Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 60 / 4

Let ${\delta }_{\mathrm{sc}}\simeq 1.686$ be the linear spherical-collapse threshold and let $D(t)$ be the linear growth factor, normalized to one at the epoch to which the random density field has been extrapolated. The time-dependent barrier is ${\delta }_c(t)={\delta }_{\mathrm{sc}}/D(t)$. The smoothed matter density variance ${\sigma }^2(M)$ is the variance of the linear density contrast smoothed with a spherical top-hat window function whose comoving mass is $M$. Both quantities must refer to the same linear normalization; equivalently use $\nu ={\delta }_{\mathrm{sc}}/\sigma (M,t)$.

Let $N$ denote the comoving number density and write $\mathcal{N}(M,t)=dN/dM$ for the positive differential Press-Schechter halo mass function. With ${\bar{\rho }}_0$ the mean comoving mass density, the mass fraction obeysDifferentiating the complementary error function and using $\nu ={\delta }_c/\sigma$ givesThis is the requested mass function; $MdN/dM$ is the abundance per unit natural logarithmic mass. For physical rather than comoving number density, multiply by $(1+z{)}^3$. The normalization in the Press-Schechter formalism includes its factor of two relative to the positive Gaussian tail, so $f(>M)$ tends to one as the smoothing mass tends to zero when $\sigma$ diverges. It is a collapse prescription, not the probability that an arbitrary already nonlinear density field stays Gaussian.

Assume matter-dominated growth over the two epochs, so $D(z)\propto (1+z{)}^{-1}$. A fixed definition of the characteristic mass selects fixed peak height ${\nu }_{\ast }$. Therefore $D(z)\sigma (M_{\ast })$ is constant. With $\sigma (M)\propto M^{-s}$, the observed ratios giveSince a spherical top-hat window function has $M\propto R^3$, the rms slope isFor a cosmological density power spectrum $P(k)\propto k^n$, the smoothed matter density variance isThe dimensionless integral converges for $-3<n<1$. Thus $m=-(n+3)/2$, andFor reference, substituting $s=1/4$ into the differential Press-Schechter halo mass function gives a low-mass prefactor $M^{-7/4}$ and an exponential argument proportional to $M^{1/2}$, so its displayed-form exponents are $\alpha =-7/4$, $\beta =1/2$. The prefactor normalization varies with epoch.

The effective galactic-scale index $-3/2$ need not equal the primordial nearly scale-invariant index one. The cosmological transfer function modifies perturbations between the primordial epoch and galaxy formation: modes entering the horizon in the radiation-dominated era grow differently from larger modes. Schematically $P_m(k,z)\propto D(z{)}^2{k}^{n_s}T(k{)}^2$, with $n_s\simeq 1$, while the cold-dark-matter transfer function decreases toward small scales. The result is a much redder late-time cosmological density power spectrum, with a scale-dependent slope. The inferred $-3/2$ is a local power-law approximation to the processed matter spectrum, not a measurement of the primordial index.

Interpret the requested abundance as comoving number density and use this question's mean density, rather than importing the different cosmology of Question 2. The mass-to-radius relation for a spherical top-hat window function givesThe normalization radius $7h^{-1}\mathrm{Mpc}$ is $10\mathrm{Mpc}$, not $4.9\mathrm{Mpc}$. Using $m=-3/4$,These are linearly extrapolated rms amplitudes, even when the numerical value at late times exceeds one. The high-redshift matter-dominated approximation is appropriate to the crossing sought below.

For $s=1/4$, the Press-Schechter halo mass function yieldsEquating this to $10^{-4}\mathrm{Mp}{\mathrm{c}}^{-3}$ gives $\nu e^{-{\nu }^2/2}=3.5555\times 10^{-5}$. On the rare-object branch, solving ${\nu }^2/2-\ln \nu =\ln (2.8125\times 10^4)$ gives $\nu \simeq 4.8634$. HenceThis is the first crossing as the Universe evolves from very early times. It retains the supplied scale-free extrapolation across a substantial range of scales; it is not a precision prediction using a full matter spectrum.

The abundance factor $\nu e^{-{\nu }^2/2}$ peaks at $\nu =1$. Therefore the same target has a second mathematical root, $\nu \simeq 3.56\times 10^{-5}$, corresponding to $z\simeq -0.99967$ if matter-dominated growth is extrapolated indefinitely. This extremely remote future branch is not the formation epoch and would not be physically justified by that growth approximation in a universe with late vacuum domination. This illustrates the two abundance crossings of the Press-Schechter mass function.

There is also an input-normalization qualification. If the characteristic cutoff is defined exactly by ${\nu }^2/2=(M/M_{\ast }{)}^{1/2}$, the variance normalization here gives $M_{\ast }(3)=10^{10}[\sqrt{2}(3.897)/1.68647{]}^4\simeq 1.14\times 10^{12}{M}_{\odot }$, rather than the earlier approximate quoted cutoff. The numerical estimate uses the explicit rms normalization in this part and the mass-scaling exponent from the preceding part; both cutoff normalizations cannot be imposed as exact simultaneously.