# Cosmology in a locally scale invariant gravity

Meir Shimon1
1 School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

A ‘bouncing’ cosmological model is proposed in the context of a Weyl-invariant
scalar-tensor (WIST) theory of gravity. In addition to being Weyl-invariant
the theory is U(1)-symmetric and has a conserved global charge. The entire
cosmic background evolution is accounted for by a complex scalar field that
evolves in the static ‘comoving’ frame. Its (dimensional) modulus $\chi$
regulates the dynamics of masses and the apparent space expansion.
Cosmological redshift is essentially due to the cosmic evolution of the
Rydberg constant in the comoving frame. The temporal evolution of $\chi$ is
analogous to that of a point particle in the presence of a central potential
$V(\chi)$. The scalar field sources the spacetime curvature; as such it can
account for the (cosmological) Dark Sector. An interplay between the energy
density of radiation and that of the kinetic energy associated with the phase
$\alpha$ of the scalar field (which are of opposite signs) results in a
classical non-singular stable and nearly-symmetric bouncing dynamics deep in
the radiation-dominated era. This encompasses the observed redshifting era
which preceded by a ‘bounce’ that follows a blushifting era. The model is
essentially free of the horizon or flatness problems. Big Bang nucleosynthesis
sets a lower 1-10 MeV bound on the typical energy scale at the ‘bounce’.

###### keywords:

cosmology: theory, dark energy

††pubyear: 2022††pagerange: Cosmology in a locally scale invariant
gravity–References

##  1 Introduction

General relativity (GR), the backbone of the standard cosmological model, has
successfully passed numerous tests within our solar system. However, it is not
comparably successful on larger, galactic and supergalactic scales, unless
cold dark matter (CDM) and dark energy (DE) are introduced into the cosmic
energy budget. The latter are foreign to the standard model (SM) of particle
physics and only appear in our cosmological model as nearly perfect fluids
with characteristic equations of state (EOS).

In addition, this classical field theory of gravitation is genuinely plagued
by singularities. A few singularity theorems imply that curvature is
inevitably singular unless certain plausible ‘energy conditions’ are violated.
For the latter to take place some form of exotic matter is needed. Curvature
or energy density singularities are encountered either at the centers of black
holes or at the Big Bang, even in the presence of a very early inflationary
phase in the latter case Borde et al. (2003). It is widely hoped that a would-
be quantum theory of gravity will ameliorate these unwelcome singularities,
thereby constituting a major thrust behind the quest for a quantum theory of
gravity.

The largest physical scales, over which the short-range nuclear interactions
and the highly screened electromagnetic interaction are irrelevant, are an
ideal testbed for alternative theories of gravitation, and indeed a few major
persistent anomalies of $\Lambda$CDM may possibly indicate that GR requires
modifications on cosmological scales.

The GR-based $\Lambda$CDM, with an early inflationary phase and a dominant
dark sector [the latter contains CDM and DE components that determine the
background evolution, large scale structure (LSS) formation history, and
gravitational potential wells on galactic and supergalactic scales], has
clearly proved to be a very successful paradigm that provides a compelling
interpretation of essentially all currently available cosmic microwave
background (CMB) and LSS observational data, as well as of light element
abundances based on Big Bang nucleosynthesis (BBN). It is remarkable that
$\Lambda$CDM provides a very good fit to this extensive observational data,
that sample a wide range of phenomena over a vast dynamical range, using less
than a dozen free parameters.

However, $\Lambda$CDM also lacks in a few ways. A major drawback is that the
essence of DE and CDM remains elusive. Another problem is that, the currently
leading inflationary scenario, ‘eternal inflation’, seems to lack predictive
power as it is most naturally realized in the multiverse. In addition,
conceptually, the existence of the Big Bang, which essentially signals the
breakdown of the underlying theory of gravitation, is also a major problem.

Moreover, a few mild to strong inconsistencies between various datasets
comparison to $\Lambda$CDM have been found, e.g. Addison et al. (2016). These
include the existence of a persistent relative deficit in power of density
perturbations on superhorizon scales, e.g. Hinshaw et al. (1996), Copi et al.
(2009), Schwarz et al. (2016), an anomalously large weak lensing of the CMB
anisotropy by the intervening large scale structure between the present and
the last scattering surface [Planck Collaboration et al. (2020)], a
statistically significant ‘Hubble tension’ between local and high-z inferences
of the local expansion rate, e.g. Freedman (2017), Riess et al. (2018), Riess
et al. (2019), Wong et al. (2020), and others.

The main objective of the present work is to demonstrate the viability of an
alternative, classical, non-singular ‘bouncing’ cosmological model within a
physical framework based on a globally U(1)-symmetric and locally Weyl-
invariant scalar-tensor (WIST) theory of gravity, while the SM of particle
physics is left unchanged. Unlike in quantum gravity-inspired bouncing models
classical bounces may take place at typical energies that are well within the
range of well-established physics.

The terminology that will be used in this work is somewhat different from the
one used in the standard cosmological model. In the latter, the observed
redshift on cosmological scales is synonymous to space expansion, and in
bouncing cosmological models it is actually meant that space contraction
itself momentarily halts at the ‘bounce’ followed by space expansion. Since in
the proposed model space is static and cosmic evolution is regulated by the
temporal evolution of mass we will employ the more appropriate notions of
‘turning point’, ‘blueshifting’- and ‘redshifting-phase’ instead of the
commonly used parlance of ‘bounce’, ‘contraction’ and ‘expansion’,
respectively.

Throughout, we adopt a mostly-positive signature for the spacetime metric
$(-1,1,1,1)$. Our units convention is $\hbar=c=1$. We outline the theoretical
approach adopted in the present work in section 2, and the cosmological model
is presented in section 3. The main results are summarized in section 4.
Stability analysis at and around the ‘bounce’ is carried out in Appendix A.

##  2 THEORETICAL FRAMEWORK

The gravitational sector of the fundamental interactions is described by the
following WIST action which is in addition globally U(1) symmetric [Shimon
(2021b)]

$\displaystyle\mathcal{I}_{WIST}$ $\displaystyle=$
$\displaystyle\mathcal{I}_{gr}+\mathcal{I}_{S},$ (1)

where

$\displaystyle\mathcal{I}_{gr}$ $\displaystyle\equiv$
$\displaystyle\xi\int\left(|\phi|^{2}R+6\phi^{*}_{\mu}\phi^{\mu}\right)\sqrt{-g}d^{4}x$
(2) $\displaystyle\mathcal{I}_{S}$ $\displaystyle\equiv$ $\displaystyle\int
V(|\phi|;\\{\psi\\})\sqrt{-g}d^{4}x,$ (3)

are the ‘free gravitational’ and ‘source’ actions, respectively, $\xi$ is a
dimensionless parameter, and $\phi\equiv\chi e^{i\alpha}$.

Here and throughout, Greek indices run through spacetime coordinates,
$f_{\mu}\equiv\frac{\partial f}{\partial x^{\mu}}$ for any scalar function
$f$, and summation convention applies. The curvature scalar $R$ is calculated
from the spacetime metric $g_{\mu\nu}$ and its first and second derivatives in
the usual way.

The potential $V(\chi;\\{\psi\\})$ depends on both the modulus $\chi$ of the
scalar field and $\\{\psi\\}$. The latter collectively denotes all fields
other than $\phi$, including $g_{\mu\nu}$. Aside from being non-negative, the
potential $V$ is only required to satisfy the relevant field equations, and
may in general be unrelated to ordinary matter. Since $V$ is independent of
the phase $\alpha$, the latter is a ‘cyclical coordinate’ in field space that
appears in Eq. (1) only via its derivative coupling. Consequently, Eq. (1) has
a global U(1) symmetry and a conserved global charge.

By construction, Eqs. (2) and (3) are (Weyl-) invariant under
$g_{\mu\nu}\rightarrow\Omega^{2}(x)g_{\mu\nu}$,
$|\phi|\rightarrow|\phi|\Omega^{-1}(x)$ and $V\rightarrow V\Omega^{-4}$. For
the kinetic term to have the canonical form $\phi^{*}_{\mu}\phi^{\mu}$ (up to
a sign difference) $\xi$ must be 1/6. Hereafter, the WIST theory considered in
the present work consists of Eqs. (1)-(3) with $\xi=1/6$.

According to Eq. (1), spacetime curvature is sourced by the kinetic and
potential terms that appear in Eqs. (2) and (3), respectively. This is not a
radically different situation than in the GR-based $\Lambda$CDM model where
gravitation is not exclusively determined by $\mathcal{L}_{SM}$, the matter
lagrangian density of the SM of particle physics; in order for $\Lambda$CDM to
provide a reasonably good fit to observational data $\mathcal{L}_{SM}$ is
amended by CDM and DE which together comprise up to $\sim 95\%$ of the cosmic
energy budget at present. However, a notable conceptual difference is that in
the framework described by Eqs. (1)-(3) a single scalar field, $\chi$, is
responsible to the evolution of the Planck mass, active gravitational masses,
and potentially also to the existence of what is normally interpreted as CDM
and DE on cosmological scales. In contrast, in $\Lambda$CDM the Planck mass is
fixed and the evolution is determined by space expansion, which in turn is
driven the by energy budget of which CDM and DE are key building blocks. The
former is believed to be in the form of some exotic beyond-the-SM particles,
and the latter is believed to be the manifestation of some slow-rolling
quintessence field.

The field $\chi$ is the analog of $a(t)$, in $\Lambda$CDM, but in the latter
case the dynamics represents space expansion and energy conservation of the
individual species. The latter are described by effective perfect fluids. In
other words, the total energy density, $\rho$, in the standard $\Lambda$CDM
model is $\rho=\sum_{i}\rho_{i}$ with $\rho_{i}\propto a^{-3(1+w_{i})}$ being
the energy density of the i’th species which is characterized by an EOS,
$w_{i}$. It is assumed that they do not interact, i.e. energy-momentum is
separately conserved for the individual species, and consequently each species
evolves independently of the others. In contrast, in the WIST-based model, the
energy-momentum is not the source of curvature, yet we obtain analogous
evolution to that of $\Lambda$CDM simply because it is assumed that the
potential, $V$, is an analytic polynomial in $\chi$, which is arguably one of
the simplest possible shapes that a potential could possibly have.

Since all quantities that would be normally referred to as ‘active
gravitational masses’, as well as the ‘Planck mass’, are now governed by a
single scalar field $\chi$, the dimensionless ratios of active gravitational
masses of any two bodies, as well as ratios of active masses to the Planck
mass, are fixed to their standard values.

If the scalar field is fixed $\chi=\sqrt{\frac{3}{8\pi G}}$ and $V(\chi)$ is
renamed $\mathcal{L}_{M}$ then Eq. (1) reduces to the Einstein-Hilbert (EH)
action. The specific constant $G$ that appears in the EH action guarantees
that the resulting gravitational field equations reduce to the Poisson
equation in the weak field limit within, e.g. the solar system, where the
‘universality’ of $G$ has been reasonably established. In addition, even if we
favor the idea that DM exists in the form of some exotic, beyond-the-SM
particles, we still lack Cavendish-like experimental evidence that they
‘couple’ gravitationally via the same ‘Universal’ strength $G$ either to each
other or to baryons.

General properties of the theory described by Eqs. (1)-(3) are discussed in
Shimon (2021b). For example, the tensor
$S_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}V)}{\delta
g^{\mu\nu}}$, satisfies

$\displaystyle S^{;\nu}_{\mu\nu}=\phi_{,\mu}\frac{\partial
V}{\partial\phi}+\phi_{,\mu}^{*}\frac{\partial V}{\partial\phi^{*}},$ (4)

i.e. in general it is not conserved. This is expected since the Planck mass
varies in space and time. In addition, the requirement that $\mathcal{I}_{S}$
described by Eq. (3) is WI implies that

$\displaystyle\phi\frac{\partial V}{\partial\phi}+\phi^{*}\frac{\partial
V}{\partial\phi^{*}}=S,$ (5)

where $S\equiv S_{\mu}^{\mu}$, and assuming that only variations of
$g_{\mu\nu}$ and the scalar field are considered. In analogy with the energy-
momentum tensor $T_{\mu}^{\nu}=-\rho\cdot diag(1,-w,-w,-w)$ of a perfect fluid
characterized by an EOS $w$, that is employed in the standard cosmological
model, we have $S_{\mu}^{\nu}=-V\cdot diag(1,-w,-w,-w)$. The perfect fluid
assumption in cosmology is the embodiment of the idea that there is no energy
and momentum flow in an isotropic and homogeneous Universe. The microphysical
meaning of $w\equiv P/\rho$, where $P$ is the pressure, is irrelevant in
$\Lambda$CDM. In practice, the EOS $w$ is simply a parameter that determines
the evolution rate of the energy density, and so is the case in the model that
is proposed here.

Since $S_{0}^{0}=-V$ it immediately follows from Eq. (5) that

$\displaystyle V\propto\chi^{1-3w},$ (6)

in the case of an effective single fluid with an EOS $w$, Shimon (2021b).
Consequently, the potential is quartic in the case $w=-1$, is independent of
$\chi$, i.e. of active masses, in the case $w=1/3$, and is linear in masses in
case of non-relativistic (NR) matter, i.e. the case $w=0$. Although there is
no fundamental principle that requires $V$ to be an analytic function of the
field $\chi$ we do impose this restriction. In particular, we assume that $V$
is an analytic polynomial. This by itself corresponds to a constraint on the
EOS, $w\leq 1/3$, in case that $V$ was a monomial. As is generically the case,
there is no prescription for choosing the form of the potential. The latter is
designed, subject to certain symmetry requirements, to recover (along with the
kinetic terms) the required dynamics of the fields, as determined by
experiments/observations. Therefore, and following the foregoing discussion,
the ‘coefficients’ $V_{i}({\psi})$ in the potential

$\displaystyle
V(|\phi|;\\{\psi\\})=\sum_{i=0}^{i_{max}}V_{i}(\\{\psi\\})\chi^{i},$ (7)

are determined on cosmological scales by the observed cosmic evolution. The
effective EOS associated with the i’th contribution is

$\displaystyle w_{i}\equiv\frac{1-i}{3}.$ (8)

Consequently, the only allowed terms are characterized by parameters $w_{i}$
which are integer multiples of 1/3 subject to the constraint $w_{i}\leq 1/3$
for all i, $V$ is a polynomial in $\chi$ that contains only non-negative
powers of $\chi$ and there is no ‘anisotropy problem’. The latter is a well-
known problem Misner (1969) that is generic to bouncing models Levy (2017). In
practice, the constraint the $w_{i}\leq 1/3$ is implicitly assumed to hold in
the standard $\Lambda$CDM model as well. The ‘mixmaster’ model of Misner
(1969) typically arises from allowing for different evolutions along three
principal axes. This would require, in particular, that these three functional
degrees of freedom are manifested in the matter lagrangian density. In the
limit of small anisotropic evolution, this is represented by an effective
stiff matter contribution to the energy budget that dominates the cosmic
energy budget prior to radiation. This dynamics does not have an analog in the
present model because the potential $V$ is ‘protected’ against this ‘shearing’
dynamics by the assumed U(1) symmetry.

In the same fashion that effective energy density and pressure of the various
species in $\Lambda$CDM depend on a single $a(t)$ function, commensurate with
isotropic expansion, so does $V$ in Eq. (7) depends on $\chi$, and not on
$\chi\cos\alpha$ and $\chi\sin\alpha$. In other words, the underlying U(1)
symmetry of Eq. (7) reflects isotropic expansion in the $\Lambda$CDM model.

##  3 COSMOLOGICAL MODEL

The modulus of the scalar field, $\chi$, delineates essentially the same
dynamics that the scale factor $a(\eta)$ does in $\Lambda$CDM (insofar
$\alpha$ is dynamically insignificant). However, unlike $a(\eta)$ which is
part of the Friedmann-Robertson-Walker (FRW) metric, $\chi(\eta)$ is a scalar
field living on a static spacetime. If the time variation of $\alpha$ is
sufficiently slow, the entire observable cosmic evolution, from BBN (taking
place at typical $\sim$1 MeV energies) onward, is essentially
indistinguishable from that of $\Lambda$CDM, thereby retaining its merits.
However, the very early Universe can be much different, e.g. there is no
initial singularity in the proposed model and possibly also no primordial
phase transitions that in $\Lambda$CDM are expected to have taken place at
energy scales of O(100) MeV and O(100) GeV, and possibly also at the GUT
scale.

###  3.1 Evolution of the cosmological background

The equivalence between the FRW action and Eq. (1) in the non-vacuum
homogeneous and isotropic case, in the case of a real $\phi$ field, has been
discussed in, e.g. Shimon (2021b). As mentioned above, according to the
alternative cosmological model proposed here, invoking a complex scalar field
evades the initial singularity problem thereby resolving in effect the
cosmological horizon Rindler (1956), Misner (1969), and flatness Dicke (1969)
problems. In the following we derive the background evolution equations.

Before focusing on the case of Minkowski spacetime background we consider a
more general spacetime which is described by the static metric
$g_{\mu\nu}=diag(-1,\frac{1}{1-Kr^{2}},r^{2},r^{2}\sin^{2}\theta)$. The latter
is the FRW metric in the ‘comoving frame’ where the time coordinate is
conformal time $d\eta=dt/a$, and $K$ is the spatial curvature parameter. This
spacetime trivially satisfies the Cosmological Principle. The Ricci curvature
scalar associated with this metric is $R=6K$. Considering $\phi=\chi
e^{i\alpha}$ and assuming $\xi=1/6$, Eq. (1) then reads

$\displaystyle\mathcal{I}=\int\left[K\chi^{2}+\chi^{\prime
2}+\chi^{2}\alpha^{\prime 2}+V(\chi)\right]\sqrt{-g}d^{4}x.$ (9)

Variation with respect to $\alpha$ and $\chi$, respectively, results in

$\displaystyle\alpha^{\prime}$ $\displaystyle=$
$\displaystyle\frac{c_{\alpha}}{\chi^{2}}$ (10) $\displaystyle
2\chi^{\prime\prime}$ $\displaystyle=$ $\displaystyle 2K\chi+2\alpha^{\prime
2}\chi+\frac{dV}{d\chi},$ (11)

where $c_{\alpha}$ is an arbitrary integration constant. Employing Eq. (10) in
(11), multiplying by $\chi^{\prime}$, and integrating we obtain the analog of
the Friedmann equation

$\displaystyle\mathcal{Q}^{2}+\alpha^{\prime 2}+K=\frac{V}{\chi^{2}},$ (12)

where an integration constant has been absorbed in $V$. This contribution to
$V$ corresponds to an EOS $w\equiv P/\rho=\frac{1}{3}$ as is evident from Eqs.
(6) and (8). Here, $P$ and $\rho$ are the pressure and energy density,
respectively, and $\mathcal{Q}\equiv\frac{\chi^{\prime}}{\chi}$ the analog of
the conformal Hubble function, $\mathcal{H}\equiv\frac{a^{\prime}}{a}$, has
been defined. Making the replacement $\chi\rightarrow a$ and substituting
$\alpha=0$ in Eqs. (9) and (12) we recover the EH action and standard
Friedmann equation, respectively. Here, instead of space expansion, the entire
cosmic evolution is due to the time-dependence of $\chi$, i.e. of active
gravitational masses and the Planck mass, or equivalently due to the evolution
of inertial masses, i.e. of the Rydberg ‘constant’. The evolution of the
latter, on a static background, is then due to the monotonic growth of
inertial masses in the redshifting era. Upon combining Eq. (12) and its
derivative with Eq. (6) we obtain

$\displaystyle
2\mathcal{Q}^{\prime}+\mathcal{Q}^{2}+K=-\frac{3wV}{\chi^{2}}+3\alpha^{\prime
2}.$ (13)

Back to the ‘Friedmann equations’, Eqs. (12) and (13). These are augmented
with $\propto\alpha^{\prime 2}$ terms that effectively behave as a negative
energy with a ‘stiff’ EOS ($w_{\alpha}=1$), with an effective contribution
$-c_{\alpha}^{2}\chi^{-2}$ to the effective potential. Assuming this
contribution is negligible at present, as well as at any observationally
accessible cosmological era, then at sufficiently small $\chi$ this term
competes with the radiation term, $V_{0}\propto\chi^{0}$, and at the ‘turning
point’ $\chi_{t}=c_{\alpha}/\sqrt{V}$ the rate $\mathcal{Q}$ momentarily
vanishes and transition from $\mathcal{Q}<0$ to $\mathcal{Q}>0$ ensues.

Light element abundances set a limit on the redshift at the turning point,
$z_{t}>10^{9}$. Since $c_{\alpha}^{2}/(\chi^{2}V_{0})\propto(1+z)^{2}$ it then
follows that if $z_{t}>10^{9}$ then already by recombination
$c_{\alpha}^{2}/\chi^{2}$ dropped at best to a trillionth of $V_{0}$ and
therefore has virtually no impact on the standard $\Lambda$CDM description of
structure formation history.

Other bouncing models with similar behavior near the bounce, that contain a
negative energy effective stiff matter component (that are sourced by other
mechanisms), have been considered in Barcelo and Visser (2000) and Barrow et
al. (2004). We emphasize that in the present model the energy condition is not
violated, i.e. $V$ – the analog of $\mathcal{L}_{M}$ – still includes only
positive contributions, and it is only the contribution of the
$U(1)$-symmetric kinetic term that makes an effective negative contribution,
$-c_{\alpha}^{2}\chi^{-2}$, to the source term in Eq. (12). This case is
somewhat similar to the effective energy density associated with spatial
curvature in the standard GR-based FRW spacetime,
$\rho_{K}\equiv-K/H_{0}^{2}$; In case of a closed Universe, $K>0$, the
effective $\rho_{K}$ is negative, and so a hypothetical radiation-only closed
Universe (a legitimate solution of the Friedmann equation) will consequently
undergo a bounce incurring no violation of the energy condition. Here, the
negative kinetic energy associated with $\alpha$ is not counted as a
contribution to the source, in the same fashion that $\mathcal{Q}^{2}$, the
kinetic energy associated with $\chi$, (which in the standard Friedmann
equation is represented by $a$) is not.

Accounting for the vacuum-like, NR, radiation and effectively stiff energy
densities, Eqs. (10) and (12) combine to

$\displaystyle\mathcal{Q}^{2}=V_{4}\chi^{2}+V_{1}/\chi+V_{0}/\chi^{2}-c_{\alpha}^{2}/\chi^{4}.$
(14)

Two interesting limits of this equation will suffice for our purposes. The
first is obtained by neglecting the vacuum-like and NR terms near the turning
point. In this case, the Friedmann-like equation, Eq. (14), integrates to

$\displaystyle\chi^{2}=V_{0}\eta^{2}+c_{\alpha}^{2}/V_{0},$ (15)

where $\chi$ attains its minimum at $\eta=0$. It could be readily integrated
to yield the analog of the cosmic time around the turning point,
$t\propto\int\chi(\eta)d\eta$, and is easily verified to be non-singular as
well. In other words, the effective time coordinates of both massless and
massive particles can be extended through the turning point, i.e. spacetime is
geodesically-complete as is generically the case in non-singular bouncing
cosmological models, Ijjas and Steinhardt (2018). In the absence of turning
point (e.g. in case that $\alpha$ is a fixed constant) the scalar field would
have scaled $\chi\propto\eta$ as does the scale factor $a(\eta)$ in the
radiation-dominated (RD) era. The scalar field then vanishes at $\eta=0$.
Whereas this does not represent a curvature singularity, it is a topological
singularity and it is not entirely clear that the theory can be extended past
beyond $\eta=0$ as this would (at least formally) imply negative (active
gravitational) masses, i.e. negative modulus of the scalar field, $\chi<0$.

In the other extreme – sufficiently far from the turning point, where the
background dynamics is dominated by the quartic potential – Eq. (12)
integrates to $\chi=\left(\sqrt{\lambda}(\eta_{c}\mp\eta)\right)^{-1}$. The
integration constant $\eta_{c}>0$ represents the start and end of the
(conformal) time coordinate in the proposed nearly symmetric model, i.e.
$\eta\in(-\eta_{c},\eta_{c})$. Again, since $\eta$ is bounded from below, this
time due to the presence of the vacuum-like energy density component, then
observed radial distances are bounded at $\eta=-\eta_{c}$ where the scalar
field diverges and the model breaks down ($\eta=\mp\eta_{c}$ correspond to
$t=\mp\infty$, i.e past and future infinity). Specifically, since in the case
$K=0$ incoming null geodesics satisfy $r(\eta)=\eta_{0}-\eta$ and since
$\eta>-\eta_{c}$ then the maximal observable distance at the present time is
$r_{max}=\eta_{0}+\eta_{c}$. The latter can be much larger than the Hubble
scale if $\eta_{c}\gg\eta_{0}$, thereby potentially addressing the horizon
problem. In this regime $\mathcal{Q}=\frac{\dot{\chi}}{\chi}$ is constant and
the scalar field is $\chi\propto\exp(\mathcal{Q}t)$. Recall that $\mathcal{Q}$
flips sign at the turning point and so $\chi$ diverges at
$t\rightarrow\pm\infty$, exactly as does $a(t)$ in the standard FRW space-time
in the DE-dominated phase. Although we focus here on DE-dominated asymptotics
it is clear that a similar breakdown is expected in any model which is
asymptotically dominated by $w<-1/3$ as Eq. (10) integrates in this case to
$\chi=c\left(\eta_{c}\pm\eta\right)^{\frac{2}{1+3w}}$ (in the cases $\eta<0$
and $\eta>0$ respectively) where $c$ and $\eta_{c}$ are positive integration
constants. Note that this is also the solution of the (equivalent) Friedmann
equation, the scale factor $a(\eta)$, and this and scalar field singularity is
present in $\Lambda$CDM as well (as expected; only the small $\chi$ dynamics
is affected by introducing the phase $\alpha$). However, in the latter it is
only a future singularity (where $\chi$ diverges), whereas in the proposed
model it is also a past singularity taking place during the blueshifting
epoch, thereby rendering the causally-connected Universe finite, although
possibly much larger than naively thought based on the singular $\Lambda$CDM
model.

Regardless of whether the ‘flatness problem’ is a genuine fine-tuning problem
of the standard cosmological model or not, e.g. Helbig (2012), Carroll (2014),
Helbig (2020), we lay out the problem as it is normally presented in the
literature followed by its resolution by the symmetric ‘bouncing’ scenario in
general, and the proposed model in particular. The ‘flatness problem’ is often
claimed to arise in the hot Big Bang model due to the monotonic expansion of
space and the consequent faster dilution of the energy density of matter
(either relativistic or NR) compared to the effective energy density dilution
associated with curvature. It is thus hard to envisage how could space be
nearly flat [as is indeed inferred from observations, e.g. Planck
Collaboration et al. (2020)] if not for an enormous fine-tuning taking place
at the very early Universe, or alternatively for an early violent inflationary
era. In the proposed scenario the matter content of the Universe has always
been the same, and in particular the present ($\eta_{0}$) ratio of matter- to
curvature-energy densities has been exactly the same at time $-\eta_{0}$.
However, in the blueshifting era matter domination over curvature is actually
an attractor point as the blueshifting universe starts essentially from
$\chi\rightarrow\infty$. In other words, had the Universe been curvature-
dominated (CD) at present (as is naively expected in the standard expanding
hot Big Bang model in the absence of inflation), i.e. at $\chi_{0}$, it must
have been CD at the ‘mirror time’, $\eta=-\eta_{0}$, as well. However, since
$\rho_{M}\propto\chi$ while $\rho_{K}\propto\chi^{2}$ then a curvature
domination at $-\eta_{0}$ would amount to an extremely fine-tuned
$\rho_{M}/\rho_{K}\rightarrow 0$ at $\eta\rightarrow-\eta_{c}$, i.e. at the
pre-bounce branch of $\chi\rightarrow\infty$. As is well-known, entropy
produced in the pre-bounce era could be processed at the bounce to thermal
radiation, implying in effect that $\rho_{r}$ might somewhat change between
pre- and post-bounce but the expectation in the proposed model is that the
contracting and expanding epochs nearly mirror each other. This is especially
the case if the bounce took place at sufficiently low temperature and density.

We stress that not only does this classical ‘bouncing’ cosmological model
avoid the disturbing initial singularity problem, it also achieves this with
no recourse to ideas from quantum gravity, a theory that does not currently
exist (and in addition would probably require a hundred trillion times higher
energy than is currently available in colliders to test).

###  3.2 Linear perturbation theory

$\Lambda$CDM has successfully passed numerous tests and proved to be
remarkably effective in explaining the formation and linear growth of density
perturbations, predicting the CMB acoustic peaks, polarization spectrum, and
damping features on small scales. It also correctly describes the linear and
nonlinear evolution phases of the LSS (on sufficiently large scales) as well
as the abundance of galaxy and galaxy cluster halos. Therefore, it would seem
essential to establish equivalence of linear perturbation theory between the
model proposed here and $\Lambda$CDM.

Consider linear perturbations over the FRW spacetime in the comoving frame.
Metric perturbation variables include the scalars $\varphi$ and $\psi$, vector
mode $v_{i}$ and tensor modes $h_{ij}$, where the latter are subject to the
constraint $\gamma^{ij}h_{ij}=0$. The weakly perturbed line element is
$ds^{2}=-(1+2\varphi)d\eta^{2}+2v_{i}d\eta
dx^{i}+[(1-2\psi)\gamma_{ij}+2h_{ij}]dx^{i}dx^{j}$, with $\gamma_{ij}\equiv
diag[1/(1-Kr^{2}),r^{2},r^{2}\sin^{2}\theta]$. When stress is negligible then
$\varphi=\psi$. We define the fractional energy density and pressure
perturbations (in energy density units)
$\delta_{\rho_{M}}\equiv\delta\rho_{M}/\rho_{M}$ and
$\delta_{P_{M}}\equiv\delta P_{M}/\rho_{M}$ ($=w\delta_{\rho_{M}}$),
respectively. The matter velocity is $v$.

Transforming from the frame where $G=constant$ [in which case Eq. (1) reduces
to GR] to an arbitrary field frame with $\Omega(x)=1-\delta_{\chi}(x)$, and
assuming $\delta_{\chi}\equiv\frac{\delta\chi}{\chi}\ll 1$, implies in
particular that the scalar and metric field transform as
$\chi\rightarrow\chi/\Omega\approx\chi(1+\delta_{\chi})$ and
$g_{\mu\nu}\rightarrow\Omega^{2}g_{\mu\nu}$, respectively, i.e.
$\psi\rightarrow\psi+\delta_{\chi}$, and
$\delta_{\rho_{M}}\rightarrow\delta_{\rho_{M}}+4\delta_{\chi}$ to leading
order. Consequently, the new, ‘shifted’, perturbation variables, e.g.
$\tilde{\varphi}\equiv\varphi+\delta_{\chi}$,
$\tilde{\psi}\equiv\psi+\delta_{\chi}$,
$\tilde{\rho}_{M}\equiv\delta_{\rho_{M}}+4\delta_{\chi}$ etc., obey the same
perturbation equations that are satisfied by the old perturbation quantities.
The fact that the structure of the perturbation equations is unchanged under
Weyl transformations is crucial in the context of stability near the bounce
because perturbations of the scalar field in bouncing scenarios based on
scalar-tensor theories of gravity are a potential cause for instability near
the bounce, e.g. Gratton et al. (2004).

By virtue of the U(1) symmetry we have two new perturbation variables
$\delta_{\chi}$ and $\delta\alpha$ in addition to the standard perturbation
variables $\varphi$, $\psi$, $\delta\rho_{M}$ and $v$. However, in the long
wavelength limit, and ignoring coupling with other perturbation variables, Eq.
(10) $\alpha^{\prime}=c_{\alpha}/\chi^{2}$ implies that
$\frac{\delta\alpha^{\prime}}{\alpha^{\prime}}=-2\delta_{\chi}$, and
consequently the only new independent perturbation variable is, e.g.
$\delta_{\chi}$ in this limit. Possible implications of redefining the
gravitational potential to CDM on galactic and cosmological scales in this
framework are discussed in Shimon (2021a) and Shimon (2022a). Here we point
out that the same rational can be applied on cosmological scales to the point
that particle CDM may not be required at the perturbation level for explaining
the clustering and growth of structure because $\delta_{\chi}$ is a priori an
arbitrary function of space and time. Specifically, the spectrum of metric
perturbations at recombination, with an rms level of $O(10^{-5})$, may simply
reflect a similar perturbation in $G$, gauged by $\delta\chi$, rather than the
existence of particulate CDM. And indeed, $G$ has never been directly measured
on cosmological scales, surely not at the $O(10^{-5})$ precision, and the fact
that CDM seems at all to be required on these scales heavily relies on the
premise that $G$ is a Universal constant. All this implies that on the largest
scales, and ignoring coupling between the various perturbation variables at
the leading perturbation order both $\varphi$ and
$\frac{\delta\alpha^{\prime}}{\alpha^{\prime}}$ are perturbed at the
$O(10^{-5})$ level at the turning point and consequently the perturbations are
stable. A more detailed stability analysis is carried out in Appendix A.

It has been argued that, quite generally, modified theories of gravity that
include no CDM component ought to be unnaturally contrived, exhibiting strong
features in the transfer function Pardo and Spergel (2020). In arriving at
this conclusion it has been assumed that the energy-momentum is conserved and
that acceleration is caused by baryon-matter-only overdensities. However, none
of these assumptions apply to the framework proposed in the present work.

In any case, we emphasize that the model proposed here by no means relies on
the premise that CDM phenomena are merely a manifestation of variations of
$\chi$. CDM can be equally well taken to be particulate, exactly as in
$\Lambda$CDM [Eq. (12) is agnostic to the microphysics of CDM insofar it is
characterized by $w=0$]. Rather, it is only mentioned in passing that at both
the background and perturbation levels CDM phenomena can be accounted for by
$\chi$ and its perturbations.

In $\Lambda$CDM the linear gravitational potential has two modes in the
matter-dominated (MD) era, $\varphi=constant$ and $\varphi\propto\eta^{-5}$.
The latter dies off very quickly in an expanding Universe and is therefore
conventionally ignored in practice. However, in a contracting Universe
$\varphi\propto\eta^{-5}$ is the fastest growing mode, and if we assume that
primordial scalar perturbations are set sometime during the contracting MD
phase or even at $-\eta_{c}$, deep in the DE era of the contracting phase,
they grow by a factor $(1+z_{eq})^{5/2}$ during the MD era (and essentially
freeze during the DE and RD pre-bounce epochs). Assuming the model is
approximately symmetric around the bounce and employing the observationally
favored value for the redshift at radiation-matter equality, $z_{eq}\approx
3400$, we obtain that $\varphi$ amplifies at most by a factor billion over the
time period spanned between $-\eta_{0}$ and $-\eta_{eq}$. Consequently, scalar
perturbations which started anywhere in the range
$\varphi\in(10^{-14}-10^{-5})$ (depending on their formation epoch) would end
up at the observed $\varphi=O(10^{-5})$ level by the time of recombination,
$\eta_{rec}$.

##  4 SUMMARY

While $\Lambda$CDM has clearly been very successful in phenomenologically
interpreting a wide range of observations, it still lacks a microphysical
explanation of several key components, primarily the nature of CDM and DE. It
also suffers from a few outstanding conceptual problems such as the initial
singularity, in addition to a few ‘coincidence’ or ‘naturality’ problems.

Direct spectral information on the CMB is unavailable (due to opacity) in the
pre-recombination era ($z\gtrsim 1100$). From the observed cosmic abundance of
light elements, BBN at redshifts $O(10^{9})$ could be indirectly probed.
Earlier on, at $z=O(10^{12})$ and $z=O(10^{15})$ [energy scales of $O(200)$
MeV and $O(100)$ GeV, respectively], the quantum chromodynamics (QCD) and
electroweak phase transitions had presumably took place, although their
(indeed weak) signatures in, e.g. the CMB, have not been found.

In addition, inflation, a cornerstone of the standard cosmological model, is
clearly beyond the realm of well-established physics; its ultimate detection
via the B-mode polarization that it imprints on the CMB could be achieved only
if inflation took place at energy scales $\sim$trillion times larger than
currently achievable in colliders. Moreover, theoretical expectations for the
amplitude of this B-mode as a function of the energy scale of inflation rely
on the assumption that gravitation is genuinely quantized. The latter
assumption lacks empirical basis at present. By itself, inflation is plagued
by the the $\eta$-problem, trans-Planckian problem, and the ‘measure problem’
in the multiverse, essentially a lack of predictive power.

Ideally, an alternative cosmological model that agrees well with $\Lambda$CDM
at BBN energies and lower, i.e. $z<10^{10}$, while still addressing the
classical problems of the hot Big Bang model that inflation was originally
designed to undertake, as well as avoiding the initial curvature singularity,
and all this in the $\lesssim 10TeV$ range of energies, will be an appealing
alternative that relies on experimentally established physics. One conclusion
of the present work is that this could be in principle achieved, at least in
part, with a (relatively late classical) non-singular ‘bounce’ that also
removes the technically and conceptually undesirable initial singularity
problem of GR-based cosmological models. In order to achieve such a bounce
within GR, or a conformally-related theory, certain ‘energy conditions’ have
to be effectively violated. One specific realization of this program has been
the focus of the present work.

Symmetries play a key role in our current understanding of the inner workings
of the fundamental interactions. For example, the SM of particle physics is
based on a local $U(1)\times SU(2)\times SU(3)$ gauge group with quantized
gauge fields. In addition, our favorite theory of gravitation, GR, is
diffeomorphism-invariant. In this work we entertained the possibility that in
addition to diffeomorphism-invariance, GR is endowed with Weyl invariance,
i.e. local scale invariance, as well as an internal $U(1)$ symmetry with a
global charge. The SM of particle physics is assumed as is, with an explicitly
broken Weyl symmetry. Only the salient merits of the cosmological model based
on this alternative theory of gravitation have been discussed in the present
work.

In the proposed model spacetime is described by the FRW metric in comoving
frame, which in the absence of spatial curvature reduces to the Minkowski
spacetime. Here, the role of the scale factor in $\Lambda$CDM as the regulator
of cosmic history is played by the modulus, $\chi$, of a complex scalar field
$\phi$ that lives on a static background. The phase, $\alpha$, plays a crucial
role near the turning point and is largely irrelevant elsewhere. There is no
analog to $\alpha$ in $\Lambda$CDM. Here, $\chi$, which regulates the
evolution of (dynamical) active gravitational masses starts infinitely large,
first monotonically decreases until it ‘bounces’, then grows again without
bound. Put alternatively, the Planck length starts out infinitely small,
increases until it peaks at the turning point, then decreases again. Described
in terms of these length ‘units’ the Universe is said to undergo a
‘contraction’ phase of evolution, followed by a ‘bounce’ and ‘expansion’.
Since spacetime is described by the FRW metric in the comoving frame, i.e. the
time coordinate is conformal, then inertial masses evolve exactly as do active
gravitational masses and the Planck mass. Cosmological redshift is then a
manifestation of evolving Rydberg ‘constant’ rather than space expansion.

The alternative cosmological scenario explored in this work starts with a
deflationary evolution which culminates in a turning point when the (absolute
value of the negative) energy density associated with the effective ‘stiff
matter’ (provided by the kinetic term of $\alpha$) momentarily equals that of
radiation. In the vacuum-like-dominated epoch the energy density of the
Universe is dominated by a $\propto\chi^{4}$ term in the potential which is
genuinely classical with no quantum fluctuations. Therefore, DE according to
the present scenario is not zero-point energy but rather a manifestation of
the self-coupling of the scalar field, i.e. a term in the potential of the
form $V_{4}\chi^{4}$ with $V_{4}$ being a dimensionless parameter. This DE
contribution is characterized by a non-dynamical EOS with no recourse to, e.g.
a new quintessence, field; here, the same scalar field accounts for both ‘$G$’
and DE, and possibly also CDM.

One of the most notable achievements of inflation was the realization that a
slightly red-tilted primordial spectrum of gaussian perturbations can be
generated by quantum fluctuations in a vacuum-like expanding Universe. It
should be note that a decade before the inflationary scenario has been
proposed it has been realized by Harrison and Zeldovich, as well as by others,
that the primordial spectrum of perturbations should be nearly flat for the
rms perturbations on the wide dynamical range of cosmological scales not to
have infrared or ultraviolet divergences that mark the breakdown of linear
perturbation theory. In any case, no mechanism has been proposed in the
present work for the generation of primordial density perturbations. Whatever
this mechanism turns out to be, it does not involve quantum fluctuations of
the metric field, unlike in inflation. Again, it is assumed here that
gravitation is a genuine classical interaction.

In addition, the ‘anisotropy problem’ that in general plagues bouncing
scenarios does not exist in our construction. As discussed above, Weyl
symmetry and the consequent absence of any dimensional parameter in the
action, in addition to the postulated global U(1) symmetry, protects the model
from running into a chaotic, anisotropy-dominated, evolution phase, unless we
are willing to consider non-canonical terms in the potential, $V$, of the
form, e.g. $V_{ani}\propto(\bar{\psi}\psi)^{2}\chi^{-2}$ and
$C_{\alpha\beta}^{\ \ \gamma\delta}C_{\gamma\delta}^{\ \
\rho\sigma}C_{\rho\sigma}^{\ \ \alpha\beta}\chi^{-2}$, where $\psi$ is e.g., a
Dirac field, and $C_{\rho\sigma}^{\ \ \alpha\beta}$ is the Weyl tensor.

Conformal time is both past- and future-bounded in this scenario, i.e.
$\eta\in(-\eta_{c},\eta_{c})$. In principle, any ‘horizon problem’ could be
avoided if $\eta_{c}\gg\eta_{0}$. Specifically in this scenario, cosmic
history starts with very large (and in principle infinite) particle masses and
therefore the causal horizon is much larger than would be naively expected
from monotonically growing masses (that corresponds to the redshifting era),
i.e. essentially $\eta_{0}\ll\eta_{c}+\eta_{0}$ if $\eta_{c}\gg\eta_{0}$.
Likewise, the ‘flatness problem’ afflicting the hot Big Bang scenario stems
from the slower decay of the energy density associated with curvature as
compared to that of matter in a monotonically expanding Universe. In bouncing
scenarios the situation is reversed before the turning point; starting at
infinitely large $\chi$ (masses) one typically expects to find that the energy
density in the forms of NR matter and radiation largely exceeds that of
curvature at any finite $\chi$ value in the blueshifting era. Since this
adiabatic model is nearly symmetric in $\chi$ around the turning point
(barring possible entropy processing effects at around the RD era, the
efficacy of which depends on the redshift at which such a bounce takes place),
one generally expects the Universe to look spatially flat at any finite $\chi$
after the would-be singularity (actually a non-singular ‘bounce’). From this
perspective flatness is an attractor-, rather than an unstable-point that
requires fine-tuning of the initial conditions.

The model we considered is falsifiable in several ways: First, $w_{DE}=-1$ due
to Weyl invariance and any observationally inferred $w_{DE}\neq-1$ would
either rule out this particular model or alternatively either imply soft
breakdown of WI at very early and late times or the existence of a non-
canonical DE term in the potential of the form, e.g.
$V_{DE}\propto(\bar{\psi}\psi)^{-\varepsilon/3}\chi^{4+\varepsilon}$ (where
$\varepsilon$ is some dimensionless parameter), that involves non-integer, and
possibly irrational, powers of the fields. Second, within the framework
adopted here, if a scale-invariant B-mode polarization is ultimately measured,
it would provide a compelling evidence that gravity is quantized, in
contradiction to the assumption made here that gravity is a genuinely
classical interaction which implies that its perturbations are not subject to
the Bunch-Davies vacuum condition. Consequently, unlike the inflationary-
induced B-mode polarization of $\Lambda$CDM, it does not follow from any
fundamental principle that B-mode polarization have to be characterized by a
flat spectrum. Third, signals from primordial phase transitions as well as
leptogenesis or baryogenesis that ought to be imprinted in the CMB anisotropy
and polarization (perhaps too weak to be detected) may not have taken place at
all in the proposed model, depending on the typical temperature at the turning
point.

We believe that, in addition to addressing the cosmological horizon and
flatness problems, the framework proposed here provides important insight on
the nature of DE, and initial singularity and stability near the turning
point. Even so, the work presented here is by no means exhaustive, and indeed
a few of its basic aspects will be further elucidated in future works.

## APPENDIX A: STABILITY ANALYSIS

As discussed in section 3.2 the initial scalar metric perturbations deep in
the pre-bounce era should be at the $O(10^{-14})-O(10^{-5})$ level (depending
on their formation time in the blueshifting phase) so that by the time of
recombination they have grown to $O(10^{-5})$. In this Appendix we analyze the
dynamics of scalar perturbations at and near $\eta=0$ and show that all linear
perturbation variables smoothly transform through this point.

Near this point the energy budget is dominated by radiation and the kinetic
term associated with $\alpha$. The Friedmann-like equation, Eq. (12), then
integrates to

$\displaystyle\chi^{2}=A\eta^{2}+B,$ (16)

where according to Eq. (15) $A\equiv V_{0}$ and $B\equiv
c_{\alpha}^{2}/V_{0}$. The analog of conformal Hubble function in this case
becomes $\mathcal{Q}=A\eta/(A\eta^{2}+B)$. Linear perturbation equations
[e.g., Hwang and Noh (2001)] result in coupled equations for $\delta\alpha$
and $\varphi$

$\displaystyle\delta\alpha^{\prime\prime}+\frac{2A\eta}{A\eta^{2}+B}\delta\alpha^{\prime}+k^{2}\delta\alpha$
$\displaystyle=$
$\displaystyle-\frac{4\sqrt{AB}}{A\eta^{2}+B}\varphi^{\prime}$ (17)
$\displaystyle\varphi^{\prime\prime}+\frac{4A\eta}{A\eta^{2}+B}\varphi^{\prime}+\frac{k^{2}\varphi}{3}$
$\displaystyle=$
$\displaystyle\frac{2\sqrt{AB}}{A\eta^{2}+B}\delta\alpha^{\prime},$ (18)

where it is understood that fractional perturbations of the modulus scalar
field, $\delta_{\chi}$, are absorbed in the gravitational potential as
described in section 3.2. Eq. (18) is a generalized Bardeen equation that is
obtained from combination of the Arnowitt-Deser-Misner (ADM) energy constraint
and the Raychaudhuri equation. In the long wavelength limit this system of
equations is satisfied by

$\displaystyle\varphi$ $\displaystyle=$ $\displaystyle
c_{1}+\frac{c_{2}\eta}{(B+A\eta^{2})^{2}}+\frac{c_{3}(A\eta^{2}-B)}{(B+A\eta^{2})^{2}}$
$\displaystyle\frac{\delta\alpha^{\prime}}{\alpha^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{4c_{2}\eta}{(B+A\eta^{2})^{2}}-\frac{c_{3}(-3B^{2}+6AB\eta^{2}+A^{2}\eta^{4})}{B(B+A\eta^{2})^{2}},$
(19)

where $c_{1}$, $c_{2}$ and $c_{3}$ are three integration constants. It is
clear from Eqs. (APPENDIX A: STABILITY ANALYSIS) that both $\varphi$ and
$\delta\alpha^{\prime}/\alpha^{\prime}$ are well-behaved at $\eta=0$ due to
the non-vanishing $B$, i.e. the non-vanishing of $\alpha^{\prime}$, and
perturbation theory does not break down there.

To solve for the short-wavelength limit we differentiate Eq. (17) with respect
to $\eta$ and then substitute for $\delta\alpha^{\prime}$ from Eq. (18). The
resulting fourth-order equation for $\varphi$ is

$\displaystyle\left(\frac{Ax^{2}}{k^{2}}+B\right)\varphi+\frac{18A}{k^{2}}x\varphi_{x}+\left(\frac{60A}{k^{2}}+\frac{4A}{k^{4}}x^{2}+4B\right)\varphi_{xx}$
(20) $\displaystyle+$
$\displaystyle\frac{30A}{k^{2}}x\varphi_{xxx}+3\left(B+\frac{Ax^{2}}{k^{2}}\right)\varphi_{xxxx}=0,$

where e.g., $\varphi_{x}\equiv\frac{\partial\varphi}{\partial x}$, in the
large-$k$ limit, which has no closed-form solution. Here $x\equiv k\eta$. In
the limit $x\ll 1$, i.e. $\eta\rightarrow 0$, the potential $\varphi$
decouples from both $A$ and $B$, and the equation considerably simplifies to

$\displaystyle 3\varphi_{xxxx}+4\varphi_{xx}+\varphi=0,$ (21)

with the general solution

$\displaystyle\varphi=c_{1}\cos x+c_{2}\sin
x+c_{3}\cos\left(x/\sqrt{3}\right)+c_{4}\sin\left(x/\sqrt{3}\right),$ (22)

illustrating the fact that perturbations are manifestly stable on small scales
as well.

Slightly off $\eta=0$, $A\eta^{2}\gtrsim B$ perturbations smoothly approach
their short wavelength limit standard behavior in the RD era. In the latter,
within $\Lambda$CDM,
$\varphi=\frac{1}{x}\left[c_{j}j_{1}(x)+c_{y}y_{1}(x)\right]$ where $j_{1}$
and $y_{1}$ are the spherical Bessel and the modified spherical Bessel
functions, respectively, of the first kind. In $\Lambda$CDM, $c_{y}=0$ since
$y_{1}(0)$ diverges, and the phase of the oscillating potential is 0, i.e.
$\varphi\propto\sin x$. This needs not be the case in the proposed model as is
evident from this discussion insofar $B\neq 0$. The implications for CMB
observables of not neglecting the ‘diverging’ mode have been discussed
recently by Kodwani et al. (2019).

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