Using the Ly \alpha mass assignment scheme ( LyMAS ) , we make theoretical predictions for the 3-dimensional 3-point correlation function ( 3PCF ) of the Ly \alpha forest at redshift z = 2.3 .
We bootstrap results from the ( 100 h ^ { -1 } \mbox { Mpc } ) ^ { 3 } Horizon hydrodynamic simulation to a ( 1 h ^ { -1 } Gpc ) ^ { 3 } N -body simulation , considering both a uniform UV background ( UVB ) and a fluctuating UVB sourced by quasars with a comoving n _ { q } \approx 10 ^ { -5 } h ^ { 3 } Mpc ^ { -3 } placed either in massive halos or randomly .
On scales of 10 - 30 h ^ { -1 } \mbox { Mpc } , the flux 3PCF displays hierarchical scaling with the square of the 2PCF , but with an unusual value of Q \equiv \zeta _ { 123 } / ( \xi _ { 12 } \xi _ { 13 } + \xi _ { 12 } \xi _ { 23 } + \xi _ { 13 } \xi _ { 23 } ) \approx% -4.5 that reflects the low bias of the Ly \alpha forest and the anti-correlation between mass density and transmitted flux .
For halo-based quasars and an ionizing photon mean free path of \lambda = 300 h ^ { -1 } \mbox { Mpc } comoving , UVB fluctuations moderately depress the 2PCF and 3PCF , with cancelling effects on Q .
For \lambda = 100 h ^ { -1 } \mbox { Mpc } or 50 h ^ { -1 } \mbox { Mpc } , UVB fluctuations substantially boost the 2PCF and 3PCF on large scales , shifting the hierarchical ratio to Q \approx - 3 .
We scale our simulation results to derive rough estimate of the detectability of the 3PCF in current and future observational data sets for the redshift range z = 2.1 - 2.6 .
At r = 10 h ^ { -1 } \mbox { Mpc } and 20 h ^ { -1 } \mbox { Mpc } , we predict a signal-to-noise ( SNR ) of \sim 9 and \sim 7 , respectively , for both BOSS and eBOSS , and \sim 37 and \sim 25 for DESI .
At r = 40 h ^ { -1 } \mbox { Mpc } the predicted SNR is lower by a factor of \sim 3 - 5 .
Measuring the flux 3PCF would provide a novel test of the conventional paradigm of the Ly \alpha forest and help separate the contributions of UVB fluctuations and density fluctuations to Ly \alpha forest clustering , thereby solidifying its foundation as a tool of precision cosmology .