Sub-millimeter galaxies in hierarchical models: revisiting the need for a top-heavy stellar initial mass function with Bayesian optimisation

TL;DR

The paper tackles whether hierarchical galaxy formation models can simultaneously reproduce high-redshift SMG observations and local galaxy properties under a universal IMF. It introduces Bayesian optimisation to explore a 15-dimensional GALFORM parameter space, testing two IMF scenarios: a universal solar-neighbourhood IMF and a burst IMF with slope $x$ as a free parameter. The results show that a universal IMF cannot fit all three calibration datasets (K-band LF, SMG counts, SMG redshift distribution), while allowing a top-heavy IMF in bursts yields excellent simultaneous fits, with a best-fit slope $x \approx 0.7$. This demonstrates, within GALFORM, that IMF variations in starbursts are necessary to reconcile diverse galaxy populations, and that Bayesian optimisation provides a fast, automated route to robust parameter calibration, typically converging in $\lesssim 2\times 10^2$ full model evaluations.

Abstract

The properties of high-redshift sub-millimetre galaxies (SMGs) remain controversial within hierarchical structure formation models. We revisit whether a top-heavy stellar initial mass function (IMF) in starbursts is required to reproduce both SMG observations and local galaxy properties. Using Bayesian optimisation, we perform an extensive search of the 15-dimensional parameter space of the GALFORM semi-analytical model. This efficient approach converges to optimal parameter values in fewer than 200 model evaluations, representing orders of magnitude fewer runs than traditional methods. We test whether GALFORM can simultaneously match three key observational constraints: the $z=0$ $K$-band luminosity function, the SMG number counts at 850~$μ$m, and the SMG redshift distribution. We consider two model variants: one with a universal solar neighbourhood IMF for all star formation, and another allowing the IMF slope in starbursts to vary as a free parameter. When assuming a universal Chabrier IMF, we find no parameter combination that simultaneously reproduces all three datasets. The model either matches the SMG constraints while grossly overpredicting the local $K$-band luminosity function, or matches the local luminosity function while severely underpredicting SMG counts by factors of 3--100. In contrast, allowing a top-heavy IMF in starbursts enables excellent simultaneous fits to all constraints. The best-fitting model prefers an IMF slope parameter $x \approx 0.7$ (where d$n$/dlog$m \propto m^{-x}$), somewhat more top-heavy than recent models but less extreme than early proposals. Our comprehensive parameter space exploration definitively confirms that, within the GALFORM framework, a top-heavy IMF in starbursts is necessary to reconcile high-redshift dusty star-forming galaxies with local galaxy populations.

Figures (7)

• Figure 1: A demonstration of the effect of the length scale adopted in the kernel function on the appearance of a Gaussian process (GP). Each panel shows several realisations or draws from a GP. In each case, the process has zero mean. However, the hyperparameter that governs the scales over which values of $f$ are correlated varies between panels. The left panel shows the shortest correlation length scale, with $\theta=0.1$, the middle panel shows 1.0, and the right panel 10.0. A shorter length scale corresponds to a function which changes rapidly with small changes to the input parameters.
• Figure 2: An illustration of one iteration of the expected improvement (EI) algorithm. The objective is to find the minimum value of the target function shown by the blue curve. Two iterations are plotted, $n=3$ (left panel) and $n=4$ (right panel). The Gaussian process (GP) posterior (orange solid line) is an estimate of the target function. The orange-shaded region shows the $3\sigma$ confidence interval of the GP. The left panel shows the GP after $3$ evaluations of the function (shown by the black solid points). The green curve shows the EI (right axis), which corresponds to the expectation integral of the GP posterior below the minimum evaluation so far (i.e. how much we expect to improve upon the current minimum evaluation at each point $x$). The EI suggests that a new evaluation just below $x=2$ will bring the biggest improvement in our knowledge of the target function. The right panel shows the updated GP posterior and EI curve after evaluating the function for $n=4$ at the point of maximum expected improvement, as shown by the black dashed line in the left panel. At this point, the next evaluation would be chosen to be below $x=-4$.
• Figure 3: Comparison between the redshift distribution for SMGs brighter than 4 mJy inferred by Ugne2020 (black solid line) and Wardlow2011 (blue histogram, which only includes SMGs with robust optical counterparts). Both redshift distributions have been normalized to unit area under the curve. Here, we calibrate GALFORM to the redshift distribution estimated by Ugne2020.
• Figure 4: Performance of the Expected Improvement (EI) Bayesian Optimisation algorithm on a neural network emulator of GALFORM. The solid blue line shows the median over 30 separate runs. The shaded region shows the minimum to maximum range for these runs. The dashed horizontal line shows the global minimum found using MCMC, with 20 chains of 10 000 steps each.
• Figure 5: A comparison of the model predictions with the three calibration datasets under consideration (the parameters of these models are given in Table 2). Left: The $z = 0$$K$-band LF. Center: the SMG number counts at $870~\mu$m. Right: the normalized SMG redshift distribution for $S_{870}>4$ mJy. The spikes in the model predictions for the redshift distribution are artefacts due to the limited number of halos simulated. In each case, the black points with error bars show the observational data. For the SMG redshift distribution, we calibrate to data from Ugne2020, using $z>0.8$ as indicated by the vertical dotted grey line. For the local $K$-band LF, we calibrate to data from Kochanek2001, and for the SMG number counts, we calibrate to data from Stach2018 at the bright end, and Chen2013 at the faint end. The orange solid curves show the model which assumes a universal Chabrier IMF in all modes of star formation. The green lines show the predictions from a model that also adopts a universal Chabrier IMF, but which is calibrated to give an improved fit to the low-redshift $K$-band LF by increasing the weight given to this dataset in the parameter optimisation. The blue lines show a model in which the IMF slope in bursts is allowed to vary according to ${\mathrm{d}}n/{\mathrm{d}}lnm \propto m^{-x}$, where $x$ is an adjustable parameter. For reference, the black dashed line shows the GALFORM model from Baugh2019: this model was calibrated using an earlier measurement of the SMG redshift distribution from Wardlow2011, which has a lower median redshift than the Ugne2020 data.