Universe as Klein-Gordon Eigenstates

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 $\beta$-times $t_\beta:=\int^t a^{-2\beta}$, where $a$ is the scale factor. In particular, it turns out that Friedmann's equations are equivalent to the eigenvalue problems $$ O_{1/2} \Psi=\frac{\Lambda}{12}\Psi \ , \qquad O_1 a =\frac{\Lambda}{3} a \ , $$ which is suggestive of a measurement problem. $O_{\beta}(\rho,p)$ are space-independent Klein-Gordon operators, depending only on energy density and pressure, and related to the Klein-Gordon Hamilton-Jacobi equations. The $O_\beta$'s are also independent of the spatial curvature, labeled by $k$, and absorbed in $$ \Psi=\sqrt a e^{\frac{i}{2}\sqrt{k}\eta} \ . $$ 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 $\eta\equiv t_{1/2}$ among the $t_\beta$'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.