Testing classical-quantum gravity with geodesic deviation

Abstract

A novel semiclassical gravity model proposed by Oppenheim et al., that consistently describes interactions between quantum systems and a classical gravitational field, has recently attracted considerable attention. However, the limitations and phenomenological viability of this model have not yet been thoroughly investigated. In this work, based on the model, we study quantum fluctuations of geodesic deviation coupled with a classical gravitational field. We analytically derive the strain spectrum expected from the fluctuations and show that the original Oppenheim et al. model can be tested with the current observational sensitivity of gravitational-wave experiments. Furthermore, motivated by the novel semiclassical model, we construct two additional models: a modified Oppenheim et al. model that is manifestly consistent with Einstein equation, and a classical-quantum model with environment-induced noise. We analyze the strain spectra predicted by these two models through comparison with those of the original Oppenheim et al. model and perturbative quantum gravity.

Figures (6)

• Figure 1: Left panel: the spectra of the original Oppenheim et al. model $S^h_{x,\text{ori}}$ and its minimum. The green solid line shows $S^h_{x,\text{ori}}$ for $\beta = 0.1$, $D_0^{\mathrm{\,ori}} = 10^{-85}\,\mathrm{Hz}^{-4}$, $L = 4\,\mathrm{km}$, and $\epsilon = 10^{-18}\,\mathrm{Hz}$. The parameter $1/\epsilon$ corresponds to the age of the universe. The green dot-dashed line shows the minimum of the spectrum, which is independent of $D_0^{\mathrm{\,ori}}$; the remaining parameters are fixed to the same values as in $S^h_{x,\text{ori}}$. Further, $S^D_{x,\text{ori}}$ contributed from $\Delta^D_{abcd}$ (shown in dashed purple) and $S^N_{x,\text{ori}}$ derived from $\Delta^N_{abcd}$ (shown in dashed gray) are plotted separately. The parameters used in the plot are the same as those used for the green solid line. Right panel: the $\beta$ dependence of $S^h_{x,\text{ori}}$ is shown for $D_0^{\mathrm{\,ori}} = 10^{-85}\,\mathrm{Hz}^{-4}$, $L = 4\,\mathrm{km}$, $\epsilon = 10^{-18}\,\mathrm{Hz}$ and $\omega = 100\,\mathrm{Hz}$. As observed in Eq. \ref{['N_Spectrum_Oppenheim_original']}, $S^N_{x,\text{ori}}$ diverges at $\beta = 1/4$ and vanishes at $\beta = 1/3$. In the range $1/4 < \beta < 1/3$, it becomes complex.
• Figure 2: Strain spectrum $S^h_{x,\text{Ein}}$ given in Eq. \ref{['Spectrum_white_Ein']} for the Einstein-consistent model. The cyan solid line shows $S^h_{x,\text{Ein}}$ plotted with $D_0^{\mathrm{\,Ein}} = 10^{-85}\,\mathrm{Hz}^{-4}$, $L = 4\,\mathrm{km}$, and $\epsilon = 10^{-18}\,\mathrm{Hz}$. The green solid line corresponding to the strain $S^h_{x,\text{ori}}$ obtained in the original Oppenheim et al. model is plotted with $\beta = 0.1$, while all other parameters are set to be the same as those used for the cyan line. In Eq. \ref{['Spectrum_white_Ein']}, the contributions from decoherence $S^D_{x,\text{Ein}}$ (shown in dashed purple) and from noise $S^N_{x,\text{Ein}}$ (shown in dashed gray) are presented separately. The parameters used in the plot are the same as cyan solid line.
• Figure 3: Spectra predicted from the environmental CQ model. The red solid line shows $S^h_{x,\text{env}}$ given by Eq. \ref{['Spectrum_Environmental_Oppenheim']} plotted with $D_0^{\mathrm{\,env}} = 10^{-80}\,\mathrm{Hz}^{-4}$, and $\mu = 10^{-18}\,\mathrm{Hz}$. The $\mu$ corresponds to a scale given by the inverse of the age of the universe. The red dot-dashed line represents the minimum spectrum, $s^h_{x,\text{env}}$, which is independent of $D_0^{\mathrm{\,env}}$. The blue line corresponds to the strain predicted from perturbative quantum gravity \ref{['PSD_QG']}. In Eq. \ref{['Spectrum_Environmental_Oppenheim']}, the contributions $S^D_{x,\text{env}}$ from the decoherence kernel (shown in dashed purple) and $S^N_{x,\text{env}}$ from the noise kernel (shown in dashed gray) are plotted separately. The parameters used in the plot are the same as the red solid line.
• Figure 4: This figure shows the spectra $S_{x}^h$ for each model plotted together with $D_0 = 10^{-85}\,\mathrm{Hz}^{-4}$. The green solid line represents the spectrum of the original Oppenheim et al. model, and the green dot-dashed line indicates its minimum value with the parameter values $\beta = 0.1$, $\epsilon = 10^{-18}\,\mathrm{Hz}$, and $L = 4\,\mathrm{km}$. The red solid line represents the spectrum of the environmental CQ model, and the red dot-dashed line shows the minimum strain given by Eq. \ref{['mini_ENmodel']} with $\mu = 10^{-18}\,\mathrm{Hz}$. The cyan line corresponds to that of the Einstein-consistent model plotted with the same parameter values as the original Oppenheim et al. model, and the blue line corresponds to the strain predicted from perturbative quantum gravity (vacuum state).
• Figure 5: Constraints on the parameter $D_0$ obtained from the sensitivities of various current gravitational-wave detectors. The spectrum $S^h_{x,\text{ori}}$ is plotted in green with the parameter $\beta = 0.1$ and $\epsilon = 10^{-18}\,\mathrm{Hz}$. The environment-induced spectrum $S^h_{x,\text{env}}$ is shown in red and is plotted with $\mu = 10^{-18}\,\mathrm{Hz}$. The spectrum $S^h_{x,\text{Ein}}$ is plotted in cyan using the same parameter as $S^h_{x,\text{ori}}$. The gray shaded region and the black horizontal line indicate the observable regions and the corresponding detection thresholds for each gravitational-wave interferometer. In other words, for a given model to remain viable, its predicted spectrum must lie below these regions. Since mainly $S^h_{x,\text{ori}}$ and $S^h_{x,\text{Ein}}$ are very close to each other, the corresponding values of $D_0$ at which each strain spectrum intersects the boundary of the detectable region are approximately indicated by the black vertical lines. Similarly, the red vertical lines indicate the intersection with the environmental CQ model. Left panel: the constraints from LIGO experiment. The typical sensitivity used here is $10^{-23} \mathrm{Hz}^{-\frac{1}{2}}$ at 100 Hz LIGO_sens, and the mean separation $L$ is set to be $4\,\text{km}$. Right panel: the constraints from LISA Pathfinder experiment. The sensitivity here is $10^{-12}\,\mathrm{Hz}^{-\frac{1}{2}}$ at 0.01 Hz LISApath_sens, and the mean separation $L$ is assumed to be $37.6\,\text{cm}$.