= Solution
Let $\delta=(\rho-\bar\rho)/\bar\rho$ and define the <cosmological density power spectrum> by $\langle\delta_{\mathbf k}\delta_{\mathbf k'}^*\rangle=(2\pi)^3\delta_{\rm D}(\mathbf k-\mathbf k')P(k)$. The contribution per logarithmic wave-number interval is the <dimensionless cosmological power spectrum> $\Delta^2(k)=k^3P(k)/(2\pi^2)$. The linear <cosmological mass variance> on mass scale $M$ is
$$
\sigma^2(M,z)=\int_0^\infty\Delta^2(k,z)|W(kR)|^2d\ln k,\qquad M=\frac{4\pi}{3}\bar\rho_{m,0}R^3,
$$
with $R$ a comoving smoothing radius and $W$ the <top-hat filter> in Fourier space. On halo scales the observed spectrum is consistent with greater variance at smaller masses. Growth therefore brings smaller objects to the collapse threshold earlier, statistically; larger structures assemble by accretion and <dark-matter halo mergers>. \b[This statistical growth from smaller bound systems to larger ones is <hierarchical galaxy formation>.] It is an ordering of dark-matter assembly, not a rule that every small visible galaxy must precede every large visible galaxy.
For linear modes the <linear growth factor> gives $P(k,z)=D^2(z)P(k,0)$. During matter domination $D\propto a$, so $P\propto a^2$ and the characteristic nonlinear mass increases. Once modes become nonlinear, coupling between scales and halo formation change the shape, so the same multiplication cannot be used for the entire late-time spectrum. At low redshift accelerated expansion suppresses linear growth. Galaxy clustering traces the matter spectrum with <galaxy bias>; it should not be equated directly with an unbiased matter measurement.
The broad linear shape was set by the <cosmological transfer function> before and around <matter-radiation equality>, with baryonic acoustic structure also imprinted before recombination. A nearly scale-invariant primordial curvature spectrum has $\mathcal P_{\mathcal R}(k)\propto k^{n_s-1}$, with $n_s$ close to one. After converting curvature perturbations to matter-density perturbations,
$$
P_{\rm lin}(k,z)\propto D^2(z)k^{n_s}T^2(k).
$$
Thus nearly scale-invariant primordial curvature does not mean a constant density $P(k)$. Modes with $k\ll k_{\rm eq}$ enter the horizon after equality and have $T\simeq1$. Modes with $k\gg k_{\rm eq}$ enter during radiation domination, when radiation controls the expansion and cold-matter perturbations grow only slowly. The <cold-dark-matter transfer function> behaves approximately as $T\propto\ln(k/k_{\rm eq})/(k/k_{\rm eq})^2$ at large $k$. Hence the density spectrum turns over near $k_{\rm eq}=a_{\rm eq}H_{\rm eq}/c$:
$$
P(k)\propto k^{n_s}\quad(k\ll k_{\rm eq}),\qquad
P(k)\propto k^{n_s-4}\ln^2(k/k_{\rm eq})\quad(k\gg k_{\rm eq}).
$$
The turnover records the equality horizon, while the late nonlinear excess records gravitational clustering. On galactic scales the effective slope is greater than $-3$, giving the growing small-scale variance needed for <hierarchical galaxy formation>.
<Cold dark matter> has negligible primordial thermal velocities and a very short <collisionless free streaming> length on galactic scales. <Warm dark matter> has appreciable residual velocities while structure is being seeded; particles stream across small fluctuations and reduce their contrast. Its <cosmological transfer function> is consequently cut off below a characteristic length, suppressing low-mass halos and delaying their formation. Above that cutoff its assembly can still be hierarchical. \b[The important distinction is the free-streaming scale, rather than the present temperature or an arbitrary particle-mass label.]
Linear evolution assumes $|\delta|\ll1$. When $\delta$ becomes of order unity, overdense regions depart strongly from the Hubble flow and can turn around and collapse. Collisionless <dark matter> develops multistream motion after trajectories cross; gravitational mixing redistributes energy and produces a bound <dark-matter halo>. The formal infinite-density collapse of an ideal spherical pressureless solution is not the physical endpoint. A roughly virialized halo has $2K+W\simeq0$ and a characteristic <virial velocity> $V_{\rm vir}^2=GM_h/r_{\rm vir}$. Its gas <virial temperature> is conventionally
$$
\boxed{k_BT_{\rm vir}\simeq\frac{\mu m_p}{2}V_{\rm vir}^2\simeq\frac{\mu m_pGM_h}{2r_{\rm vir}}.}
$$
It measures the thermal energy associated with the gravitational potential; the numerical factor depends on the velocity-dispersion convention. It does not imply that the collisionless dark matter has a thermodynamic gas temperature.
The unheaded request about <baryon conversion efficiency of a halo> is also answered here. Define $f_*=M_{\rm stars}/(f_bM_h)$ using the <cosmic baryon fraction> $f_b$. In small halos, shallow potentials let <stellar feedback> drive outflows or repeatedly heat star-forming gas; supernova energy per stellar mass is roughly fixed while binding energy per gas mass scales as $V_{\rm vir}^2$. Photoheating during <reionization> also prevents very small halos from retaining or accreting cool gas. Molecular/atomic cooling thresholds further reduce star formation in the smallest systems. These effects make $f_*$ fall toward low mass.
Near $M_h\sim10^{12}M_\odot$, gas can cool efficiently and the potential is deep enough to retain more of it, while a long-lived hot atmosphere and maintenance heating are less effective than in larger systems. At high mass, higher <virial temperature> and lower cooling efficiency let a substantial hot atmosphere persist. <Active-galactic-nucleus feedback> can prevent that atmosphere from supplying cold gas and can expel some gas; the cooling-time bottleneck alone is not an adequate explanation for the low stellar fractions of massive groups and clusters. \b[The peak reflects a competition between gas supply/cooling and feedback, rather than complete conversion of all baryons at a sharply universal mass.] Its exact location and height depend on epoch, metallicity, gas history and the stellar population included.
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