Читать книгу Origin and Evolution of the Universe - Группа авторов - Страница 11

The discovery by Hubble (1929) that distant galaxies are moving away from us with a velocity, V, that is proportional to their distance, D, was the first evidence for an evolving Universe. Observations show that this recessional velocity is

where the coefficient H₀ is known as the Hubble constant and Vp the peculiar velocity of the galaxy, which is peculiar in the sense that it is different for each galaxy. Typical values of Vp are 500 km/s, and the recessional velocities V range to several times 100,000 km/s for the most distant visible objects.

The Hubble law does not define a center for the Universe, although all but a few of the nearest galaxies seem to be receding from our Milky Way galaxy. Observers on a different galaxy, A, would measure velocities and distances relative to themselves, so we replace V by V–VA and D by D–DA. The Hubble law for A is

which is exactly the same as the Hubble law seen from the Milky Way (except for slightly different peculiar velocities), provided that VA = H₀DA. Thus, every galaxy whose velocity satisfies the Hubble law will also observe the Hubble law. An observer on any galaxy sees all other galaxies receding; thus, the Big Bang model is “omnicentric.”

The idea that the Universe looks the same from any position is codified in:

The Cosmological Principle: The Universe is homogeneous and isotropic.

Note that the Hubble law does not define a new physical interaction that leads to an expansion of the Universe. Instead, it is only an empirical statement about the observed motions of galaxies. Each individual object moves on a path dictated by its initial trajectory and the forces that act on it. Since a uniform distribution of matter produces no net force by symmetry, the only forces felt by an object are caused by the structures around it, and these forces can be computed to good accuracy using the equations of Newtonian mechanics and gravity. On small scales, local forces dominate the motions of objects. For example, the orbit of an electron in an atom is determined by electrostatic forces, and the distance between the electron and the nucleus does not increase with time. The orbits of the planets in our solar system are determined by the gravitational force of the Sun, and the distance between a planet and the Sun does not follow the Hubble law. Note that the individual galaxies in Figure 1.1 do not expand. But the motion of galaxies more distant than 30 million light years is well described by the Hubble law, and this observed fact tells us that the density distribution in the early Universe was almost uniform and that the initial peculiar velocities were small.

Figure 1.1 Schematic evolution of our expanding universe and its contents. In the earlier stage, at the top, a given volume of space has a high density of matter (yellow objects) and background photons. The latter are shown in blue because they have short wavelengths and high energies. In a later stage — middle — this volume has expanded so that its density of matter has dropped. At the same time, the density of photons has dropped, as their average wavelengths have increased (now shown in green). The photons have moved. However, the relative positions of the matter (yellow galaxies) are preserved because they follow a pure uniform Hubble expansion. In a still later stage of the expansion — bottom frame — this region has further expanded. The space density of matter is still lower, and the density of photons has dropped further. The average wavelength of the photons has increased, corresponding to a lower temperature of this background radiation, so that they are now shown in dark red.

The recessional velocity of an object is easily measured with use of the Doppler shift, which causes the length of electromagnetic waves received from a receding object to be larger than the wavelength at which they were emitted by the object. Because the long wavelength end of the visible spectrum is red light, the Doppler shift of a receding object is called its redshift. The observed wavelength λ_obs is larger than the emitted wavelength λ_em and the ratio

defines the redshift z used by astronomers. For velocities that are small compared with the speed of light, the approximation v = cz can be used. The most distant known quasar has z = 7.54. For this object, the ultraviolet (UV) Lyman α line of hydrogen with λ_em = 122 nm is seen at λ_obs = 1 μm [10⁻⁶m] in the near infrared. For such large z, corrections to the Doppler shift formula are needed for velocities approaching the speed of light. Figure 1.2 compares the absorption line spectra of a star and three galaxies at progressively larger distances moving up the figure. The characteristic pattern of absorption lines seen in a local star (with no redshift), can be seen to shift further and further to the red wavelengths. This is predicted by Hubble’s law, which says that the more distant a galaxy is, the higher its recession velocity from Earth is. Through the Dopper effect, these increasing velocities are observed as increasing shifts of the spectrum to the red — the increase of redshift with distance.

The Hubble law in Equation (1) applies to the relative velocity between any pair of galaxies. For example, the velocity of galaxy A with respect to galaxy B is V_AB(t₀) = H₀D_AB(t₀), where DAB(t₀) is the separation now (the time “now” is denoted t₀) between galaxies A and B. If we consider the separation between A and B after a small time interval Δt, it is

The time interval Δt must be a small fraction of the age of the Universe, and yet the distance light travels in Δt must be larger than structures like clusters of galaxies in which local forces produce large peculiar velocities. Observations of the Universe show that it is smooth enough on medium-to-large scales for Equation (4) to be valid. The factor (1 + H₀Δt) is independent of which pair of galaxies A and B is chosen, so it represents a universal scale factor that describes the expansion of every distance between any pair of objects in the Universe. This means that the patterns of galaxies in the Universe retain the same shape while the Universe expands, seen schematically in Figure 1.1. We call the universal scale factor a(t), so

Figure 1.2 Schematic illustration of visible spectra of several objects. The star has a characteristic set of absorption lines (black), which are at almost the wavelengths we see in the laboratory (bottom), because the star has such a small Doppler shift with respect to Earth. The relatively nearby galaxy (middle) shows the same pattern of dark absorption lines as in the star, but all shifted to longer wavelengths. The more distant galaxy, following Hubble’s Law, has all of its wavelengths shifted further to the right (to the red). And the most distant galaxy (top) has the largest redshifts. For example the indicated absorption line which is at 440 nanometers in the laboratory (not moving), is shifted all the way to 580 nanometers in the very distant galaxy, and all of its other absorption lines are shifted by this same ratio of 1.32. Source: scienceconnected.org.

for times close to the present. Note that a(t₀) = 1 by definition.

If there is no acceleration because of gravity, objects will move with constant velocity and Equation (5) is true even if Δt is not small. In this case, when Δt = −1/H₀, a(t₀ + Δt) = 0, where a negative Δt denotes an epoch earlier than the present. Thus, all distances in the Universe go to zero at a time 1/H₀ ago. (By its definition, the Hubble constant has the units of the inverse of time. Therefore it is common to refer to its inverse, 1/H₀, as the “Hubble Time.”) We normally simplify discussions by defining the moment with a(t) = 0 (the “Big Bang”) to be t = 0. This definition makes the age of the Universe equal to the current time, t₀. For the no-acceleration case, a(t) = t/t₀ and the product of the Hubble constant and the age of the Universe is H₀t₀ = 1, then the Hubble time is expressed in the same units as the current age of the Universe. In other words, the age of a Universe with no acceleration is always equal to its current Hubble Time. This implies that observers who lived earlier in the history of the Universe, with a smaller t₀, would find a larger Hubble constant H₀. Thus, the Hubble constant is not a physical constant like the electron charge e, because, although the Hubble constant is the same everywhere in the Universe, it changes with time. We call this changing value the Hubble parameter H(t) and define H₀ = H(t₀).

The exact formula for the redshift of an object is 1 + z = a(t₀)/a(t_em), where t_em is the time the light was emitted. This states that wavelengths of light expand by exactly the same scale factor that applies to the separations between pairs of galaxies.

The acceleration caused by gravity vanishes only if the Universe is empty, with no mass. When masses are present, gravity provides an attractive force that causes the expansion to slow down. This means that velocities were greater in the past; thus, for a given expansion rate now (H₀), the time since a = 0 is smaller than it would have been without any deceleration. In the most likely case, the density of the Universe is very close to the critical density that divides underdense Universes that expand forever from overdense universes that will eventually stop expanding and recollapse.

When a small object of mass m is moving under the influence of gravity near a large mass M, the equation that relates its velocity V and distance r from the large mass is

where E is the total energy, which is conserved, ½mV² is the kinetic energy, and −GMm/r is the gravitational potential energy. Here G is the constant of gravitational force. We can use this simple equation in cosmology, with m being a galaxy and M being the mass of the Universe within radius r, which is the density ρ times the volume of a sphere (4π/3)r³. The sphere is centered at r = 0, and the galaxy m is located on its surface. (Proving that we can use this equation requires general relativity.) Because all matter at larger distances than r has larger velocities than H₀r, the matter outside the sphere stays outside. Newton showed that the gravitational force on m from matter outside the sphere is zero, and this is still true under general relativity. Because all matter at smaller distances than r has smaller velocities than H₀r, the matter inside the sphere stays inside. Thus, the mass of the sphere is constant. For a body to just barely escape from r to ∞ requires a total energy E = 0. This gives the formula for the escape velocity, v_esc = √(2GM/r). When the Universe has the critical density, the Hubble velocity H₀r is equal to the escape velocity, which gives an equation for the mass M leading to the critical density as follows:

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.