C Duval
email: duval@cpt.univ-mrs.fr

G W Gibbons
email: g.w.gibbons@damtp.cam.ac.uk

P A Horváthy
email: horvathy@lmpt.univ-tours.fr

Conformal and projective symmetries in Newtonian cosmology

Definitions of non-relativistic conformal transformations are considered both in the Newton-Cartan and in the Kaluza-Klein-type Eisenhart/Bargmann geometrical frameworks. The symmetry groups that come into play are exemplified by the cosmological, and also the Newton-Hooke solutions of Newton's gravitational field equations. It is shown, in particular, that the maximal symmetry group of the standard cosmological model is isomorphic to the 13-dimensional conformal-Newton-Cartan group whose conformal-Bargmann extension is explicitly worked out. Attention is drawn to the appearance of independent space and time dilations, in contrast with the Schrödinger group or the Conformal Galilei Algebra.

Introduction

One of the cardinal properties of the standard Friedmann-Lemaître-Robertson-Walker (FLRW) model of relativistic cosmology is its maximal conformal symmetry, arising from the vanishing of the Weyl conformal curvature. The purpose of the present article is to investigate the maximal symmetry group of an older, often unjustly dismissed, description of the non-relativistic universe at a large scale, namely Newtonian cosmology, briefly reviewed in Section 2. We found it natural to relate this search to the symmetries of general time dependent mechanical systems in the framework of Newton-Cartan (NC) geometry [START_REF] Cartan | Sur les variétés à connexion affine et la théorie de la relativité généralisée[END_REF].

The surprising feature of a non-relativistic spacetime structure is the very specific independence of the NC connection with respect to the degenerate Galilei metric of spacetime [START_REF] Cartan | Sur les variétés à connexion affine et la théorie de la relativité généralisée[END_REF][START_REF] Trautman | Sur la théorie newtonienne de la gravitation[END_REF][START_REF] Künzle | Galilei and Lorentz structures on spacetime: Comparison of the corresponding geometry and physics[END_REF]. This requires that, unlike conformal Lorentzian symmetries, conformal-NC symmetries be specified on the one hand by the preservation of the space and time directions of the Galilei "metric", and, on the other hand, by the preservation of the projective structure associated with the symmetric NC connection ruling free fall. This procedure, spelled out in Section 3, has proven critical in devising a purely geometrical definition [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Duval | Nonrelativistic Conformal Symmetries and Bargmann Structures[END_REF] of the otherwise well-known Schrödinger group (re)discovered in the quantum framework in the early seventies [START_REF] Niederer | The maximal kinematical symmetry group of the free Schrödinger equation[END_REF][START_REF] Hagen | Scale and conformal transformations in Galilean-covariant field theory[END_REF] (see also [START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF] for an account about the (pre)history of the Schrödinger symmetry).

The main upshot of this article is the proof that the standard Newtonian cosmological model admits a 13-dimensional group of (local) conformal-NC symmetries isomorphic to the formerly named "chronoprojective group" [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Burdet | Cartan Structures on Galilean Manifolds: the Chronoprojective Geometry[END_REF][START_REF] Burdet | Time-Dependent Quantum Systems and Chronoprojective Geometry[END_REF] of the flat canonical NC structure (see Section 3.2) contains the 12-dimensional (centreless) Schrödinger group as a subgroup and is different from the usual centrally extended Schrödinger group.

We prove, en passant, that the Cosmological Principle here is strictly equivalent to the projective flatness of the Newtonian cosmological model under investigation.

Likewise, the conformal-NC group of the model of empty space with a cosmological constant is shown to be a 13-dimensional which contains the Newton-Hooke group [START_REF] Gibbons | Newton-Hooke spacetimes, Hpp-waves and the cosmological constant[END_REF], as shown in Section 3.2. The case of the vacuum static solution of the Newton field equations is treated along the same lines, highlighting the 5-dimensional "Virial group" as the conformal-NC group.

In the second part of the article, namely in Section 4, we have recourse to the socalled Eisenhart-lift [START_REF] Eisenhart | Dynamical trajectories and geodesics[END_REF][START_REF] Cariglia | Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux[END_REF][START_REF] Bekaert | Embedding nonrelativistic physics inside a gravitational wave[END_REF] of a NC structure. The latter allows us to deal properly with (R, +)-extensions of non-relativistic conformal symmetry groups mentioned above. This uses the full machinery of Bargmann extended spacetime structures [START_REF] Duval | Bargmann structures and Newton-Cartan theory[END_REF] above NC structures. In a nutshell, a Bargmann manifold is defined by a Lorentz metric and a null parallel vector field ξ, and is thus akin to a (generalized) pp-wave [START_REF] Duval | Celestial mechanics, conformal structures, and gravitational waves[END_REF].

Those structures enable us to easily redefine the notion of non-relativistic conformal symmetries as bona fide conformal sub-symmetries of the Lorentz metric. This time, we introduce conformal-Bargmann flatness via the vanishing of the conformal Weyl curvature, C = (C λµνρ ), subject to the condition

C λµνρ ξ ρ = 0 (1.1)
put forward some time ago [START_REF] Duval | Nonrelativistic Conformal Symmetries and Bargmann Structures[END_REF][START_REF] Burdet | Time-Dependent Quantum Systems and Chronoprojective Geometry[END_REF][START_REF] Perrin | Chronoprojective invariance of the fivedimensional Schrödinger formalism[END_REF]. The previously studied models are thus readily Eisenhart-lifted to yield their conformal-Bargmann symmetries as non-central extensions of their conformal-NC ones.

2 Newtonian cosmology

Newton-Cartan structures

Let us first recall that the quadruple (N, γ, θ, ∇) is a Newton-Cartan structure if N is a smooth 4-dimensional manifold (spacetime), γ a nowhere vanishing twice-contravariant tensor, θ a closed 1-form generating the kernel of γ, and ∇ a symmetric affine connection that parallel transports γ and θ; see, e.g., [START_REF] Trautman | Sur la théorie newtonienne de la gravitation[END_REF][START_REF] Künzle | Galilei and Lorentz structures on spacetime: Comparison of the corresponding geometry and physics[END_REF][START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF][START_REF] Duval | Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes[END_REF]. The "clock", θ, descends to the time axis, T = N/ ker(θ). As to the tensor γ, it gives rise to a Riemannian metric on the fibres of the projection M → T , that is on all copies of space at fixed time. 1 Following [START_REF] Künzle | Galilei and Lorentz structures on spacetime: Comparison of the corresponding geometry and physics[END_REF], we call such a triple (N, γ, θ) a Galilei structure.

For our present purpose, N will be an open subset of R 4 , and γ = δ ij ∂ i ⊗∂ j and θ = dt where t = x 4 is the absolute time, in a Galilean coordinate system (x α ) = (x 1 , x 2 , x 3 , x 4 ); spatial indices i, j, k, . . . will run from 1 to 3. The NC connection, ∇, of interest to us is given by its only non-vanishing coefficients

Γ i 44 = -g i (2.1)
which represent the components of the gravitational acceleration, g, in the chosen coordinate system. The NC gravitational field equations [START_REF] Künzle | Galilei and Lorentz structures on spacetime: Comparison of the corresponding geometry and physics[END_REF] are given geometrically in terms of the curvature tensor, R, of ∇ by 2

R αβ = (4πG̺ + Λ) θ α θ β & γ κγ R δ ακβ = γ κδ R γ βκα (2.2)
for all α, β, γ, δ = 1, . . . , 4. The Ricci tensor, Ric, has components R αβ = R κ καβ . In (2.2), we denote by ̺ the mass density of the sources and by Λ the cosmological constant. These equations imply that R ij = 0, i.e., space at time t is Ricci-flat, hence (locally) flat.

In view of (2.1) and (2.2), we have

∇ • g = -(4πG̺ + Λ) & ∇× g = 0 (2.3)
where ∇• and ∇× are the standard divergence and curl on R 3 .

1 In this Section devoted to cosmology, we restrict consideration to 4-dimensional NC structures; almost all geometric definitions and results can be easily generalized to any spacetime dimension d + 1.

2 Our convention is

R δ αβγ = 2∂ [α Γ δ β]γ + 2Γ δ κ[α Γ κ β]γ .

Newtonian cosmology: a review

One can trace back the beginning of Newtonian cosmology to work of Kelvin [START_REF] Lord | On the Clustering of Gravitational Matter in any part of the Universe[END_REF]. We will also refer to [START_REF] Milne | A Newtonian expanding Universe[END_REF][START_REF] Mccrea | Newtonian Universes and the curvature of space[END_REF][START_REF] Heckmann | Bemerkungen zur Newtonschen Kosmologie I[END_REF][START_REF] Heckmann | Bemerkungen zur Newtonschen Kosmologie II[END_REF] for the original modern expositions of non-relativistic cosmology. See also the recent contributions [START_REF] Ellis | Discrete Newtonian Cosmology[END_REF][START_REF] Ellis | Discrete Newtonian Cosmology: Perturbations[END_REF] using homothetic solutions of the N-body problem.

The Cosmological Principle

Following [START_REF] Souriau | Mécanique statistique, groupes de Lie et cosmologie[END_REF][START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF], we formulate the cosmological hypothesis of spatial homogeneity and isotropy by demanding that the curvature tensor be rotation-invariant for some freely chosen origin, x = 0, namely that its nonzero components be given by

R i j44 = -∂ j g i = a δ i j . (2.4)
for all i, j = 1, 2, 3, and for some function a of time.

Condition (2.4) is strictly equivalent to the vanishing of the projective Weyl curvature tensor,

W δ αβγ = R δ αβγ - 1 3 δ δ α R βγ -δ δ β R αγ =
0 of the NC connection ∇ defined by eqn (2.1). The associated projective geodesic equations (3.3) are thus invariant under the local action of the projective group, SL(5, R), on 4-dimensional spacetime. 3 Indeed, we see that W δ αβγ = 0 for the connection (2.1) iff eqn (2.4) holds true. Remarkably enough, this implies that the Ricci tensor is actually of the form (2.2) corresponding to the NC field equations. Besides, the fact that da ∧ θ = 0, i.e., that a in (2.4) is an otherwise arbitrary function of time only is a direct consequence of the Bianchi identities.

Returning to eqn (2.4), we choose g = 0 at x = 0 to end up with g = -a x.

(2.5)

Note that (2.3) is compatible with uniform expansion velocity, v = dx/dt (= ẋ), of the galaxies (Hubble law ), namely v = Hx where H = H(t) is the time dependent Hubble parameter.

3 Generalization to any spacetime dimension n = d + 1 > 2 is straightforward [START_REF] Cariglia | Eisenhart lifts and symmetries of time-dependent systems[END_REF]. If n = 2, projective flatness is guaranteed by the vanishing of the projective Cotton tensor, ∇ α R βγ -∇ α R βγ = 0; this provides a direct justification of the sl(3, R) symmetry Lie algebra for the 1-dimensional damped harmonic oscillator found in [START_REF] Cerverò | SL(3, R) realisations and the damped harmonic oscillator[END_REF]. This local SL(n + 1, R) symmetry will eventually be broken by the additional requirement that the Galilei structure, (γ, θ), be conformally preserved as in (3.1).

Assuming that Λ = 0 for simplicity, we see that eqns (2.3) and (2.5) readily yield 3a = 4πG̺.

(2.6)

As to the principle of geodesics, v = g, applied to freely falling galaxies, it gives then

Ḣ + H 2 + a = 0. (2.7) At last mass conservation, ∂ t ̺ + ∇ • (̺v) = 0, furthermore implies ȧ + 3Ha = 0. (2.8)
The general solution of this system will be worked out in the next section.

The scale factor in terms of cosmological time

Let us introduce the function Θ(t) > 0 on T , defined by 3 (2.9)

a(t) = B Θ(t)
where B = const. > 0 is related to the density of the dust of galaxies. Note that Θ plays, in Newtonian cosmology, the rôle of the scale factor of the universe in the standard model of relativistic cosmology. 4It is useful and classical to redefine time as follows

τ = dt Θ (2.10)
which we call "cosmological time" (akin to "conformal time" of FLRW cosmology).

Putting Θ ′ = dΘ/dτ , etc., we can transcribe the previous set (2.6)-(2.8) of coupled equations ruling Newtonian cosmology as

H = Θ ′ Θ 2 , Θ ′2 -ΘΘ ′′ = BΘ, 2B Θ - Θ ′ Θ 2 = K. (2.11)
These equations are strictly equivalent to the Friedmann equations in the absence of radiation (where K was interpreted as the spatial constant curvature); here, K is a mere first-integral of the above differential system (2.7) and (2.8). Straightforward integration with the proviso that the scale factor vanishes at the big bang, τ = 0, yields

Θ = B(1 -cos(kτ )) K & k = √ K if K ≥ 0 i √ -K if K < 0. (2.
12)

The scale factor in terms of Newtonian time

We note for further use that (2.11) implies

Θ 2 Θ = -B (2.13)
in terms of absolute Newtonian time, t, where Θ = dΘ/dt. This implies that the gravitational potential arising from (2.5) can be chosen as given by

U(x, t) = 1 2 a(t)|x| 2 (2.14)
where the function a is as in (2.9).

Conformal Newton-Cartan symmetries

In this section we review and substantially extend several results [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF] related to nonrelativistic "conformal symmetries" of classical mechanics in a Newton-Cartan framework.

See also [START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF][START_REF] Duval | Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes[END_REF].

General definitions

Conformal-Galilei symmetries Newton-Cartan structures possess "conformal-projective" symmetries in the following sense.

Just as in the relativistic framework where the group of conformal diffeomorphisms (strictly speaking a pseudogroup because it consists in general of local diffeomorphisms) consists of those spacetime transformations preserving the direction of the metric, we will define conformal-Galilei transformations as those (local) diffeomorphisms ϕ of N which preserve independently the direction of the "spatial-metric", γ, and that of the clock, θ, namely such that

ϕ * γ = f γ & ϕ * θ = g θ (3.1)
for some strictly positive functions f and g of N depending on ϕ. These transformations form an infinite-dimensional group which we call the conformal-Galilei group, Conf(N, γ, θ). 5

Projective symmetries

At this stage, we need to appeal to the symmetries of the equations of free fall according to Cartan and forerunners.

If we denote by Γ γ αβ with α, β, γ = 1, . . . , 4 the components of a NC connection, ∇, in some coordinate system, the equations of free fall read

d 2 x γ ds 2 + Γ γ αβ dx α ds dx β ds = 0
for all γ = 1, . . . , 4, and for some choice of affine parameter, s. The choice of an affine parameter is arbitrary, and we may choose another non-affine parameter ŝ = f (s) with f ′ (s) = 0 to describe the same physics. This reparametrization leaves us with the same geodesics (as embedded one-dimensional submanifolds of M) but with another differential equation for them to satisfy, namely

0 = d 2 x γ dŝ 2 + Γ γ αβ dx α dŝ dx β dŝ + 2Π α dx α dŝ dx γ dŝ = d 2 x γ dŝ 2 + Γγ αβ dx α dŝ dx β dŝ (3.3)
where Γγ

αβ = Γ γ αβ + δ γ α Π β + δ γ β Π α (3.4)
with Π a 1-form of M; in the above calculation, we find explicitly

2Π α dx α dŝ = d 2 ŝ ds 2 ds dŝ 2 .
We see from (3.3) that, in terms of the old connection coefficients, Γ γ αβ , the new covariant acceleration is proportional to the velocity; this happens to be reminiscent of friction, whose systematic study is at the root of the article [START_REF] Cariglia | Eisenhart lifts and symmetries of time-dependent systems[END_REF]. The connections ∇ and ∇ in (3.4) yielding the same (unparametrised) geodesics are called "projectively equivalent" [START_REF] Matveev | Geodesically equivalent metrics in general relativity[END_REF]. Let us refer to, e.g., [START_REF] Aminova | Projective geometry of systems of differential equations. General conceptions[END_REF] for an account on projective geometry applied to systems of ODE, and also to [START_REF] Nurowski | Is dark energy meaningless ?[END_REF][START_REF] Gibbons | Dark energy and projective symmetry[END_REF] for a discussion about projective symmetries in FLRW cosmology.

SV(N, γ, θ) ⊂ Conf(N, γ, θ) that preserves the direction of γ ⊗ θ, the special conformal-Galilei structure. The group SV(N, γ, θ) is thus defined by the additional constraint ϕ * (γ ⊗ θ) = γ ⊗ θ or, in view of (3.1),

f g = 1 (3.2)
which is the hallmark of Schrödinger symmetry (cf. eqn (3.15)).

From the physical standpoint, considering unparametrised geodesics as the worldlines of (spinless) particles is the natural way to deal with the Equivalence Principle in the sense of universality of free fall. Therefore the equivalence (3.4) will be considered fundamental, notably for the definition below.

Conformal-NC symmetries

To comply with the above mentioned Equivalence Principle, and hence restrict ourselves to a finite-dimensional group of spacetime symmetries, we impose that ϕ ∈ Conf(N, γ, θ) be also a projective transformation of NC spacetime, i.e., a transformation that permutes unparametrised geodesics, namely such that, according to (3.4), we have

(ϕ * Γ) γ αβ = Γ γ αβ + δ γ α Π β + δ γ β Π α (3.5)
for some 1-form Π depending on ϕ. One proves easily that necessarily Π = h θ, where h is some function of T .

These transformations satisfying (3.1) and (3.5) generate what we choose to call the "conformal-NC group", and denote by Chr(N, γ, θ, ∇); cf. [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Burdet | Cartan Structures on Galilean Manifolds: the Chronoprojective Geometry[END_REF][START_REF] Burdet | Time-Dependent Quantum Systems and Chronoprojective Geometry[END_REF][START_REF] Duval | Nonrelativistic Conformal Symmetries and Bargmann Structures[END_REF]. 6This supergroup of the Schrödinger group has recently been studied and named the enlarged Schrödinger group [START_REF] Gundry | Higher Symmetries of the Schrödinger Operator in Newton-Cartan Geometry[END_REF].

The infinitesimal version of the preceding definitions (3.1) and (3.5) is the following: we will call conformal-NC vector field, X ∈ chr(N, γ, θ, ∇), any vector field X of N, solution of the following system [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Duval | Nonrelativistic Conformal Symmetries and Bargmann Structures[END_REF]

L X γ = λ γ, L X θ = µ θ, L X Γ γ αβ = ν δ γ α θ β + δ γ β θ α (3.6)
for some functions λ, µ, ν of the time axis, T , depending on X. Let us also refer to [START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF][START_REF] Duval | Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes[END_REF]. We find that, quite generally,

λ + µ = const. & ν = 1 2 μ . (3.7)

Some examples

We will now determine the Lie algebras spanned by the solutions X = X i (x)∂ i + X 4 (x)∂ 4 of (3.6) in various instances of NC structures. Using standard notation, we recall that

L X γ αβ = -2γ κ(α ∇ κ X β) (3.8a) L X θ α = θ κ ∇ α X κ (3.8b) L X Γ γ αβ = ∇ α ∇ β X γ + R γ καβ X κ . (3.8c)
The flat Galilean case

Let us start with the case of a flat affine connection, Γ γ αβ = 0, of R 4 . In view of (3.8a)-(3.8c), the only non-trivial eqns (3.6) are

∂ i X j + ∂ j X i = -λ δ ij , ∂ 4 X 4 = µ, ∂ 4 ∂ j X k = ν δ k j . (3.9)
We find that ν = 1 2 μ = -1 2 λ, so that (3.9) and ∂ 4 ∂ 4 X k = 0 imply μ = 0. We are thus left with λ = -2αt -2χ, µ = 2αt + η, ν = α where α, χ, η ∈ R. At last, the general solution of (3.6) is given in this case by

X i = ω i j x j + β i t + γ i + αx i t + χx i (3.10a) X 4 = αt 2 + ηt + ε (3.10b)
for all i = 1, 2, 3, where ω ∈ so(3); β, γ ∈ R 3 ; α, χ, η, ε ∈ R. 7Therefore chr(R 4 , γ, θ, ∇) is generated (with a slight abuse of notation) by the Lie algebra of matrices

X =   ω β γ 0 η -χ ε 0 -α -χ   ∈ chr(R 4 ). ( 3

.11)

We have thus proved that the "conformal-NC algebra" is isomorphic to

chr(R 4 ) ∼ = (so(3) × gl(2, R)) ⋉ (R 3 × R 3 ). (3.12)
Generalization of (3.12) to n-dimensional NC spacetime is immediate. As to the "Schrödinger Lie algebra"

sch(R 4 ) ∼ = (so(3) × sl(2, R)) ⋉ (R 3 × R 3 ) (3.13)
it corresponds, in (3.11), to the choice

η = 2χ. (3.14)
Let us emphasize that preservation of the equations of free fall (which in essence do not involve the mass of the test particle) leaves the dilation factors χ and η totally independent. But eqn (3.14) is in fact designed to guarantee mass conservation in the Schrödinger equation [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Duval | Nonrelativistic Conformal Symmetries and Bargmann Structures[END_REF][START_REF] Duval | Celestial mechanics, conformal structures, and gravitational waves[END_REF]. Thus, the infinitesimal Schrödinger dilation is of the form

X = x i ∂ i + 2t ∂ t

Newtonian cosmology

Taking into account eqn (2.1) defining our NC spacetime structure, we see that the general solution of (3.6) is again given by (3.9) but with ∂ 4 ∂ 4 X k = 0 replaced by

∂ 4 ∂ 4 X k -X 4 ∂ 4 g k -2g k ∂ 4 X 4 + g j ∂ j X k -X j ∂ j g k = 0 (3.16)
where the expression (3.8c) of the Lie derivative of a connection has been used. From ∂ i X 4 = 0 and using eqns (3.16) and (2.5) we get ...

X 4 + 2 ȧ X 4 + 4a Ẋ4 = 0.
Expressing this in term of the new parameter τ , cf. (2.10), and with the help of (2.11), we obtain

Θ(X 4 ) ′′′ + (9B -2KΘ)(X 4 ) ′ -3Θ ′ (X 4 ) ′′ -6B(Θ ′ /Θ)X 4 = 0.
More effort is needed to deduce that (X 4 /Θ) ′′′ + K(X 4 /Θ) ′ = 0, hence that

X 4 = Θ 2α (1 -cos(kτ )) K + η sin(kτ ) k + ε (3.17)
where Θ is as in (2.12) and α, η, ε ∈ R.

We note that the expression (3.17) admits an analytical extension to the case K < 0, and is well-defined in the limit K → 0, see (2.12).

The equation ∂ i ∂ j X k = 0 then tells us that x i = Ω i j x j + Γ i , for some Ω and Γ depending on t only. Now (3.9) implies that

Ω i j = ω i j + ( 1 2 µ + a)δ i j
with ω ∈ so(3) and a ∈ R. Then, eqn (3.16) gives

∂ 4 ∂ 4 X i = 1 2 μ x i -a Γ i = 1 2 μ x i + Γi from the preceding expression of X i . We thus find Γi + a Γ i = 0, hence (Γ i ) ′′′ + K(Γ i ) ′ = 0.
Bearing in mind that Θ = Θ ′ = 0 at τ = 0, we get

Γ i = β i (1 -cos(kτ )) K + γ i sin(kτ ) 2k
for some vectorial constants of integration β and γ. The next step consists in showing, via (3.17), that eqn (3.9) yields the clock dilation factor in (3.6), namely

µ = η (1 + 2 cos(kτ )) + 4α sin(kτ ) k + ε k tan(kτ /2)
.

Redefining several constants of integration allows us to conclude that

X i = ω i j x j + η cos(kτ ) + 2α sin(kτ ) k + ε k/2 tan(kτ /2) + χ x i +β i (1 -cos(kτ )) K + γ i sin(kτ ) 2k (3.18)
for all i = 1, 2, 3, where ω ∈ so(3); β, γ ∈ R 3 ; α, χ, η, ε ∈ R. Again, this expression can be extended to the case K ≤ 0; cf. (2.12).

The vector fields

X = X i (x)∂ i + X 4 (t)∂ 4 ∈ chr(N, γ, θ, ∇)
defined by (3.17) and (3.18) span the 13-dimensional conformal-NC Lie algebra of the cosmological models which has been shown [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF] to be isomorphic to the Lie algebra (3.12) of the matrices (3.11).

Vacuum with cosmological constant (Newton-Hooke symmetry)

We start, in this section, with the general spherical-symmetric vacuum solution of the field equations (2.3), i.e., ̺ = 0, and Λ = 0, namely

g = - K r 3 + Λ 3 x
with K = GM = const. > 0 (with M the mass of the pointlike source), and r = |x| > 0.

Here N = R 3 \ {0} × R, while the Galilei tensors γ and θ are as before.

Putting k = Λ/3 we find, via (3.6), that

µ = α sin(2kt) k + η cos(2kt) (3.19)
with α, η ∈ R. This expression is still well-defined in the case Λ ≤ 0. Using (3.7), we will write λ = -µ + η -2χ with χ ∈ R. We find moreover that

X i = ω i j x j + 1 2 (f + 2χ -η)x i + Γ i ,
where ω ∈ so(3) and Γi + (Λ/3)Γ i = 0. • If K = 0 & Λ = 0 (no Newtonian gravitational source, M = 0, and non-vanishing cosmological constant), we easily find

Γ i = β i sin(kt)/k + γ i cos(kt)
for some vectorial constants β and γ. From the preceding expression of X i we finally obtain

X i = ω i j x j + 1 2 η (cos(2kt) -1) + α sin(2kt) 2k + χ x i +β i sin(kt) k + γ i cos(kt) (3.20)
and, with help of (3.9), we end up with

X 4 = α (1 -cos(2kt)) 2k 2 + η sin(2kt) 2k + ε (3.21)
where ω ∈ so(3); β, γ ∈ R 3 ; α, χ, η, ε ∈ R.

These vector fields generate chr(N, γ, θ, ∇), i.e., the 13-dimensional conformal-NC extension of the 10-dimensional Newton-Hooke Lie algebra [START_REF] Gibbons | Newton-Hooke spacetimes, Hpp-waves and the cosmological constant[END_REF]. 8To recover the standard notation of the literature, let us introduce the following basis of our Lie algebra, viz.,

J i = ǫ ℓ ij x i ∂ ℓ K i = sin(kt) k ∂ i P i = cos(kt) ∂ i H = ∂ t I = sin(2kt) 2k x i ∂ i + (1 -cos(2kt)) 2k 2 ∂ t T = 1 2 (cos(2kt) -1)x i ∂ i + sin(2kt) 2k ∂ t S = x ℓ ∂ ℓ
where i = 1, 2, 3. Those correspond respectively to infinitesimal rotations, boosts, spacetranslations, time-translation, inversion, and independent time-dilation, and space-dilation. The non-trivial Lie brackets are as follows:

[J i , J j ] = -ǫ ℓ ij J ℓ , [J i , K j ] = -ǫ ℓ ij K ℓ , [J i , P j ] = -ǫ ℓ ij P ℓ , [K i , H] = -P i , [K i , T ] = -K i , [K i , S] = K i , [P i , H] = -k 2 K i , [P i , I] = K i , [P i , S] = P i , [H, I] = 2T + S, [H, T ] = -2k 2 I + H, [I, T ] = -I.
• If Λ = 0 & K > 0 (in the presence of a gravitational source located at r = 0, with a vanishing cosmological constant), eqn (3.19) gives µ = 2αt + η. Straightforward calculation then yields

2Γ i + µ -3(2χ -η) -6 Γ j x j r 2 x i = 0;
but we have seen that Γi = 0 since Λ = 0, hence Γ i = β i t + γ i . Compatibility of these expressions implies that Γ i = 0. We moreover find, using µ -3(2χ -η) = 0, that α = 0 and 3χ = 2η. This leaves us with

X i = ω i j x j + χ x i & X 4 = 3 2 χ t + ε (3.22)
where ω ∈ so(3); χ, ε ∈ R.

The Lie algebra chr(N, γ, θ, ∇) defined by (3.22), which may be called the "Virial Lie algebra", is 5-dimensional and contains the "Keplerian dilations", generated by

X = x i ∂ i + 3 2 t ∂ t
featuring a dynamical exponent z = 3 2 . These spacetime dilations are precisely the transformations behind Kepler's third law. In the symplectic framework, they generate canonical similitudes [START_REF] Souriau | Structure of Dynamical Systems. A Symplectic View of Physics[END_REF] ; as such, they do not generate any Noetherian conserved quantity.

Conformal-Bargmann symmetries

We rephrase now the above results, in a somewhat simpler fashion, on (4 + 1)-dimensional Bargmann extended spacetime over a 4-dimensional Newton-Cartan spacetime. This strategy has been called the Eisenhart lift of NC structures in, e.g., [START_REF] Cariglia | Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux[END_REF][START_REF] Bekaert | Embedding nonrelativistic physics inside a gravitational wave[END_REF][START_REF] Cariglia | Null lifts and projective dynamics[END_REF].

General definitions

Recall that a general Bargmann structure is a triple (M, g, ξ) where M is a m-dimensional smooth manifold endowed with a Lorentz metric, g, together with a distinguished nowhere zero vector field, ξ, which is null and parallel [START_REF] Duval | Bargmann structures and Newton-Cartan theory[END_REF]. These structures fall into Brinkmann's classification [START_REF] Brinkmann | Einstein spaces which are mapped conformally on each other[END_REF]. It has also been shown that a Bargmann structure actually projects onto a NC structure (N, γ, θ, ∇) where N = M/(Rξ) is interpreted as n-dimensional spacetime, where n = m -1 = d + 1.

We will use this fact to advantageously lift à la Eisenhart [START_REF] Eisenhart | Dynamical trajectories and geodesics[END_REF] the above studied NC symmetries to the corresponding Bargmann structure. This will help us desingularise the Galilei "metric" structures on N, while recovering the familiar Lorentzian geometry on an extended spacetime M. This procedure has been systematically exploited in, e.g., [START_REF] Duval | Celestial mechanics, conformal structures, and gravitational waves[END_REF][START_REF] Burdet | Time-Dependent Quantum Systems and Chronoprojective Geometry[END_REF][START_REF] Cariglia | Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux[END_REF][START_REF] Bekaert | Embedding nonrelativistic physics inside a gravitational wave[END_REF][START_REF] Duval | On the Schrödinger-Newton equation and its symmetries: a geometric view[END_REF].

The conformal-NC group admits a (R, +)-extension as the "conformal-Bargmann group" defined as the "group" Chr(M, g, ξ) of all (local) diffeomorphisms, Φ, of M such that

Φ * g = λ g & Φ * ξ = ν ξ (4.1)
for some smooth, positive, real functions λ, ν of M [START_REF] Burdet | Time-Dependent Quantum Systems and Chronoprojective Geometry[END_REF][START_REF] Duval | On the Schrödinger-Newton equation and its symmetries: a geometric view[END_REF].

One easily checks that necessarily

dλ ∧ θ = 0 & dν = 0
which means that λ is actually a function of time, T = M/ξ ⊥ , and ν a constant. Since Φ permutes the fibres of the principal (R, +)-bundle M → N, it descend to NC spacetime, N, as a conformal-NC transformation, ϕ, as in (3.1) and (3.5). We thus have the exact sequence of group homomorphisms

1 → R → Chr(M, g, ξ) → Chr(N, γ, θ, ∇) → 1.
Let us emphasize that Chr(M, g, ξ) is a non-central extension of Chr(N, γ, θ, ∇) as long as ν = 1 in (4.1). 9Given a Bargmann structure, if C = (C λµνρ ) is the conformal Weyl tensor of (M, g), using the extra vector field, ξ, we can form the tensor P defined by

P λµν = C λµνρ ξ ρ . (4.2)
This tensor is clearly invariant under the rescalings (4.1) of the metric, g, and the "wavevector" ξ. Let us mention that the vanishing (4.9) of the tensor (4.2) will in fact constitute the definition (4.11) of the conformal-Bargmann Weyl tensor [START_REF] Duval | Nonrelativistic Conformal Symmetries and Bargmann Structures[END_REF][START_REF] Perrin | Chronoprojective invariance of the fivedimensional Schrödinger formalism[END_REF].

Conformal-Bargmann flatness means that (M, g, ξ) can be locally conformally mapped to (R d+2 , g 0 , ξ 0 ) where

g 0 = δ ij dx i ⊗ dx j + dt ⊗ ds + ds ⊗ dt. & ξ 0 = ∂ ∂s (4.3)
with the usual notation, i.e., there exists a local diffeomorphism Ψ :

M → R d+2 such that g = Θ 2 Ψ * g 0 & ξ = Ψ * ξ 0 (4.4)
for some function Θ > 0 necessarily defined on T . Under these circumstances, one is easily convinced that the conformal-Bargmann groups of these Bargmann structures are isomorphic, namely Chr(M, g, ξ) ∼ = Chr(R d+2 , g 0 , ξ 0 ) and that the isomorphism Chr(R d+2 , g 0 , ξ 0 ) → Chr(M, g, ξ) is given by

Φ 0 → Φ = Ψ -1 • Φ 0 • Ψ. ( 4.5)
Let us recall that the conformal-Bargmann group (alias the "chronoprojective group") acts on flat Bargmann space (4.3) as follows [START_REF] Duval | On the Schrödinger-Newton equation and its symmetries: a geometric view[END_REF]. The action Φ : (x, t, s) → (x, t, s) of Φ ∈ Chr(R d+2 , g 0 , ξ 0 ) reads

x = Ax + bt + c f t + g (4.6) t = dt + e f t + g (4.7) s = 1 ν s + f 2 |Ax + bt + c| 2 f t + g -b, Ax - 1 2 |b| 2 t + h (4.8)
where

A ∈ O(d), (b, c) ∈ R d × R d (boosts & space-translations), also d e f g ∈ GL(2, R)
is a projective transformation of the time-axis, and h ∈ R (extension parameter). We note that that λ = (f t + g) -2 and ν = dg -ef . 10According to an old suggestion of Souriau's, the subgroup defined by the constraints f = 0, d = g = ν = 1 is now called the Bargmann group after [START_REF] Bargmann | On unitary ray representations of continuous groups[END_REF]; it is the centrally extended Galilei group geometrically defined as the group of strict automorphisms of a Bargmann structure (M, g, ξ).

Let us now emphasize that the following statements are equivalent [START_REF] Perrin | Chronoprojective invariance of the fivedimensional Schrödinger formalism[END_REF] (see also [START_REF] Burdet | Cartan Structures on Galilean Manifolds: the Chronoprojective Geometry[END_REF] and [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF] for an approach based on NC theory).

1. The Bargmann structure admits a second-order normal Cartan connection for the conformal-Bargmann group. Putting F = 4πG ̺ + Λ enables us to interpret (4.10) as the Newton-Cartan gravitational field equations, see (2.2), which show up in this remarkable guise in the Bargmann framework.

It has been proved in [START_REF] Duval | Nonrelativistic Conformal Symmetries and Bargmann Structures[END_REF][START_REF] Perrin | Chronoprojective invariance of the fivedimensional Schrödinger formalism[END_REF], using the fact that ξ is null and parallel (implying R λµνρ ξ ρ = 0 and R µν ξ ν = 0), that the above condition (4.9) yields

C λµνρ = R λµνρ - 1 n -2 g λν R µρ -g λρ R µν -g µν R λρ + g µρ R λν & R µν ξ ρ = R µρ ξ ν

Bargmann-conformally flat examples

Let us consider the case d = 3 already dealt with in Section 2.

The time-dependent harmonic oscillator

Start with the Bargmann structure (M, g U , ξ), where M ⊂ R 5 , and

g U = g 0 -2Udt ⊗ dt & ξ = ∂ ∂s (4.13)
with U the gravitational potential. One proves [START_REF] Duval | Nonrelativistic Conformal Symmetries and Bargmann Structures[END_REF] that eqn (4.12) holds whenever

U(x, t) = 1 2 a(t)|x| 2 + b(t) • x + c(t)
for some arbitrary functions a, b, c of the time-axis, T . More general situations were studied in [START_REF] Duval | Conformal properties of Chern-Simons vortices in external fields[END_REF].

Let us stick henceforth to the interesting case

U(x, t) = 1 2 a(t)|x| 2 (4.14)
and posit, a priori, Ψ(x, t, s) = (x, t, ŝ) where

x = x Θ , t = dt Θ 2 , ŝ = s + 1 2 Θ Θ |x| 2 (4.15)
with Θ an arbitrary function of time t.

A short calculation leads us to eqn (4.4) for g = g U and ξ = ∂/∂s (in the present case) where the potential (4.14) is such that [START_REF] Burdet | Time-Dependent Quantum Systems and Chronoprojective Geometry[END_REF] Let us now illustrate these findings with the help of several special example, namely the cosmological Newtonian models, and the Newton-Hooke spacetime models.

a(t) = - Θ(t) Θ(t) . ( 4

Newtonian cosmology

We have seen that eqns (2.11) readily imply (2.13). In view of (4.16), this implies that the gravitational potential of Newtonian cosmology in the metric (4.13) is given by (4.14) with a(t) = B Θ 3 in accordance with (2.9). We have just confirmed the conformal-Bargmann symmetry of the Newtonian cosmological model -without cosmological constant.

We now exhibit the (inverse of the) diffeomorphism (4.15) that brings (locally) conformally the cosmological Bargmann model to flat Bargmann space (4.4). In view of (2.12) and (2.10), we find

t = Θ dτ = Bτ - B k 3 sin(kτ )
up to a constant. And (4.15) yields

t = dt Θ 2 = dτ Θ = - k B cot kτ 2
modulo a constant; see also [START_REF] Hassaïne | Symmetries of fluid dynamics with polytropic exponent[END_REF]. This shows that

τ = - 2 k arccot B t k (4.18)
which helps us express the scale factor Θ = (B/k 2 )(1 -cos(kτ )) given by (2.12) as the function

Θ = 2B 2B 2 t2 + k 2 (4.19)
of "time" t. At last, the sought diffeomorphism Ψ -1 (see (4.15)) is given by

x = Θ x, t = Bτ - B k 3 sin(kτ ), s = ŝ + B t 2 Θ |x| 2
where τ is as in (4.18), and Θ as in (4.19). The above action (4.6)-(4.8) of the conformal-Bargmann group on flat space (R 4,1 , ξ 0 ) coordinatized by (x, t, ŝ) is thus transported to (M, g U , ξ) as an action of Chr(M, g U , ξ).

This proves that the conformal symmetry group of Newtonian cosmology lifted to Bargmann's spacetime is (locally) isomorphic to the chronoprojective group Chr(R 5 , g 0 , ξ 0 ). Interestingly, using again (2.13), we discover that the Schwarzian derivative of the diffeomorphism φ : t → t given by (4.15) is12

S(φ) = φ ′′′ φ ′ - 3 2 φ ′′ φ ′ 2 = 2B Θ 3 = 8πG̺ 3 (4.20)
where ̺ is the time dependent mass density given by the Poisson equation, ∆U = 4πG̺; see (2.2) with Λ = 0. Now, strictly speaking, the Schwarzian derivative is the quadratic differential S(φ) = S(φ) dt

a(t) = k 2 & k = Λ 3 .
Using (4.16), we find Θ = A cos(kt -φ) for some integration constants A > 0 and φ, and (4.15) yields t = tan(kt -φ)/(kA 2 ). Now we want t → t if k → 0, whence the choice Θ = cos(kt)

which leads to x = x cos(kt) , t = tan(kt) k , ŝ = s - 1 2 k tan(kt)|x| 2 . (4.22)
Again (M, g U , ξ) is conformally-Bargmann flat, and the action of Chr(M, g U , ξ) is as in (4.17) together with (4.22).

We note also that the formula (4.22) is consistent with the one obtained by turning off the friction term in the "Arnold" expression [START_REF] Aldaya | The quantum Arnold transformation[END_REF] in eqn (III.76) of [START_REF] Cariglia | Eisenhart lifts and symmetries of time-dependent systems[END_REF].

The infinitesimal version of the corresponding projected action of Chr(M, g U , ξ) on NC spacetime reproduces eqns (3.20) and (3.21). See also [START_REF] Niederer | The maximal kinematical invariance group of the harmonic oscillator[END_REF].

At last, the Schwarzian derivative of the diffeomorphism φ : t → t of the time axis given in (4.22) for the Hooke solution, is

S(φ) = 2 3 Λ (4.23)
as anticipated in (4.20) and Newton's field equations (2.2); compare also (4.21). Thus, eqn (4.23) reveals the striking relationship between the Schwarzian derivative and dark energy advertised in [START_REF] Gibbons | Dark Energy and the Schwarzian Derivative[END_REF].

Conclusion

Our work on the definition and the characterization via eqns (3.1) and (3.5) of the chronoprojective group, Chr(R n ), of conformal non-relativistic symmetries, as well as its illustration through various examples of spacetime models has shown the richness of the geometric structures involved, namely a mix of Galilei and projective structures in the NC framework. This contrasts with the well-known geometric incompatibility (except in the 1-dimensional case) between projective and metric conformal structures where the symmetry groups are respectively SL(n+1, R) and SO(p+1, q +1) in dimension n = p+q.

The fundamental difference with the familiar extended Schrödinger group is that the latter has fixed dynamical exponent z = 2, and that the central extension subgroup generates (conserved) mass. In contrast, the conformal-NC group [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Burdet | Cartan Structures on Galilean Manifolds: the Chronoprojective Geometry[END_REF][START_REF] Burdet | Time-Dependent Quantum Systems and Chronoprojective Geometry[END_REF] has no central extension; however space and time dilations are independent: there is no fixed dynamical exponent, z. It is worth mentioning that the recently considered conformal Galilei groups [START_REF] Duval | Non-relativistic conformal symmetries and Newton-Cartan structures[END_REF] allow for any z = 2/N, where N = 1, 2, . . . . The case z = 1 can actually be obtained by contraction from the relativistic conformal group [START_REF] Barut | Conformal Group → Schrödinger Group → Dynamical Group -The Maximal Kinematical Group of the Massive Schrödinger Particle[END_REF][START_REF] Niederer | The connections between the Schrödinger group and the conformal group[END_REF].

This survey has also helped us reveal the power and usefulness of the Eisenhart-lift to promote these NC symmetries to the realm of Lorentzian conformal geometry through the definition of Bargmann extended spacetime structures as generalized pp-waves. Such a procedure has therefore provided a natural geometric framework for the conformal-Bargmann extensions of the NC spacetime conformal symmetry groups we started with.

Let us finish by recalling two extra examples where these non-relativistic conformal groups occur in a specific manner. The prototype of conformally-NC flat spacetime is N ∼ = (R d × S 1 )/Z 2 ; this Möbius manifold is in fact a Newton-Hooke spacetime with cosmological constant Λ = d; cf. [START_REF] Duval | Quelques procédures géométriques en dynamique des particules[END_REF][START_REF] Duval | Nonrelativistic Conformal Symmetries and Bargmann Structures[END_REF][START_REF] Duval | Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes[END_REF]. Also does the symmetry group of the Schrödinger-Newton equation [START_REF] Diósi | Gravitation and quantum-mechanical localization of macro-objects[END_REF] appear as a subgroup of the conformal-Bargmann group Chr(R d+2 ) featuring a specific dynamical exponent z = 1 3 (d + 2), as shown in [START_REF] Duval | On the Schrödinger-Newton equation and its symmetries: a geometric view[END_REF] and in agreement with [START_REF] Stichel | Self-gravitating darkon fluid with anisotropic scaling[END_REF].

and features the dynamical exponent z = 2 . ( 3 . 15 )

 2315 The Schrödinger Lie algebra (3.13) is a finite-dimensional Lie subalgebra of the Schrödinger-Virasoro Lie algebra, defined by λ + µ = 0 (cf. (3.7)), expressing infinitesimally the constraint (3.2).

2 .

 2 The Bargmann structure is of Petrov type N (see (1.1)), i.e., holdsR µν = F ξ µ ξ ν (4.10)with F some arbitrary function of spacetime, N.11

(4. 11 )

 11 where n = d + 2 is the dimension of M. This tensor C (where the Ricci tensor hence satisfies (4.10)) is called the Bargmann-conformal Weyl tensor.The Bargmann structure is therefore (locally) conformally-Bargmann flat iff C λµνρ = 0.(4.12)

Note that Θ(t) is interpreted by Souriau[START_REF] Souriau | Mécanique statistique, groupes de Lie et cosmologie[END_REF] as the "co-temperature" of the cosmological model.

It is related to the so-called (centreless) Schrödinger-Virasoro group[START_REF] Henkel | Schrödinger invariance and strongly anisotropic critical systems[END_REF][START_REF] Unterberger | The Schrödinger-Virasoro Algebra: Mathematical structure and dynamical Schrödinger symmetry[END_REF] which is the subgroup

We find that this group projects to the time axis as a group of projective transformations, i.e., diffeomorphisms of T with vanishing Schwarzian derivative.

These infinitesimal generators are interpreted as follows: ω is a rotation, β a Galilei boost, γ a space translation, α a time inversion, χ a space dilation, η a time dilation, and ε a time translation. We notice that space and time dilations are de facto independent.

Note that we recover (3.10a) and (3.10b), i.e., the Lie algebra chr(R 4 ), in the limit Λ → 0.

The (R, +) subgroup (of "vertical translations") is central in the Schrödinger subgroup, Sch(M, g, ξ), defined by ν = 1 in (4.1).

We find that dim Chr(R d+2 , g 0 , ξ 0 = 1 2 (d 2 + 3d + 10), which is 14 if d = 3.

Note that F is a priori a function of M ; now the Einstein tensor being divergence free, we get ξ(F ) = 0, and F is hence (the pull-back of) a function of spacetime, N .

The expression(4.20) is no surprise and quite general; it results from the transformation of the Ricci tensor under conformal Bargmann rescalings[START_REF] Duval | On the Schrödinger-Newton equation and its symmetries: a geometric view[END_REF].

Acknowledgments: Thanks are due to M. Cariglia for his constant interest and for helpful suggestions. GWG is grateful to the Laboratoire de Mathématiques et de Physique Théorique de l'Université de Tours for hospitality and the Région Centre for a "Le Studium" research professorship. PAH acknowledges partial support from the Chinese Academy of Sciences' Presidential International Fellowship (Grant No. 2010T1J06), and the Institute of Modern Physics of the CAS at Lanzhou for hospitality.