Massive Neutrinos and Magnetic Fields in the Early Universe

TL;DR

The paper develops a comprehensive 3+1 gauge-invariant Boltzmann framework to study primordial magnetic fields in the early Universe with massive neutrinos, deriving leading $m^2$ corrections to neutrino perturbations and incorporating them into the Boltzmann hierarchy for scalar, vector, and tensor modes. It computes full CMB power spectra sourced by inhomogeneous magnetic fields, including cross-correlations between the magnetic density perturbation $ riangle_B$ and anisotropic stress $ abla_B$, and distinguishes compensated and passive magnetic modes. A key finding is that neutrino mass modestly alters large-scale magnetic sourcing, but the effect is far smaller than previously claimed, while cross-correlations can raise magnetic contributions by roughly 15–25% across scales. The work also refines numerical treatments (tight-coupling and early-time instabilities) and showcases that passive magnetic modes are typically the strongest magnetic signature, setting robust constraints on primordial fields in light of current and upcoming CMB data. Overall, the results temper earlier expectations of dramatic large-scale enhancements due to massive neutrinos and provide a more reliable basis for using CMB observations to bound primordial magnetic fields, with $B_ ext{lambda} oughly{ ext{a few nG}}$ sensitivity in realistic scenarios.

Abstract

Primordial magnetic fields and massive neutrinos can leave an interesting signal in the CMB temperature and polarization. We perform a systematic analysis of general perturbations in the radiation-dominated universe, accounting for any primordial magnetic field and including leading- order effects of the neutrino mass. We show that massive neutrinos qualitatively change the large- scale perturbations sourced by magnetic fields, but that the effect is much smaller than previously claimed. We calculate the CMB power spectra sourced by inhomogeneous primordial magnetic fields, from before and after neutrino decoupling, including scalar, vector and tensor modes, and consistently modelling the correlation between the density and anisotropic stress sources. In an appendix we present general series solutions for the possible regular primordial perturbations.

Figures (5)

• Figure 1: The scalar power spectra with and without the cross-correlation between $\Delta_B$ and $\Pi_B$. Inclusion of it in calculations gives a consistent increase in power of around 15--25 percent at all scales.
• Figure 2: The four CMB power spectra plotted for a realistic neutrino mass $\sum m_\nu = 0.47 \:\mathrm{eV}$, with a magnetic field $B_\lambda = 4.7 \:\mathrm{nG}$. We include the scalar primary contribution for the TT,EE and TE power spectra, and the tensor primary (with a tensor to scalar ratio of 0.1) and for the BB power spectrum. The shaded regions represent the regions we would expect the passive modes to lie within for production between the reheating and the electroweak transition.
• Figure 3: The compensated vector contributions to angular power spectra of the temperature and polarization of the CMB. For each spectrum we plot three different cases, for purely massless neutrinos (dashed), and for massive neutrinos ($\sum m_\nu = 1.8 \mathrm{eV}$) calculated using the CAMB defaults (dotted), or our modified version (solid). In all cases we use a magnetic field of $B_\lambda = 4.7 \mathrm{nG}$. We also include the primary contribution to the spectrum in each case (thick solid), scalar perturbations for the TT, EE, TE plots, and the gravitational wave contribution to BB. Whilst both massive neutrino cases contain significant large scale power compared to the massless neutrinos, our modifications avoid the artificial increase at very low $l$ given by the CAMB default.
• Figure 4: The compensated tensor mode to the four CMB angular power spectra of temperature and polarization. This is the tensor equivalent of Fig. \ref{['fig:cmb_vector']}. The solid line is our modified version, dotted the CAMB default and dashed the massless case. The CAMB default exhibits the same small $l$ excess as in the vector case, and as before our modified version avoids this.
• Figure 5: The evolution of the tensor metric perturbation $H^{{ \left(2\right)}}$ (left panels), and the total anisotropic stress $\Pi^{{ \left(2\right)}}$ plotted against the scale factor $a$, at various wavenumbers. In the top panel we show the evolution with massless neutrinos. The middle panels illustrate the behaviour when we instead use three massive neutrinos $\sum m_\nu = 0.18 \mathrm{eV}$, with the default behaviour of CAMB. The problems stemming from the integration accuracy are readily apparent at early times. The bottom panels show the correct evolution of the massive neutrinos with our modifications. The degenerate evolution at small $k$ is apparent.