What you are discussing is a conformal model of cosmology.
You can have two models conformally related to each other with two time systems
The important question is does the time parameter "t" have physical significance; that is, does it relate to some clock in each model that is usable in physical experiments?
Forms of coordinate time can be anything we want them to be through conformal transformations - that is why it is important to connect any particular definition of time with a physical process to ensure that it is physically significant.
So, for example, we can define two physically significant times as follows.
Sample two photons, one emitted by a caesium atom the other sampled from the CMB radiation.
First time system: In the standard LCDM model of an expanding universe, with regular atomic clocks and fixed atomic meter rulers, the standard of time is the "atomic" second defined as the duration of exactly 9.19263177x109
periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
Second time system: A "photonic" second, may be defined as the duration of exactly 1.604x1011
periods of the radiation corresponding to the peak of the CMB black body spectrum.
Both systems of time measurement are physically significant and agree with each other in the present epoch, although they will diverge from each other at other times.
When compared to the "atomic" clock system, the "photonic" clock system model, extrapolated back to the earliest moments of the BB, diverges to (-) inifinity as atomic time t->0.
Thus in this model we can recover "an infinitely old universe" within an apparently finite (as measured by an "atomic" clock) BB paradigm. Furthermore it will be static in size as measured by the "photonic" metre.
To give this "photonic" system of clocks and rulers a physical reality (i.e. physically consistent) you will have to make atomic masses grow exponentially with time, this will require an alternative theory of gravitation and cosmology.
However in such a model, if ephemeris time is that measured in the "photonic" system, (as it was in SCC
), then the Pioneer spacecraft will appear to have an anomalous sunwards acceleration of cH or 6.69×10−8
; such an acceleration aP
is actually observed. (Allowing for drag and radiation non-inertial forces and uncertainty in H.)