Ultracompact object within a nonlocal Tolman VII model

Tolman VII (TVII) is an analytical model for the nonrotating perfect fluid sphere with a simple quadratic density profile where its value is zero at the surface and a finite critical density value at the center. Therefore, compared to another analytical model like the constant density star, TVII is more realistic. Except for the dominant energy condition (DEC), which is violated in the region near the star's center, the TVII satisfies all energy conditions. However, the causal condition is also violated for C>0.26, and the maximum compactness of the TVII is restricted, i.e., Cmax≈0.38. Here, we investigate the impacts of nonlocal gravity on the properties of the star of TVII within the range of the compactness of an ultracompact star (0.33≤C≤0.44). This nonlocal gravity version of TVII (NGTVII) is parametrized by the nonlocal parameters (β) and the compactness (C). We have found that NGTVII can reach Cmax=0.43 with βmax=3 which is significantly more compact than TVII. The nonlocal density and pressure profile differs from the TVII, depending on the stars' compactness and nonlocal parameter. We have also found that for the relatively small value of β and the compactness, i.e., C≲0.31, the causality condition and the DEC are not violated. We have also found that the NGTVII's effective potential in the interior can be larger and deeper than that of the TVII model, indicating the deceleration of the echo time. Moreover, using the effective potential of NGTVII, the quasinormal mode and gravitational echo are calculated using Bohr-Sommerfeld fitting and solving the time-dependent Reggae-Wheeler equation. We can infer that the NGVTII with the maximum compactness and nonlocal parameter values enables the existence of the ultracompact star with more trapped modes.