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unscientific
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Homework Statement
(a)Find how ##\rho## varies with ##a##.
(b) Show that ##p = \frac{2}{\lambda^2}##. Find ##B## and ##t_0##.
(c) Find ##w## and ##q_0##. What values of ##\lambda## makes the particle horizon infinite? Find the event horizon and age of universe.
(d) Find luminosity distance ##D_L## in terms of redshift ##z##. Find ##q_0## by expanding.[/B]
Homework Equations
The Attempt at a Solution
[/B]
Part(a)
For ##V_0 = 0##, we can see that ##\rho = P = \frac{1}{2} \dot \phi^2##. Thus ##w=1##. For dependency on ##a##:
[tex]\ddot \phi + 3(\frac{\dot a}{a})\dot \phi = 0[/tex]
[tex]\dot \rho + 12(\frac{\dot a}{a}) \rho = 0[/tex]
[tex]a^{-12}\frac{d}{dt}(\rho a^{12}) = 0 [/tex]
[tex] \rho \propto a^{-12}[/tex]
Part (b)
I'll replace the ##p## by ##x## to avoid confusion with pressure ##P##. Given ##a(t) = t^x## and ##\phi = BM ln(\frac{t}{t_0})##, we have ##\frac{\dot a}{a} = \frac{x}{t}## and ##\dot \phi = \frac{BM}{t}## and ##\ddot \phi = -\frac{BM}{t^2}##.
Substituting into equation of motion:
[tex]\frac{-BM}{t^2} + 3\left(\frac{x}{t}\right)\left(\frac{BM}{t}\right) - \frac{\lambda V_0}{M} \left( \frac{t}{t_0} \right)^{-\lambda B}[/tex]
[tex]BM(3x - 1) - \frac{\lambda}{M} V_0 t^2 \left( \frac{t}{t_0} \right)^{-\lambda B} = 0 [/tex]
Substituting into FRW equation:
[tex]\frac{x^2}{t^2} = \frac{8\pi G}{3} \left[ \frac{2}{2}\left(\frac{BM}{t}\right)^2 + V_0 \left( \frac{t}{t_0} \right)^{-\lambda B} \right] [/tex]
[tex]x^2 = \frac{B^2}{6} + \frac{8 \pi G}{3}V_0 t^2 \left( \frac{t}{t_0} \right)^{-\lambda B} [/tex]
Using our result from the equation of motion:
[tex]x^2 = \frac{B^2}{6}+ \frac{8 \pi G}{3} \left[ \frac{BM^2}{\lambda} (3x-1) \right] [/tex]
[tex]x^2 - \left(\frac{B}{\lambda}\right)x + \left( \frac{B}{3\lambda} - \frac{B^2}{6} \right) = 0 [/tex]
Can't seem to get ##x## solely in terms of ##\lambda##, am I doing something wrong?
Part(d)
The metric for a flat, isotopic and homogeneous universe is given by
[tex]ds^2 = -c^2 dt^2 + a(t)^2 \left[ d\chi^2 + S^2(\chi) \left( d\theta^2 + sin^2\theta d\phi^2 \right) \right] [/tex]
Flux is given by ##F = \frac{L}{4\pi D_L^2}##. From the metric, proper area is given by ##A = 4\pi(a_0 \chi)^2 = 4\pi \chi^2##. But due to redshift, photons are delayed by ##\nu_0 = \frac{\nu_e}{1+z}##. Thus we have
[tex]D_L = \chi(1+z)[/tex]
where ##\chi## is the comoving distance.
This is only in first order, how do I expand it in 2nd order?!
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