In this thesis, we study the role of canonical real scalar field in cosmology (represents dark energy) in a non-minimum coupling with gravity, then we study its effects on black holes by using the modified deSitter–Schwarzschild metric and by using McVittie metric. This thesis includes three main parts. In first part, we introduce a non-minimum coupling of scalar field with gravity, represented by the term √ −gRL(φ). In this section, we introduce a global scaling symmetry Lagrangian of non-Minimally coupled scalar field to gravity, in the minimum space (a, φ). Then we search for the conditions of breaking that global symmetry. We obtain a conserved quantity (charge) which can be used for global classification of cosmological solutions, i.e, two solutions with unequal charges can not be related to each other by any coordinates transformation. We see the role of that charge in the solutions of φ, we find that by the universal expanding (increasing a), the field φ is always exponential decreasing until reaching a critical point φ˙ = 0, φ = φ0 ̸= 0, in which, the global scaling symmetry breaks and the universal expanding is approximately in constant rate H0. As a result, we can relate the cosmological constant and gravity constant to a same identity, which is the scaling symmetry breaking in the space (a, φ). And relate the cosmological constant to the vacuum expectation value of the potential energy of φ. In second part of this thesis, we study the dynamical systems in a non-minimum coupling of scalar field with gravity, represented by the term √ −gRL(φ), for an arbitrary potential of the field (without specifying any potential). We study all possible critical points of the system and find the first order approximation of the velocities nearby that critical points. As a result, the non-minimum coupling can be regarded as a source (creation and annihilation) for dark matter. And by universal expanding, the negative pressure leads to increasing the non-minimum interaction energy ratio Ωnon−min (in certain critical points x′ = y′ = 0). therefore, according to the energy constraint (2.37), matter will lose energy, so its energy density decreases (oppositely is true). In third part of the thesis, we study the perturbation of interaction of a black hole with a real scalar field, in cosmology background, by using the modified deSitter–Schwarzschild metric and by using McVietti metric which describe some mass M(t) (call it McVittie mass) in the centre r = 0, that is immersed in an expanding universe, and that expansion is described by an expansion rate H(t). The mass M(t) is assumed to be time dependent, so we get the effects of the universal expansion on that mass (increasing, decreasing it,...) by solving Einstein equation of McVittie metric with using real scalar field with some potentials, the role of that field is representing the universal dark energy, but also we let it interact with the McVittie mass M(t). We relate the locality of φ(t, r) to its interaction with the black hole which centred in r = 0, as a result, the black hole mass MBH will vary by time, so according to conservation of energy, the decreasing (or increasing) of MBH corresponds to increasing (or decreasing) in the local energy of the field φ. We find that the scalar field φ is concentrated around r = 0 (the centre of the black hole), so inside of the black hole, due to the gravity
