Magneto-hydrodynamics in Population III star formation

The gravitational potential of DM halos facilitated the formation of the coolant H2 within baryonic gas, allowing it to cool and collapse. The evolution of the collapse is dominated by the chemistry involved. The dominant chemical processes that complicate the collapse are three-body H2 formation heating, collision-induced emission cooling and H2 collisional dissociation. The collapse becomes adiabatic at the formation of the pre-stellar core at 10−4 g cm−3, although this may happen at 10−6 g cm−3 if stellar feedback is considered (Machida & Nakamura, 2015). Due to the lack of dust and metals in the early Universe, the Jeans length continues to decrease as the gas collapses, all the way down to the formation of the pre-stellar core. This corresponds to an increase in density of 10 orders of magnitude from the first stellar core in present-day star formation when the gas becomes stable to fragmentation. The Pop III IMF is disputed among authors, but most recent studies agree that the gas fragments to form a group of stars within the halo. In Prole et al. (2022a) we performed a resolution study to show that these discrepancies are largely due to the difference in resolution adopted by these studies. We have shown that failure to resolve the Jeans length at the pre-stellar core density (0.01-0.1 au) results in underestimated fragmentation behaviour in the gas. The lower core masses that we produce when we increase our resolution effects how these stars interact with their environments through increased main sequence lifetimes, reduced contribution to reionisation, and ending their lives as Type II SN instead of a pair-instability explosion. We also noted that a number of cores are ejected from the system with masses low enough that they should have survived until the present day, and may be observable. This raises questions into the pollution of Pop III stars with metals throughout their lifetimes as an explanation for the lack of Pop III observations.

Magnetic fields have been shown to prevent fragmentation in present-day star formation simulations, increasing stellar masses. It is natural to expect that this may be the case in primordial star formation also. If this is the case, it would soften the resolution criteria discovered in Prole et al. (2022a). The main differences between magnetic fields in primordial times and the present day is their structure. The galactic magnetic field present today is uniform over the scales concerned with star formation. These highly ordered fields can provide a coherent magnetic tension force over large scales. Primordial magnetic fields go through the small-scale turbulent dynamo and are therefore expected to have small-scale structure described by a k3/2 power spectrum, where the magnetic energy increases towards smaller spatial scales. Despite this, the magnetic pressure is isotropic and can therefore provide extra support against the fragmentation caused by gravitational collapse at Jeans scales at the highest densities. In Prole et al. (2022b) we investigated whether primordial magnetic fields could reduce Pop III disc fragmentation. Since amplification via the small-scale turbulent dynamo is dependent on resolution (which is inherently finite in simulations), we opted to bypass the amplification process and introduce a fully saturated (maximum strength) magnetic field late into a non-magnetised collapse. We created the field by generating a random vector field from the k3/2 power spectrum and applied it to the central region of the collapse before the formation of the disc. We have shown that the small-scale fields do not reduce disc fragmentation or prevent the degree of fragmentation from increasing when the maximum density of the simulations is increased. As the inclusion of magnetic fields into these simulations greatly increases the computational cost without changing the results, we concluded that magnetic fields are not necessary in Pop III star formation simulations

The radiation that Pop III stares emits interacts with their environments in different ways. Photons with energies above the hydrogen ionisation limit go towards forming a HII region around the star, while photons below this limit are free to escape the halo and penetrate nearby halos. Lyman-Werner (LW) band photons photodissociate H2 in nearby halos, preventing star formation until the halo gains sufficient mass through mergers so that the outer layers can self-shield the inner H2 and form stars. While cosmological simulations have confirmed that the mass of halos required to form stars increases as the strength of the LW field is increased, these simulations lack sufficient resolution to resolve star formation within the halo. How the increasing halo masses effects the Pop III IMF is therefore unclear from cosmological simulations. In Prole et al. (2023) we performed zoom-in simulations around halos from the cosmological simulations of Schauer et al. (2021), resolving the formation of the pre-stellar core and following the fragmentation of the gas for hundreds of years after (see figure). While the mass accreted onto sink particles by the end of the simulations was not correlated with initial mass of the halo or the mass of gas that managed to initially form enough H2 to collapse, the accreted mass was correlated to the mass of the inner molecular core. Crucially, the mass of the molecular core was not correlated with the initial mass of the halo. The IMFs were almost identical across the LW field strengths, suggesting that the Pop III IMF may be invariant with halo mass and is therefore the same wherever we look in the early Universe, until the formation of Pop II stars.

Tracing the small-scale structure of the cosmic web over 12 billion years

A science highlight in 2022 was the development of the first hydrodynamical simulations of the intergalactic medium (IGM) to incorporate heating by dark photons.  Photons are the mediator of the electromagnetic force between charged particles, such as electrons and protons. We expect particles in the dark sector to interact with each other through such forces too, exchanging new particles called “dark photons” with each other. The dark photon can be massive, and if they are very light, can be produced in the early Universe to make up the dark matter. Dark photons can also mix with the regular photon, leading to possible cosmological signatures of dark photons.  In regions of space where the mass of the dark photon matches the effective plasma mass of the photon, conversions from dark photons to photons can occur. The converted photons are then rapidly absorbed by the IGM in those regions, heating the gas up.   If these dark photons are indeed the missing dark matter, then a measurable effect on the temperature of intergalactic gas is predicted. 

We have therefore compared our simulations to observations of the Lyman-alpha forest from the Comic Origins Spectrograph (COS) on the Hubble Space Telescope, finding that some heating from dark photons is indeed consistent with the measured Lyman-alpha line widths.  The image shows the main result, which was published in Physical Review Letters (Bolton et al. 2022, PRL, 129, 211102).   The top panel shows the results of fitting the Doppler parameter distribution and column density distribution function of the Lyman-alpha forest at z=0.1 assuming a maximal contribution of dark photon heating to the line widths.  The contours show the projection of the 68% and 95% intervals for the mass and mixing parameter of the dark photon. The colours correspond to different assumptions about the uncertainty of the intergalactic medium temperature at z = 2.   The bottom panel shows the corresponding best-fit models compared to the COS observational data.  The solid grey curve shows a result with no dark photon heating.

A study of quantum Lyapunov exponents and their numerical analytic continuation in low-dimensional quantum systems using exact diagonalization

The Banks-Fischler-Shenker-Susskind (BFSS) model is one of the popular microscopic models of black hole dynamics, capable of explaining how black holes scramble information. Studies of the BFSS model suggest that a microscopic mechanism of information scrambling by black holes is related to their intrinsically chaotic dynamics. In particular, BFSS model at high temperatures becomes very similar to dimensionally reduced Yang-Mills theory, a strongly nonlinear theory of strong interactions between nuclei. Strong nonlinearity of Yang-Mills theory gives rise to chaotic classical dynamics, with the distance between any two trajectories diverging exponentially with time – a generic phenomenon known as Lyapunov instability.

In quantum mechanics, a counterpart of classical-mechanical Lyapunov instability is the exponential growth of the so-called out-of-time-order correlators (OTOCs) of quantum-mechanical operators.

In this project, we studied OTOCs in a supersymmetric quantum-mechanical system very similar to the BFSS model. Our particular focus was the low-temperature regime, where classical-mechanical description becomes inaccurate, and the non-supersymmetric Yang-Mills theory is known to be in a trivial gapped phase.

Using exact diagonalization techniques, we were able to demonstrate the chaotic behaviour OTOCs of our BFSS-like model down to lowest temperatures, in agreement with the expected duality with black holes. Furthermore, we demonstrated an expected transition between the graviton gas, the Schwarzschild black hole, and the D0-brane regimes in the model. To this end, we looked at fluctuations in the spacing ΔEi between energy level, considered as a statistical system for a large number of levels. Characterizing these fluctuations in terms of r-ratios ri = min(ΔEi-1, ΔEi)/max(ΔEi-1, ΔEi), with 0 < r < 1, we identified 3 different regimes as a function of energy E: a) the graviton gas regime at low energies, with no signatures of chaos in level spacing (E < 1 on the plot above),b) the intermediate Schwarzschild black hole regime with 1 < E < 100, where statistical fluctuations of r set in, and c) the high-energy black-brane regime with a fully developed classical chaos.

Structure formation in fuzzy dark matter

PI: Anastasia Fialkov

Filamentary network in FDM. Figure from Dome et al. 2023b.

The fuzzy dark matter (FDM) scenario has received increased attention in recent years due to the small-scale challenges of the LCDM cosmological model and the lack of any experimental evidence for any candidate particle. In this DiRAC project, we use cosmological N-body simulations to investigate large-scale structure in FDM cosmologies. In Dome et al. 2023 we study halo density profiles, shapes and alignments in FDM-like cosmologies (the latter two for the first time) by providing fits and quantifying departures from LCDM as a function of the FDM particle mass m. We explore halo shapes finding that in FDM-like cosmologies halos are more elongated around the virial radius than their LCDM counterparts. We also consider intrinsic alignment correlations, stemming from the deformation of initially spherically collapsing halos in an ambient gravitational tidal field, and find that they become stronger with decreasing m. 

A follow-up project considers cosmic web structure. On large cosmological scales, anisotropic gravitational collapse is manifest in the dark cosmic web. Its statistical properties are well known for the standard CDM cosmology, yet become modified for alternative dark matter models such as FDM. In a paper submitted in February 2023 to MNRAS (Dome et al. 2023b) we assess for the first time the relative importance of cosmic nodes, filaments, walls, and voids in cosmology with small-scale suppression of power (such as FDM). We explore cosmic web by applying the NEXUS+ Multiscale Morphology Filter technique to cosmological N-body simulations. We quantify the mass and volume filling fractions of cosmic environments (nodes, filaments, walls, voids) at redshifts z=3.4-5.6.

Cosmological Simulation of Fuzzy Dark Matter with Self-Interaction

In this DiRAC project we investigate cosmological structure formation in Fuzzy Dark Matter (FDM) with an attractive self-interaction (SI) using numerical simulations. Such a SI would arise if the FDM boson were an ultralight axion, which has a strong CP symmetry-breaking scale (decay constant). Although weak, the attractive SI may be strong enough to counteract the quantum `pressure’ and alter structure formation. We find in our simulations that the SI can enhance small-scale structure formation, and soliton cores above a critical mass undergo a phase transition, transforming from dilute to dense solitons. The results are summarized in Mocz et al. accepted for publication in MNRAS in February 2023.

Figure from Mocz et al. 2023. Projected dark matter densities at z = 2, with blue arrows denoting formed solitons.

Exploring Star-Forming Regions in Fuzzy Dark Matter Models

To complement the large-scale simulations done in the framework of ACSP253 project we have run a set of small-scale hydrodynamical simulations of Fuzzy Dark Matter (FDM) using a Schrodinger-Poisson solver (developed by Mocz). We will use the complete set of simulations to explore first galaxy formation for different FDM masses and make predictions for the 21-cm signal of neutral hydrogen at cosmic dawn.

Spontaneous Symmetry Breaking in the 3d Thirring Model

PI: Simon Hands
Institution: University of Liverpool

The Thirring Model describes relativistic fermions moving in a two-dimensional plane and interacting via a contact term between conserved currents. The physical system it most resembles is low-energy electronic excitations in graphene. For free electrons at half-filling on a honeycomb lattice, conduction and valance bands form cones just touching at their vertices at two “Dirac points” lying within the first Brillouin zone. Since the density of states vanishes, and the effective fine structure constant is boosted by a factor vF/c≈1/300, where the pitch of the cones vF is the Fermi velocity, the resulting physics is described by a strongly-interacting relativistic quantum field theory, with equal likelihood of exciting electrons or holes.

Besides applications in layered condensed matter systems, the Thirring model is of interest in its own right as possibly the simplest relativistic theory of fermions requiring a computational solution. In the most recent simulation campaign we have for the first time studied the model’s spectrum in the vicinity of the phase transition, exploring both bound fermion – anti-fermion “meson” channels, with spin-0, and the spin-1/2 fermion channel. The results confirm a significant spectral gap between states with quantum numbers expected of Goldstone bosons, and non-Goldstone channels, consistent with the spontaneous breakdown of U(2) global symmetry to U(1)xU(1) accompanied by the generation of a fermion bilinear condensate <ψψ>≠0 signalling dynamical mass generation in the fermion channel.

A major highlight arises from the fermion timeslice correlator evaluated at vanishing bare mass in the vicinity of the critical interaction strength β=a/g2≈0.28, shown (left) on a log-log plot. The results are consistent with fermion propagation of the form


with the fermion anomalous dimension ηψ (plotted in the inset) taking the unexpectedly large value ≈3. This first estimate of a fermionic critical exponent suggests the emergent conformal field theory at the critical point is very strongly-interacting.

S. Hands and J. Ostmeyer, Spectroscopy in the 2+1D Thirring model with N=1 domain wall fermions

Phys.Rev.D 107 (2023) 1, 014504   2210.04790 [hep-lat]

DOI: 10.1103/PhysRevD.107.014504 (publication)

The impact of dynamical friction in gas on structure formation in the Universe

Project: dp134
Author: Ben Morton (
Team members: Prof. Sadegh Khochfar, Dr. Jose Onorbe

Dynamical friction is the process by which a massive perturber, moving through some

background medium, gravitationally interacts with that medium, producing a net retarding

force to its motion. When the background medium is gaseous, the pressure forces present in

the gas must be included in modelling the response of the medium, and so impact the

resultant force. We have run a high resolution, gravo-hydrodynamics zoom simulation of a

M~1E8 Msolar halo (at redshift z=10) embedded in a cosmological simulation box, using the

standard LambdaCDM cosmology, using the multi-physics simulation code GIZMO. We are in

the process of studying the impact of dynamical friction from dark matter substructure within

this host halo, looking specifically for the signal of density enhancement, which could

potentially trigger molecular hydrogen formation, and so spark early star formation. The

figure below shows an example of the density enhancement we might expect for such sub

structure, taken from idealised simulations of an extended perturber in a gaseous medium,

also using GIZMO.

Figure 1: Overdensity [top] in wake from extended perturber (Plummer sphere), moving at Mach 2,
produced using idealised tests using GIZMO. Analytic prediction from Ostriker 1999 shown in white.
Difference between numerical and analytic solution [bottom].

3D high resolution (10243) hydrodynamics simulation of a type-II supernova explosion

This is a 3D high resolution (10243) hydrodynamics simulation of a type-II supernova explosion

using the code AREPO. The outer boundary seen is the forward shock which results

from the supernova explosion and which is expanding into a uniform interstellar medium. A

number of instabilities develop at the supernova shock front creating the structure seen in this

image. The left/right side of the image shows the projection of gas temperature/density. The

shock compresses and heats the gas it encounters to high densities and temperatures. We are

post-processing these simulations to investigate how dust is destroyed by the forward shock as

well as see how dust may be able to form in the ejecta of the supernova. The simulations were

performed on the DIRAC Data Intensive Service (Cambridge) as part of the dp130 project.

The Gravitational Afterglow of Boson Stars

The theoretical framework of our current understanding of the Universe rests on two main pillars, the Standard Model (SM) of particle physics and Einstein’s theory of General Relativity (GR). This SM+GR framework provides us with an incredible power to explain and predict a plethora of phenomena in observations and experiment, ranging from the quantum behaviour of computers, high-energy particle collisions at colliders like CERN to the gravitational-wave (GW) symphony of neutron stars and black holes, and the evolution of the Universe as a whole. In spite of this incredible success, there exist gaps in this beautiful picture of theoretical physics. Observations of galactic dynamics and the cosmic microwave background cannot be explained in terms of the expected gravitational effects of visible matter. We either need to modify the laws of gravity on extreme length scales or assume a form of dark matter that we cannot satisfactorily explain with the standard model of particle physics. Likewise, the accelerated expansion of the Universe calls for an exotic substance dubbed dark energy or the introduction of a cosmological constant with a value many orders of magnitude below the zero-point energy estimates of quantum field theory. Further challenges to the SM+GR model of the Universe arise in the form of the hierarchy problem (the extraordinary feebleness of gravity) and the lack of renormalization of GR in a quantum theory sense. It is tempting to compare the current conundrum about gravity with irregularities observed in the 19th century in the orbital motion of Uranus and Mercury. The former led to the discovery of a previously unknown or dark matter source, namely Neptune, while the latter was eventually explained in terms of a modified theory of gravity — Einstein’s GR instead of Newtonian gravity. Clearly the lesson is to remain open to all possibilities and search for observational signatures that discriminate between the scenarios.

Figure: Snapshots of the scalar-field amplitude in the orbital plane for a grazing collision of two boson stars of equal mass moving with 10% of the speed of light. The initially spherical stars merge into a single star with complex structure that emits a long-lived afterglow of gravitational radiation. See

In our search for explaining the present anomalies in terms of dark matter or modified laws of gravity (or both), we have an unprecedented new tool available, the Nobel Prize winning detection of Gravitational Waves by LIGO. This new channel to observing the universe is tailor-made to probe dark matter or possible modifications of the laws of gravity – both of which are fundamentally gravitational (rather than electromagnetic) phenomena. In our work on the “Gravitational Afterglow of Boson Stars”, we explore the first hypothesis, dark matter in the form of a light scalar field akin to the type of fields predicted by candidate theories for quantum gravity. Through the gravitational effect of their own energy, these light scalar fields can form highly compact star-like configurations known as Boson Stars. These hypothetical stars do not emit any form of light or other electromagnetic radiation, but as binary systems, they will generate GWs analogous to the black-hole binaries routinely observed with the LIGO-Virgo-KAGRA GW detector network.

The search for dark matter in future GW observations crucially hinges on our ability to distinguish between the GW signals from black-hole binaries and those from boson stars. In our work, we have identified a key feature that arises in boson-star collisions but not in the merger of black holes: a long-lived GW afterglow. Black holes, in contrast, have a comparatively short post-merger phase known as the quasi-normal ringdown. The boson-star afterglow arises from the complicated shape of the scalar field matter after merger (see figure). The most surprising result of our simulations is the exceptional longevity of this GW afterglow, which significantly improves our chances to either detect boson stars or rule them out as dark matter candidates in future observations.

A method to reconstruct magnetic fields from multi-spacecraft measurements

Project title: Heating and Acceleration through Magnetic Reconnection in Space Plasma Turbulence
PI: Daniel Verscharen (MSSL, University College London)
DiRAC Resource: Data Intensive at Leicester (DIaL)

Jeffersson A. Agudelo Rueda, Daniel Verscharen

Advances in space technology have led to the dawn of a new era for multi-spacecraft science missions to explore our solar system. Missions such as THEMIS, CLUSTER, and MMS have proven extremely valuable for our understanding of plasma physics in the Earth’s space environment. Building on this strong heritage, newer multi-spacecraft mission concepts such as HelioSwarm and MagneToRE have been developed that will take the multi-spacecraft approach further by deploying even greater numbers of scientific spacecraft than in these earlier missions. The success of these new missions requires the development of new multi-point techniques for data analysis.

Multi-point measurements of the complex magnetic field in space plasmas are of particular interest for our understanding of the transport and transfer of energy in the near-Earth space environment. In this context, the reconstruction of the magnetic field topology from a finite ensemble of measurements is a specific goal of these new missions. MagneToRE is a dedicated mission concept designed to address this question and to shed light on the geometry and topology of the magnetic field with unprecedented multi-point measurement resolution. MagneToRE will use a mission configuration consisting of one main spacecraft (hub) and a large number (∼24) of 6U cubesats (probes) that will carry magnetometers to sample the magnetic field at the locations of the probes. MagneToRE will study and characterize the interplanetary magnetic field at scales corresponding to the inertial range of space-plasma turbulence.

In preparation for MagneToRE, we develop a new method for magnetic field reconstruction from multi-point measurements. We use our DiRAC particle-in-cell (PIC) simulations of anisotropic Alfvénic plasma turbulence as a numerical testbed for this novel reconstruction method.

Panel a) in Figure 1 shows the presence of magnetic structures that form in our simulation domain. Panel b) depicts one of four spatial configurations that we use to sample the simulated magnetic field data. We trace artificial spacecraft trajectories through the simulation box to emulate the paths of spacecraft crossing through the plasma volume. We record the local fields from the simulation along these trajectories and apply our reconstruction method to the artificial dataset. Panel c) shows traced magnetic field lines based on the information available from the entire domain of our DiRAC simulations. Panel d) shows the reconstructed magnetic field lines based on the limited information taken along the trajectories of the artificial spacecraft.

Tracing artificial spacecraft trajectories through our DiRAC simulation allows us to evaluate the quality of our and other reconstruction methods against the ground truth of our 3D simulation data.

Figure 1. DiRAC particle-in-cell simulation of plasma turbulence. a) Our simulation domain, colour-coded with the strength of the magnetic field showing magnetic structures of different sizes. b) Configuration of our spacecraft swarm. c) Magnetic field lines from the simulation (ground truth). d) Reconstructed magnetic field lines based on artificial spacecraft data.  

Impact crater scaling and inner solar system bombardment

DiRAC resource used: DiAL 2.5

PI: G.S. Collins

The relationship between impact crater size and impactor properties, such as size and speed, is key to comparing impactor and crater populations on different planets and dating planetary surfaces. Most of our understanding of this relationship, however, comes from numerical simulations of vertical-incidence impacts, and laboratory impact experiments at relatively low speed, which are comparatively rare in nature.

In this project, we have run shock physics simulations of large crater formation on Earth and the Moon, for a range of oblique impact angles and speeds more typical of planetary scale impacts. We have generated an extensive dataset of high-resolution, oblique-impact complex crater simulations which would not have been possible without the DiRAC facility.

We find that while crater size decreases as the impact angle becomes shallower, crater diameter, depth and volume are all affected by impact angle in different ways. Most importantly, we find that crater diameter depends less on impact angle than previously thought, especially for steeply inclined impacts. This implies that typical asteroid impacts on planetary surfaces form larger craters, and large craters are formed more frequently, than is currently assumed.

Davison, T. M., & Collins, G. S. (2022). Complex crater formation by oblique impacts on the Earth and Moon. Geophysical Research Letters, 49, e2022GL101117.

Fig 1: Snapshots from a simulation of a 14 km diameter projectile, hitting at 20 km/s and an angle of 45° from the horizontal, on Earth. The frames are slices along the symmetry plane of the simulation, and show (a) the initial conditions, (b) the crater at its maximum depth (c) maximum volume and (d) final morphology. The arrow in (a) shows the projectile’s trajectory. Transient (e) and final (f) crater surface profiles taken along the symmetry plane, for simulations with a projectile diameter of 14 km and velocity of 20 km/s on Earth, for impact angles 30–90°. Note in (f), the z-axis scale is exaggerated by a factor of 2.