Direct collapse of supermassive stars ( SMSs ) is a possible pathway for generating supermassive black holes in the early universe .
It is expected that an SMS could form via very rapid mass accretion with \dot { M } _ { * } \sim 0.1 - 1 ~ { } M _ { \odot } ~ { } { yr } ^ { -1 } during the gravitational collapse of an atomic-cooling primordial gas cloud .
In this paper we study how stars would evolve under such extreme rapid mass accretion , focusing on the early evolution until the stellar mass reaches 10 ^ { 3 } ~ { } M _ { \odot } .
To this end we numerically calculate the detailed interior structure of accreting stars with primordial element abundances .
Our results show that for accretion rates higher than 10 ^ { -2 } ~ { } M _ { \odot } ~ { } { yr } ^ { -1 } , stellar evolution is qualitatively different from that expected at lower rates .
While accreting at these high rates the star always has a radius exceeding 100 ~ { } R _ { \odot } , which increases monotonically with the stellar mass .
The mass-radius relation for stellar masses exceeding \sim 100 ~ { } M _ { \odot } follows the same track with R _ { * } \propto M _ { * } ^ { 1 / 2 } in all cases with accretion rates \gtrsim 10 ^ { -2 } ~ { } M _ { \odot } ~ { } { yr } ^ { -1 } ; at a stellar mass of 10 ^ { 3 } ~ { } M _ { \odot } the radius is \simeq 7000 ~ { } R _ { \odot } ( \simeq 30 AU ) .
With higher accretion rates the onset of hydrogen burning is shifted towards higher stellar masses .
In particular , for accretion rates exceeding \dot { M } _ { * } \gtrsim 0.1 ~ { } M _ { \odot } ~ { } { yr } ^ { -1 } , there is no significant hydrogen burning even after 10 ^ { 3 } ~ { } M _ { \odot } have accreted onto the protostar .
Such “ supergiant ” protostars have effective temperatures as low as T _ { eff } \simeq 5000 K throughout their evolution and because they hardly emit ionizing photons , they do not create an HII region or significantly heat their immediate surroundings .
Thus , radiative feedback is unable to hinder the growth of rapidly accreting stars to masses in excess of 10 ^ { 3 } ~ { } M _ { \odot } , as long as material is accreted at rates \dot { M } _ { * } \gtrsim 10 ^ { -2 } ~ { } M _ { \odot } ~ { } { yr } ^ { -1 } .