To have a more convenient scale for H0, astronomers use the mixed units of km/s/Mpc. A parsec is 3.26 light years, or 3.09 × 1013 km; a megaparsec (Mpc) is 3.09 × 1019 km. Data by Riess et al. (2011) indicate H0 = 73.8 ± 2.4 km/s/Mpc. The measured ages of the Universe using several methods average to t0 = 12.9 ± 0.9 Gyr (12.9 × 109 years). Because it takes 978 Gyr to travel 1 Mpc at 1 km/s, these values together give H0t0 = (73.8 × 12.9/978) = 0.97 ± 0.08, which is not consistent with the relation H0t0 = 2/3 for a critical-density Universe.
One solution to this problem would be to hypothesize that the expansion of the Universe is accelerating instead of decelerating. This hypothesis requires something that acts like antigravity on large scales, and the cosmological constant introduced by Einstein to cancel gravity in his early model of a static Universe could provide the required effect. But because the Universe is not static, the cosmological constant was regarded as an unnecessary complication by most cosmologists. However, in 1998, the Universe was in fact found to have an accelerating expansion, so the cosmological constant is back in a more modern guise called dark energy. This is a form of density that remains constant as the Universe expands, unlike matter or radiation.
The greatest difficulty in cosmology today is in determining the true distances to objects, as opposed to simply using their recessional velocities in the Hubble law. But to measure the Hubble constant, true distances as well as recessional velocities must be measured. Hubble tried this in 1929, but the distances he used were 5 to 10 times too small, and his value for H0 was 8 times too large. For H0t0 = 1, this gave an age for the Universe of t0 = 1.8 Gyr, which was less than the well-known age of the Earth. This discrepancy motivated the development of the steady-state model of the Universe, in which a(t) = exp(H0(t − t0)). The steady-state model has an accelerating expansion and a large effective cosmological constant. Because exp(H0(t − t0))→ 0 only for t → −∞ the steady-state model gives an infinite age for the Universe. However, the steady-state model made definite predictions about the expected number of faint radio sources, and observations made during the 1950s showed that the predictions were wrong.
The critical density is very low — only six hydrogen atoms per cubic meter for H0 = 74 km/s/Mpc. A very good laboratory vacuum (10−13 atmospheres) has 3 × 1012 atoms per cubic meter. While the critical density is low, the apparent density of the mass contained in visible stars in galaxies, when smoothed out over all space, is at least 100 times smaller! Thus, the Universe appears to be underdense, which means that E in Equation (6) is positive and the Universe will expand forever. However, this situation is unstable. Consider what will happen as the Universe gets 10 times older. If the density is really only 1% of the critical density now, the Universe will expand at essentially constant velocity, and thus will become 10 times larger. As a result, the density will become 1,000 times smaller, since the same amount of matter is spread over 103 times more volume. The critical density will also change because the Hubble parameter, H(t), is a function of time. When the Universe is 10 times older, the value for H will be approximately 10 times smaller. This gives a critical density that is 100 times smaller than the present density. Thus, the ratio of density to critical density becomes 0.1%. But we can start our calculations of the Universe when t = 10−43s, and t0 = 1018s. If the density were 99% of the critical density at t = 10−43s, it would be 90% of the critical density at t = 10−42s, 50% of the critical density at t = 10−41s, 10% of the critical density at t = 10−40s, and so on. For the actual density to be between 10% and 200% of the critical density now, the ratio of density to critical density had to be
at t = 10−43s. This ratio ρ/ρcrlt is known as Ω, and we see that Ω has to be almost exactly 1 early in the evolution of the Universe. Figure 1.3 shows three scale factor curves computed for three slightly different densities 10−9s after the Big Bang. The middle curve has the critical density of 447 sextillion g/cm3, but the upper curve is a universe that had only 1 g/cm3 of 447 sextillion g/cm3 less density and now has a density lower than the observed density of the Universe; the lower curve is a universe that had 1 g/cm3 more and is now at the “Big Crunch.” To get a universe like the one we see requires either very special initial conditions or a mechanism to force the density to equal the critical density. Any physical mechanism that sets the density close enough to the critical density to match the present state of the Universe will probably set the actual density of the Universe to precisely equal the critical density. But most of the density in the Universe cannot be stars, planets, plasma, molecules, or atoms. Instead, most of the Universe must be made of dark matter that does not emit light, absorb light, scatter light, or interact with light in any of the ways that normal matter does, except by gravity.
Figure 1.3. Scale factor a(t) for three different values of the density of the Universe at t = l0−9 seconds after the Big Bang. Note how a very tiny change in the density produces huge differences now.
Cosmic Background
Penzias and Wilson (1965) reported the discovery of a microwave background with a brightness at a wavelength of 7 cm equivalent to that radiated by an opaque, nonreflecting object with a temperature of 3.7 ± 1 degree Kelvin. Further observations at many wavelengths from 0.05 to 73 cm show the brightness of the sky is equivalent to the brightness of an opaque, nonreflecting object (a blackbody) with a temperature of T0 = 2.725 ± 0.001 K. The spectrum of the sky as a function of wavelength differs from an exact blackbody spectrum by less than ±60 parts per million. This shows that the Universe was once very nearly opaque and very nearly isothermal (the same temperature everywhere). By contrast, the Universe now has galaxies scattered about and separated by vast stretches of transparent space. Because the conditions necessary to produce the microwave background radiation are so different from the current conditions, we know that the Universe has evolved a great deal over its history. Since the steady-state model predicted that the Universe did not evolve, its predictions are not consistent with the observed microwave background.
Observations of the microwave background toward different parts of the sky show a small variation in the temperature, with one side of the sky being 3.36 mK (0.00336 Centigrade degrees) hotter and the opposite side of the sky being 3.36 mK colder than the average. The pattern of a hot pole and a cold pole is called a dipole. It is a measure of the peculiar velocity of our solar system at 369 ± 1 km/s relative to the Hubble law. The velocity is the sum of the motions caused by the revolution of the Sun around the Milky Way, the orbit of the Milky Way around the center of mass of the local group of galaxies, and the motion of the local group caused by the gravitational forces from the Virgo Supercluster, the Great Attractor, and other clumps of matter. The local group contains about 30 galaxies, of which the Milky Way and the Andromeda nebula are the biggest; the Virgo Supercluster contains thousands of galaxies. The reader will find more details on galaxies and their clusters in Alan Dressler’s Chapter 2.
After the dipole pattern is accounted for, the remaining temperature fluctuations are very small, only 11 parts per million. These tiny temperature differences were detected by NASA’s Cosmic