Question

2. Early Density Perturbations The density perturbations in non-relativistic matter om (above the Jeans length) has different time dependence in the matter-dominated era vs. the radiation-dominated era. We derived the former in class; here we will derive the behavior of in the radiation-dominated era. Assume a flat model with a,m 1 and Ω0.A 0 for simplicity. For clarity, we add subscript m to δ to denote perturbations in the matter component, and subscript r' to to denote perturbations in the radiation component (a) Assume that the perturbation in radiation, δ?: is negligible. Show that the linearized evolution equation for δm can be written as 3 dy 2y(1 +y) dy 2y(1) where y pm/pr a/aeg and ae is the mass-radiation equality time. (b) At late time (y1), compare your solutions for 6m with those derived in class. (c) Verify that o y + 2/3 is a solution to eq. (3) in general. Can Sm grow much in the radiation- dominated era? You don't have to compare part b answer with the one in class. Just come up with a solution for part b. Do your best in other parts 3 sinh (sinh 6- 0) (cosh 1)2

Answer 1

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som the equation 2 . have used 七 exlobHon ens Co sound speed iven point. dヒ

3 Tt Geo 3oom the iven equation Hese we have aten em a eyam Hence we have a2 hese we got dam23 dm3

니 03 とM . e ineaǐ3ed e80ldion equation