Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 3

During matter domination, pressureless matter has ${\bar{P}}_m=\delta P_m=0$. On subhorizon scales the time derivative of the Newtonian gauge in cosmology potential is negligible, so the continuity and Euler equations reduce toTaking the divergence of the second equation, differentiating the first, and using the Poisson equation givesSince $d/d\tau =a,d/dt$, one hasDivision by $a^2$ yields the standard linear cosmological density perturbation equation

During matter domination, $a\propto t^{2/3}$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. Trying ${\delta }_m\propto t^p$ giveswhose roots are $p=2/3$ and $p=-1$. Thus the cosmic-time matter density modes are

During asymptotic cosmological constant energy domination, $H$ is constant and ${\bar{\rho }}_m$ is negligible, soThe matter density modes during cosmological-constant domination are thereforeThe growing matter-era solution freezes to a constant.

Use $d/dt=aH,d/da$. Thenwhere now $H^{\prime }=dH/da$. AlsoSubstitution into part (a) and division by $a^2{H}^2$ gives the matter growth equation as a function of scale factor

For matter plus a cosmological constant,Set ${\delta }_m=Hu$. Substitution into the equation from part (c) makes the coefficient of $u$ vanish by the Friedmann relation and leavesorTherefore the two independent modes can be writtenThe first is the decaying mode. Normalizing the second to $D_{+}(a)\sim a$ at early times gives the integral linear growth factor in a matter-Lambda universe

Yes. A perturbation of amplitude $A$ at CMB decoupling must grow by at least $A^{-1}$ before becoming nonlinear. During matter domination ${\delta }_m\propto a$, but once ${\rho }_{\Lambda }$ dominates, the suppression of matter growth by smooth accelerated expansion makes the growth approach a finite limit. Increasing ${\rho }_{\Lambda }$ moves this freeze-out to an earlier scale factor and eventually prevents ${\delta }_m$ from reaching unity.

For an order-of-magnitude bound, matter growth would reach unity atRequiring matter still to dominate then gives the galaxy-formation bound on the cosmological constantThe exact upper limit follows by imposing $A{D}_{+}(\infty )/D_{+}(a_{\mathrm{dec}})\gtrsim 1$ with the integral growth factor and an appropriate nonlinear-collapse threshold.