Deflection of massive neutrinos by gravitational fields

The angular deflection undergone by massive neutrinos near astrophysical objects is described by formula (11). This result led us to two main conclusions: a) that ultrarelativistic neutrinos are deflected almost as massless particles; more precisely for g¥ º 1/Ö{1 - b¥ 2 } >> 1\gamma _\infty \equiv 1/\sqrt {1 - \beta _\infty ^2 } \gg 1 , one has Dj º \frac4mb\frac( 1 - 1/2g¥ 2 )( 1 - 1/g¥ 2 ) @ \frac4mb( 1 + \frac12g¥ 2 )\Delta \varphi \equiv \frac{{4m}}{b}\frac{{\left( {1 - 1/2\gamma _\infty ^2 } \right)}}{{\left( {1 - 1/\gamma _\infty ^2 } \right)}} \simeq \frac{{4m}}{b}\left( {1 + \frac{1}{{2\gamma _\infty ^2 }}} \right) (1) ,i.e. a deviation from 4m/b by a small factor: 1/2g¥ 2 ( 1/2g¥ 2 << 1 )1/2\gamma _\infty ^2 \left( {1/2\gamma _\infty ^2 \ll 1} \right) ; b) that nonrelativistic neutrinos (as the slow cosmological ones) suffer large deflection because of the factor b¥ 2\beta _\infty ^2 at denominator at formula (11); therefore, even for large impact parameterb, slow neutrinos could be deflected and could transfer their momentum to the gravitating objects. This effect has relevant consequences in astrophysical problems (5).