory.

In summary, at the leading order of the gradient expansion, using Eq. (2.5),
correlation functions of $\varphi^{a}$ can be computed as follows

$\displaystyle\langle\varphi^{a_{1}}(t_{1},\,\hbox{\boldmath{$x$}}_{1})\cdots\varphi^{a_{n}}(t_{n},\,\hbox{\boldmath{$x$}}_{n})\rangle$
$\displaystyle=\int\prod_{i=1}^{n}D\bar{\varphi}^{a^{\prime}_{i}}_{*}\,P(\bar{\varphi}^{a^{\prime}_{1}}_{*},\,\cdots,\,\bar{\varphi}^{a^{\prime}_{n}}_{*})\big{|}_{\bar{\varphi}^{a^{\prime}}_{*}=\varphi^{a^{\prime}}(t_{*},\,\hbox{\boldmath{\scriptsize$x$}})}$
$\displaystyle\qquad\qquad\times\bar{\varphi}^{a_{1}}(t_{1};\\{\bar{\varphi}^{a^{\prime}_{1}}_{*}\\}^{\prime})\cdots\bar{\varphi}^{a_{n}}(t_{n};\\{\bar{\varphi}^{a^{\prime}_{n}}_{*}\\}^{\prime})\,.$
(2.35)

The probability distribution of the initial condition,
$P(\bar{\varphi}^{a^{\prime}_{1}}_{*},\,\cdots,\,\bar{\varphi}^{a^{\prime}_{n}}_{*})$,
should be determined by solving all the equations ${\cal E}_{\rm all}$, which
also include ${\cal G}_{\epsilon}$, until $t_{*}$. As a result, in general,
$P(\bar{\varphi}^{a^{\prime}_{1}}_{*},\,\cdots,\,\bar{\varphi}^{a^{\prime}_{n}}_{*})$
has correlations among different patches. Meanwhile, as we have shown, the
quantities in the second line of Eq. (2.4.2) can be determined by solving
${\cal E}$, the local field equations for $\\{\varphi^{a}\\}$, correspondingly
to the independent evolution of each local patch in the limit $\epsilon\to 0$.
This verifies the separate universe evolution $(\star)$. The separate universe
evolution $(\star)$ only ensures the locality of its evolution for a given
initial condition, but not the locality of the initial distribution. This
point becomes important in discussing the existence of WAM, the second arrow
in Fig. 1. At higher orders of the gradient expansion, an influence from other
patches can be taken into account order by order in the expansion with respect
to $\epsilon$.

We have introduced three different sets of the fields, whose mutual relation
can be summarized as follows,

$\displaystyle\\{\varphi^{a}\\}:\mbox{All metric and matter fields after
solving}~{}{\cal C}\cap{\cal G}^{c}~{}\&~{}\mbox{imposing gauge conditions}$
$\displaystyle\xrightarrow[\mbox{solving}~{}({\cal G}\cap{{\cal
G}}_{\epsilon}^{c})]{}~{}~{}~{}\\{\varphi^{a^{\prime}}\\}^{\prime}:\mbox{variables
that specify initial conditions of separate universe}~{}~{}~{}$
$\displaystyle\xrightarrow[\mbox{solving}~{}{\cal G}_{\epsilon}~{}\mbox{and
removing residual gauge DOFs}]{}~{}~{}~{}\\{\varphi^{a_{\rm phys}}\\}_{\rm
phys}\,.$

If there still exist residual gauge degrees of freedom after imposing gauge
conditions that are not ${\cal X}_{\epsilon}$, we also need to remove them to
obtain the set of the physical degrees of freedom. As discussed around Eq.
(2.25), to remove the residual gauge degrees of freedom which remain after
employing $N_{i}=0$, one may want to impose the transverse condition at
$t=t_{*}$ as

$\displaystyle\partial_{i}\gamma^{ij}(t_{*},\,\hbox{\boldmath{$x$}})=0\,,$
(2.36)

which is ${\cal X}_{\epsilon}$. Unlike the gauge condition
$\partial_{i}\gamma^{ij}(t,\,\hbox{\boldmath{$x$}})=0$ for an arbitrary $t$,
whether we employ Eq. (2.36) or not is irrelevant in computing the evolution
of each patch for a given initial condition, i.e., in obtaining the quantities
in the second line of Eq. (2.4.2). It only affects the initial probability
distribution, the first line. Alternatively, one may want to employ the axial
gauge conditions, $\gamma_{iz}=0$ at $t=t_{*}$, avoiding ${\cal
X}_{\epsilon}$. The condition $(\star)$ holds for both gauge conditions.

####  2.4.3 Previous works

Before closing this section, let us summarize the previous works, highlighting
the difference from our argument. In the conventional gradient expansion, the
asymptotic solution in the limit $\epsilon\to 0$ is restricted by imposing the
additional conditions (see, e.g., Ref. [5]),

$\displaystyle[{\rm removed~{}assumptions}]:\qquad N_{i}={\cal
O}(\epsilon)\,,\qquad\dot{\gamma}_{ij}={\cal O}(\epsilon^{2})\,.$ (2.37)

Since $N_{i}$ can be set to 0 by performing the gauge transformation (even at
the linear order in perturbation), the first condition is just a choice of the
gauge. By contrast, it is obvious that the second condition does not hold when
we consider general models of gravity, e.g., the massive gravity [72, 73], in
which the graviton is gapped, or in the presence of non-zero spin fields which
can source the shear on a large scale. Our argument, which is solely based on
the $[$sDiff$]~{}$and $[$locality$]~{}$conditions, does not restrict the
asymptotic behaviour in the limit $\epsilon\ll 1$. In particular, the
condition (2.37) requires that the shear (${A^{i}}_{j}$, whose definition is
given in the next section) should be suppressed by $\epsilon$. However, as
will be shown, the decaying solution of $\zeta$, which is also known as the
Weinberg’s second mode, cannot be properly reproduced without keeping the
shear at the leading order of the gradient expansion.

The gradient expansion has been applied only to a scalar field system, with a
few exceptions [44, 74], which addressed a system with a U(1) gauge field and
inflaton as matter components. In these papers, several non-trivial conditions
such as $\gamma_{ij}={\cal O}(\epsilon)$ for $i\neq j$, which restrict large
scale anisotropy, were employed. Our argument shows that the separate universe
$(\star)$ and the $\delta N$ formalism can apply to general matter contents,
as long as the $[$sDiff$]~{}$and $[$locality$]~{}$conditions are fulfilled,
including models with non-zero spin fields, such as anisotropic inflation
models [20, 21, 22, 23]. This generalization becomes possible by solving the
evolution for the set of fields $\\{\varphi^{a}\\}$ not for the independent
degrees of freedom $\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$. This enables us
to confine the potential non-locality which emerges by solving ${\cal
G}_{\epsilon}$ only in the choice of the initial condition, ensuring the
locality of the evolution for a given initial condition.

Specifically for a scalar field system, where the momentum constraints ${\cal
H}_{i}$ are the only ${\cal G}_{\epsilon}$ in the system, it is not very
important whether we solve the evolution for $\\{\varphi^{a_{\rm
phys}}\\}_{\rm phys}$ or $\\{\varphi^{a}\\}$. Sugiyama et al. showed that,
when the other equations of motion than ${\cal H}_{i}$ are fulfilled, the
violation of the momentum constraints, even if it exists, decays with the
inverse of the physical spatial volume under the slow-roll approximation [45].
This result was extended to a more general time evolution by removing the
slow-roll approximation in Ref. [46] (see also Ref. [14]). A lesson from Refs.
[45, 46, 14] is that as far as all the other equations are properly solved,
the momentum constraints are also satisfied, leaving aside the error that
decays in an expanding universe as $1/\sqrt{g}$, i.e., $\sqrt{g}{\cal
H}_{i}=\,$const.

In Refs. [45, 46], the approximate validity of the momentum constraints was
shown based on a brute force computation in a specific model. This can be also
shown generically by using $[$sDiff$]~{}$condition as follows333In a $d+1$
Diff invariant theory, the Bianchi identity and the conservation law of the
energy momentum tensor relate the divergence of the momentum constraints to
the time derivative of the Hamiltonian constraint. Therefore, one projected
component of the momentum constraints is automatically satisfied (without the
error) as long as the Hamiltonian constraint is satisfied. This single
relation is sufficient for a system composed of scalar fields at linear order,
but obviously not so when one wants to consider non-linear orders in
perturbation or a model with a non-zero spin field, e.g., a vector field,
which does not immediately decay at large scales. . Evaluating the variation
of the action under the spatial coordinate transformation, we obtained Eq.
(2.14). From this relation, we derived Eq. (2.16), using the equations
$\\{{\cal E}\\}$. Namely, when the momentum constraints are the only ${\cal
G}_{\epsilon}$ in the system, as long as the equations ${\cal E}$ are solved
properly, Eq. (2.16) holds. This implies that the momentum constraints ${\cal
H}_{i}$ are automatically satisfied, if we accept the "error" which decays as
$\propto 1/\sqrt{g}$ in an expanding universe. In fact, as will be discussed
in the next section, the non-local components that appear in the shear by
solving ${\cal H}_{i}$ decay as $\propto 1/\sqrt{g}$, which implies
exponential decay, especially, during inflation. This fact becomes important
when we consider the evolution of the physical degrees of freedom
$\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$, which is determined by solving all
the field equations in the system including ${\cal G}_{\epsilon}$. Meanwhile,
if there is another ${\cal G}_{\epsilon}$, e.g., for a system with neutral
scalar fields and U(1) gauge field, we need to solve the evolution for
$\\{\varphi^{a}\\}$ (see Sec. 4).

##  3 Weinberg’s adiabatic mode

The approximate validity of ${\cal G}_{\epsilon}$, discussed at the end of the
previous section, is closely related to the existence of WAM444In cosmology,
the matter content is called adiabatic (even without referring to their
thermodynamic property), when the pressure $P$ is non-perturbatively given by
a function of the energy density $\rho$ without depending on any other
quantities. Moving to perturbation theory this leads to
$\displaystyle\frac{\delta
P}{\dot{\bar{P}}}=\frac{\delta\rho}{\dot{\bar{\rho}}}\,,$ (3.1) where the dot
denotes the time derivative with respect to the cosmological time. For a
scalar field system, this condition should be imposed only in the long
wavelength limit. Here, $\delta P$ and $\delta\rho$ denote the perturbations
and $\bar{P}$ and $\bar{\rho}$ denote their background values. A solution
which satisfies the adiabatic condition is called the adiabatic mode. The
adiabatic mode can still exist, even when the non-adiabatic mode also exists.
Equation (3.1) states that the time step measured by the change of the
pressure coincides with the one measured by the change of the energy density,
i.e., there is only one clock. If there is only a scalar field, this can be
rephrased as being on an attractor solution. As is discussed in this section,
Weinberg proposed one way to single out the adiabatic mode by extending the
change under the dilatation into soft modes $\hbox{\boldmath{$k$}}\neq 0$
[49]. The WAM can be absorbed by a coordinate transformation in a local patch
[75]. However, this property is not exclusive of the WAM. Following the
convention, we call a mode which is locally identical to a coordinate
transformation an adiabatic mode, even if Eq. (3.1) is not satisfied. Other
types of adiabatic modes were reported in Refs. [76, 77].. In this section, we
show that when a system under consideration satisfies $[$locality$]~{}$and
$[$sDiff$]~{}$conditions and also a condition $[$approx${\cal
G}_{\epsilon}$$]~{}$, there approximately exists the WAM as a solution beyond
the horizon scale. Here, we define

  * •

$[$approx${\cal G}_{\epsilon}$$]~{}$: When the equations for
$\\{\varphi^{a}\\}$, ${\cal E}$, are satisfied, the gauge constraints that
trivially vanish in the limit $\epsilon\to 0$, ${\cal G}_{\epsilon}$, are
approximately satisfied at the leading order of $\epsilon$.

As discussed in the previous section, a system which only contains scalar
fields satisfies the $[$approx${\cal G}_{\epsilon}$$]~{}$condition [45, 46].
By contrast, a model with a U(1) gauge field requires a more careful
consideration, particularly when the kinetic term factor ${\cal F}_{Y}$
rapidly decreases in time, making the approximate validity of ${\cal H}_{\rm
U(1)}$ non-trivial (see Eq. (2.27)).

As discussed in Sec. 2.1, when the Fourier mode expansion is possible, the
regime with $\epsilon\ll 1$ directly corresponds to considering the
superhorizon limit. In this section, in the regime with $\epsilon\ll 1$ we
show the approximate existence of the Weinberg’s adiabatic mode (WAM), the
constant solution of $\psi$, under the three conditions. Unlike the argument
in Ref. [5], which relies on the energy conservation, our argument also
applies to the case in which the $(d+1)$-dim Diff invariance is broken down to
the $d$ dim spatial Diff invariance. For the non-projectable HL gravity, in
Refs. [59, 60], the existence of the constant solution was manifestly shown in
linear perturbation. (See Refs. [78, 47, 48] about the existence of the
constant solution in the projectable HL gravity).

The existence of the constant solution should be clearly distinguished from
the conservation in time. Even if the constant solution of $\psi$, the WAM,
exists, if it is dominated by other modes, $\psi$ is not conserved.

###  3.1 WAM and [approx${\cal G}_{\epsilon}$] condition

When all the equations to be solved remain invariant under the shift of
$\psi$,

$\displaystyle\psi(t,\,\hbox{\boldmath{$x$}})\to\psi(t,\,\hbox{\boldmath{$x$}})+c({\hbox{\boldmath{$x$}}})\,,$
(3.2)

where $c({\hbox{\boldmath{$x$}}})$ satisfies

$c_{\rm
max}|\partial_{i}c(\hbox{\boldmath{$x$}})|/(e^{\psi}K)\ll|c(\hbox{\boldmath{$x$}})|\,,$

there exists a time independent solution of $\psi$ at the leading order of the
$\epsilon$ expansion. In the Weinberg’s argument [49], the invariance under
Eq. (3.2) is ensured by extending the transformation of $\psi$ under the
dilatation, $\hbox{\boldmath{$x$}}\to\hbox{\boldmath{$x$}}^{c}\equiv
e^{-c}\hbox{\boldmath{$x$}}$, which is one of the large gauge
transformations555In cosmological perturbation theory, we usually consider
only the perturbed variables around an FLRW spacetime which can be expanded in
terms of the Fourier modes with finite wavenumbers. Notice that the change due
to the dilatation does not satisfy this condition. In fact, the Fourier
transformation of $\delta x^{i}\propto cx^{i}$ reads
$\partial\delta(\hbox{\boldmath{$k$}})/(\partial\hbox{\boldmath{$k$}})$, which
includes the delta function for the $\hbox{\boldmath{$k$}}=0$ mode. Therefore,
the dilatation is out of consideration in the conventional cosmological
perturbation theory..

Under the dilatation $\hbox{\boldmath{$x$}}\to\hbox{\boldmath{$x$}}_{c}\equiv
e^{-c}\hbox{\boldmath{$x$}}$, the spatial metric $g_{ij}$ transforms as

$\displaystyle
g^{c}_{ij}(t,\,\hbox{\boldmath{$x$}}_{c})=e^{2c}g_{ij}(t,\,\hbox{\boldmath{$x$}})\,,$
(3.3)

and $\psi(x)\equiv\ln g(x)/(2d)$ as

$\displaystyle\psi^{c}(t,\,\hbox{\boldmath{$x$}}_{c})=\psi(t,\,\hbox{\boldmath{$x$}})+c\,.$
(3.4)

If only $\psi$ changes under the dilatation at ${\cal O}(\epsilon^{0})$, the
dilatation invariance implies the invariance of the action under the
introduction of the additive constant shift of $\psi$ as
$\psi_{\hbox{\boldmath{\scriptsize$k$}}=0}\to\psi_{\hbox{\boldmath{\scriptsize$k$}}=0}+c+\cdots$.
Since this is a (large) spatial gauge transformation, with the
$[$sDiff$]~{}$condition, the system remains invariant without leaving any
change in the physical configuration. Meanwhile, let us extend this dilatation
transformation to the one with a time-independent but inhomogeneous parameter
$c(\hbox{\boldmath{$x$}})$, which has comoving wave number
$\hbox{\boldmath{$k$}}_{L}\neq 0$. Here, the change of the $k=0$ mode under
the dilatation with the constant parameter $c$ is extended to that of the soft
mode $k_{L}\neq 0$, just by replacing the parameter $c$ with an inhomogeneous
one $c_{\hbox{\boldmath{\scriptsize$k$}}_{L}}$. Namely,

$\psi_{\hbox{\boldmath{\scriptsize$k$}}_{L}}\to\psi_{\hbox{\boldmath{\scriptsize$k$}}_{L}}+c_{\hbox{\boldmath{\scriptsize$k$}}_{L}}+{\cal
O}(\epsilon)\,.$

Since this extended transformation is no longer a gauge transformation, it
alters the physical configuration. The continuity at $\hbox{\boldmath{$k$}}=0$
ensures that the additive shift induced by this inhomogeneous dilatation still
solves the field equations at the leading order of $\epsilon$, although the
solution at higher orders is modified. This solution is called the WAM. In
Ref. [79], this argument was extended to a non-linear quantum theory.

Since the $[$locality$]~{}$condition ensures that the equations ${\cal E}$ are
local, ${\cal E}$ are all continuous at $k=0$. Therefore, the WAM should be a
solution of ${\cal E}$. However, this does not immediately ensure the
continuity of physical solutions at $k=0$. In fact, the configuration after
the transformation does not satisfy ${\cal G}_{\epsilon}$ in general at the
leading order of $\epsilon$. As a result, computing the physical solutions
$\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$ by solving ${\cal G}_{\epsilon}$ can
give rise to a singular pole at $k=0$. When the $[$approx${\cal
G}_{\epsilon}$$]~{}$condition is satisfied in addition to the
$[$locality$]~{}$and $[$sDiff$]~{}$conditions, all the solutions of ${\cal E}$
should approximately satisfy ${\cal G}_{\epsilon}$ at the leading order of the
$\epsilon$ expansion. Therefore, the WAM approximately solves the whole set of
equations, ${\cal E}_{\rm all}$ in the limit $\epsilon\to 0$.

###  3.2 Momentum constraints

In the rest of this section, we address when the $[$approx${\cal
G}_{\epsilon}$$]~{}$condition holds, considering models that satisfy the
$[$locality$]~{}$and $[$sDiff$]~{}$conditions. Let us express the total action
as

$\displaystyle S=\int d^{d+1}xN\sqrt{g}{\cal L}=\int
d^{d+1}xN\sqrt{g}\left[{\cal L}_{\rm g}+{\cal L}_{\rm matter}\right]\,,$ (3.5)

where ${\cal L}_{\rm g}$ is the Lagrangian density for the gravitational field
and ${\cal L}_{\rm matter}$ denotes the matter Lagrangian density. The time
derivative of the spatial metric, $g_{ij}$ which remains invariant under the
spatial coordinate transformation
$x^{i}\to\tilde{x}^{i}(t,\hbox{\boldmath{$x$}})$ is given by the extrinsic
curvature and its trace part, defined as

$\displaystyle
K_{ij}=\frac{1}{2N}(\dot{g}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i})\,,\qquad
K\equiv g^{ij}K_{ij}.$ (3.6)

Using the extrinsic curvature, the kinetic term of gravity is given by
666Since the shift vector $N_{i}$ is related to the $(d+1)$-dim metric
component, $g_{0i}$, as $g_{0i}=N_{i}$, it does not transform as a vector
field under $x^{i}\to\tilde{x}^{i}(t,\hbox{\boldmath{$x$}})$ (while it does
under $x^{i}\to\tilde{x}^{i}(\hbox{\boldmath{$x$}})$). Therefore, a
contraction of $N_{i}$, $N^{i}N_{i}$, is not invariant under the time
dependent spatial coordinate transformation.

$\displaystyle{\cal L}_{\rm g}$ $\displaystyle=\frac{F}{16\pi
G}\biggl{[}K^{i}\\!_{j}K^{j}\\!_{i}-\lambda_{\rm
HL}K^{2}+\lambda_{1}K^{3}-\lambda_{2}KK^{i}\\!_{j}K^{j}\\!_{i}+\lambda_{3}K^{i}\\!_{j}K^{j}\\!_{l}K^{l}\\!_{i}+\cdots+{\cal
O}(\epsilon)\biggr{]}\,,$ (3.7)

where the coefficients $F$, $\lambda_{\rm HL}$, $\lambda_{i}$ with
$i=1,\,2,\,3$ are arbitrary functions of the matter fields777The action (3.7)
only includes the terms which preserve the time reparametrization invariance,
i.e., $t\to\tilde{t}(t)$. However, as argued in Ref. [56], without this
invariance, more terms can appear. For instance, the coefficients can depend
on the lapse function $N$.. Potential terms of gravity such as the spatial
Ricci scalar ${{}^{s}R}$ are ${\cal O}(\epsilon^{2})$, since they have at
least two more spatial gradient compared to the terms shown in Eq. (3.7). With
the $[$locality$]~{}$condition, the non-local contributions are absent in the
above expression.

For an illustrative purpose, we have written down several possible terms of
${\cal L}_{\rm g}$ in Eq. (3.7). However, as far as the $[$sDiff$]~{}$and
$[$locality$]~{}$conditions hold, the explicit form is not important. The
above Lagrangian density includes general relativity $(\lambda_{\rm
HL}=1,\,\lambda_{1}=\lambda_{2}=\lambda_{3}=0)$, the projectable and non-
projectable versions of the HL gravity888In the HL gravity, ${\cal L}_{\rm g}$
can include the spatial Riemann tensor, the spatial Ricci tensor, and the
spatial Ricci scalar with the spatial covariant derivative operators (see,
e.g., [60]). In the non-projectable version of the HL gravity, the gradient
instability in the IR can be avoided by introducing a term proportional to
$g^{ij}\partial_{i}\ln N\partial_{j}\ln N$ [80]. This term is also included in
${\cal O}(\epsilon^{2})$. ($\lambda_{1}=\lambda_{2}=\lambda_{3}=0$), and
beyond Horndeski ($\lambda_{\rm
HL}=1,\lambda_{2}=3\lambda_{1},\,\lambda_{3}=2\lambda_{1}$ for $d=3$) [81].

Taking the functional derivative of the total action with respect to the
Lagrange multipliers $N_{i}$, we obtain the momentum constraints as

$\displaystyle\nabla_{j}\left[F\left\\{(1-\lambda_{2}K)K^{j}\\!_{i}+\frac{3}{2}\lambda_{3}K^{j}\\!_{l}K^{l}\\!_{i}-\delta^{j}\\!_{i}\left(\lambda_{\rm
HL}K-\frac{3}{2}\lambda_{1}K^{2}+\frac{1}{2}\lambda_{2}K^{i}\\!_{j}K^{j}\\!_{i}\right)\right\\}\right]$
(3.8) $\displaystyle\qquad\qquad\qquad\qquad+\cdots+8\pi GN\frac{\partial{\cal
L}_{\rm matter}}{\partial N^{i}}={\cal O}\left(\epsilon^{2}\right)\,.$ (3.9)

Here, the terms expressed by $\cdots$ denote contributions that potentially
appear from other kinetic terms of gravity, e.g., $K^{4}$ and
${K^{i}}_{j}{K^{j}}_{l}{K^{l}}_{m}{K^{m}}_{i}$. As discussed in the previous
section, not only the geometrical contributions, the matter contribution in
the second line should be also suppressed by ${\cal O}(\epsilon)$.

Following Refs. [3, 2, 82], let us introduce the tracelsss part of $K_{ij}$,
the shear $A_{ij}$, as

$\displaystyle K_{ij}=\frac{K}{d}g_{ij}+A_{ij}\,,\qquad g^{ij}A_{ij}=0\,.$
(3.10)

The tensor indices of $A_{ij}$ are raised and lowered by $g_{ij}$. Using the
expansion $K$ and the shear ${A^{i}}_{j}$, the momentum constraints are
rewritten as

$\displaystyle\partial_{i}\left[F\left\\{\left(\frac{1}{d}-\lambda_{\rm
HL}\right)K+\frac{3}{2}\left(\lambda_{1}-\frac{\lambda_{2}}{d}+\frac{\lambda_{3}}{d^{2}}\right)K^{2}-\frac{\lambda_{2}}{2}{A^{k}}_{l}{A^{l}}_{k}\right\\}\right]$
(3.11)
$\displaystyle\\!+\\!\nabla_{j}\\!\left[F\\!\left\\{\\!{A^{j}}_{i}+\\!\left(\frac{3\lambda_{3}}{d}-\lambda_{2}\right)\\!K{A^{j}}_{i}+\frac{3\lambda_{3}}{2}A^{j}\\!_{l}A^{l}\\!_{i}\right\\}\\!\right]\\!+\cdots+8\pi
GN\frac{\partial{\cal L}_{\rm matter}}{\partial N^{i}}\\!={\cal
O}\left(\epsilon^{2}\right)\\!.\,\,\,\,\,$ (3.12)

In $N_{i}=0$ gauge, $K$ and ${A^{i}}_{j}$ are given by

$\displaystyle K=\frac{d}{N}\dot{\psi}\,,$ (3.13)
$\displaystyle{A^{i}}_{j}=\frac{1}{2N}\left(\delta^{i}_{k}\delta^{l}_{j}-\frac{1}{d}\delta^{i}_{j}\delta^{l}_{k}\right)\gamma^{km}\dot{\gamma}_{ml}=\frac{1}{2N}\gamma^{im}\dot{\gamma}_{mj}=-\frac{1}{2N}\dot{\gamma}^{im}\gamma_{mj}\,.$
(3.14)

In this gauge, $K$ and ${A^{i}}_{j}$ directly correspond to the derivative of
$\psi$ and $\gamma_{ij}$ with respect to the proper time $\tau$, which is
related to the cosmological time as $d\tau|_{N_{i}=0}=Ndt$. In this gauge, by
integrating Eq. (3.13), $\psi$ directly corresponds to the $e$-folding number
as

$\displaystyle\psi(t,\,\hbox{\boldmath{$x$}})=\frac{1}{d}\int^{\tau(t)}d\tau^{\prime}K(\tau^{\prime},\,\hbox{\boldmath{$x$}})=\frac{1}{d}\int^{t}dt^{\prime}N(t^{\prime},\,\hbox{\boldmath{$x$}})K(t^{\prime},\,\hbox{\boldmath{$x$}})\,.$
(3.15)

This is the basic equation in the $\delta N$ formalism, which will be
discussed in Sec. 4.

###  3.3 Evolution of shear on large scales

In the next subsection, we point out that the shear, ${A^{i}}_{j}$, can be
schematically solved as

$\displaystyle
F{A^{i}}_{j}(t,\,\hbox{\boldmath{$x$}})\sim\frac{{D^{i}}_{j}(\hbox{\boldmath{$x$}})}{\sqrt{g}}+({\rm
contributions~{}sourced~{}by~{}matter~{}fields})^{i}\\!_{j}\,,$ (3.16)

where ${D^{i}}_{j}(\hbox{\boldmath{$x$}})$ is time-independent. In the absence
of the source, Eq. (3.16) simply states that the shear decays in an expanding
universe. This property of the shear has been confirmed in many examples [3],
but it is not (immediately) clear what ensures it. In the next subsection, we
will show that the precise version of Eq. (3.16) can be derived, making use of
the Noether charges of the large gauge transformations.

The first term of Eq. (3.16), ${D^{i}}_{j}(\hbox{\boldmath{$x$}})/\sqrt{g}$,
corresponds to the geometrical degrees of freedom, whose evolution can be
determined without referring to the detail of matter fields. Inserting this
expression into the momentum constraints (3.12), we can determine
${D^{i}}_{j}(\hbox{\boldmath{$x$}})$ so that the momentum constraints are
properly solved at a certain time, e.g., at the horizon crossing time of the
scales of interest, $t=t_{\rm hoc}$. The symmetric and traceless tensor
${D^{i}}_{j}(\hbox{\boldmath{$x$}})$ has $(d/2+1)(d-1)$ degrees of freedom,
among which $d$ components are determined by solving the momentum constraints
and the remaining $(d/2-1)(d+1)$ components correspond to the decaying modes
of gravitational waves. Once ${D^{i}}_{j}(\hbox{\boldmath{$x$}})$ is chosen to
satisfy the momentum constraints at one time, as discussed in the previous
section, we can verify that the momentum constraints hold at any time. As is
clear from Eq. (3.12), the resultant expression of
${D^{i}}_{j}(\hbox{\boldmath{$x$}})$ in terms of matter fields becomes non-
local999If ${\cal H}_{i}=0$ were to be written in the longitudinal type form
$\partial_{i}(\cdots)=0$, they can be solved without introducing non-local
integral expression. At linear perturbation around the FLRW spacetime, whose
spatial metric is maximally symmetric, the momentum constraints for the scalar
type perturbation can be written in the form, $\partial_{i}(\cdots)=0$ [83]
(see the momentum constrains derived in Sec. 4.3). Nevertheless, in general,
the transverse part of ${\cal H}_{i}=0$ does not vanish.. However, since the
non-local terms are multiplied by the inverse of the physical volume,
$1/\sqrt{g}$, which decays in an expanding universe, they die off after a
while. After that, whether the momentum constraints are solved or not becomes
insignificant. In particular, if the momentum constraints are the only ${\cal
G}_{\epsilon}$ in the system, the $[$approx${\cal
G}_{\epsilon}$$]~{}$condition holds, since all the solutions of ${\cal E}$
satisfy ${\cal H}_{i}$ approximately, just allowing the errors which decay
with $1/\sqrt{g}$. In this approximation, solving the momentum constraints is
equivalent to solving the manifestly local equations obtained by setting
${D^{i}}_{j}(\hbox{\boldmath{$x$}})=0$ in Eq. (3.16).

Before deriving the precise version, taking a detour, let us derive Eq. (3.16)
in a specific example also by solving the field equations straightforwardly.
As an example, we consider a theory of gravity whose action is given by

$\displaystyle({\rm ex})\qquad\quad{\cal L}_{\rm g}$
$\displaystyle=\frac{F}{16\pi G}\biggl{[}K^{i}\\!_{j}K^{j}\\!_{i}-\lambda_{\rm
HL}K^{2}+{\cal O}(\epsilon)\biggr{]}\,,$ (3.17)

which includes, e.g., general relativity and the Horava-Lifshitz gravity.
Taking the derivative of the action with respect to the spatial metric,
$g_{ij}$, we obtain 101010With our definition of the extrinsic curvature
$K_{ij}$, $K$ becomes positive for an expanding universe. In Ref. [2], the
definition of $K_{ij}$ has the opposite signature, which has made Eq. (3.19)
slightly different from theirs.

$\displaystyle\frac{1}{N\sqrt{g}F}\partial_{t}\left(\sqrt{g}FP^{ij}\right)+2(K^{il}{K^{j}}_{l}-\lambda_{\rm
HL}K^{ij}K)-\frac{1}{2}g^{ij}({K^{l}}_{m}{K^{m}}_{l}-\lambda_{\rm HL}K^{2})$
(3.18)
$\displaystyle\quad-\frac{1}{N}D_{k}(P^{ij}N^{k})+\frac{1}{N}D_{k}(P^{ik}N^{j})+\frac{1}{N}D_{k}(P^{kj}N^{i})+{\cal
O}(\epsilon^{2})=\frac{8\pi G}{F}T^{ij}\,,$ (3.19)

where $T^{ij}$ is (the spatial component of) the energy-momentum tensor and
$P^{ij}$ is given by

$\displaystyle P^{ij}\equiv K^{ij}-\lambda_{\rm
HL}Kg^{ij}=A^{ij}+(1-\lambda_{\rm HL}d)\frac{K}{d}g^{ij}\,.$ (3.20)

Using Eqs. (3.13) and (3.14), the traceless part of the field equation (3.19)
can be given by a compact expression as 111111The trace part of Eq. (3.19)
reads
$\displaystyle\frac{1}{N\sqrt{g}F}\partial_{t}(F\sqrt{g}K)-\frac{1}{2}K^{2}-\frac{d}{2(1-\lambda_{\rm
HL}d)}{A^{i}}_{j}{A^{j}}_{i}=\frac{d}{1-\lambda_{\rm HL}d}\frac{8\pi
G}{F}P+{\cal O}(\epsilon)\,,$ (3.21) where the isotropic pressure $P$ is
defined as $P\equiv{T^{i}}_{i}/d$. In deriving Eq. (3.21), we have eliminated
the time derivative of ${A^{i}}_{j}$ by using Eq. (3.23). Using the
Hamiltonian constraint, we can rewrite Eq. (3.21) as
$\displaystyle\frac{1}{NF}\partial_{t}(FK)-\frac{d}{1-\lambda_{\rm
HL}d}{A^{i}}_{j}{A^{j}}_{i}=\frac{d}{1-\lambda_{\rm HL}d}\frac{8\pi
G}{F}(\rho+P)+{\cal O}(\epsilon)\,.$ (3.22)

$\displaystyle\frac{1}{N\sqrt{g}}\partial_{t}\left(\sqrt{g}F{A^{i}}_{j}\right)=8\pi
G\,{\Pi^{i}}_{j}+{\cal O}(\epsilon)\,,$ (3.23)

where ${\Pi^{i}}_{j}$ is the anisotropic pressure, defined by

$\displaystyle\Pi^{ij}\equiv\left(\delta^{i}\\!_{k}\delta^{j}\\!_{l}-\frac{1}{d}g^{ij}g_{kl}\right)T^{kl}\,.$
(3.24)

The quadratic term of $A_{ij}$ does not appear in Eq.(3.22), when we express
the left hand side as the time derivative of ${A^{i}}_{j}$ instead of
$A_{ij}$. This is the fully non-perturbative equation for the leading order of
the gradient expansion. Integrating Eq. (3.23) over time from $t_{\rm hoc}$,
we obtain

$\displaystyle F{A^{i}}_{j}(t,\,\hbox{\boldmath{$x$}})=\frac{\sqrt{g_{\rm
hoc}(\hbox{\boldmath{$x$}})}}{\sqrt{g(t,\,\hbox{\boldmath{$x$}})}}\left[F_{\rm
hoc}{A^{i}}_{j\,{\rm hoc}}(\hbox{\boldmath{$x$}})+\int^{\tau(t)}_{\tau_{\rm
hoc}}d\tau^{\prime}\frac{\left(\sqrt{g}\,8\pi
G{\Pi^{i}}_{j}\right)_{(\tau^{\prime},\,\hbox{\boldmath{\scriptsize$x$}})}}{(\sqrt{g})_{(\tau_{\rm
hoc},\,\hbox{\boldmath{\scriptsize$x$}})}}+{\cal O}(\epsilon)\right],$ (3.25)

where the $\tau$ integral is performed for a given $x$. The variables with the
subscript “hoc” denote those evaluated at $t=t_{\rm hoc}$. The first term in
the right hand side generically decays in an expanding universe, being
inversely proportional to the physical spatial volume as $\propto
1/(\sqrt{g})$ [3]. Integrating Eq. (3.25) over time, we obtain

$\displaystyle\gamma_{ij}(t,\,\hbox{\boldmath{$x$}})=\gamma_{ij}(t_{\rm
hoc},\,\hbox{\boldmath{$x$}})+2\int^{\tau(t)}_{\tau_{\rm
hoc}}d\tau^{\prime}\gamma_{il}(\tau^{\prime},\,\hbox{\boldmath{$x$}})\frac{(F\sqrt{g})_{(\tau_{\rm
hoc},\,\hbox{\boldmath{\scriptsize$x$}})}}{(F\sqrt{g})_{(\tau^{\prime},\,\hbox{\boldmath{\scriptsize$x$}})}}$
(3.26)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\left[{A^{i}}_{j\,{\rm
hoc}}(\hbox{\boldmath{$x$}})+\int^{\tau^{\prime}}_{\tau_{\rm
hoc}}d\tau^{\prime\prime}\frac{\left(\sqrt{g}\,8\pi
G{\Pi^{i}}_{j}\right)_{(\tau^{\prime\prime},\,\hbox{\boldmath{\scriptsize$x$}})}}{(F\sqrt{g})_{(\tau_{\rm
hoc},\,\hbox{\boldmath{\scriptsize$x$}})}}\right]+{\cal O}(\epsilon)\,,$
(3.27)

which is a formal solution of $\gamma_{ij}$ at the leading order of the
gradient expansion for the theory of gravity, (3.17).

When the large scale anisotropic pressure is suppressed as

$\displaystyle|8\pi G\,{\Pi^{i}}_{j}|={\cal O}(\epsilon)\,,$ (3.29)

e.g., in a scalar field system, Eq. (3.25) immediately provides the key
expression (3.16) we need to show. By choosing
${D^{i}}_{j}(\hbox{\boldmath{$x$}})$ or equivalently $F_{\rm
hoc}{A^{i}}_{j\,{\rm hoc}}(\hbox{\boldmath{$x$}})$ properly, the momentum
constraints ${\cal H}_{i}$ can be satisfied at $t=t_{\rm hoc}$. Although
${D^{i}}_{j}(\hbox{\boldmath{$x$}})$ determined by solving ${\cal H}_{i}$
becomes non-local, it soon decays as $\propto 1/\sqrt{g}$ in an expanding
universe. Since the shear decays exponentially for general relativity in an
inflationary spacetime, Salopek and Bond set ${A^{i}}_{j}$ to 0 [1]. As was
pointed out in Ref. [9], while the shear ${A^{i}}_{j}$ decays in time, we need
to take into account the contribution of
${A^{i}}_{j}\propto\sqrt{g}_{*}/\sqrt{g}$ to reproduce the decaying solution
of the curvature perturbation $\zeta$, a.k.a., the Weinberg’s second mode, at
the leading order of the gradient expansion (see also Refs. [45, 46] and the
discussion in Sec. 3.6). Meanwhile, when the anisotropic pressure is not
suppressed by $\epsilon$ in the limit $\epsilon\ll 1$, the relation between
Eq. (3.16) and Eq. (3.25) may be less clear, since ${{\Pi}^{i}}_{j}$ also
depends on ${A^{i}}_{j\,{\rm hoc}}(\hbox{\boldmath{$x$}})$ through a matter
source, e.g., when their equations of motion depend on the shear.

###  3.4 Noether charge and fall-off of non-local contributions

In this subsection, we show the key expression, (3.16), using a set of the
Noether charge (densities) for the spatial large gauge transformations.

####  3.4.1 Noether charge of large gauge transformations

Gauge transformations, which are parametrized by functions of the spacetime,
are in general classified into two categories: small gauge transformations and
large gauge transformations. In Ref. [84], the former are defined as local
transformations with bounded support and the latter as local transformations
with support (as well) on the boundary. When the support of the theory extends
to the infinity, the change under the former falls off in the limit
$|x|\to\infty$, while the change under the latter does not.

Among the infinite number of large spatial gauge transformations [85, 86] (see
also Ref. [87]), we focus on a global transformation,

$\displaystyle x^{i}\to\tilde{x}^{i}=x^{i}+{M^{i}}_{j}\,x^{j}\,,$ (3.30)

where ${M^{i}}_{j}$ denotes an infinitesimal constant rank $(1,\,1)$ tensor
with $d^{2}$ independent components. The trace part of ${M^{i}}_{j}$ describes
the scale transformation and the traceless part describes the shear
transformation and the rotation.

Following the textbook argument, the Noether charges can be derived by writing
down the action evaluated at two coordinates related by the large gauge
transformations, (3.30) and setting the difference to 0 as

$\displaystyle 0$ $\displaystyle=\int
d^{d+1}x\left[\det\left({\partial\tilde{x}^{i}\over\partial
x^{j}}\right)\tilde{N}\sqrt{\tilde{g}}\tilde{\cal L}-N\sqrt{g}{\cal L}\right]$
(3.31) $\displaystyle=\int d^{d+1}x\biggl{[}{M^{i}}_{i}N\sqrt{g}{\cal
L}+\sqrt{g}{\cal H}_{i}\delta_{M}N^{i}+\frac{\partial(N\\!\sqrt{g}{\cal
L})}{\partial g_{ij}}\delta_{M}g_{ij}+\pi^{ij}\delta_{M}\dot{g}_{ij}$ (3.32)
$\displaystyle\qquad\qquad\qquad+\sum_{\alpha}\frac{\partial(N\\!\sqrt{g}{\cal
L})}{\partial\varphi_{\rm matter}^{\alpha}}\delta_{M}\varphi_{\rm
matter}^{\alpha}+\sum_{\alpha}\pi^{\alpha}_{\rm
matter}\delta_{M}\dot{\varphi}_{\rm matter}^{\alpha}\biggr{]}\,,$ (3.33)

where $\alpha$ labels different species of matter fields and $\delta_{M}$
denotes the change of each field under the transformation, (3.30), e.g.,

$\displaystyle\delta_{M}g_{ij}=\tilde{g}_{ij}(t,\,\tilde{x}^{i})-g_{ij}(t,\,x^{i})=-{M^{k}}_{i}g_{kj}(t,\,x^{i})-{M^{k}}_{j}g_{ik}(t,\,x^{i})+{\cal
O}(M^{2})\,.$ (3.34)

The first term in the right hand side of Eq. (3.33) appears from the Jacobian
for the large gauge transformation, i.e., $\det(\partial\tilde{x}^{i}/\partial
x^{j})=1+{M^{k}}_{k}$. For a notational brevity, tensor indices of matter
fields (even if any) are ignored. The conjugate momenta are defined as

$\displaystyle\pi^{\alpha}_{\rm matter}\equiv\frac{\partial(N\\!\sqrt{g}{\cal
L})}{\partial\dot{\varphi}_{\rm matter}^{\alpha}}\,.$ (3.35)

Using the equations of motion and also Eq. (3.34), we can rewrite Eq. (3.33)
as

$\displaystyle 0={M^{i}}_{j}\int d^{d+1}x\left[N\sqrt{g}{\cal
L}\,{\delta^{j}}_{i}+\partial_{t}\left(-2{\pi^{j}}_{i}+\sum_{\alpha}\pi^{\alpha}_{\rm
matter}\frac{\partial\delta_{M}\varphi_{\rm
matter}^{\alpha}}{\partial{M^{i}}_{j}}\right)+{\cal O}(\epsilon)\right],$
(3.36)

with ${\pi^{j}}_{i}=g_{ik}\pi^{jk}$ defined in Eq. (2.19). Since
$\delta_{M}N^{i}=M^{i}_{~{}j}N^{j}$, we eliminate the term with the momentum
constraints in rewriting Eq. (3.33) into Eq. (3.36), employing the gauge
$N^{i}=0$. To determine the matter contributions, we need to specify the
tensor structure of the existing matter fields. For instance, a scalar field
$\phi$ and a vector field $A_{\mu}$ transform under Eq. (3.30) as

$\displaystyle\delta_{M}\phi=\delta_{M}A_{0}=0\,,\qquad\delta_{M}A_{i}=-{M^{k}}_{i}A_{k}+{\cal
O}(M^{2},\,\epsilon)\,.$ (3.37)

Therefore, $\phi$ does not contribute to Eq. (3.36), while $A_{\mu}$
contributes as $-\pi^{j}_{A}A_{i}$, where $\pi^{i}_{A}$ denotes the conjugate
momentum of $A_{i}$. Here, higher order terms of ${M^{i}}_{j}$ are ignored,
considering infinitesimally small $|{M^{i}}_{j}|$. Since ${\cal G}_{\epsilon}$
is higher order in $\epsilon$, Eq. (3.36) does not require ${\cal
G}_{\epsilon}$ to hold at all.

Since ${M^{i}}_{j}$ is an arbitrary constant matrix, we obtain $d^{2}-1$
Noether charge densities for the shear transformation and the rotation in Eq.
(3.30) as

$\displaystyle{Q^{i}}_{j}\equiv 8\pi
G\left(\left[-2{\pi^{i}}_{j}+\sum_{\alpha}\pi^{\alpha}_{\rm
matter}\frac{\partial\delta_{M}\varphi_{\rm
matter}^{\alpha}}{\partial{M^{j}}_{i}}\right]^{\rm TL}+{\cal
O}(\epsilon)\right)\,,$ (3.38)

that satisfy

$\displaystyle\frac{d}{dt}\int d^{d}\hbox{\boldmath{$x$}}\,{Q^{i}}_{j}=0\,.$
(3.39)

Here and hereafter, the superscript TL indicates the operation to pick up the
traceless part. For our later convenience, we have inserted the constant
factor $8\pi G$ in the definition of ${Q^{i}}_{j}$. The trace part, which
corresponds to the scale transformation, does not give a conservation because
of the contribution of the Jacobian factor. Instead, the trace part of Eq.
(3.36) gives the equation which corresponds to the trace part of Eq. (3.19).

The global conserved charges, given in Eq. (3.39), do not give any relation to
the local quantity directly. Nevertheless, together with the
$[$locality$]~{}$condition, Eq. (3.39) ensures the approximate time
conservation of the Noether charge densities ${Q^{i}}_{j}$ at the leading
order of $\epsilon$. To show this, let us first consider a specific example
where all the fields in $\\{\varphi^{a}\\}$ become globally homogeneous. In
this case, since the spatial integral in Eq. (3.39) operates trivially, we
obtain $d{Q^{i}}_{j}/dt=0$. This can be also derived by using the equations of
motion for the homogeneous fields $\\{\bar{\varphi}^{a}(t)\\}$, i.e.,

$\displaystyle\frac{d}{dt}{Q^{i}}_{j}=0\quad\Longleftarrow\quad\mbox{set of
equations for}~{}\\{\bar{\varphi}^{a}(t)\\},\,{\rm i.e.},{\cal E}\,.$ (3.40)

The equations used to show the conservation of ${Q^{i}}_{j}$ are nothing but
${\cal E}$, which do not include ${\cal G}_{\epsilon}$, since ${\cal
G}_{\epsilon}$ become trivial for homogeneous fields.

Next, we turn our attention to the coarse-grained fields
$\\{\bar{\varphi}^{a}_{\hbox{\boldmath{\scriptsize$x$}}}(t)\\}$, which are
approximately homogeneous within each causal patch. The
$[$locality$]~{}$condition ensures that
$\\{\bar{\varphi}^{a}_{\hbox{\boldmath{\scriptsize$x$}}}(t)\\}$ satisfies the
same set of the equations as Eq. (3.40), allowing ${\cal O}(\epsilon)$
corrections. Therefore, using these equations, we can also show

$\displaystyle\frac{d}{dt}{Q^{i}}_{j\,{\hbox{\boldmath{\scriptsize$x$}}}}={\cal
O}(\epsilon)\quad\Longleftarrow\quad\mbox{set of the equations
for}~{}\\{\bar{\varphi}_{\hbox{\boldmath{\scriptsize$x$}}}^{a}(t)\\}\,,{\rm
i.e.},{\cal E}\,,$ (3.41)

where ${Q^{i}}_{j\,{\hbox{\boldmath{\scriptsize$x$}}}}$ denotes the Noether
charge density computed by using
$\\{\bar{\varphi}^{a}_{\hbox{\boldmath{\scriptsize$x$}}}(t)\\}$. Likewise
$\\{\bar{\varphi}^{a}_{\hbox{\boldmath{\scriptsize$x$}}}(t)\\}$,
${Q^{i}}_{j\,{\hbox{\boldmath{\scriptsize$x$}}}}$ is homogeneous within each
patch in the limit $\epsilon\to 0$. Since
$\\{\bar{\varphi}^{a}_{\hbox{\boldmath{\scriptsize$x$}}}(t)\\}$ is not exactly
homogeneous, the constraints ${\cal G}_{\epsilon}$ are no longer trivially
satisfied. However, the approximate conservation of
${Q^{i}}_{j\,{\hbox{\boldmath{\scriptsize$x$}}}}$ (3.41) can be shown without
using ${\cal G}_{\epsilon}$, as one can understand from the fact that Eq.
(3.40) is derived by using only ${\cal E}$.

In the absence of charged matter fields, repeating the same argument for the
U(1) gauge transformation (2.30) with $\theta=M_{i}\,x^{i}$ ($M_{i}$: a
constant vector), which is still allowed in the gauge $A_{0}=0$, we simply
obtain the Maxwell equation, given by

$\displaystyle\partial_{t}\pi^{i}_{A}={\cal O}(\epsilon)\,,$ (3.42)

where the conjugate momentum $\pi^{i}_{A}$ corresponds to the Noether charge
density. Meanwhile, in the presence of charged matter fields, the same
argument does not apply, since the change of the matter fields depends on the
spatial coordinates even for a constant $M_{i}$. Remember that since
$\pi^{i}_{A}$ contains the factor $\sqrt{g}$, the time derivative of $A_{i}$
decrease in time unless there is some enhancement factor which compensates
$1/\sqrt{g}$.

When the Lagrangian density ${\cal L}_{\rm g}$ is given by Eq. (3.17), we
obtain121212For a theory of gravity with the Lagrangian density, (3.7),
${\pi^{j}}_{i}$ is given by  $\displaystyle{\pi^{j}}_{i}$
$\displaystyle=\frac{F}{16\pi
G}\sqrt{g}\Biggl{[}\left\\{1+\left(\frac{3\lambda_{3}}{d}-\lambda_{2}\right)K\right\\}{A^{j}}_{i}+\frac{3\lambda_{3}}{2}\left[{A^{j}}_{l}{A^{l}}_{i}\right]^{\rm
TL}$
$\displaystyle\qquad\qquad\qquad\quad+\frac{{\delta^{j}}_{i}}{d}\left\\{(1-d\lambda_{\rm
HL})K+\frac{3}{2d}(d^{2}\lambda_{1}-\lambda_{2}+\lambda_{3})K^{2}+\frac{1}{2}(3\lambda_{3}-d\lambda_{2}){A^{k}}_{l}{A^{l}}_{k}\right\\}+\cdots\Biggr{]}\,.$

$\displaystyle({\rm ex})\qquad\quad{\pi^{j}}_{i}=\frac{F}{16\pi
G}\sqrt{g}\left[{A^{j}}_{i}+\frac{{\delta^{j}}_{i}}{d}(1-d\lambda_{\rm
HL})K\right]\,.$ (3.43)

As one can see from this example, the symmetric part of Eq. (3.38) provides
the precise version of Eq. (3.16). Here, the symmetric part of
${Q^{i}}_{j}(\hbox{\boldmath{$x$}})$ roughly corresponds to
${D^{i}}_{j}(\hbox{\boldmath{$x$}})$. In the next subsection, we will show
that the symmetric part of ${Q^{i}}_{j}$, which is accompanied by (a positive
power of) $(1/\sqrt{g})$, can be generically used to solve the momentum
constraints at $t=t_{\rm hoc}$, just accepting the non-local contributions
that decay rapidly in an expanding universe.

####  3.4.2 Decay of non-local contributions

As argued around Eq. (2.18), by using the conjugate momentum $\pi^{ij}$, the
momentum constraints can be expressed in the following compact form,

$\displaystyle
2\nabla_{j}\left(\frac{{\pi^{j}}_{i}}{\sqrt{g}}\right)+N\left(\frac{\partial{\cal
L}_{\rm matter}}{\partial N^{i}}\right)_{\\!\\!K_{ij}:{\rm fixed}}={\cal
O}\left(\epsilon^{2}\right)\,,$ (3.44)

where the second term is given by taking the derivative of the matter
Lagrangian with respect to $N_{i}$, while keeping $K_{ij}$ fixed. The
variation with respect to $N^{i}$ through $K_{ij}$ contained in ${\cal L}_{\rm
matter}$ (if exists) is included in the conjugate momentum $\pi^{ij}$.
Eliminating the traceless part of ${\pi^{j}}_{i}$ in Eq. (3.44) by using the
symmetric part of Eq. (3.38), we can further rewrite the momentum constraints
as

$\displaystyle\frac{1}{\sqrt{g}}{\nabla_{j}}{Q^{j}}^{~{}{\rm
sym}}_{i}-\frac{16\pi G}{d\sqrt{g}}\partial_{i}{\pi^{k}}_{k}$ (3.45)
$\displaystyle=8\pi
G\left[\frac{1}{\sqrt{g}}{\nabla_{\\!j}}\sum_{\alpha}\pi^{\alpha}_{\rm
matter}\left[\frac{\partial\delta_{M}\varphi_{\rm
matter}^{\alpha}}{\partial{M^{i}}_{j}}\right]^{\rm
symTL}\\!\\!\\!\\!\\!\\!\\!+N\left(\frac{\partial{\cal L}_{\rm
matter}}{\partial N^{i}}\right)_{\\!\\!K_{ij}:{\rm fixed}}\right]+{\cal
O}\left(\epsilon^{2}\right)\\!,$ (3.46)

where $Q^{{\rm sym}}_{ij}$ is the symmetric part of $Q_{ij}$. We put the
superscript “symTL” to indicate that only the symmetric and traceless part is
extracted.

The momentum constraints (3.46) relate the symmetric part of the Noether
charge densities ${Q^{i}}_{j}{}^{\rm sym}$ to matter fields in the right hand
side. As discussed in Sec. 2.4.2, when all other equations are satisfied and
Eq. (3.46) are properly solved for ${Q^{i}}_{j}{}^{\rm sym}$, e.g., at
$t=t_{\rm hoc}$, the momentum constraints (3.46) hold also at an arbitrary
time. Since Eq. (3.46) has $d$ components, we can solve it to determine $d$
components among the symmetric part of ${Q^{i}}_{j}$. Since we need to
integrate the spatial derivative to determine the $d$ components, the obtained
expression becomes non-local. Nevertheless, the non-local terms
${Q^{i}}_{j}{}^{\rm sym}$ in the expression of ${A^{i}}_{j}$ become smaller
and smaller as the universe expands, being suppressed by $1/\sqrt{g}$. By
looking at the expression, say, in (3.43), one may wonder what if $F$
decreases too rapidly to compensate the growth of $\sqrt{g}$. A temporal
decrease of $F$ is not a problem, since we just have to wait further for the
non-local terms to start decaying. Meanwhile, a long-lasting decrease of $F$
can lead to a strong coupling regime. (As will be discussed below Eq. (4.21),
a rapid change of $F$ also violates the slow-roll condition.) Once these terms
become negligible, whether ${\cal H}_{i}$ is solved or not should be
irrelevant. Therefore, when ${\cal H}_{i}$ are the only ${\cal G}_{\epsilon}$
in the system, the $[$approx${\cal G}_{\epsilon}$$]~{}$condition holds at a
late time after these non-local terms become negligible.

The momentum constraints determine how to glue the neighboring local patches
at each time, resulting in the appearance of non-local contributions. As
discussed in Sec. 2.4.2, as far as the neighboring patches are glued properly
at a certain time, satisfying the momentum constraints, they remain so also at
later times. The decay of the non-local contributions indicates that even if
the neighboring patches are not properly glued, violating the momentum
constraints or choosing wrong values for the symmetric part of ${Q^{i}}_{j}$,
the error becomes less and less important as time goes on.

One may wonder if the same argument applies for the U(1) gauge constraint
${\cal H}_{U(1)}$ in the absence of charged matter fields, by choosing
$\pi^{i}_{A}$, which corresponds to the Noether charge, in such a way that
${\cal H}_{U(1)}$ is satisfied at a certain time. This is the case when the
gauge fields become negligible at large scales. Nevertheless, when they do not
decay at large scales, e.g., being sourced by a decreasing ${\cal F}_{Y}$, the
non-local terms with $\pi^{i}_{A}$ are not negligible.

####  3.4.3 Usage of conserved charges

Equation (3.38) can be useful also in solving the shear. To illustrate this,
let us consider an example where ${\cal L}_{\rm g}$ is given by Eq. (3.17) and
the matter Lagrangian density ${\cal L}_{\rm matter}$ is given by Eq. (2.20).
Inserting Eqs. (2.27) and (3.43) into the symmetric part of Eq. (3.38), we
obtain a similar expression to Eq. (3.25) as

$\displaystyle F{A^{j}}_{i}=-\frac{{Q^{j}}^{~{}{\rm sym}}_{i}}{\sqrt{g}}-8\pi
G{\cal F}_{Y}\left[g^{jl}\frac{\dot{A}_{l}}{N}A_{i}\right]^{\rm symTL}+{\cal
O}(\epsilon)\,.$ (3.47)

The main difference between Eqs. (3.25) and (3.47) is in that the matter
contributions in the latter are not in the form of the time integral.
Integrating Eq. (3.47) formally, we obtain

$\displaystyle\gamma_{ij}(t,\,\hbox{\boldmath{$x$}})$
$\displaystyle=\gamma_{ij}(t_{\rm
hoc},\,\hbox{\boldmath{$x$}})-2\int^{\tau(t)}_{\tau_{\rm
hoc}}d\tau^{\prime}\frac{\gamma_{il}(\tau^{\prime},\,\hbox{\boldmath{$x$}})}{(F\sqrt{g})_{(\tau^{\prime},\,\hbox{\boldmath{\scriptsize$x$}})}}{Q^{l}}^{~{}{\rm
sym}}_{j}(\hbox{\boldmath{$x$}})$ (3.48) $\displaystyle\qquad-8\pi
G\int^{\tau(t)}_{\tau_{\rm
hoc}}d\tau^{\prime}e^{-2\psi(\tau^{\prime},\,\hbox{\boldmath{\scriptsize$x$}})}\frac{{\cal
F}_{Y}}{F}\left[(\partial_{\tau}A_{i})A_{j}\right]^{\rm
symTL}_{(\tau^{\prime},\,\hbox{\boldmath{\scriptsize$x$}})}+{\cal
O}(\epsilon).\,\,$ (3.49)

The second term corresponds to the decaying solution and the term in the
second line is the sourced contributions. Compared to Eq. (LABEL:Sol:gammaij),
one time integral is already performed in Eq. (3.49).

###  3.5 Approximate existence of WAM

In this section, we have shown that the Weinberg’s argument can apply much
more generically, extending it to an anisotropic universe with a large scale
anisotropic pressure and modified theories of gravity. In fact, when the
$[$approx${\cal G}_{\epsilon}$$]~{}$condition holds in addition to the
$[$locality$]~{}$and $[$sDiff$]~{}$conditions, the WAM exists as an
approximate solution in the large scale limit. Based on the argument in the
previous subsection, let us summarize cases when the additional condition
$[$approx${\cal G}_{\epsilon}$$]~{}$holds.

When the momentum constraints are the only ${\cal G}_{\epsilon}$ in the
system, as discussed in the previous subsection, the $[$approx${\cal
G}_{\epsilon}$$]~{}$holds since the non-local contributions inevitably become
negligible at late times. This includes the cases when the system only
includes scalar fields or includes also a gauge field with a charged field.

The WAM is not the exact solution even at the leading order of the gradient
expansion, because it does not fulfill the leading order of ${\cal
G}_{\epsilon}$, which is ${\cal O}(\epsilon)$, due to the decaying
contribution. This can be seen explicitly by focusing on the first term in the
second line of Eq. (3.12), which includes $(\partial_{j}\psi){A^{j}}_{i}$.
This term contributes as the leading order in the gradient expansion since
${A^{i}}_{j}={\cal O}(\epsilon^{0})$ (see the discussion in the next
subsection), while it does not remain invariant under Eq. (3.2) for an
inhomogeneous $c(\hbox{\boldmath{$x$}})$.

Let us consider an anisotropic universe with a $d$-dimensional rank $(m,\,n)$
tensor field with $m$ upper indices and $n$ lower indices,
${\varphi_{i_{1}\cdots i_{n}}}^{j_{1}\cdots j_{m}}$, which transforms under
the dilatation as

$\displaystyle\varphi_{i_{1}\cdots i_{n}}^{c~{}~{}~{}~{}~{}j_{1}\cdots
j_{m}}(t,\,\hbox{\boldmath{$x$}}^{c})=e^{(n-m)c}{\varphi_{i_{1}\cdots
i_{n}}}^{j_{1}\cdots j_{m}}(t,\,\hbox{\boldmath{$x$}})\,,$ (3.50)

satisfying the $[$approx${\cal G}_{\epsilon}$$]~{}$condition in addition to
the other two conditions. Since the system remains approximately invariant
under Eq. (3.2), when
$\varphi^{a}(t,\,\hbox{\boldmath{$x$}})=f^{a}(t,\,\hbox{\boldmath{$x$}})$ is a
solution for $\\{\varphi^{a}\\}$,

$\varphi^{a}(t,\,\hbox{\boldmath{$x$}})=\begin{cases}f^{a}(t,\,\hbox{\boldmath{$x$}})+c(\hbox{\boldmath{$x$}})&(a=\psi),\\\
e^{(n-m)c(\hbox{\boldmath{\scriptsize$x$}})}f^{a}(t,\,\hbox{\boldmath{$x$}})&(a={\rm
a~{}rank~{}}(m,\,n){\rm~{}tensor}),\end{cases}$

also satisfies the equations ${\cal E}$ at the leading order of $\epsilon$ and
${\cal G}_{\epsilon}$ approximately as well, owing to $[$approx${\cal
G}_{\epsilon}$$]~{}$131313When one wants to consider a solution where only
$\psi$ is shifted, the rank-$(m,\,n)$ tensor fields should be projected by the
tetrad in $d$-dim. space ${e^{(\alpha)}}_{i}$ as
$\displaystyle\varphi_{(\alpha_{1})\cdots(\alpha_{n})(\beta_{1})\cdots(\beta_{m})}(t,\,\hbox{\boldmath{$x$}})\equiv{e_{(\alpha_{1})}}^{i_{1}}\cdots{e_{(\alpha_{n})}}^{i_{n}}e_{(\beta_{1})\,j_{1}}\cdots
e_{(\beta_{n})\,j_{m}}\,{\varphi_{i_{1}\cdots i_{n}}}^{j_{1}\cdots
j_{m}}(t,\,\hbox{\boldmath{$x$}})\,,$ (3.51) where the indices with the
parentheses are the tetrad basis coordinates. For example, in the projected
tetrad bases, the kinetic term for a gauge field,
$g^{ij}\dot{A}_{i}\dot{A}_{i}$, is given as
$g^{ij}\dot{A}_{i}\dot{A}_{i}=(\dot{A}_{\alpha}-{\dot{e}_{\alpha}\\!}^{i}A_{i})(\dot{A}^{\alpha}-\dot{e}^{\alpha
j}A_{j})\,,$ where we have used $g^{ij}={e_{\alpha}}^{i}e^{\alpha j}$.  . As
it is clear from the definition, $\gamma_{ij}$ does not change under the
dilatation. We express the spatial metric perturbation as
$\delta\gamma^{i}_{~{}j}=\log(\delta^{ik}\gamma_{kj})$, which transforms like
a (1,1) tensor under the scale transformation. When there exists a non-
decaying tensor field with $n\neq m$, the change of these fields also should
be taken into account in the definition of WAM.

Meanwhile, when a system includes gauge fields, the gauge constraint can
become ${\cal G}_{\epsilon}$. As one can see from Eq. (2.26), the gauge
constraint ${\cal H}_{\rm U(1)}$ does not remain invariant under the
dilatation with an inhomogeneous parameter
$c(\hbox{\boldmath{$x$}})$141414Recall that the dilatation with an
inhomogeneous parameter $c(\hbox{\boldmath{$x$}})$ should be understood as
simply replacing a constant $c$ with $c(\hbox{\boldmath{$x$}})$ in the
transformation law of the dilatation with a constant $c$, which is definitely
not a coordinate transformation. This point was emphasized in Ref. [88].,
since there appears a term with
$\partial_{i}c(\hbox{\boldmath{$x$}})A^{i}(t,\,\hbox{\boldmath{$x$}})$. While
this term is suppressed by the spatial gradient, this is a leading order
contribution in the gradient expansion for ${\cal H}_{\rm U(1)}$, which is
${\cal G}_{\epsilon}$, similarly to $\partial_{j}\psi{A^{j}}_{i}$ in ${\cal
H}_{i}$. Therefore, when the gauge fields do not decay at large scales, e.g.,
being enhanced by a coupling with the inflaton, the gauge constraint ${\cal
H}_{\rm U(1)}$ does not hold even approximately, resulting in the absence of
the WAM. Exceptionally, when the background value of $A^{i}$ vanishes, ${\cal
H}_{\rm U(1)}$ remains to be valid also after the dilatation at the linear
order of perturbation, since
$\partial_{i}c(\hbox{\boldmath{$x$}})A^{i}(t,\,\hbox{\boldmath{$x$}})$ becomes
second order in perturbation. Then, the WAM approximately exists at the linear
perturbation.

Let us emphasize again that when $\psi$ is dominated by other modes, the
(approximate) existence of the constant solution is not particularly useful to
shortcut the time evolution on super-horizon scales. The constant solution
becomes more important as an asymptotic solution after the period of all the
non-trivial dynamics, which potentially generates growing modes of $\psi$.

In this paper, we have focused on the adiabatic solution of the scalar
perturbation, but the same argument also applies to the tensor perturbations
[79]. The conservation of the tensor perturbations in a scalar field system
was first pointed out in the historic paper by Starobinsky [63]. Under the
shear transformation,

$\displaystyle x^{i}\to
x^{i}_{C}\equiv\bigl{[}e^{-\frac{C}{2}}\bigr{]}{}^{i}_{\,j}\,x^{j}\,,$ (3.52)

with $C^{i}_{\,j}$ being a constant traceless and symmetric tensor, the
anisotropic spatial metric transforms as

$\displaystyle\gamma_{ij\,C}(t,\,\hbox{\boldmath{$x$}}_{C})=\bigl{[}e^{\frac{C}{2}}\bigr{]}{}^{k}_{~{}i}\,\bigl{[}e^{\frac{C}{2}}\bigr{]}{}^{l}_{~{}j}\,\gamma_{kl}(t,\,\hbox{\boldmath{\scriptsize$x$}})\,.$
(3.53)

Similarly to Eq. (3.4), this transformation shifts $\delta{\gamma^{i}}_{j}$,
defined as $\gamma_{ij}\equiv\bar{\gamma}_{ik}{[e^{\delta\gamma}]^{k}}_{j}$ by
using the (background) homogeneous contribution $\bar{\gamma}_{ij}$, as
follows

$\displaystyle\delta{\gamma^{i}}_{j\,C}(t,\,\hbox{\boldmath{$x$}}_{C})=\delta{\gamma^{i}}_{j}(t,\,\hbox{\boldmath{$x$}})+{C^{i}}_{j}+{1\over
2}\left({C^{i}}_{k}\delta{\gamma^{k}}_{j}+\delta{\gamma^{i}}_{k}{C^{k}}_{j}\right)+\cdots\,.$
(3.54)

Unlike the dilatation, the change under the shear transformation is not a
simple additive one, but is given by the above expression. When we promote the
shear transformation by replacing ${C^{i}}_{j}$ with an inhomogeneous
parameter $C^{i}\\!_{j}(\hbox{\boldmath{$x$}})$, repeating the same argument
as the one for dilatation, we find that the change under the promoted
transformation provides a solution of ${\cal E}$ at the leading order of the
gradient expansion. When the $[$approx${\cal G}_{\epsilon}$$]~{}$condition
holds in addition to the $[$locality$]~{}$and $[$sDiff$]~{}$conditions, the
solutions of ${\cal E}$ approximately satisfy all the equations in the system,
${\cal E}_{\rm all}$. If one imposes the transverse condition (2.36) at
$t=t_{*}$, which is ${\cal X}_{\epsilon}$, to remove the residual degrees of
freedom in the spatial coordinates, ${C^{i}}_{j}(\hbox{\boldmath{$x$}})$
should be chosen so that Eq. (2.36) is satisfied both before and after the
transformation. At the linear perturbation, this requires
$\partial_{i}{C^{i}}_{j}(\hbox{\boldmath{$x$}})=0$. The remaining degrees of
freedom in ${C^{i}}_{j}(\hbox{\boldmath{$x$}})$ correspond to the decaying two
solutions of the tensor perturbation.

As an example, let us consider a model with a U(1) gauge field and neutral
fields. When we perform the shear transformation with an inhomogeneous
parameter $C^{i}\\!_{j}(\hbox{\boldmath{$x$}})$, there appears a term with
$[e^{C/2}]_{~{}i}^{j}(\partial_{j}[e^{-C/2}]^{i}_{~{}k})\,\pi_{A}^{k}$ in the
gauge constraint ${\cal H}_{\rm U(1)}$. With
$\partial_{i}{C^{i}}_{j}(\hbox{\boldmath{$x$}})=0$, we find that the violation
of ${\cal H}_{\rm U(1)}$ vanishes exceptionally at the linear perturbation
even when $\pi_{A}^{k}$ has a non-zero background value. For the WAM to exist
approximately also at higher orders of perturbation, the violation of ${\cal
H}_{\rm U(1)}$ should be approximately negligible, i.e, satisfying the
$[$approx${\cal G}_{\epsilon}$$]~{}$condition. When the gauge fields survive
also at the large scales, the $[$approx${\cal G}_{\epsilon}$$]~{}$condition
can be violated at the higher orders of perturbation. Meanwhile, when the U(1)
gauge constraint is not ${\cal G}_{\epsilon}$ owing to the presence of charged
fields, once ${\cal H}_{i}$, which is the only ${\cal G}_{\epsilon}$, start to
hold approximately, the WAM exists as an approximate solution. Earlier studies
of the gravitational waves in Bianchi background can be found, e.g., in Refs.
[89, 90, 91].

###  3.6 Weinberg’s second solution

In previous works, the shear has been treated as a sub-leading contribution in
the gradient expansion. However, as we will see in this subsection, if we
assume ${A^{i}}_{j}={\cal O}(\epsilon)$, the other solution of $\psi$, which
corresponds to the Weinberg’s second mode, does not appear at the leading
order of the gradient expansion. This was shown in Ref. [9], considering a
scalar field system (see also Ref. [46], where a more general class of scalar
field models was discussed).

In this subsection, for simplicity, let us consider a single field model of
inflation, whose matter Lagrangian density is given by

$\displaystyle{\cal L}_{\rm matter}=P(X,\,\phi)\,,\qquad
X\equiv-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\,,$ (3.55)

in general relativity (with $F=1$). In this case, the energy density, defined
as

$\displaystyle\rho\equiv-\frac{\partial}{\partial N}\left(N{\cal L}_{\rm
matter}\right)\,,$ (3.56)

is given by

$\displaystyle\rho=2P_{X}X-P+{\cal O}(\epsilon^{2})\,,$ (3.57)

with $P_{X}\equiv\partial P/\partial X$ and the shear simply decays as
${A^{i}}_{j}=-Q^{i}_{\,j}/\sqrt{g}$, since there is no anisotropic pressure on
large scales.

In general relativity, the Hamiltonian and momentum constraints are given by

$\displaystyle\frac{1-d}{d}K^{2}+A^{i}\\!_{j}A^{j}\\!_{i}+16\pi G\rho={\cal
O}(\epsilon)\,,$ (3.58)

and

$\displaystyle\left(\frac{1}{d}-1\right)\partial_{i}K+\nabla_{j}{A^{j}}_{i}-{8\pi
G}P_{X}\frac{\dot{\phi}}{N}\partial_{i}\phi={\cal O}(\epsilon^{2})\,.$ (3.59)

For our convenience, let us introduce the flat FLRW spacetime as the
background and consider the deviation from there. Once we ignore
${A^{i}}_{j}$, setting ${A^{i}}_{j}={\cal O}(\epsilon)$, the constraint
equations imply that the uniform Hubble slicing with

$\displaystyle K(t,\,\hbox{\boldmath{$x$}})=dH(t)\,,$ (3.60)

the uniform density slicing, and the uniform field slicing all agree with each
other. Here, $H(t)$ is the Hubble parameter for the FLRW background and we
have noticed $P_{X}\dot{\phi}\neq 0$ in a sensible model of inflation.
Evaluating $\delta\rho={\cal O}(\epsilon)$ with $\delta\phi={\cal
O}(\epsilon)$, we obtain $\delta N=\delta(\dot{\psi}/H)={\cal O}(\epsilon)$,
i.e., the fluctuation of $\psi$ is time independent at the leading order of
$\epsilon$.

Therefore, in order to properly reproduce the Weinberg’s second mode, we need
to take into account the shear as the leading order of the gradient expansion,
i.e., ${A^{i}}_{j}={\cal O}(\epsilon^{0})$. In Ref. [46], this was pointed out
for the curvature perturbation $\zeta$ at linear perturbation, taking the
transverse gauge. It is natural that the same story follows for $\psi$ in our
gauge with Eqs. (2.24) and (2.36), because our $\psi$ and $\zeta$ agree when
${A^{i}}_{j}$ is negligible all the time151515At linear perturbation,
${A^{i}}_{j}\sim 0$ can be recast into $\dot{\gamma}_{ij}\sim 0$, which in
turn implies that the transverse condition (2.36) also holds at an arbitrary
time..

In particular, in a scalar field system, one may think that the local
spacetime metric approaches the FLRW metric if we take the limit $\epsilon\to
0$. As we have argued here, this is not correct, in contrast to the
observation made from the definition of the Bardeen potential in Ref. [92].
Instead, only after we take both the $\epsilon\to 0$ limit and the late time
limit, the local spacetime metric approaches the FLRW metric because of the
shear contribution.

##  4 Generalized $\delta N$ formalism

As is widely known, the $\delta N$ formalism [6, 7, 8, 5, 9] provides a
powerful tool to calculate the large scale evolution of the perturbations
generated during inflation (for a review, see Refs. [93, 94]). In this paper,
we have shown that the gradient expansion can be verified for a general theory
which satisfies the $[$sDiff$]~{}$and $[$locality$]~{}$conditions. This
enables us to apply the $\delta N$ formalism also to models of anisotropic
inflation. Furthermore, we can also calculate non-zero spin fields such as the
gravitational waves in this generalized formalism, which we dub the
generalized $\delta N$ formalism (the g$\delta N$ formalism).

###  4.1 User’s guide of g$\delta N$ formalism

In this subsection, we summarize the basic procedure of the g$\delta N$
formalism. Let us consider a theory which satisfies the $[$sDiff$]~{}$and
$[$locality$]~{}$conditions, including scalar fields
$\phi^{I}~{}(I=1,\,2,\,\cdots)$ and a vector field $A^{\mu}$ and so on, i.e.,
$\\{\varphi_{\rm mat}^{a}\\}=\phi^{I},\,A^{\mu},\,\cdots$.

####  4.1.1 With redundant fields

As discussed in Sec. 2.4.2, the separate universe evolution $(\star)$ can be
verified by considering the evolution of $\\{\varphi^{a}\\}$, which still
contains redundant fields to be removed. The g$\delta N$ formalism applies as
long as the $[$locality$]~{}$and $[$sDiff$]~{}$conditions hold. The initial
conditions can be set around the horizon crossing time, i.e.,
$t_{\star}=t_{\rm hoc}$. By solving the evolution of homogeneous cosmological
models with various initial conditions, we obtain the map from
$\\{\varphi^{a}(t_{\star})\\}$ to $\\{\varphi^{a}(t_{f})\\}$, where $t_{f}$ is
the final time. We dub this mapping, given by solving ${\cal E}$, as g$\delta
N$ formalism. The initial condition can be set before the $[$approx${\cal
G}_{\epsilon}$$]~{}$condition starts to hold. In order to apply this g$\delta
N$ formalism, the initial distribution needs to be specified also for the
redundant fields. (When we have only the initial conditions for physical
degrees of freedom, we need to determine the initial conditions for the
remaining variables, using ${\cal G}_{\epsilon}$.)

For a theory with the $(d+1)$-dim Diff invariance, the equations to be solved,
${\cal E}$, are the Hamiltonian constraint, Eq. (3.22), the evolution
equations for matter fields, and the expression of the shear, Eq. (3.38) or
alternatively Eq. (3.23). For non-projectable version of the HL gravity, the
Hamiltonian constraint is a non-gauge constraint ${\cal C}\cup{\cal G}^{c}$
that reduces the variables in the original Lagrangian to $\\{\varphi^{a}\\}$.
When the action is given by Eq. (3.7), the Hamiltonian constraint reads

$\displaystyle\frac{1-d\lambda_{\rm
HL}}{d}K^{2}+2\left(\lambda_{1}-\frac{\lambda_{2}}{d}+\frac{\lambda_{3}}{d^{2}}\right)K^{3}+\left\\{1+2\left(-\lambda_{2}+\frac{3\lambda_{3}}{d}\right)K\right\\}A^{i}\\!_{j}A^{j}\\!_{i}$
(4.1)
$\displaystyle\qquad\qquad\quad+2\lambda_{3}A^{i}\\!_{j}A^{j}\\!_{l}A^{l}\\!_{i}+\cdots+\frac{16\pi
G}{F}\rho={\cal O}(\epsilon)\,.$ (4.2)

As shown in Eq. (3.15), in $N_{i}=0$ gauge, $\psi$ directly corresponds to the
non-perturbative $e$-folding number. Equation (3.15) states that the the
difference of $\psi$ between $t_{*}$ and the final time $t_{f}$ is computed by
integrating the expansion between these two time slices as

$\displaystyle\psi(t_{f},\,\hbox{\boldmath{$x$}})-\psi(t_{*},\,\hbox{\boldmath{$x$}})=\frac{1}{d}\int^{t_{f}}_{t_{*}}dt^{\prime}N(t^{\prime},\,\hbox{\boldmath{$x$}})K(t^{\prime},\,\hbox{\boldmath{$x$}})\,.$
(4.3)

In this gauge, $\psi$ does not directly correspond to the curvature
perturbation $\zeta$161616Here, $\zeta$ is the curvature perturbation in
$\phi=$const. time slice, which is at linear order given by
$\displaystyle\zeta\equiv{\cal
R}-\frac{H}{\dot{\phi}}\delta\phi=\delta\psi-\frac{1}{2d}\chi-\frac{H}{\dot{\phi}}\delta\phi\,,$
(4.4) where we have introduced $\dot{\chi}$ as the scalar part of the shear,
given by
$\displaystyle{A^{i}}_{j}\sim\frac{1}{2}\delta^{im}\dot{\gamma}_{mj}\sim\frac{1}{2}\delta^{im}\left(\partial_{m}\partial_{j}\partial^{-2}-\frac{1}{d}\delta_{mj}\right)\dot{\chi}\,.$
(4.5) , since we also have to include the longitudinal part of $\gamma_{ij}$
(see Eq. (4.30)).

Let us summarize a couple of key differences between the conventional $\delta
N$ formalism and the g$\delta N$ formalism. First, in the former we can only
calculate the curvature perturbation, i.e., a scalar perturbation, while, in
the latter, we can calculate an arbitrary field in the theory under
consideration, $\\{\varphi^{a}\\}$, unless it trivially vanishes in the limit
$\epsilon\to 0$. For example, we can also compute gravitational waves,
evaluating $\gamma_{ij}$ for different values of
$\\{\varphi^{a^{\prime}}_{*}\\}^{\prime}$. Second, all the fields are
systematically incorporated through the initial condition
$\\{\varphi_{\hbox{\boldmath{\scriptsize$x$}}\,*}^{a^{\prime}}\\}^{\prime}$,
including non-zero integer spin fields and also purely geometrical degrees of
freedom, such as the gravitational waves $\gamma_{ij}$ and the Khronon in the
Horava-Lifshitz gravity. This has become possible by getting rid of
unnecessary assumptions such as the absence of anisotropic pressure at the
leading order of $\epsilon$.

####  4.1.2 Without redundant fields

When the $[$approx${\cal G}_{\epsilon}$$]~{}$condition is satisfied and the
residual gauge degrees of freedom are removed without employing ${\cal
X}_{\epsilon}$ or become trivially negligible, we can also apply the g$\delta
N$ formalism even when we give the initial conditions only for the physical
degrees of freedom $\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$, excluding all
the redundant fields. In this case, $t_{*}$ should be set by requiring that
${Q^{i}}_{j}$ is negligible in all the equations in ${\cal E}$. Typically,
$t_{*}$ can be set to several $e$-foldings after the horizon crossing in an
inflationary universe. In this case, the initial conditions at $t=t_{*}$, can
be given by solely specifying $\\{\varphi^{a_{\rm phys}}\\}_{\rm phys}$.

###  4.2 Implications on anisotropic inflation

In this subsection, we apply the g$\delta N$ formalism to a model of
anisotropic inflation [20, 21, 22, 23], considering the gravity Lagrangian
density (3.17) with $\lambda_{\rm HL}=1$ and the matter Lagrangian density
(2.20). The Hamiltonian constraint and the trace part of the $(i,\,j)$
component of Einstein’s equation give

$\displaystyle\frac{1-d}{d}K^{2}+A^{i}\\!_{j}A^{j}\\!_{i}+\frac{16\pi
G}{F}\rho={\cal O}(\epsilon^{2})\,,$ (4.6)
$\displaystyle\frac{1}{NF}\partial_{t}(FK)-\frac{d}{1-d}{A^{i}}_{j}{A^{j}}_{i}=\frac{d}{1-d}\frac{8\pi
G}{F}(\rho+P)+{\cal O}(\epsilon^{2})\,.$ (4.7)

The shear is given by Eq. (3.47). In our gauge with $N_{i}=0$ and $A_{0}=0$,
$Y$, $\tilde{Y}$, and $X^{IJ}$ are given by

$\displaystyle Y=\frac{1}{2N^{2}}g^{ij}\dot{A}_{i}\dot{A}_{j}+{\cal
O}(\epsilon^{2})\,,\qquad\tilde{Y}=-\frac{1}{2N\sqrt{g}}\epsilon^{ijk}\dot{A}_{i}F_{jk}\,,$
$\displaystyle X^{IJ}=\frac{1}{4N^{2}}\left(\dot{\phi}^{I}\dot{\phi}^{J*}+{\rm
c.c.}\right)-\frac{1}{4}g^{ij}A_{i}A_{j}e^{(I)}e^{(J)}\left(\phi^{I}\phi^{J*}+{\rm
c.c.}\right)+{\cal O}(\epsilon)\,.$ (4.8)

Using these expressions and Eq. (2.20), the energy density $\rho$ and the
isotropic and asisotropic pressures are given by

$\displaystyle\rho=2X^{IJ}{\cal F}_{IJ}+2Y{\cal F}_{Y}-{\cal
F}+\frac{1}{2}g^{ij}A_{i}A_{j}|\phi|^{2}+{\cal O}(\epsilon)\,,$ (4.9)
$\displaystyle P={\cal F}-\frac{2}{d}Y{\cal
F}_{Y}+\frac{1}{2d}g^{ij}A_{i}A_{j}|\phi|^{2}+{\cal O}(\epsilon)\,,$ (4.10)
$\displaystyle{\Pi}_{ij}=-\frac{1}{N^{2}}{\cal
F}_{Y}\left[\dot{A}_{i}\dot{A}_{j}\right]^{\rm
TL}+\frac{1}{2}|\phi|^{2}\left[A_{i}A_{j}\right]^{\rm TL}+{\cal
O}(\epsilon)\,,$ (4.11)

where we have used $T_{ij}=g_{ij}{\cal L}_{\rm matter}-2\partial{\cal L}_{\rm
matter}/\partial g^{ij}$. Here, we have introduced the amplitude of the
charged scalar fields, given by

$\displaystyle|\phi|^{2}\equiv{\cal
F}_{IJ}e^{(I)}e^{(J)}(\phi^{I}\phi^{J*}+{\rm c.c.})\,.$ (4.12)

The Maxwell equation gives

$\displaystyle\partial_{t}\pi^{i}_{A}=-J^{i}\,,$ (4.13)

where the current $J^{i}$ is given by

$\displaystyle J^{i}\equiv\frac{\partial N\sqrt{g}{\cal L}_{\rm
matter}}{\partial A_{i}}=-\frac{1}{2}N\sqrt{g}|\phi|^{2}A^{i}+{\cal
O}(\epsilon)\,.$ (4.14)

The current $J^{\mu}$ satisfies the conservation $\partial_{\mu}J^{\mu}=0$.
Sending the charge $e^{(I)}$ to 0, we can also consider neutral scalar fields.

####  4.2.1 Neutral scalar fields

First, let us consider the case where all the scalar fields are neutral. The
Klein-Gordon equation for neutral scalar fields is given by

$\displaystyle\frac{1}{N\sqrt{g}}\partial_{t}\left({\cal
F}_{IJ}\sqrt{g}\frac{\dot{\phi}^{J}}{N}\right)-\frac{\partial{\cal
F}}{\partial\phi^{I}}={\cal O}(\epsilon)\,.$ (4.15)

In this case, using Eq. (4.13), we find Eq. (3.42). Using this solution, $Y$
can be recast into

$\displaystyle Y(t,\,\hbox{\boldmath{$x$}})=\frac{1}{2({\cal
F}_{Y})^{2}}\frac{g_{ij}}{g}\pi_{A}^{i}(\hbox{\boldmath{$x$}})\pi_{A}^{j}(\hbox{\boldmath{$x$}})+{\cal
O}(\epsilon)\,.$ (4.16)

Suppose that ${\cal F}$ is dominated by a term which is proportional to
$Y^{n}$ with an almost time independent coefficient. Then, we have

$Y\propto(g_{ij}/g)^{1/(2n-1)}\propto e^{-\frac{2(d-1)}{2n-1}\psi}.$

Since $Y$, ${\cal F}$, and ${\cal F}_{Y}$ all decrease in time for $n>1/2$, if
${\cal F}$ is given by a polynomial of $Y$ with almost time independent
coefficients, the lowest power term starts to dominate ${\cal F}$ as time goes
on. Therefore, in the following, we focus on the linear term of $Y$, assuming
that $F_{Y}$ is a function of the scalar fields.

Integrating Eq. (2.27), we obtain the solution of $A_{i}$ as

$\displaystyle
A_{i}(t,\,\hbox{\boldmath{$x$}})=A_{i*}(\hbox{\boldmath{$x$}})+\pi_{A}^{j}(\hbox{\boldmath{$x$}})\int^{\tau(t)}_{\tau_{*}}\frac{d\tau^{\prime}}{\sqrt{g}{\cal
F}_{Y}}g_{ij}(\tau^{\prime},\,\hbox{\boldmath{$x$}})+{\cal O}(\epsilon)\,.$
(4.17)

Inserting this solution (4.17) into Eq. (3.47), we obtain the shear as

$\displaystyle{A^{j}}_{i}$
$\displaystyle=-\frac{1}{F\sqrt{g}}\left[{Q^{j}}_{i}+4\pi
G\Biggl{[}{\delta^{j}}_{k}{\delta_{i}}^{l}+\gamma^{jl}{\gamma}_{ik}-\frac{2}{d}{\delta^{j}}_{i}{\delta^{l}}_{k}\right]\pi^{k}_{A}$
(4.18)
$\displaystyle\qquad\qquad\qquad\qquad\times\left\\{A_{l*}+d\pi^{m}_{A}\int^{\psi(t)}_{\psi_{*}}\frac{d\psi^{\prime}}{\sqrt{g}K{\cal
F}_{Y}}g_{ml}(\psi^{\prime},\,\hbox{\boldmath{$x$}})\right\\}\Biggr{]}+{\cal
O}(\epsilon).\,$ (4.19)

In Refs. [20, 95, 22], considering a single canonical scalar field coupled
with a U(1) gauge field in general relativity ($F=1$) with $d=3$, Watanabe,
Kanno, and Soda constructed a model in which the shear survives without being
diluted by the exponential expansion. As argued in these papers, when the
coupling ${\cal F}_{Y}$ is proportional to $e^{-4\hat{c}\psi}$ with
$\hat{c}=1$ or $\hat{c}>1$, the energy density of the gauge field becomes
almost constant or grows exponentially during a slow-roll inflation.
Especially for $\hat{c}>1$, it enters the anisotropic inflation phase, where
the gauge fields slows down the inflaton.

In the following, using the formulae derived in this paper, let us consider
the anisotropic inflation model more closely for a more general matter
Lagrangian density (2.20) with ${\cal F}_{Y}(\phi^{I},\,X^{IJ})$ in a general
spacetime dimension. Considering an inflationary solution, we assume that the
expansion $K$ remains almost constant, i.e.,

$\displaystyle\varepsilon\equiv-d\frac{\dot{K}}{NK^{2}}\ll 1\,,$ (4.20)

which corresponds to the usual slow-roll parameter, when we take the
homogeneous and isotropic limit. This condition does not restrict quantities
that correspond to other slow-roll parameters. Using Eqs. (4.6) and (4.7), we
can compute $\varepsilon$ as

$\displaystyle\varepsilon=d\frac{\partial_{t}\ln
F}{NK}+\frac{d^{2}}{d-1}\frac{{A^{i}}_{j}{A^{j}}_{i}}{K^{2}}+\frac{d}{2}\frac{\rho+p}{\rho}\frac{1}{1+F{A^{i}}_{j}{A^{j}}_{i}/(16\pi
G\rho)}+{\cal O}(\epsilon)\,.$ (4.21)

Unless we accept a fine tuning, the variation of $F$ in the time scale of the
cosmic expansion violates the slow-roll condition.

When we ignore the time variation of $K$ and $F$, the time integral in the
shear, i.e., the second term in the second line of Eq. (4.19), evolves as

$\frac{g_{ij}}{g{\cal F}_{Y}}\propto\frac{e^{-2(d-1)\psi}}{{\cal F}_{Y}}\,.$

Therefore, when ${\cal F}_{Y}$ evolves as ${\cal F}_{Y}\propto
e^{-2\hat{c}(d-1)\psi}$ with $\hat{c}>1$ or $\hat{c}=1$,
${A^{i}}_{j}{A^{j}}_{i}/K^{2}$ grows or approaches to a constant. Meanwhile,
the energy density of the gauge fields can be expressed as $Y{\cal F}_{Y}$,
which also scales in the same way as the above shear contribution. This is
just a hand-waving argument to look for a model where the shear can survive in
an expanding universe. To build a concrete model, one needs to start with
${\cal F}$ which is a function of $\phi^{I}$ and/or $X^{IJ}$ as in Refs. [20,
95, 22].

Equation (4.21) immediately gives an upper bound on the shear as
$\sqrt{{A^{i}}_{j}{A^{j}}_{i}/K^{2}}\leq{\cal O}(\sqrt{\varepsilon})$ when the
null energy condition is satisfied and the time variation of $F$ is negligible
[96]. In Ref. [20], a more stringent upper bound,
$\sqrt{{A^{i}}_{j}{A^{j}}_{i}/K^{2}}\leq{\cal O}(\varepsilon)$, was obtained
under the slow-roll approximation. In what follows, considering a more general
matter Lagrangian density (2.20), we show that this inequality holds at the
leading order of the gradient expansion, when $\varepsilon\ll 1$, $F$
satisfies $\partial_{t}F\geq 0$, and the scalar fields are not ghosts. While
we assume $\varepsilon\ll 1$, other slow-roll parameters can be large likewise
in the ultra slow-roll model, which will be discussed in the next subsection.

Using Eq. (4.19) and ignoring the decaying contributions, which are suppressed
by $1/(F\sqrt{g})$, we obtain

$\displaystyle\sqrt{\frac{{A^{i}}_{j}{A^{j}}_{i}}{K^{2}}}=\frac{8\pi
G}{F\sqrt{g}K}\sqrt{d(d-1)}\pi_{A}^{i}\pi_{A}^{j}\int^{\psi(t)}_{\psi_{*}}\frac{d\psi^{\prime}}{\sqrt{g}K{\cal
F}_{Y}}g_{ij}(\psi^{\prime},\,\hbox{\boldmath{$x$}})\,,$ (4.22)

at the leading order of the gradient expansion, where we have ignored the time
variation of $\gamma_{ij}=e^{-2\psi}g_{ij}$ during the time scale of the
expansion, ${\cal O}(1)/K$, based on ${A^{i}}_{j}{A^{j}}_{i}/K^{2}<{\cal
O}(\varepsilon)\ll 1$. Using Eqs. (4.6), (4.9), (4.10), and (4.21), we obtain

$\displaystyle\frac{16\pi G}{Fd}\frac{Y{\cal
F}_{Y}}{K^{2}}\leq\frac{d}{2}\frac{\rho+P}{\rho}\leq\varepsilon\,.$ (4.23)

To show the first $\leq$, we have used $X^{IJ}{\cal F}_{IJ}\geq 0$, which is
ensured from the absence of the ghost instability. The equality holds, in the
absence of the scalar fields, while in such cases there is no enhancement of
the gauge fields either. Using

$\displaystyle\frac{8\pi G}{F}\frac{g_{ij}\pi^{i}_{A}\pi^{j}_{A}}{dK^{2}g{\cal
F}_{Y}}\leq\varepsilon\,,$ (4.24)

which can be obtained from Eqs. (4.16) and (4.23), in Eq. (4.22), we arrive at

$\displaystyle\sqrt{\frac{{A^{i}}_{j}{A^{j}}_{i}}{K^{2}}}\leq\sqrt{d^{3}(d-1)}\,\varepsilon\int^{\psi}_{\psi_{*}}d\psi^{\prime}\frac{(F\sqrt{g}K\varepsilon)_{\psi^{\prime}}}{(F\sqrt{g}K\varepsilon)_{\psi}}\,,$
(4.25)

where the numerator and the denominator are evaluated for $\psi^{\prime}$ and
$\psi$, respectively. When $(F\sqrt{g}K\varepsilon)$ increases in time, the
integral is dominated by the upper end, giving the upper bound of ${\cal
O}(\varepsilon)$. In particular, for a slow-roll inflation, using
$F\sqrt{g}K\varepsilon\propto e^{d\psi}$, we can compute the upper bound on
the shear more accurately as

$\displaystyle\sqrt{\frac{{A^{i}}_{j}{A^{j}}_{i}}{K^{2}}}\leq\sqrt{d(d-1)}\,\varepsilon\,.$
(4.26)

Meanwhile, when $(F\sqrt{g}K\varepsilon)$ decreases in time likewise in the
ultra slow-roll model, the integral is dominated by the lower end, giving the
upper bound of ${\cal
O}(\varepsilon_{*})\times(F\\!\sqrt{g}K)_{*}/(F\\!\sqrt{g}K)$.

Finally, let us discuss an application of the g$\delta N$ formalism. As has
been argued, when the $[$approx${\cal G}_{\epsilon}$$]~{}$condition does not
hold, e.g., in the absence of charged matter fields, the initial condition for
the g$\delta N$ formalism should be given by using
$\\{\varphi_{*}^{a^{\prime}}\\}^{\prime}$. Perturbing Eq. (4.3), we can
compute $\delta\psi(t_{f},\,\hbox{\boldmath{$x$}})$ as

$\displaystyle\delta\psi(t_{f},\,\hbox{\boldmath{$x$}})=\sum_{a^{\prime}}\frac{\partial
N}{\partial\bar{\varphi}^{a^{\prime}}_{*}}\delta\varphi^{a^{\prime}}_{*}+\frac{1}{2}\sum_{a^{\prime},\,b^{\prime}}\frac{\partial^{2}N}{\partial\bar{\varphi}^{a^{\prime}}_{*}\partial\bar{\varphi}^{b^{\prime}}_{*}}\delta\varphi^{a^{\prime}}_{*}\delta\varphi^{b^{\prime}}_{*}+\cdots\,.$
(4.27)

Since the gauge constraint ${\cal H}_{\rm U(1)}$ should be left unsolved, the
longitudinal mode of $\pi^{i}_{A*}$ should be also included in computing the
coefficients of the expansion,
$\partial^{n}N/\partial\bar{\varphi}^{a^{\prime}_{1}}\cdots\partial\bar{\varphi}^{a^{\prime}_{n}}$.
However, whether the longitudinal mode is kept or not does not affect the
computation of the coefficients, since there is no distinction between the
longitudinal and transverse modes in the homogeneous limit. Meanwhile, the
initial fluctuation $\delta\varphi^{a^{\prime}}_{*}$ should be given,
satisfying also ${\cal G}_{\epsilon}$ properly.

To compute the curvature perturbation, we also need to include the traceless
part of $\gamma_{ij}$ (see Eq. (4.4)) by solving the equations ${\cal E}$ in
the homogeneous limit. This can yield a difference from the result in Ref.
[44], especially when the shear is not negligible. A more detailed analysis is
left elsewhere.

####  4.2.2 Charged scalar fields

Next, let us consider a model with charged scalar fields, whose equations of
motion are given by

$\displaystyle\frac{1}{N\sqrt{g}}\partial_{t}\left({\cal
F}_{IJ}\sqrt{g}\frac{\dot{\phi}^{J}}{N}\right)+g^{ij}A_{i}A_{j}{\cal
F}_{IJ}e^{(I)}e^{(J)}\phi^{J}-2\frac{\partial{\cal
F}}{\partial\phi^{I*}}={\cal O}(\epsilon)\,.$ (4.28)

As is known and also as one can confirm by using Eq. (4.13), through the Higgs
mechanism, a non-zero vacuum expectation value of the charged scalar fields
can induce the time dependent mass term of $A^{i}$, breaking the conformal
symmetry [18]. Equation (4.14) indicates that as long as the field space
metric ${\cal F}_{IJ}$ is positive definite, the acquired mass of the gauge
field is positive. Once the gauge symmetry is spontaneously broken, the gauge
constraint ${\cal H}_{\rm U(1)}$ can be solved locally, likewise the
corresponding equation for a massive vector field. Namely, the U(1) gauge
constraint no longer belongs to ${\cal G}_{\epsilon}$. Naively, this is
because with the mass gap, the gauge fields are screened in the large scale
limit. Therefore, when there exist non-vanishing charged scalar fields in the
limit $\epsilon\to 0$, the $[$approx${\cal G}_{\epsilon}$$]~{}$condition holds
even in the presence of gauge fields. As a result, the initial conditions for
the g$\delta N$ formalism can be also given by using only the physical degrees