Dynamics of Non-minimally Coupled Scalar Fields in the Jordan Frame
- Figueroa, Daniel G. et al
- arXiv:2112.08388
 | Time evolution of the effective potential of the non-minimally coupled field $\phi$ as a function of the Ricci scalar. Oscillations between an unbounded and bounded potential occur once the Ricci scalar begins oscillating at around one e-fold. |
 | Resulting power spectrum of the non-minimally coupled spectator field from the linear analysis performed in \cref{sec:lin-analysis}. Colored lines illustrate the spectrum at the indicated number of e-folds post inflation. |
 | Power spectra of the non-minimally coupled spectator field $\phi$ at the end of inflation defined by $N = N_{*}$ and $N-N_*$ e-folds thereafter. The solid/dashed lines are the results of the lattice/linear simulations. \textbf{Left:} Power spectrum for the case of $\xi = 50$ where the spectator field has no potential, $V=0$. Initially the spectrum is well described by the linear analysis until the growth becomes large enough that backreaction occurs at approximately $N-N_{*}=1.75$. \textbf{Right:} Power spectrum for $\xi = 100$ where the spectator field has a small quartic coupling $\lambda = \SI{E-5}{}$. Deviations from the linear analysis begin almost immediately where we see the non-linear effect of the quartic term transferring power to higher $k$-modes. |
 | Power spectra of the non-minimally coupled spectator field $\phi$ at the end of inflation defined by $N = N_{*}$ and $N-N_*$ e-folds thereafter. The solid/dashed lines are the results of the lattice/linear simulations. \textbf{Left:} Power spectrum for the case of $\xi = 50$ where the spectator field has no potential, $V=0$. Initially the spectrum is well described by the linear analysis until the growth becomes large enough that backreaction occurs at approximately $N-N_{*}=1.75$. \textbf{Right:} Power spectrum for $\xi = 100$ where the spectator field has a small quartic coupling $\lambda = \SI{E-5}{}$. Deviations from the linear analysis begin almost immediately where we see the non-linear effect of the quartic term transferring power to higher $k$-modes. |
 | Power spectra of the non-minimally coupled spectator field $\phi$ at the end of inflation defined by $N = N_{*}$ and $N-N_*$ e-folds thereafter. The solid/dashed lines are the results of the lattice/linear simulations. \textbf{Left:} Power spectrum for the case of $\xi = 50$ where the spectator field has no potential, $V=0$. Initially the spectrum is well described by the linear analysis until the growth becomes large enough that backreaction occurs at approximately $N-N_{*}=1.75$. \textbf{Right:} Power spectrum for $\xi = 100$ where the spectator field has a small quartic coupling $\lambda = \SI{E-5}{}$. Deviations from the linear analysis begin almost immediately where we see the non-linear effect of the quartic term transferring power to higher $k$-modes. |
 | \textbf{Left:} Evolution of the Ricci scalar normalized to Hubble squared $H(t)^2$. The dashed-black line is the evolution of the linear free-field analysis which is used as an input to the lattice simulation. The colored lines $\xi = \{10,50,100\}$, $\{$yellow, blue, green$\}$ are the lattice simulation results which illustrate a strong deviation once the energy density of the non-minimally coupled field is comparable to that of the inflaton. Shaded in red is the region where the Ricci scalar is negative, driving tachyonic growth of the spectator field. \textbf{Right:} Expectation value $\langle \phi^2\rangle$, see \cref{eq:expphi2}, of the non-minimally coupled spectator field for the same values of $\xi$ as before. |
 | \textbf{Left:} Evolution of the Ricci scalar normalized to Hubble squared $H(t)^2$. The dashed-black line is the evolution of the linear free-field analysis which is used as an input to the lattice simulation. The colored lines $\xi = \{10,50,100\}$, $\{$yellow, blue, green$\}$ are the lattice simulation results which illustrate a strong deviation once the energy density of the non-minimally coupled field is comparable to that of the inflaton. Note that the two peaks extend to values outside the range of the figure, $R/H^2 = 20.8\, (49.1)$ for $\xi = 50\, (100)$, respectively. Shaded in red is the region where the Ricci scalar is negative, driving tachyonic growth of the spectator field. \textbf{Right:} Expectation value $\langle \phi^2\rangle$, see \cref{eq:expphi2}, of the non-minimally coupled spectator field for the same values of $\xi$ as before. |
 | \textbf{Left:} Evolution of the Ricci scalar normalized to Hubble squared $H(t)^2$. The dashed-black line is the evolution of the linear free-field analysis which is used as an input to the lattice simulation. The colored lines $\xi = \{10,50,100\}$, $\{$yellow, blue, green$\}$ are the lattice simulation results which illustrate a strong deviation once the energy density of the non-minimally coupled field is comparable to that of the inflaton. Shaded in red is the region where the Ricci scalar is negative, driving tachyonic growth of the spectator field. \textbf{Right:} Expectation value $\langle \phi^2\rangle$, see \cref{eq:expphi2}, of the non-minimally coupled spectator field for the same values of $\xi$ as before. |
 | \textbf{Left:} Evolution of the Ricci scalar normalized to Hubble squared $H(t)^2$. The dashed-black line is the evolution of the linear free-field analysis which is used as an input to the lattice simulation. The colored lines $\xi = \{10,50,100\}$, $\{$yellow, blue, green$\}$ are the lattice simulation results which illustrate a strong deviation once the energy density of the non-minimally coupled field is comparable to that of the inflaton. Note that the two peaks extend to values outside the range of the figure, $R/H^2 = 20.8\, (49.1)$ for $\xi = 50\, (100)$, respectively. Shaded in red is the region where the Ricci scalar is negative, driving tachyonic growth of the spectator field. \textbf{Right:} Expectation value $\langle \phi^2\rangle$, see \cref{eq:expphi2}, of the non-minimally coupled spectator field for the same values of $\xi$ as before. |
 | Evolution of the total equation of state as a function of e-folds post inflation. Here we define $w \equiv p(\eta) / \rho(\eta)$ using the non-minimally coupled fields contributions from \cref{eq:nmcrho} and \cref{eq:nmcp} plus the standard minimally coupled contributions from the inflaton. We show the results for $p=4$ ($p=6$) on the left (right), respectively. |
 | Evolution of the total equation of state as a function of e-folds post inflation. Here we define $w \equiv p(\eta) / \rho(\eta)$ using the non-minimally coupled fields contributions from \cref{eq:nmcrho} and \cref{eq:nmcp} plus the standard minimally coupled contributions from the inflaton. We show the results for $p=4$ ($p=6$) on the left (right), respectively. In the right-hand panel the $w$ peaks extends to $w=-2\,(-5.1)$ for $\xi=50\,(100)$. |
 | Evolution of the total equation of state as a function of e-folds post inflation. Here we define $w \equiv p(\eta) / \rho(\eta)$ using the non-minimally coupled fields contributions from \cref{eq:nmcrho} and \cref{eq:nmcp} plus the standard minimally coupled contributions from the inflaton. We show the results for $p=4$ ($p=6$) on the left (right), respectively. |
 | Evolution of the total equation of state as a function of e-folds post inflation. Here we define $w \equiv p(\eta) / \rho(\eta)$ using the non-minimally coupled fields contributions from \cref{eq:nmcrho} and \cref{eq:nmcp} plus the standard minimally coupled contributions from the inflaton. We show the results for $p=4$ ($p=6$) on the left (right), respectively. In the right-hand panel the $w$ peaks extends to $w=-2\,(-5.1)$ for $\xi=50\,(100)$. |
 | Time evolution of the energy density of both the inflaton (red) and non-minimally coupled spectator field (black). Parameter values for the inflationary potential are $c = 0.1$, $\Lambda \simeq \SI{1.79E+16}{GeV}$, while the energy density is normalized as in \cref{eq:dim-less-vars}. We show the absolute value of the energy density but change the line style to dashed to illustrate when the energy density turns negative. \textbf{Top row:} Three values of the non-minimal coupling $\xi$ for the inflationary potential with $p=4$. In the top-right panel we also show the effect of a Hubble scale mass (thin purple) and a small quartic coupling of the spectator field (thin gray). \textbf{Bottom row:} Same three values of $\xi$ but for $p=6$, where the energy density of the inflaton redshifts faster than radiation. |
 | Time evolution of the energy density of both the inflaton (red) and non-minimally coupled spectator field (black). Parameter values for the inflationary potential are $c = 0.1$, $\Lambda \simeq \SI{1.79E+16}{GeV}$, while the energy density is normalized as in \cref{eq:dim-less-vars}. We show the absolute value of the energy density but change the line style to dashed to illustrate when the energy density turns negative. \textbf{Top row:} Three values of the non-minimal coupling $\xi$ for the inflationary potential with $p=4$. In the top-right panel we also show the effect of a Hubble scale mass (thin purple) and a small quartic coupling of the spectator field (thin gray). \textbf{Bottom row:} Same three values of $\xi$ but for $p=6$, where the energy density of the inflaton redshifts faster than radiation. |
 | Numerial convergence accessed using the Hubble constraint equation \cref{eq:HubConstraint}. \textbf{Left:} shows the cases we considered for $p=4$ for the inflationary potential, see \cref{eq:inf_pot}. \textbf{Right:} same as left, except we consider the steeper potential, $p=6$. |
 | Numerial convergence accessed using the Hubble constraint equation \cref{eq:HubConstraint}. \textbf{Left:} shows the cases we considered for $p=4$ for the inflationary potential, see \cref{eq:inf_pot}. \textbf{Right:} same as left, except we consider the steeper potential, $p=6$. |