© ESO, 2014

1. Introduction

Our understanding of the formation and evolution of galaxies across cosmic time has been greatly improved in the last decade thanks to deep photometric and spectroscopic surveys. Some examples of these successful deep surveys are SDSS (Sloan Digital Sky Survey, Abazajian et al. 2009), GOODS (Great Observatories Origins Deep Survey, Giavalisco et al. 2004), AEGIS (All-Wavelength Extended Groth Strip International Survey, Davis et al. 2007), ELAIS (European Large-Area ISO Survey, Rowan-Robinson et al. 2004), COSMOS (Cosmological Evolution Survey, Scoville et al. 2007), MGC (Millennium Galaxy Catalogue, Liske et al. 2003), VVDS (VIMOS VLT Deep Survey, Le Fèvre et al. 2005, 2013), DEEP (Deep Extragalactic Evolutionary Probe, Newman et al. 2013), zCOSMOS (Lilly et al. 2009), GNS (GOODS NICMOS Survey, Conselice et al. 2011), SXDS (Subaru/XMM-Newton Deep Survey, Furusawa et al. 2008), or CANDELS (Cosmic Assembly NIR Deep Extragalactic Legacy Survey, Grogin et al. 2011; Koekemoer et al. 2011).

One fundamental uncertainty in any observational measurement derived from galaxy surveys is the cosmic variance (σ_v), arising from the underlying large-scale density fluctuations and leading to variances larger than those expected from simple Poisson statistics. The most efficient way to tackle the cosmic variance is to split the survey in several independent areas in the sky. This minimises the sampling problem and is better than increasing the volume in a wide contiguous field (e.g., Driver & Robotham 2010). However, observational constraints (depth vs. area) lead to many existing surveys that have observational uncertainties dominated by the cosmic variance. Thus, a proper estimation of σ_v is needed to fully describe the error budget in deep cosmological surveys.

The impact of the cosmic variance in a given survey and redshift range can be estimated using two basic methods: theoretically, by analysing cosmological simulations (e.g., Somerville et al. 2004; Trenti & Stiavelli 2008; Stringer et al. 2009; Moster et al. 2011), or empirically, by sampling a larger survey (e.g., Driver & Robotham 2010). Unfortunately, previous studies only estimate the cosmic variance affecting number density measurements and do not tackle the impact of σ_v in other important quantities as the merger fraction. Merger fraction studies based on close pair statistics measure the correlation of two galaxy populations at small scales (≤100 h^-1 kpc), so the amplitude of the cosmic variance and its dependence on galaxy properties, probed volume, etc. should be different than those in number density studies. In the present paper, we take advantage of the unique design, depth, and photometric redshift accuracy of the ALHAMBRA¹ (Advanced, Large, Homogeneous Area, Medium-Band Redshift Astronomical) survey (Moles et al. 2008) to estimate empirically, for the first time, the cosmic variance that affects close pair studies. The ALHAMBRA survey has observed eight separate regions of the northern sky, comprising 48 sub-fields of ~180 arcmin² each that can be assumed as independent for our purposes. Thus, ALHAMBRA provides 48 measurements of the merger fraction across the sky. The intrinsic dispersion in the distribution of these merger fractions, which we characterise in the present paper, is an observational estimation of the cosmic variance σ_v.

The paper is organised as follows. In Sect. 2, we present the ALHAMBRA survey and its photometric redshifts, and in Sect. 3, we review the methodology to measure close pair merger fractions when photometric redshifts are used. We present our estimation and characterisation of the cosmic variance for close pair studies in Sect. 4. In Sect. 5, we summarise our work and present our conclusions. Throughout this paper, we use a standard cosmology with Ω_m = 0.3, Ω_Λ = 0.7, H₀ = 100h km s^-1 Mpc^-1, and h = 0.7. Magnitudes are given in the AB system.

2. The ALHAMBRA survey

The ALHAMBRA survey provides a photometric data set over 20 contiguous, equal-width (~300 Å), non-overlapping, medium-band optical filters (3500 Å−9700 Å) plus 3 standard broad-band near-infrared (NIR) filters (J, H, and K_s) over 8 different regions of the northern sky (Moles et al. 2008). The survey has the aim of understanding the evolution of galaxies throughout cosmic time by sampling a large enough cosmological fraction of the universe, for which reliable spectral energy distributions (SEDs) and precise photometric redshifts (z_p) are needed. The simulations of Benítez et al. (2009), which relates the image depth and the accuracy of the photometric redshifts to the number of filters, have demonstrated that the filter set chosen for ALHAMBRA can achieve a photometric redshift precision that is three times better than a classical 4 − 5 optical broad-band filter set. The final survey parameters and scientific goals, as well as the technical properties of the filter set, were described by Moles et al. (2008). The survey has collected its data for the 20+3 optical-NIR filters in the 3.5 m telescope at the Calar Alto observatory, using the wide-field camera LAICA (Large Area Imager for Calar Alto) in the optical and the OMEGA2000 camera in the NIR. The full characterisation, description, and performance of the ALHAMBRA optical photometric system were presented in Aparicio-Villegas et al. (2010). A summary of the optical reduction can be found in Cristóbal-Hornillos et al. (in prep.), while that of the NIR reduction is in Cristóbal-Hornillos et al. (2009).

The ALHAMBRA survey has observed eight well-separated regions of the northern sky. The wide-field camera LAICA has four chips with a 15′ × 15′ field-of-view each (0.22 arcsec/pixel). The separation between chips is also 15′. Thus, each LAICA pointing provides four separated areas in the sky (black or red squares in Fig. 1). Six ALHAMBRA regions comprise two LAICA pointings. In these cases, the pointings define two separate strips in the sky (Fig. 1). In our study, we assumed the four chips in each strip as independent sub-fields. The photometric calibration of the field ALHAMBRA-1 is currently on-ongoing, and the fields, ALHAMBRA-4 and ALHAMBRA-5, comprise of one pointing each (see Molino et al. 2013, for details). We summarise the properties of the seven ALHAMBRA fields used in the present paper in Table 1. At the end, ALHAMBRA comprises 48 sub-fields of ~180 arcmin², which we assumed to be independent, in which we measured the merger fraction following the methodology described in Sect. 3. When we searched for close companions near the sub-field boundaries, we did not consider the observed sources in the adjacent fields to keep the measurements independent. We prove the independence of the 48 ALHAMBRA sub-fields in Sect 4.6.

2.1. Bayesian photometric redshifts in ALHAMBRA

We rely on the ALHAMBRA photometric redshifts to compute the merger fraction (Sect. 3). The photometric redshifts used all over the present paper are fully presented and tested in Molino et al. (2013), and we summarise their principal characteristics below.

The photometric redshifts of ALHAMBRA were estimated with BPZ2.0, a new version of BPZ (Benítez 2000). The BPZ is a SED-fitting method based in a Bayesian inference, where a maximum likelihood is weighted by a prior probability. The library of 11 SEDs (4 ellipticals, 1 lenticular, 2 spirals, and 4 starbursts) and the prior probabilities used by BPZ2.0 in ALHAMBRA are detailed in Benítez (in prep.). The ALHAMBRA photometry used to compute the photometric redshifts is PSF-matched aperture-corrected and based on isophotal magnitudes. In addition, a recalibration of the zero point of the images was performed to enhance the accuracy of the photometric redshifts. Sources were detected in a synthetic F814W filter image, as noted i in the following, defined to resemble the HST/F814W filter. The areas of the images affected by bright stars, as well as those with lower exposure times (e.g., the edges of the images), were masked following Arnalte-Mur et al. (2013). The total area covered by the ALHAMBRA survey after masking is 2.38 deg². Finally, a statistical star/galaxy separation is encoded in the variable Stellar_Flag of the ALHAMBRA catalogues, and throughout this paper, we keep those ALHAMBRA sources with Stellar˙Flag≤ 0.5 as galaxies.

The photometric redshift accuracy, as estimated by comparison with spectroscopic redshifts (z_s), is δ_z = 0.0108 at i ≤ 22.5 with a fraction of catastrophic outliers of η = 2.1%. The variable δ_z is the normalized median absolute deviation of the photometric versus spectroscopic redshift distribution (Ilbert et al. 2006; Brammer et al. 2008), ${\mathit{\delta }}_{\mathit{z}}\mathrm{=}\mathrm{1.48}\mathrm{\times }\mathrm{median}\hspace{0.17em}\left(\frac{\mathrm{|}{\mathit{z}}_{\mathrm{p}}\mathrm{-}{\mathit{z}}_{\mathrm{s}}\mathrm{|}}{\mathrm{1}\mathrm{+}{\mathit{z}}_{\mathrm{s}}}\right)\mathrm{·}$(1)The variable η is defined as the fraction of galaxies with | z_p − z_s | / (1 + z_s) > 0.2. We illustrate the high quality of the ALHAMBRA photometric redshifts in Fig. 2. We refer to Molino et al. (2013) for a more detailed discussion.

The odds quality parameter, as noted , is a proxy for the photometric redshift accuracy of the sources and is also provided by BPZ2.0. The odds is defined as the redshift probability enclosed on a ± K(1 + z) region around the main peak in the probability distribution function (PDF) of the source, where the constant K is specific for each photometric survey. Molino et al. (2013) find that K = 0.0125 is the optimal value for the ALHAMBRA survey. The parameter is related with the confidence of the z_p, making it possible to derive high quality samples with better accuracy and lower rate of catastrophic outliers. For example, a selection for i ≤ 22.5 galaxies yields δ_z = 0.0094 and η = 1%, while δ_z = 0.0061 and η = 0.8% for (see Molino et al. 2013, for further details). We explore the optimal odds selection in ALHAMBRA for close pair studies in Sect. 4.3.

Reliable photometric redshift errors (σ_zp) are needed to compute the merger fraction in photometric samples (Sect. 3). In addition to the z_p, we have the ${\mathit{z}}_{\mathit{\sigma }}^{\mathrm{+}}$ and the ${\mathit{z}}_{\mathit{\sigma }}^{\mathrm{-}}$ of each source, which are defined as the redshifts that enclose 68% of the PDF of the source. We estimated the photometric redshift error of each individual source as ${\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}}}\mathrm{=}\mathit{C}\mathrm{\times }\mathrm{(}{\mathit{z}}_{\mathit{\sigma }}^{\mathrm{+}}\mathrm{-}{\mathit{z}}_{\mathit{\sigma }}^{\mathrm{-}}\mathrm{)}$. The constant C is estimated from the distribution of the variable ${\mathrm{\Delta }}_{\mathit{z}}\mathrm{=}\frac{{\mathit{z}}_{\mathrm{p}}\mathrm{-}{\mathit{z}}_{\mathrm{s}}}{{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}}}}\mathrm{=}\frac{{\mathit{z}}_{\mathrm{p}}\mathrm{-}{\mathit{z}}_{\mathrm{s}}}{\mathit{C}\mathrm{\times }\mathrm{(}{\mathit{z}}_{\mathit{\sigma }}^{\mathrm{+}}\mathrm{-}{\mathit{z}}_{\mathit{\sigma }}^{\mathrm{-}}\mathrm{)}}\mathrm{·}$(2)The variable Δ_z should be normally distributed with a zero mean and unit variance if the values of σ_zp from ALHAMBRA are a good descriptor for the accuracy of the photometric redshifts (e.g., Ilbert et al. 2009; Carrasco Kind & Brunner 2013). We find that Δ_z is described well by a normal function when C = 0.49 (Fig. 3, see also Molino et al. 2013). With the definition of ${\mathit{z}}_{\mathit{\sigma }}^{\mathrm{+}}$ and ${\mathit{z}}_{\mathit{\sigma }}^{\mathrm{-}}$, note that C = 0.5 was expected. This result also implies that the Gaussian approximation of the PDF assumed in the estimation of the merger fraction (Sect. 3) is statistically valid, even if the actual PDF of the individual sources could be multimodal and/or asymmetric at faint magnitudes. We estimated C for different i-band magnitudes and odds selections, finding that the C values are consistent with the global one within ± 0.1. Thus, we conclude that σ_zp provides a reliable photometric redshift error for every ALHAMBRA source.

2.2. Sample selection

Throughout this paper, we focus our analysis on the galaxies in the ALHAMBRA first data release². This catalogue comprises ~500k sources and is complete (5σ, 3″ aperture) for i ≤ 24.5 galaxies (Molino et al. 2013). We explored different apparent luminosity sub-samples from i ≤ 23 to i ≤ 20. This ensures excellent photometric redshifts and provides reliable merger fraction measurements (Sect. 4.3) because the PDFs of i ≤ 23 sources are defined well by a single Gaussian peak (Molino et al. 2013). In Sect. 4.7, we also study the cosmic variance in luminosity- and stellar mass-selected samples. The B-band luminosities and the stellar masses of the ALHAMBRA sources were also provided by BPZ2.0 and are included in the ALHAMBRA catalogue (see Molino et al. 2013, for further details). The mass-to-light ratios from Taylor et al. (2011) and a Chabrier (2003) initial mass function were assumed in the estimation of the stellar masses.

3. Measuring of the merger fraction in photometric samples

The linear distance between two sources can be obtained from their projected separation, r_p = φd_A(z₁), and their rest-frame relative velocity along the line of sight, Δv = c | z₂ − z₁ | / (1 + z₁), where z₁ and z₂ are the redshift of the principal (more luminous or massive galaxy in the pair) and the companion galaxy, respectively; φ is the angular separation in arcsec of the two galaxies on the sky plane; and d_A(z) is the angular diameter distance in kpc arcsec^-1 at redshift z. Two galaxies are defined as a close pair if ${\mathit{r}}_{\mathrm{p}}^{\min }\mathrm{\le }{\mathit{r}}_{\mathrm{p}}\mathrm{\le }{\mathit{r}}_{\mathrm{p}}^{\max }$ and Δv ≤ Δv^max. The PSF of the ALHAMBRA ground-based images is ≲1.4″ (median seeing of ~1″), which corresponds to 7.6 h^-1 kpc in our cosmology at z = 0.9. To ensure well de-blended sources and to minimise colour contamination, we fixed ${\mathit{r}}_{\mathrm{p}}^{\min }$ to 10 h^-1 kpc (φ> 1.8″ at z< 0.9). We left ${\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\le }\mathrm{50}{\mathit{h}}^{-1}$ kpc as a free parameter and estimate its optimal value in Sect. 4.3. Finally, we set Δv^max = 500 km s^-1 by following spectroscopic studies (e.g., Patton et al. 2000; Lin et al. 2008). With the previous constraints, 50%−70% of the selected close pairs finally merge (Patton & Atfield 2008; Bell et al. 2006; Jian et al. 2012).

To compute close pairs, we defined a principal and a companion sample. The principal sample comprises the more luminous or massive galaxy of the pair, and we looked for those galaxies in the companion sample that fulfil the close pair criterion for each galaxy of the principal sample. If one principal galaxy has more than one close companion, we took each possible pair separately (i.e., if the companion galaxies B and C are close to the principal galaxy A, we study the pairs A–B and A–C as independent). In addition, through this paper, we do not impose any luminosity or mass difference between the galaxies in the close pair unless noted otherwise.

With the previous definitions, the merger fraction is ${\mathit{f}}_{\mathrm{m}}\mathrm{=}\frac{{\mathit{N}}_{\mathrm{p}}}{{\mathit{N}}_{\mathrm{1}}}\mathit{,}$(3)where N₁ is the number of sources in the principal sample and N_p the number of close pairs. This definition applies to spectroscopic volume-limited samples, but we rely on photometric redshifts to compute f_m in ALHAMBRA. In previous work, López-Sanjuan et al. (2010a) develop a statistical method to obtain reliable merger fractions from photometric redshift catalogues like those from the ALHAMBRA survey. This methodology has been tested with the MGC (López-Sanjuan et al. 2010a) and the VVDS (López-Sanjuan et al. 2012) spectroscopic surveys, and has been successfully applied in the GOODS-South (López-Sanjuan et al. 2010a) and the COSMOS fields (López-Sanjuan et al. 2012). We recall the main points of this methodology below, and we explore how to apply it optimally over the ALHAMBRA data in Sect. 4.3.

We used the following procedure to define a close pair system in our photometric catalogue (see López-Sanjuan et al. 2010a, for details): first, we search for close spatial companions of a principal galaxy with redshift z₁ and uncertainty σ_z1, assuming that the galaxy is located at z₁ − 2σ_z1. This defines the maximum φ possible for a given ${\mathit{r}}_{\mathrm{p}}^{\max }$ in the first instance. If we find a companion galaxy with redshift z₂ and uncertainty σ_z2 at ${\mathit{r}}_{\mathrm{p}}\mathrm{\le }{\mathit{r}}_{\mathrm{p}}^{\max }$, we study both galaxies in redshift space. For convenience, we assume below that every principal galaxy has, at most, one close companion. In this case, our two galaxies could be a close pair in the redshift range $\mathrm{\left[}{\mathit{z}}^{\mathrm{-}}\mathit{,}{\mathit{z}}^{\mathrm{+}}\mathrm{\right]}\mathrm{=}\mathrm{[}{\mathit{z}}_{\mathrm{1}}\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{1}}}\mathit{,}{\mathit{z}}_{\mathrm{1}}\mathrm{+}\mathrm{2}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{1}}}\mathrm{]}\mathrm{\cap }\mathrm{[}{\mathit{z}}_{\mathrm{2}}\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{2}}}\mathit{,}{\mathit{z}}_{\mathrm{2}}\mathrm{+}\mathrm{2}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{2}}}\mathrm{]}\mathit{.}$(4)Because of the variation in the range [ z⁻,z⁺ ] of the function d_A(z), a sky pair at z₁ − 2σ_z1 might not be a pair at z₁ + 2σ_z1. We thus impose the condition ${\mathit{r}}_{\mathrm{p}}^{\min }\mathrm{\le }{\mathit{r}}_{\mathrm{p}}\mathrm{\le }{\mathit{r}}_{\mathrm{p}}^{\max }$ at all z ∈ [ z⁻,z⁺ ] and redefine this redshift interval if the sky pair condition is not satisfied at every redshift. After this, our two galaxies define the close pair system k in the redshift interval $\mathrm{\left[}{\mathit{z}}_{\mathit{k}}^{\mathrm{-}}\mathit{,}{\mathit{z}}_{\mathit{k}}^{\mathrm{+}}\mathrm{\right]}$, where the index k covers all the close pair systems in the sample.

The next step is to define the number of pairs associated to each close pair system k. For this, and because all our sources have a photometric redshift, we suppose in the following that a galaxy i in whatever sample is described in redshift space by a Gaussian probability distribution, ${\mathit{P}}_{\mathit{i}}\hspace{0.17em}\mathrm{(}{\mathit{z}}_{\mathit{i}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}{\mathit{z}}_{\mathrm{p}\mathit{,i}}\mathit{,}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}\mathit{,i}}}\mathrm{)}\mathrm{=}\frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathit{\pi }}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}\mathit{,i}}}}\exp \left[\mathrm{-}\frac{\mathrm{(}{\mathit{z}}_{\mathit{i}}\mathrm{-}{\mathit{z}}_{\mathrm{p}\mathit{,i}}{\mathrm{)}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}\mathit{,i}}}^{\mathrm{2}}}\right]\mathrm{·}$(5)With the previous distribution, we are able to statistically treat all the available information in redshift space and define the number of pairs at redshift z₁ in system k as ${\mathit{\nu }}_{\mathit{k}}\hspace{0.17em}\mathrm{\left(}{\mathit{z}}_{\mathrm{1}}\mathrm{\right)}\mathrm{=}{\mathrm{C}}_{\mathit{k}}\hspace{0.17em}{\mathit{P}}_{\mathrm{1}}\mathrm{(}{\mathit{z}}_{\mathrm{1}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}{\mathit{z}}_{\mathrm{p}\mathit{,}\mathrm{1}}\mathit{,}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}\mathit{,}\mathrm{1}}}\mathrm{\left)}{\mathrm{\int }}_{{\mathit{z}}_{\mathrm{m}}^{\mathrm{-}}}^{{\mathit{z}}_{\mathrm{m}}^{\mathrm{+}}}{\mathit{P}}_{\mathrm{2}}\mathrm{\right(}{\mathit{z}}_{\mathrm{2}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}{\mathit{z}}_{\mathrm{p}\mathit{,}\mathrm{2}}\mathit{,}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}\mathit{,}\mathrm{2}}}\mathrm{)}\hspace{0.17em}\mathrm{d}{\mathit{z}}_{\mathrm{2}}\mathit{,}$(6)where ${\mathit{z}}_{\mathrm{1}}\mathrm{\in }\mathrm{\left[}{\mathit{z}}_{\mathit{k}}^{\mathrm{-}}\mathit{,}{\mathit{z}}_{\mathit{k}}^{\mathrm{+}}\mathrm{\right]}$, the integration limits are $\begin{array}{ccc}{\mathit{z}}_{\mathrm{m}}^{\mathrm{-}}\mathrm{=}{\mathit{z}}_{\mathrm{1}}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\Delta }{\mathit{v}}^{\max }\mathit{/}\mathit{c}\mathrm{)}\mathrm{-}\mathrm{\Delta }{\mathit{v}}^{\max }\mathit{/}\mathit{c,}& & \\ {\mathit{z}}_{\mathrm{m}}^{\mathrm{+}}\mathrm{=}{\mathit{z}}_{\mathrm{1}}\mathrm{(}\mathrm{1}\mathrm{+}\mathrm{\Delta }{\mathit{v}}^{\max }\mathit{/}\mathit{c}\mathrm{)}\mathrm{+}\mathrm{\Delta }{\mathit{v}}^{\max }\mathit{/}\mathit{c,}& & \end{array}$the subindex 1 [2] refers to the principal [companion] galaxy in the system k, and the constant C_k normalises the function to the total number of pairs in the interest range, $\mathrm{2}{\mathit{N}}_{\mathrm{p}}^{\mathit{k}}\mathrm{=}{\mathrm{\int }}_{{\mathit{z}}_{\mathit{k}}^{\mathrm{-}}}^{{\mathit{z}}_{\mathit{k}}^{\mathrm{+}}}{\mathit{P}}_{\mathrm{1}}\mathrm{(}{\mathit{z}}_{\mathrm{1}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}{\mathit{z}}_{\mathrm{p}\mathit{,}\mathrm{1}}\mathit{,}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}\mathit{,}\mathrm{1}}}\mathrm{)}\hspace{0.17em}\mathrm{d}{\mathit{z}}_{\mathrm{1}}\mathrm{+}{\mathrm{\int }}_{{\mathit{z}}_{\mathit{k}}^{\mathrm{-}}}^{{\mathit{z}}_{\mathit{k}}^{\mathrm{+}}}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{\mathit{z}}_{\mathrm{2}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}{\mathit{z}}_{\mathrm{p}\mathit{,}\mathrm{2}}\mathit{,}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}\mathit{,}\mathrm{2}}}\mathrm{)}\hspace{0.17em}\mathrm{d}{\mathit{z}}_{\mathrm{2}}\mathit{.}$(9)Note that ν_k = 0 if ${\mathit{z}}_{\mathrm{1}}\mathit{<}{\mathit{z}}_{\mathit{k}}^{\mathrm{-}}$ or ${\mathit{z}}_{\mathrm{1}}\mathit{>}{\mathit{z}}_{\mathit{k}}^{\mathrm{+}}$. The function ν_k tells us how the number of pairs in the system k, as noted by ${\mathit{N}}_{\mathrm{p}}^{\mathit{k}}$, are distributed in redshift space. The integral in Eq. (6) spans those redshifts in which the companion galaxy has Δv ≤ Δv^max for a given redshift of the principal galaxy. This translates to ${\mathit{z}}_{\mathrm{m}}^{\mathrm{+}}\mathrm{-}{\mathit{z}}_{\mathrm{m}}^{\mathrm{-}}\mathrm{~}\mathrm{0.005}$ in our redshift range of interest.

With the previous definitions, the merger fraction in the interval z_r = [ z_min,z_max) is ${\mathit{f}}_{\mathrm{m}}\mathrm{=}\frac{{\sum }_{\mathit{k}}{\mathrm{\int }}_{{\mathit{z}}_{\min }}^{{\mathit{z}}_{\max }}{\mathit{\nu }}_{\mathit{k}}\hspace{0.17em}\mathrm{\left(}{\mathit{z}}_{\mathrm{1}}\mathrm{\right)}\hspace{0.17em}\mathrm{d}{\mathit{z}}_{\mathrm{1}}}{{\sum }_{\mathit{i}}{\mathrm{\int }}_{{\mathit{z}}_{\min }}^{{\mathit{z}}_{\max }}{\mathit{P}}_{\mathit{i}}\hspace{0.17em}\mathrm{(}{\mathit{z}}_{\mathit{i}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}{\mathit{z}}_{\mathrm{p}\mathit{,i}}\mathit{,}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}\mathit{,i}}}\mathrm{)}\hspace{0.17em}\mathrm{d}{\mathit{z}}_{\mathit{i}}}\mathrm{·}$(10)If we integrate over the whole redshift space, z_r = [ 0,∞), Eq. (10) becomes ${\mathit{f}}_{\mathrm{m}}\mathrm{=}\frac{{\sum }_{\mathit{k}}{\mathit{N}}_{\mathrm{p}}^{\mathit{k}}}{{\mathit{N}}_{\mathrm{1}}}\mathit{,}$(11)where ${\sum }_{\mathit{k}}{\mathit{N}}_{\mathrm{p}}^{\mathit{k}}$ is analogous to N_p in Eq. (3). To estimate the observational error of f_m, as noted by σ_f, we used the jackknife technique (Efron 1982). We computed partial standard deviations, δ_k, for each system k by taking the difference between the measured f_m and the same quantity with the kth pair removed for the sample, ${\mathit{f}}_{\mathrm{m}}^{\mathit{k}}$, such that ${\mathit{\delta }}_{\mathit{k}}\mathrm{=}{\mathit{f}}_{\mathrm{m}}\mathrm{-}{\mathit{f}}_{\mathrm{m}}^{\mathit{k}}$. For a redshift range with N_p systems, the variance is given by ${\mathit{\sigma }}_{\mathit{f}}^{\mathrm{2}}\mathrm{=}\mathrm{\left[}\mathrm{\right(}{\mathit{N}}_{\mathrm{p}}\mathrm{-}\mathrm{1}\mathrm{\left)}{\sum }_{\mathit{k}}{\mathit{\delta }}_{\mathit{k}}^{\mathrm{2}}\mathrm{\right]}\mathit{/}{\mathit{N}}_{\mathrm{p}}$.

3.1. Border effects in redshift and in the sky plane

When we search for a primary source’s companion, we define a volume in the sky plane-redshift space. If the primary source is near the boundaries of the survey, a fraction of the search volume lies outside of the effective volume of the survey. López-Sanjuan et al. (2010a) find that border effects in the sky plane are representative (i.e., 1σ discrepancy) only at ${\mathit{r}}_{\mathrm{p}}^{\max }\gtrsim \mathrm{70}{\mathit{h}}^{-1}$ kpc. Thus, we restricted the search radius in our study to ${\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\le }\mathrm{50}{\mathit{h}}^{-1}$ kpc.

We avoid the incompleteness in redshift space by including the sources in the samples inside the redshift range [ z_min,z_max) under study and those sources with either z_p,i + 2σ_zp,i ≥ z_min or z_p,i − 2σ_zp,i<z_max.

3.2. The merger rate

The final goal of merger studies is the estimation of the merger rate R_m, defined as the number of mergers per galaxy and Gyr^-1. The merger rate is computed from the merger fraction by close pairs as ${\mathit{R}}_{\mathrm{m}}\mathrm{=}\frac{{\mathit{C}}_{\mathrm{m}}}{{\mathit{T}}_{\mathrm{m}}}\hspace{0.17em}{\mathit{f}}_{\mathrm{m}}\mathit{,}$(12)where C_m is the fraction of the observed close pairs that finally merge after a merger time scale T_m. The merger time scale and the merger probability C_m should be estimated from simulations (e.g., Kitzbichler & White 2008; Lotz et al. 2010a,b; Lin et al. 2010; Jian et al. 2012; Moreno et al. 2013). On the one hand, T_m depends mainly on the search radius ${\mathit{r}}_{\mathrm{p}}^{\max }$, the stellar mass of the principal galaxy, and the mass ratio between the galaxies in the pair with a mild dependence on redshift and environment (Jian et al. 2012). On the other hand, C_m depends mainly on ${\mathit{r}}_{\mathrm{p}}^{\max }$ and environment with a mild dependence on both redshift and the mass ratio between the galaxies in the pair (Jian et al. 2012). Despite the efforts in the literature to estimate both T_m and C_m, different cosmological and galaxy formation models provide different values within a factor of two-three (e.g., Hopkins et al. 2010). To avoid model-dependent results in the present paper, we focus in the cosmic variance of the observational merger fraction f_m.

4. Estimation of the cosmic variance for merger fraction studies

4.1. Theoretical background

In this section, we recall the theoretical background and define the basic variables involved in the cosmic variance definition and characterisation. The relative cosmic variance (σ_v) arises from the underlying large-scale density fluctuations and leads to variances larger than those expected from simple Poisson statistics. Following Somerville et al. (2004) and Moster et al. (2011), the mean ⟨ N ⟩ and the variance ⟨ N² ⟩ − ⟨ N ⟩ ² in the distribution of galaxies are given by the first and second moments of the probability distribution P_N(V_c), which describes the probability of counting N objects within a volume V_c. The relative cosmic variance is defined as ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{=}\frac{\mathrm{⟨}{\mathit{N}}^{\mathrm{2}}\mathrm{⟩}\mathrm{-}\mathrm{⟨}\mathit{N}{\mathrm{⟩}}^{\mathrm{2}}}{\mathrm{⟨}\mathit{N}{\mathrm{⟩}}^{\mathrm{2}}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{⟨}\mathit{N}\mathrm{⟩}}\mathrm{·}$(13)The second term represents the correction for the Poisson shot noise. The second moment of the object counts is $\mathrm{⟨}{\mathit{N}}^{\mathrm{2}}\mathrm{⟩}\mathrm{=}\mathrm{⟨}\mathit{N}{\mathrm{⟩}}^{\mathrm{2}}\mathrm{+}\mathrm{⟨}\mathit{N}\mathrm{⟩}\mathrm{+}\frac{\mathrm{⟨}\mathit{N}{\mathrm{⟩}}^{\mathrm{2}}}{{\mathit{V}}_{\mathrm{c}}^{\mathrm{2}}}{\mathrm{\int }}_{{\mathit{V}}_{\mathrm{c}}}\mathit{\xi }\mathrm{\left(}\mathrm{\right|}{\mathit{r}}_{\mathrm{a}}\mathrm{-}{\mathit{r}}_{\mathrm{b}}\mathrm{\left|}\mathrm{\right)}\hspace{0.17em}\mathrm{d}{\mathit{V}}_{\mathrm{c}\mathit{,}\mathrm{a}}\hspace{0.17em}\mathrm{d}{\mathit{V}}_{\mathrm{c}\mathit{,}\mathrm{b}}\mathit{,}$(14)where ξ is the two-point correlation function of the sample under study (Peebles 1980). Combining this with Eq. (13), the relative cosmic variance can be written as ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{=}\frac{\mathrm{1}}{{\mathit{V}}_{\mathrm{c}}^{\mathrm{2}}}\hspace{0.17em}{\mathrm{\int }}_{{\mathit{V}}_{\mathrm{c}}}\mathit{\xi }\mathrm{\left(}\mathrm{\right|}{{r}}_{\mathrm{a}}\mathrm{-}{{r}}_{\mathrm{b}}\mathrm{\left|}\mathrm{\right)}\hspace{0.17em}\mathrm{d}{\mathit{V}}_{\mathrm{c}\mathit{,}\mathrm{a}}\hspace{0.17em}\mathrm{d}{\mathit{V}}_{\mathrm{c}\mathit{,}\mathrm{b}}\mathit{.}$(15)Thus, the cosmic variance of a given sample depends on the correlation function of that population. We can approximate the galaxy correlation function in Eq. (15) by the linear theory correlation function for dark matter ξ_dm, ξ = b²ξ_dm, where b is the galaxy bias. The bias at a fixed scale depends mainly on both redshift and the selection of the sample under study. With this definition of the correlation function, we find that ${\mathit{\sigma }}_{\mathit{v}}\mathrm{\propto }\frac{\mathit{b}}{{\mathit{V}}_{\mathrm{c}}^{\mathrm{1}\hspace{0.17em}\mathrm{-}\hspace{0.17em}\mathit{\alpha }}}\mathit{,}$(16)where the power law index α takes into account the extra volume dependence from the integral of the correlation function ξ_dm in Eq. (15).

The bias of a particular population is usually measured from the analysis of the correlation function and is well established in that the bias increases with luminosity and stellar mass (see Zehavi et al. 2011; Coupon et al. 2012; Marulli et al. 2013; Arnalte-Mur et al. 2013, and references therein). The estimation of the bias is a laborious task, so we decided to use the redshift and the number density n of the population under study instead of the bias to characterise the cosmic variance. The number density is an observational quantity that decreases with the increase of the luminosity and the mass selection, so a b ∝ n^{− β} relation is expected. This inverse dependence is indeed suggested by Nuza et al. (2013) results.

In summary, we expect ${\mathit{\sigma }}_{\mathit{v}}\mathrm{\propto }\frac{\mathit{b}}{{\mathit{V}}_{\mathrm{c}}^{\mathrm{1}\hspace{0.17em}\mathrm{-}\hspace{0.17em}\mathit{\alpha }}}\mathrm{\propto }\frac{{\mathit{z}}^{\mathit{\gamma }}}{{\mathit{n}}^{\mathit{\beta }}\hspace{0.17em}{\mathit{V}}_{\mathrm{c}}^{\mathrm{1}\hspace{0.17em}\mathrm{-}\hspace{0.17em}\mathit{\alpha }}}\mathrm{·}$(17)This equation shows that the number density of galaxies, the redshift, and the cosmic volume can be assumed as independent variables in the cosmic variance parametrisation. Equation (17) and the deduction above apply to the cosmic variance in the number of galaxies. We are interested on the cosmic variance of the merger fraction by close pairs, instead, so a dependence on V_c, redshift, and the number density of the two populations under study, as noted by n₁ for principal galaxies and by n₂ for the companion galaxies, is expected. We therefore used these four variables (n₁, n₂, z, and V_c) to characterise the cosmic variance in close pair studies (Sect. 4.4).

The power-law indices in Eq. (17) could be different for luminosity- and mass-selected samples, as well as for flux-limited samples. In the present paper, we use flux-limited samples selected in the i band to characterise the cosmic variance. This choice has several benefits, since we have a well-controlled selection function, a better understanding of the photometric redshifts and their errors, and we have access to larger samples at lower redshift that in the luminosity and the stellar mass cases. That improves the statistics and increases the useful redshift range. At the end, future studies will be interested on the cosmic variance in physically selected samples (i.e., luminosity or stellar mass). Thus, we compare the results from the flux-limited i-band samples with the actual cosmic variance measured in physically selected samples in Sect. 4.7.

Finally, we set the definition of the number density n. In the present paper, the number density of a given population is the cosmic average number density of that population. For example, if we are studying the merger fraction in a volume dominated by a cluster, we should not use the number density in that volume, but the number density derived from a general luminosity or mass function work instead. Thanks to the 48 sub-fields in ALHAMBRA we have direct access to the average number densities of the populations under study (Sect. 4.4.1).

4.2. Distribution of the merger fraction and σ_v estimation

In this section, we explore which statistical distribution reproduces the observed merger fractions better and how to reliably measure the cosmic variance σ_v. As representative examples, we show the distributions of the merger fraction f_m in the 48 ALHAMBRA sub-fields for i ≤ 22 and i ≤ 21 galaxies in Fig. 4. The merger fraction was measured from close pairs with 10 h^-1 kpc ≤ r_p ≤ 30 h^-1 kpc. Unless noted otherwise, the principal and the companion samples in the following comprise the same galaxies. We find that the observed distributions are not Gaussian but follow a log-normal distribution instead, ${\mathit{P}}_{\mathit{LN}}\hspace{0.17em}\mathrm{(}{\mathit{f}}_{\mathrm{m}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}\mathit{\mu ,\sigma }\mathrm{)}\mathrm{=}\frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathit{\pi }}\hspace{0.17em}\mathit{\sigma }{\mathit{f}}_{\mathrm{m}}}\hspace{0.17em}\exp \hspace{0.17em}\left[\mathrm{-}\frac{\mathrm{(}\ln {\mathit{f}}_{\mathrm{m}}\mathrm{-}\mathit{\mu }{\mathrm{)}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma }}^{\mathrm{2}}}\right]\mathit{,}$(18)where μ and σ are the median and the dispersion of a Gaussian function in log-space ${\mathit{f}}_{\mathrm{m}}^{\mathrm{\prime }}\mathrm{=}\ln {\mathit{f}}_{\mathrm{m}}$. This is, ${\mathit{P}}_{\mathrm{G}}\hspace{0.17em}\mathrm{(}{\mathit{f}}_{\mathrm{m}}^{\mathrm{\prime }}\hspace{0.17em}\mathrm{|}\hspace{0.17em}\mathit{\mu ,\sigma }\mathrm{)}\mathrm{=}\frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathit{\pi }}\hspace{0.17em}\mathit{\sigma }}\hspace{0.17em}\exp \hspace{0.17em}\left[\mathrm{-}\frac{\mathrm{(}{\mathit{f}}_{\mathrm{m}}^{\mathrm{\prime }}\mathrm{-}\mathit{\mu }{\mathrm{)}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma }}^{\mathrm{2}}}\right]\mathrm{·}$(19)The 68% confidence interval of the log-normal distribution is [ e^μe^{− σ},e^μe^σ ]. This functional distribution was expected for two reasons. First, the merger fraction cannot be negative, implying an asymmetric distribution (Cameron 2011). Second, the distribution of overdense structures in the universe is log-normal (e.g., Coles & Jones 1991; de la Torre et al. 2010; Kovač et al. 2010) and the merger fraction increases with density (Lin et al. 2010; de Ravel et al. 2011; Kampczyk et al. 2013). We checked that the merger fraction follows a log-normal distribution in all the samples explored in the present paper.

The variable σ encodes the relevant information about the dispersion in the merger fraction distribution, including the dispersion due to the cosmic variance. The study of the median value of the merger fraction in ALHAMBRA, as estimated as e^μ, and its dependence on z, stellar mass, or colour, is beyond the scope of the present paper, and we will address this issue in a future work.

A best least-squares fit with a log-normal function to the distributions in Fig. 4 shows that σ increases with the apparent brightness from σ = 0.33 for i ≤ 22 galaxies to σ = 0.62 for i ≤ 21 galaxies. However, the origin of the observed σ is twofold: (i) the intrinsic dispersion due to the cosmic variance σ_v (i.e., the field-to-field variation in the merger fraction because of the clustering of the galaxies) and (ii) the dispersion due to the observational errors σ_o (i.e., the uncertainty in the measurement of the merger fraction in a given field, including the Poisson shot noise term). Thus, the dispersion σ reported in Fig. 4 is an upper limit for the actual cosmic variance σ_v. We deal with this limitation by applying a maximum likelihood estimator (MLE) to the observed distributions. In Appendix A, we develop a MLE that estimates the more probable values of μ and σ_v, assuming that the merger fraction follows a Gaussian distribution in log-space (Eq. (19)) that is affected by known observational errors σ_o. We prove that the MLE provides an unbiased estimation of μ and σ_v and reliable uncertainties of these parameters. Applying the MLE to the distributions in Fig. 4, we find that σ_v is lower than σ, as anticipated, and that the cosmic variance increases with the apparent brightness from σ_v = 0.25 ± 0.04 for i ≤ 22 galaxies to σ_v = 0.44 ± 0.08 for i ≤ 21 galaxies.

We constrain the dependence of σ_v on the number density of the populations under study in Sects. 4.4.1 and 4.4.4 on the probed cosmic volume in Sect. 4.4.2 and on redshift in Sect. 4.4.3. That provides a complete description of the cosmic variance for merger fraction studies. We stress that our definition of σ_v differs from the classical definition of the relative cosmic variance presented in Sect. 4.1, which is equivalent to e^σ_v. However, σ_v encodes the relevant information needed to estimate the intrinsic dispersion in the measurement of the merger fraction due to the clustering of galaxies.

4.3. Optimal estimation of σ_v in the ALHAMBRA survey

In the previous section, we have defined the methodology to compute the cosmic variance from the observed distribution of the merger fraction. However, as shown by López-Sanjuan et al. (2010a), we need a galaxy sample with either small photometric redshift errors or a large fraction of spectroscopic redshifts to avoid projection effects. In the present study, we did not use information from spectroscopic redshifts, so we should check that the photometric redshifts in ALHAMBRA are good enough for our purposes. A natural way to select excellent photometric redshifts in ALHAMBRA is by a selection in the odds parameter. On the one hand, this selection increases the accuracy of the photometric redshifts of the sample and minimises the fraction of catastrophic outliers (Molino et al. 2013), improving the merger fraction estimation. On the other hand, our sample becomes incomplete and could be biased toward a population of either bright galaxies or galaxies with marked features in the SED (i.e., emission line galaxies or old populations with a strong 4000 Å break). In this section, we study how the merger fraction in ALHAMBRA depends on the selection and derive the optimal one to estimate the cosmic variance.

Following the methodology from spectroscopic surveys (e.g., Lin et al. 2004; de Ravel et al. 2009; López-Sanjuan et al. 2011, 2013), if we have a population with a total number of galaxies N_tot in a given volume and we observe a random fraction f_obs of these galaxies, the merger fraction of the total population is ${\mathit{f}}_{\mathrm{m}}\mathrm{=}{\mathit{f}}_{\mathrm{m}\mathit{,}\mathrm{obs}}\mathrm{\times }{\mathit{f}}_{\mathrm{obs}}^{-1}\mathit{,}$(20)where f_m,obs is the merger fraction of the observed sample. In ALHAMBRA, we applied a selection in the parameter , so Eq. (20) becomes ${\mathit{f}}_{\mathrm{m}}\mathrm{=}{\mathit{f}}_{\mathrm{m}}\hspace{0.17em}\mathrm{(}\mathrm{\ge }{\mathrm{𝒪}}_{\mathrm{sel}}\mathrm{)}\mathrm{\times }\frac{{\mathit{N}}_{\mathrm{tot}}}{\mathit{N}\hspace{0.17em}\mathrm{(}\mathrm{\ge }{\mathrm{𝒪}}_{\mathrm{sel}}\mathrm{)}}\mathit{,}$(21)where is the number of galaxies with odds higher than (i.e., galaxies with ), N_tot is the total number of galaxies (i.e., galaxies with ), and is the merger faction of those galaxies with . Because f_m must be independent of the selection, the study of f_m as a function of provides the clues about the optimal odds selection for merger fraction studies in ALHAMBRA. We show f_m as a function of for galaxies with i ≤ 22.5 at 0.3 ≤ z< 0.9 in Fig. 5. We find that

• the merger fraction is roughly constant for . This is the expectedresult if the merger fraction is reliable and measured in anon-biased sample. In this particular case, the (0.6) samplecomprises 98% (66%) of the total number of galaxies with i ≤ 22.5;

  Fig. 5 Merger fraction fm as a function of the odds selection for i ≤ 22.5 galaxies at 0.3 ≤ z< 0.9. The filled triangles, circles, and squares are for ${\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{=}\mathrm{30}\mathit{,}\mathrm{40}$, and 50 h-1 kpc close pairs, respectively. The open triangles are the observed merger fractions for ${\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{=}\mathrm{30}{\mathit{h}}^{-1}$ kpc to illustrate the selection correction from Eq. (21). In several cases, the error bars are smaller than the points. The dotted, dashed, and solid lines mark the average fm at for ${\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{=}\mathrm{30}\mathit{,}\mathrm{40}$, and 50 h-1 kpc close pairs, respectively. A colour version of this plot is available in the electronic edition.

  Fig. 6 Cosmic variance σv as a function of ${\mathit{r}}_{\mathrm{p}}^{\max }$ for i ≤ 22.5,21.5, and 21 galaxies at 0.3 ≤ z< 0.9 (circles, stars, and triangles, respectively). The horizontal lines mark the error-weighted average of the cosmic variance in each case, and the coloured areas mark their 68% confidence intervals. A colour version of this plot is available in the electronic edition.

• the merger fraction is overestimated for . Even if only a small fraction of galaxies with poor constrains in the photometric redshits are included in the sample, the projection effects become important;

• the merger fraction is overestimated for . This behaviour at high odds (i.e., in samples with high quality photometric redshifts) suggests that the retained galaxies are a biased sub-sample of the general population under study.

In the analysis above, we only accounted for close companions of i ≤ 22.5 galaxies with 10 h^-1 kpc ≤ r_p ≤ 30 h^-1 kpc, but we can use other values of ${\mathit{r}}_{\mathrm{p}}^{\max }$ or searching over different samples. On the one hand, we repeated the study for ${\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{=}\mathrm{40}$ and 50 h^-1 kpc, finding the same behaviour than for ${\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{=}\mathrm{30}{\mathit{h}}^{-1}$ kpc (Fig. 5). The only differences are that the merger fraction increases with the search radius and that the point starts to deviate from the expected value (the search area increases with ${\mathit{r}}_{\mathrm{p}}^{\max }$ and more accurate photometric redshifts are needed to avoid projection effects). On the other hand, we explored a wide range of i-band magnitude selections from i ≤ 23 to 20 in the three previous ${\mathit{r}}_{\mathrm{p}}^{\max }$ cases. We find again the same behaviour. That reinforces our arguments above and suggests as acceptable odds limits to select samples for merger fraction studies in ALHAMBRA.

The merger fraction increases with the search radius (Fig. 5). However, the merger rate R_m (Sect. 3.2) is a physical property of any population, and it cannot depend on ${\mathit{r}}_{\mathrm{p}}^{\max }$. Thus, the increase in the merger fraction with the search radius is compensated with the increase in the merger time scale (e.g., de Ravel et al. 2009; López-Sanjuan et al. 2011). This is, ${\mathit{R}}_{\mathrm{m}}\mathrm{\propto }{\mathit{f}}_{\mathrm{m}}\mathrm{\left(}{\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\right)}\mathit{/}{\mathit{T}}_{\mathrm{m}}\mathrm{\left(}{\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\right)}$. For the same reason, the cosmic variance of the merger rate cannot depend on ${\mathit{r}}_{\mathrm{p}}^{\max }$. In other words, the 68% confidence interval of the merger rate, [ R_me^{− σ_v},R_me^σ_v ], should be independent of the search radius. Expanding the previous confidence interval, we find that $\begin{array}{ccc}& & \\ & & \mathrm{\left[}{\mathit{f}}_{\mathrm{m}}\mathrm{\right(}{\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\left)}\hspace{0.17em}{\mathit{T}}_{\mathrm{m}}^{-1}\mathrm{\right(}{\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\left)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{-}{\mathit{\sigma }}_{\mathit{v}}}\mathit{,}{\mathit{f}}_{\mathrm{m}}\mathrm{\right(}{\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\left)}\hspace{0.17em}{\mathit{T}}_{\mathrm{m}}^{-1}\mathrm{\right(}{\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\left)}\hspace{0.17em}{\mathrm{e}}^{{\mathit{\sigma }}_{\mathit{v}}}\mathrm{\right]}\mathrm{=}\\ & & \mathrm{\left[}{\mathit{f}}_{\mathrm{m}}\mathrm{\right(}{\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\left)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{-}{\mathit{\sigma }}_{\mathit{v}}}\mathit{,}{\mathit{f}}_{\mathrm{m}}\mathrm{\right(}{\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\left)}\hspace{0.17em}{\mathrm{e}}^{{\mathit{\sigma }}_{\mathit{v}}}\mathrm{\right]}\hspace{0.17em}{\mathit{T}}_{\mathrm{m}}^{-1}\mathrm{\left(}{\mathit{r}}_{\mathrm{p}}^{\max }\mathrm{\right)}\mathit{.}\end{array}$(22)Note that the dependence on ${\mathit{r}}_{\mathrm{p}}^{\max }$ is encoded in the median merger fraction and in the merger time scale. Thus, the cosmic variance σ_v of the merger fraction should not depend on the search radius. We checked this prediction by studying the cosmic variance as a function of the search radius for i ≤ 22.5,21.5, and 21 galaxies with at 0.3 ≤ z< 0.9. We find that σ_v is consistent with a constant value irrespective of ${\mathit{r}}_{\mathrm{p}}^{\max }$ in the three populations probed, as desired (Table 2 and Fig. 6). This supports σ_v as a good descriptor of the cosmic variance and our methodology to measure it. In the previous analysis, we have omitted the merger probability C_m, which mainly depends on ${\mathit{r}}_{\mathrm{p}}^{\max }$ and environment (Sect. 3.2). The merger fraction correlates with environment, so the merger probability could modify the factor e^σ_v in Eq. (22). Because a constant σ_v with ${\mathit{r}}_{\mathrm{p}}^{\max }$ is observed, the impact of C_m in the f_m to R_m translation should be similar in the range of ${\mathit{r}}_{\mathrm{p}}^{\max }$ explored. Detailed cosmological simulations are needed to clarify this issue.

Finally, we studied the dependence of σ_v on the odds selection for i ≤ 22.5 galaxies at 0.3 ≤ z< 0.9. Following the same arguments than before, the cosmic variance should not depend on the odds selection. We find that (i) σ_v is consistent with a constant value as a function of ${\mathit{r}}_{\mathrm{p}}^{\max }$ for any , reinforcing our results above; and (ii) σ_v is independent of the odds selection at (Fig.7). As for the merger fraction, we checked that different populations follow the same behaviour. We therefore set as the optimal odds selection to measure the cosmic variance in ALHAMBRA. This selection provides excellent photometric redshifts and ensures representative samples.

In summary, we estimate the cosmic variance σ_v from the merger fractions measured in the 48 ALHAMBRA sub-fields with 10 h^-1 kpc ≤ r_p ≤ 50 h^-1 kpc close pairs (the σ_v uncertainty is lower for larger search radii) and in samples with in the following. That ensures reliable results in representative (i.e., non-biased) samples.

4.4. Characterisation of σ_v

At this stage, we have set both the methodology to compute a robust cosmic variance from the observed merger fraction distribution (Sect. 4.2) and the optimal search radius and odds selection to estimate σ_v in ALHAMBRA (Sect. 4.3). Now, we can characterise the cosmic variance as a function of the populations under study (Sects. 4.4.1 and 4.4.4), the probed cosmic volume (Sect. 4.4.2), and the redshift (Sect. 4.4.3).

4.4.1. Dependence on the number density of the principal sample

In this section, we explore how the cosmic variance depends on the number density n₁ of the principal population under study. For that, we took the same population as principal and companion sample. We study the dependence on the companion sample in Sect. 4.4.4. To avoid any dependence of σ_v on either the probed cosmic volume and z and to minimise the observational errors, we focus on the redshift range 0.3 ≤ z< 0.9 in this section. This range probes a cosmic volume of V_c ~ 1.4 × 10⁵ Mpc³ in each ALHAMBRA sub-field. To explore different number densities, we measured the cosmic variance for different i-band selected samples from i ≤ 20 to i ≤ 23 in 0.5 magnitude steps. We estimated the average number density n₁ in the redshift range z_r as the median number density in the 48 ALHAMBRA sub-fields with ${\mathit{n}}_{\mathrm{1}}^{\mathit{j}}\hspace{0.17em}\mathrm{\left(}{\mathit{z}}_{\mathrm{r}}\mathrm{\right)}\mathrm{=}\frac{{\sum }_{\mathit{i}}{\mathrm{\int }}_{{\mathit{z}}_{\min }}^{{\mathit{z}}_{\max }}{\mathit{P}}_{\mathit{i}}^{\mathit{j}}\hspace{0.17em}\mathrm{(}{\mathit{z}}_{\mathit{i}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}{\mathit{z}}_{\mathrm{p}\mathit{,i}}\mathit{,}{\mathit{\sigma }}_{{\mathit{z}}_{\mathrm{p}\mathit{,i}}}\mathrm{)}\hspace{0.17em}\mathrm{d}{\mathit{z}}_{\mathit{i}}}{{\mathit{V}}_{\mathrm{c}}^{\mathit{j}}\mathrm{\left(}{\mathit{z}}_{\mathrm{r}}\mathrm{\right)}}$(23)being the number density in the sub-field j and ${\mathit{V}}_{\mathrm{c}}^{\mathit{j}}$ the cosmic volume probed by it at z_r. In the measurement of the number density, all the galaxies were taking into account; i.e., any odds selection was applied (). We stress that our measured number densities are unaffected by cosmic variance, and they can be used therefore to characterise σ_v. We report our measurements in Table 3.

We find that the cosmic variance increases as the number density decreases (Fig. 8), as expected by Eq. (17). The error-weighted least-squares fit of a power-law to the data is ${\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathrm{\right)}\mathrm{=}\mathrm{(}\mathrm{0.45}\mathrm{\pm }\mathrm{0.04}\mathrm{)}\mathrm{\times }(\frac{{\mathit{n}}_{\mathrm{1}}}{{\mathrm{10}}^{-3}{\mathrm{Mpc}}^{-3}}{)}^{\mathrm{-}\mathrm{0.54}\hspace{0.17em}\mathrm{\pm }\hspace{0.17em}\mathrm{0.06}}\mathit{.}$(24)In this section and in the following ones, we used i-band selected samples to characterise σ_v. We show that the results obtained with these i-band samples can be applied to luminosity- and stellar mass-selected samples in Sect. 4.7.

4.4.2. Dependence on the cosmological volume

In this section, we explore the dependence of the cosmic variance on the cosmic volume probed by the survey. We defined ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}$ as ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}\mathrm{=}{\mathit{\sigma }}_{\mathit{v}}\mathit{/}{\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathrm{\right)}$. This erased the dependence on the number density of the population and only volume effects were measured. We explored smaller cosmic volumes than in the previous section by studying (i) different redshift ranges over the full ALHAMBRA area (avoiding redshift ranges smaller than 0.1) and (ii) smaller areas, centred in the ALHAMBRA sub-fields at 0.3 ≤ z< 0.9. All the cases, as summarised in Table 4, are for i ≤ 23 galaxies. At the end, we explored volumes from V_c ~ 0.1 × 10⁵ Mpc³ to V_c ~ 1.4 × 10⁵ Mpc³. The power-law function that better describes the observations (Fig. 9) is ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}\hspace{0.17em}\mathrm{\left(}{\mathit{V}}_{\mathrm{c}}\mathrm{\right)}\mathrm{=}\mathrm{(}\mathrm{1.05}\mathrm{\pm }\mathrm{0.05}\mathrm{)}\mathrm{\times }(\frac{{\mathit{V}}_{\mathrm{c}}}{{\mathrm{10}}^{\mathrm{5}}{\mathrm{Mpc}}^{\mathrm{3}}}{)}^{\mathrm{-}\mathrm{0.48}\hspace{0.17em}\mathrm{\pm }\hspace{0.17em}\mathrm{0.05}}\mathit{.}$(25)We tested the robustness of our result by fitting the two sets of data (variation in redshift and area) separately. We find ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}\mathrm{\propto }{\mathit{V}}_{\mathrm{c}}^{\mathrm{-}\mathrm{0.43}\hspace{0.17em}\mathrm{\pm }\hspace{0.17em}\mathrm{0.08}}$ for the redshift data, while ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}\mathrm{\propto }{\mathit{V}}_{\mathrm{c}}^{\mathrm{-}\mathrm{0.48}\hspace{0.17em}\mathrm{\pm }\hspace{0.17em}\mathrm{0.05}}$ for the area data.

4.4.3. Dependence on redshift

The redshift is an expected parameter in the parametrisation the cosmic variance. However, Fig. 9 shows that the results at different redshifts are consistent with those from the wide redshift range 0.3 ≤ z< 0.9. As a consequence, the redshift dependence of the cosmic variance should be smaller than the typical error in our measurements. We tested this hypothesis by measuring σ_v in different, non-overlapping redshift bins. We summarise our measurements, as performed for i ≤ 23 galaxies, in Table 5. We defined ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }}\mathrm{=}{\mathit{\sigma }}_{\mathit{v}}\mathit{/}{\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathit{,}{\mathit{V}}_{\mathrm{c}}\mathrm{\right)}$ to isolate the redshift dependence of the cosmic variance. We find that ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }}$ is compatible with unity, ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }}\mathrm{=}\mathrm{1.02}\mathrm{\pm }\mathrm{0.07}$, and that no redshift dependence remains after accounting for the variation in n₁ and V_c (Fig. 10). This confirms our initial hypothesis and we assume, therefore, γ = 0 in the following.

4.4.4. Dependence on the number density of the companion sample

As we show in Sect. 3, two different populations are involved in the measurement of the merger fraction: the principal sample and the sample of companions around principal galaxies. In the previous sections, the principal and the companion sample were the same, and here we explore how the number density n₂ of the companion sample impacts the cosmic variance. We set i ≤ 20.5 galaxies at 0.3 ≤ z< 0.9 as principals and varied the i-band selection of the companion galaxies from i ≤ 20.5 to i ≤ 23 in 0.5 steps. As in Sect. 4.4.2, the variable ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}\mathrm{=}{\mathit{\sigma }}_{\mathit{v}}\mathit{/}{\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathrm{\right)}$ was used.

We find that the cosmic variance decreases as the number density of the companion sample increases (Table 6 and Fig. 11). We fit the dependence with a power-law, forcing it to pass for the point ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathit{,}{\mathit{n}}_{\mathrm{1}}\mathrm{\right)}\mathrm{=}\mathrm{1}$. We find that ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathit{,}{\mathit{n}}_{\mathrm{2}}\mathrm{\right)}\mathrm{=}(\frac{{\mathit{n}}_{\mathrm{2}}}{{\mathit{n}}_{\mathrm{1}}}{)}^{\mathrm{-}\mathrm{0.37}\hspace{0.17em}\mathrm{\pm }\hspace{0.17em}\mathrm{0.04}}\mathit{.}$(26)We checked that it is consistent with unity if we leave free the intercept, as we assume ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathit{,}{\mathit{n}}_{\mathrm{1}}\mathrm{\right)}\mathrm{=}\mathrm{1.04}\hspace{0.17em}\mathrm{\pm }\hspace{0.17em}\mathrm{0.12}$. In addition, the power-law index changes slightly, ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }}\mathrm{\propto }\mathrm{(}{\mathit{n}}_{\mathrm{2}}\mathit{/}{\mathit{n}}_{\mathrm{1}}{\mathrm{)}}^{\mathrm{-}\mathrm{0.39}\hspace{0.17em}\mathrm{\pm }\hspace{0.17em}\mathrm{0.08}}$.

4.4.5. The cosmic variance in merger fraction studies based on close pairs

In the previous sections, we have characterised the dependence of the cosmic variance σ_v on the basic parameters involved in close pair studies (Sect. 4.1): the number density of the principal (n₁, Sect. 4.4.1) and the companion sample (n₂, Sect. 4.4.4), the cosmic volume under study (V_c, Sect. 4.4.2), and the redshift (Sect. 4.4.3). We find that $\begin{array}{ccc}& & {\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathit{,}{\mathit{n}}_{\mathrm{2}}\mathit{,}{\mathit{V}}_{\mathrm{c}}\mathrm{\right)}\mathrm{=}\\ & & \mathrm{0.48}\mathrm{\times }(\frac{{\mathit{n}}_{\mathrm{1}}}{{\mathrm{10}}^{-3}{\mathrm{Mpc}}^{-3}}{)}^{-0.54}\mathrm{\times }(\frac{{\mathit{V}}_{\mathrm{c}}}{{\mathrm{10}}^{\mathrm{5}}{\mathrm{Mpc}}^{\mathrm{3}}}{)}^{-0.48}\mathrm{\times }(\frac{{\mathit{n}}_{\mathrm{2}}}{{\mathit{n}}_{\mathrm{1}}}{)}^{-0.37}\mathit{.}\end{array}$(27)This is the main result of the present paper. We estimated through Monte Carlo sampling that the typical uncertainty in σ_v from this relation is ~15%. The dependence of σ_v on redshift should be lower than this uncertainty. In addition, σ_v is independent of the search radius used to compute the merger fraction as we demonstrated in Sect. 4.3.

4.5. Cosmic variance in spatially random samples

In this section, we further test the significance of our results by measuring both the merger fraction and the cosmic variance in samples randomly distributed in the sky plane. For this, we created a set of 100 random samples with each random sample comprising 48 random sub-samples (one per ALHAMBRA sub-field). We generated each random sub-sample by assigning a random RA and Dec to each source in the original catalogue but retained the original redshift of the sources. This erases the clustering signal inside each ALHAMBRA sub-field (i.e., at ≲15′ scales), but the number density fluctuations between sub-fields because of the clustering at scales larger than ~15′ remains. We estimated the merger fraction and the cosmic variance for each random sample at 0.3 ≤ z< 0.9 as in Sect. 4.4.1, computed the median merger fraction, ⟨ f_m ⟩, and determined the median cosmic variance, ⟨ σ_v ⟩, in the set of 100 random samples to compare them with the values measured in the real samples. To facilitate this comparison, we defined the variables F_m = f_m/ ⟨ f_m ⟩ and Σ_v = σ_v/ ⟨ σ_v ⟩. We estimated F_m and Σ_v for different selections in n₁ following Sect. 4.4.1, and we have show our findings in Fig. 12.

On the one hand, the merger fraction in the real samples is higher than in the random samples by a factor of three–four, F_m = 4.25 − 0.27 × n₁ (Fig. 12, top panel). This reflects the clustering present in the real samples that we erased when we randomised the positions of the sources in the sky, as well as the higher clustering of more luminous galaxies. This result is consistent with previous close pair studies that compare real and random samples (e.g., Kartaltepe et al. 2007). On the other hand, the cosmic variance measured in the random samples is higher than the cosmic variance in the real ones, ⟨ Σ_v ⟩ = 0.81 ± 0.04 (Fig. 12, bottom panel). This implies that most of the variance between sub-fields is unrelated with the clustering inside these sub-fields and that the σ_v measured in the present paper is a real signature of the relative field-to-field variation of the merger fraction.

4.6. Testing the independence of the 48 ALHAMBRA sub-fields

Hitherto, we have assumed that the 48 ALHAMBRA sub-fields are independent. However, only the seven ALHAMBRA fields are really independent and correlations between adjacent sub-fields should exists. This correlations could impact our σ_v measurements, and in this section, we test the independence assumption.

We defined two groups of seven independent pointings with one per ALHAMBRA field. The first group comprises of the pointings f02p01, f03p02, f04p01, f05p01, f06p01, f07p03, and f08p02, where f0? refers to the ALHAMBRA field and p0? to the pointing in the field. The second group comprises the pointings f02p02, f03p01, f04p01, f05p01, f06p02, f07p04, and f08p01. Note that fields f04 and f05 have only one pointing in the current ALHAMBRA release. Each of the previous pointings probe a cosmic volume four times higher than our fiducial sub-fields with a median V_c = (54.49 ± 0.59) × 10⁴ Mpc³ for the first group and V_c = (55.24 ± 0.50) × 10⁴ Mpc³ for the second one at 0.3 ≤ z< 0.9. Then, we measured the merger fraction in the seven independent pointings of each group, and we obtained σ_v applying the MLE. We repeated this procedure for different selections from i ≤ 23 to i ≤ 20.5 in 0.5 magnitude steps. Finally, we defined ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }}\mathrm{=}{\mathit{\sigma }}_{\mathit{v}}\mathit{/}{\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathit{,}{\mathit{V}}_{\mathrm{c}}\mathrm{\right)}$, so the values of ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }}$ would be dispersed around unity if the cosmic variance measured from the seven independent areas is described well by the cosmic variance measured from the 48 sub-fields. We summarise our results in Table 7 and in Fig. 13.

We find that the cosmic variance from the seven independent fields nicely agree with our expectations from Eq. (27) with an error-weighted average of ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }}\mathrm{=}\mathrm{1.01}\mathrm{\pm }\mathrm{0.10}$. Thus, assuming the 48 ALHAMBRA sub-fields as independent is an acceptable approximation to study σ_v. In addition, the uncertainties in σ_v are lower by a factor of two when we use the 48 sub-fields, improving the statistical significance of our results.

4.7. Expectations for luminosity- and mass-selected samples

Throughout this paper, we have focused our analysis in (apparent) bright galaxies with i ≤ 23. This ensures excellent photometric redshifts and provides reliable merger fraction measurements (Sect. 4.3). However, one should be interested on the merger fraction of galaxies selected by their luminosity, stellar mass, colour, etc. Because the bias of the galaxies with respect to the underlying dark-matter distribution depends on the selection of the sample, our prescription to estimate σ_v could not be valid for physically selected samples (Sect. 4.1). In this section, we compare the expected cosmic variance from Eq. (27) with the actual cosmic variance of several luminosity- and stellar mass-selected samples to set the limits and the reliability of our suggested parametrisation.

We defined the variable ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }\mathrm{\ast }}\mathrm{=}{\mathit{\sigma }}_{\mathit{v}}\mathit{/}{\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathit{,}{\mathit{n}}_{\mathrm{2}}\mathit{,}{\mathit{V}}_{\mathrm{c}}\mathrm{\right)}$, so the values of ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }\mathrm{\ast }}$ would be dispersed around unity if no extra dependence on the luminosity or the stellar mass exists. Throughout the present paper, we imposed neither luminosity nor mass ratio constraints between the galaxies in the close pairs. However, merger fraction studies impose such constraints to study major or minor mergers. This ratio is defined as ℛ = M_⋆,2/M_⋆,1, where M_⋆,1 and M_⋆,2 are the stellar masses of the principal and the companion galaxy in the pair, respectively. The definition of ℛ in the B-band luminosity L_B case is similar. Major mergers are usually defined with 1 / 4 ≤ ℛ ≤ 1, while minor mergers with ℛ ≤ 1 / 4. We explored different ℛ cases and estimated n₂ as the number density of the L_B ≥ ℛL_B,1 or the M_⋆ ≥ ℛM_⋆,1 population. The properties of all the studied samples are summarised in Tables 8 and 9. The redshift range probed in each case was chosen to ensure volume-limited companion samples. We stress that the samples in Tables 8 and 9 mimic typical observational selections and ℛ values from the literature.

On the one hand, we find that the error-weighted average of all the luminosity-selected samples is ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }\mathrm{\ast }}\mathrm{=}\mathrm{1.01}\mathrm{\pm }\mathrm{0.03}$, which is compatible with unity as we expected, if no (or limited) dependence on the selection exists (Fig. 14). We obtained ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }\mathrm{\ast }}\mathrm{=}\mathrm{1.03}\mathrm{\pm }\mathrm{0.05}$ from samples with the luminosity ratio ℛ applied, while ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }\mathrm{\ast }}\mathrm{=}\mathrm{1.00}\mathrm{\pm }\mathrm{0.03}$ from samples without it. On the other hand, we find ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }\mathrm{\ast }}\mathrm{=}\mathrm{1.02}\mathrm{\pm }\mathrm{0.03}$ for the stellar mass-selected samples (Fig. 15). As noted previously, the value is compatible with unity. We obtained ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }\mathrm{\ast }}\mathrm{=}\mathrm{0.98}\mathrm{\pm }\mathrm{0.05}$ from samples with the mass ratio ℛ applied, while ${\mathit{\sigma }}_{\mathit{v}}^{\mathrm{\ast }\mathrm{\ast }\mathrm{\ast }}\mathrm{=}\mathrm{1.03}\mathrm{\pm }\mathrm{0.03}$ from samples without it.

We conclude that our results based on i-band selected samples provide a good description of the cosmic variance for physically selected samples with a limited dependence (≲ 15%) on both the luminosity and the stellar mass selection. Thus, only n₁, n₂, and V_c are needed to estimate a reliable σ_v for merger fractions studies based on close pairs.

5. Summary and conclusions

We use the 48 sub-fields of ~180 arcmin² in the ALHAMBRA survey (total effective area of 2.38 deg²) to empirically estimate the cosmic variance that affects merger fraction studies based on close pairs for the first time in the literature. We find that the distribution of the merger fraction is log-normal, and we use a maximum likelihood estimator to measure the cosmic variance σ_v unaffected by observational errors (including the Poisson shot noise term).

We find that the better parametrisation of the cosmic variance for merger fraction studies based on close pairs is (Eq. (27)), $\begin{array}{ccc}{\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathrm{\left(}{\mathit{n}}_{\mathrm{1}}\mathit{,}{\mathit{n}}_{\mathrm{2}}\mathit{,}{\mathit{V}}_{\mathrm{c}}\mathrm{\right)}\mathrm{=}& & \\ & & \mathrm{0.48}\mathrm{\times }(\frac{{\mathit{n}}_{\mathrm{1}}}{{\mathrm{10}}^{-3}{\mathrm{Mpc}}^{-3}}{)}^{-0.54}\mathrm{\times }(\frac{{\mathit{V}}_{\mathrm{c}}}{{\mathrm{10}}^{\mathrm{5}}{\mathrm{Mpc}}^{\mathrm{3}}}{)}^{-0.48}\mathrm{\times }(\frac{{\mathit{n}}_{\mathrm{2}}}{{\mathit{n}}_{\mathrm{1}}}{)}^{-0.37}\mathit{,}\end{array}$where n₁ and n₂ are the cosmic average number density of the principal and the companion populations under study, respectively, and V_c is the cosmological volume probed by our survey in the redshift range of interest. We stress that n₁ and n₂ should be estimated from general luminosity or mass function studies and that measurements from volumes dominated by structures (e.g., clusters or voids) should be avoided. In addition, σ_v is independent of the search radius used to compute the merger fraction. The typical uncertainty in σ_v from our relation is ~15%. The dependence of the cosmic variance on redshift should be lower than this uncertainty. Finally, we checked that our formula provides a good estimation of σ_v for luminosity- and mass-selected samples and for close pairs with a given luminosity or mass ratio ℛ between the galaxies in the pair. In the latter case, n₂ is the average number density of those galaxies brighter or more massive than ℛL₁ or ℛM_⋆,1, respectively.

Equation (27) provides the expected cosmic variance of an individual merger fraction measurement f_m at a given field and redshift range. The 68% confidence interval of this merger fraction is [ f_me^{− σ_v},f_me^σ_v ]. This interval is independent of the error in the measurement of f_m, so both sources of uncertainty should be added to obtain an accurate description of the merger fraction error in pencil-beam surveys. If we have access to several independent fields j for our study, we should combine the cosmic variance ${\mathit{\sigma }}_{\mathit{v}}^{\mathit{j}}$ of each single field with the following formula (see Moster et al. 2011, for details): ${\mathit{\sigma }}_{\mathit{v,}\mathrm{tot}}^{\mathrm{2}}\mathrm{=}\frac{{\sum }_{\mathit{j}}\hspace{0.17em}\mathrm{(}{\mathit{V}}_{\mathrm{c}}^{\mathit{j}}\hspace{0.17em}{\mathit{\sigma }}_{\mathit{v}}^{\mathit{j}}{\mathrm{)}}^{\mathrm{2}}}{\mathrm{(}{\sum }_{\mathit{j}}{\mathit{V}}_{\mathrm{c}}^{\mathit{j}}{\mathrm{)}}^{\mathrm{2}}}\mathit{,}$(28)where ${\mathit{V}}_{\mathrm{c}}^{\mathit{j}}$ is the cosmic volume probed by each single field in the redshift range of interest.

Thanks to the Eqs. (27) and (28), we can estimate the impact of cosmic variance in close pair studies from the literature. For example, Bundy et al. (2009) measure the major merger fraction in the two GOODS fields. We expect σ_v ~ 0.42 for massive (M_⋆ ≥ 10¹¹M_⊙) galaxies, while σ_v ~ 0.16 for M_⋆ ≥ 10¹⁰M_⊙ galaxies. The studies of de Ravel et al. (2009) and López-Sanjuan et al. (2011) explore the merger fraction in the VVDS-Deep. We expect σ_v ≲ 0.09 for major mergers and σ_v ≲ 0.07 for minor mergers in this survey. Lin et al. (2008) explore the merger properties of M_B ≤ − 19 galaxies in three DEEP2 fields. We estimate σ_v ~ 0.03 for their results. Several major close pair studies have been conducted in the COSMOS field (e.g., de Ravel et al. 2011; Xu et al. 2012). Focusing in mass-selected samples, we expect σ_v ~ 0.17 for massive galaxies, while σ_v ~ 0.07 for M_⋆ ≥ 10¹⁰M_⊙ galaxies. In addition, we estimate σ_v ~ 0.13 for the minor merger fractions reported by López-Sanjuan et al. (2012) in the COSMOS field. Regarding local merger fractions (z ≲ 0.1), the expected cosmic variance in the study of De Propris et al. (2005) in the MGC is σ_v ~ 0.03, while it is σ_v< 0.03 in the study of Patton et al. (2000). Finally, studies based in the full SDSS area are barely affected by cosmic variance with σ_v ≲ 0.005 (e.g., Patton & Atfield 2008).

Extended samples over larger sky areas are needed to constrain the subtle redshift evolution of the comic variance and its dependence on the selection of the samples. Future large photometric surveys such as J-PAS³ (Javalambre – Physics of the accelerating universe Astrophysical Survey, Benítez et al. 2014), which will provide excellent photometric redshifts with δ_z ~ 0.003 over 8500 deg² in the northern sky, are fundamental to progress on this topic.

In the present paper, we have studied the intrinsic dispersion of the merger fraction measured in the 48 ALHAMBRA sub-fields in detail. In future papers, we will explore the dependence of the median merger fraction, as estimated as e^μ, on stellar mass, colour, or morphology (see Pović et al. 2013, for details about the morphological classification in ALHAMBRA), and we will compare the ALHAMBRA measurements (both the median and the dispersion) to the expectations from cosmological simulations.

Online material

Appendix A: Maximum likelihood estimation of the cosmic variance σ_v

Maximum likelihood estimators (MLEs) have been used in a wide range of topics in astrophysics. For example, Naylor & Jeffries (2006) used a MLE to fit colour-magnitude diagrams, Arzner et al. (2007) to improve the determination of faint X-ray spectra, Makarov et al. (2006) to improve distance estimates using red giant branch stars, and López-Sanjuan et al. (2008, 2009a,b, 2010b) to estimate reliable merger fractions from morphological criteria. The MLEs are based on the estimation of the most probable values of a set of parameters, which define the probability distribution that describes an observational sample.

The general MLE operates as follows. Throughout this Appendix, we denote the probability to obtain the values a given the parameters b as P (a | b). Being x_j the measured values in the ALHAMBRA field j, and θ the parameters that we want to estimate, we may express the joined likelihood function as $\mathit{L}\mathrm{(}{\mathit{x}}_{\mathit{j}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}\mathit{\theta }\mathrm{)}\mathrm{\equiv }\mathrm{-}\ln \left[\underset{\mathit{j}}{\mathrm{􏽙}}\mathit{P}\hspace{0.17em}\mathrm{(}{\mathit{x}}_{\mathit{j}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}\mathit{\theta }\mathrm{)}\right]\mathrm{=}\mathrm{-}\sum _{\mathit{j}}\ln \mathrm{[}\mathit{P}\hspace{0.17em}\mathrm{(}{\mathit{x}}_{\mathit{j}}\hspace{0.17em}\mathrm{|}\hspace{0.17em}\mathit{\theta }{\mathrm{\left)}}^{\mathrm{\right]}}\mathit{.}$(A.1)If we are able to express P (x_j | θ) analytically, we can minimise Eq. (A.1) to obtain the best estimation of the parameters θ, as denoted as θ_ML. In our case, x_j is the observed value of the merger fraction in log-space for the ALHAMBRA sub-field j, where ${\mathit{x}}_{\mathit{j}}\mathrm{\equiv }{\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\mathrm{=}\ln {\mathit{f}}_{\mathrm{m}\mathit{,j}}$. We decided to work in log-space because that makes the problem analytic and simplifies the implementation of the method without losing mathematical rigour.

The ALHAMBRA sub-fields are assumed to have a real merger fraction (not affected by observational errors) that define a Gaussian distribution in log-space, ${\mathit{P}}_{\mathrm{G}}\hspace{0.17em}\mathrm{(}{\mathit{f}}_{\mathrm{real}\mathit{,j}}^{\mathrm{\prime }}\hspace{0.17em}\mathrm{|}\hspace{0.17em}\mathit{\mu ,}{\mathit{\sigma }}_{\mathit{v}}\mathrm{)}\mathrm{=}\frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathit{\pi }}\hspace{0.17em}{\mathit{\sigma }}_{\mathit{v}}}\hspace{0.17em}\exp \hspace{0.17em}\left[\mathrm{-}\frac{\mathrm{(}{\mathit{f}}_{\mathrm{real}\mathit{,j}}^{\mathrm{\prime }}\mathrm{-}\mathit{\mu }{\mathrm{)}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}}\right]\mathrm{·}$(A.2)Observational errors cause the observed ${\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}$ to differ from their respective real values ${\mathit{f}}_{\mathrm{real}\mathit{,j}}^{\mathrm{\prime }}$. The observed ${\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}$ are assumed to be extracted for a Gaussian distribution with mean ${\mathit{f}}_{\mathrm{real}\mathit{,j}}^{\mathrm{\prime }}$ and standard deviation σ_o,j (the observational errors), ${\mathit{P}}_{\mathrm{G}}\hspace{0.17em}\mathrm{(}{\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\hspace{0.17em}\mathrm{|}\hspace{0.17em}{\mathit{f}}_{\mathrm{real}\mathit{,j}}^{\mathrm{\prime }}\mathit{,}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}\mathrm{)}\mathrm{=}\frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathit{\pi }}\hspace{0.17em}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}}\hspace{0.17em}\exp \hspace{0.17em}\left[\mathrm{-}\frac{\mathrm{(}{\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\mathrm{-}{\mathit{f}}_{\mathrm{real}\mathit{,j}}^{\mathrm{\prime }}{\mathrm{)}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}^{\mathrm{2}}}\right]\mathrm{·}$(A.3)We assumed that the observational errors are Gaussian in log-space, or, that they are log-normal in observational space. This is a good approximation of the reality because we are dealing with fractions that cannot be negative and that have asymmetric confidence intervals, as shown by Cameron (2011). In our case, we estimated the observational errors in log-space as σ_o = σ_f/f_m. We checked that the values of σ_o derived from our jackknife errors are similar to those estimated from the Bayesian approach in Cameron (2011) with a difference between them ≲15%.

We obtained the probability P (x_j | θ) of each ALHAMBRA sub-field by the total probability theorem: $\begin{array}{ccc}\mathit{P}\hspace{0.17em}\mathrm{\left(}{\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\hspace{0.17em}\mathrm{\right|}\hspace{0.17em}\mathit{\mu ,}{\mathit{\sigma }}_{\mathit{v}}\mathit{,}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}\mathrm{)}\mathrm{=}& & \\ & & \end{array}$(A.4)where ${\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\mathrm{=}{\mathit{x}}_{\mathit{j}}$ and (μ,σ_v,σ_o,j) = θ in Eq. (A.1). Note that the values of σ_o,j are the measured uncertainties for each ALHAMBRA sub-field, so the only unknowns are the variables μ and σ_v, which we want to estimate. Note also that we integrate over the variable ${\mathit{f}}_{\mathrm{real}\mathit{,j}}^{\mathrm{\prime }}$, so we are not be able to estimate the real merger fractions individually, but only the underlying Gaussian distribution that describes the sample.

The final joined likelihood function, Eq. (A.1) after integrating Eq. (A.4), is $\mathit{L}\hspace{0.17em}\mathrm{(}{\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\hspace{0.17em}\mathrm{|}\hspace{0.17em}\mathit{\mu ,}{\mathit{\sigma }}_{\mathit{v}}\mathit{,}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}\mathrm{)}\mathrm{=}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\sum _{\mathit{j}}\ln \hspace{0.17em}\mathrm{(}{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}^{\mathrm{2}}\mathrm{)}\mathrm{+}\frac{\mathrm{(}{\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\mathrm{-}\mathit{\mu }{\mathrm{)}}^{\mathrm{2}}}{{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}^{\mathrm{2}}}\mathrm{·}$(A.5)With the minimisation of this function, we obtain the best estimation of both μ and the cosmic variance σ_v, which are unaffected by observational errors.

In addition, we can analytically estimate the errors in the parameters above. We can obtain those via an expansion of the function $\mathit{L}\hspace{0.17em}\mathrm{(}{\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\hspace{0.17em}\mathrm{|}\hspace{0.17em}\mathit{\mu ,}{\mathit{\sigma }}_{\mathit{v}}\mathit{,}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}\mathrm{)}$ in a Taylor’s series of its variables θ = (μ,σ_v,σ_o,j) around the minimisation point θ_ML. The previous minimisation process made the first L derivative null, and we obtain $\mathit{L}\mathrm{=}\mathit{L}\mathrm{\left(}{\mathit{\theta }}_{\mathrm{ML}}\mathrm{\right)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{(}\mathit{\theta }\mathrm{-}{\mathit{\theta }}_{\mathrm{ML}}{\mathrm{)}}^{{T}}{{H}}_{\mathit{xy}}\mathrm{(}\mathit{\theta }\mathrm{-}{\mathit{\theta }}_{\mathrm{ML}}\mathrm{)}\mathit{,}$(A.6)where H_xy is the Hessian matrix, and T denotes the transpose matrix. The inverse of the Hessian matrix provides an estimate of the 68% confidence intervals of μ_ML and σ_ML, as well as the covariance between them. The Hessian matrix of the joined likelihood function L is defined as ${{H}}_{\mathit{xy}}\mathrm{=}\left(\begin{array}{c}\\ \frac{{\mathit{\partial }}^{\mathrm{2}}\mathit{L}}{\mathit{\partial }{\mathit{\mu }}^{\mathrm{2}}}& \frac{{\mathit{\partial }}^{\mathrm{2}}\mathit{L}}{\mathit{\partial \mu }\hspace{0.17em}\mathit{\partial }{\mathit{\sigma }}_{\mathit{v}}}\\ \frac{{\mathit{\partial }}^{\mathrm{2}}\mathit{L}}{\mathit{\partial }{\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathit{\partial \mu }}& \frac{{\mathit{\partial }}^{\mathrm{2}}\mathit{L}}{\mathit{\partial }{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}}\end{array}\right)\mathit{,}$(A.7)with $\begin{array}{ccc}& & \frac{{\mathit{\partial }}^{\mathrm{2}}\mathit{L}}{\mathit{\partial }{\mathit{\mu }}^{\mathrm{2}}}\mathrm{=}\mathrm{-}\sum _{\mathit{i}}\frac{\mathrm{1}}{{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}^{\mathrm{2}}}\mathit{,}\\ & & \frac{{\mathit{\partial }}^{\mathrm{2}}\mathit{L}}{\mathit{\partial \mu }\hspace{0.17em}\mathit{\partial }{\mathit{\sigma }}_{\mathit{v}}}\mathrm{=}\frac{{\mathit{\partial }}^{\mathrm{2}}\mathit{L}}{\mathit{\partial }{\mathit{\sigma }}_{\mathit{v}}\hspace{0.17em}\mathit{\partial \mu }}\mathrm{=}\mathrm{-}\mathrm{2}\sum _{\mathit{i}}\frac{{\mathit{\sigma }}_{\mathit{v}}\mathrm{(}{\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\mathrm{-}\mathit{\mu }\mathrm{)}}{\mathrm{(}{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{2}}}\mathit{,}\end{array}$and $\frac{{\mathit{\partial }}^{\mathrm{2}}\mathit{L}}{\mathit{\partial }{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}}\mathrm{=}\sum _{\mathit{i}}\frac{\mathrm{(}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}^{\mathrm{2}}\mathrm{-}\mathrm{3}{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{)}\mathrm{\times }\mathrm{(}{\mathit{f}}_{\mathrm{m}\mathit{,j}}^{\mathrm{\prime }}\mathrm{-}\mathit{\mu }{\mathrm{)}}^{\mathrm{2}}}{\mathrm{(}{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{3}}}\mathrm{-}\frac{\mathrm{(}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}^{\mathrm{2}}\mathrm{-}{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{)}}{\mathrm{(}{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{\mathrm{o}\mathit{,j}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{2}}}\mathrm{·}$(A.10)Then, we computed the inverse of the minus Hessian, h_xy = ( − H_xy)^-1. Finally, we estimated the variances of our inferred parameters as ${\mathit{\sigma }}_{\mathit{\mu }}^{\mathrm{2}}\mathrm{=}{\mathit{h}}_{\mathrm{11}}$ and ${\mathit{\sigma }}_{{\mathit{\sigma }}_{\mathit{v}}}^{\mathrm{2}}\mathrm{=}{\mathit{h}}_{\mathrm{22}}$ because maximum likelihood theory states that ${\mathit{\sigma }}_{{\mathit{\theta }}_{\mathit{x}}}^{\mathrm{2}}\mathrm{\le }{\mathit{h}}_{\mathit{xx}}$.

We tested the performance and the limitations of our MLE through synthetic catalogues of merger fractions. We created several sets of 1000 synthetic catalogues with each of them composed by a number n of merger fractions randomly drawn from a log-normal distribution with μ_in = log 0.05 and σ_v,in = 0.2 and affected by observational errors σ_o. We explored the n = 50,250 and 1000 cases for the number of merger fractions and varied the observational errors from σ_o = 0.1 to 0.5 in 0.1 steps. That is, we explored observational errors in the measurement of the merger fraction from Δσ ≡ σ_o/σ_v = 0.5 to 2.5 times the cosmic variance that we want to measure. We checked that the results below are similar for any value of σ_v,in. We find that

• 1.

  The median value of the recovered μ, as noted ${\overline{\mathit{\mu }}}_{\mathrm{ML}}$, in each set of synthetic catalogues is similar to μ_in, with deviations lower than 0.5% in all cases under study. However, we find that ${\overline{{\mathit{\sigma }}_{\mathit{v}}}}_{\mathit{,}\mathrm{ML}}$ for n = 50 catalogues overestimates σ_v,in more than 5% at Δσ ≳ 2.0, while we recover σ_v,in well even with Δσ = 2.5 (Fig. A.1, top panel) for n = 1000. This means that larger data sets are needed to recover the underlying distribution as the observational errors increase.

• 2.

  We also study the values recovered by a best least-squares (BLS) fit of Eq. (18) to the synthetic catalogues. We find that (i) the BLS fit recovers the right values of μ_in. This was expected, since the applied observational errors preserve the median of the initial distribution. (ii) The BLS fit overestimates σ_v,in in all cases. The recovered values depart from the initial one as expected from a convolution of two Gaussians with a variance σ_v,in and σ_o, where ${\mathit{\sigma }}_{\mathit{v,}\mathrm{BLS}}\mathit{/}{\mathit{\sigma }}_{\mathit{v,}\mathrm{in}}\mathrm{=}\sqrt{\mathrm{1}\mathrm{+}\mathrm{(}\mathrm{\Delta }\mathit{\sigma }{\mathrm{)}}^{\mathrm{2}}}$. The MLE performs a de-convolution of the observational errors, recovering accurately the initial cosmic variance (Fig. A.1, top panel).

• 3.

  The estimated variances of μ and σ_v are reliable. That is, the median variances ${\overline{\mathit{\sigma }}}_{\mathit{\mu }}$ and ${\overline{\mathit{\sigma }}}_{{\mathit{\sigma }}_{\mathit{v}}}$ estimated by the MLE are similar to the dispersion of the recovered values, as noted s_μ and s_σv, in each set of synthetic catalogues. The difference between both variances for μ is lower than 5% in all the probed cases. However, we find that ${\overline{\mathit{\sigma }}}_{{\mathit{\sigma }}_{\mathit{v}}}$ for n = 50 catalogues overestimates s_σv, which is more than 5% at Δσ ≳ 1.5: this is the limit of the MLE to estimate reliable uncertainties with this number of data (Fig. A.1, bottom panel). Because the estimated variance tends asymptotically to s_σv for a large number of data, ${\overline{\mathit{\sigma }}}_{{\mathit{\sigma }}_{\mathit{v}}}$ for n = 1000 catalogues deviates less from the expected value than for n = 50 synthetic catalogues. Note that the value of σ_v is still unbiased as such large observational errors (Fig. A.1, top panel), when the estimated variance σ_σv deviates from the expectations at large Δσ, and we can roughly estimate σ_σv through realistic synthetic catalogues as those in this Appendix.

• 4.

  The variances of the recovered parameters decreases with n and increases with σ_o. That reflects the loss of information due to the observational errors. Remark that the MLE takes these observational errors into account to estimate the parameters and their variance.

We conclude that the MLE developed in this Appendix is not biased, providing accurate variances, and we can recover reliable uncertainties of the cosmic variance σ_v in ALHAMBRA (n = 48) for Δσ ≲ 1.5. Note that reliable values of σ_v in ALHAMBRA are recovered at Δσ ≲ 2.0. We checked that the average Δσ in our study is 0.60 (the average observational error is ${\overline{\mathit{\sigma }}}_{\mathrm{o}}\mathrm{=}\mathrm{0.18}$), and the maximum value is Δσ = 0.85. Thus, the results in the present paper are robust against the effect of observational errors.

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Acknowledgments

We dedicate this paper to the memory of our six IAC colleagues and friends who met with a fatal accident in Piedra de los Cochinos, Tenerife, in February 2007, with a special thanks to Maurizio Panniello, whose teachings of python were so important for this paper. We thank the comments and suggestions of the anonymous referee, that improved the clarity of the manuscript. This work has mainly been funding by the FITE (Fondo de Inversiones de Teruel) and the projects AYA2006-14056 and CSD2007-00060. We also acknowledge the financial support from the Spanish grants AYA2010-15169, AYA2010-22111-C03-01 and AYA2010-22111-C03-02, from the Junta de Andalucia through TIC-114 and the Excellence Project P08-TIC-03531, and from the Generalitat Valenciana through the project Prometeo/2009/064. A.J.C. (RyC-2011-08529) and C.H. (RyC-2011-08262) are Ramón y Cajal fellows of the Spanish government.

References

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