ON THE LACK OF EVOLUTION IN GALAXY STAR FORMATION EFFICIENCY

We show the main output of the method in Behroozi et al. (2012), the average SFR in dark matter halos as a function of virial mass (Bryan & Norman 1998) and time, in the top-left panel of Figure 1. The SFR depends strongly on time, yet there is also a distinct halo mass threshold, as may be seen by normalizing to the maximum SFR as a function of time (Figure 1, top-right panel). To understand the implications for gas physics in halos, it is necessary to consider the baryon accretion rate as well. We calculate this as the dark matter halo mass accretion rate (Behroozi et al. 2012) times the cosmic baryon fraction; see van de Voort et al. (2011) for a comparison with hydrodynamical simulations.

Zoom In Zoom Out Reset image size

The baryon accretion rate increases with halo mass and look-back time, as shown in Figure 1, middle-left panel. This trend combines with trends in the SFR to reveal a clear picture of star formation efficiency (SFE) in halos (Figure 1, bottom panel). This efficiency, defined as the SFR divided by the baryon accretion rate, shows a prominent maximum near a characteristic mass of 10^11.7 M_☉ (see also Figure 2). Indeed, the SFE over 90% of the history of the universe (z < 4) is strongly dependent on halo mass; by comparison, it has a weak dependence on time.

Zoom In Zoom Out Reset image size

The peak in the SFE at 10^11.7 M_☉ represents observationally constrained evidence for a characteristic mass for galaxy formation. This characteristic mass matches longstanding theoretical predictions in both its value and in its lack of evolution since z = 3–4. The steep efficiency cutoff above the characteristic mass (SFE∝M^−4/3_h, where M_h is halo mass) suggests that a strong physical mechanism prevents incoming gas from reaching galaxies in massive halos. Besides the effect of hot-mode accretion, this slope coincides with the mass and luminosity scaling for supermassive black holes (L_BH∝M_BH; scalings for M_BH vary from M_BH∝σ⁴∝M^4/3_h to M_BH∝σ⁵∝M^5/3_h; Ferrarese & Merritt 2000; McConnell et al. 2011), which may prevent residual cooling flows in massive clusters from forming stars. Below the characteristic mass, the efficiency is not a perfect power law; between M_h ∼ 10¹⁰ and ∼10^11.5 M_☉, the average slope is SFE∝M^2/3_h. This may seem to be consistent with semi-analytic galaxy formation models that use supernova feedback (most commonly scaling as V²_circ∝M_h^2/3; Hatton et al. 2003; Somerville et al. 2008; Lu et al. 2012) to expel most of the gas in low-mass halos. However, these models often assume that the expelled gas re-accretes onto the halo after a dynamical time (Lu et al. 2011); this extra incoming gas would result in a steeper mass dependence for the SFE at low halo masses.

The weak time dependence of the SFE is unexpected given the different environments of 10^11.7 M_☉ halos at z = 4 and z = 0. At z = 4, the background matter density was ∼125 times higher, mass accretion rates were ∼40 times higher, galaxy–galaxy merger rates were ∼20 times higher, and the UV background from star formation was ∼500 times more intense than at the present day (Behroozi et al. 2012). None of these differences significantly influenced average SFE (unless they conspired to cancel each other out), strongly constraining possible physical mechanisms for star formation in galaxies and halos.

While the time dependence of the SFE is weak, it is not absent. As seen in Figure 1, the characteristic halo mass evolves from a peak of 10¹² M_☉ at z = 3 to 10^11.5 M_☉ at z = 0. The peak SFE also evolves around its average value of 0.35, reaching a maximum of 0.55 at z = 0.8 and a minimum of 0.22 at z = 0. However, observational constraints on SFRs and stellar masses are uncertain at the 0.3 dex level (Behroozi et al. 2010, 2012) especially for z > 1; these are larger than the observed deviations (±0.2 dex) in the peak SFE. The variations in the characteristic mass are likely more significant; while observational biases can be stellar mass-dependent (Behroozi et al. 2012) in a way that changes the location of the peak halo mass, this effect (<0.1 dex; Leauthaud et al. 2012) cannot account for the 0.5 dex change from z = 3 to z = 0. Nonetheless, these concerns do not alter the fact that the trends with mass (four decades of variation) in the SFE are stronger than the trends with time.

One way to eliminate the residual time dependence in the characteristic mass is to use a different mass definition. For example, using M_200b (i.e., 200 times the background density) would cancel some of the evolution from z = 1 to z = 0. However, this would also raise the mass accretion rate at z = 0, which would increase evolution in the star formation efficiency's normalization. Using the maximum circular velocity (V_circ) or the velocity dispersion (σ) instead would also lead to more evolution in the SFE (at fixed V_circ or σ): due to the smaller physical dimensions of the universe at early times, both these velocities increase with redshift at fixed virial halo mass.

The nearly constant characteristic mass scale is robust to our main assumption that the baryon accretion rate is proportional to the halo mass accretion rate, because this mass scale is already present in the conditional SFR (Figure 1). A baryon accretion rate which scales nonlinearly with the dark matter accretion rate would change the width of the most efficient halo mass range, but it would not change the location. However, as discussed previously, the baryon accretion rate for small halos (M_h < 10¹² M_☉) can differ from the dark matter accretion rate through recooling of ejected gas; the changing virial density threshold can also introduce non-physical evolution in the halo mass which affects the accretion rate (Diemer et al. 2012). Properly accounting for these effects may change the low-mass slope of the SFE; we will investigate this in future work.

Note that the level of consistency seen in the SFE is not possible to achieve using other common specific ratios, e.g., the specific SFR (SFR to stellar mass ratio; Figure 1, middle-right panel) or the SFR to halo mass ratio. The stellar mass to halo mass ratio (i.e., the integrated formation efficiency) does show somewhat similar features (Conroy & Wechsler 2009; Behroozi et al. 2010, 2012; Yang et al. 2012; Leauthaud et al. 2012; Moster et al. 2012; Wang et al. 2012); however, the integrated efficiency is several steps removed from the actual physics of star formation. Galaxy stellar mass is influenced by stellar death, galaxy–galaxy mergers, and ejection of merging stellar mass into the intracluster light (Conroy & Wechsler 2009; Behroozi et al. 2012; Moster et al. 2012), complicating the interpretation of the integrated efficiency. Moreover, the shape of the integrated efficiency is influenced by star formation along the entire halo mass accretion history. Intuitively, the integrated efficiency tends to lag behind changes in the instantaneous SFE, leading to a peak at a larger halo mass and a gentler falloff in the high-mass slope, as shown in Figure 2.

Going further, it is interesting to approximate the SFE for individual halos as completely time-independent. In this case, the stellar mass formed at a given halo mass is

$$\begin{eqnarray} &&{\rm SM} = \int _0^{t_\mathrm{final}} \frac{f_b dM_h}{dt} \frac{d{\rm SM}}{f_b dM_h} dt = \int _0^{M_{h,\mathrm{final}}} \frac{d{\rm SM}}{f_b dM_h}f_b dM_h, \nonumber\\ \end{eqnarray} \tag{ 1 }$$

(where SM is the stellar mass, M_h(t) is the halo mass accretion history, and dSM/f_bdM_h is the SFE). The total stellar mass formed then becomes a function of only the final halo mass (M_{h, final}) and not of time.

The specific choice of redshift for the instantaneous SFR does not matter greatly, as shown in Figure 1. We nonetheless marginalize the instantaneous SFR over time; the resulting functional form is shown in Figure 2. Using this as the time-independent efficiency, we calculate the total stars formed as a function of halo mass using Equation (1) and reduce the resulting value by 50%, corresponding to the stellar population remaining for a 6 Gyr old starburst (Conroy & Wechsler 2009). (For comparison, a 1 Gyr old starburst would have 60% of its original stars remaining.) This allows us to calculate the stellar mass to halo mass ratio, as shown in the left panel of Figure 3. Similarly, we may use halo mass accretion rates and number densities along with the same SFE to calculate the cosmic SFR (Figure 3, right panel).

Zoom In Zoom Out Reset image size

The real universe is more complicated, of course; the stellar mass–halo mass relation must evolve weakly to accurately reproduce galaxy number counts (Conroy & Wechsler 2009; Moster et al. 2010, 2012; Behroozi et al. 2010, 2012; Leauthaud et al. 2012; Wang et al. 2012). However, integrating a time-independent SFE with respect to halo mass reproduces the z = 0 stellar mass–halo mass relation to within observational systematics over nearly five decades in halo mass (10¹⁰–10¹⁵ M_☉). Similarly, integrating the SFE times the mass accretion rate and number density of halos gives a precise match to the observed cosmic SFR from z ∼ 4 to the present.

Furthermore, the prediction in time-independent SFE models of fixed stellar mass formed at a given halo mass is not far off from observational constraints at z = 0 (0.2 dex scatter in stellar mass at fixed halo mass; Reddick et al. 2012). The evolution in the median stellar mass–halo mass relation with time, corresponding to an evolution in the SFE, may then set a lower bound on the scatter in stellar mass at fixed halo mass at the present day. Conversely, the scatter in stellar mass at fixed halo mass today sets an upper bound on the possible evolution of the median stellar mass to halo mass ratios at earlier times.

When considering the cosmic SFR, the time-independent efficiency model may imply more success matching galaxy formation physics than is warranted. In fact, a model with an SFE of 7% independent of halo mass or time also matches the decline in cosmic SFRs (Figure 3, right panel), but would not match the stellar mass to halo mass ratio or galaxy number counts. For that reason, the decline in the cosmic SFR since z = 2 is more related to declining dark matter accretion rates than changes in how galaxies form stars. This may explain past successes in reproducing the cosmic SFR with a variety of incompatible physical models (e.g., Hernquist & Springel 2003; Bouché et al. 2010; Krumholz & Dekel 2012; Davé et al. 2012)—the cosmic SFR for z < 2 alone is a poor discriminant between models. That said, the rise in the cosmic SFR from early times to z = 2 is much steeper than a mass-independent efficiency model predicts. Matching this rise is much more closely tied to galaxy formation physics, as it requires an increase in the average SFE with time. In the mass-dependent model, this is provided by an increasing number of halos reaching the characteristic mass.

The SFE leaves a distinct imprint on the star formation history of the universe: as halos pass through the characteristic mass (10^11.7 M_☉), they form most of their stars. Equivalently, most stars were formed in halos between 10^11.5 M_☉ and 10^12.2 M_☉ (Figure 4, left panel). Furthermore, because of the tight correlation between stellar mass and halo mass, most stars formed in galaxies with stellar masses between 10^9.9 and 10^10.8 M_☉ (Figure 4, right panel). This same narrow range of halo and stellar masses (which includes the stellar and halo masses of the Milky Way; Klypin et al. 2002; Flynn et al. 2006; Smith et al. 2007; Busha et al. 2011) is responsible for most star formation since at least z = 4, due to the constancy of the SFE with time. Given current observational limits (Figure 4, right panel), surveys have probed a stellar mass and redshift range corresponding to 90% of the star formation in the Universe.

Zoom In Zoom Out Reset image size