A description of random fluctuations is fundamental to extract information hidden in the background of images and to provide a quantitative measure for the comparison with sim- ulations.
The spatial power spectrum is the key of our method, allowing us to determine the amplitude of surface brightness fluctuations and to distinguish the sources of noise from the fluctuations due to the unresolved galaxies.
It should be noticed that the power spectrum we are focusing on is not the primordial one considered first by Harrison (1970) and Zeldovich (1972), and then by White (1994) to study, e.g., the cosmic microwave background anisotropies. We are, instead, interested in the spatial distribution of the faint z ∼ 6 galaxies, so as to be able to extend the faint-end of the luminosity function. This power spectrum is related to the primordial one, but departs from it due to non-linear evolutions, gas physics , and conversion of gas to stars. For this reason we did not rely on theoretical expectation, but we used the Fourier Transform to derive the spatial power spectrum of the luminosity fluctuation due to these galaxies. For point sources only their spatial distribution contributes to the power spectrum, but, for extended objects, this depends on both their size and their distribution. On the basis of above we are able to derive the typical angular scale of the faint galaxies (see, however, comments at the end of Section 2.2.3).
A technique similar to ours was used for the first time by Tonry and Schneider (1988). They discovered that the distance of a galaxy is inversely proportional to the amplitude of the luminosity fluctuations due to unresolved red giant stars and they used the spatial power spectrum to directly measure these fluctuations. After computing the Fourier trans- form and the power spectrum of the data, they fitted the power spectra with the sum of a constant value and the power spectrum of the point spread function (PSF):
Pimage= P0+ P1· PP SF (2.1)
The resulting fit is shown in Figure 2.2 using the data obtained for M32.
The main difference between their work and ours is the density of sources per pixel. They were studying nearby galaxies whose regions have an average projected density of stars of one hundred per pixel so the fluctuations between adjacent pixels have rms variations equal to 10% of the mean signal. On the contrary, we are dealing with images where the projected density of faint galaxies is much less then one per pixel, therefore we have smaller fluctuations and our results are more sensitive to the presence of spurious signals.
Our approach consists in applying the IDL1Fast Fourier Transform (FFT) routine to the masked images to compute the two-dimensional Fourier transform and, then, to derive the spatial power spectrum of the signal. Therefore, the power spectrum image obtained is two-dimensional, but to plot it and to do all the calculations in the following it is conve- nient to derive the radial trend.
Figure 2.2: Power spectrum obtained from the data of M32 (Tonry and Schneider, 1988). The observations, plotted as diamonds, are well reproduced by the sum of a constant value and the power spectrum of the PSF.
The power spectra for the two bands can be compared by calculating their difference or ratio. The difference is the most natural choice, but relies on an extremely accurate color calibration of the power spectra in order to properly subtract the contribution of lower redshift objects from the z850-band power spectrum. The ratio of the power spectra could
introduce a loss of sensitivity of the method, because it does not allow us to detect faint galaxies with the same power spectrum shape as the bright ones. On the other hand, the ratio allows us to avoid the need of a very accurate and challenging color calibration of the power spectra. Thus, from here on, we focus our analysis on the ratio of power spectra.
First of all, we obtained the power spectra for both the i775 and z850-band images (Fig-
ure 2.3).
The units of measurement for each single power spectrum can be derived as follow. First of all we should consider the definition of magnitude and AB magnitude:
mag = −2.5 · log10(signal[DN/s]) + Zpt (2.2)
ABmag = −2.5 · log10(f lux[erg/s/cm2/Hz]) − 48.6 (2.3)
Equalizing equation 2.2 and 2.3 we obtain
−2.5 · log10(signal[DN/s]) + Zpt= −2.5 · log10(f lux[erg/s/cm2/Hz]) − 48.6 (2.4)
log10 f lux signal = −29.7016 (2.5)
Since 1erg/s= 102nW and 1cm2 = 0.0001m2, we can derive the flux in units of nW/m2/Hz
f lux = signal · 10−29.7016· 102
Figure 2.3: Power spectra with error bars of the background signal of the i775 (purple top
line) and z850 images (blue bottom line).
Considering the band width of the filter, in this case ∆ν = 7.11 · 1013Hz for the i
775one, we
obtain the flux in units of nW/m2
f lux = signal · 7.11 · 10−29.7016· 106+ 1013= signal · 7.11 · 10−10.7016 (2.7)
We should also consider the pixel scale, which permits to convert the flux from nW/m2 to nW/m2/sr 0.03 206264.8 2 = 2.11 · 10−14 (2.8) 0.09 206264.8 2 = 1.9 · 10−13 (2.9)
Finally, to derive the calibrated image in physical units, i.e. nW/m2/sr, it is simply neces- sary to multiply the non calibrated one by a conversion factor
f lux = signal · CF (2.10) The final conversion factor CF depends on the pixel scale and the zero point. The value for CF is 6698.65 for the i775-band image with a pixel scale of 30 mas/pix, 743.9 for the i775-
band image with a pixel scale of 90 mas/pix, 9578.86 for the z850-band image with a pixel
scale of 30 mas/pix, and 1063.76 for the z850-band image with a pixel scale of 90 mas/pix.
When IDL is computing the Fast Fourier Transform, a normalizing factor 1/√N is in- troduced, being N the total number of pixels in the image. To derive the real FFT from the IDLoutput it is, indeed, necessary to multiply the output by√N .
Figure 2.4: Ratio between the z850 and i775 power spectra. The gray shaded area represents
the errors associated to the ratio. The region between k = 100 and k = 400, where the signal from the faint galaxies was measured, respectively, is highlighted in orange.
Considering the Fourier Transform of the calibrated image and the power spectrum de- rived from it, calling them FFT and PS, respectively
F F T ·√N · √
scale
206264.8 (2.11)
P S · N ·206264.8scale 2 (2.12) Finally we have the power spectrum in units of nW2/m4/sr, which is consistent with the finding of Kashlinsky et al. (2004, 2005, 2007, 2012).
After obtaining the power spectrum for each band, to highlight the light contribution coming from the undetected galaxies, which are bona fide z ∼ 6 candidates, we plotted the ratio between the z850 and i775-band power spectra (Figure 2.4). The undetected galaxies
are responsible for the peak visible in Figure 2.4. We tested the reliability of the peak using a χ2 statistics. Comparing the values of the ratio between the power spectra in the range
of the peak, i.e. 100 . k . 400, and in the range 1400 . k . 1700 dominated only by the noise, we obtained χ2 = 3.08. This value made us confident that the peak is a real feature and it is can not be ascribed to random noise since its confidence level is basically equal to zero.
The angular scale θ, plotted on the top x-axis of the figures, is derived from k as follows: θ [arcsec] = Dframe[pixel] · scale [arcsec pixel−1]
where Dframeis the dimension of the frame in pixels and scale is the pixel scale of the image.
Following our technique, we can derive the number of faint galaxies from the height of the signal excess they produce in the ratio between the power spectra (Figure 2.4), com- paring it with the one obtained from our detailed simulations (see Section 2.3).
It must be noticed that our result can be affected by a fraction of interlopers at lower redshift (for a general overview on the topic see Stanway et al. 2008 and Chapter 6). This is an issue affecting all studies based on the dropout selection and it is due to the aliasing between the Lyman break and the 4000 ˚A break, as discussed by Dahlen et al. (2010). Su et al. (2011), on the basis of accurate simulations, estimated that the fraction of z ∼ 1.2 galaxies that can be selected as z ∼ 6 population could be as high as 24%. This value is a pessimistic one since Malhotra et al. (2005) found half that number of contaminant galax- ies. More recently Pirzkal et al. (2013) estimated the fraction of low redshift interlopers to be on average 21%.
A preliminary study on interlopers affecting high-z galaxy samples at z ∼ 4 − 5 − 6 is de- scribed in Chapter 6.