Gravitational Perturbation in Nonlocal Modified Tolman VII

Modified Tolman VII (MTVII), in the presence of an additional parameter, can increase the compactness of compact object. If the compactness is in the ultracompact regime, the quasinormal modes~(QNMs) of the trapped mode as well as the gravitational echoes can be more viable. Starting with the MTVII model, we introduce nonlocality into the matter sector and analyze the effective potential, the QNMs, and the gravitational echoes of the compact and ultracompact object in the nonlocal model. The nonlocal gravity version of MTVII~(NGMTVII) is parametrized by the nonlocal parameters~($\tilde{\beta} $), free parameter ($ \alpha $), and the compactness ($ \mathcal{C}$). We have found that NGMTVII can reach $ \mathcal{C}_{max}=0.414 $ with $ \tilde{\beta}_{max} = 3 $ and $ \alpha=0, $ which is significantly more compact than the MTVII model. We have also found that for relatively small value of $\tilde{\beta}$ and the compactness $ \mathcal{C} \lesssim 0.277$~(with $M=2.15$ solar masses, $R=11.5$ km at $\mathcal{C}=0.277$), the causality condition and the dominant energy condition~(DEC) are satisfied. For the perturbation, the quasinormal modes are calculated using Bohr-Sommerfeld (BS) fitting and it is found that the nonlocality produces more trapped modes than the original (MTVII) counterpart. At high compactness, gravitational echoes are simulated numerically.