We extend the notion of phantom energy, which is generally accepted for homogeneously distributed matter with $w<-1$ in the universe, on inhomogeneous spherically symmetric spacetime configurations. A spherically symmetric distribution of phantom energy is shown to be able to support the existence of static wormholes. We find an exact solution describing a static spherically symmetric wormhole with phantom energy and show that a spatial distribution of the phantom energy is mainly restricted by the vicinity of the wormhole’s throat. The maximal size of the spherical region, surrounding the throat and containing the most part of the phantom energy, depends on the equation-of-state parameter $w$ and cannot exceed some upper limit.

A fundamental ingredient in wormhole physics is the flaring-out condition at the throat which, in classical general relativity, entails the violation of the null energy condition. In this work, we present the most general conditions in the context of modified gravity, in which the matter threading the wormhole throat satisfies all of the energy conditions, and it is the higher order curvature terms, which may be interpreted as a gravitational fluid, that support these nonstandard wormhole geometries. Thus, we explicitly show that wormhole geometries can be theoretically constructed without the presence of exotic matter but are sustained in the context of modified gravity.

We consider a gravitational theory of a scalar field $\varphi$ with nonminimal derivative coupling to curvature. The coupling terms have the form ${\kappa }_{1}R{\varphi }_{,\mu }{\varphi }^{,\mu }$ and ${\kappa }_2{R}_{\mu \nu }{\varphi }^{,\mu }{\varphi }^{,\nu }$, where ${\kappa }_1$ and ${\kappa }_2$ are coupling parameters with dimensions of length squared. In general, field equations of the theory contain third derivatives of $g_{\mu \nu }$ and $\varphi$. However, in the case $-2{\kappa }_1={\kappa }_2\equiv \kappa$, the derivative coupling term reads $\kappa G_{\mu \nu }{\varphi }^{,\mu }{\varphi }^{,\nu }$ and the order of corresponding field equations is reduced up to second one. Assuming $-2{\kappa }_1={\kappa }_2$, we study the spatially-flat Friedman-Robertson-Walker model with a scale factor $a(t)$ and find new exact cosmological solutions. It is shown that properties of the model at early stages crucially depend on the sign of $\kappa$. For negative $\kappa$, the model has an initial cosmological singularity, i.e., $a(t)\sim (t-t_i{)}^{2/3}$ in the limit $t\to t_i$; and for positive $\kappa$, the Universe at early stages has the quasi-de Sitter behavior, i.e., $a(t)\sim e^{Ht}$ in the limit $t\to -\infty$, where $H=(3\sqrt{\kappa }{)}^{-1}$. The corresponding scalar field $\varphi$ is exponentially growing at $t\to -\infty$, i.e., $\varphi (t)\sim e^{-t/\sqrt{\kappa }}$. At late stages, the Universe evolution does not depend on $\kappa$ at all; namely, for any $\kappa$ one has $a(t)\sim t^{1/3}$ at $t\to \infty$. Summarizing, we conclude that a cosmological model with nonminimal derivative coupling of the form $\kappa G_{\mu \nu }{\varphi }^{,\mu }{\varphi }^{,\nu }$ is able to explain in a unique manner both a quasi-de Sitter phase and an exit from it without any fine-tuned potential.

We investigate cosmological scenarios with a nonminimal derivative coupling between the scalar field and the curvature, examining both the quintessence and the phantom cases in zero and constant potentials. In general, we find that the universe transits from one de~Sitter solution to another, determined by the coupling parameter. Furthermore, according to the parameter choices and without the need for matter, we can obtain a big bang, an expanding universe with no beginning, a cosmological turnaround, an eternally contracting universe, a big crunch, a big rip avoidance, and a cosmological bounce. This variety of behaviors reveals the capabilities of the present scenario.

We consider the structure and physical properties of specific classes of neutron, quark, and “exotic” stars in Eddington-inspired Born-Infeld (EiBI) gravity. The latter reduces to standard general relativity in vacuum, but presents a different behavior of the gravitational field in the presence of matter. The equilibrium equations for a spherically symmetric configuration (mass continuity and Tolman-Oppenheimer-Volkoff) are derived, and their solutions are obtained numerically for different equations of state of neutron and quark matter. More specifically, stellar models, described by the stiff fluid, radiationlike, polytropic and the bag model quark equations of state are explicitly constructed in both general relativity and EiBI gravity, thus allowing a comparison between the predictions of these two gravitational models. As a general result it turns out that for all the considered equations of state, EiBI gravity stars are more massive than their general relativistic counterparts. Furthermore, an exact solution of the spherically symmetric field equations in EiBI gravity, describing an exotic star, with decreasing pressure but increasing energy density, is also obtained. As a possible astrophysical application of the obtained results we suggest that stellar mass black holes, with masses in the range of $3.8M_{\odot }$ and $6M_{\odot }$, respectively, could be in fact EiBI neutron or quark stars.

We investigate cosmological scenarios in the theory of gravity with the scalar field possessing a nonminimal kinetic coupling to the curvature. It is shown that the kinetic coupling provides an essentially new inflationary mechanism. Namely, at early cosmological times the domination of coupling terms in the field equations guarantees the quasi\char21{}de Sitter behavior of the scale factor: a(t)$\propto${}eH$\kappa${}t with H$\kappa${}=1/9$\kappa${}, where $\kappa${}$\simeq${}10-74\,\,sec2 is the coupling parameter. The primary inflationary epoch driven by nonminimal kinetic coupling comes to the end at tf$\simeq${}10-35\,\,sec. Later on, the matter terms are dominating, and the Universe enters into the matter-dominated epoch which lasts approximately 0.5H0-1$\sim${}0.5\ifmmode\times\else\texttimes\fi{}1018\,\,sec. Then, the cosmological term comes into play, and the Universe enters into the secondary inflationary epoch with a(t)$\propto${}eH$\Lambda${}t, where H$\Lambda${}=$\Lambda${}/3. Thus, the cosmological model with nonminimal kinetic coupling represents the realistic cosmological scenario which successfully describes basic cosmological epochs and provides the natural mechanism of epoch change without any fine-tuned potential.