Exponential amplification of the magnetic field in the primordial star-forming cloud

In the study of the initial mass function associated with the first generation of stars, known as Population III (Pop III) stars, a fundamental yet unresolved question pertains to the ultimate destiny of the secondary protostars emerging within the accretion disk – specifically, their likelihood of either merging or persisting as distinct entities. Our research concentrates on the magnetic influences affecting the genesis of these first stars under the conditions set by the cosmological initial magnetic field strength. We employ ideal magnetohydrodynamic simulations, utilizing a stiff equation-of-state (EOS) model, to accurately depict the magnetic field structure interconnecting these protostars. We observe that the magnetic field experiences rapid intensification due to the gas near the protostar completing multiple tens of orbital rotations in the initial decade following the formation of the protostar. Concurrently, as mass accretion continues, the region influenced by the significant magnetic field expands outward. This process of magnetic braking effectively curtails the disk fragmentation that would typically occur without a magnetic field. The resulting exponential augmentation of the magnetic field is posited to facilitate the formation of supermassive first stars.


Introduction
In contemporary cosmological research, a paramount and unresolved issue pertains to elucidating the formation mechanisms of the initial stellar generation, designated as Population III (Pop III) stars.To date, empirical observation of these primordial stars still needs to be explored.However, advancements in theoretical astrophysics, most notably the employment of progressively sophisticated numerical simulations, have substantially contributed to our understanding of the intrinsic characteristics of these earliest stars [1].
In the theoretical framework, a crucial yet unresolved question centers on the influence of magnetic phenomena, such as magnetic braking, on this process.Prior investigations, premised on the hypothesis of substantial magnetic fields within the primordial star-forming gas clouds, have highlighted several consequential outcomes: the retardation of gas contraction within the dark matter (DM) minihalo, thereby impacting the timing of initial star formation [2]; the inhibition of disk fragmentation owing to effective angular momentum transmission via magnetic fields [3,4]; and the diminution of protostellar rotational velocity, which in turn may 2 dictate the ultimate fate of Population III (Pop III) stars [5].Nevertheless, it is recognized that the universe's primordial magnetic field, measured at a mere 10 −18 G [6], is significantly weaker compared to the magnetic fields observed in contemporary star-forming regions, which are approximately 10 −6 G. Consequently, the amplification of the magnetic field through flux freezing during the cloud collapse phase, described by B ∝ n 2/3 , is deemed inadequate to substantially impact the formation of the first stars.
In the context of first star formation, preceding investigations have primarily focused on the intensification of the magnetic field during the collapse phase [7,8].However, the potential augmentation of this field throughout the subsequent accretion phase and its consequential impact on the growth of protostars still need to be explored.
To address this gap, we conducted three-dimensional ideal magnetohydrodynamic (MHD) simulations centered on the early stages of primordial star formation and employed the stiffequation-of-state (EOS) methodology.Following the inception of the first Population III (Pop III) protostar, these simulations revealed that a period of exponential magnetic field amplification occurs in the immediate vicinity of the protostar, notably within the initial three years.This amplified field region effectively inhibits disk fragmentation, thereby facilitating the formation of massive stars.Additionally, our study extends to exploring magnetic field amplification in metal-free and metal-enriched atomic-cooling halos (ACHs), environments conducive to forming intermediate-mass black holes (IMBHs).Our findings validate a consistent mechanism of magnetic field amplification during the accretion phase associated with the formation of direct-collapse black holes (DCBHs) in these ACHs.We deduce that exponential magnetic field amplification is a viable process in metal-enriched ACHs, which moderates the criteria necessary for DCBH formation.

Numerical methodology
In our research, we undertake three-dimensional (3D) ideal magnetohydrodynamic (MHD) simulations to probe the star formation process in the early universe.To accurately depict the thermal progression within the star-forming nebula, we incorporate a barotropic equationof-state (EOS) table, meticulously derived from chemical reaction computations [9].As an alternative to the conventional sink particle technique, our approach utilizes a stiff EOS framework to authentically represent the magnetic field structure linked to regions of dense gas. Figure 1 delineates the derived EOS tables, showcasing models of H2-cooling with varying threshold densities at nth = 10 16 cm −3 and 10 19 cm −3 (left), alongside H-cooling models differing in metallicity levels, with Z/Z ⊙ = 0, 10 −5 , and 10 −4 (right).
This paper elucidates selected model outcomes, initially detailed in the foundational paper.For an exhaustive discourse on the calculation methodology and a comprehensive comparison of the results across all models, please refer to the original publications ([11] for H 2 -cooling models and [12,13] for H-cooling models).

Results
3.1.First star formation in the H 2 -cooling gas cloud [11] We impose a uniform magnetic field B 0 in alignment with the initial cloud's rotational axis within our computational domain.Our analysis focuses on how varying magnitudes of B 0 = 0, 10 −20 , 10 −15 , and 10 −10 G (labelled as B00, B20, B15, and B10) influence the formation of the first stars.The B20 model, with B 0 = 10 −20 G, is chosen as the fiducial model in this study, as it is below the cosmological magnetic field estimation of approximately 10 −18 G. Demonstrating the impact of magnetic fields on forming the first stars in this model would substantiate that magnetic fields invariably influence these primordial stars.We conduct simulations across four models with different threshold densities, continuing until t ps = 100 yr for n th = 10 19 cm −3 and 1000 yr for n th = 10 ) Thermal evolution models of star-forming clouds as a function of the gas number density.(Left panel) H 2 -cooling models with zero-metallicity.In the left panel, we explore H2-cooling models under the condition of zero-metallicity.The fundamental theoretical model, represented by the black line, is derived from comprehensive chemical reaction calculations [9,10].The colored lines in this figure, integral to our study, represent adaptations of this base model.These adaptations are achieved through the application of the stiff equation of state (EOS) technique, with specific threshold densities of n th = 10 16 cm −3 (in red) and 10 19 cm −3 (in blue), respectively.In the right panel, we examine H-cooling models that incorporate varying levels of metallicity, specifically Z/Z ⊙ = 0, 10 −5 , and 10 −4 .The base models for these scenarios, illustrated by dashed lines, are theoretically formulated based on chemical reaction calculations, taking into account the influence of strong Lyman-Werner radiation at J 21 = 10 3 [9,10].The solid lines, representing the models adopted in our study, are variations of these dashed line models, utilizing the stiff-EOS approach.These variations are characterized by a set threshold density of n th = 10 16 cm −3 .
Figure 2 contrasts the simulation results of the fiducial magnetized model (B20) with those of the non-magnetized model (B00) throughout the first 1000 years of the protostar accretion phase.Initially, at the inception of the first protostar (depicted in the left panels), the density structure surrounding the protostar is identical in both models.This similarity arises because the magnetic field strength near the protostar is too weak (at most pico-gauss, or 10 −12 G) to influence the collapsing gas cloud.However, a notable divergence appears after a decade, as shown in the middle column of Figure 2. At this point, the magnetic field strength on the surface of the primary protostar (around 30 R ⊙ ) escalates to kilo-gauss levels.This strong "seed" magnetic field causes the magnetic field in a region with a 10 au radius surrounding the protostar to amplify significantly.Within this region of intense magnetic field strength, the density and velocity structures of the accreting gas are markedly influenced by the magnetic field.After 1000 years, as indicated in the right column of Figure 2, the area affected by the amplified magnetic field extends to a radius of approximately 500 au.In this expanded region, multiple protostars in the fiducial model have disappeared in the unmagnetized model.The emergence of a global spiral structure within the gas, inside the area of the amplified magnetic field, is attributed to the angular momentum transport by magnetic braking.This process allows for more efficient accretion.

Figure 2.
Crosssectional view on the z = 0 plane around the most massive protostar at t ps = 0, 10, and 1000 yr after the first protostar formation from left to right.The uppermost panel delineates the gas number density in the B00 model, the central panel similarly portrays the gas number density in the B20 model, and the lowermost panel illustrates the absolute magnetic field strength in the B20 model.The spatial dimensions of the depicted regions are 20 astronomical units (au) for the left and middle panels, in contrast to a significantly broader scope of 1000 au for the panels on the right.
The remarkable escalation of the magnetic field strength from 10 −12 G to 10 3 G near protostars originates from their rapid rotational motion, which effectively winds up magnetic fields.Within the first three years following protostar formation, the magnetic field strength near the protostar undergoes significant amplification, as depicted in Figure 3(a).In the area less than or equal to 10 au, the number of orbital rotations surpasses one at t ps = 0 yr and escalates to several dozen by t ps = 3 yr (Figure 3b).Consequently, the region affected by magnetic field amplification progressively expands, with its boundary aligning with the radius where the orbital rotation rate exceeds one (N rot = 1).This observation indicates that gravitational energy is effectively transformed into magnetic energy via kinetic (or rotational) energy following the formation of the first protostar.By the end of the accretion phase of the first stars (around 10 5 yr, as indicated by the dotted line), the region influenced by the amplified magnetic field could extend to approximately 10 4 au ∼ 0.05 pc, encompassing a total gas mass of about 500 M ⊙ in this model.
Regarding the influence of this exponentially amplifying magnetic field on the formation process of the first stars, the amplified magnetic field prevents fragmentation of the gravitationally unstable accretion disk, forming a single protostar at the cloud's center.However, the stellar masses remain consistent across both magnetized and unmagnetized models.Invariably, any fragments that form near the protostar will likely merge promptly.The rotation velocities, a critical parameter in stellar evolution theory, maintain a nearly constant rate at approximately 0.05 times the Keplerian velocity, irrespective of the model or evolutionary timeline.Given the low degree of rotation, employing a non-rotational model for stellar evolution seems plausible.
Our analysis first focuses on how metallicity affects magnetic influences.As metallicity rises, gas temperature decreases, leading to a lower mass accretion rate in clouds with higher metallicity.This trend is evident in unmagnetized models, where the mass growth rate of the primary protostar diminishes with increasing metallicity.At the end of simulations at t ps = 500 yr, the masses of primary protostars in these models reach a few 10 3 M ⊙ , aligning with previous unmagnetized simulations.
Incorporating magnetic effects, we find that the masses of primary protostars at t ps = 500 yr in magnetized models are about 10 times larger than those in unmagnetized models across all metallicity levels examined.While the mass infall rate in distant regions is similar in magnetized and unmagnetized models, the infall rate near the center is higher in magnetized models, indicating that magnetic fields influence the mass accretion rate in these areas.In the early accretion phase (t ps < 20 yr), the magnetic field strength amplifies rapidly with decreasing metallicity.This early difference is mainly due to the "seed" magnetic field, amplified by protostellar spin, as lower metallicity leads to a higher accretion rate and fragmentation of the central high-density gas into multiple protostars.Eventually, all magnetized models experience significant magnetic field amplification, with angular momentum transport by the amplified field Crosssectional views on the z = 0 and y = 0 planes surrounding the most massive protostar at t ps = 500 yr following the formation of the first protostar.
The panels include: (a, b, d, and e) showcasing the gas number density in models Z4B00, Z4B12, Z0B00, and Z0B12, and (c and f) displaying the absolute magnetic field strength in models Z4B12 and Z0B12.
enhancing gas accretion and protostellar mergers.After just 500 years, the primary protostars in these models have a mass with ∼ 10 4 M ⊙ .
Figure 4 presents two-dimensional maps of density and magnetic field strength around the primary protostar at the end of the simulation (t ps = 500 yr).In the magnetized models, numerous protostars form during a fragmentation burst at t ps = 400 yr.These protostars tend to escape farther from the cloud center in magnetized models than unmagnetized ones due to more efficient N -body interactions, as magnetic effects lead to a more significant influx of protostars into the central region.We conclude that in magnetized models, the rapid accretion flow displaces protostars from the cloud's center more than in unmagnetized models.
Next, we focus on the impact of initial magnetic field strength, labeled as B 0 .Figure 5 summarizes all models' B-n (magnetic field density) diagrams at a uniform time frame, t ps = 100 yr.This overview clearly shows the amplification of the magnetic field across all models.We observe that this amplification primarily occurs in two different density regions: a high-density area in the circumstellar disk (n ≥ 10 13 cm −3 ) and a lower-density region above the disk (n ∼ 10 11 cm −3 ).

Discussion
The magnetic field amplification post-protostar formation follows a nuanced progression, as illustrated in Figure 6: Phase diagrams of the absolute magnetic field strength for all models at t ps = 100 yr after the first protostar formation.
However, MHD simulations employing the sink particle methodology fall short in simulating the primary amplification of the "seed" magnetic field near the protostar and its outward propagation.This shortfall is linked to the lack of interaction between the rotation within high-density regions and the magnetic field in these specific simulations.
In atomic hydrogen (H) cooling halos, the elevated accretion rate results in several fragments, boosting the magnetic field through their rotational activity.Our observations in molecular hydrogen (H 2 ) cooling halos indicate initial disk fragmentation, which rapidly subsides as these fragments integrate into the primary protostar.The orbital rotation around the protostar fosters magnetic field amplification without triggering further fragmentation.This phenomenon, exclusive to early Universe star formation, is absent in nearby star-forming areas, where magnetic fields peak before the protostar's accretion phase, underscoring the importance of considering strong magnetic field environments.
This magnetic field amplification mechanism plays a crucial role in preventing the fragmentation of the accretion disk.When a star-forming gas cloud exhibits substantial rotational motion, it forms a gravitationally unstable accretion disk that initially fragments.However, this fragmentation is swiftly curtailed by the augmented magnetic field.Conversely, in gas clouds with minimal rotation, magnetic field amplification is less pronounced, inhibiting the formation of a gravitationally unstable accretion disk and subsequent fragmentation.In both scenarios, the result is the emergence of a singular first star.

Conclusion
We propose a mechanism that increases magnetic field strength exponentially during the accretion stage of initial stellar genesis.This process remarkably enhances a minuscule magnetic seed, even under the mere influence of a cosmological magnetic field, elevating it to kilo-gauss magnitudes in under a decade from the birth of the protostar.During the accretion phase, the area of the intensified magnetic field progressively broadens, extending to approximately 10,000 astronomical units and covering a mass of roughly 500 solar masses by about 100,000 years.This is the juncture when the protostar evolves into a zero-age main-sequence star.The formidable magnetic field plays a pivotal role in preventing the fragmentation of the accretion disk, leading Figure 6.A schematic overview of the rotation-driven magnetic field amplification process following protostar formation includes the following stages: (1) The spinning motion of each protostar leads to the amplification of magnetic fields in their vicinity.(2) The enhanced "seed" magnetic field, a result of the initial amplification, further strengthens the magnetic fields around it due to the orbital motion of the protesters.(3) As gas continues to accrete, its rotation causes the region of the amplified magnetic field to expand outward gradually.It is important to note that this mechanism of magnetic field amplification is not typically observed in contemporary star formation processes, as the magnetic field tends to dissipate significantly within the disk in these scenarios.
to the exclusive formation of a singular protostar within each disk.We infer that magnetic fields significantly impact the formation of the earliest stars, regardless of their initial magnetic field strength being as diminutive as about 10 −18 G. Furthermore, our analysis corroborates the rotation-induced magnetic field amplification during the accretion phase in direct-collapse black hole (DCBH) formation within metal-rich atomic-cooling halos (ACHs).Although heightened metallicity dampens the mass accretion rate, the central region undergoing collapse is gravitationally unstable, culminating in fragmentation and the emergence of numerous protostars.These protostars play a crucial role in magnifying the magnetic field, facilitating additional gas accretion and the merging of stars.Notably, this magnetic enhancement is effective regardless of the initial intensity of the studied regime of dynamically weak magnetic fields.This amplification process remains operational even in metalrich ACHs with an initial magnetic field strength of B 0 / G = 10 −12 at n H = 10 4 cm −3 .The rate of mass growth peaks in extremely metal-poor ACHs with Z/Z ⊙ = 10 −5 , marking a distinct deviation from trends observed in unmagnetized simulations.We conclude that exponential magnetic field amplification is feasible in metal-rich ACHs, thereby moderating the conditions required for DCBH formation.
Looking ahead, we aim to undertake a comprehensive parameter survey of MHD simulations, focusing on the primordial star-forming gas clouds ascertained from cosmological simulations.Although the current study indicates the complete elimination of disk fragmentation, future inquiries will assess whether gas clouds with diverse physical attributes, such as varying accretion rates and degrees of rotation, lead to comparable results.Moreover, our ongoing simulations terminate at t ps = 1000 yr, necessitating additional research to determine the ultimate stellar mass at t ps ∼ 10 5 yr, coinciding with the first star's arrival at the zero main sequence stage.Our goal is to thoroughly revise the theory of first star formation, incorporating MHD influences, and to ascertain the formation rates of observable counterparts, including low-mass surviving stars and massive black hole binaries.

15thFigure 3 .
Figure 3. Radial profiles for model B20 are shown at different times after the first protostar formation: t ps = 0, 1, 2, 3, 10, 30, 100, 300, and 1000 yr.The panels include (a) the magnetic field strength and (b) the number of orbital rotations at each time point t ps , expressed as N rot = (v rot t ps )/(2πR).Notably, the line for t ps = 0 yr is not included in panel (b) since N rot = 0 equals zero for every R at t ps = 0 yr.Panel (b) also features a dotted line that estimates the expected N rot at t ps = 10 5 yr, when the first star ends its accretion phase.This estimate uses the radial rotational velocity profile at t ps = 1000 yr.The horizontal lines in panel (b) indicate where N rot = 1.

Figure 4 .
Figure 4.Crosssectional views on the z = 0 and y = 0 planes surrounding the most massive protostar at t ps = 500 yr following the formation of the first protostar.The panels include: (a, b, d, and e) showcasing the gas number density in models Z4B00, Z4B12, Z0B00, and Z0B12, and (c and f) displaying the absolute magnetic field strength in models Z4B12 and Z0B12.

Figure 5 .
Figure 5.Phase diagrams of the absolute magnetic field strength for all models at t ps = 100 yr after the first protostar formation.