Matter power spectrum induced by primordial magnetic fields: from the linear to the non-linear regime

TL;DR

This paper investigates how primordial magnetic fields (PMFs) influence the matter power spectrum across linear and non-linear regimes by performing dedicated MHD simulations in a post-recombination universe. It confirms linear theory on large scales but reveals that the baryon power spectrum saturates at an O(1) level on small scales, contrary to prior expectations, and shows that earlier extrapolations overestimate total matter power near the non-linear transition. The authors extend the analysis to Batchelor-spectrum PMFs, provide semi-analytical fits for the PMF-induced power spectra, and demonstrate that previous PMF constraints from structure formation may need revision. The results offer a more accurate framework for using LSS and related tracers to constrain PMFs and highlight the importance of non-linear MHD dynamics in early-universe magnetism.

Abstract

Linear theory predicts that primordial magnetic fields (PMFs) enhance the matter power spectrum on small scales. However, the linear approximation breaks down on sufficiently small scales where PMF-induced baryon perturbations back-react onto the magnetic fields. Previous studies assumed that the baryon power spectrum would be sharply suppressed in this non-linear regime, based on arguments related to the magnetic Jeans scale. For the first time, we perform dedicated magnetohydrodynamic (MHD) simulations to investigate the transition from the linear to the non-linear regime. Our simulations confirm the expected linear behavior on large scales. In the non-linear regime, however, we find that the dimensionless baryon power spectrum saturates to an $\mathcal{O}(1)$ value, which contrasts with previous analytical expectations. Additionally, our results show that several past studies overestimated the total matter power spectrum by orders of magnitude near the transition to non-linearity. Thus, the results presented in this work are useful to obtain more accurate constraints on PMFs from structure formation processes and/or different tracers of cosmic structures.

Figures (12)

• Figure 1: Left: Evolution of baryon (solid) and dark matter (dashed) density perturbations. The perturbations have been normalised with $3M_{\rm Pl}^2S_B/[a^3\rho_{\rm m}]$, which parameterises the Lorentz force. The evolution of $\xi$ is identical for all modes whose wave numbers are much smaller than the thermal Jeans scale near recombination, $k_{\rm th}$. Right: Dimensionless baryon power spectrum for linearized solution at $a=0.01$ using eq. \ref{['eq:delta_an']}. The actual baryon power spectrum is expected to deviate from the shown plot of $\Delta_{\rm b}$ on scales where they are suppressed.
• Figure 2: A thin slice of 0.3 Mpc$/h$ is cut out of the simulation A volume and projected onto a 2D plane. This simulation has $B_{\rm 1Mpc}=0.2$ nG and $n_B=-2$. Left panel shows the map of PMF strength and the right panel shows the overdensity map of the total matter. The limits on the colour-bars are chosen manually for better visualization. These limits are not to be confused with the maximum/minimum values observed in the simulation. The square black box surrounds a region with an overdensity at $z=4$.
• Figure 3: Dimensionless power spectrum at different redshifts for $B_{\rm 1Mpc}=0.2$ nG and $n_{\rm B}=-2$. From left to right we have power spectra of primordial magnetic fields, baryon perturbations, and total matter perturbations. The dashed lines in the panels correspond to the power spectrum computed in the linear limit (eq. \ref{['eq:Delta_b']}). The vertical grey lines mark the scale below which our initial conditions for matter density perturbations are not appropriate.
• Figure 4: Left: Dimensionless magnetic field power spectrum normalized with total PMF strength, $\langle B^2\rangle$, evaluated at a redshift of 100. The solid lines show spectra at $z=100$, while the dashed lines show spectra at $z=1090$. The $x$-axis has been normalized with $\lambda_{\rm D}$ found using eq. \ref{['eq:B_B1mpc']}. The black, blue, and green lines are for different simulations but all have $n_{\rm B}=-2.0$. As simulation A, B, and C only differ in resolution, we normalize the power spectrum for simulation B and C using the value of $\langle B^2\rangle$ and $\lambda_{\rm D}$ that were found for simulation A. For simulation D, $\langle B^2\rangle$ and $\lambda_{\rm D}$ are evaluated separately. The orange dashed line corresponds to the power spectrum given in eq. \ref{['eq:PB_powerlaw3']} and is matched to have the same $\langle B^2\rangle$ as simulation A. Right: Dimensionless baryon and dark matter power spectrum for different simulations. The colour code is the same as in the left panel. The orange-dashed line is an analytical fit with an exponentially suppressed power on small scales. The thick orange line is the non-linear power spectrum obtained after inputting the analytical power spectrum in NGenIC. The figures highlight that the shape of the power spectra remains unchanged when changing $B_{\rm 1Mpc}$ but changes on small scales with a change in resolution.
• Figure 5: Same as figure \ref{['fig:slice']} but for simulation E ($n_B=2$).