Particle production of vector fields: Scale invariance is attractive

In a model of an Abelian vector boson with a Maxwell kinetic term and non-negative mass-squared it is demonstrated that, under fairly general conditions during inflation, a scale-invariant spectrum of perturbations for the components of a vector field, massive or not, whose kinetic function (and mass) is modulated by the inflaton field is an attractor solution. If the field is massless, or if it remains light until the end of inflation, this attractor solution also generates anisotropic stress, which can render inflation weakly anisotropic. The above two characteristics of the attractor solution can source (independently or combined together) significant statistical anisotropy in the curvature perturbation, which may well be observable in the near future.

Figure 1

This numerical plot demonstrates the validity of our approximation in neglecting terms proportional to $\mu$ when considering a light vector field. The plot shows a numerical solution for the evolution of the kinetic function scaling $\frac{1}{f}\frac{df}{d\alpha }=2x\Gamma$, given by Eq. (26), with respect to the number of elapsed e-folds $N$, for the particular case where the kinetic function, mass, and scalar potential are exponential. The numerical solution (thick blue line) to the full system of equations [Eqs. (17, 18, 19, 20, 21, 22)] is indistinguishable from the numerical solution (thin red line) where the terms proportional to $\mu$ have been removed until the vector field becomes heavy ($N\gtrsim 20$ in the plot). The plot is also an example of how the vector scaling solution $f_{\mathrm{att}}\propto e^{-4\alpha }$ is attained in the light and heavy vector field cases. Note that the attractors take only a handful of e-folds to be reached, corresponding to the oscillating behavior for $N\sim 5$ (light field) and $N\sim 20$ (heavy field). More examples are shown in Sec. 7.

Figure 2

This numerical plot demonstrates the flow between the three vector scaling solutions found in the light vector field case, for the particular example where the kinetic function, mass, and scalar potential are exponential. The plot shows how different initial conditions can give rise to solutions approaching different vector scaling critical points, depending on the relative magnitudes of $z$ and $s$. The model parameters are chosen so that the ${\mathcal{V}\mathcal{S}\mathcal{S}}_3$ is the late-time attractor. If a solution does not directly fall on this critical point, the small but positive eigenvalues $m_4$ for the ${\mathcal{V}\mathcal{S}\mathcal{S}}_{1,2}$ in Eqs. (73, 77), and the small but negative eigenvalue $m_4$ for the ${\mathcal{V}\mathcal{S}\mathcal{S}}_3$ in Eq. (84), drive a very weak flow of solutions along the arc (shown in red) towards the ${\mathcal{V}\mathcal{S}\mathcal{S}}_3$. However, throughout the flows, the solution $x\simeq -2/\Gamma$ is conserved and therefore so is the scaling solution attractor $f_{\mathrm{att}}\propto a^{-4}$. The figure also depicts the $\mathcal{S}\mathcal{S}\mathcal{R}$ critical point which is an unstable saddle point for the parameter space considered.

Figure 3

This numerical plot shows the evolution of the dimensionless variables $z$ and $s$ as the vector field becomes heavy. The plot demonstrates the validity of our approximation $⟨z^2⟩=⟨s^2⟩$, used when considering a heavy vector field.

Figure 4

This numerical plot shows an example of the evolution of a solution for a massive vector field. As the backreaction is initially subdominant the $\mathcal{S}\mathcal{S}\mathcal{R}$ critical point is reached, where the anisotropy $\Sigma$ vanishes. Solutions may remain there for a long period of time until the backreaction becomes important. Solutions will then move to the $\mathcal{V}\mathcal{S}\mathcal{S}$ where $\Sigma \ne 0$ and the attractor solution $x_{\mathrm{att}}=-2/\Gamma$ is obtained. The vector field may then become heavy. As it does so the anisotropy once again vanishes but the solution $x_{\mathrm{att}}=-2/\Gamma$ is conserved. This is clearly demonstrated by the plots in Fig. 5.

Figure 5

These plots show a projection of Fig. 4 in the $x-y$ plane. They show in increasing detail how the attractor solution $x_{\mathrm{att}}=-2/\Gamma$ is conserved as the solution jumps from the light to the heavy vector scaling solution.

Figure 6

These numerical plots show the evolution of the kinetic function scaling $\frac{1}{f}\frac{df}{d\alpha }=2x\Gamma$, given by Eq. (26), and anisotropy $\Sigma$, with respect to the number of elapsing e-folds $N$, for a massive vector field, for the constant dimensionless model parameters example. The plots show numerical solutions for three different values of the dimensionless model parameters and initial conditions. We observe that the $\mathcal{S}\mathcal{S}\mathcal{R}$ is obtained prior to the $\mathcal{V}\mathcal{S}\mathcal{S}$ for certain initial conditions. At the $\mathcal{S}\mathcal{S}\mathcal{R}$ the kinetic function takes on a variety of scalings given by different values of $2x\Gamma$. All solutions move then to the $\mathcal{V}\mathcal{S}\mathcal{S}$ where the same attractor $2x\Gamma =-4$ is obtained, giving $f_{\mathrm{att}}\propto a^{-4}$ and $m_{\mathrm{att}}\propto a$, when considering the connection in Eq. (56). The plots clearly show that the same attractor $2x\Gamma =-4$ is also obtained when the vector field becomes heavy. As expected the anisotropy $\Sigma$ vanishes at the $\mathcal{S}\mathcal{S}\mathcal{R}$ but not at the $\mathcal{V}\mathcal{S}\mathcal{S}$ if the vector field remains light. The anisotropy does, however, vanish once again when the vector field becomes heavy. We also observe that $\Sigma$ can be positive or negative depending on which $\mathcal{V}\mathcal{S}\mathcal{S}$ the solution approaches nearest; see Table . The plots of $\Sigma$ also show (in dashed lines) the analytic solutions obtained for the $\mathcal{V}\mathcal{S}\mathcal{S}$ attractors; see Table . We can see how well they agree with the full numerical solutions.

Figure 7

These numerical plots show the evolution of the kinetic function scaling $\frac{1}{f}\frac{df}{d\alpha }=2x\Gamma$, given by Eq. (26), and anisotropy $\Sigma$, with respect to the number of elapsing e-folds $N$, for a massive vector field, for the chaotic potential model example. The plots show numerical solutions for three different values of the dimensionless model parameters and initial conditions. We observe that the $\mathcal{S}\mathcal{S}\mathcal{R}$ is obtained prior to the $\mathcal{V}\mathcal{S}\mathcal{S}$ for certain initial conditions. At the $\mathcal{S}\mathcal{S}\mathcal{R}$ the kinetic function takes on a variety of scalings given by different values of $2x\Gamma$. All solutions move then to the $\mathcal{V}\mathcal{S}\mathcal{S}$ where the same attractor $2x\Gamma =-4$ is obtained, giving $f_{\mathrm{att}}\propto a^{-4}$ and $m_{\mathrm{att}}\propto a$, when considering the connection in Eq. (56). The plots clearly show that the same attractor $2x\Gamma =-4$ is also obtained when the vector field becomes heavy. As expected the anisotropy $\Sigma$ vanishes at the $\mathcal{S}\mathcal{S}\mathcal{R}$ but not at the $\mathcal{V}\mathcal{S}\mathcal{S}$ if the vector field remains light. We observe that $\Sigma$ is increasing for this example because the dimensionless model parameters are time-dependent. The anisotropy does, however, vanish once again when the vector field becomes heavy. We also observe that $\Sigma$ can be positive or negative depending on which $\mathcal{V}\mathcal{S}\mathcal{S}$ the solution approaches nearest; see Table . The plots of $\Sigma$ also show (in dashed lines) the analytic solutions obtained for the $\mathcal{V}\mathcal{S}\mathcal{S}$ attractors; see Table . We can see how well they agree with the full numerical solutions.

Figure 8

These numerical plots show the evolution of the kinetic function scaling $\frac{1}{f}\frac{df}{d\alpha }=2x\Gamma$, given by Eq. (26), and anisotropy $\Sigma$, with respect to the number of elapsing e-folds $N$, for a massive vector field, for the SUGRA inspired model 1 example. The plots show numerical solutions for three different values of the dimensionless model parameters and initial conditions. We observe that there is no $\mathcal{S}\mathcal{S}\mathcal{R}$ solution as the scalar potential is too steep and therefore would correspond to a period of fast-roll inflation. Despite the wide range of initial conditions, all solutions move to the $\mathcal{V}\mathcal{S}\mathcal{S}$ where the same attractor $2x\Gamma =-4$ is obtained, giving $f_{\mathrm{att}}\propto a^{-4}$ and $m_{\mathrm{att}}\propto a$, when considering the connection in Eq. (56). The plots clearly show that the same attractor $2x\Gamma =-4$ is also obtained when the vector field becomes heavy. As expected the anisotropy $\Sigma \ne 0$ at the $\mathcal{V}\mathcal{S}\mathcal{S}$ if the vector field remains light. We observe that $\Sigma$ is decreasing for this example because the dimensionless model parameters are time-dependent. The plots of $\Sigma$ also show (in dashed lines) the analytic solutions obtained for the $\mathcal{V}\mathcal{S}\mathcal{S}$ attractors; see Table . We can see how well they agree with the full numerical solutions.

Figure 9

These numerical plots show the evolution of the kinetic function scaling $\frac{1}{f}\frac{df}{d\alpha }=2x\Gamma$, given by Eq. (26), and anisotropy $\Sigma$, with respect to the number of elapsing e-folds $N$, for a massive vector field, for the SUGRA inspired model 2 example. The plots show numerical solutions for three different values of the dimensionless model parameters and initial conditions. We observe that there is no $\mathcal{S}\mathcal{S}\mathcal{R}$ solution as the scalar potential is too steep and therefore would correspond to a period of fast-roll inflation. Despite the wide range of initial conditions, all solutions move to the $\mathcal{V}\mathcal{S}\mathcal{S}$ where the same attractor $2x\Gamma =-4$ is obtained, giving $f_{\mathrm{att}}\propto a^{-4}$ and $m_{\mathrm{att}}\propto a$, when considering the connection in Eq. (56). The plots clearly show that the same attractor $2x\Gamma =-4$ is also obtained when the vector field becomes heavy. As expected the anisotropy $\Sigma \ne 0$ at the $\mathcal{V}\mathcal{S}\mathcal{S}$ if the vector field remains light. We observe that $\Sigma$ is decreasing for this example because the dimensionless model parameters are time-dependent. The plots of $\Sigma$ also show (in dashed lines) the analytic solutions obtained for the $\mathcal{V}\mathcal{S}\mathcal{S}$ attractors; see Table . We can see how well they agree with the full numerical solutions.

Figure 10

These plots show the eigenvalues of matrix $\mathcal{M}$ for the $\mathcal{V}\mathcal{S}\mathcal{S}$ critical point in the massless vector field case.

Figure 11

This plot shows the eigenvalue $m_4$ of matrix $\mathcal{M}$ for the ${\mathcal{V}\mathcal{S}\mathcal{S}}_1$ critical point in the light vector field case.

Figure 12

These plots show the eigenvalues of matrix $\mathcal{M}$ for the ${\mathcal{V}\mathcal{S}\mathcal{S}}_2$ critical point in the light vector field case.

Figure 13

These plots show the eigenvalues of matrix $\mathcal{M}$ for the ${\mathcal{V}\mathcal{S}\mathcal{S}}_3$ critical point in the light vector field case.