Chiral gravitational waves from multi-phase magnetogenesis

Abstract

Cosmological vector fields are central to many early-Universe phenomena, including inflationary dynamics, primordial magnetogenesis, and dark-matter scenarios. However, constructing models able to generate cosmological magnetic fields while avoiding strong coupling, backreaction, and cosmic microwave background constraints remains challenging. We study a novel mechanism in which brief non--slow-roll phases during inflation amplify primordial magnetic fields at small scales, while maintaining theoretical consistency and observational viability. We incorporate parity-violating interactions in the vector sector and demonstrate, for the first time in a non--slow-roll framework, that chirality can significantly boost magnetic-field amplitudes and imprint distinctive polarization-dependent spectral features. We complement detailed numerical computations with an analytical treatment yielding compact expressions for chiral vector mode functions that reproduce the main spectral properties. We then develop a systematic formalism to evaluate the stochastic gravitational-wave background naturally induced at second order by these amplified fields, identifying both an intensity component and a circularly polarized contribution with characteristic frequency profiles. We discuss detection prospects with future multiband gravitational-wave observatories, showing that chiral signatures could provide a distinctive observational probe. Our results introduce new avenues for enhancing primordial magnetic fields and their associated gravitational-wave signals, opening promising possibilities for their future detection and interpretation, both with cosmological and gravitational wave probes.

Figures (7)

• Figure 1: The magnetic power spectrum $\mathcal{P}_{_{\mathrm{B}}}(k)$ (in terms of $\mathcal{P}_{_{\mathrm{B}}}^0$) and the associated spectral index $n_{\rm B}$ arising from the three-phase model of $J(N)$ are presented across variations of the parameters characterizing the intermediate phase, namely $n_2$ and $\Delta N_2$. We compute them in the parity-preserving case of $\gamma=0$ with a representative value of $N_1=15$ (counted from the end of inflation). For reference, we mark the value of $\mathcal{P}_{_{\mathrm{B}}}(k) = \mathcal{P}_{_{\mathrm{B}}}^0$ (in dashed black) in the plots of $\mathcal{P}_{_{\mathrm{B}}}(k)$. We examine the spectrum and $n_{\rm B}$ over the range of scales where the spectrum rises towards the peak, and we inspect the behavior across across a range of large negative values of $n_2 = [-20,-26]$ (on top panels). For this analysis we set the intermediate phase of $J$ to be brief yet finite with $\Delta N_2=0.5$. We find that the spectral index $n_{\rm B} \geq 5$ close to the peak. We also examine the spectrum and $n_{\rm B}$ across a range of $\Delta N_2=[0,0.5]$, setting $n_2=-25$ (on bottom panels). Once again, $n_{\rm B}$ easily reaches values $\geq 5$ for values of $\Delta N_2 \geq 0.4$.
• Figure 2: We present $\mathcal{P}_{_{\mathrm{B}}}^{\rm L}(k)/(\mathcal{P}_{_{\mathrm{B}}}^0/2)$ and $\mathcal{P}_{_{\mathrm{B}}}^{\rm R}(k)/(\mathcal{P}_{_{\mathrm{B}}}^0/2)$ (on the left) and the total power spectrum $\sum_\lambda\mathcal{P}_{_{\mathrm{B}}}^\lambda(k)/\mathcal{P}_{_{\mathrm{B}}}^0$ (on the right) across a range of $n_2=[-3,2]$. For reference, we also plot the spectrum of the non-helical case without the intermediate phase of $J$ where $\mathcal{P}_{_{\mathrm{B}}}^{\rm L}=\mathcal{P}_{_{\mathrm{B}}}^{\rm R}=\mathcal{P}_{_{\mathrm{B}}}^0/2$ (in dashed black). We have chosen $\gamma=1$, $N_1=15$ and $\Delta N_2=3$ in the functional form of $J(N)$. This choice of parameters lead to the wavenumber corresponding to the onset of the intermediate phase of $J$ to be $k_1=9.92\times 10^{13}\,{\rm Mpc}^{-1}$. For $n_2=2$, we obtain $\mathcal{P}_{_{\mathrm{B}}}^{\rm L}(k)$ and $\mathcal{P}_{_{\mathrm{B}}}^{\rm R}(k)$ to be scale-invariant but with a significant difference in magnitude due to non-zero $\gamma$. With the decrease in $n_2$, the features of dip, rise and peak become prominent in both the spectra. The total power spectrum (on the right) is predominantly right-circular but for a brief window of scales between the asymptotic values, where the power increases and turns left-circular in polarization with decrease in $n_2$. Hence, for $n_2=-3$ with $\Delta N_2=3$, we obtain an interesting double-peaked spectrum with distinct polarization for each peak.
• Figure 3: We present $\mathcal{P}_{_{\mathrm{B}}}^{\rm L}(k)$ and $\mathcal{P}_{_{\mathrm{B}}}^{\rm R}(k)$ (on the left) and the total power spectrum $\sum_\lambda\mathcal{P}_{_{\mathrm{B}}}^\lambda(k)$ (on the right) across a range of $\Delta N_2=[0,3]$, in terms of $\mathcal{P}_{_{\mathrm{B}}}^0$ as in Fig. \ref{['fig:pb-n2-helical']}. We also plot the reference spectrum of $\mathcal{P}_{_{\mathrm{B}}}^{\rm L}=\mathcal{P}_{_{\mathrm{B}}}^{\rm R}=\mathcal{P}_{_{\mathrm{B}}}^0/2$ (in dashed black). We have chosen $\gamma=1$, $N_1=15$ and $n_2=-3$ in the functional form of $J(N)$. For $\Delta N_2=0$, we obtain $\mathcal{P}_{_{\mathrm{B}}}^{\rm L}(k)$ and $\mathcal{P}_{_{\mathrm{B}}}^{\rm R}(k)$ to be scale-invariant but with $\mathcal{P}_{_{\mathrm{B}}}^{\rm R}(k)$ dominating over $\mathcal{P}_{_{\mathrm{B}}}^{\rm L}(k)$. With the increase in $\Delta N_2$, the features of dip, rise and peak become prominent in both the spectra. Around the peak of the spectra, $\mathcal{P}_{_{\mathrm{B}}}^{\rm L}(k)$ begins to dominate over $\mathcal{P}_{_{\mathrm{B}}}^{\rm R}(k)$ with increase in $\Delta N_2$. The total power spectrum (on the right) is predominantly right-circular but for the window of scales that is peaked and left-circular in polarization (as focussed in the inset).
• Figure 4: We present the magnetic power spectrum $\mathcal{P}_{_{\mathrm{B}}}^\lambda(k)$ for a range of $\gamma=[0,1]$ for individual helicities (on the left) and for the sum of both helicities (on the right) in terms of $\mathcal{P}_{_{\mathrm{B}}}^0$ as in Figs. \ref{['fig:pb-n2-helical']} and \ref{['fig:pb-deln2-helical']}. The parameters of the model are set to be $N_1=15, n_2=-3,\, \Delta N_2=3$. Evidently, for $\gamma=0$, we have both left and right helical modes having the same spectra. The inset on the right panel zooms in on the range of scales around the peak that shall be relevant for the behavior of the spectra of induced GW. Note that the case of $\gamma=1$ generates an amplification of about $10^5$ compared to the non-helical case of $\gamma=0$ at the peak of the spectrum.
• Figure 5: The total power spectrum $\mathcal{P}_{_{\mathrm{B}}}(k)=\sum_\lambda\mathcal{P}_{_{\mathrm{B}}}^\lambda$ is presented in units of $M_{_{\mathrm{Pl}}}^4$ against the power spectrum of primary tensor perturbations $\mathcal{P}_{_{\mathrm{T}}}(k)$, for a range of $\gamma=[0,1]$. We set the model parameters to be $n_2=-3,\,N_1=15$ and $\Delta N_2=3$. The plot illustrates that the ratio of $\mathcal{P}_{_{\mathrm{B}}}(k)/(\mathcal{P}_{_{\mathrm{T}}}(k)M_{_{\mathrm{Pl}}}^4)$ is small even at the peak amplitude of $\mathcal{P}_{_{\mathrm{B}}}(k)$. At the asymptotic value over small scales, this ratio is approximately $\mathcal{P}_{_{\mathrm{B}}}(k)/(\mathcal{P}_{_{\mathrm{T}}}(k)M_{_{\mathrm{Pl}}}^4) \simeq f(\gamma)(H/M_{_{\mathrm{Pl}}})^2$, where $f(\gamma)$ is typically ${\cal O}(10^3)$ for $\gamma=1$. So, as long as $H/M_{_{\mathrm{Pl}}} \leq 10^{-3/2}$, the backreaction due to electromagnetic energy density is small in the model of interest.