General Friedmann equation - how to solve this differential equation?

  • Thread starter Thread starter beertje
  • Start date Start date
beertje
Messages
3
Reaction score
2
Homework Statement
Solve the differential equation
Relevant Equations
(adot / a)^2 = k / a^q
friedman.png


Hello fellow physicists, I am taking a course "Introduction to Cosmology" and I am asked to solve this equation called the Friedmann equation. I understand what it represents (scale factor of cosmic time) but I have no idea how to solve this differential equation, even though I took a whole course on solving those (Euler-Lagrange etc.)

Please give me a small pointer :)
 
Physics news on Phys.org
Take the square root (which sign is the right one?) and then separate variables.
 
vanhees71 said:
Take the square root (which sign is the right one?) and then separate variables.
Thanks! That was way more obvious than I thought. Summer break always breaks me up.
 
  • Haha
  • Like
Likes PhDeezNutz and vanhees71
##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).## ##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).## ##y=x-x_0 \quad x=y+x_0 \quad dy=dx.## The boundaries remain infinite, I believe. ##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).## ##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).## I then resolved the two...
Hello everyone, I’m considering a point charge q that oscillates harmonically about the origin along the z-axis, e.g. $$z_{q}(t)= A\sin(wt)$$ In a strongly simplified / quasi-instantaneous approximation I ignore retardation and take the electric field at the position ##r=(x,y,z)## simply to be the “Coulomb field at the charge’s instantaneous position”: $$E(r,t)=\frac{q}{4\pi\varepsilon_{0}}\frac{r-r_{q}(t)}{||r-r_{q}(t)||^{3}}$$ with $$r_{q}(t)=(0,0,z_{q}(t))$$ (I’m aware this isn’t...
It's given a gas of particles all identical which has T fixed and spin S. Let's ##g(\epsilon)## the density of orbital states and ##g(\epsilon) = g_0## for ##\forall \epsilon \in [\epsilon_0, \epsilon_1]##, zero otherwise. How to compute the number of accessible quantum states of one particle? This is my attempt, and I suspect that is not good. Let S=0 and then bosons in a system. Simply, if we have the density of orbitals we have to integrate ##g(\epsilon)## and we have...
Back
Top