Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 (Cosmology)

In natural units, the equilibrium phase-space distribution function iswhere the minus sign gives the Bose-Einstein distribution for bosons and the plus sign gives the Fermi-Dirac distribution for fermions.

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Summing over the $g$ internal states, the number density, energy density, and isotropic pressure areandThe factor ${\mathbf{p}}^2/(3E)$ in the kinetic pressure of an isotropic gas is the angular average of one diagonal component of the momentum flux.

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For an ultrarelativistic particle, $E\simeq |\mathbf{p}|$. Setting $\mu =0$ and writing the three-dimensional momentum integral radially givesThe standard Bose integrals yieldFor fermions, the corresponding integrals differ by the familiar factorsThus the number density scales as $T^3$, while the energy density and pressure scale as $T^4$.

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The entropy density isFor the homogeneous cosmological plasma, adiabaticity gives the cosmological entropy conservation equationor equivalently $s{a}^3=\text{constant}$ in a comoving volume.

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After thermal decoupling in cosmology, the collisionless relativistic species has $T_{X}a=\text{constant}$. The still-coupled plasma separately conserves entropy, so $g_{\ast s}{T}_{\gamma }^3{a}^3$ remains constant. At decoupling $T_X=T_{\gamma }$, while at late times the coupled plasma contains only the two photon polarizations. Therefore the temperature of a decoupled relativistic relic isHere $g_{\ast }$ is understood as the effective entropy degrees of freedom in the plasma that remains coupled after $X$ decouples.

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For the hilltop inflation potential,The two potential slow-roll parameters are consequentlyand

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Estimating the end of slow-roll inflation by ${ϵ}_V({\varphi }_e)\simeq 1$ and using the leading hilltop expression givesThis is a formal leading-order estimate. Its self-consistency requires $m^2{\varphi }_e^2\ll {\Lambda }^4$; if that condition fails, the full potential or the mechanism that ends inflation must be retained.

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The homogeneous inflaton obeysUnder the slow-roll approximation, the acceleration and kinetic-energy corrections may be neglected, so the field equation and Friedmann equation becomeThus a positive field rolls away from the hilltop while $H$ is approximately constant.

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The slow-roll e-fold count before the end of inflation isAt leading order near the hilltop, $V/V^{\prime }\simeq -{\Lambda }^4/(m^2\varphi )$, and henceThis has $N({\varphi }_e)=0$ and gives

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Comparing the stated curvature power spectrum with $P_{\mathcal{R}}=2{\pi }^2{A}_s/k^3$ at the pivot scale gives the primordial scalar amplitudeThe scalar spectral index isSufficiently close to the hilltop, ${ϵ}_V$ is suppressed by ${\varphi }^2$ and the ${\eta }_V$ term controls the spectral tilt.

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Neglecting the ${ϵ}_V$ contribution to the tilt, the observed value givesPart (d) then gives ${\varphi }_{60}=e^{-1}{\varphi }_e$. Using the formal end estimate from part (b), ${ϵ}_V({\varphi }_{60})\simeq e^{-2}$. The amplitude relation therefore yieldsThus the requested order-of-magnitude estimate isThe simultaneous approximations make ${ϵ}_V({\varphi }_{60})$ only moderately small, so this numerical result should be read as the order-of-magnitude estimate requested rather than a precision fit.

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Local energy-momentum conservation is the vanishing covariant divergence of the energy-momentum tensor:

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For a mixed tensor,Write $T=\bar{T}+\delta T$ and $\Gamma =\bar{\Gamma }+\delta \Gamma$. The homogeneous background already satisfies ${\bar{\nabla }}_{\mu }{\bar{T}}^{\mu }{}_{\nu }=0$, and discarding products of two perturbations leavesThis is simply the first variation of the tensor's covariant derivative.

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For a spatial index $j$, the background Christoffel symbols giveTogether with the $3H\delta T^0{}_j$ trace term, the first two contributions combine to $2H\delta T^0{}_j$. Varying the Levi-Civita connection and contracting it with ${\bar{T}}^{\mu }{}_{\nu }=\mathrm{diag}(-\bar{\rho },\bar{P},\bar{P},\bar{P})$ givesand the surviving combination reduces toConsequently the spatial momentum equation is

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In synchronous gauge in cosmology, $h_{0\mu }=0$, so the explicit metric terms vanish. Symmetry of the covariant energy-momentum tensor and the background metric implyHence the two expansion terms in part (c) combine to $3H\delta T^0{}_j$. Substituting the scalar velocity potential and scalar anisotropic stress definitions gives

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Every term in part (d) is a spatial gradient. Extracting the longitudinal scalar coefficient, up to a spatially homogeneous function that can be absorbed into the zero mode, yields the Cosmological Euler equation in synchronous gauge:

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Yes. Synchronous gauge in cosmology uses freely falling time lines and sets the lapse and shift perturbations to zero, so no gravitational-potential force appears explicitly in this component of momentum conservation. Metric perturbations still affect the other field equations, the evolution of matter variables, and the relation between coordinate-dependent variables and gauge-invariant observables; synchronous gauge also retains residual gauge modes.

In Newtonian gauge in cosmology, the time-time metric perturbation is a Newtonian gravitational potential. Its spatial gradient therefore appears explicitly as a force term in the Euler equation. The difference is a coordinate representation of the same covariant conservation law.

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Taking the divergence of the linear Euler equation and defining the peculiar-velocity divergence $\theta =\nabla \cdot \mathbf{v}$ givesThe assumption of negligible vorticity ensures that this longitudinal variable captures the velocity perturbation relevant to scalar density growth.

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Differentiate the linearized cosmological continuity equation, $\dot{\delta }=-\theta /a$, and use $\dot{a}/a=H$:Substitution of part (a), followed by the Poisson equation ${\nabla }^2\Phi =4\pi G{a}^2\bar{\rho }\delta$, yields

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For a barotropic equation of state, $\delta P=c_s^2\delta \rho =c_s^2\bar{\rho }\delta$. A spatial Fourier mode of comoving wavenumber $k$ therefore satisfiesThe Jeans wavenumber is defined by equality of the pressure and self-gravity terms:For $k\gg k_J$, pressure dominates and produces acoustic oscillations whose amplitude is affected by Hubble friction. For $k\ll k_J$, self-gravity dominates and a growing mode develops: this is Jeans instability.

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Cold dark matter has $c_s=0$. During matter domination,The density equation becomesTrying the power law $\delta =t^p$ giveswhose roots are $p=2/3$ and $p=-1$. Thus the cosmic-time matter density modes areand the leading growing mode is $\delta \propto t^{2/3}\propto a(t)$.

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The constant-equation-of-state density scaling givesFor a spatially flat universe dominated by this component, the Friedmann equation gives $H^2\propto a^{-1}$. Integrating $\dot{a}/a\propto a^{-1/2}$ yieldsThis is accelerated expansion because $w=-2/3<-1/3$.

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The dominant component is smooth, so only the subdominant matter density clusters. The cold-matter perturbation equation is thereforeAt late times, $a\gg 1$, the source term is negligible compared with the expansion terms. Setting $\tau =t-t_{\ast }$ and using $H=2/\tau$ givesIt integrates toThe constant mode shows the suppression of matter growth by smooth accelerated expansion; the second mode decays.

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