Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 66 (Physical Cosmology)

Write $H=\dot{a}/a$ for the Hubble parameter. Subtracting the Friedmann equation from the second equation gives the Friedmann acceleration equation,Since $\dot{H}=\ddot{a}/a-H^2$, eliminating $H^2$ with the first equation yieldsDifferentiating the first equation, with constant cosmological constant and curvature parameter, now givesThus the cosmological perfect-fluid continuity equation is $\dot{\rho }+3H(\rho +p/c^2)=0$. Multiplying by $a^3$ and using $d(a^3)/dt=3H{a}^3$ provesThis is local conservation of energy: expansion changes the energy in a comoving volume through pressure work.

With constant equation-of-state parameter $w$, the cosmological perfect-fluid continuity equation becomes $\dot{\rho }+3(1+w)H\rho =0$. ConsequentlyThe constant-equation-of-state density scaling includes $\rho \propto a^{-3}$ for pressureless matter and $\rho \propto a^{-4}$ for radiation in cosmology.

For $w=-1$, constant-equation-of-state density scaling makes ${\rho }_{\mathrm{DE}}$ constant. In the first Friedmann equation, its contribution combines with the cosmological constant asIn the second equation, $-8\pi G{p}_{\mathrm{DE}}/c^2=8\pi G{\rho }_{\mathrm{DE}}$ produces precisely the same shift of $\Lambda c^2$. Therefore homogeneous expansion measures the sum of these contributions; a constant-density fluid with $p=-\rho c^2$ is dynamically indistinguishable from cosmological constant energy in these equations.

Define the present critical density and cosmological density parameter bySpatial flatness with $\Lambda =0$ implies ${\Omega }_{M,0}=1-{\Omega }_{\mathrm{DE},0}$. Separately conserved cosmological fluids and constant-equation-of-state density scaling give ${\rho }_M={\rho }_{M,0}{a}^{-3}$ and ${\rho }_{\mathrm{DE}}={\rho }_{\mathrm{DE},0}{a}^{-3(1+w)}$. Inserting both into the Friedmann equation yieldsThis is the constant-$w$ case of the Hubble parameter for matter and dynamical dark energy.

Assume both densities are nonnegative and ${\Omega }_{\mathrm{DE},0}<1$, so a nonzero pressureless matter component exists. Their ratio obeysThus matter domination givesAt fixed $H_0$, the early expansion rate does depend on the present dark energy fraction: increasing it reduces ${\rho }_{M,0}=(1-{\Omega }_{\mathrm{DE},0})3H_0^2/(8\pi G)$. There is no contradiction with negligible early dark energy, since the leading expression is simply $H^2\sim 8\pi G{\rho }_M/3$. At fixed physical matter density, that leading rate is independent of the negligible component.

The endpoint ${\Omega }_{\mathrm{DE},0}=1$ has no matter, so the ratio argument does not apply. Its Hubble parameter is $H=H_0{a}^{-3(1+w)/2}$: it diverges for $-1<w<0$, is constant for $w=-1$, and tends to zero for $w<-1$.

For pressureless matter and zero cosmological constant, the Friedmann acceleration equation gives $\ddot{a}/a=-4\pi G\rho /3$. The deceleration parameter therefore satisfies the dust relation between deceleration and density, $q=4\pi G\rho /(3H^2)$. Evaluating today,Substitute this into the present Friedmann equation to obtain $H_0^2+k{c}^2/a_0^2=2q_0{H}_0^2$, orIn particular, $q_0>1/2$, $q_0=1/2$, and $q_0<1/2$ correspond to positive, zero, and negative spatial curvature.

Normalize the scale factor by $x=a/a_0$. Constant-equation-of-state density scaling for pressureless matter gives $\rho ={\rho }_0{x}^{-3}$. Multiplying the Friedmann equation by $x^2$ and substituting the expressions from part (i),Take the expanding branch $\dot{x}>0$ and set the origin of cosmic time at the Big Bang, $x=0$. The age of a dust universe without a cosmological constant is consequently

The stated real trigonometric substitution applies to the closed case $q_0>1/2$. Set $A=2q_0-1$, so $x=q_0(1-\cos \theta )/A$, $dx=q_0\sin \theta d\theta /A$, andThe upper limit obeys $1-\cos {\theta }_{\ast }=A/q_0$, with $0<{\theta }_{\ast }<\pi$ on the expanding branch. Substitution into the age of a dust universe without a cosmological constant yieldsOn this interval, the integrand equals $1-\cos \theta$, so it can also be evaluated:For $q_0=1/2$, the original integral instead gives $H_0{t}_0=2/3$. For $0<q_0<1/2$, use a hyperbolic substitution to obtain $H_0{t}_0=q_0(\sinh {\eta }_{\ast }-{\eta }_{\ast })/(1-2q_0{)}^{3/2}$, where $\cosh {\eta }_{\ast }=(1-q_0)/q_0$. A real $\theta$ cannot implement the printed substitution in that open case.

An empty universe with $\Lambda =0$ has ${\Omega }_{r,0}={\Omega }_{m,0}={\Omega }_{\Lambda ,0}=0$. The present Friedmann equation then requires ${\Omega }_{k,0}=1$, soThe Friedmann acceleration equation gives $\ddot{a}=0$, hence $a\propto t$ after choosing its zero at $t=0$, andThis is the Milne universe. A larger past Hubble parameter does not require negative $\ddot{a}$: even constant $\dot{a}$ produces $H=\dot{a}/a=1/t$. Equivalently, deceleration parameter from the Hubble derivative gives $q=-1-\dot{H}/H^2=-1+1=0$. The age is $t_0=H_0^{-1}$.

The Lyman-alpha forest is the collection of absorption lines blueward of a distant quasar's own Lyman-alpha line. Foreground neutral hydrogen removes photons at rest wavelength ${\lambda }_{\alpha }\simeq 1215.67$ angstroms, observed at $\lambda =(1+z){\lambda }_{\alpha }$ because of cosmological redshift. Gas at many different foreground redshifts therefore produces many lines. The weak forest traces diffuse, mostly ionized intergalactic medium, rather than a population of fully neutral clouds; much stronger, self-shielded absorbers are a separate regime. Madau's account of quasar absorption systems distinguishes these column-density regimes.

For highly ionized hydrogen in photoionization equilibrium, balance the photoionization rate against radiative recombination:Thus the neutral fraction is $n_{\mathrm{HI}}/n_H\simeq \alpha (T)n_H/{\Gamma }_{\mathrm{HI}}$ and the neutral column density across a proper path $L$ isFor an illustrative diffuse gas with $T\sim 10^4\mathrm{K}$, $\alpha \sim 4\times 10^{-13}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$, ${\Gamma }_{\mathrm{HI}}\sim 10^{-12}{\mathrm{s}}^{-1}$, and $n_H\sim 10^{-5}\mathrm{c}{\mathrm{m}}^{-3}$, the neutral fraction is only $4\times 10^{-6}$. A proper path of $100$ kiloparsecs then gives $N_{\mathrm{HI}}\sim 10^{13}\mathrm{c}{\mathrm{m}}^{-2}$, enough to produce a weak Lyman-alpha absorption feature even though the gas is almost entirely ionized.

This hydrogen number density is of the order of the cosmic mean at redshift a few. Since the weak forest is common along most high-redshift lines of sight, substantial path length passes through such diffuse gas. Much denser material would give much larger column density for comparable path lengths and ionizing flux. These arguments support near-mean or modestly overdense intergalactic medium as the origin of weak forest lines, not the claim that every absorber has exactly the cosmic mean density: path length, temperature, and ionizing background must also be constrained.

A radial null geodesic in the FLRW metric obeys $d\chi =c|dt|/a$. The redshift-time relation $dt=-dz/[(1+z)H(z)]$, with $a_0=1$, therefore givesThis is comoving radial distance. The proper distance in cosmology at the clouds' approximately common epoch is insteadThe distinction is essential when quoting a cloud separation. The approximation neglects the change of $H$ across the interval and assumes the observed spacing is dominated by cosmological redshift; peculiar velocity can perturb the inferred separation. It is the redshift version of proper absorber separation from a wavelength interval.

The quasar proximity effect is weaker Lyman-alpha forest absorption near a quasar's emission redshift. Extra photons from the quasar increase the photoionization rate and reduce the neutral fraction without necessarily reducing the total gas density.

For isotropic specific intensity $J_{\nu }$ and a steady, isotropically radiating quasar with spectral luminosity $L_{\nu }$, the background and local rates areHere $r$ is a proper separation small enough that attenuation and expansion across it can initially be neglected, and ${\sigma }_{\nu }$ is the hydrogen photoionization cross-section. At fixed density and temperature, photoionization equilibrium makes the neutral column density, and hence a given line's optical depth, scale asThe observed quasar luminosity supplies ${\Gamma }_Q(r)$; the distance at which the absorption changes significantly estimates where ${\Gamma }_Q\sim {\Gamma }_{\mathrm{bg}}$. Adopting a background spectral shape then converts the inferred rate to its specific intensity, often quoted near the hydrogen ionization energy threshold.

Important uncertainties include an enhanced gas density contrast around the host, the unobserved ionizing spectrum, anisotropic emission, source lifetime and variability, the systemic redshift and continuum placement, attenuation between source and absorber, and fluctuations between lines of sight. Higher gas density enhances recombination and can partially hide the effect, biasing a simple background estimate upward. Dall'Aglio, Wisotzki and Worseck's 2008 measurements illustrate the observed sightline scatter and the roles of variability and clustering.

The ionizing background is produced mainly by hot stars in star-forming galaxies and by accretion-powered quasars. The stellar contribution depends on star formation and the ionizing photon escape fraction; the quasar contribution depends on its evolving luminosity function. Radiative transfer through the intergalactic medium, including absorption, recombination emission, and cosmological redshifting, determines the radiation actually available for photoionization.

In the picture relevant to this 2009 paper, increasingly rare quasars cannot by themselves explain the high-redshift hydrogen ionizing field, so galaxies become especially important above $z\sim 3$. Harder quasar radiation is important for doubly ionizing helium around $z\sim 3$. Hydrogen reionization must be largely accomplished by $z\sim 6$. At lower redshift the evolution reflects declining source emissivity together with the increasing propagation distance of ionizing photons; near reionization, short absorption lengths and spatially uneven ionized regions prevent a simple uniform-background description. The background rate need not follow the quasar population alone: a roughly flat hydrogen photoionization rate over $2\lesssim z\lesssim 4$ is compatible with a changing mixture of galaxies and quasars. Faucher-Giguère and collaborators' 2009 background calculation and their preceding opacity analysis give this interpretation. These redshifts describe the broad historical picture, rather than precise universal transition times.

The comoving particle horizon is the maximum radial distance a photon could have travelled since the initial Big Bang, expressed in comoving coordinates. It describes possible past causal contact, whereas a cosmological event horizon concerns photons that can reach an observer in the future.

For a radial null geodesic in the FLRW metric, $dr/\sqrt{1-k{r}^2}=cdt/a$. Spatial flatness sets $k=0$, so with $a_0=1$,The given Hubble parameter then supplies $a^{2}H=H_0\sqrt{{\Omega }_{r,0}+{\Omega }_{m,0}a+{\Omega }_{\Lambda ,0}{a}^4}$. Since cosmological redshift obeys $a=(1+z{)}^{-1}$,The corresponding proper horizon at that epoch is $a(z)r_p(z)$.

Well before matter-radiation equality, radiation domination makes the integrand approximately constant:At equality, ${\Omega }_{r,0}{a}_{\mathrm{eq}}^{-4}={\Omega }_{m,0}{a}_{\mathrm{eq}}^{-3}$, so $a_{\mathrm{eq}}\simeq 1/3000$ and ${\Omega }_{r,0}\simeq 10^{-4}$. Extrapolating the radiation-only result gives $r_p(a_{\mathrm{eq}})\simeq 133\mathrm{Mpc}$. This overestimates the horizon because matter is no longer negligible at equality and increases the expansion rate.

Retain both matter and radiation, still neglecting the insignificant early cosmological constant. The particle horizon through radiation-matter equality isThereforeThis is a comoving distance; the proper horizon at equality is approximately $110/3000\mathrm{Mpc}\simeq 37$ kiloparsecs.

For today's horizon, extrapolating matter domination givesThis is an order-of-magnitude estimate: dark energy is significant late, and radiation matters early. Numerical integration of the complete expression with the supplied density parameters and ${\Omega }_{r,0}\simeq 10^{-4}$ givesBoth neglected positive-density terms increase $H$ at fixed $a$ and reduce the horizon relative to the matter-only extrapolation.

The distance modulus gives $m_V-M_V=5{\log }_{10}(d/10\mathrm{pc})$. At the magnitude limit, $m_V-M_V=17-(-21)=38$, soThus the stated sample reaches a typical galaxy to approximately $400$ megaparsecs. This uses the assumed absolute magnitude, neglects the specified spectral correction, and equates luminosity distance, angular diameter distance, and proper distance as permitted. It does not imply completeness for galaxies fainter than the adopted typical luminosity.

The plot uses comoving wavenumber in $\mathrm{Mp}{\mathrm{c}}^{-1}$. Using the inverse comoving particle horizon from part (ii) gives $1/r_{p,\mathrm{eq}}\simeq 9\times 10^{-3}\mathrm{Mp}{\mathrm{c}}^{-1}$. The conventional matter-radiation equality scale is insteadThese are consistent order-one conventions for locating the turnover near $10^{-2}\mathrm{Mp}{\mathrm{c}}^{-1}$. The horizon-entry condition is $k/(aH/c)\sim 1$; equating an entire Fourier wavelength $2\pi /k$ to the particle horizon would attach a different numerical factor.

For adiabatic initial conditions, the asymptotic cold-dark-matter power spectrum has $P(k)\propto k^{n_s}T(k{)}^2$, where the scalar spectral index is $n_s\simeq 1$ and $T$ is the cosmological transfer function. Modes below $k_{\mathrm{eq}}$ enter during matter domination and have little relative suppression, giving approximately $P\propto k$. Modes above $k_{\mathrm{eq}}$ enter during radiation domination, when logarithmic growth of matter perturbations during radiation domination is much slower. Their transfer is approximately $T\propto \ln (k/k_{\mathrm{eq}})/k^2$, giving $P\propto k^{-3}{\ln }^2(k/k_{\mathrm{eq}})$ well above equality. This explains the broad turnover; baryon acoustic oscillations add structure, and nonlinear evolution modifies sufficiently small scales.

The comoving distance to the Cosmic microwave background last-scattering surface is of order the present horizon, ${\chi }_{\ast }\sim 1.3\times 10^4\mathrm{Mpc}$. At recombination the particle horizon is much smaller, so replacing ${\chi }_{\ast }$ by today's horizon is adequate for the requested estimate. A transverse Fourier mode projects approximately to angular multipole $\ell \sim k{\chi }_{\ast }$. If an angular feature's width is estimated by $\theta \sim 1/\ell$, the resolution ${\theta }_{\min }\sim 10^{-3}$ givesUsing the common half-wavelength convention $\theta \sim \pi /\ell$ instead gives ${\ell }_{\max }\sim 3000$ and $k_{\max }\sim 0.2$--$0.3\mathrm{Mp}{\mathrm{c}}^{-1}$. These express the same order-of-magnitude resolution estimate with different angular-scale conventions. On the graph, a useful indicative band is therefore from about $10^{-4}\mathrm{Mp}{\mathrm{c}}^{-1}$ to order $10^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, potentially a few times $10^{-1}$; it is not a sharp instrumental boundary. The lowest useful multipoles are of order a few, with substantial cosmic variance.

The cosmic microwave background constrains the matter power spectrum indirectly through the primordial spectrum and radiation/matter cosmological transfer functions. Beam resolution, Cosmic microwave background diffusion damping, foreground emission, and projection of three-dimensional fluctuations into angular anisotropy limit the recoverable small-scale information. Resolving a given angle alone does not guarantee a precise matter-spectrum measurement there.

A survey depth $R\sim 400\mathrm{Mpc}$ makes wavelengths substantially larger than its volume poorly measurable. A characteristic inverse depth is $1/R\simeq 2.5\times 10^{-3}\mathrm{Mp}{\mathrm{c}}^{-1}$; a Fourier wavelength that fits across a dimension $L$ instead requires $k\gtrsim 2\pi /L$. For $L$ between $R$ and $2R$, this is roughly $0.008$--$0.016\mathrm{Mp}{\mathrm{c}}^{-1}$. A limited sky patch can have smaller transverse dimensions, so its survey window function raises the practical minimum in those directions. An indicative useful band for a broad survey is around $10^{-2}\mathrm{Mp}{\mathrm{c}}^{-1}$ and above, including scales near equality if its volume is large enough.

There is no upper wavenumber determined by depth alone. Mean separation of observed galaxies, galaxy power spectrum shot noise, angular/redshift resolution, and nonlinear clustering set it. Straightforward linear inference is typically restricted to roughly $k\lesssim 0.1\mathrm{Mp}{\mathrm{c}}^{-1}$ at low redshift, with smaller scales requiring modelling. The magnitude limit does not specify the galaxy comoving number density, so a more precise sampling cutoff cannot be deduced from these data.

The two measurements are complementary. The cosmic microwave background samples a large volume at an early epoch when relevant perturbations are mostly linear, and is free of galaxy bias; its matter-spectrum inference is indirect and depends on projection, cosmological parameters, foregrounds, and cosmic variance. A galaxy survey gives three-dimensional late-time information and can reach smaller scales, but observes biased, discrete tracers: galaxy bias, nonlinear gravitational growth, redshift-space distortions, selection effects, and galaxy power spectrum shot noise must be controlled. Combining both constrains the initial spectrum and its subsequent growth more strongly than either alone, while their overlapping modes test the inferred evolution.