Is there anybody out there?

The Fermi paradox is the discrepancy between the strong likelihood of alien intelligent life emerging (under a wide variety of assumptions) and the absence of any visible evidence for such emergence. We use this intriguing unlikeness to derive an upper limit on the fraction of living intelligent species that develop communication technology <\xi_{biotec}>. <...> indicates average over all the multiple manners civilizations can arise, grow, and develop such technology, starting at any time since the formation of our Galaxy in any location inside it. Following Drake, we factorize <\xi_{biotec}> as the product of the fractions in which: (i) life arises, (ii) intelligence develops, and (iii) communication technology is developed. In this approximation, the number of communicating intelligent civilizations that exist in the Galaxy at any given time is found to be N = <\zeta_{astro}> <\xi_{biotec}> L_\tau, where <\zeta_{astro}> is the average production rate of potentially habitable rocky planets with a long-lasting (~ 4 Gyr) ecoshell and L_{\tau} is the length of time that a typical civilization communicates. We estimate the production rate of exoplanets in the habitable zone and using recent determinations of the rate of gamma-ray bursts (GRBs) and their luminosity function, we calculate the probability that a life-threatening (lethal) GRB could make a planet inhospitable to life, yielding <\zeta_{astro}> ~ 2 \times 10^{-3} yr^-1. Our current measurement of N =0 then implies <\zeta_{biotec}> < 5 \times 10^{-3} at the 95\% C.L., where we have taken L_\tau > 0.3 Myr such that c L_\tau >> propagation distances of Galactic scales (~ 10 kpc), ensuring that any advanced civilization living in the Milky Way would be able to communicate with us.
Comment: To be published in Proceedings of the 35th International Cosmic Ray Conference (ICRC 2017), Bexco, Busan, Korea; v2 references added