Gravitational perturbation in nonlocal modified Tolman VII model

Abstract

In comparison to the original Tolman VII model, Exact Modified Tolman VII (EMTVII) with one additional parameter can increase the compactness of compact object. When the compactness is in the ultracompact regime, the quasinormal modes (QNMs) of the trapped mode as well as the gravitational echoes become more viable. Starting with the EMTVII 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 EMTVII (NEMTVII) is parametrized by the nonlocal parameter (β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\beta $$\\end{document}), modified Tolman VII parameter (α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\alpha $$\\end{document}), and the compactness (C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {C}}$$\\end{document}). It is found that the nonlocal profile produces the smeared surface and consequently reduce the compactness. The maximum compactness Cmax=0.4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}_{max}=0.4$$\\end{document} occurs when α=0=β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =0=\\beta $$\\end{document}, i.e., EMTVII with no smearing. For relatively small value of β=0.01\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta = 0.01$$\\end{document} and the compactness C≲0.2667\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {C}} \\lesssim 0.2667$$\\end{document} (with M=2.14\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M=2.14$$\\end{document} solar masses, R=11.835\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=11.835$$\\end{document} km at α=1.4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =1.4$$\\end{document}), the causality condition and the dominant energy condition (DEC) are satisfied. The quasinormal modes of the gravitational perturbation are calculated using Bohr-Sommerfeld (BS) fitting and we find that the nonlocality produces less trapped modes than the original (EMTVII) counterpart. At high compactness, gravitational echoes are simulated numerically. Echoes are found to exist in the parameter space where the dominant energy condition and the causality condition are violated.