Primordial magnetogenesis in loop quantum cosmology

TL;DR

Primordial magnetic fields offer a window into early-universe physics, but their origin is uncertain. This work analyzes magnetogenesis in loop quantum cosmology by treating the electromagnetic field as a test field coupled to the background via a time-dependent function, and tracks its evolution through a quantum bounce into the inflationary era. The resulting magnetic power spectra are strongly scale-dependent due to bounce dynamics, with present-day field strengths ranging from nano-gauss to micro-gauss depending on the scalar potential and initial conditions; backreaction is negligible for Starobinsky but can be sizable for a quadratic potential. The findings highlight how pre-inflationary quantum gravity effects can leave imprints in the primordial magnetic field, and show how coupling choices and initial-state timing influence both the spectra and observability under current cosmological constraints.

Abstract

Primordial magnetic fields (PMFs) are magnetic fields generated during the early universe. These fields are thought to be the seeds of extragalactic magnetic fields. The origin of PMFs is not well known. Further, if they are indeed sources of extragalactic fields, then there is a possibility that observations of extragalactic magnetic fields could provide insights into the primordial physics. With this motivation, we study the generation of the primordial magnetic field in the context of loop quantum cosmology (LQC). In LQC, inflation is preceded by a quantum bounce. In this work, we consider an electromagnetic field coupled to the background as a test field and study its evolution through the bounce and through the subsequent inflationary phase. We investigate the power spectra generated in LQC and show that it is scale-dependent. We study the power spectra with different initial conditions, discuss equivalent forms of coupling functions, investigate backreaction, and compute the amount of primordial magnetic field which can be measured today. We conclude the article with a summary and discussion of the results.

Figures (7)

• Figure 1: Plot of magnetic (left) and electric (right) power spectra generated in LQC. We have imposed the Minkowski initial condition at the bounce. We see that the power spectra of magnetic and electric fields generated in LQC are scale-dependent. Though magnetic power spectra at large wavenumbers are scale invariant, intermediate wavenumbers behave as $k^{-2}$ and infrared modes as $k^4$. The magnetic spectrum scales as $1/k$ while transitioning between intermediate and large $k$. Electric power spectra have a $k^2$ behaviour at large wavenumbers. For smaller wavenumbers, the average spectra has a $k$ dependence, for even smaller wavenumbers a scale-independent form, and for even smaller wavenumbers the spectra has a $k^{-4}$ dependence. The electric spectra behave as $k^3$ at infrared scales.
• Figure 2: Plot of comoving wavenumbers (horizontal dashed and dotted lines) and $\sqrt{|f"/f|}$, for both quadratic potential (blue) and Starobinsky potential (red), as a function of time. The peak of $\sqrt{|f"/f|}$ at the bounce sets a scale $k_b\,=\, \left(\sqrt{|f"/f|}\right)\biggl\vert_{t\,=\,0}$. Another relevant scale in the problem is $k_I$, which refers to the largest wavenumber that becomes smaller than $\sqrt{|f"/f|}$ just before the onset of inflation. Modes with $k>>k_b$ will only be excited when they cross $\sqrt{|f"/f|}$ during inflation. Hence, their spectra will be similar to that generated in a slow-roll inflationary scenario. Modes with $k_I \lesssim\,k\, \lesssim k_b$ cross $\sqrt{|f"/f|}$ during the bounce and hence are in an excited state at the onset of inflation. The spectra of these modes will be scale-dependent. Modes with $k<<k_I$ are largely not excited throughout the evolution.
• Figure 3: Comparison of magnetic and electric power spectra obtained using Eqn. (\ref{['eqn:ps-bogo']}) with exact numerical results. The two calculations agree, indicating that bounce excites the state of perturbations from their initial Minkowski vacuum. The two results are not expected to agree at infrared wavenumbers as those modes are not adiabatic at $t_{\rm mink}$ and hence we cannot approximate their behaviour by using Eqn. (\ref{['eqn:ps-bogo']}).
• Figure 4: Magnetic power spectrum generated in LQC with the quadratic potential when Minkowski initial conditions are imposed at different times.
• Figure 5: Magnetic power spectrum generated in LQC with the Starobinsky potential when Minkowski initial conditions are imposed at different times.