_{1}

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We initially look at a non singular universe representation as given by Rovelli and Vidotto, in terms of a quantum bounce, via minimum mass quark stars, as a start of how to estimate of entropy and also of the number of operations of an expanding universe. The bench mark used is, to after considering a quark star, to look at the mass of a universe, estimated, and from there, we can obtain the entropy if we look at the Schwartzshield radii of a universe, and then the radii of the universe about 380,000 years after the big bang. In the latter, we show how to get the number of operations as akin to the reasoning used by Seth Lloyd, in 2001, and also from there close with a few comments as to the “naturalness” of heavy Gravity from this formulation of entropy, which is based upon a start of considering what is a Planck star, as far as minimum quantum effects in Black hole physics, and by extension early universe cosmology.

In making the initial assumptions, we will be considering what was brought up by S. Carroll in [

Quote:

You can figure this out by looking at the Friedmann equation, which relates the Hubble parameter to the energy density and the spatial curvature of the universe. The radius of our observable universe is basically the Hubble length, which is the speed of light divided by the Hubble parameter. It’s a straightforward exercise to calculate the amount of mass inside a sphere whose radius is the Hubble length (M = 4πc^{3}H^{−3}/3), and then calculate the corresponding Schwarzschild radius (R = 2GM/c^{2}). You will find that the radius equals the Hubble length, if the universe is spatially flat. Voila!

End of quote.

Our approximation assumes that right after the Pre Planckian regime of space time, that we approach flatness.

In the Pre Planckian space time, we assume that there is a regime of space-time involving a different uncertainty principle than the usual Heisenberg, which we state below.

Start first from [

If we use the following, from the Roberson-Walker metric [

Following Unruth [

Then, the surviving version of Equation (1) and Equation (2) is, then, if

This Equation (4) is such that we can extract, up to a point the HUP principle for uncertainty in time and energy, with one very large caveat added, namely if we use the fluid approximation of space-time [

Then by [

Then,

This above is what is within the Schwartzshield radius, but also within the Pre- Planckian spacetime regime. In place of a traditional black hole, we have a break from the usual uncertainty relationship. I wish to thank the referee whom pointed out that, of course, that the early universe is not a black hole. However as far as early universe models, the following can be referenced. i.e. look at Valev, as of [

Without making too much of it, we will be assuming that in the early universe, the same equivalence shows up, i.e. for the very early universe

Within the Hubble radius, as within the presumed Schwartzhield radius, we are assuming that Equation (7) holds, and that right afterwards, the following holds, i.e. Just outside the Hubble “radius” we will be evaluating we assume that instead of Equation (7) that the usual uncertainty principle holds.

This usual treatment of the uncertainty principle holds for space-time just outside the radius as given in Equation (9) whereas Equation (7) would hold for just within the calculated radius as given by Equation (9). Having said that, we will proceed with the thermodynamics of what happens as to the boundary of space-t9me delineated by Equation (9).

We follow what Ha wrote up [

Then in the spirit of Mukhanov [

Then, we go to the Entropy, and state it is due to calculation given by Kolb and Turner [

Our article will be developed by making sense of the above formalism, and we start by getting the entropy, in its final state, pretty well bench marked, and from there scaling back to determine what entropy should be initially. In doing so, we take some arguments fromBarrau, Rovelli, and Viddotto, [

The quark star argument comes as to giving the cube of a minimum quantum wavelength for quantum gravitational effects and the linkage to quark stars, etc, as commensurate with [

We are taking a Primoridial black hole mass, in Equation (14) to be such that it would be really small, i.e. if the Earth has a 6 times 10^{24} kilogram mass, then its Schwartzhield radius would be 8.7 times 10^{−}^{3} meters. i.e. what we are thinking of is a consortium of mass for tiny black holes of about 3 times 10^{22} Kilograms, i.e. ~1/10 the mass of the Earth, as a start. So then the following roughly hold.

This is assuming that there would be a continual injection of space-matter energy into the universe, possibly utilizing the cyclical cosmological construction, i.e. a variant of Penrose Cyclic conformal cosmology as given in [

The end result, is if we use Equation (13) and also reference Equation (14) above, as a proportionality factor as far as how to obtain entropy is in having the following set up, i.e.

Furthermore, there is a linkage which can be made to Seth Lloyds number of operations, i.e. [

Note the completely different ways of charactering the number of operations, as given by Equation (16) in terms of a linkage to the radii of the universe, at Z ~ 1100, as opposed to the entropy as linked to a radii of the Schwartshield “radii”, as given in Equation (15) and part of the evolution displayed in Equation (14).

In order to do this line of reasoning, the temperature can have the following linkage [

This would then entail making the following identification. i.e. comparing the wavelength of Equation (17) with the quantum wavelength and linking it to Equation (5) so as to then make the following identification. If so, then if Entropy is identifiable with Equation (5) and then we would use the following identification, namely, if [

Also, if a Holographic relationship holds, [

And if [

which in turn may help us understand when the formation of this value occurred, i.e. [

We are supposing that Equation (21) holds at the formation of a Schwartzshield mass of the Universe radius. Also, here is our candidate as to the formation of an initial time step. As given.

Then, up to a point, if the above is in terms of seconds, and N sufficiently large, we could be talking about an initial non zero entropy, along the lines of the number of nucleated particles, at the start of the cosmological era. As given by

Initial entropy would be small, but non zero, and would be affected by

If

With an initial Graviton formation of mass occurring perhaps as early as

This frequency would be though massively redshifted down due to the onset of inflation. Also, our starting point as to wavelength, is due to the considerations given in [^{18} or so meters from the big bang, and is still congruent with [

This work is supported in part by National Nature Science Foundation of China grant No. 11375279.

Beckwith, A.W. (2016) Geddanken Experiment for Quark Star Idea, Quantum Wavelength Limit, Mini- mum Time, and Early Universe Temperature, from First Principles. Journal of High Energy Physics, Gravitation and Cosmology, 2, 478-485. http://dx.doi.org/10.4236/jhepgc.2016.24042