A symmetry of the Einstein-Friedmann equations for spatially flat, perfect fluid, universes

We report a symmetry property of the Einstein-Friedmann equations for spatially flat Friedmann-Lema\^itre-Robertson-Walker universes filled with a perfect fluid with any constant equation of state. The symmetry transformations form a one-parameter Abelian group.

The FLRW line element of spatially homogeneous and isotropic cosmology in comoving coordinates(t, x, y, z) is where the dynamics is contained in the evolution of the cosmic scale factor a(t). The Einstein-Friedmann equations for a spatially flat universe filled with a perfect fluid with energy density ρ(t) and isotropic pressure P (t) are These equations exhibit a special symmetry that maps a barotropic perfect fluid with equation of state P = P (ρ) into itself, with a rescaled energy density and pressure but with the same equation of state.

The symmetry transformation
Assume that the energy content of the universe is a single perfect fluid with constant equation of state P = wρ with w = constant; then the Einstein-Friedmann equa-tions (1.2)-(1.4) assume the form The solution of these equations is well known and easy to derive (e.g., [22]): (2.9) The change of variables leaves the Einstein-Friedmann equations (2.5)-(2.7) unchanged. In particular, the matter source, the barotropic perfect fluid, maintains the equation of state P = wρ with the same equation of state parameter w. This is in contrast with other symmetries of the same equations which change an "ordinary" fluid satisfying the weak energy condition into a phantom fluid with different equation of state parameter and are ultimately inspired by string theory dualities [21,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], and with other solutions of the EinsteinFriedmann equations obtained using methods of supersymmetric quantum mechanics [24,25,26].
To check invariance of the equations, begin from the Friedmann equation (2.5) which becomes, in terms of tilded quantities: Let us consider now the covariant conservation equation (2.6), which now reads Expanding, one has (2.20) Grouping similar terms then yields

A group of symmetry transformations
The transformations (2.10)-(2.12) form a commutative group, as shown below. First, we show that the composition of two such transformations is a change of variables of the same form. Let  where r = ps. Therefore, the composition of two transformations gives the same kind of transformation and the order of these two operations does not matter: There is a neutral element for the operation of composition of maps: the transfor-mationL 1 with s = 1 is the identity sincê L 1 : (a, dt, ρ) −→ ã, dt,ρ = (a, dt, ρ) . (3.33) Finally, each transformationL s with s = 0 has a (left and right) inverseL 1/s sincê Therefore, it is Similarly, one checks that (4.41)

Conclusions
We have reported a symmetry property of the Einstein-Friedmann equations for a spatially flat FLRW universe filled with a single barotropic perfect fluid with constant equation of state P = wρ. It is easy to check that this symmetry does not hold for spatially curved universes. As for the physical meaning of this symmetry, let us note that the scale factor and the comoving time scale as a → a s , (5.42) respectively. This scaling could be superficially interpreted by saying that the scaling of proper spatial distances and the proper time of observers comoving with the cosmic perfect fluid by the same power points to some scale-invariance property of the Einstein-Friedmann equations for spatially flat sections, with this property failing when a spatial scale associated with the curvature of the 3-dimensional spatial sections is present. However, these equations are definitely not scale invariant, and this is the meaning of the completely different scaling of the energy density (2.12) (in the units used, in which energy and mass have the dimensions of a length, energy density should scale as the inverse square of a length ℓ −2s , but it does not). This fact simply reflects the lack of scale invariance of the Einstein equations even in vacuo or in the presence of conformally invariant matter (such as, e.g., a radiation fluid with w = 1/3).
Next, one could be tempted to view the symmetry of the Einstein-Friedmann equations (2.5)-(2.7) as deriving from a conformal transformation of the spacetime metric ds 2 → ds 2 = Ω 2 ds 2 for some conformal factor Ω(x µ ), followed by a suitable redefinition of the comoving time coordinate, but this is not possible in general, as is easy to check.
The symmetry map (2.10)-(2.12) for the special case of a radiation fluid with equation of state parameter w = 1/3 was already noted in the context of an analogy between the cosmic radiation era and the freezing of bodies of water in environmental physics [27].