Galaxy Formation as a Cosmological Tool. I: The Galaxy Merger History as a Measure of Cosmological Parameters
Christopher J. Conselice, Asa F.L. Bluck, Alice Mortlock, David Palamara, Andrew J. Benson
aa r X i v : . [ a s t r o - ph . GA ] J u l Mon. Not. R. Astron. Soc. , 1–22 (2002) Printed 2 August 2018 (MN L A TEX style file v2.2)
Galaxy Formation as a Cosmological Tool. I: The GalaxyMerger History as a Measure of Cosmological Parameters
Christopher J. Conselice ⋆ , Asa F.L. Bluck , Alice Mortlock , , David Palamara , ,Andrew J. Benson University of Nottingham, School of Physics & Astronomy, Nottingham, NG7 2RD UK University of Victoria, Department of Physics and Astronomy, Victoria, British Columbia, V8P 1A1, Canada Royal Observatory Edinburgh School of Physics, Monash University, Clayton, Victoria 3800, Australia Monash Center for Astrophysics (MoCA), Monash University, Clayton, Victoria 3800, Australia Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA 91101, USA
Accepted ; Received ; in original form
ABSTRACT
As galaxy formation and evolution over long cosmic time-scales depends to a largedegree on the structure of the universe, the assembly history of galaxies is potentially apowerful approach for learning about the universe itself. In this paper we examine themerger history of dark matter halos based on the Extended Press-Schechter formalismas a function of cosmological parameters, redshift and halo mass. We calculate howmajor halo mergers are influenced by changes in the cosmological values of Ω m , Ω Λ , σ , the dark matter particle temperature (warm vs. cold dark matter), and the valueof a constant and evolving equation of state parameter w ( z ). We find that the mergerfraction at a given halo mass varies by up to a factor of three for halos formingunder the assumption of Cold Dark Matter, within different underling cosmologicalparameters. We find that the current measurements of the merger history, as measuredthrough observed galaxy pairs as well as through structure, are in agreement with theconcordance cosmology with the current best fit giving 1 − Ω m = Ω Λ = 0 . +0 . − . . Toobtain a more accurate constraint competitive with recently measured cosmologicalparameters from Planck and WMAP requires a measured merger accuracy of δf m ∼ .
01, implying surveys with an accurately measured merger history over 2 - 20 deg ,which will be feasible with the next generation of imaging and spectroscopic surveyssuch as Euclid and LSST. Key words:
Galaxies: Evolution, Formation, Structure, Morphology, Classification
One of the major goals in science is determining the past his-tory and future evolution of the universe. The determinationof this in a quantitative way has a long history, starting withthe work of Hubble (1929) who determined that the uni-verse was expanding based on radial velocity and distancemeasurements of galaxies. This has continued using variousapproaches, including the use of Cosmic Microwave Back-ground (CMB) measurements, with the most recent workusing e.g., WMAP, Planck and BICEP2 (e.g., Komatsu etal. 2011; Ade et al. 2013, 2014). Currently, the use of theCMB and type Ia supernova are the most common and influ-ential methods for measuring cosmological parameters (e.g., ⋆ E-mail: [email protected]
Kessler et al. 2009), along with baryonic acoustic oscilla-tions and clustering measurements (e.g., Eisenstein et al.2005; Blake et al. 2011).One of the dominant features of the current popular cos-mological model is that the universe’s energy budget is per-haps dominated by a cosmological constant - the so-calledDark Energy. The major evidence for this Dark Energy islargely based on observations of the luminosities of super-nova at various redshifts (e.g., Riess et al. 1998; Perlmutteret al. 1999). Other evidence for Dark Energy comes frombaryonic acoustic oscillations (e.g,. Eisenstein et al. 2005),and fluctuations in the cosmic background radiation (e.g.,Komatsu et al. 2011). The major result of this is that theuniverse appears to be accelerating since z <
1, and perhapsundergoes a deceleration phase at higher redshifts (Reiss etal. 2004). c (cid:13) C.J. Conselice et al.
What is currently lacking within this cosmologicalparadigm is physical evidence for the existence of Dark En-ergy, which in principle can significantly change the evolu-tion of the constituents of the universe, of which galaxiesare the fundamental component. The basic idea is that ifthe universe is undergoing an acceleration phase, then therate of structure formation will decline with time, haltingthe growth of massive structures, such as galaxy clusters(e.g., Allen et al. 2004; Vikhlinin et al. 2009). In fact, ob-servations of the number densities of galaxy clusters can beused as an alternative method for constraining Dark Energyproperties (e.g., Vikhlinin et al. 2009).Recently, with a basic but firm understanding of galaxyformation and evolution it is now possible to go beyond ob-servations of clusters, supernova, and the cosmic backgroundradiation to use galaxies themselves as a new probe of cos-mology. We explore in this paper how cosmological proper-ties affect the formation of galaxies throughout their historybased on examining the formation histories of dark matterhalos. This lets us examine both how galaxy formation canbe used as a probe of cosmology, and how cosmology affectsthe formation of galaxies. In this sense the formation historyof galaxies in the universe is potentially another probe of theenergy and kinematics of the universe.The use of galaxies for cosmology is not a new idea, andearly attempts to measure cosmological properties, largelythe measurement of the Hubble constant, relied on luminosi-ties and properties of stars and globular clusters in externalgalaxies. Several cosmological tests were also proposed in the1920-1930s that used the angular sizes, counts, and surfacebrightness evolution of galaxies (e.g., Tolman 1930; Sandage1988). However, these approaches were largely abandonedonce it was realized that galaxies evolve significantly throughtime, and that the properties of nearby galaxies are not nec-essarily the same as more distant galaxies.Since we are now becoming confident in the measure-ments of galaxy properties, and how at least massive galax-ies evolve and form over time, especially since z ∼ q , new approaches us-ing the evolution of galaxy properties can potentially be usedto derive features of the dominant cosmological paradigm.We specifically investigate in this paper whether theevolution of galaxies is consistent with the currently ac-cepted ideas concerning a Λ-dominated universe with a tran-sition from deceleration to acceleration occurring sometimearound z ∼ . § § § § § § m = 0.3, Ω Λ =0.7; σ = 0.9; H = 70 km s − Mpc − as the concordancecosmology. In the dominant theory for galaxy formation, based on aΛCDM cosmology, galaxies assemble by merging with oneanother over time (e.g., White & Rees 1978). The basis forthis merging is the dark matter assembly history, and howdark matter halo masses grow by merging with one another.Using Newtonian dynamics plus a simple expanding uni-verse model it is now possible to predict the total halo massfunctions of galaxies across a large range in mass to within5%, comparing different computer simulation results. Withthe small discrepancies based on the differences betweenmethods of the various calculations rather than fundamen-tal physics (e.g., Reed et al. 2007). In particular, differentgroup finding algorithms are largely the cause of the smalldifferences in masses, rather than fundamental physics (e.g.,Knebe et al. 2013).Predictions for how structure assembles is the backboneof any theory of galaxy formation. Since galaxies are believedto form at the cores of dark matter halos, then the formationof galaxies should follow in some way how the dark mat-ter assembles. Dark matter halos and large scale structureare created through these halos hierarchically. This processcan be predicted based on the basic physics of gravitationalcollapse of matter in an expanding universe, and its laterevolution, and therefore does not involve uncertain baryonicphysics. The details of how dark matter halos assemble is c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe now predicted in detailed N-body and semi-analytical sim-ulations (e.g., Fakhouri & Ma 2008). These simulations es-sentially predict when two existing dark matter halos mergetogether to form a large halo within the standard ΛCDMcosmologies assumed.It is fairly straightforward to use simulations of struc-ture formation to predict how dark matter halos withdescendant masses between 10 M ⊙ < M halo < M ⊙ assemble with time (e.g., Fakhouri & Ma 2008). We in-vestigate in this paper what various models predict for halomergers. Our primary method is to use a generalized codefor calculating dark matter halo mergers within a given cos-mology. To do this we calculate the merger history for darkmatter halos through using the ‘growl’ algorithm by Hamil-ton (2001) using a power-spectrum calculated by Eisenstein& Hu (1999). Using this numerical formalism it is possible todetermine the assembly history of dark matter halos usingbasic gravitational collapse physics.To calculate this we use the results of Hamilton (2001),and a modified form of the growl code to compute the lineargrowth factor g = Da (1)for structure in the universe as a function of time, where D is the amplitude of the growth mode, and a is the scalefactor. The linear growth rate, f = dlnDdln a (2)is the derivative of g , and relates to peculiar motions withinthe universe. Using a Friedmannn-Robertson-Walker (FRW)universe, the growth factor g can then be written asg(Ω m , Ω Λ ) = Da = 5 × Ω m Z d aa H ( a ) (3)where a is the scale-factor normalized to unity, and H ( a )is the Hubble parameter normalized to unity when a = 1,where H ( a ) = (Ω m a − + Ω k a − + Ω Λ ) / . (4)The value of the growth factor and the Hubble constantH( a ) depend upon the value of cosmological parameters. Thelinear growth rate f can then be written as f (Ω m , Ω Λ ) = − − Ω m Λ + 5Ω m g (5)Analytical solutions to the above growth rate are presentedin detail in Hamilton (2001) for different cosmological pa-rameter ratios. The growl code then implements these fittingformula for various scenarios to predict what the growth fac-tor is during the history of the universe, as a function ofcosmology and time.The power spectrum used within this code originatesfrom Eisenstein & Hu (1999), who calculate fitting formulafor the matter transfer function as a function of wavenum-ber, time, the massive neutrino density, number of neu-trino species, the Hubble parameter today, the cosmologicalconstant, baryon density and the spatial curvature. Merg-ers occur via the set excursion methodology from Press &Schechter (1974) but using the extended formalism. As aresult, we measure the halo merger history as a function ofthe mass ratio of the halo mergers: η = M − M M , (6)where M is the sum of the halo masses of the two mergercomponents (or the resulting halo mass) and M is the halomass of the more massive progenitor.We use these models with a variety of different cos-mological parameters to investigate how the halo mergerhistory varies with cosmology. The cosmological parametersthat we vary are: the matter density Ω m , the dark energydensity Ω Λ , the neutrino density Ω µ , the Hubble constant, H , the baryonic density Ω B , the temperature of the CMBT CMB , the number of neutrino species, N ν , value of σ , andthe spectral index n . We define the cosmology henceforth asthe quantityΘ = (Ω m , Ω Λ , Ω µ , H , Ω B , T CMB , N ν , σ , n ) (7)Our method uses an altered version of the publicly available growl code. We measure the halo merger history through aparticular type of ‘major’ halo merger. These cosmologicalbased halo mergers are designed to match as much as pos-sible the merger criteria used to find mergers occurring inactual galaxies ( § § In this paper we only discuss mergers which are major forboth the observational data and the theoretical results. Thecriteria for finding a major merger is that a halo at a givenredshift must have had a merger with another halo of mass1:4 or less within the past ∼ . ∼ . c (cid:13) , 1–22 C.J. Conselice et al. et al. 2010b; Moreno et al. 2013). It is found in these papersthat very few properties beyond mass ratio and gas massfraction affect the derived merger time-scales.These simulations show that mergers are identifiedwithin both CAS at the first pass of the merger, as wellas when the systems finally merge together to form a rem-nant (Lotz et al. 2008). However, merging galaxies are notfound in the merger area of the non-parametric structuralparameters for the entire merger, as was found by Conselice(2006b). This however allows the time-scales for structuralmergers to be calculated. Lotz et al. (2008, 2010a) find thatthe asymmetry time-scales for gas-rich major mergers are0.2-0.6 Gyr and 0.06 Gyr for minor mergers (Lotz et al.2010a). While the individual time-scale for a pair of galax-ies within a dark matter halo to merge will vary, based onthe variety of models the average is 0.4 Gyr and we use thisthroughout this paper as our measured merger time-scale.An issue that we have to address is that these time-scales are for gas rich mergers, and would not necessarilyapply for gas poor or dry mergers. However, at the redshiftswe are investigating here, nearly all galaxy mergers will havesome gas, as pure dry mergers are relatively rare (e.g., Lin etal. 2008; Conselice et al. 2009; De Propris et al. 2010). Fur-thermore, the massive galaxies that we examine in this paperhave gas fractions which are on average ∼
10% at z = 1 − f halo , at a variety of redshifts. f halo = N merger ( M halo ,η,z,τ, Θ) N tot ( M halo ,η,z,τ, Θ) (8)Later we investigate how the merger history of halos, andthe value of f halo changes as a function of different values ofthe equation of state parameter ω , and for a varying ω ( z )as a function of redshift. These calculations originate fromthe GALACTICUS code of Benson (2012). Within our cal-culations of the merger fraction we also take into accountdouble mergers whereby a halo or galaxy undergoes morethan one merger in a given time-scale. The merger fractionincludes the total number of mergers the examined popu-lation (selected by mass in this case) undergoes divided bythe total number of galaxies in that selection. Therefore if asingle galaxy/halo undergoes more than a single merger itis accounted for explicitly.
GALACTICUS is a semi-analytic model which is easilyadapted to differing physical and initial conditions. We uti-lized the same frame-work presented in the Hamilton (2001)structure formation model, but through using an equation ofstate parameter, ω , as well as through the use of an evolvingform as a function of redshift ( ω ( z )).We also use the results from the Millennium simulations(Springel et al. 2005) for both the merger history of galaxieswhich was discussed in depth in Bertone & Conselice (2009),as well as the merger history of halos. We furthermore com-pare with Warm Dark Matter semi-analytical models from Menci et al. (2012) to test how different dark matter particletemperatures can affect the galaxy merger history. Finally,we also compare with abundance matched merger historiesfrom Hopkins et al. (2010). One of the major goals of this study is to compare the ob-served galaxy merger history with predictions from simula-tions for the halo and predicted galaxy merger history. Assuch, the data we use for this comparison are from a diver-sity of sources and different surveys of distant galaxies. Mostof these are deep Hubble Space Telescope imaging surveyswhich have accurately measured stellar masses, redshifts andmerger fractions out to these redshifts.The galaxy merger data we use in this study come fromseveral studies of the merger history using the CAS struc-tural method (Conselice 2003, 2014), as well as galaxies inpairs (Lopez-Sanjuan et al. 2010; Bluck et al. 2009; 2012,Man et al. 2012). The surveys we take our merger datafrom include the GOODS NICMOS Survey (Conselice etal. 2011), NICMOS imaging of the COSMOS field (Manet al. 2012), and the Hubble Ultra Deep and Deep Fields(e.g., Williams et al. 1996; Conselice et al. 2008). We alsotake results from Bluck et al. (2009, 2012) for M ∗ > M ⊙ galaxies from the GOODS NICMOS Survey to z = 3(Mortlock et al. 2011). For the most massive galaxies withM ∗ > M ⊙ at z < z < .
5. We also use new CANDELS observations ofthe merger history from asymmetries calculated within theCANDELS area of the Ultra Deep Survey field (Mortlock etal. 2013).There are many other potential merger histories thatwe can use, but do not, due to time-scale and major/minormerger sensitivity, we only use galaxies in pairs and thosemeasured with the CAS system, which has a well definedmerger time-scale for gas rich major mergers discussed in § § <
30 kpc c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe mergers have a similar time-scale as the CAS selected merg-ers (e.g., Conselice et al. 2009; Bluck et al. 2012) of about τ = 0 . ⊙ . The fitting method for our stellar masses consistsof fitting a grid of model SEDs constructed from Bruzual& Charlot (2003) (BC03) stellar population synthesis mod-els, using a variety of exponentially declining star formationhistories, with various ages, metallicities and dust contentsincluded. The models we use are parameterized by an age,and an e-folding time for parameterizing the star formationhistory, where SFR(t) ∼ e − tτ .The values of τ are randomly selected from a range be-tween 0.01 and 10 Gyr, while the age of the onset of starformation ranges from 0 to 10 Gyr. The metallicity rangesfrom 0.0001 to 0.05 (BC03), and the dust content is pa-rameterized by τ v , the effective V-band optical depth forwhich we use values τ v = 0 , . , . , . , . , . , , . , . δz/ (1 + z ) ∼ .
03 (Hartley et al.2013; Mortlock et al. 2013). Details of how these photometricredshifts are computed are included in the above cited pa-pers (e.g., Conselice et al. 2009; Mortlock et al. 2011; Blucket al. 2012; Mortlock et al. 2013). The errors in the mergerfractions which we later use to fit to the predicted halo andgalaxy mergers fully take into account the uncertainties inthe stellar masses and redshifts for these samples. This is infact the largest source of uncertainty when calculating themost likely cosmological model based on the merger fractionevolution.
In the following sections we investigate the halo and galaxymerger histories both predicted, for halos ( § § § § § § § We first describe the merger histories of halos using the con-cordance cosmology, defined as Ω Λ = 0 .
7, Ω m = 0 . σ = 0 .
9, as a function of halo mass. The merger histo-ries for galaxy halos has been studied previously, but mostlyonly within the standard cosmology. Early results (e.g., Got-tlober et al. 2001) found that the merger history for haloscan be described as a power-law increase with redshift as ∼ (1 + z ) . More detailed predictions have been provided bymodern simulations such as the Millennium simulation (e.g.,Bertone & Conselice 2009; Fakhouri & Ma 2008; Fakhouri etal. 2010; Hopkins et al. 2010) where both the merger historyfor halos and galaxies are simulated and predicted. Thesesimulations find that the merger history of halos increasesas a power-law (e.g., Fakhouri & Ma 2008). These simula-tions also find that galaxy mergers are less common thanhalo mergers, and that the predicted galaxy merger fractionis much lower than what is observed (e.g., Conselice et al.2003; Bertone & Conselice 2009; Jogee et al. 2009; Hopkinset al. 2010).However, there is a better match between galaxy merg-ers and halos when using HOD models which match darkmatter halos predicted to exist at a given redshift to ob-served galaxies selected by stellar mass and clustering (e.g.,Hopkins et al. 2010). In this case the galaxy merger fractionbased on stellar mass is determined by the merger fractionof the halos in which these galaxies are located ( § Here we investigate the predicted merger history for halosusing the basic CDM model predictions with the methodsoutlined in §
2. The predicted merger histories of halos ofa given halo mass limit (M halo ) using the formalism from § halo > halo >
13. We only investigate here, as explained in de-tailed in §
2, the merger fraction for these halos of a givenmass which merged with another halo, with a mass at leasta fourth of the mass of its halo, and within 0.4 Gyr.Figure 1 shows a general trend such that the mergerfractions of halos are very high in the early universe, upto redshifts z ∼
6, with halos of the highest masses at c (cid:13) , 1–22 C.J. Conselice et al.
Figure 1.
The halo merger history for galaxies of a given halomass in the concordance cosmology, with the lines for the differenthalo masses defined on the upper left. The halo merger fractionplotted here is the fraction of galaxies at a given halo mass whichhas merged with another halo of at least a factor of 0.25 or greaterthan the mass of the progenitor halo, and within the past 0.4 Gyr.The inset shows the detailed merger history for the same systemsat z <
3. We also show as the line with open circles the best-fitpower-law to the merger history from the Millennium simulation(e.g., Fakhouri et al. 2010). Furthermore, the lower solid line withbox points is the Millennium simulation prediction for galaxymergers (as opposed to halo mergers) with log M ∗ >
10. Thered crosses are model predictions from Stewart et al. (2009) forgalaxies with halo masses of M halo = 10 M ⊙ . M halo > M ⊙ having a merger fraction close to f halo ∼ § f halo ( z, M halo ) = f halo (0 , M halo ) × (1 + z ) m , (9)where f halo ( z = 0 , M halo ) is the halo merger fraction at z = 0and m is the power-law index for the merger history. Highervalues of m are fit for steeper merger fractions. Merger his-tories have often been fit with these power-law forms forgalaxies at various stellar mass and luminosity cuts usingreal data for some time, although this is one of the firsttimes this has been done for dark matter halos as a functionof cosmology. Previously, Gottloeber et al. (2001) applied Mass limit f m log M halo > ± ± halo >
10 0.029 ± ± halo >
11 0.033 ± ± halo >
12 0.030 ± ± halo >
13 0.036 ± ± Table 1.
The best fitting power-law fits from eq. (9) to the mergerhistory for dark matter halos using halos of different masses.These merger histories are listed at the halo mass M halo limit.These fits are for the merger history up to z = 4. this fitting method to merger histories in the standard cos-mology.We list the results of this fitting for CDM halos of var-ious masses in Table 1. These results show that the halomerger history is similar at z = 0 at all halo masses, dif-fering by only a small amount. However, the power-law in-crease is such that the more massive galaxies have a steeperrise in their halo mergers, and thus a larger merger fractionat earlier times than lower mass systems. This is a demon-stration of the hierarchical nature of structure formation inhalos. This is also opposite to what is seen in the galaxypopulation where the more massive systems appear to endtheir merger and formation processes before lower mass ones(e.g., Conselice et al. 2003; Bundy et al. 2006; Conselice etal. 2009; Mortlock et al. 2011). Another remarkable aspect of Figure 1 is that the differentialbetween the halo mass merger histories amongst the variousmass selections is smallest at the lower redshifts, and highestat z >
4. This shows that the merger properties for halosof different halo masses is similar at lower redshifts, butdiverges more at the highest redshifts. Therefore the haloassembly history is more distinct earlier, as a function ofM halo , rather than later, in the universe.Currently the observed merger history is largely limitedto studies at relatively modest redshifts, those at z < z <
3. At theseredshifts the merger fractions for halos over four orders ofmagnitude in mass differ by a maximum of δf halo ∼ . z ∼
3. To quantify this, we fit the change in the halomerger history as a function of halo mass at two redshifts, z = 2 . z = 1. We later use these to determine theuncertainty in the matching of halo and stellar masses, andtherefore to go from the halo merger history to the galaxymerger history. At all redshifts we find that this relation islinear and is well described by a function of the form: f halo ( z, M halo ) = a × logM ∗ + b. (10)We find for z = 2 . a = 0 . ± .
001 and b =0 . ± .
01. This is a very shallow slope, and shows thatthe halo merger fraction at a given redshift does not change c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe very much between halo masses of log M halo = 9-13. Atlower redshifts, the relation becomes even flatter, with fitsfor z = 1, a = 0 . ± .
001 and b = 0 . ± . z = 3 as this is currently where we have the mostcertainty in our ability to measure masses and the mergerhistory in actual galaxies ( § Ω m and Ω Λ One of the features we investigate with our halo merger his-tories is how the merger history changes within differentunderlying cosmological parameters. This allows us to inves-tigate, among other things, whether the merger history ofgalaxies is consistent with the dominant cosmological model.In Figure 2 we show the merger history for halos of massesM halo > M ⊙ and for M halo > M ⊙ using our pre-scribed method for finding and defining merging halos ( § m = 1 with a zero cosmologicalconstant (Cosmo4 in Table 2). Furthermore, there is a clearcorrelation with higher merger fractions for higher values ofΩ tot . This is an indication that the merger history is tracingto some degree the geometry of the universe.This figure however shows that even for a total den-sity of Ω = 1 there are variations within the merger his-tory. Three extreme models are shown in Figure 2 – onein which the energy density is completely in the form ofmatter (Cosmo-4; the short dashed line), one which has thecurrently accepted cosmological model (Cosmo-7; Ω m = 0.3,Ω Λ = 0.7; σ = 0 . m and Ω Λ (Cosmo5; long dashed line).In general, the higher the matter density, the higherthe merger fraction within the halo merger models. This isparameterized in § Λ . The halomerger fraction begins to turn over when there is a higherΛ term in the cosmology. This is as expected, given thatdetailed numerical models of galaxy formation have shownfor many years that the large scale-structure of the universedepends strongly upon the assumed cosmology, as well asthe temperature of the dark matter particle (e.g., Jenkins etal. 1998; Menci et al. 2012).We later examine in § § f do not vary significantly between the different cos-mologies, and that the most variation is within the values of Model Ω Λ Ω m Ω tot σ f m Cosmo-1 0.0 0.1 0.1 0.9 0.027 1.73Cosmo-2 0.7 0.3 1.0 0.9 0.032 1.92Cosmo-3 0.0 0.3 0.3 0.9 0.035 1.82Cosmo-4 0.0 1.0 1.0 0.9 0.052 1.73Cosmo-5 0.5 0.5 1.0 0.9 0.041 1.88Cosmo-6 0.7 0.3 1.0 0.7 0.038 1.93Cosmo-7 0.7 0.3 1.0 0.8 0.034 1.97Cosmo-8 0.7 0.3 1.0 1.0 0.032 1.94Cosmo-9 0.7 0.3 1.0 1.1 0.033 1.88
Table 2.
The cosmological models we use in this paper tocompare with the observed merger fractions and their variousfitted parameters. These are for galaxies with halo masses ofM halo > M ⊙ . Listed are each model’s Ω Λ , Ω m and σ values.We also show the best fitting f and m values for the power-lawfits as discussed in § f m < logM ∗ <
10 CDM 0.007 ± ± ∗ >
10 CDM 0.005 ± ± ∗ >
11 CDM 0.030 ± ± ∗ > . ± ± < logM ∗ <
10 WDM 0.010 ± ± ∗ >
10 WDM 0.006 ± ± ∗ > . ± ± ∗ >
11 WDM 0.040 ± ± Table 3.
The best fit power-law parameters in the form f m = f × (1 + z ) m based on the output from the galaxy merger simulationsbased on the Millennium Simulation and the WDM simulationsof Menci et al. (2012). m . In general we find that the higher the value of Ω Λ , thehigher the fitted value of the power-law slope, m . The aver-age m value for Ω Λ = 0 . m = 1 .
93, while for Ω Λ = 0 . m = 1 .
76. The decline in mergers is therefore steeper forhigher values of Λ due to the accelerated expansion, loweringthe number of mergers at a faster rate. ω f m − .
33 0.0380 ± ± − .
40 0.0374 ± ± − .
50 0.0360 ± ± − .
60 0.0344 ± ± − .
70 0.0331 ± ± − .
80 0.0320 ± ± − .
90 0.0312 ± ± − .
00 0.0307 ± ± − .
10 0.0304 ± ± − .
20 0.0302 ± ± Table 4.
The best fit power-law parameters in the form f m = f × (1 + z ) m based on fits to dark halo mergers with differentvalues of ω . These fits are for halos with M halo > .c (cid:13) , 1–22 C.J. Conselice et al.
Figure 2.
The halo merger histories for a) halos with masses M halo > M ⊙ and b) halos with masses M halo > M ⊙ . The variouslines show how the merger history varies with redshifts for halos of these given masses and using the cosmologies shown in the upperleft. In general we find that cosmologies with the highest matter densities have the highest merger fractions, although higher values ofΛ produce a smaller merger fraction at a given redshift. σ One cosmological parameter that can vary, and depends onlarge scale structure, and therefore also the halo and galaxyformation history, is the value of σ , the normalization of thematter power spectrum, as measured in the RMS dispersionof total mass density within 8 Mpc spheres.We show the variation of the halo merger histories us-ing the standard cosmology of Ω m = 0.3, Ω Λ = 0.7 but withdifferent variations of the value of σ in Figure 3. The dif-ferences in the predicted halo merger history between thesevalues of σ are smallest at z <
1, where the merger fractiononly varies by δf halo ∼ .
02 over the range of σ = 0.7 to1.1. For the σ = 0.7 models we find at z = 1 that the halomerger fraction is f halo = 0 .
15, while for σ = 1.1 the halomerger fraction is f halo = 0 .
12. This implies that at a givenredshift the accuracy of our merger fractions would haveto be better than a few percentage, which is easier to ac-complish than that required to distinguish between variousvalues of likely different Ω cosmologies ( § σ . Wefind a strong linear relationship between the halo mergerfraction and the value of σ , such that, f halo = ( − . ± . × σ + (0 . ± . , (11)at a redshift of z = 2 .
5. This demonstrates that there arehigher values of both f and m for universes with highervalues of σ . This implies that when the RMS fluctuationsof spheres of dark matter with a radius of 8 h − Mpc are σ f m ± ± ± ± ± ± ± ± Table 5.
The best fitting power-law fits to the merger history fordark matter halos at masses M halo > M ⊙ at z = 2 . σ . smaller, there is a higher rate of merging within the universe.This result is likely due to the fact that the value of σ is directly tied to the halo mass function. That is, when σ increases this effectively shifts the halo mass function tohigher values. This then increases the normalization neededto reach the same vale of the merger rate as at a lower valueof σ . From Table 1 we can see that lower mass halos have alower merger rate and thus a higher σ pushes the effectivescaling lower, thus creating a lower merger rate at a givenhalo mass. c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe Figure 3.
The evolution of the halo merger fraction for haloswith M halo > M ⊙ as a function of the value of σ . The rangewe show here is for σ = 0.7 to σ = 1 .
1, with the highest halomerger fractions for those evolving within a universe that has thelowest σ . In this section we discuss the comparison of our observationsof the galaxy merger history with the theoretically simulatedgalaxy merger histories. We do this before we examine thecomparison of halo merger histories to galaxies as comparingwith simulated galaxy mergers is a more direct comparison,although it largely fails as we show.
While the galaxy merger history as predicted in various sim-ulations has been discussed in detail elsewhere (e.g., Jogeeet al. 2009; Bertone & Conselice 2009; Hopkins et al. 2010),we give a short summary here, as well as a comparison tothe halo models. We show on Figure 1 as the solid darkline towards the lower part of the plot the galaxy mergerhistory for systems selected by a stellar mass limit of logM ∗ >
10 as predicted in the Millennium simulation. Thisroughly corresponds to a halo mass a factor of ten higher,based on the results from Twite et al. (2014) who investigatethe ratios of stellar and halo masses for galaxies (see § ∼ . z ∼ § ∗ > M ⊙ galaxies, but widely disagreefor those with masses in the M ∗ > M ⊙ range. The rea-son for this is due to these models not agreeing on whichgalaxies belong in which halo, which becomes more of an im-portant issue at higher halo masses. These massive galaxiesare on the exponential tail of the mass function and thereforeany differences in the star formation history or feedback canproduce significant differences in the merger history throughmatching which galaxy is in which halo (see also Bertone &Conselice 2009). With one exception, the observed galaxy merger historieswith redshift start at low values, peak at redshifts of around z ∼
2, and then decline thereafter at higher redshifts. Thisis similar to what is found within the Millennium simulationpredictions for galaxies, although often at a lower level. Incontrast, the merger history for halos continues to steeplyincrease at higher redshifts at all masses (Figure 1).This difference is very likely due to the way sub-halosare dealt with within the Millennium simulation. When agalaxy is accreted into a larger halo it losses all of its coldgas, and therefore cannot produce new stars and thus whenthe merger occurs after some dynamical friction time-scale,the mass ratio of the merger is low. These mergers therebyend up as minor mergers, although in some ways in baryonsthese would still be major mergers if the striped gas wasincluded. In fact, the high minor-merger fraction predictedin the Millennium simulation suggests that this might beoccurring (Bertone & Conselice 2009). Furthermore, simplecomputational differences, such as using different methodsto calculate the feedback from supernova, can dramaticallychange the measured galaxy merger history (Bertone & Con-selice 2009). Investigating in more detail these differences isimportant, but we do not discuss this issue further in thispaper.To compare these different observed merger histories,for halos and galaxies and the actual merger histories, in amore quantitative way, we characterize the merger historiesby fitting the predicted galaxy merger histories with thepower-law form given by equation (9). The results of these c (cid:13) , 1–22 C.J. Conselice et al.
Figure 4.
The predicted merger history for galaxies using different realizations of the semi-analytical Millennium I simulation. Shownhere are models from Bertone et al. (2007) (solid line), Bower et al. (2006) (dashed red line), De Lucia et al. (2006) (long-dashed blueline), and Guo et al. (2010) (cyan dotted line). The large variation in models seen for the M ∗ > M ⊙ galaxies is solely due to theidentification of galaxies with halos, as the underlying merger history for the halos in these simulations are identical. The points witherror bars shown are actual measures of the merger fraction for galaxies as a function of redshift. The blue points on the left show themerger history for M ∗ > M ⊙ systems including results from Conselice et al. (2009) (solid boxes at z < .
2; Bluck et al. (2009, 2012)(solid circles at z > . z > ∗ > M ⊙ galaxies,including results from Conselice et al. (2003) at z > z < . z > <
30 kpc: (Man et al. 2012;crosses at z > fits are shown in Table 3. Fits such as these typically onlyprovide a good fit to the data up to some redshift where themerger fraction begins to turn over to lower values. However,with some exceptions, these are good fits to the data at z < m ∼ m = 3 to m = 1 (Tables 1-3; e.g., Conselice et al.2009; Lotz et al. 2011). Galaxy mergers selected in the man-ner we use for the halo mergers generally always find that m ∼ § § m , Ω Λ ,h, σ , n s ) = (0.3, 0.7, 0.7, 0.9, 1.0) for the concordance cos-mology, (Ω m , Ω Λ , h, σ , n s ) = (0.27, 0.73, 0.71, 0.84, 0.96)for WMAP1, (Ω m , Ω Λ , h, σ , and n s ) = (0.268, 0.732, 0.704,0.84, 0.96) for WMAP3, (Ω m , Ω Λ , h, σ , n s ) = (0.274, 0.726,0.705, 0.776, 0.95) for WMAP5.The different lines in Figure 6 show the different cos-mologies used to measure the merger history. This is slightlydifferent from our approach, which is to see how the mergerhistory varies as a function of cosmology, as opposed to de- c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe Figure 5.
The merger history for galaxies using two simulations utilizing different dark matter particle temperatures out to z = 3. Theleft panel shows the Semi-analytical merger history predictions for galaxies with various stellar masses as taken from the Millenniumsimulation results of Bertone et al. (2007). The various lines show the merger fraction, as measured using the same criteria as used toselect the merger histories for the halos shown in in Figure 2. The right panel shows the merger history at the same stellar mass limitsbut with using a simulation with Warm Dark Matter (Menci et al. 2012). The solid red line is the best fit power-law to the observedevolution in galaxy mergers for the M ∗ > M ⊙ galaxies, while the solid blue line shows the corresponding best fit for the M ∗ > M ⊙ galaxies. The data points used for comparison are the same as in Figure 4. termining how the merger history changes due to differentmeasured cosmological parameters.As can be seen, even within these models there is a slightdifference in the calculated merger history from abundancematching. The merger histories here are still lower than whatwe find observationally, however the highest merger fractionsare for the cosmologies with the highest value of Ω m . As therecent Planck cosmology has an even higher Ω m than theconcordance model (Planck collaboration, Ade et al. 2013),the merger history using this cosmology would be higherthan those shown here, and would better match the data.The important take away message here is that the halomerger history does a much better job of matching the ob-served galaxy merger history than any galaxy merger his-tory prediction. This is a fundamental insight and one inwhich we now expand on in §
5, with the use of the halomerger models as a measure of cosmology. The idea we ex-plore in this paper is to not use the predicted galaxy for-mation merger histories, but the halo merger histories andto do ‘inverse modeling’ whereby the galaxy sample’s halomass is derived, and then compared with halo mass mergerhistories. This side-steps the need for understanding the de-tailed physics in simulations of galaxy formation, but doesrequire knowledge of how to convert from observed galaxyto inferred halo mass. This is essentially what is done whenusing galaxy cluster as observational probes – the cluster’sdark matter is measured and compared with theory, espe-cially for cosmological tools such as galaxy cluster numberdensity (e.g., Vikhlinin et al. 2009).
We also investigate how the galaxy merger history varies asa function of the temperature of the dark matter particle.This is a different topic in some ways from the variation instructure formation depending on cosmological parameters,yet this is another cosmological feature that does influencestructure, and thus we include it here. Since the structureformation models we examined before all depend on ColdDark Matter, we utilize the Menci et al. (2012) dark mat-ter semi-analytical models to make the comparison with theCDM models from the Millennium simulation. In Figure 5we show the merger history for the Cold Dark Matter basedMillennium simulation as well as a Warm Dark Matter sim-ulation from Menci et al. (2012).Figure 5 shows that the Warm Dark Matter simulationsdo a better job of matching the observations, which is alsoseen in comparisons to observed galaxy number densities inthe CDM and WDM simulations (e.g., Menci et al. 2012).Effectively the merger fraction in WDM galaxy simulationsis a factor of 1.3 higher for systems with M ∗ > M ⊙ , anda factor of three higher at masses of M ∗ > . M ⊙ . It isclearly the more massive systems (Figure 5) which show thegreatest difference between the CDM and WDM predictions.As CDM halo mergers, as well as those predicted in theMillennium simulation itself, predict a higher halo mergerrate similar to the observed galaxy rate, then it is likelythat the problem with the CDM galaxy mergers is not dueto CDM itself, but due to how baryons are handled in thesesimulations. c (cid:13) , 1–22 C.J. Conselice et al.
Figure 6.
Plot of the observed and predicted galaxy merger his-tory. The predicted lines are from the WMAP1, WMAP3 ANDWMAP5 cosmologies used to calculated the merger history inHopkins et al. (2010). The data with error-bars are the same asused in Figure 4. The dashed line at the upper end of the figureis from numerical models by Maller et al. (2006).
The previous section showed that observed galaxy mergershave a different evolution than that seen in predictions ofgalaxy mergers. In this section we discuss in detail how tocompare simulated halo mergers with observed galaxy merg-ers. This is a non-trivial comparison to make, and dependson several assumptions that we discuss in detail. One ofthese issues is the correspondence between the halo and stel-lar masses which is required to match halo histories to thatof galaxies, or on just a practical level matching a halo massto a stellar mass. The relationship between these two massesis ideally well defined and has a small scatter to minimizemismatches between the halos and galaxies. Other issues ina comparison to halos and galaxies have to do with relatingthe time-scales for halo merging to those of galaxy mergers.We previously discussed briefly how there is a betteragreement between predicted halo mergers and the observedgalaxy merger fractions. One of our major conclusions isthat it is better to compare the observed galaxy merger his-tory with these predicted halo mergers rather than with thepredicted galaxy mergers. In this section we present a newmethod for comparing the observed merger history to thepredicted halo merger history as a measurement of cosmol-ogy. We show that this can be done through several steps,each containing a certain amount of uncertainty. This pro-cess is such that we convert the observed stellar mass se-lects sample into a halo mass using relations that we dis-cuss based on halo abundance matching and kinematic data ( § § A major issue that needs to be addressed in any paper thatcompares halo properties to galaxies is how to relate thestellar mass of a galaxy to its underlying halo mass. Thisrelates back to our idea of comparing with CDM, and otherdark matter based models. What we therefore need is an ac-curate way to relate the observed stellar mass to the inferredhalo/total masses of galaxies.This can be done in a number of ways, including kine-matics and gravitational lensing to derive the total massesof galaxies. This problem has been investigated at high red-shift in several studies, including Conselice et al. (2005),Foucauld et al. (2010) and Twite et al. (2014). All of thesestudies conclude that the dark matter to stellar matter ra-tio evolves together such that the different components ofthe assembly history are increasing at a similar rate. Thisis an important aspect, as it allows us to compare the ob-served merger history of galaxies which is based on a stellarmass selection, to that of halo selection, which is predictedin models. Yet another approach towards understanding therelationship between stellar and halo masses is halo abun-dance matching (e.g., Conroy et al. 2007; Twite et al. 2014)which we also investigate for relating stellar and halo masses.We calculate the abundance matching derived relationbetween stellar and halo masses using the stellar mass func-tions from Mortlock et al. (2011). We match number densi-ties from galaxies with measured stellar masses to dark mat-ter halo abundances at the same redshifts. This allows us toassociate each stellar mass range with a halo mass range.This is described in more detail in Twite et al. (2014).In summary, to compute this relation the mass functionof dark matter halos (including sub-halos) is assumed to bemonotonically related to the observed stellar mass functionof galaxies with zero scatter. This relation is given by, n g ( > M star ) = n h ( > M halo ) (12)where, the values n g and n h are the number density of galax-ies and dark matter halos, respectively.We derive these values for the halos from the Jenkinset al. (2001) modification to the Sheth & Tormen (1999)halo mass function using the analytic halo model of Seljak(2000). We also generate the linear power spectrum usingthe fitting formulae of Eisenstein & Hu (1998), the samewe use to predict the halo mergers. The predicted numberdensity of dark matter halos is then given by, n h ( > M halo ) = ¯ ρ Z inf M min < N >M halo f ( ν ) dν. (13)Where f ( ν ) is the scale independent halo mass function, ν = [ δ c /σ ( M halo )] ( δ c = 1 .
68 is the value for spherical c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe over-density collapse). σ ( M halo ) is the variance in spheres ofmatter in the linear power spectrum, ¯ ρ is the mean densityof the Universe, and h N i is the average number of halos,including sub-halos where we assume the fraction of sub-halos (f sub ) is described by, f sub = 0 . − . z, (14)as in Conroy & Wechsler (2009). This method of halo abun-dance matching does a good job at matching observationsat multiple epochs (e.g., mass-to-light ratios, clustering mea-surements).The halo abundance matching we use for our mainLambda CDM cosmology include errors that incorporatethe difference in the abundance matching at the redshiftbounds of each bin, and the uncertainty due errors in thestellar mass functions. We investigate the same abundancesusing the mass function and galaxy bias as that of Tinkeret al.(2008) and Tinker et al. (2010). These results are veryclose to that of the Jenkins et al. (2001) halo mass function,and its resulting bias. We also find that the halo to stel-lar mass relationship from abundance matching, using ourmain Lambda CDM cosmology, and the cosmology used inConroy & Wechsler (2009) extracted using DEXTER are al-most identical at higher redshifts. Our results also match theredshift z = 1 Conroy & Wechsler (2009) work well consid-ering our redshift bin is slightly higher, and we use differentgalaxy stellar mass functions. Similarly to Conroy & Wech-sler (2009) who do not trust their z > z results below the relevantstellar mass limits. e.g., for z = 2 at log M ∗ > . M ⊙ .The evolution in shape at the high mass end is also in agree-ment with Conroy & Wechsler (2009). Therefore the range ofmerger histories for different cosmologies in the abundancematching will not vary by much.Although this paper is not focused on abundancematching, which will be described in more detail in Twiteet al. (2014), another issue that we investigate is how wellour abundance matching can reproduce the angular corre-lation function of galaxies to check for halo bias within ourhalo abundance matching models. It should be noted thatthis is intended as a rough check and not a robust fit of thegalaxy HOD as this is not the focus of this work. We pro-duce angular correlation functions using our HOD code witha simplified version of the 5-parameter HOD model (e.g.,Zheng et al. 2005), where N centrals = 1 if M h > M h (M min ∗ )and M h < M h (M max ∗ ), and N sat = [( M − M ) /M ′ ] α .Here we have set α = 1, log ( M ) = log ( M ) − . ( M ′ ) = log ( M ) + 1 .
0, which correspond to averageHOD fit parameters (e.g., Zehavi et al. 2011).We also compare the resulting correlation function withthe power-law fits from Foucaud et al (2010). Here wehave assumed a Gaussian redshift distribution (whereby theGaussian was centered on the middle of the redshift bin andhad σ = (bin − width / /
3, and we then use the Limber(1954) equation to transform to the angular correlation func-tion. We follow the formalism of the Tinker et al. (2005) n g matched method to obtain the galaxy correlation function.We find that our large scale clustering is in good agreementwith the power law for the z = 1 samples, however it islikely that the power-law should be steeper for such massivegalaxies at high redshift, and the small scale clustering is Figure 7.
Figure showing the relationship between the stellarmass and halo mass. The solid blue round points show this re-lation derived through stellar mass measurements and measure-ments from kinematics (Twite et al. 2014). The other lines arefrom our halo abundance matching fitting, while the dashed linewith two solid lines around it are from abundance matching fromHopkins et al. (2010), and Conroy & Weschler (2009). at about the right amplitude. We can therefore confidentlysay that the abundance matching reproduces the ball parkcorrelation functions for galaxies, at a similar level to whatis stated in Conroy & Wechsler (2009).We show the comparison between halo masses derivedin this way and the stellar masses at the same limit in Fig-ure 7 from redshifts from z = 1 to z = 3 using two differentmethods. There is clearly little evolution in the halo to stel-lar mass ratio as a function of redshifts within our massesof interest, which is also what we find when we investigatethis relationship using the observable relations from internalmotions. Figure 7 shows the relationship between the stellarand halo masses for galaxies up to z = 3 calculated in twodifferent ways.There is some disagreement at the lower mass range,however this is due to the fact that there is unlikely a 1:1galaxy - halo ratio at these masses. Hence we get an overesti-mate of the halo masses for these systems. This is expectedto some degree within this formalism, and this effect hasbeen seen before by e.g., Conroy & Wechsler (2009). Notethat we only use Figure 7 in the stellar mass range where ourgalaxies are found (log M ∗ = 10 − . tot ) = ( α ) log(M ∗ ) + ( β ) , (15) c (cid:13) , 1–22 C.J. Conselice et al.
Figure 8.
The halo merger histories compared with data for a) halos with masses M halo > M ⊙ and b) halos with masses M halo > M ⊙ . The various lines show how the merger history varies with redshifts for halos of this given mass, and using the cosmologies listedin the upper left. The solid symbols with errors bars are data on the merger fraction evolution taken from Conselice et al. (2009) forthose at z < .
5, merger fractions from pairs from the Ultra Deep Survey (circles with inner error bars) and from Bluck et al. (2009) for z > . ∗ > M ⊙ . For panel b) the solid symbols with errors bars are taken from Conselice et al.(2009) for those at z < . z > .
5, all selected to have M ∗ > M ⊙ . For both the open symbolsare from asymmetries from Mortlock et al. (2013) using CANDELS data. The average error weighted values of the merger fractions fromthe various surveys are shown as open black stars. where the values of α and β are a function of the stellarmass. We calculate (see also Twite et al. 2014) that for logM ∗ < . α = 0 . ± . β = 5 . ± .
27, while for systemswith log M ∗ > . α = 1 . ± .
06 and β = − . ± .
61. InFigure 7 we plot this relation along side the relation for theabundance matching. In general there is a correlation suchthat M halo /M ∗ ∼
10, with a relatively small scatter.This implies that we are able to match the halo massto our measured stellar masses for our galaxies in whichwe measure the stellar masses and merger histories fromup to z = 3. When we do these comparisons we find thatthe log M ∗ = 10 limit corresponds to log M halo = 11.3 andlog M ∗ = 11 (see § halo = 12 . With the caveats explained above, and using the halo massto stellar mass comparisons in § The key to performing the comparison between the observedmergers and the predicted halo mergers is using the M halo vs. M ∗ relation described in § > > ∼ ∼ > > > > c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe per. We determine the merger fraction at this stellar massration by using the observations from Bluck et al. (2012).We use this to correct other pair fractions from Man et al.(2012) and Lopez-Sanjuan et al. (2010), with the assump-tion that the relative fractions change in the same way as inthe Bluck et al. (2012) study. Regardless the vast majorityof our comparisons are done using the CAS mergers wherethe total mass ratios are already > Using the information above we show in Figure 8 a di-rect comparison between the halo merger history predictionswhich we have discussed throughout this paper, and the ob-served galaxy merger histories based on our stellar mass se-lection. Figure 8 shows that there is a general agreementbetween the shape and normalization of the merger historyfor the predicted halos and the observed galaxy merger his-tories. To make a quantitative comparison we use the conver-sion factors from eq. (15), with its associated uncertaintiesto determine the halo mass for the corresponding selectionof stellar mass. For log M ∗ >
10 this corresponds to a se-lection of log M halo = 11 . ± .
4. We then use the relationsdiscussed in § § z ∼ .
5, the merger frac-tion with an uncertainty given by the models and conversionuncertainty is f halo = 0 . +0 . − . .Our relatively large observational errors on the mergerfraction evolution cannot easily distinguish between thesevarious models currently, an issue we discuss in more de-tail below in terms of future surveys that can improve thiscomparison with data. We however carry out a full analysisof the best fit model using using a reduced χ approach,and using the uncertainties calculated using the methodsdescribed above across all redshifts. We do this by matchingthe model with the corresponding halo mass to our stellarmass derived galaxy merger measurements. We then utilizethe best fitting power-laws to determine the χ of each fit.The reduced χ values range from 4.8 to 14.43, with the bestfitting model the concordance cosmology with Ω Λ = 0 . m = 0 .
3. The data also rules out that the universe has alow matter density of Ω m = 0 .
1, although in terms of thecomparison there is a similarity between the merger historyfor models with Ω m = 0.3, Ω Λ = 0.7 and Ω m = 0.3, Ω Λ = 0.We furthermore show in Figure 10 the variation of thehalo merger fraction as a function of Ω Λ at z = 2 . Λ + Ω m = 1 holds throughout. The bestfit relation between the merger fraction and the value of Ω Λ is given by: f halo = α × Ω Λ + β = α × (1 − Ω m ) + β (16)Where the values for the fit for Ω Λ < .
55 is α = − . ± . β = 0 . ± .
01. For Ω Λ > .
55 the best fit is given by α = − . ± .
02 and β = 0 . ± .
01. The relation betweenthe halo merger values and Ω Λ is such that at Ω Λ < . Λ and themerger fraction. This relation becomes steeper for valuesΩ Λ > .
6, making it a more sensitive measurement of Ω Λ and Ω m . Figure 9.
The relationship between the value of Ω Λ and theresulting merger fraction for systems with stellar masses > M ⊙ at z = 2.5. This is for the condition such that Ω Λ + Ω m = 1.The solid lines are the best fitted relationship for this relationshipas described in the text. Using our best measured merger fraction of f m = 0 . ± .
07 at z = 2 .
5, we find that this formally leads to a mea-sured Ω Λ = 0 . +0 . − . , which is very uncertain compared toother leading methods of finding Ω Λ , but still demonstratesconsistency with previous work, and that there is at least abroad agreement between cosmology and galaxy formationas seen through mergers. The error bars on this measure-ment come from the uncertainty in the theoretical fit and theuncertainty in the measured merger fraction from Mortlocket al. (2013) based on CANDELS data and incorporatingstellar mass and redshift uncertainties.We can also now compare the merger history to the pre-dicted halo mergers with differing values of the σ parame-ter. Figure 9 shows this comparison of our merger historiesfrom observed galaxies in comparison to the halo mergers fordiffering values of σ . What we find is a relatively high valueof the merger fraction in comparison to the halo merger his-tories. This suggests that the value of σ , as derived from ourobservations would be low, with the value of σ = 0.7 in bestagreement with our observations. In the next sub-sections wediscuss how to use observed mergers as a competitive toolto constrain cosmology. c (cid:13) , 1–22 C.J. Conselice et al.
Figure 10.
The evolution of the halo merger fraction for haloswith M halo > M ⊙ as a function of the value of σ as in Fig-ure 3 except that here we show a comparison to data. The rangeshown is from σ = 0.7 to σ = 1 .
1, The points shown here areactual measures of the merger fraction for galaxies as a functionof redshift. The red points are for M ∗ > M ⊙ galaxies, in-cluding results from Conselice et al. (2003) at z > z < . z > <
30 kpc:(Man et al. 2012; crosses at z > ∗ > M ⊙ systems including results from Conselice et al.(2009) (solid boxes at z < .
2; Bluck et al. (2009, 2012) (solidcircles at z > . z > In the previous sections we investigate the merger historyand how it compares to dark matter halo mergers for a va-riety of cosmologies. However, it is clear that the concor-dance cosmology is the most likely cosmology with the largeamount of supporting observations in the past few decades.What is not known for certain is the role of an evolvingform of the dark energy, and how this may affect the galaxyformation process. In this section we investigate how themerger history for halos changes for a variety of evolvingequations of states of the universe itself.In fact, one of the major goals in cosmology is to con-strain the equation of state of the universe, which relates thepressure (P) and density ( ρ ) of the dark energy, or ω = P/ρ .The value of ω = − ω give values ω = − . ± .
08 (Ander-son et al. 2012). The dark energy equation of state can alsobe written as ω ( z ) = ω + ω a (1 − a ), although the measure- ments of the parameters ω and ω a are not well constrained,with the latest measurements giving ω = − . ± . ω a = − . ± .
09 (Sullivan et al. 2011).In this section we investigate how the merger historyvaries with different values of the equation of state of theuniverse. These merger predictions are from the GALACTI-CUS models of Benson et al. (2012) using the equations fromPercival (2005). First we evaluate how the merger fractionvaries for halos with masses M halo > M ⊙ for values of ω between ω = − .
33 to ω = − . ω a = 0 for both. We also investigate this at two different halomasses, those with log M halo >
11 and for log M halo > ω influence the halo merger history. We makethe note that, as clearly shown in Figure 11, it will be verydifficult to utilize this type of figure to constrain the valueof ω with galaxy merger histories given that there is not alarge range of values of the halo merger fraction at differentredshifts.However, there are a few trends we can see. The firstis that higher negative values of the equation of state pa-rameter, ω , gives a higher merger fraction down to z = 1 . ω merger histories arewell fit by a power-law of the form (1 + z ) m up to z = 3.We can use this to investigate the likelihood of being ableto detect a difference in the merger history which may helpconstrain cosmology. We find that the difference betweenthe effective merger histories is roughly δf halo ∼ .
005 fromvalues of ω = − . ω = 1 .
1. We discuss how this couldbe useful in cosmological investigations in § ω ( z ) = ω + ω a (1 − a ) (17)where there is currently very little constraint based on ob-servations of supernova (Sullivan et al. 2011). We howeverfind that the merger histories for all reasonable combinationsof ω and ω a are very close to one another in merger space(Figure 12), such that the differences in δf m ∼ . ω values. This however becomeslarger at higher redshifts such that the difference betweendifferent values for varying equations of state is highest. Inthe future observing the merger fraction at z ∼ One of the issues that we have to address when comparinggalaxy observables, such as the merger history to models ofvarying cosmologies, is that the observables we use dependson the cosmology assumed when calculating derived valuessuch as the stellar mass or merger rate. For example, themergers we find are based on measuring a stellar mass andthe value of this stellar mass can change depending on thecosmology used to calculate it.Other features which change with cosmology are the c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe Figure 11.
The change in the halo merger fraction as a function of redshift for differing values of the cosmological constant, as given inthe equation of state for the dark energy, ω ( § halo >
12, while theright is for halos of mass M halo >
11. The green line and blue lines show the range of our probe from ω = − . ω = − .
33. Thered line shows the values for the fiducial ω = − Λ Ω m Ang. Size Dist Ratio ∆ M Table 6.
How galaxy observables change as a function of cosmol-ogy, most notably the angular size distance and the luminositydistance, which has a direct effect on the measured stellar masses.Shown here is the ratio of the angular size distance, defined asthe angular size distance at the given cosmology divided by theangular size distance in the concordance cosmology. Likewise thestellar mass differences between these two cosmologies is shown. physical separation between two galaxies, and whether ornot this is a physical pair or not according to our definitionbased on the angular size distance. We therefore investigatein this section how the observed merger fractions and pairscan depend upon the cosmology assumed. We investigatethese differences in terms of the concordance cosmology.We show in Table 6 the values of the difference in de-rived stellar mass ∆ M ∗ within the various cosmologies inreference to the coordinate cosmology. The biggest change isthat at a given stellar mass selection, the galaxies examinedare up to δM ∗ = 0 .
67 more massive depending on the cos-mology examined. Likewise we find that the angular scale can vary by 40%, but mostly around 20% for cosmologiesdifferent from the concordance cosmology.We investigate how the merger fraction changes whenthe measurements of stellar masses and galaxy separationsare affected by different cosmologies. Note that the cosmol-ogy has more of an effect on the measurements for mergerswhich are found within pairs of galaxies. For morphologicalmergers, there is no angular separation to worry about andthus it is only the change in the distance modulus with cos-mology that affects the choice of which bin a galaxy belongin. For this reason we utilize merger fractions, f(M ∗ ) whichare a pure observable, as a function of stellar mass, which isas noted a function of the cosmology. We could use mergerrates, but the time-scales for these merger rates depends tosome degree on cosmology, and on the nature of the darkmatter. By using merger fractions we are able to side-stepthis issue to some degree as it is a purely observational quan-tity. To understand this issue we examine how the mergerfraction varies with pairs when changing the cosmologicalparameters, which results in the different selections of un-derlying halos. We find that the observed merger fractionsincrease at larger separations, as long noted in pair studiesof galaxies, and increases for higher mass selections at z > m = 0 .
1, Ω Λ = 0 will sample at agiven angular separation, and a given flux cut lower massgalaxies and smaller separations as both the luminosity dis- c (cid:13) , 1–22 C.J. Conselice et al.
Figure 12.
The relationship between the merger history as afunction of redshift and different values of ω and ω for an evolv-ing form of the dark energy equation of state (see § tance and angular size distance is larger in this cosmologythan in the concordance one. To account for this cosmol-ogy we would have to examine intrinsically fainter galaxiesand look for pairs which have a smaller angular separationon the sky to match the intrinsic conditions used to findmergers in the coordinate cosmology. Both of these effectswill decrease the merger fraction measured. If we take the∆ M = 0 .
67 mass difference and the factor of 1.4 larger an-gular size separation we would measure the merger fractionas δf m ∼ .
05 lower than what is plotted on Figure 8. Thisis within the errors of the merger fractions themselves, butwould decrease the measurement towards the model valuesfurther than the observed.Likewise, universe’s with a high mass density, i.e., withΩ m = 1 have smaller angular size and luminosity distances.Therefore at a given flux, and at a given angular size mea-sure on the sky, the galaxies sampled within this cosmologywill be of lower mass and larger intrinsic separations. Thisgives the opposite effect from above, and to measure the cor-rect merger fraction would require going to brighter systemsand at larger angular separations. These both would resultin a higher merger fraction based on measurements of howthe observed merger fraction changes at fainter limits andsmaller separations. We calculate for this cosmology thatthe true comparison merger fraction would be δf m ∼ . Figure 13.
The stellar mass and total mass relation using theabundance matching methodology described in § z = 1 − .
5. The dashed lines are for the concordance cosmologybut using different values of σ . The solid lines are for Ω = 1 . m = 0 . Λ = 0. The bluepoints are the kinematic relations for the concordance cosmologydescribed in § There is also the issue that our mapping of stellar andhalo masses will depend on the cosmology assumed in thecalculation. We show these different stellar vs. halo massesusing different cosmologies in Figure 13. For the concordancecosmologies with differing values of σ from 0.7 to 0.9 thereis a variation of the total mass for a halo mass at log M ∗ ∼ m = 0 . m = 1. Thesedifferences lead to a factor of ten difference in the halo massat a given stellar mass from the concordance cosmology atmasses around log M ∗ ∼
11 in the most extreme examples.However, as the halo merger fraction dependence on massis quite shallow, this only leads to an addition small uncer-tainty in the halo merger fraction of f halo ∼ . c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe In this subsection we investigate using the merger history asa cosmological tool. We preface this by stating that merg-ers may never be as good of a method as some to measurecosmological properties. The reason that methods that usestandard candles such as Type Ia supernova are so success-ful is that fundamentally there are only two observations -the flux and redshift, which can then be compared directlyto cosmological predictions. However, our method does al-low us to measure cosmology by the effect it has on objectswithin the universe, rather than its effects of the expansionwhich most methods use.The use of the galaxy merger history as a competi-tive cosmological tool will require that we know more aboutthe properties of galaxies, namely accurate measurementsof galaxy masses and the ability to understand the mergerhistory without significant systematic biases. Although themerger history has some agreement using different methodssuch using galaxy pairs and through morphology (e.g., Con-selice et al. 2009; Lotz et al. 2011), there still remains workto be done to determine whether or not we can actually mea-sure the merger fraction or merger rate to within 1% up to z ∼
3. This means we will have to measure the merger his-tory accurately over all environments, such that the effectsof cosmic variance produces less than a 1% systematic erroron the measured merger values.Even if we can measure the merger history accuratelywithout significant systematic errors, there is still the issueof shot noise within these measurements, requiring largerarea surveys than what we currently have. We can investi-gate how good our merger fraction measurements will needto be to measure the history of mergers to distinguish be-tween cosmological models.To measure accurately the differences in the cosmologi-cal models presented in this paper requires that the accuracyin measuring merger fractions for galaxies at a given mass is δf m ∼ .
01. To distinguish between δf m ∼ .
01, for a mergerfraction of f m ∼ .
3, i.e., a 30% fraction of the populationundergoing mergers, which we see at z > or 1.2 deg . Surveys this large do now ex-ist, such as the CFHT legacy survey and the UKIDSS Ultra-Deep Survey (UDS), and the next generation of photomet-ric and spectroscopic redshifts within these should make anattempt at measuring merger fractions by using pairs. How-ever, the use of morphology for the measurement of mergerfractions will be difficult given that only a very small frac-tion of this area has been imaged at high enough resolutionin the near-infrared in the CANDELS survey (e.g., Groginet al. 2011; Koekemoer et al. 2011).We can further investigate the accuracy we need toget a 3 σ accuracy on statistical measures of merger frac-tions. If we want to obtain a 3 σ measurement at δf m ∼ .
01 we will need a larger area of 8.1 deg . This is muchlarger than any deep surveys, yet in the future, those withEuclid or WFIRST will be able to measure the struc-ture/morphologies over this area.Attempting to use the merger history for measurementsof different values of the equation of state parameter, ω , andfor an evolving form of ω ( z ), require a precision about twice as good. As briefly discussed in § ω to within ± . ω = −
1, or to constrain betterthan current estimates of ω and ω a requires that we mea-sure the merger fraction to within f halo ∼ .
005 or better.This will require deep surveys of at least 2.5 deg in area,which is currently possible by combining existing surveys.To obtain 3 σ statistical errors will require an area of ∼ . This also assumes that other errors, such as systemat-ics with redshifts, stellar masses, etc. are small, which theyare not. Along side obtaining deeper and wider surveys wewill also need to be more certain that accurate stellar massesand photometric/spectroscopic redshifts can be measured.As these improve to measure galaxy evolution so will ourability to use galaxies as a cosmological tool. There are a few important implications within our results,for cosmology, structure assembly, galaxy formation, as wellas for our ability to measure the merger history. We brieflydetail these here, although a further analysis of these willbe addressed in later papers. The first issue is that by ex-amining the dark matter assembly itself, rather than relyingon the baryonic physics behind semi-analytical simulationswe get a much better agreement with the merger data, mak-ing reasonable assumptions about the relation between theobserved stellar mass of galaxies and their total halo massfrom known scaling relations. This is an important point tomake, as it reflects the well known problem of matching theabundances of galaxies with their total halo mass functions.It is clear that the galaxy mass function has a shape whichdiffers significantly from the predicted halo mass functionand processes such as feedback, photoionization, conduc-tion, gas cooling among other baryonic processes must beimplemented to correctly model the observed galaxy massfunction (e.g., Benson et al. 2003).However, for whatever reason the implementation ofthe baryonic physics in semi-analytical models does not ac-curately reproduce the merger fraction and rate history ofgalaxies. This implies that the baryonic physics implementedin these models is missing ingredients necessary to reproducethe observed history of galaxy assembly, or is implement-ing them in an incorrect way. This is perhaps also reflectedin the problem of semi-analytical models in predicting thenumber of massive galaxies at high redshift (e.g., Conseliceet al. 2007; Guo et al. 2011). While the Warm Dark Mattermodels do better, this may reflect less structure on smallerscales, and thus more likely to produce similar mass mergers.We also demonstrate that the merger history dependson cosmology such that cosmologies with higher matter den-sities will contain a high merger fraction and rate, and there-fore that there will be more massive halos in these cosmolo-gies found in the nearby universe. We furthermore show, ina limited way, that there is also a good agreement betweenthe merger history and the generally accepted cosmologicalmodel, such that Ω m = 0.3 and Ω Λ = 0.7 is the best fit modelbased on a wide range of possible cosmologies. However, itis worth pointing out that this constraint is not very use-ful in itself given that the standard model of cosmology isgenerally accepted. To be a useful indicator for cosmology,this method would have to be able to distinguish suitable c (cid:13) , 1–22 C.J. Conselice et al. differences in the merger history for the limited range ofcosmologies that are now predicted in CMBR experimentssuch as WMAP and Planck (e.g., Komatsu et al. 2011; Adeet al. 2013). However, it is reassuring that the agreement wefind demonstrates a consistency between the dominant andgenerally accepted cosmology, and how the dark matter onthe level of galaxies is assembling through mergers.It is possible to distinguish between various cosmologieswith various Ω m and Ω Λ values, however we conclude thatthe accuracy in the merger fraction needed to do this isvery high, on the order of 1% to distinguish between variouscosmologies with similar values. This will require surveysfrom 1.2 deg in area up to 20 deg , but the real challengewill be in making sure that all systematics are accounted forwhen carrying out this comparison. One of these challengesis understanding the merger time-scale, that is how long anasymmetric galaxy, or a galaxy seen in a pair, takes to merge.This is essential when comparing with any predictions. Themerger time-scale is estimated to be 0.4 Gyr in this paper,which is what is found empirically by observing the changein the observed merger fraction (Conselice 2009), as wellas an average over simulations (e.g., Conselice 2006; Lotzet al. 2010). However, the merger time-scale does dependupon the galaxy gas mass fraction (e.g., Lotz et al. 2010)and other properties such as viewing angle, bulge to diskratio and orbital type. While the average of such propertiesmay be calculable from models, it remains to be determinedwhether time-scales seen at high redshift are similar to thosepredicted in models. Future detailed simulations with forexample Eagle and
Illustris will help reveal time-scales forthese mergers which will eliminate another key systematicto create these comparisons.To avoid the time-scale issue completely the mergerfraction comparison could be carried out using galaxieswithin close pairs – i.e., systems which are just about tomerge. Simulations can predict this often as well as theycan an actual merger, and observationally, these pairs arenot as ambiguous at times as deciding if a galaxy is anactive merger or not. Likely, a combination of pre-mergerclose pairs and post-merger morphological signatures arebest used together as a check on systematics of both meth-ods. We also note that the agreements with the theory forsome of our merger histories is not perfect. The most obvi-ous case of this is the comparison between the halo mergerhistory and galaxy mergers selected with stellar massesM ∗ > M ⊙ (Figure 8). Here we can see that at z < z < z < This paper explores the idea of using galaxy mergers as aprobe of cosmology. While the idea that galaxy merging canbe used to probe the properties of the universe is not new(e.g., Carlberg 1991), the use of it as a cosmological probehas never been fully characterized, or explored. In this paperwe have shown that using a variety of assumptions regardinghow the halo merger history can be matched to some degreewith the galaxy merger history, the current observations ofthe galaxy merger history are in relative agreement with aconcordance cosmological model.Our other major findings are:1. The halo merger history varies as a function of halomass, such that systems with larger halo masses have highermerger fraction at all redshifts. This is in direct contrast tothe observations of galaxy mergers whereby the most mas-sive systems (as measured in stars) have a higher mergerfraction at z >
2, but tend to show little merging at latertimes.2. Semi-analytical models based on CDM from the Millen-nium simulation under-predict the galaxy merger history,but that those using Warm Dark Matter with a particletemperature of ∼ m cosmologies, and highest for cosmologieswhere Ω m = 1. The difference between these is large, around δf halo ∼ .
25 at z ∼ Λ = 0 . m = 0 . Λ at a single redshift, z = 2 .
5, and how this can be furtherused to calculate the best fit value of Ω Λ = 0 . +0 . − .
5. The halo merger history is also strongly dependent on thevalue of σ , such that lower values of σ give a higher mergerfraction.6. We also examine how the merger history changes for dif-fering values of the dark energy equation of state ω , andhow a varying ω ( z ) changes the calculated halo merger his-tory. We find that accurate merger fractions on the levelof δf ∼ .
005 are required to distinguish between compet-ing models. The difference between the predictions for themerger histories is highest at the highest redshifts and inthe future JWST and the E-ELT can provide these mea-surements.The use of the galaxy merger history to probe cosmol-ogy in a competitive way with other techniques such as su-pernova, baryonic acoustic oscillations, and CMBR work willrequire several improvements. The first is that we must beable to match better the stellar masses of galaxies and their c (cid:13) , 1–22 alaxy Formation as a Cosmological Probe halo masses, as well as have a firmer idea of the mass rangesthat produce merging and the time-scale for halo mergers,and how these relate to the time-scale for galaxy mergers.While halo occupation is one way to do this, and can producesuccessfully matched merger histories, we are able to derivesimilar results using a calibration based on the kinematicsof galaxies.Overall, we find that the accuracy of merger fractionswould have to exceed δf m ∼ .
01 to be able to differentiatebetween current uncertainties in the value of cosmologicalparameters. As explained in this paper this will requiredsurveys on the sky of area up to several 10 deg . Thesetypes of surveys must also have accurate stellar masses andreliable redshifts with uncertainties, along with the mergeruncertainties, which do not add up to a merger fraction er-rors that are larger than 1%. This will be difficult, but bycombining results at various redshifts, and various masses,some of these limits on uncertainties could be relaxed a bitand systematics better understood. Future surveys such asEuclid and LSST will be ideal for carrying out this type ofanalysis and has a potential to be competitive with othertechniques such as those listed above.We thank Mike Santos, Aisyah Sahdan and Jack John-son for their early contributions to this work, and EdCopeland for illuminating discussions. CJC, AM and AFLBacknowledge support from the STFC and the LeverhulmeTrust. D.P. acknowledged the support of an Australian Post-graduate Award and an Endeavour Research Fellowship.A.F.L.B. acknowledges support from NSERC (National Sci-ence and Engineering Research Council) of Canada. REFERENCES
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