Phase space factors for $β^+β^+$ decay and competing modes of double-$β$ decay

A complete and improved calculation of phase space factors (PSF) for $2νβ^+β^+$ and $0νβ^+β^+$ decay, as well as for the competing modes $2νECβ^+$, $0νECβ^+$, and $2νECEC$, is presented. The calculation makes use of exact Dirac wave functions with finite nuclear size and electron screening and includes life-times, single and summed positron spectra, and angular positron correlations.