Читать книгу Origin and Evolution of the Universe - Группа авторов - Страница 14

We can see only a finite piece of the Universe. The naive estimate for how far we can see is ct₀, the speed of light times the age of the Universe. This is, in fact, the distance traveled by photons coming from the most distant visible parts of the Universe, as measured by the photons. But when one defines distances in an expanding universe, the convention is to measure all intervals at the current time, t₀. Because the Universe has expanded since t = 0, the earlier parts of the photon’s journey get extra credit. We can compute the distance we can see in a critical density universe by dividing the age of the Universe into more and more intervals. With one interval, we get ct₀. With two intervals, we get 0.5 ct₀/0.5^2/3 + 0.5 ct₀ = 1.29 ct₀ because the first half of the journey has expanded by the factor 1/a(t₀/2) = 1/0.5^2/3. With four intervals, we obtain 1.58 ct₀. With a very large number of intervals, we get 3 ct₀ which is the distance to the horizon. For t₀ ~ 13 Gyr, this is 40 billion light years.

Consider now an observer 400,000 years after the Big Bang. The distance to the horizon is 3 ct, or about 1.2 million light years. This observer (really just a cloud of gas) will try to get in thermal equilibrium with the region it can see, which extends to a 1.2-million-light year radius. If thermal equilibrium can be achieved, a patch of constant temperature 1.2 million light years in radius can be created. This patch will grow to 1 billion light-years in radius as the Universe expands from 400,000 years after the Big Bang until now. But our horizon now is 40 billion light years in radius. Thus, the constant temperature patch subtends an angle of only 1/40 radian, which is only three times the diameter of the full moon. But we see an almost constant temperature over the entire sky. For a universe to be as isotropic (identical appearance in all directions) as the one we live in requires either very special initial conditions or a mechanism to force the temperature to be constant over the entire observable Universe.

The Big Bang model described above is in good agreement with the observed Universe, but it required very special initial conditions such as the following to explain two different facts:

1.The fact that ρ/ρ_crit = Ω is close to 1 today means that Ω was nearly exactly 1 initially.

2.The microwave background temperature is nearly identical in patches that could not have communicated with each other before the Universe became transparent 400,000 years after the Big Bang.

Guth (1981) proposed the inflationary scenario, which attempts to make these initial conditions less special (see Guth and Steinhardt, 1984). The inflationary scenario supposes that at some time during the early history of the Universe, a very large dark energy density existed, which led to a rapidly accelerating expansion of the Universe. In Russia, Starobinsky (1979) began to study Universes in which a rapidly accelerating expansion preceded the normal decelerating expansion of the Big Bang. During this inflationary epoch, the Universe was like the steady-state model, but only temporarily. A suggested prologue of the inflationary scenario is a normal Big Bang expansion until a time ts. In Linde’s (1994) model of perpetual inflation, this prologue is absent because the Universe begins during inflation. At ts after the Big Bang, Act I, the inflationary phase, begins. During this time, the Hubble constant is H = 0.5/ts. During the inflationary epoch, the Universe expands by a factor of 10⁴³ or more. At about 200ts, the inflationary epoch ends, and Act II begins. Act II is a standard Big Bang model, but with initial conditions set during the inflationary epoch. For example, a very small patch can become isothermal, and inflation makes the small isothermal patch into a huge isothermal patch, which expands to become much bigger than the observable Universe.

But why does inflation make Ω = 1 almost exactly? The answer lies in the steady-state nature of the inflationary epoch. When the Universe expands, one expects the density to go down, but in a steady-state model, the density must remain constant. Thus, there must be a continuous creation of matter during a steady-state epoch. This means that the mass M in Equation (6) gets bigger, like r³, instead of staying constant. Then the potential energy term grows increasingly negative as the Universe inflates, in proportion to r². To conserve energy, the kinetic energy term mV²/2 must get bigger, so V gets bigger, and the expansion accelerates as expected in a steady-state situation. As noted in the beginning of this chapter, in the first section, on the expansion of the Universe, this acceleration is equivalent to introducing Einstein’s cosmological constant. But note that

Therefore, if before inflation the Universe had 0.5 mV² = 2, and GMm/r = 1, so E = 0.5 mV² − GMm/r = 1, and Ω = 0.5, then after inflating by a factor of 10⁴³, we have GMm/r = 10⁸⁶. Then 0.5 mV² must be 10⁸⁶ + 1 to preserve E = 1. Thus, after inflation, Ω = 1 − 10⁻⁸⁶, which is well within the tight limits given in Equation (8).

Thus, inflation solves two problems in the Big Bang model, but creates another question: why does the Universe have a large cosmological constant during the inflationary epoch? The answer to this question lies in high energy particle physics, under the topic of unified field theories. The Weinberg-Salam model that unifies the electromagnetic and weak nuclear interactions into a single electroweak theory requires a large vacuum energy density. A vacuum energy density acts just like a cosmological constant, and in the Weinberg-Salam theory, the Universe makes a phase transition from a state with a large cosmological constant when T is greater than 10¹⁵K (or when the energy density is equivalently high) to the normal state with small or zero cosmological constant at lower temperatures. A similar unification of the strong nuclear force with the electroweak force gives a grand unified theory, or GUT. In GUTs, the transition from high-to-low cosmological constant occurs when T is greater than 10²⁸K. Either of these phase transitions could cause an inflationary epoch.

Inflation produces such a tremendous enlargement that even tiny objects such as the quantum fluctuations that occur on subatomic scales get blown up to be the size of the observable Universe. But while the fluctuations are being inflated, new small ones are always being created. Because the time for the Universe to double in size is constant during inflation, the power in the fluctuations in each factor of two size bins is constant. Let us measure time in units of the doubling time, and size in units of the speed of light times the doubling time. Then at t = 10, the fluctuations created between t = 9 and t = 10 are all about size r = 1 because they have existed for less than 1 doubling time. At t = 1, there should have been the same amount of fluctuations at size r = 1. But these fluctuations now have size r = 512 and t = 10. Hence, at t = 10, the amount of fluctuations at r = 512 and r = 1 should be the same. The same argument, applied at t = 2, 3, 4, … , shows that amount of fluctuations at sizes r = 256, 128, 64, … should all be equal to the amount at r = 512 and r = 1.

These fluctuations become temperature variations, and the equality of the amount of variations on different angular scales is a prediction of the inflationary scenario. In 1992, the COBE team announced the discovery of temperature variations with a pattern that is consistent with equal variations in angular size bins centered at 10°, 20°, 40°, and 80°. Figure 1.4 compares a predicted sky map produced using equal power on all scales to the actual sky map measured by COBE. The two maps look quite similar, and a detailed statistical comparison shows that the equal power on all scales prediction of inflation is quite consistent with the observations.

Figure 1.4. Top: The temperature fluctuations measured by the COBE DMR without subtracting the Milky Way signal. Bottom: A model sky constructed using an equal power on all scales random process.