Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 48 / 3

Differentiate the linearized cosmological continuity equation, remembering the derivative of $1/a$:Insert the linear cosmological Euler equation for the peculiar velocity, givingThus the linear matter perturbation growth equation isThe factor $2H$ includes one expansion term from continuity and one from momentum dilution. This equation uses sub-Hubble, pressureless linear perturbations and the stated matter-only source for the gravitational potential.

Change independent variable from proper time to the scale factor. Then $\dot{\delta }=\dot{a}{\delta }_a$ and $\ddot{\delta }={\dot{a}}^2{\delta }_{aa}+\ddot{a}{\delta }_a$. Under radiation domination, $\ddot{a}=-{\dot{a}}^2/a$ and $H^2\simeq 8\pi G{\bar{\rho }}_r/3$. Since ${\bar{\rho }}_m/{\bar{\rho }}_r=a/a_{\mathrm{eq}}$, the linear matter perturbation growth equation becomesPutting $\delta =au$ givesThis keeps the matter self-gravity term in a radiation-dominated background. It is not an exact background equation through radiation-matter equality.

At $a/a_{\mathrm{eq}}\ll 1$, neglecting that small term gives ${\delta }_{aa}+{\delta }_a/a=0$. Integration yieldsThus matter has at most logarithmic growth during this leading radiation approximation, together with a constant independent mode. The constant is non-growing, not a mode that literally falls as $a^{-1}$; that power belongs to $u$ rather than ${\delta }_m$.

For an increasing/decreasing basis of the displayed equation with matter self-gravity retained, put $x=\sqrt{6a/a_{\mathrm{eq}}}$. Its equation becomes $x^2{\delta }_{xx}+x{\delta }_x-x^2\delta =0$. HenceThe Modified Bessel function of the first kind gives $I_0(x)=1+3a/(2a_{\mathrm{eq}})+O((a/a_{\mathrm{eq}}{)}^2)$, which increases slowly. The Modified Bessel function of the second kind gives $K_0(x)=-\frac{1}{2}\ln (a/a_{\mathrm{eq}})+\frac{1}{2}\ln (2/3)-{\gamma }_E+O((a/a_{\mathrm{eq}})|\ln (a/a_{\mathrm{eq}})|)$, where ${\gamma }_E$ is Euler's constant; this mode decreases as $a$ increases. Their leading span is precisely the constant/logarithmic pair above. Mode labels depend on the chosen basis and normalization; no rapid matter-era growth $\delta \propto a$ occurs here. Neglected background corrections can change subleading terms, so the Bessel basis should not be extrapolated through equality.

Let $u={\delta }_m/H$. The supplied equation has integrating factor $(aH{)}^3$:Consequently the integral linear growth factor in a matter-Lambda universe gives the two independent solutionsThe first is the conventional matter-era decaying mode. A finite lower limit in the second fixes a decaying admixture; adding a multiple of $H$ can choose a pure growing normalization.

During matter domination, write $H=h{a}^{-3/2}$. The integral solution isRemoving the second term by the independent $H$ solution gives the matter-era linear growth factorFor positive cosmological constant at sufficiently late times, $H\to H_{\Lambda }>0$. The integrand approaches $H_{\Lambda }^{-3}{a}^{-3}$, whose tail is integrable. ThereforeIts remaining approach has leading $a^{-2}$ behavior for the usual matter-plus-Lambda background. Growth of structure freezes rather than continuing as $a$. The basis function $H$ also approaches a constant; subtracting a suitable multiple of it isolates a genuinely decaying late-time solution $\propto a^{-2}$. The name “decaying mode” for $H$ refers to its matter-era behavior.