Phase transition and shadow behaviors of quintessential black holes in M-theory/superstring inspired models

We study d-dimensional black holes surrounded by dark energy (DE), embedded in D-dimensional M-theory/superstring inspired models having AdS$_d × 𝕊^{d+k}$ space–time where $D = 2d + k$. We focus first on the thermodynamical Hawking–Page phase transitions, whose microscopical origin is linked to $N$ coincident ($d − 2$)-branes supposed to live in such inspired models. Interpreting the cosmological constant as the number of colors, we compute various thermodynamical quantities in terms of the brane number, the entropy and the DE contribution. Calculating the ordinary chemical potential conjugated to the number of colors, we show that a generic black hole is more stable for a larger number of branes in lower dimensions. In the presence of DE, however, we find that the DE state parameter $ω_q$ takes particular values, for ($D,d,k$) models, providing nontrivial phase transitions. Then, we examine some optical properties. Concretely, we investigate shadow behaviors of quintessential black holes in terms of ($d − 2$)-brane physics. In terms of certain ratios, we find similar behaviors for critical quantities and shadow radius.