# Search results

18. ### Differential equation

Can anyone help me solve \dot{x} = Hx_0 \left( \sqrt{B} \frac{x_0}{x} + \sqrt{1-A-B} \right)
19. ### Vacuum energy density

Homework Statement Show that, in natural units h=c=1, an energy density may be expressed as the fourth power of a mass. If the vacuum energy contributed by a cosmological constant is now of order of the critical density, what is the mass to which this density corresponds? 2. The attempt...
20. ### Statistical physics

Homework Statement The result n_{0 \gamma} = \left( \frac{k_BT_{0r}}{hc} \right)^3 \int_0^{\infty} \frac{8 \pi x^2 dx}{e^x-1} = 2 \frac{\zeta(3)}{\pi^2} \left( \frac{k_BT_{0r}}{hc} \right)^3 is obtained for photons by integrating over the Planck distribution appropriate for bosons. In...
21. ### Perfect square

Homework Statement \left(\frac{\dot{x}}{a}\right)^2 = K\left[b\frac{a}{x} + c\left(\frac{a}{x}\right)^2 + (1-b-c)\right] Show that, for b<1, there is a value of c that makes the right hand side a perfect square of a function of x. 2. The attempt at a solution I guess that a perfect...
22. ### Nucleocosmochronology: hydrogen/helium ratio and its change

Homework Statement Assume that the mass-to-light ratio, M/L, for the galaxy is, and has always been, 10 in solar units. What is the maximum fraction of the total mass that could have been burnt into helium from hydrogen over 10^{10} years? (The mass deficit for the reaction 4H \rightarrow ^4He...
23. ### Surface brightness

Homework Statement Show that the Hubble profile of surface brightness I(r) = I_0 \left(1+\frac{r}{R}\right)^{-2} leads to an infinite total luminosity, while the law I = I_0 exp[-(r/a)^{1/4}], with a a constant, does not. Here I_0 and R are constants and r is the distance from the centre...
24. ### Singularities and Kasner solution

Homework Statement Investigate the possible behaviour of the singularity as t \rightarrow 0 in the Kasner solution. Homework Equations The metric for the Kasner solution is given by ds^2 = c^2dt^2 - X_1^2(t)dx_1^2 - X_2^2(t)dx_2^2 - X_3^2(t)dx_3^2 The Attempt at a Solution I...
25. ### Rotation curve

How can I calculate the rotation curve, v(R), for test particles in circular orbits of radius R around a point mass M?
26. ### Luminosity physics question

Homework Statement Show that the Hubble profile of surface brightness I(r) = I_0 \left(1+\frac{r}{R}\right)^{-2} leads to an infinite total luminosity. r is the distance from the center and R is a constant. 2. The attempt at a solution For large r this is related as I(r) \propto...
27. ### Coordinate transformation

How can I identify the coordinate transformation that turns ds^2 = \left(1+\frac{\epsilon}{1+c^2t^2}\right)^2c^2dt^2 - \left(\frac{\epsilon}{1+x^2}\right)^2x^2 - \left(\frac{\epsilon}{1+y^2}\right)^2y^2 - \left(\frac{\epsilon}{1+z^2}\right)^2z^2 into the Minkowski metric ds^2 = c^2dt^2...
28. ### Conformal time closed Friedmann universe

Homework Statement A closed Friedmann universe contains a single perfect fluid with an equation of state of the form p=w\rho c^2. Transforming variables to conformal time \tau using dt=a(t)d\tau, show that the variable y=a^{(1+3w)/2} is described by a simple harmonic equation as a function of...
29. ### Friedmann Equation

Homework Statement By substituting in \left( \frac{\dot{a}}{a_0} \right)^2 = H^2_0 \left(\Omega_0 \frac{a_0}{a} + 1 - \Omega_0 \right) show that the parametric open solution given by a(\psi)=a_0 \frac{\Omega_0}{2(1-\Omega_0)}(\cosh{\psi} - 1) and t(\psi)=\frac{1}{2H_0} \frac{\Omega_0}{(1 -...
30. ### Coordinate distance

Homework Statement For a universe with k=0 and in which (a/a_0) = (t/t_0)^n where n<1, show that the coordinate distance of an object seen at redshift z is r=\frac{ct_0}{(1-n)a_0}[1-(1+z)^{1-1/n}]. 2. The attempt at a solution I have used r=f(r)=\int_{t}^{t_0}...