Current and Future Constraints on Primordial Magnetic Fields

We use a compendium of current measurements of the CMB temperature and polarization anisotropies from ground-based and satellite experiments. We use Planck data from the 2015 release to constrain the TT, TE, EE, and lensing power spectra. Specifically, these are the "plik_dx11dr21_HM_v18_TT," "lowTEB," and "lensing" Planck likelihood modules.

In addition to the Planck data, we use a number of ground-based CMB B-mode polarization measurements. First, we include measurements of the TE, EE, and BB power spectra from the SPTpol experiment (Crites et al. 2015; Keisler et al. 2015). The BB bandpowers cover the angular multipoles ${\ell }\in [500,2000]$. We also add the BB bandpowers from Polarbear that cover the multipoles from 500 to 2500 (Ade et al. 2015). At these angular scales, both the SPTpol and Polarbear bandpowers primarily constrain the vector modes of a PMF. Finally, we include the latest BICEP2 and Keck Array joint analysis (Keck Array & BICEP2 Collaborations, Ade et al. 2016). This last data set also places limits on the tensor modes of a PMF due to its coverage of lower multipoles. During the writing of this paper, ACTpol polarization power spectra became available (Naess et al. 2014; Louis et al. 2016). We have not included them as the public likelihoods do not include the B-mode power spectrum; we do not expect adding the ACTpol bandpowers to significantly change the PMF limits based on a visual comparison of the current BB bandpowers from different experiments (see Figure 1). In all chains, we marginalize over the recommended foreground models for each data set. We do not, however, require consistency between these foreground models as the data do not have identical flux cuts for extragalactic sources and so on. These data are plotted against the expected B-mode power spectrum for the standard cosmological model in Figure 1.

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We use MCMC methods to determine the parameter constraints reported in this work. The results are calculated using the CosmoMC³ package (Lewis & Bridle 2002). CosmoMC invokes CAMB⁴ (Lewis et al. 2000) to calculate the CMB power spectrum for each set of cosmological parameters. Although CosmoMC and CAMB have a partial implementation of PMFs, we choose to adopt a simpler, fast, template-based calculation for the PMF-sourced power. We have adapted CosmoMC to add a scaled version of a PMF template to all four CMB power spectra: TT, TE, EE, and BB, where the scale factor is ${A}_{\mathrm{PMF}}$:

$$\begin{eqnarray}&&{C}_{{\ell }}={C}_{{\ell }}^{\mathrm{CAMB}}+{A}_{\mathrm{PMF}}\left[{C}_{{\ell }}^{\mathrm{PMF},\mathrm{vec}}+{\left(\displaystyle \frac{\beta }{20}\right)}^{1.9}{C}_{{\ell }}^{\mathrm{PMF},\mathrm{tens}}\right].\end{eqnarray} \tag{ 1 }$$

The calculation of the PMF templates, ${C}_{{\ell }}^{\mathrm{PMF},\mathrm{vec}}$ and ${C}_{{\ell }}^{\mathrm{PMF},\mathrm{tens}}$, and the motivation for the β scaling are described in Section 2.3. The parameter $\beta =\mathrm{ln}({a}_{\nu }/{a}_{\mathrm{PMF}})$ relates the timing of neutrino decoupling and the generation of the PMF, where a_x represents the scale factor at the respective events. Note that no other effect of PMFs is considered; this is one reason why we only use CMB data.

For the real data, unless noted, we assume the six-parameter, spatially flat ΛCDM model with a single massive neutrino of 60 meV and two parameters, ${A}_{\mathrm{PMF}}$ and β, describing the power due to PMFs. We adopt flat priors on all parameters. There are two points to note about the PMF priors. First, this is a flat prior on the observed PMF power, ${A}_{\mathrm{PMF}}$, not the rms magnetic field strength, ${B}_{1\mathrm{Mpc}}$, that has often been used in the literature. In the current era of upper limits, this prior choice has some impact on the resulting PMF limits. Second, the exact range of the uniform prior on β matters. Here we follow Planck Collaboration et al. (2016) and Zucca et al. (2016) and use $\beta \in [11.513,41.447]$.⁵ However, the constraints, especially those from Planck alone, weaken substantially if the lower bound on β is lowered further. If we follow Ade et al. (2015) who used $\beta \in [0,39]$, the upper limits are relaxed by a factor of roughly eight for Planck-only and by a factor of two for the combined data set. We sometimes also add ${A}_{\mathrm{lens}}$ as a simple way of marginalizing over uncertainty in the predicted lensed BB power spectrum for any extension to the ΛCDM model. Finally, we sometimes allow non-zero tensors, parameterized by the tensor-to-scalar ratio, r.

The density and stress perturbations introduced by PMFs give rise to vorticity and gravitational waves in the early photon–baryon plasma, i.e., vector and tensor modes, as well as scalar modes. The exact amplitude of these modes can be calculated by modifying the normal Boltzmann equations to include sources due to the PMFs. The primary PMF anisotropy is assumed to be Gaussian distributed with a power-law power spectrum, ${A}_{{\rm{B}}}{k}^{{n}_{{\rm{B}}}}$, where n_B is the spectral index and A_B is the PMF power normalization. We can translate the normalization, A_B, into more physically motivated units by calculating the rms of the PMF field strength over a length scale of 1 Mpc, which we will call ${B}_{1\mathrm{Mpc}}$. Finally, the timing of when PMFs are generated relative to neutrino decoupling matters for the tensor component because the PMF-induced stress anisotropy can be compensated for by decoupled or partially decoupled neutrinos. The timing is parameterized by $\beta =\mathrm{ln}({a}_{\nu }/{a}_{\mathrm{PMF}})$, the natural logarithm of the ratio of the scale factors at neutrino decoupling and PMF generation.

Given that we are still in the era of upper limits, we fix the value of n_B instead of exploring the full three-dimensional PMF parameter space. For the real data, we focus on a nearly scale-invariant (n_B = −2.9) PMF template in this work due to its connection to inflation. Quantum mechanical zero-point fluctuations during inflation can generate an approximately scale-invariant PMF with a spectral index close to n_B = −3 (e.g., Turner & Widrow 1988; Mack et al. 2002; Kahniashvili et al. 2008). In contrast, causal PMFs with bluer spectra require later phase transitions or other new physics after inflation (e.g., Durrer & Caprini 2003; Subramanian 2016). We also consider n_B = −1 and n_B = 2 for the forecasts in Section 4. For each value of n_B, we use publicly released modifications to CAMB from Zucca et al. (2016) to calculate a PMF template for the CMB TT, TE, EE, and BB power spectra at fixed parameters: ${B}_{1\mathrm{Mpc}}$ = 2.5 nG and β = 20.72 (${a}_{\nu }/{a}_{\mathrm{PMF}}={10}^{9}$). The calculated templates include tensor and vector PMF contributions; scalar contributions are not implemented. The lack of scalar terms should not matter for CMB polarization data. Temperature data are not the main focus of this work, and we find our Planck results to be consistent with Planck Collaboration et al. (2016), suggesting the impact of the scalar terms is minor. The PMF templates are plotted against other signals in the CMB B-mode power spectrum in Figure 2 and against each other in Figure 3.

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As noted in the previous section, the MCMC includes a scaling parameter, ${A}_{\mathrm{PMF}}$, for the PMF power spectrum template. We can translate the amplitude constraint on this template to a constraint on the rms magnetic field strength on 1 Mpc scales, ${B}_{1\mathrm{Mpc}}$, using the expected scaling:

$$\begin{eqnarray}&&{C}_{{\ell }}^{\mathrm{vec}}\propto {B}_{1\mathrm{Mpc}}^{4}.\end{eqnarray} \tag{ 2 }$$

An unfortunate consequence of this steep fourth-power scaling is that substantial reductions in the observationally allowed power lead to only modestly better limits on the magnetic field strength. However, if a PMF is detected, the steep scaling between ${B}_{1\mathrm{Mpc}}$ and the observed CMB B-mode power spectrum would allow excellent constraints on the PMF properties.

The power in the tensor modes also depends on the timing of PMF generation relative to neutrino decoupling, and is expected to scale approximately as β² (Lewis 2004; Shaw & Lewis 2010). We double-check this expectation for β in the range 2 to 20 using the Zucca et al. (2016) code, and find the computed tensor power scales as β^1.9 We use this β power-law form to handle the PMF timing dependence. Thus, the tensor PMF power will scale as

$$\begin{eqnarray}&&{A}_{\mathrm{PMF}}^{\mathrm{tens}}\propto {B}_{1\mathrm{Mpc}}^{4}{\beta }^{1.9}.\end{eqnarray} \tag{ 3 }$$

We use Fisher matrices to forecast the potential constraints on PMFs possible for each generation of experiments. The Fisher matrix, ${{ \mathcal F }}_{{ij}}$, can be defined as

$$\begin{eqnarray}&&{{ \mathcal F }}_{{ij}}=\displaystyle \frac{1}{2}\displaystyle \frac{{\partial }^{2}{\chi }^{2}}{\partial {p}_{i}\partial {p}_{j}},\end{eqnarray} \tag{ 4 }$$

for model parameters ${\boldsymbol{p}}$. The inverse of the Fisher matrix is the covariance matrix of the parameters for Gaussian likelihoods. While Fisher matrices can underestimate the true uncertainties for non-Gaussian likelihoods (see, e.g., Wolz et al. 2012), they make it easy to combine information from different experiments and estimate the final parameter constraints.

We include two external data sets in all forecasts. The first data set is the expected measurements of the TT, TE, and EE spectra from the Planck satellite. We include Planck TT information in the multipole range $2\leqslant {\ell }\leqslant 3000$. Due to the importance of galactic foreground removal at large scales in polarization, we only use TE and EE information starting from ℓ = 30. A prior on the optical depth of 0.005 is added to account for the optical depth constraint expected from the missing multipoles. Second, we include a 1% external measurement of the Hubble constant, such as is expected from the Taipan experiment (Kuehn et al. 2014).

To calculate the Fisher matrix for a CMB experiment, we need the bandpower covariance matrix and partial derivatives of the CMB spectra. The partial derivatives are calculated numerically from CAMB spectra (Lewis et al. 2000). We calculate the bandpower covariance matrix using the Knox formula (Knox 1997) for the survey parameters in Table 2. The sample variance contribution is calculated assuming the Planck best-fit ΛCDM cosmology with a small gravitational wave contribution (r = 0.01) added. This is the black line in Figure 1. We use a bandpower bin size of ${\rm{\Delta }}{\ell }=50$. We assume a 5% uncertainty on the beam FWHM and a 1% power calibration uncertainty. In Section 4.3.5, we test relaxing or tightening the beam and calibration uncertainty and find the beam and calibration uncertainties have a negligible impact on the PMF constraints.

One concern with combining Fisher matrices from different experiments is double-counting modes due to overlapping sky coverage. We avoid this problem in different ways in each of the CMB power spectra. For the TT and TE spectra, we only use the Planck measurements. We do not expect future experiments to substantially improve upon Planck in these spectra which are already cosmic variance limited out to fairly high multipoles. We do the opposite for the EE and BB spectra. We do not include Planck BB information in any forecast, and we discard Planck EE data in the overlap region by appropriately scaling the Planck EE uncertainties. We ignore overlaps between the stage III experiments on the basis that the sky overlap between a Chilean experiment like the Simons Array and South Pole experiment like SPT-3G will be small. The overlap issue is also the rationale behind using only one of the Simons Array and AdvACTpol as the two experiments are likely to have substantial sky overlap. The overlap in sky coverage will almost certainly be substantial between stage III and IV experiments; thus we do not include the stage III experiments, except as a prior on the polarized Poisson power, in the CMB-S4 constraints. This prior is only relevant in practice when the assumed CMB-S4 configuration has a larger beam size than SPT-3G. With these measures in place, this analysis should not double-count any modes.

We consider the constraints on PMFs in two cosmologies. Our fiducial cosmology is a 12-parameter model that extends ΛCDM with four commonly considered extensions as well as PMFs: ΛCDM + r + ${n}_{\mathrm{run}}$ + ${N}_{\mathrm{eff}}$ + $\sum {m}_{\nu }$ + ${A}_{\mathrm{PMF}}$ + β. Here, ${n}_{\mathrm{run}}$ is running of the scalar index, ${N}_{\mathrm{eff}}$ is the effective number of relativistic species, and $\sum {m}_{\nu }$ is the sum of the neutrino species. This 12-parameter model is our default cosmological model when forecasting future PMF constraints as these extensions are all well-motivated theoretically. We have examined the degree to which ${A}_{\mathrm{PMF}}$ is degenerate with the 10 parameters unrelated to PMFs—the only strong degeneracy is in the n_B = −2.9 case and is with r. The reason for this degeneracy is illustrated in Figure 2, which shows that the tensor mode power due to inflationary gravitational waves and a PMF with n_B = −2.9 is nearly identical on large angular scales. To test the degree to which parameter degeneracies limit the inferred constraints, we also quote constraints from a "minimal" eight-parameter model in which the PMF power is the only extension to ΛCDM: ΛCDM + ${A}_{\mathrm{PMF}}$ + β. In both cases, we always marginalize over unknown Poisson EE and BB terms due to polarized extragalactic sources. We assume the galactic foregrounds are removed by a judicious combination of data from multiple frequencies.

In this work, we restrict ourselves to power spectrum (i.e., two-point estimators) searches for PMFs. Currently the power spectrum limits from Planck or Polarbear are better than the four-point upper limits. Although outside the scope of this work, it would be interesting to extend this analysis to four-point estimators in the future. First, the four-point limits should improve faster as the noise level falls. Second, in the case of a detection, the four-point estimators would almost certainly come into play to learn more about the vector modes of the PMFs. The detection (or non-detection) of the vector PMF signal in the four-point estimators could then be used to argue for whether it is more likely that any observed tensor power is due to inflationary gravitational waves or PMFs. Finally, a lensing estimator might be used to "de-lens" the B-mode power spectrum, thereby suppressing the lensed B-mode signal and allowing better limits on PMFs. Such de-lensing techniques have long been proposed for inflationary gravitational wave searches (e.g., Kesden et al. 2002; Knox & Song 2002; Seljak & Hirata 2004; Simard et al. 2015) and more recently been demonstrated on real Planck data (Larsen et al. 2016).

The experiments that will begin taking data in 2017 will dramatically improve our knowledge of PMFs. In the minimal cosmological model, the 1-σ forecast for Planck+H₀ is $\sigma ({A}_{\mathrm{PMF}})=0.38$ for a best-motivated, nearly scale-invariant PMF spectrum (i.e., n_B = −2.9). We have set a Gaussian prior on β with σ_β chosen such that the Fisher forecast matches the actual upper limit from the Planck data. Adding EE and BB bandpowers from two stage III experiments, SPT-3G and Simons Array, reduces the uncertainty by more than an order of magnitude in the eight-parameter model to $\sigma ({A}_{\mathrm{PMF}})=0.020$. The relative improvement is larger in the more realistic 12-parameter model, as parameter degeneracies substantially weaken (by a factor of five) the Planck+H₀ constraints on ${A}_{\mathrm{PMF}}$, while weakening the stage III constraint by a more modest 40%. Thus within the 12-parameter model, the addition of stage III CMB experiments improves the ${A}_{\mathrm{PMF}}$ uncertainty by a factor of ∼35 to $\sigma ({A}_{\mathrm{PMF}})=0.022$. The parameter degeneracies largely disappear for steeper PMF spectra (i.e., n_B = −1 or 2) as the PMF B-mode spectra then peak at very small scales and this small-scale power cannot be mimicked by any of the other parameters. The improvement from adding the SPT-3G and Simons Array experiments to Planck remains impressive for these values of n_B. For the 12-parameter model and ${n}_{{\rm{B}}}=-1$, adding SPT-3G and Simons Array to Planck leads to a 17-fold reduction in $\sigma ({A}_{\mathrm{PMF}})$ from 0.38 to 0.023. For n_B = 2, there is a 20-fold reduction from $\sigma ({A}_{\mathrm{PMF}})=3.0\times {10}^{-6}$ to $1.5\,\times \,{10}^{-7}$. We can expect substantially tighter constraints on PMFs by the end of the decade.

The primary motivations behind the proposed CMB-S4 experiment are to search for inflationary gravitational waves and to measure the neutrino masses. However, CMB-S4 would also enable extremely sensitive searches for PMFs. We begin by considering our fiducial CMB-S4 configuration, as laid out in Table 2. For this fiducial configuration and the 12-parameter cosmological model, we find a threefold reduction over the stage III experiments for all three PMF template shapes. The expected uncertainties for CMB-S4 are $\sigma ({A}_{\mathrm{PMF}}^{{n}_{{\rm{B}}}=-2.9})=6.3\times {10}^{-3}$, $\sigma ({A}_{\mathrm{PMF}}^{{n}_{{\rm{B}}}=-1})=7.4\times {10}^{-3}$, and $\sigma ({A}_{\mathrm{PMF}}^{{n}_{{\rm{B}}}=2})=5.2\times {10}^{-8}$. These represent more than a 50-fold improvement on current limits.

Given that CMB-S4 is being designed, it is worth considering how design decisions would affect the final PMF constraints. We look at five aspects of the experiment: the instrumental sensitivity (as reflected by the final noise level in the maps), the telescope size or beam FWHM, the low-ℓ noise performance, the amount of sky surveyed, and how well the beam shape and calibration must be known. Of these five aspects, we find the first two to be very important to PMF searches and the second two to be somewhat important. As might be expected, in some cases the optimal design depends on the shape of the PMF spectrum (n_B) as this sets the relative power between large and small angular scales. However, as was discussed in Section 2.3, there are theoretical reasons to expect the PMF spectrum to be close to a scale-invariant spectrum. While for completeness we present results for three values of n_B, we recommend favoring the optima for n_B = −2.9.

4.3.1. Instrumental Sensitivity (Map Noise Levels)

Reducing the noise in CMB maps will monotonically improve power spectrum measurements. The goal of this section is to quantify the magnitude of that improvement: has the information gain saturated, or will further reductions in noise substantially improve PMF searches? Figure 4 shows the uncertainty on ${A}_{\mathrm{PMF}}$ as a function of the map noise level for the three PMF templates in either the 12- or eight-parameter cosmological models. For this figure, we have fixed the rest of the experimental configuration (i.e., everything except the polarized noise levels) to the values of the CMB-S4 row in Table 2. Clearly, continuing to improve the sensitivity of CMB experiments to CMB-S4 and beyond will be a major boon to PMF searches. In the general, 12-parameter cosmological model, the PMF uncertainty does not plateau in any of the three PMF templates considered until the map noise is at or below 0.3 μK arcmin (a factor of four lower than the fiducial CMB-S4 configuration).

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4.3.2. Size of Telescope

Larger telescopes can recover more modes on the sky, and should therefore always improve the PMF constraints. As shown in Figure 5, we find the gains due to resolution at fixed mapping speed to be substantial for all three power-law indices considered. We see a local plateau between FWHMs of 4' to 6'; improving the angular resolution in this range does little to aid PMF searches. We can understand this plateau by looking on the predicted PMF power in Figure 3. All three templates have two peaks across this range of angular multiples, and we would expect to see such a plateau when, for instance, the resolution is adequate to resolve the first peak, but not yet sufficient to resolve the second peak. The observed plateau positions are consistent with this hypothesis. Overall, a beam size of FWHM = 1' instead of 10' reduces the expected upper limit fivefold for n_B = −2.9 and 30- to 35-fold for the steeper PMF indices which are more heavily weighted toward small angular scales. The model only matters in the nearly scale invariant PMF case (${n}_{{\rm{B}}}=-2.9$), where the information on large angular scales dominates and the telescope size is relatively unimportant.

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However, larger telescopes are also more expensive to build, which means for a fixed experimental budget, they would necessitate less ambitious focal planes and lower instantaneous mapping speeds. We very crudely approximated a cost-neutral setup with a noise level of 1.3 μK arcmin for a FWHM of 4', a noise level of 2.8 μK arcmin for a FWHM of 2', and a noise level of 4.0 μK arcmin for a FWHM of 1'. We find that improving the mapping speed rather than telescope size leads to somewhat better potential limits for these experimental configurations, by a factor of 1.6, 1.1, or 1.1 for n_B = −2.9, −1, +2 respectively. However the results are close enough that the crudeness of the cost-neutral estimates used here is a worry, and the preference might flip for more accurately costed setups. In short, the tradeoff between larger telescopes or more detectors is largely a wash, although one might lean toward adding more detectors.

4.3.3. Survey Area

A third question is how much sky to observe with CMB-S4. Unsurprisingly, we find opposing preferences for the nearly scale-invariant n_B = −2.9 (which peaks at large angular scales) and the bluer n_B = −1 or 2 spectra which peak at smaller scales. As shown in Figure 6, observing at least 15% of the sky is very important to a search for nearly scale-invariant PMFs, and it is best to cover as much sky as possible. A caveat to this analysis is that it is likely easier to remove galactic foregrounds to a specified level on targeted, "clean" patches as opposed to a substantial fraction of the sky, and we would expect galactic foregrounds to be important for the large angular scales. That said, the PMF constraints improve by a factor of 1.4–1.5 (depending on the cosmological model) going from 10% to 25% of the full sky, and are nearly flat (a 6% improvement) from 25% to 70% of the sky (the widest area likely to be possible after galactic cuts). Conversely for ${n}_{{\rm{B}}}=-1$ or +2, the optimal survey area is around 15% of the full sky. Note, however, that the minimum is extremely broad with only a 10% worsening of the expected uncertainty as the survey area is increased from 10% to 70% of the full sky. A survey covering at least 25% of the sky would perform well for all three considered PMF models.

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4.3.4. Noise Performance at Large Angular Scales

Next we turn to the recovery of large angular scales on the sky, and the required noise performance at these low frequencies. For this, we look at the impact of shifting the 1/f knee of the map-space noise. Specifically, we are multiplying the noise power, ${N}_{{\ell }}$, which is a constant for white noise, by a function of angular multipole:

$$\begin{eqnarray}&&f({\ell })=1+{\left(\displaystyle \frac{{{\ell }}_{\mathrm{knee}}}{{\ell }}\right)}^{8/3}.\end{eqnarray} \tag{ 5 }$$

The exponent, 8/3, was selected based on a Kolmogorov spectrum of turbulence within a thin plane (Lay & Halverson 2000). Note that this knob serves as a placeholder for several effects, including a signal-to-noise hit due to galactic foregrounds or the methods used to clean these foregrounds, atmospheric noise, or actual instrumental low-frequency noise.

We generally find the forecasts to be insensitive to the 1/f knee or galactic foreground removal at the default beam size of 4'. As Shown in Figure 7, it appears that even in the nearly scale-invariant case, the PMF constraint is coming primarily from smaller angular scales due to the degeneracy at large angular scales with both the tensor-to-scalar ratio and the PMF timing parameter β. For the bluer PMF spectra (n_B = −1 or 2), the forecasts show no dependence on the knee frequency. For the fiducial 12-parameter model, our ability to separate a PMF from other physics is coming from sub-degree angular scales. The low-frequency noise performance of CMB-S4 is not crucial to PMF searches.

4.3.5. Beam and Calibration Uncertainties