Tilt in quadratic gravity II

TL;DR

This work analyzes how a tilted fluid in a Bianchi V geometry behaves under General Relativity and Quadratic Gravity, focusing on how matter properties influence future attractors and the regions of initial conditions that lead to them. The authors formulate the dynamics using expansion-normalized variables, derive the field equations for GR and the higher-derivative system for QG, and perform fixed-point analyses alongside extensive numerical simulations. They show that the RR slow-roll inflation solution persists as a primary attractor in QG, while the isotropic singularity attractor can occur in some data, sometimes with divergent vorticity; tilt generally increases transiently for ultra-radiative equations of state $w$ in the range $1/3<w<1$, and inflationary evolution tends to decouple from tilt once inflation begins. Realistic parameter choices (notably $eta$ inferred from CMBR data) render tilt effects on RR basins negligible, supporting the robustness of Starobinsky-like inflation in anisotropic settings, while higher-derivative gravity introduces richer fixed-point structure and potential recollapse scenarios toward isotropic singularities. The results have implications for the stability of inflationary dynamics in anisotropic cosmologies and highlight the nuanced role of tilt and vorticity in quadratic gravity.

Abstract

We investigate a tilted fluid component on a Bianchi V geometry in the theories of General Relativity (GR) and Quadratic Gravity (QG). The main objective of this work is the study of how the properties of matter can modify the future evolution of the attractors and their consequences on the regions of initial conditions of the solutions. As is well known, QG contains the Ruzmaikina-Ruzmaikin (RR) solution. This solution describes the slow-roll regime of Starobinsky's inflationary model, which is currently the best one due to the excellent agreement with Cosmic Microwave Background Radiation (CMBR) data. In QG, we found universes that can be attracted to the RR solution or recollapse toward the isotropic singularity attractor. If the Equation of State (EoS) parameter is ultra-radiative w>1/3, the tilt variable increases both in RR and Milne for QG or GR, respectively. In both cases, the fluid expansion and acceleration diverge, while the vorticity initially increases and then decreases to zero.

Figures (9)

• Figure 1: The dynamic time evolution to the future for Einstein-Hilbert GR with an initial condition near the Kasner exact solution, Eqs. \ref{['kasner']}-\ref{['kasner1']}, is addressed. The initial conditions are given by $H=3.33333\,m_p$, $\Omega_K=9.0\times 10^{-4}$, $\Phi_3=-4.02001\times 10^{-4}$, $\Sigma_+=2.00330\times 10^{-3}$, $\Sigma_-=9.98040\times 10^{-1}$, and a small amount of tilted matter $\Omega_m=3.0\times 10^{-3}$ with $r=1.0\times 10^{-1}$ and $\eta=1.0\times 10^{-1}$. The EoS parameter $w$, defined by $p=w\rho$, is chosen as $w=-0.6$, and the Kasner parameter is set to $\phi=1.5$. The solution is attracted to the FLRW orbit, as described by Eq. \ref{['FLRW']}. a) The graph plots the isotropization of the shear components $\Phi_3$, $\Sigma_+$, and $\Sigma_-$ in blue, red, and green, respectively. In the inset, it is shown in orange the Hubble parameter $H$, which approaches zero toward the universe expansion. b) The graph in blue shows the matter density $\Omega_m$, which approaches $1$, while the plot in the red inset shows the curvature density $\Omega_K$, which tends to zero, also in agreement with Eq. \ref{['FLRW']}. c) The graph shows the decrease of the tilt variable $r$ and the vorticity, both approaching zero, with $r$ plotted in blue and the vorticity in red inset. d) The blue graph shows the decrease of the matter expansion $\hat{\Theta}$, while the red inset displays one of the matter components of the acceleration, $\hat{\dot{u}}_0$, which also decreases to zero
• Figure 2: The future dynamics evolution for GR, with initial conditions near the Kasner solution $H=3.33333\,m_p$, $\Omega_K=9.0\times 10^{-4}$, $\Phi_3=-1.60800\times 10^{-3}$, $\Sigma_+=8.01321\times 10^{-3}$, $\Sigma_-=9.97991\times 10^{-1}$, and a small amount of tilted matter $\Omega_m=3.0\times 10^{-3}$, $r=1.0\times 10^{-1}$, and $\eta=1.0\times 10^{-1}$, with the EoS parameter $w=0.6$ and Kasner parameter $\phi=1.5$, shows that the solution is attracted to the Milne orbit. a) The isotropization of the shear variables $\Phi_3$, $\Sigma_+$, and $\Sigma_-$ is illustrated in blue, red, and green, respectively. The plot inset in orange shows the constraint $E_{00}$ with fluctuations smaller than $10^{-13}$. b) The matter density is displayed in blue and approaches zero, while in red in the inset shows the curvature density $\Omega_K$, which tends to 1, in accordance with Eq. \ref{['Milne']}. c) The evolution of the tilt direction $\eta$ is shown in blue, with an initial increase followed by a decrease. d) The plot in blue shows the increase in the tilt variable $r$. The inset in red shows the increase of the vorticity $\hat{\omega}^2$ in absolute value, which initially increases along with the tilt $r$ in the transient regime and then decreases to zero as the solution is attracted to the Milne universe. e) The matter expansion $\hat{\Theta}$ plotted in blue shows it decreasing followed by an increase during the transient regime. After this regime, $\hat{\Theta}$ continues to increase and diverges. While the matter component of acceleration $\hat{\dot{u}}_0$ increase followed by a divergence is displayed in red. f) The remaining matter acceleration components $\hat{\dot{u}}_1$ and $\hat{\dot{u}}_3$, which also increase indefinitely, are shown in green and in orange, respectively
• Figure 3: QG shows the solution with the initial condition: $H=1.0\,m_p$, $\dot{H}=-9.25926\times 10^{-3}\,m_p^2$, $\ddot{H}=-8.18805\times 10^{-2}\, m_p^3$, $\Omega_K=1.0\times 10^{-2}$, $\Phi_3=1.0\times 10^{-1}$, $\Phi_{3,0}=1.0\times 10^{-1}$, $\Phi_{3,1}=-7.33847\times 10^{-1}$, $\Sigma_+=5.0\times 10^{-1}$, $\Sigma_{+1}=1.0\times 10^{-1}$, $\Sigma_{+2}=-1.26245$, $\Sigma_-=3.0\times 10^{-1}$, $\Sigma_{-1}=1.0\times 10^{-1}$, $\Sigma_{-2}=1.0\times 10^{-1}$, near the RR asymptotic solution 1969JETP...30..372R with a small amount of tilted matter $\Omega_m=3.33333\times 10^{-2}$, $r=1.0\times 10^{-1}$ and $\eta=1.0\times 10^{-1}$. The numeric evolution is to the future, and the solution is attracted to the Ruzmaikina's regime. The EoS parameter is chosen as $w=0.6$, and the renormalization parameters are $\alpha=-30\, m_p^{-2}$ and $\beta=3 \,m_p^{-2}$. a) The blue plot shows the transient behavior of the Hubble parameter $H$, followed by the slow-roll inflationary regime. The inset graph shows the isotropization of the shear variables: $\Phi_3$ in red, $\Sigma_+$ in green, and $\Sigma_-$ in orange. b) The deceleration parameter $q$ is shown in blue, approaching $-1$, as expected. In the inset, the curvature scalar $R_{ab}R^{ab}$ is plotted in red, tending to zero as the universe expands. c) The blue plot shows the increase in the tilt variable $r$, while the inset in red shows the decrease of the tilt direction $\eta$. d) The blue graph shows the initial increase in the absolute value of the vorticity during the transient regime and its decrease to zero during the onset of the slow-roll inflationary regime. The inset in red displays the numerical checks for the constraint $E_{00}$, with fluctuations smaller than $10^{-11}$. e) The increase, followed by divergence, in the matter expansion $\hat{\Theta}$ and in the matter acceleration component $\hat{\dot{u}}_0$ are shown in blue and in red, respectively. f) The remaining matter acceleration components, $\hat{\dot{u}}_1$ and $\hat{\dot{u}}_3$, are plotted in green and in orange, respectively
• Figure 4: A basin plot is shown for different initial conditions of the diagonal shear components $\Sigma_+$ and $\Sigma_-$, with the white points showing initial conditions attracted to the RR orbit, while the black points represent solutions attracted, which evolve to a recollapse and are asymptotically attracted to the isotropic singularity. The dynamical evolution is towards the future for QG. The renormalization parameters are set as $\alpha=-30\, m_p^{-2}$ and $\beta=3 \,m_p^{-2}$ and the EoS parameter $w=1/3$ for the radiation fluid. The initial condition are given by $H=9.98148\times 10^{-1}\,m_p$, $\dot{H}=-9.25926\times 10^{-3}\,m_p^2$, $\Omega_K=1.00371\times 10^{-2}$, $\Phi_3=1.00186\times 10^{-1}$, $\Phi_{3,0}=1.00371\times 10^{-1}$, $\Sigma_{+1}=1.00371\times 10^{-1}$, $\Sigma_{-1}=1.00371\times 10^{-1}$, $\Sigma_{-2}=1.00558\times 10^{-1}$, and a small amount of matter $\Omega_m=3.34571\times 10^{-2}$. a) The graph plots a basin for a non-tilted source with $r=0$ and $\eta=0$. b) The plot shows a zoom near the origin shown in the left panel. c) Here, a small amount of tilted matter with $r=4.8$ and $\eta=1.0\times 10^{-1}$ is considered. The plot shows how the attraction regions for RR shift when the tilt is introduced. The farther regions of Ruzmaikina's attraction are moved to the left, while the region near the origin shifts to the right. d) The plot shows a not small initial condition for the tilt, $r=7.0$ and $\eta=1.0\times 10^{-1}$. In this case, the farther regions of attraction for RR are mostly relocated to the left, leading to a superposition with the attraction region near the origin. The plot is generated considering $200\times200$ points
• Figure 5: a) A basin of attraction is plotted for the same initial conditions shown in the previous figure: $H=9.98148\times 10^{-1}\,m_p$, $\dot{H}=-9.25926\times 10^{-3}\,m_p^2$, $\Omega_K=1.00371\times 10^{-2}$, $\Phi_3=1.00186\times 10^{-1}$, $\Phi_{3,0}=1.00371\times 10^{-1}$, $\Sigma_{+1}=1.00371\times 10^{-1}$, $\Sigma_{-1}=1.00371\times 10^{-1}$, $\Sigma_{-2}=1.00558\times 10^{-1}$, but with different values of tilted matter $\Omega_m=6.69143\times 10^{-1}$ for $r=4.8$ and $\eta=1.0\times 10^{-1}$. The time numerical evolution is towards the future for QG, and the EoS parameter is $w=1/3$ for the radiation fluid. The renormalization parameters are $\alpha=-30\, m_p^{-2}$ and $\beta=3 \,m_p^{-2}$. The graph shows that an increase in the tilted matter density $\Omega_m$ behaves similarly to an increase in the tilt variable $r$. b) Now, the plot considers the value for the renormalization parameter $\beta=1.305\times 10^{9}\,m_p^{-2}$, which is inferred by CMBR measurements. The other renormalization parameter is set to $\alpha=-3.0\times 10^{8}\,m_p^{-2}$, with the EoS parameter $w=1/3$ for the radiation source. The initial conditions are given by $H=1.0\, m_p$, $\dot{H}=-2.12857\times 10^{-11}\, m_p^2$, $\Omega_K=1.0\times 10^{-2}$, $\Phi_3=1.0\times 10^{-1}$, $\Phi_{3,0}=1.0\times 10^{-1}$, $\Omega_m=3.33333\times 10^{-2}$, $\Sigma_{+1}=1.0\times 10^{-1}$, $\Sigma_{-1}=1.0\times 10^{-1}$, $\Sigma_{-2}=1.0\times 10^{-1}$ with $r=4.8$ and $\eta=1.0\times 10^{-1}$. The basin to RR is changed when compared to the arbitrary values of the renormalization parameters shown in the previous figure and in the left panel. Plot made considering $200\times200$ points