A Summing Up — Question 3

Question 3

A redshifted decay line connects a particle mass to a cosmic epoch

A hypothetical nonrelativistic neutral particle X decays into two photons. A narrow line from a source at z=2 is observed at photon energy 3.0 keV. Neglect source motion and line broadening.

1. Infer the emitted photon energy and X rest energy.

2. If the line rest-frame width is .030 keV, find its observed width and fractional width, using pure cosmological stretching.

Question 3 - Solution

A redshifted decay line connects a particle mass to a cosmic epoch

(a) Method and calculation

$$\boxed{E_e=(1+z)E_o=3(3.0)=9.0\ {\mathrm{keV}}}.$$$$\boxed{M_Xc^2=2E_e=18.0\ {\mathrm{keV}}}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta E_o=\Delta E_e/(1+z)=.010\ {\mathrm{keV}}}.$$$$\boxed{\Delta E_o/E_o=\Delta E_e/E_e=.00333333}.$$

Physical check

This is a conditional mass inference if the two-photon decay interpretation is correct; an isolated spectral line does not establish dark-matter decay.