Figure 1

Time evolution of the inflaton’s mean $⟨\chi ⟩/{\chi }_c$ (blue) and of the Higgs’ root mean squared $⟨|\phi |⟩/v$ (red) in a hybrid model with $\lambda =g^2/2=0.125$, $e=0.3$, and $V_c=0.024$. One can clearly see the growth of the Higgs towards the true vacuum, while the inflaton rolls down to the bottom of the potential. Once the Higgs is sufficiently close to the true vacuum, its self-interactions compensate the tachyonic mass and the field oscillates close to the VEV, while the inflaton oscillates around $\chi =0$.Reuse & Permissions

Figure 2

Time evolution of the covariant energy density of the Higgs $⟨D_i\phi (D_i\phi {)}^{*}⟩$ (black), the electric $⟨E_i{E}_i⟩$ (red), and magnetic $⟨B_i{B}_i⟩$ (blue) energy densities, and the inflaton’s gradient energy density $⟨{\partial }_i\chi {\partial }_i\chi ⟩$ (green), all normalized to the total energy of the system. The parameters are the same as in Fig. 1. A fast growth of the energy components is experienced during tachyonic preheating, first in the Higgs field and then in the rest of the fields due to the their interactions with the Higgs.Reuse & Permissions

Figure 3

Contribution of different source terms to GW production from wavelike fields. Left panel: contribution associated to the source term in ${\partial }_i\varphi {\partial }_j\varphi$ for scalar fields, corresponding to the interaction between two scalar waves and a graviton (forbidden by helicity conservation). Middle panel: contribution associated to the terms in (25), corresponding to the interaction between two vector waves and a graviton (allowed if the vector field is massless). Right panel: contribution associated to the second term in (26), corresponding to the interaction between several scalar and vector waves and a graviton.Reuse & Permissions

Figure 4

Example of GW spectra calculated with $\mathcal{O}(dx)$ (red) and $\mathcal{O}(d{x}^2)$ (blue) accuracy (see Appendix  for details), for the model (1, 2). The two spectra were obtained with the same model and lattice parameters and they were output at the same moment of time ($mt=100$).Reuse & Permissions

Figure 5

Evolution with time of the fraction of energy density in GW during preheating, ${\rho }_{\mathrm{gw}}/{\rho }_{\mathrm{tot}}$, for the model (1, 2) with $e/\sqrt{\lambda }=6$ (red), 0.5 (blue). and 0 (black). The last case simply corresponds to the model without gauge field. The other parameters were $v={10}^{-3}{M}_{\mathrm{Pl}}$, $\lambda =g^2/2={10}^{-4}$, and $V_c=0.024$.Reuse & Permissions

Figure 6

Spectra $k^3|\phi {|}^2$ and $k^3|B{|}^2$ of the Higgs and magnetic fields at $mt=250$ for $\lambda =g^2/2={10}^{-4}$, $V_c=0.024$, and $e/\sqrt{\lambda }=8$. Here the normalization of the amplitude of the two spectra is arbitrary and has been chosen only for convenience. The results were checked with $N=256$ simulations with $k_{\mathrm{IR}}/m=0.075$, 0.1, and 0.15.Reuse & Permissions

Figure 7

Spectrum of magnetic energy density per logarithmic frequency interval divided by the total energy density, $\frac{1}{{\rho }_t}\frac{d{\rho }_{\mathrm{mag}}}{d\ln k}\propto k^3|B{|}^2$, at $mt=250$ for different values of $e/\sqrt{\lambda }$. From left to right, $e/\sqrt{\lambda }=0.2$ (black), 0.5 (blue), 2 (red), 4 (black, dashed), and 8 (green). The other parameters are the same as in Fig. 6.Reuse & Permissions

Figure 8

GW spectra (44) for $e/\sqrt{\lambda }=6$ (red) and 8 (blue) at $mt=250$. The other parameters are the same as in Fig. 6. The results were checked with $N=256$ simulations with $k_{\mathrm{IR}}/m=0.075$, 0.1, and 0.15.Reuse & Permissions

Figure 9

Same as Fig. 8 for $e/\sqrt{\lambda }=0$ (black), 0.2 (green), 0.5 (blue), 2 (red), and 4 (black, dashed).Reuse & Permissions

Figure 10

Evolution with time of the spectrum of covariant gradient energy density of $\phi$ (45) (left panels) and of the spectrum of energy density in GW (44) (right panels) for $\lambda =g^2/2={10}^{-4}$, $V_c=0.024$, and $e/\sqrt{\lambda }=6$. The spectra are shown between $mt=5$ and 15.5 on the first line, between $mt=15.5$ and 19 on the second line, between $mt=19$ and 23 on the third line, between $mt=23$ and 26.5 on the fourth line, between $mt=26.5$ and 29 on the fifth line, and between $mt=29$ and 50 on the sixth line. In each case, the spectra are output every $mt=0.24$ and red spectra correspond to earlier moments of time while blue spectra correspond to later moments of time (the colors go from red to yellow, green and blue as time evolves).Reuse & Permissions

Figure 11

GW spectra for $\lambda ={10}^{-2}$ (red) and $\lambda ={10}^{-6}$ (blue) at $mt=300$. The other parameters are $V_c=0.024$, $g^2=2\lambda$, and $e=3\sqrt{\lambda }$.Reuse & Permissions

Figure 12

GW spectra for $V_c=0.024$ (red) and $V_c=0.003$ (blue) at $mt=250$. The other parameters are $\lambda =g^2/2={10}^{-4}$ and $e/\sqrt{\lambda }=3$. The results were checked with $N=256$ simulations with $k_{\mathrm{IR}}/m=0.05$, 0.075, and 0.1.Reuse & Permissions

Figure 13

Time evolution of the inflaton’s mean $⟨\varphi ⟩/{\varphi }_c$ (blue) and of the Higgs’ root mean squared $⟨|\phi |⟩/v$ (red) for $\lambda /g^2=16$, $e/\sqrt{\lambda }=0.5$, $\lambda ={10}^{-4}$, and $V_c=0.024$. Notice the large oscillations of $⟨\varphi ⟩$ as compared to Fig. 1.Reuse & Permissions

Figure 14

GW spectra for $\lambda /g^2=16$ and $e/\sqrt{\lambda }=0$ (red, no gauge field) and $e/\sqrt{\lambda }=0.5$ (blue) at $mt=400$. The other parameters are $\lambda ={10}^{-4}$ and $V_c=0.024$. The results were checked with $N=256$ simulations with $k_{\mathrm{IR}}/m=0.03$, 0.04, and 0.075.Reuse & Permissions

Figure 15

GW spectra for $\lambda /g^2=8$ and $e/\sqrt{\lambda }=0.5$ (blue), 1 (black), and 2 (red) at $mt=350$. The other parameters are $V_c=0.024$ and $\lambda ={10}^{-4}$. The results were checked with $N=256$ simulations with $k_{\mathrm{IR}}/m=0.045$ and 0.06.Reuse & Permissions

Figure 16

GW spectra for $\lambda /g^2=0.5$ (black), 8 (red), and 16 (blue) at $mt=400$. The other parameters are $V_c=0.024$, $\lambda ={10}^{-4}$, and $e/\sqrt{\lambda }=0.5$. The results were checked with $N=256$ simulations with $k_{\mathrm{IR}}/m=0.03$, 0.04, and 0.075.Reuse & Permissions

Figure 17

Time evolution of the spatial distribution of the modulus of the Higgs field, $|\phi |(\mathbf{x},t)=\sqrt{{\phi }_1^2+{\phi }_2^2}$, during the process of symmetry breaking. The images have been obtained with a $N=256$ lattice simulation with an IR cutoff $k_{\mathrm{IR}}=0.15m$, and parameters $g^2=2\lambda =0.25$, $V_c=0.024$, and $e=6\sqrt{\lambda }$. From left to right, top to bottom, the snapshots correspond to $mt=5.5$, 11.0, 17.3, and 23.0.Reuse & Permissions

Figure 18

Snapshots at time $mt=17$ of the spatial distribution of the magnetic energy density $B^2$ (in units of $m^4$). The images have been obtained for a $N=256$ lattice with an IR cutoff $k_{\mathrm{IR}}=0.1m$, and parameters $g^2=2\lambda =0.25$, $V_c=0.024$, and $e=6\sqrt{\lambda }$. In the left we see clearly how the stringlike configurations of the magnetic fields are localized where the minima of the Higgs are, as described by the colored transverse plane which plots the Higgs amplitude (normalized to the VEV) at that moment. On the right, the analogous figure where now the transverse colored plane shows the phase of the Higgs, thus clearly demonstrating the existence of nontrivial winding in the Higgs around the positions of the magnetic strings.Reuse & Permissions

Figure 19

Time evolution of the spatial distribution of the magnetic energy density $B^2$ (in units of $m^4$) along the process of the Higgs symmetry breaking. The images have been obtained with a $N=256$ lattice simulation with an IR cutoff $k_{\mathrm{IR}}=0.1m$, and parameters $g^2=2\lambda =0.25$, $V_c=0.024$, and $e=6\sqrt{\lambda }$. From left to right, top to bottom, the snapshots correspond to $mt=5.5$, 11.0, 17.3, 19.0, 21.0, and 23.0. At early times, before the Higgs bubbles percolate, the magnetic field is still very small and has not acquired yet the distinctive shape of topological string configurations. At times $mt\sim 17–19$, the stringlike spatial distributions of the magnetic energy density have finally developed, following the locus of points which corresponds to the intermediate regions between Higgs bubbles. The stringlike distributions are most clearly seen at time $mt=19$. Later, due to the time evolution of the gauge field’s mass, the string segments fatten and start shedding away the magnetic field, see the main text.Reuse & Permissions

Figure 20

Evolution in time of both the histograms of the Higgs field normalized to its VEV (left) and the magnetic energy $B^2$ normalized to $m^4$ (right). The first row corresponds to the initial times, from $mt=5.05$ to $mt=19$. The second row corresponds to times from $mt=19$ to $mt=32$. The third row corresponds to times from $mt=32$ to $mt=50$. It can be clearly distinguished that the Higgs moves very fast towards the true VEV in the initial stages of hybrid preheating and later oscillates close to VEV (see the left tails of the Higgs histograms). At any moment, even when the Higgs is oscillating in the broken phase with small amplitude compared to the VEV, there remains a significant fraction of points in the lattice where the amplitude of the Higgs is much smaller than the VEV, corresponding to the spatial regions located in between the Higgs bubbles.Reuse & Permissions

Figure 21

Time evolution of the spatial distribution of GW $|{\dot{h}}_{ij}(\mathbf{x},t){|}^2$ (in arbitrary units) along the process of the symmetry breaking of the Higgs. Note that we use the same arbitrary units for all the plots, so the relative amplitude between one snapshot and another tells about the physical growth of the GW energy density. The images have been obtained with a $N=256$ lattice simulation with IR cutoff $k_{\mathrm{IR}}=0.1m$ and parameters $g^2=2\lambda =0.25$, $V_c=0.024$, and $e=6\sqrt{\lambda }$. From left to right, top to bottom, the snapshots correspond to $mt=5.5$, 11.0, 17.3, 19.0, 21.0, and 23.0. At early times, before the Higgs bubbles percolate, the GW energy density is still very small and distributed in lumps over the lattice, with maximum values in the regions where the gradients of the Higgs are maximum. At times $mt\sim 17–19$, however, the stringlike configurations of the gauge and Higgs fields induce similar stringlike distributions of GW. The tubes of highest energy in GW can be seen most clearly at times $mt=17–19$. Later, due to the time evolution of the strings, i.e. their fattening and shedding away of small-scale structures, the distribution of GW seems also to follow a similar pattern. Particularly noticeable here is the figure corresponding to time $mt=21$, where inside one of the concentric tubes one can see another stringlike configuration in the core of the tube, very similar to what we observed in the magnetic energy density. Finally, note that the orientation of the box has been chosen such that one can clearly see as best as possible some of the features developed by the spatial distribution of GW. Consequently, there is no specific correlation between the particular magnetic strings shown in Fig. 19 and the ones shown here, since the boxes are being observed from very different orientations.Reuse & Permissions

Figure 22

Here we show the magnetic field energy density $B^2$ in units of $m^4$ (top) and the GW spatial distribution ${\dot{h}}_{ij}^2$ in arbitrary units (bottom), for times $mt=29$ (left) and $mt=35$ (right). Note that there are in fact three components in the magnetic field spatial distribution, the very core of the strings (barely seen here), as well as lumps of a small-scale structure which has been shedded away from the initial stringlike configuration, plus a diffuse background (in green in the figures) which we interpret as radiation. The spatial distribution of GW follows a similar pattern.Reuse & Permissions

Figure 23

The predicted stochastic background of GW from preheating in the Abelian-Higgs model for two different sets of parameters, $\lambda \sim {10}^{-3}$, $g\sim {10}^{-8}$, $e\sim 0.1$, $v\sim {10}^{11}\mathrm{GeV}$ (red curve) and $g\sim \sqrt{\lambda }\sim {10}^{-6}$, $e\sim {10}^{-4}$, $v\sim {10}^{13}\mathrm{GeV}$, and negligible initial velocity (blue curve), together with the expected sensitivity from future GWO like Advanced LIGO/VIRGO, LISA, ET, BBO, and DECIGO. Note that in order to reach GWO sensitivity we had to extrapolate the position of the IR peaks using the expressions in Eq. (47), and make an educated guess for the shape of the spectra between the peaks (dashed lines). Also plotted are the expected GWB from a global phase transition and that from an inflationary model with tensor-to-scalar ratio $r=0.1$.Reuse & Permissions