### Hypersphere Cosmology in a Nutshell

Abstract. Hypersphere Cosmology presents an alternative to the expanding universe of the standard LCDM Big Bang model. In Hypersphere cosmology, a universe finite and unbounded in both space and time has a small positive curvature and a form of rotation. The positive curvature appears as an acceleration and explains the red-shifting and dimming of light from distant galaxies.

1 & 6 together constitute an Anti-Singularity Theorem.

7) Hyperspherical Lensing. Observers within a Glome Hypersphere will observe it in an apparent Stereographic Projection.

Apparent Magnitudes and Redshifts of Type 1A Supernovae taken from the line of best fit.

Where Apparent Distance means the distance obtained from calculating Flux from Apparent Magnitude and then Apparent Distance from the inverse square principle.

This hyperspherical lensing equation matches the observational results out to redshift 1 at 50% antipode distance and explains the dimming of supernovae without recourse to the hypothesis of dark energy. Thereafter the equation predicts a considerably greater dimming with full extinction at antipode distance.

Stereographic projection of flux for an observer at O.

Hyperspherical Lensing arises through the Stereographic Projection of the Flux of a Glome. This causes distant objects to have reduced brightness and large apparent distances. The antipode would have an infinite apparent distance based on apparent flux.

The Lensing Formula when applied to the Actual Size/Angular Diameter of a far distant galaxy gives its Apparent Size/Angular Diameter. This will decrease up to Z = 1, and then increase thereafter, tending toward infinity as Z does.

10) Pixilation. If the Beckenstein-Hawking conjecture applies to the hypersphere and we can equate entropy with information, then the universe has only one bit per the cube root of U Planck volumes or one bit per Planck length.

11) Commentary. The Hypersphere Cosmology seeks to replace the standard LCDM Big Bang cosmological model with something approaching its exact opposite.

In HC the universe does not expand, it remains as a finite but unbounded structure in both space and time with a spatial and temporal horizon of about 13bn light years and 13bn yr. The HC universe does not collapse because its major gravitationally bound structures all rotate back and forth to their antipode positions over a 26bnyr period about randomly aligned axes, giving the universe no overall angular momentum.

The small positive spacetime curvature of the Glome type Hypersphere of the universe has many effects, it redshifts light traveling across it, it prevents smaller hyperspheres or singularities forming within black holes and it gradually causes black holes to eject mass and energy.

In HC the CMBR simply represents the temperature of the universe, and it consists of redshifted radiation that has reached thermodynamic equilibrium with the thin intergalactic medium***

The universe will appear to observers as having the flux from distant sources in stereographic projection due to the geometry of the small positive spacetime curvature.

In terms of the enormously dimmed flux from very distant sources the antipode will appear to lie an infinite distance away, and beyond direct observation, even though the antipode of any point in the universe lies about 13bnlyr distant.

The antipode thus in a sense plays the anti-role of the Big Bang Singularity in LCDM cosmology. We can never observe either, but instead of an apparently infinitely dense and infinitely hot singularity a finite distance away in universe undergoing an accelerating expansion in space and time, Hypersphere Cosmology posits an Antipode that will appear infinitely distant in space and time and infinitely diffuse and cold, even though actual conditions at the antipode of any point will appear broadly similar on the large scale for any observer anywhere in space and time within the hypersphere.

Both HC and LCDM-BB can both model many of the important cosmological observations but in radically different ways. HC has more economical concepts, as a small positive spacetime curvature alone can account for redshift without expansion, the dimming of distant sources of light without an accelerating expansion driven by dark energy, and it also offers a singularity free universe. Hypersphere cosmology does not presently address the anomalous galactic rotation curves which has led to the hypothesis of dark matter, but it seems more likely that some form of modified Newtonian dynamics will eventually explain them.

Neither model really explains where the universe ‘came from’, but we have no reason to regard non-existence as somehow more fundamental than existence.

The evidence for one-way cosmological evolution remains mixed. The entropy of a vast Glome Hypersphere may remain constant as a function of its hypersurface area. On the very large scale the universe needs only the ability to break neutrons to maintain constant entropy. Very distant parts of the universe appear to contain structures far to large to have evolved in the BB timescale.

Hypersphere addenda. Notes and Further Speculations.

2) Gödel derived an exact solution of General Relativity in which ‘Matter everywhere rotates relative to the compass of inertia with an angular velocity of twice the square root of pi times the gravitational constant times the density’. This solution became largely ignored because of the apparent lack of observational evidence for an axis of rotation. However, in a hypersphere the galaxies can rotate back and forth to their antipode positions about randomly aligned axes (most probably around the circles of a Hopf Fibration of the hypersphere, thus resulting in a universe with no net overall angular momentum. Such a ‘Vorticitation’ would stabilise a hypersphere against implosion under its own gravity and result in the galaxies rotating at a mere fraction of an arcsecond per century – well below levels that we can currently observe.

3) Mach’s Principle can only work in a universe of constant size and density. Strong evidence exists to show that the gravitational constant and inertial masses have remained constant for billions of years.

***The Cosmic Microwave Background Radiation (CMB/CMBR) may consist of redshifted trans-antipodal light that has reached thermodynamic equilibrium with the thin intergalactic medium, but Hyperspherical Lensing raises another possibility: -

Widely spatially separated observers within the universe may see a quite different CMBR or perhaps none at all, because the CMBR originates from near their antipodes.

Consider that a galaxy like our own lies near to our antipode point. Such a galaxy will have a spherical Hot Gas Halo extending for several hundred thousand light years around it, accounting for about half of its mass, and at a temperature of about I million Kelvin.

The angular size of such a distant galaxy in Euclidean space would come out at a paltry e^{21}/e^{26} radians, about 1/10,000^{th} of a radian.

However, lensing at a redshift of 300,000 would give it an apparent angular size of about 6 radians (it would thus fill the whole sky) and reduce its temperature to around 3 Kelvin.

The lower temperature light from the starry part of the distant galaxy would become redshifted far beyond observability, but all the light from the Hot Gas Halo would end up here, coming in from every direction, providing a microwave background radiation that remains location dependent rather than cosmically uniform.

Created by Peter J Carroll

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