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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.

Use one Fourier mode, with $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{n}}$ and unperturbed ray $\mathbf{x}(\tau )=\hat{\mathbf{n}}({\tau }_0-\tau )$. Put $E(\tau )=e^{ik\mu ({\tau }_0-\tau )}$, and let ${\tau }_{\ast }$ denote last scattering. In the supplied branch $h^{\prime }=h_s^{\prime }$, the scalar decomposition givesSince $E^{\prime \prime }=-k^2{\mu }^{2}E$, two integrations by parts give the exact endpoint decomposition of the metric contribution:For the matter-era growing mode, $h_s$ is quadratic in $\tau$, so the last integral vanishes. With $E^{\prime }=-ik\mu E$, the emission endpoint isand the observer endpoint is $-h_s^{\prime \prime }({\tau }_0)/(2k^2)-i\mu h_s^{\prime }({\tau }_0)/(2k)$. The observer terms are only a angular monopole and angular dipole; they are removed when studying the anisotropy multipoles $\ell \ge 2$.

For an adiabatic growing mode outside the horizon at last scattering, ${\delta }_{\gamma }=4{\delta }_m/3$, ${\delta }_m=-h/2$ after fixing the constant trace convention, and $h\propto {\tau }^2$. Hence the intrinsic emission term is smaller than $h_s^{\prime \prime }/(2k^2)$ by order $(k{\tau }_{\ast }{)}^2$. The derivative emission endpoint is smaller by order $k{\tau }_{\ast }$. Stress-energy conservation also makes the peculiar emission velocity subleading; its leading synchronous contribution cancels in ${\delta }_{\gamma }^{\prime }+2h^{\prime }/3$, with gradients supplying the remaining small term. Keeping the Fourier phase, which need not be small over the observer's distance, givesHere $\Theta =\delta T/T$, and the sum over modes is understood. The derivative is the second derivative of $h_s$, as printed in the original PDF. This approximation assumes matter-era growing adiabatic modes and neglects late evolving potentials; large wavelength alone does not remove the last integral in a general cosmology. The equality $h^{\prime }=h_s^{\prime }$ is used as the supplied synchronous condition, rather than as a general identity for every fluid solely from vanishing anisotropic stress.

A spherical harmonic of degree $\ell$ is an eigenfunction of the angular Laplacian on the unit sphere. In three dimensions, with the sign convention inherited from $\Delta ={\sum }_j{\partial }_j^2$,One chooses an orthonormal basis for each $(2\ell +1)$-dimensional eigenspace. With surface measure $d\Omega$, examples are $Y_0^0=1/\sqrt{4\pi }$ and $Y_1^0=\sqrt{3/(4\pi )}\cos \theta$.

The degree-zero mode is an angular monopole, the degree-one modes form an angular dipole, and higher degrees resolve finer angular structure. Separation of variables in the Laplace equation pairs these angular modes with the solid harmonics $r^{\ell }{Y}_{\ell }^m$ and $r^{-\ell -1}{Y}_{\ell }^m$. This is why the same basis organizes gravitational potential expansions, atomic orbitals and sky-map multipoles.