The Lensing Equation.
Astronomical observers treat incoming light from distant objects as though it has come to them in âstraightâ lines on the assumption that space has no overall curvature, and that we inhabit a gravitationally âflatâ universe, with Euclidian space.
However if space does have the small positive curvature to make it a closed hypersphere then it will act as a vast lens which will effectively magnify or diminish distant objects at various distances as follows:
Hyperspherical lensing = x^2 / sqrt(-x^2 + x) - x
where x = d/L where d = astronomical distance, and L = antipode distance.
Now as L, the antipode distance, (to the other side of the universe), equals about 11 billion light years, the lensing does not become apparent until the astronomical distance becomes a significant fraction of this.
Objects fairly close to the observer do not become significantly lensed; we see nearby galaxies at almost their actual size, allowing for distance, Lensing ~ 0.
Objects at less than half antipode distance become negatively lensed; at about a quarter of antipode distance (~3 billion light years) objects have maximum negative lensing of about -0.1 which reduces them to about 90% of their actual size in the visual field, thus making them appear slightly brighter than one might expect for their distance.
At half antipode distance objects will appear at the expected size and brightness for their distance.
Beyond the half antipode distance objects become positively lensed, and spread out over a greater angle of the visual field, thus making them appear fainter than one might expect for their actual distance.
Towards the antipode distance the lensing effect exponentially increases the angle of the visual field that an object would have in flat space at that distance.
Thus type 1A supernovae beyond the half antipode distance appear fainter than they should for distance estimates based on redshift. This has led to the conclusion that they actually have a greater distance from us than their redshifts suggest and that the hypothesized expansion of the universe has actually accelerated since it began.
The application of this lensing equation to the type 1A supernovae data changes the interpretation of their distances based on brightness measurements, which has led to the hypothesis of a universe with an accelerating expansion and the dubious hypothesis that some form of dark energy propels it.
The huperspherical lensing hypothesis suggests that the curvature of the universe distorts our vision of it like a giant fishbowl to create a vast optical illusion, and that we perhaps need to correct for this.
Notes on the graph.
The green line represents curvature of the hypersphere of the universe from the observer at the point of origin to the antipode at 11 billion light years distance. As light has to follow the hyperspherical geodesic, observers actually see (receive light) along this path.
The red line then shows the imagined line of sight because observers always receive light 'as if' it had come to them in straight lines.
The black line then represents the expected size (red), minus the actual size (green), and shows the amount of lensing at various distances, so objects closer than halfway to the other side of the universe will seem slightly reduced, at the halfway mark they have the proper apparent size (adjusted for distance of course) whilst objects beyond the halfway point become increasingly lensed and spread out over the visual field, thus making them seem fainter.
This has led some cosmologists to conclude that the universe has a bizarrely accelerating expansion, for which they can propose no credible mechanism.
