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/ρcrit = 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.
Acoustic Scale
Observations of the CMB made since 2000 have shown a preferred angular scale of 0.8°. This scale is about 10 times smaller than the beam size of the COBE experiment. The Wilkinson Microwave Anisotropy Probe, launched by NASA in 2001, had a beam size of 0.2° and could accurately measure this preferred scale. The Planck satellite, launched by the European Space Agency (ESA) in 2009, further refined these measurements with a beam size of 0.08°. This preferred scale is related to the horizon angle discussed above. At times earlier than 400,000 years after the Big Bang, the Universe was ionized and the ionized plasma strongly scattered the photons of the CMB (cosmic microwave background). The pressure of the photons led to sound waves, or acoustic oscillations, that traveled at a large fraction of the speed of light. Thus, the two parts of a density perturbation will split up. The dark matter density perturbation will stay fixed, but the ionized gas perturbation will move away, traveling as a sound wave, due to the pressure of the CMB photons. Then, 400,000 years after the Big Bang, the Universe cools to the point where the plasma recombines into transparent gases. This leads