 Differential sky map monopole degeneracy

An ideal differential measurement constrains two sky temperatures through $d_{ij}=T_i-T_j$. Represent every observed pair by a row of a matrix $A$, with $+1$ in column $i$, $-1$ in column $j$ and zeros elsewhere. Then $d=AT$ and $A1=0$: adding any constant to the entire map changes no measurement. This null space is the angular angular monopole ambiguity, explaining why a directional comparison alone cannot measure the absolute blackbody temperature established by FIRAS.

For an original three-pixel example choose $T=(30,-10,-20)\mu \mathrm{K}$ relative to an unspecified mean. Measurements $d_{12}=40\mu \mathrm{K}$ and $d_{23}=10\mu \mathrm{K}$ determine the third difference $d_{31}=-50\mu \mathrm{K}$. The zero-mean condition recovers the chosen map, but adding $100\mu \mathrm{K}$ to every pixel gives identical differences. The closure sum $d_{12}+d_{23}+d_{31}=0$ checks consistency without establishing the mean.

More generally, the graph joining measured pixel pairs must be a connected graph. If $AT=0$, each edge forces equal endpoint temperatures; a path propagates that equality throughout each connected component. Thus a connected observation graph leaves exactly one undetermined constant, whereas disconnected coverage leaves one per component. Fixing the mean makes an ideal connected reconstruction unique; real noise calls for least-squares solution fitting and uncertainty analysis.

The figure chooses a cosmic microwave background quadrupole sky $T(\theta )=Q{P}_2(\cos \theta )$, with $Q=30\mu \mathrm{K}$ and Legendre polynomial $P_2(u)=(3u^2-1)/2$. On a great-circle scan, with $\varphi$ measured from the chosen symmetry axis and ideal beams separated by $\alpha =60^{\circ }$,Expanding ${\cos }^2\varphi =(1+\cos 2\varphi )/2$ proves the formula; its peak magnitude is $45\sqrt{3}/2\simeq 38.9711\mu \mathrm{K}$. A constant sky gives zero. This deliberately restricted scan also misses some angular patterns, so rejecting the angular monopole is not a proof of complete sky reconstruction. Broad, overlapping coverage supplies the missing constraints; finite antenna beam patterns additionally smooth small angular features.