Consider the following FLRW spacetime: t² ds² = -dt² + tz (dx² + dy² + dz²), where t is a constant. a) State whether this universe is spatially open, closed or flat. b) c) d) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function of time t, starting at t = 0. Taking galaxy A to be located at (x, y, z) = (0, 0, 0), determine the proper distance to galaxy B located at (x, y, z) = (L,0,0). Determine the recessional velocity of galaxy B with respect to galaxy A. The Friedmann equations are 2 k + 8πG 3 ä 4TG (p+3p). a 3 Use these equations to determine the energy density p(t) and the pressure p(t) for the FLRW spacetime specified at the top of the page. e) Given the result of question B3.d, state whether the FLRW universe in question is (i) radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv) none of the previous. Justify your answer. f) A conformally flat spacetime is one that admits coordinates such that ds² = (x) dxdx², where (x) is any function of the coordinates xt, and nμ is the usual Minkowski metric. Determine whether the FLRW spacetime in question is conformally flat. Hint: Consider a transformation of the coordinate t.

Consider the following FLRW spacetime: t² ds² = -dt² + tz (dx² + dy² + dz²), where t is a constant. a) State whether this universe is spatially open, closed or flat. b) c) d) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function of time t, starting at t = 0. Taking galaxy A to be located at (x, y, z) = (0, 0, 0), determine the proper distance to galaxy B located at (x, y, z) = (L,0,0). Determine the recessional velocity of galaxy B with respect to galaxy A. The Friedmann equations are 2 k + 8πG 3 ä 4TG (p+3p). a 3 Use these equations to determine the energy density p(t) and the pressure p(t) for the FLRW spacetime specified at the top of the page. e) Given the result of question B3.d, state whether the FLRW universe in question is (i) radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv) none of the previous. Justify your answer. f) A conformally flat spacetime is one that admits coordinates such that ds² = (x) dxdx², where (x) is any function of the coordinates xt, and nμ is the usual Minkowski metric. Determine whether the FLRW spacetime in question is conformally flat. Hint: Consider a transformation of the coordinate t.