Current and Future Constraints on Primordial Magnetic Fields

TL;DR

This paper provides new constraints on primordial magnetic fields using CMB data, and forecasts future experimental capabilities to further tighten these limits, potentially ruling out large portions of parameter space.

Contribution

It offers the most stringent current limits on PMFs from CMB measurements and predicts how upcoming experiments will improve these constraints significantly.

Findings

Current 95% CL upper limit on PMF amplitude is halved.

Future CMB experiments could reduce uncertainties by up to two orders of magnitude.

CMB-S4 has the potential to exclude much of the viable PMF parameter space.

Abstract

We present new limits on the amplitude of potential primordial magnetic fields (PMFs) using temperature and polarization measurements of the cosmic microwave background (CMB) from Planck, BICEP2/Keck Array, POLARBEAR, and SPTpol. We reduce twofold the 95% CL upper limit on the CMB anisotropy power due to a nearly-scale-invariant PMF, with an allowed B-mode power at $\ell=1500$ of $D_{\ell=1500}^{BB} < 0.071 \mu K^2$ for Planck versus $D_{\ell=1500}^{BB} < 0.034 \mu K^2$ for the combined dataset. We also forecast the expected limits from soon-to-deploy CMB experiments (like SPT-3G, Adv. ACTpol, or the Simons Array) and the proposed CMB-S4 experiment. Future CMB experiments should dramatically reduce the current uncertainties, by one order of magnitude for the near-term experiments and two orders of magnitude for the CMB-S4 experiment. The constraints from CMB-S4 have the potential to rule out much of the parameter space for PMFs.

Figures

Tables

Table 1. Table 1 Parameter Constraints

Model	$A_{\mathrm{PMF}}$	$A_{\mathrm{lens}}$	$r$
$\Lambda$CDM
+ $A_{\mathrm{PMF}}$		–	–
+ $A_{\mathrm{PMF}}$ + $A_{\mathrm{lens}}$		$1.123\pm 0.064$	–
+ $A_{\mathrm{PMF}}$ + $A_{\mathrm{lens}}$ + $r$		$1.127\pm 0.063$

Table 2. Table 2 Assumed survey parameters

Experiment	Sky coverage	Polarized Noise level	1/$f$ knee	Beam FWHM
		($\mu$K-arcmin)		(arcmin.)
CMB Stage III
SPT-3G	6%	3.0	200	1.2
Simons Array	36%	9.5	200	3.5
CMB Stage IV	55%	1.3	100	4.0

Equations

$${\color[rgb]{0,0,0}C_{\ell}=C_{\ell}^{\rm{CAMB}}+A_{\rm PMF}\left[C_{\ell}^{\rm PMF,\,vec}+\left(\frac{\beta}{20}\right)^{1.9}C_{\ell}^{\rm PMF,\,tens}\right]}$$

$$C_{\ell }^{\mathrm{vec}}\propto B_{1\mathrm{Mpc}}^4.$$

$$A_{\mathrm{PMF}}^{tens}\propto B_{1\mathrm{Mpc}}^4{\beta }^{1.9},$$

$${\mathcal{F}}_{ij}=\frac{1}{2}\frac{{\partial }^2{\chi }^2}{\partial p_i\partial p_j},$$

$$f(\ell )=1+{\left(\frac{{\ell }_{\mathrm{knee}}}{\ell }\right)}^{8/3}.$$

Full text

1. Introduction

Measurements of the cosmic microwave background (CMB) temperature anisotropy have provided some of the most powerful tests of cosmology. We are now entering a new era as experiments begin to measure polarized “B-modes” in the CMB for the first time (Hanson et al., 2013; The Polarbear Collaboration: Ade et al., 2014; Ade et al., 2014; Naess et al., 2014; Keisler et al., 2015; Keck Array and BICEP2 Collaborations: Ade et al., 2016; Louis et al., 2016). Precision measurements of CMB polarization promise new tests of the standard cosmological model. The best known of these tests are the searches for inflationary gravitational waves in B-modes at large angular scales and plans to measure the sum of the neutrino masses through the lensing B-modes on small angular scales (for a recent review see, Abazajian et al., 2016).

CMB B-mode measurements can also be used to constrain more exotic models, such as the possible existence of cosmic birefringence (Carroll, 1998; Lue et al., 1999) or primordial magnetic fields (PMFs) (Kosowsky & Loeb, 1996; Seshadri & Subramanian, 2001). Both effects lead to the rotation of E-modes into B-modes, thereby increasing the amplitude of the BB power spectrum. By equating the magnitude of the resulting B-modes, parity-violating processes can be translated into an equivalent PMF strength. It is therefore common to simply quote effective limits on PMFs. In this work, we present new upper limits from current CMB polarization data on the possibility of PMFs or parity-violating physics.

Magnetic fields are ubiquitous in astronomy and are found almost universally in collapsed objects from stars to galaxies and galaxy clusters (for a review, see Ryu et al., 2012; Widrow et al., 2012). There is even some evidence for magnetic fields in intergalactic space from Fermi-LAT data (Neronov & Vovk, 2010). High energy $\gamma$-rays from blazars should produce electron-positron pairs when the $\gamma$-rays collide with IR or optical photons. These pairs should later annihilate at GeV energies, but the expected GeV flux is missing in the Fermi-LAT observations. The missing flux could be explained by the deflection of particles due to intergalactic magnetic fields. There are, however, alternative explanations involving instabilities in the electron-positron plasma, leading pairs to deposit their energy in the intergalactic medium in the form of heat (Broderick et al., 2012). If the loss of flux is indeed due to magnetic fields, then the GeV results set lower limits on the intergalactic magnetic field strength of $10^{-9}-10^{-6}$ nG (Tavecchio et al., 2010; Taylor et al., 2011; Dermer et al., 2011; Vovk et al., 2012).

The mechanism to create large-scale magnetic fields, especially in intergalactic space, remains unclear. One popular proposal is that the observed fields are the product of PMFs, which are predicted by several theories of the early Universe (e.g., Turner & Widrow, 1988; Grasso & Riotto, 1998; Ichiki et al., 2006). Adiabatic compression and turbulent shocks during later structure formation amplify these initial seed PMFs into the stronger magnetic fields observed today. Of course, this amplification process may be seeded through other means, such as AGN or galactic dynamos (for a review, see Giovannini, 2004). However, the possibility of PMFs opens up the intriguing idea that observations of large-scale magnetic fields may offer insights into the physics of the very early Universe.

PMFs would have observational consequences for Big Bang nucleosynthesis (e.g., Kahniashvili et al., 2010), large scale structure (e.g., Battaner et al., 1997), the ionization history of the Universe (Kunze & Komatsu, 2015), and the black body spectrum of the CMB (e.g., Kunze & Komatsu, 2014), as well as the CMB anisotropies. The CMB anisotropies have yielded some of the strongest constraints on PMFs and are the focus of this work. There have been three recent results of note. Planck Collaboration et al. (2016) have used the Planck 2015 release of temperature and polarization data to set limits on a variety of PMF models. With the CMB power spectrum data that will be the focus of this paper, Planck Collaboration et al. (2016) find 95% CL upper limits ranging from $B_{\rm 1\,Mpc}<5.6$ nG to $<0.7$ nG depending on the exact model. Here $B_{\rm 1\,Mpc}$ is the rms magnetic field strength on a 1 Mpc length scale. The Polarbear collaboration also recently announced limits on PMFs from the Polarbear data using either a 4-point estimator or B-mode power spectrum measurement (Ade et al., 2015). The strongest constraints were from the B-mode spectrum; the observed upper limit was $B_{\rm 1\,Mpc}<3.9$ nG. Two other experiments, SPTpol and BICEP2/Keck Array, have also reported B-mode power spectrum measurements (Keisler et al., 2015; Keck Array and BICEP2 Collaborations: Ade et al., 2016), and Zucca et al. (2016) reported constraints from the combination of the Planck and SPTpol bandpowers. In this paper, we will combine the data from all four experiments to determine upper limits on PMFs, and find that the BICEP2/Keck Array data in particular improves these upper limits. We then present Fisher matrix forecasts on PMF models for the stage-III and stage-IV CMB experiments being built or designed right now.

The outline of this paper is as follows. In §2, we lay out the data and how we use the data to put limits on PMFs. We present the results of this analysis on current data in §3. In §4, we forecast constraints from future experiments using a Fisher matrix formulism. We conclude in §5.

2. Data and Methods

We use Markov chain Monte Carlo (MCMC) methods to study constraints on PMFs. In this section, we first describe the CMB temperature and polarization anisotropy data used, and then discuss the MCMC implementation.

2.1. Data

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 and BICEP2 Collaborations: Ade et al., 2016). This last dataset also places limits on the tensor modes of a PMF due to its coverage of lower multipoles. During the writing of this work, 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 Fig. 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 don’t 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 Fig. 1.

2.2. Methods

We use MCMC methods to determine the parameter constraints reported in this work. The results are calculated using the CosmoMC111http://cosmologist.info/cosmomc (July 2015 version) package (Lewis & Bridle, 2002). CosmoMC invokes CAMB222http://camb.info (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 $A_{\rm PMF}$.

[TABLE]

The calculation of the PMF templates, $C_{\ell}^{\rm PMF,\,vec}$ and $C_{\ell}^{\rm PMF,\,tens}$, and the motivation for the $\beta$ scaling are described in §2.3. The parameter $\beta={\rm ln}(a_{\nu}/a_{\rm{PMF}})$ 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 we only use CMB data.

For the real data, unless noted, we assume the six-parameter, spatially-flat $\Lambda$CDM model with a single massive neutrino of 60 meV and 2 parameters, $A_{\rm PMF}$ and $\beta$, 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, $A_{\rm PMF}$, not the rms magnetic field strength, $B_{\rm 1\,Mpc}$, 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 $\beta$ matters. Here we follow Planck Collaboration et al. (2016) and Zucca et al. (2016) and use $\beta\in[11.513,41.447]$.333This corresponds to $log_{10}(a_{\nu}/a_{\rm{PMF}})\in[4,17]$. However, the constraints, especially those from Planck alone, weaken substantially if the lower bound on $\beta$ 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 dataset. We sometimes also add $A_{\rm lens}$ as a simple way of marginalizing over uncertainty in the predicted lensed BB power spectrum for any extension to the $\Lambda$CDM model. Finally, we sometimes allow non-zero tensors, parameterized by the tensor-to-scalar ratio, r.

2.3. Primordial Magnetic Field Template

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, $A_{B}k^{n_{B}}$, 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 $B_{\rm 1\,Mpc}$.

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 by decoupled or partially-decoupled neutrinos. The timing is parametrized by $\beta=ln(a_{\nu}/a_{\rm{PMF}})$, 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 3-dimensional PMF parameter space. For the real data, we focus on 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 §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: $B_{\rm 1\,Mpc}$= 2.5 nG and $\beta=20.72$ ($a_{\nu}/a_{\rm{PMF}}=10^{9}$). 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 is 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 Fig. 2 and against each other in Fig. 3.

As noted in the last section, the MCMC includes a scaling parameter, $A_{\rm PMF}$, for the PMF power spectrum template. We can translate the amplitude constraint on this template to a constraint on the r.m.s. magnetic field strength on 1 Mpc scales, $B_{\rm 1\,Mpc}$, using the expected scaling:

[TABLE]

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 $B_{\rm 1\,Mpc}$ 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 $\beta^{2}$ (Lewis, 2004; Shaw & Lewis, 2010). We double-check this expectation for $\beta$ in the range 2 to 20 using the Zucca et al. (2016) code, and find the computed tensor power scales as $\beta^{1.9}$ We use this $\beta$ power law form to handle the PMF timing dependence. Thus the tensor PMF power will scale as:

[TABLE]

3. Results

We begin by investigating how the addition of ground-based CMB polarization data to Planck data affects the limits on PMFs. For the real data, we focus on the most theoretically motivated scenario of an approximately scale-invariant spectrum for the PMFs ($n_{B}=-2.9$), although we will consider other scenarios in the forecasts of the next section. The ground-based CMB polarization data substantially reduce the allowed PMF power; we see a factor of two reduction over Planck alone in the simple 8-parameter model. The 95% confidence upper limit for Planck is $A_{\rm PMF}<0.76$ for $\Lambda$CDM+PMF. Adding the ground-based experiments brings this limit to $A_{\rm PMF}<0.36$. To facilitate a physical interpretation, we can restate these limits as an allowed B-mode power at $\ell=1500$, which is near the peak of the PMF contribution. The equivalent limits on the allowed B-mode power are $D_{\ell=1500}^{BB}<0.071\mu K^{2}$ for Planck-alone and $D_{\ell=1500}^{BB}<0.034\mu K^{2}$ when the ground-based experiments are added. We test which datasets are important by adding single datasets to the Planck data, and find that the source of the improvement is evenly split between SPTpol and BICEP2/Keck Array. Adding either one to Planck yields the same 95% CL upper limit: $A_{\rm PMF}^{{\sl Planck}+BICEP2/KeckArray}<0.44$ versus $A_{\rm PMF}^{{\sl Planck}+SPTpol})<0.45$. Extending the $\beta$ prior to lower values would shift the relative weight more towards SPTpol (and vice-versa). Similarly, increasing the value of $n_{B}$ would increase the relative importance of the smaller angular scales measured by SPTpol. In short, the source of the factor of two reduction in the upper limit on $A_{\rm PMF}$ for a nearly-scale-invariant PMF spectrum ($n_{B}=-2.9$) is due to a more accurate measurement of the B-mode power spectrum at all scales.

We next look at the model dependence of these upper limits. We list the limits for the combined dataset with different parameter sets in Table 1. First, we vary the largest source of B-mode power, gravitational lensing, by allowing $A_{\rm lens}$ to float. Somewhat counterintuitively, the limits are somewhat better with $A_{\rm lens}$ allowed to vary (this is due to a mild preference for $A_{\rm lens}$ above unity which increases the lensed B-mode power). In this case for Planck alone, the 95% confidence upper limit is $A_{\rm PMF}<0.64$, dropping to $A_{\rm PMF}<0.30$ with the ground-based experiments. Next we introduce inflationary gravitational waves, as parameterized by the tensor-to-scalar ratio r. The 95% CL upper limit on $A_{\rm PMF}$ drops again. This is easily understood as a splitting of the large-scale B-mode power between two positive-definite terms: r and $A_{\rm PMF}$.

We also propagate the limits on the observed PMF power into limits on the magnetic field strength, $B_{\rm 1\,Mpc}$. We do this using the scaling of Eqn. 2, $A_{\rm PMF}\propto B_{\rm 1\,Mpc}^{4}$. Given this scaling, the observed upper limits with a flat prior on $A_{\rm PMF}$ would lead to an apparent “detection” of $B_{\rm 1\,Mpc}$. We therefore importance sample the chains to apply a flat prior on $B_{\rm 1\,Mpc}$. We find a 95% CL upper limit of $B_{\rm 1\,Mpc}<1.8$ nG for Planck alone, similar to the limit found by Planck Collaboration et al. (2016) of $B_{\rm 1\,Mpc}<2.0$ nG with $n_{B}=-2.9$ held fixed as we have done. Adding the ground-based polarization measurements significantly reduces the 95% CL upper limit to $B_{\rm 1\,Mpc}<0.91$ nG. Due to the inclusion of the BICEP2/Keck Array and POLARBEAR data, these results are slightly lower than the Planck + SPTpol upper limit of $<1.2\,$nG quoted by Zucca et al. (2016).

4. Forecasts

The sensitivity of CMB experiments is increasing rapidly due to the continued growth in the number of detectors. Abazajian et al. (2016) thus defines four stages of CMB experiments. Current experiments are called stage II. The B-mode measurements from these stage II experiments were used for PMF constraints in the last section and are plotted in Fig. 1. Stage III experiments such as SPT-3G, the Simons Array, or AdvACTPol (Benson et al., 2014; Suzuki et al., 2016; Henderson et al., 2016) have approximately ten times more detectors, and generically will start collecting data around 2017 and finish three to four years later. In this section, we forecast the expected constraints from the EE and BB power spectrum measurements of the stage III experiments by combining forecasts for SPT-3G and the Simons Array. There is also a proposal to build a stage IV experiment, CMB-S4, that would increase the detector counts by another order of magnitude and hopefully begin taking data in the early 2020s. A pathfinder to CMB-S4, the Simons Observatory, was funded in 2016. We examine the likely PMF constraints from the EE and BB power spectrum measurements with CMB-S4, and look at how to design CMB-S4 to maximize its potential for PMF searches.

4.1. Methods and Experimental Parameters

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:

[TABLE]

for model parameters 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 datasets in all forecasts. The first dataset is the expected measurements of the TT, TE and EE spectra from the Planck satellite. We include Planck TT information in the multipole range $2\leq\ell\leq 3000$. Due to the importance of galactic foreground removal at large scales in polarization, we only use TE and EE information starting from $\ell=30$. 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 $\Lambda$CDM cosmology with a small gravitational wave contribution ($r=0.01$) added. This is the black line in Fig. 1. We use a bandpower bin size of $\Delta\ell=50$. We assume a 5% uncertainty on the beam FWHM and a 1% power calibration uncertainty. In §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 the 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 throw away 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 $\Lambda$CDM with four commonly considered extensions as well as PMFs: $\Lambda$CDM + $r$ + $n_{\rm run}$ + $N_{\rm eff}$ + $\sum m_{\nu}$ + $A_{\rm PMF}$ + $\beta$. Here, $r$ is the tensor-to-scalar ratio, $n_{\rm run}$ is running of the scalar index, $N_{\rm eff}$ 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 $A_{\rm PMF}$ is degenerate with the 10 parameters unrelated to PMFs – the only strong degeneracy is in the $n_{B}\,=\,-2.9$ case and is with the tensor-to-scalar ratio, $r$. The reason for this degeneracy is illustrated in Fig. 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’ 8-parameter model in which the PMF power is the only extension to $\Lambda$CDM: $\Lambda$CDM + $A_{\rm PMF}$ + $\beta$. 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. 2-point estimators) searches for PMFs. Currently the power spectrum limits from Planck or Polarbear are better than the 4-point upper limits. Although outside the scope of this work, it would be interesting to extend this analysis to 4-point estimators in the future. First, the 4-point limits should improve faster as the noise level falls. Second, in the case of a detection, the 4-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 4-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-lense’ 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. Knox & Song, 2002; Kesden et al., 2002; Seljak & Hirata, 2004; Simard et al., 2015) and more recently been demonstrated on real Planck data (Larsen et al., 2016).

4.2. Stage III forecasts

The experiments that will begin taking data in 2017 will dramatically improve our knowledge of PMFs. In the minimal cosmological model, the 1-$\sigma$ forecast for Planck+H0 is $\sigma(A_{\rm PMF})=0.38$ for a best-motivated, nearly-scale invariant PMF spectrum (i.e. $n_{B}=-2.9$). We have set a Gaussian prior on $\beta$ with $\sigma_{\beta}$ 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 8-parameter model to $\sigma(A_{\rm PMF})=0.020$. 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 $A_{\rm PMF}$, 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 $A_{\rm PMF}$ uncertainty by a factor of $\sim$ 35 to $\sigma(A_{\rm PMF})=0.022$. The parameter degeneracies largely disappear for steeper PMF spectra (i.e. $n_{B}=-1$ or 2) as the PMF B-mode spectra then peaks at very small scales and this small-scale power can not 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 $n_{B}=-1$, adding SPT-3G and Simons Array to Planck leads to a 17-fold reduction in $\sigma(A_{\rm PMF})$ from 0.38 to 0.023. For $n_{B}=2$, there is a 20-fold reduction from $\sigma(A_{\rm PMF})=3.0\times 10^{-6}$ to $1.5\times 10^{-7}$. We can expect substantially tighter constraints on PMFs by the end of the decade.

4.3. Stage IV forecasts

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 3-fold reduction over the stage III experiments for all three PMF template shapes. The expected uncertainties for CMB-S4 are $\sigma(A_{\rm PMF}^{n_{B}=-2.9})=6.3\times 10^{-3}$, $\sigma(A_{\rm PMF}^{n_{B}=-1})=7.4\times 10^{-3}$, and $\sigma(A_{\rm PMF}^{n_{B}=2})=5.2\times 10^{-8}$. 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-$\ell$ 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 §2.3, there are theoretical reasons to expect the PMF spectrum to be close to a scale-invariant spectra. 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 $A_{\rm PMF}$ as a function of the map noise level for the three PMF templates in either the 12 or 8-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 $0.3\,\mu{\rm K-arcmin}$ (a factor of four lower then the fiducial CMB-S4 configuration).

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 Fig. 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 Fig. 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 five-fold for $n_{B}=-2.9$ and 30- to 35-fold for the steeper PMF indices which are more heavily weighted towards small angular scales. The model only matters in the nearly-scale invariant PMF case ($n_{B}=-2.9$), where the information on large angular scales dominates and the telescope size is relatively unimportant.

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 $\mu$K-arcmin for a FWHM of 4′; a noise level of 2.8 $\mu$K-arcmin for a FWHM of 2′; and a noise level of 4.0 $\mu$K-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 towards 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 Fig. 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. With 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 $n_{B}=-1$ 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.

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:

[TABLE]

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′. 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 $\beta$. 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

Finally, we consider if searches for PMFs introduce new requirements on the accuracy to which the beam or calibration of an experiment must be known. We find they do not. We parameterize the beam uncertainty as a fractional uncertainty on the FWHM of a Gaussian beam, and calibration uncertainty as an overall power uncertainty. We find negligible, sub-percent shifts in the forecasted PMF uncertainties for calibration uncertainties from 0.2 to 5% and beam FWHM uncertainties from 2 to 12.5%. The PMF constraints for all three power law indices considered are insensitive to the beam and calibration uncertainties.

5. Conclusions

In this work, we have improved the current upper limits on the strength of primordial magnetic fields by including more CMB B-mode polarization data. By adding BICEP2/Keck Array, Polarbear, and SPTpol to Planck we have found that the 95% CL upper limit on the PMF power falls nearly two-fold from $A_{\rm PMF}<0.76$ to $A_{\rm PMF}<0.36$. The two major contributors to this improvement are the low-$\ell$ B-mode data from BICEP2/Keck Array and the high-$\ell$ B-mode data from SPTpol.

We have also shown that the next generation of experiments should dramatically reduce these limits, with the potential to detect PMFs for the first time. The so-called stage III experiments, which will begin taking data in 2017, can be expected to set upper limits at the level of $A_{\rm PMF}<0.02$ even after marginalizing over a six-parameter extension to $\Lambda$CDM and foregrounds.

The potential for detection increases even further with planned experiments like the Simons Observatory or CMB-S4. We have shown that an ideal version of CMB-S4 might decrease the 95% CL upper limits eight-fold, to $A_{\rm PMF}<0.0025$, and more realistic versions can still set upper limits on order of $A_{\rm PMF}<0.006$ in a 12-parameter cosmological model with $\Lambda$CDM + $r$ + $n_{\rm run}$ + $N_{\rm eff}$ + $\sum m_{\nu}$ + $A_{\rm PMF}$ +$\beta$. This represents a three-fold improvement over the forecasts for the stage III experiments and a more than 100-fold improvement over the current limits for this cosmological model. We have considered how the design of future experiments will impact the resulting PMF limits, looking at the experimental sensitivity, angular resolution, survey area, low-frequency noise performance, and beam or calibration uncertainties. We have found an experiment’s sensitivity followed by telescope size to be the most important factors in predicting the PMF limits. CMB-S4 will be a very exciting probe of PMFs and other sources of cosmic birefringence.

We thank the referee as well as Srinivasan Raghunathan and Federico Bianchini for valuable feedback on the manuscript. CR is the recipient of an Australian Research Council Future Fellowship (FT150100074), and also acknowledges support from the University of Melbourne. CF acknowledges support from NASA grants NASA NNX16AJ69G and NASA NNX16AF39G. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science.