Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 45 / 4

Let $\delta (\mathbf{x})$ be a mean-zero density contrast. For a statistically homogeneous and isotropic field, its two-point correlation function is $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$, depending only on $r=|\mathbf{r}|$. For a galaxy point process the corresponding distinct-pair definition is $d{P}_{12}={\bar{n}}^2[1+\xi (r)]d{V}_{1}d{V}_2$, the excess pair probability relative to independent uniform positions. Self-pair shot noise is separated when using the density-field version.

Specify the Fourier convention byThe diagonal covariance follows from statistical homogeneity, and isotropy makes the matter power spectrum a function of $k$ alone. Substitute the inverse transforms into the correlation and integrate the delta function:Choose the polar axis parallel to $\mathbf{r}$. Angular integration gives $\int d{\Omega }_k{e}^{ikr\cos \theta }=4\pi \sin (kr)/(kr)$. HenceConversely, when the transform is defined, $P(k)=4\pi {\int }_0^{\infty }{r}^2\xi (r)\sin (kr)/(kr)dr$. The stated convention fixes every $2\pi$ factor.

Write $K=k_{\max }r$. Using the transform in part (a), the band-limited linear density correlation isTwo integrations by parts give the antiderivative $-k^2\cos (kr)/r+2k\sin (kr)/r^2+2\cos (kr)/r^3$. Subtract its value at zero to obtain, for $r>0$,The origin is a removable limit. The bracket starts with $K^4/4$, soThis is also the density variance obtained directly by setting $\sin (kr)/(kr)=1$ in the integral. The sharp spectral edge produces oscillatory correlations, with leading large-$r$ term $-A{k}_{\max }^2\cos (k_{\max }r)/(2{\pi }^2{r}^2)$. A negative correlation at some separation is compatible with a nonnegative power spectrum.

Interpret the requested scale-dependent rms as the contribution from a logarithmic band of wavelengths, or equivalently a band centered at $k\sim R^{-1}$. A global uncut rms is not finite for this scale-free spectrum. The dimensionless density power per logarithmic wavenumber isThe cosmological Poisson equation gives ${\phi }_{\mathbf{k}}=-(4\pi G{a}^2\bar{\rho }){\delta }_{\mathbf{k}}/k^2$. ThereforeThis is the potential fluctuations per logarithmic wavenumber scaling. A simple dimensional argument gives the same answer: the density amplitude on scale $R$ is proportional to $R^{-(n+3)/2}$, and its potential amplitude is of order $G\bar{\rho }(aR{)}^2{\delta }_R$, supplying two additional powers of $R$.

For $n=1$, the Harrison-Zeldovich spectrum has scale-independent potential amplitude per logarithmic interval. If the density-spectrum amplitude grows as $A(t)=b^2(t)A_{\mathrm{in}}$, then $4\pi G{a}^2\bar{\rho }=C/a$, so the potential amplitude evolves as $b/a$. It is constant in the pure growing matter mode of an Einstein-de Sitter universe, where $b=a$.

A literal total variance would be proportional to ${\int }_0^{\infty }{k}^{n-2}dk$. Infrared convergence requires $n>1$ and ultraviolet convergence requires $n<1$, so no index makes both ends finite. At $n=1$ both divergences are logarithmic. Thus cutoffs or a band specification are essential for a numerical rms; the boxed statement describes the intended scale-local fluctuations rather than claiming a finite unsmoothed potential variance.

The Press-Schechter formalism starts from a statistically homogeneous Gaussian initial density contrast with variance specified by its linear matter power spectrum. Average it on mass scale $M=(4\pi /3){\bar{\rho }}_{m0}{R}^3$, usually with a spherical top-hat. Let $\sigma (M,t)=b(t){\sigma }_0(M)$, normalizing $b$ consistently. Assume that a region becomes a collapsed halo when its linearly extrapolated smoothed density exceeds a mass-independent linear spherical-collapse threshold ${\delta }_{msc}\simeq 1.686$ in the Einstein-de Sitter approximation. This extrapolates a linear Gaussian statistic to a nonlinear collapse criterion, and ignores the detailed nonspherical dynamics.

The one-scale Gaussian upper tail only assigns half the mass to overdense regions when the variance grows without limit. The conventional factor of two is inserted to account for material in larger collapsed regions even if it falls below the threshold on a smaller scale, the cloud-in-cloud issue. A first-crossing excursion-set argument can justify that factor for an ideal sharp-k filter; it is not a theorem that an arbitrary top-hat one-scale probability alone supplies it. With this prescription, the mass fraction in objects above $M$ isConvert mass fraction to comoving number density by assigning mass $M$ to each halo, using $dn/dM=({\bar{\rho }}_{m0}/M)|dF/dM|$. This yields the Press-Schechter halo mass functionEquivalently one can use a present-extrapolated threshold ${\delta }_c(t)={\delta }_{msc}/b(t)$ and present variance; using both that threshold and an already grown variance would double-count growth. The model assumes approximate mass conservation, a smoothing-to-halo-mass assignment and a spherical universal barrier. It is an approximation to gravitational halo formation, not an exact partition of the real cosmic web.

In hierarchical formation, ${\sigma }_0(M)$ decreases with mass: smaller regions become nonlinear first and are subsequently incorporated into larger haloes. Define a characteristic mass by $b(t){\sigma }_0(M_{\ast })={\delta }_{msc}$. It increases as growth proceeds, moving the exponential cutoff towards larger masses. For a scale-free spectrum with well-defined top-hat variance, $-3<n<1$, ${\sigma }_0\propto M^{-(n+3)/6}$ and hence $M_{\ast }\propto b^{6/(n+3)}$. At a fixed mass, the derivative of $\ln (dn/dM)$ with respect to $\ln b$ is ${\nu }^2-1$: rare high-mass abundances grow, while the small-mass abundance can decline as objects merge into larger systems. Thus one must not claim that every fixed-mass bin increases forever. The growth of $M_{\ast }$ is the scale-free characteristic mass in the Press-Schechter formalism.

For the temperature test, virial equilibrium and gas hydrostatic equilibrium give $k_{B}T\sim \mu m_{p}GM/R$. Defining the halo radius by $M=(4\pi /3){\Delta }_v{\rho }_c(z)R^3$ yieldsA calibrated monotone mass-temperature relation converts the mass function to $dn/dT=(dn/dM)|dM/dT|$, with temperature scatter treated by convolution when necessary. The cluster temperature function normalization is strongly sensitive to the linear fluctuation amplitude: at fixed mass and spectral shape, $\partial \ln n/\partial \ln {\sigma }_8={\nu }^2-1$. Rare hot clusters occupy the exponential tail, so even a modest amplitude change can produce a large abundance change. Here sigma eight is the rms linear density contrast today in an $8h^{-1}\mathrm{Mpc}$ top-hat, not the full spectrum.

Compare observed cluster-temperature counts and their redshift evolution with these predictions to constrain that normalization jointly with matter density and growth. The mass-temperature calibration, selection volume, temperature scatter, nonthermal support and deviations from the ideal halo mass function limit a temperature-only inference. The abundance of rare hot clusters is a sensitive amplitude probe once their mass calibration and cosmology are specified.