_{1}

^{*}

The purpose of this paper is to show how one can use the FSC model of gravitational entropy to calculate cosmic radiation temperature anisotropy for any past cosmic time
*t* since the Planck scale. Cosmic entropy follows the Bekenstein-Hawking definition, although in the correct-scaling form of
, which scales 60.63 logs of 10 from the Planck scale. In the FSC model, cosmic radiation temperature anisotropy
*A _{t}* = (

*t/t*

_{o}). The derived past anisotropy value can be compared to current co-moving anisotropy defined as unity (

*t*). Calculated in this way, current gravitational entropy and temperature anisotropy have maximum values, and the earliest universe has the lowest entropy and temperature anisotropy values. This approach comports with the second law of thermodynamics and the theoretical basis of the Sachs-Wolfe effect, gravitational entropy as defined by Roger Penrose, and Erik Verlinde’s “emergent gravity” theory.

_{o}/t_{o}In the July 2018 issue of Journal of Modern Physics, the paper entitled, “How the CMB Anisotropy Pattern Could Be a Map of Gravitational Entropy” [

Furthermore, the FSC CMB anisotropy paper and its companion paper [

The purpose of this paper is to show how the FSC CMB anisotropy paper opens the way for a definition of cosmic radiation temperature anisotropy at any cosmic temperature T, whether it is in the form of “Universal Temperature” T_{U}, as defined in FSC reference [

The rationale for generalizing the radiation temperature anisotropy calculations to be presented herein can be summarized as follows: Sachs and Wolfe used a gravitational redshift theoretical argument that radiation temperature anisotropy could be a result of inhomogeneous gravitational particle clustering. So, while they applied their argument in anticipation of refined CMB anisotropy observations, there was nothing particularly special about the CMB emission event with respect to their gravitational redshift argument. The recombination/decoupling event concerned photon emission, making the event observable. However, the coupling of electrons with protons to form the first hydrogen atoms should have no impact whatsoever on gravitational particle clustering. As explained in reference [

The reader will maximally benefit in reading the present paper after first reading FSC references [

1) The cosmic model is an ever-expanding sphere such that the cosmic horizon always translates at speed of light c with respect to its geometric center at all times t. The observer is operationally-defined to be at this geometric center at all times t .

2) The cosmic radius R t and total mass M t follow the Schwarzschild formula R t ≅ 2 G M t / c 2 at all times t .

3) The cosmic Hubble parameter is defined by H t ≅ c / R t at all times t .

4) Incorporating our cosmological scaling adaptation of Hawking’s black hole temperature formula, at any radius R_{t}, cosmic temperature T t is inversely proportional to the geometric mean of cosmic total mass M t and the Planck mass M p l . R p l is defined as twice the Planck length (i.e., as the Schwarzschild radius of the Planck mass black hole). With subscript t for any time stage of cosmic evolution and subscript p l for the Planck scale epoch, and, incorporating the Schwarzschild relationship between M t and R t ,

k B T t ≅ ℏ c 3 8 π G M t M p l ≅ ℏ c 4 π R t R p l { M t ≅ ( ℏ c 3 8 π G k B T t ) 2 1 M p l ( A ) R t ≅ 1 R p l ( ℏ c 4 π k B ) 2 ( 1 T t ) 2 ( B ) R t T t 2 ≅ 1 R p l ( ℏ c 4 π k B ) 2 ( C ) t ≅ R t c ( D ) } (1)

5) Total entropy of the cosmic model follows the Bekenstein-Hawking black hole formula [

S t ≅ π R t 2 L p 2 (2)

The first two assumptions are based upon a literal interpretation of the Hawking-Penrose conjecture as it would pertain to a smoothly-expanding Schwarzschild black hole. The third assumption (Hubble parameter) treats maximally redshifted radial photons at the cosmic model horizon as moving with speed of light c relative to the geometric center at a distance of horizon radius R_{t}. This is a stipulation of relativity. The fourth assumption is a cosmic temperature scaling assumption. While it shows similarity to the static Hawking black hole temperature formula, the FSC cosmic model is treated as scaling in Planck mass increments. This allows for dynamic cosmic expansion modeling from the Planck scale epoch. Finally, the fifth assumption utilizes the Bekenstein-Hawking entropy definition, which seems appropriate for a model of the Hawking-Penrose conjecture. The numerous observational correlations of FSC are given in references [

Equation (11) from reference [^{11.58} seconds at the beginning of the recombination/decoupling epoch (3000 K). This equation is repeated here as Equation (3):

T 2 t s = 3.426525959553982 × 10 18 K 2 ⋅ s (3)

wherein cosmic temperature T is in degrees K and cosmic time t_{s} is in seconds since the Planck epoch. For reasons given in the reference [^{12.6} seconds (at a temperature of 924.63 K) after the Planck epoch. The Planck epoch is the cosmic time of the Planck-scale universe and is often considered to be the approximate moment of the “Big Bang” in standard cosmology. FSC reference [

S = c π L p t (4)

showing the direct proportionality relationship between gravitational entropy S and cosmic time t. Speed of light c and Planck length L_{p} are assumed to be constants over cosmic time. Thus, if we normalize the proportionality constant to unity and operationally define S in terms of seconds,

S = 10^{−42.965} at 10^{−42.965} second of cosmic time at Planck temperature 5.6 × 10^{30} K;

S = 10^{11.58} at 10^{11.58} seconds of cosmic time at CMB beginning temperature 3000 K;

S = 10^{12.6} at 10^{12.6} seconds of cosmic time at end-of-decoupling temperature 924.63 K;

S = 10^{17.66} at 10^{17.66} seconds of cosmic time at current temperature 2.72548 K.

The above CMB emission epoch gravitational entropies (10^{11.58} and 10^{12.6}) can then be related to current cosmic entropy (10^{17.66}), in ratio form, as follows:

[ S at the beginning of CMB emission]/[ S at current time] = 8.25 × 10^{−7} (0.825 × 10^{−6});

[ S at the ending of CMB emission]/[ S at current time] = 8.69 × 10^{−6} (0.869 × 10^{−5}).

These derived values of the beginning and ending CMB radiation temperature anisotropy are the same as those calculated in reference [^{−5}, as well as with the WMAP and Planck report anisotropy estimates of “approximately” 10^{−5}.

Thus, if this process for calculating the CMB radiation temperature anisotropy as gravitational entropy ratios can be generalized and extended all the way back to the Planck epoch (and conceivably beyond), the formula for doing such calculations is:

A t = S t / S o = t / t o (5)

wherein A_{t} is the radiation temperature anisotropy at cosmic time t, S t is the gravitational entropy (cosmic entropy) at cosmic time t, S o is the gravitational entropy (cosmic entropy) at current time t_{o}, and t/t_{o} is the ratio of these time values. This radiation temperature anisotropy formula can also be substituted by other FSC parameters correlated to S t , as seen in reference [

A t = R t / R o (6)

wherein R_{t} is the cosmic radius at cosmic time t and R_{o} is the current observed cosmic radius.

A t = T o 2 / T t 2 (7)

wherein T_{o} is the current cosmic radiation temperature in degrees Kelvin (2.72548 K) and T_{t} is the cosmic radiation temperature in degrees Kelvin at cosmic time t.

A t = T U o / T U t (8)

wherein T_{Uo} is the current cosmic radiation temperature in Universal Temperature units and T_{Ut} is the cosmic radiation temperature in Universal Temperature units at cosmic time t. As indicated in reference [_{Ut} is defined by T U t = T t 2 .

Based upon this calculation method, the following

_{U}. CMB starting and ending values are denoted by the white circles.

Extensive comparisons between FSC and standard inflationary cosmology are given in reference [

for predicting and calculating radiation temperature anisotropy at any given cosmic time. Thus, standard model practitioners can only guess as to the possible reasons why the CMB anisotropy pattern has a dT/T RMS anisotropy value of 18 micro-Kelvins/2.725 K = 0.66 × 10^{−5}, as measured in the COBE DMR experiment [Wright (1996)]. Current speculation seems to favor a CMB pattern produced by a “quantum fluctuation” Big Bang event smoothed out by cosmic inflation and splayed out across the sky. It is even proposed that a “quantum fluctuation” CMB pattern somehow must provide important clues to the nature of gravity at the quantum scale (i.e., “quantum gravity”).

In contrast, the FSC model, as detailed in references [_{o}/t_{o}) = 0].

Note that, by this definition of current co-moving anisotropy, current gravitational entropy and temperature anisotropy have maximum values, and the earliest universe has the lowest entropy and temperature anisotropy values. This approach comports with the second law of thermodynamics and the theoretical basis of the Sachs-Wolfe effect, gravitational entropy as defined by Roger Penrose, and Erik Verlinde’s “emergent gravity” theory.

The purpose of this paper has been to show how one can use the FSC model of gravitational entropy to calculate cosmic radiation temperature anisotropy for any past cosmic time t since the Planck scale. In the FSC model, cosmic radiation temperature anisotropy A_{t} = (t/t_{o}). The derived past anisotropy value can be compared to current co-moving anisotropy defined as unity (t_{o}/t_{o}). Calculated in this way, current gravitational entropy and temperature anisotropy have maximum values, and the earliest universe has the lowest entropy and temperature anisotropy values. This approach comports with the second law of thermodynamics and the theoretical basis of the Sachs-Wolfe effect, gravitational entropy as defined by Roger Penrose, and Erik Verlinde’s “emergent gravity” theory.

This paper is dedicated to the late Dr. Stephen Hawking and Dr. Roger Penrose for their groundbreaking work on black holes and their possible application to cosmology. Dr. Tatum also thanks Dr. Rudolph Schild of the Harvard Center for Astrophysics for his past support and encouragement.

The author declares no conflicts of interest regarding the publication of this paper.

Tatum, E.T. (2018) Calculating Radiation Temperature Anisotropy in Flat Space Cosmology. Journal of Modern Physics, 9, 1946-1953. https://doi.org/10.4236/jmp.2018.910123