Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 75 / 2

Use physical momentum $p_{\mathrm{phys}}$ and conserved-background comoving momentum $q=a{p}_{\mathrm{phys}}$. At fixed $q$, the homogeneous collisionless phase-space distribution function is $f_0(q)$, with no spatial or angular dependence and no explicit background time derivative. The printed relation $p_i=aq{n}_i$ cannot have this meaning: freely propagating physical momentum redshifts as $a^{-1}$, so it should read $p_i=q{n}_i/a$. Keeping the printed relation literally would give $q=p_{\mathrm{phys}}/a$ and a nonzero zeroth-order redshift, inconsistent with the target equation. The standard definition is positively confirmed by the phase-space convention in Ma and Bertschinger's perturbation treatment.

With this conserved $q$, the orders of the factors are:

Thus the linear Collisionless Boltzmann equation isFor photons $v_0^i=n^i/a$. For a particle of mass $m$, $v_0^i=q{n}^i/[a\sqrt{q^2+a^2{m}^2}]$. The order counting does not assume that the first-order angular deflection itself vanishes; only its multiplication by an already perturbed angular distribution makes it higher order.

To match the positive source printed in the question, choose synchronous gauge metricThe commonly used convention with ${\delta }_{ij}+h_{ij}$ gives the opposite source sign; switching conventions means replacing $h$ and $h_S$ by their negatives together. This distinction is necessary when comparing with the usual synchronous photon brightness equation.

Let $p^{\alpha }=d{x}^{\alpha }/d\lambda$ be the photon tangent for an affine parameter. The covariant geodesic equation isFor $\nu =0$, the homogeneous scale-factor contributions cancel by the null geodesic condition, leaving $d{\tilde{p}}_0/d\lambda =-a^2{h}_{ij}^{\prime }{p}^i{p}^j/2$. Since ${\tilde{p}}_0=-a^2{p}^0=-q$ and $d\tau /d\lambda =p^0$, use the background $p^i=p^0{n}^i$ in this first-order term to obtainUsing the affine form also avoids confusing a covariant $p_0$ with the contravariant factor $p^0$ required when changing the geodesic parameter to coordinate time.

In conformal time, the equation from part(a) becomes $f_1^{\prime }+n^i{\partial }_i{f}_1+q^{\prime }{f}_0^{\prime }=0$. A Fourier mode proportional to $e^{i\mathbf{k}\cdot \mathbf{x}}$ has streaming term $ik\mu f_1$. Multiply by $-4/(q{f}_0^{\prime })$; its prefactor is background-time independent. With $\Delta =-4f_1/(q{f}_0^{\prime })$,Contracting the scalar decomposition explicitly givessoThis is the photon brightness equation for a negative spatial perturbation. For a perturbation of blackbody-temperature form, $f_1=-q{f}_0^{\prime }\Theta$, so $\Delta =4\Theta$ is the photon brightness perturbation. A more general initial spectral distortion can retain $q$ dependence; the gravitational forcing is still independent of $q$ in this linear massless equation.

The supplied matter-era solution has $h^{\prime }=h_S^{\prime }=A{k}^{1/2}\tau /2$, so the source reduces to $A{k}^{1/2}{\mu }^2\tau$. Put $D={\tau }_0-{\tau }_{\ast }$ and $W=e^{-ik\mu D}$. The integrating factor for the streaming equation gives the exact matter-era photon endpoint solutionFor $b=k\mu \ne 0$, the integral is $({\tau }_0-{\tau }_{\ast }W)/(ib)+(1-W)/b^2$. Substitution and collection of endpoint terms givesAt $\mu =0$ the original source is zero and the expression has the continuous limit ${\Delta }_0={\Delta }_{\ast }$. The integral form is numerically preferable near that limit, where the separated endpoint terms cancel.

The last two terms are respectively a local observer monopole and dipole. For measured higher angular multipoles, remove them. The remaining emission bracket contains initial brightness, an emission velocity-type term, and a gravitational Sachs-Wolfe effect term. Under ordinary adiabatic initial conditions for the growing synchronous matter mode, ${\Delta }_{\ast }\sim A{k}^{1/2}{\tau }_{\ast }^2$, while the velocity endpoint scales as $A{k}^{-1/2}{\tau }_{\ast }$. Relative to $A{k}^{-3/2}$ these are suppressed by $(k{\tau }_{\ast }{)}^2$ and $k{\tau }_{\ast }$ respectively. Thus for modes outside the horizon at last scattering,Its $k^{-3/2}$ amplitude is the scale-invariant large-angle contribution. In Newtonian language it is the ordinary endpoint Sachs-Wolfe effect, not a late integrated effect in a constant-potential matter universe. With completely arbitrary prescribed ${\Delta }_{\ast }$, no unconditional dominance theorem is possible: an independently large initial anisotropy could dominate. The exact solution above remains valid without that adiabatic assumption.