Beyond the Boost: Measuring the Intrinsic Dipole of the Cosmic Microwave Background Using the Spectral Distortions of the Monopole and Quadrupole

We present a general framework for the accurate spectral modeling of the low multipoles of the cosmic microwave background (CMB) as observed in a boosted frame. In particular, we demonstrate how spectral measurements of the low multipoles can be used to separate the motion-induced dipole of the CMB from a possible intrinsic dipole component. In a moving frame, the leakage of an intrinsic dipole moment into the CMB monopole and quadrupole induces spectral distortions with distinct frequency functions that, respectively, peak at 337 and 276 GHz. The leakage into the quadrupole moment also induces a geometrical distortion to the spatial morphology of this mode. The combination of these effects can be used to lift the degeneracy between the motion-induced dipole and any intrinsic dipole that the CMB might possess. Assuming the current peculiar velocity measurements, the leakage of an intrinsic dipole with an amplitude of $\mathrm{\Delta }T=30\mu \mathrm{K}$ into the monopole and quadrupole moments will be detectable by a PIXIE-like experiment at $\sim 40\mathrm{nK}$ ($2.5\sigma$) and $\sim 130\mathrm{nK}$ ($11\sigma$) level at their respective peak frequencies.

Figure 1

The CMB dipole constituents observed in a moving frame with $\beta =0.00128$ and $\widehat{\mathit{\beta }}=(264°,48°)$. The intrinsic dipoles $d30$ and $d60$ have identical frequency functions as the kinematic dipole. The average $T=2.725$ blackbody spectrum (solid black curve) is depicted in all plots for reference.

Figure 2

The motion-induced spectral distortions of the observed CMB monopole.

Figure 3

The motion-induced spectral distortions of the observed CMB quadrupole. Since the leakage of the octupole has a different frequency function compared to the other components, we have assumed that this term can be identified and subtracted and therefore is not shown here. In this specific direction in the sky, the leakage of the $d30$ is not distinguishable from a $d60$ with the same projection along $\widehat{\mathit{\beta }}$ (short dashed blue curve). However, the amplitude of these two leakage components are different at other lines of sight (see Fig. 4).

Figure 4

Mollweide projection of the leakage of an intrinsic dipole $d60$ with $\widehat{\mathit{d}}·\widehat{\mathit{\beta }}=1/2$ into the observed quadrupole. The solid (dashed) black lines are the $-40$ (80) nK contour lines for the leakage of a smaller intrinsic dipole $d30$ with a different orientation $\widehat{\mathit{d}}=\widehat{\mathit{\beta }}$. Even though the two dipole leakage components have the same amplitude along the $\widehat{\mathit{\beta }}$ direction (black dot), their spatial morphology is different over the whole sky.