Foreground simulations for the LOFAR - Epoch of Reionization Experiment
Future high redshift 21-cm experiments will suffer from a high degree of contamination, due both to astrophysical foregrounds and to non-astrophysical and instrumental effects. In order to reliably extract the cosmological signal from the observed data, it is essential to understand very well all data components and their influence on the extracted signal. Here we present simulated astrophysical foregrounds datacubes and discuss their possible statistical effects on the data. The foreground maps are produced assuming windows that match those expected to be observed by the LOFAR Epoch-of-Reionization (EoR) key science project. We show that with the expected LOFAR-EoR sky and receiver noise levels, which amount to at 150 MHz after 400 hours of total observing time, a simple polynomial fit allows a statistical reconstruction of the signal. We also show that the polynomial fitting will work for maps with realistic yet idealised instrument response, i.e., a response that includes only a uniform uv coverage as a function of frequency and ignores many other uncertainties. Polarized galactic synchrotron maps that include internal polarization and a number of Faraday screens along the line of sight are also simulated. The importance of these stems from the fact that the LOFAR instrument, in common with all current interferometric EoR experiments has an instrumentally polarized response.
keywords:cosmology: theory, diffuse radiation, observation, radio lines: general, instrumentation: interferometers, radio continuum: general
The Epoch of Reionization (hereafter, EoR), which marks the end of the Universe’s ‘Dark Ages’, is one of the least explored epochs in cosmic evolution. Currently, there are two main observational constraints on the EoR. The first is the sudden jump in the Lyman- optical depth in the Gunn-Peterson troughs (Gunn & Peterson, 1965) observed in the Sloan Digital Sky Survey quasar spectra (Becker et al., 2001; Fan et al., 2001; Pentericci et al., 2002; White et al.q, 2003; Fan et al., 2006), marking a lower limit to the redshift at which the Universe became completely ionized. The second constraint comes from the fifth year WMAP satellite data on the temperature and polarization anisotropies of the cosmic microwave background (CMB) (Spergel et al., 2007; Page et al., 2007) which gives an integral constraint on the Thomson optical depth experienced by the CMB photons since the EoR. However, both of these observational methods provide limited information on the reionization process.
The redshifted 21-cm hyperfine transition line of neutral hydrogen is the most promising and immediately accessible method for probing the intergalactic medium (IGM) during reionization (e.g. Field, 1958, 1959; Scott & Rees, 1990; Kumar, Subramanian, & Padmanabhan, 1995; Madau, Meiksin, & Rees, 1997). Recent years have witnessed a flurry of theoretical activities to predict reionization sources and their impact on the IGM (e.g. Barkana & Loeb, 2001; Loeb & Barkana, 2001; Ciardi, Ferrara, & White, 2003; Ciardi, Stoehr, & White, 2003; Bromm & Larson, 2004; Iliev et al., 2007; Zaroubi et al., 2007; Thomas & Zaroubi, 2007). Measurements of the 21-cm signal can also help to constrain the cosmological parameters independently (McQuinn et al., 2006).
Future telescopes like LOFAR111http://www.lofar.org, MWA222http://www.haystack.mit.edu/ast/arrays/mwa, 21CMA333http://web.phys.cmu.edu/ past/ and SKA444http://www.skatelescope.org are being designed to study the redshifted 21-cm signal from the EoR. A successful detection of this signal will help us derive the nature of the first sources and their impact on the surrounding IGM.
Unfortunately however, the cosmological EoR signal is contaminated by a slew of astrophysical and non-astrophysical components. Typically, the contamination level is orders of magnitude larger than the cosmological 21-cm signal. Thus, the primary challenge of the EoR observations will be the accurate modelling of the various data components – foregrounds, instrumental response, ionospheric disturbances, to name a few – which is essential to develop a robust signal extraction scheme.
For the foregrounds, there are currently no available data in the 115-180 MHz frequency range and 4 arcmin resolution at high Galactic latitude that would allow accurate modelling of the LOFAR-EoR foregrounds. Therefore, one has to rely on the available relevant data and extrapolate, based on theoretical arguments, into the frequency range and resolution observed by LOFAR. However, recently Ali, Bharadwaj, & Chengalur (2008) used obervations with Giant Meter Wave Radio Telescope to characterize the statistical properties – visibility correlation function – of the foregrounds. This paper focuses on simulating the galactic and extragalactic foregrounds that dominate the sky at frequencies of interest for the LOFAR-EoR experiment (–). The main foreground components are: Galactic synchrotron emission from diffuse and localised sources, Galactic thermal (free-free) emission and integrated emission from extragalactic sources (like radio galaxies and clusters). The dominant component of the foregrounds is the Galactic synchrotron emission (70 per cent). The extragalactic emission contributes 27 per cent and Galactic free-free emission 1 per cent (Shaver et al., 1999). Although the difference between the mean amplitude of the EoR signal and the foregrounds is expected to be 4-5 orders of magnitude, an interferometer like LOFAR measures only the fluctuations which in this case are expected to be different by ‘only’ three orders of magnitude.
Various authors have studied the foregrounds in the context the EoR measurements. Shaver et al. (1999) have studied the diffuse synchrotron and free-free emission from our Galaxy and extragalactic sources; Di Matteo et al. (2002) and Di Matteo, Ciardi, & Miniati (2004) have considered emission from unresolved extragalactic sources at low radio frequencies; and Oh & Mack (2003) and Cooray & Furlanetto (2004) studied the effect of free-free emission from extragalactic haloes. Over the years, several methods have been explored to filter out the foregrounds. Most of the methods rely on the relative smoothness in the frequency of the foregrounds, with respect to the signal (Shaver et al., 1999; Di Matteo et al., 2002; Zaldarriaga, Furlanetto, & Hernquist, 2004; Morales, Bowman, & Hewitt, 2006; Wang et al., 2006; Gleser, Nusser, & Benson, 2007).
Santos, Cooray, & Knox (2005) have studied the foregrounds for the EoR experiment and their influence on the measurement of the 21-cm signal. In their multi-frequency analysis of the power spectra, they considered four types of foregrounds: Galactic diffuse synchrotron emission; Galactic free-free emission; extragalactic free-free emission; and extragalactic point sources. They showed that foregrounds cleaning is aided by the large scale angular correlation, especially of the extragalactic point sources, which facilitates signal extraction to a level suitable for the EoR experiments.
The current study is part of the general effort undertaken by the LOFAR-EoR key science project to produce simulated data cubes. The pipeline under construction will simulate the LOFAR-EoR data cube that includes the simulated cosmological 21-cm signal, the galactic and extragalactic foregrounds, ionospheric effects, radio frequency interferences (RFIs) and the instrumental response. These datacubes will be used to design the observational strategy and test our signal-processing methods. Our main concern in this paper is the simulation of the galactic and extragalactic foregrounds.
Recently, a study by Gleser, Nusser, & Benson (2007) has been conducted along lines similar to parts of the current paper. The authors test a certain signal extraction algorithm on simulated foregrounds maps in which they take most of the relevant foregrounds into account. However, there are many important differences between the two papers. First, in the Gleser, Nusser, & Benson (2007) study the assumption for the noise level in the LOFAR-EoR project, as well as the other experiments, is at least an order of magnitude too low. They assume 1 and 5 mK noise models whereas in reality the noise for the LOFAR-EoR experiment is about 50 mK. They also present a simplified model of the Galactic foregrounds that does not take into account all the spatial and frequency correlations of the Galactic diffuse synchrotron emission and underestimates that of the Galactic free-free emission, both of which are very important. In contrast to them, we also present polarized maps and introduce the LOFAR instrumental response and noise in a realistic manner.
In the foregrounds simulations presented in this paper we choose a different approach from previous groups, since our main aim is to produce the simulations that will be part of the LOFAR-EoR data pipeline. In this context our main aim is to produce foregrounds maps in the angular and frequency range of the LOFAR-EoR experiment, i.e. 3D datacubes, and then use those simulations for testing the accuracy of removal of the foregrounds. Section 5 outlines the importance of the polarized character of the foregrounds and how to model the Stokes I, Q, and U polarization maps of the Galactic synchrotron emission. Section 6 presents simulated instrumental effects of the LOFAR telescope and their influence on the foregrounds maps, and Section 7 discusses a method to extract the EoR signal from the foregrounds. The paper concludes with a discussion and outlook (Section 8).
2 The Cosmological 21-cm signal
In radio astronomy, where the Rayleigh-Jeans law is applicable, the radiation intensity, is expressed in terms of the brightness temperature , such that:
where is the frequency, is the speed of light and is Boltzmann’s constant. The predicted differential brightness temperature deviation of the cosmological 21-cm signal from the cosmic microwave background radiation is given by (Field, 1958, 1959; Ciardi & Madau, 2003):
Here is the spin temperature, is the neutral hydrogen fraction, is the matter density contrast, and are the mass and baryon density in units of the critical density and 555We assume a CDM Universe with , , and .
In his seminal papers, Field (1958, 1959) used the quasi-static approximation to calculate the spin temperature, as a weighted average of the CMB, kinetic and colour temperature (Wouthuysen, 1952; Field, 1958):
where is the CMB temperature and and are the kinetic and Lyman- coupling terms, respectively. We have assumed that the color temperature, , is equal to . The kinetic coupling term increases with the kinetic temperature, whereas the coupling term is due to the Lyman- pumping, known also as the Wouthuysen-Field effect (Wouthuysen, 1952; Field, 1958). The two coupling terms are dominant under different conditions and in principle could be used to distinguish between ionization sources, e.g., between first stars, for which Lyman- pumping is dominant, vs. first mini-quasars for which X-ray photons and therefore heating is dominant (see e.g., Nusser, 2005; Kuhlen, Madau, & Montgomery, 2006; Zaroubi et al., 2007; Thomas & Zaroubi, 2007).
The brightness temperature of the cosmological signal used in this study is produced from a dark-matter-only N-body simulation. This simulation is used to produce a cube of the cosmological signal, i.e., the density as a function of right ascesion, declination, and redshift (for more details see Thomas et al., , in preparation). Although is calculated according to Eq. 3, we assume that . The reason for this assumption is that towards the redshifts of interest for the experiment (6–12), the abundance of Ly photons in the Universe is sufficient to couple to which is obviously much greater than (Ciardi & Madau, 2003). Hence from Eq. 2, follows the cosmological density and . We further assume that along each sight-line the neutral fraction follows the function , where for each pixel (or line of sight) is set to and where is the density contrast at redshift 10. We used this approach to randomize the reionization histories along different lines of sight while preserving the spatial correlations of the cosmological signals. In principle, this randomization could be drawn out of a Gaussian distribution function. Redshift 10 here is an arbitrary choice. along each line of sight varies in accordance with the cosmological density along that line-of-sight at z=10 and has a variance of unity centred at 8.5. Fig. 1 shows the signal data cube that we use in order to test our foregrounds filtering procedure.
Currently, a number of experiments (e.g., LOFAR, 21CMA, MWA and SKA) are being designed to directly measure of the HI 21-cm hyperfine line and probe the physics of the reionization process by observing the neutral fraction of the IGM as a function of redshift. In this study, we focus on predictions for LOFAR, but our conclusions could be easily applied to the other telescopes.
The LOFAR-EoR666For more information, see the LOFAR web site: www.lofar.org and the LOFAR-EoR web site: www.astro.rug.nl/ LofarEoR. key project plans to measure the brightness fluctuations in the frequency range of 115–190 MHz, corresponding to redshift range 6-11.5 with spectral resolution of and angular resolution of about arcmin. A more detailed discription of the LOFAR array will be given later in the paper when the instrumental effects are discussed (Section 6).
3 Galactic foregrounds
The Galactic foregrounds have three main contributions. The first and largest component is the Galactic diffuse synchrotron emission (GDSE), which is the dominant foreground component in the frequency range of the LOFAR-EoR experiment. The second component is radio synchrotron emission from discrete sources, mostly supernova remnants (SNRs). The third and last component is the free-free radio emission from diffuse ionized gas. This component is the weakest of the three, yet it still dominates over the cosmological component. Moreover, it has a different spectral dependence, making it very imporant in testing the signal extraction schemes that we have. In this section we describe how we simulate the contribution of each of these components to the total intensity. The polarized intensity simulations are described later on.
3.1 Galactic diffuse synchrotron emission (GDSE)
The GDSE originates from the interaction between the free electrons in the interstellar medium and the Galactic magnetic field. Therefore the observed GDSE intensity as a function of frequency, , depends on the number density of emitting electrons, , and the Galactic magnetic field component perpendicular to the line of sight, :
where is the electron spectral energy distribution power law index (Pacholczyk, 1970). The intensity of the synchrotron emission as expressed in terms of the brightness temperature varies with position and frequency and its spectrum is close to a featureless power law , where is the brightness temperature spectral index, related to by .
Observational data that are relevant to the LOFAR-EoR project are scarce. Landecker & Wielebinski (1970) have produced an all sky map of the total intensity of the GSDE at low radio frequencies at 150 MHz with 5 resolution. The other Galactic survey relevant to the LOFAR-EoR experiment is the 408 MHz survey of Haslam et al. (1982) with a resolution of and of Reich & Reich (1988) at with resolution. In the Reich & Reich (1988) paper the authors also assume a smooth power law change in the intensity as a function of frequency which they calculate from their 1420 MHz and 408 MHz maps.
At high Galactic latitudes the minimum brightness temperature of the GDSE is about 20 K at 325 MHz with variations of the order of 2 per cent on scales from 5–30 arcmin across the sky (de Bruyn et. al, 1998). At the same Galactic latitudes, the temperature spectral index of the GDSE is about at 100 MHz and steepens towards higher frequencies (e.g. Reich & Reich, 1988; Platania et al., 1998). Furthermore, the spectral index gradually changes with position on the sky. This change appears to be caused by a variation in the spectral index along the line of sight. An appropriate standard deviation in the power law index, , in the frequency range 100–200 MHz appears to be of the order of (Shaver et al., 1999). Recent data, collected around a galaxy cluster Abell 2255 using the WSRT telescope at 350 MHz, indicate that the rms of the brightness temperature at 3 arcmin resolution could be as low as 0.1–0.3 K (Pizzo and de Bruyn, private communication). If extrapolated to 150 this result implies that the rms in that region could be 1–2 , which is an order of magnitude smaller than the low resolution data suggest.
For the purpose of this paper we assume that the GDSE as a function of frequency is well approximated by a power law within the limited frequency range of 115–180 MHz. This is a central assumption in our simulation which is consistent with the general trend shown by the available data, namely that the change in the frequency power law index is gradual. The values we choose for the power law index are based on the high Galactic latitude regions in the Haslam et al. (1982) and Reich & Reich (1988) maps. The second assumption we make is that both the intensity and power law index of the GDSE can be spatially modelled as Gaussian random fields (GRFs). For the power spectrum of GRFs we assume a power law with 2D index . The standard deviation of the GRFs is normalized to 0.4, assuming an angular scale corresponding roughly to the field of view (). This is consistent with the value adopted by Tegmark et al. (2000), Giardino et al. (2002) and Santos, Cooray, & Knox (2005) for the angular power spectrum index , where , varies from to , and is the harmonic number.
In contrast to the previous authors (Tegmark et al., 2000; Giardino et al., 2002; Santos, Cooray, & Knox, 2005) who directly used the angular and frequency power spectrum of the GDSE for their analysis, we simulate GDSE in four dimensions (three spatial and one frequency), produce maps at each frequency and then do our analysis on them. The four dimensional realisation approach has the added benefit of enabling us to account for the amplitude and temperature spectral index variations of the GDSE along the line of sight (-coordinate). We obtain the final map of the GDSE at each frequency, , by integrating the GDSE amplitude () along the z-coordinate:
where is the brightness temperature of the GDSE as a function of position and frequency and is a normalization constant. is dimensionless and at each frequency is defined by power law:
where is the reference frequency at which the normalisation is done and is the temperature spectral index as a function of 3D position and frequency . The power law index has a weak frequency dependence, also as a power law.
and of the GDSE at the normalization frequency are modelled spatially as two Gaussian random fields with 3D power law spectrum . Note that the absolute value of the 3D power law index is where is the 2D power law index mentioned above. and are normalized according to observations (the Galactic surveys mentioned above).
For clarity, the steps we followed to produce the GDSE maps are listed below:
Generate the same 3D Gaussian random field for both and . The assumption here is that both fields have a correlated spatial distribution, which is supported by visual inspection of the high Galactic latitude portions of the Reich & Reich (1988) maps. We have also explored the possibility that and are independent; this has led to results very similar to the correlated case, and therefore we show only maps in which and are correlated.
Normalize the mean and standard deviation of by integrating along the direction and setting the mean and standard deviation of to match the observations (the Galactic surveys mentioned above). In other words we set the integration constant in Eq. 5, in a way that the properties of the field after integration match the observed properties of .
Normalize the mean and standard deviation of according to observations.
Use Eq. 6 to calculate at each frequency.
Integrate along the -coordinate to get the two-dimensional maps of the GDSE brightness temperature at each frequency (Eq. 5).
For the purpose of this paper we simulate the GDSE on grid, where the x,y plane corresponds to angular size of and z direction scales between 0–1 in dimensionless units. The amplitude, , of the GDSE is normalized in the way described above to match (de Bruyn et. al, 1998), while is normalized at : (Shaver et al., 1999).
Fig. 2 shows a simulated map of the Galactic diffuse synchrotron emission according to the procedure described above, at a frequency of MHz with an angular size of on a grid. The mean brightness temperature of the map is with .
In contrast to Fig. 2, which shows the angular variations of the GDSE at one frequency, Fig. 3 shows the amplitude variations of GDSE as a function of frequency for a number of lines of sight. Each line of sight has a slightly different power law index along the frequency direction as a result of the spatial variations in the temperature spectral index. Furthermore, the brightness temperature variation for one line of sight is not a single power law but superposition of many power laws, due to the spectral index variations both spatially and in the frequency direction. Note that is still a very smooth function of frequency.
3.2 Emission from SNRs
Supernova remnants are composed of expanding shells that have strong magnetic fields which are able to produce cosmic rays. As the particles escape the expanding shell, their energy decreases due to synchrotron cooling and we detect them at radio frequencies. The majority of the Galactic SNRs are within the Galactic plane but their distribution exponentially decreases with distance from the Galactic plane, , (e.g. Caswell & Lerche, 1979; Xu, Zhang, & Han, 2005), that is, . Moreover, due to the interaction of SNRs with the interstellar medium their radio surface brightness decreases with an increase of their diameter and with an increase of their height , (e.g. Caswell & Lerche, 1979), namely .
Our goal is to calculate the expected number of known SNRs within a LOFAR-EoR observational window at high Galactic latitudes, using the known number of observed radio SNRs from the Green (2006) catalogue and assuming that their distribution follows . On average, we obtain between one and two known SNRs in each observational window. Given the extended nature of the SNRs we include two of them in each window in order to examine the influence of bright extended sources on the calibration process and foreground removal.
The simulated SNRs assume a power law spectrum:
where is the flux density of a SNR at frequency , is its value at normalization frequency , and is the spectral index.
The simulated SNRs are placed randomly on the map and their angular size, flux density and spectral index are arbitrary chosen from the Green (2006) catalogue. The SNRs are added on the map as disks with uniform surface brightness.
Properties of the two SNRs included in our foreground simulations are shown in Table 1.
|angular size||position on the map|
3.3 Diffuse free-free emission
The diffuse thermal (free-free) emission contributes only per cent of the total foregrounds within the frequency range of the LOFAR-EoR experiment (Shaver et al., 1999). It arises due to bremsstrahlung radiation in very diffuse ionized gas, with a total emission measure of about at high Galactic latitudes and (Reynolds, 1990). This gas is optically thin at frequencies above a few MHz, so its spectrum is well determined and has a temperature spectral index of .
At high Galactic latitudes, H and free-free emission of the diffuse ionized gas are both proportional to the emission measure. Therefore, the Galactic H survey is generally used as a tracer of the Galactic diffuse free-free emission (Smoot, 1998). However, some groups also find significant correlation between free-free emission and dust emission (Kogut et al., 1996; de Oliveira-Costa et al., 1997) which can also be used as another independent tracer of the Galactic free-free emission.
In our simulations we followed Tegmark et al. (2000) and Santos, Cooray, & Knox (2005) who included the Galactic diffuse free-free emission as a separate component of the Galactic foregrounds with an angular power spectrum and frequency . Despite its small contribution to the foregrounds, the free-free emission is important for two reasons. Firstly, the amplitude of its angular fluctuations is much larger than that of the EoR signal. Secondly, and more importantly, its spectral index along the frequency direction is quite different from the other foreground components and could be important in testing the algorithms for the EoR signal extraction.
To obtain the Galactic free-free emission maps we followed the same procedure as for the Galactic synchrotron emission with the additional simplification of fixing the power law index to across the map. is normalized according to the relation between H and free-free emission (see review by Smoot, 1998) whereby the intensity of H emssion, , is:
where is total emission measure and temperature. For the value of is . Combining Eq. 8 with the free-free equations in Smoot (1998), one finds a relation between and brightness temperature of free-free emission, :
Fig. 4 shows a simulated map of Galactic diffuse free-free emission at MHz. The angular size of the map is on grid, with the mean brightness temperature of and .
4 Extragalactic foregrounds
4.1 Radio galaxies
At the frequency range of the LOFAR-EoR experiment, bright radio sources are dominated by radio-loud galaxies, quasars and BL Lac objects (an AGN class of objects). However, at sub-mJy flux densities the contribution of late-type (star forming) galaxies, whose radio synchrotron emission originates from supernovae rather than AGN, becomes significant (Prandoni et al., 2001; Magliocchetti et al., 2002; Sadler et al., 2002)
The bright extragalactic radio sources are normally divided into two classes based on the relative physical position of their high and low surface brightness area within the lobes. These two classes are called FRI and FRII radio sources (Fanaroff & Riley, 1974; Jackson & Wall, 1999). Our simulations of radio galaxies are based on the tables by Jackson (2005) of extragalactic radio source counts at 151 MHz. Jackson (2005) has used CDM based models to calculate the evolution of the radio luminosity function of these sources, from which was predicted the source distributions and their number densities.
In obtaining these tables, Jackson (2005) assumed that the radio sky consists of three population types of radio sources: FRI, FRII and star forming galaxies. Moreover, it is assumed that the local radio luminosity function of star forming galaxies can be determined from the 2dFGRS-NVSS (Sadler et al., 2002) galaxy sample at 1.4 GHz. The parameterized number density and luminosity evolution of star forming galaxies is adopted from Haarsma et al. (2000). For the local radio luminosity function of FRI and FRII radio galaxies, Jackson (2005) assumed that it can be determined by exponential fitting of luminosity-dependent density evolution to the observed source counts at (Jackson & Wall, 1999). Given the three evolving radio luminosity functions, Jackson (2005) simulated the sky at different frequencies by randomly positioning and orientating each source on the sky. The intrinsic size is also selected randomly, assuming redshift independence. The FRI and FRII sources were modelled as double-lobe structures, and the star forming galaxies as circular discs.
In our simulations of radio galaxies we adopt the three types of radio sources from Jackson (2005) and use the predicted source surface densities per deg for 10, 5, 2, 1 and 0.1 mJy flux density limit in order to obtain the number of sources with certain flux density per deg. However, and in contrast to the simulations by Jackson where each source is randomly positioned, we introduce a angular clustering of the sources. The clustering is motivated by the results of Di Matteo, Ciardi, & Miniati (2004) in which they showed that the contribution of the angular clustering of extragalactic radio sources to the angular fluctuations of the foregrounds, at scales , is dominated by bright sources. Hence, in order to detect angular fluctuations in the cosmological 21-cm emission, efficient source removal should be carried out.
For angular clustering of the radio galaxies we used the particularly elegant procedure of Rayleigh-Lèvy random walk proposed by Mandelbrot (1975, 1977). Starting from any arbitrary position, one places the next galaxy in a randomly chosen direction at angular distance , drawn from the distribution:
This is repeated many times until the correlation function of the distribution converges to the one desired. However, to compare the introduced correlation with obervational results and set the right values of and , the two point correlation function needs to be calculated.
The two point correlation function, , of the radio galaxy population is defined as the excess probability, over that expected for a Poisson distribution, of finding a galaxy at an angular distance from a given galaxy (e.g. Peebles, 1980):
where is probability, the mean surface density and a surface area element. Given the very large number of galaxies that can be simulated, we adopted the simplest form for estimating , defined as:
where is the number of pairs of galaxies with separation in the correlated sample of galaxies and is the number of pairs with the same separation but in a randomly distributed uncorrelated sample of galaxies. The total number of galaxies of the two samples must be the same.
Recent results on the angular clustering of radio sources in the NRAO VLA Sky Survey (NVSS) and Faint Images of the Radio Sky at Twenty centimetres (FIRST) (Overzier et al., 2003) showed that the two point correlation function is best fitted by a double power law with slopes of , and amplitudes , . However, in this study we adopt the simpler single power law correlation function with only two parameters, and . The reason for doing this is that steeper power law component () of the observed two point correlation function is dominant on angular scales smaller than those resolved by the LOFAR-EoR experiment. Note that in the two point correlation function is the same one as in Eq. 11, while .
Fig. 5 shows a simulated map of radio galaxies with angular power law distribution. All sources are point like as the angular resolution of the LOFAR-EoR project will not be sufficient to resolve most of them. The angular extent of the map is . On the map there are in total 20690 radio galaxies: per cent SF radio galaxies, per cent FRI and per cent FRII radio galaxies. The flux density distribution at of the simulated radio galaxies are shown in the Table 2, while the two point correlation fuction is (, ). The simulated radio galaxies assume a power law spectrum (see Eq. 7) with spectral index -0.7 (e.g. Jackson, 2005).
|number of sources with flux density limit|
|10 mJy||5 mJy||2 mJy||1 mJy||100 Jy|
4.2 Radio clusters
Galaxy clusters as radio sources are classified into cluster radio haloes and cluster radio relics. The former are morphologically regular diffuse sources, typically centred inside the cluster and mostly unpolarized, whereas the radio relics are typically irregular, located at the periphery of the cluster and consist mostly of polarized radio diffuse sources. Both types of cluster radio source have steep frequency spectra with (see e.g. Feretti, 2002, for review).
The emission in radio haloes is due to synchrotron radiation by relativistic electrons with energies of GeV in G magnetic fields. The distribution of the radio haloes seems to follow closely the large scale distribution of the free-free driven X-ray emission of clusters (Govoni et al., 2001). This association is also supported by a strong correlation between the radio halo luminosity and the host cluster X-ray luminosity (e.g. Enßlin & Röttgering, 2002). However, not all clusters host radio haloes. Statistically, it is found that roughly 30–40 per cent of galaxy clusters with X-ray luminosity do host radio haloes.
In our simulations of extragalactic foregrounds maps, as a starting point for simulating radio clusters, we used the CDM deep wedge cluster catalogue of The Hubble Volume Project777The cluster catalogue is part of simulations carried out by the Virgo Supercomputing Consortium using computers based at the Computing Centre of the Max-Planck Society in Garching and at the Edinburgh parallel Computing Centre. The data are publicly available at http://www.mpa-garching.mpg.de/galform/virgo/hubble.. The catalogue was obtained from an N-body simulation with one billion dark matter particles. This catalogue provides a list of clusters up to redshift with angular coverage of 100 deg (Colberg et al., 2000; Jenkins et al., 2001; Evrard et al., 2002).
In order to translate the cluster mass into X-ray luminosity and then into radio luminosity , we used the empirical mass–X-ray luminosity relation of Reiprich & Böhringer (2002):
and the X-ray–radio luminosity relation of Enßlin & Röttgering (2002):
where , , and (Enßlin & Röttgering, 2002). It is important to note that Eq. 14 is derived for the radio frequency at 1.4 GHz. Note also that in Eq. 13 the mass, , is related to the cluster real mass, , by .
Since is derived for 1.4 GHz, we extrapolate the luminosity of each cluster to lower frequencies, relevant to the LOFAR-EoR experiment, according to:
where (Kempner et al., 2004).
The angular size of the radio clusters was estimated from their physical radius and redshift. For the physical radius we used the virial radius calculated from the cluster mass according to (Bryan & Norman, 1998),
where is the mean density and is the critical density at redshift .
In order to obtain the maps of radio clusters at the angular and frequency ranges desired we first randomly choose 30 per cent of the catalogue’s clusters (note that observations show that only 30–40 per cent of clusters have radio properties). The cluster masses are then used to estimate the radio luminosity of each cluster (Eq. 13, 14 & 15) and its virial radius (Eq. 16). Finally, the cluster is projected onto the simulated map according to its survey coordinates in the Hubble Volume simulation, its redshift and estimated size. Note that the radio clsuters are added on the map as disks with uniform surface brightness.
Fig. 6 shows a map of radio clusters simulated at . The colour bar represents the brightness temperature of the clusters in logarithmic units. The size of each cluster has been scaled by a factor of 10 for visual clarity.
The need for understanding the polarization response stems from two main factors. One is the geometry of the LOFAR configuration, and the other is the cross-talk contamination between the two dipoles of a LOFAR antenna. In order to detect the EoR signal, which has at best a signal to noise ratio (SNR) of per beam on most angular scales – assuming 400 hours of observation under the configuration specified in Section 6 – we need to fully understand the response of the LOFAR system in total intensity and polarization. As discussed before, this is vital for us in order to be able to span a dynamic range of 4–5 orders of magnitude.
Since the LOFAR antennae are fixed to the ground, the sources are tracked only by beam-forming and not by steering the antennae as in traditional radio astronomy. This implies that, depending on the position of the source on the sky, only a certain projection of the two dipoles is visible. This projection changes as the source is tracked over time. Therefore, at most times the sources in the sky see different projections of each of these dipoles. Now, if the sources/foregrounds are polarized, we immediately see that the power output from the pair of dipoles (which is the sum of the two polarized components) will vary dramatically. This has to be taken in account almost exactly during the inversion and calibration processes in order to achieve the desired dynamic range.
On the other hand, a leakage in the electronics can cause the power that is supposed to go through one of the dipoles to show up in the other (referred to as cross-talk). Although the cross-talk is small compared to the effect due to projection, we need to take it into account to eliminate any systematics.
In this paper we show a first order simulation of the Galactic diffuse polarized emission and defer a more advanced discussion on the topic to a future paper.
There are several polarization surveys of the Galactic synchrotron emission between MHz and GHz (see review, Reich, 2006). The most recent one was done with the Westerbork telescope at MHz, with arc minute angular resolution (Wieringa et al., 1993; Haverkorn, Katgert, & de Bruyn, 2000, 2002, 2003). Its low frequency maps reveal a large amount of unusually shaped polarized small-scale structures, which have no counterpart in the total intensity. These structures are normally attributed to the coexistence of magnetic fields and thermal gas in the interstellar medium, which produce Faraday rotation at each line of sight.
The Faraday rotation depends on the observing frequency and rotation measure (RM) of the structure and it is defined along the line of sight. In order to measure Faraday rotation, observations at two or more frequencies are required. However, full understanding of the observed results could be quite difficult due to the possibility of multiple Faraday rotation layers (screens) along the line of sight and depolarization effects. Brentjens & de Bruyn (2005) introduced a new method (Faraday Rotation Measure Synthesis) that is able to cope with multiple screens and analyzes the contribution of each of these screens separately.
The Galactic diffuse synchrotron emission is linearly polarized and its polarized intensity can be expressed in terms of Stokes parameters and :
where is the polarization angle.
In order to simulate the polarization maps of the Galactic synchrotron emission at low radio frequencies, we use a simple model of the Galactic synchrotron polarization at high frequencies (Giardino et al., 2002) in combination with Faraday screens that are introduced to account for the effects of rotation and depolarization at low frequencies. Note that the assumption about the correlation between the polarized and total intensity going into the Galactic synchrotron polarization model at high frequencies is not valid at low frequencies, since the observations mentioned above show polarized structures that have no counterpart in the total intensity. However, this assumption can be acceptable for a first estimate. Note that the correlation assumption should not be used in the analysis of the redshifted 21 cm data as it is mostly invalid and can lead to wrong interpretations. In the future, we will improve on the model itself and use results from real polarization data obtained by the LFFE888LFFE (Low Frequency Front End) are receivers at the Westerbork Synthesis Radio Telescope (WSRT) and cover the frequency range from 115 to 170 MHz.
The basic assumptions of the Giardino et al. (2002) model of Galactic synchrotron polarization at high frequencies are:
The polarized component of Galactic synchrotron emission is proportional to the unpolarized intensity, which in terms of brightness temperature is:
where is the fraction of polarized emission (or polarization degree) and is the polarization angle;
The fraction of polarized radiation is related to the temperature spectral index (Cortiglioni & Spoelstra, 1995):
The polarization angle is given by:
where are 2D random Gaussian fields with the mean zero and characterized by a power law spectrum, while . The power law spectral index is and its value is driven by observations, e.g. the Parkes 2.4 GHz polarimetric survey (Duncan et al., 1995).
The Faraday screens are modelled as 2D fields of rotation angles defined by (Rybicki & Lightman, 1986):
where is the wavelength of radiation and is the rotation measure modelled as a 2D Gaussian random field (GRF) with a power law spectrum of spectral index . Note that for the demonstrative purpose of this simulation we introduced only two Faraday screens, with mean zero and standard deviation 0.3, and arbitrarily set the value of to -2.
Therefore, in order to generate polarization maps of Galactic synchrotron emission at given frequencies, first we take the GDSE maps of total intensity () and temperature spectral index () from Sec. 3.1 and calculate the fraction of polarized radiation according to Eq. 20. Then, using Eqs. 21 & 22 we obtain polarization angle and Faraday rotation angle . Finally, we use Eq. 19 to get polarization maps Q and U. The angle in Eq. 19 is the sum of over all Faraday screens and .
6 Instrumental effects
In this section we give a basic overview of the simulations of LOFAR antenna response and show how the foreground maps are seen by LOFAR. More detailed discussion on the LOFAR response and data model for the LOFAR-EoR experiment will appear in Labropoulos et al. (, in prep.). For the LOFAR-EoR experiment we plan to use the LOFAR core, which will consist of approximately 25 stations. However, in this paper we set the number of LOFAR core stations to 24. Each station is further split into two substations which are separated by a few tens of metres (see Fig. 9). Each substation consists of 24 tiles, with each tile having 44 crossed dipoles. For our goals we assume that each of the forty-eight substations is a circular array with a radius of thirty-five metres. The stations are distributed in a randomized spiral layout and span a baseline coverage from 40 to 2000 metres. The total effective collecting area for the LOFAR-EoR experiment is at . The instantaneous bandwidth of the LOFAR telescope is and the aim for the LOFAR-EoR experiment is to observe in the frequency range between 115–180, which is twice the instantaneous bandwidth. To overcome this, multiplexing in time has to be used (for more details see, de Bruyn et. al, 2007). For the purpose of this paper we ignore this complication and assume 400 hours of integration time for the hole frequency range. The LOFAR specifications are not final yet and they might slightly change in the near future. A detailed description of the telescope together with the data model will be provided in forthcoming papers.
In order to compute the true underlying visibilities, we make some simplifying assumptions. We assume that the narrow bandwidth condition holds and that the image plane effects have been calibrated to a satisfactory level. This includes station complex gain calibration, a stable primary beam, and adequate compensation for the ionospheric effects, such that the ionospheric phase introduced during the propagation of electromagnetic waves in the ionosphere and the ionospheric Faraday rotation are corrected for. For an interferometer, the measured spatial correlation of the electric fields between two antennae is called ‘visibility’ and is approximately given by Taylor, Carilli & Perley (1999):
where is the primary beam, is the intensity map corresponding to frequency , are the coordinates, as seen from the source, of the tracks followed by an interferometer as the Earth rotates, and are the direction cosines.
We further treat each pixel of the map as a point source with the intensity corresponding to intensity of the pixel. Note that the equation above takes into account the sky curvature. The visibilities are sampled for all substation pairs and also at different pair positions, as the Earth rotates.
We calculate the Fourier transform of the foreground model for each frequency in the above range. For every baseline and frequency, the uv tracks sample different scales of the Fourier transform of the sky at that frequency. Thus, the sampling function becomes
where the summation is carried over all the pixels .
We compute those tracks for each interferometer pair for 4 hours of synthesis with an averaging interval of and we then grid them on a regular grid in the uv plane. The maximum baseline assumed for the LOFAR core is and the station diameter is , the number of independent elements in the uv plane is . If the uv plane is oversampled by a factor of four, this yields pixels 999This is the closest power of two to match the number of sampled elements. By doing this one can benefit from the speed of the FFT. in the uv plane of in size. After counting how many track points fall within each grid cell, we end up with a matrix representing the naturally weighted sampling function in the uv plane. By multiplying this sampling matrix with the Fourier transform of our model sky we get the visibilities on that grid with appropriate weights. This procedure is done for each baseline pair.
where is the Fourier transform of the input image and is the sampling function.
The LOFAR visibility densities per resolution element at 150 MHz, for the LOFAR-EoR experiment, are shown in Fig. 10. The total integration time is 400 hours () with averaging time of and observing declinations (left panel) and (right panel).
The inverse Fourier transform of the sampled visibilities is called the ‘dirty’ map. It is actually the sky map convolved with the Fourier transform of the sampling function, which is called the ‘dirty’ beam or the ‘PSF’. This is a simple-minded approach to estimating the sky brightness as it uses linear operations. The approximation of the underlying brightness with the ‘dirty’ map is not always satisfactory, as side lobes from bright features will obscure fainter ones. In cases of low signal to noise, however – such as during the observation of the redshifted 21-cm transition line of HI – one might choose not to proceed further than this first approximation. To go beyond that we need extra information like the positivity of the intensity and compact support. The discussion of such issues is beyond the scope of this paper. This incomplete sampling of the uv plane also means that we do not measure the complete power at all scales, due to the holes in the uv coverage and its finite extent.
An example of a ‘dirty’ map of the diffuse components in the foregrounds is shown in the Fig. 11, together with the ‘original’ simulated foreground map with no interferometric effects and noise. The corresponding total integration time is 400 hours, with an averaging time of at . Note that the ‘dirty’ maps are generated without the inclusion of noise.
The ultimate sensitivity of a receiving system is determined principally by the system noise. The discussion of the noise properties of a complex receiving system like LOFAR can be lengthy, so we concentrate for our purposes on some basic principles. The theoretical rms noise level in terms of flux density on the final image is given by
where is the system efficiency that accounts for electronic, digital losses, N is the number of substations, is the frequency bandwidth and is the total integration time. SEFD is the System Equivalent Flux Density in . The system noise we assume has two contributions. The first comes from the sky and is frequency dependent () and the second from receivers.
For the LOFAR core the SEFD will be around at , depending on the final design (de Bruyn et. al, 2007). This means that we can reach a sensitivity of at with bandwidth in one night of observations. In order to calculate the SEFD we use the following system temperature () scaling relation as function of frequency (): . Accumulating data from a hundred nights of observations brings the sensitivity down to . We further assume that the distribution of noise over the map at one frequency is Gaussian. The noise contribution to each pixel in a map is drawn independently from a Gaussian distribution. The EoR signal is extracted from two different scenarios. The first scenario involves the extraction of the signal from the ‘original’ maps – simulated maps that are not convolved with the dirty beam – after adding the noise directly to the ‘original’ maps. In the other scenario, the EoR signal is extracted from ‘dirty’ maps to which we do not add noise but convolve the ‘original’ maps of the EoR signal plus the foregrounds with a simplified dirty beam.
As the uv coverage scales linearly with frequency, one has to be careful in using the ‘dirty’ maps for extraction. This is because a pixel sampled at a given frequency need not be sampled at a later frequency. Since the analysis performed in this paper involves data across the frequency domain, we need a good uv coverage. If the uv sampling functions are not scaled accordingly, we will introduce additional difficulties arising from the mixing of spatial scales. To overcome spatial scale mixing, one of the strategies in the data analysis is to use only the uv points that are present at all frequencies. In other words one can construct the uv plane mask that only contains the uv points that are sampled at every frequency. This step of course results in substantial data loss.
The uv coverage for the LOFAR-EoR experiment changes in scale by between the frequency range of observation (–). By choosing only those uv points that are present at all frequencies, of the total data is lost in the frequency range specified above. Since with decreasing bandwidth of observation the amount of data lost decreases, one of the strategies could be to observe in windows of smaller bandwidth. However the observational strategy of the LOFAR-EoR experiment is not yet finalized and will be discussed in detail in upcoming papers of the project. A detail discussion on the scaling of the uv coverage with frequency and its influence on the number of discarded baselines and the amount of data loss will be discussed in (Labropoulos et al., , in prep.).
In the following section we will show our ability to statistically detect the EoR signal from the ‘original’ maps that include realistic level for the noise and from ‘dirty’ maps that do not include the noise but are sampled with the uv mask that contains only uv points present at each frequency. In both cases the statistical detection of the signal is done on total intensity maps only. Moreover, perfect calibration is assumed and any other systematics that might influence the data are ignored. Those issues will be addressed in a follow-up paper.
7 Detection of the EoR signal from the FG
This section presents the results on the statistical detection of the EoR signal from ‘original’ and ‘dirty’ LOFAR-EoR data maps that include the cosmological 21cm signal, diffuse components of the foregrounds and realistic noise. By ‘original’ maps we mean maps before inversion or in other words maps with no calibration errors or interferometric effects. ‘Dirty’ maps include only simplified uv coverage effect as an interferometric effect, but no calibration errors.
By using only diffuse components of the foregrounds (Galactic diffuse synchrotron and free-free emission and integrated emission from unresolved extragalactic sources) we assume that all resolved discrete and extended sources have been successfully removed from the observed maps, without any subtraction residuals. Also note that our analysis is done on total intensity maps only. The polarized case will be considered in the follow-up paper.
The foreground and noise maps are simulated in the frequency range between and in steps of . The original maps simulated for a field on a grid are re-binned to a grid, so that each pixel corresponds to which is the resolution attained by the core of the LOFAR telescope.
The EoR maps are simulated between the frequencies of and in steps of , corresponding to redshifts between 11.5 and 6.5.
The mean of the EoR signal, foreground and noise maps at each frequency are set to zero since LOFAR is an interferometric instrument and measures only fluctuations around the mean. The typical variations, , over the map at 150 MHz are for the EoR signal, for the foregrounds and for noise. Hereafter, these values are considered fiducial values for the EoR signal, foregrounds and noise.
The analysis on the LOFAR-EoR data maps can be done in two ways: firstly along the frequency direction where the foregrounds are assumed to be smooth in contrast to the EoR signal; and secondly in the spatial domain where the EoR signal and some components of the foregrounds are spatially correlated, but the noise is not. In this paper we will demonstrate statistical detection of the signal by analysis along the frequency direction, taking lines of sight (map pixels) one by one.
Fig. 12 shows one line of sight (one pixel along frequency) of the ‘original’ LOFAR-EoR data cube (upper black solid line) without interferometric effects. The first step in the extraction of the EoR signal is to take out the smooth foreground component (dotted black line). It is important to note, however, that the smooth component of the foregrounds is not a simple power law but a superposition of three power laws (Galactic synchrotron and free-free emission and integrated emission from unresolved extragalactic sources) including the fact that one of the power law indices (Galactic synchrotron emission) changes slightly with frequency.
The simplest method for foreground removal is a polynomial fitting in logarithmic scale (). The dashed green line on Fig. 12 represents the foregrounds fitted with a 3 order polynomial in the logarithmic scale.
Fig. 13 shows a comparison between the detected and original EoR signal for randomly chosen lines of sight in the case of the fiducial foreground level and without noise for ‘original’ maps. As one can see, there is an almost exact agreement: this confirms that our approach when applied to noiseless data does not introduce any systematic biases.
After taking out the fitted foregrounds from the ‘original’ data maps, the residuals should contain only the noise and the EoR signal (lower solid black line on Fig. 12). However, the assumption here is that we have fitted well enough such that the residuals between the fitted and the ‘real’ foregrounds are smaller than the EoR signal. Otherwise the EoR signal could be fitted out if we are over-fitting, or be dominated by the foreground fitting residuals if we are under-fitting the foregrounds.
The extraction of the EoR signal from the residuals along one line of sight is an impossible task, since the level of the noise is order of magnitudes larger than EoR signal and its value is unknown for a certain pixel. However, general statistical properties of the noise (standard deviation as a function of frequency) might be determined from the experiment and be used to statistically detect the EoR signal. By statistical detection we mean determination of the standard deviation of the EoR signal over the entire map as an excess variance over the variance of the noise. The general statistical properties of the noise might be determined in two ways. The first method is based on the difference between measured fluxes of a discrete point source, with well know properties, at two consecutive frequency channels. The second one is based on the difference between the measured flux in total and polarized intensity at the same frequency. However, the accuracy of the both methods need to be tested for the LOFAR-EoR experiment and we leave further discussion on this topic for a forthcoming paper.
Fig. 14 shows the standard deviation of residuals as a function of frequency (dotted green line), after taking out the smooth component of the foregrounds, by polynomial fitting in logarithmic scale to each line of sight of the ‘original’ maps. The most satisfactory result we get with a 3 order polynomial. The dashed-dotted black line represents the standard deviation of the noise. By subtracting (in quadrature) the from , we get the excess variance () of the EoR signal. However, in order to determine the error on the detection of the EoR signal, we conducted a Monte-Carlo simulation of the extraction of the signal. We made 1000 independent noise cube realisations and applied the signal extraction algorithm to each. The results of the simulation are shown in Fig. 14. The grey shaded zone shows the 2 detection, whereas the dashed white line shows the mean of the detection. As one can see the mean of the detected EoR signal is in good agreement with the original (solid red line) up to 165 MHz. The disagreement for higher frequencies is due to over-fitting and low EoR signal level. Remember that for most of the sightlines our simulated Universe has already been ionized at this frequency, corresponding to (see section 2).
In order to see the influence of the foreground and noise level on the EoR extraction and detection scheme, we repeated the same analysis on ‘original’ maps of the four different models. The first model has a foreground level two times bigger than fiducial and the second four times; the third has the fiducial foreground level but smaller noise level by ; and the last one has a normal foreground level and no noise (see Table 3). Note that by fiducial foreground level and noise level we mean and . The results are shown in Figs. 13 & 15.
|case (a)||case (b)||case (c)||case (d)||case (e)|
Comparing Figs. 14 & 15, we see the higher foreground levels decrease the quality of the EoR detection. Lower quality in the EoR detection is due to over-fitting at higher frequencies. However, even for the four times bigger foreground level we are able to detect the EoR signal up to .
Comparing Figs. 13, 14 & 15, we can see that a lower noise level increases the quality of the EoR detection, as expected. Better precision in the EoR detection with lower noise level also confirms that our foreground removal procedure works well.
Finally, in Fig. 17, we show the statistical detection of the EoR signal from the ‘dirty’ foregrounds + EoR signal maps without any noise. Note that the ‘dirty’ maps are produced with a sampling function (uv mask) that contains only uv points present at each frequency, in order to overcome additional difficulties from the mixing of angular scales in the frequency direction introduced by the linear frequency variation of the uv coverage and incomplete sampling in the frequency direction.
The smooth component of the foregrounds is removed by polynomial fitting to each line of sight. The most satisfactory result we get with a 6 order polynomial. A different order of polynomial from the case of the ‘original’ maps is required due to the angular scale mixing over each map introduced by convolution of the map with a ‘dirty’ beam.
Fig. 16 shows a comparison along the frequency direction of the same pixel from the ‘original’ (solid line) and ‘dirty’ (dashed line) foregrounds + signal maps. Note that the foregrounds are still smooth along the frequency direction of the ‘dirty’ maps, but the shape of the function is slightly different. The difference is due to incomplete uv coverage sampling and its finite extent, determined by the shortest and longest baselines.
The dashed green line in Fig. 17 shows the standard deviation, as a function of frequency, of the extracted EoR signal from ‘dirty’ foregrounds + EoR signal maps. The dashed-dotted black line shows the standard deviation of the ‘original’ EoR signal maps, while the solid black line shows the standard deviation of the ‘dirty’ EoR signal maps. The agreement between the standard deviations of the extracted and ‘dirty’ EoR signals is satisfactory, while their slightly lower levels than the ‘original’ signal are due to the smoothing effect of the instrumental response.
8 Discussion and Outlook
This paper presents foreground simulations tailored for the LOFAR-EoR experiment that is set to study the redshifted 21-cm hyperfine line of neutral hydrogen from the Epoch of Reionization. The foreground simulations include Galactic diffuse synchrotron and free-free emission, synchrotron emission from Galactic supernova remnants and extragalactic emission from radio galaxies and clusters. For each of the foreground components, we generate the field in the frequency range approximately between 115 and 180 MHz pertaining to the LOFAR-EoR.
Since the diffuse Galactic synchrotron emission is the dominant component ( per cent) we include all its observed characteristics: spatial and frequency variations of brightness temperature and its spectral index, and brightness temperature variations along the line of sight. Discrete sources of Galactic synchrotron emission are included as observed emission from supernovae remnants.
Despite the minor contribution of the Galactic free-free emission ( per cent), it is included in our simulations of the foregrounds as an individual component. It has a different temperature spectral index from Galactic synchrotron emission.
Integrated emission from extragalactic sources is decomposed into two components: emission from radio galaxies and from radio clusters. Simulations of radio galaxies are based on the source count functions at low radio frequency by Jackson (2005), for three different types of radio galaxies, namely FRI, FRII and star forming galaxies. Correlations obtained by radio galaxy surveys are used for their angular distribution. Simulations of radio clusters are based on a cluster catalogue from the Virgo consortium and observed mass–X-ray luminosity and X-ray–radio luminosity relations.
Under the assumption of perfect calibration, LOFAR-EoR data maps that include the simulated cosmological 21cm signal (), diffuse components of the foregrounds () and realistic noise () are produced. We refer to this set of parameters as our fiducial model. For noise we assume it has two components, the sky noise and receiver noise. The former varies with frequency as whereas the latter is roughly frequency independent.
The extraction of the EoR signal is performed along the frequency direction, taking lines of sight (map pixels) one by one. The first step in the EoR extraction is removal of the smooth foregrounds component for each line of sight (see Fig. 12). In our analysis we fit a 3 order polynomial in the logarithmic scale. However, one should be careful in choosing the order of the polynomial to perform the fitting. If the order of the polynomial is too small, the foregrounds will be under-fitted and the EoR signal could be dominated and corrupted by the fitting residuals, while if the order of the polynomial is too big, the EoR signal could be fitted out.
After foreground removal, the residuals are dominated by instrumental noise. Since the noise is unknown for each line of sight and is an order of magnitude larger than the EoR signal, it is an impossible task to directly extract the EoR signal for each line of sight. However, assuming that the statistical properties of the noise () will be known, we can use it to statistically detect the EoR signal. The statistical detection of the EoR signal is the measure of the excess variance over the entire map, , that should be obtained by subtracting the variance of the noise, , from that of the residuals, .
Fig. 14 shows the results of a successful statistical detection of the EoR signal in the fiducial model of the foregrounds and noise. The detected standard deviation of the signal is in a good agreement with original signal up to 165 MHz. The disagreement for higher frequencies is due to over-fitting caused by the very weak cosmological signal at these frequencies.
In order to see the influence of the foreground and noise level on the EoR extraction and detection, the same analysis was done for models with two and four times bigger foreground levels than in the fiducial model, for a model where the noise is smaller by , and for a model without noise (see Tabel 3). The results are shown in Figs. 15 & 13.
In the case of higher foreground levels, the EoR signal detection is hampered by over-fitting. In the case of lower noise levels, however, the proposed EoR detection algorithm performs extremely well.
For the diffuse components of simulated foregrounds, a ‘dirty’ map with realistic but idealised instrumental response of LOFAR is produced (see Fig. 11). However, the signal extraction scheme we apply only to the ‘dirty’ maps that have been produced with a uniform uv coverage as a function of frequency and have no noise. This is due to the additional difficulties introduced by mixing of angular scales in the frequency direction. Those issues will be discussed in a follow-up paper.
In addition to the simulations of the total brightness temperature, the polarized Galactic synchrotron emission maps are produced. Here, we follow a simple model that includes multiple Faraday screens along the line of sight (see Figs. 7 & 8). The motivation behind these simulations is that improper polarization calibration could severely contaminate the EoR signal, so future robust extraction algorithms have to take this into account.
Fig. 18 shows the angular power spectra of the simulated EoR signal (dotted red line), simulated diffuse component of the foregrounds (solid black line) and three levels of noise (blue lines) at . The lines are drawn as the best fit to the coresponding points. The dashed blue line represents the level of the noise () after one year of the LOFAR-EoR experiment (h of total observing time) with a single beam. For this case of instrumental noise and inclusion of realistic diffuse foregrounds we have shown that we are able to statistically detect the EoR signal despite the small signal to noise ratio. However, the current observing plan of the LOFAR-EoR experiment (de Bruyn et. al, 2007) is to observe with five independent beams, which reduces the by a factor of (dashed-dotted blue line). After four years of observations (h) with five beams the is reduced by a factor of (dashed-dotted-dotted blue line), which means that the signal to noise ratio is roughly 0.5.
Finally we would like to emphasize that this paper is just a first step in testing and developing the analysis tools for the LOFAR-EoR experiment. Future papers will use the foregrounds simulations developed in this paper together with simulations of the EoR signal (Thomas et al., , in prep.), instrumental response (Labropoulos et al., , in prep.), and other non-astrophysical effects (e.g. ionosphere, RFIs, …) in order to test all aspects of the pipeline in the LOFAR-EoR experiment.
The authors thank to the anonymous referee for his illustrative and constructive comments. As LOFAR members, the authors are partially funded by the European Union, European Regional Development Fund, and by ‘Samenwerkingsverband Noord-Nederland’, EZ/KOMPAS.
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