How to read the decaying dark matter limits of Lucca and colleagues

Open arXiv:1910.04619v3, dated 5 February 2020 and published as JCAP 02 (2020) 026: The synergy between CMB spectral distortions and anisotropies, by Matteo Lucca, Nils Schöneberg, Deanna C. Hooper, Julien Lesgourgues and Jens Chluba. This guide uses the arXiv page numbers: equation 2.60 on page 24, section 3.3 on pages 32 to 33, section 4.3 and Figure 8 on pages 37 to 38, and section 2.4.3 on pages 17 to 18. You will learn to read the axes and shaded regions, identify what each region assumes, and connect the result to the energy releases in the Lab.

What is being constrained: equation 2.60

On page 24 the heating rate from a decay is written as

$\rho_{\mathrm{cdm}}$ is the cold dark matter density, $f_{\mathrm{frac}}$ the fraction of it that decays, $\Gamma_{\mathrm{dec}}=1/\tau_{\mathrm{dec}}$ the decay rate and $f_{\mathrm{eff}}$ the fraction of decay energy deposited in the plasma. Section 4.3 sets $f_{\mathrm{eff}}=1$: all decay products interact electromagnetically and deposit their energy on the spot. The energy release peaks near $t\simeq\tau_{\mathrm{dec}}$ and then dies away exponentially.

How it maps onto the Lab: the Lab releases a burst with a Gaussian profile in $\ln(1+z)$, not an exponential decay. Its slider amplitude is the total $\Delta\rho/\rho_\gamma$, not $f_{\mathrm{frac}}$. To convert a point on Figure 8 into a Lab setting you would need to integrate equation 2.60 over time; this guide does not do that conversion. What does carry over is the timing. In the Lab's teaching background, the universe is 19 years old, or $5.9\times10^{8}\,\mathrm s$, at redshift $2\times10^{5}$; $9.9\times10^{10}\,\mathrm s$ at $1.5\times10^{4}$; and $1.2\times10^{13}\,\mathrm s$ at 1090, when the light was released.

Figure 8: axes and shaded regions

The horizontal axis is the lifetime $\tau_{\mathrm{dec}}$ in seconds, from $10^{6}$ to $10^{30}$. The vertical axis is $f_{\mathrm{frac}}$, from $10^{-13}$ to $10^{-1}$. Both are logarithmic. Shaded regions are excluded at 95 per cent confidence; the text on page 38 defines this as $\chi^2>\chi^2_{\mathrm{min}}+4$.

There are three shaded combinations: Planck with FIRAS in blue, Planck with a PIXIE-like mission, and LiteBIRD with CMB-S4 and a PRISM-like mission. Black lines repeat earlier bounds from Poulin, Lesgourgues and Serpico, labelled P17, as a dotted line for spectral distortions and a dash-dotted line for anisotropies. The red dashed line, labelled K05, is the bound from big-bang nucleosynthesis, the production of light elements in the first minutes, taken from Kawasaki, Kohri and Moroi. A second nucleosynthesis line is labelled P17.

Common misreading: reading the figure as a detection or a preferred value. Every curve is an upper limit on $f_{\mathrm{frac}}$; points above a boundary are excluded, points below are allowed.

Two regimes in one figure

Section 4.3 states why the figure has two parts. Lifetimes between $10^{5}$ and $10^{12}\,\mathrm s$ change the thermal history before recombination and so mainly create spectral distortions. Lifetimes between $10^{12}$ and $10^{25}\,\mathrm s$ are constrained strongly by the anisotropies, because energy released during and after recombination changes the free-electron history.

That split follows the two halves of the Lab. Drag the release epoch to redshift $1.5\times10^{4}$ and the spectrum changes while the visibility function stays put. Choose the Delayed last scattering preset at redshift 1100 and the visibility peak moves from 1081 to 1037 while $y$ is only $6.4\times10^{-10}$. Anisotropy experiments detect exactly that kind of shift.

How the limits were made

The model is standard cosmology with six parameters plus $\log_{10}f_{\mathrm{frac}}$ and $\log_{10}\tau_{\mathrm{dec}}$. Because the allowed region in those two parameters has an awkward shape, the authors fixed $\tau_{\mathrm{dec}}$ at many values from $10^{4}$ to $10^{27}\,\mathrm s$ and sampled the rest at each value, with a flat prior on $\log_{10}f_{\mathrm{frac}}$ from $-13$ to $0$. The plotted boundary interpolates between those slices.

For the spectrum, the distortion is computed from the full Green's function and its branching ratios, the method of section 3.2. Their contours are therefore smoother than the earlier P17 lines, which combined separate $\mu$ and $y$ bounds. At long lifetimes the new bounds are slightly tighter than P17. The authors attribute this to their choice $f_{\mathrm{eff}}=1$, against P17's smaller values for specific decay channels, and to likelihood differences: a mock likelihood approximating Planck 2018 here, against the full Planck 2015 likelihood in P17.

Forecasts and their assumptions: section 3.3

The future regions use mock experiments described on pages 32 to 33. A PIXIE-like mission is assumed to have equal 15 GHz channels from 30 GHz to 1 THz, limited only by uncorrelated instrumental noise of about $5\times10^{-26}\,\mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$ per channel, with all foregrounds removed using its higher-frequency channels. A PRISM-like mission has the same channels and ten times lower noise. The paper states that these are idealised scenarios with nearly optimal foreground removal, that it does not marginalise over the $y$ from reionization, and that a more realistic foreground treatment could degrade the sensitivity to $\mu$ in particular.

The quoted two-sigma sensitivities are $3\times10^{-5}$ in $\mu$ for FIRAS, $9\times10^{-9}$ for PIXIE and $9\times10^{-10}$ for PRISM. Common misreading: comparing their FIRAS $3\times10^{-5}$ with the published FIRAS 95 per cent limit of $9\times10^{-5}$. Fixsen and colleagues, 1996, abstract The authors themselves call their levels optimistic for this reason; the two numbers come from different treatments.

A scale check with the Lab's units: $5\times10^{-26}\,\mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$ is $5\,\mathrm{Jy\,sr^{-1}}$ per channel. From the Lab's chemical-potential shape, $\mu=9\times10^{-9}$ changes the intensity at 300 GHz by about $2.6\,\mathrm{Jy\,sr^{-1}}$. A signal near the single-channel noise, spread over some sixty channels, is the right order for a detection threshold. This estimate is ours, made with the Lab's shape.

Recombination lines: the guaranteed signal not in Figure 8

Section 2.4.3, on pages 17 to 18, describes the cosmological recombination radiation. Atoms forming emit and absorb photons: hydrogen between redshifts 500 and 2000, neutral helium between 1600 and 3500, and singly ionized helium between 5000 and 8000. The result is a pattern of small peaks on top of the $y$ distortion, adding roughly $0.01$ to $1\,\mathrm{Jy\,sr^{-1}}$. The paper calls it the smallest standard contribution, possibly within reach of futuristic detectors, and notes two things. Its shape stays unchanged from the end of hydrogen recombination to today, and exotic energy injection could change that shape. The authors did not include it in their code.

The Lab does not model these lines either. Its spectrum shows only the $\mu$ and $y$ shapes. A real signal would be the sum of those, the recombination lines, the $y$ from late hot gas, and foregrounds from our own Galaxy.

Checklist for the next limit plot

1. Write down the injection model and every efficiency factor set to one.
2. Identify the axes, the confidence definition and whether each region is a measurement or a forecast.
3. For each forecast, list the noise, frequency range and foreground assumption.
4. Find which observable dominates each part of the axis, and why it is that one.
5. Check whether quoted sensitivities match the published limits of the same instrument, and if not, why.
6. Note any guaranteed signal the analysis left out.