Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 56 / 4

Normalize the linear growth factor to $D(0)=1$. The collapse overdensity at the collapse epoch is ${\delta }_{\mathrm{sc}}\simeq 1.686$ for the matter-dominated spherical-collapse model. The present-extrapolated spherical-collapse barrier isIt is the initial linear overdensity, extrapolated to today, required to collapse by $t$; it is not the nonlinear density contrast of a virialized halo. The smoothed matter density variance ${\sigma }^2(M,z)$ is the variance of the linear density contrast smoothed on a Lagrangian comoving scale containing mass $M$. For a spherical top-hat filter,Consequently $\sigma (M,z)=D(z)\sigma (M,0)$ for scale-independent linear growth. The equivalent halo peak height conventions are $\nu ={\delta }_c(t)/\sigma (M,0)={\delta }_{\mathrm{sc}}/\sigma (M,z)$; using both an evolved barrier and an evolved variance would count growth twice.

To calculate a number-density growth time, use a narrow fixed-mass bin, not the collapsed mass fraction itself. Differentiate the Press-Schechter formalism mass fraction and divide the mass density in the resulting interval by $M$. This gives the Press-Schechter halo mass functionAt fixed $M$, the mass factor and logarithmic slope do not depend on time. Since $\dot{\nu }/\nu =-\dot{D}/D$, the Press-Schechter abundance growth at fixed mass isThe prefactor $\nu$ matters here: the exponential-only rare-peak approximation drops the minus one and is accurate only for $\nu \gg 1$.

Read the supplied variance relation as a mass-scale calibration using the numerical velocity label given for the selected population. It gives $\sigma (M,3)=0.85\sqrt{250/10}=4.25$. Matter domination between the two high-redshift epochs gives $D(19)/D(3)\simeq 4/20$, so $\sigma (M,19)\simeq 0.85$ and $\nu \simeq 1.984$. At the epoch in question,Thus the requested fixed-mass-bin growth estimate with that calibration isAn exponential-only approximation gives about $8.1\times 10^7\mathrm{yr}$, which is somewhat shorter because this is only a roughly two-sigma population.

The wording leaves two sample conventions worth distinguishing. First, an actual virial velocity of a spherical-overdensity halo is epoch-dependent at fixed mass. If the variance fit is instead calibrated using physical virial velocities at redshift three, the same mass whose velocity is $10\mathrm{km}{\mathrm{s}}^{-1}$ at redshift nineteen has $v(3)\simeq 10\sqrt{4/20}=4.47\mathrm{km}{\mathrm{s}}^{-1}$. The halo virial-velocity conversion between epochs then gives $\sigma (M,19)\simeq 1.271$, $\nu \simeq 1.327$ and $t_d\simeq 4.2\times 10^8\mathrm{yr}$ for a fixed-mass bin. The two numerical answers reflect the velocity-label convention in the supplied fit, not two ways of differentiating one fixed fit.

Second, a cumulative number density is $n(>M)={\int }_M^{\infty }(dn/d{M}^{\prime })d{M}^{\prime }$, not simply ${\bar{\rho }}_{m,0}f(>M)/M$: the latter is a mass fraction divided by a threshold mass, not the number of objects. The cumulative derivative is an abundance-weighted average of $(\nu (M^{\prime }{)}^2-1)\dot{D}/D$ over that integral. If the supplied power-law variance fit is extended over all larger masses, with the same velocity-label convention, direct integration gives a cumulative-number growth time of about $8.4\times 10^7\mathrm{yr}$ instead. A sample maintained at fixed physical velocity at successive epochs additionally moves its mass boundary and needs a selection convention. The numerical estimate above explicitly uses the ordinary fixed-mass differential interpretation. These distinctions are important when an exact growth time rather than a rare-tail estimate is intended.

For fixed epoch and the increasing branch $\nu >1$, increasing mass lowers the smoothed matter density variance, hence raises the halo peak height $\nu$. The rate $({\nu }^2-1)\dot{D}/D$ therefore increases and its inverse decreases. Rarer, more massive haloes have a shorter fractional abundance-growth time, even though their actual number density is much smaller. They lie farther into the exponential tail of the Press-Schechter halo mass function, so a small change in the linear growth factor causes a large fractional change. This statement compares fixed-mass bins in the same cosmology and concerns relative growth, not the time for one halo to assemble all its mass.

At fixed mass, decreasing redshift increases the linear growth factor and reduces the halo peak height $\nu$. During matter domination $\dot{D}/D=H$ also falls as the universe expands. As long as $\nu >1$, both effects reduce the positive abundance-growth rate, so $t_d$ grows towards lower redshift. In the very rare-tail limit $t_d\propto D^2/H\propto (1+z{)}^{-7/2}$ provides the approximate trend.

There is a limit to calling this an increase time. At $\nu =1$, the fixed-mass Press-Schechter halo mass function reaches its maximum as a function of time; its logarithmic growth rate vanishes and $t_d$ diverges. For $\nu <1$, the derivative becomes negative, corresponding to net transfer of small objects into more massive systems, so the signed inverse is no longer a positive doubling or increase time. At still later epochs the cosmological constant further suppresses the linear growth factor. These qualifications prevent extrapolating the positive high-redshift growth trend indefinitely.