Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 67 / 3

In the Maxwell-Boltzmann velocity distribution, each Cartesian component has mean-square velocity $k_B{T}_e/m_e$. Thus $⟨v^2⟩=3k_B{T}_e/m_e$. The original PDF calls this quantity an rms velocity, but the rms speed is its square root. Also, its $n_{\gamma }$ must mean the dimensionless mean occupation number $n$, since its equilibrium value is a Planck photon distribution; it is not the frequency-integrated photon number density.

Take $x=h\nu /(k_B{T}_{\gamma })$ at a fixed local radiation temperature. Because $x$ is proportional to frequency,The hot-electron Kompaneets equation therefore becomesFor traversal length $d\ell =cdt$, the Compton y parameter isIt measures electron pressure integrated along the line of sight, or the Thomson scattering optical depth weighted by $k_B{T}_e/(m_e{c}^2)$. In this hot-electron limit the mean fractional energy gain is approximately $4y$. The derivation assumes nonrelativistic electrons, Thomson-limit photon energies and $T_e\gg T_{\gamma }$. The original PDF uses $h\nu$, as required for ordinary frequency; the converted TeX incorrectly substitutes $\hslash \nu$.

Write $n=n_0+y\mathcal{L}{n}_0+O(y^2)$, with $n_0=(e^x-1{)}^{-1}$ and $\mathcal{L}=x^{-2}{\partial }_x(x^4{\partial }_x)$. Differentiating the Planck photon distribution givesDividing by $n_0$ proves the first-order thermal Sunyaev-Zeldovich effect:In the Rayleigh-Jeans law limit, $x{e}^x/(e^x-1)\to 1$ and $x\coth (x/2)\to 2$, so $\Delta n/n_0\to -2y$. In the Wien approximation, the prefactor approaches $x$ and the bracket approaches $x-4$, so $\Delta n/n_0\sim yx(x-4)\sim y{x}^2$. The last limit is an asymptotic description of the first-order coefficient: a perturbative prediction at large $x$ still requires $y{x}^2\ll 1$.

Hot electrons transfer energy to the cosmic microwave background, removing low-frequency photons and enhancing the high-frequency tail. The first-order distortion crosses zero where $x\coth (x/2)=4$, at $x\simeq 3.830$; for today's CMB this is near $217\mathrm{GHz}$. Photon conservation follows directly from the boundary termIn contrast, integration by parts gives $\int x^3\Delta ndx=4y\int x^3{n}_{0}dx$, so radiation gains a fraction $4y$ of its original energy. The thermal Sunyaev-Zeldovich effect is thus a spectral distortion, not simply a new blackbody temperature.

A galaxy cluster with bulk peculiar velocity along the line of sight additionally produces the kinetic Sunyaev-Zeldovich effect. Define $v_{\parallel }>0$ for recession and $\tau =\int n_e{\sigma }_{T}d\ell$. To first order in velocity and optical depth, $\Delta T/T=-\tau v_{\parallel }/c$: receding gas gives a decrement and approaching gas an increment. Its occupation number perturbation is $(\Delta T/T)x{e}^x/(e^x-1{)}^2$, the spectrum of a small blackbody temperature change. It remains nonzero at the thermal null and therefore complicates measurements near that frequency.

There is a genuine flux typo in the original PDF. A received bolometric radiative flux must have inverse-square luminosity distance, so the physically consistent thermal bremsstrahlung expression is $S_X\propto d_L^{-2}\int n_e^2{T}_e^{1/2}dV$. The printed single power of $d_L$ cannot imply the requested Hubble constant scaling.

For an approximately uniform, spherical galaxy cluster let its depth be $\ell =q\theta d_A$, where $q$ is a fixed geometrical factor. With fixed profile coefficients $A,B$, the Rayleigh-Jeans law limit of the thermal Sunyaev-Zeldovich effect and the bolometric X-ray radiative flux giveEliminate the electron number density and use $d_L=(1+z{)}^2{d}_A$:The cluster distance from Sunyaev-Zeldovich and X-ray signals is consequentlyAt low cosmological redshift, $d_A\simeq cz/H_0$, sowhen the other observables and profile parameters are held fixed. With the PDF's literal $d_L^{-1}$ instead, $S_X/D^2\propto \ell /(T_e^{3/2}{d}_L)=q\theta /[T_e^{3/2}(1+z{)}^2]$, independent of distance: this explicitly shows why the correction is necessary.

Real galaxy clusters are neither uniform nor generally spherical. The inferred depth depends on the ratio of line of sight depth to transverse extent; elongation and orientation selection therefore bias the distance. density clumps enhance X-ray emission, which weights $n_e^2$, more than the thermal Sunyaev-Zeldovich effect, which weights $n_e$. An unmodelled gas clumping factor greater than one makes the inferred distance too small and the inferred Hubble constant too large in this simple model. Temperature gradients and multiple thermal components bias the X-ray spectral temperature relative to the pressure-weighted Compton y parameter. Metals alter the cooling spectrum, and finite X-ray bands require a spectral and redshift correction to the bolometric approximation. Relativistic thermal corrections, the kinetic Sunyaev-Zeldovich effect, radio sources and primary Cosmic microwave background anisotropy contaminate the signal. Finally, calibration, angular-profile fitting and low-redshift peculiar velocities affect the observables and the relation between measured redshift and expansion distance.