Efficient Highly-Subsonic Turbulent Dynamo and Growth of Primordial

Eﬃcient highly-subsonic turbulent dynamo and growth of primordial magnetic ﬁelds

Radhika Achikanath Chirakkara1,2,3,∗ Christoph Federrath2,† Pranjal Trivedi3,4,5,‡ and Robi Banerjee3§ 1Department of Physics, Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pune 411008, India 2 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia 3Hamburger Sternwarte, UniversitätHamburg, Gojenbergsweg 112, 21029 Hamburg, Germany. 4UniversitätHamburg, II. Institut fürTheoretische Physik, Luruper Chaussee 149, 22761 Hamburg, Germany. 5Department of Physics, Sri Venkateswara College, University of Delhi 110020 India

We present the first study on the amplification of magnetic fields by the turbulent dynamo in the highly subsonic regime, with Mach numbers ranging from 10−3 to 0.4. We find that for the lower Mach numbers the saturation efficiency of the dynamo, (Emag/Ekin)sat, increases as the Mach number decreases. Even in the case when injection of energy is purely through longitudinal forcing −2 −3 modes, (Emag/Ekin)sat & 10 at a Mach number of 10 . We apply our results to magnetic field amplification in the early Universe and predict that a turbulent dynamo can amplify primordial −16 −13 magnetic fields to & 10 Gauss on scales up to 0.1 pc and & 10 Gauss on scales up to 100 pc. This produces fields compatible with lower limits of the intergalactic magnetic field inferred from blazar γ-ray observations.

I. INTRODUCTION 10−29 Gauss at the electroweak phase transition and field strengths of ∼ 10−20 Gauss at the QCD phase transi- Magnetic fields are ubiquitous on all scales in the Uni- tion. Turner and Widrow [18] predict magnetic fields −34 −10 verse, from the surface of stars to galaxies to the voids in with strengths ∼ 10 –10 Gauss on a scale of 1 Mpc the large-scale structure of the Universe. The turbulent may be produced during inflation. Otherwise, the un- small-scale dynamo (SSD) amplifies small seed magnetic avoidable presence of vorticity in the primordial plasma fields, by converting turbulent kinetic energy into mag- leads to the generation of weak magnetic fields in the ra- netic energy [1, 2]. The turbulent dynamo has a wide diation era [19, 20]. Studies by [21, 22] investigate the range of applications as it can operate in a variety of as- properties of hydrodynamic turbulence in the primordial trophysical situations and has been studied in the super- plasma at the QCD phase transition. Upper limits of −9 sonic and transonic regime of turbulence [3, 4], however, ∼ 10 Gauss [16, 23–31] and recent stricter limits of −11 it remains unexplored in the extremely subsonic regime. ∼ 5 × 10 Gauss [32] have been placed on PMF from This regime is important for studies on magnetohydro- cosmic microwave background anisotropies. dynamic turbulence and is relevant for many processes in The observed magnetic fields, in many cases, are or- astrophysics and cosmology, including the amplification ders of magnitude greater than the initially generated of primordial magnetic fields (PMF). fields. To explain the magnitude of the observed strong Several studies have inferred the presence of inter- magnetic fields in the voids of the Universe, Wagstaff galactic magnetic fields (IGMFs) through γ-ray obser- et al. [33] showed that the SSD can amplify the mag- vations of TeV blazars and have predicted a lower limit netic field seeds present in the early Universe. Turbu- of 10−16–10−18 Gauss for the IGMF on Mpc scales [5–12]. lence in the early Universe is unavoidably generated by The inferred lower bounds have been questioned due to gravitational acceleration due to the primordial density the possible effect of plasma instabilities in the intergalac- perturbations (PDP), which gives rise to longitudinal (ir- tic medium [13]. However, recent studies have taken into rotational) driving modes. From Wagstaff et al. [33], we account the effect of plasma instabilities in the observa- expect the turbulent dynamo in the early Universe to arXiv:2101.08256v1 [astro-ph.HE] 20 Jan 2021 tions and have shown that a lower limit on the IGMF can have operated under very subsonic conditions with Mach be placed from the blazar γ-ray observations [14, 15]. numbers (M) ∼ 10−5–10−4. Understanding the origin of these magnetic fields is Motivated by these predictions, we study the be- an unsolved problem. Magnetic fields can be generated haviour of the SSD in the very subsonic regime with a during various phases in the early Universe [16]. Sigl purely compressive driving of the turbulence. Further- et al. [17] predict the generation of magnetic fields ∼ more, it has been shown that the SSD operating during the collapse of gas clouds in minihalos can give rise to rather strong magnetic fields during the formation of ∗ [email protected] the first stars [34, 35]. Xu and Lazarian [36] present a † [email protected] consolidated study on the kinematic and the non-linear ‡ [email protected] growth phases of the SSD. The authors discuss the dy- § [email protected] namo mechanism during primordial star formation and 2

⊥ 2 in the first galaxies and find that during early star forma- and Pij = δij −kikj/k is the divergence-free (solenoidal) tion, magnetic fields on the Jeans scale cannot be easily projection. The parameter, ζ, defines the nature of the generated [36]. Thus, Xu and Lazarian [36] show that projection and lies in the range [0,1]. ζ = 0 corresponds more work is needed to understand the initial generation to injection of purely compressive (or longitudinal) modes of magnetic fields in the early Universe, which may play in the velocity field and ζ = 1 implies injection of purely an important role during early star formation. Recent solenoidal (or rotational) modes. The purely compres- studies [37, 38] have also investigated the role of mag- sive forcing models the turbulent acceleration field, f~, netic fields in the formation of the first stars. such that ∇ × f~ = 0 and the purely solenoidal forcing A previous study by Federrath et al. [3] has examined has ∇ · f~ = 0 [42]. The amplitude of the turbulent driv- the properties of the dynamo as a function of the Mach ing controls the amount of kinetic energy injected into number and the nature of turbulent driving. They in- the plasma and therefore the Mach number, M = v/cs. vestigated the case when the turbulent dynamo is driven We perform a systematic study wherein we vary the solely by longitudinal modes for Mach numbers in the Mach number and the nature of the turbulent driving range M ∼ 0.1–20, thus not reaching sufficiently far into to determine their effects on the properties of the SSD. the very subsonic regime relevant for the amplification of We run our simulations on uniform grids with 1283 cells PMF. In this paper, we determine the properties of the and set up a turbulent initial seed field with an initial SSD with non-helical magnetic fields in the very subsonic 10 14 −3 plasma beta, βi ∼ 10 − 10 . In addition to the above regime for Mach numbers in the range M = 10 –0.4 mentioned ideal-MHD simulations, we solve the non-ideal and for a wide range of turbulent driving conditions. MHD equations on 2563 grid cells to estimate the effective Reynolds number (Re) and magnetic Prandtl number (Pm) of the ideal MHD simulations(see Figure 2). In II. METHODS agreement with earlier work [3], we find that Re ∼ 1500 and Pm ∼ 2 are good approximations for the effec- We solve the following compressible, three- tive Reynolds and magnetic Prandtl number in the ideal dimensional, ideal magnetohydrodynamical(MHD) MHD simulations with 1283 grid cells. While in the early equations with the FLASH code on a periodic computa- Universe, we expect much higher Re and Pm [43], the sat- tional grid [39–41] uration level of the dynamo, which is our main concern, is converged to within a factor of 2 compared to the limit ∂ρ of very high Re and Pm [4]. + ∇ · (ρ~v) = 0 (1) ∂t The stretch-twist-fold dynamo mechanism results in an exponential amplification of the magnetic energy, Em/Em0 = exp(Γt) where Γ is the amplification rate, ∂(ρ~v) +∇·(ρ~v ⊗~v −B~ ⊗B~ )+∇p = ∇·(2νρS)+ρf~ (2) Em0 is the initial magnetic energy and t is the time, nor- ∂t malized to the eddy-turnover time ted, which is defined as ted = L/(2Mcs) [1, 2]. The saturation efficiency of the ~ dynamo, defined as the ratio of the magnetic energy to ∂B 2 = ∇ × (~v × B~ ) + η∇ B,~ (3) kinetic energy at saturation ((Emag/Ekin)sat), is a func- ∂t tion of the Mach number and the nature of turbulent closed by the isothermal equation of state, pthermal = driving [3]. 2 cs ρ, with constant sound speed, cs, and satisfying ∇ · B~ = 0. In the above equations, ρ, ~v and B~ are the density, velocity and the magnetic field. ν and η are the III. RESULTS kinematic viscosity and the magnetic resistivity. p is the sum of the thermal and magnetic pressure of the system We assign a model name to all our simulations. In 2 the model name “M” stands for the Mach number and p = pthermal + (1/2)|B~ | . S is the traceless rate of strain “S” stands for the solenoidal fraction (ζ) in the driving tensor, Sij = (1/2)(∂ivj + ∂jvi) − (1/3)δij∇ · ~v, which field. For example, the model “M0.001S0.1” represents captures the viscous interactions and f~ is the turbulent the simulation with M ∼ 10−3 and a solenoidal fraction acceleration field used to drive the turbulence. of 0.1 in the turbulent driving. We study the proper- The acceleration field f~, is modelled using the ties of the SSD in the subsonic regime, M ∼ 10−3–0.4. Ornstein-Uhlenbeck process in Fourier space [42]. In our A dynamo driven by solenoidal forcing shows a higher simulations, we stir the turbulence continuously on large amplification rate and saturation efficiency, because in scales, i.e., wavenumbers k(2π/L) = [1 ... 3], where L is this case, the driving field injects vorticity directly into the side length of the cubic Cartesian computational do- the plasma, which is then able to drive the stretch-twist- main, as in previous studies [3, 42]. The forcing is mod- fold dynamo mechanism efficiently [3]. However, with elled by a projection operator in Fourier space, which compressive forcing, solenoidal modes are not injected ζ ~ ⊥ ~ k ~ is defined as Pij(k) = ζPij (k) + (1 − ζ)Pij(k), where directly by the turbulent driving and the plasma might k 2 Pij = kikj/k is the curl-free (compressive) projection have zero initial vorticity. 3

Before we present and discuss our numerical results, M0.001S0.1 M0.2S0.1 M0.05S0.001 M0.01S0 we brieﬂy address the basic equation for the evolution of M0.01S0.1 M0.001S0.001 M0.2S0.001 M0.05S0 0 M0.05S0.1 M0.01S0.001 M0.001S0 M0.2S0 vorticity. Vorticity, deﬁned as ~ω = ∇ × ~v , follows the 10 evolution equation [44] 10 1

∂~ω 2 1 10 2 = ∇ × (~v × ~ω) + ν∇ ~ω + 2 ∇ρ × ∇p + 2ν∇ × S∇lnρ. ∂t ρ 3 (4) 10

The vorticity equation has the same structure as the in- 107 duction equation (3) and can therefore give rise to an ex- 0 105 ponential growth of vorticity similar to the amplification m E of magnetic fields by the SSD, if the last three terms on / 3 m 10 the right hand side of equation (4) are subdominant com- E pared to the first term [45]. Considering we start with 101 2 zero initial vorticity, the baroclinic term (∇ρ × ∇p)/ρ 10 1 can not generate any vorticity, as the system is isothermal 100 with the equation of state p = c2ρ. However, if density

s n i 1 gradients are present, then through viscous interactions, k 10 E / the last term on the right-hand side of equation (4) can g 2 a 10 generate vorticity, which can then be amplified through m E the first term on the right-hand side of equation (4). 10 3 Figure 1 depicts the time evolution of the Mach num- 10 4 101 102 103 ber, Em/Em0, and Emag/Ekin as a function of time for a representative sample of our simulations (a full list of t/ted simulations is provided in the supplemental material A). We find that increasing the solenoidal fraction (ζ) of forc- FIG. 1. Mach number (top panel), magnetic energy, Em/Em0 ing enhances the amplification rate of the dynamo and (middle panel), and saturation level, Emag/Ekin (bottom panel) as a function of time normalised to the eddy turnover increases the saturation level. time (ted) for a representative sample of our low Mach number The saturation efficiency of the SSD and the solenoidal simulation models on 1283 grid cells with solenoidal fraction fraction of the kinetic energy, Esol/Etot, as a function of of 0.1, 0.001 and 0 (purely compressive) in the forcing. In the the Mach number and the turbulent driving, are shown model name “M” stands for the Mach number and “S” stands in Figure 2. The solenoidal fraction of the kinetic energy for the solenoidal fraction (ζ) in the driving field. The dotted is correlated to the amplification rate, Γ. The greater the lines in the middle panel show the fits for the amplification solenoidal modes in the velocity field, the higher the vor- rate. The dotted black lines in the bottom panel show the fits ticity of the plasma, which leads to a more efficient am- for the saturation efficiency. plification of the magnetic energy and therefore a higher amplification rate. We find that for purely compressive driving, the amplification rate and the saturation effi- the saturation efficiency converges with resolution. ciency decline with the Mach number until M ∼ 0.05. The density fluctuations in the plasma decrease with Below this Mach number, it is easier for the energy in- the Mach number, leading to a decrease in the density jected by the turbulence to drive rotational modes, thus gradients. This in turn enables the first term on the right generating relatively more vorticity in the plasma and in- hand side of equation (4) to operate more efficiently and to generate a higher fraction of vorticity modes in the creasing Esol/Etot. The dynamo is very sensitive to the very low Mach number limit. Consequently, the kinetic solenoidal fraction of the kinetic energy and as Esol/Etot increases, the amplification rate and the saturation ef- energy in the rotational (∇ × ~v) modes increases relative ficiency of the dynamo increase. In the very subsonic to the kinetic energy in the compressive (∇ · ~v) velocity modes in the very subsonic regime. This causes E /E regime, both Esol/Etot and (Emag/Ekin)sat increase as sol tot the Mach number decreases. to increase in this limit, which then leads to an efficient With a solenoidal fraction of 0.1 in the driving, we find SSD mechanism, thereby increasing the saturation effi- that the saturation efficiency approaches the results from ciency at very low Mach numbers. Federrath et al. [3] for purely solenoidal driving. This is also observed for the dynamo with solenoidal fractions of 0.01 and 0.001 in the forcing. With a solenoidal frac- IV. APPLICATIONS tion of 0.0001, we find that at M ∼ 10−3, the saturation efficiency increases by an order of magnitude com- Magnetic fields are unavoidably created in the primor- pared to the dynamo driven by purely compressive driv- dial Universe [19] and can act as a seed for the SSD. ing (ζ = 0). We also perform the low Mach number sim- Wagstaff et al. [33] show that turbulence can be es- ulations with solenoidal fractions of 0.001 and 0.01 on tablished in the early Universe between the electroweak 2563, 5123 and 5763 grid cells and show that the value of epoch and neutrino decoupling (T = 0.2–100 GeV), 4

101 Federrath et al.2011 Sol.frac 0.01 Sol.frac 0.0001 to operate. This dynamo is expected to have operated in −4 Sol.frac 0.1 Sol.frac 0.001 Sol.frac 0 very subsonic conditions, M ∼ 10 . In the radiation- 100 dominated era, the relativistic√ equation of state, p = ρ/3, t

a sets the sound speed to c/ 3. s 1 ) 10 n i In the following discussion, we use the results obtained k E

/ by Wagstaﬀ et al. [33], where the authors estimate the g 10 2 a magnetic ﬁelds generated by a SSD in the primordial m E

( Universe and follow the evolution of these ﬁelds to esti- 10 3 mate the IGMF at present day. In the aforementioned work, physical quantities like the magnetic ﬁeld strength 10 4 and their coherence length are calculated in a co-moving frame and are evaluated at the present-day epoch. We 0 10 note that the local viscosity, which determines the high Reynolds and Prandtl numbers, are set by the relativistic background in the early Universe. However, the velocity

t 1

o 10 t ﬂuctuations responsible for driving the turbulence in the E / l

o early Universe are non-relativistic, therefore, for our sim- s E 10 2 ulations of the SSD in the radiation-dominated era, the non-relativistic MHD equations are appropriate (see supplemental material B). We also note that we apply our 10 3 results to the baryon-photon ﬂuid in the early Universe prior to recombination, where using the comoving coor- 10 3 10 2 10 1 100 101 dinates with the above mentioned relativistic equation of state is a suitable approach [16, 46–50].

FIG. 2. Saturation efficiency, (Emag/Ekin)sat (top panel) and Now, we will apply our results for the SSD in the solenoidal ratio, Esol/Etot (bottom panel) as a function of subsonic regime to the early Universe dynamo. The Mach number for solenoidal fraction (ζ) of 0.1, 0.01, 0.001, turbulent dynamo action occurs on timescales substan- 3 0.0001 and 0 in the turbulent driving on 128 grid cells. tially smaller than the expansion of the early Universe Dark blue (diamond) data points show purely compressive (see supplemental material C). At M ∼ 10−3 we re- and purely solenoidal driving cases taken from Figure 3 in port the saturation efficiency to be ∼ 8.3 × 10−3. Tak- Federrath et al. [3]. The dotted black lines show the fits ing the value of the saturation efficiency at M ∼ 10−3 to the data to guide the eye. The black data points in the top panel correspond to the simulations done on 2563 grid to be a lower bound for the early Universe dynamo, we cells for ζ = 0.01 and ζ = 0.001. The grey data points at predict the generation of magnetic fields with strengths −17 M ∼ 0.01 and 0.05 in the top panel correspond to simula- & 9.1 × 10 Gauss on scales up to λc ∼ 0.1 pc through tions done on 5123 grid cells (for ζ = 0.01) and 5763 grid cells the dynamo action driven by PDP. If the SSD is driven (for ζ = 0.001). The grey plus symbols in the top panel cor- by first-order phase transitions, we predict that the dy- respond to non-ideal MHD simulations on 2563 grid cells for namo generates much higher magnetic field strengths of ζ = 0.01 and ζ = 0.001 and Mach numbers in the range, M ∼ 9.1×10−14 Gauss on scales up to λ ∼ 100 pc [33]. We −3 & c 5 × 10 –0.1 with Reynolds number Re = 1500 and magnetic note that these values are lower limits, as the magnetic Prandtl number Pm = 2, i.e., these are approximately the 3 field generated increases with the saturation efficiency, effective Re and Pm for all the simulations on 128 grid cells. which is likely to be appreciably greater in the early Uni- verse. We also note that these dynamo-amplified magnetic fields are well below the recent sub-nanogauss upper where the dissipation scale is set by neutrino damping limits placed on PMF [32] but likely too weak to alleviate and is ∼ 3 × 10−12 pc in comoving coordinates at the the Hubble tension [51]. electroweak epoch. They further describe two mecha- The conservative estimates of the lower bounds nisms for driving the turbulence in this early evolution on the IGMF from blazar γ-ray observations of the Universe: 1) through velocity fluctuations gener- are 10−17–10−14 Gauss on scales of 0.1 pc and ated by the PDP, and 2) through first-order phase tran- 10−19–10−15 Gauss on scales of 100 pc [11, 12]. The SSD sitions which may occur in this epoch. In the former mechanism driven by first-order phase transitions in case, the velocity fluctuations arise due to acceleration the early universe can therefore explain the lower-limit by the gravitational potential generated due to PDP and on the IGMFs on scales of ∼ 100 pc. Our important therefore are longitudinal or compressive velocity modes. conclusion is that the dynamo mechanism driven by They would also be driven continuously as is the case in velocity fluctuations generated by the PDP can produce our simulations. appreciable magnetic fields at shorter scales up to 0.1 pc Well developed turbulence together with the high mag- comparable to the lower bounds on the IGMF at these netic Reynolds numbers and Prandtl numbers in the scales. This raises the interesting possibility of explain- early Universe provides optimal conditions for the SSD ing the IGMF lower bounds on these scales, without 5 invoking beyond the standard model (BSM) physics, i.e, funding by the DFG through the projects BA 3706/14-1, without requiring a first-order phase transition. In case a BA 3706/15-1, BA 3706/17-1 and BA 3706/18. Compu- first-order phase transition occurs in the early Universe, tational resources used to conduct simulations presented it could generate stronger magnetic fields. However, the here were provided in part by the Regionales Rechen- possibility of such an event in the primordial Universe is zentrum at the University of Hamburg. We further ac- uncertain. knowledge high-performance computing resources pro- These primordial fields can act as seeds for galactic vided by the Leibniz Rechenzentrum and the Gauss Cen- dynamos and may influence the formation of the first tre for Supercomputing (grants pr32lo, pr48pi and GCS stars [52, 53]. The Reynolds number and the Prandtl Large-scale project 10391), the Australian National Com- number in the early Universe are orders of magnitude putational Infrastructure (grant ek9) in the framework higher than what we achieve in our simulations. In this of the National Computational Merit Allocation Scheme limit, the growth rate increases with the Reynolds num- and the ANU Merit Allocation Scheme. The simulation ber as Γ ∝ Re1/2 [45]. Therefore, the growth rate of the software FLASH was in part developed by the DOE- early-Universe dynamo will be much higher than what is supported Flash Center for Computational Science at the predicted from our simulations [4, 33]. University of Chicago.

V. CONCLUSIONS SUPPLEMENTAL MATERIAL A

In this exploratory study of the highly subsonic MHD We assign a model name to our simulations in which regime, we find that the small-scale dynamo amplifies “M” stands for the Mach number and “S” stands for the magnetic fields efficiently for all the turbulent forcing solenoidal fraction (ζ) in the driving field. Tabulated be- models we have studied and the saturation efficiency in- low, in increasing order of ζ, are the values for the Mach creases with decreasing Mach number in the highly sub- number (M), solenoidal fraction in the turbulent forc- sonic limit. Our results in this previously unexplored ing, saturation efficiency of the dynamo ((Emag/Ekin)sat), regime may be regarded as a proof-of-concept and can amplification rate of the magnetic energy (Γ) and the have wide-ranging applications for systems governed by solenoidal ratio in the kinetic energy (Esol/Etot) for our MHD turbulence. ideal MHD simulations on 1283, 2563, 5123 and 5763 grid The results of this study can be used to estimate the cells (see table I) and for the non-ideal MHD simulations magnetic field strengths produced in the early Universe with a kinetic Reynolds number, Re = 1500 and mag- by using the purely compressively-driven dynamo model. netic Prandtl number Pm = 2 on 2563 gird cells (see We find the small-scale dynamo action in the early Uni- table II). verse can generate magnetic fields with strength greater than ∼ 10−16 Gauss on scales up to 0.1 pc when the turbulence is forced by primordial density perturbations and SUPPLEMENTAL MATERIAL B field strengths greater than ∼ 10−13 Gauss on scales up to 100 pc when forced by first-order phase transitions. This We simulate the non-relativistic baryon fluctuations in prediction produces fields compatible with lower limits a relativistic background plasma, which drives the small- of the intergalactic magnetic field inferred from blazar scale dynamo in the radiation-dominated Universe. The γ-ray observations on these scales. non-relativistic MHD equations in Minkowski space-time with the relativistic equation of state, p = ρ/3, can be used to model these fluctuations in the primordial plasma ACKNOWLEDGMENTS of the early Universe [16, 46–48]. This approach has been used by studies investigating the evolution of MHD tur- We thank Amit Seta for discussions on the SSD in bulence in the radiation-dominated Universe through nu- the early Universe. R. A. would like to thank the Aus- merical simulations [49, 50]. tralian National University for the Future Research Tal- ent award and is grateful to the University of Ham- ∂ρ 4 burg and the Australian National University for hosting + ∇ · (ρ~v) − E~ · J~ = 0 (5) her during the course of this project. C. F. acknowl- ∂t 3 edges funding provided by the Australian Research Coun- cil (Discovery Project DP170100603 and Future Fellow- ∂ 1 3 ship FT180100495), and the Australia-Germany Joint (ρ~v) + (~v · ∇)(ρ~v) +~v ∇ · (ρ~v) = − ∇ρ + J~× B~ (6) Research Cooperation Scheme (UA-DAAD). R. B. and ∂t 4 4 P. T. acknowledge support by the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation) un- ∂B~ der Germany’s Excellence Strategy – EXC 2121 “Quan- = ∇ × (~v × B~ ) (7) tum Universe” – 390833306. R. B. is also thankful for ∂t 6

3 −1 Model (Resolution 128 ) M ζ (Emag/Ekin)sat Γ(ted )Esol/Etot M0.001S0 (9.5 ± 1.4) × 10−4 0 (8.3 ± 3.6) × 10−3 (5.5 ± 0.9) × 10−2 (4.9 ± 0.6) × 10−2 M0.005S0 (3.9 ± 0.3) × 10−3 0 (2.4 ± 0.8) × 10−3 (2.1 ± 0.2) × 10−2 (1.1 ± 0.2) × 10−2 M0.01S0 (1.0 ± 0.1) × 10−2 0 (1.3 ± 0.2) × 10−3 (4.8 ± 0.3) × 10−2 (9.7 ± 1.7) × 10−3 M0.02S0 (1.9 ± 0.2) × 10−2 0 (6.0 ± 1.2) × 10−4 (2.2 ± 0.1) × 10−2 (2.8 ± 0.1) × 10−3 M0.05S0 (4.8 ± 0.4) × 10−2 0 (2.3 ± 0.5) × 10−4 (1.1 ± 0.1) × 10−2 (1.5 ± 0.3) × 10−3 M0.1S0 (9.7 ± 0.9) × 10−2 0 (5.7 ± 1.7) × 10−4 (2.3 ± 0.2) × 10−2 (4.8 ± 0.8) × 10−3 M0.2S0 (1.9 ± 0.2) × 10−1 0 (3.7 ± 0.9) × 10−3 (8.5 ± 0.4) × 10−2 (3.3 ± 0.7) × 10−2 M0.4S0 (4.3 ± 0.3) × 10−1 0 (1.6 ± 0.3) × 10−2 (2.4 ± 0.1) × 10−1 (1.1 ± 0.1) × 10−1 M0.001S0.0001 (1.0 ± 0.1) × 10−3 0.0001 (7.6 ± 2.3) × 10−2 (2.2 ± 0.2) × 10−1 (2.7 ± 0.1) × 10−1 M0.005S0.0001 (4.6 ± 0.4) × 10−3 0.0001 (1.2 ± 0.4) × 10−2 (9.2 ± 1.2) × 10−2 (5.5 ± 0.9) × 10−2 M0.01S0.0001 (8.0 ± 0.6) × 10−3 0.0001 (6.6 ± 1.0) × 10−3 (6.0 ± 0.3) × 10−2 (2.0 ± 0.4) × 10−2 M0.02S0.0001 (2.1 ± 0.2) × 10−2 0.0001 (2.8 ± 0.4) × 10−3 (6.6 ± 0.5) × 10−2 (1.4 ± 0.2) × 10−2 M0.05S0.0001 (4.6 ± 0.4) × 10−2 0.0001 (1.1 ± 0.2) × 10−3 (3.1 ± 0.3) × 10−2 (4.4 ± 1.0) × 10−3 M0.1S0.0001 (1.3 ± 0.1) × 10−1 0.0001 (1.9 ± 0.6) × 10−3 (6.4 ± 0.5) × 10−2 (1.0 ± 0.1) × 10−2 M0.2S0.0001 (1.9 ± 0.2) × 10−1 0.0001 (4.7 ± 1.1) × 10−3 (8.0 ± 0.4) × 10−2 (3.1 ± 0.3) × 10−2 M0.4S0.0001 (4.4 ± 0.4) × 10−1 0.0001 (1.7 ± 0.4) × 10−2 (2.5 ± 0.1) × 10−1 (9.1 ± 1.8) × 10−2 M0.001S0.001 (1.1 ± 0.1) × 10−3 0.001 (3.5 ± 0.5) × 10−1 (4.4 ± 0.3) × 10−1 (8.8 ± 0.3) × 10−1 M0.005S0.001 (5.7 ± 0.2) × 10−3 0.001 (1.7 ± 0.3) × 10−1 (4.8 ± 0.2) × 10−1 (4.0 ± 0.7) × 10−1 M0.01S0.001 (1.2 ± 0.1) × 10−2 0.001 (1.0 ± 0.2) × 10−1 (5.2 ± 0.2) × 10−1 (2.6 ± 0.7) × 10−1 M0.02S0.001 (2.3 ± 0.1) × 10−2 0.001 (5.9 ± 1.0) × 10−2 (3.5 ± 0.1) × 10−1 (1.1 ± 0.0) × 10−1 M0.05S0.001 (4.1 ± 0.3) × 10−2 0.001 (2.8 ± 0.5) × 10−2 (1.9 ± 0.1) × 10−1 (6.5 ± 0.5) × 10−2 M0.1S0.001 (9.9 ± 0.9) × 10−2 0.001 (1.2 ± 0.2) × 10−2 (1.5 ± 0.1) × 10−1 (4.0 ± 0.6) × 10−2 M0.2S0.001 (1.8 ± 0.2) × 10−1 0.001 (6.6 ± 1.5) × 10−3 (9.2 ± 0.5) × 10−2 (3.5 ± 0.6) × 10−2 M0.4S0.001 (3.9 ± 0.3) × 10−1 0.001 (1.4 ± 0.3) × 10−2 (1.7 ± 0.1) × 10−1 (1.0 ± 0.1) × 10−1 M0.001S0.01 (1.1 ± 0.2) × 10−3 0.01 (6.1 ± 1.3) × 10−1 (5.8 ± 0.5) × 10−1 (1.0 ± 0.0) × 10+0 M0.005S0.01 (5.1 ± 0.2) × 10−3 0.01 (4.3 ± 1.0) × 10−1 (5.5 ± 0.3) × 10−1 (9.6 ± 0.1) × 10−1 M0.01S0.01 (1.4 ± 0.1) × 10−2 0.01 (5.0 ± 1.0) × 10−1 (1.1 ± 0.0) × 10+0 (7.8 ± 0.2) × 10−1 M0.02S0.01 (2.2 ± 0.1) × 10−2 0.01 (3.5 ± 0.6) × 10−1 (8.6 ± 0.5) × 10−1 (6.7 ± 0.8) × 10−1 M0.05S0.01 (4.6 ± 0.2) × 10−2 0.01 (2.4 ± 0.3) × 10−1 (5.9 ± 0.4) × 10−1 (5.2 ± 0.1) × 10−1 M0.1S0.01 (9.3 ± 0.6) × 10−2 0.01 (1.6 ± 0.3) × 10−1 (5.7 ± 0.3) × 10−1 (3.9 ± 0.6) × 10−1 M0.2S0.01 (2.1 ± 0.2) × 10−1 0.01 (8.3 ± 1.6) × 10−2 (5.1 ± 0.1) × 10−1 (1.9 ± 0.3) × 10−1 M0.4S0.01 (3.8 ± 0.3) × 10−1 0.01 (3.4 ± 0.7) × 10−2 (2.8 ± 0.3) × 10−1 (1.4 ± 0.1) × 10−1 M0.001S0.1 (1.2 ± 0.1) × 10−3 0.1 (5.4 ± 1.0) × 10−1 (5.2 ± 0.6) × 10−1 (1.0 ± 0.0) × 10+0 M0.005S0.1 (4.4 ± 0.3) × 10−3 0.1 (6.4 ± 1.9) × 10−1 (6.1 ± 0.4) × 10−1 (1.0 ± 0.0) × 10+0 M0.01S0.1 (9.3 ± 0.5) × 10−3 0.1 (5.4 ± 1.0) × 10−1 (7.6 ± 0.2) × 10−1 (1.0 ± 0.0) × 10+0 M0.02S0.1 (1.9 ± 0.1) × 10−2 0.1 (5.3 ± 1.1) × 10−1 (7.5 ± 0.2) × 10−1 (9.8 ± 0.1) × 10−1 M0.05S0.1 (4.9 ± 0.2) × 10−2 0.1 (4.7 ± 0.8) × 10−1 (8.9 ± 0.6) × 10−1 (9.2 ± 0.1) × 10−1 M0.1S0.1 (1.0 ± 0.0) × 10−1 0.1 (4.2 ± 0.7) × 10−1 (9.1 ± 0.4) × 10−1 (8.9 ± 0.2) × 10−1 M0.2S0.1 (2.1 ± 0.1) × 10−1 0.1 (3.4 ± 0.6) × 10−1 (9.5 ± 0.4) × 10−1 (8.2 ± 0.3) × 10−1 M0.4S0.1 (4.3 ± 0.1) × 10−1 0.1 (2.5 ± 0.4) × 10−1 (8.6 ± 0.5) × 10−1 (7.3 ± 0.4) × 10−1 3 −1 Model (Resolution 256 ) M ζ (Emag/Ekin)sat Γ(ted )Esol/Etot M0.005S0.001 (6.3 ± 0.2) × 10−3 0.001 (2.0 ± 0.2) × 10−1 (8.8 ± 0.7) × 10−1 (6.0 ± 0.4) × 10−1 M0.01S0.001 (1.3 ± 0.1) × 10−2 0.001 (1.2 ± 0.2) × 10−1 (8.4 ± 0.3) × 10−1 (3.8 ± 0.2) × 10−1 M0.02S0.001 (2.4 ± 0.1) × 10−2 0.001 (7.3 ± 1.1) × 10−2 (5.7 ± 0.3) × 10−1 (2.0 ± 0.4) × 10−1 M0.05S0.001 (4.2 ± 0.3) × 10−2 0.001 (3.2 ± 0.5) × 10−2 (2.7 ± 0.1) × 10−1 (8.5 ± 1.1) × 10−2 M0.1S0.001 (9.8 ± 0.8) × 10−2 0.001 (1.5 ± 0.3) × 10−2 (2.5 ± 0.1) × 10−1 (5.0 ± 0.2) × 10−2 M0.005S0.01 (5.4 ± 0.2) × 10−3 0.01 (4.9 ± 0.5) × 10−1 (1.0 ± 0.0) × 10+0 (9.9 ± 0.0) × 10−1 M0.01S0.01 (1.7 ± 0.1) × 10−2 0.01 (4.1 ± 0.4) × 10−1 (1.6 ± 0.1) × 10+0 (8.5 ± 0.6) × 10−1 M0.02S0.01 (2.6 ± 0.1) × 10−2 0.01 (3.4 ± 0.4) × 10−1 (1.4 ± 0.1) × 10+0 (6.8 ± 0.6) × 10−1 M0.05S0.01 (4.7 ± 0.2) × 10−2 0.01 (2.4 ± 0.4) × 10−1 (9.4 ± 0.6) × 10−1 (5.0 ± 0.0) × 10−1 M0.1S0.01 (9.4 ± 0.5) × 10−2 0.01 (1.9 ± 0.3) × 10−1 (8.5 ± 0.6) × 10−1 (3.3 ± 0.3) × 10−1 Model (Resolution 5123) M0.01S0.01 (1.7 ± 0.0) × 10−2 0.01 (5.7 ± 0.4) × 10−1 (2.7 ± 0.2) × 10+0 (8.8 ± 0.1) × 10−1 M0.05S0.01 (4.6 ± 0.2) × 10−2 0.01 (1.8 ± 0.4) × 10−1 (1.4 ± 0.1) × 10+0 (5.7 ± 0.3) × 10−1 Model (Resolution 5763) M0.01S0.001 (1.3 ± 0.1) × 10−2 0.001 (9.5 ± 1.2) × 10−2 (1.3 ± 0.2) × 10+0 (3.1 ± 0.1) × 10−1 M0.05S0.001 (4.2 ± 0.3) × 10−2 0.001 (2.8 ± 0.5) × 10−2 (4.9 ± 0.7) × 10−1 (8.1 ± 1.1) × 10−2

TABLE I. Table of all the ideal-MHD simulations with the corresponding Mach number (M), solenoidal fraction (ζ) in the forcing of turbulent driving, saturation eﬃciency of the dynamo ((Emag/Ekin)sat), ampliﬁcation rate of the magnetic energy (Γ) and the solenoidal ratio in the kinetic energy (Esol/Etot). 7

3 −1 Model (Resolution 256 ) M ζ (Emag/Ekin)sat Γ(ted )Esol/Etot M0.005S0.001 (6.0 ± 0.2) × 10−3 0.001 (1.6 ± 0.2) × 10−1 (6.9 ± 0.4) × 10−1 (5.6 ± 0.4) × 10−1 M0.01S0.001 (1.2 ± 0.1) × 10−2 0.001 (1.2 ± 0.2) × 10−1 (5.2 ± 0.4) × 10−1 (2.8 ± 0.5) × 10−1 M0.02S0.001 (2.3 ± 0.1) × 10−2 0.001 (5.2 ± 0.7) × 10−2 (3.8 ± 0.1) × 10−1 (2.2 ± 0.3) × 10−1 M0.05S0.001 (4.0 ± 0.3) × 10−2 0.001 (1.7 ± 0.5) × 10−2 (1.1 ± 0.0) × 10−1 (8.4 ± 0.8) × 10−2 M0.1S0.001 (9.7 ± 0.8) × 10−2 0.001 (7.5 ± 1.9) × 10−3 (8.9 ± 0.6) × 10−2 (5.8 ± 1.3) × 10−2 M0.005S0.01 (5.1 ± 0.4) × 10−3 0.01 (4.6 ± 0.7) × 10−1 (7.7 ± 0.5) × 10−1 (9.6 ± 0.1) × 10−1 M0.01S0.01 (1.5 ± 0.1) × 10−2 0.01 (4.8 ± 0.9) × 10−1 (1.3 ± 0.1) × 10+0 (8.4 ± 0.4) × 10−1 M0.02S0.01 (2.4 ± 0.2) × 10−2 0.01 (3.0 ± 0.5) × 10−1 (9.6 ± 0.6) × 10−1 (6.9 ± 0.4) × 10−1 M0.05S0.01 (4.5 ± 0.2) × 10−2 0.01 (1.8 ± 0.3) × 10−1 (5.7 ± 0.3) × 10−1 (5.3 ± 0.4) × 10−1 M0.1S0.01 (9.3 ± 0.5) × 10−2 0.01 (1.3 ± 0.3) × 10−1 (5.3 ± 0.2) × 10−1 (3.9 ± 0.4) × 10−1

TABLE II. Table of all the non-ideal MHD simulations with Reynolds number, Re = 1500 and magnetic Prandtl number Pm = 2 on 2563 grid cells with the corresponding Mach number (M), solenoidal fraction (ζ) in the forcing of turbulent driving, saturation eﬃciency of the dynamo ((Emag/Ekin)sat), ampliﬁcation rate of the magnetic energy (Γ) and the solenoidal ratio in the kinetic energy (Esol/Etot).

MHD Simulations Sol.frac 0 Modified MHD Simulations Sol.frac 0 tion we solve accordingly and ﬁnd that the properties of the low-Mach number small-scale dynamo are consistent 1 10 to within 1-sigma with the results obtained from solving t

a the usual MHD equations (Figure 3). s ) n i 2 k 10 E / g

a SUPPLEMENTAL MATERIAL C m

E 3 ( 10 The small-scale dynamo amplification of seed magnetic fields present in the primordial plasma occurs in 10 4 the radiation-dominated early Universe. Wagstaff et al. 3 2 1 10 10 10 [33] discuss two mechanisms for generating turbulence in the early Universe: (i) Turbulence driven by primor- 10 1 dial density perturbations and (ii) Turbulence from first- order phase transitions, and they show that the kinetic t o t and magnetic Reynolds numbers in the early Universe E / l

o are higher than the critical values required for dynamo s 10 2 E action. The authors of the aforementioned study assume a Kolmogorov spectrum for the turbulence, however, the rapid growth and saturation of magnetic ﬁelds is attained in the early Universe independent of the nature of tur- 10 3 bulence as the kinetic and magnetic Reynolds numbers 10 3 10 2 10 1 are very high in the radiation-dominated Universe [45]. The rapid exponential ampliﬁcation of magnetic energy

FIG. 3. Saturation efficiency, (Emag/Ekin)sat (top panel) and is followed by a slower linear growth phase leading up solenoidal ratio, Esol/Etot (bottom panel) as a function of to the saturation of the SSD [36]. Neutrino damping Mach number for solenoidal fraction 0 in the turbulent driv- sets the viscous dissipation scale, which is the small- ing for our standard MHD simulations (on 1283 grid cells; est length scale at which turbulence can be maintained, black data points) and MHD simulations with the modified 3 in the radiation-dominated Universe (at temperatures, momentum equation (on 144 grid cells; blue data points). T ' 0.2 − 100 GeV). The ratio of the physical timescales The dotted black lines show the fits to the MHD simulation for the expansion (Hubble) time, τ , to the eddy turn- data to guide the eye. H over time at the neutrino damping scale, τl, for the primordial density perturbations are The relativistic ideal MHD equations for non- rms relativistic velocity fluctuations in co-moving coordinates τH 1 vl 4 −12 are equations (5)-(7), where the relativistic equation of = ∼ 2 × 10 (lc ' 3 × 10 pc,T ' 100 GeV) τl H (a lc) state, p = ρ/3, is used. These equations resemble the τH 3 −10 usual MHD equations (1)-(3) albeit with some constant and ∼ 5 × 10 (lc ' 10 pc,T ' 15 GeV). τl factors being introduced in the equations as the pressure of the relativistic plasma is significant compared to its Here the physical length scale for turbulent driving l = energy density. We have modified the momentum equa- alc, where lc is the comoving length scale and a is the 8 cosmological scale factor. In the case of energy injection in the high Prandtl number limit P m 1. Using this into the primordial plasma due to a first-order PT for Kolmogorov (ϑ = 1/3) and Burgers (ϑ = 1/2) turbulence, we find Γ ∝ Re1/2 and Γ ∝ Re1/3, respectively. τ H ∼ 3 × 107 for T ' 100GeV and τl τ H ∼ 107 for T ' 15GeV. τl We note that for length scales in between the neutrino damping scale and the largest possible driving Therefore, for a range of length scales (1 − 100) l, at scale (given by vrmsτH ), τH /τeddy ∝ a in the radiation- epochs close to T ' 100 GeV, the exponential ampli- dominated early Universe [33, 47]. For purely compres- fication and saturation is reached appreciably faster in sive turbulence driving (ζ = 0), we find the amplifica- the early Universe compared to what we find in our tion of magnetic energy, including the exponential and simulations at Re ' 1500 and the growth and satura- the slower linear growth phase, and the saturation of the tion of magnetic fields from the SSD action occurs on SSD occurs after approximately 500 turn-over times at time scales substantially smaller than the expansion of M ∼ 10−3 in our simulations with Reynolds number the Universe. Wagstaff et al. [33] also derive the net- Re ' 1500. However, the Reynolds numbers in the early amplification factor of the primordial magnetic fields and Universe are much higher than what we obtain in our conclude that amplification of magnetic fields until satu- simulations; Re ' 107 at T ' 100 GeV and Re ' 105 at ration is achieved by the turbulent dynamo in the early T ' 15 GeV [33]. Schober et al. [45] find the amplifica- Universe. While the growth rates are strongly dependent tion rate, Γ, of SSD-amplified magnetic fields for turbu- on the kinetic Reynolds number, the saturation levels are lence with different velocity scaling, following v(`) ∼ `θ, not, in the large magnetic Prandtl number limit, P m > 1 to be [4]. Thus, the saturation levels we find in our simulations provide reasonable estimates of the saturation levels that Γ ∝ Re(1−ϑ)/(1+ϑ), (8) would be obtained in the early Universe, where P m 1.

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