ASTR 3740 Spring 2007 Homepage
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1. Condition for an accelerating Universe
Suppose that the Universe contains only matter energy (M) and vacuum energy (a cosmological constant L), and that it is geometrically flat
| (1.1) |
| (1.2) |
2. Solutions to Friedmann's equations in a Flat Universe
Suppose that the Universe is flat, k = 0, so that Friedmann's energy equation reduces to
| (2.1) |
| (2.2) |
(a) Case n ¹ 0
Solve Friedmann's equation to show that, for n ¹ 0,
| (2.3) |
(b) Deceleration or acceleration?
For what range of n is the Universe decelerating (da/dt < 0) or accelerating (da/dt > 0)? Is the Universe decelerating or accelerating in the particular cases of a matter-dominated (n = 3) or radiation-dominated (n = 4) Universe?
(c) Case n = 0
The case n = 0 corresponds to vacuum density, which remains constant as the Universe expands. Solve Friedmann's equation for this case to show that
| (2.4) |
(d) For your information (no credit)
You may be wondering whether there is a relation between the index n in this question and the pressure p in the Anti-Gravity question. The answer is yes. It is straightforward to show (but I'm not asking you to do this) from the energy equation d (ra^{3} ) + p d (a ^{3}) = 0 (which you may recognize as the equation dE + p dV = 0 of thermodynamics) that
| (2.5) |
3. Flatness Problem
An amusing statement of this cosmological problem can be found on Ned Wright's graph.
Suppose that the temperature at the moment of the Big Bang was about the Planck temperature ~10^{32} K, and suppose for simplicity that from that time to the present, when the temperature is about the CMB temperature 3 K, the Universe has been radiation-dominated, so that the density r has been declining with cosmic scale factor a as r µ a^{-4}. As you have learned in the lectures, curvature can be described by a “curvature density” r_{K} which declines as r_{K} µ a^{-2}. Observations of the CMB today indicate that the curvature today is quite small, so r_{K}/r is a small number, at most 0.01 or so. Given the above rough assumptions, roughly how big, at most, was the ratio r_{K}/r of curvature to total density at the moment of the Big Bang? What value of W_{K} º r_{K}/(r + r_{K}) does that translate into? [Hint: recall that temperature T varies with cosmic scale factor a as T µ a^{-1}.]