2.1. De Sitter effect
In Einstein’s cosmological model [
3], the density of matter
, with the following relationship between
and
:
where
is the Einstein constant. The time-related metric coefficient
in (
3) is unity. So, time is universal for the whole Universe in Einstein’s model. By contrast, in de Sitter’s model,
(there is no matter!) and
. The
component of the metric (
4) is
implying that, from the observer’s perspective, time at the source’s location is dilated, arriving to the complete halt at the maximal distance, at which
and
.
This time dilatation (called the de Sitter effect) causes “the frequency of light-vibrations to diminish with increasing distance from the origin of coordinates. The lines in the spectra of very distant stars or nebulae must therefore be systematically displaced towards the red, giving rise to a spurious positive radial velocity” [
2]. This de Sitter’s prediction of the cosmological redshift phenomenon was made a decade before its observational discovery in 1927 by Lemaître [
6] and in 1929 by Hubble [
9].
In the current standard cosmological model,
CDM, the cosmological redshift is interpreted in terms of the expanding space paradigm encoded by the Robertson-Walker’s time-dependent scaling factor
in the Friedmann-Lemaître-Robertson-Walker (FLRW) metric [
6,
8,
10,
11], with no possible alternatives. The Hubble diagram based on the FLRW metric fits pretty well the observational distance moduli from the
Pantheon+ catalogue of 1701 type-Ia supernova data.
2.2. Luminosity distances.
Luminosity distances and distance moduli based on the de Sitter effect and the metric coefficient (
8) do not match observational data. For example, the comparison of the
CDM-model with the observational data from [
13] gives a pretty small value of the goodness-of-fit parameter
, whereas a similar parameter for de Sitter’s model is very large (many thousands). This comparison is made by using the
CDM parameters from [
13]:
and
km s
−1 Mpc
−1, assuming a flat cosmology with
.
For devising a better relationship between luminosity distances and redshifts of remote sources within the elliptical space framework, one can resort to the main property of this space – the topological connectivity between its antipodal points.
The connectivity of points separated by some distance can be viewed as their short-circuiting, which is sometimes associated with quantum entanglement of subatomic particles separated by large distances [
14] or even with the possibility of instantaneous travels between remote regions of space [
15]. A general-relativistic description of such a feature was discovered in 1935 by Einstein and Rosen [
16], and it is called the Einstein-Rosen bridge or, more frequently, a “wormhole”.
Morris, Thorne & Yurtsever [
17] found that wormhole creation must be accompanied by extremely large energies aka spacetime curvatures. This means that wormholes are likely to be microscopic, on a scale-length of the order of the Planck-Wheeler length,
cm. A static wormhole can be described by the Schwarzschild metric (in spherical coordinates
):
which is what we need for calculating redshifts in elliptical space if the mathematical connectivity of its antipodal points is physically interpreted as the connectivity via microscopic wormholes. The metric coefficients
and
of the Schwarzschild metric
are distinct from those of the de Sitter metric (
4). Thence, in this approach, the redshifts of remote sources are not due to the de Sitter effect, but due to the gravitational redshift corresponding to the metric coefficient
in (
10). So, in addition to the initial free parameter
R, there is another parameter
corresponding to the Schwarzschild radius, which is unknown.
Originally, the Schwarzschild metric (
9) was devised for asymptotically flat spacetimes (
as
). In our case, finite values of
R are allowed, so we keep
R as a free parameter. For this reason, in our calculations we need a modified Schwarzschild metric residing in a curved space. Such metric was devised in 1939 by R.C. Tolman [
18], who proposed a general method for finding exact solutions of Einstein’s field equations. Among various solutions described by him in [
18], the suitable one for our case is the solution No.II with the Swarzschild-de Sitter (SdS) metric. It differs from the Schwarzschild solution (
9) by an additional term in the metric coefficient
, which accounts for the possibility of finite values of
R:
the metric coefficient related to radial-coordinates being
, like in (
9).
We assume that two wormhole’s throats are located at two antipodal points separated by a very large distance from each other (
). Yet they are also short-circuited via a microscopic wormhole structure (in
Figure 1 this is shown as a connection between the north and south poles marked with a dashed line). The observer is located near one of the throats (called hereafter the near-throat), the source being somewhere in between the observer and the far-throat of the observer’s antipodal-point.
In
Figure 1, distances
r and
are measured from the origin at the observer’s location (
o) towards a remote source (
s). As we know from the de Sitter’s model case, this measurement does not lead to an agreement with observations. One can subdue this problem by transferring the origin of coordinates to the observer’s antipodal point. In this new set-up, the distances to the source (
) and observer (
) are measured from the wormhole’s far-throat, which is unusual, but we shall see that this provides a better agreement between the model and observations (hereinafter, we drop the prime indices previously used for denoting quantities in the natural space
).
A diagram corresponding to this alternative choice of the origin is presented in
Figure 2 where the entire sphere
is conformally mapped to the Poincaré disk.
For clarity, the observer’s location (
o) in this diagram is shown slightly shifted from the center of the Poincaré disk. The antipodal point of the disk’s center is transformed in this map in such a way that it appears as the disk’s circumference (the outermost dash-dotted circle in
Figure 2). In the three-dimensional space
, this circle corresponds to a 2-sphere
surrounding the observer at a very large distance (
) from the observer.
And yet, it is a zero-size point near the observer – see our comments in
Section 1 about the unusual property of elliptical space, which makes the objects in the vicinity of the observer to be visible in all directions across the celestial sphere. One would need to keep in mind that this circle (a sphere
) and the central point of the disk are not distinguished in elliptical space.
In concordance with our choice of the origin of coordinates, any distance is now measured from this 2-sphere towards the source (
) or towards the observer (
). These two distances are indicated in
Figure 2 by the arrows from the left edge of the diagram. Our purpose here is to find a relationship between the source redshift (
z) and the source-to-observer distance (
d):
Any point slightly offset from the center of the Poincaré disk has its corresponding antipodal sphere at a small distance from the disk border towards the interior of the disk. For example, if we draw a small 2-sphere around the center of the disk near the observer (a small central dashed circle in the scheme which denotes the wormhole’s near-throat), then its corresponding antipodal 2-sphere (the wormholes far-throat) will be very large, which is shown in
Figure 2 as a large dashed circle at some distance
from the disk border. Although the observer is nearby and
outside of this small sphere, yet it is
inside of it because the antipodal image of this sphere completely surrounds the observer at far distances.
Spherical symmetry.
An essential aspect of this construct is its spherical symmetry for any arbitrary location. This follows from the topology of elliptical space. When looking at the near-throat of a microscopic wormhole, the observer localises it within a very narrow (nearly zero) solid angle of cm across. But when the observer looks at the far throat of the same microscopic wormhole, the solid angle spans steradians (a sphere around the observer), as we have discussed above.
The remote horizons corresponding to locations within some vicinity of the observer are at extremely large distances from the observer. From the observer’s perspective, this neighbourhood region might appear large, including some nearby galaxies of even clusters of galaxies. But this size is likely to be negligible as compared with the distances to the remote horizons. Therefore, the average collective horizon around any point of space must be almost ideally spherically symmetric, thus, making all points of space equivalent to each other.
Redshift-distance relationship.
In our set-up shown in
Figure 2 there are two unknown distances,
and
. The latter can be replaced with the source-to-observer distance (
12). The gravitational radius
and the global radius
R are also unknowns, as was mentioned before. So, in total, our model has three free parameters,
,
and
R, to be determined by using observational data.
Both source and observer are located within the Schwartzschild-de Sitter metric (
9) with its redshift-defining coefficient (
11). So, the source’s redshift with respect to the observer is
or, by taking into account (
11),
At this point, we can define the parameter
as unit distance for measurements,
. Later on, it can be converted to some common distance units, like Mpc. Thus,
From (
12)
which gives
Then the source-to-observer distance reads
which is a recursive expression [in units of
] for finding the source-to-observer distance as a function of the source redshift. It has to be multiplied yet by the scaling factor
in order to obtain the source luminosity distance:
with one of the
-factors accounting for the loss of luminosity due to the cosmological redshift
z and also for the lower rate at which the photons reach the observer because of the cosmological time dilatation caused by the non-unit metric coefficient
, and another accounting for the photon path distortion (the
coefficient in the Schwarzschild-de Sitter metric). This relationship can be used for determining the unknown parameters of our model by fitting the theoretical distance moduli
(in stellar magnitudes) to the observational distance moduli of type Ia supernovae. The numerical coefficients in (
20) correspond to the luminosity distances
expressed in Mpc.