Origin and Evolution of the Universe. Группа авторов
alt=""/>
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 λem = 122 nm is seen at λobs = 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 VAB(t0) = H0DAB(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(tem), where tem 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, vesc = √(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