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The concentration of a spherical dark-matter halo compares its outer radius with an inner scale radius. For a Navarro--Frenk--White profile, $c=r_v/r_s$.

Interpret the model comparison at the exam's 2008 epoch. A simulation's failure depends on its treatment of baryons, resolution, and adopted stellar feedback; it is not a theorem that the Lambda-CDM model cannot produce the observation under any physical implementation. The following were important tensions for then-standard realizations, with different implications for monolithic galaxy formation.

Old, alpha-enhanced massive spheroids and downsizing. The most massive quiescent elliptical galaxies tend to contain older stellar populations and elevated alpha element abundances relative to iron, indicating that much of their star formation occurred early and rapidly. Prompt core-collapse enrichment followed by shutdown before substantial delayed iron enrichment accounts for the abundance pattern. Models that spread a massive galaxy's star formation over many late progenitor-merger episodes can make its stars too young or insufficiently alpha enhanced. A rapid early dissipative collapse naturally gives a short formation interval and is therefore more immediately consistent with this particular observation. However, downsizing in galaxy formation is about stellar formation, not necessarily final assembly: late dry galaxy mergers can assemble already old alpha-enhanced stars. Early quenching and an appropriate chemical-enrichment model can consequently reconcile hierarchical assembly with the fossil record.

Thin, high-angular-momentum stellar discs. The existence of extended thin galactic disks, including systems with small galactic bulges, challenged simulations that produced compact central stellar concentrations or overly thick discs. Premature gas cooling into small progenitors, transfer of specific angular momentum to the halo, and repeated dynamical perturbations could cause the galactic angular-momentum catastrophe; satellite encounters also increase galactic disk heating. A comparatively quiet dissipative collapse with approximate conservation of baryonic specific angular momentum more readily produces an extended rotating disc, and avoiding subsequent strong perturbations helps retain thinness. Nevertheless, rapid collapse alone does not explain a disc's prolonged star formation and gas replenishment. Hierarchical histories with quieter late accretion, better feedback, delayed cooling, and dissipative disc regrowth can reduce these tensions; a problem in early implementations is not an unavoidable property of the cosmological framework.

Cores inferred in low-mass galaxies. Resolved inner galaxy rotation curves in a number of low-surface-brightness and dwarf systems favoured shallower central mass densities than the cusps of collisionless cold-dark-matter simulations. For a roughly constant-density dark-matter core, $M(<r)\propto r^3$ and $V_c\propto r$; an inner Navarro--Frenk--White profile has $\rho \propto r^{-1}$, hence $M(<r)\propto r^2$ and $V_c\propto r^{1/2}$. The differing slopes explain the cusp--core problem. Beam smearing, inclination, and noncircular motions complicate inference, but do not make the issue merely a normalization mismatch. A simple early dissipative collapse does not automatically solve it: baryonic contraction can deepen the central concentration. Potential fluctuations from gas loss and stellar feedback can instead redistribute dark matter and modify a cusp. Thus this observation tests the joint halo-and-baryon model rather than preferentially vindicating the old collapse picture.

Rapid formation of the stars and hierarchical assembly of the final galaxy are compatible histories. The comparison must keep those clocks separate, and the historical modelling tensions should not be presented as universal failures of all later simulations.

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.

Put $x=r/r_s$. The enclosed mass of the Navarro--Frenk--White profile isSince $y/(1+y{)}^2=(1+y{)}^{-1}-(1+y{)}^{-2}$,The circular-orbit condition $v_c^2/r=GM(r)/r^2$ then gives the galaxy rotation curve

For $x\ll 1$,and thereforeFor $x\gg 1$,soThe NFW curve thus rises from the center, peaks, and eventually declines slowly. Typical spiral-galaxy rotation curves remain approximately flat over the observed outer disk, although an NFW halo combined with baryons can approximate such a plateau over a finite interval.

At the maximum, $x_m=2.16$, so the scale radius of a Navarro--Frenk--White profile isWriting $f(x)=\log (1+x)-x/(1+x)$, the characteristic density of a Navarro--Frenk--White profile isWith $G=4.3009\times 10^{-6}\mathrm{kpc}(\mathrm{km}{\mathrm{s}}^{-1}{)}^2{M}_{\odot }^{-1}$ and $f(2.16)=0.4670$, this is

An NFW cusp has $\rho \propto r^{-1}$ and hence $v_c\propto r^{1/2}$. Many dwarf and low-surface-brightness galaxies instead favor a nearly constant-density core, for which $M(r)\propto r^3$ and $v_c\propto r$; after matching the outer speed, this gives the more slowly rising observed inner curve. This is the cusp--core problem. Repeated burst-driven gas outflows can fluctuate the central potential and transfer orbital energy to dark matter, while self-interacting dark matter provides another possible core-forming mechanism. Beam smearing and noncircular motions are observational systematics but do not explain every case.

For an exponential galactic disk,soSince $7\mathrm{kpc}\simeq 2.3R_d$ is near the disk maximum, assigning the full $220\mathrm{km}{\mathrm{s}}^{-1}$ there givesIf the disk mass is increased while the observed total curve is fixed, the halo contribution must decrease in the inner galaxy. An NFW fit therefore needs a lower characteristic density, a larger scale radius and lower concentration, or a correlated combination of these changes.

A maximum disk model raises the stellar mass-to-light ratio as far as the rotation curve permits, conventionally making the disk contribute about $85\%$ of the speed near $2.2R_d$. High stellar surface densities, population-synthesis mass-to-light ratios, microlensing optical depths, and fast bars that have suffered little dynamical-friction braking can support a large disk contribution. Directly measured vertical stellar dispersions often favor submaximal disks, especially in low-surface-brightness galaxies, while stability and cosmological halo constraints may also disfavor an excessively massive disk.

The disk--halo degeneracy follows fromOver a finite radial range, raising the stellar mass-to-light ratio and lowering or broadening the halo can leave the same sum. The degeneracy can be reduced by measuring the disk surface density dynamically from vertical velocity dispersion and scale height, or by constraining the stellar mass-to-light ratio with resolved populations, colors, and a specified initial mass function. Strong or weak gravitational lensing, gas-layer flaring, stellar streams, and rotation data far beyond the optical disk provide further independent halo constraints.