1. Wed. 17th Sept. 20081Hamburg LOFAR Workshop

2.  Extract a cosmological signal from a datacube, the three axes of which are x and y positions, and frequency.  Consider three components:  The cosmological signal itself;  Iliev et al. (2008)  Thomas et al. (2008)  Astrophysical foregrounds;  Jelić et al. (2008)  Noise.  The foregrounds are expected to be smooth in frequency, which allows them to be fitted out.  Does the signal also have any special properties which should enable us to tease it out?  Seek statistics which are robust with respect to making incorrect assumptions about the foregrounds and noise. Wed. 17th Sept. 2008Hamburg LOFAR Workshop2

3. Wed. 17th Sept. 2008Hamburg LOFAR Workshop3 100 h -1 Mpc 5° Tile Instrumental response Noise Subtract foregrounds Compare Signal ForegroundsDirty map Residuals

4.  Foregrounds are assumed smooth as a function of frequency.  Fit with e.g. a polynomial, or use some sort of non- parametric fit. Wed. 17th Sept. 2008Hamburg LOFAR Workshop4  If the fitting is biased, how does this relate to the statistics used for signal extraction?  Are fitting errors correlated in the same way as noise?  Better to fit uv points rather than image pixels?

5.  δ T b = (stuff)∙x HI ∙(1+ δ )∙(1+ z ) 1/2  Skewness= μ 3 / σ 3  At early times, the brightness temperature follows the cosmological density field which is positively skewed.  Reionization generates ionized bubbles (‘Swiss cheese’), which show up as a peak at zero emission in the one-point distribution, reducing the skewness.  At the late stages of reionization, the few remaining areas with emission form a high- δ T b tail. Wed. 17th Sept. 2008Hamburg LOFAR Workshop5 f250C simulation

6. Wed. 17th Sept. 2008Hamburg LOFAR Workshop6 f250C - Iliev et al. (2008) T-QSO T-star Three simulations: Thomas et al. (2008)

7.  Convolve images of the cosmological signal with the instrumental response, and add realistic noise.  This is equivalent to achieving perfect foreground subtraction (and very similar to the case when fitting errors have exactly the same characteristics as the noise).  Look at the skewness of the one- point distribution in these residual images as a function of redshift. Wed. 17th Sept. 2008Hamburg LOFAR Workshop7  Some features of the signal recovered, but at low redshift, where the signal is very small, the noise dominates.

8.  Fit the foregrounds with a third-order polynomial in log ν.  Skewness in the residual maps dominated by fitting errors and noise.  Can no longer recover the redshift evolution of skewness. Wed. 17th Sept. 2008Hamburg LOFAR Workshop8  If the noise on each image pixel was independent, then smoothing the images would help (because of correlations in the signal).  The real noise is more structured, so we require something more sophisticated.

9.  For the residual map at each frequency, attempt to reconstruct the cosmological signal with a Wiener deconvolution.  Optimal in a least- squares sense, BUT...  Requires knowledge of the correlation properties of the signal and noise, though in fact we ignore the contribution from the fitting errors.  Recovers the main features of the evolution of the skewness in the cosmological signal simulations.  The dip followed by the rise may be hard for noise and bias to mimic. Wed. 17th Sept. 2008Hamburg LOFAR Workshop9 (Original simulations)

10.  Might the real foregrounds be more skewed and will this require a more sophisticated foreground subtraction algorithm?  Test the level of accuracy to which the correlation properties of the cosmological signal must be estimated to make the Wiener deconvolution feasible (relation here to power spectrum estimation).  Can the whole process be carried out in the uv-plane?  Generate larger simulations of the signal which don’t require tiling.  Effect of exotic reionization scenarios – does this wipe out the ‘generic’ features?  Other statistics. Wed. 17th Sept. 2008Hamburg LOFAR Workshop10

11.  We test the LOFAR EoR signal extraction algorithms using datacubes including the cosmological signal, foregrounds, noise and instrumental effects.  Subtracting foregrounds which are smooth in frequency leaves a cube with three components: cosmological signal, noise and fitting errors, which have different correlation properties.  We exploit these properties to differentiate the cosmological signal from the fitting errors and noise.  The skewness of residual maps ‘denoised’ in this way shows similar (generic) features as a function of redshift to the cosmological signal simulations.  The difficulty comes from a combination of having to accurately subtract foregrounds and deal with structured noise. Wed. 17th Sept. 2008Hamburg LOFAR Workshop11

12. Wed. 17th Sept. 2008Hamburg LOFAR Workshop12

13. Wed. 17th Sept. 2008Hamburg LOFAR Workshop13

14. Wed. 17th Sept. 2008Hamburg LOFAR Workshop14 Density field Neutral fraction Differential brightness temperature Skewness Third moment (Variance) 3/2