Magnetic fields are present at many levels of structure in the Universe. They have been observed in low redshift galaxies with strengths of O(10)  μG over galactic scales of O(kpc) [1,2]. On larger scales, O(1)  μG fields with O(Mpc) coherence lengths have been observed in galaxy clusters [3–5]. While the exact origin of magnetic fields over galactic length scales is not fully understood, a commonly accepted scenario is that they are the result of a cosmological seed field which gets exponentially amplified by the action of a dynamo operating during structure formation [6,7]. Even though no direct evidence exists for such extragalactic scale magnetic seed fields, this possibility currently cannot be excluded. The seed field hypothesis can accommodate varying strengths of magnetic fields over different length scales, since within a galaxy the seed is assumed to be strongly amplified compared to intracluster regions where the dynamo action is less efficient. There is a vast body of literature on the possible cosmological origin of large-scale seed fields, such as from inflation [8,9], phase transitions [10], axion electrodynamics [11], and magnetic field transfer from a hidden sector via kinetic mixing [12]—for a review see, e.g., Ref. [13]. These various realizations of the large-scale seed field hypothesis provide an economical cosmological explanation for the observation of magnetic fields across intergalactic, intracluster, and intercluster distances. However, since the existence of magnetic fields on all these length scales is itself still an open question, it is interesting to consider the possibility that magnetic fields of different coherence lengths have different origins.

This work investigates how very light, i.e., fuzzy, bosonic dark matter (DM) fields could contribute to the generation of galactic magnetic fields. Fuzzy dark matter (FDM) was originally proposed as a resolution to certain shortcomings found when comparing simulations of cold dark matter (CDM) [14,15] with observations, specifically dwarf galaxies with cores [16] and missing satellite galaxies [17–19]. The current understanding of these issues is that they are solved by improved observations [20] and including baryonic effects in galaxy formation simulations [21]. Nevertheless, FDM presents an interesting paradigm where DM could affect the large-scale structure of galaxies (see Refs. [22,23] for a review). Although FDM is typically modeled by an ultralight scalar field, which we define as having a mass of O(10-15)  eV or lighter, vector FDM models have also been studied [24,25]. These models also result in novel large-scale gravitational phenomena that can be distinguishable from scalar FDM, such as inducing deterministic timing delays in pulse arrival times detectable through pulsar timing arrays (PTAs) [26–31] and astrometry [32]. They also act as a long-range fifth force that can potentially modify the dynamics of binary systems [33].

By introducing nongravitational couplings through a kinetic mixing between the dark photon and the Standard Model (SM) photon [34], the dark photon can source observable magnetic fields, particularly on galactic and extragalactic scales [35–37]. Upon halo virialization in the early Universe, these dark photons can act as seed fields required for dynamo amplification. We will soon show that these ULDPs with masses of ≲O(10-21)  eV can naturally produce magnetic fields coherent over galactic scales. The fields are sourced in a similar way to well-known mechanisms for astrophysical galactic seed fields, which can be produced through the separation of ions and electrons in protogalactic plasma—an effect known as the Biermann battery [38]. This mechanism is distinct from the cosmological seed field hypothesis, which posits a seed field coherent over length scales larger than Mpcs that are generated well before structure formation.

In our scenario, magnetic fields are generated only after an overdense region of the universe virializes to form a protogalaxy. The overall strength of the ULDP seed field is set by fraction of DM energy density of the protogalaxy in the form of ULDPs,¹

For reasons that will be explained later, we will generally assume that DM is an admixture of CDM and ULDPs, the latter not comprising more than 1% of all the DM in the Universe.

The remainder of this work is organized as follows. In Sec. II we introduce ULDPs and show how their coupling to SM photons can give rise to electric and magnetic fields. In Sec. III we consider magnetic fields from ULDPs during the formation of galaxies, in which they can provide the seed field which would eventually be amplified through the dynamo mechanism. In Sec. IV we make concluding remarks and discuss limitations and future considerations. Throughout this paper, we use natural units where ℏ=c=kB=1.

The model we consider requires a simple extension of the SM by a U(1) symmetry, which means that it has another abelian gauge boson, A′μ—the ULDP—in addition to the gauge bosons in the SM. For dynamical processes occurring in the sub-GeV range, the physical A′μ field mixes predominantly with the SM photon Aμ, and the resulting Lagrangian can be written as L=-14FμνFμν-14Fμν′F′μν+mγ′2Aμ′A′μ-e(Aμ+εAμ′)Jμ.(1)Here ε is the kinetic mixing parameter, mγ′ the mass of the dark photon, and Fμν=∂μAν-∂νAμ is the usual field strength tensor for the photon with Fμν′ similarly defined for the dark photon. We work in the canonical mass basis, where the mixing with the Z boson is negligible for ULDPs. The interaction part of this Lagrangian consists of the photon and the dark photon, both coupled to the electromagnetic 4-current Jμ, with the latter coupling suppressed by a factor of ε. We comment that although other couplings between dark photons and the SM are possible (for example through B or B-L couplings), we only consider the kinetic mixing in this work.

Many possibilities have been studied for the cosmological production of ULDPs that ultimately comprise the DM of the universe [24,35,39–42]. Since we are interested in late-time effects, the particular realization in question is not of significant importance to us. However for concreteness, we will focus on the well-studied misalignment mechanism for vector DM [24,35], which we now give a brief account of, providing some comments on its difficulties for producing 100% of the DM for the mass range we are interested in [35,39] toward the end of this section.

Assuming the Universe is described on large scales by a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric, ds2=dt2-a2(t)dx2,(2)where a(t) is the scale factor of the Universe, the equation of motion for a nonrelativistic dark photon field is given by d2A′dt2+3HdA′dt+mγ′2A′=0,(3)where H≡a˙/a is the Hubble parameter and we have neglected the dark photon momentum and any backreaction of charged baryonic matter on the dark photon field. The latter is justified for the dark photon mass range and small value of the kinetic mixing parameter we consider in this work. In particular, for dark photons masses below 10-17  eV, the in-medium conversion of dark photons to photons that can deplete the abundance of dark photons is negligible since the plasma frequency of the medium never resonantly matches the dark photon mass. In the early Universe, H(t)≫mγ′ and the field evolution in Eq. (3) is dominated by the Hubble friction term. The field is overdamped and frozen, such that dA′/dt≈0. In this case, the ULDP field does not oscillate: it remains at a fixed value until it can enter the horizon, which occurs when the Hubble term becomes comparable to the dark photon mass, H(t)≃mγ′. Afterward, the dark photon field begins to oscillate and the different modes of this oscillation represent a population of nonrelativistic massive vector bosons that constitute (at least partially) the DM. For a benchmark mass of 10-21  eV, the transition from a over-damped and frozen field to an underdamped and oscillatory once occurs at a redshift z∼O(106), or when the temperature of photons was about 200 eV. This is still during the radiation-dominated era, albeit well after big bang nucleosynthesis (BBN) and when electrons have become nonrelativistic.

Once H<mγ′, since the Hubble friction term is a decreasing function of time, it becomes subdominant and Eq. (3) can be solved using the WKB approximation under the initial condition dA′/dt|t=tm=0 to give A′(t)=(ama)3/2Am′cos(mγ′(t-tm)),(4)where tm is the time at which the misalignment condition is fulfilled, am=a(tm). The coherence length ℓc of this field is given by ℓc=2πmγ′v,(5)where v≡|v| is the amplitude of the velocity vector. The corresponding electric and magnetic fields are defined in the usual way, E′≡-∇A′0-A˙′B′≡∇×A′.(6)The amplitude of the field is set by the requirement that the energy density of the ULDPs match the corresponding fraction f of the DM energy density ρDM. At the misalignment time and over a length scale ℓc, 12mγ′2|Am′|2(1+zm)-3≃fρDM(z),(7)where zm is the value of the redshift at misalignment [we have replaced the scale factor with redshift using a(t)=1/(1+z)]. The electric field over a coherence length is then given by E′=E0(z)sin(mγ′(t-tm)),(8)with |E0(z)|≡E0(z)=2fρDM(z).(9)The magnetic field, which is not present at zeroth order in the DM velocity, is given by B′=vE0(z)cos(mγ′(t-tm)),(10)where v is the velocity of the dark photon field. To obtain this expression, we note the fact that taking the curl of the A introduces a characteristic length scale ∼1/ℓc.

We will assume throughout this work that the DM consists primarily O(99)% of CDM and that the density of ULDPs closely follows the density of CDM, such that ρULDP(z)=fρCDM(z),(11)with f∼1%. Equation (11) should be a valid approximation for such small f as long as the dynamical scales are larger than the coherence length. Although one may expect these ULDPs to have a significantly different picture of structure formation, particularly on small scales [43], we assume that their evolution is predominantly determined by the more-abundant CDM component. In general, considering a smaller fraction of ULDPs is expected to relax constraints which arise from kinematical observations of, e.g., dwarf galaxies [44], since the gravitational potential is dominated by the abundant component of DM.

Indeed, the behavior of this field over lengths much greater than ℓc, or times larger than the coherence time, τc≃2π/mγ′v2 is much more complicated. In particular, for virialized dark photons, the velocity distribution induces a phase decoherence at times t≫τC, when interference between different modes become relevant and cause the net field to fluctuate stochastically. When considering cosmological structure, one must also take into account the spatial decoherence over length scales larger than ℓc. Such a complete study incorporating all these effects is only possible with a proper cosmological simulation, which is beyond the scope of this work. We will therefore assume, for simplicity, that the dark photon has a mass of roughly ∼10-21  eV, corresponding to de Broglie wavelength of O (kpc), and represents a minute fraction of the DM, as previously mentioned. In such a scenario, the dark electric and magnetic fields are well described by Eqs. (8) and (10) over the length and timescales relevant for this work.

Before closing this section, we offer some additional remarks regarding our assumption that f∼(1%). We assume that the ULDPs only provide a fraction of the observed dark matter density, because this assumption addresses several details: (i) Constraints on the kinetic mixing ε are generally relaxed due to ULDPs not making up all of the DM. For instance, note that Ref. [45] found that Lyman-α constraints on ultralight bosons are lifted for f≲0.2. (ii) As previously mentioned, the dynamics of halo formation and evolution for ULDPs are expected to follow the dynamics of the more-abundant CDM component. (iii) Finally, the difficulties in producing a sufficient abundance of low mass dark photons [35,39,46] can be alleviated if it is not all of the DM.

The mechanism we propose for seeding magnetic fields from ULDPs can be summarized as follows: after the Universe is populated with cold ULDPs that behave like a background dark electromagnetic field, by virtue of the Lorentz force law, charged particles in the cosmic fluid are displaced by the dark electric and magnetic fields. Because of the mobility difference between electrons and ions, a net current is sourced which generates a magnetic field from Ampère’s law. As the dark magnetic field depends on the DM velocity, which is random prior to virialization, no net magnetic field can be generated before that point. The idea is similar to that first proposed by Refs. [47,48] for generating primeval magnetic fields during the radiation domination era through vorticity. This was later applied to rotating protogalaxies postrecombination [49]. As we will show, the presence of the dark photon field can generate a seed field comparable to that generated through vorticity effects in a protogalaxy.

In this work, we have studied how the observed galactic-scale magnetic fields today can potentially arise from a subdominant, ultralight, dark vector boson that kinetically mixes with the Standard Model photon. We demonstrated that ULDPs can source a seed magnetic field while the galaxy virializes, which can eventually be amplified to explain the galactic magnetic fields observed today. Although the complexity of galactic dynamos requires simulations for a complete picture and is therefore an ongoing effort, we have shown that ULDPs can produce seed fields comparable to other astrophysical seed fields, such as one arising from the Biermann battery [38]. While we have focused primarily on the possibility that the seed fields from ULDPs are sufficient for dynamo amplification to observed field values, another possible scenario is where the magnetic field from ULDPs is a contributing field in the initial seed fields.

In calculating the magnetic field from ULDPs, we note that the observable magnetic field amplitude is highly sensitive to the free electron fraction Xe due to plasma screening, and therefore whether a magnetic field from ULDPs can solely be the seed field required for the dynamo effect is also dependent on Xe. Although the usual assumption is that the free electron fraction prior to reionization is around Xe∼10-4, more general model-independent constraints on Xe from the CMB [62,63] typically place upper bounds on Xe. While suppressing the abundance of free electrons seems to be difficult in standard cosmological scenarios [64], it is nevertheless an interesting possibility (e.g., [65]). The upcoming Square Kilometer Array (SKA) [66] is expected to shed light on the epochs of reionization and cosmic dawn, which will help solidify the constraints on the allowed free electron abundance at high redshifts.

Finally, we comment that in order to truly model the effects of dynamo amplification, simulations will be necessary. Furthermore, the amplification of the seed field can be significantly different in the case of small-scale and large-scale dynamos [13,67]. While in this work we assumed a simplified amplification due to a large-scale dynamo, it would be interesting to see if the seed fields from ULDPs can also be sufficiently enhanced in other dynamo scenarios. We leave these considerations for future studies.