Equations of Motion in an Expanding Universe Sergei M. Kopeikin and Alexander N. Petrov Abstract We make use of an effective field-theoretical approach to derive post-Newtonian equations of motion of hydrodynamical inhomogeneities in cosmology. The matter Lagrangian for the perturbed cosmological model includes dark matter, dark energy, and ordinary baryonic matter. The Lagrangian is expanded in an asymptotic Taylor series around a Friedmann-Lemeître-Robertson-Walker background. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the expansion series is gauge-invariant and all of them together form a basis for the successive post-Newtonian approximations around the background metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Newtonian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background metric. Temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large-scale inhomogeneities of matter as those generated by the primordial cosmological perturbations in these two components with δρ/ρ ≤ 1. The small scale inhomogeneities are generated by the baryonic matter which is considered as a bare perturbation of the background gravitational field, dark matter and energy. Mathematically, the large scale structure inhomogeneities are given by the homogeneous solution of the post-Newtonian equations while the small scale inhomogeneities are described by a particular solution of these equations with the stress-energy tensor of the baryonic matter that admits δρ/ρ ≫ 1. We explicitly work out the field equations of the first post-Newtonian approximation in cosmology and derive the S.M. Kopeikin (B) Department of Physics & Astronomy, University of Missouri, 322 Physics Bldg., Columbia, MO 65211, USA e-mail: kopeikins@missouri.edu S.M. Kopeikin Siberian State Academy of Geodesy, 10 Plakhotny Stz., Novosibirsk 630108, Russia A.N. Petrov Sternberg Astronomical Institute, Lomonosov Moscow State University, Universitetskii Prospect 13, Moscow 119992, Russia e-mail: alex.petrov55@gmail.com © Springer International Publishing Switzerland 2015 D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, DOI 10.1007/978-3-319-18335-0_21 689 690 S.M. Kopeikin and A.N. Petrov post-Newtonian equations of motion of the large and small scale inhomogeneities which generalize the covariant law of conservation of stress-energy-momentum tensor of matter in asymptotically-flat spacetime. 1 Introduction 1.1 Brief Overview of Perturbation Techniques The multiwavelength satellite observations of cosmic microwave background (CMB) radiation have opened a new chapter in cosmology [1]. The standard cosmological model has been worked out to fit the model parameters to CMB [2–5]. The agreement between the standard theory and CMB observations was achieved at the decisive confidence level of 95 % [6]. The Planck satellite observations provide further evidences in robustness of the standard model [7, 8] (see http://www.sciops.esa.int/index.php? project=PLANCK for a comprehensive list of Planck collaboration papers) though it might be still difficult to discern between various scenarios of the early universe [9]. Study of the formation and evolution of the large scale structure in the universe is a key for understanding the present state of the universe and for prediction its uttermost fate [10]. It is extensively researched but, as of today, remains yet unsolved problem in physical cosmology. It is dark matter which plays a key role in structure formation. The dark matter consists of weakly interacting massive particles and interacts with baryons only by the force of gravity. The baryonic matter forms galaxies which, at early stage of structure formation, simply follow the evolution of dark matter. Therefore, the large scale distribution of galaxies is to trace the distribution of dark matter. At the linear stage the density contrast δρ = ρ − ρ̄ is much smaller than the background (mean) density ρ̄ of the universe: δρ/ρ̄ ≪ 1. At later stages of cosmological evolution the structure formation enters non-linear regime when δρ/ρ̄ ≃ 1 and caustics are formed. Further growth of the perturbations leads to formation of small scale structures like nuclei of galaxies, dwarf galaxies, globular clusters, stars and more compact relativistic objects which have δρ/ρ̄ ≫ 1. Gravitational field and matter of these super-dense baryonic objects counteract with the gravitational potential of dark matter and dark energy but details of this process are still unknown because it involves fluid’s magnetohydrodynamics, turbulence and the physics of strong gravity field that implies a general relativistic approach taking into account the non-linear interaction of gravitational field with itself and the surrounding matter. Presumably, some insight to the solution of this problem can be gained by exploring exact Lemaître-Tolman cosmological solution of Einstein’s equations admitting spatially inhomogeneity along radial coordinate [11–13]. This purely geometric approach is mathematically sound but physically unrealistic as it describes a pressureless, spherically-symmetric accretion of dust to a single point on the cosmological manifold while the real universe has a continuous set of the accretion points Equations of Motion in an Expanding Universe 691 determined by the initial spectrum of the primordial density fluctuations [3–5] and the baryonic fluid pressure cannot be ignored at non-linear regime. It is supposed that non-linear gravitational effects in the formation of small-scale structure in the universe can be treated with the help of the general relativistic postNewtonian approximations. They had been developed in asymptotically-flat spacetime by a number of researchers [14–18] and were very successful in describing non-linear gravitational effects of fluids, for example, in derivation of equations of motion of binary stars [19–22], in calculation of equilibrium models of rapidly rotating neutron stars [23–25], etc. Post-Newtonian approximations in the baryonic component of the cosmological matter are more entrapped because the background manifold is not flat and we have to take into account not only the perturbations of the background metric tensor, ḡαβ , but those of the background stress-energy tensor of dark matter and dark energy as well. Furthermore, the formalism of the postNewtonian approximations in cosmology is to separate the contribution of bare, small-scale perturbations of the baryonic matter from the large-scale perturbations of the background matter and the metric tensor of spacetime manifold. A number of theoretical attempts was undertaken to work out the first postNewtonian approximation and equations of motion of perfect fluid on cosmological manifold [26–29]. These works provide a good insight to the possible solution of the problem but are insufficiently consistent in separation of perturbations from their background values. They do not suggest a systematic approach for extending the calculations of the linear perturbation theory to the second, and higher, post-Newtonian approximations either. We also point out a rather surprising result by Oliynyk [30, 31] who analysed the general structure of the post-Newtonian expansions on cosmological manifold and arrived to the conclusion that the post-Newtonian series are analytic with respect to the small parameter ε = v/c, where v is a peculiar velocity of fluid with respect to the Hubble flow and c is the constant fundamental speed [32]. The Oliynyk’s conclusion disagrees with that established for the post-Newtonian expansions in asymptotically-flat spacetime [33–35]. Recently, a new interest for developing a self-consistent theory of post-Newtonian approximations in precision cosmology was triggered by a lively discussion [36–40] on whether the small-scale structure of the universe affects its Hubble expansion rate and, thus, can explain the cosmic acceleration of the universe discovered in 1998–99 [41, 42] without invoking a dark energy. This is a, so-called, backreaction problem which intimately relates to the procedure of averaging the small-scale matter perturbations on a curved cosmological manifold [43, 44]. A certain progress in this direction was achieved but many mathematical aspects of the backreaction problem are still poorly understood [45]. The task is to build a rigorous mathematical formalism being able to describe on equal footing both the large-scale perturbations of the background matter of cosmological manifold with the density contrast δρ/ρ̄ ≪ 1 and the small-scale perturbations at present epoch caused by small-scale structures (galaxy, globular cluster, star) having the density contrast δρ/ρ̄ ≫ 1. Einstein’s equations tell us that the density perturbation, δρ/ρ̄, is proportional to the second derivatives of the metric tensor perturbation which can be very large if δρ/ρ̄ ≫ 1. At the same time, the metric tensor perturbation, καβ = gαβ − ḡαβ , and its first derivatives, καβ,γ , 692 S.M. Kopeikin and A.N. Petrov can still remain small enough in order to apply a perturbation technique for solving the Einstein equations. This is similar to the situation in the solar system where the matter density contrast is huge but, nonetheless, the gravitational weak-field approximation for solving Einstein’s equations is fully applicable [46]. It supports the idea that the perturbation technique in cosmology (under above assumptions) is valid for calculating physical effects of inhomogeneities on both the large and small scales [39, 47, 48]. The question is what mathematical technique is the most adequate to deal with physical applications. Historically, the very first perturbation scheme in cosmology was worked out by Lifshitz [49, 50]. It is technically convenient for calculating time evolution of matter’s large scale structure inhomogeneities and gravitational field perturbations [4, 51] but is unsuitable for discussing the process of formation and time evolution of the small scale structures in universe. This is because the Lifshitz approach uses the synchronous gauge where the time-time component of the metric tensor is fixed, g00 = −1, at any order of approximation. The small scale structure in cosmology corresponds to a localized astronomical system having a large density contrast, δρ/ρ̄, and governed by the Newtonian law of gravity which demands the presence of the Newtonian potential, U , making g00 = −1+2U/c2 . It breaks down the synchronous gauge condition at the regime of δρ/ρ̄ ≫ 1. Bardeen’s perturbation approach [52] (see also [53]) is more flexible as it admits a rather large freedom in choosing a particular gauge condition for solving cosmological problems [3, 5]. In the framework of this approach, the longitudinal (conformalNewtonian) gauge is the most appropriate for discussing the small scale structure formation and its physical effects [54]. Some mathematical disadvantage of Bardeen’s approach is in imposing a scalar-vector-tensor decomposition on the metric tensor. It requires application of the Helmholtz theorem [55] that demands to foliate spacetime by a family of spacelike hypersurfaces and to integrate the metric tensor over these hypersurfaces. It makes Bardeen’s approach non-local as contrasted to Lifshitz’s perturbation scheme. Moreover, in order to preserve the gauge-invariance of the Bardeen perturbation scheme one has to decompose the gauge functions in the same fashion as the metric tensor. As the universe evolves the gauge-invariance of the overall Bardeen’s scheme can be preserved, if and only if, one maintains the mapping of spatial points on the foliations along the vector field of time coordinate world lines. Evidently, this approximation scheme differs significantly from the post-Newtonian approximations in asymptotically flat spacetime [14, 46, 56] which are more similar to Lifshitz’s approach but use a different gauge condition (harmonic gauge). A gauge-invariant alternative to Bardeen’s approach was suggested by Ellis and Bruni [57] (see also [58, 59]), who developed a perturbation scheme based on a reduction of full Einstein’s equations down to a system of field equations that are linear around a particular background. The Ellis-Bruni approach uses gauge-invariant variables which are spatial projections on the local comoving-observer frame threading the entire space-time of a real universe. Thus, the Bardeen’s foliation has been replaced in Ellis-Bruni approach with the frame threading and, thus, observer dependent. This is not convenient, and is not used, for developing the post-Newtonian approximations in asymptotically-flat spacetime. Equations of Motion in an Expanding Universe 693 At the epoch of precise cosmology we need more transparent theoretical scheme of the post-Newtonian approximations for handling the iterative calculation of cosmological perturbations and derivation of their equations of motion. This iterative scheme must satisfy a number of well-established criteria like to be covariant, gaugeinvariant, operate with locally defined quantities, be systematic and self-consistent in improving the order of approximations, clearly separate the large-scale from smallscale matter perturbations, be independent of the mathematical ambiguities introduced by the averaging procedures, etc. Some steps in developing such a scheme were done by Green and Wald [39, 40]. However, their work was focused mainly on the discussion of the averaging procedure in cosmology, on the proof that the small-scale, post-Newtonian inhomogeneities do not produce a noticeable backreaction and on finding mathematical evidences that the Newtonian approximation is sufficient in numerical N-body simulations of large scale structure formation [60]. No doubt, theoretical questions about how to perform the averaging in cosmology and whether it produces any backreaction at all, are important for understanding the mathematics of averaging of differential operators in non-linear equations and for clarification of the true nature of dark matter and dark energy. However, the postNewtonian approximation scheme in cosmology has broader implications that are going beyond the discussion of averaging and backreaction problems and relates to the problem of interpretation of precise measurement of cosmological parameters by the advanced gravitational wave detector’s technique [61, 62] and formation of small-scale structures in the universe at the non-linear regime. The formalism of the post-Newtonian approximations can be also helpful in better understanding of the influence of cosmological expansion on celestial mechanics of isolated astronomical systems like binary pulsars which are currently the best laboratories for testing nonlinear regime of general relativity [63, 64]. These tests will be made significantly more precise with advent of gravitational-wave astronomy and Square Kilometer Array (SKA) radio telescope [65]. Recently, we have started a systematic investigation of the dynamics of smallscale inhomogeneities moving on the FLRW background manifold. We have set up a Lagrangian formalism to derive the post-Newtonian field equations for linearised cosmological perturbations [48] and analysed the Newtonian limit of these equations [66]. The present paper goes beyond the linear regime and explores some nonlinear effects. In particular, we derive the post-Newtonian hydrodynamic equations of motion of the background matter (dark matter and dark energy) along with the equations of motion of the baryonic matter forming a small-scale structure with high-density contrast like a star, or galaxy or a cluster of galaxies. We discuss the concept of the covariant and Lie derivatives on manifold in Sect. 2. Geometric theory of Euler-type variational perturbations of arbitrary background manifold is set up in Sect. 3. This theory is applied to the FLRW universe, governed by dark matter and dark energy, in Sect. 4. Section 5 derives the stress-energy tensors for perturbations of the gravitational field, dark matter and dark energy. Finally, we derive equations of motion of the small-scale (bare) perturbations in Sect. 6. Appendix outlines some particular mathematical aspects of our derivation. 694 S.M. Kopeikin and A.N. Petrov Before going into details of our presentation we explain the notations adopted in the present paper. 1.2 Notations We use G to denote the universal gravitational constant and c for the ultimate speed in Minkowski spacetime. Every time, when there is no confusion about the system of units, we use a geometrized system of units where G = c = 1. We put a bar over any function that belongs to the background manifold of the FLRW cosmological model. Any function without such a bar belongs to the perturbed manifold. The other notations used in the present paper are as follows: • T and X i = {X, Y, Z } are the coordinate time and isotropic spatial coordinates on the background manifold; • X α = {X 0 , X i } = {cη, X i } are the conformal coordinates with η being a conformal time; • x α = {x 0 , x i } = {ct, x i } is an arbitrary coordinate chart on the background manifold; • Greek indices α, β, γ, . . . run through values 0, 1, 2, 3, and label spacetime coordinates; • Roman indices i, j, k, . . . take values 1, 2, 3, and label spatial coordinates; • Einstein summation rule is applied for repeated (dummy) indices, for example, P α Q α ≡ P 0 Q 0 + P 1 Q 1 + P 2 Q 2 + P 3 Q 3 , and P i Q i ≡ P 1 Q 1 + P 2 Q 2 + P 3 Q 3 ; • gαβ is a full metric on the cosmological spacetime manifold; • ḡαβ is the FLRW metric on the background spacetime manifold; √ • gμν = √−gg μν —the metric tensor density of a unit weight +1; μν • ḡ = −ḡ ḡ μν —the background metric tensor density of a unit weight +1; • fαβ is the metric on the conformal spacetime manifold; • ηαβ = diag{−1, +1, +1, +1} is the Minkowski metric; • the scale factor of the FLRW metric is denoted as R = R(T ), or as a = a(η) = R[T (η)]; • the Hubble parameter, H = R −1 d R/dT ; • the conformal Hubble parameter, H = a ′ /a; • F denotes a geometric object on the manifold. It can be either a scalar, or a vector, or a tensor field, or a corresponding tensor density; • a bar, F̄ above a geometric object F, denotes the unperturbed value of F on the background manifold; • the tensor indices of geometric objects on the background manifold are raised and lowered with the background metric ḡαβ , for example Fαβ = ḡαμ ḡβν F μν ; • the tensor indices of geometric objects on the conformal spacetime are raised and lowered with the conformal metric fαβ ; • symmetry of a geometric object  to two indices is denoted with round  with respect parenthesis, F(αβ) ≡ (1/2) Fαβ + Fβα ; Equations of Motion in an Expanding Universe 695 • antisymmetry of a geometric object  with respect  to two indices is denoted with square parenthesis, F[αβ] ≡ (1/2) Fαβ − Fβα ; • a prime F ′ = dF/dη denotes a total derivative with respect to the conformal time η; • a dot Ḟ = dF/dT denotes a total derivative with respect to the coordinate time T ; • ∂α = ∂/∂x α is a partial derivative with respect to the coordinate x α ; • a comma with a following index F,α ≡ ∂α F is an other designation of a partial derivative with respect to a coordinate x α which is more convenient in some cases. In some cases which may not cause confusion, the comma as a symbol of the partial derivative is omitted. For example, we denote the partial derivatives of the perturbations of matter variables as φα ≡ φ,α , ψα ≡ ψ,α , etc.; • a vertical bar, F|α denotes a covariant derivative of a geometric object F with with connection referred to the background metric ḡαβ . Covariant derivatives of scalar fields coincide with their partial derivatives; • a semicolon, F;α denotes a covariant derivative of a geometric object F with the connection referred to the conformal metric fαβ ; • a covariant derivative with the connection referred to the full metric, gαβ is denoted with the help of ∇α ; •  A —a multiplet of A = {1, 2, . . . , a} matter fields. In other words,  A denotes a set of fields  A ≡ {1 , 2 , . . . , a } where each of the fields 1 , 2 , . . . is a tensor density of its own weight m 1 , m 2 , . . . respectively. These fields generate the full metric gμν of FLRW universe via the Einstein equations; ¯ A —the background value of the fields  A . These fields generate the background •  metric ḡμν of FLRW universe via the unperturbed Einstein equations; •  B —a multiplet of B = {1, 2, . . . , b} matter fields,  B = {1 , 2 , . . . , b } which can be also tensor densities. They generate the stress energy tensor of the bare perturbation of the metric tensor gμν and that of the fields  A ; ¯ B —the background value of the fields  B . •  ¯ A —the perturbation of the field  A . Fields  A and  ¯ A refer to the • φA ≡ A −  same point on the manifold; ¯ B —the perturbation of the field  B caused by the counteraction • τ B ≡ B −  of the metric tensor perturbations lμν and those of the dynamic fields φ A on the stress-energy tensor of the bare perturbations; • κμν ≡ gμν − ḡμν —the metric tensor perturbation. Fields gμν and ḡμν refer to the same point on the manifold; • hμν ≡ gμν −√ḡμν —the perturbation of the metric density caused by  B ; • l μν ≡ hμν / −ḡ. In a linear approximation, l μν = −κ μν + 21 ḡ μν κ α α , where κ α α = ḡ αβ καβ ;   1 • the Christoffel symbols, Ŵ α βγ = g αν gνβ,γ + gνγ,β − gβγ,ν ; 2 • the Riemann tensor, R α βμν = Ŵ α βν,μ − Ŵ α βμ,ν + Ŵ α μγ Ŵ γ βν − Ŵ α νγ Ŵ γ βμ ; • the Ricci tensor, Rαβ = R μ αμβ ; • the Ricci scalar, R = g αβ Rαβ . 696 S.M. Kopeikin and A.N. Petrov We shall often employ the term on-shell. By on-shell we mean satisfying the equations of motion. For instance, Noether’s theorem links conserved quantities to symmetries of the system on-shell. It is invalid off-shell. We shall introduce and explain other notations as they appear in the main text of the paper. 2 Derivatives on the Geometric Manifold 2.1 Variational Derivative Theory of perturbations of physical fields on manifolds rely upon the principle of the least action of a functional S called action. Variational derivative arises in the problem of finding solutions of the gravitational field equation that extremize the action  (1) S = Fd 4 x,   √ where F ≡ f −g, is a scalar density of weight +1. Let F = F Q, Q α , Q αβ depend on the field variable Q, its first—Q α ≡ Q ,α and second—Q αβ ≡ Q ,αβ partial derivatives that play here a similar role as velocity and acceleration in the Lagrangian mechanics of point-like particles. The field variable Q can be a tensor field of an arbitrary type with the covariant and/or contravariant indices. For the time being, we suppress the tensor indices of Q as it may not lead to a confusion. Function F depends on the determinant g of the metric tensor and can also depend on its derivatives. We shall discuss this case in the sections that follow. A certain care should be taken in choosing the dynamic variables of the Lagrangian formalism in case when the variable Q is a tensor field. For example, if we choose a covariant vector field Aμ as an independent variable, the corresponding “velocity” and “acceleration” variables must be chosen as Aμ,α and Aμ,αβ respectively. On the other hand, if the independent variable is chosen as a contravariant vector Aμ , the corresponding “velocity” and “acceleration” variables must be chosen as Aμ ,α and Aμ ,αβ . The same remark is applied to any other tensor field. The reason behind is that Aμ and Aμ are interrelated via the metric tensor, Aμ = g μν Aν . Therefore, derivative of Aμ differs from that of Aμ by an additional term involving the derivative of the metric tensor which, if being improperly introduced, can bring about spurious terms to the field equations derived from the principle of the least action. Variational derivative, δF/δ Q, taken with respect to the variable Q relates a change, δS, in the functional S to a change, δF, in the function F that the functional depends on,  δS = where δFd 4 x, (2) Equations of Motion in an Expanding Universe δF = 697 ∂F ∂F ∂F δQ + δ Qα + δ Q αβ . ∂Q ∂ Qα ∂ Q αβ (3) This is a functional increment of F. The variational derivative is obtained after we single out a total divergence in the right side of (3) by making use of the commutation relations, δ Q α = (δ Q),α and δ Q αβ = (δ Q),αβ . The total divergence is reduced to a surface term in the integral (2) which vanishes on the boundary of the volume of integration. Thus, the variation of S with respect to Q is given by δS =  δF δ Qd 4 x, δQ (4) where δF ∂F ∂ ∂F ∂F ∂2 ≡ − α + α β . δQ ∂Q ∂x ∂ Q α ∂x ∂x ∂ Q αβ (5) Similar procedure can be applied to S by varying it with respect to Q α and Q αβ . In such a case we get the variational derivatives of F with respect to Q α δF ∂F ∂ ∂F ≡ − β , δ Qα ∂ Qα ∂x ∂ Q αβ (6) and that of F with respect to Q αβ , ∂F δF ≡ δ Q αβ ∂ Q αβ (7)   Let us assume that there is another geometric object, T Q, Q α , Q αβ , which  differs from the original one F Q, Q α , Q αβ by a total divergence     T Q, Q α , Q αβ = F Q, Q α , Q αβ + ∂β H β (Q, Q α ) . (8) It is well-known [67, 68] that taking the variational derivative (5) from T and F yields the same result δF δT ≡ , (9) δQ δQ because the variational derivative from the divergence is zero identically. In fact, it is straightforward to prove a more general result, namely, that the variational derivative (5), after it applies to a partial derivative of an arbitrary smooth function, vanishes identically   δ ∂F ≡ 0. (10) δ Q ∂x α 698 S.M. Kopeikin and A.N. Petrov However, this property does not hold for a covariant derivative in the most general case [68]. The variational derivatives are covariant geometric object that is they do not depend on the choice of a particular coordinates on manifold [46, 68]. In case, when the dynamic variable Q is not a metric tensor, this statement can be proved by taking the first, Qα ≡ Q ;α , and second, Qαβ ≡ Q ;αβ , covariant derivatives of Q as independent dynamic variables instead of its partial derivatives, Q α and Q αβ . In this case the procedure of derivation of variational derivatives (5), (6) remains the same and the result is     ∂F ∂F ∂F δF = − + . (11) δQ ∂Q ∂Qα ;α ∂Qαβ ;βα The order, in which the covariant derivatives are taken, is imposed by the procedure of the extracting the total divergence from the variation of the action in (2). The order of the derivatives is important because the covariant derivatives do not commute. Variational derivative of F with respect to the metric tensor gμν is defined by the same Eqs. (5)–(7) where we identify Q ≡ gμν , Q α ≡ gμν,α , and Q μν ≡ gμν,αβ . It yields δF ∂F ∂F ∂ ∂2 ∂F ≡ − α + α β δgμν ∂gμν ∂x ∂gμν,α ∂x ∂x ∂gμν,αβ δF ∂F ∂F ∂ ≡ − β , δgμν,α ∂gμν,α ∂x ∂gμν,αβ δF ∂F ≡ δgμν,αβ ∂gμν,αβ (12) (13) (14) Covariant generalization of (12)–(14) is not quite straightforward because the covariant derivative of the metric tensor gμν;α = 0, and we cannot use it as a dynamic variable. In this case, we consider the set of the metric tensor, gμν , the Christoffel varisymbols Ŵ α μν , and the Riemann tensor R α βμν as a set of independent dynamic  √ ables. The action is given by (1) where F ≡ −g f gμν , Ŵ α μν , R α βμν is a scalar density of weight +1. Variation of F is δF = ∂F ∂F ∂F δgμν + δŴ α μν + δ R α βμν , α ∂gμν ∂Ŵ μν ∂ R α βμν (15) where variations of the Christoffel symbols and the Riemann tensor are tensors that can be expressed in terms of the variation δgμν of the metric tensor [51] 1 ασ (δgσμ );ν + (δgσν );μ − (δgμν );σ , g 2 α = (δŴ βν );μ − (δŴ α βμ );ν . δŴ α μν = δR α βμν (16) (17) Equations of Motion in an Expanding Universe 699 Now, we replace (16), (18) in (15) and single out a total divergence.1 It yields δF = δF δgμν + B α ,α , δgμν (18) where the total divergence vanishes on the boundary of integration of the action, and the covariant variational derivative is δF ∂F = δgμν ∂gμν   1 σμ ∂F σν ∂F σα ∂F g − +g −g 2 ∂Ŵ σ να ∂Ŵ σ μα ∂Ŵ σ μν ;α   ∂F ∂F σα ∂F + g σν − g + g σμ σ ∂ R αβν ∂ R σ μβα ∂ R σ μβν ;βα (19) Variational derivative with respect to the contravariant metric tensor is ∂gαβ δF δF δF = = −gαμ gβν . δg μν ∂g μν δgαβ δgαβ (20) The variational derivatives are not linear operators. For example, they do not obey Leibniz’s rule [69, Sect. 2.3]. More specifically, for any geometric object,  H = FT , that is a corresponding product of two other geometric objects, F = F Q, Q α , Q αβ   and T = T Q, Q α , Q αβ , the variational derivative δ (FT ) δF δT = T +F , δQ δQ δQ (21) in the most general case. The chain rule with regard to the variational derivative is preserved  in a limitedsense. More specifically, let us consider a geometric object F = F Q, Q α , Q αβ where Q is a function of a variable P, that is Q = Q(P). Then, the variational derivative δF ∂ Q δF = , δP δQ ∂ P (22) that can be confirmed by inspection [70]. On the other  hand, if we have a singledvalued function H = H(Q), and Q = Q P, Pα , Pαβ , the chain rule ∂F δ Q δH = , δP ∂Q δP 1 The fact that F (23) is a scalar density is essential for the transformation of covariant derivatives to the total divergence. The total divergences can be converted to surface integrals which vanish on the boundary of integration and, hence, can be dropped off the calculations. 700 S.M. Kopeikin and A.N. Petrov is also valid. The chain rule (23) will be often used in calculations of the present paper. 2.2 Lie Derivative Lie derivative on the manifold can be viewed as being induced by a diffeomorphism x ′α = x α + ξ α (x), (24) such that a vector field ξ α has no self-intersections, thus, defining a congruence of curves which provides a natural mapping of the manifold into itself. Lie derivative of a geometric object F is denoted as £ξ F. It is defined by a standard rule £ξ F = F ′ (x) − F(x), (25) where F ′ is calculated by doing its coordinate transformation induced by the change of the coordinates (24) with subsequent pulling back the transformed object from the point x ′α to x α along the congruence ξ α [46]. In particular, for any tensor density μ ...μ F = Fν11...νq p of type ( p, q) and weight m one has μ ...μ μ ...μ μ ...μ £ξ Fν11...νq p = ξ α Fν11...νq p ,α + mξ α ,α Fν11...νq p (26) μ1 ...μ p α μ1 ...μ p α + Fα...ν q ξ ,ν1 + · · · + Fν1 ...α ξ ,νq α...μ ...α μ p ξ ,α . − Fν1 ...νqp ξ μ1 ,α − · · · − Fνμ11...ν q We notice that all partial derivatives in the right side of Eq. (26) can be simultaneously replaced with the covariant derivatives because the terms containing the Christoffel symbols cancel each other. The Lie derivative commutes with a partial (but not a covariant) derivative   ∂α £ξ F = £ξ (∂α F) . (27) This us to prove that a Lie derivative from a geometric object   property allows F Q, Q α , Q αβ can be calculated in terms of its variational derivative. Indeed, £ξ F = ∂F ∂F ∂F £ξ Q α + £ξ Q αβ . £ξ Q + ∂Q ∂ Qα ∂ Q αβ (28) Now, after using the commutation property (27) and changing the order of partial derivatives in £ξ Q α and £ξ Q αβ , one can express (28) as an algebraic sum of the variational derivative and a total divergence Equations of Motion in an Expanding Universe £ξ F = ∂ δF £ξ Q + α δQ ∂x  701  δF δF £ξ Q + £ξ Q β . δ Qα δ Q αβ (29) This property of the Lie derivative indicates its close relation to the variational derivative on the manifold and will be used in the calculations that follow. It is also worth pointing out that (29) is used for derivation of Noether’s theorem of conservation of the canonical stress-energy tensor of the field Q in case when F = L is the Lagrangian density of the field for which the variational derivative vanishes on-shell, δF/δ Q = δL/δ Q = 0, and £ξ L = ∂α (ξ α L). 2.3 Partial Derivatives with Respect to the Metric Tensor Calculation of variational derivatives requires calculation of partial derivatives with respect to the metric tensor and other geometric objects like the Christoffel symbols, the Riemann tensor, etc. An example is the partial derivatives from the determinant of the metric tensor √ √ ∂ −g ∂ −g 1√ 1√ −ggμν , −gg μν , (30) =− = ∂g μν 2 ∂gμν 2 where g is the determinant of the metric tensor. α and R α is facilitated Taking partial derivatives from F with respect to gμν , Ŵμν βμν with the help of the following formulas ∂gαβ ν) = δα(μ δβ , ∂gμν ∂Ŵ σ αβ = δρσ δα(μ δβν) , ∂Ŵ ρ μν ∂ R σ γαβ = δρσ δγκ δα[μ δβν] , ∂ R ρ κμν (31) (32) (33) where we have accounted for the symmetry of the Christoffel symbols and the antisymmetry of the Riemann tensor. Variational derivative with respect to the contravariant metric tensor is achieved with the help of the derivative ∂gαβ δ δ δ = = −gαμ gνβ . μν μν δg ∂g δgαβ δgαβ (34) We will also need to calculate the variational derivative with respect to the density of the metric tensor, gμν . It relates to the variational derivative of the metric tensor as follows, 702 S.M. Kopeikin and A.N. Petrov 1 δ δ ∂g ρσ δ = = Aρσ , μν √ μν μν ρσ δg ∂g δg −g δg ρσ where Aρσ μν = 1 ρ σ δ δ + δνρ δμσ − gμν g ρσ . 2 μ ν (35) (36) Partial derivatives of any geometric object with respect to either the metric tensor or its first, or second derivatives can be easily calculated after making use of ∂g αβ = δμ(α δνβ) , ∂g μν ∂g αβ ,γ = δμ(α δνβ) δγσ , ∂g μν ,σ ∂g αβ ,γκ = δμ(α δνβ) δγ(ρ δκσ) . ∂g μν ,ρσ (37) (38) (39) 3 Geometric Perturbation Theory of Arbitrary Background Manifold Let us consider a field theory on a background pseudo-Riemannian manifold M̄ having the metric tensor ḡαβ that is a solution of Einstein’s equations with the stress¯ A , where the index A numerates the fields energy tensor T̄αβ of the physical fields  and takes values A = 1, 2, . . . , a. Let us perturb the manifold by “injecting” into it some matter with a bare stress-energy tensor Tαβ that is generated by a set of physical fields  B where the index B numerates the bare fields and takes values ¯ B . The tensor B = 1, 2, . . . , b. We denote the unperturbed value of the bare fields,  Tαβ is the source of the bare perturbations of the background values of the metric ¯ A on the background manifold M̄. We assume that the tensor ḡμν and the fields  A B fields  and  are both minimally coupled to the curvature of spacetime in the sense of the strong equivalence principle [46, Sect. 3.8.2], [71, Sect. 6.13]. We also assume for the sake of simplicity that the fields  A and  B do not directly interact one with another. The basic geometric theory of the present paper is Einstein’s general relativity. Other geometric theories beyond general relativity can be considered after making corresponding modifications in the Lagrangian formalism presented in this paper but we shall not proceed in this direction here (see, e.g. [72]). The perturbation caused by tensor Tαβ changes the metric ḡαβ to gαβ = ḡαβ + ¯ A to  A =  ¯ A +φ A , where καβ and φ A are the perturbations of καβ , and the fields  the metric and the physical fields respectively. Due to the non-linearity of Einstein’s equations the perturbations interact one with another through gravitational coupling in Einstein’s equations. It makes the geometric structure of the perturbed manifold M rather entangled. This section describes how to find out the perturbed structure Equations of Motion in an Expanding Universe 703 of the manifold M, the field equations, and the equations of motion of matter on the basis of the Lagrangian variational principle [70, 73]. 3.1 Variational Principle Lagrangian formulation of the dynamic theory of physical perturbations starts off the Hilbert-Einstein action defined on an unperturbed (background) manifold M̄ S̄ =  ¯A , d 4 x L̄ ḡμν ,  (40)   √ ¯ A is the where L̄ = −ḡ L̄ is a scalar density of weight +1, L̄ = L̄ ḡμν ,  Lagrangian depending on the metric tensor, the multiplet of matter (tensor den¯ A = { ¯ 1,  ¯ 2, . . . ,  ¯ a }, and their partial derivatives. We did not show sity) fields  explicitly (but are keeping in mind) the dependence of the Lagrangian on the deriv¯ A determine the dynamic atives to avoid superfluous notations. The tensor fields  and geometric structure of the background manifold M̄ via Einstein’s equations. For short, we shall call the Lagrangian density L̄ simply the Lagrangian. The Lagrangian L is split in two parts   ¯ A = L̄G ḡμν + L̄M ḡμν ,  ¯A , L̄ ḡμν ,  (41) where the gravitational (Hilbert) Lagrangian   1 L̄G ḡαβ ≡ − −ḡ R̄, 16π (42) depends on the background metric tensor ḡμν , its first and second derivatives. The matter Lagrangian, L̄M depends solely on the metric tensor and (for instance, in case of Yang-Mills fields) its first derivatives. Of course, it also depends on the matter ¯ A and their derivatives. fields  Dynamic equations of the gravitational field and matter are derived from the principle of the least action by varying the action (40) and equating its variation to zero. This procedure is equivalent to taking the variational derivatives (5) from the Lagrangian (41) that yields δ L̄M = 0, ¯A δ δ L̄G δ L̄M + μν = 0. μν δ ḡ δ ḡ (43) (44) 704 S.M. Kopeikin and A.N. Petrov ¯ A . Equation (44) Equation (43) describes dynamic evolution of the matter fields  can be recognized as the Einstein equations for the background gravitational field (the metric) after noticing that the variational derivatives δ L̄G 1 =− R̄μν , μν δ ḡ 16π   1 δ L̄M 1 M M , ḡ − = T̄ T̄ μν μν δ ḡμν 2 2 (45) (46) where R̄μν is the background value of the Ricci tensor calculated with the help of M ¯ A. the background metric ḡμν , and T̄μν is the stress-energy tensor of the fields  Equation (46) is just a definition of the metrical stress-energy tensor of matter [46, Sect. 3.9.5]. Equation (45) is usually derived by varying the gravitational action (see, for instance, [46, p. 310], [51, p. 364]) and extracting the total derivative that vanishes on the boundary of the volume of integration. We emphasize that the variational derivative (45) given by Eqs. (19) and (20) achieves the same goal in much more simple and attractive way. Indeed, the Lagrangian (42) is equivalent to L̄G = − 1 κλ σ ρ ḡ δρ R̄ κσλ 16π (47) This expression depends only on the metric tensor and the Riemann tensor as a linear algebraic function. Hence, partial derivatives with respect to the Christoffel symbols are automatically nil. Moreover, the covariant derivatives from the metric tensor vanish identically. Hence, the variational derivative (20) is reduced to a simple partial derivative δ L̄G 1 ∂ ḡκλ =− R̄κλ , (48) μν δ ḡ 16π ∂ ḡμν that immediately results in (45). Of course, we could calculate the variational derivative in (45) by applying a more conventional definition (12). It yields the same result but the calculations are lengthy and less obvious. We explain details of this calculation in Appendix section “Variational Derivative from the Hilbert Lagrangian”. For many astrophysical applications the background value of the metric tensor ḡμν is the Minkowski metric ημν which is a trivial solution of the field equations (44). However, this assumption is not applicable in cosmology where (44) are the dynamic Friedmann equations describing the expanding universe by means of the FLRW metric  1 d s̄ = −dT + R (T ) 1 + kr 2 4 2 2 2 −2 δi j d X i d X j , (49) whereX α = (T, X i ) are the global coordinates associated with the Hubble flow, r = δi j X i X j , T is the cosmic time, k is the curvature of space taking one of Equations of Motion in an Expanding Universe 705 the three values k = −1, 0, +1, and R(T ) is the scale factor which exact timedependence is governed by the solution of Einstein’s equations with the background ¯ A [4, 5, 51]. stress-energy tensor determined by the matter fields  Physical perturbations of the background bare mat√ manifold M̄ are caused by the ¯ B with the Lagrangian L̄P = −ḡ L̄ P where L̄ P ≡ L̄ P (ḡμν ,  ¯ B ) is a scalar ter field  ¯ B is minimally coupled with gravity but function. The present paper assumes that  ¯ A . We postulate that the absolute value does not interact directly with the fields  P of L̄ is much smaller than the Lagrangian L̄ of the background manifold, that is ¯ B can be conceived, for example, as a matter composing of an L̄P ≪ L̄. The fields  isolated astronomical system like our solar system or a galaxy, or a cluster of galaxies. ¯ A, ¯ B as a seed perturbations of the fields  However, it also admissible to consider  for example, in case of discussion of the formation of the large-scale structure of the universe from the primordial cosmological perturbations. The assumptions imposed ¯ B presume that in order to describe the dynamic on the Lagrangian of the fields  evolution of the perturbed manifold M we should add algebraically the Lagrangian L̄P of the bare perturbations to the unperturbed Lagrangian L̄ of the background manifold, write down the perturbed Einstein equations for the metric tensor perturbations ¯ A , solve them, lμν along with the equations for the perturbations φ A of the fields  and proceed to the second, third, etc. iterations if necessary. The iterative theory of the Lagrangian perturbations of the manifold is described in the following sections. 3.2 The Lagrangian Perturbations of Dynamic Fields Lagrangian formulation of the dynamic theory of physical perturbations of a manifold starts off the Hilbert-Einstein action  S = d 4 x L(gμν ,  A ,  B ), (50) √ where L = −gL is the scalar density of weight +1, and L = L(gμν ,  A ,  B ) is the Lagrangian depending on the metric tensor, the matter fields  A ={1 , 2 , . . . , a }, and the fields  B = {1 , 2 , . . . , b } representing the bare perturbation of the manifold. The Lagrangian L consists of three parts   L gμν ,  A ,  B = LG gμν + LM gμν ,  A + LP gμν ,  B , (51) where the gravitational (Hilbert) Lagrangian   1 √ −g R, LG gαβ ≡ − 16π (52) 706 S.M. Kopeikin and A.N. Petrov depends on the metric tensor gμν , its first and second derivatives. The Lagrangians of matter, LM and LP , depend solely on the metric tensor and its first derivatives. They also depend directly on the matter fields  A and  B and their partial derivatives but we did not show it explicitly to avoid tedious notations. The matter fields  A and  B are minimally coupled to gravity but we assume that they are not directly coupled to each other. Hence, the Lagrangian of the interaction between these fields does not appear explicitly in (51). This assumption can be relaxed in some alternative theories of gravity [74]. Such cases can be handled with the formalism of the present paper but the computational algebra becomes more intricate and will be considered somewhere else. It is worth noticing that LM and LP depend on the metric tensor gμν both explicitly and implicitly through the mathematical definition of the matter fields  A and  B . √ For example, consider the Lagrangian of the perfect fluid LM = ρ (1 + ) −g, where  is the specific internal energy of the fluid and ρ is the energy density. The √ metric tensor appears explicitly as −g and implicitly in ρ that is defined as the ratio of the rest energy of the fluid’s element to its comoving volume which depends on the determinant of the metric tensor [46]. We define perturbations of the gravitational and matter fields residing on the background manifold by the following equations, gμν (x) = ḡμν (x) + hμν (x), ¯ A (x) + φ A (x),  A (x) =  ¯ B (x) + θ B (x),  B (x) =  (53) (54) (55) where all functions are taken at one and the same point x ≡ x α of the unperturbed manifold. The perturbed values of the fields are assumed to be small compared with ¯ A | and |θ B | < | ¯ B |. There their background counterparts: |hμν | < |ḡμν |, |φ A | < | are no specific limitations on the rate of change of the perturbations that is on their first partial derivatives. The second partial derivatives of the fields are comparable (due to the field equations) with the magnitude of the stress-energy tensor, Tμν , of the bare perturbations that is |hμν ,αβ | ∼ |φ A ,αβ | ∼ |Tμν |. We consider the perturbations hμν , φ A along with the perturbing field θ B as a set of independent dynamic variables which propagate on the background manifold M̄ with the metric ḡμν . In order to derive differential equations governing the evolution of the perturbations we substitute the field decompositions (53), (54) to the Lagrangian L defined by Eq. (51) which yields ¯ A + φ A , ḡμν + hμν ) + LP ( ¯ B + θ B , ḡμν + hμν ). L = LG (ḡμν + hμν ) + LM ( (56) Because the perturbations hμν , φ A , θ B are linearly superimposed with the background ¯  ¯ B respectively, the perturbed (total) values of the metric tensor ḡμν and the fields , Lagrangian (56) admits the following property of the variational derivatives, Equations of Motion in an Expanding Universe δL δL = μν , μν δh δ ḡ 707 δL δL , = A ¯A δφ δ δL δL . = B ¯B δθ δ (57) These relations allow us to replace the variational derivatives with respect to the dynamic perturbation of the field for that taken with respect to the background value of the field. It turns out to be useful in calculations that follow. 3.3 The Lagrangian Series Decomposition Let us now expand the total Lagrangian (56) in a Taylor series by making use of the variational derivatives of L with respect to hμν and φ A . The Taylor expansion of L with respect to θ is not important at this stage of the calculation procedure because physical measurements yield access to the total value of the bare perturbation . We assume at the beginning that the perturbations and their derivatives are small enough to ensure the convergence of the expansion. For the Lagrangian is a function of several variables, the Taylor series has terms with mixed derivatives starting from the second order. There is a nice way of handling the mixed derivatives in the Taylor expansion of the Lagrangian by noticing the following property of the commutator of two variational derivatives [70] δ hαβ αβ δ ḡ  δ L̄ φA ¯A δ  δ − φA ¯A δ  δ L̄ hαβ αβ δ ḡ  = ∂α H α , (58) where Hα denotes a vector density of weight +1 made of the partial derivatives from the background Lagrangian L̄, and the repeated field label A denotes Einstein’s summation over all fields  A . This commutation rule is also valid for any two fields from the field multiplet  A . Equation (58) allows us to change the order of the variational derivatives to reshuffle the terms with the mixed derivatives in the Taylor expansion of the perturbed Lagrangian L. All terms representing the total divergence are omitted from the Taylor expansion since the variational derivative from them vanishes and they do not contribute to the field equations according to (8), (9). After all terms with the mixed derivatives are put in order by applying the commutator rule (58) and the total divergences are discarded, the Taylor expansion of the Lagrangian takes the following simple form L = LP + ∞  Ln . (59) n=0 Here, LP is the Lagrangian of the bare perturbation, L0 ≡ L̄ is the Lagrangian (42) describing dynamic properties of the background manifold, and for any n ≥ 1,   1 μν δLn−1 A δLn−1 h , Ln = +φ ¯A n δ ḡμν δ (60) 708 S.M. Kopeikin and A.N. Petrov represents a collection of terms of the power n with respect to the perturbations hμν and φ A . In particular, the linear and quadratic terms of the expansion in (59) read δ L̄ δ L̄ L1 = hμν μν + φ A , ¯A δ ḡ δ   1 μν δL1 A δL1 , +φ L2 = h ¯A 2 δ ḡμν δ (61) (62) and so on. Equation (60) can be proved by induction starting from the value of L1 in (61) which is apparently true, and operating with the commutation rule (58) in higher orders in order to confirm that the result is reduced to the original Taylor series. The commutation property (58) of the variational derivatives allows us to write down the Taylor expansion (59) as follows   δ δ L̄ + LP , L = exp hμν μν + φ A ¯A δ ḡ δ (63) that establishes a relation between the perturbed and unperturbed Lagrangians in the most succinct, exponential form. By applying Eq. (57) to the Taylor series (59), and making use of δLP /δhμν = P δL /δ ḡμν , we get an important relation between the variational derivatives of the consecutive terms in the series δLn−1 δLn = , δhμν δ ḡμν δLn δLn−1 = . ¯A δφ A δ (64) These relations can be confirmed directly by making use of (60) that establishes a relation between the adjacent orders of the Lagrangian expansion (59). The Lagrangian of the bare perturbation can be also expanded in the Taylor series with respect to hμν , δL δ 1 LP = L + hμν μν + hαβ αβ δ ḡ 2! δ ḡ   δ = exp hμν μν L , δ ḡ   δL hμν μν + · · · δ ḡ (65) where we have defined L ≡ LP ( B , ḡμν ). However, in practical calculations it is more convenient to hold LP unexpanded, keeping in mind that at each iteration the metric tensor gμν and the field  entering LP are known up to the order of the approximation under consideration. Equations of Motion in an Expanding Universe 709 3.4 Dynamic and Effective Lagrangians The principle of the least action tells us that the Lagrangian (56) must be stationary with respect to variations of the metric tensor gμν and the field variables  A , δL = 0, δgμν δL = 0. δ A (66) We also assume that the background Lagrangian (51) is stationary with respect to the ¯ A , and the field equations (43), (44) variations of the background variables ḡμν and  are valid. It means that the variational derivatives with respect to hμν and φ A from the background Lagrangian L0 ≡ L̄ vanish identically. Therefore, applying Eq. (64) to n = 1 yields δL1 δ L̄ = μν = 0, δhμν δ ḡ δL1 δ L̄ = = 0, ¯A δφ A δ (67) (68) due to the background field equations (43), (44). Equations (67), (68) point out that dynamics of physical perturbations are governed solely by the quadratic, cubic and higher-order polynomial terms in the Lagrangian decomposition (59). We define the dynamic Lagrangian of the dynamic perturbations as follows [70, 73] Ldyn ≡ L2 + L3 + · · · , (69) so that the total Lagrangian (59) can be written down in the following form L = L̄ + L1 + Ldyn + LP . (70) The background Lagrangian, L̄, does not depend on the dynamic variables, hμν , and θ B . Hence, the variational derivative from L̄ taken with respect to any of these variables is identically zero. On the other hand, the variational derivatives from L1 taken with respect to hμν and/or φ A vanish on-shell due to the background field equations, as shown in (67), (68). Hence, the Lagrangian perturbation theory of dynamic fields residing on the background manifold can be built on-shell with the effective Lagrangian (71) Leff ≡ Ldyn + LP . φA The effective Lagrangian is convenient for deriving the field equations of the physical perturbations and equations of motion of matter which are discussed in the rest of the present paper. 710 S.M. Kopeikin and A.N. Petrov 3.5 Field Equations for Gravitational Perturbations The Einstein field equations for the metric perturbations are obtained after taking the variational derivative (5) from the effective Lagrangian Leff with respect to hμν , and equating it to zero, δLeff = 0. (72) δhμν   Because of (67), it is equivalent to equation δ L − L̄ /δhμν = 0 or, after applying   (57), to δ L − L̄ /δ ḡμν = 0. Replacing L in this equation with expansion (70) and accounting for the background Einstein equations, δ L̄/δ ḡμν = 0, we recast (72) into the following form δL1 δLeff − μν = μν , (73) δ ḡ δ ḡ where we have used (35) in order to replace the variational derivative with respect to ḡμν for that with respect to ḡ μν . The Euler-Lagrangian equation (73) is convenient to work with. It is worth emphasizing that is fully equivalent to the first variational equation (66). By taking the variational derivatives one can reduce Eq. (73) to a more tractable tensor form M G = 8πμν , (74) + Fμν Fμν where 2 δLeff μν = √ , −ḡ δ ḡ μν (75) is the effective stress-energy tensor and the left side of (74) is a Laplace-Beltrami operator for tensor fields on curved manifolds [48] that consists of two parts G Fμν 16π δ ≡ −√ −ḡ δ ḡ μν 16π δ M Fμν ≡ −√ −ḡ δ ḡ μν   G δ L̄ hρσ ρσ , δ ḡ   M M ρσ δ L̄ A δ L̄ +φ . h ¯A δ ḡρσ δ (76) (77) G describes perturbation of the Ricci tensor and can be easily calcuOperator Fμν lated on any background manifold. Indeed, taking into account (45), we immediately get  1 δ  ρσ G =√ h R̄ρσ . Fμν μν −ḡ δ ḡ (78) Equations of Motion in an Expanding Universe 711 Now, according to the rule of rising and lowering indices of the variational derivatives, we can recast (78) to δ 1 G Fμν hργ δλκ R̄ λ ρκγ . = − √ ḡμχ gνǫ δ ḡχǫ −ḡ (79) Variational derivative in (79) is calculated with the help of the covariant definition (19) where the covariant derivatives are taken on the background manifold and are denoted with a vertical bar. We recall that hργ is an independent dynamic variable while the term under the sign of the variational derivative in (79) depends merely on the background Riemann tensor. Therefore, the derivative in (79) is taken only only with respect to the Riemann tensor in accordance with (19). It yields δ hργ δλκ R̄ λ ρκγ δ ḡχǫ   = hργ δλκ ḡ σχ δσλ δρα δκ[β δγǫ] + ḡ σǫ δσλ δρχ δκ[β δγα] − ḡ σα δσλ δρχ δκ[β δγǫ] = h α[ǫ β]χ ḡ +h χ[α β]ǫ ḡ −h |βα χ[ǫ β]α ḡ |βα 1 αχ βǫ h ḡ + hαǫ ḡ βχ − hχǫ ḡ αβ − hαβ ḡ χǫ = 2 |βα , (80) where we have taken into account that the expression enclosed in the brackets, is symmetric with respect √ to indices α and β. We substitute (80) to (79) and recollect definition of hμν = −ḡl μν along with the constancy of the background metric tensor ḡμν with respect to the covariant derivative. It results in G Fμν = 1 lμν |α |α + ḡμν l αβ |αβ − l α μ|να − l α ν|μα , 2 (81) where each vertical bar denotes √ a covariant derivative with respect to the background metric ḡμν , and lαβ ≡ hαβ / −ḡ. M Operator Fμν describes perturbation of the stress-energy tensor of the matter governing the evolution of the background manifold. Hence, it vanishes on any Ricciflat spacetime manifold in general relativity. Cosmological FLRW spacetime is not M Ricci flat. Therefore, Fμν makes a non-trivial contribution to the field equations M is for gravitational perturbations. Variational derivative in definition (77) of Fμν M A ¯ , and taken from the Lagrangian, L , characterizing the background matter fields  depends crucially on its particular form which must be specified in each individual case. We can bring (77) to a more explicit form by accounting for the definition of the metrical stress-energy tensor of the background matter [46] 2 δ L̄M M T̄μν ≡√ , −ḡ δ ḡ μν (82) 712 S.M. Kopeikin and A.N. Petrov and introducing a new function 2 δ L̄M I¯AM ≡ √ . ¯A −ḡ δ  (83) We notice that I¯AM vanishes on-shell because of the field equation (43). However, M this equation should not be applied immediately in the definition of Fμν as we have to take the variational derivative with respect to the metric tensor which is off-shell operation. With these remarks Eq. (77) takes on the following form 8π δ M = −√ Fμν −ḡ δ ḡ μν  1 M hρσ T̄ρσ − hT̄ M + 2  −ḡφ A I¯AM . (84) We shall calculate (84) later on for an ideal fluid and a scalar field in the case of the FLRW universe governed by dark matter and dark energy. The right side of Eq. (74) contains the effective stress-energy tensor consisting of two contributions where μν = Tμν + Tμν , (85) 2 δLP Tμν ≡ √ , −ḡ δ ḡ μν (86) is the stress-energy tensor of the bare perturbation, and 2 δLdyn Tμν ≡ √ , −ḡ δ ḡ μν (87) is the stress-energy tensor associated with the dynamic field perturbations hμν and φ A . We notice that tensor Tμν is defined as a variational derivative with respect to the background metric, ḡμν . Hence, it differs from 2 δLp Tαβ = √ . −g δg αβ (88) which is defined in terms of the variational derivative with respect to the full metric gμν = ḡμν + κμν + · · · Relation of the two tensors can be found by making use of equations   1 μν ∂gμν (μ ν) , ḡ ḡ − ḡ δ δ − = αβ α β 2 ∂ ḡ αβ   ∂g ρσ 1 ρσ 1 (ρ σ) δ , g δ − = g √ μν −g μ ν ∂gμν 2 (89) (90) Equations of Motion in an Expanding Universe and 713 δ ∂gμν ∂g ρσ δ = . δ ḡ αβ ∂ ḡ αβ ∂gμν δg ρσ (91)   1 1 1 Tμν = Tμν − gμν T − ḡμν ḡ αβ Tαβ − gαβ T , 2 2 2 (92) It yields an exact relation where the trace of the stress energy tensor is defined as T ≡ g αβ Tαβ . Relation (92) can be inverted leading to Tμν = Tμν   1 1 1 αβ − ḡμν T − gμν g Tαβ − ḡαβ T , 2 2 2 (93) where T = ḡ αβ Tαβ . Tensor Tμν can be split in two algebraically-independent parts Tμν = tμν + τμν , (94) where tμν is the stress-energy tensor of pure gravitational perturbations hμν while τμν is the stress-energy tensor characterizing gravitational coupling of the matter field φ A with the gravitational perturbations hμν . For example, in the second-order approximation, when Ldyn = L2 , the corresponding stress-energy tensors are given by equations tμν τμν  δ G hρσ Fρσ − =− 16π −ḡ δ ḡ μν  δ 1 M hρσ Fρσ =− − √ 16π −ḡ δ ḡ μν  1 G hF , 2  1 M A M hF + −ḡφ FA , 2 1 √ (95) (96) where FAM is defined in (101). As soon as the differential operators and the source terms in Eq. (74) are specified, it can be solved by successive iterations. It requires decomposition of the perturbations hμν = φa in the post-Friedmanian series μν μν μν hμν = h1 + h2 + h3 + . . . , A φ = φ1A + φ2A + φ3A + ..., (97) (98) where the terms with indices n = 1, 2, 3, . . . represent the successive approximations of the corresponding order of magnitude. We conjecture that the series (97), (98) are analytic and convergent for a sufficiently small magnitude of the perturbations. However, this requires a special mathematical study. 714 S.M. Kopeikin and A.N. Petrov The iteration procedure starts off the substitution of the unperturbed values of μν hμν = φa = 0 to the right side of (74) and finding the linear perturbation h1 . The solution is substituted back to the right side of (74), which is solved again to find μν h2 , and so on. However, we need additional set of differential equations for the perturbations of the matter fields φ A . 3.6 Field Equations for Matter Perturbations Equations for the matter field perturbation φ A are derived from the Lagrangian (71) by taking the variational derivative with respect to φ A . Since the Lagrangian LP does not depend on φ A it drops out from calculations. Moreover, we assume that the background field equations (43) are satisfied. Thus, the stationarity of the Lagrangian (70) with respect to the perturbations φ A yields δLdyn = 0. δφ A (99) After making use of the second relation in (57), Eq. (99) recasts to FAM = 8π M A, (100) where the linear differential operator 16π δ FA ≡ − √ ¯A −ḡ δ  M  h μν M δ L̄M A δ L̄ + φ ¯A δ ḡμν δ  , (101) and the source density 2 δLdyn M . A ≡ √ ¯A −ḡ δ  (102) All linear with respect to hαβ and φ A terms are included in the left side of Eq. (100) while the non-linear terms have been put in  M A . More explicit form of the operator FAM can be obtained with the help of (82), (83) that results in 8π δ FAM = − √ ¯A −ḡ δ   1 M hρσ T̄ρσ − hT̄ M + 2  −ḡφ A I¯AM . (103) Further specification of FAM requires a particular model of the background matter Lagrangian L̄M which will be discussed in the following section. Equations of motion of the field  B are obtained after taking the variational derivative from the Lagrangian (71) with respect to the variable  B . Because the Equations of Motion in an Expanding Universe 715 only part of the Lagrangian which depends on this filed, is LP , the equations are reduced to δLP = 0. δ B (104) Particular form of this equation depends on a specific choice of the Lagrangian of the bare perturbation. In the lowest order approximation Eqs. (100) and (104) describe evolution of the dynamic variables φ A and  B on the unperturbed cosmological background. The next-order approximations take into account the back reaction of the background on these fields. 3.7 Gauge Invariance of the Field Equations Gauge invariance of the dynamic perturbations is an important geometric property that allows us to distinguish physical degrees of freedom of gravitational and matter fields from the spurious modes generated by transformations of the local coordinates on manifold. Any self-consistent perturbation theory must clearly separate the coordinate-dependent effects from physical perturbations which do not depend on the choice of coordinates. The gauge transformation is understood in the sense of the exponential mapping of the background spacetime to itself that is induced by a non-singular vector flow with a tangent vector ξ α ≡ ξ β (x α ) that is a generator of a (finite) coordinate transformation x ′α = exp ξ β ∂β x α = x α + ξ α + 1 β ξ ∂β ξ α + . . . . 2! (105) The mapping of any geometric object Q ≡ Q(x α ) to another one Q ′ ≡ Q ′ (x α ) being induced by transformation (105) defines the gauge transformation of Q that is given by the exponential Lie transform   1 Q ′ (x α ) = exp £ξ Q(x α ) = Q(x α ) + £ξ Q(x α ) + £2ξ Q(x α ) + . . . , 2! (106) where the Lie derivative £ξ has been defined in Sect. 2.2, £2ξ ≡ £ξ £ξ , £3ξ ≡ £ξ £ξ £ξ , and so on. The gauge transformation of the metric tensor, gμν , and the matter fields,  A  B , implies that their background values do not change under the gauge transformation, only the dynamic perturbations, hμν , φ A , θ B change. Hence, the gauge transformation (106) applied to these variables read    h′μν = hμν + exp £ξ − 1 ḡμν + hμν ,   ¯ A + φA , φ′A = φ A + exp £ξ − 1  (107) (108) 716 S.M. Kopeikin and A.N. Petrov θ′B  ¯ B + θB , = θ B + exp £ξ − 1   (109) Let us consider the gauge transformation of the Lagrangian (51) induced by the gauge transformations of its arguments in more detail. The transformed Lagrangian L′ has the same functional form now on the new values of  as L that depends ¯ A + φ′A ,  ¯ B + θ′B . Replacing the the dynamic variables, L′ ≡ L ḡμν + h′μν ,  primed variables by making use of Eqs. (107)–(109), yields   ¯ B + θB . ¯ A + φ A ), exp £ξ  L′ = L exp £ξ (ḡμν + hμν ), exp £ξ ( (110) This equation can be further transformed by making use of the following commutation relation [70]   ¯ A + φ A ), exp £ξ  ¯ B + θB L exp £ξ (ḡμν + hμν ), exp £ξ ( (111) ¯ A + φA,  ¯ B + θB , = exp £ξ L ḡμν + hμν ,  that is valid modulo total divergence which is inessential in the variational calculus. Expanding the right side of (111) in a Taylor series, like in (106), and taking into account that £ξ L = ∂α (ξ α L), yields the gauge transformation of the Lagrangian,   L′ = L + ∂α ξ α L     1 1 + ∂β ξ β ∂α ξ α L + ∂γ ξ γ ∂β ξ β ∂α ξ α L 2! 3! (112) + ..., where the second, third, and all other terms in the right side of this infinite series represent a divergence. The divergence vanishes when one takes the variational derivative from it and, hence, it can be omitted from the action functional S given in (50). The conclusion is that both the action S and the Lagrangian (56) are gauge invariant with respect to the gauge transformation of their arguments. This assertion is valid both on-shell and off-shell. On the other hand, the effective Lagrangian (71) is gauge-invariant only on-shell that is only when the background equations of motion (43), (44) are satisfied. Indeed, the effective Lagrangian can be represented as a difference Leff = L − L1 − L̄. After making the gauge transformations (107)–(109) of the dynamic variables, we get a new effective Lagrangian L′eff = L′ − L′1 − L̄. The difference δLeff = L′eff − Leff is eff δL    δ L̄    μν δ L̄ A A μν ¯ +φ +  ḡ + h = δL + exp £ξ − 1 , ¯A δ ḡμν δ (113) where the terms being enclosed in the square brackets, vanish on-shell due to the background equations of motion (43), (44). Therefore, δLeff = δL ≡ L′ − L is a total divergence as follows from (112). Hence, Leff is gauge-invariant on-shell. Equations of Motion in an Expanding Universe 717 The gauge invariance of the Lagrangian suggests that the Einstein equations (74) for metric perturbations are gauge invariant as well. It is straightforward to prove it by direct but otherwise tedious calculation which technical details are given in [70]. Gauge transformations (107)–(109) applied to the Einstein equations (74) transform them as follows ′G ′M G M + Fμν − 8π′μν = Fμν + Fμν − 8πμν + exp £ξ F, Fμν (114) where function F vanishes on-shell similar to (113). Therefore, if one assumes that the field equations are valid at least in one gauge, the last term in the right side of (114) vanishes, thus, proving that the field equations are valid in any other gauge. 4 The Lagrangian Formalism in Cosmology We shall use the cosmological model that is in a close agreement with modern observational data. In this model the background manifold represents the spatially homogeneous and isotropic FLRW universe which temporal evolution is governed by a doublet of matter fields,  A = {1 , 2 }. We identify 1 ≡  with the Clebsch potential of an ideal fluid with pressure, and 2 ≡ —with a scalar field having the potential W = W () depending only on the scalar field . The ideal fluid models a self-interacting dark matter [75] while the scalar field describes dark energy in the form of quintessence [76]. The dark matter without self-interaction is included in our theoretical scheme as a pressureless ideal fluid. The overall Lagrangian of the model under consideration is given by Eq. (51) with the Lagrangian of the matter consisting of two non-directly interacting pieces LM = Lm + Lq , (115) where Lm is the Lagrangian of dark matter, and Lq is the Lagrangian of dark energy. 4.1 Lagrangian of Dark Matter Dark matter is modelled as an ideal fluid that is characterized by four thermodynamic parameters: the rest-mass density ρm , the specific internal energy per unit mass m , pressure pm , and entropy sm where the sub-index ‘m’ stands for the dark ‘matter’. We shall assume that the entropy of the ideal fluid remains constant (isentropic motion) and dissipative processes are neglected. The total energy density of the fluid ǫm = ρm (1 + m ). (116) One more thermodynamic parameter is the specific enthalpy of the fluid defined as 718 S.M. Kopeikin and A.N. Petrov μm = pm ǫm + pm = 1 + m + . ρm ρm (117) We shall consider barotropic fluid which thermodynamic equation of state is given by equation pm = pm (ρm , m ), where the specific internal energy m is related to pressure by the first law of thermodynamics dm + pm d  1 ρm  = 0. (118) It is used to derive the following thermodynamic relationships dpm = ρm dμm , dǫm = μm dρm , (119) (120) which are derived directly from (116)–(118). Equation (119) immediately tells us that the partial derivatives ∂ pm = ρm , ∂μm ∂ǫm = μm . ∂ρm (121a) (121b) Equation (121) elucidate that all thermodynamic quantities are functions of only one thermodynamic potential. We accept that this potential is the specific enthalpy μm . Equation of state, relating pressure and the energy density, becomes pm = pm (ǫm ), and it is also an implicit function of the same thermodynamic variable μm because ǫm = ǫm (μm ). Partial derivatives of the thermodynamic quantities with respect to μm can be calculated by making use of (119), (120), and the equation of state pm = pm (ǫm ) giving rise to definition of the (adiabatic) speed of sound cs propagating in the fluid ∂ pm c2 = s2 , ∂ǫm c (122) where the partial derivative is taken under a condition that the entropy, sm , does not change. Notice that the speed of sound in dark matter is not constant in the most general case of a non-linear equation of state. In this case, the speed of sound depends on the thermodynamic potential μm through the equation of state, that is cs = cs (μs ). It is also worth emphasizing that Eq. (122) is valid for any wavelength of sound wave in the ideal fluid, not only for short wavelengths. Other partial derivatives of the thermodynamic quantities can be calculated with the help of the equation of state and derivatives (121), (122) which can be inverted, if necessary, because all thermodynamic relations in the ideal fluid are single-valued. We have, for example, Equations of Motion in an Expanding Universe ∂ǫm c2 = 2 ρm , ∂μm cs 719 ∂ρm c2 ρm = 2 , ∂μm cs μm (123) where all partial derivatives are performed under the same condition of the constant entropy. Theoretical description of the ideal fluid as a dynamic system on space-time manifold is given the most conveniently in terms of the Clebsch potential,  which is also known as the velocity or the Taub potential [77]. The Clebsch potential is a continuous scalar field which can be taken as an independent dynamic variable characterizing the motion of the fluid. This description is complimentary (dual) to the Lagrangian formalism of the ideal fluid based on the coordinates and four-velocity of the fluid particles. In the case of a single-component fluid the Clebsch potential is introduced by the following relationship (124) μm wα = −α , where w α = d x α /dτ is the four-velocity of the fluid, wα = gαβ w β , τ is the proper time of the fluid element taken along its world line, and we denote α ≡ ,α from now on. Equation (124) solves the relativistic Euler equation of motion of the ideal fluid [78]. The four-velocity is normalized, gαβ w α w β = −1, so that the specific enthalpy can be expressed in the following form μm =  −g αβ α β . (125) One may also notice that μm = w α α . (126) The Clebsch potential  has no direct physical meaning as it can be changed to ˜ such that the gauge function, , ˜ is constant along another value  → ′ =  +  α ˜ the worldlines of the fluid in the sense that w α = 0. The Lagrangian of the ideal fluid is usually taken in the form of the total energy √ density, Lm = −gǫm [79]. However, this form is less convenient for applying the Lagrangian formalism on manifolds as it does not contain the kinetic energy (∼g aβ α β ) of dynamic field variables directly. For this reason, the Lagrangian √ of the ideal fluid in the form of pressure, L m = − −g pm , is used in the present paper. It differs from the Lagrangian in the form of energy by a total divergence (see [46, pp. 334–335] for more detail) which is unimportant as it vanishes when the variational derivative from the Lagrangian is taken. The Lagrangian in the form of pressure includes the kinetic energy term Lm =    √ √ −g (ǫm − ρm μm ) = −g ǫm − ρm −g αβ α β . (127) 720 S.M. Kopeikin and A.N. Petrov Metrical stress-energy tensor of the ideal fluid is obtained by taking a variational derivative of the Lagrangian (127) with respect to the metric tensor, 2 δLm m Tαβ =√ . −g δg αβ (128) In our field-theoretical approach to the description of the ideal fluid the metric tensor enters all thermodynamic quantities only through the specific enthalpy in the form of (125). Therefore, taking the variational derivative in (128) with respect to the metric tensor can be done with the help of the chain rule (23) and the results are given in Appendix section “Variational Derivatives of Dark Matter Variables”. Finally, the stress-energy tensor (128) is m = (ǫm + pm ) wα wβ + pm gαβ , Tαβ (129) that is a standard form of the stress-energy tensor of the ideal fluid [79]. 4.2 Lagrangian of Dark Energy The Lagrangian of dark energy is taken in the form of a quintessence of a scalar field    √ 1 αβ g α β + W , Lq = −g (130) 2 where W ≡ W () is a potential of the scalar field and we denote α ≡ ,α from now on. We assume that there is no direct coupling between the Lagrangian of dark energy and that of dark matter. They interact only indirectly through the gravitational field. Many various forms of the potential W are used in cosmology [80] but at the present paper we do not need to specify it further. The metrical stress-energy tensor of the scalar field is obtained by taking a variational derivative 2 δLq q Tαβ = √ , (131) −g δg αβ that yields q Tαβ = α β − gαβ   1 μν g μ ν + W () . 2 (132) One can formally reduce tensor (132) to the form similar to that of the ideal fluid by making use of the following procedure. First, we define the analogue of the specific enthalpy of the scalar field “fluid” μq =  −g αβ α β , (133) Equations of Motion in an Expanding Universe 721 and the effective four-velocity, v α , of the “fluid” μq vα = −α . (134) The four-velocity v α is normalized to gαβ v α v β = −1. Therefore, the scalar field enthalpy μq can be expressed in terms of the partial derivative from the scalar field μq = v α α . (135) We introduce the analogue of the rest mass density ρq of the scalar field “fluid” by defining,  ρq = μq = v α α = −g αβ α β . (136) As a consequence of the above definitions, the energy density, ǫq and pressure pq of the scalar field “fluid” can be introduced as follows 1 ǫq ≡ − g αβ α β + W () = 2 1 pq ≡ − g αβ α β − W () = 2 We notice that relation μq = 1 ρq μq + W (), 2 1 ρq μq − W (). 2 ǫq + pq , ρq (137) (138) (139) between the specific enthalpy μq , density ρq , pressure pq and the energy density, ǫq , of the scalar field “fluid” formally holds on the same form (117) as in the case of the barotropic ideal fluid. After substituting the above-given definitions of various “thermodynamic” quantities into Eq. (132), it formally reduces to the stress-energy tensor of an ideal “fluid”   q Tαβ = ǫq + pq vα vβ + pq gαβ . (140) It is worth emphasizing that the analogy between the stress-energy tensor (140) of the scalar field “fluid” with that of the barotropic ideal fluid (129) is rather formal since the scalar field, in the most general case, does not satisfy all required thermodynamic equations because of the presence of the potential W = W () in the energy density ǫq , and pressure pq of the scalar field. The dark energy is physically different from dark matter! 722 S.M. Kopeikin and A.N. Petrov 4.3 Lagrangian of the Bare Perturbation The Lagrangian LP of bare perturbation that appears in (51), can be chosen arbitrary. We assume that the matter of the bare perturbation is described by a variable  which may be a scalar ortensor field. The Lagrangian of the bare perturbation is  √ given by Lp = −gL p , gαβ . Stress-energy tensor of the bare perturbation, Tαβ , is defined in (88). Tensor Tαβ is a source of the bare gravitational perturbation of the background manifold which generates the large-scale structure of the universe. Its structure should be specified depending on a particular physical situation. In this paper we focus on derivation of hydrodynamic equations of motion of an ideal fluid on background manifold with taking into account the manifold back reaction. Therefore, we assume (141) Tαβ = (ǫ + p) Uα Uβ + pgαβ , where ǫ, p are the energy density and pressure of the fluid comprising the bare perturbation, and U α is its four-velocity normalized to Uα U α = −1. It is worth emphasizing that the four-velocity of the bare perturbation is not a part of the background fluid flow and is decoupled from it in the first approximation. Notice that the stress-energy tensor Tμν defined in (85), is not equal to Tμν . We have derived relation between the two tensors in (92) and (93). 4.4 Background Manifold All geometric objects on background cosmological manifold M̄ will be denoted with a bar over the object. The FLRW metric on the background manifold is given in (49). It is convenient to introduce global isotropic coordinates X α = (X 0 , X i ) by changing the cosmic time T to the conformal time η ≡ X 0 via differential equation dT = a(η)dη. The FLRW metric tensor in the isotropic coordinates reads ḡμν = a 2 (η)fμν , (142) where a cosmological scale factor, a(η) ≡ R[T (η)]. The conformal metric fμν = (−1, fi j ), where the spatial part of the metric   1 2 −2 fi j = 1 + r δi j , 4 (143) depends on the curvature of the spatial hypersurfaces, k = {−1, 0, +1}. In case, k = 0, the conformal metric fμν is reduced to the Minkowski metric, ημν . Four-velocity of the Hubble flow in the isotropic coordinates is Ū α = d X α /dT = (a −1 , 0, 0, 0, 0). Due to the maximal symmetry of FLRW spacetime, all background geometric objects depend only on time η in the isotropic coordinates. We can, and will use Equations of Motion in an Expanding Universe 723 arbitrary coordinates x α = (x 0 , x i ) which are connected to the isotropic coordinates X α by a smooth diffeomorphism x α = x α (X β ). Partial derivative of a background geometric object, Q̄ = Q̄(η), in the arbitrary coordinates is given by Q̄ ,α = − Q̄ ′ ū α = − Q̄˙ ū α , a (144) where ū α is four-velocity of the Hubble flow in arbitrary coordinates. Equation (144) applied to the scale factor, yields a,α = −ȧ ū α = −Hū α , and the partial derivative from the conformal Hubble parameter H,α = −Ḣū α . Einstein’s field equations on the background manifold are given by (44)–(46) which yield two Friedmann equations for the temporal evolution of the scale factor a, k 8π ǭ − 2 , 3 a k 2 2 Ḣ + 3H = −8π p̄ − 2 a H2 = (145) (146) where ǭ and p̄ are the effective energy density and pressure of the background matter. A consequence of the Friedmann equations (145), (146) is an equation Ḣ = −4π (ǭ + p̄) + k , a2 (147) relating the time derivative of the Hubble parameter with the sum of the overall energy density and pressure of dark matter and dark energy ǭ + p̄ = ρ̄m μ̄m + ρ̄q μ̄q . (148) The equation of continuity for the rest mass density ρ̄m of the background dark ¯ It reads ¯ A → . matter is (43) for  that is equivalent to   ρ̄m ū α |α = 0, (149) ρ̄m|α − 3H ρ̄m ū α = 0. (150) The background equation of the conservation of energy of dark matter is ǭm|α − 3H (ǭm + p̄m ) ū α = 0, (151) where we have employed the definition of energy (116), and Eqs. (118), (150). ¯ It reads ¯ is (43) for  ¯ A → . Background equation for the scalar field  724 S.M. Kopeikin and A.N. Petrov ¯ |αβ − ḡ αβ  ∂ W̄ = 0. ¯ ∂ (152) After making use of the definition of the background enthalpy of the scalar field ¯ |α , an equality μ̄q = ρ̄q , and definition (137) of the specific energy ǭq of μ̄q ≡ ū α  the scalar field, the Eq. (152) can be recast to   ǭq|α − 3H ǭq + p̄q ū α = 0, (153) that is similar to the hydrodynamic equation (150) of the conservation of energy of dark matter. Because of this similarity, the second Friedmann equation (146) is not independent and can be derived directly from the first Friedmann equation (145) by taking a time derivative and applying the energy conservation Eqs. (151) and (153). The equation of continuity for the density ρ̄q of dark energy is obtained by differentiating definition (136) of ρ̄q , and making use of (152). It yields   ∂ W̄ ρ̄q ū α |α = − , ¯ ∂ or, equivalently, ρ̄q|α − 3H ρ̄q ū α = (154) ∂ W̄ ū , ¯ α ∂ (155) which shows that the density ρ̄q is not conserved. There is no violation of physical laws here since (155) is simply another way of writing (152) and dark energy is not thermodynamically equivalent to dark matter. 4.5 Perturbations of Dynamic Variables In the present paper, FLRW background manifold is defined by the metric ḡαβ which ¯ of dark matter dynamics is governed by the two fields—the Clebsch potential  ¯ and the scalar field  of dark energy. We assume that the background metric and the fields are perturbed by the presence of a bare matter field θ. Perturbations are considered to be sufficiently small so that the perturbed metric and the matter fields can be split in their background values and the corresponding perturbations, gαβ = ḡαβ + καβ , ¯ + φ, = ¯ + ψ. = (156) These equations are exact. Perturbation of the contravariant component of the metric β is not independent and is determined from the isomorphism gαγ g γβ = ḡαγ ḡ γβ = δα , and is given by g αβ = ḡ αβ − κ αβ + κ α γ κ γβ + . . . , (157) Equations of Motion in an Expanding Universe 725 where the ellipses denote terms of the higher order. We consider perturbation of the metric—καβ , that of the potential of dark matter— φ, and that of the potential of dark energy—ψ as weak with respect to their corre¯ and , ¯ which dynamics is governed by equasponding background values ḡαβ , , tions that have been explained in Sect. 4.4. Because the field variable θ is the source of the bare perturbation, we postulate that its background value is equal to zero. The perturbations καβ , φ, and ψ have the same order of magnitude as θ. More convenient dynamic variable of gravitational field is a contravariant (Gothic) metric gαβ = √ −gg αβ . (158) The covariant Gothic metric gβγ is defined by gαβ gβγ = δγα that yields gαβ = √ αβ g√αβ / −g. The Gothic metric gαβ is expanded around its background √ value, ḡ αβ= αβ −ḡ ḡ , as shown in (53). Calculations also prompt to single out −ḡ from h , and operate with a variable hαβ l αβ ≡ √ . (159) −ḡ Tensor indices of the metric tensor perturbations, l αβ , hαβ , etc., are raised and lowered with the help of the background metric, for example, lαβ ≡ ḡαμ ḡβν l μν . The field variable l αβ relates to the perturbation καβ of the metric tensor as follows   1 2 1 αβ 1 αβ 1 αβ μν μ(α β) κ κμν − κ + . . . , l = −κ + ḡ κ + κ κ μ − κ κ − ḡ 2 2 4 2 (160) where κ ≡ κ σ σ = ḡ ρσ κρσ , and ellipses denote terms of higher orders in καβ . Perturbations of four-velocities, wα and v α , entering definitions of the stressenergy tensors (129), (140), are fully determined by the perturbation of the metric and those of the potentials of dark matter and dark energy. Indeed, according to definitions (124) and (136) the four-velocities are defined by the following equations αβ αβ wα = − α , μm vα = − α . μq (161)   where μm = −g αβ α β and μq = −g αβ α β in accordance with (125) and (133) respectively. We define perturbation of the covariant components of the four-velocities as follows wα = ū α + δwα , vα = ū α + δvα , (162) where the unperturbed values of the four-velocities coincide and are equal to the four-velocity of the Hubble flow due to the requirement of the homogeneity and isotropy of the background FLRW spacetime that is w̄α = v̄ α = ū α . Hence, we have 726 S.M. Kopeikin and A.N. Petrov ū α = − ¯α  , μ̄m ū α = − ¯α  . μ̄q (163) Making use of (161) and (163) in the left side of definitions (161), and expanding its right side by making use of expansions (156) and (157), yield δwα = − 1 β 1 P̄ α φ|β − qū α , μ̄m 2 δvα = − 1 β 1 P̄ α ψ|β − qū α , μ̄q 2 (164) where P̄αβ = ḡαβ + ū α ū β is projector onto the hypersurface orthogonal to the Hubble flow, and l (165) q ≡ −ū α ū β καβ = ū α ū β lαβ + , 2 is projection of the metric tensor perturbation on the Hubble flow. 4.6 Field Equations 4.6.1 Gravitational Field Field equations for metric tensor perturbation are given by the Euler-Lagrange equations (74) where the operator q m M , + Fμν = Fμν Fμν (166) consists of two pieces corresponding to dark matter (index ‘m’) and dark energy (index ‘q’). These pieces are defined in accordance with (77) or, more exactly, m Fμν q Fμν 16π δ ≡ −√ −ḡ δ ḡ μν 16π δ ≡ −√ −ḡ δ ḡ μν  δ L̄m +φ ¯ δ  , (167) δ L̄q δ L̄q +ψ ρσ ¯ δ ḡ δ  . (168) δ L̄m hρσ ρσ δ ḡ  h ρσ Taking variational derivatives from various functions entering the Lagrangians Lm and Lq is rather straightforward and follows from their definitions, the chain rule (23), and a set of variational derivatives of thermodynamic quantities given in Appendix section “Variational Derivatives of Dynamic Variables with Respect to the Metric Tensor”. Making use of the Lagrangians (127) and (130) taken on the background manifold and calculating variational derivatives in (167), (168), we obtain Equations of Motion in an Expanding Universe 727   m Fμν = −4π( p̄m − ǭm )lμν + 8π ρ̄m ū μ φν + ū ν φμ − ḡμν ū α φα    c2 1 ū α φα − μ̄m q ū μ ū ν , + 8π ρ̄m 1 − 2 c 2  s   q Fμν = −4π pq − ǫq lμν + 8π ρ̄q ū μ ψν + ū ν ψμ − ḡμν ū α ψα + 8π ḡμν (169) (170) ∂ W̄ ψ, ¯ ∂ ˙¯ where ρ̄q = μq ≡ /a in accordance with definition (136) projected on the back¯ is arbitrary. ground manifold. The dark energy potential function, W̄ = W̄ (), Substituting (169), (170) along with (81) to the left side of (74) yields field equations for gravitational perturbations l αβ in a covariant form [48, Eq. 161] M = 16πμν , (171) lμν |α |α + ḡμν Aα |α − 2 A(μ|ν) − 2 R̄ α (μlν)α − 2 R̄μαβν l αβ + 2Fμν where Aα ≡ l αβ |β is the gauge vector function. They can be drastically simplified by choosing the gauge condition imposed on the variable Aα ≡ l αβ |β in the following form   Aα = −2Hl αβ ū β + 16π ρ̄m φ + ρ̄q ψ ū α . (172) This gauge condition cancels a significant number of terms in the field equations (171) and allows us to decouple equations for different components of l αβ . Picking up the isotropic coordinates of the Hubble observers we bring the gravity field equations to the following form [48]   c2 q + 2H q,0 + 4kq − 4π 1 − 2 ρ̄m μ̄m q = 8π (00 + kk ) (173a) cs   c2 − 8πa ρ̄m 1 − 2 φ0 cs ∂ W̄ ψ ¯ ∂  + 32πa H ρ̄m φ + ρ̄q ψ , − 16πa 2 l0i + 2Hl0i,0 + 2kl0i = 16π0i ,   l<i j> + 2Hl<i j>,0 + 2 Ḣ − k l<i j> = 16π<i j> ,   l + 2Hl,0 + 2 Ḣ + 2k l = 16πkk . (173b) (173c) (173d) where we denoted the wave operators q ≡ fμν g;μν and lμν ≡ f̄αβ lμν;αβ . Other notations are φ0 ≡ φ,0 , q ≡ (l00 + lkk ) /2, l ≡ lkk = l11 + l22 + l33 , l<i j> = li j − (1/3)δi j lkk , and the same index notations are applied to the effective stressenergy tensor <i j> . 728 S.M. Kopeikin and A.N. Petrov 4.6.2 Dark Matter Evolution of dark matter perturbation is described by the perturbation of the Clebsch potential φ. Equation for φ is derived from a general equation (100) is, in case of dark matter, reads (174) Fm = 8π m . All terms in this equation can now be explicitly written down because the Lagrangian of dark matter is fully determined by (127). The linear differential operator Fm is derived from (103) which yields  1 8π δ m − hT̄ m + hρσ T̄ρσ Fm ≡ − √ ¯ 2 −ḡ δ   −ḡφ I¯m , (175) where   I¯m = 2 ρ̄m ū α |α . The source density (176) 2 δLdyn m ≡ √ , ¯ −ḡ δ  (177) and we shall calculate it explicitly in next section. Calculation of variational derivative in (175) requires taking variational derivative from the density ρ̄m of the ideal fluid with respect to the background value of ¯ The density implicitly depends on the potential through the Clebsch potential . the specific enthalpy, ρ̄m = ρ̄ (μ̄m ) which depends only on the derivatives α of the potential. Hence, ∂ δρm ∂ ∂ρm =− α =− α ¯ ¯ ∂x ∂ α ∂x δ  ∂ρm ∂μm ¯α ∂ μ̄m ∂   , (178) where the partial derivative of the density with respect to the specific enthalpy is calculated with the help of (123) by making use of equation of state of the ideal fluid, and the partial derivative ∂μm = ū α . (179) ¯α ∂ It reveals that Fm = 8πY α |α , (180) where the vector field    ρ̄m β c2 1 ρ̄m α αβ φ − ρ̄m l ū β + 1 − 2 ū φβ − ρ̄m q ū α . Y ≡ μ̄m cs μ̄m 2 α (181) Equations of Motion in an Expanding Universe 729 Taking covariant derivative in (180) brings about the field equations for φ φα α − μm Aα ū α − 4μ̄m H (4q − l)   c2 1 α β α ū ū φαβ − μ̄m ū qα + 1− 2 cs 2   2 ∂ ln cs 1 α ū φα − μ̄m q =  m , − 3H μ̄m ∂ μ̄m 2 (182)  where the very last term accounts for the fact that the speed of sound is not constant in inhomogeneous medium [78]. Indeed, the speed of sound, cs , relates to other thermodynamic quantities by equation of state making the speed of sound a function of the specific enthalpy. Hence, cs = cs (μ̄m ). Covariant derivative from the speed of sound is cs|α = (∂cs /∂ μ̄m ) μ̄m|α , where the covariant derivative μ̄m|α = (∂ μ̄m /∂ ρ̄m ) ρ̄m|α and, according to equation of continuity, ρ̄m|α = 3H ρ̄m ū α . It yields   ∂ ln cs2 c2 1− 2 = 3H μ̄m ū α , cs |α ∂ μ̄m (183) that explains the origin of the last term in (182). After imposing the gauge condition (172), Eq. (182) is reduced to   φα α + 16π μ̄m ρ̄m φ + ρ̄q ψ − 2μ̄m H q    c2 1 ū α ū β φαβ − μ̄m ū α qα + 1− 2 cs 2   2 ∂ ln cs 1 β ū φβ − μ̄m q =  m . −3H μ̄m ∂ μ̄m 2 (184) 4.6.3 Dark Energy Calculation of the field equation for dark energy perturbation ψ follows similar path like in previous subsection. The field equations follows from (100), and they are q F = 8π q , (185) q where F and  q are determined by the Lagrangian of dark energy (130). Making use of this Lagrangian in (103) we obtain 8π δ q F ≡ − √ ¯ −ḡ δ  where  1 q hρσ T̄ρσ − hT̄ q + 2  −ḡψ I¯q , (186) 730 S.M. Kopeikin and A.N. Petrov I¯q = 2    ∂ W̄ ρ̄q ū |α + . ¯ ∂ α  (187) The source density 2 δLdyn q ≡ √ , ¯ −ḡ δ  (188) and we shall calculate it explicitly in next section. According to Eq. (130), the Lagrangian density of the scalar field Lq depends on both the field  and its first derivative, α . For this reason, the differential operator F q is not reduced to the covariant derivative from a vector field as the partial derivative of the Lagrangian with respect to  does not vanish. We have q F l ∂ W̄ ∂ 2 W̄ ≡ 8π Z |α − −ψ ¯ ¯2 2 ∂ ∂  α  (189) where l ≡ ḡ αβ lαβ , and vector field Z α ≡ ψ |α − ρ̄q l αβ ū β , (190) ¯ |α = −ū β  ¯ |β ū α = −ρ̄q ū α . Taking covariant derivwhere we have used equation  ative in (189) yields the field equations for ψ     ∂ 2 W̄ ∂ W̄ q− ψ α + 16π μ̄m ρ̄m φ + ρ̄q ψ − 2μ̄q H + ψ = q, ¯ ¯2 ∂ ∂ α (191) where Eq. (172) has been used along with the equality ρ̄q = μ̄q . 5 Stress-Energy Tensor of Dynamic Perturbations The effective stress-energy tensor μν is defined as a variational derivative (75) taken from the effective Lagrangian (71). According to (85), it consists of two parts—the stress-energy tensor of matter of the bare perturbation Tμν , and the stress-energy tensor of dynamic perturbations Tμν . We shall keep tensor Tμν unspecified as long as theory permits and focus on calculation of Tμν which also consists of two parts, tμν and τμν , according to (287). Tensor tμν is the stress-energy tensor of gravitational field perturbations (95). Tensor τμν is the stress-energy tensor due to the coupling of matter perturbations on the background manifold and the gravitational field perturbations. General formula for calculating τμν is given in (96). In case of FLRW universe gowerved by dark matter and dark energy, tensor τμν is linearly split in two counterparts Equations of Motion in an Expanding Universe 731 q m ταβ = ταβ + ταβ , (192) q m and τ where ταβ αβ describe contributions of dark matter and dark energy respectively. 5.1 Stress-Energy Tensor of Gravitational Field Perturbations Stress-energy tensor of gravitational perturbations, tμν , has a universal presentation on any manifold. We begin its calculation from definition (95) and notice that variational derivative is insensitive to terms which are total divergences. Hence, it is reasonable to single out a total divergence in the argument of variational derivative, G − (1/2)hF G entering definition (95) of the gravitational stress-energy F G ≡ hρσ Fρσ G to calculate F G , and, then, employ the Leibniz rule tensor. We substitute (81) for Fρσ to single out the total divergence from the products of two functions. We arrive to 1 1 F G = hρσ|λlλρ|σ − hρσ|λlρσ|λ + h|λl|λ + div, 2 4 (193) √ where hρσ = −ḡl ρσ , h = ḡρσ hρσ , l = ḡ μν lμν , and div is a total divergence that can be discarded from the calculation of the variational derivative. Next step is to apply the covariant definition (19) of variational derivative to (193) in definition (95) of tαβ . After accounting for transformation (34), definition (95) can be written as follows δF G 1 16πtαβ = √ ḡαμ ḡβν , δ ḡμν −ḡ (194) that conforms with the lower position of indices in variational derivative (19). It is worthwhile to remind the reader that we consider perturbation hρσ as independent variable which has been used in derivation of (95). It means that the partial derivative ∂hαβ = 0. ∂ ḡμν (195) The covariant components of the gravitational perturbation, hαβ = ḡακ ḡβλ hκλ , contain the background metric tensor and cannot be considered as independent from the background metric tensor. We have ∂ ḡβλ ∂hαβ ∂ ḡακ (μ = ḡβλ hκλ + ḡακ hκλ = δα(μ hν) β + δβ hν) α . ∂ ḡμν ∂ ḡμν ∂ ḡμν (196) 732 S.M. Kopeikin and A.N. Petrov Let us consider now the dependence of the covariant derivative hαβ |λ on the metric tensor. We notice that the perturbation hαβ is a tensor density of weight −1. Therefore, its covariant derivative has one more term as compared with that of a tensor of a second rank. More specifically, hαβ |κ = hαβ ,κ + Ŵ α σκ hσβ + Ŵ β σκ hσα − Ŵ σ σκ hαβ . (197) It reveals that the derivative depends merely directly on the Christoffel symbols and is independent of the metric tensor ḡαβ . Hence, the partial derivative ∂hαβ |λ = 0. ∂ ḡμν (198) It confirms that the metric tensor and the Christoffel symbols are true independent variables along with the tensor density hαβ and its covariant derivative hαβ |λ . Equation (193) given in terms of the independent variables, reads ḡ κλ ḡβρ FG = √ −ḡ   1 1 ḡαλ hρσ |κ hαβ |σ − ḡασ hρσ |κ hαβ |λ + ḡασ hασ |κ hβρ |λ , (199) 2 4 where we have discarded the total divergence. Variational derivative (19) from (193) engages the partial derivatives with respect to the background metric tensor and those with respect to the Christoffel symbols. Partial derivative with respect to the metric tensor yields   1 ∂F G 1 ρσ|λ 1 μν ρσ|λ 1 |λ l lλρ|σ − l = − ḡ lρσ|λ + l l|λ √ 2 2 4 −ḡ ∂ ḡμν (200) − l μσ|ρl ν σ|ρ + l μσ|ρl ν ρ|σ 1 1 1 + l ρσ|μlρσ |ν + l μν|σ l|σ − l |μl |ν . 2 2 4 Partial derivative with respect to the Christoffel symbols taken from hρσ|κ is calculated with the help of (197). It gives ∂hρσ |κ = δαρ δκ(λ hγ)σ + δασ δκ(λ hγ)ρ − δα(λ δκγ) hρσ . ∂Ŵ α λγ (201) After making use of this formula, the partial derivative of F G with respect to the Christoffel symbols results in 1 ∂F G = 2l ργ l μ (ρ|α) + 2l ρμl γ (ρ|α) − 2lρα |(μl γ)ρ √ −ḡ ∂Ŵ α μγ 1 |(μ γ) δα − δα(μl γ)l. − 2δα(μl γ) ρ|σ l ρσ + lα (γ l |μ) + l ρσ lρσ 2 (202) Equations of Motion in an Expanding Universe 733 It allows us to calculate the linear combination of the partial derivatives entering definition of variational derivative (19), −   G G G 1 1 σν ∂F σμ ∂F σγ ∂F ḡ + ḡ − ḡ √ 2 −ḡ ∂Ŵ σ μγ ∂Ŵ σ νγ ∂Ŵ σ μν 1 = 2lρ (μl ν)ρ|γ − 2l γρ|(μl ν) ρ − lρ γ l μν|ρ − l μν l |γ 2   1 ρσ |γ 1 |γ μν ρσ γ l l ρ|σ − l lρσ + ll . + ḡ 2 4 (203) After replacing (200) and (203) into variational derivative (194) defined by the rule (19), the stress-energy tensor of gravitational field takes on the following form   1 1 1 (204) 16πtμν = − ḡμν l ρσ|γ lγρ|σ − l ρσ|γ lρσ|γ + l |γ l|γ 2 2 4 1 1 1 − lμσ|ρlν σ|ρ + lμσ|ρlν ρ|σ + lρσ|μl ρσ |ν + lμν|σ l |σ − l|μl|ν 2 4   2 1 1 + ḡμν l ρσ l γ ρ|σ − l ρσ lρσ |γ + ll |γ 2 4 |γ   1 ρ|γ |γ γρ γρ γρ . + 2lρ(μlν) − lνρl |μ − lμρl |ν − l lμν|ρ − lμν l 2 |γ It apparently depends on the second derivatives of the gravitational perturbation, lμν . Significant number of the second derivatives can be eliminated on-shell by making use of the covariant field equation (171). We take the covariant derivative (204) and express the commutator of the second-order derivatives from the metric tensor perturbation in terms of the Riemann tensor l α ρ|σβ = l α ρ|βσ − l γ ρ R̄ α γσβ + l α γ R̄ γ ρσβ . (205) A useful consequence of this equation is l α ρ|σα = Aρ|σ + l α ρ R̄σα + l α γ R̄ γ ρσα , (206) where Aα ≡ l αβ |β . Straightforward but tedious rearrangement of the second-order derivatives from the metric tensor perturbations with the help of (205), (206) allows us to put (204) into the following form 734 S.M. Kopeikin and A.N. Petrov 16πtμν = 2lμρ|σ lν (ρ|σ) − l ρσ |μlνρ|σ − l ρσ |ν lμρ|σ 1 1 + lρσ|μl ρσ |ν − l|μl|ν − l ρσ lμν|ρσ 2 4   1 ρσ|γ 1 |γ 1 ρσ|γ lγρ|σ − l lρσ|γ + l l|γ + ḡμν l 2 2 4 ρ ρ ρ +2l (μ Aν)|ρ − lμν A |ρ − lμν|ρ A    l + 8π 4l ρ (μ ν)ρ − lμν  − ḡμν l ρσ ρσ −  2 (207) + 2l ρ μl σ ν R̄ρσ + 2l αβ l ρ (μ R̄ν)αβρ where αβ = Tαβ − 1 q F m + Fαβ , 8π αβ (208) and the spur,  ≡ ḡ αβ αβ . Notice that the on-shell form (207) of the stress-energy tensor tμν for gravitational field depends on the choice of the gauge function Aα as well as on the value of the Riemann tensor of the background manifold. 5.2 Stress-Energy Tensor of Dark Matter Perturbations The part of the stress-energy tensor describing the dark matter perturbation is given m that, according to (96), is calculated as a variational derivative in (192) by τμν δF m 1 m 16πταβ = √ ḡαμ ḡβν , δ ḡμν −ḡ where 1 m − hF m + F m ≡ hρσ Fρσ 2 −ḡφFm (209) (210) Individual terms entering the right side of (210) are taken from (169) and (180). We single out the total divergence and brings (210) to the following form 1 m − hF m − 8π −ḡφα Y α , F m ≡ hρσ Fρσ 2 (211) where the total divergence has been dropped off, φα ≡ φ|α , and Y α is given in (181). After reducing similar terms Eq. (211) takes on the following form Equations of Motion in an Expanding Universe 735  ρ̄m α (212) φ φα − 3ρ̄m l αβ ū α φβ μ̄m   1 − 4π −ḡ ( p̄m − ǭm ) l αβ lαβ − l 2 2    2  α 2 3 c ρ̄m 1 2 2 α − 8π −ḡ 1− 2 ū φα − μ̄m qū φα + μm q . μ̄m cs 2 2 F m = −8π −ḡ  where again we have used notation φα ≡ φ|α . Taking variational derivative in (209) is rather straightforward but tedious procedure. Because the Lagrangian F m depends neither on the Christoffel symbols nor on the curvature tensor, the variational derivative (209) is reduced to a partial derivative with respect to the metric tensor δF m /δgμν = ∂F m /∂gμν . Calculation of the partial derivative is done with the help of the chain rule and equations in Appendix section “Variational Derivatives”. It yields the stress-energy tensor of dark matter ρ̄m φμ φν (213) 2μ̄m      c2 3 ρ̄m 3 1− 2 μ̄m q − 2ū α φα ū (μ φν) − μ̄m ū α φα lμν − 2μ̄m cs 2 4    2 c 3 ρ̄m α αβ α φ φα − 3μ̄m l ū α φβ + μ̄m l ū φα ū μ ū ν 1− 2 + 4μ̄m cs 2      2 2 2 c c ρ̄m c ∂ ln cs2 3 − 2 − ρ̄m μ̄m 2 1− 2 + 4μ̄m cs cs cs ∂ p̄m    α 2 3 1 α 2 2 × ū φα − μ̄m qū φα + μm q ū μ ū ν 2 2    2 2 c ρ̄m ḡμν φα φα + 1 − 2 ū α φα − 4μ̄m cs      c2 1 1 2 2 αβ 2qlμν − q ḡμν + l lαβ − l + 2ql ū μ ū ν − ρ̄m μ̄m 1 − 2 8 c 2 s    1 2 1 1 αβ 1 α l lαβ − l ḡμν , − ( p̄m − ǭm ) lαμlν − llμν − 2 2 4 2 m = τμν where we have used (117) to make a replacement ǭm + p̄m = ρ̄m μ̄m . 5.3 Stress-Energy Tensor of Dark Energy Perturbations The part of the stress-energy tensor describing the dark energy perturbation is given q in (192) by τμν that, according to (96), is calculated as a variational derivative 736 S.M. Kopeikin and A.N. Petrov δF q 1 q 16πταβ = √ ḡαμ ḡβν , δ ḡμν −ḡ where 1 q − hF q + F q ≡ hρσ Fρσ 2 (214) q −ḡψ F (215) Individual terms entering the right side of (215) are taken from (170) and (189). We single out the total divergence and bring (215) to the following form q F ≡h ρσ q Fρσ   1 q ∂ 2 W̄ ∂ W̄ α , (216) − hF − 8π −ḡψα Z − 4π −ḡψ l + 2ψ ¯ ¯2 2 ∂ ∂ where the total divergence has been dropped off. More explicitly,   2 3 ∂ W̄ 2 ∂ W̄ αβ α F = −8π −ḡ ψ ψα + lψ − 3μ̄q l ū α ψβ +ψ ¯ ¯2 2 ∂ ∂   1 ¯ l αβ lαβ − l 2 , + 8π −ḡ W̄ () 2 q (217) where ψα ≡ ψ,α . Taking variational derivative with respect to ḡμν , we obtain the stress-energy tensor of dark energy perturbation q τμν   2 1 3 1 ∂ W̄ α 2 ∂ W̄ ψ ψα + ψ ḡμν − lμν ψ = ψμ ψν − 2 ¯ ¯ 2 4 4 ∂ ∂     1 2 1 αβ 1 α ¯ l lαβ − l ḡμν . + W̄ () lαμlν − llμν − 2 4 2 (218) 6 Post-Newtonian Equations of Motion in Cosmology 6.1 General Formulation  ¯ A, Let us consider a general manifold with the effective Lagrangian Leff =Leff ḡ μν ,   hμν , φ A , θ B depending on five independent variables and their derivatives which we intentionally omitted from the argument of the Lagrangian to avoid a cumbersome notation. We have proved in Sect. 3.7 that the effective Lagrangian Leff is gauge-invariant modulo a total divergence. The gauge invariance of Leff suggests that its Lie derivative must be also nil modulo a total divergence of a vector field: £ξ Leff = ∂α Aα . Because the total divergences do not affect field equations we drop all of them out of the subsequent equations. We compute the Lie derivative of the effective Lagrangian by making use of (29) that reduce calculation of the Lie derivative to that of variational derivatives modulo Equations of Motion in an Expanding Universe 737 a total divergence. After dropping off the divergence, we have £ξ Leff = δLeff δLeff δLeff αβ ¯A+ £ ḡ +  £ξ hμν £ ξ ξ ¯A δhμν δ ḡ αβ δ δLeff δLeff + A £ξ φ A + £ξ θ B . δφ δθ B (219) Field equations (72), (99), (104) describing evolution of perturbations hμν , φ A , θ B on the background manifold exterminate the last three terms in the right side of (219). The first term in the right side of (219) can be written down as follows δLeff £ξ ḡ αβ = − −ḡαβ ξ α|β , δ ḡ αβ (220) where we have used definitions (75), (85)–(87), and equation £ξ ḡ αβ = −ξ α|β −ξ β|α . ¯ A depends on its geometric properties. In a The Lie derivative of the field   A μ1 ...μ p A ¯ ≡  ¯ , the Lie derivative is given by particular case of tensor density  ν1 ...νq (26) that can be written symbolically as follows A A α|β ¯ A = ξα ¯ |α + K̄ αβ ξ , £ξ  (221)  A μ1 ...μ p A = K̄ Aσ ḡ , and ¯A ≡  ¯ , K̄ αβ where  α σβ |α ν ...ν |α 1 q ¯A K̄ αAσ ≡ mδασ  ¯A − δαμ1  + δνσ1 ¯  A μ1 ...μ p ν1 ...νq σμ2 ...μ p ν1 ...νq μ1 ...μ p αν2 ...νq (222) μ ¯A − · · · − δα p  + · · · + δνσq ¯  A μ1 ...μ p−1 σ ν1 ...νq μ1 ...μ p ν1 ...νq−1 α . Making use of definition (102) and (222) we can present the second term in the right side of (219) in the following form 1 δLeff α¯ A A α|β ¯A= −ḡ M £  A ξ |α + K̄ αβ ξ ¯A ξ 2 δ (223) Substituting (220), (223) to the right side of (219) results in £ξ Leff = 1 ¯A α −ḡ M A |α ξ + 2   1 A ξ α|β . −ḡ −αβ +  M K̄ 2 A αβ (224) Applying the Leibniz rule to change the order of differentiation in the terms depending on ξ α|β , we can recast (219) to the following form 738 £ξ Leff = S.M. Kopeikin and A.N. Petrov −ḡ  1 M A 1 A ¯ + αβ |β −   M A K̄ αβ 2 A |α 2 |β  ξα + −ḡWβ |α , (225) where the vector field   1 A ξα Wβ ≡ −αβ +  M K̄ 2 A αβ (226) The last term in (225) is reduced to the total divergence of a vector density −ḡWβ |β = −ḡW β ,β (227) , where W b = ḡ αβ Wα . Therefore, the variational derivative of the effective Lagrangian can be equal to zero modulo the divergence of the vector field if, and only if, the combination of terms enclosed to the square brackets in (225) is nil. It yields the equations of motion of the matter of bare perturbation 1 1 A ¯A αβ |β = −  M M A |α + A K̄ αβ 2 2 |β (228) . It should be compared with the law of conservation of matter in flat background spacetime where it has a similar form αβ ,β = 0 of the total divergence [79]. The ¯ A makes the right side of (228) different presence of the background matter fields  from zero. This result was established in [73]. Equation (228) can be also interpreted as the integrability condition of the gravitational field equation (74). Taking a covariant derivative from both sides of the field equation (74) and applying the equations of motion (228) yields G M + Fαβ Fαβ |β  M A ¯A = −4π  M A |α −  A K̄ αβ |β  . (229) In the linear approximation, when all quadratic and higher-order terms with respect to G M |β the perturbations are discarded ( M A → 0), the covariant divergence (Fαβ + Fαβ ) = 0. It agrees with the assumption that the stress-energy tensor of the bare perturbation is conserved in the linearized perturbative order, Tαβ |β = 0. Now we are set to start calculating equations of motion of matter of the bare perturbation in FLRW universe. 6.2 Equations of Motion in the Universe Governed by Dark Matter and Dark Energy The dark matter and dark energy components of matter that governs the temporal evolution of the universe are modelled by scalar fields  and . Hence, the tensor, A ≡ 0, and, consequently, the second term in the right side of (228) is identically K̄ αβ Equations of Motion in an Expanding Universe 739 nil. Therefore, equations of motion of matter (228) can be written more explicitly in the following form m |ν q |ν + τμν = Tμν |ν + tμν |ν + τμν  1 μ̄m  m + μ̄q  q ū μ , 2 (230) ¯ and  ¯ in where we have used definitions (163) of the gradients of the scalar fields  α terms of the background four-velocity ū . Next step is to calculate the explicit form of  m and  q as well as the covariant divergences of stress-energy tensors entering (230). We split the process of calculation in three parts—for gravitational field, for dark matter, and dark energy. 6.2.1 Gravitational Field Covariant divergence from the stress-energy tensor of gravitational field, tμν , is derived from (207). We again need to employ the commutation relations (205), (206) along with a rule for the third order derivative l λ μ|ρσν = l λ μ|ρνσ − l γ μ|ρ R̄ λ γσν + l λ γ|ρ R̄ γ μσν + l λ μ|γ R̄ γ ρσν , (231) which allows us (after one more commutation of the covariant derivative in l λ μ|ρνσ ) to derive l ν μ|ρσν = Aμ|ρσ + l α μ|σ R̄αρ + l α μ|ρ R̄ασ + l α β|σ R̄ β μρα +l α β β|ρ R̄ μσα +l α β μ|β R̄ ρσα +l α μ R̄αρ|σ +l α (232) β β R̄ μρα|σ . Significant number of terms cancel out, and after long calculation we obtain a rather simple result   |ν  1 1 ρσ 1 l ρσ|μ − l|μ , − tμν |ν = l ρ ν ρμ − lμν  2 2 2 (233) where αβ was defined in (208). After taking the covariant divergence, it is convenient to algebraically split (233) in three parts    |ν 1 1 1 ρσ l Tρσ|μ − lT|μ − tμν |ν = l ρ ν Tρμ − lμν T 2 2 2   |ν  1 1 1 m 1 ρ m ρσ m m − l ν Fρμ − lμν F l Fρσ|μ − l F|μ + 8π 2 16π 2   |ν  1 1 q 1 1 q q l ρ ν Fρμ l ρσ Fρσ|μ − l F|μ , − lμν F q + − 8π 2 16π 2 (234) 740 S.M. Kopeikin and A.N. Petrov where the first line describes the coupling of gravity with the stress-energy tensor Tμν of the bare perturbation, and the second and the third lines outline the contribution of dark matter (index ‘m’) and dark energy (index ‘q’). 6.2.2 Dark Matter The source density of dark matter,  m , depends only on the derivatives of the scalar ¯ and can be written in the form of a covariant divergence field ,  m = Jνm |ν , (235) with the matter current ∂F m . ¯ν 16π −ḡ ∂  1 √ (236) Jαm = ρ̄m ū α j + ρ̄m P̄α β jβ , (237) Jνm = The current is split in two components where j=     c2  α 2 c2 α ū φ + 1 − φ φ α α cs2 cs2     c2  α  1 3 c2 αβ l ū α φβ + 1 − 2 ū φα q + 2μ̄m cs2 2 cs      2 2 c c l2 1 2 αβ 1− 2 1 + 2 q + l lαβ − − 4 cs cs 2    α  1 2 2 1 c2 ∂ ln cs2  α 2 3 , μ̄ μ̄ ū q ū φ + q φ − − m α α 2μ̄m cs2 ∂ μ̄m 2 2 m 1 2μ̄2m 1 jβ = 2 μ̄m   1− c2 1− 2 cs   α  ū φα φβ (238) (239)     α  γ 3 c2 3 1 α 1− 2 ū φα lβ ū γ + qφβ − l β φα − 2μ̄m 2μ̄m cs 2   c2 + 1 − 2 qlβ α ū α . cs Taking covariant divergence from Jαm entangles a lot of algebraic operations which number can be significantly reduced by making use of the following procedure. We notice that the first term in the right side of (230) can be transformed to Equations of Motion in an Expanding Universe 741  |ν c2 μ̄m  m ū μ = μ̄m Jνm ū μ + 3 s2 H ρ̄m μ̄m j ū μ − H ρ̄m μ̄m jν P̄μ ν . c (240) Furthermore, ∂F m ∂F m δ μ̄m + ∂ ḡρσ ∂ μ̄m δ ḡρσ m m ∂F δ μ̄m ∂F + , = ¯ν ¯ν ∂ μ̄m δ  ∂ m 16π −ḡτμν = ḡμρ ḡνσ 16π −ḡ Jνm   , (241) (242) Accounting for the variational derivatives (299), (322) we obtain   ∂F m 1 1 ∂F m 1 m τμν ū − − μ̄m Jνm ū μ = . ḡμρ ḡνσ √ μ ¯ν 2 ∂ ḡρσ 2 ∂ 16π −ḡ (243) This equation elucidates that we do not need to directly calculate the partial derivatives with respect to the specific enthalpy μ̄m when calculating the equations of motion (230). It saves us from doing a lot of algebra. We combine (243) with the divergence from the second line of (234) and denote X μν ≡ m τμν   1 1 1 m ρ m m . l ν Fρμ − lμν F − μ̄m Jν ū μ − 2 8π 2 (244) Notice that X μν is not symmetric with respect to its indices. Calculation reveals X μν   1 α (245) φμ φν − φ φα ḡμν 2      α  ρ̄m c2 1  α 2 1− 2 ū φα ḡμν ū φα φμ ū ν − 2μ̄m cs 2   1 ρ̄m φμlν ρ ū ρ + ū μlν ρ φρ 4       1 c2 3 1 ρ̄m 1 − 2 qφμ ū ν + ū α φα lμν + ū μlν ρ ū ρ cs 8 8 4      2 c 1 2 1 ρσ 2 ρ̄m μ̄m 1 − 2 q + ( p̄m − ǭm ) l lργ − l ḡμν . 8 cs 2 ρ̄m = 2μ̄m + − − + Let us denote the density of the force caused by dark matter on the motion of the matter of the bare perturbation, by f μm . After grouping all terms together the force density is defined by the following expression 742 S.M. Kopeikin and A.N. Petrov f μm ≡ −X μν |ν − +   1 1 m l ρσ Fρσ|μ − l F m |μ 16π 2 (246) 3 cs2 1 H ρ̄m μ̄m j ū μ − H ρ̄m μ̄m jν P̄μ ν , 2 c2 2 which can be split in two orthogonal components f μm = a m ū μ + aνm P̄ ν μ , (247) where a m ≡ −ū ν f νm and aμm ≡ P̄μ ν f νm . We have am = + × + + + aμm = + − + +    1  αβ ρ̄m l φαβ + Aα φα − 2 ū α φα ū β Aβ (248) 4   c2 1 ρ̄m 1 − 2 8 cs       α β γ ū ū lβ φαγ + qū α ū β φαβ − ū α φα ū β qβ + ū α φα ū β Aβ    α  1 2ρ̄m H ū φα 2q − l 2    2   1 c ρ̄m H 1 − 2 l αβ ū α φβ + ū α φα (3q − l) 8 cs   l  α  3 ∂ ln cs2 q− ū φα , ρ̄m μ̄m H 8 ∂ μ̄m 2 1  α  ρ̄m ū Aα φμ (249) 2        c2  1 ρ̄m 1 − 2 lμ α ū β φαβ − qū α φμα + ū α qα φμ + ū α φα Aμ 8 cs   1 2ρ̄m H 2q − l φμ 2     α  1 c2 1 1 ū φα lμβ ū β − qφμ + lμ α φα ρ̄m H 1 − 2 2 cs 4 4 2   ∂ ln cs 3 ū α φα lμ α ū α − qφμ . ρ̄m μ̄m H 8 ∂ μ̄m 6.2.3 Dark Energy ¯ The dark energy source,  q , depends not only on the derivatives of the scalar field  but on the field itself through the field potential W = W (). Therefore, according to definition (102) Equations of Motion in an Expanding Universe  ∂F q −  = + ¯ 16π −ḡ ∂ q 1 √ 743  ∂F q ¯ν ∂   . (250) |ν After taking the variational derivatives we obtain   3 ∂ 2 W̄ 1 ∂ W̄ αβ 1 2 1 2 ∂ 3 W̄ + lψ − l lαβ − l  = ψ ¯3 ¯2 ¯ 2 ∂ 4 ∂ 2 ∂ 2 3 αβ γ αβ γ αβ l |γ ψa ū β ū + l ψαγ ū β ū + 3Hl ū α ψβ . + 2 q (251) The force caused by dark energy on the motion of the matter of the bare perturbation is  |ν 1 1 q q |ν l ρ ν Fρμ − lμν F q f μq ≡ −τμν + 8π 2   1 1 1 q l ρσ Fρσ|μ − l F q |μ + μ̄q  q ū μ , − 16π 2 2 (252) After long but straightforward calculation we get (253) f μq = ρ̄q ū μ   3 3 9 × l αβ ψαβ + l αβ |γ ψα ū β ū γ + l αβ ψαγ ū β ū γ + Hl αβ ū α ψβ 4 4 4    α  1 α  1 + A ψα ū μ + A ū α ψμ + ρ̄q H 2q − l ψμ 2 2  2 W̄   1 1 ∂ 1 ∂ W̄  ν l l ū ū − ψ Aμ + lμ ν ψν − 2qψμ + ρ̄q ψ − μ . ¯ ¯ 2 μν 4 ∂ 4 2 ∂ 6.3 Final Form of the Equations of Motion After making use of the results of the presiding section, equations of motion (230) take on the following form Tμ ν |ν +l ρν  Tρμ|ν    1 1 l ν ν − Tρν|μ − lμ − δμ T|ν (254) 2 2 2 1 + Aρ Tρμ − Aμ T = f μm + f μq . 2 The left side of this equation can be brought to a more conventional form if we use relation (93) between the stress-energy tensor of the bare perturbation Tμν and Tμν . Let us take a covariant divergence of Tμν with respect to the full metric gμν that is ∇ν Tμ ν ≡ g νρ ∇ν Tρμ where ∇ν denotes a covariant derivative with the connection 744 S.M. Kopeikin and A.N. Petrov referred to the full metric so that ∇α gβγ ≡ 0. The covariant derivatives from the stress-energy tensor Tμν are calculated with the help of ∇α Tμν = Tμν|α − Gβαμ Tνβ − Gβαν Tμβ , β β (255) β where Gαμ ≡ Ŵαμ − Ŵ̄αμ is a tensor describing the difference between the perturbed and unperturbed values of the Christoffel symbols which we shall call the Christoffel tensor. In terms of the metric tensor perturbations Gβαμ =  1 βγ  g κγα|μ + κγμ|α − καμ|γ . 2 (256) Equation (256) is exact. In the linear approximation with respect to the lμν it can be expressed in terms of the background metric and perturbations lμν as follows   1 β 1 δ l|μ + δμβ l|α − ḡαμl |β , Gβαμ = − ḡ βγ lγα|μ + lγμ|α − lαμ|γ + 2 4 α (257) where all quadratic terms with respect to lμν have been neglected. Two contracted values of the Christoffel tensor are β Gα ≡ Gαβ = 1 l|α , 2 γ ḡ αβ Gαβ = −l γβ |β = −Aγ , (258) Making use of these notations and definitions and doing direct calculation allows us to show that   1 (259) ∇ν Tμ ν = Tμ ν |ν + l ρν Tρμ|ν − Tρν|μ 2   1 l 1 lμ ν − δμ ν T|ν + Aρ Tρμ − Aμ T. − 2 2 2 Hydrodynamic equation of motion (254) becomes ∇ν Tμ ν = f μm + f μq . (260) Were the background spacetime flat, the right side of (260) would vanish. However, in cosmology the background spacetime is given by FLRW metric, thus, making the divergence of the stress-energy-momentum tensor of the bare perturbation different from zero. The reader should not be confused here. Equation (260) is the law of conservation of the total stress-energy-momentum tensor given in the form, in which the covariant divergence of the stress-energy-momentum tensor of the bare perturbation is singled out and put to the left side. The right side represents the covariant divergence of the stress-energy-momentum tensor of the background matter in the q disguised form of the sum of two vectors, f μm and f μ . Equations of Motion in an Expanding Universe 745 Acknowledgments Sergei Kopeikin thanks the Center of Applied Space Technology and Microgravity (ZARM) of the University of Bremen for providing partial financial support for travel and Physikzentrum at Bad Honnef (Germany) for hospitality and accommodation. The work of Sergei Kopeikin has been supported by the grant 14-27-00068 of the Russian Scientific foundation. Appendix 1: Hilbert and Einstein Lagrangians We define the Christoffel symbols of the second kind as usual [46] Ŵ α βγ ≡  1 αδ  g gδβ,γ + gδγ,β − gβγ,δ . 2 (261) The Christoffel symbols of the first type Ŵαβγ = gασ Ŵ σ βγ =  1 gαβ,γ + gαγ,β − gβγ,α , 2 (262) We notice the symmetry with respect to the last two indices Ŵαβγ = Ŵα(βγ) . There is no any symmetry with respect to the first two indices. In general, Ŵαβγ = Ŵ(αβ)γ + Ŵ[αβ]γ , where Ŵ(αβ)γ = 1 gαβ,γ , 2 Ŵ[αβ]γ =  1 gγα,β − gγβ,α , 2 (263) (264) There are two, particularly useful symbols that are obtained by contracting indices of the Christoffel symbols of the first kind. They are denoted as Yα ≡ Ŵ β αβ , Y α = g αβ Yβ , (265) and Ŵ α ≡ g βγ Ŵ α βγ , Ŵα = gαβ Ŵ β , (266) Direct inspection shows that Yα = = The two symbols are interrelated  √  1 βγ g gβγ,α = ln −g ,α . 2 Ŵα = −Ya + g βγ gαβ,γ , α a Ŵ = −Y − g αβ ,β , (267) (268) (269) 746 S.M. Kopeikin and A.N. Petrov We define the Riemann tensor as follows [46] R α μβν = Ŵ α μν,β − Ŵ α μβ,ν + Ŵ α βγ Ŵ γ μν − Ŵ α νγ Ŵ γ μβ . (270) Second covariant derivatives of tensors do not commute due to the curvature of spacetime. For example, denoting the covariant derivatives with a vertical bar we have for a covector field Fα and a covariant tensor of second rank, Fαβ the following commutation relations Fα|βγ = Fα|γβ + R μ αβγ Fμ , Fαβ|γδ = Fαβ|δγ + R μ αγδ Fμβ (271) +R μ βγδ Fαμ . (272) Riemann tensor can be also expressed in terms of the second partial derivatives of the metric tensor and the Christoffel symbols  1 gμβ,αν + gνα,βμ − gαβ,μν − gμν,αβ + Ŵρμβ Ŵ ρ αν − Ŵρμν Ŵ ρ αβ . 2 (273) Contraction of two indices in the Riemann tensor yields the Ricci tensor Rαμβν = Rμν = Ŵ α μν,α − Yμ,ν + Yγ Ŵ γ μν − Ŵ α νγ Ŵ γ μα , (274) or, in terms of the second derivatives from the metric tensor and the Christoffel symbols, Rμν =  1 κǫ  g gμκ,ǫν + gνκ,ǫμ − gκǫ,μν − gμν,κǫ + g κǫ Ŵρμǫ Ŵ ρ κν − Ŵρμν Ŵ ρ . (275) 2 One more contraction of indices in the Ricci tensor brings about the Ricci scalar which we shall write down in the form suggested by Fock [14, Appendix B] R = L + Yα Ŵ α − Yα Y α + Ŵ α ,α − Y α ,α , (276)   L = g μν Ŵ α νγ Ŵ γ μα − Yα Ŵ α μν , (277) R = L + (−g)−1/2 Aα ,α , (278) where is (up to a constant factor) the gravitational Lagrangian introduced by Einstein [79] as an alternative to the gravitational Lagrangian, R, of Hilbert. The Hilbert Lagrangian is the Ricci scalar which depends on the second derivatives of the metric tensor while the Einstein Lagrangian does not. The two Lagrangians are interrelated Equations of Motion in an Expanding Universe where Aα =  √  α −g Ŵ − Y α , 747 (279) is a vector density of weight +1. After performing differentiation in (278), and accounting for (265) we can easily prove that (278) reproduces (276). One more form of relation between R and L will be useful for calculating the variational derivative in Appendix section “Variational Derivative from the Hilbert Lagrangian”. To this end we introduce a new notation Ŵ ≡ Ŵ α ,α + Yα Ŵ α , (280) g αβ Yα,β = Y α ,α + Yα Ŵ α + Yα Y α , (281) and notice that Equations (280), (281) allows us to cast (276) to the following form R = L + Ŵ + Yα Ŵ α − g αβ Yα,β , (282) that was found by Fock [14, Appendix B]. Appendix 2: Variational Derivatives Variational Derivative from the Hilbert Lagrangian The goal of this section is to prove relation (45) being valid on the background manifold M̄. We shall omit the bar over the background geometric objects as it does not bring about confusion. We notice that the Hilbert Lagrangian density, LG = √ √ −(16π)−1 −g R, differs from LE = −(16π)−1 −gL by a total derivative that is a consequence of (278). Due to relation (9) the Lagrangian derivatives from LG and LE coincides δLG δLE = , (283) δgμν δgμν thus, pointing out that we can safely operate with the Einstein Lagrangian density LE . Because of (35), we have δLE 1 δLE Aρσ =√ , μν μν δg −g δg ρσ (284) which suggests that calculation of the variational derivative with respect to the metric tensor is sufficient. Calculation of the variational derivative δLE /δg ρσ demands the partial derivatives of the contravariant metric and Christoffel symbols with respect to g μν . The partial 748 S.M. Kopeikin and A.N. Petrov derivatives of the metric are calculated with the help of (37), (30). The Christoffel symbols are given in terms of the partial derivatives from covariant metric tensor, gαβ,γ which are not conjugated with the dynamic variable g αβ . Thus, calculation of the partial derivative with respect to g μν from the Christoffel symbols demands its transformation to the form where the conjugated variables g αβ ,γ are used instead. This form of the Christoffel symbols is Ŵ α βγ =  1  ρκ ασ g ,σ g gρβ gκγ − g ασ ,β gγσ − g ασ ,γ gβσ . 2 (285) Taking the partial derivative of (285) with respect to the contravariant metric yields   ∂Ŵ α βγ = −g ασ Ŵ[σβ](μ gν)γ + Ŵ[σγ](μ gν)β + Ŵ(βγ)(μ gν)σ , μν ∂g (286) ∂Yα = −Ŵ(μν)α , ∂g μν (287) and where we have used (263). Contracting (286), (287) with the Christoffel symbols and the metric tensor results in ∂Ŵ α βγ β Ŵ σα = −2Ŵ α βμ Ŵ β να , ∂g μν ∂Yβ g σγ μν Ŵ β σγ = −Ŵ(μν)α Ŵ α , ∂g ∂Ŵ β σγ Yβ = Ŵ(μν)α Y α − Ŵαμν Y α − Yμ Yν . g σγ ∂g μν g σγ (288) (289) (290) Partial derivatives of the Christoffel symbols with respect to the metric derivatives are calculated from (285) with the help of (38). We get  ∂Ŵ α βγ 1  ρα ρ α ρ α g , = g g − δ δ g − δ δ g ν)γ ν)γ β(μ ν)β γ (μ (μ β ∂g μν ,ρ 2 ∂Yβ 1 ρ = − gμν δβ . ∂g μν ,ρ 2 (291) (292) Contracting (291), (292) with the Christoffel symbols and the metric tensor results in Equations of Motion in an Expanding Universe ∂Ŵ α βγ β 1 Ŵ σα = − Ŵ ρ μν , μν ∂g ,ρ 2 ∂Y 1 β g σγ μν Ŵ β σγ = − gμν Ŵ ρ , ∂g ,ρ 2 g σγ g σγ ∂Ŵ β σγ 1 Yβ = gμν Y ρ − δ ρ (μ Yν) . μν ∂g ,ρ 2 749 (293) (294) (295) Explicit expression for the variational derivative of the Einstein Lagrangian is   √ ∂ −g σγ √ ∂g σγ Ŵ α βγ Ŵ β σα − Yβ Ŵ β σγ (296) g + −g μν ∂g μν ∂g   √ ∂Yβ ∂Ŵ α βγ β ∂Ŵ β σγ + −gg σγ 2 Ŵ σα − μν Ŵ β σγ − Yβ μν ∂g ∂g ∂g μν    ∂Ŵ α βγ β ∂Yβ β ∂Ŵ β σγ ∂ √ σγ − −gg 2 μν Ŵ σα − μν Ŵ σγ − μν Yβ ∂x ρ ∂g ,ρ ∂g ,ρ ∂g ,ρ δLE − 16π μν = δg Replacing the partial derivatives in (296) with the corresponding right sides of Eqs. (37), (30), (293)–(295) and taking the partial derivative with respect to spatial coordinates, yields − 16π   √ δLE 1 −g R − = R , g μν μν δg μν 2 (297) where we have used expressions (274), (276) for the Ricci tensor and Ricci scalar respectively. Substituting Eq. (297) to (284) yields δLE 1 Rμν . =− μν δg 16π (298) Variational Derivatives of Dynamic Variables with Respect to the Metric Tensor Variational Derivatives of Dark Matter Variables The primary thermodynamic variable of dark matter is μm defined in (125). Variational derivative from μm is calculated directly from its definition and yields 1 δ μ̄m = μ̄m ū μ ū ν . δ ḡμν 2 (299) Variational derivative of pressure p̄m is obtained from thermodynamic relation (121a) by making use of the chain differentiation rule along with (299), that is 750 S.M. Kopeikin and A.N. Petrov δ p̄m 1 = ρ̄m μ̄m ū μ ū ν . δ ḡμν 2 (300) Variational derivative of the rest mass and energy density are obtained by making use of (299) along with equation of state that allows us to express partial derivatives of ρm and ǫm in terms of the variational derivative for μm . More specifically, δ ρ̄m 1 c2 = ρ̄m ū μ ū ν , δ ḡμν 2 cs2 (301) 1 c2 δ ǭm = ρ̄m μ̄m ū μ ū ν δ ḡμν 2 cs2 (302) where the speed of sound appears explicitly. Variational derivatives from products and/or ratios of the thermodynamic quantities are calculated my applying the chain rule of differentiation and the above equations,  δ (ρ̄m μ̄m ) 1 1+ = δ ḡμν 2    δ 1 ρ̄m 1− =− δ ḡμν μ̄m 2  δ ( p̄m − ǭm ) 1 1− = δ ḡμν 2 c2 cs2 c2 cs2 c2 cs2    ρ̄m μ̄m ū μ ū ν , (303) ρ̄m μ ν ū ū , μ̄m (304) ρ̄m μ̄m ū μ ū ν . (305) Variational Derivatives of Dark Energy Variables The primary thermodynamic variable of dark energy is μ̄q defined in (125). Variational derivative from μ̄q is calculated directly from its definition, δ μ̄q 1 = μ̄q ū μ ū ν . δ ḡμν 2 (306) Variational derivative of the mass density ρ̄q of the dark energy “fluid” follows directly from ρ̄q = μ̄q , and reads δ ρ̄q 1 = ρ̄q ū μ ū ν . δ ḡμν 2 (307) Variational derivative of pressure p̄q is obtained from definition (138) along with (306), which yields δ p̄q 1 = ρ̄q μ̄q ū μ ū ν . (308) δ ḡμν 2 Equations of Motion in an Expanding Universe 751 Variational derivative of energy density ǭq is obtained by making use of (306) along with (137). More specifically, δ ǭq 1 = ρ̄q μ̄q ū μ ū ν . δ ḡμν 2 (309) Variational derivatives from products and ratios of other quantities are calculated my making use of the chain rule of differentiation and the above equations   δ ρ̄q μ̄q = ρ̄q μ̄q ū μ ū ν , δ ḡμν   ρ̄q δ = 0, δ ḡμν μ̄q   δ p̄q − ǭq = 0. δ ḡμν (310) (311) (312) Variational Derivatives of Four-Velocity of the Hubble Flow Variational derivatives from four-velocity of the fluid are derived from the definition ¯ or  ¯ which are indepen(163) of the four-velocity given in terms of the potential  dent dynamic variables that do not depend on the metric tensor. Taking variational derivative from (163) and making use either (299) or (306) we obtain 1 δ ū α = − ū α ū μ ū ν , δ ḡμν 2 δ ū α 1 = − ū α ū μ ū ν − ḡ α(μ ū ν) , δ ḡμν 2   1 δ (ū α φα ) = −φ(μ ū ν) − ū μ ū ν ū α φα , δ ḡμν 2 (313) (314) (315) where Eq. (315) accounts for the fact that φα is an independent variable that does not depend on the metric tensor. Variational Derivatives of the Metric Tensor Perturbations αβ Variational derivatives from the metric √ tensorαβperturbations l are determined by αβ αβ taking into account that l = h / ḡ and h is an independent dynamic variable which does not depend on the metric tensor. Therefore, its variational derivative is nil, and we have 752 S.M. Kopeikin and A.N. Petrov δl αβ δ = δ ḡμν δ ḡμν  hαβ √ ḡ  = hαβ δ δ ḡμν  1 √ ḡ  1 = − l αβ ḡ μν . 2 (316) Other variational derivatives are derived by making use of tensor operations of rising and lowering indices with the help of ḡαβ and applying from (316). It gives δ δ ḡμν  l αβ lαβ δlαβ δ ḡμν δl δ ḡμν δq δ ḡμν  l2 − 2 1 ν) = − lαβ ḡ μν + 2lα (μ δβ , 2 1 = l μν − l ḡ μν , 2    1  μν 1 μν μ ν + l + l ū μ ū ν , = −q ū ū + ḡ 2 2   l2 = 2l α(μl ν) α − ll μν − ḡ μν l αβ lαβ − . 2 (317) (318) (319) (320) Variational Derivatives with Respect to Matter Variables Variational Derivatives of Dark Matter Variables ¯ directly but The dark matter variables do not depend on the Clebsch potential  ¯ α . Therefore, any variational derivative of dark matter merely on its first derivative  ¯ α ), is reduced to a total divergence variable, say, Q = Q( ∂ ∂Q δQ =− α . ¯ ¯α ∂x δ ∂ (321) ¯ α. We present a short summary of the partial derivatives with respect to  ∂ μ̄m ¯α ∂ ∂ p̄m ¯α ∂ ∂ ρ̄m ¯α ∂ ∂ ǭm ¯α ∂ = ū α , (322) = ρ̄m ū α , (323) = c2 ρ̄m α ū , cs2 μ̄m c2 ρ̄m ū α , cs2   ∂ (ρ̄m μ̄m ) c2 = 1 + 2 ρ̄m ū α , ¯α cs ∂ = (324) (325) (326) Equations of Motion in an Expanding Universe 753    ∂ ρ̄m = − 1− ¯ α μ̄m ∂  ∂ ( p̄m − ǭm ) = + 1− ¯α ∂ c2 cs2  2 c cs2 ρ̄m α ū , μ̄2m (327) ρ̄m ū α . (328) Partial derivatives of four velocity ∂ ū α P̄α β =− , ¯β μ̄m ∂ P̄ αβ ∂ ū α =− . ¯β μ̄m ∂ (329) It allows us to deduce, for example, ∂ (ū α φα ) 1 αβ P̄ φβ , =− ¯ μ̄m ∂ β ∂q 2 α μν =− P̄ μl ū ν . ¯ μ̄m ∂ α (330) (331) Variational Derivatives of Dark Energy Variables ¯ and its first derivative The dark energy variables depend on both the scalar potential  ¯ α in the most generic situation. This is because there is a potential of the scalar  ¯ that is absent in case of the dark matter. Therefore, variational derivative field W () ¯  ¯ α ), is of the dark energy variable, say, ⊣ = A(, δA ∂A ∂ ∂A = − α . ¯ ¯ ¯α ∂x ∂  δ ∂ (332) ¯ = (∂A/∂W )(∂W/∂ , ¯ and their particular form depends Partial derivatives ∂A/∂  on the shape of the potential W . As for the partial derivatives with respect to the derivatives of the field, they can be calculated explicitly for each variable, and we present a short summary of these partial derivatives below. More specifically, ∂ μ̄q ¯α ∂ ∂ p̄q ¯α ∂ ∂ ρ̄q ¯α ∂ ∂ ǭq ¯α ∂   ∂ ρ̄q μ̄q ¯α ∂ = ū α , (333) = ρ̄q ū α , (334) = ū α , (335) = ρ̄q ū α , (336) = 2ρ̄q ū α , (337) 754 S.M. Kopeikin and A.N. Petrov   ρ̄q ∂ = 0, ¯ α μ̄q ∂   ∂ p̄q − ǭq = 0. ¯α ∂ (338) (339) Partial derivatives of four velocity ∂ ū α P̄α β , =− ¯β μ̄q ∂ ∂ ū α P̄ αβ . =− ¯β μ̄q ∂ (340) It allows us to deduce, for example, ∂ (ū α ψα ) 1 = − P̄ αβ ψβ , ¯ μ̄ ∂ β q ∂q 2 α μν = − P̄ μl ū ν . ¯α μ̄q ∂ (341) (342) References 1. O.V. Verkhodanov, A.G. Doroshkevich, Advances in Machine Learning and Data Mining for Astronomy, chapter Cosmic Microwave Background Mapping (CRC Press, Taylor & Francis Group, Boca Raton, 2012), pp. 133–159 2. J. Peacock, Encyclopedia of Astronomy and Astrophysics, chapter Cosmology: Standard Model (2002) 3. V. Mukhanov, Physical Foundations of Cosmology (Cambridge University Press, Cambridge, 2005), p. 442 4. S. Weinberg, Cosmology (Oxford University Press, Oxford, 2008) 5. D.S. Gorbunov, V.A. Rubakov, Introduction to the Theory of the Early Universe (World Scientific Publishing Company, Singapore, Hackensack, 2011) 6. M. Tegmark, M.A. Strauss, M.R. Blanton, K. Abazajian, S. Dodelson, H. Sandvik, X. Wang, D.H. Weinberg, I. Zehavi, N.A. Bahcall, F. Hoyle, D. Schlegel, R. Scoccimarro, M.S. Vogeley, A. Berlind, T. Budavari, A. Connolly, D.J. Eisenstein, D. Finkbeiner, J.A. Frieman, J.E. Gunn, L. Hui, B. Jain, D. Johnston, S. Kent, H. Lin, R. Nakajima, R.C. Nichol, J.P. Ostriker, A. Pope, R. Scranton, U. Seljak, R.K. Sheth, A. Stebbins, A.S. Szalay, I. Szapudi, Y. Xu, J. Annis, J. Brinkmann, S. Burles, F.J. Castander, I. Csabai, J. Loveday, M. Doi, M. Fukugita, B. Gillespie, G. Hennessy, D.W. Hogg, Ž. Ivezić, G.R. Knapp, D.Q. Lamb, B.C. Lee, R.H. Lupton, T.A. McKay, P. Kunszt, J.A. Munn, L. O’Connell, J. Peoples, J.R. Pier, M. Richmond, C. Rockosi, D.P. Schneider, C. Stoughton, D.L. Tucker, D.E. vanden Berk, B. Yanny, D.G. York, Cosmological parameters from SDSS and WMAP. Phys. Rev. D 69(10), 103501 (2004) 7. Planck Collaboration, P.A.R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A.J. Planck Banday et al., Planck 2013 results. XVI. Cosmological parameters, ArXiv e-prints, March 2013 8. Planck Collaboration, P.A.R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A.J. Planck Banday et al., Planck 2013 results. XXIV. Constraints on primordial non-gaussianity, ArXiv e-prints, March 2013 9. J.-L. Lehners, P.J. Steinhardt, Planck 2013 results support the cyclic universe. Phys. Rev. D 87(12), 123533 (2013) Equations of Motion in an Expanding Universe 755 10. A.R. Liddle, D.H. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University Press, Cambridge, 2000) 11. A. Krasiński, C. Hellaby, Structure formation in the Lemaître-Tolman model. Phys. Rev. D 65(2), 023501 (2002) 12. A. Krasiński, C. Hellaby, Formation of a galaxy with a central black hole in the LemaîtreTolman model. Phys. Rev. D 69(4), 043502 (2004) 13. P. Jacewicz, A. Krasiński, Formation of Gyrs old black holes in the centers of galaxies within the Lemaître-Tolman model. Gen. Relativ. Gravit. 44, 81–105 (2012) 14. V.A. Fock, The Theory of Space, Time and Gravitation (Pergamon Press, Oxford, 1964) 15. A. Papapetrou, Equations of motion in general relativity. Proc. Phys. Soc. A 64, 57–75 (1951) 16. S. Chandrasekhar, Y. Nutku, The second post-Newtonian equations of hydrodynamics in general relativity. Astrophys. J. 158, 55 (1969) 17. S. Chandrasekhar, F.P. Esposito, The 2 21 -post-Newtonian equations of hydrodynamics and radiation reaction in general relativity. Astrophys. J. 160, 153 (1970) 18. J.L. Anderson, T.C. Decanio, Equations of hydrodynamics in general relativity in the slow motion approximation. Gen. Relativ. Gravit. 6, 197–237 (1975) 19. N. Spyrou, Relativistic celestial mechanics of binary stars. Gen. Relativ. Gravit. 13, 473–485 (1981) 20. S.M. Kopeikin, General relativistic equations of binary motion for extended bodies with conservative corrections and radiation damping. Sov. Astron. 29, 516–524 (1985) 21. R.A. Breuer, E. Rudolph, The force law for the dynamic two-body problem in the second post-Newtonian approximation of general relativity. Gen. Relativ. Gravit. 14, 181–211 (1982) 22. R.A. Breuer, E. Rudolph, Radiation reaction and energy loss in the post-Newtonian approximation of general relativity. Gen. Relativ. Gravit. 13, 777–793 (1981) 23. S.T. Shapiro, A.P. Lightman, Rapidly rotating, post-Newtonian neutron stars. Astrophys. J. 207, 263–278 (1976) 24. I. Ciufolini, R. Ruffini, Equilibrium configurations of neutron stars and the parametrized postNewtonian metric theories of gravitation. Astrophys. J. 275, 867–877 (1983) 25. S.L. Shapiro, S. Zane, Bar mode instability in relativistic rotating stars: A post-Newtonian treatment. Astrophys. J. Suppl. 117, 531 (1998) 26. M. Takada, T. Futamase, A post-Newtonian Lagrangian perturbation approach to large-scale structure formation. Mon. Not. R. Astron. Soc. 306, 64–88 (1999) 27. W. Petry, Cosmological post-Newtonian approximation in flat space-time theory of gravitation. Astrophys. Space Sci. 272(4), 353–368 (2000) 28. P. Szekeres, T. Rainsford, Post-newtonian cosmology. Gen. Relativ. Gravit. 32, 479–490 (2000) 29. J.-C. Hwang, H. Noh, D. Puetzfeld, Cosmological non-linear hydrodynamics with postNewtonian corrections. J. Cosmol. Astropart. Phys. 3, 10 (2008) 30. T.A. Oliynyk, Cosmological post-Newtonian expansions to arbitrary order. Commun. Math. Phys. 295, 431–463 (2010) 31. T.A. Oliynyk, The fast Newtonian limit for perfect fluids. Adv. Theor. Math. Phys. 16(2), 359–391 (2012) 32. S.M. Kopeikin, The speed of gravity in general relativity and theoretical interpretation of the Jovian deflection experiment. Class. Quantum Gravity 21, 3251–3286 (2004) 33. J.L. Anderson, R.E. Kates, L.S. Kegeles, R.G. Madonna, Divergent integrals of post-Newtonian gravity: nonanalytic terms in the near-zone expansion of a gravitationally radiating system found by matching. Phys. Rev. D 25, 2038–2048 (1982) 34. L. Blanchet, Post-Newtonian gravitational radiation, in Einstein’s Field Equations and Their Physical Implications. Lecture Notes in Physics, vol. 540, ed. by B.G. Schmidt (Springer, Berlin, 2000), p. 225 35. G. Schäfer, Post-Newtonian methods: analytic results on the binary problem. In: L. Blanchet, A. Spallicci and B. Whiting (Eds.), Mass and Motion in General Relativity. Fundam. Theor. Phys. 162, 167–210 (2011) 36. T. Buchert, Dark energy from structure: a status report. Gen. Relativ. Gravit. 40, 467–527 (2008) 756 S.M. Kopeikin and A.N. Petrov 37. E.W. Kolb, V. Marra, S. Matarrese, Description of our cosmological spacetime as a perturbed conformal Newtonian metric and implications for the backreaction proposal for the accelerating universe. Phys. Rev. D 78(10), 103002 (2008) 38. E.W. Kolb, V. Marra, S. Matarrese, Cosmological background solutions and cosmological backreactions. Gen. Relat. Gravit. 42, 1399–1412 (2010) 39. Stephen R. Green, Robert M. Wald, New framework for analyzing the effects of small scale inhomogeneities in cosmology. Phys. Rev. D 83, 084020 (2011) 40. S.R. Green, R.M. Wald, Newtonian and relativistic cosmologies. Phys. Rev. D 85(6), 063512 (2012) 41. A.G. Riess, A.V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich, R.L. Gilliland, C.J. Hogan, S. Jha, R.P. Kirshner, B. Leibundgut, M.M. Phillips, D. Reiss, B.P. Schmidt, R.A. Schommer, R.C. Smith, J. Spyromilio, C. Stubbs, N.B. Suntzeff, J. Tonry, Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998) 42. S. Perlmutter, G. Aldering, G. Goldhaber, R.A. Knop, P. Nugent, P.G. Castro, S. Deustua, S. Fabbro, A. Goobar, D.E. Groom, I.M. Hook, A.G. Kim, M.Y. Kim, J.C. Lee, N.J. Nunes, R. Pain, C.R. Pennypacker, R. Quimby, C. Lidman, R.S. Ellis, M. Irwin, R.G. McMahon, P. RuizLapuente, N. Walton, B. Schaefer, B.J. Boyle, A.V. Filippenko, T. Matheson, A.S. Fruchter, N. Panagia, H.J.M. Newberg, W.J. Couch and Supernova Cosmology Project, Measurements of omega and lambda from 42 high-redshift supernovae. Astrophys. J. 517, 565–586 (1999) 43. R. Zalaletdinov, The averaging problem in cosmology and macroscopic gravity. Int. J. Mod. Phys. A 23, 1173–1181 (2008) 44. T. Futamase, Averaging of a locally inhomogeneous realistic universe. Phys. Rev. D 53, 681– 689 (1996) 45. C. Clarkson, G. Ellis, J. Larena, O. Umeh, Does the growth of structure affect our dynamical models of the universe? The averaging, backreaction and fitting problems in cosmology. Rep. Prog. Phys. 74, 112901 (2011) 46. S. Kopeikin, M. Efroimsky, G. Kaplan, Relativistic Celestial Mechanics of the Solar System (Wiley, Weinheim, 2011) 47. G.F.R. Ellis, Inhomogeneity effects in cosmology. Class. Quantum Gravity 28(16), 164001 (2011) 48. S.M. Kopeikin, A.N. Petrov, Post-Newtonian celestial dynamics in cosmology: field equations. Phys. Rev. D 87(4), 044029 (2013) 49. E.M. Lifshitz, I.M. Khalatnikov, Special issue: problems of relativistic cosmology. Sov. Phys. Uspekhi 6, 495–522 (1964) 50. E. Lifshitz, On the gravitational stability of the expanding universe. JETP 10, 116–129 (1946). http://www.citebase.org/abstract?id=oai 51. S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972) 52. J.M. Bardeen, Gauge-invariant cosmological perturbations. Phys. Rev. D 22, 1882–1905 (1980) 53. H. Kodama, M. Sasaki, Cosmological perturbation theory. Prog. Theor. Phys. Suppl. 78, 1 (1984) 54. A. Ishibashi, R.M. Wald, Can the acceleration of our universe be explained by the effects of inhomogeneities? Class. Quantum Gravity 23, 235–250 (2006) 55. R. Baierlein, Representing a vector field: Helmholtz’s theorem derived from a Fourier identity. Am. J. Phys. 63, 180–182 (1995) 56. V.A. Brumberg, Essential Relativistic Celestial Mechanics (Adam Hilger, Bristol, 1991), p. 271 57. G.F.R. Ellis, M. Bruni, Covariant and gauge-invariant approach to cosmological density fluctuations. Phys. Rev. D 40, 1804–1818 (1989) 58. G.F.R. Ellis, J. Hwang, M. Bruni, Covariant and gauge-independent perfect-fluid RobertsonWalker perturbations. Phys. Rev. D 40, 1819–1826 (1989) 59. C.G. Tsagas, A. Challinor, R. Maartens, Relativistic cosmology and large-scale structure. Phys. Rep. 465, 61–147 (2008) Equations of Motion in an Expanding Universe 757 60. S. Shandarin, S. Habib, K. Heitmann, Cosmic web, multistream flows, and tessellations. Phys. Rev. D 85(8), 083005 (2012) 61. W. Del Pozzo, Inference of cosmological parameters from gravitational waves: Applications to second generation interferometers. Phys. Rev. D 86(4), 043011 (2012) 62. S.R. Taylor, J.R. Gair, I. Mandel, Cosmology using advanced gravitational-wave detectors alone. Phys. Rev. D 85(2), 023535 (2012) 63. M. Kramer, N. Wex, The double pulsar system: a unique laboratory for gravity. Class. Quantum Gravity 26(7), 073001 (2009) 64. J.M. Weisberg, D.J. Nice, J.H. Taylor, Timing measurements of the relativistic binary pulsar PSR B1913+16. Astrophys. J. 722, 1030–1034 (2010) 65. M. Kramer, Pulsars, SKA and time-domain studies in the future, in IAU Symposium, ed. by E. Griffin, R. Hanisch, R. Seaman, vol. 285, pp. 147–152, April 2012 66. S.M. Kopeikin, Celestial ephemerides in an expanding universe. Phys. Rev. D 86(6), 064004 (2012) 67. S. Hojman, Problem of the identical vanishing of Euler-Lagrange derivatives in field theory. Phys. Rev. D 27, 451–453 (1983) 68. N.V. Mitskevich, Physical Fields in General Relativity (Nauka, Moscow, 1969), p. 326 69. W. Greiner, J. Reinhardt, Field Quantization (Springer, Berlin, 1996). Translated from the German. With a foreword by D.A. Bromley 70. A.D. Popova, A.N. Petrov, Exact dynamic theories on a given background in gravitation. Int. J. Mod. Phys. A 3, 2651–2679 (1988) 71. B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge University Press, Cambridge, 1980) 72. A.N. Petrov, Nœther and Belinfante corrected types of currents for perturbations in the EinsteinGauss-Bonnet gravity. Class. Quantum Gravity 28(21), 215021 (2011) 73. L.P. Grishchuk, A.N. Petrov, A.D. Popova, Exact theory of the (Einstein) gravitational field in an arbitrary background space-time. Commun. Math. Phys. 94, 379–396 (1984) 74. C.M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, Cambridge, 1993), p. 396 75. E. Bertschinger, Testing dark matter models with the phase space structure of dark matter halos, in Probes of Dark Matter on Galaxy Scales, p. 10001 (2013) 76. I. Zlatev, L. Wang, P.J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant. Phys. Rev. Lett. 82, 896–899 (1999) 77. B.F. Schutz, Perfect fluids in general relativity: velocity potentials and a variational principle. Phys. Rev. D 2, 2762–2773 (1970) 78. L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1959) 79. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1975) 80. L. Amendola, S. Tsujikawa, Dark Energy: Theory and Observations (Cambridge University Press, Cambridge, 2010)