Energy Conditions in a Generalized Second-Order Scalar-Tensor Gravity

Abstract

The study of energy conditions has many significant applications in general relativistic and cosmological contexts. This paper explores the energy conditions in the framework of the most general scalar-tensor theory with field equations involving second-order derivatives. For this purpose, we use flat FRW universe model with perfect fluid matter contents. By taking power law ansatz for scalar field, we discuss the strong, weak, null, and dominant energy conditions in terms of deceleration, jerk, and snap parameters. Some particular cases of this theory like 𝑘-essence model, modified gravity theories and so forth. are analyzed with the help of the derived energy conditions, and the possible constraints on the free parameters of the presented models are determined.

1. Introduction

“The increasing rate of cosmic expansion in current phase” is one of the primal facts in modern cosmology that is supported by some sorts of energy with negative pressure as well as hidden characteristics refereed to dark energy (DE) (its existence is affirmed by the recent data of many astronomical observations) [1–4]. The investigation of this hidden unusual nature of DE has been carried out in two ways: one approach utilizes the modified matter sources, that is, different models like Chaplygin gas [5], quintessence [6–9], 𝑘-essence [10–12], cosmological constant [13, 14], and so forth which are introduced in the usual matter contents within the gravitational framework of general relativity (GR), while in second approach, GR framework is modified by the inclusion of some extra degrees of freedom [15, 16]. Examples of some well-known modified gravity theories include 𝑓(𝑅) gravity [17, 18], scalar-tensor theories like Brans-Dicke (BD) gravity [19, 20], Gauss-Bonnet gravity [21, 22], 𝑓(𝑇) theory [23, 24], and 𝑓(𝑅,𝑇) gravity [25]. Modified matter sources are rather interesting but each faces some difficulties and hence could not prove to be very promising. Modified gravitational theories being large-distance modifications of gravity have brought a fresh insight in modern cosmology. Among these, scalar-tensor theories are considered to be admirable efforts for the investigation of DE characteristics, which are obtained by adding an extra scalar degree of freedom in Einstein-Hilbert action.

Scalar field provides a basis for many standard inflationary models, leading to an effective candidate of DE. In literature [26–28], many inflationary models have been constructed like chaotic inflation, small/large inflation, 𝑘-inflation, Dirac-Born-Infeld (DBI) inflation, single field 𝑘-inflation, and so forth. All of these are some peculiar extensions of the 𝑘-essence models. Although the 𝑘-essence scalar models are considered to be the general scalar field theories described by the Lagrangian in terms of first-order scalar field derivatives, that is, 𝐿=𝐿(𝜙,∇𝜙). However, Lagrangian with higher-order scalar field derivatives (𝐿=𝐿(𝜙,∇𝜙,∇∇𝜙) can be taken into account which fixes the equations of motion (obtained by metric and scalar field variations of the Lagrangian density) to second-order [29, 30]. Horndeski [29] was the pioneer to discuss the concept of most general Lagrangian with single scalar field. Recently, this action is discussed by introducing a covariant Galilean field with second-order equations of motion [30]. Kobayashi et al. [31] developed a correspondence between these Lagrangians. This theory has fascinated many researchers and much work has been done in this context, for example, [31–35].

The energy conditions have many significant theoretical applications like Hawking Penrose singularity conjecture that is based on the strong energy condition [36], while the dominant energy condition is useful in the proof of positive mass theorem [37]. Furthermore, null energy condition is a basic ingredient in the derivation of second law of black hole thermodynamics [38]. On the cosmological grounds, Visser [39] discussed various cosmological terms like distance modulus, look back time, deceleration, and statefinder parameters in terms of red shift using energy condition constraints. These conditions are originally formulated in the context of GR and then extended to modified theories of gravity. Many authors have explored these energy conditions in the framework of modified gravity and found interesting results [40–43].

Basically, modified gravity theories contain some extrafunctions like higher-order derivatives of curvature term or some functions of Einstein tensor or scalar field, and so forth. Thus it is a point of debate that how one can constrain the added extra degrees of freedom consistently with the recent observations. The energy conditions can be used to put some constraints on these functions that could be consistent with those already found in the cosmological arena. Recently, these energy conditions have been discussed in 𝑓(𝑇) [44] and 𝑓(𝑅,𝑇) [45] theories.

In this paper, we study the energy condition bounds in a most general scalar-tensor theory. The paper is designed in the following layout. Next section defines the energy conditions in GR as well as in a general modified gravitational framework. Section 3 provides basic formulation of the most general scalar-tensor theory. In the same section, we formulate the energy conditions in terms of some cosmological parameters within such modified framework. In Section 4, we provide some specific cases of this theory and discuss the corresponding constraints. Finally, we summarize and present some general remarks.

2. Energy Conditions

In this section, we discuss the energy conditions in GR framework and then express the respective conditions in a general modified gravity. In GR, the energy conditions come from a well-known purely geometric relationship known as Raychaudhuri equation [38, 46] together with the lineament of gravitational attractiveness. In a spacetime manifold with vector fields 𝑢^𝜇 and 𝑘^𝜇 as tangent vectors to timelike and null-like geodesics of the congruence, the temporal variation of expansion for the respective curves is described by the Raychaudhuri equation as 𝑑𝜃1𝑑𝜏=−3𝜃²−𝜎_𝜇𝜈𝜎^𝜇𝜈+𝜔_𝜇𝜈𝜔^𝜇𝜈−𝑅_𝜇𝜈𝑢^𝜇𝑢^𝜈,𝑑𝜃1𝑑𝜏=−2𝜃²−𝜎_𝜇𝜈𝜎^𝜇𝜈+𝜔_𝜇𝜈𝜔^𝜇𝜈−𝑅_𝜇𝜈𝑘^𝜇𝑘^𝜈. (1) Here 𝑅_𝜇𝜈, 𝜃, 𝜎^𝜇𝜈, and 𝜔^𝜇𝜈 represent the Ricci tensor, expansion, shear, and rotation, respectively, related with the congruence of timelike or null-like geodesics.

The characteristic of the gravity that is attractive leads to the condition 𝑑𝜃/𝑑𝜏<0. For infinitesimal distortions and vanishing shear tensor 𝜔_𝜇𝜈=0, that is, zero rotation (for any hypersurface of orthogonal congruence), we ignore the second-order terms in Raychaudhuri equation, and consequently integration leads to 𝜃=−𝜏𝑅_𝜇𝜈𝑢^𝜇𝑢^𝜈=−𝜏𝑅_𝜇𝜈𝑘^𝜇𝑘^𝜈. It further implies that 𝑅_𝜇𝜈𝑢^𝜇𝑢^𝜈≥0,𝑅_𝜇𝜈𝑘^𝜇𝑘^𝜈≥0. (2) Since GR and its modifications lead to a relationship of the matter contents, that is energy-momentum tensor in terms of Ricci tensor through the field equations, therefore the respective physical conditions on the energy-momentum tensor can be determined as follows: 𝑅_𝜇𝜈𝑢^𝜇𝑢^𝜈=𝑇_𝜇𝜈−𝑇2𝑔_𝜇𝜈𝑢^𝜇𝑢^𝜈𝑅≥0,_𝜇𝜈𝑘^𝜇𝑘^𝜈=𝑇_𝜇𝜈−𝑇2𝑔_𝜇𝜈𝑘^𝜇𝑘^𝜈≥0, (3) where 𝑇_𝜇𝜈 and 𝑇 are the energy-momentum tensor and its trace, respectively. For perfect fluid with density 𝜌 and pressure 𝑝 defined by 𝑇_𝜇𝜈=𝜌+𝑝𝑢_𝜇𝑢_𝜈−𝑝𝑔_𝜇𝜈, (4) the strong and null energy conditions, respectively, are defined by the inequalities 𝜌+3𝑝≥0 and 𝜌+𝑝≥0, while the weak and dominant energy conditions are defined, respectively, by 𝜌≥0 and 𝜌±𝑝≥0.

Raychaudhuri equation being a geometrical statement works for all gravitational theories. Therefore, its interesting features like focussing of geodesic congruences as well as the attractiveness of gravity can be used to derive the energy constraints in the context of modified gravity. In case of modified gravity, we assume that the total matter contents of the universe act like perfect fluid, and consequently these conditions can be defined in terms of effective energy density and pressure (matter sources get modified, and we replace 𝑇_𝜇𝜈 and 𝑇 in (3) by 𝑇^eff_𝜇𝜈 and 𝑇^eff, resp.). These conditions can be regarded as an extension of the respective conditions in GR given by [43] NEC∶𝜌^eff+𝑝^eff≥0,SEC∶𝜌^eff+𝑝^eff≥0,𝜌^eff+3𝑝^eff≥0,WEC∶𝜌^eff≥0,𝜌^eff+𝑝^eff≥0,DEC∶𝜌^eff≥0,𝜌^eff±𝑝^eff≥0. (5) For a detailed discussion, we suggest the readers to study a recent paper [47].

The DE requires negative EoS parameter 𝜔≤−1/3, for the explanation of cosmic expansion. Indeed, for cosmological purposes, we are curious for a source with 𝜌≥0; in that case, all of the energy conditions require 𝜔≥−1 [48]. The role of possible DE candidates with 𝜔<−1 was pointed out by Caldwell, who referred to null DEC violating sources as phantom components. It is argued that DE models with 𝑤≥−1 such as the cosmological constant and the quintessence satisfy the NEC, but the models with 𝜔<−1 (predicted for instance by the phantom theory), where the kinetic term of the scalar field has a wrong (negative) sign, does not satisfy. However, quintom models can also satisfy NEC as they yield the phantom era for a very short period of time [49]. Usually, the discussions on energy conditions for cosmological constant are available in literature by introducing it in some other type of matter like electromagnetic field [50]. The cosmological constant will trivially satisfy all these energy conditions except SEC.

3. Energy Conditions in the Most General Scalar-Tensor Gravity

The most general scalar-tensor theory in 4 dimensions is given by the action [31–35] 𝑆=𝑑⁴𝑥√×𝐾−𝑔𝜙,𝑋−𝐺₃𝜙,𝑋□𝜙+𝐺₄𝜙,𝑋𝑅+𝐺_4𝑋□𝜙²−∇_𝜇∇_𝜈𝜙∇^𝜇∇^𝜈𝜙+𝐺₅𝜙,𝑋𝐺_𝜇𝜈∇^𝜇∇^𝜈𝜙−16𝐺_5𝑋×□𝜙³−3□𝜙∇_𝜇∇_𝜈𝜙∇^𝜇∇^𝜈𝜙+2∇^𝜇∇_𝛼𝜙∇^𝛼∇_𝛽𝜙∇^𝛽∇_𝜇𝜙+𝐿_𝑚, (6) where 𝜙 is the scalar field, 𝑔 is the determinant of the metric tensor, 𝐿_𝑚 denotes the matter part of the Lagrangian, and 𝑋 represents the kinetic energy term defined by 𝑋=−(1/2)𝜕^𝜇𝜙𝜕_𝜇𝜙. Moreover, 𝐺_𝜇𝜈 is the Einstein tensor, 𝑅 is the Ricci scalar, ∇_𝜇 is the covariant derivative operator, and □=∇_𝜇∇^𝜇 is the de’Alembertian operator. The functions 𝐾(𝜙,𝑋) and 𝐺_𝑖(𝜙,𝑋);𝑖=3,4,5 are all arbitrary functions and 𝐺_𝑖𝑋=𝜕𝐺_𝑖/𝜕𝑋. In this action, the term 𝐺₃(𝜙,𝑋)□𝜙 is the Galilean term, 𝐺₄(𝜙,𝑋)𝑅 can yield the Einstein-Hilbert term and 𝐺₅(𝜙,𝑋) leads to the interaction with Gauss-Bonnet term. This indicates that it covers not only several DE proposals like 𝑘-essence, 𝑓(𝑅) gravity, BD theory and Galilean gravity models, but it also contains 4-dimensional Dvali, Gabadadze, and Poratti (DGP) model (modified), the field coupling with Gauss-Bonnet term, and the field derivative coupling with Einstein tensor as its particular cases.

By varying the action (6) with respect to the metric tensor, the gravitational field equation can be written as 𝐺_𝜇𝜈=1𝐺₄Θ^eff_𝜇𝜈=1𝐺₄𝑇^𝑚_𝜇𝜈+𝑇^𝜙_𝜇𝜈, (7) where Θ^eff_𝜇𝜈 is the modified energy-momentum tensor, 𝑇^𝑚_𝜇𝜈 is the source of usual matter field that can be described by the perfect fluid, while 𝑇^𝜙_𝜇𝜈 provides the matter source due to scalar field and hence yields the source of DE, defined in the Appendix. The scalar wave equation for such modified gravity has been described in the literature [32–35].

By inverting (7), the Ricci tensor can be expressed in terms of effective energy-momentum tensor and its trace as follows: 𝑅_𝜇𝜈=𝑇^eff_𝜇𝜈−12𝑔_𝜇𝜈𝑇^eff, (8) where the effective energy-momentum tensor 𝑇^eff_𝜇𝜈 and its trace 𝑇^eff are 𝑇^eff_𝜇𝜈=𝑇_𝜇𝜈+12𝐾_𝑋∇_𝜇𝜙∇_𝜈1𝜙−2𝐺_3𝑋□𝜙∇_𝜈𝜙∇_𝜇𝜙−∇_(𝜇𝐺₃∇_𝜈)1𝜙+2𝐺_4𝑋×𝑅∇_𝜇𝜙∇_𝜈1𝜙+2𝐺_4𝑋𝑋×□𝜙²−∇_𝛼∇_𝛽𝜙²∇_𝜇𝜙∇_𝜈𝜙+𝐺_4𝑋□𝜙×∇_𝜇∇_𝜈𝜙−𝐺_4𝑋∇_𝜆∇_𝜇𝜙∇^𝜆∇_𝜈𝜙−2∇_𝜆𝐺_4𝑋∇^𝜆∇_(𝜇𝜙∇_𝜈)𝜙+∇_𝜆𝐺_4𝑋×∇^𝜆𝜙∇_𝜇∇_𝜈𝜙−2𝐺_4𝑋𝑅_𝜆(𝜇∇_𝜈)𝜙∇^𝜆𝜙−∇_(𝜇𝐺_4𝑋∇_𝜈)𝜙□𝜙−𝐺_4𝑋×𝑅_{𝜇𝛼𝜈𝛽}∇^𝛼𝜙∇^𝛽𝜙+𝐺_4𝜙∇_𝜇∇_𝜈𝜙+𝐺_4𝜙𝜙∇_𝜇𝜙∇_𝜈𝜙−2𝐺_4𝑋𝜙∇^𝜆𝜙×∇_𝜆∇_(𝜇𝜙∇_𝜈)𝜙+𝐺_4𝑋𝑋∇^𝛼𝜙∇_𝛼∇_𝜇𝜙∇^𝛽𝜙∇_𝛽∇_𝜈𝜙−𝐺_5𝑋𝑅_𝛼𝛽∇^𝛼𝜙×∇^𝛽∇_(𝜇𝜙∇_𝜈)𝜙+𝐺_5𝑋𝑅_𝛼(𝜇∇_𝜈)𝜙∇^𝛼1𝜙□𝜙+2𝐺_5𝑋𝑅_𝛼𝛽∇^𝛼𝜙∇^𝛽𝜙×∇_𝜇∇_𝜈1𝜙+2𝐺_5𝑋𝑅_{𝜇𝛼𝜈𝛽}∇^𝛼𝜙∇^𝛽𝜙□𝜙−𝐺_5𝑋𝑅_{𝛼𝜆𝛽(𝜇}∇_𝜈)𝜙∇^𝜆𝜙∇^𝛼∇^𝛽𝜙−𝐺_5𝑋𝑅_{𝛼𝜆𝛽(𝜇}∇_𝜈)∇^𝜆𝜙∇^𝛼𝜙∇^𝛽𝜙+12∇_(𝜇𝐺_5𝑋∇^𝛼𝜙∇_𝛼∇_𝜈)1𝜙□𝜙−2×∇_(𝜇𝐺_5𝜙∇_𝜈)𝜙□𝜙+∇_𝜆𝐺_5𝜙∇_(𝜇𝜙∇_𝜈)∇^𝜆𝜙−12∇_𝜆𝐺_5𝜙∇^𝜆𝜙−∇_𝛼𝐺_5𝑋∇_𝛽𝜙∇^𝛼∇^𝛽𝜙×∇_𝜇∇_𝜈𝜙−∇^𝛼𝐺₅∇^𝛽𝜙𝑅_{𝛼(𝜇𝜈)𝛽}+∇_(𝜇𝐺₅𝑅_𝜈)𝜆×∇^𝜆1𝜙−2∇_(𝜇𝐺_5𝑋∇_𝜈)𝜙×□𝜙²−∇_𝛼∇_𝛽𝜙²+∇^𝜆𝐺₅𝑅_𝜆(𝜇∇_𝜈)𝜙−∇_𝛼𝐺_5𝑋∇_𝛽𝜙∇^𝛼∇_(𝜇𝜙∇^𝛽∇_𝜈)𝜙+∇_𝛽𝐺_5𝑋×□𝜙∇^𝛽∇_(𝜇𝜙−∇^𝛼∇^𝛽𝜙∇_𝛼×∇_(𝜇𝜙∇_𝜈)𝜙−12∇^𝛼𝜙∇_𝛼𝐺_5𝑋□𝜙∇_𝜇∇_𝜈𝜙−∇_𝛽∇_𝜇𝜙∇_𝛽∇_𝜈𝜙+12𝐺_5𝑋×𝐺_𝛼𝛽∇^𝛼∇^𝛽𝜙∇_𝜇𝜙∇_𝜈1𝜙+2𝐺_5𝑋□𝜙×∇_𝛼∇_𝜇𝜙∇^𝛼∇_𝜈1𝜙−2𝐺_5𝑋□𝜙²×∇_𝜇∇_𝜈𝜙−1𝐺12^5𝑋𝑋□𝜙³−3□𝜙∇_𝛼∇_𝛽𝜙²+2∇_𝛼∇_𝛽𝜙³×∇_𝜇𝜙×∇_𝜈1𝜙−2∇_𝜆𝐺₅𝑅_𝜇𝜈∇^𝜆𝑇𝜙,_eff=𝐾+∇_𝜆𝐺₃∇^𝜆𝜙−2𝐺_4𝜙□𝜙−2𝑋𝐺_4𝜙𝜙−2−2𝐺_4𝑋𝜙∇_𝛼∇_𝛽𝜙∇^𝛼𝜙∇^𝛽𝜙+𝐺_4𝑋𝑋∇_𝛼∇_𝜆𝜙×∇_𝛽∇^𝜆𝜙∇^𝛼𝜙∇^𝛽1𝜙+2𝐺_4𝑋□𝜙²−∇_𝛼∇_𝛽𝜙²+2𝐺_4𝑋×𝑅^𝛼𝛽∇_𝛼𝜙∇_𝛽𝜙−∇_𝜆𝐺_4𝑋∇^𝜆−1𝜙□𝜙2𝑅∇_𝜆𝐺₅∇^𝜆𝜙−1−26𝐺_5𝑋□𝜙³−3□𝜙∇_𝛼∇_𝛽𝜙²+2∇_𝛼∇_𝛽𝜙³+∇_𝛼𝐺₅𝑅^𝛼𝛽∇_𝛽1𝜙−2∇_𝛼𝐺_5𝜙∇^𝛼1𝜙□𝜙+2∇_𝛼𝐺_5𝜙×∇_𝛽𝜙∇^𝛼∇^𝛽1𝜙−2∇_𝛼𝐺_5𝑋∇^𝛼1𝑋□𝜙+2∇_𝛼𝐺_5𝑋×∇_𝛽𝑋∇^𝛼∇^𝛽1𝜙−4∇^𝜆𝐺_5𝑋∇_𝜆𝜙□𝜙²−∇_𝛼∇_𝛽𝜙²+12𝐺_5𝑋𝑅_𝛼𝛽∇^𝛼𝜙∇^𝛽1𝜙□𝜙−2𝐺_5𝑋×𝑅_{𝛼𝜆𝛽𝜌}∇^𝛼𝜙×∇^𝛽𝜙∇^𝜆𝜙∇^𝜌𝜙1+𝑇+2𝐾_𝑋∇_𝜇𝜙²−12𝐺_3𝑋□𝜙∇_𝜇𝜙²−∇_𝜇𝐺₃∇_𝜇1𝜙+2𝐺_4𝑋𝑅∇_𝜇𝜙²+12𝐺_4𝑋𝑋□𝜙²−∇_𝛼∇_𝛽𝜙²∇_𝜇𝜙²+𝐺_4𝑋□𝜙∇_𝜇∇_𝜇𝜙−𝐺_4𝑋∇_𝜆∇_𝜇𝜙∇^𝜆∇_𝜇𝜙−2∇_𝜆𝐺_4𝑋∇^𝜆∇𝜙²+∇_𝜆𝐺_4𝑋×∇^𝜆𝜙∇_𝜇∇_𝜇𝜙−2𝐺_4𝑋𝑅_𝜆𝜇∇_𝜇𝜙∇^𝜆𝜙−∇_𝜇𝐺_4𝑋∇_𝜇𝜙□𝜙−𝐺_4𝑋𝑅_{𝜇𝛼𝜇𝛽}∇^𝛼𝜙∇^𝛽𝜙+𝐺_4𝑋∇_𝜇∇_𝜇𝜙+𝐺_4𝜙𝜙∇_𝜇𝜙²−2𝐺_4𝑋𝜙∇^𝜆𝜙×∇_𝜆∇_𝜇𝜙²+𝐺_4𝑋𝑋∇^𝛼𝜙∇_𝛼×∇_𝜇𝜙∇^𝛽𝜙∇_𝛽∇_𝜇𝜙−𝐺_5𝑋𝑅_𝛼𝛽∇^𝛼𝜙∇^𝛽∇_𝜇𝜙²+𝐺_5𝑋𝑅_𝛼(𝜇∇_𝜇)𝜙∇^𝛼1𝜙□𝜙+2𝐺_5𝑋𝑅_𝛼𝛽∇^𝛼𝜙∇^𝛽×𝜙∇_𝜇∇_𝜇1𝜙+2𝐺_5𝑋×𝑅_{𝜇𝛼𝜇𝛽}∇^𝛼𝜙∇^𝛽𝜙□𝜙−𝐺_5𝑋𝑅_{𝛼𝜆𝛽𝜇}∇_𝜇𝜙∇^𝜆𝜙∇^𝛼∇^𝛽𝜙−𝐺_5𝑋𝑅_{𝛼𝜆𝛽𝜇}×∇_𝜇∇^𝜆𝜙∇^𝛼𝜙∇^𝛽1𝜙+2∇_(𝜇𝐺_5𝑋∇^𝛼𝜙∇_𝛼∇_𝜇)−1𝜙□𝜙2∇_(𝜇𝐺_5𝜙∇_𝜇)𝜙□𝜙+∇_𝜆𝐺_5𝜙∇_(𝜇𝜙∇_𝜇)∇^𝜆𝜙−12∇_𝜆𝐺_5𝜙∇^𝜆𝜙−∇_𝛼𝐺_5𝑋∇_𝛽𝜙∇^𝛼∇^𝛽𝜙×∇_𝜇∇_𝜇𝜙−∇^𝛼𝐺₅∇^𝛽𝜙𝑅_{𝛼𝜇𝜇𝛽}+∇_𝜇𝐺₅𝑅_𝜇𝜆∇^𝜆𝜙−12∇_𝜇𝐺_5𝑋×∇_𝜇𝜙□𝜙²−∇_𝛼∇_𝛽𝜙²+∇^𝜆𝐺₅𝑅_𝜆𝜇∇_𝜇𝜙−∇_𝛼𝐺_5𝑋∇_𝛽𝜙∇^𝛼∇_𝜇𝜙×∇^𝛽∇_𝜇𝜙+∇_𝛽𝐺_5𝑋□𝜙∇^𝛽∇_(𝜇𝜙−∇^𝛼∇^𝛽𝜙∇_𝛼∇_(𝜇𝜙∇_𝜇)𝜙−12∇^𝛼𝜙×∇_𝛼𝐺_5𝑋□𝜙∇_𝜇∇_𝜇𝜙−∇_𝛽∇_𝜇𝜙∇_𝛽∇_𝜇𝜙+12𝐺_5𝑋𝐺_𝛼𝛽∇^𝛼∇^𝛽𝜙∇_𝜇𝜙²+12𝐺_5𝑋□𝜙∇_𝛼∇_𝜇𝜙∇^𝛼∇_𝜇𝜙−12𝐺_5𝑋□𝜙²∇_𝜇∇_𝜇1𝜙−𝐺12^5𝑋𝑋×□𝜙³−3□𝜙∇_𝛼∇_𝛽𝜙²+2∇_𝛼∇_𝛽𝜙³∇_𝜇𝜙²−12∇_𝜆𝐺₅𝑅∇^𝜆𝜙+12∇_𝛼𝐺_5𝑋∇_𝛽𝑋∇^𝛼∇^𝛽𝜙−14∇^𝛼𝐺_5𝑋∇_𝜆𝜙□𝜙²−∇_𝛼∇_𝛽𝜙²+12𝐺_5𝑋𝑅_𝛼𝛽∇^𝛼𝜙∇^𝛽1𝜙□𝜙−2𝐺_5𝑋𝑅_{𝛼𝜆𝛽𝜌}. (9)(10) Evaluating the temporal and spatial components of the effective energy-momentum tensor and its trace defined previously and using these values in (3), we can find the energy conditions for any spacetime.

Let us consider the spatially homogeneous, isotropic, and flat FRW universe model with 𝑎(𝑡) as a scale factor described by the metric 𝑑𝑠²=−𝑑𝑡²+𝑎²(𝑡)𝑑𝑥²+𝑑𝑦²+𝑑𝑧². (11) The background fluid is taken as perfect fluid given by (4) with 𝑢_𝜇=(1,0,0,0), and the null like vector is taken as 𝑘_𝜇=(1,𝑎,0,0). Furthermore, we assume that the scalar field is a function of time only. The Friedmann equations for the generalized scalar-tensor theory in terms of effective energy density and pressure are given by [32–35] 3𝐻²=𝜌^eff𝐺₄,−3𝐻²𝐻̇=𝑝+2^eff𝐺₄, (12) where 𝜌^eff=12𝜌^𝑚+2𝑋𝐾_𝑋−𝐾+6𝑋𝜙̇𝐻𝐺_3𝑋−2𝑋𝐺_3𝜙+24𝐻²𝑋×𝐺_4𝑋+𝑋𝐺_4𝑋𝑋−12𝐻𝑋𝜙̇𝐺_4𝑋𝜙−6𝐻𝜙̇𝐺_4𝜙+2𝐻³𝑋𝜙̇5𝐺_5𝑋+2𝑋𝐺_5𝑋𝑋−6𝐻²𝑋3𝐺_5𝜙+2𝑋𝐺_5𝑋𝜙,𝑝^eff=12𝑝^𝑚+𝐾−2𝑋𝐺_3𝜙+𝜙̈𝐺_3𝑋−12𝐻²𝑋𝐺_4𝑋−4𝐻𝑋̇𝐺_4𝑋−8𝐻̇𝑋𝐺_4𝑋−8𝐻𝑋𝑋̇𝐺_4𝑋𝑋+2𝜙̈+2𝐻𝜙̇𝐺_4𝜙+4𝑋𝐺_4𝜙𝜙+4𝑋𝜙̈−2𝐻𝜙̇×𝐺_4𝑋𝜙−2𝑋2𝐻³𝜙̇+2𝐻𝐻̇𝜙̇+3𝐻²𝜙̈𝐺_5𝑋−4𝐻²𝑋²𝜙̈𝐺_5𝑋𝑋+4𝐻𝑋𝑋̇−𝐻𝑋𝐺_5𝑋𝜙+22(𝐻𝑋)^.+3𝐻²𝑋𝐺_5𝜙+4𝐻𝑋𝜙̇𝐺_5𝜙𝜙. (13)(14) Here 𝐺₄, being an arbitrary function of 𝜙 and 𝑋, acts as a dynamical gravitational constant, and it should be positive for any gravitational theory. Furthermore, 𝜌^𝑚 and 𝑝^𝑚 are density and pressure, respectively, for ordinary matter. We shall discuss its different forms in the next section. Using these values in (5), it can be checked that the NEC, WEC, SEC, and DEC require the following conditions to be satisfied: NEC∶𝜌^eff+𝑝^eff1≥0⇒2𝐺₄𝜌^𝑚+𝑝^𝑚+2𝑋𝐾_𝑋+6𝑋𝜙̇𝐻𝐺_3𝑋−4𝑋𝐺_3𝜙+12𝐻²𝑋𝐺_4𝑋+24𝐻²𝑋²𝐺_4𝑋𝑋−20𝑋𝐻𝜙̇𝐺_4𝑋𝜙−2𝐻𝜙̇𝐺_4𝜙+6𝐻³𝑋𝜙̇𝐺_5𝑋+4𝐻³𝑋²𝜙̇×𝐺_5𝑋𝑋−16𝐻²𝑋²𝐺_5𝑋𝜙−12𝐻²𝑋𝐺_5𝜙−2𝑋𝜙̈𝐺_3𝑋−4𝐻𝑋̇𝐺_4𝑋−8𝐻̇𝑋×𝐺_4𝑋−8𝐻𝑋̇𝑋𝐺_4𝑋𝑋+2𝜙̈𝐺_4𝜙+4𝑋𝐺_𝜙𝜙+4𝑋𝜙̈𝐺_4𝑋𝜙−2𝑋2𝐻𝐻̇𝜙̇+3𝐻²𝜙̈𝐺_5𝑋−4𝐻²𝑋²𝜙̈𝐺_5𝑋𝑋+4𝐻𝑋𝑋̇𝐺_5𝑋𝜙+4(𝐻𝑋)^.𝐺_5𝜙+4𝐻𝑋𝜙̇𝐺_5𝜙𝜙≥0,WEC∶𝜌^eff+𝑝^eff≥0,𝜌^eff1≥0⇒2𝐺₄𝜌^𝑚+2𝑋𝐾_𝑋−𝐾+6𝑋𝐻𝜙̇𝐺_3𝑋−2𝑋𝐺_3𝜙+24𝐻²𝑋×𝐺_4𝑋+𝑋𝐺_4𝑋𝑋𝜙̇̇−12𝑋𝐻−6𝐻𝜙𝐺_4𝜙+2𝐻³𝑋𝜙̇5𝐺_5𝑋+2𝑋𝐺_5𝑋𝑋−6𝐻²𝑋3𝐺_5𝜙+2𝑋𝐺_5𝑋𝜙≥0,SEC∶𝜌^eff+𝑝^eff≥0,𝜌^eff+3𝑝^eff1≥0⇒2𝐺₄𝜌^𝑚+3𝑝^𝑚+2𝑋𝐾_𝑋+2𝐾+6𝑋𝐻𝜙̇𝐺_3𝑋−8𝑋𝐺_3𝜙−6𝑋𝜙̈𝐺_3𝑋−12𝐻²𝑋𝐺_4𝑋+24𝐻²𝑋²×𝐺_4𝑋𝑋−36𝑋𝐻𝜙̇𝐺_4𝑋𝜙+6𝐻𝜙̇𝐺_4𝜙+6𝜙̈𝐺_4𝜙−2𝐻³𝑋𝜙̇𝐺_5𝑋+4𝐻³𝑋²𝜙̇𝐺_5𝑋𝑋−24𝐻²𝑋²𝐺_5𝑋𝜙+12𝐻𝑋𝑋̇𝐺_5𝑋𝜙+12(𝐻𝑋)^.𝐺_5𝜙+12𝐻𝑋𝜙̇𝐺_𝜙𝜙−12𝐻²𝑋²𝜙̈𝐺_5𝑋𝑋−12𝑋𝐻𝐻̇𝜙̇𝐺_5𝑋−18𝑋𝐻²𝜙̈𝐺_5𝑋+12𝑋𝜙̈𝐺_4𝑋𝜙−24𝐻𝑋𝜙̇𝐺_4𝑋𝜙+12𝑋𝐺_4𝜙𝜙−24𝐻𝑋𝑋̇×𝐺_4𝑋𝑋−24𝐻̇𝑋𝐺_4𝑋−12𝐻𝑋̇𝐺_4𝑋−36𝐻²𝑋𝐺_4𝑋≥0,DEC∶𝜌^eff≥0,𝜌^eff+𝑝^eff≥0,𝜌^eff−𝑝^eff1≥0⇒2𝐺₄𝜌^𝑚−𝑝^𝑚+2𝑋𝐾_𝑋−2𝐾+6𝑋𝐻𝜙̇𝐺_3𝑋+2𝑋𝜙̈𝐺_3𝑋+36𝐻²𝑋𝐺_4𝑋+24𝐻²𝑋𝐺_4𝑋𝑋−4𝑋𝐻𝜙̇𝐺_4𝑋𝜙−10𝐻𝜙̇𝐺_4𝜙+14𝐻³𝑋𝜙̇𝐺_5𝑋+4𝐻³𝑋²𝜙̇×𝐺_5𝑋𝑋−24𝐻²𝑋𝐺_5𝜙−4(𝐻𝑋)^.𝐺_5𝜙−4𝐻𝑋𝜙̇𝐺_𝜙𝜙−8𝐻²𝑋²𝐺_5𝑋𝜙+4𝐻𝑋̇𝐺_4𝑋+8𝐻̇𝑋𝐺_4𝑋+8𝐻𝑋𝑋̇𝐺_4𝑋𝑋−2𝜙̈𝐺_4𝜙−4𝑋𝐺_4𝜙𝜙−4𝑋𝜙̈𝐺_4𝑋𝜙+2𝑋2𝐻𝐻̇𝜙̇+3𝐻²𝜙̈𝐺_5𝑋+4𝐻²𝑋²𝐺_5𝑋𝑋𝜙̈−4𝐻𝑋𝑋̇𝐺_5𝑋𝜙≥0. (15)

In a mechanical framework, the terms velocity, acceleration, jerk, and snap parameters are based on the first four time derivatives of position. In cosmology, the Hubble, deceleration, jerk, and snap parameters are, respectively, defined as 𝑎̇𝐻=𝑎1,𝑞=−𝐻²𝑎̈𝑎1,𝑗=𝐻³𝑎⃛𝑎,1𝑠=𝐻^4⃜𝑎𝑎. (16) In order to have a more precise picture of these conditions (15), we use the relations of time derivatives of Hubble parameter in terms of cosmological quantities like deceleration, snap, and jerk parameters as 𝐻̇=−𝐻²1+𝑞,𝐻̈=𝐻³,⃛𝑗+3𝑞+2𝐻=𝐻⁴𝑠−2𝑗−5𝑞−3. (17) Moreover, we assume that the scalar field evolves as a power of scale factor, that is, 𝜙(𝑡)∼𝑎(𝑡)^𝛽 [51, 52], which leads to 𝜙̇∼𝛽𝐻𝑎^𝛽,𝜙̈∼𝛽𝑎^𝛽𝐻̇+𝛽𝐻²=𝛽𝑎^𝛽𝐻²,𝛽𝛽−1−𝑞𝑋∼²𝐻²𝑎^2𝛽2,𝑋̇∼𝛽²𝐻³𝑎^2𝛽𝛽−1−𝑞. (18) Here 𝛽 is a nonzero parameter (𝛽=0 yields constant scalar field, 𝛽>0 yields expanding scalar field, and 𝛽<0 corresponds to contracting scalar field). Clearly, 𝜙 remains as a positive quantity.

Introducing these quantities in the energy conditions given by (15), it follows that 1𝐺₄𝐾_𝑋−2𝐺_3,𝜙+6𝐻²𝐺_4𝑋−6𝐻²𝐺_5𝜙+2𝐺_4𝜙𝜙×𝛽²𝐻²𝑎^2𝛽+3𝐻𝐺_3𝑋−10𝐻𝐺_4𝑋𝜙+3𝐻³𝐺_5𝑋+2𝐻𝐺_5𝜙𝜙×𝛽³𝐻³𝑎^3𝛽+6𝐻²𝐺_4𝑋𝑋−4𝐻²𝐺_5𝑋𝜙×𝛽⁴𝐻⁴𝑎^4𝛽−2𝐻²𝛽𝑎^𝛽𝐺_4𝜙+𝐻⁸𝛽⁵𝑎^5𝛽𝐺_5𝑋𝑋−𝛽³𝐻⁴𝑎^3𝛽𝛽−1−𝑞𝐺_3𝑋−4𝐻⁴𝑎^2𝛽𝛽²𝛽−1−𝑞×𝐺_4𝑋+4𝐻⁴𝑎²𝛽𝛽²1+𝑞𝐺_4𝑋−4𝐻⁶𝛽⁴𝑎^4𝛽×𝛽−1−𝑞𝐺_4𝑋𝑋+2𝐻²𝑎^𝛽𝛽−1−𝑞𝛽𝐺_4𝜙+2𝛽³𝐻⁴𝑎^3𝛽𝛽−1−𝑞𝐺_4𝑋𝜙+2𝐻⁶1+𝑞𝛽³𝑎^3𝛽𝐺_5𝑋−3𝐻⁶𝛽³𝑎^3𝛽𝛽−1−𝑞𝐺_5𝑋−𝐻⁸𝛽⁵𝑎^5𝛽×𝛽−1−𝑞𝐺_5𝑋𝑋+2𝐻⁶𝛽⁴𝑎^4𝛽×𝛽−1−𝑞𝐺_5𝑋𝜙+4𝐻⁴𝛽²𝑎^2𝛽𝛽−1−𝑞𝐺_5𝜙−2𝐻⁴1+𝑞𝛽²𝑎^2𝛽𝐺_5𝜙+𝜌^𝑚+𝑝^𝑚1≥0,𝐺₄𝜌^𝑚+3𝑝^𝑚+2𝐾+3𝛽³𝑎^3𝛽𝐻⁴𝐺3𝑋−4𝛽²𝐻²×𝑎^2𝛽𝐺_3𝜙−3𝛽³𝑎^3𝛽𝐻⁴𝛽−1−𝑞𝐺_3𝑋−6𝐻⁴𝛽²𝑎^2𝛽𝐺_4𝑋+3𝐻⁶𝛽⁴𝑎^4𝛽𝐺_4𝑋𝑋−18𝐻⁴𝛽³𝑎^3𝛽𝐺_𝑋𝜙+6𝐻²𝛽×𝑎^𝛽𝐺_4𝜙+6𝐻²𝛽𝑎^𝛽×𝛽−1−𝑞𝐺_4𝜙−𝐻⁶𝛽³𝑎^3𝛽𝐺_5𝑋+2𝐻⁸𝛽⁵𝑎^5𝛽𝐺_5𝑋𝑋−3𝐻⁶𝛽⁴𝑎^4𝛽𝐺_5𝑋𝜙+6𝐻⁶𝛽⁴𝑎^4𝛽×𝛽−1−𝑞𝐺_5𝑋𝜙−6𝐻⁴𝛽²𝑎^2𝛽1+𝑞𝐺_5𝜙+12𝐻⁴𝛽²𝑎^2𝛽𝛽−1−𝑞𝐺_5𝜙+6𝐻⁴𝛽³𝑎^3𝛽𝐺_5𝜙𝜙−3𝐻⁸𝛽⁵𝑎^5𝛽𝛽−1−𝑞×𝐺_5𝑋𝑋+𝐻⁶𝛽³𝑎^3𝛽1+𝑞𝐺_5𝑋−9𝐻⁶𝛽³𝑎^3𝛽𝛽−1−𝑞𝐺_5𝑋+6𝐻⁴𝛽³𝑎^3𝛽×𝛽−1−𝑞𝐺_4𝑋𝜙−12𝐻⁴𝛽³𝑎^3𝛽𝐺_4𝑋𝜙+6𝐻²𝛽²𝑎^2𝛽𝐺_4𝜙𝜙−12𝐻⁶𝛽⁴𝑎^4𝛽×𝛽−1−𝑞𝐺_4𝑋𝑋+12𝐻⁴𝛽²𝑎^2𝛽×1+𝑞𝐺_4𝑋−12𝐻⁴𝛽²𝑎^2𝛽𝛽−1−𝑞𝐺_4𝑋−18𝐻⁴𝛽²𝑎^2𝛽𝐺_4𝑋+𝛽²𝑎^2𝛽𝐻²𝐾_𝑋1≥0,𝐺₄𝐻²𝛽²𝑎^2𝛽𝐾_𝑋−𝐾+3𝐻⁴𝛽³𝑎^3𝛽𝐺_3𝑋−𝐻²𝛽²𝑎^2𝛽𝐺_3𝜙+12𝐻⁴𝛽²𝑎^2𝛽𝐺_4𝑋+6𝐻⁶𝛽⁴𝑎^4𝛽𝐺_4𝑋𝑋−6𝐻⁴𝛽³𝑎^3𝛽𝐺_𝑋𝜙−6𝐻²𝛽𝑎^𝛽𝐺_4𝜙+5𝐻⁶𝛽³𝑎^3𝛽𝐺_5𝑋+𝐻⁸𝛽⁵𝑎^5𝛽𝐺_5𝑋𝑋−9𝐻⁴×𝛽²𝑎^2𝛽𝐺_5𝜙−3𝐻⁶𝛽⁴𝑎^4𝛽𝐺_4𝑋𝜙+𝜌^𝑚1≥0,𝐺₄𝜌^𝑚−𝑝^𝑚+𝛽²𝐻²𝑎^2𝛽𝐾_𝑋−2𝐾+3𝐻⁴𝛽³𝑎^3𝛽𝐺_3𝑋+𝛽²𝐻³𝑎^2𝛽𝛽−𝑞−1×𝐺_3𝑋+18𝛽²𝐻⁴𝑎^2𝛽𝐺_4𝑋+12𝐻⁴𝛽²𝑎^2𝛽𝐺_4𝑋𝑋−2𝐻⁴𝛽³𝑎^3𝛽𝐺_4𝑋𝜙−10𝐻²×𝛽𝑎^𝛽𝐺_4𝜙+7𝐻⁶𝛽³𝑎^3𝛽𝐺_5𝑋+𝐻⁸𝛽⁵𝑎⁵𝛽𝐺_5𝑋𝑋−12𝐻⁴𝛽²𝑎^2𝛽𝐺_5𝜙−4𝐻⁴𝛽²×𝑎^2𝛽𝐺_5𝜙+2𝐻⁴𝛽²𝑎^2𝛽1+𝑞𝐺_5𝜙−2𝐻⁴𝛽³𝑎^3𝛽𝐺_5𝜙𝜙−4𝐻⁴𝛽²𝑎^2𝛽𝐺_𝑋𝜙+4×𝐻⁴𝛽²×𝑎^2𝛽𝛽−𝑞−1𝐺_4𝑋−4𝐻⁴𝛽²𝑎^2𝛽1+𝑞𝐺_4𝑋+4𝐻⁶𝛽⁴𝑎^4𝛽×𝛽−1−𝑞𝐺_4𝑋𝑋−2𝐻²𝑎^𝛽𝛽𝛽−1−𝑞𝐺_4𝜙−2𝐻²𝛽²𝑎^2𝛽𝐺_4𝜙𝜙−2𝐻⁴𝛽³𝑎^3𝛽𝛽−𝑞−1𝐺_4𝑋𝜙−2𝐻⁶𝛽³𝑎^3𝛽1+𝑞𝐺_5𝑋+3𝐻⁶𝛽³𝑎^3𝛽𝛽−1−𝑞𝐺_5𝑋+𝐻⁸𝛽⁵×𝑎^5𝛽𝛽−1−𝑞𝐺_5𝑋𝑋−2𝐻⁶𝛽⁴𝑎^4𝛽𝛽−1−𝑞𝐺_5𝑋𝜙≥0. (19) These are the most general energy conditions that can yield the energy conditions for various DE models like 𝑘-essence and modified theories in certain limits. In order to satisfy these conditions, it must be guaranteed that the function 𝐺₄ is a positive quantity. However, we have discussed earlier that 𝐺₄, being a gravitational constant, would be positive in all cases (if it is not so, then we impose this condition and restrict the free parameters). Clearly, these conditions are only dependent on the Hubble, deceleration parameters and arbitrary functions, namely, 𝐾, 𝐺₃, 𝐺₄, and 𝐺₅. Once these arbitrary functions are specified, the energy bounds on the selected models can be determined by using these conditions.

In order to have a better understanding of these constraints, we can use either the power law ansataz for the scale factor, for example, [45] or we can use the estimation of present values of the respective parameters available in literature. In this study, we consider the present value of the Hubble parameter 𝐻₀=0.718, the scale factor 𝑎₀=1, and the deceleration parameter 𝑞=−0.64 as suggested by Capozziello et al. [53]. Since it is well known that the energy constraints are satisfied for usual matter contents like perfect fluid, therefore we shall focus on validity of the energy constraints for the scalar field terms only (either we take vacuum case or assume that the energy conditions for ordinary matter hold). It is interesting to mention here that the respective energy conditions in GR can be recovered by taking the arbitrary functions 𝐾,𝐺₃, and 𝐺₅ zero with 𝐺₄ as constant.

4. Energy Conditions in Some Particular Cases

Now we discuss application of the derived conditions to some particular cases of this theory. The violation of energy conditions leads to various interesting results. In particular, for a canonical scalar field, violation of these conditions yields instabilities and ghost pathologies. It is important to discuss the violation of these energy conditions in order to check the existence of instabilities in Horndeski theory. The procedure for FRW universe model in most general scalar-tensor theory based on tensor and scalar perturbations is available in literature [31]. By introducing perturbed metric, it has been shown that for the avoidance of ghost and gradient instabilities, the tensor perturbations suggest ℱ_𝑇=2𝐺₄−𝑋𝜙̈𝐺_5𝑋+𝐺_5𝜙𝒢>0,_𝑇=2𝐺₄−2𝑋𝐺_4𝑋−𝑋𝐻𝜙̇𝐺_5𝑋−𝐺_5𝜙>0, (20) while scalar perturbations impose ℱ_𝑆=1𝑎𝑑𝑎𝑑𝑡Θ𝒢²_𝑇−ℱ_𝑇𝒢>0,_𝑆=ΣΘ²𝒢²_𝑇+3𝒢_𝑇>0, (21) where the quantities Σ and Θ are defined in [31]. We simply plug the values in these conditions for the following cases and show that violation of energy conditions leads to the existence of ghost instabilities.

4.1. 𝑘−Essence Models in General Relativity

The 𝑘-essence dynamical models of DE play a dominant role in the solution of various problems in cosmological context [54]. The action (6) can be reduced to the action for 𝑘-essence model in GR framework defined by 𝑆=∫√−𝑔[𝐾(𝜙,𝑋)+(𝑀²_𝑝𝑙/2)𝑅+𝐿_𝑚]𝑑⁴𝑥 with the following choice of the functions: 𝐾=𝐾𝜙,𝑋,𝐺₄=𝑀²_𝑝𝑙2,𝐺₃=𝐺₅=0, (22) where 𝑀²_𝑝𝑙 is Planck mass. The 𝑘-essence models can be classified into three forms: (i)𝐾(𝜙,𝑋)=𝐾₁(𝑋) (Kinetic case),(ii)𝐾(𝜙,𝑋)=𝐾₁(𝑋)𝐵(𝜙),(iii)𝐾(𝜙,𝑋)=𝐾₁(𝑋)+𝐵(𝜙).For the choice of arbitrary functions given by (22), the energy conditions (15) take the following forms: 1NEC∶𝑀²_𝑝𝑙2𝑋𝐾_𝑋+𝜌^𝑚+𝑝^𝑚≥0,WEC∶𝜌^eff+𝑝^eff1≥0,𝑀²_𝑝𝑙2𝑋𝐾_𝑋−𝐾+𝜌^𝑚≥0,SEC∶𝜌^eff+𝑝^eff1≥0,𝑀²_𝑝𝑙2𝑋𝐾_𝑋+2𝐾+𝜌^𝑚+3𝑝^𝑚≥0,DEC∶𝜌^eff+𝑝^eff≥0,𝜌^eff1≥0,𝑀²_𝑝𝑙2𝑋𝐾_𝑋−2𝐾+𝜌^𝑚−𝑝^𝑚≥0. (23) Here 𝐾(𝜙,𝑋) is arbitrary.

In order to see how the function 𝐾(𝜙,𝑋) can be constrained by using the previous energy conditions, we choose a particular model of 𝑘-essence [55] as follows: 𝐾=1𝜙,𝑋2𝐶₂−𝐶₁−2𝐵𝜙²+12𝐶₁+𝐶₂𝑋+12𝑀⁴₀(𝑋−1)², (24) where 𝐶₁, 𝐶₂, 𝐵, and 𝑀₀ are arbitrary constants. In this case, WEC requires the following conditions: 1𝑀²_𝑝𝑙12𝑋𝐶₁+𝐶₂+2𝑋𝑀⁴₀()−1𝑋−12𝑀⁴₀(𝑋−1)²−12𝐶₂−𝐶₁−2𝐵𝜙²+𝜌^𝑚1≥0,𝑀²_𝑝𝑙𝑋𝐶₁+𝐶₂+2𝑀⁴₀𝑋(𝑋−1)+𝜌^𝑚+𝑝^𝑚≥0, (25) where 𝑀²_𝑝𝑙>0. For the interpretation of the previous inequalities, we consider the power law ansatz for the scalar field 𝜙∼𝑎^𝛽; 𝛽≠0, which further yields 𝑋∼(𝛽²𝑎^2𝛽𝐻²/2). Consequently, the WEC (25) turns out to be 12𝛽²𝑎^2𝛽𝐻²2𝐶₁+𝐶₂𝛽+2²𝑎^2𝛽𝐻²2𝑀⁴₀𝛽²𝑎^2𝛽𝐻²2−1−12𝑀⁴₀𝛽²𝑎^2𝛽𝐻²2−1²−12𝐶₂−𝐶₁−2𝐵𝜙²+𝜌^𝑚𝛽≥0,₂𝑎^2𝛽𝐻²2𝐶₁+𝐶₂+2𝑀⁴₀𝛽²𝑎^2𝛽𝐻²2×𝛽²𝑎^2𝛽𝐻²2−1+𝜌^𝑚+𝑝^𝑚≥0. (26)(27)

It is difficult to find the admissible ranges of all constants 𝐶₁, 𝐶₂, 𝐵, 𝑀₀, and 𝛽 from the previous conditions. In order to find the constraints on these parameters, we consider that these conditions are satisfied for ordinary matter, that is, 𝜌^𝑚>0 and 𝜌^𝑚+𝑝^𝑚>0. Moreover, we take the present value of Hubble parameter and choose some particular values of the constants 𝐶₁ and 𝐶₂ to find the ranges of 𝐵, 𝑀₀, and 𝛽, consistent with the WEC. It turns out from the graphs that we can take the parameter 𝛽 as follows 𝛽>1.4, 0<𝛽<1.4 and 𝛽<0, while 𝐵 and can be positive or negative. For the consistency of condition (27), we restrict the parameter 𝑀₀ as 0<𝑀₀<1. From the condition (26), it can be observed that energy conditions are satisfied only when we take 𝛽>1.4, with arbitrary 𝐵 and 𝛽<1.4 with 𝐵>10 only. Other choices of these parameters lead to violation of WEC. Figures 1(a) and 1(b) show that WEC is satisfied with these fixed input parameters by taking 𝛽>1.4, 0<𝑀₀<1, and 0<𝐵<1.

(a) 𝜌eff+𝑝eff≥0

(b) 𝜌eff≥0

Figure 1 

Plot (a) shows 𝜌eff+𝑝eff≥0 versus 𝛽 and 𝑀0. Plot (b) represents 𝜌eff≥0 versus parameters 𝛽 and 𝐵 with 𝑀0=0.2. In both cases, we take the present values of cosmological parameters with 𝐶1=1, 𝐶2=2.

4.2. Brans-Dicke Theory

Brans-Dicke gravity with action ∫√−𝑔[(𝑀_𝑝𝑙𝑋𝜔/𝜙)−𝑉(𝜙)+(1/2)𝑀_𝑝𝑙𝑅𝜙+𝐿_𝑚]𝑑⁴𝑥 can be defined by the following choice of functions: 𝑀𝐾=^𝑝𝑙𝑋𝜔𝜙−𝑉𝜙,𝐺₃=𝐺₅𝐺=0,₄=12𝑀_𝑝𝑙𝜙. (28) Here 𝜔 is the BD parameter and 𝑉 is the field potential. The action for general scalar-tensor gravity can be obtained by taking 𝐹(𝜙) instead of 𝜙 in 𝐺₄. In this case, the energy conditions take the form 1NEC∶𝑀_𝑝𝑙𝜙̇²𝜙²𝑀_𝑝𝑙𝐻𝜙̇𝜔−𝜙𝑀_𝑝𝑙+𝜙̈𝜙𝑀_𝑝𝑙+𝜌^𝑚+𝑝^𝑚𝜙≥0,WEC∶𝜌^eff+𝑝^eff1≥0,𝑀_𝑝𝑙𝜔𝑀_𝑝𝑙𝜙̇²2𝜙²+𝑉𝜙𝜙−3𝐻𝜙̇𝜙𝑀_𝑝𝑙+𝜌^𝑚𝜙≥0,SEC∶𝜌^eff+𝑝^eff1≥0,𝑀_𝑝𝑙2𝜙̇²𝜙²𝑀_𝑝𝑙𝜙𝜔−2𝑉𝜙+3𝐻𝜙̇𝜙𝑀_𝑝𝑙𝜙̈+3𝜙𝑀_𝑝𝑙+𝜌^𝑚+3𝑝^𝑚𝜙≥0,DEC∶𝜌^eff+𝑝^eff≥0,𝜌^eff1≥0,𝑀_𝑝𝑙2𝑉𝜙𝜙𝐻𝜙̇−5𝜙𝑀_𝑝𝑙−𝜙̈𝜙𝑀_𝑝𝑙+𝜌^𝑚−𝑝^𝑚𝜙≥0. (29) A suitable choice of the field potential has always been a matter of debate in BD gravity. We use power laws for the scalar field and potential as 𝜙=𝜙₀𝑎^𝛽 and 𝑉=𝑉₀𝜙^𝑚, respectively, where 𝑚 and 𝛽 are nonzero parameters. Consequently, the WEC restricts the parameters as 𝑉₀≥3𝐻²₀𝜔𝛽−2𝛽²𝐻²₀,𝛽²𝐻²₀𝜔−𝐻²₀𝛽+𝐻²₀𝑎^𝛽𝛽²−𝑎^𝛽𝛽𝐻²₀1+𝑞₀≥0. (30)

For accelerated expanding universe, the observed range of BD parameter is −2<𝜔<−3/2 [56]. Clearly, both of these conditions are independent of the parameter 𝑚 which shows that these conditions are valid for both the positive and inverse power law potentials; however, these conditions depend on the present value of the field potential. Figure 2(a) shows that the condition 𝜌^eff≥0 leads to 𝛽<0, which further yields positive present value of the BD field potential 𝑉₀. Moreover, the second condition will be satisfied if we take 𝛽<0 with arbitrary 𝜔 and 𝛽>0 with 𝜔>0 only. Figure 2(b) indicates that 𝜌^eff+𝑝^eff≥0 is satisfied for a particular choice of 𝛽<0 and −2<𝜔<−1.5. Clearly, the energy condition 𝜌^eff+𝑝^eff≥0 is violated for 𝛽>0 with 𝜔<0. It is easy to check that this choice of parameters (𝛽=2,𝜔=−1.8) also leads to the violation of condition imposed by scalar perturbations given by the expression 2𝐻𝜙²+4𝜙𝜙̇2𝐻𝜙+𝜙̇²−2𝜙²2𝐻𝜙̇+2𝐻̇𝜙+𝜙̈2𝐻𝜙+𝜙̇²²>𝑎𝜙,4𝜙²𝑋𝜔/𝜙−3𝐻²𝜙−𝐻𝜙̇/22𝐻𝜙+𝜙̇²²+3𝜙>0 (31) and consequently yields the ghost instabilities for the model. However, the conditions imposed by tensor perturbation are trivially satisfied.

(a) 𝜌eff+𝑝eff≥0

(b) 𝜌eff≥0 (𝑉0≥0)

Figure 2 

Plots (a) and (b) show WEC versus parameters 𝜔 and 𝛽 for 𝐻0=0.718 and 𝑞0=−0.64.

4.3. 𝑓(𝑅) Gravity

The action for 𝑓(𝑅) gravity described by 𝑆=∫√−𝑔[(𝑀²_𝑝𝑙/2)𝑓(𝑅)+𝐿_𝑚]𝑑⁴𝑥 can be obtained from the action (6) for 𝑀𝐾=−^2𝑝𝑙2𝑅𝑓_𝑅−𝑓,𝐺₃=𝐺₅𝐺=0,₄=𝑀_𝑝𝑙2𝜙,𝜙=𝑀_𝑝𝑙𝑓_,𝑅. (32) Here 𝑓(𝑅) is an arbitrary function of the Rici scalar. Using these values in energy conditions (15), we obtain NEC∶𝑅̈−𝑅̇𝐻𝑓^′′(𝑅)+𝑅̇²𝑓^′′′(𝑅)+𝜌^𝑚+𝑝^𝑚≥0,WEC∶𝜌^eff+𝑝^eff𝜌≥0,_𝑚−12𝑓(𝑅)−𝑅𝑓^′(𝑅)−3𝐻𝑅̇𝑓^′′(𝑅)≥0,SEC∶𝜌^eff+𝑝^eff𝑓(𝑅)≥0,−𝑅𝑓^′(𝑅)+3𝑅̈+𝑅̇𝐻𝑓^′′(𝑅)+3𝑅̇²𝑓^′′′(𝑅)+𝜌^𝑚+3𝑝^𝑚≥0,DEC∶𝜌^eff+𝑝^eff≥0,𝜌^eff−̇𝑅̈𝑓≥05𝐻𝑅+^′′(𝑅)−𝑅̇²𝑓^′′′(𝑅)−𝑓(𝑅)−𝑅𝑓^′(𝑅)+𝜌^𝑚−𝑝^𝑚≥0, (33) where prime denotes the derivative with respect to 𝑅. Let us consider the example of logarithmic model in 𝑓(𝑅) gravity defined by [57] 𝑓(𝑅)=𝑅log(𝛼𝑅)^𝑞−𝑅;𝛼<0,𝑞>0. (34) In this case, the WEC (𝜌^eff≥0) leads to −6𝐻²₀1−𝑞₀log−6𝛼𝐻²₀1−𝑞₀^𝑞+6𝐻²₀1−𝑞₀+6𝐻²₀1−𝑞₀×log−6𝛼𝐻²₀1−𝑞₀^𝑞+𝑞log−6𝛼𝐻²₀1−𝑞₀^𝑞−1−1−36𝐻⁴₀×𝑗₀−𝑞₀×𝑞−26𝐻²₀1−𝑞₀log−6𝛼𝐻²₀1−𝑞₀^𝑞+𝑞𝑞−16𝐻²₀1−𝑞₀×log−6𝛼𝐻²₀1−𝑞₀^𝑞−2≥0, (35) where we have used 𝑅̇=−6𝐻²(1−𝑞) and the present values of respective parameters. Notice that we have taken only the condition 𝜌^eff≥0 as the other condition involves the present value of snap parameter 𝑠 which is not correctly estimated in the literature yet. Figure 3 shows that the WEC is satisfied for a suitable range of both the parameters 𝑞 and 𝛼.

Figure 3 

Plot shows WEC versus free parameters 𝑞 and 𝛼. Here we take 𝑞0=−0.64, 𝐻0=0.718, and 𝑗0=1.41.

4.4. Kinetic Gravity Braiding Model

Kinetic gravity braiding is defined by the action [58] 𝑆=𝑑⁴𝑥√−𝑔𝐾𝜙,𝑋−𝐺₃𝑀𝜙,𝑋□𝜙+^2𝑝𝑙2𝑀𝑅+^2𝑝𝑙2×□𝜙²−∇_𝜇∇_𝜈𝜙∇^𝜇∇^𝜈𝜙+𝐿_𝑚, (36) where the functions 𝐾 and 𝐺₃ are arbitrary (while other functions are taken to be zero in action (6)). A particular choice of these functions proposed by Dvali and Turner [59] is given by 𝐾=−𝑋 and 𝐺₃=𝑐𝑋^𝑛, where 𝑐 and 𝑛 are constants. For this choice of model, the respective energy conditions turn out to be 1NEC∶𝑀²_𝑝𝑙−2𝑋+6𝑛𝑐𝐻𝜙̇𝑋^𝑛−2𝑐𝑛𝑋^𝑛𝜙̈+𝜌^𝑚+𝑝^𝑚≥0,WEC∶𝜌^eff+𝑝^eff1≥0,𝑀²_𝑝𝑙−𝑋+6𝑛𝑐𝐻𝜙̇𝑋^𝑛+𝜌^𝑚≥0,SEC∶𝜌^eff+𝑝^eff1≥0,𝑀²_𝑝𝑙−4𝑋+6𝑛𝑐𝐻𝜙̇𝑋^𝑛−6𝑐𝑛𝑋^𝑛𝜙̈+𝜌^𝑚+3𝑝^𝑚≥0,DEC∶𝜌^eff+𝑝^eff≥0,𝜌^eff1≥0,𝑀²_𝑝𝑙6𝑛𝑐𝐻𝜙̇𝑋^𝑛+2𝑐𝑛𝑋^𝑛𝜙̈+𝜌^𝑚−𝑝^𝑚≥0. (37) By taking the power law evolution of the scalar field, WEC yields −𝛽²𝑎^2𝛽𝐻²2+6𝑛𝑐𝐻𝜙̇𝛽²𝑎^2𝛽𝐻²2^𝑛𝛽≥0,−2²𝑎^2𝛽𝐻²2+6𝑛𝑐𝐻𝜙̇𝛽²𝑎^2𝛽𝐻²2^𝑛𝛽−2𝑐𝑛²𝑎^2𝛽𝐻²2^𝑛𝜙̈≥0. (38) Here we have taken the present values of the ordinary density and pressure to be zero. Using the present values of the Hubble parameter and the scale factor, WEC can be satisfied only when both the parameters 𝑛 and 𝛽 remain positive as indicated in Figure 4, where we have taken 0<𝑛<1 and 1<𝛽<2. In this case, scalar perturbations lead to the following constraints: 𝐻²2+𝑞−𝑐𝑛𝐻𝑋^𝑛𝜙̇𝑎+𝑐𝑛²𝑋^𝑛−1𝑋̇𝜙̇+𝑐𝑛𝑋^𝑛𝜙̈>𝐻−𝑐𝑛𝑋^𝑛𝜙̇²,−𝑋+12𝑐𝑛𝐻𝑋^𝑛𝜙̇+6𝑛𝑐(𝑛−1)𝐻𝑋^𝑛𝜙̇−3𝐻²𝑀²_𝑝𝑙𝐻𝑀²_𝑝𝑙−𝑐𝑛𝑋^𝑛𝜙̇²×𝑀²_𝑝𝑙>−3. (39) It is easy to check that the energy conditions are violated for 0.6<𝑛<1 and negative range of 𝛽 (e.g., 𝛽=−10). For this choice of parameters, the previous constraints are also violated, and hence the ghost instabilities occur. However, constraints imposed by tensor perturbations are trivially satisfied as 𝑀²_𝑝𝑙𝜙>0.

(a) 𝜌eff≥0

(b) 𝜌eff+𝑝eff≥0

Figure 4 

Plots (a) and (b) show WEC, namely, 𝜌eff≥0 and 𝜌eff+𝑝eff≥0 versus parameters 𝑛 and 𝛽.

4.5. Covariant Galilean Model

In the absence of potential, the covariant Galilean model [59] is defined by the following choice of parameters in action (6): 𝐾=−𝑐₂𝑋,𝐺₃=𝑐₃𝑀³𝑋,𝐺₄=𝑀²_𝑝𝑙2−𝑐₄𝑀⁶𝑋²,𝐺₅=3𝑐₅𝑀⁹𝑋², (40) where 𝑐₂, 𝑐₃, 𝑐₄, and 𝑐₅ are dimensionless constants, while 𝑀 is constant with dimensions of mass. Using these values in (15), it follows that 1NEC∶2𝑀²_𝑝𝑙/2−𝑐₄/𝑀⁶𝑋²×−2𝑋𝑐₂𝜙̇𝑐+6𝑋𝐻³𝑀³−72𝐻²𝑋²𝑐₄𝑀⁶+60𝐻³𝑋²𝑐₅𝑀⁹−2𝑋𝜙̈𝑐₃𝑀³+24𝐻𝑋𝑋̇𝑐₄𝑀⁶+16𝐻̇𝑋²𝑐₄𝑀⁶−12𝑋²×2𝐻𝐻̇𝜙̇+3𝐻²𝜙̈𝑐₅𝑀⁹−24𝐻²𝑋²𝜙̈𝑐₅𝑀⁹+𝜌^𝑚+𝑝^𝑚≥0,WEC∶𝜌^eff+𝑝^eff1≥0,2𝑀²_𝑝𝑙/2−𝑐₄/𝑀⁶𝑋²×−𝑋𝑐₂𝜙̇𝑐+6𝑋𝐻³𝑀³−96𝐻²𝑋²𝑐₄𝑀⁶+84𝐻³𝑋²𝜙̇𝑐₅𝑀⁹+𝜌^𝑚≥0,SEC∶𝜌^eff+𝑝^eff1≥0,2𝑀²_𝑝𝑙/2−𝑐₄/𝑀⁶𝑋²×−4𝑋𝑐₂𝜙̇𝑐+6𝑋𝐻³𝑀³−6𝑋𝜙̈𝑐₃𝑀³+24𝑋𝐻²𝑐₄𝑀⁶−48𝐻²𝑋²𝑐₄𝑀⁶+12𝐻³𝑋²𝜙̇𝑐₅𝑀⁹−180𝐻²𝑋²𝜙̈𝑐₅𝑀⁹−72𝑋²𝐻𝐻̇𝜙̇×𝑐₅𝑀⁹+48𝑋𝑋̇𝐻𝑐₄𝑀⁶+48𝑋²𝐻̇𝑐₄𝑀⁶𝑋̇×𝑐+24𝐻𝑋₄𝑀⁶+72𝐻²𝑋²×𝑐₄𝑀⁶+𝜌^𝑚+3𝑝^𝑚≥0,DEC∶𝜌^eff+𝑝^eff≥0,𝜌^eff1≥0,2𝑀²_𝑝𝑙/2−𝑐₄/𝑀⁶𝑋²×𝜙̇𝑐6𝑋𝐻³𝑀³+2𝑋𝜙̈𝑐₃𝑀³−72𝐻²𝑋²𝑐₄𝑀⁶−48𝑋𝐻²𝑐₄𝑀⁶+108𝐻³𝑋²𝜙̇𝑐₅𝑀⁶−24𝐻𝑋𝑋̇𝑐₄𝑀⁶−16𝐻̇𝑋²𝑐₄𝑀⁶+12𝑋²2𝐻𝐻̇𝜙̇+2𝐻²𝜙̈𝑐₅𝑀⁹+24𝐻²𝑋²𝜙̈𝑐₅𝑀⁹+𝜌^𝑚−𝑝^𝑚≥0. (41)

By taking the power law ansataz for scalar field and consequently, for kinetic term 𝑋, the WEC in terms of present values of the involved parameter require the following inequalities: 11−𝑐₄𝛽⁴𝐻⁴₀/2𝑀⁶×−𝑐₂𝐻²₀𝛽²+3𝐻⁴₀𝑐₃𝛽³𝑀³−18𝐻⁶₀𝛽⁴𝑐₄𝑀⁶+15𝐻⁷₀𝛽𝑐₅𝑀⁹−𝛽³𝐻⁴₀𝛽−1−𝑞₀𝑐₃𝑀³+24𝐻⁶₀𝛽⁴𝛽−1−𝑞₀𝑐₄𝑀⁶−16𝐻⁶₀𝛽⁴×𝑐1+𝑞⁴𝑀⁶−3𝐻⁴₀𝛽⁴3𝐻⁴₀𝛽𝛽−1−𝑞₀−2𝐻⁴₀𝛽𝑐1+𝑞⁵𝑀⁹−6𝐻⁸₀𝛽⁵𝛽−1−𝑞₀𝑐₅𝑀⁹1≥0,1−𝑐₄𝛽⁴𝐻⁴₀/2𝑀⁶×−𝐻²₀𝛽²𝑐₂2+3𝑐₃𝐻⁴₀𝛽³𝑀³−24𝐻⁶₀𝛽⁴𝑐₄𝑀⁶+𝜌^𝑚₀≥0. (42) Clearly, these conditions are satisfied when both 𝐺₄ and terms inside the brackets are positive. Since 𝐺₄>0 requires 𝛽>(7.5𝑀⁶/𝑐₄)^1/4, therefore a suitable choice of all these parameters yield the consistency with WEC if parameter 𝛽 remains small and positive (𝛽<7.8), while 𝑐₂ remains negative as shown in Figure 5. Here we have taken −5<𝑐₂<−1 or −50<𝑐₂<−10, and 5.5<𝛽<8. In this case, tensor perturbations suggest the following conditions for the avoidance of ghost instabilities: ℱ_𝑇=𝑀²_𝑝𝑙−2𝑋²𝑐₄𝑀⁶−12𝑋²𝜙̈𝑐₅𝑀⁹𝒢>0,_𝑇=𝑀²_𝑝𝑙+6𝑋²𝑐₄𝑀⁶−12𝑋²̇𝑐𝜙𝐻⁵𝑀⁹>0. (43) Clearly, the energy conditions are violated for 𝛽>8 with −5<𝑐₂<−1. For this choice of parameters, the previous constraints imposed by tensor perturbations are also violated, and consequently the ghost instabilities exist. However, scalar perturbations lead to very complicated expressions, so we consider only the conditions imposed by tensor perturbations.

(a)  𝜌eff+𝑝eff≥0

(b) 𝜌eff≥0

Figure 5 

Plots (a) and (b) show WEC versus parameters 𝑐2 and 𝛽 with 𝑐3=2, 𝑐4=4, 𝑐5=3, and 𝑀=2.

5. Summary

The most general scalar-tensor theory being a combination of various DE proposals provides a vast gravitational framework for the discussion of accelerated expansion of the universe. The modified theories involve some extra degrees of freedom that are described by the models with some unknown parameters. It would be interesting to restrict these parameters on physical grounds. In cosmology, this can be done by making compatibility with local gravity tests. In a gravitational theory, energy conditions can be used as an approach to restrict these parameters. In the present paper, we consider the most general scalar-tensor gravity with the field equations involving second-order derivatives. Firstly, we have explored the effective energy-momentum tensor and its trace by inverting the generalized field equations which can be used to find the energy condition bounds for any spacetime manifold.

In order to describe these conditions for specific cases, we consider flat FRW universe model with perfect fluid. By defining the effective energy density and pressure, we have expressed the strong, weak, null, and dominant energy conditions. For the sake of convenience, we have assumed that the scalar field evolves as a power of scale factor. Also, the derivative terms are removed by expressing these conditions in terms of cosmological quantities like deceleration, snap and jerk parameters. An estimation to the present values of these parameters is available in literature [53] that can be used to find the constraints on free parameters of the model. The derived energy conditions are the most general in nature involving many arbitrary functions 𝐾, 𝐺₃, 𝐺₄, and 𝐺₅ that correspond to different DE proposals.

For the application of these energy conditions, we have taken different choices of the functions 𝐾,𝐺₃,𝐺₄, and 𝐺₅ and have deduced the energy conditions for 𝑘-essence model, BD gravity, 𝑓(𝑅) theory, kinetic gravity braiding, and covariant Galilean model. In each case, there are many free parameters. It is not possible to find their range that would be consistent with energy conditions as well. For this reason, we have specified some of these parameters and restricted the others. The results can be summarized in the following. (i)The WEC in 𝑘-essence model is satisfied only if the free parameters satisfy 0<𝑀₀<1, 𝛽>1.4 with arbitrary 𝐵 and 𝛽<1.4 with 𝐵>10. (ii)In the literature [40–42], using the equivalence between BD and 𝑓(𝑅) gravity (𝜔=0), it has been shown that 𝑉₀ should be negative. In our case (𝜔≠0), the WEC restricts the parameter 𝛽 to be negative (the scalar field should be of contractive nature) for the positive present value of BD field potential, that is, 𝑉₀>0 which is physically correct. However, the condition 𝜌^eff+𝑝^eff≥0 is satisfied only when we take 𝛽<0 with arbitrary 𝜔 and 𝛽>0 with 𝜔>0. Thus it can be concluded that expanding scalar field with negative range of BD parameter allowed for cosmic expansion is inconsistent with the WEC. (iii)In 𝑓(𝑅) gravity, many authors have used the energy conditions to find the constraints on the models like 𝑓(𝑅)=𝛼𝑅^𝑛 or 𝑅+𝛼𝑅^𝑛 [40–42]. In the present case, we have found the constraints on the logarithmic 𝑓(𝑅) model. It is seen that the restrictions on free parameters 𝛼<0 and 𝑞>0 are consistent with WEC. (iv)In kinetic gravity braiding, the WEC is satisfied for the presented model only when 𝛽>0 (that shows expanding scalar field) and 𝑛>0.(v)In covariant Galilean model of DE, the WEC is satisfied when free parameters of the model satisfy 𝛽>(7.5𝑀⁶/𝑐₄)^1/4 and 𝑐₂<0. All these results are also shown through graphs. Further, we have determined the conditions for the avoidance of ghost instabilities using the constraints based on the scalar and tensor perturbations proposed by Kobayashi et al. [31]. It is concluded that the violation of these energy conditions leads to the occurrence of ghost and gradient instabilities in the above-mentioned cases of the most general second-order scalar-tensor theory. It would be interesting to investigate the constraints on other DE models like exponential model of 𝑓(𝑅) gravity, other forms of potentials for BD gravity, and so forth by making them consistent with the energy conditions.

Appendix

𝑇^(𝜙)_𝜇𝜈=12𝐾_𝑋∇_𝜇𝜙∇_𝜈1𝜙+2𝐾𝑔_𝜇𝜈−12𝐺_3𝑋□𝜙∇_𝜈𝜙∇_𝜇𝜙−∇_(𝜇𝐺₃∇_𝜈)1𝜙+2𝑔_𝜇𝜈∇_𝜆𝐺₃∇^𝜆1𝜙+2𝐺_4𝑋𝑅∇_𝜇𝜙∇_𝜈𝜙+12𝐺_4𝑋𝑋□𝜙²−∇_𝛼∇_𝛽𝜙²×∇_𝜇𝜙∇_𝜈𝜙+𝐺_4𝑋□𝜙∇_𝜇∇_𝜈𝜙−𝐺_4𝑋∇_𝜆∇_𝜇𝜙∇^𝜆∇_𝜈𝜙−2∇_𝜆𝐺_4𝑋×∇^𝜆∇_(𝜇𝜙∇_𝜈)𝜙+∇_𝜆𝐺_4𝑋∇^𝜆𝜙∇_𝜇∇𝜈𝜙−𝑔_𝜇𝜈𝐺_4𝜙□𝜙−2𝑋𝐺_4𝜙𝜙−𝑔_𝜇𝜈×−2𝐺_4𝑋𝜙∇_𝛼∇_𝛽𝜙∇^𝛼𝜙∇^𝛽𝜙+𝐺_4𝑋𝑋∇_𝛼∇_𝜆×𝜙∇_𝛽∇^𝜆𝜙∇^𝛼𝜙∇^𝛽1𝜙+2𝐺_4𝑋□𝜙²−∇_𝛼∇_𝛽𝜙²−2𝐺_4𝑋𝑅_𝜆(𝜇∇_𝜈)𝜙∇^𝜆𝜙−∇_(𝜇𝐺_4𝑋∇_𝜈)×𝜙□𝜙+𝑔_𝜇𝜈𝐺_4𝑋𝑅^𝛼𝛽∇_𝛼𝜙∇_𝛽𝜙−∇_𝜆𝐺_4𝑋∇^𝜆𝜙□𝜙−𝐺_4𝑋𝑅_{𝜇𝛼𝛽𝜈𝛽}×∇^𝛼𝜙∇^𝛽𝜙+𝐺_{4𝜙∇𝜇∇𝜈𝜙}+𝐺_4𝜙𝜙∇_𝜇𝜙∇_𝜈𝜙−2𝐺_4𝑋𝜙∇^𝜆𝜙∇_𝜆∇_(𝜇𝜙∇_𝜈)𝜙+𝐺_4𝑋𝑋∇^𝛼𝜙∇_𝛼∇_𝜇𝜙∇^𝛽×𝜙∇_𝛽∇_𝜈𝜙−𝐺_5𝑋𝑅_𝛼𝛽∇^𝛼𝜙∇^𝛽∇_(𝜇𝜙∇_𝜈)𝜙+𝐺_5𝑋×𝑅_𝛼(𝜇∇_𝜈)𝜙∇^𝛼1𝜙□𝜙+2𝐺_5𝑋𝑅_𝛼𝛽∇^𝛼𝜙∇^𝛽𝜙∇_𝜇∇_𝜈𝜙×12𝐺_5𝑋𝑅_{𝜇𝜈𝛼𝛽}∇^𝛼𝜙×∇^𝛽𝜙□𝜙−𝐺_5𝑋𝑅_{𝛼𝜆𝛽(𝜇}∇_𝜈)𝜙∇^𝜆𝜙×∇^𝛼∇^𝛽𝜙−𝐺_5𝑋𝑅_{𝛼𝜆𝛽(𝜇}∇_𝜈)∇^𝜆𝜙∇^𝛼𝜙×∇^𝛽1𝜙+2×∇_(𝜇𝐺_5𝑋∇^𝛼𝜙∇_𝛼∇_𝜈)1𝜙□𝜙−2∇_(𝜇𝐺_5𝜙∇_𝜈)𝜙□𝜙+∇_𝜆𝐺_5𝜙∇_(𝜇𝜙∇_𝜈)∇^𝜆1𝜙−2×∇_𝜆𝐺_5𝜙∇^𝜆𝜙−∇_𝛼𝐺_5𝑋∇_𝛽𝜙∇^𝛼∇^𝛽𝜙×∇_𝜇∇_𝜈𝜙−∇^𝛼𝐺₅∇^𝛽𝜙𝑅_{𝛼(𝜇𝜈)𝛽}+∇_(𝜇𝐺₅𝐺_𝜈)𝜆∇^𝜆1𝜙−2∇_(𝜇𝐺_5𝑋∇_𝜈)𝜙×□𝜙²−∇_𝛼∇_𝛽𝜙²+∇^𝜆𝐺₅𝑅_𝜆(𝜇∇_𝜈)𝜙−∇_𝛼𝐺_5𝑋∇_𝛽𝜙∇^𝛼∇_(𝜇𝜙∇^𝛽∇_𝜈)𝜙+∇_𝛽𝐺_5𝑋×□𝜙∇^𝛽∇_𝜇𝜙−∇^𝛼∇^𝛽𝜙∇_𝛼∇_(𝜇𝜙∇_𝜈)1𝜙−2∇^𝛼𝜙×∇_𝛼𝐺_5𝑋□𝜙∇_𝜇∇_𝜈𝜙−∇_𝛽∇_𝜇𝜙∇_𝛽∇_𝜈𝜙+12𝐺_5𝑋𝐺_𝛼𝛽×∇^𝛼∇^𝛽𝜙∇_𝜇𝜙∇_𝜈1𝜙+2𝐺_5𝑋□𝜙∇_𝛼∇_𝜇𝜙∇^𝛼∇_𝜈𝜙−12𝐺_5𝑋□𝜙²∇_𝜇∇_𝜈1𝜙−𝐺12^5𝑋𝑋×□𝜙³−3□𝜙∇_𝛼∇_𝛽𝜙²+∇_𝛼∇_𝛽𝜙³×∇_𝜇𝜙∇_𝜈1𝜙−2∇_𝜆𝐺₅𝐺_𝜇𝜈∇^𝜆𝜙−𝑔_𝜇𝜈×−16𝐺_5𝑋□𝜙³−3□𝜙∇_𝛼∇_𝛽𝜙²+∇_𝛼∇_𝛽𝜙³×∇_𝛼𝐺₅𝑅^𝛼𝛽∇_𝛽1𝜙−2∇_𝛼𝐺_5𝜙∇^𝜆𝜙+1□𝜙2∇_𝛼𝐺_5𝜙∇_𝛽𝜙∇^𝛼∇^𝛽1𝜙−2∇_𝛼𝐺_5𝑋∇^𝛼+1𝑋□𝜙2∇_𝛼𝐺_5𝑋∇_𝛽𝑋∇^𝛼∇^𝛽1𝜙−4∇^𝜆𝐺_5𝑋∇_𝜆𝜙×□𝜙²−∇_𝛼∇_𝛽𝜙²+12𝐺_5𝑋𝑅_𝛼𝛽×∇^𝛼𝜙∇^𝛽1𝜙□𝜙−2𝐺_5𝑋𝑅_{𝛼𝜆𝛽𝜌}. (A.1)