The work done by pressure in an expanding fluid uses heat energy drawn from the fluid. The universe is expanding, so we expect that in the past matter was hotter as well as denser than at present. If we look far enough backward in time we come to an era when it was too hot for electrons to be bound into atoms. At sufficiently early times the rapid collisions of photons with free electrons would have kept radiation in thermal equilibrium with the hot dense matter. The number density of photons in equilibrium with matter at temperature T at photon frequency between ν and ν + d ν is given by the black-body spectrum:
nT(ν)d ν = 8π ν2d ν
exp (hν/kBT ) − 1 , (2.1.1) where h is the original Planck’s constant (which first made its appearance in a formula equivalent to this one), and kBis Boltzmann’s constant. (Recall that we are using units with c = 1.)
As time passed, the matter became cooler and less dense, and eventually the radiation began a free expansion, but its spectrum has kept the same form.
We can see this most easily under an extreme assumption, that there was a time tLwhen radiation suddenly went from being in thermal equilibrium with matter to a free expansion. (The subscript L stands for “last scat-tering.”) Under this assumption, a photon that has frequency ν at some later time t when photons are traveling freely would have had frequency
νa(t)/a(tL) at the time the radiation went out of equilibrium with matter, and so the number density at time t of photons with frequency between ν and ν + d ν would be
n(ν, t) d ν =
a(tL)/a(t) 3
nT (tL)
νa(t)/a(tL)
d (νa(t)/a(tL)) , (2.1.2) with the factor
a(tL)/a(t) 3
arising from the dilution of photons due to the cosmic expansion. Using Eq. (2.1.1) in (2.1.2), we see that the redshift factors a(t)/a(tL) all cancel except in the exponential, so that the number density at time t is given by
n(ν, t)d ν = 8π ν2d ν
exp (hν/kBT (t)) − 1 = nT (t)(ν) d ν , (2.1.3) where
T (t) = T (tL)a(tL)/a(t) . (2.1.4) Thus the photon density has been given by the black-body form even after the photons went out of equilibrium with matter, but with a redshifted temperature (2.1.4).
This conclusion is obviously unchanged if the transition from opacity to transparency occupied a finite time interval, as long as the interactions of photons with matter during this interval are limited to elastic scattering processes in which photon frequencies are not changed.
This is a very good approximation. We will see in Section 2.3 that the last interaction of photons with matter (until near the present) took place at a time when the cosmic temperature T was of order 3,000 K, when by far the most important interaction was the elastic scattering of photons with electrons, in which the fractional shift of photon frequency was of order kBT /mec2 ≈ 3 × 10−7. In the following section we shall show that, because of the large photon entropy, even the small shift of photon fre-quency in elastic scattering and the relatively infrequent inelastic interac-tions of photons with hydrogen atoms had almost no effect on the photon spectrum.
It was George Gamow and his collaborators who first recognized in the late 1940s that the universe should now be filled with black-body radiation.1 The first plausible estimate of the present temperature of this radiation was
1G. Gamow, Phys. Rev. 70, 572 (1946); R. A. Alpher, H. A. Bethe, and G. Gamow, Phys. Rev. 73, 803 (1948); G. Gamow, Phys. Rev. 74, 505 (1948); R. A. Alpher and R. C. Herman, Nature 162, 774 (1948); R. A. Alpher, R. C. Herman, and G. Gamow, Phys. Rev. 74, 1198 (1948); ibid 75, 332A (1949);
ibid 75, 701 (1949); G. Gamow, Rev. Mod. Phys. 21, 367 (1949); R. A. Alpher, Phys. Rev. 74, 1577 (1948); R. A. Alpher and R. C. Herman, Phys. Rev. 75, 1089 (1949).
2.1 Expectations and discovery of the microwave background
made in 1950 by Ralph Alpher and Robert Herman.2 On the basis of considerations of cosmological nucleosynthesis, to be discussed in Sec-tion 3.2, they found a present temperature of 5 K. This work was largely forgotten in subsequent decades, until in 1965 a group at Princeton started to search for a cosmic radiation background left over from the early uni-verse. They had only a rough idea of the temperature to be expected, based on a nucleosynthesis calculation of P. J. E. Peebles, which suggested a value of 10 K.3 Before they could complete their experiment the radiation was discovered in a study of noise backgrounds in a radio telescope by Arno Penzias and Robert Wilson,4 who published their work along with a com-panion article5by the Princeton group explaining its possible cosmological significance.6
Originally Penzias and Wilson could only report that the antenna temperature at a wavelength 7.5 cm was 3.5±1.0 K, meaning that the int-ensity of the radiation at this one wavelength agreed with Eq. (2.1.1) for this temperature. This of course did not show that they were observing black-body radiation. Then Roll and Wilkinson7 measured the radiation intensity at a wavelength of 3.2 cm, finding an antenna temperature of 3.0 ± 0.5 K, in agreement with what would be expected for black-body radiation at the temperature measured by Penzias and Wilson. In the fol-lowing few years a large number of measurements were made by other radio astronomers at other wavelengths. These measurements also gave antenna temperatures at the wavelengths being studied around 3 K, with uncertainties that gradually improved to of order 0.2 K. But this also did not establish the black-body nature of the radiation, because these mea-surements were all at wavelengths greater than about 0.3 cm, where the black-body energy distribution hνnT(ν) with T ≈ 3 K has its maximum.
For these long wavelengths the argument of the exponential is small, and Eq. (2.1.1) gives
h ν nT(ν) 8π ν2kBT , (2.1.5) This is the Rayleigh–Jeans formula of classical statistical mechanics, but it describes the long-wavelength distribution of radiant energy under
vari-2R. A. Alpher and R. C. Herman, Rev. Mod. Phys. 22, 153 (1950)
3This work was never published. According to A. Guth, The Inflationary Universe (Perseus Books, Reading, MA, 1997), Peebles’ paper was rejected by The Physical Review, apparently because of the issue of the credit to be given to earlier work by R. Alpher, G. Gamow, and R. Herman. This earlier work and the subsequent work of Peebles and others is briefly described here in Section 3.2.
4A. A. Penzias and R. W. Wilson, Astrophys. J. 142, 419 (1965).
5R. H. Dicke, P. J. E. Peebles, P. G. Roll, and D. T. Wilkinson, Astrophys. J. 142, 414 (1965).
6For a more detailed history of these developments, see A. Guth, op. cit., and S. Weinberg, The First Three Minutes (Basic Books, New York, 1977; second edition 1993).
7P. G. Roll and D. T. Wilkinson, Phys. Rev. Lett. 16, 405 (1966).
ous circumstances more general than black-body radiation. For instance, black-body radiation diluted by an expansion that preserves the energy of individual photons also has hν n(ν) ∝ ν2 for low frequencies.8 To con-firm that the cosmic microwave background radiation is really described by the black-body formula, it was necessary to see at least the beginning of the exponential fall-off of nT(ν) for wavelengths shorter than about 0.3 cm.
This was difficult because the earth’s atmosphere becomes increasingly opaque at wavelengths shorter than about 0.3 cm. However, there had been a measurement of the radiation temperature at a wavelength 0.264 cm in 1941, long before the discovery by Penzias and Wilson. In between the star ζ Oph and the earth there is a cloud of cold molecular gas, whose absorption of light produces dark lines in the spectrum of the star. In 1941 W. S. Adams,9 following a suggestion of Andrew McKellar, found two dark lines in the spectrum of ζ Oph that could be identified as due to absorption of light by cyanogen (CN) in the molecular cloud. The first line, observed at a wavelength of 3,874.62 Å, could be attributed to absorp-tion of light from the CN ground state, with rotaabsorp-tional angular momentum J = 0, leading to the component of the first vibrationally excited state with J = 1. But the second line, at 3,874.00 Å, represented absorption from the J = 1 rotationally excited vibrational ground state, leading to the J = 2 component of the first vibrationally excited state.10 From this, McKellar concluded11 that a fraction of the CN molecules in the cloud were in the first excited rotational component of the vibrational ground state, which is above the true J = 0 ground state by an energy hc/(0.264 cm). and from this fraction he estimated an equivalent molecular temperature of 2.3 K. Of course, he did not know that the CN molecules were being excited by radiation, much less by black-body radiation. After the discovery by Penzias and Wilson several astrophysicists12 independently noted that the old Adams–McKellar result could be explained by radiation with a
8The sunlight falling on the earth’s surface provides a pretty good example of dilute black-body radiation; it is described by the Planck formula (2.1.1), with T ≈ 6, 000 K the temperature of the sun’s surface, but with the right-hand side of Eq. (2.1.1) multiplied by a factor (R /r)2, where R is the radius of the sun and r is the distance from the sun to the earth.
9W. S. Adams, Astrophys. J. 93, 11 (1941)
10Today the wavelengths of these two lines are more accurately known to be 3,874.608 and 3,873.998 Å. There is another line at 3875.763 Å, produced by transitions from the J = 1 rotationally excited vibrational ground state to the J = 0 component of the first vibrationally excited state.
11A. McKellar, Publs. Dominion Astrophys. Observatory (Victoria, B.C.) 7, 251 (1941).
12G. Field, G. H. Herbig, and J. L. Hitchcock, Astron. J. 71, 161 (1966); G. Field and J. L. Hitchcock, Phys. Rev. Lett. 16. 817 (1966); Astrophys. J. 146, 1 (1966); N. J. Woolf, quoted by P. Thaddeus and J. F. Clauser, Phys. Rev. Lett. 16, 819 (1966); I. S. Shklovsky, Astronomicheskii Tsircular No.
364(1966).
2.1 Expectations and discovery of the microwave background
black-body temperature at wavelength 0.264 cm in the neighborhood of 3 K. Theoretical analysis showed that nothing else could explain the exci-tation of this roexci-tational state.13 This interpretation was then borne out by continuing observations on this and other absorption lines in CN as well as CH and CH+in the spectrum of ζ Oph and other stars.14
The black-body spectrum of the cosmic microwave radiation background began to be established by balloon-borne and rocket-borne observations above the earth’s atmosphere at wavelengths below 0.3 cm. For some years there were indications of an excess over the black-body formula at these short wavelengths. It was clearly necessary to do these observations from space, but this is difficult; to measure the absolute value of the microwave radiation intensity it is necessary to compare the radiation received from space with that emitted by a “cold load” of liquid helium, which rapidly evaporates. Finally, the Planck spectrum of the microwave background was settled in the 1990s by observations with the FIRAS radiometer carried by the Cosmic Background Explorer Satellite (COBE), launched in November 1989.15 When a slide showing the agreement of the observed spectrum with the Planck black-body spectrum was shown by J. C. Mather at a meeting of the American Astronomical Society in January 1990, it received a standing ovation. It was found that the background radiation has a nearly exact black-body spectrum in the wavelength range from 0.5 cm to 0.05 cm.16 The comparison of observation with the black-body spectrum is shown in Figure 2.1. After six years of further analysis, the temperature was given as 2.725 ± 0.002 K (95% confidence).17 Other observations at wavelengths between 3 cm and 75 cm and at 0.03 cm are all consistent with a Planck distribution at this temperature.18
The energy density in this radiation is given by
∞
0
h ν n(ν) d ν = aBT4 (2.1.6) where aBis the radiation energy constant; in c.g.s. units,
aB = 8π5k4
B
15h3c3 = 7.56577(5) × 10−15 erg cm−3deg−4 (2.1.7)
13Field et al., ref. 12; Thaddeus and Clauser, ref. 12.
14For a summary of this early work with references to the original literature, see G&C, Table 15.1.
15J. C. Mather et al., Astrophys. J. 354, 237 (1990).
16J. C. Mather et al., Astrophys. J. 420, 439 (1994).
17J. C. Mather, D. J. Fixsen, R. A. Shafer, C. Mosier, and D. T. Wilkinson, Astrophys. J. 512, 511 (1999). A 1999 review by G. F. Smoot, in Proc. 3K Cosmology Conf., eds. A. Melchiorri et al. [astro-ph/9902027], gave a temperature 2.7377 ± 0.0038 K (95% confidence), but the result of Mather et al.
seems to be the one usually quoted.
18For a review, see G. Sironi et al., in Proc. Third Sakharov Conf. – Moscow 2002 [astro-ph/0301354].
2000
1500
1000
500
0
5 10 15 20
Frequeny (cm–1)
kJY/Sr
Figure 2.1: Comparison of the intensity of radiation observed with the FIRAS radiometer carried by COBE with a black-body spectrum with temperature 2.728 K, from D. J. Fixsen et al., Astrophys. J. 473, 576 (1996) [astro-ph/9605054]. The vertical axis gives the intensity in kiloJansky per steradian (one Jansky equals 10−23 erg cm−2 s−1 Hz−1);
the horizontal axis gives the reciprocal wavelength in cm−1. The 1σ experimental uncertainty in intensity is indicated by the tiny vertical bars; the uncertainty in wavelength is negligible.
Using T = 2.725 K, this gives an equivalent mass density (reverting to c = 1)
ργ 0= aBT4
γ 0= 4.64 × 10−34g cm−3. Taking the ratio of this with the critical density (1.5.28) gives
γ ≡ ργ 0 ρ0crit
= 2.47 × 10−5h−2 (2.1.8) We will see in Section 3.1 that the photons are accompanied with neutrinos and antineutrinos of three different types, giving a total energy density in radiation (that is, in massless or nearly massless particles):
ρR0=
1 + 3
7 8
4 11
4/3
ργ 0= 7.80 × 10−34g cm−3 , (2.1.9) or in other words, using Eq. (1.5.28),
R≡ ρR0
ρ0,crit
= 4.15 × 10−5h−2 . (2.1.10)
2.1 Expectations and discovery of the microwave background
We see that ρR 0is much less than the critical mass density needed to give K = 0, and much less even than the mass density of ordinary matter seen in stars. It is for this reason that we have generally neglected Rin calculating luminosity distances as a function of redshift.
On the other hand, even at present the number density of photons is relatively very large. Eq. (2.1.1) gives
nγ 0=
∞
0
8π ν2d ν
exp (hν/kBT ) − 1 = 30 ζ (3) π4
aBT3 kB
= 0.3702 aBT3
kB = 20.28 [T (deg K )]3 cm−3, (2.1.11) where ζ (3) = 1.202057 . . . For T = 2.725 K this gives a present number density
nγ 0= 410 photons/cm3 . (2.1.12) This is much larger than the present number density nB 0of nucleons, given by
nB 0 = 3BH2
0
8π GmN
= 1.123 × 10−5Bh2 nucleons/cm3 . (2.1.13) Both nγ and nB vary with time as a−3(t), so the ratio nγ/nB has been the same at least during the whole period that photons have been traveling freely.
* * *
There is an effect of the cosmic microwave background that has long been expected but has been difficult to observe. A cosmic ray proton of moderate energy striking a photon in the cosmic microwave background can only scatter the photon, a process whose rate is proportional to the square of the fine structure constant α 1/137. However, if the proton has sufficiently high energy then it is also possible for the photon to be converted into a π meson in the reactions γ + p → π0+ p or γ + p → π++ n, processes whose rate is proportional to α, not α2. Assuming that high energy cosmic rays come to us from outside our galaxy, we therefore expect a dip in the spectrum of cosmic ray protons at an energy where the cross section for these processes becomes appreciable.19 Although some pions can be produced at lower energy, the effective threshold is at a value of the total energy W of the
19K. Greisen, Phys. Rev. Lett. 16, 748 (1966); G. T. Zatsepin and V. A. Kuzmin, Pis’ma Sh. Exsp.
Teor. Fiz. 4, 114 (1966) [transl. Sov. Phys. JETP Lett. 4, 78 (1966)]; F. W. Stecker, Phys. Rev. Lett. 21, 1016 (1968).
initial proton and photon in the center-of-mass system equal to m= 1232 MeV, the mass of the pion–nucleon resonance with spin-parity 3/2+and isospin 3/2. The center of mass energy is
W = and θ is the angle between them. The threshold condition that W > m thus requires that
qp(1 − cos θ ) > m2− m2p. (2.1.15) The typical energy of photons in black-body radiation at temperature Tγ 0= 2.725K is ργ 0/nγ 0 6 × 10−4eV, while the largest value for 1 − cos θ is 2, so the effective threshold is roughly at a proton energy
pthreshold≈ m2
− m2p
2ργ 0/nγ 0 1020 eV . (2.1.16) This effect is not easy to see. The flux of cosmic ray protons with energies between E and E + dE goes roughly as E−3dE , so there are few protons at these very high energies, roughly one per square kilometer per year above 1019 eV and 0.01 per square kilometer per year above 1020 eV. At these rates, direct observation is clearly impossible, and the cosmic rays have had to be studied indirectly by observation at ground level of the large showers of photons and charged particles that they produce. Also, there is a smooth distribution of photon energies and directions, so one is not looking for a sharp cut-off at 1020eV, but rather for a dip below the E−3curve at around this energy.20 No such effect was observed by the Akeno Giant Air Shower Array,21 but a subsequent analysis of this and other observations showed the effect.22 More recently signs of a drop appeared in the High Resolution Fly’s Eye experiment.23 In 2006 this group announced the observation of a “sharp suppression” of the primary cosmic ray spectrum at an energy of 6 × 1019GeV, just about where expected.24
20For instance, see I. F. M. Albuquerque and G. F. Smoot, Astroparticle Phys. 25, 375 (2006) [astro-ph/0504088].
21M. Takeda et al. Phys. Rev. Lett. 81, 1163 (1998).
22J. N. Bahcall and E. Waxman, Phys. Lett. B556, 1 (2003) [hep-ph/0206217].
23R. U. Abbasi et al., Phys. Rev. Lett. 92, 151101 (2004) [astro-ph/0208243]; Phys. Lett. B619, 271 (2005) [astro-ph/0501317].
24G. B. Thompson, for HiRes Collaboration, in Proc. Quarks ’06 Conf. [astro-ph/0609403]; R. U.
Abbasi et al., astro-ph/0703099.