# Gauge-invariant perturbations at a quantum gravity bounce

Steffen Gielen<EMAIL_ADDRESS>Lisa Mickel
<EMAIL_ADDRESS>School of Mathematics and Statistics, University of
Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom

###### Abstract

We study the dynamics of gauge-invariant scalar perturbations in cosmological
scenarios with a modified Friedmann equation, such as quantum gravity bouncing
cosmologies. We work within a separate universe approximation which captures
wavelengths larger than the cosmological horizon; this approximation has been
successfully applied to loop quantum cosmology and group field theory. We
consider two variables commonly used to characterise scalar perturbations: the
curvature perturbation on uniform-density hypersurfaces $\zeta$ and the
comoving curvature perturbation $\mathcal{R}\,$. For standard cosmological
models in general relativity as well as in loop quantum cosmology, these
quantities are conserved and equal on super-horizon scales for adiabatic
perturbations. Here we show that while these statements can be extended to a
more general form of modified Friedmann equations similar to that of loop
quantum cosmology, in other cases, such as the simplest group field theory
bounce scenario, $\zeta$ is conserved across the bounce whereas $\mathcal{R}$
is not. We relate our results to approaches based on a second order equation
for a single perturbation variable, such as the Mukhanov–Sasaki equation.

##  I Introduction

On large scales, the observable Universe can be described in simple terms: it
is very close to a homogeneous, isotropic (and probably spatially flat,
although that is less clear closedFLRW ; closedFLRW2 )
Friedmann–Lemaître–Robertson–Walker (FLRW) universe, with nearly scale-
invariant small scalar perturbations. The challenge for all approaches to
modern cosmology is to find an explanation for this observed structure, and to
resolve various puzzles inherent to the currently accepted $\Lambda$CDM model
of cosmology. This challenge is often seen as an opportunity for theories of
quantum gravity to connect to observations, in particular since the
$\Lambda$CDM model features the Big Bang singularity which signals its own
fundamental incompleteness. The relative simplicity of our Universe means that
one does not need to understand quantum gravity in full generality to say
something about cosmology: all that is needed is a formalism powerful enough
to deal with approximately homogeneous and isotropic universes. Various
quantum-gravity inspired cosmological scenarios can indeed describe the
evolution of perturbations across a quantum bounce BounceReview .

The description of cosmological perturbations in the standard framework is
based on gauge-invariant perturbation variables cosmopert , characterised by
their invariance under infinitesimal diffeomorphisms, which correspond to
physical perturbations. On top of this notion of gauge invariance, there are
gauge-invariant perturbation variables with a particularly direct physical
interpretation. The curvature perturbation on uniform-density hypersurfaces
$\zeta$ is directly related to cosmological observations at late times, in
particular of the cosmic microwave background (CMB) BounceReview . The notion
of what characterises a ‘good’ cosmological perturbation variable may not
extend straightforwardly to quantum gravity, where the notions of gauge
invariance and diffeomorphisms may be modified BojowaldPaily , or where one
may not have access to an effective action for perturbations. The physical
mechanism for the generation of cosmological perturbations may also be
different from the most commonly assumed framework of inflation, where they
arise from quantum fluctuations on an effectively classical background. It is
important to be clear about which assumptions from standard perturbation
theory are carried over to a particular quantum gravity formalism of interest.

In this paper we study gauge-invariant scalar perturbations in quantum gravity
bounce scenarios governed by a modified Friedmann equation which reduces to
general relativity at low energies but includes high-curvature corrections.
This starting point is familiar from the standard effective dynamics of Loop
Quantum Cosmology (LQC) LQC ; LQC_2 but clearly more general111As a basic
example, generalisations of LQC can already lead to more complicated Friedmann
equations GLQCnew .. We do not assume an effective spacetime description of
perturbations, but ask what properties of the dynamics of perturbations can be
obtained from the Friedmann equation and homogeneous matter dynamics alone.
Consequently, we work in the separate universe picture for long-wavelength
cosmological perturbations SepUniv ; SepUniv_2 ; hamSU , which suggests that
such perturbations can be well approximated by a universe consisting of many
independent, locally homogeneous patches, each governed by a local Friedmann
equation and dynamical equations for matter. The separate universe approach
has already been studied in LQC LQCsepUniv and Group Field Theory (GFT)
GFTsepUniv . Our work extends the results of LQCsepUniv to more general
modified Friedmann equations such as those appearing in GFT, while also going
beyond GFTsepUniv in that we study gauge-invariant perturbation variables.

In standard cosmology the quantity $\zeta$ is conserved on super-horizon
scales for adiabatic perturbations, $\zeta^{\prime}=0$ (see, e.g., conserved ;
here and in the following ′ refers to derivative with respect to an arbitrary
time coordinate). This property is very important: it means one only has to
follow the evolution of cosmological perturbations while they are within the
Hubble horizon. Typically, in a cosmological scenario that aims to solve the
horizon problem, perturbations are initially generated deep inside the Hubble
horizon, then leave the horizon as the Universe expands or contracts, only to
re-enter later as amplified, classical perturbations. In bounce scenarios, the
point of exiting the horizon is often in a contracting phase before the bounce
and re-entry happens in the subsequent expansion phase BounceReview . (A
subtle point which we will get back to later is that at the bounce itself the
horizon is necessarily infinite so all modes are sub-horizon.) We review the
derivation of the conservation law for $\zeta$ from which one can readily see
that it will continue to hold for adiabatic perturbations in quantum gravity
scenarios of interest, where the continuity equation remains unaltered. We
also study a related quantity, the comoving curvature perturbation
$\mathcal{R}$. This quantity usually satisfies a similar conservation law for
long-wavelength modes, $\mathcal{R}^{\prime}=0$, and in general relativity one
can show that $-\zeta=\mathcal{R}$ on super-horizon scales (with appropriate
sign conventions). This equality is no longer guaranteed if one goes beyond
general relativity as we do here. Indeed the simplest GFT bounce scenario
gives an example of quantum-gravity inspired cosmological dynamics for which
$\zeta$ is still conserved, but $\mathcal{R}$ is not.

This article is organised as follows: In sec. II, we give a brief introduction
to standard cosmological perturbation theory, including the definitions of
$\zeta$ and $\mathcal{R}$, and introduce the separate universe framework. We
then proceed to rederive the well-known conservation law for $\zeta$ in the
case of adiabatic perturbations in sec. III. The main results of this paper
are contained in sec. IV: we first derive perturbation equations for long-
wavelength modes starting from a general modified Friedmann equation without
specifying a lapse or choosing a specific gauge for perturbation variables. We
then comment on the meaning of different gauge choices in the separate
universe framework (sec. IV.1). Working in the comoving gauge, we study a
class of modified Friedmann equations, where the modification is a function of
the energy density $\rho$ only (a special case of which is LQC), in sec. IV.2.
We find that for this particular case $\mathcal{R}^{\prime}=\zeta^{\prime}=0$
continues to hold. In sec. IV.3 we consider a modified Friedmann equation
obtained from GFT as an example for which $\mathcal{R}^{\prime}\neq 0$. Our
main results are based on solving first-order equations of motion, but in sec.
V we compare this strategy to approaches based on a single second order
equation of motion for a single perturbation variable, such as the
Mukhanov–Sasaki equation. We finally conclude in sec. VI.

##  II Cosmological perturbations

The usual starting point in cosmological perturbation theory is to model the
Universe as a flat FLRW universe with inhomogeneous perturbations. We will
follow this assumption even though the observational status of spatial
curvature is not fully settled. This is because we are interested in the
behaviour of perturbations near a bounce, where spatial curvature would be
subdominant, and because the quantum gravity scenarios we are interested in
might prefer flat FLRW geometries (see, e.g., GFTscalarcosmo ;
GFTscalarcosmo_2 for the situation in GFT cosmology). The general form of the
perturbed line element at linear order for scalar perturbations, following
standard conventions cosmopert ; BaumannNotes , is 222The tilde over
$\tilde{\Phi}$ is used to distinguish this lapse perturbation variable (which
is not gauge-invariant) from its gauge-invariant analogue, the Bardeen
variable $\Phi$.

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$
$\displaystyle-N^{2}(t)\left(1+2\tilde{\Phi}(t,x^{i})\right){\rm
d}t^{2}+2N(t)a(t)\,\partial_{i}B(t,x^{i})\,{\rm d}t\,{\rm d}x^{i}$ (1)
$\displaystyle+a^{2}(t)\left[\left(1-2\psi(t,x^{i})\right)\delta_{ij}+2\partial_{i}\partial_{j}E(t,x^{i})\right]{\rm
d}x^{i}\,{\rm d}x^{j}\,.$

For concreteness we have added the functional dependence of all variables
explicitly: we have the background scale factor $a(t)$ and lapse function
$N(t)$ (which only depend on time) and perturbations $\tilde{\Phi},\psi,B$ and
$E$, which in general depend on space and time.

In cosmological bounce scenarios, departures from general relativity are
expected to become relevant near the bounce, where most modes are super-
horizon (as mentioned earlier, at the bounce itself all modes are sub-
horizon). We will therefore only be interested in a long-wavelength
approximation in which spatial gradients are neglected, so that

$\tilde{\Phi}(t,x^{i})\rightarrow\tilde{\Phi}(t)\,,\quad\psi(t,x^{i})\rightarrow\psi(t)\,,\quad\partial_{i}B\rightarrow
0\,,\quad\partial_{i}\partial_{j}E\rightarrow 0\,.$ (2)

$\tilde{\Phi}$ and $\psi$ can then be seen as perturbations in the background
quantities $N$ and $a$, which is the essence of the separate-universe idea:
one can consider a universe composed of many independent, locally homogeneous
flat FLRW patches, with local lapse and scale factor

$N_{{\rm loc}}=N(1+\tilde{\Phi})\,,\quad a_{{\rm loc}}=a(1-\psi)\,,$ (3)

where $N$ and $a$ are now considered as averages over many patches, and
$\tilde{\Phi}$ and $\psi$ as small quantities ($O(\epsilon)$ with $\epsilon\ll
1$) characterising the difference between one patch and the average. We see
that the metric for each patch would then be given by (1) with (2) (here and
throughout the paper, quantities of quadratic and higher order in perturbation
variables are $O(\epsilon^{2})$ and will be dropped, as we work within linear
perturbation theory). The separate universe picture is particularly useful in
quantum gravity scenarios that do not allow for a satisfactory description of
inhomogeneities.

If matter is described by an energy density $\rho$ and pressure $P$, one can
analogously introduce perturbation variables $\delta\rho$ and $\delta P$ with

$\rho_{{\rm loc}}=\rho+\delta\rho\,,\quad P_{{\rm loc}}=P+\delta P\,.$ (4)

In the case where the matter content of the very early universe is given by a
single scalar field $\phi$ with potential $\tilde{V}(\phi)$ (such as in
inflation, LQC or GFT), the energy density and pressure are given by

$\displaystyle\rho=\frac{\phi^{\prime 2}}{2N^{2}}+\tilde{V}(\phi)\,,\qquad
P=\frac{\phi^{\prime 2}}{2N^{2}}-\tilde{V}(\phi)$ (5)

and the dynamics of the scalar field are determined by the Klein–Gordon
equation

$\displaystyle\phi^{\prime\prime}-\frac{N^{\prime}}{N}\phi^{\prime}+N^{2}\frac{{\rm
d}\tilde{V}(\phi)}{{\rm d}\phi}+3\mathfrak{H}\phi^{\prime}=0\,,$ (6)

where we define $\mathfrak{H}=a^{\prime}/a$ (which reduces to the usual Hubble
parameter $H$ if $t$ is chosen to be proper time). For $\phi_{{\rm
loc}}=\phi+\delta\phi$ one obtains the perturbations for the energy density
and pressure by perturbing (5) at linear order:

$\displaystyle\delta\rho=\frac{\phi^{\prime
2}}{N^{2}}\left(\frac{\delta\phi^{\prime}}{\phi^{\prime}}-\tilde{\Phi}\right)+\frac{{\rm
d}\tilde{V}(\phi)}{{\rm d}\phi}\delta\phi\,,\qquad\delta P=\frac{\phi^{\prime
2}}{N^{2}}\left(\frac{\delta\phi^{\prime}}{\phi^{\prime}}-\tilde{\Phi}\right)-\frac{{\rm
d}\tilde{V}(\phi)}{{\rm d}\phi}\delta\phi\,.$ (7)

In cosmological perturbation theory, there exist two notions of gauge
invariance, one related to the choice of the lapse $N$ and another to the
choice of the coordinate system of perturbations. The first allows an
arbitrary change of the background time coordinate $t\to f(t)$; the second
encodes the invariance under _infinitesimal_ diffeomorphisms $x^{\mu}\to
x^{\mu}+\xi^{\mu}$, where $\xi^{\mu}$ is $O(\epsilon)$. It is the second gauge
freedom we refer to when we discuss gauge invariance and different gauge
choices in the following, keeping in mind that our separate universe
approximation will reduce the relevant gauge transformations to those of the
form $t\to t+\xi^{0}(t)$. For more details, see cosmopert ; BaumannNotes . The
metric and matter perturbation variables in (1) are not gauge-invariant, but
have certain transformation properties under infinitesimal diffeomorphisms,
such that gauge-invariant variables need to be obtained by combining
perturbation variables in a suitable manner. The two gauge-invariant
perturbation variables we have mentioned above are defined by (where different
sign conventions for $\zeta$ exist in the literature and we follow the one in
conserved and going back to BST1983 )

$-\zeta=\psi+\frac{\mathfrak{H}}{\rho^{\prime}}\delta\rho\,,\quad\mathcal{R}=\psi+\frac{\mathfrak{H}}{\phi^{\prime}}\delta\phi\,,$
(8)

where the second definition specifically refers to a scalar field. As such,
the variables in (8) are invariant under infinitesimal diffeomorphisms acting
on the metric and matter fields at $O(\epsilon)$ cosmopert . Importantly, we
will assume that the same notion of gauge invariance applies in our quantum
gravity scenarios of interest so that (8) are gauge-invariant, and hence
potentially observable, also for our modified gravitational dynamics. This is
certainly an assumption given that the action of diffeomorphisms might receive
quantum corrections as in LQC LQCanomaly , but in the absence of a full
spacetime picture we need to make such an assumption to proceed. Since we are
only interested in long-wavelength perturbations, we only need to assume that
diffeomorphisms act as in general relativity in that limit, which is true in
LQC and might be seen as an admissible assumption to make: naively, the
longest wavelengths should be least sensitive to any quantum gravity
corrections.

Within general relativity coupled to a scalar field, one can show that
$-\zeta=\mathcal{R}+O(k^{2})$, where $k$ is the wavenumber in a Fourier
decomposition, so that in the separate universe limit $k\rightarrow 0$ (or,
more accurately, a limit in which the physical wavelength of a mode is much
larger than the Hubble horizon: $k\ll\frac{a^{\prime}}{N}$) we have
$-\zeta=\mathcal{R}$. For this reason the two quantities are often treated as
interchangeable when long-wavelength modes are studied, but again it is not
clear whether a similar relation holds in more general cosmological models of
the type we are interested in. We would like to point out that during slow-
roll inflation one can approximate

$\rho=\frac{\phi^{\prime
2}}{2N^{2}}+\tilde{V}(\phi)\approx\tilde{V}(\phi)\quad\Rightarrow\;\rho^{\prime}\approx\phi^{\prime}\frac{{\rm
d}\tilde{V}(\phi)}{{\rm d}\phi}$ (9)

and hence in this case $-\zeta\approx\mathcal{R}$ without using any
gravitational field equations. We will not be interested in slow-roll
inflation, but consider scalar fields with general potentials such that the
high-energy regime can be dominated by kinetic energy.

##  III Simple conservation law for $\zeta$

We start by deriving the simplest conservation law for the perturbation
variables we are considering: $\zeta$ is conserved on large scales for
adiabatic matter perturbations if the continuity equation for matter is
unchanged. This is an old result in cosmology conserved which extends directly
to many quantum bounce scenarios, in particular to LQC LQC ; LQC_2 and GFT
GFTscalarcosmo ; GFTscalarcosmo_2 ; deparamcosmo which do not introduce any
alterations to the dynamics of the matter content of the universe. Rederiving
this result illustrates the philosophy behind the separate-universe approach.

The continuity equation satisfied by the background variables reads

$\rho^{\prime}+3\mathfrak{H}(\rho+P)=0\,,$ (10)

and if we introduce locally perturbed variables according to (3) and (4),
assuming that the continuity equation also holds in each local patch, we find
the perturbed continuity equation

$\delta\rho^{\prime}+3\mathfrak{H}(\delta\rho+\delta
P)-3\psi^{\prime}(\rho+P)=0\,,$ (11)

using that $\mathfrak{H}_{{\rm loc}}=\mathfrak{H}-\psi^{\prime}$. Hence,

$-\zeta^{\prime}=\psi^{\prime}+\left(\frac{\mathfrak{H}}{\rho^{\prime}}\delta\rho\right)^{\prime}=-\left(\frac{1}{3(\rho+P)}\right)^{\prime}\delta\rho+\frac{\mathfrak{H}(\delta\rho+\delta
P)}{\rho+P}=\frac{\rho^{\prime}+P^{\prime}}{3(\rho+P)^{2}}\delta\rho+\frac{\mathfrak{H}(\delta\rho+\delta
P)}{\rho+P}$ (12)

using (10) and (11). The assumption of adiabatic perturbations means that we
can write $\frac{\delta P}{\delta\rho}=\frac{P^{\prime}}{\rho^{\prime}}$ and
one finally obtains, after again using (10), that $\zeta^{\prime}=0$ for these
perturbations. This argument relies on the long-wavelength limit, since the
perturbed continuity equation (11) would in general contain terms involving
spatial derivatives (sec. V.1). It does however not use the gravitational
dynamics, and hence holds in many scenarios beyond general relativity.

One commonly introduces the equation of state parameter $w=\frac{P}{\rho}$ and
the sound speed $c_{s}^{2}=\frac{P^{\prime}}{\rho^{\prime}}$ and using these
definitions together with the continuity equation (10), one finds that

$w^{\prime}=-3\mathfrak{H}(w+1)(c_{s}^{2}-w)\,.$ (13)

In the case of a perfect fluid, $w$ is constant, $c_{s}^{2}=w$, and the
adiabaticity condition is always satisfied. A scalar field can mimic a perfect
fluid if the potential is chosen such that it has a constant equation of state
parameter (at least for the time scales one is interested in). In particular,
this is the case for a massless scalar field, where $w=1$. For more general
scalar field dynamics, there is exchange between kinetic and potential energy
in the scalar field and perturbations can not generally be assumed to be
adiabatic; however, in general relativity a single scalar field can generally
only produce adiabatic perturbations on large scales, so that
$\zeta^{\prime}=0$ GRscalarAdiabatic . As we will show in sec. IV.2,
$\zeta^{\prime}=0$ still holds for a single scalar field for specific forms of
modified gravitational dynamics. However, for general modifications to the
Friedmann equation, one cannot conclude that a single scalar field with an
arbitrary potential induces only adiabatic perturbations.

##  IV Generalised Friedmann dynamics and their perturbations

We now introduce gravitational dynamics given by generalised Friedmann
equations of the type expected in many approaches to quantum gravity. For this
generalised Friedmann equation, we write

$\frac{\mathfrak{H}^{2}}{N^{2}}=\frac{\kappa}{3}\rho\,\mathcal{F}\,,$ (14)

where $N$ is a general choice of lapse, $\kappa=8\pi G$ is a rescaled Newton’s
constant and the function $\mathcal{F}$ encodes the quantum gravity
corrections to the Friedmann equation of general relativity. ($\mathcal{F}=1$
then corresponds to the general relativistic Friedmann equation.)

The perturbation equations that follow in this section are independent of the
specific form of $\mathcal{F}$, but we would like to give two examples that we
will get back to later, namely the modified Friedmann equations of LQC and
GFT. As we will see, these two examples characterise two qualitatively
different cases, where $\mathcal{F}$ and its perturbations either depend only
on $\rho$ and $\delta\rho$ (sec. IV.2) or on other variables as well (sec.
IV.3). When deriving the effective Friedmann equation in LQC or GFT one
assumes that the matter content of the universe is given by a single massless
scalar field, such that the energy density is given by
$\rho=\frac{\pi_{\phi}^{2}}{(2a^{6})}$, where the scalar field momentum
$\pi_{\phi}$ is a constant of motion. In the standard effective dynamics of
LQC effectiveLQC

$\mathcal{F}_{\rm LQC}=1-\frac{\rho}{\rho_{{\rm c}}}\,,$ (15)

where $\rho_{{\rm c}}$ is a universal, maximal energy density, which
characterises the regime in which quantum gravity corrections become relevant.
For phenomenological applications $\mathcal{F}_{\rm LQC}$ is sometimes assumed
to hold also for massive scalar fields, even if this cannot be directly
derived from the quantum theory (see, e.g., LQCsepUniv ; LQC_genRho ;
LQC_matterBounce ). In GFT more general forms appear such as deparamcosmo

$\mathcal{F}_{\rm GFT}=1+\frac{v_{0}}{a^{3}}+\frac{\mathcal{Y}}{a^{6}}\,.$
(16)

Here, $v_{0}$ is a fixed constant that has the unit of volume and gets its
interpretation from the underlying quantum theory. $\mathcal{Y}$ on the other
hand is a constant of motion rather than a fundamental parameter and so in the
separate universe picture will vary from one patch to another,
$\delta\mathcal{Y}\neq 0$. Its value is related to the volume of the universe
at the bounce.

To obtain the dynamics of perturbations for a modified Friedmann equation
while being agnostic about the details of the underlying gravitational theory,
we proceed as follows. An equation of motion for $\mathfrak{H}$ is obtained
from the time derivative of (14)333The equation is written in this form for
convenience, however, division by $\mathfrak{H}$ and $\mathcal{F}$ is not
defined at the bounce.,

$\displaystyle\frac{\mathfrak{H}^{\prime}}{\mathfrak{H}}=\frac{N^{\prime}}{N}+\frac{1}{2}\left(\frac{\rho^{\prime}}{\rho}+\frac{\mathcal{F}^{\prime}}{\mathcal{F}}\right)\,,$
(17)

and the first and second order equations of motion for the metric perturbation
$\psi$ are obtained by perturbing (14) and (17) at linear order (or
equivalently, perturbing (14) and then taking the time derivative). Giving
different equivalent forms of these equations, we have

$\displaystyle\begin{split}\mathfrak{H}\psi^{\prime}=&-\mathfrak{H}^{2}\left(\tilde{\Phi}+\frac{\delta\rho}{2\rho}+\frac{\delta\mathcal{F}}{2\mathcal{F}}\right)\\\
=&-\frac{\kappa}{3}N^{2}\left(\rho\mathcal{F}\,\tilde{\Phi}+\frac{1}{2}\left(\mathcal{F}\delta\rho+\rho\delta\mathcal{F}\right)\right)\,,\end{split}$
(18)
$\displaystyle\begin{split}-\psi^{\prime\prime}=&\left(-\frac{N^{\prime}}{N}-\frac{\mathcal{F}^{\prime}}{2\mathcal{F}}+3\mathfrak{H}\frac{\rho+P}{\rho}\right)\psi^{\prime}+\mathfrak{H}\,\tilde{\Phi}^{\prime}+\frac{\mathfrak{H}}{2}\left(\frac{\delta\mathcal{F}^{\prime}}{\mathcal{F}}-\frac{\mathcal{F}^{\prime}}{\mathcal{F}^{2}}\delta\mathcal{F}\right)\\\
&+\frac{\kappa}{2}N^{2}\mathcal{F}(\rho+P)\left(\frac{\delta\rho}{\rho}-\frac{\delta\rho+\delta
P}{\rho+P}\right)\\\
=&-\frac{N^{\prime}}{N}\psi^{\prime}+\mathfrak{H}\,\tilde{\Phi}^{\prime}-\frac{\kappa}{2}N^{2}\mathcal{F}(P+\rho)\left(\frac{(\delta
P+\delta\rho)}{P+\rho}+\frac{\delta\mathcal{F}}{\mathcal{F}}+2\tilde{\Phi}\right)\\\
&+\frac{\mathfrak{H}}{2}\frac{\mathcal{F}^{\prime}}{\mathcal{F}}\left(-\frac{\delta\mathcal{F}}{2\mathcal{F}}+\frac{\delta\rho}{2\rho}+\tilde{\Phi}+\frac{\delta\mathcal{F}^{\prime}}{\mathcal{F}^{\prime}}\right)\,.\end{split}$
(19)

These forms can be transformed into each other by using the Friedmann equation
(14) and by using (18) to rewrite terms proportional to $\psi^{\prime}$ in
(19). For $\mathcal{F}=1$ (and hence $\mathcal{F}^{\prime}=0\,,\
\delta\mathcal{F}=0\,,\ \delta\mathcal{F}^{\prime}=0$) the above reduce to the
standard general relativistic equations of motions for perturbations in the
long-wavelength limit (spatial gradients vanish) as obtained from the Einstein
field equations.

So far, we have not assumed a specific form of matter content. Neither have we
chosen a specific form of lapse, nor made a gauge choice for perturbation
variables. A popular choice of time coordinate is conformal time, where $N=a$,
and certain gauge choices simplify the perturbation equations further. We
comment on the question of gauge in the following subsection, and then discuss
a specific class of functions $\mathcal{F}$ for which the perturbation
equations above simplify in an LQC-like fashion.

###  IV.1 Gauge choices in the separate universe approximation

Even though one is ultimately interested in the dynamics of gauge-invariant
quantities, it is often useful to carry out calculations in a specific gauge.
A popular gauge choice in cosmology is the Newtonian or longitudinal gauge, in
which all anisotropic perturbations of the metric tensor are set to zero
($B=E=0$). This gauge was also used in previous studies of the separate
universe framework in LQC LQCsepUniv and GFT GFTsepUniv . Here we will instead
work in the comoving gauge, which for scalar matter is defined by
$\delta\phi=0$ and $E=0\,$.444There exist different conventions for the
comoving gauge: in Giesel_2018 the comoving gauge is defined by $\delta\phi=0$
and $B=0$ instead. The above convention is used in, e.g., BaumannNotes . (As
the matter content in LQC and GFT is taken to be a scalar field, we focus on
this case in the following.) For scalar matter the lapse perturbation is
directly related to the perturbation of the energy density and pressure (7) in
comoving gauge,

$\displaystyle\delta\rho=-\frac{\phi^{\prime
2}}{N^{2}}\tilde{\Phi}=-(\rho+P)\tilde{\Phi}=\delta P\,$ (20)

and one can write, using (20) and the continuity equation (10),

$\displaystyle-\zeta=\mathcal{R}+\mathfrak{H}\frac{\delta\rho}{\rho^{\prime}}=\mathcal{R}+\frac{\tilde{\Phi}}{3}\,.$
(21)

The reasons we choose the comoving gauge are threefold. Firstly, the comoving
curvature perturbation takes a particularly simple form, $\mathcal{R}=\psi$.
Secondly, when working with relational settings such as GFT, where the scalar
field takes the role of a physical clock GFTscalarcosmo ; GFThamiltonian , the
comoving gauge is simply the statement that at an instant of time all patches
of the separate universe picture have the same clock value. The third reason
is more subtle and is connected to the application of gauge choices in general
cosmological perturbation theory to the separate universe picture. As already
pointed out in hamSU , fixing the Newtonian gauge $E=B=0$ does not provide an
additional prescription in the separate universe picture, as anisotropic
degrees of freedom are already absent in this approximation (see (2)). In
general relativity, a relation between the metric and the lapse perturbation
in Newtonian gauge is obtained from the off-diagonal spatial components of the
perturbed Einstein field equations ($\delta G_{i\neq j}=\kappa\delta T_{i\neq
j}$) cosmopert , which for $E=B=0$ reduce to

$\displaystyle\partial_{i}\partial_{j}\psi-\partial_{i}\partial_{j}\tilde{\Phi}=0\,.$
(22)

They imply that $\psi=\tilde{\Phi}+x^{i}h_{i}(t)+g(t)\,$, where $h$ and $g$
are arbitrary homogeneous functions. One then usually sets $\tilde{\Phi}=\psi$
arguing that perturbations should average out to zero ($\int{\rm
d}^{3}x\;\psi=0=\int{\rm d}^{3}x\;\tilde{\Phi}$) since any homogeneous
contribution to the perturbations could be absorbed in the background
cosmopert . This argument would forbid any nontrivial $h_{i}(t)$ or $g(t)\,$.
In the separate universe framework all spatial gradients automatically vanish
and the off-diagonal components of the Einstein field equations are trivially
satisfied, so there is no analogue of (22). Equivalently, requiring that
$\psi=\tilde{\Phi}+g(t)$ is not a constraint as the perturbations
$\psi,\,\tilde{\Phi}$ are in any case only functions of time in the separate
universe picture. Fixing $\tilde{\Phi}=\psi$ for the Newtonian gauge in the
separate universe framework is therefore an additional assumption, somewhat
harder to justify than in usual cosmological perturbation theory. This issue
is also discussed in 0801Wands , where the authors introduce a ‘pseudo-
longitudinal gauge’, which ensures $\psi=\tilde{\Phi}$ throughout as long as
it is assumed to hold in some limit.555 It is also discussed for the
Hamiltonian framework in hamSU . In the Hamiltonian picture the relation
$\psi=\tilde{\Phi}$ cannot be recovered due to the absence of the
diffeomorphism constraint in the separate universe framework. In hamSU the
authors recover $\tilde{\Phi}=\psi$ by redefining the Newtonian gauge, where
the redefinition relies on a relation between perturbation variables obtained
from the diffeomorphism constraint.

###  IV.2 A special case: $\mathcal{F}=\mathcal{F}(\rho)$

In LQC, the general perturbation equations take a particularly simple form due
to the specific form of $\mathcal{F}$ (15). In this section we generalise the
LQC case by considering a restricted class of corrected Friedmann equations,
namely those in which $\mathcal{F}$ is a function of the energy density $\rho$
only. In particular, this means that the perturbation of $\mathcal{F}$ is
proportional to the perturbation of the energy density,
$\delta\mathcal{F}=\frac{{\rm d}\mathcal{F}}{{\rm d}\rho}\delta\rho$. Unlike
the case of LQC, the GFT correction given in (16) does _not_ fall in this
category, $\mathcal{F}_{\rm GFT}\neq\mathcal{F}(\rho)$, as $\mathcal{Y}$ is
perturbed as well.

If we define the quantities

$\displaystyle\mathcal{F}_{\rho}:=$ $\displaystyle\frac{{\rm
d}\mathcal{F}}{{\rm d}\rho}\,,\qquad\mathcal{F}_{\rho\rho}:=\frac{{\rm
d}^{2}\mathcal{F}}{{\rm
d}\rho^{2}}\,,\qquad\mathcal{A}:=\mathcal{F}+\mathcal{F}_{\rho}\,\rho\,,$ (23)

we obtain the following relations for quantities derived from $\mathcal{F}$,
using again the continuity equation (10):

$\displaystyle\begin{split}\delta\mathcal{F}=&\,\mathcal{F}_{\rho}\delta\rho\,,\qquad\mathcal{F}^{\prime}=-3\mathfrak{H}(\rho+P)\mathcal{F}_{\rho}\,,\qquad\mathcal{A}^{\prime}=-3\mathfrak{H}(\rho+P)(2\mathcal{F}_{\rho}+\rho\mathcal{F}_{\rho\rho})\,,\\\
\frac{\delta\mathcal{F}^{\prime}}{\mathcal{F}^{\prime}}=&\,\frac{\mathcal{F}_{\rho\rho}\delta\rho}{\mathcal{F}_{\rho}}+\frac{\delta\rho+\delta
P}{\rho+P}-\frac{\psi^{\prime}}{\mathfrak{H}}\,.\end{split}$ (24)

We can then write the second Friedmann equation (17) and generalised
perturbation equations (18)-(19) for the $\mathcal{F}(\rho)$ class of modified
Friedmann equations as

$\displaystyle\mathfrak{H}^{\prime}-\frac{N^{\prime}}{N}\mathfrak{H}$
$\displaystyle=-\frac{\kappa}{2}N^{2}(\rho+P)\mathcal{A}\,,$ (25)
$\displaystyle\mathfrak{H}\psi^{\prime}$
$\displaystyle=-\mathfrak{H}^{2}\tilde{\Phi}-\frac{\kappa}{6}N^{2}\mathcal{A}\,\delta\rho\,,$
(26)
$\displaystyle\begin{split}-\psi^{\prime\prime}&=-\frac{N^{\prime}}{N}\psi^{\prime}+\mathfrak{H}\tilde{\Phi}^{\prime}-\frac{\kappa}{2}N^{2}\left(\mathcal{A}\,\delta
P+2(\rho+P)\mathcal{A}\,\tilde{\Phi}+(\mathcal{F}+\rho(\rho+P)\mathcal{F}_{\rho\rho}+(2P+3\rho)\mathcal{F}_{\rho})\delta\rho\right)\\\
&=-\frac{N^{\prime}}{N}\psi^{\prime}+\mathfrak{H}\tilde{\Phi}^{\prime}-\kappa\phi^{\prime}\mathcal{A}\,\delta\phi^{\prime}-\left(\mathcal{F}_{\rho}+\frac{\rho}{2}\mathcal{F}_{\rho\rho}\right)\kappa\phi^{\prime
2}\,\delta\rho\,,\end{split}$ (27)

where in the last line we used the fact that matter is given by by a scalar
field with $\rho+P=\frac{\phi^{\prime 2}}{N^{2}}$ (the other equations are
general and hold for any matter content). Equations (25)-(27) hold in any
gauge. They correspond to the LQC equations reported in LQCsepUniv (which are
given in conformal time $N=a$, and for a gauge in which $\psi=\tilde{\Phi}$)
for $\mathcal{A}=1-2\frac{\rho}{\rho_{{\rm c}}}$ and
$\mathcal{F}_{\rho}=-\frac{1}{\rho_{{\rm c}}}$.

To make further progress, another equation is needed. In general relativity,
this is the diffeomorphism constraint arising from the mixed time-space
components of the perturbed Einstein field equations, $\delta
G^{0}_{i}=\kappa\delta T^{0}_{i}$, or cosmopert

$\displaystyle\partial_{i}\left(\mathfrak{H}\tilde{\Phi}+\psi^{\prime}-\frac{\kappa}{2}\phi^{\prime}\,\delta\phi\right)=:\partial_{i}D$
$\displaystyle=0\,,$ (28)

which implies $D=s(t)$ where $s$ is an arbitrary homogeneous function. In the
usual formalism, one then argues that perturbations are inhomogeneous
functions over a homogeneous background, and any homogeneous contribution to
$D$ can be absorbed in the homogeneous background dynamics to justify the
requirement that the perturbation variables satisfy $D=0$. This discussion is
analogous to the gauge choice $\psi=\tilde{\Phi}$ in Newtonian gauge that we
discussed in sec. IV.1.

As in the discussion in sec. IV.1, there is no analogue of (28) in the
separate universe framework due to the absence of spatial gradients. However,
in scenarios of interest such as bouncing cosmologies, one expects to set
initial conditions in a regime where general relativity holds and one can
consider the case where spatial gradients are small, but not exactly zero.
Then, at some initial time $t_{0}$ where initial conditions are set, which can
be any time at which the strict $k\to 0$ limit has not been applied yet, (28)
implies $D=0$ for the modes of interest. If one can then show that in general
$D^{\prime}(t)=0$ together with the initial condition $D(t_{0})=0\,$, $D=0$
holds throughout the evolution. This means we obtain another effective
constraint equation for perturbations which, as we will see below, can be used
to infer conservation laws for gauge-invariant perturbation variables. For a
setting with a modified Friedmann equation, (28) represents the low-curvature
limit ($\mathcal{F}\to 1$) of a possibly modified form of $D$. The modified
form for $D$ cannot be derived directly but guessed and justified in
hindsight, as was done for LQC in LQCsepUniv . In the following, we introduce
a differential equation for suitable $D$ in comoving gauge and show that it
holds for any form of $\mathcal{F}(\rho)\,$. We work in the comoving gauge,
but a similar calculation can be carried out in the Newtonian gauge (see app.
A).

Inspired by LQCsepUniv , where the corrections to $D$ for a modified Friedmann
equation appear in the $\delta\phi$ term only, we assume that in comoving
gauge $D$ takes the same form as in general relativity,

$\displaystyle D=\psi^{\prime}+\tilde{\Phi}\mathfrak{H}\,.$ (29)

From the equation for $\psi^{\prime}$ (26) together with (20) we obtain

$\psi^{\prime}=-\mathfrak{H}\tilde{\Phi}+\frac{\kappa}{6}\mathcal{A}\frac{\phi^{\prime
2}}{\mathfrak{H}}\,\tilde{\Phi}$ (30)

and it immediately follows that

$\displaystyle D=\frac{\kappa}{6}\frac{\phi^{\prime
2}}{\mathfrak{H}}\mathcal{A}\tilde{\Phi}\,.$ (31)

We now show that $D$ satisfies

$\displaystyle\mathcal{A}D^{\prime}+W\mathcal{A}D-\mathcal{A}^{\prime}D=0$
(32)

for a certain form of $W$, and with $\mathcal{A}$ defined in (23). Using the
perturbation equation (27) for $\psi^{\prime\prime}$ in the
$\mathcal{F}(\rho)$ case in comoving gauge (so that $\delta\phi^{\prime}=0$),
$D^{\prime}$ reduces to

$\displaystyle
D^{\prime}=\frac{N^{\prime}}{N}\psi^{\prime}+\mathfrak{H}^{\prime}\,\tilde{\Phi}+\kappa\phi^{\prime
2}\left(\mathcal{F}_{\rho}+\frac{\rho}{2}\mathcal{F}_{\rho\rho}\right)\delta\rho=\frac{N^{\prime}}{N}\psi^{\prime}+\mathfrak{H}^{\prime}\,\tilde{\Phi}-\kappa\frac{\phi^{\prime
4}}{N^{2}}\left(\mathcal{F}_{\rho}+\frac{\rho}{2}\mathcal{F}_{\rho\rho}\right)\tilde{\Phi}\,,$
(33)

where we used the relation between $\delta\rho$ and $\tilde{\Phi}$ given in
(20). It then follows that

$\displaystyle\mathcal{A}D^{\prime}+W\mathcal{A}D-\mathcal{A}^{\prime}D=$
$\displaystyle\,\mathcal{A}\tilde{\Phi}\left(\mathfrak{H}^{\prime}-\frac{N^{\prime}}{N}\mathfrak{H}+\frac{\kappa\,\phi^{\prime
2}}{6\mathfrak{H}}\left(\mathcal{A}\frac{N^{\prime}}{N}-\mathcal{A}^{\prime}+W\mathcal{A}\right)-\frac{\kappa\phi^{\prime
4}}{N^{2}}\left(\mathcal{F}_{\rho}+\frac{\rho}{2}\mathcal{F}_{\rho\rho}\right)\right)$
(34) $\displaystyle=$ $\displaystyle\,\kappa\phi^{\prime
2}\mathcal{A}\,\tilde{\Phi}\left(\frac{1}{6\mathfrak{H}}\left(\mathcal{A}\frac{N^{\prime}}{N}-\mathcal{A}^{\prime}+W\mathcal{A}-3\mathfrak{H}\mathcal{A}\right)-\frac{\phi^{\prime
2}}{N^{2}}\left(\mathcal{F}_{\rho}+\frac{\rho}{2}\mathcal{F}_{\rho\rho}\right)\right)$
(35) $\displaystyle=$ $\displaystyle\,\frac{\kappa\phi^{\prime
2}\mathcal{A}^{2}\,\tilde{\Phi}}{6\mathfrak{H}}\left(\frac{N^{\prime}}{N}+W-3\mathfrak{H}\right)\,,$
(36)

using (30), the equation for $\mathfrak{H}^{\prime}\,$, (25), and then
eliminating $\mathcal{A}^{\prime}$ using (24). If we now choose
$W=3\mathfrak{H}-N^{\prime}/N\,$, we obtain (32). Hence, as long as initial
conditions are set in a regime where $D=0$ is satisfied,
$\psi^{\prime}+\tilde{\Phi}\mathfrak{H}=0$ holds at all times.

We can now proceed to study the conservation laws for the gauge-invariant
curvature perturbations $\zeta$ and $\mathcal{R}$ defined in (8) for a
modified Friedmann equation of the $\mathcal{F}=\mathcal{F}(\rho)$ type, which
is one of the main results of this paper. As established in sec. III, $\zeta$
is conserved whenever the adiabaticity condition $\frac{\delta
P}{P^{\prime}}=\frac{\delta\rho}{\rho^{\prime}}$ holds. If we recall that in
comoving gauge, $\delta\rho=-(\rho+P)\tilde{\Phi}=\delta P\,$, it follows from
(31) and $D=0$ that $\tilde{\Phi}=0$ and hence the adiabaticity equation is
always satisfied, irrespective of the explicit form of $\tilde{V}(\phi)\,$.
Therefore, in the $\mathcal{F}(\rho)$ case (which, to repeat, includes both
general relativity and standard LQC), $\zeta$ will always be conserved on
super-horizon scales, and a single scalar matter field cannot introduce non-
adiabatic perturbations. Furthermore, from (21) it follows that
$-\zeta=\mathcal{R}\,$, as in general relativity. While the intermediate steps
in this argument, like the form of the constraint equation and
$\delta\rho=0\,$, are gauge-dependent statements, the implications for the
gauge-invariant variables $\zeta$ and $\mathcal{R}$ hold in any gauge.

Another example for the $\mathcal{F}(\rho)$ case can be found in the modified
Friedmann equation that arises by considering Barrow entropy, as was done in
barrowEntropy . The authors give the modified Friedmann equation as
$\frac{8\pi}{\kappa}\frac{\pi^{\Delta/2}(2+\Delta)}{(2-\Delta)}\left(\frac{\kappa}{8\pi}H^{2}\right)^{1-\Delta/2}=\frac{\kappa}{3}\rho$,
where $N=1$, $\Delta\in[0,1]$ is the parameter of the Barrow entropy and
$\Delta\to 0$ gives the Friedmann equation of general relativity.666Here we
omit the integration constant $c$, which can be identified with the
cosmological constant, as it is subdominant in the early universe; and we
consider the flat case $k=0$. This can be rewritten as
$H^{2}=\frac{\kappa}{3}\rho\,\left(\frac{\kappa^{2}\rho}{24\pi^{2}}\right)^{\frac{\Delta}{2-\Delta}}\left(\frac{2-\Delta}{2+\Delta}\right)^{\frac{2}{2-\Delta}}$,
i.e.,
$\mathcal{F}=\left(\frac{\kappa^{2}\rho}{24\pi^{2}}\right)^{\frac{\Delta}{2-\Delta}}\left(\frac{2-\Delta}{2+\Delta}\right)^{\frac{2}{2-\Delta}}$.
Since this is of the form $\mathcal{F}=\mathcal{F}(\rho)$, we can use the
above result to conclude that also in this scenario $-\zeta=\mathcal{R}\,$ as
in general relativity, leading to the same conservation laws for a single
scalar matter field.

###  IV.3 $\mathcal{F}\neq\mathcal{F}(\rho)$ example: GFT

We now turn to the more general case of $\mathcal{F}\neq\mathcal{F}(\rho)$,
where there is no equation analogous to the diffeomorphism constraint of
general relativity. In this case, we cannot exclude non-adiabatic
perturbations in general, but if we restrict ourselves to the special case of
a scalar field satisfying the adiabaticity condition $\delta
P=c_{s}^{2}\delta\rho\,$, it follows that $\zeta^{\prime}=0\,$, as
demonstrated in sec. III. The quantity $\mathcal{R}$ on the other hand is no
longer conserved: if we insert $\delta\rho=-(1+w)\rho\,\tilde{\Phi}$ (see
(20)) into the equation of motion for $\psi$ (18), we find that in comoving
gauge $\mathcal{R}$ satisfies

$\displaystyle-\frac{\mathcal{R}^{\prime}}{\mathfrak{H}}=\tilde{\Phi}+\frac{\delta\rho}{2\rho}+\frac{\delta\mathcal{F}}{2\mathcal{F}}=(1-w)\frac{\tilde{\Phi}}{2}+\frac{\delta\mathcal{F}}{2\mathcal{F}}\,.$
(37)

Consequently, for generalised Friedmann equations with general $\mathcal{F}$,
$-\zeta=\mathcal{R}$ no longer holds: while $\zeta$ remains constant on
super–horizon scales, $\mathcal{R}$ now has non-trivial dynamics. For a
massless scalar field ($c_{s}^{2}=w=1$), which we consider in the following,
the dynamics in $\mathcal{R}$ are determined only by the expression for
$\mathcal{F}\,$.

In the remainder of this section, we investigate the dynamics of the comoving
curvature perturbation $\mathcal{R}$ in a GFT toy model as established in
deparamcosmo , which leads to an effective Friedmann equation specified by
(16). The GFT framework uses a massless scalar field $\phi$ as the only matter
content of the universe (or rather, it represents the dominant matter content
in the bounce region one is interested in when studying quantum gravitational
effects) and $\phi$ also serves as a relational matter clock. In the special
case of a massless scalar field, $\tilde{V}(\phi)=0\,$, the Klein–Gordon
equation (6) can be solved and the expressions for the energy density and its
perturbation simplify as

$\displaystyle\phi^{\prime}=\frac{\pi_{\phi}N}{a^{3}}\quad\Rightarrow\quad\rho$
$\displaystyle=\frac{\pi_{\phi}^{2}}{2a^{6}}\,,\qquad\delta\rho=2\rho\left(\frac{\delta\pi_{\phi}}{\pi_{\phi}}+3\psi\right)\,,\qquad\rho^{\prime}=-3\mathfrak{H}\frac{\pi_{\phi}^{2}}{a^{6}}=-6\mathfrak{H}\rho\,,$
(38)

where the scalar field momentum $\pi_{\phi}$ is a constant of motion.
Furthermore, for a massless scalar field, the relation between the lapse
perturbation and the energy density perturbation in comoving gauge as given in
(20) reduces to $\frac{\delta\rho}{\rho}=-2\tilde{\Phi}\,$. One also obtains

$\displaystyle\zeta=\frac{1}{3}\frac{\delta\pi_{\phi}}{\pi_{\phi}}\,,$ (39)

so that the conservation of $\zeta$ follows directly from the fact that
$\pi_{\phi}$ and its perturbation $\delta\pi_{\phi}$ are constants of motion.
We first consider the evolution of $\mathcal{R}\ (=\psi$ in comoving gauge) as
obtained from the evolution of the GFT volume operator studied separately in
each patch of the separate universe picture and then proceed to compare this
to the dynamics of $\mathcal{R}$ obtained by solving the generalised
perturbation equation (37). We limit our presentation to the main points; for
details, please see app. B.

####  IV.3.1 Evolution of $\mathcal{R}$ for exact solutions in a GFT model

The GFT corrected Friedmann equation originates from the evolution of the
expectation value of the GFT volume operator777Not to be confused with the
potential of the scalar field $\tilde{V}(\phi)$.
$V(\phi):=\langle\hat{V}(\phi)\rangle$ taken over a suitable class of
semiclassical states with respect to the clock $\phi$. The analytic solution
for the evolution of $V(\phi)$ in a non-interacting GFT and assuming a single
dominant field mode is given by deparamcosmo

$\displaystyle
V(\phi)=v_{0}Ae^{2\omega\phi}+v_{0}Be^{-2\omega\phi}-\frac{v_{0}}{2}\,,$ (40)

where $A,\,B\geq 0$ are real parameters determined by the initial conditions
(and $v_{0}$ is a fixed constant). The effective Friedmann equation is then
obtained from $\frac{V(\phi)^{\prime 2}}{V(\phi)^{2}}$ (which can be related
to the usual form using $V=a^{3}$ and rewriting $\mathfrak{H}$ in relational
time $\phi\,$, see (72)) and in order to obtain the correct late-time limit of
this Friedmann equation the fundamental parameter $\omega$ is fixed to satisfy
$\omega^{2}=\frac{3}{8}\kappa\,$.

To obtain an exemplary evolution of $\mathcal{R}$ directly from the solution
to $V(\phi)$ as given in (40), we set up an ensemble of separate universe
patches labelled by $p$, each with slightly different initial conditions
$A_{p},\ B_{p}\,$. The bounce in each patch happens at $\phi_{p,\rm
bounce}=\frac{1}{4\omega}\log\left(\frac{B_{p}}{A_{p}}\right)$, such that for
generic initial conditions each patch reaches its minimum volume at a
different value of $\phi\,$. We obtain the perturbation $\psi_{p}$ of each
patch from $V_{p}=(a_{p})^{3}=(a_{bg})^{3}-3\psi_{p}$ at linear order, such
that

$\displaystyle\psi_{p}$
$\displaystyle=\frac{1}{3}\left(1-\frac{V_{p}}{V_{bg}}\right)\,,\quad\text{where}\
V_{bg}:=\frac{1}{N_{\rm patches}}\sum_{p}V_{p}\,,$ (41)

and $N_{\rm patches}$ is the the total number of patches in the ensemble
considered. This gives an analytic expression for the perturbation $\psi_{p}$
(and hence $\mathcal{R}_{p}$) of each patch. In comoving gauge the value of
$\phi$ at a given instant of relational time is (by definition) the same in
each patch, and it is therefore straightforward to compare the evolution of
$V_{p}(\phi)$ of different patches (unlike in GFTsepUniv , where more general
gauge choices were studied).

####  IV.3.2 Evolution of $\mathcal{R}$ from separate universe perturbation
equations

We now compare the evolution of $\psi_{p}$ as given by (40) and (41) to that
obtained from the generalised perturbation equations (18) or (37) in comoving
gauge. We wish to establish if and for how long these generalised perturbation
equations correctly capture the exact evolution of $\mathcal{R}\,$.

As the matter content is given by a massless scalar field, $w=1$ in (37) and
$\mathcal{R}^{\prime}=-\mathfrak{H}\frac{\delta\mathcal{F}}{2\mathcal{F}}\,$.
We fix $\mathcal{F}$ as given in (16), so that

$\mathcal{F}=1+\frac{v_{0}}{a^{3}}+\frac{\mathcal{Y}}{a^{6}}\,,\quad\delta\mathcal{F}=3\frac{v_{0}}{a^{3}}\psi+6\frac{\mathcal{Y}}{a^{6}}\psi+\frac{\delta\mathcal{Y}}{a^{6}}\,,$
(42)

and the constant of motion $\mathcal{Y}$ is related to the coefficients in
(40) as

$\displaystyle\mathcal{Y}=\frac{v_{0}^{2}}{4}-4\,v_{0}^{2}\,A\,B\,.$ (43)

From the definition of $A$ and $B$ from the underlying quantum theory, it
follows that $A\,B\geq\frac{1}{16}$ and hence $\mathcal{Y}\leq 0$ (see app.
B). We can consider at least two inequivalent approaches of defining the
background quantity $\mathcal{Y}=\mathcal{Y}_{bg}$, namely
$\mathcal{Y}=\frac{v_{0}^{2}}{4}-4\,A_{bg}\,B_{bg}$ or
$\mathcal{Y}=\frac{1}{N_{p}}\sum_{p}\mathcal{Y}_{p}\,$, where we define
$A_{bg}:=\frac{1}{N_{\rm patches}}\sum_{p}A_{p}$ and $B_{bg}:=\frac{1}{N_{\rm
patches}}\sum_{p}B_{p}\,$. If we average over the volumes of each patch
$V_{p}$, which are given by (40) with $A,\,B\to A_{p},\,B_{p}\,$, we find that
$V_{bg}$ is obtained by replacing $A,\,B$ with their background values
$A,\,B\to A_{bg},\,B_{bg}\,$ in (40). We will therefore use
$\mathcal{Y}=\frac{v_{0}^{2}}{4}-4A_{bg}B_{bg}$ in the following. The
alternative choice would introduce nonlinear averaging effects in the
evolution of $V_{bg}$ around the bounce, but these would have no impact on the
qualitative statements made in the remainder of this section. We define
$\delta\mathcal{Y}:=\mathcal{Y}_{p}-\mathcal{Y}_{bg}=-4\,v_{0}^{2}\left(\delta
A_{p}\,B_{bg}+\delta B_{p}\,A_{bg}+\delta A_{p}\,\delta B_{p}\right)\,$.

To solve the equation of motion for $\mathcal{R}$, we first obtain an
expression for $\mathfrak{H}$ by solving the background Friedmann equation
(14), which in relational time reads

$\displaystyle\mathfrak{H}=$ $\displaystyle\frac{1}{3}\frac{{\rm d}V}{{\rm
d}\phi}\frac{1}{V}\phi^{\prime}\qquad\Rightarrow\qquad\left(\frac{{\rm
d}V}{{\rm d}\phi}\frac{1}{V}\right)^{2}=\frac{3}{2}\kappa\mathcal{F}\,,$ (44)

and is solved by

$\displaystyle
V(\phi)=\frac{\mathcal{C}}{4}e^{\sqrt{3\kappa/2}\,\phi}+\left(-\mathcal{Y}+\frac{v_{0}^{2}}{4}\right)\mathcal{C}^{-1}e^{-\sqrt{3\kappa/2}\,\phi}-\frac{v_{0}}{2}\,.$
(45)

As an example, we consider again an ensemble of patches that follow (40) with
perturbed initial conditions (different values of $A_{p},\ B_{p}$ for each
patch). These determine the value of $\mathcal{Y}_{bg}$ and the integration
constant $\mathcal{C}$ is fixed by setting the initial condition from $V_{bg}$
as given by (41) in the post-bounce regime. The solution to the modified
Friedmann equation $V(\phi)$ as given in (45) then agrees with the exact
expression for $V_{bg}$ obtained from (40) and (41).

To now obtain the evolution of $\mathcal{R}=\psi$, we solve (37). However, as
we are concerned also with the bounce region we use the following form to
avoid division by zero (since at the bounce $\mathfrak{H}=0=\mathcal{F}\,$),
and rewrite in relational time:

$\displaystyle 2\mathfrak{H}\psi^{\prime}=$
$\displaystyle-\frac{\kappa}{6}\phi^{\prime
2}\delta\mathcal{F}\qquad\Rightarrow\qquad\frac{{\rm d}V}{{\rm
d}\phi}\frac{1}{V}\frac{{\rm d}\psi}{{\rm
d}\phi}=-\frac{\kappa}{4}\delta\mathcal{F}\,.$ (46)

Note that this is independent of the explicit form of the lapse $N$, like the
relational Friedmann equation (44).

A solution to (46) (inserting the solution (45) in (46)) is given by

$\displaystyle\psi=\frac{\mathcal{C}_{\psi}\left(\mathcal{C}^{2}e^{\sqrt{6\kappa}\,\phi}-v_{0}^{2}+4\mathcal{Y}\right)+\frac{4}{3}\delta\mathcal{Y}}{\left(\mathcal{C}\,e^{\sqrt{3\kappa/2}\,\phi}-v_{0}\right)^{2}-4\mathcal{Y}}\,.$
(47)

The integration constant $\mathcal{C}_{\psi}$ is fixed by setting the initial
condition in the post-bounce regime from the exact solution obtained from (41)
for a specific patch of the ensemble. Fig. 1 shows the exact evolution of
$\mathcal{R}$ as obtained from (41) as well as the solution to the
perturbation equations (47) for a patch of an exemplary ensemble. If
perturbations are small the exact and perturbative solution agree well. The
difference in the asymptotic values in the pre-bounce regime is given by
$\frac{1}{3}\frac{\delta A_{p}}{A_{bg}}\frac{\delta B_{p}}{B_{bg}}\,$.

If $\psi$ is of order $\epsilon$, the discrepancy between the asymptotic
values is of order $\epsilon^{2}$, i.e., a quantity that is assumed negligible
in linear perturbation theory. (For details, see app. B.3.) We would like to
point out, however, that linear perturbation theory breaks down in the bounce
region, since perturbations are no longer small relative to their respective
background quantity: at the bounce, $\mathcal{F}=0$ but $\delta\mathcal{F}\neq
0$, and hence $\frac{\delta\mathcal{F}}{\mathcal{F}}\ll 1$ does not hold (fig.
1(c)). Note that in the special case where all patches reach their minimum
volume at the same value of $\phi\,$ ($\frac{B_{p}}{A_{p}}$ is the same in all
patches), the qualitative evolution of $\psi$ differs from the example in fig.
1. Then, even though $\psi$ is not constant around the bounce, $\mathcal{R}$
takes the same value in the semiclassical regimes before and after the bounce.

We can then conclude that, despite the invalidity of linear perturbation
theory in the bounce region, the generalised perturbation equations we
established accurately capture the non-trivial evolution of $\mathcal{R}$
introduced by the modified Friedmann equation across the bounce if $\psi$ is
sufficiently small. If perturbations become too large the perturbation
equations reproduce the correct qualitative behaviour of $\psi$, but lead to
different values around the bounce and in the post-bounce regime.

(a) Evolution of $\psi_{\rm pert}$ (47) in an exemplary patch. The horizontal
dashed lines represent the asymptotic values of the solution (78).

(b) The difference between the exact solution (41) and the solution obtained
from the perturbation equations (47), $\Delta\psi=\psi_{\rm exact}-\psi_{\rm
pert}$.

(c) In the immediate vicinity of the bounce
$\frac{\delta\mathcal{F}_{p}}{\mathcal{F}_{bg}}$ is large, indicating a
breakdown of linear perturbation theory.

Figure 1:  $\psi$ for a single patch of an ensemble with $N_{\rm patches}=16$
(see also fig. 2) as given in (47) ($\psi_{\rm pert}$) compared to the exact
solution (41) ($\psi_{\rm exact}$). The difference between the two solutions
increases in the bounce region and asymptotically approaches a constant value,
but remains small throughout. Initial conditions are set in the post-bounce
regime at $\phi=4$. The asymptotic values of $\psi$ are given by (77) and
(78). While the qualitative behaviour of the plots and the conclusions we draw
in the main text are independent of the specific choice of initial conditions,
we quote the numerical values of parameters in the solution of $\psi_{\rm
pert}$ for reference: $A_{bg}=200.226\,,\ B_{bg}=200.262\,,\
\mathcal{Y}=-160390\,,\ \mathcal{C}=800.903\,,\ \delta\mathcal{Y}=35.0685\,,\
\mathcal{C}_{\psi}=2.75576\times 10^{-4}\,,\ v_{0}=1\,$. The bounce time is
$\phi_{\rm bounce}=1.47351\times 10^{-5}$ and we set $\kappa=8\pi$.

We conclude this section with two remarks: Even though the far pre- and post-
bounce regime follow general relativistic dynamics, the relation
$-\zeta=\mathcal{R}$ can only hold in one of them: $\zeta$ remains conserved
even in the GFT case, but $\mathcal{R}$ has different asymptotic values (see
fig. 1). This can be understood by recalling that the dynamical equivalence of
$-\zeta$ and $\mathcal{R}$ in the $\mathcal{F}(\rho)$ case follows from
initially setting $D=0$ and the conservation law of $D$ (see sec. IV.2). If
the system evolves through a period in which the conservation law for $D$ no
longer holds as is the case here, this can introduce a shift between $-\zeta$
and $\mathcal{R}$. Hence, in the GFT bouncing scenario where $\mathcal{R}$ has
non-trivial dynamics, the far pre- and post-bounce phase must be treated as
independent general relativistic regimes. Finally, the assumption that the
background dynamics satisfy the same Friedmann equation as the individual
locally homogeneous patches is not exact, but only holds in a perturbative
regime (see app. B). The fact that averaged quantities and their perturbations
are inadequate to capture the true evolution is referred to as ‘the averaging
problem’ in standard cosmology avgProb_Zalaletdinov ; avgProb_Wiltshire ;
avgProb_Hossenfelder . It can be summarised as follows: the assumption that
the Universe (even at present) is homogeneous and isotropic, such that it can
be described by the FLRW metric, only holds on average over large scales.
Einstein’s equations are highly nonlinear and it is per se unclear whether an
average of an exact solution that takes the true matter distribution of the
Universe into account will match a solution obtained from perturbations around
an exact FLRW universe.

##  V Relation to second order perturbation equations

In the previous section we studied the dynamics of the comoving curvature
perturbation $\mathcal{R}$ by deriving an analogue to the diffeomorphism
constraint and thereby obtaining a conservation law in the $\mathcal{F}(\rho)$
case, and then considering directly a solution to the first order equation in
$\psi$ for a GFT model, where the matter content is given by a massless scalar
field and thus takes a specific, simple form. As established in sec. III,
$\zeta$ is conserved in the separate universe picture as long as the
continuity equation holds.

Solving the first order equation (18) in $\psi$ directly, as we did in the GFT
case, works only for specific forms of matter content, where one can eliminate
all perturbation variables but one. In more general cases, one can combine
perturbation equations to obtain a single second order equation of motion that
only refers to a single perturbation variable (and background quantities).
This could be an equation for $\psi$ (which is equivalent to the Bardeen
variable $\Psi$ in longitudinal gauge) as in 0801Wands , or the
Mukhanov–Sasaki equation as in 0112249 . Notably, these two approaches lead to
different results for the evolution of $\zeta$ in the separate universe
framework already in general relativity. If one obtains its evolution from a
second order equation in $\psi\,$, $\zeta$ remains constant, in agreement with
our considerations in sec. III. On the other hand, if one solves the long-
wavelength limit of the Mukhanov–Sasaki equation, the solution for $\zeta$ has
a constant and a dynamical part, where the latter is particularly important in
the contracting branch. We will discuss how this discrepancy could be
understood as a limitation of the strict separate universe limit.

In order to relate our results to some of the literature, we summarise the
above-mentioned two second order approaches of main interest in standard
cosmology and comment how they would apply to the more general types of
cosmological dynamics we consider. To simplify comparison, in this section we
use conformal time ($N=a$ and we denote the Hubble parameter as
$\frac{a^{\prime}}{a}=\mathcal{H}$) and longitudinal gauge ($E=B=0$ and
$\psi=\tilde{\Phi}\,$, where we discussed the origin of the last relation in
sec. IV.1). In this gauge, the relevant linearised Einstein equations
involving the perturbation variable $\psi$ are (see, e.g., cosmopert )

$\displaystyle-k^{2}\psi-3\mathcal{H}\left(\mathcal{H}\psi+\psi^{\prime}\right)$
$\displaystyle=\frac{\kappa}{2}a^{2}\delta\rho\,,$ (48)
$\displaystyle\psi\left(2\mathcal{H}^{\prime}+\mathcal{H}^{2}\right)+3\mathcal{H}\psi^{\prime}+\psi^{\prime\prime}$
$\displaystyle=\frac{\kappa}{2}a^{2}\delta P\,.$ (49)

###  V.1 Second order equation for the metric perturbation $\psi$

One can combine the temporal (48) and spatial-diagonal component (49) of the
Einstein equations to obtain an equation of motion for a single perturbation
variable only, using that for adiabatic perturbations $\frac{\delta
P}{\delta\rho}=\frac{P^{\prime}}{\rho^{\prime}}=c_{s}^{2}\,$. Together with
the background equation
$\mathcal{H}^{\prime}=-\frac{1}{2}\mathcal{H}^{2}(1+3w)\,$, one obtains

$\displaystyle
3\mathcal{H}^{2}\left(c_{s}^{2}-w\right)\psi+3\mathcal{H}(c_{s}^{2}+1)\psi^{\prime}+\psi^{\prime\prime}=-c_{s}^{2}k^{2}\psi\,.$
(50)

In the separate universe limit $k\rightarrow 0\,$, one neglects the right-hand
side, obtaining an equation that also follows from our earlier general
separate universe equations (18) and (19) in conformal time and for
$\mathcal{F}=1$. In this limit, one can find the general solution
$\psi(\eta)=\frac{\mathcal{H}}{a^{2}}\left(\frac{3}{2}C_{1}\int{\rm
d}\eta\left(a^{2}\left(w+1\right)\right)+C_{2}\right)\,$, where $C_{1}\,$,
$C_{2}$ are constants depending on $k$ in the range of wavenumbers covered by
this approximation (see, e.g., Bertschinger ; 0801Wands 888To see that this
expression solves (50) when $k\to 0\,$, one needs to use the background
equations for $\mathcal{H}^{\prime}$ and $\mathcal{H}^{\prime\prime}$ as well
as the continuity equation (13).).

We can obtain an expression of $\zeta$ in terms of $\psi$ by replacing the
energy density perturbation (48) and inserting the continuity equation (10) in
its definition (8):

$\displaystyle-\zeta=\frac{2k^{2}}{9\mathcal{H}^{2}(w+1)}\psi+\frac{3w+5}{3(w+1)}\psi+\frac{2\psi^{\prime}}{3\mathcal{H}(w+1)}\,.$
(51)

One can then derive an equation of motion for $\zeta$ which, after using (50)
and the background equations for $\mathcal{H}^{\prime}$ and $w^{\prime}\,$,
reads

$\displaystyle-\zeta^{\prime}=\frac{2k^{2}\left(\mathcal{H}\psi+\psi^{\prime}\right)}{9\mathcal{H}^{2}(w+1)}\,,$
(52)

so that we again find $\zeta^{\prime}=0$ when spatial gradients can be
neglected. In particular, one can verify explicitly that the long-wavelength
solution $\psi(\eta)$ derived above results in

$\displaystyle-\zeta(\eta)=C_{1}-k^{2}\frac{2C_{2}+3C_{1}\int{\rm
d}\eta\left(a^{2}(w+1)\right)}{9a^{2}\mathcal{H}(w+1)}$ (53)

and so again $\zeta$ is a constant in the separate universe limit
$k\rightarrow 0\,$: the second solution $\psi\sim\frac{\mathcal{H}}{a^{2}}$
does not contribute.

We will now investigate to what extent this approach could be illuminating for
the general cosmological scenarios we consider in this paper. Even though it
is inherent to the separate universe approach that the form of corrections to
the perturbation equations arising from spatial gradients cannot be
determined, we include a generic form of possible modifications, assuming they
appear in a similar way to $k-$dependent terms in general relativity. Recall
that in order to derive the expression for $\psi^{\prime\prime}$ in the
separate universe picture (19) we take the derivative of the first order
equation for $\psi$ (18) and insert an expression for $\delta\rho^{\prime}$
obtained by perturbing the continuity equation (11). If we follow the same
procedure when including inhomogeneities in $\psi^{\prime}\,$, we also need to
include $k-$dependent correction terms in $\delta\rho^{\prime}$ to ensure
consistency with general relativity: inhomogeneous changes to the dynamics of
metric perturbations must lead to changes in the dynamics of the matter
perturbations (whereas in the separate universe picture we assumed the matter
sector remains unaltered). Specifically, we consider modifications to the
first order equation of $\psi$ (18) and the perturbed continuity equation (11)
of the form

$\displaystyle\mathcal{H}\psi^{\prime}$
$\displaystyle=-\mathcal{H}^{2}\left(\psi+\frac{\delta\rho}{2\rho}+\frac{\delta\mathcal{F}}{2\mathcal{F}}\right)+G_{k}\,,$
(54) $\displaystyle\delta\rho^{\prime}$
$\displaystyle=3\psi^{\prime}(\rho+P)-3\mathcal{H}(\delta\rho+\delta
P)+Z_{k}\,,$ (55)

where in the classical limit $G_{k}\to-\frac{k^{2}}{3}\psi$ and
$Z_{k}\to-\frac{2k^{2}}{\kappa a^{2}}(\psi^{\prime}+\mathcal{H}\psi)\,$. One
can then compute an expression for $\psi^{\prime\prime}$ from the derivative
of (54) by inserting (55), the continuity equation and replacing $\delta\rho$
as given by (54), as well as making use of the background equation
$\mathcal{H}^{\prime}=-\frac{1}{2}\left(\mathcal{H}^{2}(1+3w)-\mathcal{H}\frac{\mathcal{F}^{\prime}}{\mathcal{F}}\right)$
and the modified Friedmann equation:

$\displaystyle\begin{split}3\mathcal{F}\mathcal{H}^{2}(c_{s}^{2}-w)\psi+\left(3\mathcal{F}\mathcal{H}(c_{s}^{2}+1)-\frac{\mathcal{F}^{\prime}}{2}\right)\psi^{\prime}+\mathcal{F}\psi^{\prime\prime}+\left(3\mathcal{H}^{2}(c_{s}^{2}-w)-\mathcal{H}\frac{\mathcal{F}^{\prime}}{\mathcal{F}}\right)\frac{\delta\mathcal{F}}{2}+\mathcal{H}\frac{\delta\mathcal{F}^{\prime}}{2}\\\
=-\frac{\kappa
a^{2}\mathcal{F}^{2}Z_{k}}{6\mathcal{H}}+G_{k}\left((3c_{s}^{2}+1)\mathcal{F}-\frac{\mathcal{F}^{\prime}}{\mathcal{H}}\right)+\frac{G_{k}^{\prime}}{\mathcal{H}}\mathcal{F}\,.\end{split}$
(56)

In the classical limit ($\mathcal{F}\to 1\,,\ \delta\mathcal{F}\to 0\,,\
\mathcal{F}^{\prime}\to 0,\,\delta\mathcal{F}^{\prime}\to 0$ and $G_{k}\,,\
Z_{k}$ as given above) this reduces to (50). In order to recover (18), (19)
and the perturbed continuity equation (11) in the separate universe limit, we
require $Z_{k}\to 0$ and $G_{k}\to 0$ for small $k\,$. However, unlike for
(50), the long-wavelength limit of (56) cannot necessarily be solved directly
to obtain a solution for $\psi\,$, as $\delta\mathcal{F}$ can depend on
perturbation variables other than $\psi$.

As in the general relativistic case, we can again replace $\delta\rho$ from
(54) and insert the continuity and Friedmann equation in the definition of
$\zeta$ (8) to obtain

$\displaystyle-\zeta=-\frac{2G_{k}}{3\mathcal{H}^{2}(w+1)}+\frac{(3w+5)\psi}{3(w+1)}+\frac{2\psi^{\prime}}{3\mathcal{H}(w+1)}+\frac{\delta\mathcal{F}}{3\mathcal{F}(w+1)}\,.$
(57)

We can again compute its derivative, where we replace $\delta\rho^{\prime}$
from (55), $\delta\rho$ from (54) and use the background equations:

$\displaystyle-\zeta^{\prime}=-\frac{Z_{k}}{3(1+w)\rho}=-\frac{\kappa
a^{2}\mathcal{F}Z_{k}}{9\mathcal{H}^{2}(1+w)}\,.$ (58)

This reduces to (52) in the classical limit and vanishes when spatial
gradients can be neglected.

In summary, following the approach used in 0801Wands ; Bertschinger , we found
a second order equation in $\psi$ similar to (50), but whether it can be
written in a form that can be solved directly in the long-wavelength limit
depends on the specific form of $\delta\mathcal{F}\,$. The analysis above is
consistent with the general statement from sec. III that $\zeta^{\prime}=0$
also for a modified Friedmann equation in the separate universe limit as long
as we have adiabatic perturbations.

###  V.2 Mukhanov–Sasaki equation

In conventional cosmological perturbation theory, one commonly works with the
Mukhanov–Sasaki variable $v$ MukhanovSasaki ; MukhanovSasaki_2 , which has the
property of evolving like a canonically normalised scalar field in an
expanding background; it appears with the standard kinetic term of a scalar
field in its action, and hence can be quantised canonically as a scalar field
(see, e.g., BaumannNotes ).

The dynamics of the Mukhanov–Sasaki variable are governed by the
Mukhanov–Sasaki equation which (after a Fourier decomposition) reads cosmopert

$\displaystyle
v^{\prime\prime}+c_{s}^{2}k^{2}v-\frac{z^{\prime\prime}}{z}v=0\,.$ (59)

For scalar matter $v=a(\delta\phi+\frac{\phi^{\prime}}{\mathcal{H}}\psi)$ and
$z=a\frac{\phi^{\prime}}{\mathcal{H}}$ so that $v=z\mathcal{R}\,$. In general
relativity and in the long-wavelength limit, one then also has $v=-z\zeta\,$.
The Mukhanov–Sasaki equation can be derived by rewriting the matter and
gravity action in terms of $v$ cosmopert  or, in the separate universe
approximation, through algebraic manipulation of the perturbation equations
LQCsepUniv . The derivation of the Mukhanov–Sasaki equation requires the
constraint equation
$D=\mathfrak{H}\tilde{\Phi}+\psi^{\prime}-\frac{\kappa}{2}\phi^{\prime}\delta\phi=0\,$,
which as discussed in sec. IV.2 originates from the spatiotemporal component
of the Einstein field equations and is generally not available in a separate
universe framework. (But as we have seen a modified version can be derived in
the $\mathcal{F}(\rho)$ case.)

Another approach to obtain a solution for $\zeta$ or $\mathcal{R}$ on super-
horizon scales is then to solve the long-wavelength limit of (59), which leads
to cosmopert ; 0112249

$\displaystyle\zeta=V+S\int\frac{d\eta}{z^{2}}\,,$ (60)

where $V$ and $S$ are ($k-$dependent) constants. Here, the dynamical part of
the solution has no clear $k-$dependence that disappears in the separate
universe limit. Imposing the long-wavelength limit of (52) (which is
equivalent to requiring that the long-wavelength limit of the first order
equation in $\psi$ is satisfied) would however recover a constant solution, by
requiring $S=0\,$. It is then unclear whether it is justified to keep the
dynamical part of $\zeta$ in (60), as this amounts to neglecting $k-$dependent
terms in (59), but not in (52). In some way, one then acknowledges that
$\zeta^{\prime}\neq 0$ for small but non-zero $k\,$, which becomes relevant in
scenarios as those discussed in 0112249 ; LQCsepUniv , where $\zeta$ contains
important information about small, but non-zero $k$ modes in the contracting
phase. The authors point out that for the contracting branch in a bouncing
universe, $\zeta^{\prime}$ can increase as one approaches the bounce
($-\eta\to 0$), namely in the cases where $\zeta^{\prime}\sim
k^{2}(-\eta)^{-|p|}\ (p\in\mathbb{R})$, leading to a growing mode. To remain
in the separate universe regime, one then has to assume that the wavelength of
perturbations is always large enough ($k$ sufficiently small) for
$\zeta^{\prime}$ to remain negligible. However, the separate universe limit
cannot be consistently applied at the bounce point, as it arises from the
requirement that wavelengths are much larger than the Hubble horizon,
$k\ll\mathcal{H}\,$. For a full treatment one would therefore need to
understand the finite theory: it seems necessary to verify any statements made
about the dynamics of perturbations in the separate universe limit around the
bounce region against the full dynamics including gradient terms. In LQC there
exists an effective Hamiltonian that allows to study perturbations also
outside the strict $k\to 0$ limit and an analogue of the Mukhanov-Sasaki
equation was derived LQCanomaly , where LQC corrections to (59) appear only in
the $k^{2}$– term. This cannot be done in general for model-independent
perturbation equations as we consider here. Furthermore, in absence of the
diffeomorphism constraint it is unclear whether a Mukhanov–Sasaki like
equation can be derived algebraically as was done in LQCsepUniv 999Note that
in the $\mathcal{F}(\rho)$ case, where $\mathcal{R}^{\prime}=0$ in the
separate universe framework, a second order equation
$(\mathcal{R}^{\prime}z)^{\prime}=0$ holds independently of the choice of $z$.
It is not clear how to justify a particular choice of $z$ as corresponding to
the $k\rightarrow 0$ limit of an equation valid more generally.. The
applicability of (59) to a scenario with a modified Friedmann equation is
therefore far from clear and an analogue needs to be established from the full
dynamics for a specific model in question.

In conclusion, second order equations as are used in standard cosmology
generally do not provide additional insight into the evolution of gauge-
invariant perturbation variables for general theories with a modified
Friedmann equation in the separate universe limit. They may nonetheless be
useful for specific theories where additional information is available.

##  VI Conclusion

In this article we investigated the evolution of scalar perturbations in
cosmological scenarios with a modified Friedmann equation, such as those that
can arise in quantum gravity. We focused on the gauge-invariant perturbation
variables $\zeta$ and $\mathcal{R}$ which are frequently studied in
conventional cosmological perturbation theory, as they have a physical
interpretation and are related to the power spectrum of the CMB.

Our starting point is a generic modified Friedmann equation, and the main body
of our analysis is agnostic with regards to the underlying theory. We do
however assume an unchanged continuity equation, from which it follows that
$\zeta$ is conserved for long wavelengths as long as perturbations are
adiabatic, independent of the gravitational dynamics. We furthermore need to
assume that the notion of gauge invariance, which ensures $\zeta$ and
$\mathcal{R}$ are physically meaningful variables to study, remains unchanged.
In cases where the underlying gravitational theory admits an effective
description of the modified cosmological dynamics, this could be investigated
explicitly (as is the case in LQC LQCanomaly ).

We work in the separate universe framework, where the Universe is modelled as
an ensemble of disconnected patches that each follow the dynamics of an FLRW
universe and all spatial gradients vanish. In this framework, the
perturbations are homogeneous in each patch and defined with respect to the
background values of the entire ensemble. The perturbation equations are
obtained by perturbing the Friedmann equation and its derivative at linear
order. We then focus on a special case, where the modification of the
Friedmann equation can be contained in a function that depends on the energy
density $\rho$ only, $\mathcal{F}=\mathcal{F}(\rho)\,$. In this case one can
show that a relation similar to the diffeomorphism constraint usually obtained
from the spatiotemporal components of the Einstein field equations holds,
which simplifies the perturbation equations. It then follows that
$\mathcal{R}$ is conserved for these types of models, as in general
relativity. Similar considerations were made for LQC in LQCsepUniv and here we
show that these results hold in general for this class of modified Friedmann
equations. We then investigate a specific example of a Friedmann equation that
does not have this property, namely the GFT Friedmann equation as established
in deparamcosmo . In this case, $\mathcal{R}^{\prime}\neq 0$ and we compare
the evolution of $\mathcal{R}$ across the bounce as obtained from the
expectation value of the GFT volume operator to analytical solutions of the
generalised perturbation equations. The difference between the solutions for
$\psi$ obtained from the two procedures is of second order and therefore
negligible for small perturbations. Finally, we consider two common approaches
in the literature that use second order equations in perturbation variables
and comment how they relate to our findings. We conclude that neither of them
can be used to make further general statements about the evolution of
perturbations in scenarios with a general modified Friedmann equation.

In summary, we established that for a general modified Friedmann equation in
the separate universe framework, the relation $\mathcal{R}=-\zeta$ no longer
holds, whereas it remains valid for a certain type of modification. As $\zeta$
remains conserved irrespective of the type of modification, the separate
universe framework alone is not suitable to establish possible imprints on the
CMB power spectrum from quantum gravitational effects. Inhomogeneous
perturbations need to be included in an analysis to obtain alterations to the
dynamics of $\zeta\,$. Also, this is the only way to rigorously establish how
sub-horizon dynamics around the bounce influence the evolution of
perturbations through the bounce. How and whether this can be done depends on
the underlying theory that generates the modified Friedmann dynamics. In LQC,
techniques have been established LQCanomaly , and first investigations have
also been initiated in the context of GFT GFTrelationalPert .

A final comment on the definition of $\zeta$ is in order: Here we have assumed
that the definition of $\zeta$ remains unchanged also in the non-general
relativistic regime. However, a modified Friedmann equation of the form (14)
can also be interpreted as a modification to the energy density $\rho_{\rm
eff}=\rho\mathcal{F}\,$, which implies a modified form of the curvature
perturbation on uniform density hypersurfaces $\zeta_{\rm eff}$, as was
considered in 0801Wands . The arguments for the conservation of $\zeta$
presented here would no longer apply to $\zeta_{\rm eff}\,$, since in that
case $\rho_{\rm
eff}^{\prime}=-3\mathfrak{H}(\rho+P)\mathcal{F}+\rho\mathcal{F}^{\prime}\,$.
It is clear that from a Friedmann equation alone, one cannot conclude whether
the modification arises in the matter sector (which then also alters the
continuity equation) or is limited to gravitational dynamics. Such an input
would originate from the theory that gives the modified Friedmann equation,
and in the examples studied here (LQC and GFT) one assumes that the matter
sector remains unaltered.

###### Acknowledgements.

The work of SG was funded by the Royal Society through a University Research
Fellowship (UF160622).

##  Appendix A Constraint equation in Newtonian gauge

In sec. IV.2 we showed that the diffeomorphism constraint can be retrieved in
the separate universe picture in the comoving gauge. Here we carry out a
similar calculation in the Newtonian gauge, which generalises what was done in
LQCsepUniv for the LQC case to any modified Friedmann equation where
$\mathcal{F}=\mathcal{F}(\rho)\,$. We work in conformal time $N=a$ (and denote
the Hubble parameter as $\mathcal{H}$) and the longitudinal gauge
$\tilde{\Phi}=\psi\,$. Inspired by the results in LQCsepUniv , we assume the
following form of the constraint:

$\displaystyle
D:=\psi^{\prime}+\mathcal{H}\psi-\mathcal{A}\frac{\kappa}{2}\phi^{\prime}\delta\phi\,.$
(61)

We again show that eq. (32) holds, and hence, if initial conditions are set in
a regime where $D=0$, it follows that $D^{\prime}=0$ and the constraint
equation holds throughout the evolution. We use the result from sec. IV.2 that
$W=3\mathfrak{H}-N^{\prime}/N=2\mathcal{H}$ in conformal time, so that we now
need to show

$\mathcal{A}D^{\prime}+2\mathcal{H}\mathcal{A}D-\mathcal{A}^{\prime}D=0\,.$
(62)

For this, we use the generalised equations for the $\mathcal{F}(\rho)$ case as
given in (25)-(27) and first rewrite $D^{\prime}$ as

$\displaystyle D^{\prime}=$
$\displaystyle\psi^{\prime\prime}+\mathcal{H}^{\prime}\psi+\mathcal{H}\psi^{\prime}-\frac{\kappa}{2}\left(\mathcal{A}^{\prime}\phi^{\prime}\delta\phi+\mathcal{A}\phi^{\prime\prime}\delta\phi+\mathcal{A}\phi^{\prime}\delta\phi^{\prime}\right)$
(63) $\displaystyle=$
$\displaystyle\psi^{\prime\prime}+\mathcal{H}^{\prime}\psi-\mathcal{H}^{2}\psi-\frac{\kappa\mathcal{H}}{6\mathcal{H}}a^{2}\mathcal{A}\,\delta\rho-\frac{\kappa}{2}\left(\mathcal{A}^{\prime}\phi^{\prime}\delta\phi-a^{2}\frac{{\rm
d}\tilde{V}}{{\rm
d}\phi}\mathcal{A}\delta\phi-2\mathcal{H}\phi^{\prime}\mathcal{A}\delta\phi+\mathcal{A}\phi^{\prime}\delta\phi^{\prime}\right)$
(64) $\displaystyle=$
$\displaystyle\psi^{\prime\prime}-\frac{\kappa}{6}a^{2}\mathcal{A}\,\delta\rho+\frac{\kappa}{2}\left(-\mathcal{A}^{\prime}\phi^{\prime}\delta\phi+a^{2}\mathcal{A}\,\delta\rho-2\phi^{\prime}\mathcal{A}\,\delta\phi^{\prime}+2\mathcal{H}\phi^{\prime}\mathcal{A}\,\delta\phi\right)\,.$
(65)

In the first step we inserted the Klein-Gordon equation
$\phi^{\prime\prime}=-a^{2}\frac{{\rm d}\tilde{V}}{{\rm
d}\phi}-2\mathcal{H}\phi^{\prime}$ and $\psi^{\prime}$ from (26) and in the
second made use of the background identity
$\mathcal{H}^{\prime}-\mathcal{H}^{2}=-\frac{\kappa}{2}\phi^{\prime
2}\mathcal{A}$ and used the expression for $\frac{{\rm d}\tilde{V}(\phi)}{{\rm
d}\phi}\delta\phi$ obtained from (7). If we then insert $\psi^{\prime\prime}$
from (27), the expression for $D^{\prime}$ can be written as

$\displaystyle
D^{\prime}=\kappa\delta\rho\left((\mathcal{F}_{\rho}+\frac{\rho}{2}\mathcal{F}_{\rho\rho})\phi^{\prime
2}+\frac{1}{3}a^{2}\mathcal{A}\right)+\frac{\kappa}{2}\delta\phi\left(-\mathcal{A}^{\prime}\phi^{\prime}+2\mathcal{H}\phi^{\prime}\mathcal{A}\right)\,.$
(66)

Furthermore, making again use of (26), we can write
$D=-\frac{\kappa}{6\mathcal{H}}a^{2}\mathcal{A}\delta\rho-\mathcal{A}\frac{\kappa}{2}\phi^{\prime}\delta\phi\,$,
such that $2\mathcal{H}\mathcal{A}D-\mathcal{A}^{\prime}D$ reads

$\displaystyle 2\mathcal{H}\mathcal{A}D-\mathcal{A}^{\prime}D=$
$\displaystyle\left(-\mathcal{A}^{2}\frac{\kappa}{3}a^{2}+\mathcal{A}^{\prime}\frac{\kappa}{6\mathcal{H}}a^{2}\mathcal{A}\right)\delta\rho+\left(-\mathcal{H}\mathcal{A}^{2}\kappa\phi^{\prime}+\mathcal{A}^{\prime}\mathcal{A}\frac{\kappa}{2}\phi^{\prime}\right)\delta\phi\,.$
(67)

Combining the above expressions, we finally obtain

$\displaystyle\mathcal{A}D^{\prime}+2\mathcal{H}\mathcal{A}D-\mathcal{A}^{\prime}D=$
$\displaystyle\kappa\left(\mathcal{F}_{\rho}+\frac{\rho}{2}\mathcal{F}_{\rho\rho}\right)\phi^{\prime
2}\mathcal{A}\,\delta\rho+\mathcal{A}^{\prime}\frac{\kappa}{6\mathcal{H}}a^{2}\mathcal{A}\,\delta\rho=0\,,$
(68)

where in the last step we inserted $\mathcal{A}^{\prime}$ as given in (24).

We have thus shown explicitly that a constraint equation holds also in
longitudinal gauge in the case where $\mathcal{F}=\mathcal{F}(\rho)$. Note
that in this gauge the conservation of $\mathcal{R}$ is not immediately
apparent from the constraint. (It can also be shown explicitly for the
$\mathcal{F}(\rho)$ case, but since $\mathcal{R}$ is gauge-invariant,
$\mathcal{R}^{\prime}=0$ follows from the comoving gauge.)

##  Appendix B Details of the GFT case

Here we report the details of the exemplary evolution of $\psi$ studied in
sec. IV.3. We first summarise briefly how the evolution of the volume of the
universe $V(\phi)$ is obtained from the quantum theory and leads to an
effective Friedmann equation. We illustrate this further by providing more
specifics of the ensemble used to obtain fig. 1. We then give some more detail
on how to solve the Friedmann equation to retrieve the same expression for
$V(\phi)$ and include expressions for the asymptotic values of the
perturbative and exact solution of $\psi\,$.

###  B.1 Exact solution

As mentioned in the main text, we use the GFT effective Friedmann equation
obtained in deparamcosmo and for the details, we refer the reader to the main
paper. To derive the effective Friedmann equation, the authors of deparamcosmo
work in the Hamiltonian formulation of GFT GFThamiltonian . We work within the
free theory, i.e., neglect any interactions between GFT quanta, and
furthermore restrict to a single field mode (in general the volume operator
can include multiple modes). The Hamiltonian for a single GFT mode can then be
written as

$\displaystyle\hat{H}=\frac{\mathchar
1406\relax}{2}\omega\left((\hat{a}^{\dagger})^{2}+\hat{a}^{2}\right)\,,$ (69)

where $\hat{a}\,,\hat{a}^{\dagger}$ are obtained from the GFT field operator
and its momentum and are ladder operators for the Fock vacuum
($\hat{a}\ket{0}=0$). They satisfy the bosonic commutation relations
$[\hat{a},\hat{a}^{\dagger}]=1\,$ and dynamics are obtained by solving the
Heisenberg equations of motion for $\hat{a}$ and $\hat{a}^{\dagger}$
($\,\partial_{\phi}\hat{a}=\frac{i}{\mathchar
1406\relax}[\hat{H}\,,\hat{a}]\,$). The volume operator is given by
$\hat{V}(\phi)=v_{0}\,\hat{a}^{\dagger}(\phi)\,\hat{a}(\phi)$ and its
evolution with respect to the relational clock $\phi$ follows directly from
the solutions to the equations of motion for $\hat{a}(\phi)$ and
$\hat{a}^{\dagger}(\phi)\,$.

In order to infer an expression for an effective Friedmann equation, one
considers the expectation value of the volume operator in a suitable
semiclassical state. In deparamcosmo the authors investigated different
choices of coherent states and found that in the free theory, Fock coherent
states can be regarded as semiclassical at late times. Fock coherent states
$\ket{\sigma}=e^{\sigma\hat{a}^{\dagger}-\sigma^{*}\hat{a}}\ket{0}$ are
eigenstates of the annihilation operator
$\hat{a}\ket{\sigma}=\sigma\ket{\sigma}$, where $\sigma\in\mathbb{C}\,$. The
volume at $\phi=0$ is then given by
$\bra{\sigma}\hat{V}(\phi=0)\ket{\sigma}=|\sigma|^{2}$ and the evolution of
the volume of the universe is (40)

$\displaystyle
V(\phi)=\bra{\sigma}\hat{V}(\phi)\ket{\sigma}=v_{0}\,Ae^{2\omega\phi}+v_{0}\,Be^{-2\omega\phi}-\frac{v_{0}}{2}\,,$
(70)

where
$A=\frac{\text{Re}(\sigma)^{2}}{2}+\frac{\text{Im}(\sigma)^{2}}{2}+\,\text{Im}(\sigma)\,\text{Re}(\sigma)+\frac{1}{4}$
and
$B=\frac{\text{Re}(\sigma)^{2}}{2}+\frac{\text{Im}(\sigma)^{2}}{2}-\,\text{Im}(\sigma)\,\text{Re}(\sigma)+\frac{1}{4}\,$.

The effective Friedmann equation for GFT is then recovered from

$\displaystyle\left(\frac{d\bra{\sigma}\hat{V}(\phi)\ket{\sigma}}{d\phi}\right)^{2}\bra{\sigma}\hat{V}(\phi)\ket{\sigma}^{-2}=4\omega^{2}\left(1+\frac{v_{0}}{\bra{\sigma}\hat{V}(\phi)\ket{\sigma}}+\frac{\mathcal{Y}}{\bra{\sigma}\hat{V}(\phi)\ket{\sigma}^{2}}\right)\,,$
(71)

where
$\mathcal{Y}=-v_{0}^{2}\left(\text{Re}(\sigma)^{4}+\text{Im}(\sigma)^{4}+\text{Re}(\sigma)^{2}+\text{Im}(\sigma)^{2}-2\text{Re}(\sigma)^{2}\text{Im}(\sigma)^{2}\right)$.

To obtain an exemplary evolution of $\psi\,$, we proceed as follows: We first
choose an arbitrary number of patches for the ensemble that we wish to study.
Then, we choose initial conditions for the parameters $A_{p}$ and $B_{p}$ in
each patch, where random fluctuations in the initial conditions
$\text{Re}(\sigma)$ and $\text{Im}(\sigma)$ are generated from a white noise
process. We then obtain $V_{p}(\phi)$ for each patch from the initial
conditions and use this to calculate $V_{bg}$ and $\psi_{p}$ as defined in
(41). Fig. 2 shows the evolution of the expectation value of the volume
operator for $N_{\rm patches}=16$. The difference in evolution is strongest in
the bounce region (where $\phi_{\rm bounce}$ refers to the bounce of the
background, i.e., the minimum of $V_{bg}$) and each patch reaches its minimum
volume at a different value of $\phi\,$. As a result, $\psi_{p}$ varies around
the bounce, but is (approximately) constant in the far pre- and post-bounce
regimes.

(a) Evolution of the volume in each patch and the background (average) volume
$V_{bg}\,$.

(b) The evolution of the volume of each patch and $V_{bg}$ close to the
bounce. The minimum volume is reached at a different value of $\phi$ in each
patch.

(c) Evolution of $\psi$ (41) as determined by $V_{p}$ and $V_{bg}$ for all
patches.

Figure 2: The evolution of the expectation value of the volume operator for
Fock coherent states $V(\phi)=\bra{\sigma}\hat{V}(\phi)\ket{\sigma}$ and the
resulting dynamics for $\psi$ in an example ensemble with $N_{\rm patches}=16$
and different values of $A_{p}\,,\ B_{p}$ in each patch 111111 The values in
different patches (generated from a white noise process) are given in the
table below. Note that the behaviour presented above is generic and the
specific values are only reported for completeness.
$\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
A_{p}&200.03&200.103&200.391&199.99&200.119&200.876&199.984&200.244&199.948&200.298&200.046&200.93&200.432&200.361&199.946&199.915\\\
B_{p}&199.981&199.934&200.053&200.395&200.035&200.146&200.405&200.317&200.554&200.29&200.192&200.382&200.242&200.453&200.733&200.079\\\
\hline\cr\end{array}$ . $v_{0}=1\,$.

###  B.2 Background dynamics from a modified Friedmann equation

In order to solve the perturbation equation for $\psi$ and compare the
solution to the exact evolution (fig. 2(c)), we first need to solve the
effective Friedmann equation (14) in relational time. To rewrite (14), we use
$V=a^{3}$ to first rewrite $\mathfrak{H}=\frac{{\rm d}a}{{\rm
d}\phi}\frac{1}{a}\phi^{\prime}=\frac{1}{3}\frac{{\rm d}V}{{\rm
d}\phi}\frac{1}{V}\phi^{\prime}$ and then insert the expression for
$\phi^{\prime}=\frac{\pi_{\phi}N}{V}$ and $\rho=\frac{\pi_{\phi}^{2}}{2V^{2}}$
(38) into the Friedmann equation

$\displaystyle\mathfrak{H}^{2}=\frac{1}{9}\left(\frac{{\rm d}V}{{\rm
d}\phi}\frac{1}{V}\right)^{2}\left(\frac{\pi_{\phi}N}{V}\right)^{2}=\frac{\kappa}{3}N^{2}\rho\mathcal{F}\qquad\Rightarrow\qquad\left(\frac{{\rm
d}V}{{\rm d}\phi}\frac{1}{V}\right)^{2}=\frac{3}{2}\kappa\mathcal{F}\,.$ (72)

In the separate universe picture, each patch then follows $\left(\frac{{\rm
d}V_{p}}{{\rm
d}\phi}\frac{1}{V_{p}}\right)^{2}=\frac{3}{2}\kappa\mathcal{F}_{p}$ and the
perturbation of the volume in each patch $\delta V_{p}$ is defined by $\delta
V_{p}=V_{p}-V_{bg}\,,\ V_{bg}=\frac{1}{N_{\rm patches}}\sum_{p}V_{p}\,$. It is
furthermore assumed that the average volume follows a Friedmann equation of
the form
$\left(\frac{dV_{bg}}{d\phi}\frac{1}{V_{bg}}\right)^{2}=\frac{3}{2}\kappa\mathcal{F}_{bg}\,$.
For $\mathcal{F}=\mathcal{F}_{\rm GFT}$, $\mathcal{F}_{bg}$ reads (we include
subscripts on all quantities for clarity)

$\displaystyle\mathcal{F}_{bg}=\frac{1}{N_{p}}\sum_{p}\mathcal{F}_{p}=1+v_{0}\sum_{p}\frac{1}{V_{bg}+\delta
V_{p}}+\sum_{p}\frac{\mathcal{Y}_{bg}+\delta\mathcal{Y}_{p}}{(V_{bg}+\delta
V_{p})^{2}}\approx
1+\frac{v_{0}}{V_{bg}}+\frac{\mathcal{Y}_{bg}}{V_{bg}^{2}}\,,$ (73)

i.e., we obtain $\mathcal{F}_{bg}$ by replacing the expressions in (16) with
background quantities, which is an approximation that holds when perturbations
are relatively small ($\frac{\delta V_{p}}{V_{bg}}\ll 1\,,\
\frac{\delta\mathcal{Y}_{p}}{\mathcal{Y}_{bg}}\ll 1\,$) and linear
perturbation theory is therefore applicable. Solving the Friedmann equation
(72) for $\mathcal{F}=\mathcal{F}_{\rm GFT}$ with $\kappa=8\pi G$ gives the
following solution

$\displaystyle
V(\phi)=\frac{\mathcal{C}}{4}e^{\sqrt{3\kappa/2}\,\phi}+\left(-\mathcal{Y}+\frac{v_{0}^{2}}{4}\right)\mathcal{C}^{-1}e^{-\sqrt{3\kappa/2}\,\phi}-\frac{v_{0}}{2}\,.$
(74)

The value of $\mathcal{Y}$ is given by $\mathcal{F}_{\rm GFT}$, whereas
$\mathcal{C}$ is an integration constant fixed by the initial condition for
$V$. If we compare to (40) we find that

$\displaystyle A=\frac{\mathcal{C}}{4}\,,\qquad
B=\left(-\mathcal{Y}+\frac{v_{0}^{2}}{4}\right)\mathcal{C}^{-1}\,.$ (75)

###  B.3 Asymptotic values of $\psi$

Recall that fig. 1 compares the exact solution obtained from (41) of one patch
of the ensemble (portrayed in fig. 2) to the perturbative solution (47). We
can obtain analytic expressions for the asymptotic values of both solutions.
The asymptotic values of the solution to the perturbation equations $\psi$
(47) in the far pre- and post-bounce regime, respectively, are

$\displaystyle\psi_{\rm
pre,\,pert}=\frac{4\delta\mathcal{Y}}{3\left(v_{0}^{2}-4\mathcal{Y}\right)}-\mathcal{C}_{\psi}\,,\qquad\psi_{\rm
post,\,pert}=\mathcal{C}_{\psi}\,.$ (76)

On the other hand, the asymptotic values of $\psi$ obtained from (41) and the
solution of $V$ given in (40) are

$\displaystyle\psi_{\rm
pre,\,exact}=\frac{1}{3}\left(1-\frac{B_{p}}{B_{bg}}\right)=-\frac{1}{3}\frac{\delta
B_{p}}{B_{bg}}\,,\qquad\psi_{\rm
post,\,exact}=\frac{1}{3}\left(1-\frac{A_{p}}{A_{bg}}\right)=-\frac{1}{3}\frac{\delta
A_{p}}{A_{bg}}\,.$ (77)

As we set the initial condition in the post-bounce regime, this fixes the
value of $\mathcal{C}_{\psi}=\psi_{\rm post,\,exact}\,$. If we insert this
into (76), we obtain

$\displaystyle\psi_{\rm
pre,\,pert}=\frac{1}{3}\frac{A_{p}}{A_{bg}}\left(1-\frac{B_{p}}{B_{bg}}\right)=-\frac{1}{3}\frac{\delta
B_{p}}{B_{bg}}\left(1+\frac{\delta A_{p}}{A_{bg}}\right)=\psi_{\rm
pre,\,exact}+\mathcal{O}(\epsilon^{2})\,.$ (78)

If the perturbations in $\psi$ are of order $\epsilon\,$, the difference in
asymptotic values will be of order $\epsilon^{2}\,$, i.e., the discrepancy is
a second order effect and does not affect the results of linear perturbation
theory. In order for (78) to match the expression in (77), $\delta\mathcal{Y}$
would have to be given by $\delta\mathcal{Y}=-4v_{0}^{2}(\delta
A_{p}B_{bg}+\delta B_{p}A_{bg})$ instead of
$\delta\mathcal{Y}=-4v_{0}^{2}\left(\delta A_{p}B_{bg}+\delta
B_{p}A_{bg}+\delta A_{p}\delta B_{p}\right)\,$. The accuracy of the solution
to the perturbed equations then depends on the magnitude of $\delta
A_{p}\,\delta B_{p}$ in a patch, which is determined by the standard deviation
of the white noise process we use to create perturbations in the quantum
picture.

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