Stacking Analysis of Infrared Galaxies in the Planck Map

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1 Stacking Analysis of Infrared Galaxies in the Planck Map Alice Giroul The University of Tokyo - The University of Glasgow June & July 2014 Abstract Infrared emission from galaxies has been detected by stacking image analysis in the Planck survey. About 10 6 galaxies have been stacked for each of 12 apparent magnitude bands. The shape of the signal corresponds to what is expected from galaxies with a clustering factor. Detection was made in four different frequency bands (217, 353, 545 and 857 GHz). Comparing the signal strength in those bands, it was possible to attempt a fit to a blackbody distribution. This showed that the spectral distribution does not strictly follow the one of a blackbody spectrum. I. Introduction and Aims The Planck mission s all-sky maps have been taken in a variety of energy bands in the infrared and microwave spectra. In these regimes, the dust emission from the Milky Way Galaxy is dominant from an earthly perspective. Indeed this dust has temperatures of about 20 K. Therefore its peak blackbody emission is 2000 GHz, while Planck s high frequency instrument covers six frequency bands between 100 and 857 GHz. [4] For most purposes, this Galactic foreground is removed from the surveys. However, one may wonder if the Planck maps could also be contaminated by other galaxies, which of course also contain the infrared-emitting dust. This emission would be tiny in comparison to the Milky Way s, and indeed the galaxies cannot be detected directly in these frequencies. This is where the idea of Stacking Analysis comes in. Stacking multiple galaxies together limits the effect of noise and other fluctuations of the infrared emission. In 2013, Kashiwagi, Yahata and Suto performed this analysis on the Galactic Extinction Map by Schlegel, Finkbeiner and Davis (SFD map). Using galactic positions from an optical survey, the Sloan Digital Sky Survey[5], they stacked 10 7 galaxies in multiple categories and obtained a signal, showing that the SFD map is contaminated by external galaxies. Their results are especially important as the map is widely used for galactic extinction correction. In some fields where precision is key, the small contamination might introduce considerable systematic errors, and should be accounted for. [1] In this experiment, we applied the same method to four of the high frequency Planck all-sky maps. This will allow an estimate of the effect of the galaxies emission on the map. As opposed to the stacking on the SFD map, this experiment studies stacking results in multiple frequency bands, which gives the possibility for spectral analysis. Therefore, an additional aim is to show that the stacking method s results can give useful information about the galaxies themselves, and could be used to study weak emissions in the future. 1

2 II. Method I. Stacking Galaxies The stacking was done using a C++ program previously used in Kashiwagi, Yahata and Suto s experiment. The program uses the Sloan Digital Sky Survey [5], which references optically observed galaxies and other objects. Objects that are suspected of not being galaxies are not used (for more details see Kashiwagi et al, 2013). Squares of arcminutes were then extracted around each of the galaxies positions in the Planck survey [2]. Galaxies were categorised by their optical apparent magnitudes, and stacked only with galaxies in their magnitude band. Pixel values were then divided by the number of galaxies stacked, giving an "average galaxy". The magnitudes range from 15 to 20 and were separated in 10 bands. There are more faint galaxies, and this should be considered when looking at the results. Figure 1 shows the number of galaxies used for each magnitude band. This was done for the four highest frequency bands of the Planck survey: 217, 353, 545 and 857 GHz. They were chosen because they are closest to the dust emission peak, and therefore give higher chances of successful detection. Number of Galaxies 1e Optical Apparent Magnitude Figure 1: Histogram of the number of galaxies used for each magnitude band. The number of galaxies rises exponentially with magnitude. II. Intensity Profile Fitting First, a circular average was taken from the stacked pictures obtained. This was done by splitting the image into 50 rings, and averaging the pixel value in these rings. Each value was assigned an error corresponding to the standard deviation of the pixels in each ring. This gives a profile function with error, which can be fitted by χ 2 minimisation. The point spread function of the Planck instrument, shown in the Planck documentation [3], can be approximated to a Gaussian: 2

3 ) PSF(θ) exp ( θ2 2σ 2, (1) where θ is the angle from the centre of the galaxy and 2 2 ln 2 σ is the Gaussian full width half maximum. It depends on the frequency band and can be either found by fitting, or obtained from the documentation (see table 1). Fitting with both fixed and free σ was tried. Frequency σ 217 GHz GHz GHz GHz Table 1: fixed σ values used The galaxies are approximated as point sources. However, the stacked images do not only contain the sum of the central galaxies. Due to clustering, there is a higher probability to find galaxies near other galaxies, which will result in a broadened profile. The stacked pictures are the result of the sum of both the central galaxies and their cluster neighbours, plus an averaged out (constant) background. This is summarised in the following equation: I(θ) = C + PSF(θ) (I 0 δ(θ) + I 1 ω(θ)) (2) Here I(θ) is the intensity profile and C is the background value. The point spread function, PSF(θ) is convoluted with the source emission. δ(θ) is a delta function, representing the central galaxy, while ω(θ) is the clustering factor. I 0 and I 1 are constants, which are found by fitting the data. The clustering factor can be approximated to a power law, i.e. we have ω(θ) θ γ, (3) where we assume the exponent γ 0.75 [6]. The convolution in equation 2 results in: I(θ) = C ) + I 0 exp ( θ2 2σ 2 ) ( + I 1 exp ( θ2 2σ 2 1F 1 1 γ 2 ; 1; θ 2 ) 2σ 2, (4) where 1 F 1 is a confluent hypergeometric function. If σ is not taken fixed, this is a fit with 4 free parameters. However, all values for the best fit C, I 0 and I 1 can be computed analytically before optimising σ. Both the analytical and numerical optimisations were done by weighted χ 2 minimisation. Errors on the best fit parameters were roughly estimated by finding their value at which χ 2 was increased by 1 of its minimal value. (4) 3

4 III. Blackbody Analysis The spectral analysis in this experiment limits itself to blackbody fitting. The aim is to obtain a fit that corresponds to the blackbody emission expected from the cold dust in the Galaxy. As our studied frequencies are in the lower range of a 20 K blackbody emission, a fit would not be possible without some additional data near the emission peak. For this purpose, the results from a stacking analysis on IRAS (The Infrared Astronomical Satellite) data has been used. This gives us a datapoint at 100µm, or 2997 GHz. The amount of radiation from a fitted galaxy profile is: B(ν 0 ) = I 0 2πσ 2 (5) were I 0 and σ are defined as before, and ν 0 is the frequency. This value can be calculated for five frequencies: 217, 353, 545, 857 and 2997 GHz. However, only the best fit parameters of good fits could be used. Hence in practice, only about three or four data points were available, depending on the magnitude band. These were then fitted to a blackbody distribution (equation (6)), using the χ 2 method. B(ν) = B 0 1hν3 c 2 1 exp ( hν k B T ) 1 Here T is the temperature, ν is the frequency of the emitted photons, and B 0 is a constant. h, c and k B have their usual meaning : Planck s constant, the speed of light and the Boltzmann constant. T was taken as a free parameter, although it is known to be around 20 K. (6) III. Results The results from the stacking can be found in figure 2. A signal can clearly be identified, although it is very noisy at low frequencies. The amplitude of the signal over the background average varies, but stays minimal: see table 2 below for more details. The fitting is accurate for the clear signals, as can be seen in figure 3a. Unfortunately, for the fainter galaxies, the clustering galaxies largely dominate and the fit sometimes resulted in an unphysical negative I 0. To avoid this, fixed values of σ have been used. This did not eliminate negative best-fit I 0 values, but made them small enough to be assumed null. Frequency Signal over Background 217 GHz ± GHz ± GHz ± GHz ± Table 2: The average values of the signal over the background value, and their standard errors. The standard errors are large because the signal also varies with apparent optical magnitude. 4

5 217 GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = GHz, m = 20 Figure 2: Results from the stacking. To spare space, only one magnitude band out of two has been included. Here m is the lower limit of the magnitude band. Higher frequencies are on the right hand side. 5

6 Profile analysis fitting Stack profile mag Best Fit: Sig = , Chi Squared = C = , I0 = , I1 = Planck map gal Intensity arcmins arcminutes (a) A very good fit at 545 GHz. The intensity in in units of MJy sr 1. Profile analysis fitting Stack profile mag Best Fit: Sig = , Chi Squared = C = , I0 = e-07, I1 = 5.112e-07 Planck map gal Intensity arcmins arcminutes (b) A low quality fit at 217 GHz, due to a noisy stacking result (see figure 2, 217 GHz, m = 16)

7 1 0.1 Flux T = for T = for T = for Frequency (GHz) Figure 4: Unsuccessfully trying to fit a blackbody spectrum to our results. The y-axis reads arbitrary units of intensity, in a logarithmic scale. To make the graph readable, only three magnitude bands have been included, and errors have not been included. More problems arose when trying to fit the blackbody spectrum: all the fits with a negative, or null I 0 had to be eliminated, leaving only few usable data points, especially at low frequencies. Therefore a blackbody fit could only be made for certain magnitude bands, where we had at least three data points. Another issue are the estimated errors: they are extremely small. When fitting the data (figure 4), one can see that the fit is of poor quality. This again shows that the errors have been underestimated. The best fit temperatures are also higher than expected, ranging from 24.8 to 51.0 K. IV. Conclusions The stacking analysis was successful on the Planck survey data, and galaxies were detected in all frequency bands. This confirms the effectiveness of the method. However, some refining on the fitting techniques would need to be done, especially in the lower frequency range where the results were quite noisy. In particular, the error estimates would need to be done more carefully. Indeed the poor errors estimates were partially responsible for making blackbody fitting so difficult. Another possibility could be that a χ 2 minimisation method was not ideal for a fit with so few data points. To get a better result, more frequency ranges would be useful and could be obtained from the planck map or other all-sky surveys. 7

8 Due to the time limitations of this project, a full analysis of the stacking results, including the suggestions above, was not possible. This will hopefully not be seen as the limitations of this type of method. Stacking analysis can not only be useful in the frame of correcting surveys, but also as a tool to observe faint objects statistically. This experiment was more of a preliminary study, yet it succeeded at detecting galaxies in the Planck all-sky surveys for the first time, and provided encouraging results that fit theory. Acknowledgements I would especially like to thank Toshiya Kashiwagi for the time he gave to guide me throughout this project, and for making it such a useful experience. Many thanks also go to Prof. Yasushi Suto for welcoming me to his laboratory, and to the University of Tokyo Research Internship Programme for giving me the chance to come to Japan and complete this project. Working on this experiment would not have been so enjoyable without the help and friendliness of all the group members, and in particular my student guide Shoya Kamiaka. Thank you all! References [1] Kashiwagi, T., Yahata, K. and Suto, Y. (2013). Detection of Far-Infrared Emission from Galaxies and Quasars in the Galactic Extinction Map by Stacking Analysis. Publications of the Astronomical Society of Japan, 65(2), p.43. [2] Based on observations obtained with Planck ( an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. Ade, P., Aghanim, N., Armitage-Caplan, C., Arnaud, M., Ashdown, M., Atrio-Barandela, F., Aumont, J., Baccigalupi, C., Banday, A., Barreiro, R. and others, (2013). Planck 2013 results. VI. High Frequency Instrument data processing. arxiv preprint arxiv: [3] Ade, P., Aghanim, N., Armitage-Caplan, C., Arnaud, M., Ashdown, M., Atrio-Barandela, F., Aumont, J., Baccigalupi, C., Banday, A., Barreiro, R. and others, (2013). Planck 2013 results. VII. HFI time response and beams. arxiv preprint arxiv: [4] The Infrared Science Archive (2014). IRSA - Planck. Available at: [5] York, D., Adelman, J., Anderson Jr, J., Anderson, S., Annis, J., Bahcall, N., Bakken, J., Barkhouser, R., Bastian, S., Berman, E. and others, (2000). The sloan digital sky survey: Technical summary. The Astronomical Journal, 120(3), p [6] Connolly, A., Scranton, R., Johnston, D., Dodelson, S., Eisenstein, D., Frieman, J., Gunn, J., Hui, L., Jain, B., Kent, S. and others, (2002). The angular correlation function of galaxies from early Sloan Digital Sky Survey data. The Astrophysical Journal, 579(1), p.42. 8