Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 62 / 5

During radiation domination, the cosmological continuity equation gives ${\rho }_R\propto a^{-4}$. The flat Friedmann equation in conformal time is $(a^{\prime }/a{)}^2=8\pi G{a}^2{\rho }_R/3$. Hence $a^{\prime 2}$ is constant, and choosing the Big Bang at $\tau =0$ yieldsFor the CDM density equation in a matter-radiation universe, neglect ${\rho }_C{\delta }_C$ while retaining ${\delta }_C$ itself. The adiabatic initial conditions give ${\delta }_R=4{\delta }_C/3$, soThis Euler-Cauchy equation has powers ${\tau }^p$ with $p(p-1)+p-4=0$, namely $p=2,-2$. Selecting the growing superhorizon adiabatic CDM mode in radiation domination gives

After cosmological horizon entry in radiation domination, the oscillating radiation source averages away. Neglecting both that source and CDM self-gravity in the CDM density equation in a matter-radiation universe givesso ${\delta }_C=C_1(k)+C_2(k)\log \tau$. The superhorizon adiabatic CDM mode in radiation domination has amplitude $A_R(k){\tau }_h^2\sim A_R(k)k^{-2}$ at entry ${\tau }_h\sim k^{-1}$, fixing this scale for the subsequent logarithmic CDM growth after radiation-era horizon entry.

More explicitly, in a sharp-transition approximation with ${\tau }_h=1/k$, continuity of ${\delta }_C$ and ${\delta }_C^{\prime }$ givesThe matching the CDM growing mode at horizon entry therefore yields the late subhorizon scalingHere $\simeq$ suppresses order-one coefficients and the nonlogarithmic term, as in the question. A smooth treatment of the radiation forcing through entry changes those coefficients; the factor two is exact only for the sharp matching model just used. This logarithmic growth, rather than growth proportional to $a$, is the Mészáros effect.

During matter domination, ${\rho }_C\propto a^{-3}$. The flat Friedmann equation in conformal time implies $a^{\prime 2}\propto a$, and henceWith radiation neglected, the CDM density equation in a matter-radiation universe reduces toFor this Euler-Cauchy equation, ${\delta }_C\propto {\tau }^p$ requires $p(p-1)+2p-6=(p-2)(p+3)=0$. Thus the matter-era growing and decaying density modes are ${\tau }^2$ and ${\tau }^{-3}$, respectively, andThere is no explicit $k$ in this pressureless equation, so the result holds on all scales within the stated CDM-comoving synchronous gauge description. The amplitude can still depend on $k$ through its earlier evolution; the result does not equate density contrasts on different time slicings.