Assessing the Predictive Power of Galaxy Formation Models: A Comparison of Predicted and Observed Rest-Frame Optical Luminosity Functions at 2.0<z<3.3
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ASSESSING THE PREDICTIVE POWER OF GALAXY FORMATION MODELS: A COMPARISON OF PREDICTED ANDOBSERVED REST-FRAME OPTICAL LUMINOSITY FUNCTIONS AT 2 . ≤ Z ≤ . D ANILO M ARCHESINI AND P IETER G. VAN D OKKUM Received 2007 April 11; accepted 2007 May 24; published 2007 June 21
ABSTRACTRecent galaxy formation models successfully reproduce the local luminosity function (LF) of galaxies by in-voking mechanisms to suppress star formation in low- and high-mass galaxies. As these models are optimizedto fit the LF at low redshift, a crucial question is how well they predict the LF at earlier times. Here we comparerecently measured rest-frame V -band LFs of galaxies at redshifts 2 . ≤ z ≤ . z ∼
3. However, all models predict an increase withtime of the rest-frame V -band luminosity density, whereas the observations show a decrease. The models alsohave difficulty matching the observed rest-frame colors of galaxies. In all models the luminosity density of redgalaxies increases sharply from z ∼ z ∼ .
2, whereas it is approximately constant in the observations. Con-versely, in the models the luminosity density of blue galaxies is approximately constant, whereas it decreasesin the observations. These discrepancies cannot be entirely remedied by changing the treatment of dust andsuggest that current models do not yet provide a complete description of galaxy formation and evolution since z ∼ Subject headings: galaxies: evolution — galaxies: formation — galaxies: fundamental parameters —galaxies: high-redshift — galaxies: luminosity function, mass function INTRODUCTION
In the current paradigm of structure formation, dark mat-ter (DM) halos build up in a hierarchical fashion through thedissipationless mechanism of gravitationally instability. Theassembly of the stellar content of galaxies is instead gov-erned by much more complicated physical processes, oftendissipative and non-linear, which are generally poorly under-stood. To counter this lack of understanding, prescriptionsare employed in the galaxy formation models. One of thefundamental tools for constraining the physical processes en-coded in these models is the luminosity function (LF), sinceits shape retains the imprint of galaxy formation and evolutionprocesses.The faint end of the LF can be matched with a combina-tion of supernova feedback and the suppression of gas cool-ing in low-mass halos due to a background of photoioniz-ing radiation (e.g., Benson et al. 2002). Matching the brightend of the LF has proven more challenging. Very recent im-plementation of active galactic nucleus (AGN) feedback insemianalytic models (SAMs) has yielded exceptionally faith-ful reproductions of the observed local rest-frame B - and K -band global LFs (Bower et al. 2006; Croton et al. 2006; seealso Granato et al. 2004), including good matches to the lo-cal rest-frame B -band LFs of red and blue galaxies (althoughwith some discrepancies for faint red galaxies; Croton et al.2006).The excellent agreement between observations and mod-els at z ∼ Department of Astronomy; Yale Center for Astronomy and Astro-physics, Yale University, New Haven, CT, USA; [email protected] to observations at 0 < z < . ≤ z ≤ .
3, using a combi-nation of the K -selected MUSYC, GOODS, and FIRES sur-veys (Marchesini et al. 2007). In this Letter we compare theobserved LF to that predicted by theoretical models in thisredshift range, in order to test the predictive power of the lat-est generation of galaxy formation models. We also comparethe observed LF to predictions from smoothed particle hy-drodynamics (SPH) simulations, which have so far only beencompared to data at z ∼ Ω M = . Ω Λ = .
7, and H =
70 km s - Mpc - . Allmagnitudes are in the AB system, while colors are on the Vegasystem. THE OBSERVED LUMINOSITY FUNCTIONS
The observed rest-frame optical LFs at z ≥ B -band (at 2 . < z ≤ . ≤ z ≤ . V -band (at 2 . ≤ z ≤ . R -band (at2 ≤ z ≤ . K -selected sample constructedfrom the MUltiwavelength Survey by Yale-Chile (MUSYC;Quadri et al. 2007), the ultradeep Faint InfraRed Extragalac-tic Survey (FIRES; Franx et al. 2003), and the Great Ob-servatories Origins Deep Survey (GOODS; Giavalisco et al.2004; Chandra Deep Field–South). This K -selected sam-ple, comprising a total of ∼
990 galaxies with K tots <
25 at2 ≤ z ≤ .
5, is unique for its combination of surveyed area Marchesini & van Dokkum( ∼
380 arcmin ) and large range of luminosities.In this Letter we limit our comparison between observedand predicted LFs to the rest-frame V band, at the two redshiftintervals 2 . ≤ z ≤ . ≤ z ≤ . THE MODEL-PREDICTED LUMINOSITY FUNCTIONS
The Bower et al. (2006) SAM is implemented on the Mil-lennium DM simulation described in Springel et al. (2005).The details of the assumed prescriptions and the spe-cific parameter choices are described in Cole et al. (2000),Benson et al. (2003), and Bower et al. (2006). We have alsoused the outputs from the SAM of Croton et al. (2006) asupdated by De Lucia & Blaizot (2007). This model differsfrom the SAM of Bower et al. (2006) in many ways. Thescheme for building the merger trees is different in detail,as are many of the prescriptions adopted to model the bary-onic physics, most notably those associated with the growth ofand the feedback from SMBHs in galaxy nuclei and the cool-ing model (see Kauffmann & Haehnelt 2000; Springel et al.2001; De Lucia et al. 2004; De Lucia & Blaizot 2007 for de-tails). Finally, we have compared the observed LFs withthe predictions from the cosmological SPH simulations ofOppenheimer & Davé (2006), already used in Finlator et al.(2007) to constrain the physical properties of z ∼ - Mpcbox simulation, combined with a 64 h - Mpc box to bettersample the bright end of the LF. We note that the key differ-ence between the AGN feedback implementation in the SAMsand the superwind feedback is that the former does not requirestar formation.Computing the SAM-predicted rest-frame V -band LFs isstraightforward, as the catalog is complete in the luminosityrange of interest and has no redshift errors . We also extractedrest-frame colors of the galaxies in order to determine the LFfor red and blue galaxies separately. RESULTS
The comparison between the observed rest-frame V -bandLFs of all galaxies at 2 . ≤ z ≤ . ≤ z ≤ . ∼ . ≤ z ≤ .
3, the global LF predicted by the SAM ofBower et al. (2006) agrees well with the observed LF, al-though the SAM slightly underpredicts the density of galaxies To derive the model-predicted LF in a specific redshift interval, we av-eraged the number of galaxies as function of rest-frame V-band magnitude(with dust modeling included) of all redshift snapshots in the targeted red-shift interval. F IG . 1.— Comparison between the rest-frame V -band observed global LFsand those predicted by models. The observed LFs are plotted with black cir-cles (1 / V max method) with 1 σ error bars (including field-to-field variance)and by the black solid line (maximum likelihood method) with 1, 2, and3 σ solutions ( gray shaded regions ). The arrow shows the observed valueof M ⋆ . Red lines show predictions from the Bower et al. (2006) SAM, bluelines from the De Lucia & Blaizot (2007) SAM, and green lines from theFinlator et al. (2007) SPH model. Poisson errors (1 σ ) are shown for theSPH model only, as they are very small for the SAMs. In the small panels,the ratio between the predicted and the observed LFs is plotted, together withthe 1, 2, and 3 σ errors for the Bower et al. (2006) SAM ( gray shaded re-gions ). The oblique line regions delimit the comparison to the luminosityrange probed by the sample of Marchesini et al. (2007). around the knee of the LF. However, while at 2 ≤ z ≤ . Φ ⋆ is ∼ . ≤ z ≤ .
3, the De Luciamodel matches the faint end but underpredicts (by a factor of ∼ ≤ z ≤ .
5, instead, the predictedLF matches the bright end but overpredicts the faint end by afactor of &
2. The SPH simulations of Finlator et al. (2007)predict LFs that are qualitatively similar to those predictedby the two SAMs, although the former are characterized bylarger uncertainties, due to the much smaller simulated vol-ume.We quantified these results by determining the luminositydensity j V (obtained by integrating the LF) for the observa-tions and models. The luminosity density is a more robustFs at z ≥
2: observations vs. predictions 3 F IG . 2.— Top panels:
Observed luminosity density ( j obsV ) as function ofredshift of all ( black circles ), red ( red circles ), and blue ( blue squares ) galax-ies, splitting the sample based on rest-frame U - V ( left panels ) and B - V ( right panels ) colors. Bottom panels:
Luminosity density predicted by theSAM of Bower et al. (2006) ( j SAMV ) as function of redshift; symbols as intop panels; the observed evolution of j V is also plotted with dashed lines forcomparison. measure than M ⋆ , Φ ⋆ , and the faint-end slope α , becausethe errors in these parameters are highly correlated. The ob-served j V ( j obsV ) has been estimated by integrating the best-fitSchechter function down to M V = - .
5, which is the faintestluminosity probed by the K -selected sample . To estimate j V from the SAM ( j SAMV ), we have fitted the predicted LFs with aSchechter function, leaving M ⋆ , Φ ⋆ , and α as free parameters,applying the same limits as to the data.The comparison between j obsV and j SAMV of Bower et al.(2006) is shown in Figure 2 ( bottom panels ) by the black linesand data points. The Bower SAM matches the observed lumi-nosity density at z ∼
3. However, the model does not matchthe evolution of j V . In the model the luminosity increaseswith cosmic time, by a factor of ∼ . z ∼ . decreases withtime, by a factor of ∼ . COLORS
We investigated the cause of the discrepancies by splittingthe sample into blue and red galaxies, using their rest-framecolors. Here we focus on the Bower model, as it provides thebest match to the shape of the global LF, and a wide rangeof rest-frame colors are available. Interestingly, the resultsdepend strongly on the choice of color: splitting the sampleby U - V color (as done in Marchesini et al. 2007) producesvery different results than splitting by B - V color.To define red galaxies, we first use the criterion U - As in Marchesini et al. (2007) the 3 σ error on j V was calculated byderiving the distribution of all the values of j V within the 3 σ solutions of theSchechter LF parameters from the maximum-likelihood analysis, including inquadrature a 10% contribution from photometric redshift uncertainties. Usinga brighter integration limit of the LF ( M V = - .
4) does not change the resultsof the comparison significantly. V ≥ z ∼ z ∼ .
2, although it predicts the correct ratio between thetwo (roughly 1:1).Next, we use the criterion B - V ≥ As can be seen inthe top panels of Figure 2, this criterion leads to very similarobserved densities of red and blue galaxies as the U - V crite-rion. However, the predicted densities are in severe disagree-ment with the observations, particularly at z ∼ bottom right panel ). The red galaxy density at z ∼ ∼
8. Qualita-tively similar results are obtained when j SAMV from the SAMof De Lucia & Blaizot (2007) is used in the comparison. Irrespective of the color criterion that is used, we find thatthe predicted evolution of the red and blue luminosity densi-ties is in qualitative disagreement with the observed evolution.In the observations, the moderate evolution of the luminositydensity is mainly driven by a decrease with cosmic time of thedensity of blue galaxies, with the red galaxies evolving muchless (see also Brammer & van Dokkum 2007). By contrast,in the SAMs, the moderate evolution seen in the global LF isin the opposite sense and dominated by a strong evolution inthe red galaxy population. DISCUSSION
The main results of our comparison between the observedand the model-predicted rest-frame V -band LFs of galaxiesat z ≥ z ∼
3; (2) the models predict an in-crease with time of the rest-frame V -band luminosity density,whereas the observations show a decrease; (3) the models pre-dict strong evolution in the red galaxy population, whereas inthe observations most of the evolution is in the blue popu-lation; (4) the models greatly underpredict the abundance ofgalaxies with B - V ≥ . z ∼ U - V and B - V colorsare interesting, as they may hint at possible ways to improvethe models. We further investigate the disagreement betweenobserved and predicted colors in the SAM of Bower et al.(2006) in Figure 3, which shows the comparison of observa-tions and predictions in the B - V versus U - B diagram. Whilethe SAM seems to broadly reproduce the observed U - B dis-tribution, it predicts galaxies that are systematically bluer in B - V than the observed galaxies. We have plotted evolution-ary tracks of stellar population synthesis models constructedwith the Bruzual & Charlot (2003) code, assuming three dif-ferent prescriptions for the star formation history (SFH): aconstant SFH (CSF), an exponentially declining in time SFHcharacterized by the parameter τ =
300 Myr ( τ ⊙ and 100 M ⊙ , and modeled the extinction by dustusing the attenuation law of Calzetti et al. (2000). A newburst of star formation lasting 100 Myr and contributing 20%to the mass is also added at t = . × yr ( t = . × yr)at z ∼ z ∼ .
2) to explore more complex SFHs.As can be deduced from Fig. 3, the differences between ob- For observed galaxies in the Marchesini et al. (2007) sample, U - V = .
25 roughly corresponds to B - V = . The De Lucia model provides B - V colors, but no U - V colors. Marchesini & van Dokkum F IG . 3.— B - V vs U - B comparison between observations ( open circles ) and predictions from the SAM of Bower et al. (2006, gray shaded regions ) in the twotargeted redshift intervals. The error bars in the top left corner represent the median errors on the observed colors. The filled triangles with error bars representthe mean colors of the observed galaxies and the error on the mean. The yellow, orange, and red lines show the evolutionary tracks described in § 6 (CSF, τ A V =0, 1, and 2, respectively. The tracks are plotted from 50 Myr to the age of the universe at the lower limit of thetargeted redshift range. The cyan, blue, and purple lines show the evolution of the colors after a burst of star formation, for the three values of A V , respectively.The arrow indicates the extinction vector for A V =1. The dashed lines correspond to B - V =0.5 and U - B = ( U - V ) - ( B - V ) = . - . = - .
25. Observedgalaxies have redder B - V colors than predicted, possibly due to additional dust and/or secondary star bursts. served and predicted colors could be due to larger amount ofdust and/or to more complex SFHs in the observed galaxies.The ad hoc treatment of dust absorption is a significant andwell-known source of uncertainty in the models. Modifica-tions to the specific dust model could partly resolve the differ-ences between observations and SAM predictions. By simplymultiplying the A V in the SAM by a fixed factor, we were ableto better reproduce the observed LFs at z ∼ . z ∼ B - V ≥ . z ∼3.While our ability to simulate galaxy formation has greatlyimproved in the past few years, our results imply that thepresent understanding of the physical processes at work in galaxy formation and evolution is still far from being satis-factory. On the observational side, more accurate redshift andcolor estimates would benefit studies of this kind. The lack ofspectroscopic redshifts is particularly worrying, as systematicerrors in redshift will lead to systematic errors in colors andluminosities (e.g., Kriek et al. 2006; M. Kriek et al. 2007, inpreparation).We are grateful to G. De Lucia (the referee) and G. Lemsonfor assistance with the Millennium Simulation database andhelpful clarifications. We thank K. Finlator and R. Davé formaking available their SPH simulations, and R. Bower for hishelp with obtaining the Bower et al. (2006) predictions. D.M.is supported by NASA LTSA NNG04GE12G. The authors ac-knowledge support from NSF CARRER AST 04-49678. TheMillennium Simulation databases used in this Letter and theWeb application providing online access to them were con-structed as part of the activities of the German AstrophysicalVirtual Observatory.