Abstract
The cosmological candidate fields for dark energy as quintessence, phantom and cosmological constant are studied in terms of an entropic hypothesis imposed on the McVittie solution surrounded by dark energy. We certify this hypothesis as “D-bound-Bekenstein bound identification” for dilute systems and use it as a criterion to determine which candidate of dark energy can satisfy this criterion for a dilute McVittie solution. It turns out that only the cosmological constant can pass this criterion successfully while the quintessence and phantom fields fail, as non-viable dark energy fields for this particular black hole solution. Moreover, assuming this black hole to possess the saturated entropy, the entropy-area law and the holographic principle can put two constraints on the radius R of the cosmological horizon. The first one shows that the Hubble radius is discrete such that for any arbitrary value of the black hole mass \(m_{0}\), the value of R is determined up to an integer number. The latter one shows that when a black hole is immersed in a cosmological background, the radius of the cosmological horizon is constrained as \(R<\frac{1}{H}\).
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Notes
In the modern language, Eq. (4) represents the constancy of the Hawking–Hayward mass \(m_H\), i.e. \(\dot{m}_H=0\) [74]. Indeed, m(t) here stands just as a metric coefficient in a particular coordinate system. In fact, one has to identify the Hawking–Hayward mass \(m_H\) [74] as the physically relevant mass, which eventually is related to the physical size of the central object or its corresponding horizon, in order to avoid making any coordinate-dependent statements on the mass and size [50, 74] or the temperature [75] of the central object.
Note that in all of the other formulas we have put \(l_{p}=1\) but for the clarification here we have inserted the \(l_{p}\) in Eq. (28).
There is also an interesting method for calculation of the entropy of black hole by the method of graph theory [103].
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We appreciate the anonymous referee for the constructive comments that significantly improved the quality of the paper.
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Hadi, H., Heydarzade, Y., Darabi, F. et al. D-bound and Bekenstein bound for McVittie solution surrounded by dark energy cosmological fields. Eur. Phys. J. Plus 135, 584 (2020). https://doi.org/10.1140/epjp/s13360-020-00601-7
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DOI: https://doi.org/10.1140/epjp/s13360-020-00601-7