On the cosmological abundance of magnetic monopoles

We demonstrate that Debye shielding cannot be employed to constrain the cosmological abundance of magnetic monopoles, contrary to what is stated in the previous literature. Current model-independent bounds on the monopole abundance are then revisited for unit Dirac magnetic charge. We find that the Andromeda Parker bound can be employed to set an upper limit on the monopole flux at the level of $F_M\lesssim 5.3\times 10^{-19}\,\text{cm}^{-2}\text{s}^{-1}\text{sr}^{-1}$ for a monopole mass $10^{13}\,\text{GeV}/c^2\lesssim m\lesssim 10^{16}\,\text{GeV}/c^2$, which is more stringent than the MACRO direct search limit by two orders of magnitude. This translates into stringent constraints on the monopole density parameter $\Omega_M$ at the level of $10^{-7}-10^{-4}$ depending on the mass. For larger monopole masses the scenarios in which magnetic monopoles account for all or the majority of dark matter are disfavored.


Introduction
Magnetic monopole is one of the most fascinating objects in theoretical physics [1][2][3][4][5][6][7][8][9][10].It stems from the natural requirement of extending the symmetry between electric and magnetic fields beyond the source-free Maxwell's equations.Its consistency with quantum mechanics is demonstrated in 1931 by Dirac, which has led to an intriguing explanation for the charge quantization observed in nature [11].In 1974 't Hooft and Polyakov demonstrated that magnetic monopoles necessarily appear as topological defects in particle theories of grand unification [12,13], with the topological charge identified with the magnetic charge.It was soon realized that the cosmological abundance of magnetic monopoles poses a serious problem for such scenarios [14][15][16][17], the solution of which gave a strong boost to the birth of the inflation theory [18,19].Because magnetic monopole is deeply connected with the fundamental symmetries of nature at very high energy, the discovery of even a single magnetic monopole would have far-reaching implications 1 .It is thus intriguing that the recent Amaterasu cosmic ray particle found by the Telescope Array Group [24] might be interpreted as a magnetic monopole with extremely high energy [25,26].
Although it is possible that cosmic inflation has diluted away magnetic monopoles so that they end with an abundance too small to have observable consequences, it is not necessarily the case [1,27].Relatively light magnetic monopoles might be produced after inflation through various mechanisms [28][29][30][31][32][33][34][35][36] so that their abundance is not affected by the dilution.Even if for superheavy magnetic monopoles that are indeed affected by inflation, their final abundance still depends on its initial value and the actual amount of dilution and which can be quite model-dependent and not well-constrained [37][38][39][40][41][42].Alternative scenarios to inflation could also be envisioned [43].Therefore we choose to be open-minded and inquire what model-independent constraints can be obtained on the cosmological monopole abundance, which is taken as a free parameter determined by some unknown production mechanism in the very early universe.Here by "model-independent constraints" we refer to constraints that can be derived from the electromagnetic and gravitational properties of the magnetic monopole only.Other properties that depend on the model detail (such as catalysis of baryon number violation [44][45][46]) will not be assumed.
Recently it has been suggested in ref. [47] that Debye shielding in the magnetic monopole plasma leads to significant constraints on the cosmological monopole abundance, especially for superheavy monopoles which are not well-constrained by conventional means.Requiring the Debye length of the magnetic monopole plasma be larger than the scale of the largest coherent magnetic structure so far observed, ref. [47] found that the cosmological density parameter of the magnetic monopole should satisfy Ω M ≲ 3 × 10 −4 .This would certainly exclude the scenario in which magnetic monopoles account for all of the dark matter, and would also be more stringent than the conventional Parker bound for relatively heavy nonrelativistic monopoles.It was expected that future detection of magnetic fields of even larger coherent length would improve this bound by a few orders of magnitude.
In this work we shall however demonstrate that the Debye shielding bound suggested by ref. [47] cannot be used to obtain meaningful bounds on the cosmological monopole abundance.The reason is that, as we shall see, the concept of Debye shielding simply does not hold for magnetic monopole plasma in the presence of magnetic fields that are not curl-free.A similar issue in the context of solar physics was discussed in Appendix B of ref. [48], where the authors commented that "Debye shielding will not shield out a transverse electric field."In our analysis we note that Debye shielding is implicitly based on the requirement of thermal and mechanical equilibrium2 which leads to inconsistencies in the magnetic case with test electric currents.
For non-relativistic magnetic monopoles that do not catalyze baryon number violation, the existing bounds on their cosmological abundance then become quite limited.We revisit the mass density constraint, the bound from direct search in cosmic rays and the conventional Parker bounds.We find that the Andromeda Parker bound which was once proposed to constrain the abundance of magnetic black holes [49], can be employed to constrain the abundance of ordinary magnetic monopoles effectively.
This paper is organized as follows.In Sec. 2 we demonstrate the problem with Debye shielding in the magnetic monopole plasma, which implies that in an astrophysical context no meaningful bounds can be obtained from this effect.We then turn to current modelindependent bounds on cosmological monopole abundance in Sec. 3, disregarding the bound from Debye shielding.We present our discussion and conclusions in Sec. 4.
2 Problem with Debye shielding in the magnetic monopole plasma

Debye shielding in an electron-proton plasma
We first give a concise review of Debye shielding in an ordinary (electron-proton) plasma, paying special attention to the assumptions underlying the derivation.Following ref. [50], a fixed test charge Q is introduced into a plasma of electrons and protons at temperature T .It is expected that the plasma particles will redistribute in space until an equilibrium is reached.Suppose the equilibrium state is characterized by a stationary proton number density distribution n p (r) and a stationary electron number density distribution n e (r), and also an electrostatic potential Φ(r).Taking the location of the test charge Q to be the origin, the Poisson equation for Φ(r) reads where e > 0 denotes the proton charge and the equation is written in Gaussian units 3 .
When the equilibrium at temperature T is reached, the number density distributions n p (r) and n e (r) should be controlled by the corresponding Boltzmann factor, that is ) Here k B denotes the Boltzmann constant, and n denotes the proton (or electron) number density in the plasma at spatial infinity, which is just the mean proton (or electron) number density averaged over a sufficiently large volume.We may now plug Eqs.(2.3) into Eq.(2.1) for a solution of Φ, assuming spherical symmetry so that Φ is a function of the radial distance r only.Analytical solutions can be found by considering eΦ ≪ k B T , so that The linearized Poisson equation then becomes which has a spherically symmetric solution with the Debye length λ D given by The solution Eq. (2.6) has the desired property that (assuming eΦ ≪ k B T still holds) while the appearance of the exponential suppression factor e − √ 2r/λ D implies that the electric field of the test charge Q is screened beyond r ∼ O(λ D ).In an intuitive picture the test charge Q repels plasma particles of the same sign while attracts plasma particles of the opposite sign, leading to a net charged cloud of the opposite sign around Q, screening electric fields beyond a few Debye lengths.

Potential issues for Debye shielding in the magnetic monopole plasma
At first sight the above considerations carry over to the case of the magnetic monopole plasma (composed of monopoles and anti-monopoles) without difficulty, the only changes coming from the magnetic coupling strength and the monopole mass.In fact this is true for test magnetic charges put into a quasi-neutral magnetic monopole plasma.Nevertheless, in a cosmological or astrophysical setting, magnetic fields are likely to originate from electric currents rather than magnetic charges.For example, it is demonstrated in ref. [51] that the magnetic field of our own Galaxy is not associated with monopole oscillations.Therefore, to place bounds on monopole abundance, one should consider test electric currents rather than test magnetic charges.Now, both electric currents and magnetic charges are able to produce magnetic fields.However, they produce magnetic fields of distinct mathematical properties.Electric currents are associated with magnetic fields that are divergencessless but not curl-free, referred to as "transverse magnetic fields" hereafter.On the other hand, magnetic charges are associated with magnetic fields that are curl-free but not divergenceless, referred to as "longitudinal magnetic fields" hereafter.The key problem is therefore analyzing Debye shielding in the magnetic monopole plasma in the presence of transverse magnetic fields produced by test electric currents.In this context, a number of issues arise compared to Debye shielding in an ordinary electron-proton plasma: 1. Nonexistence of a scalar potential.
The derivation of Debye shielding in an electron-proton plasma relies on the use of the electrostatic potential Φ, without which one cannot use a Boltzmann factor exp[∓eΦ/(k B T )] to characterize the equilibrium proton and electron number density distribution.However, the counterpart of Φ, which should be a magnetic scalar potential, cannot be introduced since the magnetic fields from electric currents are not curl-free.One might argue that a magnetic scalar potential can still be introduced in a simply connected region4 of space free of electric currents.However, this is certainly unsatisfactory if one considers electric current densities that fill out a finite volume.This is also unsatisfactory when the region of space free of electric currents is not simply-connected and thus needs to be covered by two or more simply-connected regions.A single-valued magnetic scalar potential cannot be assigned, prohibiting the use of a Boltzmann factor.
2. Screening of fields of distinct mathematical properties.According to the discussion above, magnetic charges produce longitudinal magnetic fields while electric currents produce transverse magnetic fields.Debye shielding then require transverse magnetic fields be screened by longitudinal magnetic fields, which is quite counterintuitive.
3. Lack of an intuitive physical picture.Debye shielding in an electron-proton plasma can be intuitively understood as originating from attractive forces between opposite-sign charges and repulsive forces between same-sign charges.However, in the case of magnetic monopole plasma with test electric currents, no such simple intuitive understanding is known.4. Two different interpretation of magnetic field neutralization.Debye shielding provide an alternative interpretation of magnetic field neutralization in the presence of magnetic monopoles, which is different from the usual interpretation underlying the Parker bound [52].The conventional Parker bound relies on the fact that magnetic monopoles drain energy from galactic magnetic fields.The complete physical picture should be understood as monopole plasma oscillation with strong Landau damping [53].This is distinct from the Debye shielding interpretation, leading to different predictions of magnetic field neutralization.
Although the above four issues strongly question the validity of the concept of Debye shielding in a magnetic monopole plasma with test electric currents, they do not lead to definite conclusions.

Anatomy of the magnetic monopole plasma with a two-fluid model
In order to present a solid argument, let us show that the assumption that Debye shielding is achieved in a magnetic monopole plasma with test electric currents leads to inconsistencies, and thus cannot be used to obtain a meaningful constraint on the cosmological monopole abundance.We start by considering the magnetic monopole plasma using a twofluid model, with one component fluid composed of magnetic monopoles, and the other component fluid composed of magnetic anti-monopoles.This means that the two fluids can be characterized via the usual fluid equations and equations of state.Such a two-fluid treatment is routine practice in the context of electron-ion plasmas in laboratory and space, while in the case of a magnetic monopole plasma one might worry about the validity of such a fluid approximation.In ref. [47] the number of magnetic monopoles within a Debye sphere is estimated to be much larger than unity in parameter space of phenomenological relevance 5 , so collective effects associated with the monopoles and anti-monopoles should be significant.On the other hand, the magnetic monopole plasma is found to be highly collisionless [47], since the collision frequency due to magnetic Coulomb scattering is very small.Whether collisionless systems can be treated within a fluid approximation is a subtle issue.Zero collision frequency implies an inifinite mean free path, leading to absurd results of divergent coefficients of viscosity and heat conduction.There are known circumstances in which a fluid approximation is inadequate for capturing the essential physics of such collisionless systems [54,55]. 5Using notations and some benchmark values in Sec. 3, we may compute the number of magnetic monopoles within a Debye sphere as , which is certainly much larger than unity in all parameter space of interest.
In the problem at hand the essence of the fluid approximation is the use of a temperature and an equation of state (based on an isotropic pressure) to characterize the magnetic monopole plasma.Then the fluid equations can be simply obtained from integrating the collisionless Boltzmann equation (and its first velocity moment) over the monopole velocity distribution.The equation of state plays the role of a closure relation so that we do not need to consider higher moments of the collisionless Boltzmann equation [50].The notion of a temperature and the associated equation of state seems to hinge on a state of thermal equilibrium that can only be achieved through binary collisions.On the other hand, astrophysical plasmas are often collisionless on scales of interest, and collisionless (astrophysical) plasmas are routinely observed or assumed to exhibit collisional, fluid-like behaviors (e.g.'Langmuir's Paradox') [56].It is conceivable that a number of mechanisms exist to achieve collisionless thermalization.There can be wave-particle scattering due to plasma instabilities. 6For self-gravitating systems phase mixing and violent relaxation can settle the system quickly into a stationary state [57].Though out of thermal equilibrium in the strictest sense, in the context of dark matter physics it is possible [58,59] that the dark matter velocity distribution can be described by Tsallis distributions derived from nonextensive statistical mechanics, which permits the notion of a temperature [60].
As to the magnetic monopole plasma, although the effect of Coulomb collisions between monopoles is negligible, motion of magnetic monopoles is subject to the influence of the environmental gravitational field and wave-particle interaction which are yet to be studied.There are two possibilities regarding the outcome of these collisionless thermalization mechanisms.The first possibility is that they are inadequate to bring the magnetic monopole plasma into a state of quasi-equilibrium7 that can be characterized by the notion of a temperature and the associated equation of state.Since the concept of Debye shielding hinges on such a state of quasi-equilibrium and the notion of a temperature, in this first possibility Debye shielding cannot be used to yield a meaningful constraint on the cosmological monopole abundance, so the plasma constraint obtained by ref. [47] does not apply.The second possibility is that the collisionless thermalization mechanisms can bring the magnetic monopole plasma into a state of quasi-equilibrium that can be characterized by the notion of a temperature and the associated equation of state.In this work, we will however show that even if this second possibility is realized, Debye shielding still cannot be used to yield a meaningful constraint on the cosmological monopole abundance.
Keeping the second possibility in mind, we therefore proceed with the two-fluid model, and write down the Euler equations for the magnetic monopole fluid and the anti-monopole fluid: ) (2.10) In the above two equations, m is the mass of the magnetic monopole (which is the same for the magnetic anti-monopole), g is the magnetic charge of the monopole (so the antimonopole carries magnetic charge −g).n + , u + , P + are the number density, velocity, and the pressure of the magnetic monopole fluid, with the corresponding quantities for magnetic anti-monopole fluid denoted by a subscript − instead of +.Ψ is the gravitational potential which we put in for generality.B, E are the macroscopic magnetic and electric field strengths, respectively.There are also equations of continuity for the two fluids: Furthermore, the macroscopic mganetic and electric fields should satisfy Maxwell's equations in the presence magnetic charge and currents: ∇ × B = 4πJ e + ∂E ∂t . (2.16) Here ρ e , J e are the electric charge and current densities respectively, while ρ m , J m are the magnetic charge and current densities respectively.Note that ρ m , J m can be expressed in terms of n + , n − , u + , u − as follows: while J e is related to the macroscopic fields via Ohm's law: in which v is the fluid velocity associated with the electric currents 9 , and σ denotes the electric conductivity which may be a function of the position and time.Finally, there should be equations of state which express the pressure as a function of the number density and temperature: If the monopole and anti-monopole fluids can be viewed as dilute gases, these equations of state should be well-approximated by the ideal gas law P = nk B T .Here we allow for general equations of state of the Onnes type The key assumptions underlying the above system of equations include: 1.The magnetic monopole plasma can be viewed using a two-fluid model, with one fluid composed of magnetic monopoles and the other fluid composed of magnetic anti-monopoles.The two fluids interact with each other only through macroscopic electric and magnetic fields.
2. The fluid velocities are assumed to be non-relativistic.
Similar assumptions are made in the usual treatment of electron-proton plasma in a twofluid model [50].Besides the subtlety discussed at the beginning of this subsection, the first assumption implies that we are effectively working at scales much larger than the mean separation between magnetic monopoles and the magnetic monopole plasma is not very dense so that the monopole annihilation can be neglected.
The system of equations from Eq. (2.9) to Eq. (2.21) can then be solved given appropriate initial or boundary conditions.Since we wish to discuss Debye shielding in analogy with the corresponding effect in an ordinary electron-proton plasma, let us consider an equilibrium configuration characterized by and ) Moreover, in an astrophysical setting we will make the following simplified assumption: the electric conductivity σ is very large while E + v × B ≃ 0 so as to maintain a given finite J e .Then the two Maxwell's equations associated with the magnetic field become and the two Euler equations for the monopole and anti-monopole fluids become Eqs. (2.26) and (2.27) simply imply that for n + − n − and J e that tend to zero sufficiently rapidly at infinity (which is true for discussing Debye shielding), the magnetic field strength can be expressed as (according to the Helmholtz theorem) On the other hand, the Euler equations (2.28) and (2.29) can be cast as ) Now, for equations of state which are of the Onnes type (c.f.Eq. (2.22)), one may write Eqs. (2.31) and (2.32) thus become ) (2.36) If we take the curl of these two equations, the left-hand side becomes 4πgJ e while the right-hand side vanishes, thus leading to inconsistencies.We therefore conclude that the concept of Debye shielding does not hold for the astrophysical magnetic monopole plasma.The above argument should hold obviously for volume current density J e which is nonzero in a macroscopic volume across which the two-fluid description is valid for the magnetic monopole plasma.One might want to check the more singular case when the test electric currents are given as a surface current density or simply a line current.In such cases the length scale associated with the electric currents is smaller than the minimum scale at which the fluid equations are supposed to hold.Nevertheless, one can still find solid arguments that invalidate the concept of Debye shielding.For example, consider a steady line current running through a loop L. Now the region R 3 \L is not simply connected, prohibiting the use of a single-valued magnetic scalar potential even if ∇ × B = 0 inside this region.However, this problem can be circumvented if we choose an arbitrary finite surface S bounded by L. Then the region R 3 \S is simply connected, which allows us to introduce a single-valued magnetic scalar potential Φ m .By the integral form of Ampère's law, Φ m should exhibit a discontinuity across the surface S.This discontinuity should be proportional to the electric current I running through the loop L. The existence of such a discontinuity does not pose problem for defining the magnetic field strength B, as the value of B at the surface S can be found by considering a different surface S ′ bounded by L. However, it does pose a problem for discussing Debye shielding as we need to use the Boltzmann number density distributions such as (2.37) with nm being the mean number density of the magnetic monopole (or aniti-monopole).These Boltzmann number density distributions are solutions to the steady state Boltzmann equations, and the discontinuity in Φ m across the surface S results in discontinuities in the number density distributions n ± across S. Now the inconsistency is manifest: The surface S can be chosen arbitrarily and thus by choosing a different surface bounded by L one ends up with a different continuity behavior for n ± .This is certainly not acceptable since n ± is physical.Moreover, discontinuity in n ± across the surface S in general also predicts a discontinuity in the pressure P ± , leading to problem with mechanical equilibrium if we consider a small volume of fluid across the surface.
Inconsistencies also arise if we consider a surface current density flowing through a surface S 1 of finite area and bounded by a loop L 1 .The region R 3 \S 1 is now simply connected, in which we can introduce a single-valued magnetic scalar potential Φ m .However, if we construct an Ampèrian loop that crosses S 1 once at some point P , then according to the integral form of Ampère's law, Φ m will in general exhibit a discontinuity across the surface S 1 at P .The amount of discontinuity depends on the total electric current enclosed by the Ampèrian loop.According to the Boltzmann number density distributions, the discontinuity in Φ m results in discontinuity in n ± , which in turn results in discontinuity in the pressure P ± .The pressure discontinuity implies that the mechanical equilibrium cannot be maintained if we consider a small volume of fluid across the surface S 1 around the point P .This argument can be easily extended to the case that S 1 is a closed surface (such as the surface of a ball).
We therefore conclude that a stationary configuration of fluids and fields cannot be maintained at thermal and mechanical equilibrium under the assumptions of Eqs.(2.23)--(2.25).These assumptions are obviously natural if we require magnetic Debye shielding be in parallel with its electric counterpart.One assumption that one might wish to give up is the vanishing of the fluid velocities Eq. (2.23).If we allow for time-independent but spatially inhomogeneous fluid velocity fields u ± , we may get extra terms in Euler equations which could save their compatibility with the Maxwell's equations.That is, Eqs.(2.35) and (2.36) become ) Taking the curl of these equations will not lead to inconsistencies.Furthermore, equations of continuity should be satisfied Note that since B should be viewed as given by Eq. (2.30), Eqs.(2.39)--(2.42)then become a system of 8 equations for 8 functions u + , u − , n + , n − , once the test electric current J e is specified.Given the compatibility and independence of these equations, we expect a unique solution for a given J e .However, if we attempt to construct the magnetic current density J m according to Eq. (2.18), in general we would expect a nonzero J m .By the electric Ampère's law Eq.(2.15) this generates an electric field which decays in a powerlaw manner (rather than exponentially) at spatial infinity.Since a large-scale electric field is incompatible with current observations, we conclude such a solution is not of practical interest and thus nonzero fluid velocities do not save the concept of Debye shielding in an astrophysical context.Let us comment that if we introduce an ideal test magnetic dipole (rather than a test electric current loop enclosing a finite area) into the magnetic monopole plasma, the concept of Debye shielding can be made valid.Suppose the test magnetic dipole is located at some point P 1 , then the region R 3 \P 1 is simply connected, which allows the introduction of a single-valued magnetic scalar potential.The number density distributions then follow a well-defined Boltzmann rule, with no ambiguities or unacceptable discontinuities.For a test electric current loop enclosing a finite area the same argument does not go through.

Revisiting current bounds on the cosmological monopole abundance
In this section we revisit current bounds on the cosmological monopole abundance.We shall consider magnetic monopoles with a Dirac charge g D = 1 2e and mass m ≳ 10 11 GeV so that they have not been accelerated to relativistic velocities by magnetic fields in galaxies or galaxy clusters.Since it is demonstrated that Debye shielding cannot yield a meaningful constraint, we shall collect and compare the following bounds: 1. Mass density constraint.
2. Direct search in cosmic rays.

Parker bounds.
All these bounds have been discussed to various extent in previous literature [8-10, 27, 62, 63]. 10 The emphasis of the discussion here is to examine whether magnetic monopoles may account for all (or the majority) of the dark matter, and the assumptions and uncertainties involved in the analysis.Moreover, motivated by previous studies on magnetic black holes [49,66,67], we will improve the existing Parker bound by considering magnetic fields from the Andromeda galaxy.
Bounds on the cosmological monopole abundance can be represented in various manners.In many cases one considers magnetic monopoles (including anti-monopoles) populated with the number density n M and an isotropic average velocity v in some volume defined in a reference frame S.Here the reference frame and the volume depend on the type of local measurement under consideration.For example, for direct search in cosmic rays, the reference frame is fixed on the Earth and the volume is associated with the size of the solar system, while for the Parker bound from the galactic magnetic field, the reference frame is fixed on the Milky way galaxy and the volume is associated with the galactic magnetic field region in the Milky way.The local isotropic flux of magnetic monopoles is defined as Eq. (3.1) means that if one considers an area element dS with its normal direction n, then during a short time dt the number of monopoles (and anti-monopoles) that cross dS with velocity direction in a small solid angle dΩ around n is given by F M dtdSdΩ.Therefore the unit of F M is written as cm −2 s −1 sr −1 .To gain an intuitive feeling of the flux, we may compute the mean separation of magnetic monopoles as a function of F M and v as which implies monopoles saturating the flux upper limit of the MACRO experiment [68] (see Eq. (3.14) below) and moving with a virial velocity v ∼ 10 −3 c the mean separation is about 300 km.Alternatively, the cosmological monopole abundance is embodied by the density parameter Ω M of magnetic monopoles (including anti-monopoles), which is defined as Here ρM denotes the energy density of magnetic monopoles averaged on cosmological scales, and is the current critical density of the universe [27], with h = 0.674 being the scaling factor for Hubble expansion rate [69].In order to discuss the conversion between F M and Ω M , we introduce γ ≡ (1−v/c) −1/2 which is the Lorentz boost factor of the magnetic monopoles measured in the reference frame S associated with the local measurement under consideration, and ρ M = n M mc 2 γ which is the local energy density of the magnetic monopoles in the S frame.The conversion between F M and Ω M is then given by ) where the overdensity ∆ is defined by Strictly speaking ρM is defined in the rest frame of the cosmic microwave background (CMB), while ρ M is defined in the S frame.In practice the relative motion of S with respect to CMB is always non-relativistic, and thus we ignore this small difference.Moreover, the use of the relation does not coincide with the velocity that appears in Eq. (3.1) due to an extended velocity distribution.Again we ignore this small difference as we work in the non-relativistic regime.
We will consider two benchmark values of ∆.The first is ∆ = 1, applicable when magnetic monopoles are distributed uniformly in the universe.Such a uniform distribution is possible only if magnetic monopoles are accelerated by cosmic or astrophysical magnetic fields so that they do not bind with galaxies or galaxy clusters.The second is ∆ = 2.86 × 10 5 , which is the ratio between the local dark matter density ρ ⊙ = 0.36 GeV cm −3 [70] and the average dark matter density of the universe ρ DM = 1.26 × 10 −6 GeV cm −3 [27].The use of ∆ = 2.86 × 10 5 is based on the assumption that magnetic monopoles cluster in the same way as dark matter, and even with this assumption in mind the value ∆ = 2.86 × 10 5 is strictly speaking only applicable to direct searches in cosmic rays which indeed depend on the local monopole density.For Parker bounds derived from astrophysical magnetic fields the relevant region is on a much larger scale and thus could be associated with a different ∆.For Parker bounds based on the magnetic field of the Milky Way we ignore this difference for simplicity.Based on the above considerations, we will express ∆ as a simple function of the monopole mass 2.86 × 10 5 , m > 10 18 GeV/c 2 . (3.8) Here the boundary between clustered and unclustered monopoles is taken to be m = 10 18 GeV/c 2 , as suggested by ref. [53,63] as an approximate estimate.In practical situations we shall expect a smooth transition from ∆ ≃ 1 at large monopole mass values to ∆ ≃ O(10 5 ) to small monopole mass values, with the transition range spanning 2 ∼ 3 orders of magnitude.Since a definite prediction of the transition behavior is lacking, we opt for a simple step function as Eq.(3.8), keeping in mind the large uncertainties in ∆ associated with the transition range.The conversion between F M and Ω M critically depends on the velocity v of the magnetic monopoles.For monopoles clustered with the galaxy (m ≳ 10 18 GeV/c 2 ), v ≃ 10 −3 c which is the virial velocity of the galaxy.Due to the presence of galactic and intracluster magnetic fields lighter magnetic monopoles can be accelerated to higher (and even relativistic) velocities.It is estimated that magnetic monopoles with masses around 10 16 --10 17 GeV can be accelerated to v ≃ 10 −2 c [27, 47] which is the escape velocity of large galaxy clusters [47], while magnetic monopoles with masses smaller than 10 11 --10 13 GeV are expected to be in the relativistic regime [27,47,62,71].The dependence of the average monopole velocity on its mass is associated with its complicated acceleration history [71,72] and thus cannot be predicted precisely.
To facilitate comparison between different constraints, let us express Eqs.(3.5) and (3.6) using various benchmark values (we work in the non-relativistic regime here so that γ ≃ 1; for ∆ we consider two benchmark values ∆ = 1 and ∆ = 2.86 × 10 5 so there are two equations each for F M and Ω M ) ) (3.12) We are then ready to revisit a variety of bounds on the cosmological abundance of non-relativistic magnetic monopoles.
1. Mass density constraint.This is simply the requirement that the density parameter of non-relativistic magnetic monopoles should not exceed that of dark matter, i.e. [27] Ω M ≤ 0.27. ( This can be converted into a constraint on F M using Eqs.(3.9) and (3.10).
2. Direct search in cosmic rays.For non-relativistic magnetic monopoles with magnetic charge g = g D , the most stringent bound from direct searches in cosmic rays come from the MACRO experiment [68], which provides 90%C.L. upper bounds on the local monopole flux.Precise values of the bounds depend on the monopole velocity.For our purposes we adopt which can be converted into a bound on Ω M using Eqs.(3.11) and (3.12).The MACRO bound does not apply to the highly relativistic case with a Lorentz boost factor γ ≳ 10 [68], while for moderately or extremely relativistic cases there are bounds from other direct search experiments such as IceCube [22] and Pierre Auger [73] which turn out to be more stringent.
3. Parker bounds.Parker bounds are based on the neutralization of cosmological or astrophysical magnetic fields [74][75][76] by magnetic monopoles.There are various versions of Parker bounds [33,52,53,[77][78][79], based on different assumptions on the evolution history of magnetic fields.The most reliable one is the original version with refinement by M. S. Turner et al. [53] (referred to as TPB hereafter), where B, R, l c , t reg is the averaged magnetic field strength, the scale of the magnetized region, the coherent length of the magnetic field, and the regeneration time of the magnetic field of the Milky Way respectively, with their TPB benchmark values displayed in the denominators.The threshold mass value m is given by The Parker bound can be improved by considering astrophysical magnetic fields of larger coherence length and longer regeneration time.One promising candidate is the magnetic field of the Andromeda galaxy, whose parameters can be estimated as [80][81][82]   Here for B we consider only the regular magnetic field [81].The Parker bound from magnetic fields of the Andromeda galaxy was employed by ref. [49,66,67] to constrain the abundance of extremal magnetic black holes.It was then pointed out in ref. [63] that whether such magnetic black holes remain clustered with the Andromeda galaxy is uncertain.Here we use the Andromeda version of the Parker bound to constrain the abundance of ordinary magnetic monopoles with g = g D .The boundary between clustered and unclustered monopoles in the case of the Andromeda galaxy is expected to be at m ≃ 10 19 GeV/c 2 [63], again with a transition range spanning 2 ∼ 3 orders of magnitude.Therefore somewhat large uncertainty on the overdensity ∆ is involved if one wants to convert the Andromeda Parker bound on the monopole flux at large monopole mass m ∼ 10 19 GeV/c 2 to a corresponding bound on the monopole density parameter.For much lighter magnetic monopoles with m ≲ 10 19 GeV/c 2 it is likely that they are unclustered with ∆ ≃ 1.
Using the benchmark values in Eq. (3.17) we may express the Andromeda Parker bound as Strictly speaking the process of extracting energy from galactic magnetic fields by magnetic monopoles should be viewed as belonging to the first quarter cycle of a magnetic monopole plasma oscillation [53].If magnetic monopoles spend more than the first quarter cycle in a coherent domain of the magnetic field, it might be possible that their kinetic energy is returned to the magnetic field again [84], invalidating the Parker bound obtained thereof.Such a situation can be avoided if the monopole plasma oscillation is subject to strong Landau damping, which occurs for non-relativistic monopoles when where v is the thermal velocity of monopoles, k is the minimum wavenumber associated with the magnetic field, and ω M is the monopole plasma (angular) frequency Eq. (3.20) means that strong Landau damping occurs when the phase velocity of the monopole plasma oscillation becomes comparable or less than the monopole thermal velocity [51].The minimum wavenumber k is given by where l c is the coherence length of the magnetic field.Using Eq. (3.1) we may then express the condition for strong Landau damping as Therefore for monopoles with m ≳ 10 16 GeV/c 2 and v ≃ 10 −3 c corresponding to the virial velocity, if the flux is at or below the MACRO direct search limit, then since for the Andromeda galaxy l c ≃ 10 kpc, it is guaranteed that the monopole plasma oscillation is subject to strong Landau damping, so the Andromeda Parker bound is valid.Lighter monopoles with m ≲ 10 16 GeV/c 2 are expected to be accelerated by galactic and intracluster magnetic fields to higher velocities so as to achieve larger values of mv 3 .We therefore expect their oscillation is also subject to strong Landau damping.Furthermore we note that magnetic monopole plasma oscillation is incompatible with the observed features of the magnetic fields in spiral galaxies [51].We therefore do not expect the oscillation effect to invalidate the Parker bound obtained thereof.We summarize our findings in Figs. 1 and 2. The two figures contain the same information, but are displayed with different vertical axes to facilitate interpretations.The conversion between two figures can be made by employing Eqs.(3.9)--(3.12).In the figures, "Parker bound (MW)" is obtained from Eqs. (3.15) and (3.15) using benchmark values of the Milky Way shown in these equations, while "Parker bound (Andromeda)" is obtained from Eqs. (3.18) and (3.19) using benchmark values of the Andromeda galaxy.We do not display bounds from Debye shielding proposed in ref. [47] since we have argued in Sec. 2 that the concept of Debye shielding is not valid for magnetic monopole plasma in the presence of test electric currents.We focus on the case in which g = g D while results for other values of the magnetic charge can be inferred from simple scalings.
Despite the uncertainties associated with the transition region for ∆ and the relativistic boundary, important information can be extracted from these figures.First consider the parameter region around m ≃ 10 20 GeV/c 2 .This value of monopole mass is intriguing as it is the mass predicted in Kaluza-Klein theory [1,85,86] and also the maximum mass allowed by the weak gravity conjecture (WGC) [87].Moreover, it is not constrained by the Milky Way Parker bound so at early times magnetic monopoles with masses around m ≃ 10 20 GeV/c 2 are conjectured to make up all or the majority of dark matter [1], despite the challenge in finding a convincing production mechanism.The MACRO direct search experiment [68] can however constrain such ultraheavy magnetic monopoles to an abundance one order of magnitude smaller than that of dark matter.This direct search bound is subject to the astrophysical uncertainty in predicting the monopole density at scales comparable to the solar system.This astrophysical uncertainty is important since the expected number of signal events in a detector module in the total collecting period for a monopole flux saturating the MACRO bound is small (say 2 ∼ 3).For direct detection of weakly interacting massive particle (WIMP) dark matter the corresponding astrophysical uncertainty is not a big concern since simulations of dark matter fine-grained structures suggest a very smooth distribution [88,89].For magnetic monopoles the situation is much less clear considering the plasma effects and astrophysical magnetism involved.The Andromeda Parker bound nevertheless also disfavors the scenario in which such ultraheavy magnetic monopoles constitute all or the majority of dark matter.
Concerning the constraints on lighter monopoles with 10 13 GeV/c 2 ≲ m ≲ 10 18 GeV/c 2 (so that they remain non-relativistic), the MACRO bound improves over the Milky Way Parker bound by roughly one order of magnitude, though the comparison is subject to astrophysical uncertainties to some extent.Moreover, both the Milky Way Parker bound and the MACRO experiment do not constrain the mass range 10 16 GeV/c 2 ≲ m ≲ 10 18 GeV/c 2 well.The MACRO bound does not seem to exclude monopoles within this mass range to account for all or the majority of dark matter.This situation is remedied by the Andromeda Parker bound, which excludes such a possibility.Although this mass range is subject to uncertainties in ∆, the conclusion remains unaffected since the actual mass density constraint line in this mass range is supposed to be higher than what is shown in Fig. 1.For lighter monopoles satisfying 10 13 GeV/c 2 ≲ m ≲ 10 16 GeV/c 2 the uncertainties in ∆ is much less relevant and we see that the Andromeda Parker bound improves over its Milky Way counterpart by roughly three orders of magnitude, setting an upper limit on the monopole density parameter Ω M to be about 10 −7 -10 −4 depending on the monopole mass.

Conclusion
Magnetic monopoles, if existing, would reflect fundamental symmetries of nature at very high energy, with profound implications for both particle physics and cosmology.In this work we revisited a number of issues on their cosmological abundance.First we demonstrated that Debye shielding cannot be employed to yield a meaningful constraint on the monopole abundance, contrary to what is stated in the previous literature [47].The concept of Debye shielding relies on a state of thermal and mechanical equilibrium, which cannot be satisfied in the case of a magnetic monopole plasma in the presence of test electric currents.This is shown explicitly by analyzing the corresponding fluid and Maxwell's equations.We then proceeded to analyze three model-independent bounds on the cosmological monopole abundance, namely the mass density constraint, the direct search constraint from MACRO experiment, and the Parker bounds.We have improved the Parker bound on magnetic monopoles by considering magnetic fields of the Andromeda galaxy, which was previously used to constrain the abundance of magnetic black holes.It is found that for non-relativistic monopoles with masses 10 13 GeV/c 2 ≲ m ≲ 10 16 GeV/c 2 and unit Dirac magnetic charge, the Andromeda Parker bound is more stringent by the MACRO bound by two orders of magnitude, setting an upper limit on monopole flux F M at the level of 5.3 × 10 −19 cm −2 s −1 sr −1 , or an equivalent upper limit on the monopole density parameter Ω M at the level of 10 −7 -10 −4 depending on the mass.For heavier monopoles with masses 10 16 GeV/c 2 ≲ m ≲ 10 20 GeV/c 2 and unit Dirac magnetic charge the Andromeda Parker bound disfavors scenarios in which such monopoles account for all or the majority of dark matter.Around m ≃ 10 20 GeV/c 2 which is the largest mass allowed by the WGC, the Andromeda Parker bound is as competitive as the direct search limit from MACRO, albeit the comparison between different bounds thereof hinges on important astrophysical uncertainties which deserve further investigation.
Magnetic monopoles can be produced in the early universe through symmetry-breaking phase transitions, which could be of first-order, second-order, or a smooth crossover.For the monopole masses considered in this work, these phase transitions generically lead to an unacceptably large monopole abundance, even if one takes into monopole annihilation into account [15].Inflation theory is the standard way to get rid of the unwanted monopoles, but if the inflationary energy scale is high, lighter monopoles might still be produced, for example in partially-unified gauge theories [90][91][92][93][94][95][96].The subsequent evolution of their abundance depends on a number of issues such as monopole annihilation, capture by primordial black holes [97][98][99][100], and interaction with other topological defects [101,102].Gravitational waves might result from the symmetry-breaking phase transition if it is strongly first-order [103], and solition isocurvature perturbations [104,105].Improving the model-independent bound on the cosmological monopole abundance may thus have interesting implications for the early universe, which we leave for future study.

Figure 1 .
Figure 1.Bounds on the local flux F M of magnetic monopoles with g = g D as a function of monopole mass.The region below each line is excluded by the corresponding bound.The grey shaded area is the estimated transition range from clustered to unclustered monopoles with large uncertainties in predicting the overdensity ∆.The pink area is associated with the uncertainty in predicting the relativistic boundary.See the text for details.

Figure 2 .
Figure 2. Same as Fig. 1, but presented as bounds on the density parameter Ω M of magnetic monopoles as a function of monopole mass.