_{1}

^{*}

This investigation sets forth initial conditions for a start of the arrow of time in cosmology based upon the idea that of having initial degrees of freedom set as g
_{*}~ 1000 initially, instead of a maximum value of g
_{*}~ 100-120 for the number of degrees of freedom during the electro weak era.

Recently, Beckwith asked [

Usual treatments of the arrow of time, i.e. the onset of entropy. The discussion below makes the point that expansion of the universe in itself does not “grow” entropy.

The entropy density s of a radiation field of temperature T is s ~ T^{3}. The entropy S in a given comoving volume V is S = sV. Since the commoving volume V increases as the universe expands, we have V ~ R^{3}. And since the temperature of the microwave background goes down as the universe expands: T ~ 1/R, we have the result that the entropy of a given comoving volume of given space S ~ R^{−3} * R^{3} = constant. Thus the expansion of the universe by itself is not responsible for any entropy increase. There is no heat exchange between different parts of the universe. The expansion is adiabatic and isentropic: d S expansion = 0. i.e. a process has to be initiated in order to start entropy production.

This discussion above is to emphasize the importance of an initial process for the onset and the growth of entropy. We will initiate candidates for making sense of the following datum.

To measure entropy in cosmology we can count photons. If the number of photons in a given volume of the universe is N, then the entropy of that volume is S ~ kN where k is called here Boltzmann’s constant.

Note that Y. Jack Ng. has [

This counting procedure is different from traditional notions. To paraphrase them, one can state that “The reason why entropy is increasing is because there are stars in that ‘box’ (unit of phase space used for counting contributions to entropy). Hydrogen fuses to helium and nuclear energy is transformed into heat.” i.e. the traditional notion would be akin to heat production due to, initially start BBN nucleosynthesis, and then, frankly, star production/nuclear burning. i.e. one would need to have nuclear processes to initiate heat production. This idea of heat production is actually similar to setting S ∝ T 3 , with heat production due to either BBN/ hydrogen burning leading to an increase in temperature, T. In this manuscript, we make use of, if S ≡ [ E − μ N ] / T → S ∝ T 3 by setting the chemical potential μ → 0 with initial entropy S ~ 10 5 at the beginning of inflation. This entails, as we will detail, having increased number of degrees of freedom, initially, with re setting the degrees of freedom of about g ∗ ~ 100 - 120 of the electro weak era, to g ∗ ~ 1000 at the onset of inflation, I.e. what will be examined will be the feasibility of the following: S ≡ [ E − μ N ] / T → μ → 0 S ∝ T 3 ≈ n , with n an initial “quantum unit” count in phase space of Planckian dimensions, where S ~ 10 5 at the beginning of inflation. Let us now look at how to initiate such a counting algorithm if one is looking at, say, highly energized gravitons, initially, as part of a counting “algorithm”.

As suggested earlier by Beckwith [

m graviton | RELATIVISTIC < 4.4 × 10 − 22 h − 1 eV / c 2 ⇔ λ graviton ≡ ℏ m graviton ⋅ c < 2.8 × 10 − 8 meters (1.1)

For looking at the onset of creation, with a bounce; if we look at ρ max ∝ 2.07 ⋅ ρ planck for the quantum bounce with a value put in for when ρ planck ≈ 5.1 × 10 99 grams/ meter^{3}, where [

E e f f ∝ 2.07 ⋅ l Planck 3 ⋅ ρ planck ~ 5 × 10 24 GeV (1.2)

Then, taking note of this, one is obtaining having a scaled entropy of S ≡ E / T ~ 10 5 when one has an initial Planck temperature T ≈ T Planck ~ 10 19 GeV . One needs, then to consider, if the energy per given graviton is, if a frequency ν ∝ 10 10 Hz and E graviton-effective ∝ 2 ⋅ h v ≈ 5 × 10 − 5 eV , then [

S ≡ E e f f / T ~ [ 10 38 × E graviton-effective ( v ≈ 10 10 Hz ) ] / [ T ~ 10 19 GeV ] ≈ 10 5 (1.3)

Having said that, the [ E graviton-effective ∝ 2 ⋅ h v ≈ 5 × 10 − 5 eV ] is 10^{22} greater than the rest mass energy of a graviton if E ~ m graviton [ red-shift ~ 0.55 ] ~ ( 10 − 27 eV ) grams is taken when applied to Equation (1.2) above.

A typical value and relationship between an inflaton potential V [ ϕ ] , and a hubble parameter value, H is [

H 2 ~ V [ ϕ ] / m Planck 2 (1.4)

Also, if we look at the temperature T ∗ occurring about the time of the Electro weak transition, if T ≤ T ∗ when T ∗ = T c was a critical value, (of which we can write v ( T c ) / T c > 1 , where v(T_{c}) denotes the Higgs vacuum expectation value at the critical temperature T_{c}., i.e. v ( T c ) / T c > 1 according to C. Balazc et al. (2005) [

H ~ 1.66 ⋅ [ g ˜ ∗ ] ⋅ [ T 2 / m Planck 2 ] (1.5)

Here, the factor put in, of g ˜ ∗ is the number of degrees of freedom. Kolb and Turner [

S ~ 3 m Plank 2 [ H = 1.66 ⋅ g ˜ ∗ ⋅ T 2 / m planck ] 2 T ~ 3 ⋅ [ 1.66 ⋅ g ˜ ∗ ] 2 T 3 (1.6)

Should the degrees of freedom hold, for temperatures much greater than T ∗ , and with g ˜ ∗ ≈ 1000 at the onset of inflation, for temperatures, rising up to, say T ~ 10^{19} GeV, from initially a very low level, pre inflation, then this may be enough to explain how and why certain particle may arise in a nucleated state, without necessarily being transferred from a prior to a present universe.

Furthermore, if one assumes that S ∝ T 3 [

H. de La Vega, in conversations with the author in Colmo, Italy, 2009 [

n ~ sinh 2 [ m 0 η 1 ] (1.7)

If so, then one can also ask if there is a linkage between the initial conditions, as pertinent to early inflation, and Beckwith’s model of re acceleration of the universe one billion years ago.

The following is speculative, and if confirmed through additional research would be a major step toward a cosmological linkage between initial inflation, and re acceleration of the universe one billion years ago [

Potential in both cases chaotic inflation of the type [^{ }

V = m ↔ 2 Φ ∗ Φ (1.8)

The “mass” term would be, then, as Beckwith understands it, for early universe versions of the Friedman equation

m ↔ ≈ 3 8 ⋅ [ 3 H 2 4 π G | time ~ 10 − 35 sec + 3 H 2 4 π G | time ~ 10 − 44 sec ] (1.9)

Furthermore, its bound would be specified by having

| m ↔ | ≤ [ l 2 4 ] (1.10)

The term, l would be an artifact of five dimensional space time, as provided in a metric as given by Maarten’s [

d S 2 | 5 - dim = l 2 z 2 ⋅ [ η u v d x μ d x v + d z 2 ] (1.11)

The 2^{nd} scalar fields as Yurov [^{nd} inflation, which Beckwith represents [

φ 0 , − = 2 / 3 ⋅ m ↔ ⋅ [ t 1 st-EXIT ~ 10 − 35 sec ] (1.12)

And

φ + = [ φ 0 , + 3 − 3 / 2 ⋅ 3 M 2 t m ↔ ] 1 / 3 (1.13)

As Beckwith sees it, making a full linkage between Yurov’s formalism [

H 2 = 1 6 ⋅ [ φ ˙ 2 + m ↔ 2 φ 2 + M 2 φ 2 ] ↔ ( κ ˜ 2 3 [ ρ + ρ 2 2 λ ] ) + m a 4 (1.14)

As well as having:

H ˙ = V − 3 H ↔ H ˙ ≅ 2 m a 4 (1.15)

The left hand side of both Equation (1.14) and Equation (1.15) are Yurov’s [

3 H 2 4 π G ≫ V ( t ) | time ~ 10 − 44 sec (1.16)

i.e. that the potential energy, V, of initial inflation is initially over shadowed by the contributions of the Friedman equation, H, at the onset of inflation.

We should note, that the potential energy as stated would be assuming that Equation (1.16) has consistency with Equation (1.5), for very large temperatures.

A way to obtain traces of information exchange, from prior to present universe cycles is finding a linkage between information and entropy. If such a parameterization can be found and analyzed, then Seth Lloyd’s [

I = S total / k B ln 2 = [ # operations ] 3 / 4 = [ ρ ⋅ c 5 ⋅ t 4 / ℏ ] 3 / 4 (1.17)

could be utilized as a way to represent information which can be transferred from a prior to the present universe. The question is to ask, if Equation (1.17) permits a linkage of gravitons as information carriers, can there be a linkage of information, in terms of the appearance of gravitons in the time interval of, say 0 < t < t Planck either by vacuum nucleation of gravitons/information packets. Appropriate values/inputs into ρ are being considered along the lines of graviton mass/contributions along the lines brought up in this paper already.

An alternative to Equation (1.7) if one sees no way of implementing what Ng. suggested via his infinite quantum statistics [

n f = [ 1 / 4 ] ⋅ [ v ( a initial ) v ( a ) − v ( a ) v ( a final ) ] (1.9)

As well as, if h 0 ~ 0.75

Ω g w ( v ) ≅ 3.6 h 0 2 ⋅ [ n f 10 37 ] ⋅ ( v 1 kHz ) 4 (1.10)

If we take into consideration having a ~ a final , then Equation (1.14) above will, in most cases be approximately

n f = [ 1 / 4 ] ⋅ [ v ( a initial ) v ( a ) − 1 ] ~ [ 1 / 4 ] ⋅ [ v ( a initial ) v ( a ) ] (1.11)

For looking at Ω g ≈ 10 − 5 - 10 − 14 , with Ω g ≈ 10 − 5 in pre-big bang scenarios, with initial values of frequency set for v ( a initial ) ≈ 10 8 - 10 10 Hz, as specified by Grishkuk [

S ≈ n (1.12)

using Equation (1.10) directly, or if S ≠ n , using Equation (1.10), but instead uses S ∝ T 3 , with temperature rapidly increasing from a low value to T Planck ≈ 10 19 GeV in about a time interval during the onset of inflation, for the beginning of the arrow of time, in cosmology. Beckwith views determining if the degrees of freedom initially could go as high as g ˜ ∗ ≈ 1000 or even higher even if T ~ 10^{19} GeV as essential in determining the role of S ∝ T 3 , as temperatures go from an initial low point, to T ~ 10^{19} GeV for understanding the role of thermal heat transfer in the arrow of time issue.

Note, very importantly, any coupling between Equation (1.5) and Equation (1.16) probably requires that there is no initially low temperature behavior, pre-inflation, prior to the rise of temperature of the Quantum Planck Temperature of ~10^{19} GeV.

In a colloquium presentation done by Dr. Smoot in Paris [

1) Physically observable bits of information possibly in present Universe―10^{180}

2) Holographic principle allowed states in the evolution/development of the Universe―10^{120}

3) Initially available states given to us to work with at the onset of the inflationary era―10^{10}

4) Observable bits of information present due to quantum/statistical fluctuations―10^{8}

Our guess is as follows. That the thermal flux from a prior to the present universe may account for up to 10^{10} bits of information. These could be transferred from a prior universe to our present, and that there could be, perhaps 10^{120} minus 10^{10} bytes of information temporarily suppressed during the initial changing of fermion states of matter to a bosonic phase of matter right at the onset of the big bang itself.

Then after the degrees of freedom dramatically drops during the beginning of the descent of temperature from about T ≈ 10 32 Kelvin to at least three orders of magnitude less, as we move out from an initial red shift

z ≈ 10 25 (1,13)

To [

T ≈ ε V × 10 28 Kelvin ~ T Hawkings ≅ ℏ ⋅ H initial 2 π ⋅ k B (1.14)

Whichever model we can come up with that does this is the one we need to follow, experimentally. And it gives us hope in confirming if or not we can eventually analyze the growth of structure in the initial phases of quantum nucleation of emergent space time [

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

The authors declare no conflicts of interest regarding the publication of this paper.

Beckwith, A.W. (2018) Initial Conditions for Defining an Arrow of Time at the Start of Inflation? Journal of High Energy Physics, Gravitation and Cosmology, 4, 787-795. https://doi.org/10.4236/jhepgc.2018.44044