The ALHAMBRA survey: An empirical estimation of the cosmic variance for merger fraction studies based on close pairs

A&A 564, A127 (2014) DOI: 10.1051/0004-6361/201322474 c ESO 2014 Astronomy & Astrophysics The ALHAMBRA survey: An empirical estimation of the cosmic variance for merger fraction studies based on close pairs, C. López-Sanjuan1,A.J.Cenarro 1, C. Hernández-Monteagudo1, J. Varela1,A.Molino 2, P. Arnalte-Mur3, B. Ascaso2, F. J. Castander4, A. Fernández-Soto5,6, M. Huertas-Company7,8,I.Márquez 2, V. J. Martínez6,9, J. Masegosa2, M. Moles1,2,M.Povi ´ c2,J.A.L.Aguerri 10,11,E.Alfaro 2, T. Aparicio-Villegas12,2, N. Benítez2, T. Broadhurst13, J. Cabrera-Caño14,J.Cepa 10,11, M. Cerviño2,10, D. Cristóbal-Hornillos1,A.DelOlmo 2, R. M. González Delgado2, C. Husillos2,L.Infante 15, J. Perea2,F.Prada 2, and J. M. Quintana2 1Centro de Estudios de Física del Cosmos de Aragón, Plaza San Juan 1, 44001 Teruel, Spain e-mail: [email protected] 2Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la astronomía s/n, 18008 Granada, Spain 3Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK 4Institut de Ciències de l’Espai (IEEC-CSIC), Facultat de Ciéncies, Campus UAB, 08193 Bellaterra, Spain 5Instituto de Física de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain 6Observatori Astronòmic, Universitat de València, C/Catedrático José Beltrán 2, 46980 Paterna, Spain 7GEPI, Paris Observatory, 77 av. Denfert Rochereau, 75014 Paris, France 8University Denis Diderot, 4 rue Thomas Mann, 75205 Paris, France 9Departament d’Astronomia i Astrofísica, Universitat de València, 46100 Burjassot, Spain 10 Instituto de Astrofísica de Canarias, vía Láctea s/n, La Laguna, 38200 Tenerife, Spain 11 Departamento de Astrofísica, Facultad de Física, Universidad de la Laguna, 38200 La Laguna, Spain 12 Observatório Nacional-MCT, Rua José Cristino 77, CEP 20921-400, Rio de Janeiro-RJ, Brazil 13 Department of Theoretical Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain 14 Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, 41012 Sevilla, Spain 15 Departamento de Astronomía y Astrofísica, Facultad de Física, Pontificia Universidad Católica de Chile, 782-0436 Santiago, Chile Received 12 August 2013 /Accepted 17 January 2014 ABSTRACT Aims. Our goal is to estimate empirically the cosmic variance that affects merger fraction studies based on close pairs for the first time. Methods. We compute the merger fraction from photometric redshift close pairs with 10 h−1kpc ≤rp≤50 h−1kpc and Δv≤500 km s−1and measure it in the 48 sub-fields of the ALHAMBRA survey. We study the distribution of the measured merger fractions that follow a log-normal function and estimate the cosmic variance σvas the intrinsic dispersion of the observed distribution. We develop a maximum likelihood estimator to measure a reliable σvand avoid the dispersion due to the observational errors (including the Poisson shot noise term). Results. Thecosmicvarianceσvof the merger fraction depends mainly on (i) the number density of the populations under study for both the principal (n1) and the companion (n2) galaxy in the close pair and (ii) the probed cosmic volume Vc. We do not find a significant dependence on either the search radius used to define close companions, the redshift, or the physical selection (luminosity or stellar mass) of the samples. Conclusions. We have estimated the cosmic variance that affects the measurement of the merger fraction by close pairs from observations. We provide a parametrisation of the cosmic variance with n1,n2,andVc,σv∝n−0.54 1V−0.48 c(n2/n1)−0.37. Thanks to this prescription, future merger fraction studies based on close pairs could properly account for the cosmic variance on their results. Key words. galaxies: interactions – galaxies: fundamental parameters – galaxies: statistics 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 Based on observations collected at the German-Spanish Astronomical Center, Calar Alto, jointly operated by the Max-Planck-Institut für Astronomie (MPIA) at Heidelberg and the Instituto de Astrofísica de Andalucía (IAA-CSIC).  Appendix is available in electronic form at http://www.aanda.org 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), zCOS- MOS (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 Article published by EDP Sciences A127, page 1 of 17 A&A 564, A127 (2014) 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 (σv), 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 σvis 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 σvin 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−1kpc), 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 arcmin2each 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 σv. The paper is organised as follows. In Sect. 2,wepresent the ALHAMBRA survey and its photometric redshifts, and in Sect. 3, we review the methodologyto 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 Ωm=0.3, ΩΛ=0.7, H0= 100 hkms−1Mpc−1,andh=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,andKs)over8dif- ferent 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 1http://alhambrasurvey.com 213.5 214.0 214.5 215.0 RA (deg) 52.2 52.4 52.6 52.8 Dec (deg) Fig. 1. Schematic view of the ALHAMBRA field’s geometry in the sky plane. We show the eight sub-fields (one per LAICA chip) of the field ALHAMBRA-6. The black and red squares mark the two LAICA pointings in this particular field. The geometry of the other seven fields is similar. A colour version of this plot is available in the electronic edition. energy distributions (SEDs) and precise photometric redshifts (zp) 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 (LargeArea 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 A127, page 2 of 17 C. López-Sanjuan et al.: The ALHAMBRA survey. An empirical estimation of the cosmic variance for merger fraction studies Table 1. ALHAMBRA survey fields. Field Overlapping RA DEC sub-fields /area name survey (J2000) (J2000) (# /deg2) ALHAMBRA-2 DEEP2 01 30 16.0 +04 15 40 8 /0.377 ALHAMBRA-3 SDSS 09 16 20.0 +46 02 20 8 /0.404 ALHAMBRA-4 COSMOS 10 00 00.0 +02 05 11 4 /0.203 ALHAMBRA-5 GOODS-N 12 35 00.0 +61 57 00 4 /0.216 ALHAMBRA-6 AEGIS 14 16 38.0 +52 24 50 8 /0.400 ALHAMBRA-7 ELAIS-N1 16 12 10.0 +54 30 15 8 /0.406 ALHAMBRA-8 SDSS 23 45 50.0 +15 35 05 8 /0.375 Total 48 /2.381 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 F814Wfilter image, as noted iin 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 (zs), is δz=0.0108 at i≤22.5 with a fraction of catastrophic outliers of η=2.1%. The variable δzis the normalized median absolute deviation of the photometric versus spectroscopic redshift distribution (Ilbert et al. 2006;Brammer et al. 2008), δz=1.48 ×median |zp−zs| 1+zs·(1) The variable ηis defined as the fraction of galaxies with |zp−zs|/(1 +zs)>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 O, 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 Kis specific for each photometric survey. Molino et al. (2013)findthatK=0.0125 is the optimal value for the ALHAMBRA survey. The parameter O∈[0,1] is related with the confidence of the zp, making it possible to derive high quality samples with better accuracy and lower rate of catastrophic outliers. For example, a O≥0.5 selection for i≤22.5 galaxies yields δz=0.0094 and η=1%, while δz=0.0061 and η=0.8% 0.20.40.60.81.01.21.41.6 zs 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 zp δz=0.011 η=2.1% Fig. 2. Photometric redshift (zp) versus spectroscopic redshift (zs)for the 3813 galaxies in the ALHAMBRA area with i≤22.5 and a measured zs. The solid line marks identity. The sources above and below the dashed lines are catastrophic outliers. The accuracy of the photometric redshifts (δz) and the fraction of catastrophic outliers (η) are labelled in the panel. A colour version of this plot is available in the electronic edition. for O≥0.9(seeMolino 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 (σzp) are needed to compute the merger fraction in photometric samples (Sect. 3). In addition to the zp,wehavethez+ σand the z− σof 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 σzp=C×(z+ σ−z− σ). The constant Cis estimated from the distribution of the variable Δz=zp−zs σzp =zp−zs C×(z+ σ−z− σ)·(2) The variable Δzshould be normally distributed with a zero mean and unit variance if the values of σzpfrom 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 Δzis described well by a normal function when C=0.49 (Fig. 3, see also Molino et al. 2013). With the definition of z+ σ and z− σ, 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 Cfor different i-band magnitudes and odds selections, finding that the Cvalues are consistent with the global one within ±0.1. Thus, we conclude that σzpprovides a reliable photometric redshift error for every ALHAMBRA source. A127, page 3 of 17 A&A 564, A127 (2014) −4−2024 Δz 0 50 100 150 200 250 300 350 N μ=−0.07 σ=1.00 C=0.49 Fig. 3. Distribution of the variable Δzfor the 3813 galaxies in the ALHAMBRA area with i≤22.5 and a measured spectroscopic redshift. The red line is the best least-squares fit of a Gaussian function to the data. The median, dispersion, and the factor Cderived from the fit are labelled in the panel. A colour version of this plot is available in the electronic edition. 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)anda 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, rp=φdA(z1), and their rest-frame relative velocity along the line of sight, Δv=c|z2−z1|/(1 +z1), where z1and z2are the redshift of the principal (more luminous or massive galaxy in the pair) and the companion galaxy, respectively; φis the angular separation in arcsec of the two galaxies on the sky plane; and dA(z) is the angular diameter distance in kpc arcsec−1at redshift z. Two galaxies are defined as a close pair if rmin p≤rp≤rmax pand Δv≤Δvmax. The PSF of the ALHAMBRA ground-based images is 1.4 (median seeing of ∼1), which corresponds to 7.6h−1kpc in our cosmology at z=0.9. To ensure well de-blended sources and to minimise colour contamination, we fixed rmin pto 10 h−1kpc (φ>1.8 at z<0.9). We left rmax p≤50 h−1kpc as a free parameter and estimate its optimal value in Sect. 4.3. Finally, we set Δvmax =500 kms−1by 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). 2http://cloud.iaa.es/alhambra/ 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 fm=Np N1,(3) where N1is the number of sources in the principal sample and Npthe number of close pairs. This definition applies to spectroscopic volume-limited samples, but we rely on photometric redshifts to compute fmin 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-Sanjuanet 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 z1and uncertainty σz1, assuming that the galaxy is located at z1−2σz1. This defines the maximum φpossible for a given rmax pin the first instance. If we find a companion galaxy with redshift z2and uncertainty σz2at rp≤rmax p, 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 [z−,z+]=[z1−2σz1,z1+2σz1]∩[z2−2σz2,z2+2σz2].(4) Because of the variation in the range [z−,z+] of the function dA(z), a sky pair at z1−2σz1might not be a pair at z1+2σz1. We thus impose the condition rmin p≤rp≤rmax pat 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 kin the redshift interval [z− k,z+ k], where the index kcovers 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 iin whatever sample is described in redshift space by a Gaussian probability distribution, Pi(zi|zp,i,σ zp,i)=1 √2πσzp,i exp ⎡⎢⎢⎢⎢⎢⎣−(zi−zp,i)2 2σ2 zp,i⎤⎥⎥⎥⎥⎥⎦·(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 z1in system kas νk(z1)=CkP1(z1|zp,1,σ zp,1)z+ m z− m P2(z2|zp,2,σ zp,2)dz2,(6) A127, page 4 of 17 C. López-Sanjuan et al.: The ALHAMBRA survey. An empirical estimation of the cosmic variance for merger fraction studies where z1∈[z− k,z+ k], the integration limits are z− m=z1(1 −Δvmax/c)−Δvmax/c,(7) z+ m=z1(1 +Δvmax/c)+Δvmax/c,(8) the subindex 1 [2] refers to the principal [companion] galaxy in the system k, and the constant Cknormalises the function to the total number of pairs in the interest range, 2Nk p=z+ k z− k P1(z1|zp,1,σ zp,1)dz1+z+ k z− k P2(z2|zp,2,σ zp,2)dz2.(9) Note that νk=0ifz1<z− kor z1>z+ k. The function νktells us how the number of pairs in the system k, as noted by Nk p, are distributed in redshift space. The integral in Eq. (6) spans those redshifts in which the companion galaxy has Δv≤Δvmax for a given redshift of the principal galaxy. This translates to z+ m−z− m∼0.005 in our redshift range of interest. With the previous definitions, the merger fraction in the interval zr=[zmin,zmax)is fm=kzmax zmin νk(z1)dz1 izmax zmin Pi(zi|zp,i,σ zp,i)dzi·(10) If we integrate over the whole redshift space, zr=[0,∞), Eq. (10) becomes fm=kNk p N1,(11) where kNk pis analogous to Npin Eq. (3). To estimate the observational error of fm, as noted by σf, we used the jackknife technique (Efron 1982). We computed partial standard deviations, δk, for each system kby taking the difference between the measured fmand the same quantity with the kth pair removed for the sample, fk m, such that δk=fm−fk m. For a redshift range with Np systems, the variance is given by σ2 f=[(Np−1) kδ2 k]/Np. 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 boundariesof 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., 1σdiscrepancy) only at rmax p70 h−1kpc. Thus, we restricted the search radius in our study to rmax p≤50 h−1kpc. We avoid the incompleteness in redshift space by including the sources in the samples inside the redshift range [zmin,zmax) under study and those sources with either zp,i+2σzp,i≥zmin or zp,i−2σzp,i<zmax. 3.2. The merger rate The final goal of merger studies is the estimation of the merger rate Rm, defined as the number of mergers per galaxy and Gyr−1. The merger rate is computed from the merger fraction by close pairs as Rm=Cm Tmfm,(12) where Cmis the fraction of the observed close pairs that finally merge after a merger time scale Tm. The merger time scale and the merger probability Cmshould 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, Tmdepends mainly on the search radius rmax p, 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, Cmdepends mainly on rmax pand 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 Tmand Cm,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 fm. 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 (σv) 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)andMoster et al. (2011), the mean Nand the variance N2−N2in the distribution of galaxies are given by the first and second moments of the probability distribution PN(Vc), which describes the probability of counting Nobjects within a volume Vc. The relative cosmic variance is defined as σ2 v=N2−N2 N2−1 N·(13) The second term represents the correction for the Poisson shot noise. The second moment of the object counts is N2=N2+N+N2 V2 cVc ξ(|ra−rb|)dVc,adVc,b,(14) where ξis 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 σ2 v=1 V2 cVc ξ(|ra−rb|)dVc,adVc,b.(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 ξdm,ξ=b2ξdm,wherebis 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 σv∝b V1−α c ,(16) where the power law index αtakes into account the extra volume dependence from the integral of the correlation function ξdm 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 A127, page 5 of 17 A&A 564, A127 (2014) 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 nof 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−βrelation is expected. This inverse dependence is indeed suggested by Nuza et al. (2013) results. In summary, we expect σv∝b V1−α c∝zγ nβV1−α c·(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 Vc, redshift, and the number density of the two populations under study, as noted by n1for principal galaxies and by n2for the companion galaxies, is expected. We therefore used these four variables (n1,n2,z,andVc) 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 fluxlimited samples. In the present paper, we use flux-limited samples selected in the iband to characterise the cosmic variance. This choice has several benefits, since we have a well-controlled selection function,a better understandingof the photometricredshifts 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.Inthe 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 σv estimation In this section, we explore which statistical distribution reproduces the observed merger fractions better and how to reliably measure the cosmic variance σv. As representative examples, we show the distributions of the merger fraction fmin 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−1kpc ≤rp≤30 h−1kpc. 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, PLN (fm|μ, σ)=1 √2πσfm exp −(ln fm−μ)2 2σ2,(18) 0.02 0.04 0.06 0.08 0.10 fm 0 2 4 6 8 10 12 14 16 N i≤22 0.3≤z<0.9 σ=0.33 σv=0.25 0.02 0.04 0.06 0.08 0.10 fm 0 2 4 6 8 10 12 14 16 N i≤21 0.3≤z<0.9 σ=0.62 σv=0.44 Fig. 4. Distribution of the merger fraction fmfor i≤22 (top panel) and i≤21 (bottom panel) galaxies in the 48 ALHAMBRA sub-fields, as measured from close pairs with 10 h−1kpc ≤rp≤30 h−1kpc at 0.3≤z<0.9. In each panel, the red solid line is the best least-squares fit of a log-normal function to the data. The star and the red bar mark the median and the 68% confidence interval of the fit, respectively. The black bar marks the confidence interval from the maximum likelihood analysis of the data and is our measurement of the cosmic variance σv. A colour version of this plot is available in the electronic edition. where μand σare the median and the dispersion of a Gaussian function in log-space f m=ln fm.Thisis, PG(f m|μ, σ)=1 √2πσ exp −(f m−μ)2 2σ2·(19) The 68% confidence interval of the log-normal distribution is [eμe−σ,eμeσ]. 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;Kovaˇ c 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 σencodes 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 eμ, 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. 4shows that σincreases 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 σis A127, page 6 of 17 C. López-Sanjuan et al.: The ALHAMBRA survey. An empirical estimation of the cosmic variance for merger fraction studies twofold: (i) the intrinsic dispersion due to the cosmic variance σv (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 σo(i.e., the uncertainty in the measurement of the merger fraction in a given field, including the Poisson shot noise term). Thus, the dispersion σreported in Fig. 4is an upper limit for the actual cosmic variance σv. We deal with this limitation by applying a maximum likelihood estimator (MLE) to the observed distributions. In Appendix A,wedevelopaMLE that estimates the more probable values of μand σv, assuming that the merger fraction follows a Gaussian distribution in log-space (Eq. (19)) that is affected by known observational errors σo. We prove that the MLE provides an unbiased estimation of μand σvand reliable uncertainties of these parameters. Applying the MLE to the distributions in Fig. 4,wefindthatσv is lower than σ, as anticipated, and that the cosmic variance increases with the apparent brightness from σv=0.25 ±0.04 for i≤22 galaxies to σv=0.44 ±0.08 for i≤21 galaxies. We constrain the dependence of σvon 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 andonredshiftin Sect. 4.4.3. That provides a complete description of the cosmic variance for merger fraction studies. We stress that our definition of σvdiffers from the classical definition of the relative cosmic variance presented in Sect. 4.1, which is equivalent to eσv. However, σvencodes 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 σv 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 galaxiesor old populationswith a strong 4000 Å break). In this section, we study how the merger fraction in ALHAMBRA depends on the Oselection 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 Ntot in a given volume and we observe a random fraction fobs of these galaxies, the merger fraction of the total population is fm=fm,obs ×f−1 obs,(20) where fm,obs is the merger fraction of the observed sample. In ALHAMBRA, we applied a selection in the parameter O,so Eq. (20) becomes fm=fm(≥Osel)×Ntot N(≥Osel),(21) Table 2. Cosmic variance σvas a function of the search radius rmax pfor O≥O sel =0.3 galaxies at 0.3≤z<0.9. rmax pσvσvσv (h−1kpc) (i≤22.5) (i≤21.5) (i≤21.0) 30 0.181 ±0.030 0.235 ±0.053 0.447 ±0.091 35 0.184 ±0.027 0.246 ±0.045 0.433 ±0.079 40 0.199 ±0.026 0.284 ±0.041 0.460 ±0.073 45 0.195 ±0.024 0.289 ±0.040 0.447 ±0.067 50 0.190 ±0.023 0.284 ±0.038 0.451 ±0.066 Average 0.190 ±0.011 0.272 ±0.019 0.448 ±0.033 0.00.20.40.60.8 Osel 0.02 0.04 0.06 0.08 0.10 0.12 0.14 fm Fig. 5. Merger fraction fmas a function of the odds selection Osel for i≤ 22.5 galaxies at 0.3≤z<0.9. The filled triangles, circles, and squares are for rmax p=30,40, and 50 h−1kpc close pairs, respectively. The open triangles are the observed merger fractions for rmax p=30 h−1kpc 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 fmat 0.3≤O sel ≤0.6forrmax p=30,40, and 50 h−1kpc close pairs, respectively. A colour version of this plot is available in the electronic edition. where N(≥Osel) is the number of galaxies with odds higher than Osel (i.e., galaxies with O≥O sel), Ntot is the total number of galaxies (i.e., galaxies with O≥0), and fm(≥Osel)isthemerger faction of those galaxies with O≥O sel. Because fmmust be independentof the Oselection, the study of fmas a function of Osel provides the clues about the optimal odds selection for merger fraction studies in ALHAMBRA. We show fmas a function of Osel for galaxies with i≤22.5at0.3≤z<0.9inFig.5.Wefind that •the merger fraction is roughly constant for 0.2≤O sel ≤0.6. This is the expected result if the merger fraction is reliable and measured in a non-biased sample. In this particular case, the Osel =0.2 (0.6) sample comprises 98% (66%) of the total number of galaxies with i≤22.5; •the merger fraction is overestimated for Osel ≤0.1. 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 Osel ≥0.7. 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−1kpc ≤rp≤30 h−1kpc, but we can use other values of rmax por searching over different samples. On the one hand, we repeated the study for rmax p=40 A127, page 7 of 17 A&A 564, A127 (2014) 30 35 40 45 50 rmax p[h−1kpc] 0.2 0.3 0.4 0.5 0.6 σv Fig. 6. Cosmic variance σvas a function of rmax pfor 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. and50h −1kpc, finding the same behaviour than for rmax p= 30 h−1kpc (Fig. 5). The only differences are that the merger fraction increases with the search radius and that the Osel =0.2 point starts to deviate from the expected value (the search area increases with rmax pand 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 rmax pcases. We find again the same behaviour. That reinforces our arguments above and suggests 0.3≤O sel ≤0.6 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 Rm(Sect. 3.2) is a physical property of any population, and it cannot depend on rmax p. 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, Rm∝fm(rmax p)/Tm(rmax p). For the same reason, the cosmic variance of the merger rate cannot depend on rmax p. In other words, the 68% confidence interval of the merger rate, [Rme−σv,Rmeσv], should be independent of the search radius. Expanding the previous confidence interval, we find that [Rme−σv,Rmeσv]∝(22) [fm(rmax p)T−1 m(rmax p)e −σv,fm(rmax p)T−1 m(rmax p)e σv]= [fm(rmax p)e −σv,fm(rmax p)e σv]T−1 m(rmax p). Note that the dependence on rmax pis encoded in the median merger fraction and in the merger time scale. Thus, the cosmic variance σvof 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 O≥O sel =0.3at0.3≤z<0.9. We find that σvis consistent with a constant value irrespective of rmax p in the three populations probed, as desired (Table 2and Fig. 6). This supports σvas a good descriptor of the cosmic variance and our methodology to measure it. In the previous analysis, we have omitted the merger probability Cm, which mainly depends on rmax pand environment (Sect. 3.2). The merger fraction correlates with environment, so the merger probability could modify the factor eσvin Eq. (22). Because a constant σvwith rmax pis observed, the impact of Cmin the fmto Rmtranslation should 0.00.20.40.60.8 Osel 0.1 0.2 0.3 0.4 0.5 σv Fig. 7. Cosmic variance σvas a function of the odds selection Osel for i≤22.5 galaxies at 0.3≤z<0.9. Triangles, circles, and squares are for rmax p=30,40, and 50 h−1kpc close pairs, respectively. The dotted, dashed, and solid lines mark the average σvat 0.1≤O sel ≤0.5for rmax p=30,40, and 50 h−1kpc close pairs, respectively. A colour version of this plot is available in the electronic edition. be similar in the range of rmax pexplored. Detailed cosmological simulations are needed to clarify this issue. Finally, we studied the dependence of σvon 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.Wefindthat(i)σvis consistent with a constant value as a function of rmax pfor any Osel, reinforcing our results above; and (ii) σvis independent of the odds selection at 0.1≤O sel ≤0.5(Fig.7). As for the merger fraction, we checked that different populations follow the same behaviour. We therefore set O≥O sel =0.3 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 σvfrom the merger fractions measured in the 48 ALHAMBRA sub-fields with 10 h−1kpc ≤rp≤50 h−1kpc close pairs (the σvuncertainty is lower for larger search radii) and in samples with O≥O sel =0.3 in the following. That ensures reliable results in representative (i.e., non-biased) samples. 4.4. Characterisation of σv 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 σvin 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 n1of 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 σvon either the probed cosmic volume and zand 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 Vc∼1.4×105Mpc3 in each ALHAMBRA sub-field. To explore different number A127, page 8 of 17 C. López-Sanjuan et al.: The ALHAMBRA survey. An empirical estimation of the cosmic variance for merger fraction studies 1234567 n1[10−3Mpc−3] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 σv σv=0.45 ×n−0.54 1 Fig. 8. Cosmic variance σvas a function of the number density n1of the principal population under study. Increasing the number density, the principal sample comprises i≤20, 20.5, 21, 21.5, 22, 22.5, and 23 galaxies, respectively. The probed cosmic volume is the same in all the cases, Vc∼1.4×105Mpc3(0.3≤z<0.9). The dashed line is the error-weighted least-squares fit of a power-law to the data, σv∝n−0.54 1. A colour version of this plot is available in the electronic edition. Table 3. Cosmic variance σvas a function of the principal sample’s number density n1. Principal n1σv sample (10−3Mpc−3) i≤23.06.88 ±0.16 0.158 ±0.019 i≤22.54.79 ±0.14 0.190 ±0.023 i≤22.03.30 ±0.11 0.245 ±0.030 i≤21.52.12 ±0.07 0.284 ±0.038 i≤21.01.28 ±0.05 0.451 ±0.066 i≤20.50.73 ±0.03 0.587 ±0.100 i≤20.00.35 ±0.01 0.695 ±0.154 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 n1in the redshift range zras the median number density in the 48 ALHAMBRA subfields with nj 1(zr)=izmax zmin Pj i(zi|zp,i,σ zp,i)dzi Vj c(zr)(23) being the number density in the sub-field jand Vj cthe cosmic volume probed by it at zr. In the measurement of the number density, all the galaxies were taking into account; i.e., any odds selection was applied (O≥0). We stress that our measured number densities are unaffected by cosmic variance, and they can be used therefore to characterise σv. 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 errorweighted least-squares fit of a power-law to the data is σv(n1)=(0.45 ±0.04) ×n1 10−3Mpc−3−0.54±0.06 .(24) In this section and in the following ones, we used i-band selected samples to characterise σv. 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. 0.20.40.60.81.01.21.4 Vc[105Mpc3] 0.5 1.0 1.5 2.0 2.5 3.0 σ∗ v σ∗ v=1.05 ×V−0.48 c Fig. 9. Normalised cosmic variance σ∗ vas a function of the probed cosmic volume Vcfor galaxies with i≤23. The circle corresponds to same data as in Fig. 8. The stars probe different redshift intervals, while triangles probe sky areas smaller than the fiducial ALHAMBRA sub-field. The dashed line is the error-weighted least-squares fit of a power-law to the data, σ∗ v∝V−0.48 c. A colour version of this plot is available in the electronic edition. 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 σ∗ vas σ∗ v=σv/σv(n1). 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 Vc∼0.1×105Mpc3to Vc∼1.4×105Mpc3. The power-law function that better describes the observations (Fig. 9)is σ∗ v(Vc)=(1.05 ±0.05) ×Vc 105Mpc3−0.48 ±0.05 .(25) We tested the robustness of our result by fitting the two sets of data (variation in redshift and area) separately. We find σ∗ v∝ V−0.43±0.08 cfor the redshift data, while σ∗ v∝V−0.48 ±0.05 cfor the area data. 4.4.3. Dependence on redshift The redshift is an expected parameter in the parametrisation the cosmic variance. However, Fig. 9shows 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 σvin different, non-overlappingredshift bins. We summarise our measurements, as performed for i≤23 galaxies, in Table 5.Wedefined σ∗∗ v=σv/σv(n1,Vc) to isolate the redshift dependence of the cosmic variance. We find that σ∗∗ vis compatible with unity, σ∗∗ v=1.02±0.07, and that no redshift dependence remains after accounting for the variation in n1and Vc(Fig. 10). This confirms our initial hypothesis and we assume, therefore, γ=0in the following. A127, page 9 of 17 A&A 564, A127 (2014) Appendix A: Maximum likelihood estimation of the cosmic variance σv 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 agiven the parameters bas P(a|b). Being xjthe measured values in the ALHAMBRA field j,andθthe parameters that we want to estimate, we may express the joined likelihood function as L(xj|θ)≡−ln ⎡⎢⎢⎢⎢⎢⎢⎣ j P(xj|θ)⎤⎥⎥⎥⎥⎥⎥⎦ =− j ln P(xj|θ).(A.1) If we are able to express P(xj|θ) analytically, we can minimise Eq. (A.1) to obtain the best estimation of the parameters θ,asdenoted as θML. In our case, xjis the observed value of the merger fraction in log-space for the ALHAMBRA sub-field j,where xj≡f m,j=ln fm,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, PG(f real,j|μ, σv)=1 √2πσ v exp ⎡⎢⎢⎢⎢⎢⎣−(f real,j−μ)2 2σ2 v⎤⎥⎥⎥⎥⎥⎦·(A.2) Observational errors cause the observed f m,jto differ from their respective real values f real,j. The observed f m,jareassumedtobe extracted for a Gaussian distribution with mean f real,jand standard deviation σo,j(the observational errors), PG(f m,j|f real,j,σ o,j)=1 √2πσ o,j exp ⎡⎢⎢⎢⎢⎢⎣− (f m,j−f real,j)2 2σ2 o,j⎤⎥⎥⎥⎥⎥⎦·(A.3) We assumed that the observational errors are Gaussian in logspace, or, that they are log-normalin 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 σo=σf/fm. We checked that the values of σoderived 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(xj|θ) of each ALHAMBRA sub-field by the total probability theorem: P(f m,j|μ, σv,σ o,j)= ∞ −∞ PG(f real,j|μ, σv)×PG(f m,j|f real,j,σ o,j)df real,j,(A.4) where f m,j=xjand (μ, σv,σ o,j)=θin Eq. (A.1). Note that the values of σo,jare the measured uncertainties for each ALHAMBRA sub-field, so the only unknowns are the variables μand σv, which we want to estimate. Note also that we integrate over the variable f real,j, 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 L(f m,j|μ, σv,σ o,j)=−1 2 j ln(σ2 v+σ2 o,j)+ (f m,j−μ)2 σ2 v+σ2 o,j·(A.5) With the minimisation of this function, we obtain the best estimation of both μand the cosmic variance σv, 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 L(f m,j|μ, σv,σ o,j) in a Taylor’s series of its variables θ=(μ, σv,σ o,j) around the minimisation point θML. The previous minimisation process made the first Lderivative null, and we obtain L=L(θML)+1 2(θ−θML)THxy(θ−θML),(A.6) where Hxyis the Hessian matrix, and Tdenotes the transpose matrix. The inverse of the Hessian matrix provides an estimate of the 68% confidence intervals of μML and σML,aswellasthe covariance between them. The Hessian matrix of the joined likelihood function Lis defined as Hxy=⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂2L ∂μ2∂2L ∂μ ∂σv ∂2L ∂σv∂μ ∂2L ∂σ2 v ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,(A.7) with ∂2L ∂μ2=− i 1 σ2 v+σ2 o,j ,(A.8) ∂2L ∂μ ∂σv =∂2L ∂σv∂μ =−2 i σv(f m,j−μ) (σ2 v+σ2 o,j)2,(A.9) and ∂2L ∂σ2 v = i (σ2 o,j−3σ2 v)×(f m,j−μ)2 (σ2 v+σ2 o,j)3−(σ2 o,j−σ2 v) (σ2 v+σ2 o,j)2·(A.10) Then, we computed the inverse of the minus Hessian, hxy= (−Hxy)−1. Finally, we estimated the variances of our inferred parameters as σ2 μ=h11 and σ2 σv=h22 because maximum likelihood theory states that σ2 θx≤hxx. 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 nof merger fractions randomly drawn from a log-normal distribution with μin =log0.05 and σv,in =0.2and affected by observational errors σo. We explored the n=50,250 and 1000 cases for the number of merger fractions and varied the observational errors from σo=0.1 to 0.5 in 0.1 steps. That is, we explored observational errors in the measurement of the merger fraction from Δσ≡σo/σv=0.5 to 2.5 times the cosmic A127, page 16 of 17 C. López-Sanjuan et al.: The ALHAMBRA survey. An empirical estimation of the cosmic variance for merger fraction studies 0.51.01.52.02.5 Δσ 1.0 1.5 2.0 2.5 3.0 σv,out/σv,in 0.51.01.52.02.5 Δσ 0.8 1.0 1.2 1.4 1.6 1.8 2.0 σσv/sσv Fig. A.1. Recovered cosmic variance over input cosmic variance (top panel) and median σσvover the dispersion of the recovered cosmic variance (bottom panel) as a function of Δσ. In both panels, triangles, circles, and squares are the results from synthetic catalogues with n=50,250, and 1000, respectively. White symbols show the results from the BLS fit to the data (σv,BLS), while those coloured show the ones from the MLE (σv,ML). The n=50 and 1000 points are shifted to avoid overlap. The dashed lines mark identity, and the solid line in the top panel shows the expectation from a convolution of two Gaussians in log-space, σv,BLS/σv,in =1+(Δσ)2. variance that we want to measure. We checked that the results below are similar for any value of σv,in.Wefindthat 1. The median value of the recovered μ, as noted μML, in each set of synthetic catalogues is similar to μin, with deviations lower than 0.5% in all cases under study. However, we find that σv,ML for n=50 catalogues overestimates σv,in more than 5% at Δσ2.0, while we recover σv,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 μin.This was expected, since the applied observational errors preserve the median of the initial distribution. (ii) The BLS fit overestimates σv,in in all cases. The recovered values depart from the initial one as expected from a convolution of two Gaussians with a variance σv,in and σo,whereσv,BLS/σv,in = 1+(Δσ)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 μand σvare reliable. That is, the median variances σμand σσvestimated by the MLE are similar to the dispersion of the recovered values, as noted sμand sσv, in each set of synthetic catalogues. The difference between both variances for μis lower than 5% in all the probed cases. However, we find that σσvfor n=50 catalogues overestimates sσv, 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 sσvfor a large number of data, σσvfor n=1000 catalogues deviates less from the expected value than for n=50 synthetic catalogues. Note that the value of σvis still unbiased as such large observational errors (Fig. A.1, top panel), when the estimated variance σσvdeviates from the expectations at large Δσ,and we can roughly estimate σσvthrough realistic synthetic catalogues as those in this Appendix. 4. The variances of the recovered parameters decreases with n and increases with σo. 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 σvin ALHAMBRA (n=48) for Δσ1.5. Note that reliable values of σvin ALHAMBRA are recovered at Δσ2.0. We checked that the average Δσin our study is 0.60 (the average observational error is σo=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. A127, page 17 of 17