GALAXY DISKS DO NOT NEED TO SURVIVE IN THE ΛCDM PARADIGM: THE GALAXY MERGER RATE OUT TO z ∼ 1.5 FROM MORPHO-KINEMATIC DATA

In the frame of the IMAGES survey, a sample of 35 emission-line galaxies at z = [0.4–0.75], i.e., about six billion years ago (mean/median redshift of z = 0.6/0.65), were observed in the CDFS (Yang et al. 2008). This subsample of IMAGES is of particular interest since it benefits from homogeneous imaging and three-dimensional data (Yang et al. 2008; Neichel et al. 2008). IMAGES targets were selected to have I_AB ⩽ 23.5 and ${\rm EW}_0([{\rm O}\,\mathsc{ ii}])\ge 15{\rm \,\AA}$ to guarantee their detection by the multi-IFU spectrograph FLAMES/GIRAFFE at Very Large Telescope (VLT; Ravikumar et al. 2007; Flores et al. 2006). These galaxies were selected to be intermediate-mass galaxies (M_stellar = 10¹⁰–10¹¹ M_☉ using a diet Salpeter IMF and simplified prescriptions to convert J-band luminosity and B − V colors into stellar mass, following Bell et al. 2003). This range of mass is of particular interest since most of the star formation at z ⩽ 1 occurred in such galaxies (Hammer et al. 2005; Bell et al. 2005; Zheng et al. 2007). Since the fraction of E/S0 did not raise between z ∼ 0.6 and z = 0 (Delgado-Serrano et al. 2010), z ⩽ 1 intermediate-mass galaxies are the likeliest progenitors of local spirals (see also Appendix B).

From the full IMAGES sample, one galaxy was rejected because it turned out not to be a starburst galaxy (Hammer et al. 2009a; Yang et al. 2009), and another one because the HST/ACS images were corrupted and no morphological analysis could be conducted. The resulting sample is fully representative of the J-band luminosity function at z = 0.5–1.0 (Yang et al. 2008). At these redshifts, the near-infrared luminosity functions of the global population and of the blue sub-population of galaxies are very similar (Cirasuolo et al. 2007). At first order, emission-line and blue galaxies sample the same population (Delgado-Serrano et al. 2010), which in principle guarantees that the IMAGES-CDFS sample is representative of the star-forming population of galaxies at these epochs.

The total star formation rate (SFR) of all 33 remaining galaxies in the IMAGES-CDFS sample was estimated as the sum of the unobscured SFR measured from the rest-frame UV luminosity at 2800 Å combined with the dust-reprocessed contribution, which was estimated from the IR luminosity as measured using 24 μm Spitzer fluxes (see Puech et al. 2010 for details). We find a mean SFR of ∼12 M_☉ yr⁻¹, which compares well with other UV+IR measurements in larger samples at a similar range of mass and redshift (e.g., Jogee et al. 2009). In Figure 1, the location of the IMAGES-CDFS galaxies in the SFR-M_stellar plan is compared to the sequence of starbursting galaxies observed by Noeske et al. (2007) at z = 0.45–0.7. As expected, the IMAGES-CDFS sample comprises luminous infrared galaxies, which populates the high-SFR tail of the distribution at SFR ⩾ 14 M_☉ yr⁻¹ (Le Floc'h et al. 2005), and less obscured (UV) starbursts, which populate the low-SFR tail of the distribution. This figure reveals that most (91%) IMAGES-CDFS galaxies lie well within the observed SFR-M_stellar sequence at these redshifts. The SFR density in the IMAGES-CDFS sample is found to be log (ρ_sfr) ∼ −1.4 (where ρ_sfr is expressed in M_☉ yr⁻¹ Mpc⁻³), also in good agreement with expectations from larger surveys that include IR data at 24 μm (Zheng et al. 2007).⁷ This confirms that the IMAGES-CDFS sample is representative of star-forming galaxies at z ∼ 0.6.

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Gas masses were estimated from the total SFR by inverting the Schmidt–Kennicutt law (Puech et al. 2010). All galaxies were found in a range of total baryonic mass log (M_b/M_☉) = 10.2 to 11, or log (M_stellar/M_☉) in the range 10 to 11. The stellar-mass density in the IMAGES-CDFS sample is found to be log (ρ_stellar) ∼8.1 (where ρ_stellar is expressed in M_☉ Mpc⁻³), which is in relatively good agreement with expectations from larger surveys of blue galaxies (Arnouts et al. 2007; Bell et al. 2007), within expected uncertainties.

Detailed color morphology (at a ∼500 pc resolution scale) and large-scale kinematics (at a ∼7 kpc resolution scale) were obtained thanks to the HST and three-dimensional spectroscopy using the GIRAFFE multi-IFU spectrograph, respectively. Galaxies were classified independently according to their morphology (Neichel et al. 2008) and kinematics (Yang et al. 2008), based on similarities with the properties of local galaxies.

The kinematical maps allowed us to classify the IMAGES-CDFS galaxies into three distinct kinematic classes (see Yang et al. 2008). RDs were those for which the velocity field (VF) showed an order gradient, a dynamical axis aligned with the morphological axis, and a velocity dispersion peak located at the dynamical center. For these objects, it was required that the sigma peak location and amplitude can be reproduced by modeling the velocity dispersion map from the velocity gradient observed in the VF. At the GIRAFFE spatial resolution, it is indeed expected that most of the rotation curve gradient falls into the GIRAFFE pixel, which results in a velocity dispersion peak at the dynamical center of the object. Conversely, galaxies with complex kinematics (CKs) were those for which no ordered gradient was observed in the VF, or when a significant misalignment between the kinematical rotation axis and the morphological principle axis was detected. Perturbed rotators (PRs) gathered objects with an order velocity gradient in the VF, but for which the velocity dispersion peak (location and/or amplitude) could not be reproduced from the VF, which indicates that this peak cannot be attributed to rotation. This classification was tested against quantified criteria and considering all uncertainties associated with kinematic measurements, which makes it objective and reproducible, as described extensively in Yang et al. (2008).

The detailed color morphology of all 33 galaxies in the IMAGES-CDFS sample was studied by Neichel et al. (2008). Using a reproducible decision tree relying on parametric parameters and visual inspection of images and color maps, galaxy morphologies were classified into six different classes: Compact Systems, Mergers, Peculiar/Possible Mergers, Peculiar Tadpoles, Peculiar Irregulars, and Spiral Disks. In particular, it was required for RDs to show redder colors toward the center as observed in local galaxies. This classification was done by three independent classifiers using a decision tree, which limits the subjectivity and makes it reproducible. The comparison between the morphological and kinematical classes shows a very good agreement, with more than 80% of spiral disks having RDs and more than 90% of peculiar galaxies having complex or perturbed kinematics.

Since perturbations in the (ionized) gaseous phase generally correspond to perturbations in the stellar phase, the result of both the morphological and kinematical classifications can be robustly synthesized into a morpho-kinematic classification (Hammer et al. 2009a). With this classification, rotating spiral disks are those classified both as RDs from the kinematic point of view and as spiral disks from the morphological point of view. Non-relaxed systems are those that show CKs and a peculiar morphology. Semi-relaxed systems are galaxies that show either some rotation from their kinematics (being classified as either a RD or either a perturbed rotation) but with a peculiar morphology or galaxies that do not show rotation in their kinematics (being classified as having CKs) but with a spiral disk morphology. According to this classification, 80% of z ∼ 0.6 starburst galaxies are non-relaxed or semi-relaxed systems (Hammer et al. 2009a), which corresponds to ∼50% of all intermediate-mass, z ∼ 0.6 galaxies, since 60% of intermediate-mass, z ∼ 0.6 galaxies are indeed starbursts (Hammer et al. 1997; Mignoli et al. 2005). The HST/ACS images and kinematic maps for a representative sub-sample of the IMAGES-CDFS can be found in Figure 6 (see Appendix A).

In the IMAGES sample, kinematic perturbations are detected over a ⩾7 kpc spatial scale, which corresponds to the GIRAFFE-IFU spatial resolution element (i.e., two spaxels). They generally affect the galaxies across a significant fraction of their diameter, which, in principle, excludes de facto any mechanism resulting in local kinematic disturbances. In order to quantify this, we derived the Jeans scale length L_J (Bournaud & Elmegreen 2009) in the IMAGES-CDFS, which is found to be 0.7 ± 0.1 kpc. L_J increases to 1.0 ± 0.4 kpc in the sub-sample of clumpy galaxies that show rotation (Puech 2010). Thus, such Jeans instabilities should remain largely undetected at the GIRAFFE IFU spatial resolution. In the bottom panel of Figure 2 we show J033230.57–274518.2, one of the clumpiest galaxies in the IMAGES-CDFS sample. This galaxy shows at least seven clumps spread over its disks and is classified as CK. However, the kinematic perturbations are mostly large scale, with a misalignment between the morphological and kinematic axes as defined by the location of the maximal and minimal rotation velocities, and with a velocity dispersion peak that cannot be reproduced by the rotation model inferred from the VF (Yang et al. 2008; Puech et al. 2010). This confirms that even in the most disturbed clumpy galaxies, the kinematic perturbations cannot be associated with the position of the clumps, i.e., they are not local but global. In addition, clump instabilities are limited to a fraction of 20% of intermediate-mass galaxies at z ∼ 0.6 that are effectively clumpy (Puech 2010), which also rules out clump instabilities as a driver for the strongest kinematic perturbations.

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Regarding minor mergers, such processes are also expected to lead to local kinematic perturbations only. An obvious case of minor merger with a mass ratio of ∼1:18 was studied in detail by Puech et al. (2007a): only a local signature is found in the velocity dispersion map (see top panel in Figure 2). Even if we cannot exclude that in very specific cases such local processes could result in large kinematic disturbances such as those observed in IMAGES galaxies, we can safely conclude that they cannot be the main driver for the majority of galaxies, at least for the CK galaxies. This does not mean that minor mergers are not ongoing in the IMAGES sample, but given the spatial resolution of GIRAFFE-IFU data, they probably remain mostly undetected.

Regarding the impact of processes resulting in global disturbances, it is instructive to consider the range of the relative variations of the gravitational potential (compared to the dynamical equilibrium state) Δϕ/ϕ, in which galaxies can be considered as isolated and subject to secular evolution, as a function of mass and redshift (Hopkins et al. 2010b). We estimated Δϕ/ϕ for galaxies in the IMAGES-CDFS sample using rotation kinematic models (Puech et al. 2008). For this, we derived the pixel-by-pixel relative difference between the observed VF and each galaxy model and considered the sum ΔV/V of this map over all pixels, weighted by the observed signal-to-noise ratio (S/N) in each pixel. The quantity 2.ΔV/V can then be used as a proxy for Δϕ/ϕ. We also derived the dynamical timescale t_dyn for each galaxy in the IMAGES-CDFS sample using as a proxy the ratio between the gas radius and the rotation velocity as determined by Puech et al. (2010). In galaxies showing rotation (RDs+PRs), we found that Δϕ/ϕ = 1.2 ± 0.4 (median value) and t_dyn = 60 ± 3 Myr, while for CK galaxies, Δϕ/ϕ = 3.1 ± 1.0 and t_dyn = 60 ± 24 Myr. Uncertainties correspond to a 1σ error bar and were estimated using bootstrap resampling.

In this regime (i.e., t_dyn  ≳  10 Myr and Δϕ/ϕ  ≳  1), the two dominant physical processes expected to drive gravitational perturbations in this range of mass and redshifts are secular evolution (i.e., instabilities occurring in a galaxy) and the direct dynamical evolution of individual substructures due to some previously initiated perturbation, which can eventually dominate the evolution of the whole system (Hopkins et al. 2010b). Large global perturbations (i.e., Δϕ/ϕ  ≳  1) are typically triggered by major mergers. The characteristic timescale for secular evolution can be estimated using the circular disk period, i.e., P_d = 2πR/V_c, with R the disk radius and V_c the circular velocity. We used again gas radius and the rotation velocity as determined by Puech et al. (2010) as proxies for these two quantities. The timescale for dynamical friction associated with the dynamical evolution of substructure was estimated following the simplified prescription given by Hopkins et al. (2010b):

$$\begin{equation*} t_{{\rm df}}=\frac{0.2}{\delta _0 ln(\Lambda)}P_d, \end{equation*}$$

where δ₀ = Δϕ/ϕ and Λ = 1+1/δ₀. This timescale quantifies how fast the dynamical evolution of substructures evolves on its own, once they have been triggered. For instance, a merging galaxy will trigger instabilities in the main progenitors that will subsequently evolve on their own, on a timescale given by t_df (Hopkins et al. 2010b). The dominant process at given amplitude of perturbation is the one associated with the shortest timescale.

In Figure 3, we plotted the expected range of values of these two regimes at z ∼ 0.6 in the IMAGES-CDFS sample. Secular evolution is found to be dominant at small relative variations of the gravitational potential Δϕ/ϕ, while the influence of (major) mergers dominates above Δϕ/ϕ  ⩾  0.3. More specifically, isolated galaxies should be found in the upper left region of Figure 3, close to the gray region (as well as large local disks such as M31), while ongoing mergers and merger remnants should be found at Δϕ/ϕ  ∼  1 and close to the red region. Comparing with these theoretical expectations, CK galaxies lie closer to the individual dynamical perturbation regime (i.e., the merger-driven regime), which is consistent with (major) merger-driven kinematic disturbances in these galaxies. Galaxies showing rotation also lie in this regime, although with a smaller range of gravitational perturbation, which is consistent with these galaxies being more (temporally) advanced merger remnants, as suggested by their other morpho-kinematic properties (see Hammer et al. 2009a and Section 4.1). Overall, z ∼ 0.6 rotators (RDs+PRs) lie closer to the expected merger-driven regime, which probably reflects the fact that they are closer to dynamical equilibrium than CK galaxies, in which gravitational perturbations are about three times larger. This suggests that any other known process (gas accretion or secular evolution) is very unlikely to be responsible for the high level of kinematic disturbance observed on large spatial scales, at least for the majority of the IMAGES-CDFS sample.

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In this context, it was explored whether hydrodynamical major merger models for the 27 galaxies in the IMAGES-CDFS sample that are not relaxed morphologically and/or kinematically could generally account for their perturbations (Hammer et al. 2009a, 2009b; Puech et al. 2009; Peirani et al. 2009; Fuentes-Carrera et al. 2010). Morphological and kinematic maps from these simulations were produced and degraded to the spatial resolution of the observations. Such maps were then explored visually as a function of time and viewing angles to determine the best match to observations. The so-determined best model was graded by three different examiners (independent of the one who determined the best model) using specific constraints (Hammer et al. 2009a). Secure major merger candidates (18/27) were identified as galaxies having a model that fitted observations with a good level of confidence (i.e., grades ⩾4/6). Examples of such models are shown in Appendix A. These candidates can be used as a basis to estimate the fraction of mergers and the merger rate at different epochs as detailed in Section 4.1. Interestingly, the modeling exercise led to a median merger mass ratio μ = 3.03 ± 0.34 in the IMAGES-CDFS sample, while theoretical expectations from variations of the gravitational potential give consistent predictions with Δϕ/ϕ ∼ μ = 3.1 ± 1.0 in CK galaxies (see above). We refer the reader to Hammer et al. (2009a) for more details about the comparison between simulations and observations. This comparison suggests that at least 33% of z ∼ 0.6 galaxies (50% of galaxies not relaxed morpho-kinematically according to the morpho-kinematic classifications × the ratio of secured models, which is 18/27 = 66%) are involved in major mergers with a median mass ratio ∼3.

All the merger phases that are expected to lead to a significant SFR enhancement, i.e., the pre-fusion, fusion/post-fusion, and relaxation phases (Larson & Tinsley 1978), are homogeneously populated as a function of time by the IMAGES-CDFS galaxies, as evidenced by Figure 4. In this figure, SFRs were homogenized by dividing the observed SFR by the estimated mass of gas in each galaxy and multiplying each derived number by the average mass of gas in the sample. The time sequence reproduces quite well expectations from modeling the star formation during a typical merger (see insert). This shows that morpho-kinematic data such as those provided by the IMAGES survey are sensible to all the phases of a gas-rich major merger event.

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Among the galaxies populating the relaxation phase, six of them are rotating, but they are clearly distinct of local RDs because of their high average SFR and higher gas velocity dispersion (Puech et al. 2007a; Epinat et al. 2010). The latter suggests that these galaxies are possibly observed during a post-major-merger relaxation phase (Robertson & Bullock 2008; Monreal-Ibero et al. 2010; see also Figure 3 and Section 3.2). Most of the star formation activity is indeed concentrated at the disk outskirts, which suggests an ongoing inside-out growth of their disks (Neichel et al. 2008; Hammer et al. 2009a). The time sequence in Figure 4 empirically accounts for a cosmological distribution of merger parameters, since it is constructed using a representative sample. It can thus be used to estimate accurately the evolution of the merger fraction and merger rate as follows.

Theory usually gives the merger rate at the time when halos start merging, i.e., where galaxies that inhabit these halos are still in pairs. The epoch when a given merging system was still a pair is dated using individual numerical models from Hammer et al. (2009a). A homogeneous and self-consistent determination of the merger rate as a function of look-back time can be estimated by simply counting the fraction of galaxies in each phase and dividing this number by the corresponding average visibility timescale provided by the modeling of IMAGES galaxies. This method assumes that the visibility timescales do not evolve with redshift. Such an assumption is consistent with semi-analytic models, which predict almost no evolution of the merger visibility timescale as a function of look-back time, at least within uncertainties (Kitzbichler & White 2008). Another possible caveat is the observed evolution of the galaxy gas fraction with redshift (Puech et al. 2010; Daddi et al. 2010; Rodrigues et al. 2012; Erb et al. 2006), which could potentially result in evolving timescales depending on the merging phase probed (Lotz et al. 2010b). However, hydrodynamical simulations of binary mergers (Lotz et al. 2010b) predict that unequal mass ratio mergers (which dominate the empirical distribution of merger mass ratios; see Section 3.2) generally show no evolution of the pair visibility timescale with gas fraction.

4.2.1. Pre-fusion and Post-fusion Phases

4.2.2. Relaxation Phase

The merger rate r [Gyr⁻¹] corresponding to the three merger phases considered above (i.e., the pre-fusion, post-fusion, and relaxation phases) can be estimated using the ratio between the corresponding merger fraction and visibility timescale (see Table 1). Error bars due to random fluctuations in the sub-samples were estimated by propagating the random uncertainties on the fraction of mergers and on the visibility timescale. The derived merger rates and their associated uncertainties are shown in Figure 5.

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For selection criteria similar to those of the IMAGES-CDFS sample, a fraction of galaxies in pairs ∼2%–6% at z ∼ 0.6 is generally found (Rawat et al. 2008; Kartaltepe et al. 2007; Bundy et al. 2009). In the present study, the fraction of galaxies involved in major mergers is found to be 10% ± 3% at z ∼ 0.7. Such a fraction is larger than the fraction of galaxies in pairs since the pair technique only picks up progenitors when they are approaching for the first time and does not consider progenitors between the first and second passage and/or when they are getting too close to each other, while morpho-kinematic data are sensitive to all phases. This is reflected in the visibility timescale associated with pair fractions, which is often assumed to be ∼0.5 Gyr from analytic arguments or independent simulations (i.e., which do not necessarily reflect the exact properties of the observations; see also Lotz et al. 2011). Using such a timescale and a fraction of galaxies in pairs ∼2%–6% indeed leads to a merger rate ∼4%–12% Gyr⁻¹, which is consistent with our z ∼ 0.7 estimate of 5.5%. When no lower limit to the distance between the two progenitors is imposed, the pair technique gives very similar results, with ∼9% of galaxies in pairs at z ∼ 0.75 and ∼15% at z ∼ 1 (Ryan et al. 2008).

The morphological technique provides estimates of the fraction of morphologically disturbed mergers of 7%–11% at z ∼ 0.7, 11%–17% at z ∼ 1.1, and 27% ± 11% at z ∼ 1.4 for similar mass ranges and depending on the method used (CAS/GM system; see Conselice et al. 2008; Lotz et al. 2008b; Conselice et al. 2009; or visual inspection; see Bridge et al. 2010; Jogee et al. 2009). Within uncertainties, this is consistent with the fraction of mergers found using morpho-kinematic data (see Table 1).

We conclude that alternative techniques generally give consistent estimates, although with a resulting large range of values (Hopkins et al. 2010a), which reflects systematic effects due to their limited sensitivity to the merging process and different sample selection effects (see also Lotz et al. 2011).

A first possible cause of systematic effects is linked to the selection criteria of the IMAGES-CDFS sample, and in particular to the criterion ${\rm EW}_0([{\rm O}\,\mathsc{ ii}])\ge 15{\rm \,\AA}$. Indeed, we assumed that all quiescent (${\rm EW}_0([{\rm O}\,\mathsc{ ii}])< 15{\rm \,\AA}$) galaxies are not major mergers, whereas a significant fraction of them (25%) were found to have peculiar morphologies (Delgado-Serrano et al. 2010), which could be linked to less active mergers. Indeed, numerical simulations have revealed that major mergers do not always lead to strong enhancements of the SFR (Di Matteo et al. 2008), and therefore these peculiar quiescent galaxies might also be undergoing major merger events. This could potentially raise the fraction of mergers by 10.0% since quiescent galaxies represent 40% of intermediate-mass galaxies at z ∼ 0.6 (Delgado-Serrano et al. 2010). Since we do not have kinematic data for these objects, this could potentially affect any of the three merger phases considered in this study. We therefore considered that, on average, the merger fraction in each phase could be affected by a ∼3.3% systematic effect upward (see Table 1).

The number of mergers in the pre-fusion phase could also be overestimated due to flyby interlopers. We estimated that 4 out of the 11 objects in this phase could be possible flybys and escape a subsequent merger (J033217.62-274257.4, J033219.32-274514.0, J033220.48-275143.9, and J033238.60-274631.4). These objects correspond to cases where the two potential progenitors are well separated and photometric redshifts are consistent, but no reliable spectroscopic information was available to confirm whether or not the two galaxies were indeed gravitationally bound. This can potentially have an impact on the fraction of mergers in the pre-fusion phase by 3.7% downward, and on the corresponding merger rate by 2.0% (see Table 1).

Six galaxies in the IMAGES-CDFS are consistent with expectations from clumpy galaxies (Puech 2010), i.e., gas-rich Toomre-unstable disks. These six galaxies are found to be in the post-fusion phase (3 out of 6). Since the numerical merger models do not consider small-scale perturbations such as clumps, these galaxies might correspond to Toomre-unstable disks driven by internal instabilities, rather than major mergers. The post-fusion fraction of mergers and merger rate might therefore be affected by a 5.5% and a 3.3% effect downward, respectively (see Table 1). The impact on the pre-fusion and relaxation phases is marginal (one object) compared to other uncertainties, so we did not consider it further.

Regarding the relaxation phase, we added six objects corresponding to the post-fusion phase but that were not safely identified as mergers. This could be due to insufficiently long evolution times, so that the objects could not be safely identified, or to a too-limited exploration of the parameter space. Considering these six galaxies as possible interlopers for the relaxation phase, this would impact the corresponding fraction of mergers and merger rate by a 9.1% and a 5.1% effect downward, respectively (see Table 1).

Systematic uncertainties are shown as a gray region in Figure 5.

We used the Hopkins et al. (2010a) semi-empirical model⁹ to predict the merger rate in a given range of baryonic mass [M^min_b(z), M_b^max(z)], gas fraction [f^min_gas(z), f^max_gas(z)], and baryonic mass ratio between the two progenitors [μ^min_b(z), μ_b^max(z)] as a function of redshift z. Indeed, Hopkins et al. (2009a, 2010a) emphasized that the most appropriate mass would be the tightly bound material that survives stripping to strongly perturb the primary, which is found to be the baryonic plus tightly bound dark matter mass within a few disk scale lengths. The best observable proxy for this quantity is the baryonic mass (Hopkins et al. 2010a), which was therefore used in this study. In most studies, the galaxy merger rate is estimated in a fixed range of stellar or baryonic mass, constant with redshift, i.e., [M^min_b(z), M_b^max(z)] = [M^min_b, M_b^max]. Here, we need to estimate the proper range of baryonic mass [M^min_b(z), M_b^max(z)] at all z in order to sample the progenitors and descendants of the IMAGES-CDFS galaxies.

Given [M^min_b(z), M_b^max(z)], we determined the corresponding range of halo mass [M^min_h(z), M_h^max(z)] using the relationship between baryonic and halo masses M_b–M_h as provided by the Hopkins et al. (2010a) halo occupation model (HOD) using the mass function from Perez-Gonzalez et al. (2008). Using alternative mass functions changes the merger rate by less than 1%. We then estimate the average mass of the progenitor halos at z+dz following Neistein et al. (2006, 2008), i.e., [M^min_h(z + dz), M_h^max(z + dz)], which is then converted back to [M^min_b(z + dz), M_b^max(z + dz)] using the M_b–M_h relationship. The HOD model is used to estimate the galaxy merger rate at z+dz in the updated range of baryonic mass. This ensures that the derived galaxy merger rate is integrated over the population of the progenitors of all galaxies within [M^min_b(0), M_b^max(0)] at z = 0 only, and not over the entire mass spectrum, which would overestimate the merger rate compared to estimates from observations. In practice, we set [M^min_b(0)/M_☉, M^max_b(0)/M_☉] ≃ [10^10.4, 10^11.1] in order to sample the range [M^min_b(z = 0.6)/M_☉, M^max_b(z = 0.6)/M_☉] = [10^10.2, 10^11.0], which corresponds to the baryonic mass range of the IMAGES-CDFS sample (see Section 2). We tested that following the mean progenitor mass instead of the average progenitor mass between the upper and lower limits of the mass range does not change the resulting merger rate significantly.

To test the consistency of the adopted HOD model with the IMAGES sample, we checked that, according to the HOD model, the range of baryonic mass sampled at z = 0.6 corresponds to a range in log (M_stellar/M_☉) = [10^10.0, 10^11.0], which also matches the range of observed stellar mass in the IMAGES sample (see Section 2.1). At z = 0, the HOD model predicts that the z = 0.6 progenitors as defined by their range of baryonic mass as above will end up in a stellar mass range of log (M_stellar/M_☉) = [10^10.3, 10^11.1], corresponding to the baryonic mass range of log (M_b/M_☉) = [10^10.4, 10^11.1] as quoted above. The evolution in stellar mass as predicted by the HOD model is therefore ∼0.1–0.3 dex.

The median gas fraction in the progenitors of the IMAGES-CDFS sample was estimated using their observed SFRs (Hammer et al. 2009a). We used this to parameterize the expected gas fraction in the IMAGES-CDFS progenitors and descendants, using f_gas = {15%, 32%, 50%, 75%, 90%} at z = {0, 0.6, 0.72, 1.12, 1.55}. Estimates at z = 0 and 0.6 were derived using local H i measurements and inversion of the Schmidt–Kennicutt relation, respectively (Puech et al. 2010; Rodrigues et al. 2012). Estimates at z = 0.72, 1.12, and 1.55 were derived from the estimated median gas fractions in the pre-fusion, post-fusion, and relaxation phase, respectively (Hammer et al. 2009a). The HOD model predicts that the progenitors of z ∼ 0.6 ongoing mergers in the IMAGES-CDFS sample are galaxies with stellar mass in the range log (M_stellar/M_☉) ∼ [10^9.2, 10^10.4]. Gas fractions of the order of ∼90% in such a range of mass at z ∼ 1.5 are consistent with observational estimates of the gas fraction in more z ∼ 1.5 massive galaxies (see discussion in Section 6.2). We added a ±1σ scatter of 0.15 dex, which appears to represent well the observed scatter at all z (Hopkins et al. 2009b, 2010a; Rodrigues et al. 2012). At given z, the range of gas fraction [f^min_gas(z), f_gas^max(z)] over which the merger rate was integrated is interpolated from the f_gas(z) relationship as defined above, with an added ±2σ scatter. We explored the impact of a larger ±3σ scatter with no qualitative change in the results. Finally, we set [μ^min_b(z), μ_b^max(z)] = [μ^min_b, μ_b^max] = [0.25, 1.0], as found in the IMAGES-CDFS sample (Hammer et al. 2009a).

Figure 5 compares the derived estimates of the merger rate with state-of-the-art semi-empirical ΛCDM models (Hopkins et al. 2010a). This theoretical model is found to be in remarkable agreement with the observed estimates without any particular fine-tuning. At the redshifts and masses probed by the IMAGES-CDFS sample, semi-empirical (HOD-based) models can predict the merger rate within a factor of ∼1.5 (Hopkins et al. 2010c). HOD models are in relatively good agreement with each other because they rely on observational constraints (e.g., the galaxy stellar mass function) that are well constrained at z ⩽ 1 and for stellar masses around M* (Hopkins et al. 2010c). As stated above, changing the galaxy stellar mass function changes the resulting merger rate by less than ∼1%. The dominant source of random uncertainty is therefore due to variations from one (HOD) model to another, which is illustrated in Figure 5 as a red region. This is the same order of magnitude that random uncertainties associated with the observed merger rate (see Table 1). This implies that random uncertainties limit the comparison between observations and theory at a level of a factor of ∼2.

In this comparison, we adopted a diet Salpeter IMF for all observables at any epoch. Several studies suggest that the IMF might become top-heavy under extreme starburst conditions (Weidner et al. 2010). This could result in an evolution of the IMF with redshift, which would impact the estimation of stellar mass by up to ∼ −0.18 dex (rising almost linearly with redshift from z = 0 to z ∼ 1.5) compared to estimates at z = 0 (van Dokkum & van der Marel 2007). This could result in a systematic shift of the merger rate with redshift. Several systematic effects associated with the stellar population models can also affect the determination of the stellar mass. The most significant effect is the influence of the thermally pulsing asymptotic giant branch phase, omission of which in stellar population models such as those used in the present study can lead to overestimating the stellar mass by 0.14 dex (Pozzetti et al. 2007). In addition, using simplified prescriptions for deriving the stellar mass instead of full spectral energy distribution modeling (Bell et al. 2003), when used at high redshifts, can also potentially lead to overestimating stellar masses by 0.2 dex (Puech et al. 2008; Gallazzi & Bell 2009).

In order to bracket the upper limit of the predicted merger rate at all redshifts, we combined quadratically the systemic uncertainty associated with a possible evolution of the IMF and the effects of stellar population models, since they are independent. This results in a total systematic uncertainty of −0.27 dex on the baryonic mass. In Figure 5, we plotted theoretical predictions with [M^min_b(z), M_b^max(z)] − 0.27, using a dashed red line. In doing so, [f^min_gas(z), f_gas^max(z)] was updated accordingly (upward). All these systematic uncertainties associated with the estimation of the stellar mass tend to decrease the galaxy merger rate at all redshifts. However, the theoretical predictions remain consistent with the observational estimates within the corresponding range of observational systematic uncertainties. Quantitatively, this could decrease the integrated merger rate by ∼20%.

We conclude that the major merger rate is determined from both theory and observations within a factor of 2–3, accounting for random and systematic uncertainties. Such an agreement between the merger rate determined using morpho-kinematic data and theoretical predictions without any particular fine-tuning is remarkable. This is an improvement of a factor of ∼3 compared to previous comparisons between observations and theory (Hopkins et al. 2010a).

Using morpho-kinematic data of z ∼ 0.6 galaxies from the IMAGES-CDFS survey, we derived the evolution of the major merger rate out to z ∼ 1.5, i.e., nine billion years ago. Using comparisons of the spatially resolved morpho-kinematic properties between observations and a grid of hydrodynamical simulations of galaxy mergers, we constrain their time evolution, which allowed us to accurately date back when they were still in pairs. In this comparison, the estimate of the merger rate at z ∼ 1.5 is much more uncertain since it relies on the relaxation phase, during which it is more difficult and/or degenerate to interpret morpho-kinematic features as merger relics. However, it makes sense to interpret such galaxies as post-mergers in the relaxation phase as detailed above, and Figures 3 and 4 indeed suggest a plausible causal link between distant merging galaxies and such dynamically hot RDs. We found a remarkable agreement between these new observational estimates and predictions from state-of-the-art semi-empirical ΛCDM models within a factor of 2–3, accounting for both random and systematic uncertainties.

As a test of consistency, the ΛCDM model predicts a large fraction of peculiar galaxies at z = 0.4–0.75, which is found to be 〈r〉 (i.e., the average merger rate at z = 0.65, ∼0.06 Gyr⁻¹; see Figure 5) × τ (i.e., the total duration of the average major merger at z = 0.65, ∼5.5 Gyr; see Figure 4) ∼ 33%. This corresponds exactly to the fraction of observed galaxies in which the peculiar morphologies and/or kinematics can be reproduced by a major merger model (see Section 3.2). The ΛCDM model therefore predicts that ∼33% of the progenitors of local disks were destroyed over the past 9 Gyr during major (i.e., baryonic mass ratio ⩾1:4) merger events, as observed.

The occurrence of 33% of gas-rich major mergers in z ∼ 0.6 galaxies is also consistent with an increase of the scatter of the TF relation at high redshifts (Flores et al. 2006; Kassin et al. 2007; Puech et al. 2008, 2010; Covington et al. 2010)¹⁰ and the observed random walk in the specific angular momentum of z ∼ 0.6 galaxies compared to local galaxies (Puech et al. 2007a).

The agreement found between observations and theoretical predictions (see Figure 5) evidences that 50% (65%) of local galaxies with stellar mass in the range 10^10.3–10^11.1 M_☉ (see Section 5.1) have undergone a major merger (defined here as baryonic mass ratio ⩾0.25) since z = 1.5 (z = 2.0). The Hopkins et al. (2010a) model takes into account the specific physics involved in gas-rich major mergers, which was shown to reproduce the observed local relation between bulge-to-disk ratio and stellar mass (Hopkins et al. 2009b, 2010a). This suggests that the time at which local galaxies have undergone their last major merger could be an important factor in determining how they acquired their internal structure (B/T ratio and angular momentum).

Depending on the gas fraction at fusion time, major mergers are well known to result in ellipticals (through dry mergers), but also in spirals (through gas-dominated mergers). Indeed, hydrodynamical simulations of isolated merging disk galaxies have shown that with a fraction of gas at the fusion time larger than 50%, a significant disk can reform in the remnant (Springel & Hernquist 2005; Robertson et al. 2006; Hopkins et al. 2009a). Note that the required fraction could be smaller with significant cosmological gas accretion (Brooks et al. 2009), but we will adopt 50% as a conservative limit.

The median gas fraction at z = 0.6 is found to be ∼30% (Puech et al. 2010; Rodrigues et al. 2012). It is therefore likely that major mergers occurring between z ∼ 0.6 and z = 0 (which involved 20% of local galaxies according to Figure 5) were relatively dry, probably resulting in galaxies with significant bulges.

The major mergers sampled by the IMAGES-CDFS sample are those occurring between z ∼ 1.5 and z ∼ 0.6 (see Table 1). Figure 5 shows that 50% of local galaxies were involved in a major merger since z = 1.5, of which 30% occurred between z = 0.6 and z = 1.5. Hammer et al. (2009a) have extrapolated the gas fraction in the progenitors of these mergers and found gas fractions well in excess of the required value of 50% to reform a disk (∼90%; see Section 5.1). Several measurements of the gas fraction were attempted at z ∼ 1.5 (Daddi et al. 2010; Weinzirl et al. 2011). Most measurements involve relatively massive galaxies (∼M_stellar = 10^10.7–10^11.1 M_☉) and are limited to their half-light radius. However, the HOD model predicts that the progenitors of the ongoing z ∼ 0.6 major mergers at z = 1.5 should be in a lower range of stellar mass ∼10^9.2–10^10.4 M_☉. Rodrigues et al. (2012) homogenized gas estimates from the literature to parameterize the evolution with redshift and stellar mass of the gas fraction within the full gas radius. Extrapolating their gas-fraction–stellar-mass relation at z ∼ 1.5 in the range of mass 10^9.2–10^10.4 M_☉, one finds that gas fractions at a level of ∼80%–100% are probably close to average conditions (see Figure 7 of Rodrigues et al. 2012). The progenitors of all the z ∼ 0.6 ongoing major mergers at z = 0.75–1.5 should therefore be rich enough in gas for a significant disk reform after fusion, as also evidenced by observations of young dust-enshrouded disks at z ∼ 0.6 (Hammer et al. 2009b; Puech et al. 2009).

We emphasize that we have considered here only the "average" results of major mergers since z ∼ 2 and such statements have to be taken at the lowest order only, since one expects significant scatter in the B/T of a major merger remnant as a function of orbital parameters, gas fraction, and feedback (Hopkins et al. 2008, 2010a; Governato et al. 2009). We also emphasize that we considered here only a specific range of mass and that the impact of more modest mergers, i.e., with baryonic mass ratio smaller than 0.25, was not considered.

Results of semi-analytic models suggest that the likeliest channel for the formation of disk-dominated galaxies is a relatively quiet merger history (e.g., see discussion and references in Weinzirl et al. 2009). Of particular interest are local "bulgeless galaxies" (which we define here as galaxies with B/T ⩽ 0.2, as generally considered), which are often thought to have undergone their last major merger above z ∼ 2. The relatively large fraction of such galaxies is often considered to be in tension with the fraction of major mergers as inferred from predictions of ΛCDM models (e.g., Stewart et al. 2008). It was argued in Section 6.2 that most descendants of IMAGES-CDFS galaxies should lie within early-type spirals. We now examine whether the fraction of mergers found in Figure 5 is in tension with the observed fraction of local bulgeless galaxies.

Weinzirl et al. (2009) used bulge+bar+disk decompositions on H-band images and found that two-thirds of local spirals with M_stellar = 10^10.1–10^11.6 M_☉ are bulgeless. Their determination maximizes the fraction of bulgeless galaxies since they removed the contribution of the bar to the central light and followed the common assumption that bars are pure products of secular evolution and not merging (see discussion in Athanassoula 2008). Since the fraction of spirals among local galaxies is ∼70% (see Appendix B), this would translate into a maximal fraction of bulgeless galaxies in the local universe of ∼46%. As Weinzirl et al. (2009) argued, their sample appears to be representative of the B-band luminosity function at M_stellar ⩾10^10.1 M_☉ (see their Figure 2). However, their Figure 3 evidences that their sample can be representative of stellar mass function only above M_stellar ⩾10^10.6 M_☉. In this range of mass, the maximal fraction of local bulgeless galaxies is found to be 61% or 43% of local galaxies. Assuming that such galaxies cannot be formed by mergers, this would therefore limit the fraction of major mergers in local galaxies since z = 2 to 57%. We re-derived Figure 5 for the range of mass M_stellar = 10^10.6–10^11.6 M_☉. Indeed, the HOD model used in the present study predicts that ongoing major mergers in the z ∼ 0.6 IMAGES-CDFS sample should end up in a stellar mass range of 10^10.3–10^11.1 M_☉, which is shifted to smaller mass relative to the range 10^10.6–10^11.6 M_☉ in which the Weinzirl et al. (2009) sample is representative. In this higher range of mass, we found that the fraction of local galaxies that have undergone a major merger since z = 2 decreases from 65% to 53%. Within uncertainties, this matches quite well the maximal fraction of bulgeless galaxies found by Weinzirl et al. (2009) in the same range of mass (57%).

We conclude that there is no tension between the fraction of mergers inferred from morpho-kinematic data and the fraction of local bulgeless galaxies. In other words, there is no need for disks to survive major mergers in the ΛCDM paradigm, at least at z ⩽ 2.

The results of this paper point toward a re-building of half of local disks during the past 9.4 Gyr after their last gas-rich major merger (Hopkins et al. 2009b, 2010a; Hammer et al. 2005; Stewart et al. 2009). This does not imply that the evolution of the star formation and stellar mass densities are primarily driven by such processes over the same period (Robaina et al. 2009; Hopkins et al. 2010d). These results are consistent with recent developments of the ΛCDM model at earlier look-back times (i.e., z ⩾1.5), according to which galaxy formation would be mainly driven by streams of cold gas, rather than by major mergers (Förster-Schreiber et al. 2009; Agertz et al. 2009; Weinzirl et al. 2011). Indeed, cosmological simulations predict that cold flows are strongly suppressed below z ∼ 1–1.5 (Keres et al. 2009). Such an epoch might correspond to a transition between two different mechanisms driving galaxy formation, i.e., cold gas streams and gas-rich major mergers, as suggested by the strong evolution of the fraction and properties of clumpy galaxies as a function of redshift (Puech 2010; Contini et al. 2012a). In such simulations, halos with a relatively quiet merger history are usually selected as the best probes for the formation of local thin disks (e.g., Brook et al. 2012). The present results suggest that gas-rich major mergers could be an equally plausible way of forming large thin disks. Further work will be needed to quantify what is the main reservoir of gas from which such thin disks can reform. Potential sources of gas include gas expelled during the merger and that falls back onto the remnant (Barnes 2002), gas coming from external gas accretion (Brooks et al. 2009), and/or gas recycled through stellar feedback (Martig & Bournaud 2010). This emphasizes the need to further explore how such disks can (re-)form in halos with a more recently active merger history (Governato et al. 2009). Interestingly, recent cosmological simulations conducted with state-of-the-art hydrodynamical codes reveal cold gas disks that have larger extent and specific angular momentum compared to disks formed in former smoothed particle hydrodynamics cosmological simulations (Torrey et al. 2011; Keres et al. 2011). These simulations show that massive halos (i.e., M_halo ⩾ 10¹¹ M_☉) have significantly larger SFRs at z = 0–2, compared to previous simulations (Keres et al. 2011).