The morphology of the large-scale Galactic magnetic field (GMF) is surprisingly poorly understood for such an important component of the Milky Way’s interstellar medium (ISM). There is a long list of topics in Galactic astrophysics that currently depend on an incomplete understanding of the GMF, such as disk dynamics, cosmic-ray propagation, the turbulent ISM, molecular cloud collapse, star formation, supernova remnant evolution, etc. There is also a list of studies in the literature modeling the GMF using a variety of observational tracers and parametrized morphological forms. For earlier reviews, see, e.g., [1
] and references therein. In this review, I will discuss modeling work that is either relatively recent or still being used.
In addition to its importance in its own right, the GMF has a significant impact on several extragalactic observations because of effects it can have on the observables or because of the foreground confusion it adds. The cosmic microwave background (CMB) is the most obvious example, since we observe the CMB in the foreground minimum between the synchrotron emission that dominates in the radio (Section 2.3
) and the dust emission that dominates the higher frequencies (Section 2.4
). Another cosmological problem impacted by the Galaxy foregrounds is the study of the recombination epoch using redshifted 21 cm emission, which may be contaminated by Galactic synchrotron emission that is orders of magnitude brighter ([3
] Section 5.5). The search for the sources of the highest energy particles in the Universe, ultra-high-energy cosmic rays (UHECRs) is also complicated by the fact that these particles are deflected by magnetic fields as they propagate to the Earth, so back-tracing them requires an accurate GMF model (see Section 3.4
). These needs have driven some of the modeling work in the field and will continue to do so.
The variety of observables and the variety of contexts where the GMF is important explain the variety of modeling efforts in the literature. Some of these analyses focus on only one part of the problem and, as in the case of the proverbial elephant, find an answer that is incomplete. All of the analyses require certain assumptions about the distributions of particles associated with the observables, whether thermal electrons, relativistic cosmic rays, or dust grains. These assumptions mean that there remain hidden degeneracies in the parameter space and that different analyses come up with different measures even for fundamental quantities such as the average strength or degree of ordering of the field. However, the information we can gain from these disparate efforts can also help us to tackle the problem from different angles.
I will summarize the observables in Section 2
and the components of a physical model of the Galaxy needed to simulate them in Section 3
. I will review some of the different models, their origins and contexts, their advantages and disadvantages, what common features they have, and what we have learned in Section 4
. There are a variety of challenges faced by all such modeling efforts, which I will summarize in Section 5
. However, there is also a prospect of making significant progress in the next few years, which I will speculate on in Section 6
There are many physical processes affected by the GMF that allow us to observe it indirectly. None of these phenomena give complete and unambiguous information about the large-scale GMF by themselves. This section will briefly summarize the principle phenomena that have been used so far to model the Galactic-scale GMF, with the analyses and results summarized in the next section. A summary of this section is also presented in Table 1
There are other tracers not discussed here, from Zeeman splitting of masers ([4
]) to HI velocity gradients ([7
]). For a thorough review of observations and analyses from small-scale turbulence to the intergalactic medium (IGM), see Han [8
]. Here we will focus on probes that probe the large-scale magnetic field over large portions of the Galaxy.
2.1. Polarized Starlight
One of the longest-known observational signatures of the GMF is the polarization of starlight. Amorphous dust grains tend to align their long axes perpendicular to the local magnetic field, and therefore absorption leaves the starlight partly linearly polarized parallel to that local field as projected onto the sky from the point of view of the observer (perpendicular to the line of sight, i.e., ).
The catalog of Heiles [9
], for example, provides measurements to over 9k individual stars over a significant fraction of the sky. The advantage of starlight polarization is that one can in principle use the distance to the star as well as multiple stars in a given direction to extract 3D information about the magnetic field. This analysis depends on knowledge of the distribution of the relevant dust grains, of course, which can be measured via the reddening. Its main disadvantage for modeling the large-scale GMF is sampling, since we cannot make such measurements, much less get accurate distances, for stars in the more distant regions of the disk. We also run out of stars away from the Galactic plane. However, Santos et al. [10
] demonstrate how useful they can be for studying local features, particularly the North Polar Spur (NPS, aka Loop I), and Pavel [11
] explore how near infra-red observations can be used to constrain large-scale field properties. See Section 5.2
. Panopoulou et al. [12
] demonstrate how combining polarization measurements with Gaia1
distances allows a detailed tomography of the ISM.
2.2. Faraday Rotation Measures (RMs) of Point Sources
The Faraday rotation of polarized emission can probe the line-of-sight (LOS) component of the magnetic field () through the wavelength-dependent rotation of the orientation of the linear polarization vector of emission originating from a source behind a Faraday rotating medium. For each polarized point source observed at multiple frequency bands, a single RM value can be fit to the polarization orientation as a function of frequency, and this RM represents the integrated product of the thermal electron density and the LOS component of the magnetic field between the observer and the source. This observable is unique in that it probes not only the orientation of the magnetic field but also its direction: a positive (negative) RM implies a field pointed toward (away from) the observer. RMs of Galactic pulsars can give us 3D information if we have a measured distance to each pulsar, but our sampling is currently very limited. RMs of extragalactic point sources are far more plentiful, but each measurement represents the full LOS through the Galaxy (as well as the intergalactic medium) in that direction as well as the Faraday rotation intrinsic to the source itself.
We currently have roughly 1k RMs for Galactic pulsars [13
] and 42k for extragalactic polarized sources [14
] and more all the time [16
]. The sampling of the Galactic pulsars allows some analysis of the field morphology (see, e.g., work by Han et al. [13
]) but is not currently sufficient to use for robust model fitting and to take full advantage of the 3D information this tracer has the potential to offer. (That situation will change, however, with the Square Kilometer Array (SKA)2
and its associated pathfinder surveys; see Section 6
) Extragalactic sources, however, are plentiful over the full sky and particularly well sampled over the Southern and Canadian Galactic Plan Surveys (SGPS and CGPS [17
]. Many catalogs of extragalactic RMs have been collected by Oppermann et al. [19
] and synthesized into a reconstructed sky map of observed total RM through the Galaxy as well as an uncertainty based on the sampling and the variations among the data used in any given region of the sky.
2.3. Diffuse Polarized Synchrotron Emission
Diffuse synchrotron emission dominates the sky in maps from the radio frequencies to the microwave bands. It depends both on the strength and orientation of the magnetic field projected onto the plane of the sky (
). One of the first full-sky maps and one that remains useful today is that of Haslam et al. at 408 MHz [20
] giving total synchrotron intensity, I
, from a combination of ground-based single-dish data. Polarization (in the form of Stokes parameters Q
, or polarized intensity as
) has been measured over the full sky in the radio at, for example, 1.4 GHz by Reich et al. [22
] and Testori et al. [23
], showing large-scale coherent polarization signals in the NPS and Fan regions as well as significant depolarization in the Galactic plane compared to the high latitude sky. This is the main disadvantage of probing the polarization at radio frequencies, where it is easier to do from the ground: there is a so-called polarization horizon [24
], which is a function of telescope beam size and observation frequency, beyond which little polarization signal can be observed due to Faraday effects in the turbulent ISM.
Space-based microwave observations began with WMAP
that showed us the 23 GHz sky in polarized emission free of Faraday effects [26
], and similarly at 30 GHz by Planck
]. These data allow us to study the apparent morphology of the magnetic fields projected onto the sky. One of the most important observables, however, that of the polarization fraction,
, remains unavailable to us. This is because the total intensity sky at microwave bands is dominated by other emission processes, principally thermal Bremsstrahlung emission and the anomalous microwave emission (AME) believed to arise from spinning dust grains. Both processes are thought to produce only unpolarized emission, but their presence makes it difficult to map the synchrotron total intensity in the microwave bands and therefore to estimate the degree of polarization and the field ordering. An additional complexity is the calibration of the zero-level of these maps; see Wehus et al. [28
] for a recent analysis. Most of these radio and microwave observations are not absolutely calibrated, which means that there is an unknown net offset in the datasets, and though this offset is not important for the fitting of morphological features, it is again vital for the polarization fraction and inferred field ordering.
If we understood the energy spectrum of the cosmic-ray lepton population that produces the synchrotron emission from the radio to the microwave bands, we could combine the low-frequency total intensity maps (where contamination is minimal) with the high-frequency polarization maps (where Faraday effects are minimal) to measure the polarization fraction. Unfortunately, the shape of the spectrum and its likely turnover around a few GeV are not well understood. Since direct measures of the cosmic-ray electron (CRE) spectrum near the Earth are additionally complicated by solar modulation, and the local spectrum may not be typical of the Galaxy, the synchrotron emission itself may be the best indirect probe of that region of the CRE spectrum [29
] if we can combine information from enough different frequencies around a few GHz. This situation is improving because of the C-Band All Sky Survey (C-BASS) at 5 GHz [30
], precisely the region most interesting for probing not only the synchrotron spectral turnover but also the regime where Faraday effects go from dominant to negligible in different regions of the plane.
2.4. Diffuse Polarized Thermal Dust Emission
The same dust grains that polarize background starlight through absorption also produce thermal emission that is polarized perpendicular to the local magnetic field. The observed dust polarization is then another tracer of the orientation of the magnetic field projected onto the sky. It is not thought to be a strong function of the magnetic field strength, but the degree to which the grains tend to align depends on the grain properties and environment in ways not well understood. This was first measured by Archeops [32
] and more recently with the full-sky high-resolution and multi-frequency data of the Planck
Though the geometric dependence is similar to that of synchrotron emission (i.e., polarization perpendicular to the
orientation), this observable is complementary to the synchrotron emission, because it arises from a different region of the ISM, the cold dusty ISM close to the Galactic plane. The Planck
data showed [33
] that though the two tracers correlated in some regions such as the Fan and NPS, they were uncorrelated over much of the sky. This means that we can use the different particle distributions to probe the field along the LOS. Though detailed models of the distribution of CREs are lacking, the Gaia mission is providing new 3D dust models from the extinction measurements of millions of stars [34
]. The combination of Gaia and Planck
data will allow us to probe the local magnetic field in 3D using the dust emission. This in turn will then help us to determine which aspects of the large angular-scale structure in the synchrotron sky are local (see Section 5.2
) and let us account for them. This in turn would then let us focus on the vertical structure of both dust and synchrotron emission to probe from the disk into the Galactic halo out to a few kpc.
2.5. Diffuse -ray Emission
-ray emission has long been used to study the cosmic-ray population in the Galaxy and is therefore crucial to interpreting the synchrotron emission and studying the GMF. The Fermi3
data in particular provide both direct measurements of the local population of CRs as well as the diffuse inverse Compton
-ray emission mapped over the full sky. Both have been used to constrain the CR population [35
], to probe the cosmic-ray spectrum by combining
-ray and microwave band observables [36
], and to fit models of the magnetic field and measure the scale height of the CRE population [37
]. It is now being recognized that the question of CR propagation cannot be solved without the combination of
-ray and synchrotron emission, as demonstrated in recent work by Orlando [38
2.6. Faraday Tomography/RM Synthesis
With enough spectral resolution, one can do better than fit a single RM value to the variation of the polarization angle with frequency in a given direction. A full Fourier analysis converts the emission as a function of frequency into a measure of the polarized intensity along the LOS as a function of Faraday depth. This allows us to study diffuse emission where the synchrotron-emitting regions and the Faraday rotating regions are mixed. The Faraday depth, though it is not a physical distance scale, provides 3D information about the distribution of emitting regions along the LOS. It therefore probes the magnetic fields not only through their Faraday effects but also from the synchrotron emission itself. This sort of analysis avoids the sampling restrictions of pulsar RMs, though in return, it links the Faraday information to the cosmic-ray density distribution.
See Ferriere [39
] for a brief review of Faraday tomography and its prospects.
2.7. Supernova Remnants
The morphology of supernova remnants (SNRs) can be a complementary probe of the GMF as shown by West et al. [40
]. When the SNR expands into the ambient ISM, it will compress the local magnetic field component that is tangent to the shell (perpendicular to the expansion direction). This will in turn produce synchrotron emission that is strongest where the field is most compressed, implying that the morphology of the remnant in the radio is in part determined by the orientation of the ambient field relative to the line of sight. Though there are not many SNRs with such regular morphology, West et al. [40
] showed that one large-scale GMF model predicted a significantly better agreement with the available observations than another model. This analysis is therefore a useful addition to the toolbox of informative probes of the GMF.
4. Models and Analyses
There have been several studies of the GMF over the past few decades, and though there are some common features (e.g., spirals in disks), their morphologies have a surprising variety in the details. Some analyses include an exploration of the parameter space and a quantification of statistical uncertainties, but few have accounted for systematic uncertainties or have quantitatively compared the different parametrized models that have be explored independently. These models can be roughly grouped into those that are largely ad hoc constructions built to be compared with specific observations, and those that arise from theoretical work. Clearly, some physical properties such as the divergence-free condition can be enforced on the ad hoc models, and equally clearly, the observations inform the theoretical work. However, the two approaches can be considered to be complementary, and if they do not meet up in the middle, they will each continue to be useful to compare.
The first half of this section reviews the analyses that have been done recently, with emphasis on their pros and cons compared to the others. This collection of models contains many common morphological features, and since it is those physical features that are of interest, and how well they are constrained, we summarize them in the second half of this section.
4.1. Current Magnetic Field Model Fits
This section presents an incomplete but representative sample of some of the current models in the literature, describing what datasets were used to constrain the models and what particular advantages or disadvantages each analysis had. Some of these are shown in Figure 2
to illustrate the varieties of morphologies. (Please note that this is a biased subset of only those that could be visualized in a consistent way using the modeling code hammurabi5
Sun et al. [42
] (refined in [64
], “Sun10”) first used the combination of synchrotron total and polarized intensity (at 408 MHz and 23 GHz respectively) along with RMs to compare several 3D models of the GMF. They used the NE2001 model for thermal electrons and a simple exponential disk with a power law spectrum of index
for the cosmic rays. This work included an analysis of the impact of the filling factor of the ionized gas in the ISM and examined several models from the literature, both axi-symmetric and bisymmetric spirals. The model they concluded was favored by the data was the “ASS+RING” based on an axisymmetric spiral disk with field reversals defined in several regions to match the data. The turbulence was modeled with a single-scale random field. This was the first such analysis, though it was not quantitative model fit, and it assumed a very high local cosmic-ray density to fit the data without an ordered random field component.
Jaffe et al. [41
] (refined in [29
], “Jaffe13”) used these same synchrotron observables and the SGPS and CGPS extragalactic RMs to perform a systematic likelihood exploration in the plane of a 2D model based loosely on previous work by [65
]. It used the NE2001 model for thermal electrons and a Galprop cosmic-ray model from [35
]. It included an exponential disk to which is added four Gaussian-profiled spiral arms as well as a ring around the Galactic center. This analysis first included realizations of the random components, both isotropic and ordered, based on a Kolmogorov-like GRF in an MCMC likelihood space optimization, but only in 2D. The update in Jaffe et al. [29
] simultaneously constrained the CR lepton break at low energies in one of the first attempts to model the CRs and GMF simultaneously. Then [60
] added the polarized dust information and saw how the different distributions of particles means that the two observables can perhaps constrain the GMF in different regions of the ISM. These analyses, though, remain limited by the systematic uncertainties of the particle distributions.
Jansson and Farrar [43
] (refined in [59
], “JF12”) used the synchrotron total and polarized intensity from WMAP
23 GHz as well as the 40k extragalactic RMs to perform a systematic likelihood exploration in 3D of a model with both thin and thick disk components, eight spiral arm or inter-arm segments, and an x-shaped halo field. It was based on the NE2001 thermal electron density model (with the scale height correction from [51
]) and a CR model based on the “71Xvarh7S” from Galprop. It used an analytical method to treat the random field components, and the measured pixel variance was used in the likelihood. (See Section 4.2.6
) This was the first 3D model optimization with an MCMC likelihood analysis, but the use of the WMAP
synchrotron map at 23 GHz meant that the extra total intensity foregrounds biased their estimate of the random field component. See also Unger & Farrar below for updates.
Han et al. [13
] used both Galactic and extragalactic radio sources to model the RM reversals in the Galactic plane with a set of spiral arm and inter-arm segments. The analysis used the YMW16 model for thermal electrons. This is not a global GMF model for the Galaxy, but an analysis specifically focused on where the field reversals lie using the distance information from pulsars.
Terral and Ferriere [66
] (“TF17”) used the spiraling x-shaped field models derived in [67
] to fit the RM data. They explore both axisymmetric and bisymmetric possibilities. This work represents the first quantitative fitting to models of spiraling x-shaped fields in theoretically derived forms (rather than ad hoc).
Unger & Farrar [68
] built on the JF12 work by replacing the ad hoc x-shaped halo field by the models of Ferriere and Terral [67
]. They also compared the results of fits based on different datasets (WMAP
synchrotron total intensity versus 408 MHz), different thermal electron models (NE2001 vs. YMW16), and CR distributions from the original work compared to those of [35
Shukurov et al. [69
] derived eigenfunctions of the mean-field dynamo equation that can be used to construct any model consistent with those assumptions. Though this analysis does not present one model fit to the data, it provides a framework for fitting more physically realistic models in future with a publicly available software package.
4.2. Magnetic Field Morphological Features
The rest of this section discusses the morphological features that are astrophysically interesting and are common among many if not all the different models, such as magnetic arms, reversals, and x-shaped vertical fields.
4.2.1. Axisymmetric Spirals
Simple models began with axisymmetric spirals (e.g., [42
]) with, e.g., exponential disks, and though one of these alone cannot reproduce all the observables, the morphology remains a component of many different models. Sun10, for example, uses such a model as the basis on which reversals are added in an annular region and/or a spiral arm segment. Jaffe13 adds spiral arms to an axisymmetric spiral base model. JF12 uses multiple disk components to model the GMF in the thin and thick disks, where the spiral arm segments are imposed on the former. Until the GMF can be modeled without parametrized ad hoc models (e.g., using the eigenfunctions of Shukurov et al. [69
]), the basic axisymmetric spiral will remain useful.
The parameters of the axisymmetric spiral, however, are not yet well constrained because of the uncertainties in the equivalent parameters of the particle distributions. In the case of CRs, for example, the thick disk scale height is not known even to within a factor of two [37
], and this translates into a corresponding uncertainty in the magnetic field strength as a function of height above the disk. The thermal electron density is better known, and so for example, the Sun10 model was corrected by a factor of two change in that model, but as discussed by those authors, there remains some uncertainty. Recent analysis by Sobey et al. [71
] of pulsar data from LOFAR6
estimate the GMF scale height assuming the Yao et al. [52
] model for thermal electrons, but the paper also discusses how these systematic uncertainties may affect their estimates of both the scale height of the coherent magnetic field and its overall strength. The pitch angle of the spiral is often assumed to be
° in the disk (Sun10, JF12, Jaffe13) following the NE2001 electron density model, and the RMs are consistent with this. Steininger et al. [72
] showed that when allowed to vary, this pitch angle is not constrained by full-sky maps of RM and synchrotron emission, though this may be due to a pitch angle that has one value in the Galactic plane on average and another value in the local neighborhood that dominates the measures that are at higher latitudes.
Though the model fits discussed above provide numbers for these parameters, and some of them with small statistical error bars, these systematic uncertainties make it difficult to conclude that they are really constrained.
4.2.2. Spiral Arms
Though we have mapped out the spiral structure of our Galaxy’s stellar component, it is harder to determine whether the GMF has a similar structure because of the difficulty determining distances to the corresponding measurements. In several external galaxies that we can see face-on in synchrotron emission, the fields appear to be strongly ordered in spiral arm structures that may or may not be coincident with the material spiral arms. Observations of these so-called magnetic arms are reviewed by Beck and Wielebinski [73
]. The question is of particular interest for what the magnetic arms may say about the mean-field dynamo, as discussed by Moss et al. [74
], for example. For this reason, most models of the GMF include magnetic arms of some sort, whether as explicit bisymmetric spirals, ad hoc but continuously defined spiral arms [60
], or as discontinuous segments [13
]. If the extragalactic RM features along the plane are indeed tracing these large-scale arms, then the models largely agree where the arms with the strongest coherent fields are. However, since the models so far generally constrain the magnetic arms to follow the disk field pitch angle, the same systematic uncertainties above apply.
Comparison of the polarized emission of synchrotron and dust has the potential to probe whether the two components come from different spiral arm regions, as discussed by Jaffe et al. [60
], but again, this depends on a better understanding of the field ordering.
As reviewed in Haverkorn [1
], one of the most obvious features of the sky traced by RMs toward Galactic pulsars as well as extragalactic radio sources is the clear evidence of reversals in the large-scale GMF along the Galactic plane. Though it remains difficult to determine the distance to these reversals, Han et al. [13
] use Galactic pulsars with distance estimates to model the reversals with alternating magnetic arms and inter-arm regions. Ordog et al. [75
] recently added the analysis of the RM gradient in the diffuse polarized radio emission. Their findings highlight the importance of modeling these reversals in 3D and connecting them to dynamo theory.
If indeed the GMF reversals are a feature of the large-scale Galactic structure and can be defined as reversals between magnetic arms, then we can perhaps learn about such structures from external galaxies. See Beck and Wielebinski [73
] for a review of how observations of the RM of the diffuse emission can be combined with the rotational velocity information but are not yet sensitive enough to confirm the sorts of reversals we see in our own Milky Way.
The modeling projects discussed above include large-scale GMF reversals as either magnetic spiral arms or as annuli or both. However, only Han et al. [13
] use the distance information from Galactic pulsars that can constrain where there are. Models that use only extragalactic RMs and assume that the field along an entire magnetic arm is oriented the same way (Sun10, JF12, Jaffe13) do agree on which segments are reversed. However, the Han et al. analysis considers the direction in different sections of each arm, and these are not required, nor found empirically, to be uniform in each arm. Increased pulsar sampling will help, as will adding information from other observables such as Zeeman splitting of masers [4
4.2.4. Vertical (Poloidal) Field
Early models fit to the GMF included no vertical component to the field, because starlight polarization observations showed clearly a field that remained parallel to the Galactic plane in the disk, and the strongest synchrotron emission in the plane likewise. Observations of external galaxies seen edge-on in radio polarization initially showed a field largely parallel to the disk. However, more sensitive radio observations of external galaxies where fainter emission from the “halo” or “thick disk” component could be observed showed an x-shaped magnetic field structure. (Again, see Beck and Wielebinski [73
] for a review.) Such a vertical component is again connected with the Galactic dynamo and winds.
The Sun and Reich [64
] model included such a vertical component, as did Jansson and Farrar [43
], both cases of ad hoc x-shaped components. Ferriere and Terral [67
] more generally classified the different field morphologies that could have such a vertical component. Both Jansson and Farrar [43
] and the fitting of Terral and Ferriere [66
] conclude that such a component is likely favored by observations of our Milky Way. However, again, the systematic uncertainties in the particle distributions keep this question open for the time being.
4.2.5. Beyond the Ad Hoc
Ferriere and Terral [67
] began the work of looking beyond ad hoc parametric models by using the Euler method to define convenient field configurations that could reproduce a spiral and an x-shaped vertical field. Then next step was taken by Shukurov et al. [69
]: determining the eigenfunctions of the mean-field dynamo equation. These functions can then reproduce any GMF that is physically possible within those assumptions. Though parametrized models will always be useful for studying specific identified features of the large-scale GMF, we should increasingly move beyond them and exploit these more physical representations of the possible morphologies. Terral and Ferriere [66
] fit the Ferriere and Terral [67
] models to the RM data, and this work can now be combined with other tracers, as partly begun by Unger and Farrar [68
], who combined the Ferriere and Terral [67
] models with the ad hoc JF12 model.
4.2.6. Turbulent Field
The treatment of the small-scale fluctuations in the GMF is one of the thornier questions that must be addressed in any modeling, even when the large-scale GMF is the only goal. One reason is that small but local structures project to large angular-scale structures on the sky and can have a large effect on the model fitting if not properly taken into account. Furthermore, there may be systematic correlations between fields and particles on small scales that must be accounted for in modeling the large scales, whether explicitly or statistically. Lastly, when comparing models to data, it must be quantified how far the latter are expected to deviate from the former. The Milky Way should be considered one realization of a galaxy model we are looking for, and that model includes some “galactic variance” due to the expected small-scale fluctuations that are model dependent.
The ISM is known to be turbulent at a range of scales (again, see [1
] and references therein), and this turbulence is neither expected nor observed to be Gaussian. Both properties present a challenge for generating simulated galaxy models and for comparing the data to the simulated observables. Some modeling efforts simply ignore the stochasticity by fitting mean-field models to observables such as averaged RMs that depend only on that coherent field component. This is effectively what Han et al. [13
] do fitting RM vs distance plots for different regions of the Galaxy, and the error bars include the scatter that is partially due to the ISM turbulence. The JF12 model includes an analytic expression for the average contribution to each observable from the turbulence under a few simplifying assumptions. This allows the average contribution to, e.g., synchrotron polarization to be correctly reproduced. To compare that average to our Galaxy that itself is a single realization of a field with a random component, JF12 used the data essentially to bootstrap this statistically, so that the optimized model did take this measured variance into account in the likelihood. However, as they point out and as further discussed in [63
], the variance itself is an observable that should be used to improve the constraints on the degree of ordering in the magnetic field.
The next lowest order approximations are to create realizations of the random component during the simulation process by simply adding a randomly drawn number (usually from a Gaussian distribution), or a set of three for a random vector. This can be done either to every pixel of a simulated sky map, to every point along a simulated LOS, or to every 3D voxel over which the simulated observables are integrated. The first approach is in a sense effectively similar to the JF12 method of simulating the ensemble average observed sky and comparing with a bootstrapped estimate of the variance. The second approach was used by O’Dea et al. O’Dea et al. [76
] (with a further refinement discussed in the next paragraph), which then takes into account LOS averaging effects (i.e., depth depolarization) for each observed pixel. The third approach was used in the modeling of Sun10 and (effectively) in TF17, which then includes some effects of averaging within the observing angle (beam depolarization).
Adding information about the two-point correlation function of the turbulence is the next step, i.e., simulating a GRF with a given, e.g., Kolmogorov, power spectrum. O’Dea et al. used such a prescription for generating the 1D turbulence for each LOS. Jaffe13 used this in a 2D analysis restricted to the Galactic plane. The full 3D approach was used in the Planck
] (without any quantitative parameter optimization) and in Steininger et al. [72
] (with a full MCMC likelihood exploration). This last result shows that it is now computationally feasible to do this.
The next step will be to include more information than simply the two-point statistics of the magnetic field. Studies of the ISM turbulence have begun to characterize a variety of its statistical properties based on the diffuse synchrotron emission in total [77
] or polarized intensity [78
]. Likewise, for dust, the Planck
collaboration has opened a new window into the turbulence in the colder phase of the ISM with high-resolution maps of the polarized dust emission ([81
] and references therein). These studies make use of MHD simulations with known physical parameters to study how to infer the physics from the observables. Those simulations in turn can encode different assumptions about the turbulence and about the correlations among the relevant physical quantities such as the field strength and direction and the particle distributions (whether thermal electrons, CRs, or dust etc.). See, e.g., Stepanov et al. [82
] for comparisons of data and simulations focusing on the cosmic rays in MHD simulations, or Planck Collaboration XX [83
] or Kandel et al. [84
] for discussions of the dust. With these studies, we can then use the information learned from MHD simulations to define physical parameters of the turbulence that the data may constrain.
As always, the way forward certainly includes continuing to gather more of the traditional observational tracers, e.g., more pulsar RMs and distances, more starlight polarization measurements and distances, more synchrotron frequencies between the radio and microwave bands, etc. Gaia has already demonstrated how orders of magnitude more stellar distance and extinction measurements can impact our understanding of the local ISM [34
]. Combining this with dust polarization measurements from Planck
and additional starlight polarization surveys will likewise help us to map the magnetic fields in the solar neighborhood. Next-generation radio surveys are poised to do the same for Galactic pulsars and extragalactic radio sources. The GALFACTS7
team have completed their survey to improve our sampling of radio sources by an order of magnitude over the full sky visible from Arecibo. The LOFAR project is now underway to demonstrate the power of not only the idea of linking large numbers of radio antennae in a software telescope but also of opening the low-frequency window onto the Universe. The latter will allow us to probe the more tenuous regions of the ISM. This is one of the first SKA-pathfinder projects testing technology and algorithms for the planned SKA that is expected to detect every pulsar in our Galaxy (that sweeps in our direction [100
]) as well as orders of magnitude more extragalactic sources. This improved sampling and distance information will help us to isolate the field reversals that we see in the RMs as well as the local features such as bubbles and loops. A new survey of OH Masers [6
] will increase our sampling of Zeeman splitting measurements so that it can also tighten constraints on the large-scale field.
Constraints on Galactic cosmic rays will also be improved by additional direct measurements of the local interstellar spectrum by Voyager [87
] and future constraints in the medium-energy
-rays by, e.g., the proposed AMEGO8
mission. When combined with additional synchrotron maps at intermediate (a few GHz) frequencies such as from S-PASS, C-BASS, and QUIJOTE 9
, we should be able to constrain the CR and synchrotron spectra at large scales and therefore the magnetic field ordering, at least on average in the disk. If we then combine this information with the variations in the RMs from the vastly improved sampling from the SKA pathfinders, we may be able to start studying its variations in different regions of the Galaxy in 3D.
In addition to more data from the traditional tracers, we also have the prospect of new tracers and methods. UHECRs are also deflected in the magnetic field, and though we cannot back-trace the particles to their sources, the anisotropy in the distribution of their arrival directions is a statistical probe of the local magnetic field (where the meaning of “local” depends on the particle rigidity). The HAWC Collaboration [101
] have recently measured the direction of the local interstellar magnetic field from the anisotropy in the HAWC and IceCube data at 10 TeV. Multi-messenger astronomy has the potential to identify the sources of these particles by associating them with photons, neutrinos, and gravitational wave events unaffected by the GMF, so the individual UHECR deflections could be used as an additional tracer of the GMF. New techniques include analyses of synchrotron data in new ways, such as the family of tools based on the polarization gradient of Lazarian and Yuen [102
]. See [103
] for how these methods compare to Faraday tomography, for example. This review has not discussed Zeeman splitting, because generally the number of measurements is not sufficient for probing the large-scale GMF. However, these data can provide crucial independent probes in the regions where we do have them, and the sampling is soon to be improved by new projects such as MAGMO by Green et al. [104
]. Though this is not a new magnetic field tracer, using a large sampling of measurements to trace the GMF may indeed soon have a new discriminatory power.
However, in making progress on understanding the GMF, more data may not be more important than putting together information from different fields and from asking different questions. The IMAGINE project [105
] aims not merely to add observables such as UHECRs to the mix but more importantly to provide a common framework for analysis. Just as a decade ago, significant progress was made by several teams combining three complementary observables, so we can expect the next advances to come from a yet more holistic approach. Initial attempts have already been made to fit the CR spectrum simultaneously with the large-scale GMF [29
] and to add the dust polarization [60
], but it was not computationally feasible with the tools available at the time to go beyond a very simple analysis with too many assumptions. Steininger et al. [72
] have published a more efficient platform into which we can plug in the information from the many disparate tracers and explore the likelihood space in a rigorous Bayesian analysis. It is also important to move beyond the ad hoc models and simple field components described in Section 3.1
to include higher-order moments of the random components as well as properties such as helicity.
It is also worth mentioning that simulations of Milky Way-like galaxies are becoming more informative about the amplification of magnetic fields during galaxy formation. Pakmor et al. [106
] for example show how the Auriga simulations reproduce spiral galaxies with magnetic fields with similar exponential disks to what we observe in our Galaxy and with similar strengths. Whether these simulations will inform GMF modeling work or vice versa is an interesting question.
The bottom line for the time being is that modeling the GMF has now reached an interesting and challenging point, where there are both degeneracies and contradictions in the parameter space, but still too many systematic uncertainties to know what to make of them yet. However, the uncertainties are being attacked in several ways, with new techniques, new observables, and new combinations of old observables, and the prospects are correspondingly bright for improving our models.