TITLE:
CMB Temperature as Geometric Mean of Maximum and Minimum Unruh Temperature: T cmb = T U,min T U,max
AUTHORS:
Espen Gaarder Haug
KEYWORDS:
CMB Temperature, Unruh Temperature, Rindler Horizon, Hawking Temperature, Geometric Mean
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.8,
August
13,
2026
ABSTRACT: We demonstrate that the CMB temperature can be described as the geometric mean of the minimum and maximum Unruh temperatures:
T
cmb
=
T
U,min
T
U,max
. The geometric-mean relation is an algebraic consequence of the limiting scales adopted below; its interpretation as a possible origin of the CMB temperature is a separate physical hypothesis. One possible interpretation of this result is that the CMB temperature corresponds to the geometric mean of the horizon-induced temperature spectrum associated with vacuum excitations in the observable universe. In this framework, the Unruh effect links proper acceleration to effective temperature over a finite range. The lower bound is fixed by the smallest physically relevant acceleration determined by the cosmic horizon, while the upper bound is set by the largest admissible acceleration near the Planck scale. The minimum corresponds to an observer whose Rindler horizon is comparable to the Hubble radius, whereas the maximum corresponds to a horizon of Planckian size. Viewed this way, the observed CMB temperature emerges as the geometric midpoint of these two extreme scales.