Context : The cosmological lithium problem , that is , the discrepancy between the lithium abundance predicted by the Big Bang nucleosynthesis and the one observed for the stars of the ’ Spite plateau ’ , is one of the long standing problems of modern astrophysics .
Recent hints for a possible solution involve lithium burning induced by protostellar mass accretion on Spite plateau stars .
However , to date , most of the protostellar and pre-main sequence stellar models that take mass accretion into account have been computed at solar metallicity , and a detailed analysis on the impact of protostellar accretion on the lithium evolution in the metal-poor regime , which is relevant for stars in the Spite plateau , is completely missing .

Aims : The purpose of this paper is to fill this gap , analysing , in detail , for the first time the effect of protostellar accretion on low metallicity low-mass stars with a focus on pre-main sequence lithium evolution .

Methods : We computed the evolution from the protostar to the main-sequence phase of accreting models with final masses equal to 0.7 and 0.8 M _ { \sun } , and three metallicities Z = 0.0001 , Z = 0.0010 , and Z = 0.0050 , corresponding to [ Fe/H ] \sim - 2.1 , -1.1 ( typical of Spite plateau stars ) , and [ Fe/H ] \sim - 0.42 , respectively .
We followed the temporal evolution of the chemical composition by considering nuclear burning , convective mixing , and diffusion .
The effects of changing some of the main parameters affecting accreting models , that is the accretion energy ( i.e .
cold versus hot accretion ) , the initial seed mass M _ { \mathrm { seed } } and radius R _ { \mathrm { seed } } , and the mass accretion rate \dot { m } ( also considering episodic accretion ) , have been investigated in detail .

Results : As for the main stellar properties and in particular the surface ^ { 7 } Li abundance , hot accretion models converge to standard non-accreting ones within 1 Myr , regardless of the actual value of M _ { \mathrm { seed } } , R _ { \mathrm { seed } } , and \dot { m } .
Also , cold accretion models with a relatively large M _ { \mathrm { seed } } ( \gtrsim 10 ~ { } M _ { \mathrm { J } } ) or R _ { \mathrm { seed } } ( \gtrsim 1 ~ { } R _ { \sun } ) converge to standard non-accreting ones in less than about 10-20 Myr .
However , a drastically different evolution occurs whenever a cold protostellar accretion process starts from small values of M _ { \mathrm { seed } } and R _ { \mathrm { seed } } ( M _ { \mathrm { seed } } \sim 1 ~ { } M _ { \mathrm { J } } , R _ { \mathrm { seed } } \lesssim 1 ~ { } R _ { \sun } ) .
These models almost entirely skip the standard Hayashi track evolution and deplete lithium before the end of the accretion phase .
The exact amount of depletion depends on the actual combination of the accretion parameters ( \dot { m } , M _ { \mathrm { seed } } , and R _ { \mathrm { seed } } ) , achieving in some cases the complete exhaustion of lithium in the whole star .
Finally , the lithium evolution in models accounting for burst accretion episodes or for an initial hot accretion followed by a cold accretion phase closely resemble that of standard non-accreting ones .

Conclusions : To significantly deplete lithium in low-mass metal poor stars by means of protostellar accretion , a cold accretion scenario starting from small initial M _ { \mathrm { seed } } and R _ { \mathrm { seed } } is required .
Even in this extreme configuration leading to a non-standard evolution that misses almost entirely the standard Hayashi track , an unsatisfactory fine tuning of the parameters governing the accretion phase is required to deplete lithium in stars of different mass and metallicity – starting from the Big Bang nucleosynthesis abundance – in such a way as to produce the observed Spite plateau .