We present the first limits on the Epoch of Reionization 21 cm H I power spectra, in the redshift range z=7.9-10.6, using the Low-Frequency Array (LOFAR) High-Band Antenna (HBA). In total, 13.0 hr of data were used from observations centered on the North Celestial Pole. After subtraction of the sky model and the noise bias, we detect a non-zero 56 13 mK D < ( ) at k=0.053 h cMpc −1 in the range z=9.6-10.6. The excess variance decreases when optimizing the smoothness of the direction-and frequency-dependent gain calibration, and with increasing the completeness of the sky model. It is likely caused by (i) residual side-lobe noise on calibration baselines, (ii) leverage due to nonlinear effects, (iii) noise and ionosphere-induced gain errors, or a combination thereof. Further analyses of the excess variance will be discussed in forthcoming publications.

The epoch of reionization (EoR) 21-cm signal is expected to be highly non-Gaussian in nature and this non-Gaussianity is also expected to evolve with the progressing state of reionization. Therefore the signal will be correlated between different Fourier modes (k). The power spectrum will not be able capture this correlation in the signal. We use a higherorder estimator -the bispectrum -to quantify this evolving non-Gaussianity. We study the bispectrum using an ensemble of simulated 21-cm signal and with a large variety of k triangles. We observe two competing sources driving the non-Gaussianity in the signal: fluctuations in the neutral fraction (x H ) field and fluctuations in the matter density field. We find that the non-Gaussian contribution from these two sources vary, depending on the stage of reionization and on which k modes are being studied. We show that the sign of the bispectrum works as a unique marker to identify which among these two components is driving the non-Gaussianity. We propose that the sign change in the bispectrum, when plotted as a function of triangle configuration cos θ and at a certain stage of the EoR can be used as a confirmative test for the detection of the 21-cm signal. We also propose a new consolidated way to visualize the signal evolution (with evolvingx H or redshift), through the trajectories of the signal in a power spectrum and equilateral bispectrum i.e. P(k) − B(k, k, k) space.

We present a detailed comparison of three different simulations of the epoch of reionization (EoR). The radiative transfer simulation (C 2 -RAY) among them is our benchmark. Radiative transfer codes can produce realistic results, but are computationally expensive. We compare it with two semi-numerical techniques: one using the same halos as C 2 -RAY as its sources (SemNum), and one using a conditional Press-Schechter scheme (CPS+GS). These are vastly more computationally efficient than C 2 -RAY, but use more simplistic physical assumptions. We evaluate these simulations in terms of their ability to reproduce the history and morphology of reionization. We find that both Sem-Num and CPS+GS can produce an ionization history and morphology that is very close to C 2 -RAY, with Sem-Num performing slightly better compared to CPS+GS.We also study different redshift space observables of the 21-cm signal from EoR: the variance, power spectrum and its various angular multipole moments. We find that both seminumerical models perform reasonably well in predicting these observables at length scales relevant for present and future experiments. However, Sem-Num performs slightly better than CPS+GS in producing the reionization history, which is necessary for interpreting the future observations. The CPS+GS scheme, however, has the advantage that it is not restricted by the mass resolution of the dark matter density field.

We have used semi-numerical simulations of reionization to study the behaviour of the power spectrum of the EoR 21-cm signal in redshift space. We have considered two models of reionization, one which has homogeneous recombination (HR) and the other incorporating inhomogeneous recombination (IR). We have estimated the observable quantities -quadrupole and monopole moments of H I power spectrum at redshift space from our simulated data. We find that the magnitude and nature of the ratio between the quadrupole and monopole moments of the power spectrum (P s 2 /P s 0 ) can be a possible probe for the epoch of reionization. We observe that this ratio becomes negative at large scales forx H i ≤ 0.7 irrespective of the reionization model, which is a direct signature of an inside-out reionization at large scales. It is possible to qualitatively interpret the results of the simulations in terms of the fluctuations in the matter distribution and the fluctuations in the neutral fraction which have power spectra and cross-correlation P ∆∆ (k), P xx (k) and P ∆x (k) respectively. We find that at large scales the fluctuations in matter density and neutral fraction is exactly anti-correlated through all stages of reionization. This provides a simple picture where we are able to qualitatively interpret the behaviour of the redshift space power spectra at large scales with varyingx H i entirely in terms of a just two quantities, namelyx H i and the ratio P xx /P ∆∆ . The nature of P ∆x becomes different for HR and IR scenarios at intermediate and small scales. We further find that it is possible to distinguish between an inside-out and an outside-in reionization scenario from the nature of the ratio P s 2 /P s 0 at intermediate length scales.

In this paper we establish the accuracy and robustness of a fast estimator for the bispectrum -the 'FFT bispectrum estimator'. The implementation of the estimator presented here offers speed and simplicity benefits over a direct-measurement approach. We also generalise the derivation so it may be easily be applied to any order polyspectra, such as the trispectrum, with the cost of only a handful of FFTs. All lower order statistics can also be calculated simultaneously for little extra cost. To test the estimator we make use of a non-linear density field, and for a more strongly non-Gaussian test case we use a toy-model of reionization in which ionized bubbles at a given redshift are all of equal size and are randomly distributed. Our tests find that the FFT estimator remains accurate over a wide range of k, and so should be extremely useful for analysis of 21-cm observations. The speed of the FFT bispectrum estimator makes it suitable for sampling applications, such as Bayesian inference. The algorithm we describe should prove valuable in the analysis of simulations and observations, and whilst we apply it within the field of cosmology, this estimator is useful in any field that deals with non-Gaussian data.