$\mu$ distortions or running: A guaranteed discovery from CMB spectrometry

We discuss the implications of a PIXIE-like experiment, which would measure $\mu$-type spectral distortions of the cosmic microwave background (CMB) at a level of ${\sigma }_{\mu }=(1/n)\times {10}^{-8}$, with $n\ge 1$ representing an improved sensitivity (e.g. $n=10$ corresponds to PRISM). Using Planck data and considering the six-parameter $\mathrm{\Lambda }\mathrm{CDM}$ model, we compute the posterior for ${\mu }_8\equiv \mu \times {10}^8$ and find ${\mu }_8=1.57_{-0.13}^{+0.11}$ (68% C.L.). This becomes ${\mu }_8=1.28_{-0.52}^{+0.30}$ (68% C.L.) when the running ${\alpha }_{\mathrm{s}}$ of the spectral index is included. We point out that a sensitivity of about $3\times \mathrm{PIXIE}$ implies a guaranteed discovery: $\mu$ distortion is detected or ${\alpha }_{\mathrm{s}}\ge 0$ is excluded (both at 95% C.L. or higher). This threshold sensitivity sets a clear benchmark for CMB spectrometry. For a combined analysis of PIXIE and current Planck data, we discuss the improvement on measurements of the tilt $n_{\mathrm{s}}$ and the running ${\alpha }_{\mathrm{s}}$ and the dependence on the choice of the pivot. A fiducial running of ${\alpha }_{\mathrm{s}}=-0.01$ (close to the Planck best fit) leads to a detection of negative running at $2\sigma$ for $5\times \mathrm{PIXIE}$. A fiducial running of ${\alpha }_{\mathrm{s}}=-0.02$, still compatible with Planck, requires $3\times \mathrm{PIXIE}$ to rule out ${\alpha }_{\mathrm{s}}=0$ (at 95% C.L.). We propose a convenient and compact visualization of the improving constraints on the tilt, running and tensor-to-scalar ratio.

Figure 1

This cartoon plot shows the scales which are probed by $\mu$- and $y$-type spectral distortions, using the “window function” approximation of Eq. (5).

Figure 2

The figure shows the one-dimensional posteriors for ${\mu }_8$ predicted by Planck $TT$, $TE$, $EE+\mathrm{lowP}$ data, for the $\mathrm{\Lambda }\mathrm{CDM}$ model (orange curve) and the $\mathrm{\Lambda }\mathrm{CDM}+{\alpha }_{\mathrm{s}}$ model (purple curve). The posteriors have been obtained through the idistort code developed by Khatri and Sunyaev [20, 46].

Figure 3

This figure shows the 68% C.L. and 95% C.L. contours in the $n_{\mathrm{s}}\text{−}\mu$ (left panel) and the ${\alpha }_{\mathrm{s}}\text{−}{\mu }_8$ plane (right panel) for the Planck $TT$, $TE$, $EE+\mathrm{lowP}$ data set for $\mathrm{\Lambda }\mathrm{CDM}+{\alpha }_{\mathrm{s}}$, together with the $1\sigma$ detection limits for PIXIE and possible improvements.

Figure 4

$(1-n_{\mathrm{s}})/{\sigma }_{n_{\mathrm{s}}}$ (left panel) and $|{\alpha }_{\mathrm{s}}|/{\sigma }_{{\alpha }_{\mathrm{s}}}$ (right panel) as a function of the pivot scale $k_{\star }$, for a vanishing fiducial distortions ${\mu }_8^{(\mathrm{fid})}=0$. A dependence on the pivot scale is always present for $n_{\mathrm{s}}$ (left panel), while for ${\alpha }_{\mathrm{s}}$ the dependence becomes appreciable only for significant improvements over PIXIE’s sensitivity. The optimal choice of $k_{\star }$ shifts towards $k<0.05{\mathrm{Mpc}}^{-1}$ for $n_{\mathrm{s}}$ and $k>0.05{\mathrm{Mpc}}^{-1}$ for ${\alpha }_{\mathrm{s}}$, when the information from spectral distortions is included.

Figure 5

Same as Fig. 4, but in this case we consider a fiducial ${\mu }_8$ amplitude of 1.3 (see Sec. 4 for details). The behavior is qualitatively similar to the case of a vanishing fiducial ${\mu }_8$.

Figure 6

Left panel: 68% C.L. and 95% C.L. contours in the ${\alpha }_{\mathrm{s}}\text{−}{\mu }_8$ plane, for Planck alone (yellow contour) and including in the analysis the likelihood with ${\alpha }_{\mathrm{s}}^{(\mathrm{fid})}=-0.01$ (i.e. ${\mu }_8^{(\mathrm{fid})}=1.06$) for a $2\times$ and $3\times$ improvement over PIXIE (orange and purple contours). Right panel: Same as left panel, but with the fiducial ${\alpha }_{\mathrm{s}}$ equal to $-0.02$.

Figure 7

This contour plot shows ${\alpha }_{\mathrm{s}}$ as a function of $\epsilon$ for different values of the NLO slow-roll parameters. Notice that the uncertainty in $n_{\mathrm{s}}$ is smaller than the thickness of the lines in the plot. In red we show $\alpha (\epsilon )$ of Eq. (17) for $\mathrm{NLO}=0$, while the blue line is its asymptotic value $(1-n_{\mathrm{s}}{)}^2\approx 0.0013$. The black line shows the predictions of the Starobinsky model [71] (with $N$ going from 20 to 70), with the yellow dot being its prediction for $N=56$ (chosen to reproduce the observed value of $n_{\mathrm{s}}$). The gray bands show the values of ${\alpha }_{\mathrm{s}}$ excluded (at 95% C.L.) by Planck $TT$, $TE$, $EE+\mathrm{lowP}$ data, while the gray dashed vertical line shows the current bound on $\epsilon =r/(16c_{\mathrm{s}})$ from Eq. (13), considering $c_{\mathrm{s}}=1$.

Figure 8

This plot shows the spectral shapes (normalized at the maximum) $I(\nu )$ for $\mu$ and $y$ distortions, together with the spectra for $i$-type distortions at redshifts $z=\mathcal{O}(2\times {10}^5)$, $z=\mathcal{O}(1\times {10}^5)$ and $z=\mathcal{O}(5\times {10}^4)$ and the spectral shape of the monopole of temperature anisotropies $\mathrm{\Theta }$. We see that for increasing redshift, the maximum, minimum and zero of the occupation numbers are moved towards lower frequencies.