### The Theoretical Significance of a Logarithmic Distance-Redshift Relationship

by Fred Vaughan

There is something very compelling about a logarithmic functional
form for the distance-redshift relation in observational
cosmology. In fact, it is so compelling as to seem logically
necessary as the form of that relationship —
whether that fact is generally acknowledged or not, which of
course…it is *not*.

To adequately understand this, let us look at what is involved in
light being redshifted along a propagation path between emission
and observation. Suppose there is an observer at point A for
which a telescope on earth would suffice as an instance. And
suppose that there is an ensemble of atoms in a star in a distant
galaxy that we will refer to as point C that emit light at a
specific wavelength associated with the spectra of a particular
element. These atoms emit photons of light that can ultimately be
observed by the telescope at A. If there is a distance-related
redshift in the spacetime where all this takes place, then the
wavelength of the light λ_{A}observed at A will
be related to the emission wavelength
λ_{C}emitted at location C according to the
redshift definition:

ZThis is true no matter what the separation between A and C or anything else. It's just a definition. For physical reasons Z_{AC}= ( λ_{A}− λ_{C}) / λ_{C}

_{AC}must be a continuously increasing function of the separation AC. So, let us define the redshift-related parameter ζ(d) as a continuous function of the separation d = AC as follows:

ζ(d) = Z_{AC}+ 1 = λ_{A}/ λ_{C}

Since ζ(d) applies to for any
separation, we should be able to place an observer at any point B
along the light path from C to A, where d_{1}= AB and
d_{2} = BC, with the observed radiation exhibiting
redshifts as follows:

&zeta(dTherefore, over the total distance for which d = d_{1}) = λ_{A}/ λ_{B}and &zeta(d_{2}) = λ_{B}/ λ_{C}

_{1}+ d

_{2}the following relation must apply:

ζ(dAnd as a necessary consequence of this relation, we must have that:_{1}+ d_{2}) = ζ(d_{1}) . ζ(d_{2}),

ζ(d) = eAnd, of course, the inverse functionality must be:^{αd}= e^{α ( d1 + d2 )}.

d(ζ) = ln (ζ)

The "standard model" embraces a broad class of disparate alternatives loosely associated by adherence to Hubble's hypothesis and one form or another of Einstein's theory of general relativity. The Einstein — de Sitter model is but one of the simpler of these alternatives that exhibits a "flat" spacetime, because of which it is frequently discussed for didactic purposes, although it is generally disparaged as a somewhat naïve candidate for serious consideration. This short shrift seems ill-advised to the author in light of the interesting fact that a key feature of the Einstein — de Sitter model (unlike the others that are considered more viable) is that the distance-redshift relation is given by the logarithmic form.

Although the Einstein — de Sitter model is virtually never considered a viable contender by current cosmologists for the ultimate acceptance, its logarithmic form of the distance-redshift relation is generally used for convenience in analyzing associated phenomena because it so closely fits the actual data as distances to observed objects increase. Strange isn't it?

The preceding discussion explains the situation depicted in the figure below.

It is worth considering what would be implied by a relationship other than one involving the logarithm: What is involved is whether or not homogeneity applies to this relationship.

The seeming improbability of, but nonetheless presumed, failure
of the logic we have described above is what has contributed so
substantially to presumptions of the supposed evolution of
developments in our universe. But if one decouples redshifting as
an observed phenomenon (*whatever its cause*) from
constraints imposed by *whatever causes it* according to one
cosmological theory or other, then the logarithmic relationship
to distance continues to make logical sense as we have shown
above. We will be told, of course, that to presume that distances
could be linearly additive if space itself is nonlinearly
distorted would itself be an improbability. But would it? Even
along a curved path the distance along that path is linearly
additive as the basis for the integration of distance along
infinitesimal line segments.

In the next figure we have drawn a situation similar to that shown in figure 1 except that space is such that line of sight distance is curved along a light path through space. In this case, in addition to observers A and B, we have observer B capable of emitting light from a separate source at the moment of his observation of the light from C which is set to resonate at precisely the same frequency (wavelength) as the radiation he observes. Let us analyze the possibilities here.

As before, we must now have that ζ(d_{1}) =
λ_{A} / λ_{B}and
ζ(d_{2}) =
λ_{B1} / λ_{C}.
This would seem to apply by reason of the definition of redshift,
if the source of the radiation of wavelength
λ_{B1}is indeed set up to equal that
of λ_{B}. This can be verified by the digital
communication from B to A independent of the redshift impact on
that link if A's antenna is properly tunable. Then as long as
there is a general formula applicable throughout space and time
relating redshift and distance,

d ←If the peculiar functionality of the inversesf(Z+1) and Z+1 ← ζ(r)

*f*(x) and ζ(y) were independent of position in spacetime (i. e., if spacetime is indeed homogeneous), then the logarithmic/exponential relationships must apply. It is the

*ad hoc*denial of this cosmological principle that had reigned supreme since Copernicus that empowers the standard model with the freedom to deny an otherwise logical premise.

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