In　this　paper,　a　critical　Galton-Watson　branching　process　{Zn}　is　considered.　Large　deviation　rates　of　SZ:=∑i=1ZnXi　are　obtained,　where　{Xi,　i　≥　1}　is　a　sequence　of　independent　and　identically　distributed　random　variables　and　X1　is　in　the　domain　of　attraction　of　an　α-stable　law　with　α　∈　(0,　2).　One　shall　see　that　the　convergence　rate　is　determined　by　the　tail　index　of　X1　and　the　variance　of　Z1.　Our　results　can　be　compared　with　those　ones　of　the　supercritical　case.　©　The　Editorial　Office　of　AMAS　&　Springer-Verlag　GmbH　Germany　2024.