Universe as Klein–Gordon eigenstates

We formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects theβ-times tβ: = ∫ ta-2β, where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems O1/2Ψ=Λ12Ψ,O1a=-Λ3a,which is suggestive of a measurement problem. Oβ(ρ, p) are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The Oβ’s are also independent of the spatial curvature, labeled by k, and absorbed in Ψ=aei2kη.The above pair of equations is the unique possible linear form of Friedmann’s equations unless k= 0 , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time η≡ t1 / 2 among the tβ’s, which is the key to absorb the curvature term. An immediate consequence of the linear form is that it reveals a new symmetry of Friedmann’s equations in flat space.