# Proper time reparametrization in cosmology:
Möbius symmetry and Kodama charges

Jibril Ben Achour 1, 2

(1 Arnold Sommerfeld Center for Theoretical Physics, Munich, Germany
2 Yukawa Center for Theoretical Physics, Kyoto University, Kyoto, Japan)

###### Abstract

It has been noticed that for a large class of cosmological models, the gauge
fixing of the time-reparametrization invariance does not completely fix the
clock. Instead, the system enjoys a surprising residual Noether symmetry under
a Möbius reparametrization of the proper time, which maps gauge-inequivalent
solutions to the Friedman equations onto each other. In this work, we provide
a unified treatment of this hidden conformal symmetry and its realization in
the homogeneous and isotropic sector of the Einstein-Scalar-$\Lambda$ system.
We consider the flat Friedmann-Robertson-Walker (FRW) model, the (A)dS
cosmology and provide a first treatment of the model with spatial constant
curvature. We derive the general condition relating the choice of proper time
and the conformal weight of the scale factor, and give a detailed analysis of
the conserved Noether charges generating this physical symmetry. Our approach
allows us to identify new realizations of this symmetry while recovering
previous results in a unified manner. We also present the general mapping onto
the conformal particle and discuss the solution-generating nature of the
transformations beyond the Möbius symmetry. Finally, we show that, at least in
a restricted context, this hidden conformal symmetry is intimately related to
the Kodama charges of spherically symmetric gravity. This new connection
suggests that the Möbius invariance of cosmology is only the corner of a
larger symmetry structure which could be relevant beyond cosmological models.

###### Contents

  1. 1 Introduction
  2. 2 The cosmological model
    1. 2.1 Reduced action
    2. 2.2 Field dependent reparametrization
    3. 2.3 Hamiltonian formulation
  3. 3 Diff$(S^{1})$ proper time reparametrization of the action
    1. 3.1 Symmetries
      1. 3.1.1 Flat FRW cosmology
      2. 3.1.2 (A)dS cosmology
      3. 3.1.3 The closed universe
    2. 3.2 Conformal bridge, Niederer transformation and Schwarzian curvature
    3. 3.3 Mapping on the conformal particle
  4. 4 Noether charges
    1. 4.1 Charges for the flat model
    2. 4.2 Charges for the closed universe
      1. 4.2.1 First clock
      2. 4.2.2 Second clock and Niederer transformations
    3. 4.3 Charges for the (A)dS cosmology
  5. 5 Relation to volume preserving diffeomorphisms
    1. 5.1 Review of the Kodama machinery
    2. 5.2 From the Kodama to the CVH generators
  6. 6 Discussion
  7. A Resolution of the differential equation

##  Introduction

Symmetry reduced models of General Relativity (GR) play a pivotal role in our
understanding of cosmological and black hole geometries. Investigating the
gravitational dynamics in this simplified context, where a large number of
degrees of freedom are frozen, turns out to have many advantages as one can
have a better control on the resulting system to classify its solutions as
well as to investigate its quantization. Moreover, the process of symmetry
reduction can give rise to a rich underlying structure through the emergence
of powerful enhanced symmetries [1]. In turn, such a hidden symmetry can give
access to interesting solution-generating techniques to explore the solution
space while revealing surprising properties of the underlying gravitational
system.

When restricting the gravitational field to a homogeneous geometry, the gauge
invariance of the full theory reduces to a simple invariance under time-
reparametrization. Fixing this gauge through a given choice of clock breaks
the gauge symmetry and leaves us with a mechanical system for which the clock
should, a priori, inherit the status of an absolute mechanical time. However,
the situation turns out to be more subtle. For a large class of cosmological
systems, the reduced action actually enjoys a residual Noether symmetry under
the 1d conformal group SL$(2,\mathbb{R})$ even after fixing the gauge. As a
result, the clock is only defined as an equivalence class of clocks which can
be mapped to each other under appropriate symmetry transformations. This
residual symmetry is a global one, which acts as a Möbius reparametrization of
the proper time. As a physical symmetry of the system, it maps gauge-
inequivalent solutions to the Friedmann equations onto each other. The goal of
this work is to provide a complete picture of this hidden symmetry for the
class of homogenous and isotropic cosmological models, completing and
generalizing previous findings.

The existence of such residual conformal symmetry in cosmology was first
noticed long time ago by Pioline and Waldron [2]. It was observed that the
dynamics of a (higher dimensional) vacuum (A)dS universe can be recast as the
one of the conformal particle, an SO$(2,1)$ invariant mechanical system
introduced in [3]. However, the mapping was restricted to a specific example
and the explicit symmetries acting on the reduced cosmological action were not
presented. Much later, an in-depth analysis of this conformal invariance was
presented in [4] in the context of the flat Friedmann-Robertson-Walker (FRW)
model filled with a massless scalar field. See also [7, 6, 5] for earlier
investigations of this symmetry in the context of quantum cosmology. Starting
from this analysis, the fate of this symmetry in cosmological systems was
explored in different directions, revealing interesting generalizations and
suggesting the existence of a larger underlying symmetry. On the one hand, the
generalization of the $\text{SL}(2,\mathbb{R})$ transformations to the whole
Diff$(S^{1})$ transformations was investigated in [8] and shown to provide a
solution-generating map, using the property of the Schwarzian derivative,
which allows one to reconstruct the (A)dS cosmology from the flat FLRW model.
See also [9, 10] for related discussions on this point. On the other hand, it
was shown in [11, 12] that the SL$(2,\mathbb{R})$ symmetry stands as a
subsector of a larger SO$(3,2)$ group of symmetry which mixes the
gravitational and matter degrees of freedom. Finally, this symmetry was found
to hold for more general systems, such as the Schwarzschild-(A)dS black hole
mechanics (described by the Kantowski-Sachs midi-superspace model) for which
the symmetry is upgraded to the group
SL$(2,\mathbb{R})\ltimes\mathbb{R}^{3}$[13, 14]. These different
generalizations therefore suggest the existence of a deeper structure to be
uncovered and beg for additional investigations.

In this work, we shall not focus on the recent developements mentioned above,
but reconsider instead the class of homogeneous and isotropic cosmological
systems. Indeed, despite the previous investigations, a complete picture of
how this conformal symmetry is realized in such simple models is still missing
and several questions have remained unanswered:

  * •

Being realized as a Möbius reparametrization of the proper time, the symmetry
discussed in [4, 8] involves a gauge fixing. How does the choice of clock
affect the realization of the residual symmetry ? What are the conditions
selecting the clocks for which the symmetry is realized ?

  * •

Moreover, under such a reparametrization of the proper time, the scale factor
transforms as a primary field and not as a scalar. How does the conformal
weight of the scale factor get modified when different clocks are used ? Can
we derive some general conditions relating the choice of proper time and the
value of the conformal weight ?

  * •

Previous works have focused on the flat FRW cosmology as well as on its (A)dS
extension [4, 8]. In the framework of homogeneous and isotropic cosmological
models, one can also consider models with a homogeneous spatial curvature
being either positive, i.e. the closed universe, or negative, with hyperbolic
spatial section. These models have not been treated yet. Do they also enjoy a
residual symmetry similar to the flat and (A)dS models ? What is the fate of
the symmetry when both sources of homogeneous curvature are turned on ?

  * •

Following the initial work by Pioline and Waldron [2], can we understand this
symmetry through a unified mapping of these cosmological models into the
conformal particle ?

  * •

Finally, it is worth pointing out that up to now, the investigations on this
residual conformal symmetry have been restricted to homogeneous gravitational
fields describing either cosmology or the black hole interior. Can we find
some hints towards a possible generalization to inhomogeneous spacetime such
as inhomogeneous spherically symmetric gravity ?

In the following, we shall address these questions step by step. We will
consider the homogeneous and isotropic sector of the Einstein-Scalar-$\Lambda$
system and allow the geometry to admit a homogeneous constant spatial
curvature. The model is therefore parametrized by two parameters
$\left(L_{c},L_{\Lambda}\right)$ which encode respectively the spatial
constant curvature and the length scale associated to the cosmological
constant. In order to investigate how the choice of proper time affects the
realization of the symmetry, we futhermore introduce a one-parameter field
dependent reparametrization of the lapse given by (2.7) which encodes the
freedom in choosing the clock. Starting from this general set-up, we compute
the general variation of the cosmological action under a Diff$(S^{1})$
transformation and identify the constraint relating the choice of proper time
and the value of the conformal weight. Then, this general condition (3.6)
allows us to identify in a unified manner the different realizations of the
residual conformal symmetry. In addition to rederiving the known cases treated
in [4, 8], we identify new realization of the symmetry for the flat FRW and
(A)dS models, while providing a first treatment of the model with spatial
constant curvature. This first part thus provides a unified treatment of this
residual conformal symmetry in homogeneous and isotropic cosmological models.
This is the first result of this work.

In a second part, we investigate the specific Diff$(S^{1})$ transformation
associated with a constant Schwarzian derivative known as Niederer’s
transformations [15]. We show that such a conformal reparametrization is
solution-generating: from the flat FRW model, they allow one not only to
generate the potential term associated with the cosmological constant, but
also the one associated with a constant spatial curvature. Thus, they provide
non-trivial deformation of the CVH generators and allow one to turn on or off
the potential contribution associated with a constant curvature in the
symmetry reduced action, while preserving the ${\mathfrak{sl}}(2,\mathbb{R})$
structure. This generalizes the findings presented in [8] for the (A)dS case.
Then, we turn to the mapping onto the conformal particle. We show that the
seminal work by Pioline is a subcase, associated to a specific choice of
clock, belonging to a more general duality relating the different isotropic
cosmological systems to the conformal particle with or without a Newton-Hooke
interaction [16]. We use this to comment on the possible consequences of such
mapping at the quantum level.

In the third part of this work, we perform a detailed analysis of the
conserved charges generating the Möbius invariance of each cosmological
models. These charges are evolving constants of motion111The name was
introduced in [17]. In our case, it allows to stress the difference between
the role of the CVH generators, which are not conserved along the evolution,
and the Noether charges which are built as suitable time-dependent
combinations of the CVH generators such that they are conserved. See [18] for
a discussion on the notion of evolving constant of motion. built up from a set
of phase space functions which form an ${\mathfrak{sl}}(2,\mathbb{R})$ Lie
algebra, known as the CVH algebra [7, 6, 5]. We show explicitly how this
charge algebra is modified when using different clocks to express the evolving
constants of motion. Finally, we use them to solve the cosmological dynamics
in an algebraic way, which provides an efficient shortcut to capture the
cosmological dynamics, especially for the closed universe.

The last part of this work aims at providing a first hint toward the last
question raised above. Using the fact that our set-up is spherically
symmetric, we compute the associated Kodama charges [19]. The Kodama vector is
a divergence-free vector field defined for any spherically symmetric
background. Being divergence free, it naturally defines two conserved currents
which are covariantly defined again for any spherically symmetric geometries.
This fact is known as the Kodama’s miracle and has triggered interesting
investigations [24, 26]. However, to our knowledge, the charges associated
with these Kodama currents have not been analysed so far, and in particular
the type of transformations they generate has remained elusive. We point, at
least for a specific choice of clock and only for the flat FRW and (A)dS
models, that the Kodama generators coincide with two of the phase space
functions forming our CVH algebra, while the third piece is obtained through
the Poisson bracket of the first two. It follows directly that the Kodama
generators can be combined to form again a set of evolving constants of
motion, which satisfies our time-dependent ${\mathfrak{sl}}(2,\mathbb{R})$ Lie
algebra. As a consequence, these conserved Kodama charges can be recognized as
generating the residual symmetry of flat cosmology discussed in this work. To
our knowledge, this is the first observation that the well known Kodama
generators are associated with a conformal symmetry of the underlying
gravitational system, at least in the specific example discussed here. From a
more general perspective, this coincidence suggests that our symmetry might
find some links with the volume-preserving diffeomorphism, to which the Kodama
vector belongs. However, the result discussed here is restricted to a specific
example and we have not been able to find a general mapping yet. More work
would be required to go further. It nevertheless provides an interesting
starting point to investigate possible connections beyond homogeneous
spacetimes.

Having summarized our approach and our findings, let us briefly comment on the
relevance of this symmetry for quantum cosmology. Independently of what should
be the ultimate theory of quantum gravity, it is expected that cosmological
geometries should emerge from some hydrodynamical approximation of this yet-
to-be-constructed theory. Such emergent geometries would then result from a
suitable mean field approximation of the quantum dynamics of the fundamental
degrees of freedom of both quantum geometry and matter, capturing their
collective behavior in the effective dynamics of a single collective wave
function. This provides an interpretation for the wave function of the
universe as obtained in the standard canonical Wheeler-de Witt approach. This
point of view raised several key questions: given some candidate theory for
quantum gravity, how one should select the suitable mean field approximation
to extract semi-classical cosmology ? Can we find some criteria based on
symmetry arguments ? We expect that the conformal symmetry discussed in this
work might play a role in this direction. Concretely, our result might allow
one to connect recent findings on quintic non-linear Schrödinger equation and
its conformal invariance, as discussed in [27, 28], with the current efforts
in the group field theory approach attempting to extract cosmology as a
quantum gravity condensate [29, 30, 31, 32, 33]. Another interesting road
would be to follow the more developed fluid/gravity correspondence and seek
for a dual description of cosmology in terms of conformal fluid dynamics [34].
Besides these on-going investigations, the residual symmetry discussed in this
work might also be relevant for the problem of time in quantum cosmology [35].
Whether this residual symmetry under proper time reparametrization is only a
corner of a much larger Noetherian symmetry describing internal change of
clock, and associated with non-vanishing conserved charges, appears as a
fascinating possibility. Current efforts are devoted towards exploring this
possibility. Finally, we point out that the duality with the conformal
particle opens the possibility to import and adapt results and methods
developed for the quantization of the conformal particle to quantum cosmology,
thus providing a new point of view to discuss long standing issues from a
symmetry point of view. See for examples along this line [36, 37, 38, 39, 40,
41, 42, 43, 44, 45].

This work is organized as follows. Section 2 is devoted to describe our set-
up. We present the Einstein-Scalar-$\Lambda$ reduced action and the geometry
on which we shall focus. In Section 2.2, we introduce our one-parameter field
dependent reparametrization of the lapse and recast the action in the suitable
form for our purpose. Section 3 is devoted to the unified treatment of the
Möbius symmetry for all isotropic models and the derivation of the general
conditions for this symmetry to be realized. In Section 3.2, we present the
solution-generating map based on the Niederer transformation, while Section
3.3 is devoted to the general mapping onto the conformal particle. Section 4
is devoted to a detailed analysis of the Noether charges, their algebra and
the resolution of the dynamics. Finally, we discuss the relation to the Kodama
generators in Section 5 and summarize our results and open directions in
Section 6.

##  The cosmological model

In this section, we specify the family of cosmological models we shall
investigate in this work. They correspond to homogeneous and isotropic gravity
filled with a massless scalar field and a cosmological constant. We present
the symmetry reduced action and discuss the different choices of clocks to
describe the dynamics. Finally, we briefly describe the phase space of our
system before investigating its hidden symmetries.

###  Reduced action

Consider the line element describing the homogeneous and isotropic curved FRW
cosmological background given by

$\textrm{d}s^{2}=-N^{2}(t)\textrm{d}t^{2}+a^{2}(t)\left[\left(1-\frac{r^{2}}{L^{2}_{c}}\right)^{-1}\textrm{d}r^{2}+r^{2}\textrm{d}\Omega^{2}\right]\;\;$
(2.1)

where as usual, $N(t)$ is the lapse function, $a(t)$ is the scale factor and
$L_{c}$ is the length scale encoding the constant spatial curvature of the
background. One can switch from the closed universe with spherical spatial
section to the open universe with hyperbolic spatial section by the mapping
$L_{c}\rightarrow iL_{c}$. The flat case is recovered in the limit
$L_{c}\rightarrow+\infty$. The symmetry reduced action of the Einstein-
Scalar-$\Lambda$ system reads

$\displaystyle S[g,\varphi]$
$\displaystyle=\int\textrm{d}^{4}x\sqrt{|g|}\left[\frac{\mathcal{R}-2\Lambda}{L^{2}_{P}}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\right]$
(2.2)
$\displaystyle=\frac{3\mathcal{V}_{0}}{L^{2}_{P}}\int\textrm{d}t\left[\frac{\left(a^{2}a^{\prime\prime}+a(a^{\prime})^{2}\right)}{N}-\frac{a^{2}a^{\prime}N^{\prime}}{N^{2}}+\frac{Na}{L^{2}_{c}}-\frac{Na^{3}}{L^{2}_{\Lambda}}+L^{2}_{P}\frac{a^{3}(\phi^{\prime})^{2}}{6N}\right]\;\;\;\;$
(2.3)

where a prime refers to a derivative w.r.t the time coordinate $t$. The length
scale $L_{P}=\sqrt{8\pi G}$ is the Planck length,
$L_{\Lambda}=\sqrt{3/2\Lambda}$ and we have introduced an homogeneous scalar
field $\phi:=\phi(t)$ without a self-interacting potential as a non-trivial
matter source. Notice that we have restricted the spatial integration to a
fiducial volume $\mathcal{V}_{0}$ given by

$\mathcal{V}_{0}=4\pi\int^{r_{\ast}}_{0}r^{2}\left(1-\frac{r^{2}}{L^{2}_{c}}\right)^{-1/2}\textrm{d}r\;,$
(2.4)

where $r_{\ast}$ sets the scale at which we restrict the integration in the
radial direction. Thus $\mathcal{V}_{0}$ plays the role of the IR cut-off in
our system and $[\mathcal{V}_{0}]=L^{3}$. We find convenient to introduce the
rescaling

$V_{0}=3\mathcal{V}_{0},\qquad\varphi=\phi/3,$ (2.5)

Finally, integrating by part and upon implementing the above rescaling, the
reduced action reads

$\displaystyle S[a,\varphi,N]$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}t\left[L^{2}_{P}\frac{a^{3}(\varphi^{\prime})^{2}}{2N}-\frac{a(a^{\prime})^{2}}{N}+\frac{Na}{L^{2}_{c}}-\frac{Na^{3}}{L^{2}_{\Lambda}}+\left(\frac{a^{2}a^{\prime}}{N}\right)^{\prime}\right]$
(2.6)

Our goal is to investigate the realization of hidden symmetry of this
cosmological system and in particular the role of proper time
reparametrization. In that regard, it will reveal useful to find a suitable
parametrization for the choice of proper time. This can be achieved by a field
dependent reparametrization of the action which we now describe.

###  Field dependent reparametrization

Consider the following field-dependent reparametrization where the lapse, and
thus the time coordinate, are switched to

$N\textrm{d}t=\frac{\mathcal{N}\textrm{d}\tau}{a^{n}}$ (2.7)

where $n\in\mathbb{Z}$. The line element becomes

$\textrm{d}s^{2}=-\frac{\mathcal{N}^{2}(\tau)}{a^{2n}(\tau)}\textrm{d}\tau^{2}+a^{2}(\tau)\left[\left(1-\frac{r^{2}}{L^{2}_{c}}\right)^{-1}\textrm{d}r^{2}+r^{2}\textrm{d}\Omega^{2}\right]$
(2.8)

and in term of this new time coordinate, the reduced action reads

$\displaystyle S_{n}[a,\varphi,\mathcal{N}]$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\tau\left[L^{2}_{P}\frac{a^{3+n}\dot{\varphi}^{2}}{2\mathcal{N}}-\frac{a^{1+n}\dot{a}^{2}}{\mathcal{N}}+\mathcal{N}a^{1-n}\left(\frac{1}{L^{2}_{c}}-\frac{a^{2}}{L^{2}_{\Lambda}}\right)+\frac{\textrm{d}}{\textrm{d}\tau}\left(\frac{a^{3+n}\dot{a}}{\mathcal{N}}\right)\right]$
(2.9)

where a dot refers to a derivative w.r.t. the new coordinate time $\tau$. Let
us make two remarks at this stage. First, notice that the system is still
gauge invariant and the gauge transformations are given by the time-
reparametrization

$\displaystyle\tau\rightarrow\tilde{\tau}$ $\displaystyle=f(\tau)\;,$ (2.10)
$\displaystyle\mathcal{N}\rightarrow\tilde{\mathcal{N}}(\tilde{\tau})$
$\displaystyle=\dot{f}^{-1}(\tau)\mathcal{N}(\tau)$ (2.11) $\displaystyle
a\rightarrow\tilde{a}(\tilde{\tau})$ $\displaystyle=a(\tau)$ (2.12)
$\displaystyle\varphi\rightarrow\tilde{\varphi}(\tilde{\tau})$
$\displaystyle=\varphi(\tau)$ (2.13)

which is the gauge invariance descending from the full diffeomorphism
invariance of the theory after imposing homogeneity and isotropy. Finally, let
us briefly discuss the notion of boundary in our set-up. Having consider a
four-dimensional finite region of spacetime $\mathcal{M}$ and restricted the
spatial integration to a fiducial cell with boundary $\partial\mathcal{M}$, we
can split this boundary as
$\partial\mathcal{M}=\Gamma\cup\Sigma_{+}\cup\Sigma_{-}$ where $\Gamma$ stands
as the finite spatial boundary while $\Sigma_{\pm}$ are respectively the
future/past spacelike boundaries. The assumption of homogeneity implies that
all the fields evolve everywhere the same in that bulk region and on its
boundary $\partial\mathcal{M}$. As a result, the only information about
$\partial\mathcal{M}$ is contained in the fiducial volume $V_{0}$ which
encodes the spatial integration. Therefore, in this homogeneous set-up, the
boundary term reduces to a simple total time derivative for our mechanical
action and it will not play any role in the following. However, notice that it
should become important when relaxing homogeneity.

Before diving in the exploration of the hidden symmetries of this system, let
us write down its equations of motion. They reads

$\displaystyle\mathcal{E}_{\mathcal{N}}$
$\displaystyle=\frac{1}{\mathcal{N}^{2}}\left[a^{1+n}\dot{a}^{2}-\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}^{2}\right]+\frac{a^{1-n}}{L^{2}_{c}}-\frac{a^{3-n}}{L^{2}_{\Lambda}}\simeq
0$ (2.14) $\displaystyle\mathcal{E}_{a}$
$\displaystyle=2\frac{\textrm{d}}{\textrm{d}\tau}\left(a^{1+n}\frac{\dot{a}}{\mathcal{N}}\right)-(\beta+1)^{2}(1+n)\frac{\dot{a}^{2}}{\mathcal{N}}+(3+n)\;a^{2+n}\frac{L^{2}_{P}}{2}\frac{\dot{\varphi}^{2}}{\mathcal{N}}$
$\displaystyle+\mathcal{N}\left(\frac{1-n}{L^{2}_{c}}a^{-n}-\frac{3-n}{L^{2}_{\Lambda}}a^{2-n}\right)\simeq
0$ (2.15) $\displaystyle\mathcal{E}_{\varphi}$
$\displaystyle=\frac{\textrm{d}}{\textrm{d}\tau}\left(a^{3+n}\frac{\dot{\varphi}}{\mathcal{N}}\right)\simeq
0$ (2.16)

The first equation (2.14) is a constraint which enforces the invariance under
time-reparametrization and forces the hamiltonian of the system to vanish.
Together with the second equation (2.15), they correspond to the Friedmann
equations. The last equation (2.16) is the continuity equation for the scalar
matter field. Notice that depending on the choice of $n$, which encodes a
choice of proper time to express the dynamics, the equations of motion can
take very different forms as kinetic or potential terms can be removed. The
question we shall investigate is whether this set of equations enjoy some
hidden invariance for some specific choice of proper time, i.e. of parameter
$n$. This question has been partially investigated in some specific cases in
previous works, but a unified picture is still missing in the context of
homogeneous and isotropic cosmological models. The goal of the next section is
to derive the conditions of existence for these hidden symmetries for the
family of models parametrized by $L_{c}$ and $L_{\Lambda}$. It will allow us
to recover previous results found in [4, 8] as well as to identify new
realizations of these hidden symmetries, in particular in the case of the
closed universe which is treated for the first time here.

###  Hamiltonian formulation

Before discussing the hidden symmetries, let us complete the presentation of
our cosmological system by writing down its phase space. The conjugated
momenta to the gravitational and matter configuration d.o.f $(a,\varphi)$ are
given by

$\pi_{a}=-\frac{2V_{0}}{L^{2}_{P}}\frac{a^{1+n}\dot{a}}{\mathcal{N}}\;,\qquad\pi_{\varphi}=V_{0}a^{3+n}\frac{\dot{\varphi}}{\mathcal{N}}$
(2.17)

such that $[\pi_{a}]=[a]=1$, while $[\pi_{\varphi}]=L$ and $[\varphi]=L^{-1}$
and we have

$\\{a,\pi_{a}\\}=\\{\varphi,\pi_{\varphi}\\}=1$ (2.18)

The hamiltonian reads

$\displaystyle H_{n}[\mathcal{N}]$
$\displaystyle=\frac{\mathcal{N}}{V_{0}}\left[\frac{\pi^{2}_{\varphi}}{2a^{3+n}}-\frac{L^{2}_{P}}{4}\frac{\pi^{2}_{a}}{a^{1+n}}-\frac{V^{2}_{0}}{L^{2}_{P}L^{2}_{c}}a^{1-n}+\frac{V^{2}_{0}}{L^{2}_{P}L^{2}_{\Lambda}}a^{3-n}\right]\simeq
0$ (2.19)

Notice that we have $[H]=L^{-1}$. In the following, it will be useful to
introduce the alternative canonical pair

$v=a^{3+n}\;,\qquad\pi_{v}=\frac{1}{(3+n)}\frac{\pi_{a}}{a^{2+n}}\;,\qquad\text{such
that}\qquad\\{v,\pi_{v}\\}=1$ (2.20)

where we have assumed $n\neq-3$. This choice of clock corresponds to setting
the volume to unity and therefore to select the unimodular clock. In term of
the new canonical variables, the hamiltonian reads

$\displaystyle H_{n}[\mathcal{N}]$
$\displaystyle=\frac{\mathcal{N}}{V_{0}}\left[\frac{\pi^{2}_{\varphi}}{2v}-\frac{(3+n)^{2}}{4}L^{2}_{P}v\pi^{2}_{v}-\frac{V^{2}_{0}}{L^{2}_{P}L^{2}_{c}}v^{\beta_{1}}+\frac{V^{2}_{0}}{L^{2}_{P}L^{2}_{\Lambda}}v^{\beta_{2}}\right]\simeq
0$ (2.21)

where

$\beta_{1}=\frac{1-n}{3+n}\;,\qquad\beta_{2}=\frac{3-n}{3+n}$ (2.22)

It will also reveal useful to define the dilatation generator as

$\mathcal{C}_{n}:=\frac{2V_{0}}{(3+n)^{2}L^{2}_{P}}\;\\{v,H_{n}\\}=-v\pi_{v}$
(2.23)

which coincides with the (rescaled) speed of $v$.

##  Diff$(S^{1})$ proper time reparametrization of the action

The conformal symmetry found in [4, 8] is realized as suitable
reparametrization of the proper time under which the gravitational
configuration variable transforms as a primary field. Following these works,
we introduce the proper time coordinate

$\textrm{d}\eta=\mathcal{N}\textrm{d}\tau$ (3.1)

and we write the reduced action as

$\displaystyle S_{n}[a,\varphi]$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}^{2}-a^{1+n}\dot{a}^{2}+a^{1-n}\left(\frac{1}{L^{2}_{c}}-\frac{a^{2}}{L^{2}_{\Lambda}}\right)\right]$
(3.2)

where we use again a dot to refer now to the time derivative w.r.t the proper
time coordinate $\eta$.

Writing the action in term of the proper time $\eta$ has important
consequences on the underlying symmetry of the system. The gauge invariance
under time-reparametrization (2.10-2.13) which requires the presence of the
lapse is no longer realized such that the equation of motion (2.14) is not
imposed anymore as a first class constraint. As it stands, the action (3.2)
encodes the dynamics of a mechanical system whose hamiltonian can have any
positive real values, i.e. $H=E$. In order to encode the same dynamics as
before, we have to restrict the system such that its hamiltonian vanishes. The
only difference is that now, this constraint is not enforced dynamically but
fixed by hand. Keeping in mind this point, the action (2.2) supplemented with
the restriction $H=0$ nevertheless encodes the very same cosmological dynamics
as before. At this stage, one might misleadingly conclude that the gauge
invariance (2.10-2.13) being broken, the proper time $\eta$ stands as an
absolute time. The situation is however more subtle as the action (2.2) turns
out to enjoy some surprising hidden symmetries.

In order to see this, consider a general Diff$(S^{1})$ transformation given by

$\displaystyle\eta\rightarrow\tilde{\eta}$ $\displaystyle=f(\eta)$ (3.3)
$\displaystyle\varphi\rightarrow\tilde{\varphi}(\tilde{\eta})$
$\displaystyle=\varphi(\eta)$ (3.4) $\displaystyle
a\rightarrow\tilde{a}(\tilde{\eta})$
$\displaystyle=\left[\dot{f}(\eta)\right]^{\lambda}a(\eta)$ (3.5)

such that the scalar field $\varphi$ transforms as a scalar while the scale
factor $a$ transforms as a primary field with conformal weight $\lambda$.
Notice that the transformation of the scale factor (3.5) is rather different
from its standard behavior under standard time-reparametrization given by
(2.12) where it transforms instead as a scalar. Performing this finite
Diff$(S^{1})$ transformation on the cosmological action, one can show that
provided

$\lambda\left(3+n\right)-1=0\qquad\text{and}\qquad n\neq-3$ (3.6)

the variation of the cosmological action reads

$\displaystyle\Delta S_{n}$
$\displaystyle=-\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[2\lambda^{2}\frac{\textrm{d}}{\textrm{d}\eta}\left(\frac{\ddot{f}}{\dot{f}}a^{3+n}\right)\right.$
(3.7)
$\displaystyle\left.\qquad\qquad\qquad-2\lambda^{2}\text{Sch}[f]a^{3+n}-\frac{a^{1-n}}{L^{2}_{c}}\left(\dot{f}^{2-2\lambda(n+1)}-1\right)+\frac{a^{3-n}}{L^{2}_{\Lambda}}\left(\dot{f}^{2-2\lambda
n}-1\right)\right]$ (3.8)

where we have introduced the Schwarzian derivative of the function $f(\eta)$
defined as

$\text{Sch}[f]:=\frac{\dddot{f}}{\dot{f}}-\frac{3}{2}\left(\frac{\ddot{f}}{\dot{f}}\right)^{2}$
(3.9)

This variation is the first result of this work. It allows to identify in a
clean way the hidden symmetries of the cosmological action. The first term is
a total derivative term which does not modify the dynamics. The second term
encodes the conformal anomaly of the transformation, given by the Schwarzian
cocycle. As it does not depend on the length scale $(L_{c},L_{\Lambda})$, this
term will survive in the flat case. We shall refer to it as the flat
cosmological anomaly. The two last terms encode respectively the corrections
to the flat anomaly coming from i) the constant curvature of the spatial
section given by $1/L^{2}_{c}$ or ii) the presence of a non-vanishing
cosmological constant $1/L^{2}_{\Lambda}$. Notice that the condition (3.6)
shows that the choice of proper time $\eta$ dictates the value of the
conformal weight $\lambda$ entering in the transformation (3.5). Using the
general result (3.12), we shall now identify the possible hidden symmetries
associated to the different families of cosmological models parametrized by
the length scales $(L_{c},L_{\Lambda})$.

###  Symmetries

As a first result, notice that when both $(L_{c},L_{\Lambda})$ are present, no
symmetry can be realized. This is manifest from the fact that there is no
choice for $n$ which would allow one to factorized the powers of the scale
factor involved in the last three terms of (3.12). Let us now discuss the
different families of cosmological models.

####  Flat FRW cosmology

The flat FRW cosmology is obtained in the limit $L_{c}\rightarrow+\infty$ and
$L_{\Lambda}\rightarrow+\infty$. In that case, we see that the variation of
the action under a Diff$(S^{1})$ transformation reduces to

$\displaystyle\Delta S_{n}$
$\displaystyle=\frac{2\lambda^{2}V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[\text{Sch}[f]a^{3+n}-\frac{\textrm{d}}{\textrm{d}\eta}\left(\frac{\ddot{f}}{\dot{f}}a^{3+n}\right)\right]$
(3.10)

This result shows that at each value of $n\in\mathbb{Z}/\\{-3\\}$, there is a
solution for $\lambda$, given by (3.6) such that the transformations
satisfying $\text{Sch}[f]=0$ are Noether symmetries of our cosmological
system, i.e the action varies only by a total derivative term. These
transformations are given by

$f(\eta)=\frac{a\eta+b}{c\eta+d}\;,\qquad\text{with}\qquad ad-bc\neq 0$ (3.11)

where $(a,b,c,d)$ are real constant parameters. One can show that among these
transformations, the non trivial ones can be accounted by considering the
subset satisfying $ad-bc=1$ while quotienting by the transformations
$(a,d,b,c)\rightarrow(-a,-b,-c,-d)$. As a result, the symmetry group is given
by SL$(2,\mathbb{R})/\mathbb{Z}_{2}$ which corresponds to the group of Möbius
transformations. This provides an extension of the result found in [4] which
corresponds to $n=0$ and $\lambda=1/3$. Before going to the next model, we
point that the specific value $n=-3$ is forbidden because the transformation
generates a global factor which cannot be removed by any choice of the
conformal weight, such that this case does not enjoy any symmetry. Let us now
investigate the case where a source of constant curvature is turned on.

####  (A)dS cosmology

Consider the (A)dS cosmological system which corresponds to
$L_{c}\rightarrow+\infty$ and let us focus on the de Sitter model with
$L_{\Lambda}>0$. The AdS case can be obtained by sending
$L_{\Lambda}\rightarrow iL_{\Lambda}$. The transformation of the reduced
action becomes

$\displaystyle\Delta S_{n}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[2\lambda^{2}\text{Sch}[f]a^{3+n}-\frac{a^{3-n}}{L^{2}_{\Lambda}}\left(\dot{f}^{2-2\lambda
n}-1\right)-2\lambda^{2}\frac{\textrm{d}}{\textrm{d}\eta}\left(\frac{\ddot{f}}{\dot{f}}a^{3+n}\right)\right]$
(3.12)

In general, the second term prevents the symmetry to be realized. However,
there are two special values for $n$ for which a symmetry emerges.

Case 1: A first possibility is realized for the special case $n=3$ which
imposes $\lambda=1/6$. In that case, the contribution of the cosmological
constant reduces to a pure constant in the initial lagrangian, similar to the
choice used for the Kantowschi-Sachs model in [13]. Then, the variation of the
action reads

$\displaystyle\Delta S_{3}$
$\displaystyle=\frac{V_{0}}{3L^{2}_{P}}\int\textrm{d}\eta\left[\text{Sch}[f]a^{6}-\frac{\textrm{d}}{\textrm{d}\eta}\left(\frac{\ddot{f}}{\dot{f}}a^{6}-\frac{3(f-\eta)}{L^{2}_{\Lambda}}\right)\right]$
(3.13)

The Noether symmetry is then realized by imposing only $\\{f,\eta\\}=0$ which
coincides with the condition imposed in the flat case, i.e.

$f(\eta)=\frac{a\eta+b}{c\eta+d}\;,\qquad\text{with}\qquad ad-bc\neq 0$ (3.14)

This alternative realization of the symmetry provides a new result for (A)dS
cosmology and shows how the Möbius invariance of the system can be realized in
different forms depending on the choice of the clock.

Case 2: For $n=0$, one obtains from (3.6) that $\lambda=1/3$. The volume
$a^{3}$ entering in the two first terms can then be factorized and the
variation of the action becomes

$\displaystyle\Delta S_{0}$
$\displaystyle=\frac{2V_{0}}{9L^{2}_{P}}\int\textrm{d}\eta\left[\left(\text{Sch}[f]-\frac{9}{2L^{2}_{\Lambda}}\left(\dot{f}^{2}-1\right)\right)a^{3}-\frac{\textrm{d}}{\textrm{d}\eta}\left(\frac{\ddot{f}}{\dot{f}}a^{3}\right)\right]$
(3.15)

This variation reduces to a pure total derivative term and thus to a Noether
symmetry of the system provided $f(\eta)$ satisfies the differential equation

$\text{Sch}[f]-\Omega_{\Lambda}^{2}\dot{f}^{2}=-\Omega_{\Lambda}^{2}\;,\qquad\Omega_{\Lambda}=\frac{3}{\sqrt{2}L_{\Lambda}}$
(3.16)

This differential equation can be recast using the composition law for the
Schwarzian. See Appendix A for detail. The reparametrization function
$f(\eta)$ solving this differential equation is given by

$f(\eta)=F^{-1}\circ M\circ F\;,\qquad\text{with}\qquad
M:=\frac{a\eta+b}{c\eta+d}\;,\qquad
F:=\tanh{\left[\frac{\Omega_{\Lambda}\eta}{\sqrt{2}}\right]}$ (3.17)

The conserved charges and the transformations of the solution space of this
system have been discussed previously and we shall not expand further on this
specific case in what follows. See [8] for details.

####  The closed universe

Finally, we consider the closed universe which has not been treated yet. The
closed universe is obtained by taking the limit
$L_{\Lambda}\rightarrow+\infty$ and choosing $L_{c}>0$ (The model with
homogeneously curved hyperbolic spatial section is obtained by the mapping
$L_{c}\rightarrow iL_{c}$). For this system, the transformation of the action
reduces to

$\displaystyle\Delta S_{n}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[2\lambda^{2}\text{Sch}[f]a^{3+n}+\frac{a^{1-n}}{L^{2}_{c}}\left(\dot{f}^{2-2\lambda(n+1)}-1\right)-2\lambda^{2}\frac{\textrm{d}}{\textrm{d}\eta}\left(\frac{\ddot{f}}{\dot{f}}a^{3+n}\right)\right]$
(3.18)

There are again two special values for $n$ for which a Noether symmetry is
realized:

Case 1: A first possibility corresponds to $n=1$ which implies $\lambda=1/4$.
This choice allows one to recast the contribution from the constant curvature
to a pure constant in the action. The transformation reduces then to

$\displaystyle\Delta S_{1}$
$\displaystyle=\frac{V_{0}}{8L^{2}_{P}}\int\textrm{d}\eta\left[\text{Sch}[f]a^{4}-\frac{\textrm{d}}{\textrm{d}\eta}\left(\frac{\ddot{f}}{\dot{f}}a^{4}+\frac{8(f-\eta)}{L^{2}_{c}}\right)\right]$
(3.19)

and the symmetry is again obtained by imposing $\text{Sch}[f]=0$, selecting
the Möbius transformation (3.14). This choice parallels the second possibility
found for the (A)dS case.

Case 2: The second possibility is to choose $n=-1$ which imposes
$\lambda=1/2$. Then, the variation of the action becomes

$\displaystyle\Delta S_{-1}$
$\displaystyle=\frac{V_{0}}{2L^{2}_{P}}\int\textrm{d}\eta\left[\left(\text{Sch}[f]+\frac{2}{L^{2}_{c}}(\dot{f}^{2}-1)\right)a^{2}-\frac{\textrm{d}}{\textrm{d}\eta}\left(\frac{\ddot{f}}{\dot{f}}a^{2}\right)\right]$
(3.20)

Interestingly, the condition to have a Noether symmetry is the same as for the
(A)dS case up to a sign and the definition of the pulsation. It reads

$\text{Sch}[f]+\Omega^{2}_{c}\dot{f}^{2}=\Omega^{2}_{c}\;,\qquad\Omega_{c}=\frac{\sqrt{2}}{L_{c}}$
(3.21)

The resolution of the third order differential equation is discussed in
Appendix A. Therefore the closed universe enjoys a Möbius invariance under the
reparametrization

$f(\eta)=F^{-1}\circ M\circ F\;,\qquad\text{with}\qquad
M:=\frac{a\eta+b}{c\eta+d}\;,\qquad
F:=\tan{\left[\frac{\Omega_{c}\eta}{\sqrt{2}}\right]}$ (3.22)

It is intriguing to see that the closed universe and the (A)dS cosmology,
despite encoding radically different dynamics, share the very same symmetry,
provided we identify the right clocks. This is made possible by the specific
values of the parameters $(n,\lambda)$ allowing one to recast the reduced
action for the closed universe model in a form similar to the (A)dS case.

Let us now summarize our findings. We have shown that under a general
Diff$(S^{1})$ transformation in proper time, the cosmological action
parametrized by the two length scales $(L_{c},L_{\Lambda})$ transforms as
(3.12). This result has allowed us to identify in a unified manner the hidden
symmetries of the different cosmological models of homogeneus and isotropic
Einstein-Scalar-$\Lambda$ gravity: the flat FRW cosmology, the (A)dS cosmology
and finally to treat the closed universe model:

  * •

We have recovered known results presented in [4] and [8], namely the Möbius
invariance of the flat model for $\left(n,\lambda\right)=(0,1/3)$ and the one
of (A)dS cosmology for $\left(n,\lambda\right)=(0,1/3)$.

  * •

We have also obtained new results for these two models : namely that the
Möbius invariance of the flat model is realized for any choice of proper time,
i.e. for any choice of $n\in\mathbb{Z}/\\{-3\\}$. Moreover, we have identified
a second realization of the Möbius symmetry for (A)dS cosmology given by
$\left(n,\lambda\right)=(3,1/6)$.

  * •

Finally, we have treated for the first time the spatially homogeneously curved
model. We have shown that this model admits also two specific choices of
proper time for which a Möbius invariance is realized, namely
$(n,\lambda)=(1,1/2)$ and $(n,\lambda)=(-1,1/4)$.

Therefore, we have shown that, upon identifying the suitable proper time, the
above homogeneous and isotropic cosmological systems enjoy a hidden symmetry
under the one dimensional conformal group $\text{SL}(2,\mathbb{R})$. This
symmetry is residual in the sense that it exists after having gauge fixed the
time-reparametrization invariance of the system. The existence of this global
Noether symmetry provides thus a second layer of relativeness in the status of
the clock used to follow the cosmological dynamics which can thus be defined
only as an equivalence class of clocks.

Let us now make few remarks regarding the more general Diff$(S^{1})$
transformations. While they are not symmetries of the action, they
nevertheless play an interesting role. Under these transformations, the action
gets shifted by the standard cocycle term given by the Schwarzian of the
reparametrization function. This cocyle governs the conformal anomaly of the
action under these general conformal transformations and the constant term in
front of it can be interpreted as a central charge. In order to be
dimensionless, one can rescale the time coordinate such that
$\eta\rightarrow\eta/L_{P}$. One can then associate to the cosmological system
a central charge given by

$c=\frac{V_{0}}{L^{3}_{P}}$ (3.23)

which encodes the ratio between the IR cut-off of our system, namely the
fiducial volume $V_{0}$, and the UV cut-off given by the Planck length
$L_{P}$. This result fits with the standard interpretation of the central
charge as encoding the effective number of degrees of freedom of the system.
One can therefore view the Schwarzian as a new interaction term and the
central charge as its coupling constant. While in general, this new
contribution to the potential is time-dependent, one can select a subset of
Diff$(S^{1})$ transformations which generate a constant potential such that it
allows to generate the contribution of a constant curvature in the reduced
action. Let us demonstrate this explicitly.

###  Conformal bridge, Niederer transformation and Schwarzian curvature

In this section, we show that, starting from the flat model, one can generate
the potential terms encoding the presence of a homogeneous curvature using
suitable Diff$(S^{1})$ transformation. This mapping is known as the Niederer
transformations [15]. Consider thus the reduced action for the flat model
written in proper time

$\displaystyle S_{n}[a,\varphi]$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}^{2}-a^{1+n}\dot{a}^{2}\right]$
(3.24)

The Niederer’s transformation are given by

$\displaystyle\eta\rightarrow\tilde{\eta}$
$\displaystyle=\frac{\sqrt{2}}{\Omega}\tan{\left(\frac{\Omega\eta}{\sqrt{2}}\right)}$
(3.25) $\displaystyle a\rightarrow\tilde{a}$
$\displaystyle=\frac{a}{\left[\cos{\left(\frac{\Omega\eta}{\sqrt{2}}\right)}\right]^{2\lambda}}$
(3.26) $\displaystyle\varphi\rightarrow\tilde{\varphi}$
$\displaystyle=\varphi$ (3.27)

It is easy to see that if $\Omega\in\mathbb{R}^{\ast}$, the transformation
(3.25) compactifies the clock $\tilde{\eta}\in]-\infty,+\infty[$ to
$\eta\in]-\pi/(\sqrt{2}\Omega),\pi/(\sqrt{2}\Omega)[$. However, if
$\Omega\in\mathbb{R}^{\ast}$, this map decompactifies instead the clock
$\tilde{\eta}$. As we shall see, this feature reflects a physical property of
the cosmological models generated by this transformation.

Now, notice that the above transformation belongs to the Diff$(S^{1})$
transformation with $f(\eta)=\sqrt{2}\tan{[\Omega\eta/\sqrt{2}]}/\Omega$.
Using that the Schwarzian derivative of this function is a constant, i.e.

$\\{f,\eta\\}=\Omega^{2}$ (3.28)

the general variation of the action (3.10) gives us

$\displaystyle\tilde{S}_{n}=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}^{2}-a^{1+n}\dot{a}^{2}+\frac{2\Omega^{2}}{(3+n)^{2}}a^{3+n}-\frac{\textrm{d}}{\textrm{d}\eta}\left(\frac{\ddot{f}}{\dot{f}}a^{3+n}\right)\right]$
(3.29)

The Niederer’s transformation generates therefore a potential term linear in
$a^{3+n}$ which modifies the dynamics. This transformation allows one to
reconstruct the (A)dS cosmological system as well as the model with
homogeneous spatial curvature. To see that, consider the two following cases:

  * •

For $(n,\lambda)=(0,1/3)$, and upon setting

$\Omega=\left\\{\begin{array}[]{ll}i\Omega_{\Lambda}&\mbox{for de Sitter
cosmology $\Lambda>0$}\\\ \Omega_{\Lambda}&\mbox{for Anti de Sitter cosmology
$\Lambda>0$}\\\ \end{array}\right.$ (3.30)

where $\Omega_{\Lambda}=3/(\sqrt{2}L_{\Lambda})$, the resulting action becomes

$\displaystyle\tilde{S}_{0}=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[\frac{L^{2}_{P}}{2}a^{3}\dot{\varphi}^{2}-a\dot{a}^{2}-\frac{a^{3}}{L^{2}_{\Lambda}}\right]$
(3.31)

which corresponds to the (A)dS cosmological action in proper time.

  * •

For $(n,\lambda)=(-1,1/2)$, and upon setting

$\Omega=\left\\{\begin{array}[]{ll}\Omega_{c}&\mbox{for closed universe}\\\
i\Omega_{c}&\mbox{for hyperbolic universe}\\\ \end{array}\right.$ (3.32)

where $\Omega_{\Lambda}=\sqrt{2}/L_{c}$, the resulting action becomes

$\displaystyle\tilde{S}_{0}=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[\frac{L^{2}_{P}}{2}a^{3}\dot{\varphi}^{2}-a\dot{a}^{2}+\frac{a}{L^{2}_{c}}\right]$
(3.33)

which corresponds to the reduced action for the closed universe in proper
time.

Therefore, the Niederer’s transformation transforms the reduced action of the
flat cosmological model to either the (A)dS action or to the universe with
homogeneous spatial curvature. It follows that the equations of motion for
these systems are also mapped to each other, providing therefore a solution-
generating method for these family of cosmological models. Moreover, the
compactifiction of the time coordinate $\tilde{\eta}$ of the flat model when
$\Omega\in\mathbb{R}^{\ast}$ can be understood from the fact that the
curvature can stop the cosmic expansion in AdS cosmology and in the closed
universe model, such that the universe in both models reaches a finite size
which can be monitored by a compact time coordinate $\eta$.

While this result was already known for the (A)dS case [8], the present
treatment shows that it also generalizes to the case where a spatial constant
curvature is turned on. Interestingly, the two sources of homogeneous
curvatures, despite having radically different effects on the cosmological
dynamics, can be both encoded using a constant Schwarzian cocyle. Moreover, in
general, this Schwarzian term is a time-dependent function and for $n=0$, the
new interaction term comes with the right power of the scale factor, i.e.
$a^{3}$, to mimic a self-interaction potential for the scalar field. Whether
this Schwarzian can be used to generate inflationary dynamics out of the free
case is an interesting direction which begs for more investigations. See [28]
for interesting results in this direction.

Coming back to the case of a constant Schwarzian transformation discussed
here, it is interesting to note that such mappings are in close analogy with
the one discussed for example in [46, 47] which relates the conformal particle
with its Newton-Hooke version. This analogy is actually not a mere accident,
as one can show that the isotropic cosmological models discussed here can be
recast into the conformal particle using different appropriate clocks. This is
the subject of the next section.

###  Mapping on the conformal particle

Now, let us show how the different cosmological models can be recast, upon
using the right choice of clock, as a conformal particle with or without a
Newton-Hooke interaction [3]. The first step is to recast the kinetic matter
term as the potential contribution of the action. The reason is that for a
scalar field without self-interacting potential, one has
$\dot{\pi}_{\varphi}=0$. Performing a Legendre transform on the matter sector,
we obtain

$\displaystyle\bar{S}[a]$
$\displaystyle=\int\textrm{d}\eta\pi_{\varphi}\dot{\varphi}-S_{n}[a,\varphi]$
(3.34)
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[a^{1+n}\dot{a}^{2}+\frac{L^{2}_{P}}{2V^{2}_{0}}\frac{\pi^{2}_{\varphi}}{a^{3+n}}+\frac{a^{3-n}}{L^{2}_{\Lambda}}-\frac{a^{1-n}}{L^{2}_{c}}\right]$
(3.35)

where we have ignored the boundary term. At this stage, one can derive
different mapping onto the conformal particle depending on the choice of
clock, namely on the value of the parameter $n$. First, we introduce the
following configuration variable

$q(\eta)=\frac{2}{(3+n)}\sqrt{a^{3+n}(\eta)}\;,\qquad\frac{\textrm{d}q}{\textrm{d}\eta}=\sqrt{a^{1+n}}\frac{\textrm{d}a}{\textrm{d}\eta}$
(3.36)

together with a coupling constant, which also plays the role of the
${\mathfrak{sl}}(2,\mathbb{R})$ Casimir of the conformal particle, given by

$\mathcal{G}=-\frac{2L^{2}_{P}\pi^{2}_{\varphi}}{(3+n)^{2}V^{2}_{0}}$ (3.37)

Using these redefinitions, a first possibility consists in recasting the
cosmological action as

$\displaystyle\bar{S}[q]$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[\dot{q}^{2}-\frac{\mathcal{G}}{q^{2}}+\Omega^{2}q^{2}\right]$
(3.38)

where we have

$\displaystyle\Omega=\left\\{\begin{array}[]{ll}\frac{3}{2L_{\Lambda}}&\mbox{if
}L_{c}\rightarrow+\infty\;,\;\;n=0\\\ \frac{1}{L_{c}}&\mbox{if
}L_{\Lambda}\rightarrow+\infty\;,\;\;n=-1\\\ 0&\mbox{if
}L_{\Lambda},L_{c}\rightarrow+\infty\;,n\in\mathbb{Z}\\\ \end{array}\right.$
(3.42)

The action (3.38) is the mechanical action of the conformal particle in a
Newton-Hooke spacetime [16]. Notice that this mapping is valid only for a
specific set of values of $n$. Moreover, it fails when both sources of
curvature are present. This first case associated to $n=0$ corresponds to the
original work of Pioline and Waldron [2]. As one can see, the duality with the
conformal particle turns out to be more general and is valid for different
isotropic models such as the closed universe. Moreover, one can notice that
there is a second possibility which corresponds to the two other realizations
of the conformal symmetry for the (A)dS case when $n=3$ and for closed
universe when $n=1$. In that case, the cosmological action can be recast into

$\displaystyle\bar{S}[q]$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[\dot{q}^{2}-\frac{\mathcal{G}}{q^{2}}+\frac{\textrm{d}}{\textrm{d}\eta}\left(\Omega^{2}\eta\right)\right]$
(3.43)

where we have

$\displaystyle\Omega=\left\\{\begin{array}[]{ll}\frac{3}{2L_{\Lambda}}&\mbox{if
}L_{c}\rightarrow+\infty\;,\;\;n=3\\\ \frac{1}{L_{c}}&\mbox{if
}L_{\Lambda}\rightarrow+\infty\;,\;\;n=1\end{array}\right.$ (3.46)

and where the last term is a total derivative which does not affect the
dynamics. This result shows that the (A)dS and closed universes can be mapped
either onto the standard conformal particle or on its Newton-Hooke extension
depending on the choice of clock used to describe their dynamics. This
reflects the two possible realizations of the Möbius symmetry for each models
identified in Section 3. In particular, notice that the initial result by
Pioline and Waldron [2] corresponds only to one of them. Finally, we stress
that in order for this general mapping to faithfully encode the cosmological
dynamics, one has to assume that the conformal particle has a vanishing
energy.

Before going to the next section, let us stress that this mapping onto the
conformal particle might allow one to import and adapt the techniques
developed for conformal quantum mechanics to quantum cosmology. Preliminary
results along that line have already been reported in [4], following the
techniques developed in [39, 40]. See also [36, 37, 38] for more recent works
on conformal quantum mechanics and the structure of its correlation functions.
Another series of earlier works on conformal quantum mechanics (and more
generally on systems with long-range inverse square interaction and SO$(2,1)$
symmetry) can also be found in [41, 42, 43, 44, 45]. The results and methods
developed there for these systems might therefore be adapted to investigate
the quantum theory of homogeneously curved FRW geometries. In particular, the
renormalization methods and the consequences of the breaking of the conformal
symmetry in the strong-coupling regime might provide a new framework to
discuss the physics in the nearly singular regime as well as to discuss the
singularity resolution for these cosmological models. Having reviewed the
different mapping and dualities, we now turn to a detailed analysis of the
conserved charges generating this Möbius symmetry.

##  Noether charges

In this section, we focus on the $\text{SL}(2,\mathbb{R})$ sector of the
Noether symmetry and we compute the associated Noether charges. Besides being
the generators of the Noether symmetry, these charges play several key roles:
on one hand, they allow one to solve for the dynamics in an algebraic manner,
and on the other hand, their algebra provide a key structure to guide us when
quantizing these cosmological systems. We shall come back on this point in the
discussion. Consider therefore the action

$\displaystyle S_{n}[a,\varphi]$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left[\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}^{2}-a^{1+n}\dot{a}^{2}+a^{1-n}\left(\frac{1}{L^{2}_{c}}-\frac{a^{2}}{L^{2}_{\Lambda}}\right)\right]$
(4.1)

and let us perform a infinitesimal transformation (3.3 \- 3.5) where
$f(\eta)=\eta+\epsilon(\eta)$. Under this infinitesimal diffeomorphism, the
fields vary as

$\displaystyle\delta a$
$\displaystyle=\tilde{a}(\eta)-a(\eta)=-\epsilon\dot{a}+\lambda\dot{\epsilon}a$
(4.2) $\displaystyle\delta\varphi$
$\displaystyle=\tilde{a}(\varphi)-a(\varphi)=-\epsilon\dot{\varphi}$ (4.3)

Upon imposing the condition $\lambda(3+n)-1=0$ derived above, we obtain

$\displaystyle\delta S_{n}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\int\textrm{d}\eta\left\\{-\frac{\textrm{d}}{\textrm{d}\eta}\left[\epsilon\left(\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}^{2}-a^{1+n}\dot{a}^{2}+\frac{a^{1-n}}{L^{2}_{c}}-\frac{a^{3-n}}{L^{2}_{\Lambda}}\right)+\frac{2\lambda}{3+n}a^{3+n}\ddot{\epsilon}\right]\right.$
(4.4)
$\displaystyle\left.\qquad\qquad\qquad+\;2\dot{\epsilon}\;\left[\frac{(1-2\lambda)}{L^{2}_{c}}a^{1-n}-\frac{(1-3\lambda)}{L^{2}_{\Lambda}}a^{3-n}\right]+\;\frac{2\lambda}{3+n}a^{3+n}\dddot{\epsilon}\right\\}$
(4.5)

Depending on the parameters $(L_{c},L_{\Lambda})$ and on the specific values
of $n$ (and thus of $\lambda$) identified in the previous section, the
infinitesimal variation of the action reduces then to a total derivative

$\delta S_{n}=\int\textrm{d}\eta\frac{\textrm{d}F_{n}}{\textrm{d}\eta}$ (4.6)

Identifying the quantity $F_{n}$ allows one to compute the conserved charges
as follows. Using the pre-symplectic potential of the system given by

$\displaystyle\Theta_{n}$
$\displaystyle=\frac{2V_{0}}{L^{2}_{P}}\left[\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}\delta\varphi-a^{1+n}\dot{a}\delta
a\right]$ (4.7) $\displaystyle=\frac{2V_{0}}{L^{2}_{P}}\left[-\lambda
a^{2+n}\dot{a}\dot{\epsilon}-\epsilon\left(\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}^{2}-a^{1+n}\dot{a}^{2}\right)\right]$
(4.8)

the Noether charge generating the infinitesimal transformations (4.2 \- 4.3)
is given by

$Q^{(n)}=F_{n}-\Theta_{n}$ (4.9)

We can now evaluate them in the different cases identified above.

###  Charges for the flat model

For the flat FRW case where $L_{c}\rightarrow+\infty$ and
$L_{\Lambda}\rightarrow+\infty$, we have found that we do not need to impose
any restriction on the parameter $n$ encoding the choice of proper time. The
conserved charge is given by

$\displaystyle Q^{(n)}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\left[\epsilon\left(\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}^{2}-a^{1+n}\dot{a}^{2}\right)+\frac{2}{3+n}a^{2+n}\dot{a}\dot{\epsilon}-\frac{2}{(3+n)^{2}}a^{3+n}\ddot{\epsilon}\right]$
(4.10)

At this stage, it is instructive to consider the phase space functions

$\displaystyle H_{n}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\left(\frac{L^{2}_{P}}{2}a^{3+n}\dot{\varphi}^{2}-a^{1+n}\dot{a}^{2}\right)\;,$
(4.11) $\displaystyle C_{n}$
$\displaystyle=\frac{2V_{0}}{L^{2}_{P}}\frac{a^{2+n}\dot{a}}{3+n}\;,$ (4.12)
$\displaystyle v_{n}$ $\displaystyle=a^{3+n}\;,$ (4.13)

which correspond to the hamiltonian (which we assume to be vanishing), the
dilatation operator introduced in (2.23) and finally the 3d dimensionless
volume. They satisfy the so called CVH algebra

$\\{v_{n},H_{n}\\}=\frac{(3+n)^{2}L^{2}_{P}}{2V_{0}}\mathcal{C}_{n}\;,\qquad\\{C_{n},H_{n}\\}=-H_{n}\;,\qquad\\{C_{n},v_{n}\\}=v_{n}$
(4.14)

which was first discussed in [7, 6, 5] in the specific case $n=0$. This
structure is the key ingredient allowing to recognize directly the existence
of the Möbius symmetry. Since its first appearance in [7, 6, 5], and its
analysis in [4], it has been noticed that it actually appears in a large class
of symmetry reduced models of GR, such as in Bianchi or in black hole interior
models such as the Kantowski-Sachs midi-superspace [13]. In term of the CVH
generators, the conserved charge reads

$\displaystyle Q^{(n)}(v,\pi_{v},\pi_{\varphi},\eta)$ $\displaystyle=\epsilon
H_{n}+\dot{\epsilon}\;\mathcal{C}_{n}-\frac{2V_{0}}{(3+n)^{2}L^{2}_{P}}\ddot{\epsilon}v$
(4.15)

Working with the standard Möbius reparametrization (3.14), we obtain for the
function $\epsilon(\eta)$ the three possibilities

$\displaystyle\epsilon=\left\\{\begin{array}[]{ll}\sigma&\mbox{for
}\;a=d=1,\;c=0,\;b=\sigma\;,\;\text{translation}\\\
\sigma\eta&\mbox{for}\;b=c=0,\;d=1,\;a=\sigma\;,\;\text{dilatation}\\\
\sigma\eta^{2}&\mbox{for}\;b=a=0,\;d=1,\;c=\sigma\;,\;\text{special conformal
transformation}\\\ \end{array}\right.$ (4.19)

such that $\epsilon(\eta)$ is at most quadratic in the proper time $\eta$. We
have therefore $\dddot{\epsilon}=0$. Omitting the parameter $\sigma$ which
comes as global factor, the three charges generating these transformations are
given by

$\displaystyle Q^{(n)}_{t}$ $\displaystyle=H_{n}$ (4.20) $\displaystyle
Q^{(n)}_{d}$ $\displaystyle=\eta H_{n}+C_{n}$ (4.21) $\displaystyle
Q^{(n)}_{s}$ $\displaystyle=\eta^{2}H_{n}+2\eta
C_{n}-\frac{4V_{0}}{(3+n)^{2}L^{2}_{P}}v_{n}$ (4.22)

Using the CVH brackets, it is then direct to check that these charges are
indeed conserved in time

$\frac{\textrm{d}Q^{(n)}}{\textrm{d}\eta}=\\{Q^{(n)},H_{n}\\}+\frac{\partial
Q^{(n)}}{\partial\eta}=\frac{2v}{(3+n)^{2}}\;\dddot{\epsilon}=0$ (4.23)

and that they satisfy at any time an ${\mathfrak{sl}}(2,\mathbb{R})$ charge
algebra

$\displaystyle\\{Q^{(n)}_{d},Q^{(n)}_{s}\\}=Q^{(n)}_{s}\;,\qquad\\{Q^{(n)}_{d},Q^{(n)}_{t}\\}=-Q^{(n)}_{t}\;,\qquad\\{Q^{(n)}_{t},Q^{(n)}_{s}\\}=2Q^{(n)}_{d}$
(4.24)

These charges provide an example of evolving constants of motion for flat
cosmology and their algebra reflects the fact that the proper time is defined
as an ${\mathfrak{sl}}(2,\mathbb{R})$ equivalence class of clocks. This result
generalizes the findings of [4] for any choice of the parameter
$n\in\mathbb{Z}/\\{-3\\}$. Noticed that the CVH phase space functions forming
the CVH algebra (4.14) play the role of the initial conditions for the
conserved charges $(Q^{(n)}_{t},Q^{(n)}_{d},Q^{(n)}_{s})$. Having computed the
conserved charges and their algebra for the flat model, we can express the
dynamics of the volume configuration variable in term of them. Remembering
that the hamiltonian has to vanish, i.e. $Q^{(n)}_{t}=0$, we obtain

$v_{n}(\eta)=\frac{(3+n)^{2}L^{2}_{P}}{2V_{0}}C_{n}\;\eta-Q^{(n)}_{s}$ (4.25)

which shows that for any choice of clock, the configuration variable evolves
always linearly with the chosen clock. We can now turn to the closed universe.

###  Charges for the closed universe

Consider now the model describing the closed universe.

####  First clock

In the following we shall treat the first choice of proper time corresponding
to $(n,\lambda)=(+1,1/4)$. The associated charge is given by

$\displaystyle
Q^{+}_{c}=\frac{V_{0}}{L^{2}_{P}}\left[\epsilon\left(\frac{L^{2}_{P}}{2}a^{4}\dot{\varphi}^{2}-a^{2}\dot{a}^{2}\right)+\frac{\dot{\epsilon}}{2}a^{3}\dot{a}-\frac{a^{4}}{8}\ddot{\epsilon}\right]$
(4.26)

For this model and this specific choice of proper time, the CVH phase space
functions read

$\displaystyle H_{+}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\left(\frac{L^{2}_{P}}{2}a^{4}\dot{\varphi}^{2}-a^{2}\dot{a}^{2}-\frac{1}{L^{2}_{c}}\right)$
(4.27) $\displaystyle C_{+}$
$\displaystyle=\frac{V_{0}}{2L^{2}_{P}}a^{3}\dot{a}$ (4.28) $\displaystyle
v_{+}$ $\displaystyle=a^{4}$ (4.29)

and the CVH algebra is realized as

$\\{v_{+},H_{+}\\}=\frac{8L^{2}_{P}}{V_{0}}\mathcal{C}_{+}\;,\qquad\\{C_{+},H_{+}\\}=-H_{+}-\frac{V_{0}}{L^{2}_{c}L^{2}_{P}}\;,\qquad\\{C_{+},v_{+}\\}=v_{+}$
(4.30)

Notice that in this case, the homogeneous spatial curvature is appearing as a
central extension of the flat CVH algebra. Using the CVH generators, (4.26)
can be expressed as

$Q^{+}_{c}=\epsilon\left(H+\frac{V_{0}}{L^{2}_{P}L^{2}_{c}}\right)+\dot{\epsilon}\;\mathcal{C}-\frac{V_{0}}{8L^{2}_{P}}\ddot{\epsilon}\;v$
(4.31)

For this choice of proper time, the symmetry of the action is realized as
standard Möbius transformation (3.14) and one has again

$\displaystyle\epsilon(\eta)=\left\\{\begin{array}[]{ll}\sigma&\mbox{for
}\;a=d=1,\;c=0,\;b=\sigma\;,\;\text{translation}\\\
\sigma\eta&\mbox{for}\;b=c=0,\;d=1,\;a=\sigma\;,\;\text{dilatation}\\\
\sigma\eta^{2}&\mbox{for}\;b=a=0,\;d=1,\;c=\sigma\;,\;\text{special conformal
transformation}\\\ \end{array}\right.$ (4.35)

for the three infinitesimal operations building up the Möbius transformations.
Then, the three charges generating these transformations are given by

$\displaystyle Q^{+}_{t}$
$\displaystyle=H_{+}+\frac{V_{0}}{L^{2}_{P}L^{2}_{c}}$ (4.36) $\displaystyle
Q^{+}_{d}$
$\displaystyle=\eta\left(H_{+}+\frac{V_{0}}{L^{2}_{P}L^{2}_{c}}\right)+C_{+}$
(4.37) $\displaystyle Q^{+}_{s}$
$\displaystyle=\eta^{2}\left(H_{+}+\frac{V_{0}}{L^{2}_{P}L^{2}_{c}}\right)+2\eta
C_{+}-\frac{V_{0}}{4L^{2}_{P}}v_{+}$ (4.38)

The conservation of the charge is straightforward and reads

$\frac{\textrm{d}Q^{+}_{c}}{\textrm{d}\eta}=-\frac{V_{0}}{L^{2}_{P}L^{2}_{c}}\dddot{\epsilon}=0$
(4.39)

Moreover, one can show that these three charges satisfy also an
${\mathfrak{sl}}(2,\mathbb{R})$ Lie algebra at any time given by

$\displaystyle\\{Q^{+}_{d},Q^{+}_{s}\\}=Q^{+}_{s}\;,\qquad\\{Q^{+}_{d},Q^{+}_{t}\\}=-Q^{+}_{t}\;,\qquad\\{Q^{+}_{t},Q^{+}_{s}\\}=2Q^{+}_{d}$
(4.40)

With this first choice of proper time, the constant curvature does deform the
CVH structure but the charge algebra remains unchanged. Moreover, the dynamics
is given by

$\displaystyle
v_{+}(\eta)=\frac{4}{L^{2}_{c}}\eta^{2}+\frac{8L^{2}_{P}}{V_{0}}C_{+}\;\eta-\frac{4L^{2}_{P}Q^{+}_{s}}{V_{0}}$
(4.41)

Notice that presence of the curvature through the length scale $L_{c}$ deforms
the trajectory from a linear evolution in $\eta$ into a quadratic one.

####  Second clock and Niederer transformations

Consider now the second choice of proper time corresponding to
$(n,\lambda)=(-1,1/2)$. We have shown in Section 3.2 that upon choosing this
clock, the reduced action of the flat model can be mapped to the one of the
closed universe using Niederer’s transformations. Therefore, the dynamics as
well as the conserved charges of the two models can also be mapped via such
transformation. In the following, we shall therefore derive the Noether
charges generating the second realization of the Möbius invariance using the
Niederer’s transformations. Consider thus the charges generating the
dilatation and special conformal transformation for the flat model:

$\displaystyle\tilde{Q}^{-}_{d}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\left[\tilde{\eta}\left(\frac{L^{2}_{P}}{2}\tilde{a}^{2}\left(\frac{\textrm{d}\tilde{\varphi}}{\textrm{d}\tilde{\eta}}\right)^{2}-\left(\frac{\textrm{d}\tilde{a}}{\textrm{d}\tilde{\eta}}\right)^{2}\right)+\tilde{a}\frac{\textrm{d}\tilde{a}}{\textrm{d}\tilde{\eta}}\right]$
(4.42) $\displaystyle\tilde{Q}^{-}_{s}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\left[\tilde{\eta}^{2}\left(\frac{L^{2}_{P}}{2}\tilde{a}^{2}\left(\frac{\textrm{d}\tilde{\varphi}}{\textrm{d}\tilde{\eta}}\right)^{2}-\left(\frac{\textrm{d}\tilde{a}}{\textrm{d}\tilde{\eta}}\right)^{2}\right)+2\tilde{a}\frac{\textrm{d}\tilde{a}}{\textrm{d}\tilde{\eta}}\tilde{\eta}-\tilde{a}^{2}\right]$
(4.43)

and let us apply the transformations (3.25), (3.26) and (3.27). For the clock
$(n,\lambda)=(-1,1/2)$ and with $\Omega=\Omega_{c}=\sqrt{2}/L_{c}$, they read

$\displaystyle\eta\rightarrow\tilde{\eta}$
$\displaystyle=L_{c}\tan{\left[\frac{\eta}{L_{c}}\right]}\qquad
a\rightarrow\tilde{a}=a/\cos{\left[\frac{\eta}{L_{c}}\right]}$ (4.44)

while $\varphi$ transforms as a scalar. Under this transformation, the
conserved charges become

$\displaystyle Q^{-}_{d}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\left\\{\frac{L_{c}}{2}\sin{\left[\frac{2\eta}{L_{c}}\right]}\left(\frac{L^{2}_{P}}{2}a^{2}\dot{\varphi}^{2}-\dot{a}^{2}\right)+\cos{\left[\frac{2\eta}{L_{c}}\right]}a\dot{a}+\sin{\left[\frac{2\eta}{L_{c}}\right]}\frac{a^{2}}{2L^{2}_{c}}\right\\}$
(4.45) $\displaystyle Q^{-}_{s}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\left\\{L^{2}_{c}\sin^{2}{\left[\frac{\eta}{L_{c}}\right]}\left(\frac{L^{2}_{P}}{2}a^{2}\dot{\varphi}^{2}-\dot{a}^{2}\right)+L_{c}\sin{\left[\frac{2\eta}{L_{c}}\right]}a\dot{a}-\cos^{2}{\left[\frac{\eta}{L_{c}}\right]}a^{2}\right\\}$
(4.46)

We can now express them in term of the CVH phase space functions of the closed
universe. For this specific choice of clock $(n,\lambda)=(-1,1/2)$, they read

$\displaystyle H_{-}$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\left(\frac{L^{2}_{P}}{2}a^{2}\dot{\varphi}^{2}-\dot{a}^{2}-\frac{V_{0}}{L^{2}_{P}L^{2}_{c}}a^{2}\right)$
(4.47) $\displaystyle C_{-}$ $\displaystyle=\frac{V_{0}}{L^{2}_{P}}a\dot{a}$
(4.48) $\displaystyle v_{-}$ $\displaystyle=a^{2}$ (4.49)

and the CVH algebra is realized as

$\\{v_{-},H_{-}\\}=\frac{2L^{2}_{P}}{V_{0}}\mathcal{C}_{-}\;,\qquad\\{C_{-},H_{-}\\}=-H_{-}-\frac{2V_{0}}{L^{2}_{c}L^{2}_{P}}v\;,\qquad\\{C_{-},v_{-}\\}=v_{-}$
(4.50)

Notice that the homogeneous spatial curvature shows up as a deformation of the
bracket $\\{C_{-},H_{-}\\}$. From the point of view of the dynamics, this
deformation implies a second source of acceleration in the hamiltonian, as
expected when turning on the homogeneous spatial curvature. In term of the CVH
phase space function, the charges (4.45) and (4.46) take the simpler form

$\displaystyle Q^{-}_{t}$ $\displaystyle=H_{-}$ (4.51) $\displaystyle
Q^{-}_{d}$
$\displaystyle=\frac{L_{c}}{2}\sin{\left[\frac{2\eta}{L_{c}}\right]}H_{-}+\cos{\left[\frac{2\eta}{L_{c}}\right]}C_{-}+\sin{\left[\frac{2\eta}{L_{c}}\right]}\frac{V_{0}}{L^{2}_{P}L_{c}}v_{-}$
(4.52) $\displaystyle Q^{-}_{s}$
$\displaystyle=L^{2}_{c}\sin^{2}{\left[\frac{\eta}{L_{c}}\right]}H_{-}+L_{c}\sin{\left[\frac{2\eta}{L_{c}}\right]}C_{-}-\cos{\left[\frac{2\eta}{L_{c}}\right]}\frac{V_{0}}{L^{2}_{P}}v_{-}$
(4.53)

where $[Q^{-}_{t}]=L^{-1}$, $[Q^{-}_{d}]=L$ and $[Q^{-}_{s}]=L^{2}$. It is
then straightforward to show that these charges are conserved, i.e.

$\frac{\textrm{d}Q^{-}_{d}}{\textrm{d}\eta}=\\{Q^{-}_{d},H_{-}\\}+\frac{\partial
Q^{-}_{d}}{\partial\eta}=0\;,\qquad\frac{\textrm{d}Q^{-}_{s}}{\textrm{d}\eta}=\\{Q^{-}_{s},H_{-}\\}+\frac{\partial
Q^{-}_{s}}{\partial\eta}=0$ (4.54)

One can again solve for the evolution of the configuration variable $v_{-}$ in
term of this clock to obtain

$v_{-}(\eta)=\frac{L^{2}_{P}}{V_{0}}\left[L_{c}Q^{-}_{d}\sin{\left[\frac{2\eta}{L_{c}}\right]}-Q^{-}_{s}\cos{\left[\frac{2\eta}{L_{c}}\right]}\right]$
(4.55)

which provides a periodic motion.

Let us make few remarks regarding the closed universe. A priori, the effect of
a spatially homogeneous curvature through the length scale $L_{c}$ in the line
element (2.1) is radically different from the effects of a cosmological
constant in the dynamics. However, we have seen that both models share the
very same algebraic structure and thus enjoy the same hidden symmetry when
using the clocks identified in the Section 2. In particular, both source of
homogeneous curvature can be generated using a constant cocycle, i.e a proper
time reparametrization function with a constant Schwarzian derivative. Then,
the closed universe ($L_{c}>0$) shared the same structure as the AdS cosmology
($\Lambda<0$) while the universe with hyperbolic negatively curved spatial
section ($L_{c}<0)$ shares the same algebraic structure as the de Sitter
universe ($\Lambda>0$). To our knowledge, such relation between these
cosmological models has not appeared elsewehere. The reason for such dualities
comes from the existence of this hidden Möbius invariance which emerges only
for some specific choices of clocks.

###  Charges for the (A)dS cosmology

Consider now the (A)dS cosmology. The case $(n,\lambda)=(0,1/3)$ has already
been treated in detail in [8] and we shall therefore only provide the details
on the new realization of this symmetry found in this work, namely the case
$(n,\lambda)=(3,1/6)$. The conserved charge is given by

$\displaystyle
Q=\frac{V_{0}}{L^{2}_{P}}\left\\{\epsilon\left(\frac{L^{2}_{P}}{2}a^{6}\dot{\varphi}^{2}-a^{4}\dot{a}^{2}\right)+\frac{\dot{\epsilon}}{3}a^{5}\dot{a}-\frac{1}{18}a^{6}\ddot{\epsilon}\right\\}$
(4.56)

For this model and this choice of proper time, the CVH observables read

$\displaystyle H$
$\displaystyle=\frac{V_{0}}{L^{2}_{P}}\left[\frac{L^{2}_{P}}{2}a^{6}\dot{\varphi}^{2}-a^{4}\dot{a}^{2}+\frac{1}{L^{2}_{\Lambda}}\right]$
(4.57) $\displaystyle C$ $\displaystyle=\frac{V_{0}}{3L^{2}_{P}}a^{5}\dot{a}$
(4.58) $\displaystyle v$ $\displaystyle=a^{6}$ (4.59)

such that the conserved charge can be recast as

$\displaystyle
Q=\epsilon\left(H-\frac{V_{0}}{L^{2}_{p}L^{2}_{\Lambda}}\right)+\dot{\epsilon}\;C-\ddot{\epsilon}\;\frac{V_{0}}{18L^{2}_{P}}v$
(4.60)

As we have seen in Section 3.1.2, the symmetry is realized as standard Möbius
reparametrization of the proper time, such that one can again extract the
three charges

$\displaystyle Q_{t}$ $\displaystyle=H-\frac{V_{0}}{L^{2}_{P}L^{2}_{\Lambda}}$
(4.61) $\displaystyle Q_{d}$
$\displaystyle=\eta\left(H-\frac{V_{0}}{L^{2}_{P}L^{2}_{\Lambda}}\right)+C$
(4.62) $\displaystyle Q_{s}$
$\displaystyle=\eta^{2}\left(H-\frac{V_{0}}{L^{2}_{P}L^{2}_{\Lambda}}\right)+2\eta
C-\frac{V_{0}}{9L^{2}_{P}}v$ (4.63)

generating respectively translation, dilatation and special conformal
transformations of the proper time. The conservation of the charges is
straightforward as

$\frac{\textrm{d}Q}{\textrm{d}\eta}=-\frac{V_{0}}{L^{2}_{P}L^{2}_{\Lambda}}\dddot{\epsilon}=0$
(4.64)

and one can show that these three charges satisfy also an
${\mathfrak{sl}}(2,\mathbb{R})$ Lie algebra at any time given by

$\displaystyle\\{Q_{d},Q_{s}\\}=Q_{s}\;,\qquad\\{Q_{d},Q_{t}\\}=-Q_{t}\;,\qquad\\{Q_{t},Q_{s}\\}=2Q_{d}$
(4.65)

This provides therefore an alternative realization of the Möbius invariance of
(A)dS cosmology than the one discussed first in [2] and later on in [8].
Moreover, it parallels the discussion of Section 3.3 on the exietsnce of two
different mappings onto the conformal particle. Finally, the dynamics of (A)dS
cosmology can be written in term of this clock and the associated conserved
charges as

$v(\eta)=\frac{18L^{2}_{p}}{V_{0}}C\eta-\frac{9}{L^{2}_{\Lambda}}\eta^{2}-\frac{9L^{2}_{P}}{V_{0}}Q_{s}$
(4.66)

where we have set $H=0$. Notice that for this clock, $C$ is a constant of
motion. This parallels the structure found for the closed universe for which
the trajectory is given by Eq (4.41). Having discussed in detail the Noether
charges generating the Möbius invariance w.r.t the different choices of clock,
we now turn to the last result of this work, namely the relation of our
charges with the ones introduced by Kodama in [19].

##  Relation to volume preserving diffeomorphisms

As discussed in the introduction, the Möbius symmetry discussed here has been
investigated in different gravitational systems and shown to belong to a
larger structure still to be explored. Up to now, all the investigated models
are restricted to homogeneous gravitational fields, describing either
cosmological or black hole interior spacetimes. In order to go beyond, it
would be desirable to see if the Möbius symmetry have some connection with
more general objects which can be covariantly defined in inhomogeneous
spacetimes. The goal of this section is to provide a first hint in this
direction.

In that regard, our set-up given by the line element (5.9) is especially
useful. Indeed, our cosmological model is described by an inhomogeneous and
time-dependent metric, even if the radial dependency can be fully integrated
out at the level of the action. For such spherically symmetric geometry, it is
well known since the early eighties and the seminal work by Kodama that this
geometry is naturally equipped with a divergence-free vector field known as
the Kodama vector [19]. Out of this vector field, one can build up two locally
conserved currents from which one can extract the so called Kodama generators.
In the following, we show that, at least for a specific choice of clock and
only for the flat FRW and (A)dS models : i) one can define a third charge out
of the two known Kodama charges and ii) that these three Kodama charges are
nothing else than the phase space functions forming the CVH algebra (given by
Eq (4.14) and $n=0$ in the flat case). Below, we first review the Kodama’s
machinery and then present this interesting coincidence.

###  Review of the Kodama machinery

Consider a spherically symmetric line element given by

$\textrm{d}s^{2}=\gamma_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}+R^{2}(\tau,r)\textrm{d}\Omega^{2}$
(5.1)

where $\gamma_{\mu\nu}$ is the metric of the 2d base space with coordinates
$(\tau,r)$ and $R(\tau,r)$ is the physical radius of the spacetime. By
definition, the Kodama vector is given by

$k^{\mu}\partial_{\mu}:=\epsilon_{\perp}^{\mu\nu}\nabla_{\nu}R(t,r)\partial_{\mu}$
(5.2)

where $\epsilon^{\perp}_{\mu\nu}$ is the volume form of the 2d base space with
coordinates $(\tau,r)$ [19]. A key property of the Kodama vector is to be
self-conserved

$\nabla_{\mu}k^{\mu}=0$ (5.3)

Therefore, this vector field generates volume preserving diffeomorphisms of
the geometry. Because of this, it defines automatically two conserved currents
given by

$J^{\mu}_{+}=k^{\mu}\;,\qquad J^{\mu}_{-}=G^{\mu\nu}k_{\nu}$ (5.4)

Indeed, it is direct to check that
$\nabla_{\mu}J^{\mu}_{+}=\nabla_{\mu}J^{\mu}_{-}=0$. Notice that these
conservations laws are purely kinematical and do not require a choice of
dynamics, such as the Einstein equations. They are known as the Kodama
currents and they were introduced in the early 80’s [19]. Consider now the
spacelike hypersurface $\Sigma$ with unit normal $n^{\mu}n_{\mu}=-1$ and
induced metric $h_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu}$. Integrating the
conserved currents on this spacelike slice, we obtain the Kodama generators

$\displaystyle K_{+}$
$\displaystyle=\frac{1}{L^{2}_{P}}\int_{\Sigma}\textrm{d}^{3}x\sqrt{h}\;n_{\mu}k^{\mu}$
(5.5) $\displaystyle K_{-}$
$\displaystyle=\int_{\Sigma}\textrm{d}^{3}x\sqrt{h}\;n_{\mu}k_{\nu}G^{\mu\nu}$
(5.6)

These generators are canonical quantities associated to any spherically
symmetric vacuum backgrounds. Having presented briefly the Kodama machinery,
it is worth pointing that since its introduction, the Kodama vector has been
used for different purposes: it provides in particular an efficient tool to
capture the causal structure of a given spherically symmetric spacetime and
its (possibly dynamical) apparent horizons [20], formulate the thermodynamics
of time-dependent spacetimes [21, 22, 23], and it can also serve to select a
preferred foliation where interesting simplifications occur [24, 25, 26].
Finally, we stress that while we should focus on the two standard generators
$(K_{+},K_{-})$ in the following, more general conserved currents can be
constructed out of the Kodama vector, as discussed in detail in [26].

Let us now come back to the two generators $(K_{+},K_{-})$. In order to make
contact with the cosmological system we are interested in, and therefore take
into account a geometry filled with matter fields, one has to extend the
generator $K_{-}$ to take into account the matter sector. We consider
therefore the full current
$J^{\mu}_{-}=\left(L^{2}_{P}T^{\mu\nu}-G^{\mu\nu}\right)k_{\nu}$. Notice that
now, the local conservation of this generator requires the imposition of the
Einstein equations, such that its conservation is no longer kinematical.
Integrating this current leads to the extended generator which we call again
$K_{-}$ given by

$K_{-}=\int_{\Sigma}\textrm{d}^{3}x\sqrt{h}\;n_{\mu}k_{\nu}\left(L^{2}_{P}T^{\mu\nu}-G^{\mu\nu}\right)$
(5.7)

Notice that one could also add the contribution of the cosmological constant
though a term $\Lambda g^{\mu\nu}k_{\nu}$ to the current, without spoiling its
conservation. In the following, we shall restrict ourselves to $\Lambda=0$,
while keeping in mind that the conclusion can be easily extended to the (A)dS
case. Now, our goal is to compute the two generators $K_{\pm}$ for the flat
FRW cosmology.

###  From the Kodama to the CVH generators

Consider therefore the line element (2.9) and let us take the limit
$L_{c}\rightarrow+\infty$ to remove the spatial constant curvature, giving

$\textrm{d}s^{2}=-\frac{\mathcal{N}^{2}(\tau)}{a^{2n}(\tau)}\textrm{d}\tau^{2}+a^{2}(\tau)\left[\textrm{d}r^{2}+r^{2}\textrm{d}\Omega^{2}\right]$
(5.8)

At this level, we do not make a specific choice of the parameter $n$ to remain
as general as possible. We denote the metric of the three-dimensional
spacelike hypersurface $\Sigma$ as

$\textrm{d}s^{2}_{\Sigma}=h_{ij}\textrm{d}x^{i}\textrm{d}x^{j}=a^{2}(\tau)\left[\textrm{d}r^{2}+r^{2}\textrm{d}\Omega^{2}\right]$
(5.9)

such that its timelike unit normal $n^{\mu}n_{\mu}=-1$ is given by

$n_{\mu}\textrm{d}x^{\mu}=\frac{\mathcal{N}}{a^{n}}\textrm{d}\tau\;,\qquad
n^{\mu}\partial_{\mu}=-\frac{a^{n}}{\mathcal{N}}\partial_{\tau}$ (5.10)

Then, the physical radius being $R(\tau,r)=a(\tau)r$, the Kodama vector reads

$\displaystyle k^{\mu}\partial_{\mu}$
$\displaystyle=\frac{a^{n}}{\mathcal{N}}\partial_{\tau}-\frac{\dot{a}}{\mathcal{N}}a^{n-1}r\partial_{r}\;,\qquad
k_{\mu}\textrm{d}x^{\mu}=-\frac{\mathcal{N}}{a^{n}}\textrm{d}\tau-\frac{\dot{a}}{\mathcal{N}}a^{n+1}r\textrm{d}r$
(5.11)

for which the divergence-free property, i.e $\nabla_{\mu}k^{\mu}=0$, is easily
checked. Using these expressions, it is straightforward to notice that
$n_{\mu}k^{\mu}=+1$ for any values of $n\in\mathbb{Z}$. We now can use this
result to compute the first Kodama charge.

Computing the first Kodama charge, and working w.r.t the proper time
$\textrm{d}\eta=\mathcal{N}\textrm{d}\tau$, one obtains

$\displaystyle K_{+}$
$\displaystyle=\frac{1}{L^{2}_{P}}\int_{\Sigma}\textrm{d}^{3}x\sqrt{h}\;n_{\mu}k^{\mu}=\frac{\mathcal{V}_{0}}{L^{2}_{P}}a^{3}$
(5.12)

where $\mathcal{V}_{0}$ is the fiducial volume introduce in Eq (2.4) (with
$L_{c}\rightarrow+\infty$). As a result, this Kodama generator coincides with
the volume configuration variable, given by Eq (4.13), but only for the choice
of clock $n=0$, i.e. $v_{(0)}=a^{3}$. We can therefore write

$K_{+}=\frac{V_{0}}{3L^{2}_{P}}v_{(0)}$ (5.13)

To compute the second Kodama charge, we restrict ourselves to the case $n=0$.
Since our matter content is a massless scalar field, one has

$G^{\tau\tau}n_{\tau}k_{\tau}=\frac{3}{\mathcal{N}^{2}}\frac{\dot{a}^{2}}{a^{2}}\;,\qquad
T^{\tau\tau}n_{\tau}k_{\tau}=\frac{\dot{\varphi}^{2}}{2\mathcal{N}^{2}}$
(5.14)

Working with in term of the proper time
$\textrm{d}\eta=\mathcal{N}\textrm{d}\tau$, one obtains that

$\displaystyle
K_{-}=\frac{3\mathcal{V}_{0}}{L^{2}_{P}}\left[\frac{L^{2}_{P}}{6}a^{3}\dot{\phi}^{2}-a\dot{a}^{2}\right]=\frac{V_{0}}{L^{2}_{P}}\left[\frac{L^{2}_{P}}{2}a^{3}\dot{\varphi}^{2}-a\dot{a}^{2}\right]=H_{(0)}$
(5.15)

where we have used the rescaling (2.5). Therefore, the generators associated
to the two currents introduced in [19] match with the volume and hamiltonian
phase space functions for the specific clock $(n,\lambda)=(0,1/3)$. From this
observation, the identification with the CVH algebra is straightforward. Using
the brackets (4.14) at $n=0$, one can define a third Kodama generator through
the Poisson bracket of the first two generators given by

$K_{0}:=\gamma\\{K_{-},K_{+}\\}=C_{(0)}$ (5.16)

where $\gamma=2/3$. By construction, they satisfy the following
${\mathfrak{sl}}(2,\mathbb{R})$ algebra

$\\{K_{+},K_{-}\\}=\frac{K_{0}}{\gamma}\;,\qquad\\{K_{0},K_{-}\\}=-K_{+}\;,\qquad\\{K_{0},K_{+}\\}=K_{+}$
(5.17)

and one can construct evolving constants of motion in terms of the clock
$\eta$ given by

$\displaystyle\mathcal{K}_{t}$ $\displaystyle=K_{-}$ (5.18)
$\displaystyle\mathcal{K}_{d}$ $\displaystyle=\eta K_{-}+K_{0}$ (5.19)
$\displaystyle\mathcal{K}_{s}$ $\displaystyle=\eta^{2}K_{-}+2\left(\eta
K_{0}-\frac{2}{3}K_{+}\right)$ (5.20)

These Kodama charges
$\left(\mathcal{K}_{t},\mathcal{K}_{d},\mathcal{K}_{s}\right)$ are conserved
in time and generate respectively the proper time translation, dilatation and
special conformal transformations for the underlying spherically symmetric
cosmology. They satisfy

$\\{\mathcal{K}_{s},\mathcal{K}_{t}\\}=2\mathcal{K}_{d}\qquad\\{\mathcal{K}_{d},\mathcal{K}_{t}\\}=-\mathcal{K}_{t}\;,\qquad\\{\mathcal{K}_{d},\mathcal{K}_{s}\\}=\mathcal{K}_{s}$
(5.21)

Therefore, the Kodama generators $\left(K_{0},K_{+},K_{-}\right)$ can be
understood, just as the CVH observables, as the initial conditions of the
Noether charges $\left(\mathcal{K}_{t},\mathcal{K}_{d},\mathcal{K}_{s}\right)$
generating the Möbius symmetry for this specific cosmological model and clock.
As emphasized above, the inclusion of a cosmological constant is
straightforward and this structure generalizes to the (A)dS cosmology for the
specific choice of clock $n=0$. The associated charges can be derived using
again the Niederer transformation used in Section 4.2. To the knowledge of the
author, this is the first time the Kodama generators
$\left(K_{+},K_{-}\right)$ are related to an underlying conformal symmetry of
the associated spherically symmetric gravitational system and that well
defined conserved Kodama charges
$\left(\mathcal{K}_{t},\mathcal{K}_{d},\mathcal{K}_{s}\right)$ are constructed
out of them.

This coincidence suggests that the Möbius symmetry could be related in a
subtle way to volume preserving diffeomorphisms which are generated by
divergence-free vector fields. However, it is worth recalling that the Möbius
symmetry discussed here is a physical symmetry which maps gauge-inequivalent
solutions of the equations of motion and thus is different from standard gauge
transformations. Understanding the general picture requires more works and the
present result raises more questions than it can answer. It might nevertheless
provide an interesting starting point to investigate how the Möbius symmetry
could be realized in inhomogeneous spacetimes. Up to now, the different
generalizations we have attempted to reproduce the whole structure uncovered
here for any $n\in\mathbb{N}/\\{-3\\}$ have not been successful, justifying a
posteriori the restrictions used in this section. It might be that more
general divergence-free vector fields can be constructed which allow one to
reproduce our conserved charges for any value of $n$ and in presence of a
spatial curvature, but this possibility remains to be explored and no evident
solutions have appeared to us. It will be the subject of future
investigations.

##  Discussion

We conclude with a brief summary of our results and a perspective on the
future research directions. Motivated by previous results demonstrating the
existence of a residual global symmetry under proper time reparametrization in
different cosmological systems [4, 8] , we have presented a unified treatment
of this symmetry in a large class of homogeneous and isotropic cosmological
models, completing and generalizing previous findings.

Concretely, we have considered the homogeneous and isotropic class of
geometries described by the line element (2.1). This set up generalizes the
previous investigations by allowing for a spatial constant curvature
parametrized by the length scale $L_{c}$. We have focused on the Einstein-
Scalar-$\Lambda$ system where the scalar matter has no self-interaction such
that the symmetry reduced action is given by Eq (2.2). This set up allows us
to treat at once the flat FRW model, the (A)dS cosmology as well as the closed
universe (and its counterpart with negative constant spatial curvature). In
order to provide a complete picture of the residual symmetry in this family of
cosmological models, and in particular how the choice of clocks affects its
realization, we have introduced a one-parameter field dependent
reparametriztion of the lapse, given by Eq (2.7), where the parameter
$n\in\mathbb{Z}$ encodes the infinite freedom to pick up a given internal
clock to describe the dynamics. Starting from this set-up, we have shown that:

  * •

Under a finite Diff$(S^{1})$ transformation given by Eq (3.3) - Eq (3.5), the
symmetry-reduced Einstein-Scalar-$\Lambda$ action transforms as Eq (3.8). This
transformation of the action provided the conformal weight of the scale
factor, namely $\lambda$, is related to the parameter $n$ through the
condition (3.6). Therefore, the choice of clock dictates the value of the
conformal weight. Moreover, the general formula (3.8) allows us to identify in
a unified manner the different realizations of the residual symmetry. From
this formula, we have shown that

    * –

The flat FRW model corresponding to $L_{c}\rightarrow+\infty$ and
$L_{\Lambda}\rightarrow+\infty$ possesses a residual symmetry under Möbius
reparametrization for any value of $n\in\mathbb{Z}/\\{-3\\}$. Hence, for any
choice of internal clock parametrized by a given value of $n$, this symmetry
is realized. This generalizes the result found in [4] for $n=0$.

    * –

When turning on the cosmological constant while keeping the spatial curvature
vanishing, i.e. $L_{c}\rightarrow+\infty$, which corresponds to the (A)dS
model, we have found that the residual symmetry is realized only for two
values of $n$. For the value $n=3$, the (A)dS cosmology enjoys a Möbius
symmetry similar to the flat case, while for $n=0$, the system is invariant
under a reparametrization of the proper time given by Eq (3.17). This last
case corresponds to the one treated earlier in [8]. We have therefore
identified a new realization of the symmetry while recovering our previous
result for the (A)dS model.

    * –

When turning on only the spatial curvature while working with a vanishing
cosmological constant, namely $L_{\Lambda}\rightarrow+\infty$, we find again
that a residual symmetry is realized only for two specific values of the
parameter $n$, and thus for two specific clocks. For $n=1$, the system is
invariant under standard Möbius transformation while for $n=-1$, the system is
invariant under proper time reparametrization given by Eq (3.22). The
existence of this symmetry for the model with homogeneous spatial curvature,
as well as its two possible realizations, is new.

    * –

Finally, the symmetry cannot be realized when both sources of (three
dimensional and four dimensional) curvature are present together.

Therefore, turning on a homogeneous curvature, being three dimensional or four
dimensional, reduces the number of realizations of the residual symmetry to
only two clocks. What role these clocks are playing ? One can notice that, in
each case, the first choice of clock allows one to recast the potential
contribution as a mere constant in the Lagrangian, while the second one allows
us to reabsorb it in the flat conformal anomaly (given by the cocycle). Beyond
this observation, it would be interesting to understand if these specific
clocks (and thus if these observers) have some additional properties. These
findings show that, provided one identifies the right clocks, the residual
conformal symmetry found in [4, 8] is also a property of the model with
constant spatial curvature, and that it enjoys several different realizations
depending on the source of curvature which we turn on. The different
possibilities are compactly captured by our approach which thus provides a
unified treatment of this residual conformal symmetry for the class of
homogeneous and isotropic cosmological models considered here222Other
symmetries of the flat model have been recently discussed at the level of the
equations of motion in [50] and it would be interesting to see how they are
related to the present structure..

  * •

This surprising residual symmetry can be understood through a duality between
our cosmological systems and the conformal particle introduced in [3]. We have
shown that the duality uncovered initially by Pioline and Waldron [2] is only
a specific case of a more general mapping: provided one selects the right
clock, the different cosmological models can be recast into the action of
conformal mechanics at vanishing energy. Interestingly, the two possible
realizations of the Möbius symmetry identified for each homogeneously curved
models translate into two different mappings onto the conformal particle given
by Eq (3.38) and Eq (3.43). This interesting duality opens up the possibility
to import and adapt techniques developed for conformal quantum mechanics in
quantum cosmology, and reconsidered standard key issues of the field from a
new perspective based on symmetry arguments [36, 37, 38, 39, 40, 41, 42, 43,
44, 45]. Moreover, this mapping could be used as a starting point to explore
how cosmological models could be related to more involved 1d conformally
invariant mechanical systems, such as the well known Schwarzian mechanics
[48].

  * •

Besides the Möbius symmetry transformation which leaves the cosmological
action invariant, the more general finite Diff$(S^{1})$ transformation shifts
the action by a cocycle term which can be viewed as a new interaction term in
the reduced action. The coupling constant in front of this term stands as a
central charge of the system, i.e. $c=V_{0}/L^{3}_{P}$, which governs the
conformal anomaly. It coincides with the ratio between the IR cut-off $V_{0}$
and the UV cut-off $L_{P}$, thus encoding the effective size of the
homogeneous region we have considered. In the particular case where the
reparametrization function has a constant Schwarzian, which corresponds to
Niederer transformation, the new interaction term mimics the potential
contribution of an homogeneous curvature, which allows one to generate out of
the flat system the (A)dS or the spatially curved cosmological model.
Therefore, this transformation can be viewed as a solution-generating map for
homogeneous and isotropic cosmology. This generalizes the result of [8] and
shows that there is an intriguing duality between the (A)dS model and the
spatially curved one: upon choosing the suitable clock, the dynamics of both
systems can be recast in the very same form thanks to their underlying
conformal symmetry.

  * •

We have presented in detailed the Noether charges generating the different
realization of the Möbius symmetry for each systems. These charges are
evolving constants of motion of the system and their algebra is a key
structure which fully encodes the gravitational cosmological dynamics. In
particular, it allows one to solve in an algebraic manner the evolution of the
configuration variable $v$. Preserving this structure at the quantum level
shall provide a key guide to narrow (and even remove) standard quantum
ambiguities related to factor ordering in the quantum theory. See [4, 5] for
preliminar results in this direction. In each case, the CVH algebra built up
from the hamiltonian, the volume configuration variable and the dilatation
generator (which coincides with the trace of the extrinsic curvature) can be
viewed as the initial conditions for the charge algebra.

  * •

Finally, we have used the spherically symmetric nature of our set-up to
explore the status of the well known Kodama generators [19]. We have shown
that these generators are associated with charges which generate the residual
conformal symmetry of the flat (and (A)dS) cosmological system uncovered in
[4, 8] and studied further in this work. This finding is based on the
observation that the two standard Kodama generators coincide with the
hamiltonian and the volume configuration variable entering in the CVH algebra,
in the specific case where there is no spatial curvature and only for the
choice of clock $n=0$. This observation allows one to introduce a third Kodama
generator which coincides with the dilatation operator $C$ and to build via
them well defined conserved charges given by Eq (5.18)-(5.20). While this
result is restricted to a specific case, it suggests that the residual
symmetry discussed here might find a generalization in term of volume
preserving diffeomorphisms (which are generated by divergence-free vector
fields to which the Kodama vector belongs). This further suggests possible
connections with the incompressible fluid mechanics which is known to possess
a similar conformal symmetry as the one discussed in this work [51, 52, 53,
54]. Whether a deeper connection exists and can be formulated in the context
of the fluid/gravity correspondence is an interesting direction we expect to
explore in the near future. Another interesting startegy would be to
understand the structure enclosed by the Kodama generators at the level of the
full theory. Results in this direction will be presented elsewhere.

In summary, our set-up has allowed us to present a complete picture of this
new Möbius symmetry for a large class of homogeneous and isotropic
cosmological models, completing and generalizing previous investigations [4,
8]. As pointed out in the introduction, it has been understood that this
conformal symmetry is only a corner of a deeper structure which still needs to
be explore further. On the one hand, it would be interesting to identify the
largest group of symmetry in these cosmological models, in which the present
SL$(2,\mathbb{R})$ symmetry group is only a subsector. On the other hand, it
would be interesting to understand how these symmetries generalize to more
general symmetry-reduced gravitational systems such as black holes. See [11,
12, 13, 14] for recent results in these two directions. In the end, we expect
this structure to play a guiding role when attempting to describe these
simplified gravitational fields from a suitable hydrodynamical limit of a more
fundamental theory of quantum gravity.

Finally, let us comment on some questions left opened. The Möbius invariance
discussed here stands as a residual Noether symmetry only at the level of the
gauge-fixed action. It would be interesting to understand if the Möbius
symmetry can be also realized at the level of the action prior to the gauge-
fixing and what is the role of the lapse in this transformation. If so, what
is the status of the Möbius reparametrization in term of the ungauge fixed
time coordinate ? What is the role played by the lapse field ? From that
perspective, it was noticed in [4] that the Möbius transformations can be
written in a different manner involving different transformations of both the
scale factor and the lapse (see Eq (2.13) in [4]). It would be interesting to
clarify the status of these alternative transformations which have remained
unexplored so far, as well as the status of the charges which generate them.
We expect to clarify this point in the near furture. Finally, it would also be
useful to understand how this symmetry can be realized in the deparametrized
picture, where the massless scalar field is used as a clock.

## Acknowledgments

The work of J. Ben Achour is supported by the Alexander Von Humboldt
foundation. I am grateful to Etera Livine for the nice and enjoyable
discussions over the last years. I would like also to thank Shinji Mukohyama
for his constant support and Ibrahim Akal and Mohammad Ali Gorji for endless
scientific and non-scientific discussions during my wonderful time in YITP.

##  Appendix A Resolution of the differential equation

In this appendix, we summarize the resolution of the third order differential
equation encountered in Sections 3.1.2 and 3.1.3. Consider the ODE given for
example by Eq (3.21), i.e.

$\text{Sch}[f]+\Omega^{2}_{c}\dot{f}^{2}=\Omega^{2}_{c}$ (A.1)

where

$\text{Sch}[f]:=\frac{\dddot{f}}{\dot{f}}-\frac{3}{2}\left(\frac{\ddot{f}}{\dot{f}}\right)^{2}$
(A.2)

In order to solve this equation, we use two properties of the Schwarzian
derivative. First, the Schwarzian derivative of a function $f(\eta)$ remains
invariant under a Möbius transformation such that

$\text{Sch}[M\circ g]=\text{Sch}[g]\;,\qquad\text{with}\qquad
M(\eta):=\frac{a\eta+b}{c\eta+b}\;,\qquad ad-bc\neq 0$ (A.3)

Second, the Schwarzian derivative of a composed function $f=f_{1}\circ f_{2}$
reads

$\text{Sch}[g_{1}\circ g_{2}]=\text{Sch}[g_{2}]+\left(\text{Sch}[g_{1}]\circ
g_{2}\right)\;\dot{g}^{2}_{2}$ (A.4)

Using these properties of the Schwarzian derivative, solving (A.1) amounts at
looking for a couple of functions $(F,f)$ such that

$\displaystyle\text{Sch}[F]=\text{Sch}[M\circ F]$
$\displaystyle=\Omega^{2}_{c}$ (A.5) $\displaystyle\text{Sch}[F\circ f]$
$\displaystyle=\Omega^{2}_{c}$ (A.6)

where $\Omega_{c}$ is a constant pulsation. These conditions give us

$\text{Sch}[M\circ F]=\text{Sch}[F\circ f]$ (A.7)

such that the solution reads

$f=F^{-1}\circ M\circ F\;,$ (A.8)

where $M$ is a Möbius transformation and $F$ is a function admitting a
constant Schwarzian derivative. For $\Omega_{c}>0$ which corresponds to the
closed universe case, one finds

$F(\eta)=\tan{\left[\frac{\Omega_{c}\eta}{\sqrt{2}}\right]}\;,$ (A.9)

while for the universe with hyperbolic homogeneous curved spatial section,
i.e. $\Omega_{c}<0$, the solution is given by the hyperbolic tangante.
Therefore, the reparametrization function satisfying our condition for the
closed universe reads explicitly

$f(\eta):=\text{Arctan}\left\\{\frac{\Omega_{c}}{\sqrt{2}}\left[\frac{a\tan{\left[\frac{\Omega_{c}\eta}{\sqrt{2}}\right]}+b}{c\tan{\left[\frac{\Omega_{c}\eta}{\sqrt{2}}\right]}+d}\right]\right\\}\qquad\text{with}\qquad
ad-bc\neq 0$ (A.10)

## References

  * [1] G. A. Alekseev, “Thirty years of studies of integrable reductions of Einstein’s field equations,” arXiv:1011.3846
  * [2] B. Pioline and A. Waldron, “Quantum cosmology and conformal invariance,” Phys. Rev. Lett. 90, 031302 (2003) arXiv:0209044
  * [3] V. de Alfaro, S. Fubini and G. Furlan, “Conformal Invariance in Quantum Mechanics,” Nuovo Cim. A 34, 569 (1976). doi:10.1007/BF02785666
  * [4] J. Ben Achour and E. R. Livine, “Cosmology as a CFT1,” JHEP 1912, 031 (2019) arXiv:1909.13390
  * [5] J. Ben Achour and E. R. Livine, “Protected $SL(2,\mathbb{R})$ Symmetry in Quantum Cosmology,” JCAP 1909, 012 (2019) arXiv:1904.06149
  * [6] J. Ben Achour and E. R. Livine, “Polymer Quantum Cosmology: Lifting quantization ambiguities using a $SL(2,\mathbb{R})$ conformal symmetry,” Phys. Rev. D 99, no. 12, 126013 (2019) arXiv:1806.09290
  * [7] J. Ben Achour and E. R. Livine, “Thiemann complexifier in classical and quantum FLRW cosmology,” Phys. Rev. D 96, no. 6, 066025 (2017) arXiv:1705.03772
  * [8] J. Ben Achour and E. R. Livine, “The cosmological constant from conformal transformations: Möbius invariance and Schwarzian action,” Class. Quant. Grav. 37, no. 21, 215001 (2020) arXiv:2004.05841
  * [9] G. W. Gibbons, “Dark Energy and the Schwarzian Derivative,” arXiv:1403.5431
  * [10] A. E. Faraggi and M. Matone, “The Geometrical Origin of Dark Energy,” Eur. Phys. J. C 80, no. 11, 1094 (2020) arXiv:2006.11935
  * [11] J. Ben Achour and E. R. Livine, “Conformal structure of FLRW cosmology: spinorial representation and the $\mathfrak{so}$ (2, 3) algebra of observables,” JHEP 2003, 067 (2020) arXiv:2001.11807
  * [12] J. Ben Achour and E. R. Livine, “Cosmological spinor,” Phys. Rev. D 101, no. 10, 103523 (2020) arXiv:2004.06387
  * [13] M. Geiller, E. R. Livine and F. Sartini, “Symmetries of the Black Hole Interior and Singularity Regularization,” SciPost Phys. 10, 022 (2021) arXiv:2010.07059.
  * [14] J. Ben Achour and E. R. Livine, “Symmetries and conformal bridge in Schwarzscild-(A)dS black hole mechanics” (2021) arXiv:2110.01455
  * [15] U. Niederer, ”The maximal kinematical invariance group of the harmonic oscillator,” Helv. Phys. Acta 46 (1973) 191
  * [16] A. Galajinsky, “Conformal mechanics in Newton-Hooke spacetime,” Nucl. Phys. B 832, 586 (2010) arXiv:1002.2290.
  * [17] C. Rovelli, ”Time in Quantum Gravity: A hypothesis”, Phys. Rev. D43, 442 (1991).
  * [18] A. Anderson, “Evolving constants of motion,” arXiv:9507038.
  * [19] H. Kodama, “Conserved Energy Flux for the Spherically Symmetric System and the Back Reaction Problem in the Black Hole Evaporation,” Prog. Theor. Phys. 63, 1217 (1980). doi:10.1143/PTP.63.1217.
  * [20] S. A. Hayward, “Unified first law of black hole dynamics and relativistic thermodynamics,” Class. Quant. Grav. 15, 3147-3162 (1998) arXiv:9710089.
  * [21] R. G. Cai, L. M. Cao and Y. P. Hu, “Hawking Radiation of Apparent Horizon in a FRW Universe,” Class. Quant. Grav. 26, 155018 (2009) arXiv:0809.1554.
  * [22] Y. X. Chen, J. L. Li and Y. Q. Wang, “Thermodynamics for Kodama observer in general spherically symmetric spacetimes,” (2010) arXiv:1008.3215.
  * [23] Q. J. Cao, Y. X. Chen and K. N. Shao, “Clausius relation and Friedmann equation in FRW universe model,” JCAP 05, 030 (2010) arXiv:1001.2597.
  * [24] I. Racz, “On the use of the Kodama vector field in spherically symmetric dynamical problems,” Class. Quant. Grav. 23, 115-124 (2006) arXiv:0511052.
  * [25] P. Csizmadia and I. Racz, “Gravitational collapse and topology change in spherically symmetric dynamical systems,” Class. Quant. Grav. 27, 015001 (2010) arXiv:0911.2373.
  * [26] G. Abreu and M. Visser, “Kodama time: Geometrically preferred foliations of spherically symmetric spacetimes,” Phys. Rev. D 82, 044027 (2010) arXiv:1004.1456.
  * [27] P. K. Ghosh, “Conformal symmetry and the nonlinear Schrodinger equation,” Phys. Rev. A 65, 012103 (2002) arXiv:0102488.
  * [28] J. E. Lidsey, “Inflationary Cosmology, Diffeomorphism Group of the Line and Virasoro Coadjoint Orbits,” arXiv:1802.09186.
  * [29] S. Gielen, D. Oriti and L. Sindoni, “Homogeneous cosmologies as group field theory condensates,” JHEP 1406, 013 (2014) arXiv:1311.1238.
  * [30] D. Oriti, L. Sindoni and E. Wilson-Ewing, “Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates,” Class. Quant. Grav. 33, no. 22, 224001 (2016) arXiv:1602.05881.
  * [31] D. Oriti, L. Sindoni and E. Wilson-Ewing, “Bouncing cosmologies from quantum gravity condensates,” Class. Quant. Grav. 34, no. 4, 04LT01 (2017) arXiv:1602.08271.
  * [32] D. Oriti, “The universe as a quantum gravity condensate,” Comptes Rendus Physique 18, 235 (2017) arXiv:1612.09521.
  * [33] A. G. A. Pithis and M. Sakellariadou, “Group field theory condensate cosmology: An appetizer,” Universe 5, no. 6, 147 (2019) arXiv:1904.00598.
  * [34] M. Rangamani, “Gravity and Hydrodynamics: Lectures on the fluid-gravity correspondence,” Class. Quant. Grav. 26, 224003 (2009) arXiv:0905.4352.
  * [35] P. A. Hoehn, “Effective relational dynamics,” J. Phys. Conf. Ser. 360, 012014 (2012) arXiv:1110.5631.
  * [36] R. d. Ardón, “Conformal quantum mechanics as a Floquet-Dirac system,” arXiv:2103.15248.
  * [37] S. Khodaee and D. Vassilevich, “Note on correlation functions in conformal quantum mechanics,” Mod. Phys. Lett. A 32, no.29, 1750157 (2017) arXiv:1706.10225
  * [38] K. Andrzejewski, “Quantum conformal mechanics emerging from unitary representations of SL(2,$\mathbb{R}$),” Annals Phys. 367, 227-250 (2016) arXiv:1506.05596
  * [39] C. Chamon, R. Jackiw, S. Y. Pi and L. Santos, “Conformal quantum mechanics as the CFT1 dual to AdS2,” Phys. Lett. B 701, 503 (2011) arXiv:1106.0726
  * [40] R. Jackiw and S.-Y. Pi, “Conformal Blocks for the 4-Point Function in Conformal Quantum Mechanics,” Phys. Rev. D 86, 045017 (2012) Erratum: [Phys. Rev. D 86, 089905 (2012)] arXiv:1205.0443
  * [41] H. E. Camblong, L. N. Epele, H. Fanchiotti, C. A. Garcia Canal and C. R. Ordonez, “Effective field theory program for conformal quantum anomaly,” Phys. Rev. A 72, 032107 (2005) arXiv:0501193
  * [42] H. E. Camblong and C. R. Ordonez, “Anomaly in conformal quantum mechanics: From molecular physics to black holes,” Phys. Rev. D 68, 125013 (2003) arXiv:0303166
  * [43] H. E. Camblong and C. R. Ordonez, “Renormalization in conformal quantum mechanics,” Phys. Lett. A 345, 22 (2005) arXiv:0305035
  * [44] G. N. J. Ananos, H. E. Camblong and C. R. Ordonez, “SO(2,1) conformal anomaly: Beyond contact interactions,” Phys. Rev. D 68, 025006 (2003) arXiv:0302197
  * [45] G. N. J. Ananos, H. E. Camblong, C. Gorrichategui, E. Hernadez and C. R. Ordonez, “Anomalous commutator algebra for conformal quantum mechanics,” Phys. Rev. D 67, 045018 (2003) arXiv:0205191
  * [46] K. Andrzejewski, “Conformal Newton-Hooke algebras, Niederer’s transformation and Pais-Uhlenbeck oscillator,” Phys. Lett. B 738, 405 (2014) arXiv:1409.3926
  * [47] A. Galajinsky and I. Masterov, “On dynamical realizations of l-conformal Galilei and Newton-Hooke algebras,” Nucl. Phys. B 896, 244 (2015) arXiv:1503.08633
  * [48] S. Filyukov and I. Masterov, “On the Schwarzian counterparts of conformal mechanics,” arXiv:2004.03304
  * [49] G. Lindblad and B. Nagel, “Continuous Bases For Unitary Irreducible Representations Of Su(1,1),” PRINT-69-1210.
  * [50] S. Dussault and V. Faraoni, “A new symmetry of the spatially flat Einstein-Friedmann equations,” Eur. Phys. J. C 80, no. 11, 1002 (2020) arXiv:2009.03235
  * [51] L. O’Raifeartaigh and V. V. Sreedhar, “The Maximal kinematical invariance group of fluid dynamics and explosion - implosion duality,” Annals Phys. 293, 215 (2001) arXiv:0007199
  * [52] M. Hassaine and P. A. Horvathy, “Field dependent symmetries of a nonrelativistic fluid model,” Annals Phys. 282, 218-246 (2000) arXiv:9904022
  * [53] S. Bhattacharyya, S. Minwalla and S. R. Wadia, “The Incompressible Non-Relativistic Navier-Stokes Equation from Gravity,” JHEP 0908, 059 (2009) arXiv:0810.1545
  * [54] P. A. Horvathy and P.-M. Zhang, “Non-relativistic conformal symmetries in fluid mechanics,” Eur. Phys. J. C 65, 607 (2010) arXiv:0906.3594