Energy conditions of bouncing solutions in quadratic curvature gravity coupled with a scalar field

Abstract

We examine the validity of classical energy conditions in nonsingular bouncing cosmological solutions arising in quadratic curvature gravity minimally coupled to a scalar field. Focusing on the null, weak, strong, and dominant energy conditions, we perform a systematic analysis under two distinct formulations of the energy-momentum tensor. In the first approach, the energy-momentum tensor is assumed to be sourced solely by the scalar field, whereas in the second, an effective energy-momentum tensor is constructed that incorporates the higher-curvature corrections characterizing deviations from general relativity. Our results reveal that, in the scalar-field description, the null, weak, and dominant energy conditions remain satisfied throughout the cosmological evolution, while the strong energy condition is necessarily violated during the bounce phase, enabling the avoidance of the initial singularity. In contrast, when the effective energy-momentum tensor is considered, all four energy conditions are violated near the bounce, highlighting the intrinsically non-Einsteinian nature of the underlying gravitational dynamics. These findings clarify the role of higher-order curvature terms in facilitating nonsingular cosmological bounces, providing important insights into the energy condition violations required in modified theories of gravity.

Figures (5)

• Figure 1: Evolution of the scalar fields in the FLRW spacetime. The horizontal axis is the dimensionless time $\tau\equiv m(t-t_{\mathrm{b}})$, and the vertical dashed line marks the bounce at $\tau=0$. Top panel: $\varphi/M_{\mathrm{Pl}}$ as a function of $\tau$ (solid curve). Bottom panel: $\psi/M_{\mathrm{Pl}}^{2}$ as a function of $\tau$ (solid curve). We use the parameter set $M_{\mathrm{Pl}}=1$, $m=10^{-6}M_{\mathrm{Pl}}$, $\lambda=10^{-12}$, $\beta_{\mathrm{fac}}=4.49$, $\beta=-\sqrt{\beta_{\mathrm{fac}}\lambda}\,m$, $\alpha=10^{-3}$, $A=10^{12}M_{\mathrm{Pl}}^{-2}$, $K=m^2$, $a_{\mathrm{i}}=10^2$ and the initial conditions $\varphi_\mathrm{i} = \frac{-\beta + \sqrt{\beta^2 -4\lambda m^2}}{2\lambda}$, $\psi_{\mathrm{i}}=\frac{8A\,V(\varphi_{\mathrm{i}})}{F(\varphi_{\mathrm{i}})}+\frac{1}{2}F(\varphi_{\mathrm{i}})$, $H_{\mathrm{i}}^2=\frac{V(\varphi_{\mathrm{i}})}{3F(\varphi_{\mathrm{i}})}$ with $H_{\mathrm{i}}<0$, $\dot\varphi_{\mathrm{i}}=0$, $\dot\psi_{\mathrm{i}}=0$, and $a(t_{\mathrm{i}})=a_{\mathrm{i}}$, where $V(\varphi)$ is given by eq:potential and $F(\varphi)=M_{\mathrm{Pl}}^2-\alpha\varphi^2$.
• Figure 2: Evolution of the background variables using the same parameters and initial conditions as in fig:scalar_fields. The horizontal axis is $\tau\equiv m(t-t_{\mathrm{b}})$, and the vertical dashed line indicates the bounce ($\tau=0$). Top panel: $H/M_{\mathrm{Pl}}$ as a function of $\tau$ (solid curve). Middle panel: $\dot H/M_{\mathrm{Pl}}^{2}$ as a function of $\tau$ (solid curve), where the dot '$\cdot$' denotes $d/dt$. Bottom panel: $a$ as a function of $\tau$ (solid curve).
• Figure 3: Evolution of $w_{\mathrm{eff}}(\tau)$ defined in eq:w-eff-def, using the same parameters and initial conditions as in fig:scalar_fields. The horizontal axis is $\tau\equiv m(t-t_{\mathrm{b}})$, and the vertical dashed line indicates the bounce ($\tau=0$). The solid curve shows $w_{\mathrm{eff}}$. The horizontal reference lines show $w=-1$ (dashed) and $w=-1/3$ (dotted).
• Figure 4: Evolution of the scalar-field energy-condition indicators, using the same parameters and initial conditions as in fig:scalar_fields. The horizontal axis in all panels is $\tau\equiv m(t-t_{\mathrm{b}})$, and the vertical dashed line marks the bounce ($\tau=0$). Panels (a)--(d) correspond to NEC, SEC, WEC, and DEC diagnostics, respectively. All quantities are normalized as indicated on the plot axes.
• Figure 5: Evolution of the effective-fluid energy-condition indicators, using the same parameters and initial conditions as in fig:scalar_fields. The horizontal axis in all panels is $\tau\equiv m(t-t_{\mathrm{b}})$, and the vertical dashed line marks the bounce ($\tau=0$). Panels (a)--(d) correspond to NEC, SEC, WEC, and DEC diagnostics, respectively. All quantities are normalized as indicated on the plot axes.