Universe as Klein–Gordon eigenstates - Marco Matone (SpringerOpen, 2021)
Abstract We formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the $$β $$ β -times $$t_β :=\int ^t a^{-2β }$$ 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 $$\begin{aligned} O_{1/2} Ψ =\frac{Λ }{12}Ψ , \quad O_1 a =-\frac{Λ }{3} a , \end{aligned}$$ O 1 / 2 Ψ = Λ 12 Ψ , O 1 a = - Λ 3 a , which is suggestive of a measurement problem. $$O_{β }(ρ ,p)$$ 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 $$O_β $$ O β ’s are also independent of the spatial curvature, labeled by k, and absorbed in $$\begin{aligned} Ψ =\sqrt{a} e^{\frac{i}{2}\sqrt{k}η } . \end{aligned}$$ Ψ = a e i 2 k η . The above pair of equations is the unique possible linear form of Friedmann’s equations unless $$k=0$$ k = 0 , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $$η \equiv t_{1/2}$$ η ≡ t 1 / 2 among the $$t_β $$ 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.