Consistent Initial Conditions for Early Modified Gravity in Effective Field Theory

Abstract

Precise initial conditions (ICs) are crucial for accurate computation in cosmological perturbation theory. We derive the consistent ICs for Horndeski theory in the Effective Field Theory (EFT) approach, assuming constant EFT functions at early times. We implement the ICs into the public Boltzmann code \texttt{EFTCAMB}, and demonstrate that the expected early-time behavior of perturbations and Weyl potential can be obtained with theory-consistent MG ICs. We identify significant deviations when comparing Cosmic Microwave Background angular power spectra in MG models obtained with consistent MG ICs versus inconsistent GR ICs. Our findings underline the importance of using accurate, theory-consistent MG ICs to ensure robust cosmological constraints on early MG models.

Figures (4)

• Figure 1: Fractional difference in the MG Weyl potential relative to its GR initial condition, i.e., $((\Psi+\Phi)-(\Psi+\Phi)^{\mathrm{GR}})/(\Psi+\Phi)^{\mathrm{GR}}$. The figure is color-coded according to this ratio for different values of $\Omega_0$ and $\gamma_{30}$, with $\gamma_{20}$ fixed at zero. Adopting consistent MG ICs can produce $\sim1.4\%$ differences in the Weyl potential during the radiation-dominated epoch. The red dot indicates the GR value. The grey region indicates parameter space excluded by stability conditions: ghost instability (negative kinetic term) or gradient instability (sound speed $c_s^2 < 0$).
• Figure 2: Evolution of the absolute photon density perturbation, $|\delta_\gamma(a)|$, for MG models. From left to right, the panels vary the EFT functions $\Omega_0$, $\gamma_{20}$, and $\gamma_{30}$, respectively, while the remaining functions are fixed at their baseline values ($\Omega_0 = 0.001$ and $\gamma_{10} = \gamma_{20} = \gamma_{30} = 0$). All cases use identical MG equations of motion and differ only in their ICs. Solid lines correspond to the evolution using the correct MG ICs, whereas the dashed lines represent the evolution with GR ICs. An $a^2$ scaling function is plotted to represent the expected $\delta_\gamma$ evolution, which agrees with the $|\delta_\gamma(a)|$ evolution using MG ICs.
• Figure 3: Evolution of the Weyl potential, $-(\Psi + \Phi)$, with varying $\Omega_0$, $\gamma_{20}$, and $\gamma_{30}$, respectively. The remaining EFT functions remain fixed at their baseline values ($\Omega_0 = 0.001$ and $\gamma_{10} = \gamma_{20} = \gamma_{30} = 0$). Solid lines (using MG ICs) demonstrate that $-(\Psi + \Phi)$ is conserved at early times, whereas dashed lines (GR ICs) show significant deviations from it.
• Figure 4: Absolute value of the relative difference in the CMB TT angular power spectrum, $|\Delta C_\ell/\sigma_{\mathrm{CV}}| = |(C_{\ell,\text{MG ICs}} - C_{\ell,\text{GR ICs}})/\sigma_{\mathrm{CV}}|$, for various MG models specified by $\Omega_0$, $\gamma_{20}$, and $\gamma_{30}$. Solid lines indicate positive relative differences, and dashed lines are negative differences. In the left panel, $\gamma_{20}=0$ and $\gamma_{30}=0.07$ are held fixed, while $\Omega_0$ is varied; these choices lie close to the instability boundary, and the difference between the GR and MG ICs is small. In the right panel, $\Omega_0=0.02$ and $\gamma_{20}=0$ are fixed, while $\gamma_{30}$ is varied; these choices stay away from the instability boundary, and have a large discrepancy between the GR and MG ICs. $C_{\ell,\text{MG ICs}}$ and $C_{\ell,\text{GR ICs}}$ only differ in using the consistent MG ICs and the inconsistent GR ICs. The relative difference is scaled with cosmic variance per multiple of the $C_{\ell,\text{MG ICs}}$.