Codex @codex  0

Let $\delta =(\rho -\bar{\rho })/\bar{\rho }$ and define the cosmological density power spectrum by $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_{\mathrm{D}}(\mathbf{k}-{\mathbf{k}}^{\prime })P(k)$. The contribution per logarithmic wave-number interval is the dimensionless cosmological power spectrum ${\Delta }^2(k)=k^{3}P(k)/(2{\pi }^2)$. The linear cosmological mass variance on mass scale $M$ iswith $R$ 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. 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,Thus nearly scale-invariant primordial curvature does not mean a constant density $P(k)$. Modes with $k\ll k_{\mathrm{eq}}$ enter the horizon after equality and have $T\simeq 1$. Modes with $k\gg k_{\mathrm{eq}}$ 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 $T\propto \ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ at large $k$. Hence the density spectrum turns over near $k_{\mathrm{eq}}=a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$: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. The important distinction is the free-streaming scale, rather than the present temperature or an arbitrary particle-mass label.

Linear evolution assumes $|\delta |\ll 1$. 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 $2K+W\simeq 0$ and a characteristic virial velocity $V_{\mathrm{vir}}^2=G{M}_h/r_{\mathrm{vir}}$. Its gas virial temperature is conventionallyIt 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 $f_{\ast }=M_{\mathrm{stars}}/(f_b{M}_h)$ 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 $V_{\mathrm{vir}}^2$. 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 $f_{\ast }$ fall toward low mass.

Near $M_h\sim 10^{12}{M}_{\odot }$, 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. 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.

Let $P_m(k,z)$ be the dimensional cosmological density power spectrum and ${\Delta }_m^2=k^3{P}_m/(2{\pi }^2)$ its dimensionless power spectrum. In linear theory,where $D$ is the linear growth factor and $\mathcal{T}$ is the cosmological transfer function. An approximately scale-invariant primordial curvature spectrum corresponds, with the conventional matter-spectrum normalization, to $P_{\mathrm{prim}}\propto k^{n_s}$ with $n_s\simeq 1$. This does not say that the dimensional density spectrum is white noise or that the early curvature and density perturbations are the same variable.

The crucial scale dependence develops around horizon entry and matter-radiation equality. Modes entering during radiation domination undergo only logarithmic cold-matter growth, whereas modes entering later avoid that suppression. For cold dark matter, the large-scale transfer tends to a constant and the small-scale transfer behaves schematically as $\mathcal{T}(k)\propto \ln (k/k_{\mathrm{eq}})/k^2$. ThusFor $n_s\simeq 1$, the dimensional spectrum turns from an approximately $k$ rise to an approximately $k^{-3}{\ln }^{2}k$ decline. Multiplying by $k^3$ shows that the dimensionless small-scale power still increases slowly. The resulting variance of density fluctuations smoothed on mass $M$ generally increases as $M$ decreases, over the scales relevant to galaxy assembly.

After equality, and over scales where linear growth is scale-independent, the transfer shape is approximately preserved while the amplitude grows. During matter domination, $D\propto a$; late accelerated expansion slows that growth. A region collapses when its linearly evolved density contrast reaches the linear spherical-collapse threshold. Smaller mass scales, having larger variance, typically reach this threshold earlier. Small haloes form first, then accrete matter and merge into larger haloes; galaxies form from cooling baryons in those wells and themselves merge. This small-to-large assembly is hierarchical galaxy formation. It is a statistical trend: rare high peaks can produce unusually massive early objects, and gas cooling and feedback prevent a one-to-one identification of halo collapse with star formation.

For cold dark matter, random particle velocities are small enough that collisionless free streaming erases little power on galactic scales. Warm dark matter retains a larger early velocity dispersion and a larger free-streaming length, suppressing power below a finite scale. There are consequently fewer low-mass haloes, delayed formation of the smallest galaxies, and a lower limit to the hierarchy that can develop. Above the cutoff, warm-matter structure can still assemble hierarchically. The distinction concerns the particles' velocity history and transfer function, not the thermal temperature of gas in a galaxy today. The broad late-time shape was largely established by early horizon-entry physics, with subsequent nonlinear collapse and merging altering the spectrum at high wavenumber.

Interpret $P(k)$ in the variance integral as the unsmoothed cosmological density power spectrum of $\delta$. A linear smoothing window gives $P_R(k)=P(k)|\tilde{W}(kR){|}^2$ for the smoothed field; the PDF's description of $P(k)$ as already belonging to ${\delta }_R$ would otherwise count the window twice.

For a scale-free matter power spectrum $P(k)=A{k}^{n_{\mathrm{eff}}}$, substitute $q=kR$ to obtainWhenever the dimensionless integral is finite and nonzero, this proves the scale-free smoothed density varianceThe printed condition $n_{\mathrm{eff}}<3$ is insufficient by itself. For a normalized window with $\tilde{W}(0)=1$, infrared convergence requires $n_{\mathrm{eff}}>-3$. Ultraviolet convergence depends on the filter: a spherical top-hat window function requires $n_{\mathrm{eff}}<1$, a Gaussian filter suppresses every ultraviolet power, and a sharp-k smoothing filter has finite Fourier support. In particular, the monotone relation $R\to \infty \Rightarrow S={\sigma }_R^2\to 0$ requires $n_{\mathrm{eff}}>-3$ in the scale-free model.

At a single variance $S_{\ast }$, the Gaussian random field has ${\delta }_{S_{\ast }}\sim N(0,S_{\ast })$. Thus its endpoint tail isHere $\mathrm{erfc}$ is the complementary error function.

For the crossing probability, the key assumption is independent, symmetric increments as the smoothing scale changes. A sharp-k smoothing filter, $\tilde{W}(kR)=1_{k<1/R}$, supplies this property: decreasing $R$ adds independent Fourier shells of the underlying Gaussian field. Parameterizing their accumulated variance by $S$ gives a Brownian motion with covariance $\min (S,S^{\prime })$. An arbitrary real-space window gives correlated increments and does not justify this argument.

Reflect a trajectory after its first hit of ${\delta }_c$. The Strong Markov property and symmetric future increments preserve its probability, and reflection maps an endpoint below the barrier to one above it. Every endpoint above the barrier has already crossed; the reflected paths give an equally probable set that crossed but ended below. The Brownian reflection principle therefore yields the excursion-set description of halo formation:Its derivative is the Brownian first-passage time densityThe factor of two resolves trajectories that crossed a larger-scale barrier but finish below it on the chosen smaller scale.

Define the halo peak height $\nu ={\delta }_c/{\sigma }_R$. The cumulative probability $\mathcal{P}(>M)$ is a dimensionless mass fraction. With $V_M=M/\bar{\rho }$, its conversion to halo number density givesConsequently the Press-Schechter halo mass function isIt is positive when ${\sigma }_R$ decreases with mass. A chosen mass-radius convention $M\propto R^3$ gives $d\log {\sigma }_R/d\log M=-(n_{\mathrm{eff}}+3)/6$. For a sharp Fourier filter that mass assignment needs a convention, since its real-space kernel is not a localized top-hat volume.

For warm dark matter, the cutoff introduces a new length and invalidates the pure power-law variance scaling near that length. With the same sharp-k filter and $n_{\mathrm{eff}}>-3$, the variance is exactlyThe Brownian walk runs only up to $S_{\max }$ and then stops. The cutoff does not destroy independence of the Fourier shells that are still present. Thus the first-crossing density and derivative mass-function formula remain valid on the invertible part of $S(M)$, using the actual cutoff variance. For $R<k_{\mathrm{WDM}}^{-1}$, $d{\sigma }_R/dM=0$ and this idealized sharp-filter model has no new crossings or new low-mass haloes. Its total collapsed fraction isThis is halo first crossing with a finite variance cutoff: some trajectories never cross, so one must not force the collapsed fraction to one. The uncut power-law abundance cannot be extrapolated to small masses. For other windows, a variance plateau still occurs but the Markov/reflection derivation is not exact; the same mass function is then a modelling approximation, rather than a result established by this calculation.

The definition of the virial radius of a dark-matter halo immediately gives the virial mass of a dark-matter haloIn an Einstein-de Sitter universe, spherical collapse gives $v=18{\pi }^2\simeq 178$, conventionally rounded to about $200$.

For the Navarro--Frenk--White profile, put $x=r/r_s=cs$. Direct integration givesUsing $r_s=r_v/c$ and ${\delta }_{\mathrm{char}}=v{c}^{3}g(c)/3$ therefore yieldsAt small radius the bracket is $(cs{)}^2/2+O((cs{)}^3)$, so $M(r)\propto r^2$, consistent with the central $\rho \propto r^{-1}$ cusp. At large radius it is $\log (cs)-1+o(1)$, so the mass diverges logarithmically unless the halo is truncated.

The primordial free streaming of warm dark matter erases small-scale density fluctuations and lowers the central phase-space density, tending to replace the smallest, earliest cold-dark-matter cusps by shallower central profiles or cores.

Since $V_v^2=G{M}_v/r_v$, the spherical potential that vanishes at infinity isIndeed, $rd\Phi /dr=GM(r)/r$, and normalization at $s=1$ requiresThe circular speed is consequentlyIt rises as $s^{1/2}$ near the centre, peaks at $r\simeq 2.16r_s$, and then declines approximately as $\sqrt{\log r/r}$. Increasing the concentration of a dark-matter halo moves the peak inward in units of $r_v$ and raises it relative to $V_v$. Thus an NFW curve can be fairly broad but is not exactly a flat galaxy rotation curve; stellar and gas contributions matter when comparing with an observed galaxy rotation curve.

Lower-mass haloes typically collapse earlier, when the cosmic background density is larger. Their characteristic inner densities are consequently larger relative to the present virial density, producing the mass-concentration relation of dark-matter haloes in which concentration decreases weakly with mass.

For a mass smaller by $1024=2^{10}$,and $c=101024^{0.1}=20$. HenceA halo of roughly $10^9{M}_{\odot }$ and kiloparsec scale naturally hosts a dwarf galaxy, possibly an extremely faint one if star formation is inefficient.