  We analyze the time evolution of the luminosity of a cluster of Population
III protostars formed in the early universe. We argue from the Jeans criterion
that primordial gas can collapse to form a cluster of first stars that evolve
relatively independently of one another (i.e., with negligible gravitational
interaction). We model the collapse of individual protostellar clumps using
2+1D nonaxisymmetric numerical hydrodynamics simulations. Each collapse
produces a protostar surrounded by a massive disk (i.e., $M_{\rm disk} / M_{*}
\gtrsim 0.1$), whose evolution we follow for a further 30--40 kyr.
Gravitational instabilities result in the fragmentation and the formation of
gravitationally bound clumps within the disk. The accretion of these fragments
by the host protostar produces accretion and luminosity bursts on the order of
$10^6\,\LSun$. Within the cluster, we show that a simultaneity of such events
across several protostellar cluster members can elevate the cluster luminosity
to 5--10${\times}$ greater than expected, and that the cluster spends
$\sim15\%$ of it's star-forming history at these levels. This enhanced
luminosity effect is particularly enabled in clusters of modest size with
$\simeq$ 10--20 members. In one such instance, we identify a confluence of
burst events that raise the luminosity to nearly $1000{\times}$ greater than
the cluster mean luminosity, resulting in $L > 10^8\,\LSun$. This phenomenon
arises solely through the gravitational-instability--driven episodic
fragmentation and accretion that characterizes this early stage of protostellar
evolution.
