Another Look at the Cosmic Background Radiation

In my study of the cosmic background radiation to see what knowledge we can glean from it, I calculated a time for the decoupling of the primordial radiation from matter and got a result almost two orders of magnitude larger than what others have calculated. Clearly someone made an error. Before we can find that error we need to know how astrophysicists calculated the lower estimate of the time elapsed between the origin of the Universe and the decoupling event.

Nobody could have made any calculations pertaining to the cosmic background radiation prior to 1929, when Edwin Hubble revealed that his study of galaxies implied that the Universe originated in a super-powerful explosion. It took almost twenty years after that for physicists to come to understand that they could derive calculations pertaining to the afterglow of that explosion. In 1948 George Gamow calculated the anticipated current temperature of that afterglow. Coming seventeen years before Arno Penzias and Robert Wilson first detected the cosmic background radiation and made measurements of it, Gamow’s calculation contained no data from the afterglow itself.

One key equation that Gamow used related the energy
density of the radiation (ρ_{r})
and the mass density of the matter (ρ_{m})
in the Universe on the largest scale,

(Eq’n 1)

in which K represents a constant. Alternatively, he could have written

(Eq’n 2)

The fact that we have in that equation the energy density in the radiation proportional to the fourth power of the cube root of the mass density of the matter in the Universe tells us two of the major assumptions that Gamow incorporated into his derivation of Equation 1.

Setting the energy density of the radiation equal to the fourth power of something implies that the something stands in direct proportion to the absolute temperature of the something’s environment and, consequently, that Equation 2 represents one form of the equation expressing the Stefan-Boltzmann law, which we usually have as

(Eq’n 3)

That equation expresses the energy density in the radiation emanating from a perfectly black body at the absolute temperature T. Ludwig Boltzmann deduced that law by considering radiation inside a cavity whose black walls have a certain absolute temperature. By considering that system at equilibrium, he could assign a temperature to the radiation.

But why would temperature stand in direct proportion to the cube root of the Universe’s mass density? To answer that question consider the work of Wilhelm Wien in radiation thermodynamics. Wien imagined a radiation field confined inside a cavity bounded by perfectly reflective walls. Like Boltzmann, he had gained inspiration from the work of Adolfo Bartoli, who used a similar imaginary experiment to deduce the fact that light and heat radiation exert pressure, using the laws of thermodynamics in that deduction. Exploiting that fact, Wien imagined the radiation in his imaginary experiment undergoing an adiabatic expansion or contraction as the cavity grows or shrinks. In that process the total entropy of the radiation,

(Eq’n 4)

does not change. That fact means that the cube of the temperature stands in reciprocal proportion to the volume of the cavity.

We assume that the total mass M of the Universe does not change with the elapse of time. Further, the Universe has spherical symmetry, so its volume stands in cubic proportion to its radius,

(Eq’n 5)

For the mass density we have ρ_{m}=M/V,
a number that we can, in concept, determine from measurements. Correlating the
information in Equations 4 and 5 gives us

(Eq’n 6)

Combining that result with Equation 3 gives us the equation that Gamow used.

But one assumption underlying that derivation does not apply to the Universe. Wien, and Boltzmann to a lesser extent, assumed that the radiation in his imaginary cavity bounced off the cavity’s walls, which moved slowly enough for the interaction to proceed at a quasi-static pace, thereby maintaining thermal equilibrium. But the radiation in the cosmic background that comes into our telescopes has never touched a wall since it came free of the primordial plasma.

To make matters worse, the derivation makes no reference to Relativity, even though Gamow and his collaborators knew that distant galaxies fly away from Earth at relativistic speeds and, presumably, that the source of the cosmic background moves even faster. That seems like a remarkable oversight.

But for all its illegitimacy can Equation 1 enable us to calculate a time at which the cosmic background radiation decoupled from the primordial plasma? In Equation 6 we have a relation between the radius of the Universe and the temperature of the cosmic background, so we could exploit Hubble’s law, which says that the radius of the Universe stands in direct proportion to the time elapsed since the origin of the Universe. That plan gives us

tT=constant,

(Eq’n 7)

in which the lower-case tee represents the elapsed time and the upper-case tee represents the absolute temperature of the cosmic background. Using modern values for the variables in that equation, we calculate

(13.7x10^{9} years)(2.725 Kelvin)=(t)(2900 Kelvin),

(Eq’n 8)

which gives us t=12.9x10^{6} years for the age of the Universe at
decoupling rather than the 389,000 years that astrophysicists use. Ironically,
that number equals what we get from our relativistic calculation.

But Gamow used a different equation to relate the temperature of the cosmic background to the age of the Universe, one based on General Relativity; specifically, he used an equation derived, in part, from the first of the Friedmann equations describing an homogeneous and isotropic model of the Universe. He started with Equation 6, relating the temperature of the Universe to the radius of the Universe, and then used the Friedmann equation to relate the radius of the Universe to the Universe’s age.

Assuming, reasonably, that spacetime is flat and that the cosmological constant equals zero, Gamow had Friedmann’s equation in the form

(Eq’n 9)

in which H represents the Hubble parameter (or Hubble constant), G represents the Newtonian gravitational constant, and ρ represents the average energy density of the Universe. Alexander Alexandrovich Friedmann (1888 Jul 16 – 1925 Sep 16) derived the more general form of that equation in 1922 from Einstein’s Equation and taught it to his students, one of whom was George Gamow. Familiar with Friedmann’s model of the Universe, Gamow had only to determine the relationship between the energy density and the radius of the Universe in order to derive the relationship between the radius and the age of the Universe.

For that step of his derivation Gamow assumed that the energy density of the Universe was dominated by the energy in radiation. In that case the Stefan-Boltzmann law would provide a relatively accurate description of the energy density, making it proportional to the fourth power of the temperature. But Equation 6 makes the energy density stand in reciprocal proportion to the fourth power of the radius. Given that the Hubble parameter stands in reciprocal proportion to the age of the Universe, we can rewrite Equation 9 as

(Eq’n 10)

That relation gives us

(Eq’n 11)

which we use to rewrite Equation 6 as

(Eq’n 12)

Now we need only determine the value of the constant.

In the late 1940's Gamow and his collaborators worked out a description of primordial nucleosynthesis. They began with the reasonable assumption that the first matter to come into existence consisted solely of a proton-electron plasma, which Gamow called Ylem, a Medieval term derived from the Greek hyle (matter). Once the plasma had cooled to a certain temperature, the protons could fuse into the nuclei of helium and lithium. The fusions continued until the plasma cooled and thinned to a state in which the fusions would no longer occur at a substantial rate. With a mathematical description of that process in hand, Gamow correlated it with data from astronomical observations. One of the figures that he derived told him that the Universe had a temperature of 15 billion Kelvin when it was one second old. Thus he wrote Equation 12 as

(Eq’n 13)

in which we use degrees Kelvin for the temperature and seconds for the time.

On the assumption that the temperature at which the radiation decoupled from the primordial plasma stands at 2900 Kelvin, that equation tells us that decoupling occurred when the Universe had attained an age of 849,000 years, over twice the currently accepted figure. However, if we make the constant of Equation 13 equal to ten billion, then we get a decoupling age of 377,500 years, which implies that the currently accepted date of decoupling comes from a tweaking of Gamow’s model of primordial nucleosynthesis.

Thus we have the thermodynamic calculation of the date of decoupling. But we also have a relativistic calculation of the date of decoupling, one that conflicts significantly with the thermodynamic date.

We assume that the low temperature of the cosmic background radiation comes from the relativistic Doppler shift between the inertial frames occupied by the primordial plasma that emitted the radiation and (approximately) by Earth. We also assume that, over cosmological distances, the second postulate of Special Relativity remains valid and that light obeys the conservation laws (which means that the light does not get stretched or tired). We use Wien’s displacement law to relate the temperature of the radiation to the frequency of the radiation’s maximum amplitude, so we have

(Eq’n 14)

in which the subscript zero marks the values that we would measure in the proper frame and the subscript E marks the values that we measure in Earth’s frame. The relativistic Doppler shift gives us

(Eq’n 15)

in which β represents, as usual, the fraction of the speed of light at which the proper frame moves relative to the Earth frame. Thus we have

(Eq’n 16)

which we can solve for β. We then get the Lorentz factor between the frames as

(Eq’n 17)

Using the modern value for the proper temperature at which decoupling occurs (2900 Kelvin) and the temperature of the radiation that we receive from the cosmic background (2.725 Kelvin), we have η=1064.22. The corresponding Lorentz factor comes out of Equation 17 as L=532.11. A frame that differs from ours by a Lorentz factor that big moves so close to the speed of light that if we want to calculate the distance that an object occupying that frame moves in a given interval of time, we can simply multiply the elapsed time by the speed of light and get a distance negligibly different from the actual distance.

Our calculation of η from the theoretical value of the temperature of decoupling and the temperature of the cosmic background radiation inferred from measurements implies that the source of the radiation does, indeed, occupy that rapidly moving frame. That fact means that the part of the primordial plasma that emitted the radiation that we detect today moves away at near the speed of light from the point we all occupied when the Universe emerged into existence; it did so for about half the age of the Universe (6.85 billion years) before emitting the radiation; and then the radiation took the other half of the Universe’s age to come back to us. We see the relativistic nature of this phenomenon in the understanding that the 6.85 billion years that elapsed between the origin of the Universe and the emission of the radiation, time elapsed in our frame, consists of dilated time. We calculate the proper time corresponding to that dilated time by dividing it by the Lorentz factor between the two frames, so we get 6,850,000,000 ÷ 532.11=12,870,000 years.

Thus, from essentially the same datum, the proper temperature of decoupling, we have calculated two grossly different dates for the separation of radiation from the primordial plasma. But both calculations stand on assumptions that conform to Reality as we currently understand it. The disagreement thus means that we don’t fully understand Reality. We don’t have an either/or choice in this matter; both calculations stand on sound assumptions as far as we can tell. So we must infer that we have missed something. We will know that we have discovered that something when we recalculate the date of decoupling both ways and the answers that we get agree with each other.

On the thermodynamic side we must almost certainly modify Gamow’s assumption of a radiation-dominated Universe: at decoupling the density of the radiation had to be low enough to allow the plasma to decay into neutral matter. We might also want to re-examine Gamow’s calculation of primordial nucleosynthesis, in light of what we have learned since 1948, to see whether it might have occurred later than Gamow calculated. And we should also recalculate the proper temperature of decoupling, based on the observation that radiation and matter decouple at temperatures higher than 2900 Kelvin in the atmospheres of the stars.

On the relativistic side we want to re-examine our assumption that we can apply the rules of Special Relativity over cosmological distances. That means that we must redo the entire calculation using the rules of General Relativity to see whether the cosmological-scale warping of space and time has an effect on the propagation of light. Note that while I assumed that light obeys the conservation laws, especially the conservation of energy, we know that energy is not conserved locally in situations described by General Relativity.

Those recalculations will appear in other essays on this website when I have them ready.

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