Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 65 / 2 / iv

Measure the logarithmic surface brightness slope in resolved radial annuli after accounting for instrumental blurring and foreground emission. Between a halo core and its outer boundary, dark-matter decay radiation predicts $d\ln I/d\ln R=-1$, whereas dark-matter annihilation radiation predicts $-3$; doubling radius therefore reduces the respective surface brightness by factors $2$ and $8$. These are quantitative templates to compare with independently mapped stellar and gaseous components.

An exponential galactic disk has $I_{\ast }(R)\propto e^{-R/R_d}$, so $d\ln I_{\ast }/d\ln R=-R/R_d$ and $I_{\ast }(2R)/I_{\ast }(R)=e^{-R/R_d}$. At $R=5R_d$ the latter is about $0.0067$, much faster decline than either halo template, and the projected emission should also be flattened along the galactic disk. A galactic bulge described by a Sérsic profile has $d\ln I_{\ast }/d\ln R=-(b_n/n)(R/R_e{)}^{1/n}$; for $n=4$ and $b_4\simeq 7.67$, this is about $-3.4$ at $10R_e$ and continues to steepen, whereas a pure halo power law keeps a fixed slope.

For a stellar halo with outer luminosity density proportional to $r^{-p_{\ast }}$, the projection of a spherical power-law emissivity gives $I_{\ast }\propto R^{1-p_{\ast }}$. Illustratively, $p_{\ast }=3$--$4$ gives projected slopes $-2$--$-3$: then $I_d/I_{\ast }\propto R^{p_{\ast }-2}$ rises strongly outward, but annihilation can resemble the steeper stellar template. Gas-related gamma emission must be compared with observed gas columns and the cosmic ray distribution; it is not always proportional to optical light.

A shallow, extended, approximately spherical $R^{-1}$ component is particularly distinctive; an $R^{-3}$ component needs additional evidence. Fit the stellar, gas and halo templates together over a broad radial range, including morphology and emission spectra. A measured slope alone cannot uniquely identify dark matter, since stellar or gas profiles can coincide with a halo slope over a limited interval.