We consider the contribution of 3rd and 4th order terms to the power spectrum of 21 cm brightness temperature fluctuations during the epoch of reionization ( EoR ) , which arise because the 21 cm brightness temperature involves a product of the hydrogenic neutral fraction and the gas density .
The 3rd order terms vanish for Gaussian random fields , and have been previously neglected or ignored .
We measure these higher order terms from radiative transfer simulations and estimate them using cosmological perturbation theory .
In our simulated models , the higher order terms are significant : neglecting them leads to a \gtrsim 100 \% error in 21 cm power spectrum predictions on scales of k \gtrsim 1 h { Mpc ^ { -1 } } when the neutral fraction is \langle x _ { H } \rangle \sim 0.5 .
At later stages of reionization , when the ionized regions are bigger , the higher order terms impact 21 cm power spectrum predictions on still larger scales , while they are less important earlier during reionization .
The higher order terms have a simple physical interpretation .
On small scales they are produced by gravitational mode coupling .
Small scale structure grows more readily in large-scale overdense regions , but the same regions tend to be ionized and hence do not contribute to the 21 cm signal .
This acts to suppress the influence of non-linear density fluctuations and the small-scale amplitude of the 21 cm power spectrum .
In alternate models , where the voids are reionized before over-dense regions ( ‘ outside-in ’ reionization ) , the effect should have the opposite sign , and lead to an enhancement in the 21 cm power spectrum .
These results modify earlier intuition that the 21 cm power spectrum simply traces the density power spectrum on scales smaller than that of a typical bubble , and imply that small scale measurements contain more information about the nature of the ionizing sources than previously believed .
On large scales , higher order moments are not directly related to gravity .
They are non-zero because over-dense regions tend to ionize first and are important in magnitude at late times owing to the large fluctuations in the neutral fraction .
Finally , we show that 2nd order Lagrangian perturbation theory approximately reproduces the statistics of the density field from full numerical simulations for all redshifts and scales of interest , including the mode-coupling effects mentioned above .
It can , therefore , be used in conjunction with semi-analytic models to accurately , and rapidly , explore the broad regions of parameter space relevant for future 21 cm surveys .