AbstractWe formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the$$\beta $$
β
-times $$t_\beta :=\int ^t a^{-2\beta }$$
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 $$\begin{aligned} O_{1/2} \Psi =\frac{\Lambda }{12}\Psi , \quad O_1 a =-\frac{\Lambda }{3} a , \end{aligned}$$
O
1
/
2
Ψ
=
Λ
12
Ψ
,
O
1
a
=
-
Λ
3
a
,
which is suggestive of a measurement problem. $$O_{\beta }(\rho ,p)$$
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 $$O_\beta $$
O
β
’s are also independent of the spatial curvature, labeled by k, and absorbed in $$\begin{aligned} \Psi =\sqrt{a} e^{\frac{i}{2}\sqrt{k}\eta } . \end{aligned}$$
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 $$\eta \equiv t_{1/2}$$
η
≡
t
1
/
2
among the $$t_\beta $$
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.
Nehari’s univalence criteria, pre-Schwarzian derivative and applications
