SOLVED Physics 396 Homework Set 10

1. A distant galaxy has a redshift of z = 0.2. According to Hubble’s law, how far away was the galaxy, in light years, when the light was emitted if the Hubble parameter is 72 km/s/Mpc? 2. When the flat Robertson-Walker line element, given by Eq. (18.1) in your text, and the perfect fluid energy-momentum tensor are inserted into the Einstein equations of GR, one arrives at a set of non-linear, ordinary di↵erential equations of the form 8⇡⇢ = 3 a˙ 2 a2 (1) 8⇡p = ✓ 2 a¨ a + a˙ 2 a2 ◆ (2) 0= ˙⇢ + 3 a˙ a (⇢ + p) (3) where ⇢ and p are the density and pressure, respectively. It is noted that a dot represents a time derivative, a double-dot represents two time derivatives. Also notice that Eq. (3) is not a field equation, but rather a conservation statement. These di↵erential equations represent the Standard Model of Cosmology. a) Show that Eq. (3) is a redundant equation by inserting Eqs. (1) and (2) for ⇢ and p into Eq. (3). Hint: when performing derivatives, notice that a˙ 2 a2 = a˙ a 2 . Having 2 equations and 3 unknowns (⇢, p, and a), we have an underdetermined system of equations. At this stage one usually employs an equation of state (EoS) of the form p = w⇢ (4) where w is called the equation of state parameter.1 In general, w can depend on time, however, here we will treat it as a constant. We now have 3 equations and 3 unknowns. At this point, Eq. (3) will yield an expression relating the density to the scale factor. 1Notice: w = 0 corresponds to pressureless matter, w = 1/3 corresponds to radiation, and w = 1 corresponds to vacuum energy. 1 8Tp This will allow us to decouple Eqs. (1) and (2), thus yielding two di↵erential equations involving only the scale factor. b) Using Eqs. (4) and (3), find ⇢ = ⇢(a). Hint: eliminate the pressure from Eq. (3) in favor of the density, separate variables, and integrate. Answer: ⇢(t) = ⇢0a(t) 3(1+w) (5) c) Using Eqs.(1)-(3) and (4), obtain an expression for the acceleration of the Universe, a/a ¨ , in terms of the density of the Universe. d) For what values of w does one obtain an accelerated expansion of the Universe? 3. Reconsider the flat Friedman-Robertson-Walker field equations of cosmology, given by Eqs. (1) - (3). a) Show that a(t) = a0 t 2/3(1+w) is a solution to Eq. (1). b) Show that a(t) = a0 t 2/3(1+w) is a solution to Eq. (2). c) Solve Eqs. (1) and (2) for when w = 1. Hint: Use the fact that d dt ✓a˙ a ◆ = a¨ a a˙ 2 a2 . (6) 4. In class, we arrived at an equation of motion describing the scale factor of the form 1 2H2 0 ✓da dt ◆2 + Uef f (a)=0, (7) where Uef f (a) ⌘ 1 2 ✓ ⌦va2 + ⌦m a + ⌦r a2 ◆ (a(t0) = 1). (8) It is noted that Eq. (7) is of the form of a 1D conservation of energy expression for a particle with zero energy in Newtonian mechanics. Using your favorite plotting software (i.e. Maple, Matlab, Wolframalpha, etc.), plot Uef f (a) vs a for a) Matter-dominated flat FRW spacetime, ⌦m = 1, ⌦r = 0, ⌦v = 0, b) Radiation-dominated flat FRW spacetime, ⌦m = 0, ⌦r = 1, ⌦v = 0, 2 c) Vacuum-dominated flat FRW spacetime, ⌦m = 0, ⌦r = 0, ⌦v = 1, d) Our flat FRW spacetime, ⌦m = 0.3, ⌦r = 5 ⇥ 105, ⌦v = 0.7. Your a-axis should have a range of 0.5