Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 (Field Theory in Cosmology)

Write $\varphi =\bar{\varphi }+\phi$ and evaluate derivatives of $P$ on the homogeneous backgroundThe first- and second-order changes in $X$ areAfter using the background equation to remove the linear action, the quadratic action of the P(X, phi) scalar field theory iswhereVarying gives

The Sound speed of a P(X, phi) scalar perturbation isAt leading slow variation, take $H$, $c_s$, and the kinetic coefficients as nearly constant and neglect the effective mass and their logarithmic derivatives. The Fourier equation then becomesFor $u_k=a{\phi }_k$ and de Sitter spacetime $a^{\prime \prime }/a=2/{\tau }^2$, this isTwo independent solutions areThe Bunch-Davies vacuum selects the positive-frequency behavior $e^{-i{c}_{s}k\tau }$ as $-c_{s}k\tau \to \infty$. Canonical normalization of $v=a\sqrt{A}\phi$ gives the general amplitude; with the field normalization $P_X\simeq 1$, so $A\simeq c_s^{-2}$, it reduces, up to an overall phase, to

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At cubic order, the expansion of $P(X,\varphi )$ produces the schematic operatorsThe derivative self-interactions involving $P_{XX}$ and $P_{XXX}$ are generally largest when $c_s\ll 1$, while coefficients containing explicit $\varphi$ derivatives of a slowly varying $P$ are commonly slow-variation suppressed. The former therefore tend to dominate primordial non-Gaussianity.

The terms contributing to $\phi {\phi }^{\prime 2}$ areThusEquivalently, if the complete time-dependent coefficient is called $\lambda$, then $\lambda (\tau )=a^2{\lambda }_0$.

Treat ${\lambda }_0$ as constant at leading slow variation. To cubic order the corresponding interaction Hamiltonian isFor $q_i=c_s{k}_i$, the mode function above satisfiesWriting $K=k_1+k_2+k_3$, the needed regulated integral isThere are three choices for the undifferentiated leg and two contractions interchanging the differentiated legs. The stated in-in formalism formula therefore giveswithThe powers of $c_s$ cancel for this vertex with the normalization specified in part (i). Reversing the convention for the sign of $H_I$ reverses the displayed overall sign but not the momentum shape of the primordial bispectrum.

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Use the Fourier convention $f(\mathbf{x})={\int }_{\mathbf{k}}f(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}$ with ${\int }_{\mathbf{k}}=\int d^{3}k/(2\pi {)}^3$. Neglect anisotropic stress and assume the pressureless velocity is irrotational, soFourier transforming $\nabla \cdot (\delta \mathbf{v})$ in the nonlinear continuity equation giveswhere the alpha mode-coupling kernel and its symmetrization areTaking the divergence of the Euler equation giveswith the symmetric beta mode-coupling kernel

In the Einstein-de Sitter universe, $a^{\prime }=\mathcal{H}a$ and ${\mathcal{H}}^{\prime }=-{\mathcal{H}}^2/2$. At linear order the ansatz gives ${\tilde{\theta }}^{(1)}={\tilde{\delta }}^{(1)}$. At second order, the continuity and Euler equations becomewhere each integral includes the momentum-conserving measure above. Eliminating ${\tilde{\theta }}^{(2)}$ yieldswith the standard perturbation theory density kernel

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Expand $\delta =a{\delta }_1+a^2{\delta }_2+a^3{\delta }_3+\cdots$. For a Gaussian linear density field, odd linear correlators vanish and Wick theorem reduces the fourth-order terms to products of the linear power spectrum. HenceThe two contractions of the two quadratic fields giveThe three choices for which argument of $F_3$ carries the external momentum giveThus the contribution conventionally called $P_{13}+P_{31}$ is $2P_{13}=6P(k)\int F_{3}P$. These are the two one-loop diagrams of the one-loop matter power spectrum.

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Set $\mathbf{q}={\mathbf{k}}_1$, $\mathbf{k}-\mathbf{q}={\mathbf{k}}_2$, $q=|\mathbf{q}|$, and $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{q}}$. Substitution into $F_2=5{\alpha }_s/7+2\beta /7$ givesThe azimuthal integral in $d^{3}q=q^{2}dqd\mu d\varphi$ and the factor two in $P_{22}$ then give

For the scale-free matter power spectrum $P(k)=A{k}^n$, put $r=q/k$ andAll dimensional factors separate:where

As $r\to 0$, $K_{22}\sim r^n{\mu }^2/(8{\pi }^2)$, so the soft-$q$ integral converges exactly when $n>-1$. For $r\to \infty$, $K_{22}\sim r^{2n-2}$ times an angular function, requiring $n<1/2$. At fixed $\mu <1$, $r\to 1$ is regular. The corner $(r,\mu )=(1,1)$ makes $|\mathbf{k}-\mathbf{q}|$ soft; after resolving that corner in the local soft momentum, its radial behavior is again $p^{n}dp$, requiring $n>-1$. Thereforeis the infrared- and ultraviolet-convergence window for $P_{22}$.

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For a Gaussian smoothed density contrast, the Press-Schechter factor of two gives the collapsed mass fractionThe fraction in the interval $[M,M+dM]$ is $-(dF/dM)dM=(M/\bar{\rho })(dn/dM)dM$. Since $d\nu /dM=-\nu d\log \sigma /dM$,or, equivalently, the same expression with $|d\log \sigma /d\log M|$. The minus sign is needed because $\sigma (M)$ decreases with $M$; this is the positive Press-Schechter halo mass function.

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In the peak-background split, a long-wavelength overdensity changes the local threshold from ${\delta }_c$ to ${\delta }_c-{\delta }_l$. At fixed mass, the Press-Schechter halo mass function depends on the threshold through $\nu e^{-{\nu }^2/2}$. ThereforeIt follows that the linear Lagrangian halo bias is

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On wavelengths much larger than halos, all matter is partitioned among halos. The mass-weighted halo overdensity must therefore equal the matter overdensity. Since ${\delta }_h(M)=b(M){\delta }_m$, mass conservation requires

For the Press-Schechter formalism, the mass-fraction measure becomesIt is normalized and is a half-normal distribution, soUsing the linear Eulerian halo biastherefore giveswhich verifies the consistency relation.

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Write the matter density as a sum over normalized halo profiles,For $\mathbf{k}\ne 0$, its density contrast isIf halo locations form an uncorrelated Poisson process, only equal-halo terms survive after subtracting the homogeneous contribution. Replacing the sum per unit volume by the halo abundance gives the one-halo term

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For distinct halos, insert their linearly biased correlationinto the double sum. Fourier transformation converts ${\xi }_{\mathrm{lin}}$ into $P_{\mathrm{lin}}$, while the two independent mass integrals factorize. The result is the halo modelwhereThus the requested function isAt small $k$, profile normalization gives $\tilde{u}\to 1$ and the bias consistency relation makes the square bracket tend to one, so the two-halo term approaches the linear matter spectrum.

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Project the photon Boltzmann equation onto Legendre polynomials. The angular average uses $⟨\Theta ⟩={\Theta }_0$ and $⟨\mu ⟩=0$, so the collision term has no monopole andFor the dipole, the recurrence relation makes free streaming couple ${\Theta }_1$ to ${\Theta }_0$ and ${\Theta }_2$. The gravitational term $-ik\mu \Psi$ contributes $k\Psi /3$, while projection of $i\mu v_b$ gives the baryon-velocity source. ThusNeglecting the photon quadrupole gives

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In the tight-coupling approximation, finiteness of the photon Euler equation as $\Gamma \to \infty$ requiresLet $S=3{\Theta }_1+v_b$. Rearranging the baryon Euler equation and substituting the leading relation only on its right-hand side givesInsert this slip into the photon Euler equation:The photon continuity equation gives ${\Theta }_1=-({\Theta }_0^{\prime }-{\Phi }^{\prime })/k$. Eliminating the dipole yieldswhere the photon-baryon sound speed is

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For $R=0$, $c_s=1/\sqrt{3}$. If the Newtonian potentials are constant and equal, the equation in part (b) becomesThe Sachs-Wolfe combination $D={\Theta }_0+\Psi$ therefore obeysAdiabatic initial conditions have vanishing initial fluid velocity, hence $D^{\prime }(0)=0$. With the sound horizon $r_s(\eta )={\int }_0^{\eta }{c}_s,d{\eta }^{\prime }$, the solution is

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The $\ell =2$ projection of the Boltzmann hierarchy isIn tight coupling, ${\Theta }_2=O(k/\Gamma )$ and ${\Theta }_3$ is higher order. Neglecting ${\Theta }_2^{\prime }$ relative to $\Gamma {\Theta }_2$ at leading order givesso $C=2/5$.

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The continuity equation and the cosine solution givewhen the potentials are constant. Part (d) then impliesSince Cosmic microwave background polarization is sourced by the local quadrupole at last scattering,The monopole contribution to the CMB temperature oscillates as $\cos (k{r}_{s\ast })$, so its acoustic extrema occur near $k{r}_{s\ast }=m\pi$, whereas polarization extrema occur near $k{r}_{s\ast }=(m+\frac{1}{2})\pi$. The principal polarization peaks are therefore interleaved with the principal temperature peaks. Projection, baryon loading, time-varying potentials, and the temperature Doppler contribution shift the observed angular peaks from this idealized phase relation.

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