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Clustering of Lyman alpha emitters at z ~ 4.5

We present the clustering properties of 151 Lyman alpha emitting galaxies at z ~ 4.5 selected from the Large Area Lyman Alpha (LALA) survey. Our catalog covers an area of 36' x 36' observed with five narrowband filters. We assume that the angular correlation function w(theta) is well represented by a power law A_w = Theta^(-beta) with slope beta = 0.8, and we find A_w = 6.73 +/- 1.80. We then calculate the correlation length r_0 of the real-space two-point correlation function xi(r) = (r/r_0)^(-1.8) from A_w through the Limber transformation, assuming a flat, Lambda-dominated universe. Neglecting contamination, we find r_0 = 3.20 +/- 0.42 Mpc/h. Taking into account a possible 28% contamination by randomly distributed sources, we find r_0 = 4.61 +/- 0.6 Mpc/h. We compare these results with the expectations for the clustering of dark matter halos at this redshift in a Cold Dark Matter model, and find that the measured clustering strength can be reproduced if these objects reside in halos with a minimum mass of 1-2 times 10^11 Solar masses/h. Our estimated correlation length implies a bias of b ~ 3.7, similar to that of Lyman-break galaxies (LBG) at z ~ 3.8-4.9. However, Lyman alpha emitters are a factor of ~ 2-16 rarer than LBGs with a similar bias value and implied host halo mass. Therefore, one plausible scenario seems to be that Lyman alpha emitters occupy host halos of roughly the same mass as LBGs, but shine with a relatively low duty cycle of 6-50%.


Introduction
Galaxy clustering provides a powerful tool for testing cosmological models and galaxy formation models, through quantitative comparisons between predicted and observed clustering statistics. The galaxy two-point correlation function is the most widely used of these statistics, thanks to its straightforward calculation and its direct relationship to the galaxy power spectrum. It has been established for decades that the two-point correlation function is reasonably well described by a power law over a range of distances between the observed galaxies (see pioneering works of Totsuji &Kihara 1969 andPeebles 1974).
Large redshift surveys of galaxies, such as the two-degree Field Galaxy Redshift Survey (2dFGRS; Colless et al. 2001) and the Sloan Digital Sky Survey (SDSS; Loveday 2002) provide precise measurements of the clustering signal for redshift z ≈ 0. Their size makes it possible to investigate the dependence of the clustering signal on intrinsic galaxy properties, such as morphology or luminosity. Red galaxies are clustered more strongly, and their power law is steeper, compared to the power law which describes the clustering properties of blue galaxies (e.g. Norberg et al. 2002;Zehavi et al. 2002Zehavi et al. , 2004. This conclusion is in agreement with the results from surveys at intermediate redshifts about the clustering properties of galaxies of different color (Le Févre et al. 1996;Carlberg et al. 1997). These surveys also detect redshift evolution in galaxy clustering. Recently, surveys have achieved a sufficient size and uniformity to detect the small deviations between real correlation functions and pure power law fits (Zehavi et al 2004;Zheng 2004).
Identification of large high-redshift galaxy samples using multiband color selection techniques (Meier 1976;Madau et al. 1996;Steidel et al. 1996Steidel et al. , 1998 has opened the way for studies of luminosity functions and correlation functions in the distant universe Adelberger et al. 1998;Adelberger et al. 2000;Ouchi et al. 2003;Shimasaku et al. 2003;Hamana et al. 2003;Brown et al. 2005;Allen et al. 2005;Lee et al. 2006). Galaxies selected in these broad band photometric surveys are expected to have broadly similar properties and lie in a restricted redshift interval (∆z ∼ 1).
Lyman-α emission offers an alternative method for finding high redshift galaxies. These are typically star-forming galaxies with smaller bolometric luminosities than the usual continuumselected objects. These samples do not appear to contain substantial numbers of active galactic nuclei Wang et al. 2004;Dawson et al. 2004).
In the modern picture of galaxy formation, based on the Cold Dark Matter (CDM) model, galaxies form in dark matter halos which evolve in a hierarchical manner. Here, the clustering pattern of galaxies is determined by the spatial distribution of dark matter halos and the manner in which dark matter halos are populated by galaxies (Benson et al. 2000;Peacock & Smith 2000;Seljak 2000;Berlind & Weinberg 2002). Galaxy surveys provide constraints on the galaxy distribution. The dark matter distribution is estimated using N-body simulations or an analytical approach, generally based on the Press-Schechter formalism (Press & Schechter 1974) and its extensions (Sheth et al. 2001;Sheth & Tormen 2002). The statistical relation between galaxies and the dark matter halos where they are found can be described empirically using a "halo occupation function" (e.g. Moustakas & Somerville 2002), which describes the probability of an average number N galaxies being found in a halo as a function of halo mass.
In this article we describe the clustering properties of galaxies selected through their Lyman-α emission at z ≈ 4.5. In section 2 we present the data used in this paper and describe the selection of the Lyman-α candidates. In section 3 we present the correlation function analysis and results. We compare these results to the prediction of CDM theory in section 4. A discussion and a summary of our main conclusions are given in section 5. For all calculations we adopt a ΛCDM cosmology with Ω M = 0.3, Ω Λ = 0.7, H 0 = 70 km s −1 Mpc −1 and the power-spectrum normalization σ 8 = 0.9. We scale our results to h = H 0 /(100 km s −1 Mpc −1 ).

The LALA sample
The Large Area Lyman Alpha (LALA) survey started in 1998 as a project to identify a large sample of Lyα-emitting galaxies at high redshifts (Rhoads et al 2000). Over 300 candidates have been identified so far at z ≈ 4.5 (Malhotra & Rhoads 2002), with smaller samples at z ≈ 5.7 (Rhoads & Malhotra 2001;Rhoads et al. 2003) and z ≈ 6.5 (Rhoads et al. 2004). The search for Lyman-α emitters is realized through narrowband imaging using the wide-field Mosaic camera at Kitt Peak National Observatory's 4m Mayall telescope. Two fields of view of 36' × 36' are observed, covering a total area of 0.72 deg 2 . In this article we discuss the properties of the Lyman-α emitters selected from Boötes field, centered at 14h25m57s, +35 0 32' (2000.0) at z ≈ 4.5. Full details about the survey and data reduction are given in Rhoads et al. (2000) and Malhotra & Rhoads (2002). Five overlapping narrowband filters of width FWHM ≈ 8 nm are used. The central wavelengths are 655.9, 661.1, 665.0, 669.2, and 673.0 nm, giving a total redshift coverage 4.37 < z < 4.57. This translates into a surveyed volume of 7.3 × 10 5 comoving Mpc 3 per field (Rhoads et al. 2000).
vey (Jannuzi & Dey 1999) in a custom B w filter and the Johnson-Cousins R and I filters.
Candidates are selected using the following criteria. In narrowband images candidates have to be 5 σ detections where σ is the locally estimated noise. The flux density in narrowband images has to exceed that in the broadband images by a factor of two. This corresponds to a minimum equivalent width (EW) of Lyα of 80Å in the observer frame, which helps cut down foreground emitters. Additionally, the narrowband flux density must exceed the broad band flux density at the 4σ level or above. Finally, candidates that are detected in B w band image at ≥ 2σ are rejected, as such blue flux should not be present if the source is really at high redshift.
These selection criteria were followed by visual inspection. In the overlapping area of all 5 narrowband filters, we selected a total of 151 candidate Lyα emitters. More information about the sample is summarized in Table 1, where we give the number of candidates as detected in each of the filters. Because the filters overlap in wavelength, many objects were selected in more than one filter. Thus, the total of the sample sizes for the five individual filters exceeds the size of the merged final sample.

The w(θ) estimation
The angular correlation function w(θ) is defined such that the probability of finding two galaxies in two infinitesimal solid angle elements of size δΩ, separated by angle θ, is (1 + w(θ)) Σ 2 δΩ 2 , where Σ is the mean surface density of the population. Typically, w(θ) is measured by comparing the observed number of galaxy pairs at a given separation θ to the number of pairs of galaxies independently and uniformly distributed over the same geometry as the observed field. A number of statistical estimators of w(θ) have been proposed (Landy & Szalay 1993;Peebles 1980;Hamilton 1993).
We calculate the angular correlation function using the estimator w(θ) proposed by Landy & Szalay (1993) where DD(θ) is the number of pairs of observed galaxies with angular separations in the range (θ, θ + δθ), RR(θ) is the number of random pairs for the same range of separations and DR(θ) is the analogous number of observed-random cross pairs. Each of these parameters: DD(θ), RR(θ) and DR(θ) is normalized with the total number of pairs in the observed, random and cross-correlated observed-random sample respectively.
Due to the small number of galaxies detected in the individual filters we perform w(θ) calculations for the total sample consisting of 151 galaxies (numbers are given in Table 1). We are not able to resolve galaxies which are separated by less than 1 arcsecond from each other, thus we used this value as the smallest distance in the calculation of number of pairs. The random sample consists of 1000 individual catalogs, which have been generated to have the same number of objects and the same geometry as the observed field. Formal errors are estimated for every bin using the relation (1 + w(θ))/ √ DD as an approximation of the Poisson variance, which is very good estimation of the noise in the case of w(θ) estimator (Landy & Szalay 1993). Our data show a strong correlation in the innermost bins, but the estimated w(θ) value approaches zero rapidly at θ 40".
It is generally assumed that w(θ) is well represented by a power law A w θ −β . From the estimated w(θ) values for our data set, we conclude that there are not enough bins with significant power for us to estimate both the amplitude and the slope of the correlation law. For further calculations we therefore adopt the fiducial slope β = 0.8. This value is within the range for published Lyman break samples (see, e.g., Giavalisco et al. 1998), and is moreover consistent with results for a flux limited sample of over 10 5 low redshift galaxies from the SDSS (Zehavi et al. 2004).
We use the χ 2 method to obtain the amplitude of the power law fitted to the estimated w(θ) points, using the assumed slope of β = 0.8. The best-fit amplitude is A w = 6.73 ± 1.80 for θ in arcseconds (Figure 1), obtained with χ 2 =1.90 total (weighting the points with the modeled values). The confidence interval for the derived amplitude is estimated from the Monte Carlo simulations in the following manner. We create a set of 10000 random realizations of w(θ) values modeling them with a power law with the above estimated amplitude A w and slope β = 0.8 assuming normal errors (Press et al. 1992). For every realization of w(θ) values we obtain the best-fit amplitude using the χ 2 minimization process, fitting a power law with the fiducial value of the slope β. The resulting distribution of the estimated amplitudes is given on the left panel of Figure 2.
Estimates of w(θ) require an estimate of the background galaxy density. We base our density estimate on the survey itself. We therefore need to account for uncertainty in the background density due to cosmic variance in the local number density in our survey volume. This bias, known as the "integral constraint", reduces the value of the amplitude of the correlation function by the amount (see e.g. Peebles 1980) Here Ω corresponds to the solid angle of the survey. The last integral can be approximated with the expression (Roche et al. 2002) Summing over the observed field we calculate C = 0.00456. This value is small and we neglect it in further calculations.

The real-space correlation length r 0
In the previous subsection we presented the measurement of the correlation signal between galaxies projected on the sky. If the redshift distribution of the observed galaxies N(z) is known, the spatial correlation function can be obtained from the angular correlation function using the inverse Limber transformation (Peebles 1980;Efstathiou et al. 1991). In the case of the power law representation of the angular correlation function, the spatial correlation function is also in power law form and it can be written as The slope γ is related to the slope β by γ = β + 1. The amplitudes of the power law representation of angular and spatial correlation functions are related by the equation : Here D θ is the angular diameter distance, and C is a numerical factor given by where Γ stands for the Gamma function. The function F (z) describes the redshift dependence of ξ(r), and we take F = constant given the small redshift range covered in our survey. For the assumed cosmological model and the galaxy redshift distribution described with a tophat function in the redshift interval 4.37 < z < 4.57, we calculate the correlation length r 0 of the Lyman-α galaxies to be r 0 = 3.20 ± 0.42 h −1 Mpc. The 1 σ confidence interval is estimated using synthetic values of A w created in simulations. The distribution of correlation lengths shows smaller scatter than the corresponding distribution of amplitudes ( Figure 2).
The observed clustering signal may be diluted if our sample is contaminated by foreground sources. From the spectroscopic follow-up of selected Lyman-α emitters at z ≈ 4.5 the fraction of the contaminants is f cont ≈ 28% ). Presence of foreground sources can reduce A ω by a maximum factor of (1 − f cont ) 2 assuming no correlation between the contaminants. Following this assumption (i.e., no correlation between the contaminants) the contamination-corrected spatial correlation length for our sample is r 0 = 4.61 ± 0.60 h −1 Mpc. The corrected r 0 value corresponds to the maximum correlation length permitted for the sample studied. All results discussed in the following text based on the contaminationcorrected correlation lengths should be therefore understood as the upper limits. Figure 3 shows the observed correlation length r 0 (in comoving units) of Lyman-α galaxies at redshift z ≈ 4.5 in our sample, together with r 0 values for a number of surveys covering the redshift interval 0 < z < 5. Two points represented with circles in Figure 3 are measures of the correlation strength from the two samples of Lyman-α galaxies. The correlation length estimated from our sample at z ≈ 4.5 (filled circle in Figure 3) is in very good agreement with the correlation length r 0 = 3.5 ± 0.3 h −1 Mpc for the sample of Lyman-α galaxies at z = 4.86 (empty circle in Figure 3) obtained by Ouchi et al. (2003).
A discrepancy arises when comparing the contamination corrected correlation lengths from these two samples. In the following we address exactly this issue in more detail. Ouchi et al. (2003) use Monte Carlo simulations to assess the contamination of the sample by foreground sources. Briefly, by generating the large number of sources created to correspond to the detected sources, distributing them randomly into the two real broadband and one narrowband images, and consequently using the same detecting criteria as for the real sources, Ouchi et al. (2003) find that the maximum fraction of contaminants is about 40%. The contamination by foreground sources increases the correlation length up to the maximum permitted value of 6.2 ± 0.5 h −1 Mpc, quoted in Ouchi et al. 2003, much larger than our maximum permitted correlation length of 4.61 ± 0.6 h −1 Mpc. Even though the sample of Lyman-α emitters studied by Ouchi et al. (2003) is peculiar -galaxies studied in the discussed paper belong to a large-scale structure of Lyman-α emitters discussed into detail in Shimasaku et al. (2003) -we believe that the reason for the discrepancy between the contamination corrected correlation lengths lies in the different methods used to estimate the fraction of foreground sources. While our estimate is based on the spectroscopic followup, the fraction of contaminants derived in Ouchi et al. (2003) is based purely on the photometric data. Shimasaku et al. (2003) discuss the sample of Lyman-α emitters at z = 4.86, extending the sample presented in Ouchi et al. (2003) with additional Lymanα emitters. These emitters are detected in the field which partially overlays and partially extends in the direction of the observed overdensity of Lyman-α emitters studied by Ouchi et al. (2003). Shimasaku et al. (2003) use the same criteria as Ouchi et al. (2003) to define the Lyman-α emitters, except the limiting magnitude of the Lyman-α candidates in the narrowband images, which is half a magnitude lower. Shimasaku et al. (2003) include the spectroscopic follow-up to test their photometric selection (the spectroscopic sample contains 5 Lyman-α candidates). The fraction of foreground contaminants estimated by Shimasaku et al. (2003) using both the photometric and spectroscopic data is about 20 %, two times lower than the fraction of low-z contaminants estimated in Ouchi et al. (2003). Using the updated fraction of contaminants to be valid also for the sample of Lyman-α emitters discussed in Ouchi et al. (2003), the maximum permitted correlation length of that sample would be r 0 = 4.5 ± 0.4 h −1 Mpc, assuming no correlation between the contaminants. This value is again in very good agreement with our estimate of the maximum correlation length of r 0 = 4.61 ± 0.60 h −1 Mpc corrected for the dilution of the sample of Lyman-α emitters with the low-z galaxies.
However, one should bare in mind that the correlation properties of the sample of Lyman-α emitters studied by Shimasaku et al. (2003) differs from the correlation properties of the sample presented in Ouchi et al. (2003). The angular correlation function of Lyman-α emitters at z = 4.86 is no longer well described by the power law of the angular distance: it is practically flat taking values w ∼ 1-2 at distances ≤ 8 arcmin, except for the point at 0.5 arcmin . The authors claim that the constant amplitude of the angular correlation function is largerly implied by the large-scale structure and the large void regions in the observed field (see Figure 3 in Shimasaku et al. 2003 or slightly modified Figure 3 in Shimasaku et al. 2004). Moreover, Shimasaku et al. (2004) searched the same field for the Lyman-α emitters at redshift z = 4.79 (using the imaging in the additional narrowband filter) and find only weak clustering of these Lyman-α emitters on any scale. These results point out that there exists a large cosmic variance of clustering properties of Lyman-α emitters on scales of ∼ 35 h −1 Mpc ).
The measured r 0 values of Lyman-α emitters (presented in Figure 3) are comparable to the r 0 values of LBGs. More generally, the correlation length of sources observed at high redshifts are smaller for about a third of the r 0 values measured for the nearby galaxies. When corrected for the contamination by low-z objects, the maximum permitted correlation lengths of the samples studied at high redshifts are practically consistent with the value of the correlation length at zero redshift. Groth & Peebles (1977) proposed a theoretical model to describe the redshift evolution of the correlation length, the so called "ǫ-model". In comoving units this model has the following form For the fiducial slope of the correlation power law γ = 1.8, the parameter ǫ = 0.8 corresponds to the evolution of correlation function as expected in linear perturbation theory for a Universe with Ω = 1. For ǫ = −1.2, the clustering pattern is fixed. We use normalization r 0 (z = 0) = 5.3 h −1 Mpc to calculate the modeled redshift evolution of the correlation length. The measurements of the correlation length of the Lyman-α emitters do not follow the redshift evolution of correlation length predicted by the 'ǫ-model' (short-and long-dashed lines in Figure 3) . We conclude that there is no value of ǫ for which equation 8 can fit the observed correlation lengths measured for the full range 0 ≤ z 5. Similar conclusions have been presented by a number of authors Connolly et al. 1998;Matarrese et al. 1997;Moscardini et al. 1998).
This implies that the population of Lyman-α galaxies at 4 z 5 is much more strongly biased than the low redshift galaxy samples shown in Figure 3. Figure 3 can not be straightforwardly interpreted as the redshift evolution of the correlation length, given that the correlation length of the Lyman-α emitters (and similarly of the LBGs) does not necessarily track that of the general population of galaxies. Typically, high redshift systems have been selected using the Lyman-break or Lyman-α techniques, which are sensitive to detect galaxies actively forming stars. Proper comparison of the values of correlation lengths of galaxies at low and high redshifts would require to select the local sample using the same criteria as to detect high redshift sources. For example, Moustakas & Somerville (2002) study three populations of galaxies (local giant ellipticals, extremely red objects and LBGs) observed at the three different redshifts (z ∼ 0, z ∼ 1.2 and z ∼ 3, respectively) with clustering lengths of similar values. The masses of the host dark matter haloes, obtained from the clustering analysis, of populations observed at different epochs were different, implying that these populations of objects do not have the same origin. Therefore the values of the clustering strength measured for the population of galaxies residing at low and high redshifts (possibly corrected for the contaminants) can not be used to make definite conclusions about the evolution of the clustering properties of all galaxies. More understanding of the evolution of galaxies can be gained by comparing the clustering properties of haloes which can host this type of galaxies at a specific epoch.

Comparison with CDM
Using the correlation length and the comoving number density estimated from the observations of Lyman-α emitters at z ≈ 4.5, we can constrain the possible masses of the host dark matter halos of the observed population. We compute the implied 'bias' of the Lyman-α emitters, i.e. how clustered they are relative to the underlying dark matter in our assumed cosmology. Readers should be cautioned that there are different definitions of bias used in the literature, and bias is also a non-trivial function of spatial scale. Quoted numerical bias values depend on these assumptions. We define the bias as the square root of the ratio of the galaxy and dark matter real-space correlation functions: where we have assumed that both the galaxy and dark matter correlation functions ξ are represented by a power-law, with slope γ g = 1.8 for the galaxies and γ DM = 1.2 for the dark matter (as measured in N-body simulations of Jenkins et al. 1998). We compute our bias values at a comoving spatial scale of 3.6 h −1 Mpc, which corresponds to an angular separation of 100 arcsec at z = 4.5, approximately the largest scale where we obtain a robust signal in our measured correlation function, and is the same scale used in several other recent analyzes (e.g. Lee et al. 2006).
In order to predict the clustering properties of an observed galaxy population, we must consider both (a) the expected clustering of the underlying dark matter halos at a given redshift and in a given cosmology, and (b) the halo occupation function, or the number of objects residing within dark halos of a given mass. This function is dependent on the survey redshift and sample selection method. The halo occupation function (or distribution) may be parameterized with varying levels of complexity. Here, we use a very simple formulation, following Wechsler at al. (2001), Bullock et al. (2002), and Moustakas & Somerville (2002). We define N g (M) to be the average number of galaxies found in a halo with mass M, and parameterize this via a three-parameter function: The parameter M min represents the smallest mass of a halo that can host an observed galaxy (N g = 0 for M < M min ). The normalization M 1 is the mass of a halo that will host, on average, one galaxy. The slope α describes the dependence of the number of galaxies per halo on halo mass. Though extremely simple, this functional form has been widely used and has been found to be a reasonably good approximation to the halo occupation function predicted by semi-analytic models and hydrodynamic simulations (e.g. Wechsler et al. 2001;White et al. 2001).
We compute the halo mass function using the analytic expression provided by Sheth & Tormen (1999): Here, the parameters a = 0.707, p = 0.30 and c = 0.163 are chosen to match the halo number density from N-body simulations. The parameter ν is defined by ν ≡ δ c /σ, where δ c ≃ 1.686 is the critical overdensity for the epoch of collapse and σ is the linear rms variance of the power spectrum on the mass scale M at redshift z. Sheth & Tormen (1999) also give the halo bias b h in the form Now, the integral of the halo mass function weighted by the halo occupation function gives the comoving number density of galaxies: Similarly, the integral of the halo bias as a function of mass weighted by the occupation function gives the average bias for galaxies: We first consider the simplest case, in which each dark matter halo above a minimum mass contains a single Lyman-α emitter (i.e., N g = 1 for M > M min ). The comoving number density and bias values for the Lyman-α sample, both uncorrected and corrected for contamination, are shown in Figure 4, along with the relation between number density and average bias for dark matter halos as a function of the minimum mass. The number density and bias values for Lyman-break galaxies (LBGs) at z ∼ 3.8 (B-dropouts) and z ∼ 4.9 (Vdropouts) and for three different observed magnitude limits (z 850 = 26, 26.5, 27.0) from the recent study of Lee et al. (2006) are also shown. We recalculate the bias values from the Lee et al. (2006) sample using our definition of bias (Equation 9); Lee et al. (2006) define the bias using the angular correlation function. From Figure 4 it is apparent that there is a clear trend for Lyman-α emitters to be less common than halos that are as strongly clustered at their observed redshift. This may imply that Lyman-α is detected in only a fraction of the halos that host the objects producing the emission. It is also interesting that the Lyman-α emitters have similar bias values to the LBG samples at similar redshifts, but again have much smaller number densities. This suggests a picture in which the host halos for these two populations may have a similar distribution of masses, but in which Lyman-α emission is seen only a fraction of the time.
We now consider the general halo occupation function given above, and invert the equations for b g and n g to solve for the parameters M min and M 1 . As noted by Bullock et al. (2002), and exploited by several recent studies such as Lee et al. (2006), we can only constrain the value of the halo occupation function slope α if we have information on the clustering of objects on rather small angular scales. Here we do not have this information (we have only one measurement of the correlation function on scales smaller than 10 arcsec), so our solutions are degenerate in this parameter. We give the values of our obtained halo occupation parameters for three values of α in Table 2: α = 0 (one galaxy per halo), α = 0.5, and α = 0.8. We note that Bullock et al. (2002) found a best-fit value of α = 0.8 for LBGs at z ∼ 3, while Lee et al. (2006) found best fit values of α = 0.65 and α = 0.8 for z ∼ 3.8 (B-dropout) and z ∼ 4.9 (V-dropout) LBGs, respectively. We see from Table 2 that the minimum host halo masses range from ∼ 1.6-4×10 10 h −1 M ⊙ using the uncorrected values of number density and bias, and larger values ∼ 1.3-2.5 × 10 11 h −1 M ⊙ for the values obtained when we corrected for possible contamination of our sample by foreground objects. In general, M 1 is much larger than M min , again reflecting that the Lyman-α emitters' number densities are low relative to the halos that cluster strongly enough to host them.

Discussion and Conclusions
We have estimated the correlation properties of Lyman-α emitters from the LALA sample at z ≈ 4.5. From the observed data we measure the amplitude of the angular twopoint correlation function A ω = 6.73 ± 1.80 assuming a fiducial value of the slope of modeled power law β = 0.8. Using the inverse Limber transformation for the given cosmology and the top-hat redshift distribution of the analyzed galaxies in the interval 4.37 < z < 4.57 we calculate the spatial correlation length to be r 0 = 3.20 ± 0.42 h −1 Mpc. After correcting for the possible contamination of the sample by uncorrelated sources (assuming a contaminant fraction of 28% based on spectroscopic surveys), we obtain r 0 = 4.61 ± 0.60 h −1 Mpc. This is the maximum permitted value of the correlation length for our sample.
While large scale structure in the form of voids and filaments is seen in Lyman-α emitters (Campos et al.1999;Møller & Fynbo 2001;Ouchi et al. 2005, Venemans et al. 2002Palunas et al. 2000;Steidel at al. 2000), the measurement of the correlation function is finely balanced between detection (this paper and Ouchi et al. 2003) and non-detection ). Similar to this work, Murayama et al. (2007) measure the weak clustering of Lymanα emitters on small scales (less than 100 arcsec), which can be well fitted by a power law. Ouchi et al. (2004) find correlation at scales of θ > 50 arcsec in a field where they see a well defined clump of Lyman-α emitters. The distribution of Lyman-α emitters from the survey of Palunas et al. (2004), targeted on a known cluster at z = 2.38, show a weak correlation (significant excess of close pairs with separation less than 1 arcmin) and an excess of large voids (size of 6 -8 arcmin). Our detection is at a smaller scale ( θ < 50 arcsec) in a field with no noticeable clumping. The spatial correlation length we derive agrees within the 1σ error with the estimate at z = 4.86 by Ouchi et al. (2003), who measured r 0 = 3.5 ± 0.3 h −1 Mpc. On the other hand, the maximum permitted r 0 value of Lyman-α emitters in our sample is significantly lower than the maximum permitted value estimated by Ouchi et al. (2003) of 6.2± 0.5 h −1 Mpc. The 40% fraction of low-z contaminants in the mentioned work was derived using only the photometric data. Shimasaku et al. (2004) included the data of the spectroscopic follow-up of the enlarged field observed by Ouchi et al. (2003), and derived a lower fraction of contaminants of 20%. Using this value for the contamination by low-z galaxies, the maximum permitted correlation length discussed in Ouchi et al. (2003) would be r 0 = 4.5 ± 0.4 h −1 Mpc, assuming no correlation between the contaminants. This fraction of the low-z contaminants brings our and Ouchi et al. (2003) maximum permitted correlation length back into agreement.
The r 0 values of Lyman-α emitters measured at high redshifts are about 2/3 of the measured spatial correlation length of galaxies in the nearby Universe, or almost equal when comparing the contamination corrected correlation lengths of the discussed Lyman-α. The high values of the correlation length at high redshifts, measured for the specifically selected samples of galaxies, which are as high as the correlation length measured at the low redshift, for more general populations of galaxies, do not imply the absence of the evolution in correlation length.
We compare the measured clustering values with the expected clustering of dark matter and dark matter halos in the CDM paradigm. We find that the Lyman-α emitters are strongly biased, b ≃ 2.5-3.7, relative to the dark matter on scales of 3.6h −1 Mpc at z = 4.5.
These bias values imply that the Lyman-α emitters must reside in halos with minimum masses of 1.6-4 × 10 10 h −1 M ⊙ (uncorrected) or ∼ 1.3-2.5 × 10 11 h −1 M ⊙ using the results after correction for contamination. Interestingly, the observed number density of Lyman-α emitters is a factor of ∼ 2-16 lower than that of dark matter halos that cluster strongly enough to host them. We further notice that the observed bias of Lyman-α emitters is similar to that of Lyman-break galaxies at z ∼ 3.8 and z ∼ 4.9, but again, the number density of the Lyman-α emitters is much lower. This suggests a picture in which the parent population of Lyman-α emitters may occupy dark matter halos with a similar distribution of masses as those that host LBGs, but are detectable in Lyman-α with a finite duty cycle in the range of 6 to 50%. Malhotra & Rhoads (2002) estimated this duty cycle by combining stellar population modelling with the extrapolated luminosity function of LBGs at z = 4 (Pozzetti et al. 1998;Steidel et al. 1999). The Lyman-α emitters were modeled with different stellar population models to match the observed EW distribution. To match the number density of Lymanα emitters, only a small fraction of the inferred number of faint objects from the LBG luminosity function need to be active in Lyman-α emission. This fraction is derived to be 7.5% -15% , depending on the stellar population model, the lower number corresponding to a zero-metallicity stellar population with an IMF slope of α = 2.35 and whose spectra at the age of 10 6 yr is derived by Tumlinson & Shull (2000). This is very consistent with the range of allowed duty cycles inferred from the clustering analysis presented here. However, the field-to-field variance in the number density of Lyman-α emitters is large, and analysis of more fields is needed before we can pin this value down further. Measurement of the correlation of Lyman-α emitters on smaller angular scales would allow us to better constrain the parameters of the halo occupation function, in particular its mass dependence α.      Adelberger (2000) and at z = 4 determined by Ouchi et al. (2004). The last point is for a sample of the selected LBGs with i ′ < 26.0. The filled triangles are r 0 values by Lee et al. (2005) calculated when both β and A w were allowed to vary. The point at z = 3.8 is the r 0 value for B-dropouts and the point at z = 4.9 is the corresponding value for V-dropouts, both with the magnitude limit z 850 ≤ 27 . The low-and intermediate-redshift measurements of r 0 's are represented by empty star (Loveday et al. 1995; data from Stromlo-APM redshift survey), filled square (Zehavi et al. 2002; SDSS galaxies), empty square (Hawkins et al. 2003; 2dFGRS galaxies), hexagons (Carlberg et al. 2000; data from Canadian Network for Observational Cosmology field galaxy redshift survey) and crosses (Brunner, Szalay, & Connoly 2000; data from field located at 14:20, +52:30, covering approximately 0.054 deg 2 , with photometrically measured redshifts). The dashed lines are r 0 values as predicted by the "ǫ-model" at different redshifts: the short-dashed line corresponds to parameter ǫ = 0.8 and long-dashed line corresponds to parameter ǫ = 0. For comparison the solid line shows the redshift evolution of the spatial correlation length of dark matter given by equation A1 in Moustakas & Somerville (2002). Having the bias defined by equation 9 we conclude that high redshift galaxies are biased more strong than the galaxies from nearby samples and samples at intermediate redshifts.