= Solution
Take regular <adiabatic initial conditions> and a common scale-invariant primordial amplitude, $k^{3/2}\Phi_{\rm prim}(k)=\text{constant}$. Let $k_{\rm eq}=\mathcal H_{\rm eq}$. The three histories are distinguished by their time of <cosmological horizon crossing>:
* For $k\ll k_{\rm eq}$, the mode remains outside the <Hubble radius> throughout <radiation domination>. Its potential remains nearly constant through equality, changing to $9/10$ of the primordial radiation value for a purely adiabatic transition. It stays constant after later horizon entry in <matter domination>.
* For $k\gg k_{\rm eq}$, horizon entry occurs during <radiation domination>. Radiation pressure produces oscillations and a rapidly decreasing potential. The small <cold dark matter> component eventually supports a much smaller constant matter-era potential; the pure-radiation oscillations are not continued forever after equality.
* For $k\sim k_{\rm eq}$, the transition and horizon entry overlap, giving an intermediate suppression before the potential approaches a constant.
The $9/10$ factor follows from conserving the large-scale adiabatic curvature: the constant potential is proportional to $3(1+w)/(5+3w)$, whose matter-to-radiation ratio is $(3/5)/(2/3)=9/10$. Define the <cold-dark-matter transfer function> by $\Phi_{\rm late}=(9/10)T(k)\Phi_{\rm prim}$. Then $T\to1$ on large scales, while $T\sim(k_{\rm eq}/k)^2\ln(k/k_{\rm eq})$ far below the equality length, as explained by the density growth below.
\b[Late amplitudes are $\Phi_{\rm late}=(9/10)T(k)\Phi_{\rm prim}$, with $T\simeq1$ for $k\ll k_{\rm eq}$ and $T\ll1$ for $k\gg k_{\rm eq}$.]
The upper panel sketches $k^{3/2}\Phi$ for the three modes. The lower panel provides the corresponding <density contrast> histories used in the next part. A negative common primordial potential was chosen so the growing density is positive; this arbitrary phase has no effect on a <cosmological density power spectrum>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-49-mode-evolution.png]
{title=Evolution of scale-invariant gravitational-potential and cold-dark-matter density modes across radiation–matter equality}
{height=760}
The curves integrate the ideal coupled radiation-fluid and pressureless-matter equations, rather than patching a pure-radiation solution onto a matter solution. They neglect baryons, free-streaming anisotropic stress, dark energy and nonlinear evolution, consistently with the stated mixture. Dots mark $k=\mathcal H$; equality is the dashed vertical line.
Back to article page
