We perform numerical simulations on the merger of multiple black holes ( BHs ) in primordial gas at early cosmic epochs .
We consider two cases of BH mass : M _ { \mathrm { BH } } = 30 ~ { } M _ { \odot } and M _ { \mathrm { BH } } = 10 ^ { 4 } ~ { } M _ { \odot } .
Attention is concentrated on the effect of the dynamical friction by gas in a host object .
The simulations incorporate such general relativistic effects as the pericentre shift and gravitational wave emission .
As a result , we find that multiple BHs are able to merge into one BH within 100 Myr in a wide range of BH density .
The merger mechanism is revealed to be categorized into three types : gas-drag-driven merger ( type A ) , interplay-driven merger ( type B ) , and three-body-driven merger ( type C ) .
We find the relation between the merger mechanism and the ratio of the gas mass within the initial BH orbit ( M _ { \mathrm { gas } } ) to the total BH mass ( \sum M _ { \mathrm { BH } } ) .
Type A merger occurs if M _ { \mathrm { gas } } \gtrsim 10 ^ { 5 } \sum M _ { \mathrm { BH } } , type B if M _ { \mathrm { gas } } \lesssim 10 ^ { 5 } \sum M _ { \mathrm { BH } } , and type C if M _ { \mathrm { gas } } \ll 10 ^ { 5 } \sum M _ { \mathrm { BH } } .
Supposing the gas and BH density based on the recent numerical simulations on first stars , all the BH remnants from first stars are likely to merge into one BH through the type B or C mechanism .
Also , we find that multiple massive BHs ( M _ { \mathrm { BH } } = 10 ^ { 4 } ~ { } M _ { \odot } ) distributed over several parsec can merge into one BH through the type B mechanism , if the gas density is higher than 5 \times 10 ^ { 6 } ~ { } { cm } ^ { -3 } .
The present results imply that the BH merger may contribute significantly to the formation of supermassive BHs at high redshift epochs .