which is Hubble's law. (Remember, this approximation is for small At.)
It is important to note that equation (20.29) tells how to interpret the redshifts of distant galaxies. It is tempting to say that these galaxies are moving relative to us and their radiation is therefore Doppler-shifted. However, in computing a relative velocity for a Doppler shift, we take the difference between the velocities of the two galaxies. These two velocities must be with respect to a co-moving coordinate system. Therefore, apart from its peculiar motion, each galaxy's velocity is zero with respect to this coordinate system. Therefore, strictly speaking, there is no Doppler shift due to the expansion of the universe. The redshift arises as a result of the increase in wavelengths of all photons moving through an expanding universe. We therefore call it the cosmological redshift. Any additional motions with respect to the co-moving coordinates would produce a Doppler shift (red or blue) in addition to the cosmological redshift.
As a consequence of this, we should not directly interpret the redshift of a galaxy as giving a particular distance. The amount of redshift just tells us the amount by which the scale factor has changed between the time the photon was emitted and the time it was detected. For this reason, we often talk about the redshift of a particular galaxy, and don't bother to convert it to a distance. For example, we simply say that 3C273 is at z = 0.15. To convert a redshift to a distance we need a particular model for how R(t) has evolved.
In general relativity, the solutions for R(t) are different from the Newtonian case. The equation in general relativity, analagous to equation (20.19), is called Einstein's equation. There are two parts relevant to our discussion of cosmology:
where P is the gas pressure, usually taken to be zero, except when the matter is hot and dense. Remember, the density at any time is p0/R3. So the mathematical effects of a non-zero density and a non-zero A are different, since the density term will have an extra factor of the variable R. We will look at separate cases below.
Since H = (1/R)(dR/dt), we can write Einstein's equation as
From these, the deceleration parameter becomes qo _
Note that this reduces to the classical case (equation 20.24) when A = P = 0.
The integration of these equations to give R(t) is generally quite difficult. To simplify the situation, it is useful to look at limiting cases, namely zero cosmological constant and zero density. Zero cosmological constant would also be an approximate description of a universe with small cosmological constant, where the matter term dominates. Likewise, zero density would also approximately describe a case with low density where the cos-mological constant dominates. In each case, we must also look separately at zero, positive and negative curvature. Below, we give results for various limiting cases without deriving them. You can verify that they are solutions by plugging them into the appropriate equations (see Problem 20.18). The results for some of these models are shown in Fig. 20.9.
Models with zero A are called Friedmann models. For a flat universe (k = 0), R(t) in these models is given by
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