Astrophys Space Sci (2007) 310: 237–239DOI 10.1007/s10509-007-9507-7

O R I G I NA L A RT I C L E

On the energy density of the cosmic microwavebackground

A. Dinculescu

Received: 3 January 2007 / Accepted: 10 April 2007 / Published online: 16 May 2007© Springer Science+Business Media B.V. 2007

Abstract Starting from the assumption that the radiationsource at the origin of the cosmic microwave background(CMB) could not have a luminosity larger than the maximumenergy in ordinary matter divided by the minimum time al-lowed by causality, one arrives at an expression that givesthe energy density of CMB as a function of the main cosmo-logical parameters. Also, by defining a radiation charge asthe hypothetical charge that opposes the congregation of acloud of particles around a source of electromagnetic radia-tion, on arrives at another expression for the energy densityof CMB that agrees exactly with the measured value for avalue of the Hubble constant equal to 72.09 km s−1 Mpc−1.Both expressions are independent of the redshift.

Keywords Cosmic microwave background · Baryonicmatter · The universe

1 Introduction

The cosmic microwave background is one of the pillars ofstandard cosmology. Its discovery greatly increased our con-fidence in the hot Big Bang model and the mechanism of thelight element nucleosynthesis. However, although the CMBis one of the most important cosmological relics yet discov-ered, we still have no firm explanation as to where this ra-diation came from (Turner 1993). Its energy density can betheoretically estimated on the basis of primordial nucleosyn-thesis and the observed helium mass fraction Yp . Due to theimportance of the subject, it would be interesting to see if

A. Dinculescu (�)4148 NW 34 DR., Gainesville, FL 32605, USAe-mail: astra@ufl.edu

the CMB energy density can also be estimated starting fromother considerations.

2 A luminosity constraint and its consequences

In order to calculate the energy density of the radiationwe see today coming uniformly from all directions in thesky, we start from the simple assumption that the sourceat its origin could not have a luminosity L larger than themaximum energy in ordinary matter Mbc

2 divided by theminimum time R/c allowed by causality. (Here Mb andR are the baryonic mass and, respectively, the size of thecausally connected universe, and c is the speed of light).With 2GMb/Rc2 = �b (where G is the gravitational con-stant and �b is the baryon density parameter) one obtains

Lmax = 1

2

�bc5

G. (1)

Based on previous work (Dinculescu 2007) we assume thatall photons in the CMB are scattered photons. The probabil-ity that a photon is scattered on a path r is τ = σT ner , whereσT is the Thomson cross section, ne = (1 − Yp/2)(ρb/mp)

is the electron number density in a fully ionized matter,ρb = (3/8π)(�bH

2/G) is the baryon density, mp(e) is themass of the proton (electron), and H is the Hubble constant.The energy of the radiation scattered in a sphere of radiusr centered on the source in the average time t = (1/2)(r/c)

spent by a photon inside the sphere at a certain time from itsemission is

Wγ = Ltτ = 1

2

LσT

cner

2. (2)

With L = Lmax and σT = (8π/3)r2e (where re = e2/mec

2

is the classical radius of the electron and e is the elementary

238 Astrophys Space Sci (2007) 310: 237–239

electric charge in Gaussian units) this corresponds to an en-ergy density

uγ = 1

2�bNpe

re

r

(1 − Yp

2

)ρbc

2, (3)

where Npe = e2/Gmpme is Dirac’s (1938) large number.Using the concept of Photon Mean Free Path Sphere (Din-culescu 2005), one can write this equation in the simple form

uγ = �b

GMPhρb

r(4)

where MPh is the mass of a self-gravitating Photon MeanFree Path Sphere of maximum size with the same compo-sition as the observable universe. As can be easily seen,because the radius of a commoving volume of space andthe density of matter inside varies with the redshift as(1 + z)−1 and (1 + z)3, respectively, while the radiationenergy density varies as (1 + z)4, (3) and (4) are valid atany epoch. For a commoving volume of space of the sizeof the Hubble radius at the present epoch, and the mostrecent values of the cosmological parameters (Roos 2005;Lahav and Liddle 2006), one obtains uγ = uCMB within thecorresponding margin of errors.

3 A radiation charge and its value

Since a source of radiation exerts pressure on the surround-ing matter, one can think of it as a charge. As in the caseof assembling a sphere of electric charge with a uniformcharge density, one has to spend energy in order to assemblea spherical cloud of radius r around a source of radiationof constant luminosity L. The radiation force on a sphericalshell of radius r and thickness dr is

dFγ = L

cσT nedr. (5)

The energy required to assemble the entire sphere is

Wγ =∫ r

0rdFγ = 1

2

L

cσT ner

2. (6)

Therefore one can define a Radiation Charge Qγ ≡(Wγ r)1/2 and write

Q2γ = Lr2

e

cNe, (7)

where Ne is the number of free electrons in the sphere. Theradiation energy density in the sphere is

uγ = 3

4π

Q2γ

r4. (8)

From this equation and (3) one can easily find an expres-sion that gives Qγ as a function of the main cosmologicalparameters. However, taking into account that this charge isa characteristic of the CMB and is the same at any epoch,it might be possible to express it as a function of universalconstants only. In order to do that, let us consider the numberof standing electromagnetic waves

dNγ = −8πV

λ4dλ (9)

with wavelength λ in the range dλ that can occur in a cavityof volume V , and let us impose a cut-off at a certain wave-length λx . The number Nγ of standing waves with wave-lengths larger than λx is then

Nγ = −8πV

∫ ∞

λx

dλ

λ4dλ = 8π

3

V

λ3x

. (10)

If each mode in the cavity has an average energy ε =(1/2)e2/r , where r is a certain radius, the radiation energydensity is

uγ = 4π

3

e2

r

1

λ3x

. (11)

Now, let i and j be a pair of particles of masses mi and mj

that satisfy the equality

e2

r= Gmimj

λx

(12)

and let Nij = e2/Gmimj . One has

uγ = 4π

3

e2

r4N3

ij = 4π

3

Gmimj

r4N4

ij . (13)

If the space in the cavity expands, r and λx vary in the sameway, hence the quantity Nij is a constant. We do not knowthe value of this constant, but recently (Dinculescu 2006) itwas shown that from all possible pairs of known elementaryparticles, the only particles that fit the Dirac type equation

R0

(λiλj )1/2∼= e2

Gmimj

(14)

(where λi(j) is the corresponding Compton wavelength) arethe proton and the electron. Therefore, let us take Nij = Npe

and write (13) as

uγ = 4π

3

Gmpme

r4N4

pe. (15)

Since uγ r4 = const. this equation is also independent of theredshift. On equating it with (8) one obtains

Qγ = 4π

3N2

peγpe, (16)

Astrophys Space Sci (2007) 310: 237–239 239

where γpe = (Gmpme)1/2 is the gravitational charge of the

proton and the electron. With this simple expression for theradiation charge, (8) gives an energy density equal to that ofthe cosmic microwave background for a value of the Hub-ble constant H0 = 72.09 km s−1 Mpc−1, which is practicallyidentical to the most probable value H = 72 km s−1 Mpc−1

(Freedman 2003). Obviously, since uγ = aT 4, where a isthe radiation density constant and T is the radiation temper-ature, all the above equations can also be used to derive theCMB temperature.

4 Concluding remarks

We wrote this paper being convinced that science can onlybenefit when one looks at the same problem from a differ-ent perspective, but the fact that we were able to recover theenergy density of the CMB starting from a different premise

than primordial nucleosynthesis does not contradict in anyway the current theory. Our result might be a fluke, or mightpoint to a deeper connection between the CMB and the bary-onic matter in the universe. At this moment we simply do notknow.

References

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ph/0601168 (2006)Roos, M.: Consensus values for the cosmological parameters, astro-

ph/0509089 (2005)Turner, M.S.: Science 262, 861 (1993)