Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2012 / Paper 1 / 3

The normalized cycloid hasThe initially expanding perturbation reaches maximum radius at $\eta =\pi$, $t=t_m$, then formally collapses to zero radius at $\eta =2\pi$, $t=2t_m$. A finite virialized object is assigned the conventional formation epoch $t_f=2t_m$; random motions replace the pressureless point-collapse singularity. The cycloid alone does not describe that relaxation.

For the standard top-hat virialization estimate, write $U=-C_{G}G{M}^2/R$ and retain the same gravitational structure factor $C_G$ at turnaround and final equilibrium. The coherent kinetic energy vanishes at turnaround, whereas the virial theorem gives $2K_v+U_v=0$ in the final isolated equilibrium. conservation of energy givesThe final density is eight times the turnaround density. Meanwhile $\bar{\rho }\propto a^{-3}\propto t^{-2}$, so $\bar{\rho }(t_f)=\bar{\rho }(t_m)/4$. The Einstein-de Sitter virial overdensity follows by combining these factors with the supplied turnaround contrast:The value 200 is a convenient rounded overdensity convention, not exactly the Einstein-de Sitter result.

The virial radius of a dark-matter halo follows from $M_v=4\pi R_v^3{\rho }_v/3$:The hydrostatic temperature of an inverse-square halo follows from hydrostatic equilibrium and the ideal gas law:Since ${\rho }_g\propto r^{-2}$ and $M(r)=M_{v}r/R_v$, cancellation of ${\rho }_g/r$ gives $2k_B{T}_v/\mu =G{M}_v/R_v$. ThereforeHere $\mu$ is a dimensionful mean particle mass, as specified in the paper; a dimensionless mean molecular weight would need multiplication by a particle-mass unit. The isothermal interior is a further halo idealization, not an exact continuation of the uniform top-hat profile used in the virial-overdensity estimate. Changing its gravitational structure factor in a literal energy comparison would change the turnaround-radius relation.

The formation redshift from halo mean density uses the cluster data, which imply ${\rho }_v=3M_v/(4\pi R_v^3)=5.37148\times 10^{13}{M}_{\odot }{\mathrm{Mpc}}^{-3}$. Thus, using the requested rounded convention,Using $18{\pi }^2$ instead gives $z_f\simeq 1.00$. In the Einstein-de Sitter time scaling the rounded-convention result is $t_f/t_0=(1+z_f{)}^{-3/2}\simeq 0.376$. Directly from the mass and radius,For the estimate, take mean particle mass $\mu =0.6m_p$, appropriate approximately to a hydrogen-helium mixture with complete ionization. This gives $T_v\simeq 1.41\times 10^8\mathrm{K}$, or $k_B{T}_v\simeq 12.1\mathrm{keV}$. The composition choice is additional input, not a value supplied by the examination; the boxed expression retains the result for any $\mu$.