https://doi.org/10.1140/epjc/s10052-021-09865-4
Letter
Universe as Klein–Gordon eigenstates
1
Dipartimento di Fisica e Astronomia “G. Galilei”, Università di Padova, Via Marzolo, 8, 35131, Padua, Italy
2
Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Padua, Italy
Received:
27
October
2021
Accepted:
21
November
2021
Published online:
1
December
2021
We formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the
-times 
, where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems 
which is suggestive of a measurement problem. 
are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The 
’s are also independent of the spatial curvature, labeled by k, and absorbed in 
The above pair of equations is the unique possible linear form of Friedmann’s equations unless 
, in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time 
among the 
’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.
© The Author(s) 2021
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