PHOTON-TO-BARYON RATIO IN THE UNIVERSE AND TIME –DEPENDENT COSMOLOGICAL TERM

CORRADO MASSAVia Fratelli Manfredi 55, 42100 Reggio Emilia, Italy; E-mail: _cmass@tin.it

(Received 26 February 2002; accepted 24 April 2003)

Abstract. The photon-to-baryon ratio in the universe is much greater than unity (about 109); thisfact is unexplained in the standard model of the universe; it can be explained by a photon creationprocess related to a time-dependent cosmological term.

1. Introduction

At present cosmic time, the temperature of the cosmological black body radiationfield is 2.7 K, and the mass density of baryonic matter in the universe is of the orderof 10−31 g cm−3. The resulting value of the photon-to-baryon ratio in the Universeis

� ≡ nγ /nb ∼ 109 (1.1)

where nγ = a(kT ch)3 is the number of 2.7 K photons per unit volume (a ∼ 0.24 isa dimensionless constant factor, k is the Boltzmann constant, c is the speed of lightand h is the reduced Planck constant) and nb = µ/m is the number of baryons perunit volume (µ is the mean mass density of the baryonic matter and m is a typicalbaryon mass). We get:

� = a(k/ch̄)3(m/µV )V T 3 (1.2)

where V is the volume of any fluid element in the three space, related to the cosmicscale factor R = R(t) (t is the cosmic time) by

V/R3 = constant = χ (1.3)

(throughout this paper, constant means independent of the cosmic time t).The standard model of the universe requires � = constant. The proof is as

follows (for details see e.g. Misner et al., 1973): The Einstein gravitational fieldequations require a divergenceless energy tensor which for a perfect fluid of massdensity U and pressure P implies the first law of thermodynamics

d(UV ) + PdV = 0. (1.4)

Astrophysics and Space Science 286: 313–322, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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From U = µ + ρ (where ρ is the radiation energy density) and from P = ρ/3 (weneglect the baryonic pressure with respect the black body radiation pressure) andremembering that in any epoch t > few seconds the energy exchanged betweenmatter and radiation was negligible compared to µV and ρV individually, for (1.4)we obtain µV = constant = M (which expresses the constancy of baryon massin a comoving volume) and ρV 4/3 = constant (which expresses the constancy ofnumber of photons in a comoving volume). From the Stefan law ρ/T 4 = constantwe get V T 3 = constant which for (1.2) implies � = constant.

The standard model cannot explain why the universe began with a value of � ofthe order of 109, rather than some other value (e.g., unity). Such a lack of explicat-ive power is a flaw of the standard model. Grand unified theories (GUTs) attempt toexplain � with questionable assumptions (Narlikar, 1983). What follows suggestsa new way to explain the observed value of �. The idea is that the great valueof � might be caused by a photon creation process related to a time-dependentcosmological term in Einstein-like field equations of gravity. I will explore twopossibilities: � proportional to U (Section 2 and 3) and � proportional to 1/R2,where R is the cosmic scale factor of the RW spacetime (Section 4).

Throughout this work we employ the usual black body relations nγ /T 3 = constand ρ/T 4 = const; they are certainly valid in a photon-conserving universe, but arethey valid in the presence of a continuous photon creation phenomenon? Accordingto Wichoski and Lima (1999) in a Robertson-Walker universe with photon creationthe standard Planckian spectral distribution law has to be replaced with a moregeneral law; if T � (knγ /b)1/3 or the radiation is decoupled, the spectrum is notdestroyed as the universe evolves, and the relations nγ /T 3 = constant, ρ/T 4 =constant still hold.

I thank the anonymous referee who brought Wichowski and Lima paper to myattention.

2. Field Equations With Varying Cosmological Term

The gravitational field equations with cosmological term � read

Gik = Tik + �gik (2.1)

where Gik is the Einstein tensor, gik is the fundamental tensor, and Tik is the en-ergy tensor of matter and non-gravitational fields. Throughout the work I employgeometric units, for which c = k = h̄ = 8πG = 1 where G is the Cavendishconstant.

Is � a universal constant? A connection between � and matter in Newtonianuniverses has been pointed out by Wilkins (1986) and a Newtonian universe withvarying � has been investigated by Waga (1992). It is well known that Newto-nian cosmology leads to spatially isotropic cosmological models identical withthose derived using the general relativity theory (this can be explained with the

PHOTON-TO-BARYON RATIO 315

cosmological principle, united to the fact that Newton’s and Einstein’s theory areindistinguishable for weak fields and low velocities). Then a connection between� and the mean mass-energy density U of the universe could well exist in a generalrelativistic context.

On dimensional grounds the simplest way to relate � (dimensions [length]−2)to U and to the two fundamental parameters of the general relativity theory (thevelocity of light c and the Cavendish constant G) is

� = 8πλGU/c2 (2.2)

where λ is a dimensionless constant; other arguments suggesting Equation (2.2)are in Massa (1994). Thus, we assume the gravitational field equations

Gik = Tik + λUgik. (2.3)

In a RW universe filled with a homogeneous perfect fluid the field equations (2.3)lead to

3(dR/dt)2 + 3K = (1 + λ)UR2 (2.4)

(2/R)(d2R/dt2) + P + (U/3)(1 − 2λ) = 0. (2.5)

K is the space curvature index, normalized to −1, 0, or +1.With λ = 0 they reduce to the equations of the standard model.With λ = −1 Equation (2.4) gives dR/dt = 1 hence for Equation (2.5) U +

P = 0; we assume U > 0, P > 0, thus λ = / = −1.Differentiate (2.4) with respect to time t , take (2.5) into account and get

(1 + λ)dU/dR + (3/R)(U + P) = 0 (2.6)

which also follows directly from the field equations by calculating the covariantdivergence of the energy tensor and taking the Bianchi identities into account.

If, and only if, λ = 0, Equation (2.6) reduces to the first law of thermodynamics(1.4); in general, for Equation (1.3) i.e. for V/R3 = const, a nonzero λ violates thefirst law of thermodynamics and implies a creation process.

Clearly, equations µV = constant and ρV = constant cannot be both true.Consider the equation

(1 + λ)(dµ/dR + dρ/dR) + (1/R)(3µ + 4ρ) = 0 (2.7)

obtained from (2.6) by using U = µ+ρ and ρ = 3P ; assume constancy of numberof photons in a comoving volume, expressed by ρV 4/3 = constant, and obtain

(1 + λ)d(µV ) = λMdV/V − (4/3)ρdV − (1 + λ)V dρ (2.8)

which violates µV = constant, i.e. constancy of baryon mass in a comovingvolume. By the way, Equation (2.8) can be written in the form

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(1 + λ)dN/dt = 3λHN(1 + 4ρ/3µ) (2.9)

where H = (1/R)dR/dt is the Hubble parameter and N = µV/m is the numberof baryons of (constant) mass m in the volume V . Set λ > 0 and not far from unity;at present cosmic time, H ∼ 10−18 s−1, ρ ∼ 10−34 g cm−3, µ ∼ 10−31 g cm−3,hence dN/dt ∼ NH that is roughly the rate of the steady state cosmology. Thepossibility of uniform creation of baryon–antibaryon pairs throughout the universewith such a rate has been discussed and rejected (Chen and Wu, 1990) becausethe baryon–antibaryon annihilation would provide an isotropic cosmic γ -ray back-ground whose intensity is much higher than what has been observed. Cohen andKing (1969) searched for the appearance of hydrogen gas in mercury metal andfound

(1/N) (dN/dt) < 10−15 yr−1 ∼ 10−5 H.

This fact suggests to keep the baryon conservation law (µV = constant) unchanged.Insert µV = constant = M into Equation (2.7) and find

(1 + λ)R4dρ/dR + 4ρR3 = 3λM/χ. (2.10)

If λ = / = 1/3 integration yields

ρ = AR−4/(1+λ) + (3λM/R3χ)/(1 − 3λ) (2.11)

(where A is an integration constant), while if λ = 1/3,

ρ = BR−3 + (3/4)(M/χ)R ln R (2.12)

(B = an integration constant).In what follows we focus attention to Equation (2.11).Because V/R3 = constant, Equation (2.11) is at variance with ρV 4/3 = con-

stant and creation of photons occurs.We assume ρ > 0 at all cosmic times, namely

Lim ρ > 0 as R → 0 and Lim ρ ≥ 0 as R → ∞,

which implies (for λ = / = 1/3)

A > 0, 0 < λ < 1/3. (2.13a,b)

From (2.11), from µV = M and from V = χR3 we find

ρ/µ = (χA/M)R−(1−3λ)/(1+λ) + 3λ/(1 − 3λ). (2.14)

Note, ρ/µ → ∞ as R → 0, and ρ/µ → 3λ/(1 − 3λ) as the universe getsold (R → ∞). Thus, in an infinitely diluted universe the ratio (photon energydensity) / (matter energy density) stays constant, which is at variance with standardcosmology, and denotes a photon creation process. This process looks more evidentby remembering that the total number of photons Nγ = ρV/ε with typical energyε contained in a comoving volume V = χR3 turns out to be, for Equation (2.11),

PHOTON-TO-BARYON RATIO 317

Nγ = (χ/ε)AR(3λ−1)/(1+ + (3λ/ε)M/(1 − 3λ). (2.15)

In the beginning of next Section, just after Equation (3.4), we give a dimensionalargument supporting ε ∼ 1/R in curved space, and ε ∼ 1/L in flat space, withL = the radius of the observable, causally connected domain. To fix ideas, let usfocus attention to the curved space; from (2.15) we get

Nγ = χAR4λ/(1+λ) + 3λMR/(1 − 3λ). (2.16)

For Equation (2.13b), the exponent 4λ/(1 + λ) is positive and less than unity, thusNγ is zero when R = 0, and is proportional to R as the expansion goes on and on.

In the early (hot) epochs, this model has essentially the same thermal history asthe standard one; proof: insert Equation (2.11) into Equation (2.4) and get

3(dR/dt)2 + 3K = A(1 + λ)/Rζ + (M/R)η (2.17)

where

ζ = 2(1 − λ)/(1 + λ) > 1

and

η = (3λ/χ)(1 + λ)/(1 − 3λ) > 0.

If R → 0 Equation (2.17) reads

(dR/dt)2 = (A/3)(1 + λ)/Rζ (2.18)

and Equation (2.11) reads

ρR4/(1+λ) = A. (2.19)

Equation (2.18) integrates to

R = (Bt)(1+λ)/2 (2.20)

where

B = [(4A/3)/(1 + λ)](1+λ)/4 = constant,

and the integration constant has been put equal to zero to satisfy the ususal initialcondition R(0) = 0.

Equation (2.20), the Stefan law ρ = bT 4, and Equation (2.19) lead to

T√

t = (A/B2b)1/4. (2.21)

T scales as 1/√

t , just as in the standard model of the hot universe. The thermalhistories of the two models are then essentially the same, and the well known pre-dictions of the standard model relative to the cosmic helium and litium abundancehold unchanged.

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3. The Great Value of � Derived From A Photon Creation Process

By now on we employ Planck units.The photon-to-baryon ratio in the universe is expressed by Equation (1.2),

� = a(m/M)V T 3 (3.1)

where M = µV = constant. Remember the Stefan law ρ = bT 4 and write

� = ρ(a/b)(m/M)(V/T ). (3.2)

This, and Equations (1.3), (2.11) and (2.14), give

� = β(a/b)(m/T )[1 + (χ/β)(A/M)R−ϕ]. (3.3)

Where, for the sake of brevity, β = 3λ/(1 − 3λ), and ϕ = (1 − 3λ)/(1 + λ).From the black body distribution law we have a ∼ 0.24, and b = π2/15.Owing to Equation (2.13a,b), Equation (3.3) implies

� > β(a/b)(m/T ). (3.4)

The cosmological nature of the creation process suggests that the wavelength ofa typical newly created photon should be of the order of the only natural scale oflength available in the universe, that is the radius R of space curvature (if K=/=0)or the radius L = R

∫dt/R of the observable (causally connected) domain (if

K = 0).Subsequent interactions with matter lead the newly created photons to thermal-

ize with matter and to assume the temperature T of the cosmic fluid: their typicalwavelength becomes then ∼ 1/T according to the Wien law.

Recombination of the plasma at time t = td (the decoupling era) brings anend to the thermal interaction between matter and radiation; all photons created atepochs t > td propagate freely through space with their own original wavelengthof order R (or L, in flat space). Such ‘tardy’ photons (which in the transparentuniverse subsequent the decoupling era cannot thermalize with matter) do notcontribute to the cosmic background black body radiation we observe today. Ac-cordingly, the value of the photon-to-baryon ratio, σ , reached its maximum value�d at t = td ; since then, � stays constant. In any epoch t > td the value of � is thesame as it was at t = td , namely �d ; therefore, the present value of � is expectedto equal �d ,

� (today) = �d. (3.5)

The value of �d springs from Equation (3.3) by setting T = Td where Td is thedecoupling temperature. The value of Td can be evaluated from the Saha ionizationequation which gives the fraction of ionization Y = E/B at the temperature T ,where E and B are the total free electron and baryon number density, respectively.The Saha equation gives (Narlikar, 1983)

PHOTON-TO-BARYON RATIO 319

Y 2/(1 − Y ) = (m/µ)(meT /2π)3/2 exp(−Z/T ) (3.6)

where m is the proton rest mass, me is the electron rest mass, and Z = 13.6 eV∼ 1.6 × 105 K is the Hydrogen ionization energy.

The universe became transparent to radiation at the decoupling time t = td ,when µ = µd , T = Td , Y = Yd < 1, and 1 − Yd ∼ 1.

If this fact happened during the radiation era (when µ < ρ) then

µd < ρd = bT 4d (3.7)

which inserted into the Saha equation leads to

Td exp(2Z/5Td) > (m/b)2/5(me/2π)3/5. (3.8)

In the geometrized units for which 8πG = c = h̄ = k = 1 the proton and theelectron rest mass are 4 × 10−19 and 2.2 × 10−22 respectively and Equation (3.8)gives

Td < 4000 K. (3.9)

This and inequality (3.4) gives �d > β109 and therefore for Equation (3.5)

� (today) > β109 (3.10)

where β ∼ 1 because λ is very likely not far from unity.Thus, if we assume that the decoupling time happened before the beginning of

the matter era, the photon-to-baryon ratio in the present universe should exceed109.

What if the decoupling happened when µ > ρ? Note that owing to µV =µχR3 = M = constant the Saha equation reads

Y 2/(1 − Y ) = χ(m/M)(mεT /2π)3/2R3 exp(−Z/T ). (3.11)

The R − T relation in the matter era is easily found: for R → ∞ Equation (2.11)reads ρ = βM/χR3 which for ρ = bT 4 gives

R(T ) = (βM/bχ)1/3T −4/3 (3.12)

and therefore in the matter era the Saha equation reads:

Y 2T 5/2 exp(Z/T ) = (β/b)m(me/2π)3/2. (3.13)

With β not very far from unity, and with Y = Yd < 1, Equation (3.13) implies

Td < 4000 K. (3.14)

More precisely:

Td ∼ 4000 K if Yd ∼ 1 (3.15a)

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Td ∼ 3500 K if Yd ∼ 1/10 (3.15b)

Td ∼ 3000 K if Yd ∼ 1/100 (3.15c)

As R → ∞ Equation (3.3) leads to

� ∼ β(a/b)(m/T ) (3.16)

or, restoring conventional units,

� ∼ β(a/b)(mc2/kT )

which for T = Td and Equations (3.5), (3.15) leads to

� (today) ∼ 109. (3.17)

To conclude: the field equations (2.3) give a natural way to explain the observedhuge value of the cosmological photon-to-baryon ratio.

4. The Ansatz �R2 = Constant

From a dimensional argument in line with quantum cosmology, Chen and Wu(1990) proposed a cosmological model characterized by Einstein-like gravitationalfield equations, with varying cosmological term given by � = γ /R2; their fieldequations

Gik = Tik + (γ /R2)gik (4.1)

imply a nonzero divergence for the energy tensor Tik and a particle creation pro-cess. Can Equation (4.1) account for the observed value of �? For the identitiesGik; k = 0, the time-component of the covariant divergence of T ik in a RWspacetime turns out to be

T 0k;k = (2γ /R3)dR/dt. (4.2)

In the case of a perfect fluid (mass-energy density U = µ + ρ, pressure P = ρ/3)we have also

T 0k;k = d(µ + ρ)/dt + (3µ + 4ρ)(1/R)dR/dt. (4.3)

From (4.2), (4.3) and µV = constant, one finds

ρR4 = γ R2 + A (4.4)

where A is an integration constant.This equation and the Stefan law ρ = bT 4 lead to

R2 = (γ /2)(1/bT 4)� (4.5)

PHOTON-TO-BARYON RATIO 321

where

� = 1 + √[1 + (4AbT 4/γ 2)]. (4.6)

Equation (4.5) for V = χR3 and for Equation (1.2), gives (with M = µV , and ingeometrized units)

� = χa(m/M)(1/T 3)(γ �/2b)3/2. (4.7)

If the decoupling happened during the radiation era, then Td < 4000 K (Equa-tion 3.9); hence, for Equation (4.7) and � > 1,

� (today) > (aχ/4)(γ ch̄/πbG)3/2(c2/kTd)3(m/M) (4.8)

where c, h̄, k and G have been displayed.To fix ideas, consider a closed space (χ = 2π2) with curvature radius (at present

time) R = 1028 cm; from µ ∼ 10−31 g cm−3. We find M ∼ 2 × 1054 g, andm/M ∼ 10−78; with γ not very far from unity we have � (today) > 106.

During the matter era (R → ∞) Equation (4.4) reads

R = √(γ /ρ) = (1/T 2)

√(γ /b). (4.9)

Insert into the Saha equation (3.11) and obtain

Y 2T 9/2 exp(Z/T ) = (1 − Y )χ(m/M)(γ me/2πb)3/2. (4.10)

With Yd < 1 and 1 − Yd ∼ 1, we get

Td ∼ 5000 K (4.11)

� (today) ∼ 107. (4.12)

Final remark: the Chen and Wu ansatz �R2 = constant expresses a connectionbetween the cosmological term and the mean mass energy density U of the uni-verse. The connection however is quite different from my (2.2). For Equation (4.4)and for U = µ + ρ, and remembering µV = µχR3 = M, we get the algebraicequation

UR4 − γ R2 − (M/χ)R − A = 0 (4.13)

whose solutions give R, and then �, as function of U .Clearly, the functional link is now much more complex than (2.2), and simple

links can be obtained only in two limiting cases: in a very young, radiation dom-inated universe (R → 0), where one finds R = (A/U)1/4 and � proportional to√

U ; and in an old, matter dominated universe (R → ∞), where one finds � = U .

References

Chen, W. and Wu, Y.-S.: 1990, Phys. Rev. D 41, 695.

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Cohen, S.A. and King, J.G.: 1969, Nature 222, 1158.Massa, C.: 1994, Astrophys. Space Sci. 215, 59.Narlikar, J.V.: 1983, Introduction to Cosmology, Jones and Bartlett, Boston.Waga, I.: 1992, Gen. Relativ. Gravitation 24, 783.Wichoski, U.F. and Lima, J.A.S.: 1999, Phys. Lett. A 262, 103.Wilkins, D.: 1986, Am. J. Phys. 54, 726.