Use $c=1$ and signature $+---$. The positively curved FLRW metric in the usual radial chart isSet $r=\sin \chi$ and introduce conformal time by $dt=ad\eta$. Then $d{r}^2/(1-r^2)=d{\chi }^2$ in the chart, and the spherical polar coordinate $\chi \in [0,\pi ]$ extends the spatial description to the full three-sphere. ThusThe $r$ coordinate itself is not one-to-one across the equator, which is why the new polar coordinate is useful.

For $\Lambda =0$ and $k=1$, the Friedmann equation is ${\dot{a}}^2=(8\pi G/3)\rho a^2-1$. Since $a^{\prime }=a\dot{a}$, it becomesFor pressureless matter, the background continuity equation gives $\rho a^3={\rho }_0{a}_0^3$. Define the conserved normalization $M=(4\pi /3){\rho }_0{a}_0^3$. It makes the equation $(a^{\prime }{)}^2=2GMa-a^2$. The closed matter-dominated Friedmann solution is the cycloidIndeed $a^{\prime }=GM\sin \eta$ and $2GMa-a^2=(GM{)}^2{\sin }^2\eta$. Integrating $dt/d\eta =a$ gives the second expression, with the Big Bang at $t=\eta =0$. It expands to $a=2GM$ at $\eta =\pi$ and recollapses at $\eta =2\pi$. Here $M$ is the constant appearing in the Friedmann equation; it is not the total mass of the entire closed spatial slice, whose volume is $2{\pi }^2{a}^3$.

The cosmological principle asserts that, after averaging over sufficiently large scales, the universe is spatially homogeneous and isotropic on constant-cosmic time slices. Homogeneity means no distinguished location; isotropy means no distinguished spatial direction. These are statistical large-scale statements, not assertions that individual galaxies, clusters or the cosmic web are uniform, and they do not require the universe to be unchanging in time.

The nearly isotropic cosmic microwave background is strong evidence for large-scale isotropy about us. Its dominant dipole is explained by our peculiar velocity; the much smaller residual fluctuations provide the seeds and record of structure formation, rather than invalidating the principle. Large-area galaxy and radio-source distributions and deep galaxy surveys support statistical isotropy and a tendency towards statistical homogeneity after smoothing beyond the strongly clustered scales. An approximately isotropic Hubble flow and broadly uniform background radiation supply complementary checks.

There is an important logical distinction: isotropy seen by one observer alone does not prove homogeneity. A spherically symmetric inhomogeneous universe centered on that observer could look isotropic. Combining the observations with the assumption that our location is not privileged, so comparable observers would see comparable large-scale isotropy, motivates the homogeneous FLRW metric. Selection effects, foregrounds, finite survey volumes and the fact that observations lie on our past light cone must be accounted for. The evidence supports a large-scale statistical principle, not exact uniformity on every scale.

The sign of cluster evolution depends on the orbital energy of the captured stars, not just on their small angular momentum. Let the instantaneous capture rate be $\Gamma >0$, so the remaining stellar mass $M_s=Nm$ has ${\dot{M}}_s=-m\Gamma$, while ${\dot{M}}_H\simeq m\Gamma$. A captured star has negative specific orbital energy ${ϵ}_{\mathrm{cap}}$ measured outside the absorbing region. Removing it increases the energy of the surviving stars by $-m{ϵ}_{\mathrm{cap}}$; growth of the central black hole instead deepens their potential.

To turn this balance into a radius estimate, assume the surviving cluster evolves slowly through homologous virial equilibria of scale $R$. Write its stellar self-energy and black-hole interaction energy aswhere positive $a,b$ depend on the density shape. The virial theorem givesAt the instant of capture, the stellar-system energy budget, excluding feedback and energy carried by other escaping stars, isThe second term is the work of the changing central potential; it must not be confused with the change in the number of stars. Differentiating the virial expression and using ${\dot{M}}_H=m\Gamma$ givesThis estimates the radius response of a general homologous cluster once its structure and capture energies are specified.

For a simpler locally Keplerian subsystem, neglect its stellar self-gravity, so $E=-bG{M}_H{M}_s/(2R)$. Put ${ϵ}_{\mathrm{cap}}=\xi E/M_s$, with $\xi >0$ measuring how tightly bound a captured star is relative to the average orbital energy. The preceding balance reduces toThis is Keplerian cluster radius evolution under stellar capture. Removing representative-energy stars, $\xi =1$, leaves unchanged the characteristic orbits if $M_H$ is fixed; with black-hole growth it contracts them at rate $\dot{R}/R\simeq -{\dot{M}}_H/M_H$. Preferential removal of more tightly bound stars drives expansion if $\xi -1>M_s/M_H$. In that heating-dominated case,so inserting the empty-cone flux gives $t_R\sim t_{\mathrm{rel}}\log (1/{\mathcal{R}}_{\mathrm{lc}})/|\xi -1|$ for the one-zone estimate. Growth-dominated contraction instead has timescale $M_H/(m\Gamma )$.

For a Kepler orbit of semimajor axis $a_{\mathrm{cap}}$, ${ϵ}_{\mathrm{cap}}=-G{M}_H/(2a_{\mathrm{cap}})$ and $\xi =R/(b{a}_{\mathrm{cap}})$. Reaching $R_{\mathrm{sch}}$ does not imply that the orbit had binding energy of order $c^2$: a nearly parabolic star can reach that pericentre with almost zero orbital energy. Therefore expansion is expected when strongly bound central stars are preferentially swallowed, but the supplied information does not uniquely fix the sign or rate for the entire remaining cluster. Captured-energy selection, the density profile and any energy feedback must be specified. The balances above give both the conventional expansion estimate and the contraction case without assuming missing data.

A small loss cone does not by itself specify its supply rate. Introduce a characteristic orbital radius $r$, an orbital period $P\sim r/v$, the number $N_r$ of stars on the relevant energy range, and their local number density $n$. Weak encounters of impact parameter $b$ give velocity kicks of order $Gm/(bv)$. Summing their squared kicks over encounter rate $2\pi bdbnv$ givesHere $\log \Lambda$ is the Coulomb logarithm in stellar dynamics. This derives the stellar relaxation time relevant to angular-momentum scattering, up to the velocity-averaging constants.

Let $J_c^2\sim G{M}_{H}r$ and define the dimensionless cone size ${\mathcal{R}}_{\mathrm{lc}}=J_{\mathrm{lc}}^2/J_c^2\sim R_{\mathrm{sch}}/r$. In one period, two-body relaxation changes $J^2$ by order $J_c^{2}P/t_{\mathrm{rel}}$, so the refilling parameter isIf refilling is sufficiently rapid to keep the cone populated, an isotropic fraction of order ${\mathcal{R}}_{\mathrm{lc}}$ reaches its absorbing pericentre each orbit. The full-cone rate isIf $q\ll 1$, the cone is depleted and capture is diffusion-limited. At fixed orbital energy, the two transverse angular-momentum coordinates have an approximately two-dimensional diffusion equation. Its steady radial solution between the absorbing boundary $J_{\mathrm{lc}}$ and the reservoir $J_c$ satisfiesThe gradient supplies a constant inward flux. With diffusion coefficient of order $J_c^2/t_{\mathrm{rel}}$ and reservoir population $N_r$, this givesConstants of two depend on using $J$ or $J^2$ and on the definition of $t_{\mathrm{rel}}$. A rough interpolation uses the slower of the orbital draining and diffusion-supply rates; an accurate transition requires an orbit-averaged diffusion calculation. These are the empty and full loss-cone capture rates. In particular, multiplying $N_r/t_{\mathrm{rel}}$ by the cone's tiny solid-angle fraction would incorrectly treat diffusive feeding as independent isotropic redraws.

For a one-zone black-hole-dominated cluster with $N_r\sim N$, $n\sim N/r^3$ and $v^2\sim G{M}_H/r$,The mass ingestion rate is ${\dot{M}}_H\simeq m\Gamma$. A radially extended cluster requires summing the energy-dependent fluxes, not replacing every radius by the same $r$. No numerical capture rate follows from $N,m,M_H$ alone: its radius/density, orbital distribution and refilling regime are required. The displayed relations are the requested estimates with those physical scales explicit.

Work at $r\gg R_{\mathrm{sch}}$ in the region where the central black hole dominates the gravitational potential. Let $J$ be a star's specific angular momentum. Near its very small pericentre, the centrifugal term balances the deep gravitational potential. For an approximately parabolic passage with capture pericentre $r_c=R_{\mathrm{sch}}$,At radius $r$, a velocity making angle $\vartheta$ with the inward radial direction has $J=rv\sin \vartheta$. Therefore the loss cone half-angle isTaking the characteristic speed to be the circular speed $v^2\sim G{M}_H/r$ giveswith Newtonian circular-speed normalization ${\vartheta }_{\mathrm{lc}}^2\simeq 2R_{\mathrm{sch}}/r$. The inward cap occupies solid angle ${\Omega }_{\mathrm{lc}}\simeq \pi {\vartheta }_{\mathrm{lc}}^2$, or a fraction ${\vartheta }_{\mathrm{lc}}^2/4$ of an isotropic velocity sphere. Counting low-angular-momentum trajectories in both radial directions changes only an order-one factor; each such orbit is removed at its next pericentre.

The capture occurs in the strong-field region, so this Newtonian pericentre estimate is not an exact relativistic capture coefficient. The Schwarzschild capture calculation alters an order-one constant, while retaining the small-angle scaling ${\vartheta }_{\mathrm{lc}}\propto r^{-1/2}$ for these characteristic Keplerian speeds.

Use the usual spherical circular-shell adiabatic contraction approximation: initial dark-matter orbits are circular, slow accumulation keeps them circular, and dark shells do not cross. Angular momentum is conserved in a spherical potential, and the vanishing radial action stays zero as an adiabatic invariant. For a circular shell,Here $r_i$ is the initial shell radius and $r$ its final radius. The final flat circular speed $V=2v_{\mathrm{NFW},\mathrm{c},\max }$ gives $M_f(r)=V^{2}r/G$.

For an initial shell in the inner NFW profile, write $A=2\pi {\rho }_1{r}_1$, so $M_i(r_i)\simeq A{r}_i^2$. The invariant givesConservation of dark shell mass yieldsand differentiation givesTo express the requested ratios, set $K=F(x_{\max })\simeq 0.2162166$ and $B=8K\simeq 1.72973$, so $V^2/(GA)=B{r}_1$. Since the original cusp has ${\rho }_D(r)\simeq A/(2\pi r)$,The total density of the final spherical flat-rotation model is ${\rho }_{\mathrm{tot}}=V^2/(4\pi G{r}^2)$. Its dark fraction isSubtracting dark matter from the total gives the added-matter density, ${\rho }_B={\rho }_{\mathrm{tot}}(1-h)$, and thusThese expressions describe the leading inner approximation, not an extension to radii where the denominator becomes negative or the initial shell is outside the NFW cusp. As $r/r_1\to 0$, dark matter is enhanced relative to its original density while its fraction of the even steeper total density tends to zero. This is inner NFW contraction to a flat rotation curve.

The orbital assumptions matter. An initial density profile does not specify the phase-space distribution function. For eccentric orbits, slow evolution conserves their actions, but $rM(r)$ at an instantaneous radius is not generally their invariant. The above ratios are the intended circular-shell estimate; their numerical coefficients cannot be uniquely inferred from the density profile and slow accumulation alone.

Observed galaxy rotation curves commonly remain approximately flat across much of the measured outer luminous disk. A pure NFW profile is not an exactly flat curve: it rises in the inner cusp and ultimately declines. Nevertheless, its broad maximum can look nearly flat over a finite radial range, so observing a flat segment does not by itself exclude an NFW halo.

The measured circular speed responds to the total gravitational potential, including stars and gas as well as dark matter. For components in the same plane,For spherical components this reduces to $G{\sum }_j{M}_j(<r)/r$; a disk contribution must instead be computed from its actual disk potential. Baryonic mass can supply much of the inner rotation and modify the halo through slow contraction or more impulsive evolution. Thus the appropriate comparison is with the combined mass model, not an isolated halo-only curve. At arbitrarily large radius a finite isolated galaxy must eventually have a declining circular speed; an observed flat segment is not a claim of flat rotation to infinity.

Put $x=r/r_1$. The NFW profile becomes ${\rho }_D={\rho }_1/[x(1+x{)}^2]$. Integrating spherical shells givesDefine $F(x)=[\log (1+x)-x/(1+x)]/x$. The spherical circular speed is thereforeAt small $x$, the bracket is $x^2/2-2x^3/3+O(x^4)$, givingAt large $x$, the bracket is $\log x-1+O(1/x)$, givingThus the curve rises from zero, reaches one broad maximum, and eventually falls as $[\log r/r{]}^{1/2}$. The untruncated NFW profile has logarithmically divergent mass, but its circular speed still tends to zero.

Differentiating $F$ shows that its stationary point satisfiesThe numerator of $F^{\prime }$ has derivative $x(1-x)/(1+x{)}^3$: it increases up to one and decreases thereafter, crossing zero once at positive $x>1$. Numerical bisection gives $x_{\max }\simeq 2.16258$ and $F(x_{\max })\simeq 0.2162166$. HenceEquivalently $v_{\max }\simeq 0.4650\sqrt{4\pi G{\rho }_1{r}_1^2}$. This is the NFW mass and circular-speed profile.

At rest with all separations initially infinite, both initial kinetic and gravitational energies vanish, so the conserved total energy is zero. During an isolated collapse, $T+W=0$, rather than the equilibrium relation $2T+W=0$. Hence at the stated instantaneous gravitational potential energy,The rms collapse speed is $\sqrt{2}$ times the rms equilibrium speed for the same instantaneous $W$ and mass. For a perfectly radial cold infall this mean-square motion includes coherent inward streaming; it need not equal a local random velocity dispersion after subtracting that streaming. The quoted comparison follows the question's global mean-square convention.

Take the gravitational potential to vanish at infinity. At the position of star $i$, excluding its own point-mass field, the instantaneous test-particle escape velocity obeys $v_{\mathrm{esc},i}^2=-2{\Phi }_i$. The factor of one half in the total gravitational potential energy is important:Therefore the equal-mass average isThus the rms escape speed is twice the equilibrium rms speed. This is the virial mean escape speed relation, using the same three-dimensional averaging convention as the preceding part. It concerns escape in the instantaneous system potential, rather than a subsequent collective rearrangement of the cluster.

Work in the centre-of-mass frame and define $I={\sum }_{i}m|{\mathbf{r}}_i{|}^2$, $T=\frac{1}{2}{\sum }_{i}m|{\mathbf{v}}_i{|}^2$. Differentiation givesFor each gravitational pair, the two contributions combine toSumming pairs therefore gives the exact scalar virial theoremIn a stationary equilibrium, or a bounded long-time average with vanishing average $\ddot{I}$, $2T=-W$. With $M=Nm$, the stated gravitational radius gives $W=-G{M}^2/R_g$, henceThe rms speed itself is $[GNm/R_g{]}^{1/2}$. This is a three-dimensional mean-square speed; for an isotropic system the one-component velocity dispersion is smaller by a factor of three in its square. No assumption about a uniform spatial density was used.

Build the spherical system by adding shells from the centre outwards. The shell theorem says that a shell of mass $dM$ at radius $r$ interacts with the already assembled interior mass through potential $-GM(r)/r$. Thus its contribution is $dW=-GM(r)dM(r)/r$. Each pair is counted when its outer member is added, so no further factor of one half is needed:Integration by parts givesFor an isolated finite-mass system without a singular central point mass, the endpoint term vanishes provided the central cusp has finite binding energy. ConsequentlyThis is the spherical binding-energy mass integral. If a central singularity makes the endpoint or integral divergent, the finite-energy assumption must be checked rather than silently discarding that term.

For continuous-birth binary state fractions, conditional mass independence and the common age giveThe factor two in the middle expression allows either component to be the red giant. ThusGeometrically, the corresponding sections of the unit cube have areas proportional to $y^2$, so their volumes equal one third of their $y=1$ base areas. This is exactly the pyramid-volume calculation: the equal height and common factor $1/3$ leave the base-area ratio $1:8:16$ unchanged.

A system containing a red giant and another evolved component is either a giant–giant pair or an unordered giant–white dwarf pair. At common age $y$, these disjoint possibilities have conditional probabilities $g(y{)}^2$ and $2g(y)w(y)$. The conditional evolved-companion fraction in a coeval binary population therefore has numeratorDivide by the fraction $299/30000$ of systems containing at least one giant:Thus just over $3\%$ of giant-containing binaries have an evolved companion. The $0.03\%$ numerator is a fraction of all binaries; it is not itself the requested conditional fraction.

Condition on the common age $Y=y$ of the coeval binary population. Each component is a red giant with conditional probability $g(y)=y/100$, independently of the other component at that age. The conditional probability of at least one giant is $1-(1-g{)}^2=2g-g^2$. Integrating over the uniform age distribution givesThus the fraction is $299/30000=0.996\overline{6}\%$, just below $1\%$. The subtracted term removes the systems counted twice because both components are red giants. Using the square of the marginal giant fraction would incorrectly ignore their shared age.