Towards super-resolution simulations of the fuzzy dark matter cosmological model

ABSTRACT

AI super-resolution, combining deep learning and N-body simulations, has been shown to successfully reproduce the large-scale structure and halo abundances in the Lambda cold dark matter cosmological model. Here, we extend its use to models with a different dark matter content, in this case fuzzy dark matter (FDM), in the approximation that the difference is encoded in the initial power spectrum. We focus on redshift z = 2, with simulations that model smaller scales and lower masses, the latter by two orders of magnitude, than has been done in previous AI super-resolution work. We find that the super-resolution technique can reproduce the power spectrum and halo mass function to within a few per cent of full high-resolution calculations. We also find that halo artefacts, caused by spurious numerical fragmentation of filaments, are equally present in the super-resolution outputs. Although we have not trained the super-resolution algorithm using full quantum pressure FDM simulations, the fact that it performs well at the relevant length and mass scales means that it has promise as a technique that could avoid the very high computational cost of the latter, in some contexts. We conclude that AI super-resolution can become a useful tool to extend the range of dark matter models covered in mock catalogues.

1 INTRODUCTION

The most accepted model for structure formation, cold dark matter (CDM; see e.g. Arbey & Mahmoudi 2021 for a review), has alternatives. Many of these share the feature of a lower amplitude than CDM in the power spectrum of density fluctuations on galaxy scales (wavenumbers higher than k ∼ 10 Mpc⁻¹). This is in order to address possible small-scale problems with the CDM model in the event that (relatively poorly understood) hydrodynamics and star formation processes cannot account for them. These problems include the overabundance of substructure in CDM galaxies and cusps at their centres (see comprehensive discussions in e.g. Weinberg et al. 2015; Del Popolo & Le Delliou 2017). One of the most prominent alternatives to CDM is so-called fuzzy dark matter (FDM; see e.g. Hu, Barkana & Gruzinov 2000; Niemeyer 2020), which proposes that dark matter is extremely light, with particle mass of the order of 10⁻²¹ eV. Due to this extremely small mass, dark matter has de Broglie wavelengths that are macroscopic in size, kiloparsec scale rather than the nanometres of other particles. Other models with reduced small-scale power include warm dark matter (WDM; see e.g. Colombi, Dodelson & Widrow 1996; Iršič et al. 2017b), CDM with a small-scale inflationary cut-off (e.g. Kamionkowski & Liddle 2000; White & Croft 2000), and self-interacting dark matter (e.g. Spergel & Steinhardt 2000; Tulin & Yu 2018).

In cosmological simulations (see e.g. Vogelsberger et al. 2020 for a review), the physical processes in each of these models can be simulated more accurately using specialized numerical techniques, such as solving the Schrodinger–Poisson equation (FDM; e.g. Mocz et al. 2017, 2018; May & Springel 2023) or thermal velocities for particles (WDM; e.g. Paduroiu 2022). An approximation to all these models, which can be accurate enough for some purposes, is merely to modify the initial linear power spectrum, changing it with respect to CDM on small scales (as in Bond, Efstathiou & Silk 1980). For example, this approach was used by Ni et al. (2019, hereafter N19) for FDM, and by Bode, Ostriker & Turok (2001) and Smith & Markovic (2011) for WDM. We use this approximation here, primarily to explore techniques for AI super-resolution (AISR) simulations in a new regime.

As the FDM model has only recently become more prominent (e.g. Hui et al. 2017), there are fewer high-resolution (HR) simulations than for CDM run with this paradigm (see, for a recent example of deep zoom-in technique, Schwabe & Niemeyer 2022), which has limited the studies of small-scale structure in the FDM model. What is extremely challenging is to couple the large scales, the dynamics of which is dominated by a cut-off in the initial power spectrum (and can be solved using N-body methods) with small scales where the wave interference effects near the de Broglie scale in FDM become important. For this reason, simulations with statistically meaningful volumes able to resolve the wave-like dynamics in FDM haloes have been really limited due to their extreme computational costs. Through use of AI-assisted super-resolution (SR) techniques, it is possible that these limitations might be overcome. AISR (see e.g. Bode et al. 2021; Werhahn et al. 2019; Jiang et al. 2020) is a recently developed computational technique that shows promise in this area, and may be useful in particular for generating large mock catalogues at low computational cost.

SR is a term originally applied to image processing, where features are added below the original resolution scale (Lakshminarayanan, Calvo & Alieyva 2012). One technique for doing this involves the use of neural networks (NNs; see Goodfellow, Bengio & Courville 2016; Russell & Norvig 2020 for reviews) to generate these features, and this approach was adapted to use in cosmological simulations by Kodi Ramanah et al. (2020) and Li et al. (2021). NNs are tools used in machine learning, and in the context of SR, an architecture known as a generative adversarial network (GAN; Goodfellow et al. 2014) has proven useful (Wang, Chen & Hoi 2021). Li et al. (2021) and Ni et al. (2021) have shown that by making the output of a GAN conditional on a low-resolution (LR) cosmological simulation, an SR simulation can be generated. In those works, it was shown that statistical properties of the simulation, such as the power spectrum of mass fluctuations and halo and subhalo mass functions, are within a few per cent of their equivalents in a conventionally run HR simulation with the same particle number. The SR models, not needing to calculate the gravitational dynamics, run four orders of magnitude faster (for a resolution enhancement of a factor of 512 in particle number) than the fully N-body HR simulations.

AISR has so far been used to simulate dark matter structure formation in the CDM model on relatively large scale: the power spectrum P(k) has been compared to HR simulations up to wavenumber of k = 16 h Mpc⁻¹ (Ni et al. 2021), and predictions of AISR simulations for CDM haloes and subhaloes down to masses of |$2.5 \times 10^{11} \, {\rm M}_{\odot }$|⁠. This paper is an incremental step forward – we test AISR simulations down to smaller scales and lower masses (two orders of magnitude lower). The aim is to show how we can rapidly run numerical models with different initial matter power spectra P(k) and use them as tools for inference.

2 METHODS

2.1 Data

We work with N-body simulation data for the FDM and Lambda cold dark matter (LCDM) models. Both sets are run with initial conditions consistent with the WMAP9 cosmology with matter density Ω_m = 0.2814, dark energy density Ω_Λ = 0.7186, baryon density Ω_b = 0.0464, power spectrum normalization σ₈ = 0.82, spectral index n_s = 0.971, and Hubble parameter h = 0.697. The difference between them is solely encoded in the initial power spectrum, with the largest difference being on small scales. Our FDM simulations employ the initial power spectrum at z = 99 generated by axioncamb (Hlozek et al. 2015), also used in N19. The FDM particle mass assumed in our models is 2.5 × 10⁻²² eV, which leads to a damping scale relative to LCDM in the linear P(k) at k = 18 h Mpc⁻¹ (see fig. 2 of N19).

We use the current version of the mp-gadget cosmological N-body code (Bird et al. 2022), part of the gadget code family (Springel 2005), to evolve the initial conditions forward in time. The simulation is run in dark matter only mode, 512³ tracer particles in a cubical periodic volume of side length 20 h⁻¹ Mpc resulting in a tracer particle mass of |$4.6\times 10^{6}\, {\rm M}_{\odot }$|⁠. The simulation is evolved from redshift z = 99 to z = 2, and we use the final, z = 2 snapshot exclusively in our study. We do this because at high redshifts the differences between the evolved density distributions in FDM and LCDM have not been erased by non-linear evolution (see e.g. Iršič et al. 2017b; N19). Sixteen simulation realizations with different random seeds are run for each of the FDM and LCDM models. Each of these realizations consists of an HR simulation described above, and a matching LR simulation with only 64³ particles. The mass per particle of the LR simulation is 512 times more than that for HR, i.e. |$2.4\times 10^{9}\, {\rm M}_{\odot }$|⁠.

2.2 Deep learning models

GANs (Goodfellow et al. 2014) are a deep learning technique that consists of two separate NNs, named the generative and the discriminative network. The generative network generates candidate distributions, and the discriminative network tries to determine whether that distribution came from an HR simulation (in our case), or from the generator. In doing this, the GAN learns to generate new data with characteristics matching the training set. We use a conditional GAN (CGAN), where the input to the generator is both noise and an LR version of the simulation data. The process adds noise during convolution to add irregularity to the smaller scales that was previously absent. These irregularities become small-scale over- and under-densities in the distribution of particles of the resulting SR simulation.

The specific NN architecture used is that described fully in Ni et al. (2021) and the reader is referred to that paper for more details. Briefly, the resolution enhancement is applied to the particle distribution, through the displacement field (displacement of particles from their initial positions). The particle displacements and their velocities in an LR simulation and a noise vector are the inputs to the generator. The discriminator network receives the high-resolution particle positions and displacements, as well as an Eulerian density field representing those particles assigned to grid cells using a CIC scheme. The loss function is the total distance between the real and generated data sets using optimal transport, also known as the Wasserstein distance, meaning that the type of GAN we use is a Wasserstein GAN (WGAN; see Gulrajani et al. 2017 for the particular variant we use). This GAN variation is more stable and requires less tuning than the original GAN.

2.3 Training

We focus on training our GAN to generate FDM SR particle distributions from LR distributions. We do this to determine how well a GAN can create a valid SR particle distribution for FDM. When passing the simulation data to the network for training, we first split them into discrete chunks, due to GPU memory limitations. The chunks have a side length one-fifth of the simulation volume, and the LR simulation input includes an extra margin of the same size, making use of the periodic boundary conditions of the simulation.

After the network is fully trained, we generate three different types of SR distributions; one of LCDM from an LR LCDM distribution, one of FDM from an LR FDM distribution, and one of FDM from an LR LCDM distribution. The LCDM to LCDM SR process follows that employed by Ni et al. (2021), except that it extends to spatial scales five times smaller and a mass resolution 125 times greater. This LCDM SR model is used as a baseline for determining the success of the FDM GAN training, as well as a test of the SR procedure on smaller scales than have yet been tested. The reader is referred to section 2.3 of Ni et al. (2021) for more details.

The FDM to FDM SR distribution was our test case and the LCDM to FDM was used as a proof of concept for seeing whether our FDM training could generate an FDM output from an LCDM input. During training we analysed all of the SR distributions by visual observation and by use of the density field power spectrum, P(k), which is used as a metric to evaluate the SR distribution fidelity. We focus in particular on the region between k = 20 h Mpc⁻¹ and k = 150 h Mpc⁻¹ as it shows how much small-scale structure the SR process has added from the original LR distribution.

3 RESULTS

3.1 Visual impression

Fig. 1 shows slices of the dark matter density distributions for both LCDM and FDM at all resolutions. We show both a slice through the whole box as well as a smaller region centered on the most massive halo in the volume. Visually, the HR and SR distributions for FDM tend to display less small-scale structure as indicated by fewer smaller clumps of particles than the distributions for LCDM as expected from the different models. The LR simulations for both FDM and LCDM are very similar in appearance however, and we shall see in Section 3.2 that the most of the extra fluctuation power in LCDM with respect to FDM is above the Nyquist frequency of the LR model. Comparing the HR and SR distributions for both models, we have difficulty telling them apart in general appearance, although individual small scale clumps are different, when looked at closely. As the SR and HR models should only be statistically similar, this is to be expected.

If we look at filaments in the FDM model (for example on the right-hand side of the close-up panel), we can see that that they are broken up into regularly spaced clumps, like beads on a string. These small haloes are unphysical, as has been noted in the context of other cosmological models by authors from Melott & Shandarin (1989) onwards (including most extensively by Wang & White 2007 and Angulo, Hahn & Abel 2013). These occur most prominently when a model has little power at frequencies close to the Nyquist frequency of the initial particle grid. If the initial particle distribution is indeed set up in a grid fashion, these fake haloes will remain. They are also exacerbated when the numerical resolution is significantly smaller than the grid spacing (see e.g. Splinter et al. 1998), as is usual in cosmological simulations. We will see in Section 3.3 when we examine the halo mass function that they are found by the halo finder, and dominate the number of haloes below a certain mass. This issue has lead some papers (e.g. N19) to not use haloes below a certain mass in their analyses, using the empirically determined halo mass limit for spurious fragmentation of Wang & White (2007). In our case, we note that the SR FDM simulation is sufficiently similar to the HR model that it also includes these spurious objects.

3.2 Power spectrum

We compute the power spectra, P(k) for all simulations after assigning the particles to a grid with twice the linear spatial resolution of the initial particle grid (1024³ for the HR and SR simulations and 128³ for the LR simulations). The particles are assigned to this grid using the CIC (cloud-in-cell) resampling scheme. In Fig. 2, the dimensionless power spectra δ²(k) = k³P(k) for both dark matter models are shown, with all results being at our chosen redshift of z = 2. We plot all curves up to the Nyquist frequency of the grid, including for the LR simulations, for which this is a factor of 8 lower than for HR and SR. The LCDM model at this redshift has more power on small scales than FDM, 50  per cent less at the Nyquist frequency, the smallest scale plotted. The SR power spectrum matches the HR spectrum within 5  per cent on all scales for both FDM and LCDM indicating the GAN was successfully trained.

3.3 Halo mass function

We make halo catalogues from the simulation output, using a friend-of-friends halo finder (Davis et al. 1985), with a linking length of 0.2 times the mean interparticle separation. Following N19, we choose 300 particles as the cut-off below which haloes are not simulated reliably. Numerical artefacts strongly affect the small halo mass function in models with low power on halo scales as mentioned in Section 3.1.

In Fig. 3, we show the halo mass functions are given for both LCDM and FDM. We can see that the SR and HR mass functions match within 10  per cent for the vast majority of the mass scale above the 300 particle halo mass cut-off for both FDM and LCDM. There are slightly larger deviations just above the 300 particle cut-off, and above |$\log_{10}(M_\mathrm{ h}/\, {\rm M}_{\odot }) = 12$|⁠. The mass range of the haloes just above the 300 particle cut-off is below the mass of a single LR simulation particle, which is a mass range where the model is expected to struggle. The deviation above |$\log_{10}(M_\mathrm{ h}/\, {\rm M}_{\odot }) = 12$| falls within the Poisson error for both the HR and SR mass functions. Near the 300 particle cut-off, which corresponds to halo masses of |$1.4\times 10^{9} \, {\rm M}_{\odot }$|⁠, we can see that the HR and SR mass functions for FDM are less than those of LCDM by an order of magnitude. This corresponds to a deficit of low-mass galaxies with respect to LCDM that could be used to constrain the FDM particle mass (e.g. N19). At redshift z = 2 that we are plotting here, the deficit is smaller than at higher redshifts, but still substantial for the relatively low FDM mass we are simulating (2.5 × 10⁻²² eV).

We plot haloes down to two particles, a mass of |$9\times 10^{6}\, {\rm M}_{\odot }$| (for HR and SR). Even though most of the haloes below 300 particles are not reliably simulated haloes, due to both numerical problems with FDM/WDM (as mentioned above) and shot noise, we can see that both the SR and HR simulation results track each other closely all the way down to the lowest masses plotted. Because the SR simulation process works with the particle displacements and velocities, it effectively generates simulation data in all its details, including the spurious haloes that were present in the training set. A more in-depth analysis of the haloes and subhaloes in SR simulations created using this architecture can be found in Ni et al. (2021).

3.4 SR of FDM using LCDM LR

The LCDM and FDM simulations have very similar initial (linear) power spectra at scales larger than the damping scale (k = 18 h Mpc⁻¹ in the case of m_FDM = 2.5 × 10⁻²² eV; see fig. 2 of N19). This scale is above the Nyquist frequency of the LR simulations, and so opens the possibility of using LR simulations to make SR simulations for a different model. In our final test, we have tried using the LR LCDM simulations as an input to the FDM-trained network. The results are shown in Fig. 4(a), where we can see that the resulting SR FDM power spectrum is quite close to that of the HR FDM model, with a maximum deviation of 8  per cent. The halo mass function, in Fig. 4(b), is also within 40  per cent of the FDM HR results, with the majority of the significant deviation occurring below |$\log _{10}(M_h/\, {\rm M}_{\odot }) = 10.5$|⁠. This is the area where the FDM and LCDM halo mass functions differ most, so it is not unexpected that the model struggles in this mass range as the training data are from the wrong model by design. Performing better matched training on this task would likely mitigate this effect.

These tests show that we could in principle use the same LR simulations to generate different SR dark matter models, as long as the differences in the power spectra are on small enough scales and the accuracy is sufficient for the desired use case (e.g. mock catalogues). The NNs in the present case are trained on a specific DM model, however, and so require full HR simulations of that model. A more general tool would be able to train on a range of models and interpolate between them. The NN architecture StyleGAN2 (e.g. Karras et al. 2020) has shown some promise for field level emulation in different cosmological models (see e.g. Jamieson et al. 2023) and we hope to explore its use in SR in future work.

4 SUMMARY AND DISCUSSION

4.1 Summary

This work attempts to extend the use case of the SR architecture developed in Li et al. (2021) beyond the LCDM model. The results indicate this extension works well for the case of FDM. Other viable alternative DM models, such as self-interacting DM (Vogelsberger, Zavala & Loeb 2012) or WDM (Bode et al. 2001), also feature suppression of the linear matter power spectrum. Extensions like this indicate that the approach is ripe for further development to allow a single trained network to be able to produce simulations with different cosmological models, which would be an extremely useful tool in studying possible alternate cosmological models.

4.2 Discussion

This paper represents a small step on the way to the exploitation of AISR techniques in cosmology. We have shown that dwarf galaxy haloes and subhaloes with masses of |$10^{9} \, {\rm M}_{\odot }$| can be quite accurately simulated, even for the LCDM model, which has significant small-scale structure, with orbit crossing and virialization obviously being very important on these scales. The models still only include dark matter, and the real test will come when methods to include hydrodynamics are developed. Hydrodynamics will also break the approximate scale invariance of the LCDM model on these scales and provide an even more strenuous test of the AISR technique.

As with all deep learning methods, how well the AISR technique works is dependent on the training data. We need the NN to learn how structure forms in a way that generalizes beyond the specific examples that were trained on. This issue is particularly acute if we eventually intend to apply SR to FDM simulations that include quantum pressure (e.g. May & Springel 2023) because their computational cost is so large, and only small training sets will be feasible. A thorough study of the effect of training set size on the fidelity of the SR models is something that should be carried out in future work. In the meantime, we note that Li et al. (2021) were able to successfully reproduce summary statistics for a simulation with a volume 1000 times that of the largest training run. Even though this large SR run included rare haloes with mass |$\gt 10^{15}\, {\rm M}_{\odot }$| that were not present in the training data, they formed with reasonable morphology, substructures, and correct population density.

An application of FDM simulations is to compare to the abundance and statistical properties of galaxies and therefore constrain the FDM particle mass. At present, some of the tightest constraints on this parameter come from measurements of the Lyman-α forest power spectrum (e.g. Iršič et al. 2017a). N19 have shown that the high-redshift galaxy population can also provide competitive constraints, and with different possible systematic errors (e.g. the Lyman-α forest is also sensitive to the temperature of the intergalactic medium). In this paper, we have used simulations at the relatively high redshift of z = 2 because non-linear evolution to lower redshifts tends to erase the differences between CDM and FDM. At higher redshifts relevant for JWST observations (e.g. z = 7–12; Bradley et al. 2022; Marshall et al. 2022), N19 have shown using hydrodynamic simulations that there are significant differences in the galaxy stellar mass functions between FDM and LCDM models. In our simulations, the maximum ∼5 per cent differences between HR and SR are smaller than those between LCDM and FDM models with mass of 2.5 × 10⁻²² eV. As shown by Li et al. (2021), the accuracy of SR increases at higher redshift, meaning that heavier mass FDM models will be feasible to simulate with FDM.

Ideal applications for the AISR techniques will be studies of observational systematics using large mocks, where the uncertainties modelled are larger than the demonstrated accuracy of AISR. In a similar vein, semi-analytical models could be run on AISR FDM simulations, where the uncertainties in the model parameters are larger than AISR accuracy. The speed of AISR would allow many tens of thousands mock catalogues to be generated for the same computational cost of a single HR model. We have also shown that for models such as CDM where the large-scale P(k) is close to that of FDM, it is even possible to use the same LR simulation to make SR models with different particle masses, thereby saving on the cost of running the larger LR N-body simulation.

Because we have used a very simplified treatment of FDM, the truncation of the input power spectrum, our conclusions about the success of AISR also apply to other models when this approximation to the small-scale behaviour is used, such as WDM. There is, however, nothing in principle to prevent the use of AI SR techniques together with more physically accurate methods for simulating structure formation in these models. The methods used by e.g. Mocz et al. (2017) and May & Springel (2021) to solve the Schrodinger–Poisson equation could be used to train the NN used in AISR. This could be used to produce rapid simulations that reproduce the quantum pressure effects seen by e.g. May & Springel (2023) beyond those linked directly to the suppression of power. We note that particle discreteness problems that cause fake substructures to form in models with low small-scale power are also present in our current AIRS model results (and are visible in Fig. 1). These could be corrected with better training simulations and analysis (e.g. Angulo et al. 2013; May & Springel 2021).

ACKNOWLEDGEMENTS

MS acknowledges support from the NSF AI Institute: Physics of the Future Summer Undergraduate Research Program in Artificial Intelligence and Physics. RACC and TDM were supported by NASA ATP 80NSSC18K101, NASA ATP NNX17AK56G, and the NSF AI Institute: Physics of the Future, NSF PHY-2020295. RACC also acknowledges support from NSF AST-1909193 and TDM from NSF ACI-1614853 and NSF AST-1616168.

DATA AVAILABILITY

The SR models and the pipeline used to generate the SR fields are available at https://github.com/yueyingn/SRS-map2map. The simulation outputs and measured summary statistics are available on request from the authors.

References