R6

1. Derive the Friedman-Lemaître equations [math]\displaystyle{ \left({\dot a\over a}\right)^2=8\pi G{\rho\over 3} }[/math] and [math]\displaystyle{ \left({\ddot a\over a}\right)=-4\pi G\left({\rho\over 3}+p\right) }[/math] from Einstein's equations. What is the frame that leads to these?
2. Consider a Bogolyubov transformation for fermionic fields [math]\displaystyle{ a^-_k=\alpha_k b_k^-+\beta_k b_{-k}^+ }[/math], [math]\displaystyle{ a^+_k=\gamma_k b_k^+ +\delta_k b_{-k}^- }[/math]. What are the conditions on the coefficients?
3. Find the relation between the two bosonic vacua [math]\displaystyle{ |0;a\gt }[/math] and [math]\displaystyle{ |O;b\gt }[/math] that are related by the Bogolyubov transformation [math]\displaystyle{ a^-_k=\alpha^*_k b_k^-+\beta_k b_{-k}^+ }[/math], [math]\displaystyle{ a^+_k=\alpha_k b_k^+ +\beta_k b_{-k}^- }[/math].
4. Show that [math]\displaystyle{ v_k(\eta) }[/math] must have a Wronskian equal to [math]\displaystyle{ 2i }[/math] for the commutation relations of the fields to lead to a Hilbert space structure.
5. Write the expression of the hamiltonian of a scalar field in an expanding universe as a function of creation and annihilation operators.