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The Carmeli Cosmological Special Relativity theory (CSR) is used to study the universe at early times after the big bang. The universe temperature vs. time relation is developed from the mass density relation. It is shown that CSR is well suited to analyze the nucleosynthesis of the light elements up to beryllium, equivalent to the standard model.

Although Carmeli described the temperature of the early universe using his own Cosmological Special Relativity (CSR) theory [

In CSR, the redshift z of light emitted in the past at time t' when the size of the universe had a radius of R(t') is defined by

1 + z = R 0 R ( t ′ ) , (1)

where R_{0} is the radius of the universe at the present epoch of time. It can be shown [

1 + z = 1 + t ′ / τ 1 − t ′ / τ , (2)

where τ is the Hubble-Carmeli time constant, the age of the universe in Carmeli cosmology. Combining (1) and (2), in CSR, a distance R_{0} measured by an observer at the present epoch time t ′ = 0 , transforms to the distance R(t') for a time t' in the past given by

R ( t ′ ) = R 0 1 − t ′ / τ 1 + t ′ / τ , (3)

with the limit into the past being the big bang time t ′ = τ . Note that in the Carmeli cosmological theory, time is measured from the present at t ′ = 0 toward the past at time t ′ > 0 . The Hubble constant H 0 ≈ 1 / τ . If we take R 0 = φ 0 ( t ′ ) c τ to be the radial distance to the edge of the universe, where c is the speed of light in vacuum and φ_{0}(t') is a scaling factor, then R(t') from (3) is the universe radius at a past time t'.

For nucleosynthesis studies it is conventional to measure time from the big bang toward the present. Transforming (3) by the change of variable t = τ – t ′ , we have for the universe radius

R ( t ) = ϕ 0 ( t ) c τ 2 τ / t − 1 , (4)

wheret is the time measured from the big bang when t = 0 toward the present when t = τ and the scaling factor

ϕ 0 ( t ) = T 0 T 0 past if t < 1.98 × 10 13 sec . , ϕ 0 ( t ) = 1 otherwise . (5)

where T 0 ≈ 2.73 K is the present CMB temperature and T 0past ≈ 14 .42 K is the past radiation field temperature, for which we will give a derivation subsequently. Note that the cutoff time t c = 1.98 × 10 13 sec . is an approximate time representing the recombination time when the universe temperature was 0.3 eV, and this cutoff time should expect to be tweaked when studying that period.

The mass density ρ(t) of the universe is defined to be composed of a relativistic mass density ρ_{r}(t) and a non-relativistic mass density ρ_{m}(t), where the total mass density ρ ( t ) = ρ r ( t ) + ρ m ( t ) .

We define the non-relativistic mass density in the usual way,

ρ m ( t ) = M m V ( t ) , (6)

where M_{m} is the total non-relativistic (rest) mass of the universe and V(t) is the volume of the universe. The non-relativistic mass is defined by

M m = g t M = g t f m ( τ c 3 4 G ) , (7)

where M = f m τ c 3 / 4 G is the total mass of the universe and f_{m} and g_{t} are coefficients which depend on the baryon to photon number density ratio η and the baryon number density n_{B} at the present time. Expressions for these constant coefficients will be shown to be given by

g t = 1 ( ζ k B T 0 / η m p c 2 ) + 1 , (8)

f m = 2 Ω B [ ( ζ k B T 0 / η m p c 2 ) + 1 ] , (9)

where ζ ≈ 2.70 is the black body average energy coefficient, G is Newton’s gravitational constant, k_{B} is Boltzmann’s constant, T 0 ≈ 2.73 K is the present CMB temperature and the baryon density parameter Ω B = m p n B / ρ c , where m_{p} is the proton mass and ρ c = 3 / 8 π G τ 2 is the critical mass density. We use the standard value τ = 4.28 × 10 17 s . For η ≈ 6.61 × 10 − 10 and Ω B ≈ 0.046 , we have g t ≈ 0.999 and f m ≈ 0.093 .

The relativistic mass is defined by

M r ( t ) = f m ( 1 − g t ) ( τ c 3 4 G ) ( c τ R ( t ) ) . (10)

The volume for Euclidean flat space is given by

V ( t ) = 4 3 π R 3 ( t ) . (11)

The non-relativistic mass density (6), using (7) and (11) is given by

ρ m ( t ) = M m V ( t ) = ( f m g t ϕ 0 3 ( t ) ) ( 3 16 π G τ 2 ) [ ( 2 τ / t ) − 1 ] 3 / 2 . (12)

The relativistic mass density ρ_{r}(t), using (10) and (11), is given by

ρ r ( t ) = M r ( t ) V ( t ) = ( f m ( 1 − g t ) ϕ 0 4 ( t ) ) ( 3 16 π G τ 2 ) [ ( 2 τ / t ) − 1 ] 2 . (13)

The total mass density parameter is given by the sum of (12) and (13) divided by the critical density ρ_{c},

Ω ( t ) = ρ r ( t ) + ρ m ( t ) ρ c = [ f m ( 1 − g t ) ϕ 0 4 ( t ) ] [ ( 2 τ / t ) − 1 ] 2 ( 1 + ϕ 0 ( t ) [ g t / ( 1 − g t ) ] [ ( 2 τ / t ) − 1 ] 1 / 2 ) . (14)

Assuming that the universe expands adiabatically, the relativistic density ρ_{s}(T) of the universe is related to the temperature by [

ρ s ( T ) = ( π 2 g a ( T ) 30 ) T 4 , (15)

where g_{a}(T) is the effective number of degrees of freedom for the particles involved in the annihilation process at temperature T, where T is in MeV. Take the density (13), scale it and convert it from ergs to MeV and equate it to (15), yielding,

( ζ M e V 8 ) ( ϕ 0 4 ( t ) f m ( 1 − g t ) ) ρ r = ( 3 ζ M e V 32 π G ) ( 1 t − 1 2 τ ) 2 = ( π 2 g a ( T ) 30 ) T 4 , (16)

where M e V ≈ 6 .24 × 10 5 MeV ⋅ erg − 1 . Equation (16) can be put into the form of time (seconds) since the big bang as a function of temperature (MeV),

t ( T ) = 2 τ 2 τ ( 16 π 3 G g a ( T ) / 45 ) ( 1 / ζ M e V ) T 2 + 1 . (17)

Equation (16) can also be put into the form of temperature (MeV) as a function of time (seconds) since the big bang,

T ( t ) = ( 45 ζ M e V 16 π 3 G g a ( T ) ) 1 / 4 ( 1 t − 1 2 τ ) . (18)

Setting g a ( T ) = 3.363 (after e+, e− annihilation when the temperature T < 500 keV and the neutrinos have decoupled), setting t = τ and ignoring any further reactions such as the recombination period when the temperature was around 0.3 eV, (18) gives

T 0past T ( τ ) ≈ 14.42 K , (19)

which we used in (5).

Since ( 1 / 2 τ ) ≪ ( 1 / t ) at early times, it can be neglected so that (18) can be written more simply

T 2 ( t ) = ( 2.4255 3.363 ) ( 1 t ) , (20)

which is equivalent to the expression in [

T 2 ( t ) = ( 2.4 g a ) ( 1 t ) , (21)

for g a = 3.363 .

The equivalence of the Carmeli big bang temperature model (20) with the standard model (21), implies that light element reaction analysis using Carmeli’s model will agree with the standard model nucleosynthesis results.

Expansion rate equation H(t)

We now take the time rate of change d R ( t ) / d t of the universe radius given by (4) relative to R(t), obtaining the expansion rate H(t),

H ( t ) = ( 1 R ( t ) ) d d t R ( t ) = ( 1 R ( t ) ) d d t ( R 0 2 τ / t − 1 ) = ( 1 2 ) [ 1 t + 1 2 τ − t ] ≈ 1 2 t (22)

where the approximation is good for t ≪ τ as is the case for BBN. This is equivalent to the standard Friedmann-Lemaître-Robertson-Walker expansion rate for Euclidean space (zero curvature) [

Derivation of f_{m} and g_{t}

We derive expressions for f_{m} and g_{t} in terms of the baryon to photon number ratio η ( t ) = n B ( t ) / n γ ( t ) , where n_{B}(t) is the baryon number density and n_{γ}(t) is the photon number density, and where n B ( t ) = ρ m ( t ) / m p , where m_{p} is the proton rest mass and n γ ( t ) = ρ r ( t ) c 2 / ζ k B T ( t ) , where ζ k B T ( t ) is the average energy of the photons in the black body radiation field at temperature T(t). Using (12) and (13) we have

n B ( t ) = ρ m ( t ) m p = ( f m g t ϕ 0 3 ( t ) m p ) ( 3 16 π G τ 2 ) [ ( 2 τ / t ) − 1 ] 3 / 2 , (23)

n γ ( t ) = ρ r ( t ) c 2 ζ k B T ( t ) = [ f m ( 1 − g t ) ϕ 3 ( t ) ζ k B T 0 ] ( 3 c 2 16 π G τ 2 ) [ ( 2 τ / t ) − 1 ] 3 / 2 , (24)

η = n B ( t ) n γ ( t ) = ζ k B T 0 g t m p c 2 ( 1 − g t ) , (25)

where T(t) is the temperature (K) at time t since the big bang and is given by

T ( t ) = T 0 ( c τ R ( t ) ) = ( T 0 ϕ 0 ( t ) ) ( 2 τ / t ) − 1 , (26)

where ϕ 0 ( t ) is given by (5).

From (24) we can solve for g_{t},

g t = 1 ( ζ k B T 0 / η m p c 2 ) + 1 _{.} (27)

Using (23) for n_{B}(τ) at time t = τ and (27) for g_{t}, we can solve for f_{m},

f m = ( n B m p g t ) ( 16 π G τ 2 3 ) = 2 Ω B [ ( ζ k B T 0 / η m p c 2 ) + 1 ] , (28)

where n B = n B ( τ ) and Ω B = m p n B / ρ c .

We have shown that CSR is quite adequate for a development of a nuclear physical analysis of the very early moments of the universe, from 10^{−3} sec. to 10^{3} sec. The modeling assumes two possible final temperatures for the universe, T 0past ≈ 14 .42 K , which is the temperature the universe would attain at the present epoch if there were no recombination at time t c ≈ 1.98 × 10 13 sec . , and T 0 ≈ 2 .73 K , which is the present CMB temperature including the recombination period when the temperature was about 0.3 eV (3500 K). The details of the transition in temperature at the recombination period are not covered in this report, but we should expect some modifications to the temperature profile when that analysis is carried out.

Consider the neutron to proton ratio using CSR based on [

( n p ) = exp ( − Q T ) , (29)

where n is the number of neutrons, p is the number of protons, Q = m n − m p = 1.293 MeV , where m n and m p are the masses of the neutron and proton, respectively, and T is the temperature in MeV. The neutron mean lifetime τ n ≈ 889.1 sec . (Hikasa, et al. 1992.)

Substituting for temperature T as a function of time t from (20) into (29) yields the neutron to proton ratio as a function of time,

( n p ) = exp ( − a Q t ) , (30)

where a = ( 3.363 ) 1 / 4 / ( 2.4255 ) 1 / 2 .

The relativistic mass density ρ_{S}(T) of (15) along with (16) is equivalent to the

energy density given by Eq. (22.41) in [

Using (4), the rate of acceleration of the expanding universe is given by the time rate of change of the Hubble parameter H(t), defined in (22), and is given by

H ˙ ( t ) = R ¨ ( t ) R ( t ) − H 2 ( t ) , (31)

where · = d / d t .

The standard Lambda-Cold-Dark-Matter model [

A plot of Ω(t) for the present times from around 2.26 × 10^{9} years up to 13.6 × 10^{9} years since the big bang is shown in

shown the matter density parameter from (12), Ω m ( t ) = ρ m ( t ) / ρ c , for the first 1000 seconds since the big bang.

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

Oliveira, F.J. (2021) Big Bang Nucleosynthesis in Carmeli Cosmology—Mass Density, Temperature and Expansion Rate of the Early Universe. Journal of High Energy Physics, Gravitation and Cosmology, 7, 333-343. https://doi.org/10.4236/jhepgc.2021.71017