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 the β -times t β : = ∫ t a - 2 β , where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems O 1 / 2 Ψ = Λ 12 Ψ , O 1 a = - Λ 3 a , which is suggestive of a measurement problem. O β ( 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 O β ’s are also independent of the spatial curvature, labeled by k , and absorbed in Ψ = a e i 2 k η . 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 ≡ t 1 / 2 among the t β ’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.