Origin and Evolution of the Universe. Группа авторов

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      If the Universe has the critical density now, it must have the critical density at all times. Thus, if we can figure out how the density changes as the Universe grows, we can figure out how the Hubble parameter H(t) changes as the Universe grows. For normal matter the density drops by a factor of 8 when the Universe doubles in size. The radiation filling the Universe also contributes to the density, but this density goes down faster than the matter density due to the redshift, dropping by a factor of 16 as the Universe doubles in size. For a critical density Universe, these factors lead to a dimensionless product of the Hubble constant times the age of the Universe H₀t₀ = 2/3 for a matter-dominated Universe and ½ for a radiation-dominated Universe.

      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/ρ_crlt 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.

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