TITLE:
A Liouville Invariance Theorem for Blackbody Spectrum Preservation
AUTHORS:
Michael Aaron Cody
KEYWORDS:
Cosmological Redshift, Cosmic Microwave Background, Liouville Theorem, Kinetic Redshift, Scalar Field Dynamics, Observational Degeneracy, Blackbody Spectrum, Static Cosmology, Alternative Cosmology, CMB Thermodynamics
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.7,
July
31,
2026
ABSTRACT: The preservation of the cosmic microwave background (CMB) Planck spectrum across cosmological time is commonly interpreted as requiring metric expansion. This note demonstrates that such preservation follows from Liouville transport under a homogeneous, isotropic, frequency-independent redshift mapping, without assuming any specific cosmological model or spacetime dynamics. For collisionless photon propagation, a generic mapping of the form
ν
o
=
ν
e
/
R(
t
e
,
t
o
)
, with
R
independent of frequency, suffices to conserve the spectral shape of an initial blackbody distribution. The derivation yields the standard temperature scaling
T(
z
)=
T
0
(
1+z
)
, preserves the zero chemical potential condition (
μ=0
), and reproduces the phase space and energy-momentum conservation scalings
n
˙
=−3Hn
,
ρ
˙
=−4Hρ
, under an effective rate defined by
H≡
d(
lnR
)/
dt
. The result establishes a conditional invariance statement. Metric expansion is sufficient for blackbody preservation, but the spectrum shape alone does not uniquely identify expansion as the underlying cause.