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Matter is all the stuff in the universe stars, planets, comets, asteroids, and galaxies (and whatever else there may be). When constructing the stress-energy
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tensor, we model matter as dust. We will indicate the energy density of matter as m . The pressure Pm = 0. Just as a model to keep in your head for the moment, think of the universe as a box lled with a diffuse gas, with the particles that make up the gas representing galaxies. The expansion of the universe is represented by the volume of the box expanding. But the number of particles in the box remains the same therefore, the number density decreases with the expansion. In the real universe, we model this behavior by saying as the universe expands, the energy density of matter m decreases.
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We indicate the energy density of radiation by R and the pressure by PR and treat radiation as a perfect uid. As the universe expands, the radiation density (think in terms of photons) decreases like it does in the case of matter. However, the energy density for radiation decreases faster because photons are redshifted as the universe expands, and hence loose energy.
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Recent evidence indicates that the expansion of the universe is accelerating, and this is consistent with a nonzero cosmological constant. The cosmological constant represents vacuum energy that is modeled as a perfect uid with the condition that = p.
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The evolution of the scale factor a (t) is in uenced by whether or not the universe is matter dominated or radiation dominated. The ratio of M / R tells us whether the universe is matter or radiation dominated. In our time, observation indicates that the ratio M / R 103 and therefore the universe is matter dominated. In the early history of the universe, it was radiation dominated. We will see later that as the universe ages, it will eventually become dominated by vacuum energy.
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THE HUBBLE PARAMETER
The Hubble constant or Hubble parameter indicates the rate of expansion of the universe. It is de ned in terms of the time derivative of the scale factor as H= a a (12.7)
Cosmology
Remember that the scale factor is a measure of the size of the universe. From the equation, the units of the Hubble constant can be seen to be sec 1 . However, for convenience in astronomical applications the units are often stated as km/sec/Mpc, where Mpc are megaparsecs, which is a measure of distance (see below). In addition, there is a great deal of uncertainty about the actual value of the Hubble constant; therefore, it is often de ned as H0 = 100h km/sec/M pc where present evidence indicates h 0.7.
THE HUBBLE TIME
The inverse of the Hubble constant is the Hubble time, which is a rough estimate of the age of the universe. At the Hubble time, all galaxies in the universe were located at the same point. The Hubble length is a measure of cosmological scales, and is given by d = c/H0 .
HUBBLE S LAW
The distance of a galaxy from us is related to its velocity by Hubble s law v = H0r
THE DECELERATION PARAMETER
We can quantify the rate of change of the expansion with the deceleration parameter. It has a simple de nition q= aa 2 a
OPEN UNIVERSE
An open universe is one that expands forever. The geometry of an open universe is one of negative curvature (think potato chip or saddle, as in Fig. 12-4) and it can in principle extend to in nity. For an open universe, k = 1.
CLOSED UNIVERSE
A closed universe expands to some maximum size, then reverses, and then falls back on itself. Think of a closed universe as a sphere. From our discussion in the previous section, this corresponds to k = +1 and is illustrated in Fig. 12-2.
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