Dynamic Field Theory and Equations of Motion in Cosmology
aa r X i v : . [ g r- q c ] J u l Annals of Physics 00 (2018) 1–62
Annals ofPhysics
Dynamic Field Theory and Equations of Motion in Cosmology
Sergei M. Kopeikin ∗ Department of Physics & Astronomy, University of Missouri, 322 Physics Bldg., Columbia, MO 65211, USA
Alexander N. Petrov
Sternberg Astronomical Institute, Lomonosov Moscow State University, Universitetskij Prospect 13, Moscow 119992, Russia
Abstract
We discuss a field-theoretical approach based on general-relativistic variational principle to derive the covariant field equations andhydrodynamic equations of motion of baryonic matter governed by cosmological perturbations of dark matter and dark energy. Theaction depends on the gravitational and matter Lagrangian. The gravitational Lagrangian depends on the metric tensor and its firstand second derivatives. The matter Lagrangian includes dark matter, dark energy and the ordinary baryonic matter which plays arole of a bare perturbation. The total Lagrangian is expanded in an asymptotic Taylor series around the background cosmologicalmanifold defined as a solution of Einstein’s equations in the form of the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metrictensor. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the seriesexpansion is gauge-invariant and all of them together form a basis for the successive post-Friedmannian approximations around thebackground metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does notrequire the post-Friedmannian perturbations to be small though computationally it works the most e ff ectively when the perturbedmetric is close enough to the background FLRW metric. The temporal evolution of the background metric is governed by darkmatter and dark energy and we associate the large scale inhomogeneities in these two components as those generated by theprimordial cosmological perturbations with an e ff ective matter density contrast δρ/ρ ≤
1. The small scale inhomogeneities aregenerated by the condensations of baryonic matter considered as the bare perturbations of the background manifold that admits δρ/ρ ≫
1. Mathematically, the large scale perturbations are given by the homogeneous solution of the linearized field equationswhile the small scale perturbations are described by a particular solution of these equations with the bare stress-energy tensor ofthe baryonic matter. We explicitly work out the covariant field equations of the successive post-Friedmannian approximations ofEinstein’s equations in cosmology and derive equations of motion of large and small scale inhomogeneities of dark matter anddark energy. We apply these equations to derive the post-Friedmannian equations of motion of baryonic matter comprising stars,galaxies and their clusters.c (cid:13)
Keywords: cosmology, gravitation, hydrodynamics, dynamic field theory, Lagrangian, equations of motion
PACS: ∗ Corresponding author.
Email addresses: [email protected] (Sergei M. Kopeikin), [email protected] (Alexander N. Petrov) . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 Contents1 Introduction 4 ffi ne connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 ff ective Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.6 Field equations for gravitational perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.7 Field equations for matter perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.8 Gauge invariance of the field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 A Variational Derivatives 58
A.1 Variational derivative from the Hilbert Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.2 Variational derivatives of dynamic variables with respect to the metric tensor . . . . . . . . . . . . . . 59A.2.1 Variational derivatives of dark matter variables . . . . . . . . . . . . . . . . . . . . . . . . . 59A.2.2 Variational derivatives of dark energy variables . . . . . . . . . . . . . . . . . . . . . . . . . 60A.2.3 Variational derivatives of four-velocity of the Hubble flow . . . . . . . . . . . . . . . . . . . 60A.2.4 Variational derivatives of the metric tensor perturbations . . . . . . . . . . . . . . . . . . . . 61A.3 Variational derivatives with respect to matter variables . . . . . . . . . . . . . . . . . . . . . . . . . 61A.3.1 Variational derivatives of dark matter variables . . . . . . . . . . . . . . . . . . . . . . . . . 61A.3.2 Variational derivatives of dark energy variables . . . . . . . . . . . . . . . . . . . . . . . . . 623 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62
1. Introduction
The multiwavelength satellite observations of cosmic microwave background (CMB) radiation have opened a newchapter in cosmology [1]. The standard cosmological model [2–5] has been worked out to fit the model parametersto the CMB map and with the decisive confidence level of 95% [6]. The Planck satellite observations provide furtherevidences in robustness of the standard model [7, 8] (see for a comprehensive list of Planck collaboration papers) though it might be still di ffi cult to discern betweenvarious 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 thepresent state of the universe and for predicting 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 the large scalestructure formation. The dark matter consists of weakly interacting massive particles whose true nature is not knownso far except that they interact with baryons mainly by the force of gravity. The baryonic matter forms galaxieswhich, at early stage of structure formation, simply follow the evolution of dark matter condensations. Therefore, itis supposed that the observed large scale distribution of galaxies trace the distribution of dark matter.At the linear stage the e ff ective matter density contrast, δρ = ρ − ¯ ρ , is much smaller than the background (mean)density ¯ ρ of the universe: δρ/ ¯ ρ ≪
1. At later stages of cosmological evolution the structure formation enters a non-linear regime where δρ/ ¯ ρ ≃
1, and caustics are formed. Further growth of the perturbations leads to the developmentof small scale structures like nuclei of galaxies, dwarf galaxies, globular clusters, stars and more compact relativis-tic objects which have δρ/ ¯ ρ ≫
1. Gravitational field and matter of these super-dense baryonic objects counteractwith the gravitational potential of dark matter and dark energy but details of this process are still unclear becauseit involves rather complicated physics of fluid’s magnetohydrodynamics, turbulence and the strong gravity field thatimplies a general-relativistic approach taking into account the non-linear interaction of gravitational field with itselfand the surrounding matter. Presumably, some insight to the solution of this problem can be gained by exploringexact Lemaˆıtre-Tolman cosmological solution of Einstein’s equations admitting spatially inhomogeneity along radialcoordinate [11–13]. This purely geometric approach is mathematically sound but not very realistic as it describes apressureless, spherically-symmetric accretion of dust to a single point of a cosmological manifold while the real earlyuniverse has a continuous set of the accretion points (seeds of future galaxies) determined by the initial spectrum ofthe primordial density fluctuations [3–5]. In addition, the baryonic fluid pressure cannot be ignored at the non-linearregime.Exact solution of Einstein’s equations is unavailable for the general case of perturbed universe. Therefore, wehave to resort to approximations in order to treat non-linear gravitational e ff ects in the structure formation. Two ap-proximation schemes of solving Einstein’s equations are known in asymptotically-flat spacetime - post-Newtonianand post-Minkowskian approximations [14, pp. 340-344]. Post-Minkowskian approximations (PMA) rely on the as-sumptions that gravitational field is weak everywhere without any limitation on velocity of matter besides that it mustbe smaller than the fundamental speed, c . Post-Newtonian approximations (PNA) are made under assumption that thefield is weak and velocity of matter is much smaller than the speed of light. The PMA formalism has been basicallydeveloped in a series of papers by Damour and Blanchet for studying the mechanism of emission and propagation ofgravitational waves emitted by isolated astronomical systems [15–18]. The PNA formalism has been developed by anumber of independent researchers [19–23] for describing non-linear gravitational e ff ects in fluids, for deriving equa-tions of motion of binary stars [24–27], for calculating equilibrium models of rapidly rotating neutron stars [28–30],etc. Both PMA and PNA expand explicitly only the metric tensor of the manifold by making use either Landau-Lifshitz pseudotensor or 3 + / PNA expansion is not su ffi cientin cosmology because the background spacetime is not asymptotically flat and we have to take into account not onlythe perturbations of the metric tensor but also those of the background stress-energy tensor of the cosmological matterthat governs evolution of the cosmological spacetime. Additional non-trivial problem of the perturbation technique incosmology is to separate the contribution of bare perturbations of the baryonic matter of small-scale inhomogeneitiesfrom the large-scale perturbations of the background matter and the metric tensor of spacetime manifold. Thus, wehave to generalize PMA / PNA schemes of finding solutions of Einstein’s equations to include the case of more generalbackground manifold. We follow Tegmark [31] and use the name of post-Friedmannian approximations for such amore general, iterative procedure. 4 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 A number of theoretical attempts was undertaken to work out the first post-Friedmannian approximation andequations of motion of perfect fluid on cosmological manifold [32–35]. These works provide a good insight to thepossible solution of the problem but are insu ffi ciently consistent in separation of perturbations from their backgroundvalues. They do not suggest a systematic approach for extending the calculations of the linear perturbation theoryto the second, and higher, post-Friedmannian approximations either. We also point out a result by Oliynyk [36, 37]who analyzed the general structure of the post-Friedmannian expansions on cosmological manifold and arrived to theconclusion that the post-Friedmannian series are di ff erentiable, but not 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[38]. The Oliynyks conclusion di ff ers with the results obtained by making use of the post-Newtonian expansions inasymptotically-at spacetime [39–41] due to the compactness of the spatial slices in the cosmological manifold.Recently, a new interest for developing a self-consistent theory of post-Friedmannian approximations in precisioncosmology was triggered by a lively discussion [42–46] on whether the small-scale structure of the universe a ff ects itsHubble expansion rate and, thus, can explain the cosmic acceleration of the universe discovered in 1998-99 [47, 48]without invoking a dark energy. This is a, so-called, backreaction problem which intimately relates to the procedureof averaging the small-scale matter perturbations on a curved cosmological manifold [49, 50]. A certain progress inthis direction was achieved but many mathematical aspects of the backreaction problem are still poorly understood[51]. 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/ ¯ ρ ≪ δρ/ ¯ ρ ≫
1. Einstein’s equations tell us that the density perturbation, δρ/ ¯ ρ , is proportional to thesecond derivatives of the metric tensor perturbation which can be very large if δρ/ ¯ ρ ≫
1. At the same time, themetric tensor perturbation, κ αβ = g αβ − ¯ g αβ , and its first derivatives, κ αβ,γ , can still remain small enough in order toapply a perturbation technique for solving the Einstein equations. This is similar to the situation in the solar systemwhere the matter density contrast is huge but, nonetheless, the gravitational weak-field approximation for solvingEinstein’s equations is fully applicable [14]. It supports the idea that the perturbation technique in cosmology (underabove assumptions) is valid for calculating physical e ff ects of inhomogeneities on both the large and small scales[45, 52, 53]. 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 [54, 55]. It is techni-cally convenient for calculating time evolution of matter’s large scale structure inhomogeneities and gravitational fieldperturbations [4, 56] but is unsuitable for discussing the process of formation and time evolution of the small scalestructures in universe. This is because the Lifshitz approach uses the synchronous gauge where the time-time compo-nent of the metric tensor is fixed, g = −
1, at any order of approximation. The small scale structure in cosmologycorresponds to a localized astronomical system having a large density contrast, δρ/ ¯ ρ , and governed by the Newtonianlaw of gravity which demands the presence of the Newtonian potential, U , making g = − + U / c . It breaks downthe synchronous gauge condition at the regime of δρ/ ¯ ρ ≫ ff ects [59]. Some mathematical disadvantage of Bardeen’s approach is in imposing a scalar-vector-tensordecomposition on the metric tensor. It requires application of the Helmholtz theorem [60] that demands to foliatespacetime by a family of spacelike hypersurfaces and to integrate the metric tensor over these hypersurfaces. It makesBardeen’s approach non-local as contrasted to Lifshitz’s perturbation scheme. Moreover, in order to preserve thegauge-invariance of the Bardeen perturbation scheme one has to decompose the gauge functions in the same fashionas 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 coordinateworld lines. Evidently, this approximation scheme di ff ers significantly from the post-Newtonian approximations inasymptotically flat spacetime [14, 19, 61] which are more similar to Lifshitz’s approach but use a di ff erent gaugecondition (harmonic gauge).A gauge-invariant alternative to Bardeen’s approach was suggested by Ellis and Bruni [62] (see also [63, 64]), whodeveloped a perturbation scheme based on a reduction of full Einstein’s equations down to a system of field equationsthat are linear around a particular background. The Ellis-Bruni approach uses gauge-invariant variables which arespatial projections on the local comoving-observer frame threading the entire space-time of a real universe. Thus, the5 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 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-flatspacetime.At the epoch of precise cosmology we need more transparent theoretical scheme of the post-Friedmannian ap-proximations for handling the iterative calculation of cosmological perturbations and derivation of their equations ofmotion. This iterative scheme must satisfy a number of well-established criteria like to be covariant, gauge-invariant,operate with locally defined quantities, be systematic and self-consistent in improving the order of approximations,clearly separate the large-scale from small-scale matter perturbations, be independent of the mathematical ambiguitiesintroduced by the averaging procedures, etc. Some steps in developing such a scheme were done by Green and Wald[45, 46]. However, their work was focused mainly on the discussion of the averaging procedure in cosmology, onthe proof that the small-scale inhomogeneities do not produce a noticeable backreaction and on finding mathematicalevidences that the Newtonian approximation is su ffi cient in numerical N-body simulations of large scale structureformation [65].No doubt, theoretical questions about how to perform the averaging in cosmology and whether it produces anybackreaction at all, are important for understanding the mathematics of averaging of di ff erential operators in non-linearequations and for clarification of the true nature of dark matter and dark energy. However, the post-Friedmannian ap-proximation scheme in cosmology has broader implications that are going beyond the discussion of averaging andbackreaction problems and relates to the problem of interpretation of precise measurement of cosmological parame-ters by the advanced gravitational wave detector’s technique [66, 67] and formation of small-scale structures in theuniverse at the non-linear regime. The formalism of the post-Friedmannian approximations can be also helpful inbetter understanding of the influence of cosmological expansion on celestial mechanics of isolated astronomical sys-tems like binary pulsars which are currently the best laboratories for testing non-linear regime of general relativity[68, 69]. These tests will be made significantly more precise with advent of gravitational-wave astronomy and SquareKilometer Array (SKA) radio telescope [70].Recently, we have started a systematic investigation of the dynamics of small-scale inhomogeneities moving onthe FLRW background manifold. We have set up a Lagrangian formalism to derive the post-Friedmannian fieldequations for linearised cosmological perturbations [53] and analysed the Newtonian limit of these equations [71].The present paper goes beyond the linear regime and explores some non-linear e ff ects. In particular, we derive thepost-Friedmannian hydrodynamic equations of motion of the background matter (dark matter and dark energy) alongwith the equations of motion of the baryonic matter forming a small-scale structure with high-density contrast like astar, or galaxy or a cluster of galaxies.We explain the idea of manifold and underlying geometric objects in 2. The concept of the covariant and Liederivatives on manifold are explained in section 3. This section also defines the variational derivatives on manifoldin the context of the dynamic field theory. Geometric theory of Euler-type perturbations of arbitrary backgroundmanifold is set up in section 4. This theory is applied to the FLRW universe, governed by dark matter and darkenergy, in section 5. Section 6 derives the stress-energy tensors for perturbations of the gravitational field, darkmatter and dark energy. Finally, we derive equations of motion of the small-scale (bare) perturbations in section 7and compare our framework against other theoretical approaches in section 8. Appendix outlines some particularmathematical aspects of our derivation.Before going into details of our presentation we explain the notations adopted in the present paper. 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 cosmologicalmodel. Any function without such a bar belongs to the perturbed manifold. The other notations used in the presentpaper 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 , X i } = { c η, X i } are the conformal coordinates with η being a conformal time; • x α = { x , x i } = { ct , x i } is an arbitrary coordinate chart on the background manifold;6 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 • Greek indices α, β, γ, . . . run through values 0 , , ,
3, and label spacetime coordinates; • Roman indices i , j , k , . . . take values 1 , ,
3, and label spatial coordinates; • Einstein summation rule is applied for repeated (dummy) indices, for example, P α Q α ≡ P Q + P Q + P Q + P Q , and P i Q i ≡ P Q + P Q + P Q ; • g αβ is a full metric on the cosmological spacetime manifold; • ¯ g αβ is the FLRW metric on the background spacetime manifold; • g µν = √− gg µν – the metric tensor density of weight + §
4] di ff ers by sign from the standard definition, and is notcommonly accepted; • ¯ g µν = √− ¯ g ¯ g µν – the background metric tensor density of weight + • f αβ is the metric on the conformal spacetime manifold; • η αβ = diag {− , + , + , + } 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 − dR / dT ; • the conformal Hubble parameter, H = a − da / d η ; • F denotes a geometric object on the manifold. It can be either a scalar, or a vector, or a tensor field, or acorresponding tensor density; • a bar, ¯ F 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 backgroundmetric ¯ g αβ , for example F αβ = ¯ g αµ ¯ g βν F µν ; • the tensor indices of geometric objects on the conformal spacetime are raised and lowered with the conformalmetric f αβ ; • symmetry of a geometric object with respect to two indices is denoted with round parenthesis, F ( αβ ) ≡ (1 / (cid:16) F αβ + F βα (cid:17) ; • antisymmetry of a geometric object with respect to two indices is denoted with square parenthesis, F [ αβ ] ≡ (1 / (cid:16) F αβ − F βα (cid:17) ; • a prime F ′ = d F / d η denotes a total derivative with respect to the conformal time η ; • a dot ˙ F = d F / 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 acoordinate x α which is more convenient in some cases. In some cases which may not cause confusion, thecomma as a symbol of the partial derivative is omitted. For example, we denote the partial derivatives of theperturbations of matter variables as φ α ≡ φ ,α , ψ α ≡ ψ ,α , etc.; • a vertical bar, F | α denotes a covariant derivative of a geometric object F with respect to the background metric¯ g αβ . Covariant derivatives of scalar fields coincide with their partial derivatives; • a semicolon, F ; α denotes a covariant derivative of a geometric object F with respect to the conformal metric f αβ ; 7 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 • Φ A – a multiplet of A = { , , . . . , a } matter fields, Φ A = { Φ , Φ , . . . , Φ a } . These fields generate the full metric g µν of FLRW universe via the Einstein equations; • an operator ∇ α denotes the covariant derivative with respect to the full metric g αβ ; • ¯ Φ A – the background value of the fields Φ A . These fields generate the background metric ¯ g µν of FLRW universevia the Einstein equations; • Θ B – a multiplet of B = { , , . . . , b } matter fields, Θ B = { Θ , Θ , . . . , Θ b } . They generate the stress energytensor of the bare perturbation of the metric tensor g µν and that of the fields Φ A ; • ¯ Θ B – the background value of the fields Θ B . • φ A ≡ Φ A − ¯ Φ A – the perturbation of the field Φ A . Fields Φ A and ¯ Φ A refer to the same point on the manifold; • τ B ≡ Θ B − ¯ Θ B – the perturbation of the field Θ B caused by the counteraction of the metric tensor perturbations l µν and those of the dynamic fields φ A on the stress-energy tensor of the bare perturbations; • κ µν ≡ g µν − ¯ g µν – the metric tensor perturbation. Fields g µν and ¯ g µν refer to the same point on the manifold; • h µν ≡ g µν − ¯ g µν – the perturbation of the metric density caused by Θ B ; • l µν ≡ h µν / √− ¯ g . In a linear approximation, l µν = − κ µν + ¯ g µν κ αα , where κ αα = ¯ g αβ κ αβ ; • the Christo ff el symbols, Γ αβγ = g αν (cid:16) g νβ,γ + g νγ,β − g βγ,ν (cid:17) ; • the Riemann tensor, R αβµν = Γ αβν,µ − Γ αβµ,ν + Γ αµγ Γ γβν − Γ ανγ Γ γβµ ; • the Ricci tensor, R αβ = R µαµβ ; • the Ricci scalar, R = g αβ R αβ .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 o ff -shell. We shallintroduce and explain other notations as they appear in the main text of the paper.
2. Geometric manifold
Manifold M is a geometric arena for gravitational physics. Topologically manifold is a set of points endowed withsome particular di ff erential structure giving the manifold certain rigidity and physical properties. The basic elementof this structure is the metric tensor g αβ that allows to measure the distance between infinitesimally close points onthe manifold and the angles between two vectors attached to the same point on the manifold. In general relativity, themetric tensor represents gravitational field which is a tensor field of rank two. Alternative theories of gravity eitherintroduce other fields (scalar, vector, etc.) which yield additional contribution to the overall gravitational field besidesthe metric tensor or operate with the Lagrangian which is more complicated than the Hilbert Lagrangian of generaltheory of relativity like F ( R ) theories of gravity [73, 74]. The present paper deals exclusively with general theory ofrelativity and does not discuss the alternative theories in order to treat either gravitational field or dark matter and darkenergy. 8 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 ffi ne connection The first level of di ff erential structure of manifold is associated with the a ffi ne connection allowing us to di ff eren-tiate geometric objects and to transport them from one point of manifold to another. The a ffi ne connection consists ofthree algebraically-irreducible components which are the Christo ff el symbols, torsion and non-metricity [14]. Torsionand non-metricity do not appear in general relativity and we do not mention them from now on.We define the Christo ff el symbols of the second kind as usual [14] Γ αβγ ≡ g αδ (cid:16) g δβ,γ + g δγ,β − g βγ,δ (cid:17) . (1)The Christo ff el symbols of the first kind Γ αβγ = g ασ Γ σβγ = (cid:16) g αβ,γ + g αγ,β − g βγ,α (cid:17) , (2)We notice the symmetry with respect to the last two indices Γ αβγ = Γ α ( βγ ) . There is no any symmetry with respect tothe first two indices. In general, Γ αβγ = Γ ( αβ ) γ + Γ [ αβ ] γ , (3)where Γ ( αβ ) γ = g αβ,γ , Γ [ αβ ] γ = (cid:16) g γα,β − g γβ,α (cid:17) , (4)There are two, particularly useful symbols that are obtained by contracting indices of the Christo ff el symbols of thefirst kind. They are denoted as Y α ≡ Γ βαβ , Y α = g αβ Y β , (5)and Γ α ≡ g βγ Γ αβγ , Γ α = g αβ Γ β , (6)Direct inspection shows that Y α = = g βγ g βγ,α = (cid:16) ln √− g (cid:17) ,α . (7)The two symbols are interrelated Γ α = −Y a + g βγ g αβ,γ , (8) Γ α = −Y a − g αβ,β , (9) The second level of di ff erential structure of manifold is its curvature defined in terms of the Riemann tensor. Wedefine the Riemann tensor as follows [14] R αµβν = Γ αµν,β − Γ αµβ,ν + Γ αβγ Γ γµν − Γ ανγ Γ γµβ . (10)Riemann tensor can be also expressed in terms of the second partial derivatives of the metric tensor and the Christo ff elsymbols R αµβν = (cid:16) g µβ,αν + g να,βµ − g αβ,µν − g µν,αβ (cid:17) + Γ ρµβ Γ ραν − Γ ρµν Γ ραβ . (11)Contraction of two indices in the Riemann tensor yields the Ricci tensor R µν = Γ αµν,α − Y µ,ν + Y γ Γ γµν − Γ ανγ Γ γµα , (12)or, in terms of the second derivatives from the metric tensor and the Christo ff el symbols, R µν = g κǫ (cid:16) g µκ,ǫν + g νκ,ǫµ − g κǫ,µν − g µν,κǫ (cid:17) + g κǫ Γ ρµǫ Γ ρκν − Γ ρµν Γ ρ . (13)9 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 One more contraction of indices in the Ricci tensor brings about the Ricci scalar which we shall write down in theform suggested by Fock [19, Appendix B] R = L + Y α Γ α − Y α Y α + Γ α,α − Y α,α , (14)where L = g µν (cid:16) Γ ανγ Γ γµα − Y α Γ αµν (cid:17) , (15)is (up to a constant factor) the gravitational Lagrangian introduced by Einstein [75] as an alternative to the gravitationalLagrangian, R , of Hilbert. The Hilbert Lagrangian is the Ricci scalar which depends on the second derivatives of themetric tensor while the Einstein Lagrangian does not.The two Lagrangians are interrelated R = L + ( − g ) − / A α,α , (16)where A α = √− g ( Γ α − Y α ) , (17)is a vector density of weight +
1. After performing di ff erentiation in (16), and accounting for (5) we can easily provethat (16) reproduces (14).One more form of relation between R and L will be useful for calculating the variational derivative in AppendixA.1. To this end we introduce a new notation Γ ≡ Γ α,α + Y α Γ α , (18)and notice that g αβ Y α,β = Y α,α + Y α Γ α + Y α Y α , (19)Equations (18), (19) allows us to cast (14) to the following form R = L + Γ + Y α Γ α − g αβ Y α,β , (20)that was found by Fock [19, appendix B].
3. Derivatives on manifold
Covariant derivative on manifold is a rule of transportation of geometric objects from one point of the manifold toanother. If the geometric object is a tensor density F = F µ ...µ p ν ...ν q of type ( p , q ) and weight m , the covariant derivative isdefined by the following rule F µ ...µ p ν ...ν q ; α = F µ ...µ p ν ...ν q ,α + Γ µ αβ F β...µ p ν ...ν q + . . . + Γ µ p αβ F µ ...βν ...ν q − Γ βαν F µ ...µ p β...ν q − . . . − Γ βαν q F µ ...µ p ν ...β − m Y α F µ ...µ p ν ...ν q . (21)Second covariant derivatives of tensors do not commute due to the curvature of spacetime. For example, for acovector field F α and a covariant tensor field of second rank, F αβ the following commutation relations are hold F α ; βγ = F α ; γβ + R µαβγ F µ , (22) F αβ ; γδ = F αβ ; δγ − R αµγδ F µβ + R µβγδ F αµ . (23)It is straightforward to extend these commutation relations to tensors and tensor densities of higher rank.10 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 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 thatextremize the action S = Z F d x , (24)where F ≡ √− g f = √− g f (cid:16) Q , Q α , Q αβ (cid:17) , is a scalar density of weight +
1. Let F = F (cid:16) Q , Q α , Q αβ (cid:17) depend on thefield variable Q , its first - Q α ≡ Q ,α and second - Q αβ ≡ Q ,αβ partial derivatives that play here a similar role as velocityand acceleration in the Lagrangian mechanics of point-like particles. The field variable Q can be a tensor field of anarbitrary 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 dependon 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 whenthe variable Q is a tensor field. For example, if we choose a covariant vector field A µ as an independent variable,the corresponding “velocity” and “acceleration” variables must be chosen as A µ,α and A µ,αβ respectively. On theother hand, if the independent variable is chosen as a contravariant vector A µ , the corresponding “velocity” and“acceleration” variables must be chosen as A µ,α and A µ,αβ . The same remark is applied to any other tensor field.The reason behind is that A µ and A µ are interrelated via the metric tensor, A µ = g µν A ν . Therefore, derivative of A µ di ff ers from that of A µ by an additional term involving the derivative of the metric tensor which, if being improperlyintroduced, can bring about spurious terms to the field equations derived from the principle of the least action.Variational derivative, δ F /δ Q , taken with respect to the variable Q relates a change, δ S , in the functional S to achange, δ F , in the function F that the functional depends on, δ S = Z δ F d x , (25)where δ F = ∂ F ∂ Q δ Q + ∂ F ∂ Q α δ Q α + ∂ F ∂ Q αβ δ Q αβ . (26)This is a functional increment of F . The variational derivative is obtained after we single out a total divergence in theright side of (26) by making use of the commutation relations, δ Q α = ( δ Q ) ,α and δ Q αβ = ( δ Q ) ,αβ . The total divergenceis reduced to a surface term in the integral (25) which vanishes on the boundary of the volume of integration. Thus,the variation of S with respect to Q is given by δ S = Z δ F δ Q δ Qd x , (27)where δ F δ Q ≡ ∂ F ∂ Q − ∂∂ x α ∂ F ∂ Q α + ∂ ∂ x α ∂ x β ∂ F ∂ Q αβ . (28)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 variationalderivatives of F with respect to Q α δ F δ Q α ≡ ∂ F ∂ Q α − ∂∂ x β ∂ F ∂ Q αβ , (29)and that of F with respect to Q αβ , δ F δ Q αβ ≡ ∂ F ∂ Q αβ . (30)Let us assume that there is another geometric object, T (cid:16) Q , Q α , Q αβ (cid:17) , which di ff ers from the original one F (cid:16) Q , Q α , Q αβ (cid:17) by a total divergence T (cid:16) Q , Q α , Q αβ (cid:17) = F (cid:16) Q , Q α , Q αβ (cid:17) + ∂ β H β ( Q , Q α ) . (31)11 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 It is well-known [76, 77] that taking the variational derivative (28) from T and F yields the same result δ T δ Q ≡ δ F δ Q , (32)because the variational derivative from the divergence is zero identically. In fact, it is straightforward to prove a moregeneral result, namely, that the variational derivative (28), after it applies to a partial derivative of an arbitrary smoothfunction, vanishes identically δδ Q ∂ F ∂ x α ! ≡ . (33)However, this property does not hold for a covariant derivative in the most general case [77].The variational derivatives are covariant geometric object that is they do not depend on the choice of a particularcoordinates on manifold [14, 77]. In case, when the dynamic variable Q is not a metric tensor, this statement canbe proved by taking the first, Q α ≡ Q ; α , and second, Q αβ ≡ Q ; αβ , covariant derivatives of Q as independent dynamicvariables instead of its partial derivatives, Q α and Q αβ . In this case the procedure of derivation of variational derivatives(28), (29) remains the same and the result is δ F δ Q = ∂ F ∂ Q − " ∂ F ∂ Q α ; α + " ∂ F ∂ Q αβ ; βα . (34)The order, in which the covariant derivatives are taken, is imposed by the procedure of the extracting the total di-vergence from the variation of the action in (25). The order of the derivatives is important because the covariantderivatives do not commute.Variational derivative of F with respect to the metric tensor g µν is defined by the same equations (28)–(30) wherewe identify Q ≡ g µν , Q α ≡ g µν,α , and Q µν ≡ g µν,αβ . It yields δ F δ g µν ≡ ∂ F ∂ g µν − ∂∂ x α ∂ F ∂ g µν,α + ∂ ∂ x α ∂ x β ∂ F ∂ g µν,αβ (35) δ F δ g µν,α ≡ ∂ F ∂ g µν,α − ∂∂ x β ∂ F ∂ g µν,αβ , (36) δ F δ g µν,αβ ≡ ∂ F ∂ g µν,αβ . (37)If the geometric object F depends on the contravariant components of the metric tensor, g αβ , and / or its derivativesthe partial derivatives in (35)–(37) are calculated after making use of relations ∂ g αβ ∂ g µν = − g α ( µ g ν ) β , (38) ∂ g αβ,γ ∂ g µν,σ = − g α ( µ g ν ) β δ σγ , (39) ∂ g αβ,γκ ∂ g µν,ρσ = − g α ( µ g ν ) β δ ( ργ δ σ ) κ . (40)Variational derivatives (35)–(37) preserve covariance. The most simple way to prove it would be to express (35)–(37) in terms of the covariant derivatives like we did in transformation of variational derivative (26) to its covariantanalogue (34). However, in case of variational derivative with respect to the metric tensor this procedure is not sostraightforward because the covariant derivative of the metric tensor, g µν ; α ≡
0, and we cannot use it as a covariantdynamic variable being conjugated to the metric tensor. In this case, we consider a set of the metric tensor, g µν , theChristo ff el symbols Γ αµν , and the Riemann tensor R αβµν as independent dynamic variables. The action is given by(24) where F ≡ √− g f (cid:16) g µν , Γ αµν , R αβµν (cid:17) is a scalar density of weight +
1. Variation of F is δ F = ∂ F ∂ g µν δ g µν + ∂ F ∂ Γ αµν δ Γ αµν + ∂ F ∂ R αβµν δ R αβµν , (41)12 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where variations of the Christo ff el symbols and the Riemann tensor are tensors that can be expressed in terms of thevariation δ g µν of the metric tensor [56] δ Γ αµν = g ασ h ( δ g σµ ) ; ν + ( δ g σν ) ; µ − ( δ g µν ) ; σ i , (42) δ R αβµν = ( δ Γ αβν ) ; µ − ( δ Γ αβµ ) ; ν . (43)Now, we replace variations of the Christo ff el symbols and the Riemann tensor in (41) with (42), (44) and single out atotal divergence . It yields δ F = δ F δ g µν δ g µν + B α,α , (44)where the total divergence vanishes on the boundary of integration of the action, and the covariant variational deriva-tive is δ F δ g µν = ∂ F ∂ g µν − g σµ ∂ F ∂ Γ σνα + g σν ∂ F ∂ Γ σµα − g σα ∂ F ∂ Γ σµν ! ; α (45) + g σµ ∂ F ∂ R σαβν + g σν ∂ F ∂ R σµβα − g σα ∂ F ∂ R σµβν ! ; βα This equation can be further simplified if we shall make use of the Christo ff el symbols of the first kind, Γ αµν = g ασ Γ σµν , and R αβµν = g ασ R σβµν . The partial derivatives ∂ F ∂ Γ σµν = g σρ ∂ F ∂ Γ ρµν , ∂ F ∂ R σλµν = g σρ ∂ F ∂ R ρλµν . (46)Moreover, the cyclic permutation property of the Riemann tensor tells us that ∂ F ∂ R ρλµν = − ∂ F ∂ R ρµνλ − ∂ F ∂ R ρνλµ . (47)Employing (46), (47) in (45) transforms, and making use of antisymmetry R σβµν = − R σβνµ of the Riemann tensor,reduces (45) to a more compact form δ F δ g µν = ∂ F ∂ g µν − ∂ F ∂ Γ µνα + ∂ F ∂ Γ νµα − ∂ F ∂ Γ αµν ! ; α + ∂ F ∂ R µαβν + ∂ F ∂ R µβαν ! ; βα (48)Calculation of variational derivatives requires calculation of partial derivatives with respect to the metric tensorand other geometric objects like the Christo ff el symbols, the Riemann tensor, etc. An example is the partial derivativesfrom the determinant of the metric tensor ∂ √− g ∂ g µν = − √− gg µν , ∂ √− g ∂ g µν = √− gg µν , (49)where g is the determinant of the metric tensor. Taking partial derivatives from F = F (cid:16) g µν , Γ αµν , R αβµν (cid:17) with respect to g µν , Γ αµν and R αβµν is performed with the help of the following formulas ∂ g αβ ∂ g µν = δ ( µα δ ν ) β , (50) ∂ Γ σαβ ∂ Γ ρµν = δ σρ δ ( µα δ ν ) β , (51) ∂ R σγαβ ∂ R ρκµν = δ σρ δ κγ δ [ µα δ ν ] β , (52) The fact that F is a scalar density is essential for the transformation of covariant derivatives to the total divergence. The total divergences canbe converted to surface integrals which vanish on the boundary of integration and, hence, can be dropped o ff the calculations. . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where we have accounted for the symmetry of the Christo ff el symbols and the antisymmetry of the Riemann tensor.In case when F is a function of g µν the variational derivative with respect to the contravariant metric tensor is δ F δ g µν = ∂ g αβ ∂ g µν δ F δ g αβ = − g αµ g βν δ F δ g αβ . (53)We will also need to calculate the variational derivative with respect to the density of the metric tensor, g µν . It relatesto the variational derivative of the metric tensor as follows, δδ g µν = ∂ g ρσ ∂ g µν δδ g ρσ = A ρσµν √− g δδ g ρσ , (54)where A ρσµν = (cid:16) δ ρµ δ σν + δ ρν δ σµ − g µν g ρσ (cid:17) . (55)The variational derivatives are not linear operators. For example, they do not obey Leibniz’s rule [78, Section 2.3].More specifically, for any geometric object, H = F T , that is a corresponding product of two other geometric objects, F = F (cid:16) Q , Q α , Q αβ (cid:17) and T = T (cid:16) Q , Q α , Q αβ (cid:17) , the variational derivative δ ( F T ) δ Q , δ F δ Q T + F δ T δ Q , (56)in the most general case. The chain rule with regard to the variational derivative is preserved in a limited sense. Morespecifically, let us consider a geometric object F = F (cid:16) Q , Q α , Q αβ (cid:17) where Q is a function of a variable P , that is Q = Q ( P ). Then, the variational derivative δ F δ P = δ F δ Q ∂ Q ∂ P , (57)that can be confirmed by inspection [79]. On the other hand, if we have a singled-valued function H = H ( Q ), and Q = Q (cid:16) P , P α , P αβ (cid:17) , the chain rule δ H δ P = ∂ F ∂ Q δ Q δ P , (58)is also valid. The chain rule (58) will be often used in calculations of the present paper. Lie derivative on the manifold can be viewed as being induced by a di ff eomorphism x ′ α = x α + ξ α ( x ) , (59)such that a vector field ξ α has no self-intersections, thus, defining a congruence of curves which provides a naturalmapping of the manifold into itself. Lie derivative of a geometric object F is denoted as £ ξ F . It is defined by astandard rule £ ξ F = F ′ ( x ) − F ( x ) , (60)where F ′ is calculated by doing its coordinate transformation induced by the change of the coordinates (59) withsubsequent pulling back the transformed object from the point x ′ α to x α along the congruence ξ α [14]. In particular,for any tensor density F = F µ ...µ p ν ...ν q of type ( p , q ) and weight m one has£ ξ F µ ...µ p ν ...ν q = ξ α F µ ...µ p ν ...ν q ,α + m ξ α,α F µ ...µ p ν ...ν q (61) + F µ ...µ p α...ν q ξ α,ν + . . . + F µ ...µ p ν ...α ξ α,ν q − F α...µ p ν ...ν q ξ µ ,α − . . . − F µ ...αν ...ν q ξ µ p ,α . We notice that all partial derivatives in the right side of equation (61) can be simultaneously replaced with the covariantderivatives because the terms containing the Christo ff el symbols cancel each other.14 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 The Lie derivative commutes with a partial (but not a covariant) derivative ∂ α (cid:16) £ ξ F (cid:17) = £ ξ ( ∂ α F ) , (62)where F is actually an arbitrary geometric object rather than merely a tensor density. This property allows us to provethat a Lie derivative from a geometric object F (cid:16) Q , Q α , Q αβ (cid:17) can be calculated in terms of its variational derivative.Indeed, £ ξ F = ∂ F ∂ Q £ ξ Q + ∂ F ∂ Q α £ ξ Q α + ∂ F ∂ Q αβ £ ξ Q αβ . (63)Now, after using the commutation property (62) and changing the order of partial derivatives in £ ξ Q α and £ ξ Q αβ , onecan express (63) as an algebraic sum of the variational derivative and a total divergence£ ξ F = δ F δ Q £ ξ Q + ∂∂ x α δ F δ Q α £ ξ Q + δ F δ Q αβ £ ξ Q β ! . (64)This property of the Lie derivative indicates its close relation to the variational derivative on manifold and will be usedin the calculations that follow this section.It is also worth pointing out that (64) is used in derivation of Noether’s theorem of conservation of the canonicalstress-energy tensor of the field Q in case when F = L is the Lagrangian density of weight m = + Q which variational derivative vanishes on-shell, δ F /δ Q = δ L /δ Q =
0. The Lagrangian density has the Lie derivativein the form of total divergence, £ ξ L = ∂ α ( ξ α L ), and (64) yields the conserved Noether current J α ≡ ξ α L − δ F δ Q α £ ξ Q − δ F δ Q αβ £ ξ Q β , (65)where ξ α is a vector field defining the change of coordinates (62). This field should not be confused with the Killingvector defining isometry of the metric tensor. The Noether current is conserved, ∇ α J α =
0, independently of whetherthe manifold admit isometries or not [80].
4. Field Perturbation Theory on Spacetime Manifold
Let us consider a field theory on a background pseudo-Riemannian manifold ¯ M having the metric tensor ¯ g µν thatis a solution of Einstein’s equations ¯ G µν − π ¯ T M µν = , (66)where ¯ G µν = ¯ R µν − / g µν ¯ R is the Einstein tensor, and ¯ T M µν is the stress-energy tensor of the matter fields ¯ Φ A , wherethe index A numerates the fields and takes the values A = , , . . . , a . We assume that the solution of the backgroundEinstein’s equations (66) is known.In the simplest case the background manifold M is considered as Minkowski spacetime with the backgroundmetric ¯ g µν = η µν = diag[ − , + , + , +
1] being the Minkowski metric. In this case, both tensors ¯ G µν and ¯ T M µν vanishidentically, and the background Einstein’s equations (66) satisfy automatically. The use of the Minkowski backgroundis typical in solving the problems of post-Newtonian celestial mechanics [14, 81] and in the branch of gravitationalwave astronomy dealing with calculation of templates of gravitational waves emitted by coalescing binary systems[40, 41, 82]. Minkowski background is not appropriate in cosmology which operates with conformally-flat Friedman-Lemaˆıtre-Robertson-Walker (FLRW) metric tensor d ¯ s = − dT + R ( T ) + kr ! − δ i j dX i dX j , (67)where T is the cosmic time, X α = ( T , X i ) are the global coordinates associated with the Hubble flow, r = p δ i j X i X j , k is the curvature of space taking one of the three values k = − , , +
1, and R ( T ) is the scale factor which temporal15 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 evolution is governed by the solution of Einstein’s equations (66) with the background stress-energy tensor ¯ T M µν de-termined by the matter fields ¯ Φ A [4, 5, 56]. FLRW metric (67) is conformally flat and is not reduced to Minkowskimetric globally. The conformal nature of the cosmological metric does not allow us to apply the post-Newtonianapproximations which should be generalized to take into account the curvature of the background spacetime to solvethe Einstein equations for the field perturbations.Let us perturb the background manifold M so that the geometry of the perturbed manifold M is now described bythe metric tensor g µν that is a solution of perturbed Einstein’s equations G µν − π T M µν = π T µν , (68)with the perturbed values of the Einstein tensor G µν = R µν − / g µν R , and the stress-energy tensor T M µν of the samephysical fields Φ A . Besides the background matter we admit the presence of the stress-energy tensor T µν of the othermatter fields Θ B where the index B numerates the bare fields and takes values B = , , . . . , b in the right side ofEinstein’s equations. These fields represent the source of the bare perturbation of the background manifold whichcan be associated in cosmology with a small-scale inhomogeneities having rather large density contrast created by thepresence of baryonic matter making stars, planets, etc. or, even, black holes.The perturbed metric g µν and the perturbed matter fields Φ A can be always represented as linear combinations g µν = ¯ g µν + κ µν , and Φ A = ¯ Φ A + φ A , where κ µν and φ A are the perturbations of the metric and the matter fieldsrespectively. The fields Θ B are not present on the background manifold ¯ M , and appear only as bare perturbations“injected” to the background manifold from “outside”. In fact, the only physical system we are dealing with, isthe perturbed manifold. Mathematical formalism of the perturbation theory is based on separation of the physicalmanifold into two parts – the background and the perturbations. This is done merely for mathematical conveniencesince it allows us to set up a consistent and rigorous mathematical framework for adequate description of gravitationalphysics on perturbed manifold M . In what follows, we assume that the fields Φ A and Θ B are both minimally coupledto the curvature of spacetime in the sense of the strong equivalence principle [14, Section 3.8.2], [83, Section 6.13].We also assume, for the sake of simplicity, that the fields Φ A and Θ B do not directly interact one with another. Thisassumption can be easily relaxed but the calculations will be longer. We postpone consideration of this problem to afuture publication.In the first approximation the field perturbations κ µν and φ A can be split in two categories:1. the free perturbations, which are solutions of the homogeneous Einstein’s equations (68) with the right side T µν = T µν , ff ects which can be removed by a corresponding choice of coordinates on thebackground manifold ¯ M . This is no longer true if the background manifold is curved. For example, in cosmologythe free perturbations are basically primordial perturbations which are relics of the boundary conditions imposed atthe epoch of Big Bang without presence of any particular physical source. There are several alternative explanationsof the formation of the primordial (free) perturbations which are discussed, for example, in textbooks [3, 4, 84]. Thefree perturbations of matter fields grow to form the large-scale structure of the universe. The source of the inducedperturbations in cosmology is the stress-energy tensor T µν of the bare perturbations Θ B . Due to the non-linearityof Einstein’s equations the perturbations interact between themselves and one with another through the gravitationalcoupling in non-linear Einstein’s equations. It makes the geometric structure of the perturbed manifold M ratherentangled as we go from the first to higher-order approximation theory.The present paper describes how to find out the perturbed geometric structure of the manifold M , how to formulatethe field equations for the perturbations, and how to derive the equations of motion of perturbations on the basis ofthe Lagrangian-based variational principle [79, 85]. The Lagrangian formalism allows us to set up the perturbationtheory in a covariant and gauge-invariant representation directly, without splitting the metric tensor perturbations in thescalar, vector and tensor modes which is a rather popular approach in the studies of cosmological perturbations [3, 4].However, such splitting is usually accompanied with a specific foliation of comsological spacetime which disguisesthe covariant nature of the perturbation theory and makes the entire perturbation approach look quite di ff erent from16 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 the post-Newtonian approximation scheme elaborated on the Minkowski-flat background manifold. The Lagrangian-based variational formalism allows us to reconcile the theory of cosmological perturbations with the post-Newtonianapproximations in a natural and self-consistent way. The Lagrangian formulation of the dynamic theory of field perturbations in general relativity starts o ff the Hilbertaction defined on the unperturbed background manifold ¯ M ,¯ S = Z d x ¯ L (cid:16) ¯ g µν , ¯ Φ A (cid:17) , (69)where ¯ L = √− ¯ g ¯ L is a scalar density of weight + √− ¯ g ), ¯ L = ¯ L (cid:16) ¯ g µν , ¯ Φ A (cid:17) is the Lagrangian which is ascalar function depending on the metric tensor, the matter fields ¯ Φ A = { ¯ Φ , ¯ Φ , . . . , ¯ Φ a } , and their partial derivatives.To avoid superfluous notations we did not show explicitly in (69) but keep in mind, the dependence of the Lagrangianon the partial derivatives of the field. The matter fields ¯ Φ A determine the dynamic and geometric structure of thebackground manifold ¯ M via Einstein’s equations. For short, we shall call the Lagrangian density, ¯ L , simply theLagrangian.The Lagrangian L is split in two parts¯ L (cid:16) ¯ g µν , ¯ Φ A (cid:17) = ¯ L G (cid:16) ¯ g µν (cid:17) + ¯ L M (cid:16) ¯ g µν , ¯ Φ A (cid:17) , (70)where the gravitational (Hilbert) Lagrangian ¯ L G (cid:16) ¯ g αβ (cid:17) ≡ − π p − ¯ g ¯ R , (71)depends on the background metric tensor ¯ g µν as well as on its first and second derivatives. The matter Lagrangian,¯ L M , depends on the matter fields ¯ Φ A and their derivatives. It also depends on the metric tensor and (for instance, inthe case of Yang-Mills fields) on its first derivatives.Dynamic equations of the gravitational field and matter are derived from the principle of the least action by varyingthe action (69) and equating its variation to zero. This procedure is well-known and we shall not repeat it over here(see, for example, [14, section 3.9]). It is equivalent to taking the variational derivatives (28) from the Lagrangian (70)with the variable Q = { ¯ g µν , ¯ Φ A } . It yields the Euler-Lagrange equations δ ¯ L M δ ¯ Φ A = , (72) δ ¯ L G δ ¯ g µν + δ ¯ L M δ ¯ g µν = . (73)Equation (72) describes a dynamic evolution of the matter fields ¯ Φ A . Equation (73) can be recognized as the Einsteinequations (66) for the background gravitational field (the metric tensor) after noticing that the variational derivatives δ ¯ L G δ ¯ g µν = − π ¯ R µν , (74) δ ¯ L M δ ¯ g µν = ¯ T M µν −
12 ¯ g µν ¯ T M ! , (75)where ¯ R µν is the background value of the Ricci tensor calculated with the help of the background metric ¯ g µν , and ¯ T M µν isthe stress-energy tensor of the fields ¯ Φ A . Equation (75) is just a definition of the metrical stress-energy tensor of matter[14, Section 3.9.5]. Equation (74) is usually derived by varying the gravitational action (see, for instance, [14, page310], [56, page 364]) and extracting the total derivative that vanishes on the boundary of the volume of integration.The same result is obtained if we take the variational derivative (74) directly by making use of (54), (53) and (35).Calculations are straightforward but tedious, and are given in Appendix (A) of the present paper. The same result can17 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 be achieved in much shorter and attractive way if we use the covariant definition (45) of the variational derivative.Indeed, the Lagrangian (71) is equivalent to ¯ L G = − π ¯ g κλ δ σρ ¯ R ρκσλ (76)This expression depends only on the metric tensor and the Riemann tensor as a linear algebraic function. Hence,partial derivatives with respect to the Christo ff el symbols are automatically nil. Moreover, the covariant derivativesfrom the metric tensor vanish identically. Hence, the variational derivative (45) (along with (53)) is reduced to asimple partial derivative δ ¯ L G δ ¯ g µν = − π ∂ ¯ g κλ ∂ ¯ g µν ¯ R κλ , (77)that immediately results in (74).Physical perturbations of the background manifold ¯ M are caused either by imposing the initial spectrum of pri-mordial perturbations on the metric tensor g µν and fields Φ A or by “injecting” on the manifold the bare matter field¯ Θ B with the Lagrangian ¯ L P = √− ¯ g ¯ L P where ¯ L P ≡ ¯ L P ( ¯ g µν , ¯ Θ B ) is a scalar function. The present paper assumes that¯ Θ B is minimally coupled with gravity but does not interact directly with the fields ¯ Φ A . We postulate that the absolutevalue of ¯ L P is much smaller than the Lagrangian ¯ L of the background manifold, that is ¯ L P ≪ ¯ L . The fields ¯ Θ B canbe conceived, for example, as a baryonic matter composing an isolated astronomical system like the solar system ora galaxy, or a cluster of galaxies. However, it is also admissible to consider ¯ Θ B as a seed perturbations of the fields¯ Φ A , for example, in discussion of the formation of the small-scale structure of the universe at the latest stages of itsevolution. The assumptions imposed on the Lagrangian of the fields ¯ Θ B presume that in order to describe the dynamicevolution of the perturbed manifold M we should add algebraically the Lagrangian ¯ L P of the bare perturbations to theunperturbed Lagrangian ¯ L of the background manifold, write down the perturbed Einstein equations for the metrictensor perturbations l µν along with the equations for the perturbations φ A of the fields ¯ Φ A , solve them, and proceed tothe second, third, etc. iterations if necessary. The iterative theory of the Lagrangian perturbations of the manifold isdescribed in the following sections. Lagrangian-based formulation of the dynamic theory of physical perturbations of a manifold starts o ff the Hilbertaction S = Z d x L ( g µν , Φ A , Θ B ) , (78)where L = √− gL is the scalar density of weight +
1, and L = L ( g µν , Φ A , Θ B ) is the Lagrangian depending on themetric tensor, the matter fields Φ A = { Φ , Φ , . . . , Φ a } , and the fields Θ B = { Θ , Θ , . . . , Θ b } representing the bareperturbation of the manifold.The Lagrangian L consists of three parts L (cid:16) g µν , Φ A , Θ B (cid:17) = L G (cid:16) g µν (cid:17) + L M (cid:16) g µν , Φ A (cid:17) + L P (cid:16) g µν , Θ B (cid:17) , (79)where the gravitational (Hilbert) Lagrangian L G (cid:16) g αβ (cid:17) ≡ − π √− gR , (80)depends on the metric tensor g µν , its first and second derivatives. The Lagrangians of matter, L M and L P , depend solelyon the metric tensor and its first derivatives. They also depend directly on the matter fields Φ A and Θ B and their partialderivatives but we did not show it explicitly to avoid tedious notations. The matter fields Φ A and Θ B are minimallycoupled to gravity but we assume that they are not directly coupled to each other. Hence, the Lagrangian of theinteraction between these fields does not appear explicitly in (79). This assumption can be relaxed and successfullyhandled with the formalism of the present paper but the computational aspects become more intricate and will beconsidered somewhere else. 18 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 It is worth noticing that L M and L P depend on the metric tensor g µν both explicitly and implicitly through themathematical definition of the matter fields Φ A and Θ B . For example, consider the Lagrangian of the perfect fluid L M = ρ (1 + Π ) √− g , where Π is the specific internal energy of the fluid and ρ is the energy density. The metric tensorappears explicitly as √− g and implicitly in ρ that is defined as the ratio of the rest energy of the fluid’s element to itscomoving volume which depends on the determinant of the metric tensor [14]. Since Π = Π ( ρ ) is a thermodynamicfunction of ρ , it also depends implicitly on the metric tensor. It means that the variational derivatives of ρ and Π withrespect to the metric tensor have certain values which we shall discuss in the sections which follow and in AppendixA.2.1.We define perturbations of the gravitational and matter fields residing on the background manifold by the followingequations, g µν ( x ) = ¯ g µν ( x ) + h µν ( x ) , (81) Φ A ( x ) = ¯ Φ A ( x ) + φ A ( x ) , (82) Θ B ( x ) = ¯ Θ B ( x ) + θ B ( x ) , (83)where all functions are taken at one and the same point x ≡ x α of the unperturbed manifold ¯ M . The perturbedvalues of the fields are assumed to be su ffi ciently small compared with their background counterparts: | h µν | < | ¯ g µν | , | φ A | < | ¯ Φ A | and | θ B | < | ¯ Θ B | . There are no specific limitations on the rate of change of the perturbations that is weassume a slow-motion approximation and do not assume that the time derivatives are much smaller than spatial partialderivatives. The second partial derivatives of the fields are comparable (due to the field equations) with the magnitudeof the stress-energy tensor, T µν , of the bare perturbations that is | h µν,µν | ∼ | φ A ,µν | ∼ | T µν | . Similar assumptions areused in the method of solution of Einstein’s equations in asymptotically-flat space time called the post-Minkowskianapproximations [15, 16, 86]. The present paper extends the post-Minkowskian approximations to a more sophisticatedrealm of curved background manifolds.We consider the conjugated pairs of perturbations and its first partial derivatives (cid:8) h µν , h µν | α (cid:9) , n φ A , φ A | α o along withthe bare perturbing field n θ B , θ B | α o , where the vertical bar denotes a covariant derivative on the background manifold, asa set of independent dynamic variables which propagate on the background manifold ¯ M with the metric ¯ g µν . In orderto derive the di ff erential equations governing the evolution of the perturbations we substitute the field decompositions(81)–(83) to the Lagrangian L defined by equations (79) which yields L = L G (¯ g µν + h µν ) + L M ( ¯ Φ A + φ A , ¯ g µν + h µν ) + L P ( ¯ Θ B + θ B , ¯ g µν + h µν ) . (84)Because the perturbations h µν , φ A , θ B are linearly superimposed on the background values of the metric tensor ¯ g µν and fields ¯ Φ , ¯ Θ B respectively, the perturbed (total) Lagrangian (84) admits the following property of the variationalderivatives, δ L δ h µν = δ L δ ¯ g µν , δ L δφ A = δ L δ ¯ Φ A , δ L δθ B = δ L δ ¯ Θ B . (85)These relations allow us to replace the variational derivatives of the total Lagrangian taken with respect to the dynamicperturbation of the field for those taken with respect to the background value of the corresponding field. It turns outto be a very useful device in calculations of the variational derivatives and in building the iterative scheme of solvingthe Einstein equations by successive approximations. The perturbative theory of the dynamic fields on the background spacetime manifold ¯ M is based on the Taylorseries decomposition of the total Lagrangian with respect to the field perturbations which magnitude plays the role ofa small parameter of the theory. The formal procedure is straightforward and has been described by B. DeWitt [87].More specifically, we take the total Lagrangian (84) and expand it in a Taylor series by making use of the variationalderivatives of L with respect to the dynamic variables h µν and φ A . The Taylor expansion of L with respect to θ canbe also performed but we prefer to avoid it because physical measurements yield access to the total value of the bareperturbation Θ . The reader should keep in mind that the expansion of the Lagrangian is performed under the sign ofthe integral in the action functional (78). Therefore, all terms in this expansion which are reduced to a total divergencecan be discarded as they do not contribute to the value of the action integral.19 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 We assume at the beginning of the calculation that the perturbations and their derivatives are su ffi ciently smallto ensure the convergence of the Taylor expansion of the Lagrangian. For the Lagrangian is a function of severalvariables, the Taylor series has terms with the mixed derivatives starting from the second order. At the first glance,the presence of the mixed derivatives causes mathematical complication in ordering the higher-order terms. It isremarkable that this problem can be nicely handled after taking into account the following property of the commutatorof two variational derivatives [79] h αβ δδ ¯ g αβ φ A δ ¯ L δ ¯ Φ A ! − φ A δδ ¯ Φ A h αβ δ ¯ L δ ¯ g αβ ! = ∂ α H α , (86)where H α denotes a vector density of weight + L ,and the repeated field label A denotes Einstein’s summation over all fields Φ A . This commutation rule is also validfor any two fields from the field multiplets Φ A , Θ B , etc. Equation (86) allows us to change the order of the variationalderivatives to reshu ffl e terms with the mixed derivatives in the Taylor expansion of the perturbed Lagrangian L . Indoing this, all terms representing the total divergence can be omitted from the Taylor expansion since the variationalderivative from them vanishes identically, and they do not contribute to the field equations according to (31), (32).Using this procedure we can put all terms with the mixed derivatives in a specific order so that the Taylor expansionof the Lagrangian takes the following elegant form L = L P + ∞ X n = L n . (87)Here, L P is the Lagrangian of the bare perturbation, L ≡ ¯ L is the Lagrangian (70) describing dynamic properties ofthe background manifold, and for any n ≥ L n = n h µν δ L n − δ ¯ g µν + φ A δ L n − δ ¯ Φ A ! , (88)represents a collection of terms of the power n with respect to the perturbations h µν and φ A . In particular, the linearand quadratic terms of the expansion (87) read L = h µν δ ¯ L δ ¯ g µν + φ A δ ¯ L δ ¯ Φ A , (89) L = h µν δ L δ ¯ g µν + φ A δ L δ ¯ Φ A ! , (90)and so on. We conclude that each subsequent term in the Taylor expansion of the Lagrangian (87) can be obtained fromthe previous approximation by taking the variational derivative. The entire analytic procedure is easily computerized.Equation (88) can be proved by induction starting from the value of L in (89) which is apparently true, andoperating with the commutation rule (86) in higher orders in order to confirm that the result is reduced to the orig-inal Taylor series. The commutation property (86) of the variational derivatives allows us to write down the Taylorexpansion (87) as follows L = exp h µν δδ ¯ g µν + φ A δδ ¯ Φ A ! ¯ L + L P , (91)that establishes a mapping relation between the perturbed, L , and unperturbed, ¯ L , Lagrangians in the most succinct,exponential form.By applying equation (85) to the Taylor series (87), and making use of δ L P /δ h µν = δ L P /δ ¯ g µν , we get an importantrelation between the variational derivatives of the consecutive terms L n and L n − in the series decomposition of theLagrangian, δ L n δ h µν = δ L n − δ ¯ g µν , δ L n δφ A = δ L n − δ ¯ Φ A . (92)20 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 These relations can be confirmed directly by making use of (88) that establishes a relation between the adjacent ordersof the Lagrangian expansion (87). In doing so, we have to keep in mind that the total divergences can be alwaysdiscarded.If necessary, the Lagrangian of the bare perturbation can be also expanded in the Taylor series with respect to h µν , L P = L Θ + h µν δ L Θ δ ¯ g µν + h αβ δδ ¯ g αβ h µν δ L Θ δ ¯ g µν ! + . . . (93) = exp h µν δδ ¯ g µν ! L Θ , where we have defined L Θ ≡ L P ( Θ B , ¯ g µν ). However, in practical calculations it is more convenient to keep L P unexpanded, remembering that at each iteration the metric tensor g µν and the field Θ entering L P are known up to theorder of the approximation under consideration. ff ective Lagrangians In order to build the field perturbation theory on a curved background manifold ¯ M we have to single out the firstorder terms which represent the linear di ff erential equations for the dynamic field variables. The entire theory is builtunder assumption that the background field equations are valid exactly. In other words, the perturbation theory is valid on-shell .The principle of the least action tells us that the Lagrangian (84) must be stationary with respect to variations ofthe metric tensor g µν and the field variables Φ A , δ L δ g µν = , δ L δ Φ A = . (94)We also assume that the background Lagrangian (70) is stationary with respect to the variations of the backgroundvariables ¯ g µν and ¯ Φ A , and the field equations (72), (73) are valid. It means that the variational derivatives with respectto h µν and φ A from the background Lagrangian L ≡ ¯ L vanish identically. Therefore, applying equations (92) to theterms of the linear order, n =
1, yields δ L δ h µν = δ ¯ L δ ¯ g µν = , (95) δ L δφ A = δ ¯ L δ ¯ Φ A = , (96)due to the background field equations (72), (73).Equations (95), (96) point out that the dynamics of physical field perturbations is governed solely by the quadratic,cubic and higher-order polynomial terms in the Lagrangian decomposition (87). We define the dynamic Lagrangianof the dynamic perturbations as follows [79, 85] L dyn ≡ L + L + . . . , (97)so that the total Lagrangian (87) can be written down in the following form L = ¯ L + L + L dyn + L P . (98)The background Lagrangian, ¯ L , does not depend on the dynamic variables, h µν , φ A and θ B which represent thefield perturbations. Hence, the variational derivative from ¯ L taken with respect to any of these variables is identicallyzero, δ ¯ L δ h µν ≡ , δ ¯ L δφ A ≡ . (99)21 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 On the other hand, the variational derivative from L taken with respect to h µν or φ A vanishes on-shell due to thebackground field equations, as evident in (95), (96). Hence, the Lagrangian perturbation theory of dynamic fieldsresiding on the background manifold ¯ M can be built on-shell with the help of the e ff ective Lagrangian L e ff ≡ L dyn + L P . (100)The e ff ective Lagrangian is convenient for deriving the field equations of the physical perturbations and equations ofmotion of matter which are discussed in the rest of the present paper. By definition, the dynamic perturbations of gravitational field are the perturbations h µν of the contravariant com-ponents of the metric tensor density. The field equations for the metric perturbations are obtained after taking thevariational derivative from the total Lagrangian L with respect to h µν , and equating it to zero. Due to equations (95),(96) and (99) this derivative is reduced to that taken from the e ff ective Lagrangian L e ff , δ L e ff δ h µν = . (101)Because of (95), it is equivalent to equation δ (cid:16) L − ¯ L (cid:17) /δ h µν = δ (cid:16) L − ¯ L (cid:17) /δ ¯ g µν = L in this equation with expansion (98) and accounting for the background Einstein equations, δ ¯ L /δ ¯ g µν = − δ L δ ¯ g µν = δ L e ff δ ¯ g µν , (102)where we have used (54) in order to replace the variational derivative with respect to ¯ g µν for that with respect to ¯ g µν .The Euler-Lagrangian equation (102) is equivalent to (101) but more convenient to work with. It is worth emphasizingthat is equivalent on-shell to the first variational equation (94).By taking the variational derivatives one can reduce equation (102) to a more tractable tensor form F G µν + F M µν = π Λ µν , (103)where Λ µν ≡ √− ¯ g δ L e ff δ ¯ g µν , (104)is the e ff ective stress-energy tensor and the left side of (103) is a Laplace-Beltrami operator for tensor field h µν on thebackground manifold [53] that consists of two parts [79, 85] F G µν ≡ − π √− ¯ g δδ ¯ g µν h ρσ δ ¯ L G δ ¯ g ρσ ! , (105) F M µν ≡ − π √− ¯ g δδ ¯ g µν h ρσ δ ¯ L M δ ¯ g ρσ + φ A δ ¯ L M δ ¯ Φ A ! . (106)Operator F G µν describes the linearized perturbation of the Ricci tensor and can be easily calculated on any back-ground manifold. Indeed, taking into account (74), we immediately get F G µν = √− ¯ g δδ ¯ g µν (cid:16) h ρσ ¯ R ρσ (cid:17) . (107)Now, according to the rule of rising and lowering indices of the variational derivatives, we can recast (107) to F G µν = − √− ¯ g ¯ g µχ g νǫ δδ ¯ g χǫ (cid:16) h ργ δ κλ ¯ R λρκγ (cid:17) . (108)Variational derivative in (108) is calculated with the help of the covariant definition (45) where the covariant derivativesare taken on the background manifold ¯ M and are denoted with a vertical bar. We recall that h ργ is an independent22 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 dynamic variable while the term under the sign of the variational derivative in (108) depends merely on the backgroundRiemann tensor without explicit appearance of the Christo ff el symbols. Therefore, the variational derivative in (108)is taken only with respect to the Riemann tensor in accordance with (45). It yields δδ ¯ g χǫ (cid:16) h ργ δ κλ ¯ R λρκγ (cid:17) = h h ργ δ κλ (cid:16) ¯ g σχ δ λσ δ αρ δ [ βκ δ ǫ ] γ + ¯ g σǫ δ λσ δ χρ δ [ βκ δ α ] γ − ¯ g σα δ λσ δ χρ δ [ βκ δ ǫ ] γ (cid:17)i | βα = (cid:16) h α [ ǫ ¯ g β ] χ + h χ [ α ¯ g β ] ǫ − h χ [ ǫ ¯ g β ] α (cid:17) | βα = (cid:16) h αχ ¯ g βǫ + h αǫ ¯ g βχ − h χǫ ¯ g αβ − h αβ ¯ g χǫ (cid:17) | βα , (109)where we have taken into account that the expression enclosed in the brackets, is symmetric with respect to indices α and β . We substitute (109) to (108) and recollect definition of h µν = √− ¯ gl µν along with the constancy of thebackground metric tensor ¯ g µν with respect to the covariant derivative. It results in the di ff erential operator F G µν = (cid:16) l µν | α | α + ¯ g µν l αβ | αβ − l αµ | να − l αν | µα (cid:17) , (110)where each vertical bar denotes a covariant derivative with respect to the background metric ¯ g µν , and l αβ ≡ h αβ / √− ¯ g (the indices are raised and lowered with the background metric ¯ g αβ ). We emphasize that expression (110) is exact .Operator F M µν describes perturbation of the stress-energy tensor ¯ T µν of the background matter governing the on-shell evolution of the background manifold ¯ M . Hence, it vanishes on any Ricci-flat spacetime manifold ( ¯ R µν =
0) ingeneral relativity as a consequence of the background Einstein’s equations (66). Cosmological FLRW spacetime is notRicci flat. Therefore, F M µν makes a non-trivial contribution to the field equations (103) for gravitational perturbations.Variational derivative in definition (106) of F M µν is taken from the Lagrangian, L M , characterizing the backgroundmatter fields ¯ Φ A , and depends crucially on its particular form which must be specified in each individual case ofphysical fields under consideration. We can bring (106) to a more explicit form by accounting for the definition of themetrical stress-energy tensor of the background matter [14]¯ T M µν ≡ √− ¯ g δ ¯ L M δ ¯ g µν , (111)and introducing a new function ¯ I M A ≡ √− ¯ g δ ¯ L M δ ¯ Φ A . (112)We notice that ¯ I M A vanishes on-shell because of the field equation (72). However, this equation should not be appliedimmediately in the definition (106) of F M µν as we, first, have to take the variational derivative with respect to the metrictensor which is o ff -shell operation. With these remarks equation (106) takes on the following form F M µν = − π √− ¯ g δδ ¯ g µν h ρσ ¯ T M ρσ − h ¯ T M + p − ¯ g φ A ¯ I M A ! . (113)We shall calculate (113) later on for an ideal fluid (dark matter) and a scalar field (dark energy) in the case of theFLRW universe governed by dark matter and dark energy.The right side of equation (103) contains the e ff ective stress-energy tensor consisting of two contributions Λ µν = T µν + T µν , (114)where T µν ≡ √− ¯ g δ L P δ ¯ g µν , (115)is the stress-energy tensor of the bare perturbation, and T µν ≡ √− ¯ g δ L dyn δ ¯ g µν , (116)23 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 is the stress-energy tensor associated with the dynamic field perturbations h µν and φ A .It is important to emphasize that stress-energy tensor T µν of the bare perturbations is defined as a variationalderivative with respect to the background metric, ¯ g µν . Hence, it di ff ers from the similar tensor T αβ = √− g δ L P δ g αβ , (117)which was introduced earlier (see (68)) and is defined in terms of the variational derivative with respect to the fullmetric g µν = ¯ g µν + κ µν + ... . These two tensors, T µν and T µν , are closely related. The relation between them can befound by making use of equations ∂ g µν ∂ ¯ g αβ = p − ¯ g " δ ( µα δ ν ) β −
12 ¯ g µν ¯ g αβ , (118) ∂ g ρσ ∂ g µν = √− g " δ ( ρµ δ σ ) ν − g ρσ g µν , (119)and δδ ¯ g αβ = ∂ g µν ∂ ¯ g αβ ∂ g ρσ ∂ g µν δδ g ρσ . (120)It yields an exact relation T µν = T µν − g µν T −
12 ¯ g µν ¯ g αβ T αβ − g αβ T ! , (121)where the trace of the stress energy tensor is defined as T ≡ g αβ T αβ . Relation (121) can be inverted leading to anotherexact formula T µν = T µν −
12 ¯ g µν T − g µν g αβ T αβ −
12 ¯ g αβ T ! , (122)where T = ¯ g αβ T αβ .Tensor T µν can be split in two algebraically-independent parts T µν = t µν + τ µν , (123)where t µν is the stress-energy tensor of pure gravitational perturbations h µν while τ µν is the stress-energy tensor char-acterizing gravitational coupling of the matter field φ A with the gravitational perturbations h µν . For example, in thesecond-order approximation, when L dyn = L , the corresponding stress-energy tensors are given by equations t µν = − π √− ¯ g δδ ¯ g µν h ρσ F G ρσ − h F G ! , (124) τ µν = − π √− ¯ g δδ ¯ g µν h ρσ F M ρσ − h F M + p − ¯ g φ A F M A ! , (125)where F M A is defined below in (131).As soon as the di ff erential operators and the source terms in the field equation (103) are specified, it can be solvedby successive iterations. It requires decomposition of the perturbations h µν and φ a in the post-Friedmanian series h µν = G h µν + G h µν + G h µν + . . . , (126) φ A = G φ A + G φ A + G φ A + . . . , (127)where the terms with indices n = , , , . . . represent the successive approximations of the corresponding order ofmagnitude with respect to the universal gravitational constant G (which we showed in these equations explicitly).These series generalize analogous series in the post-Minkowskian approximation scheme applied to solve Einstein’sequations in asymptotically-flat spacetime [15, 16, 86, 88]. We conjecture that the series (126), (127) are analyticand convergent for a su ffi ciently small magnitude of the perturbations. However, the proof of this conjecture requires24 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 dedicated mathematical e ff orts and a special study which we do not pursue in the present paper because of its enormousmathematical di ffi culty.The post-Friedmannian iteration procedure starts o ff the substitution of the unperturbed values of h µν = φ a = h µν . The solution is substituted back to the right side of (103),which is solved again to find h µν , and so on. In addition to the field equations for the metric tensor perturbations, weneed additional set of di ff erential equations to find out the perturbations of the matter fields φ A . Equations for the background matter field perturbation φ A are derived from the e ff ective Lagrangian (100) bytaking the variational derivative with respect to the dynamic variable φ A . The Lagrangian L P does not depend on φ A and drops out from further calculations. Moreover, we assume that the background field equations (72) and thestationary conditions (96) are satisfied. Thus, the stationarity of the Lagrangian (98) with respect to the perturbations φ A yields δ L dyn δφ A = , (128)which is equivalent to − δ L δ ¯ Φ A = δ L dyn δ ¯ Φ A . (129)After taking the variational derivatives, equation (129) assumes the following form F M A = π Σ M A , (130)where the linear (Laplace-Beltrami) di ff erential operator F M A ≡ − π √− ¯ g δδ ¯ Φ A h µν δ ¯ L M δ ¯ g µν + φ A δ ¯ L M δ ¯ Φ A ! , (131)and the source density Σ M A ≡ √− ¯ g δ L dyn δ ¯ Φ A . (132)All linear with respect to h αβ and φ A terms are included in the left side of equation (130) while the non-linear termshave been put in Σ M A . More explicit form of the operator F M A can be obtained with the help of (111), (112) that resultsin F M A = − π √− ¯ g δδ ¯ Φ A h ρσ ¯ T M ρσ − h ¯ T M + p − ¯ g φ A ¯ I M A ! . (133)Further specification of the operator F M A requires a particular model of the background matter Lagrangian ¯ L M whichwill be discussed in section 5.Field equations for bare perturbations, Θ B , are obtained after taking the variational derivative from the Lagrangian(100) with respect to the variable Θ B . Because the only part of the Lagrangian which depends on this field, is L P , theequations are reduced to δ L P δ Θ B = . (134)Particular form of this equation depends on a specific choice of the Lagrangian L P of the bare perturbation. In thelowest order of approximation the field equations (130) and (134) describe evolution of the dynamic perturbations φ A and Θ B on the unperturbed cosmological background. The next-order approximations take into account the backreaction of the background perturbations on these fields. 25 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 Gauge invariance of the dynamic perturbations is an important geometric property that allows us to distinguishphysical degrees of freedom of gravitational and matter fields from the spurious modes generated by transformationsof the local coordinates on manifold. Any self-consistent perturbation theory must clearly separate the coordinate-dependent e ff ects from physical perturbations which do not depend on the choice of coordinates. The gauge trans-formation is generated by the exponential mapping of spacetime manifold to itself, M → M , that is induced by anon-singular vector flow having a tangent vector ξ α ≡ ξ β ( x α ) associated with a finite translation of each point of themanifold x ′ α = exp (cid:16) ξ β ∂ β (cid:17) x α = x α + ξ α + ξ β ∂ β ξ α + . . . . (135)It drags the coordinate grid on manifold along the vector field ξ α , and makes a point-wise change of any geometricobject F ( x α ) to F ′ ( x ′ α ). The transformed object F ′ ( x ′ α ) is, then, pulled back to its value F ′ ( x α ) taken at the point onthe manifold having the same coordinates. It defines the gauge transformation of F that is found to be an exponentialLie transform [79, 85], F ′ ( x α ) = (cid:16) exp £ ξ (cid:17) F ( x α ) = F ( x α ) + £ ξ F ( x α ) +
12! £ ξ F ( x α ) + . . . , (136)where the Lie derivative £ ξ F has been defined in equation (61), £ ξ ≡ £ ξ £ ξ , £ ξ ≡ £ ξ £ ξ £ ξ , and so on.For the gauge transformation of a geometric object is generated by the change of coordinates it has no real physicalmeaning and should be considered as spurious. The gauge freedom should be carefully studied in order to eliminatethe non-physical degrees of freedom. The gauge transformation of the metric tensor, g µν , and the matter fields, Φ A , Θ B , leads to appearance of the gauge-dependent perturbations which imply that the background values of g µν , Φ A , Θ B do not change under the gauge transformation – only the dynamic perturbations, h µν , φ A , θ B change. Hence, the gaugetransformation (136) applied to these fields induces the following gauge transformations of the perturbations of thesefields, h ′ µν = h µν + (cid:16) exp £ ξ − (cid:17) (¯ g µν + h µν ) , (137) φ ′ A = φ A + (cid:16) exp £ ξ − (cid:17) (cid:16) ¯ Φ A + φ A (cid:17) , (138) θ ′ B = θ B + (cid:16) exp £ ξ − (cid:17) (cid:16) ¯ Θ B + θ B (cid:17) , (139)that depend on the gauge vector field ξ α .Let us consider the gauge transformation of the total Lagrangian (79) induced by the gauge transformations of itsarguments. The transformed Lagrangian L ′ has the same functional form as L but depends now on the transformed(denoted with a prime) values of the dynamic variables, L ′ ≡ L (cid:16) ¯ g µν + h ′ µν , ¯ Φ A + φ ′ A , ¯ Θ B + θ ′ B (cid:17) . We replace thetransformed variables with their original values by making use of equations (137)–(139). It yields L ′ = L h exp £ ξ (cid:16) ¯ g µν + h µν (cid:17) , exp £ ξ (cid:16) ¯ Φ A + φ A (cid:17) , exp £ ξ (cid:16) ¯ Θ B + θ B (cid:17)i . (140)This equation can be further transformed by making use of the following relation [79] L h exp £ ξ (cid:16) ¯ g µν + h µν (cid:17) , exp £ ξ (cid:16) ¯ Φ A + φ A (cid:17) , exp £ ξ (cid:16) ¯ Θ B + θ B (cid:17)i = exp £ ξ h L (cid:16) ¯ g µν + h µν , ¯ Φ A + φ A , ¯ Θ B + θ B (cid:17)i , (141)that is valid modulo total divergence which is inessential in the variational calculus. We expand the right side of (141)in a Taylor series, like in (136), and take into account that the Lagrangian is a scalar density of weight + ξ L = ∂ α ( ξ α L ). It eventually yields the gauge transformation of the Lagrangian in the followingform L ′ = L + ∂ α (cid:16) ξ α L (cid:17) + ∂ α (cid:16) ξ α ∂ β (cid:16) ξ β L (cid:17)(cid:17) + ∂ α (cid:16) ξ α ∂ β (cid:16) ξ β ∂ γ (cid:16) ξ γ L (cid:17)(cid:17)(cid:17) + . . . , (142)where the second, third, and all other terms in the right side of this infinite series represent a divergence. The di-vergence vanishes when one takes the variational derivative from it and, hence, it can be omitted from the action26 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 functional S given in (78). The conclusion is that the action S and the Lagrangian (84) are gauge-invariant withrespect to the gauge transformation of their arguments. This assertion does not involve any background equations ofmotion and / or field equations and, thus, is valid both on-shell and o ff -shell.On the other hand, the e ff ective Lagrangian (100) is gauge-invariant only on-shell that is only when the backgroundfield equations (72), (73) are satisfied. Indeed, the e ff ective Lagrangian can be represented as a di ff erence L e ff = L − L − ¯ L . After making the gauge transformations (137)–(139) of the dynamic variables, we get a new e ff ectiveLagrangian L ′ e ff = L ′ − L ′ − ¯ L where the background value of the Lagrangian stays the same. The di ff erence δ L e ff = L ′ e ff − L e ff is δ L e ff = δ L + (cid:16) exp £ ξ − (cid:17) " (¯ g µν + h µν ) δ ¯ L δ ¯ g µν + (cid:16) ¯ Φ A + φ A (cid:17) δ ¯ L δ ¯ Φ A , (143)where the terms being enclosed in the square brackets, vanish on-shell due to the background field equations (72),(73). Therefore, δ L e ff = δ L which is a total divergence as follows from (142). Hence, L e ff it is gauge-invarianton-shell.The gauge invariance of the Lagrangian suggests that the Einstein equations (103) for metric perturbations aregauge invariant as well. It is straightforward to prove it by direct but otherwise tedious calculation which technicaldetails are given in [79]. Gauge transformations (137)–(139) applied to the Einstein equations (103) transform themas follows F ′ G µν + F ′ M µν − π Λ ′ µν = F G µν + F M µν − π Λ µν + exp £ ξ F , (144)where function F vanishes on-shell due to the background field equations (72), (73). Therefore, if the field equationsfor gravitational perturbations are valid at least in one gauge, they are valid in any other gauge as well. We havechecked that the field equations (130) for the matter perturbations are also gauge-invariant.
5. The Dynamic Field Theory in Cosmology
We shall implement the formalism of the dynamic field theory in cosmology to derive the field equations forcosmological perturbations of gravitational field and matter. We shall rely in our analysis upon the cosmologicalmodel that is in the most close agreement with modern observational data. In this model the background manifoldrepresents the spatially homogeneous and isotropic FLRW universe which temporal evolution is governed by an idealfluid with an arbitrary equations of state and a scalar field with an arbitrary potential. The ideal fluid models a self-interacting dark matter [89] while the scalar field describes dark energy in the form of quintessence [90]. The darkmatter without self-interaction is included in our theoretical scheme as a pressureless ideal fluid. The dynamic fieldvariables of the dark matter and dark energy are two scalar fields, Φ and Φ which form a doublet, Φ A = { Φ , Φ } .We identify the scalar field Φ with the (scalar) Clebsch potential Φ of the ideal fluid, and Φ with a scalar field Ψ having the potential W = W ( Ψ ) depending only on the scalar field Ψ . The third matter component in our model isthe baryonic matter making stars, galaxies, etc. as well as neutrino. The baryonic matter and neutrino make up asmall fraction ( ≃ Θ B ≡ Θ describing this perturbation. This model may bestill too simple to describe the real universe but it nicely demonstrates the richness and flexibility of the formalism ofthe dynamic field theory in doing cosmological applications without involving too many secondary details.The overall Lagrangian of the cosmological model under consideration is given by equation (79) with the La-grangian of the background matter consisting of two non-directly interacting pieces L M = L m + L q , (145)where L m is the Lagrangian of dark matter, and L q is the Lagrangian of dark energy. The Lagrangian of the baryonicmatter perturbation is L P . We describe the specific structure of the Lagrangians in next sections. Dark matter is modelled as an ideal fluid that is characterized by four thermodynamic variables [72]: the rest-massdensity ρ m , the specific internal energy per unit mass Π m , pressure p m , and entropy per unit mass s m , where the sub-index ’m’ stands for the dark matter. We shall assume that the entropy of the ideal fluid remains constant, s m = . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 and dissipative processes are neglected (isentropic motion). This assumption can be relaxed by adding some otherthermodynamic variables [91] but we do not discuss this extension in the present paper.The total energy density of the ideal fluid is ǫ m = ρ m (1 + Π m ) . (146)A physically meaningful thermodynamic variable is formed from the energy density, pressure and the rest-mass den-sity. It is called the specific enthalpy of fluid, µ m , and defined as [72] µ m ≡ ǫ m + p m ρ m = + Π m + p m ρ m . (147)We shall consider a barotropic fluid which thermodynamic equation of state is given by equation p m = p m ( ρ m , Π m ),where the specific internal energy Π m is related to pressure and the rest-mass density by the first law of thermodynam-ics d Π m + p m d ρ m ! = . (148)This equation along with the definition of the specific enthalpy and the energy density given above, allow us to derivethe following di ff erential relations d p m = ρ m d µ m , (149) d ǫ m = µ m d ρ m . (150)which immediately tells us that the partial derivatives ∂ p m ∂µ m = ρ m , (151a) ∂ǫ m ∂ρ m = µ m . (151b)Equations (151) elucidate that all thermodynamic quantities are functions of only one thermodynamic variable. Forthe reasons which are explained below, we accept that this variable is the specific enthalpy µ m . The equation of state,relating pressure and the energy density, becomes p m = p m ( ǫ m ), and it is also an implicit, single-valued function ofthe thermodynamic variable µ m because ǫ m = ǫ m ( µ m ).Partial derivatives of the thermodynamic quantities with respect to µ m can be calculated by making use of (149),(150), the equation of state p m = p m ( ǫ m ), and definition of the (adiabatic) speed of sound c s propagating in the fluid ∂ p m ∂ǫ m = c c , (152)where the partial derivative is taken under a condition that the entropy, s m , does not change. Notice that the speed ofsound in dark matter is not constant in the most general case of a non-linear equation of state. In this case, the speedof sound depends on the thermodynamic potential µ m through the equation of state, that is c s = c s ( µ s ). It is also worthemphasizing that the speed-of-sound-defining equation (152) is valid for any wavelength of sound waves in the idealfluid, not only for short wavelengths. In cosmology, the equation of state of dark matter is postulated as having thefollowing form p m = w m ǫ m , (153)where w m is an implicit function of the specific enthalpy, w m = w m ( µ m ). Taking a partial derivative from both sides ofequation (153) with respect to µ m and making use of (152) yield ∂ w m ∂µ m = − c c w m − c / c + Π m , (154)which is naturally reduced to w m = c / c in case of the constant parameter w m of the cosmological equation of state.However, in more general cosmological studies w m is not constant and changes as the universe evolves.28 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 Other partial derivatives of the thermodynamic quantities can be calculated with the help of the equation of stateand (151), (152) which can be inverted, if necessary, because all thermodynamic relations in the ideal fluid are single-valued. We have, for example, ∂ǫ m ∂µ m = c c ρ m , ∂ρ m ∂µ m = c c ρ m µ m , (155)where all partial derivatives are performed under the same condition of the constant entropy.Theoretical description of the ideal fluid as a dynamic field system evolving on space-time manifold is given themost conveniently in terms of the Clebsch scalar potential [92, 93], Φ which is also known either as a velocity potentialor the Taub potential [91]. The Clebsch potential is a scalar function on spacetime manifold which can be taken as anindependent dynamic variable characterizing dynamic evolution of the ideal fluid. This description is complimentary(dual) to the Lagrangian formulation of the ideal fluid based on the coordinates and four-velocity of the fluid particles[19, 94, 95] which is more familiar in the field of cosmology. Nonetheless, the description of the ideal fluid (darkmatter) in terms of the Clebsch vector field Φ and its derivative Φ | α considered as independent dynamic variables,makes it very similar to the description of dark energy also given in terms of (another) scalar field. It allows us toconsider physical e ff ects of dark matter and dark energy on the same fundamental level of the Lagrangian formalism.It seems the first researcher who realized the advantages of using the Clebsch potential description of the ideal fluidin cosmology, was V. N. Lukash [96].In the case of a single-component fluid the Clebsch potential Φ is introduced by the following relationship µ m w α = − Φ α , (156)where w α = dx α / d τ is the four-velocity of the fluid, w α = g αβ w β , τ is the proper time of the fluid element takenalong its world line, and we denote Φ α ≡ Φ ,α = Φ | α from now on. This type of representation of fluid’s velocity hasbeen introduced by A. Clebsch [97]. Equation (156) solves the relativistic Euler equation of motion of the ideal fluidwhich justifies the connection between the specific enthalpy, four-velocity and the Clebsch potential Φ α [92, 95]. Thefour-velocity is normalized, g αβ w α w β = −
1, so that the specific enthalpy can be expressed in terms of the the metrictensor and the derivative from the Clebsch potential, µ m = q − g αβ Φ α Φ β . (157)One may also notice that the normalization condition for the four-velocity allows us to re-write (156) in the followingform, µ m = w α Φ α . (158)The Clebsch potential Φ has no direct physical meaning as it can be changed to another value: Φ → Φ ′ = Φ + ˜ Φ such that the gauge function, ˜ Φ , is constant along the worldlines of the fluid in the sense that w α ˜ Φ α =
0. This gaugetransformation of the Clebsch potential does not change the value of the specific enthalpy µ m .The Lagrangian of the ideal fluid is usually taken in the form of the total energy density, L m = √− g ǫ m [92, 94, 95].However, this form of the Lagrangian implicitly assumes that the equation of continuity is valid and has been usedas a constraint in the form of the Lagrange multiplier [98] so that the rest mass density ρ m of the fluid is solely anexplicit function of the metric tensor g αβ . The equation of continuity is used, then, to derive the variational derivativeof the rest mass density of the fluid [19, 94, 95]. This way, however, becomes coordinate dependent as it relies uponusing coordinates and velocities of the fluid particles for doing variational analysis. It prevents us from making use ofthe full power of the dynamic field theory on the manifolds because coordinates and velocities are not field variables.An attempt to use the fluid density ρ m as a dynamic variable is not satisfactory because ρ m has no a correspondingconjugated counterpart as contrasted to the Clebsch potential, Φ , and its derivative, Φ | α , which are truly independentpair of canonically-conjugated dynamic variables on manifold. We avoid the approach based on the Lagrangian L m = √− g ǫ m by taking the Lagrangian of the ideal fluid in the form of pressure, L m = − √− gp m , and demand thatthermodynamic equations like (151a), (151b) are valid. This allows us to treat all thermodynamic quantities enteringthe Lagrangian as single-valued explicit functions of the specific enthalpy µ m . Any dependence on the metric tensorin this treatment of the ideal fluid is only through the specific enthalpy as given in (157). The equation of continuityis not a priory imposed on the dynamic system but is a consequence of the Euler-Lagrange equation for the Clebschpotential Φ considered as a dynamic variable (more details in [14, pp. 334-336]).29 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 The Lagrangian in the form of pressure di ff ers from the Lagrangian in the form of energy by a total divergence[14, pp. 334-335 ]. The Lagrangian of the ideal fluid in the form of pressure is L m = √− g ( ǫ m − ρ m µ m ) , (159)where ǫ m = ǫ m ( µ m ) and ρ m = ρ m ( µ m ) are functions of the specific enthalpy µ m = p − g αβ Φ α Φ β . It is important tonotice that the Lagrangian of dark matter is a singled-valued function of µ m and depends only on the derivative Φ α ofthe Clebsch potential. There is no explicit dependence on the field Φ whatsoever. It could appear if the ideal fluid hadsome special kind of potential interaction between the fluid’s particles like in plasma which is not electrically neutral.However, we exclude such type of fluids from further consideration.The metrical stress-energy tensor of the ideal fluid is obtained by taking a variational derivative of the Lagrangian(159) with respect to the metric tensor, T m αβ = √− g δ L m δ g αβ . (160)In our field-theoretical description of the ideal fluid the metric tensor enters all thermodynamic quantities only throughthe specific enthalpy in the form of equation (157). Therefore, taking the variational derivative in (160) with respectto the metric tensor can be done with the help of the chain rule δ L m δ g αβ = ∂ L m ∂µ m δµ m δ g αβ , (161)where the variational derivative from µ m is given in (A17) of Appendix A.2.1. Calculation shows that the stress-energytensor (160) is as follows, T m αβ = ( ǫ m + p m ) w α w β + p m g αβ , (162)which is just the standard form of the stress-energy tensor of the ideal fluid [19, 75]. Many studies in cosmology andgeneral relativity take the stress-energy tensor (162) as a starting point. However, the dynamic field theory disclosesthat there is more deep underlying structure - the Clebsch potential which drastically simplifies theoretical analysis ofhydrodynamic behaviour of the ideal fluid. The Lagrangian of dark energy is taken in the form of a quintessence of a scalar field Ψ [3, 84] L q = √− g g αβ Ψ α Ψ β + W ! , (163)where W ≡ W ( Ψ ) is the scalar field potential, and we denote the partial derivative of the field, Ψ α ≡ Ψ ,α = Ψ | α from now on. We assume that there is no direct coupling between the Lagrangian of dark energy and that of darkmatter. They interact only indirectly through the gravitational field. Many various forms of the potential W are usedin cosmology [5, 84] but at the present paper we do not need to specify it further on, and keep it arbitrary. The scalarfield Ψ does not admit the gauge transformation like that of the Clebsch potential Φ for the ideal fluid. The reason isthat the quintessence scalar field has a potential W ( Ψ ) which is not gauge invariant. This makes the true scalar fielddi ff erent from the Clebsch potential.The metrical stress-energy tensor of the scalar field is obtained by taking a variational derivative T q αβ = √− g δ L q δ g αβ , (164)that yields T q αβ = Ψ α Ψ β − g αβ (cid:20) g µν Ψ µ Ψ ν + W ( Ψ ) (cid:21) . (165)We can formally reduce tensor (165) to the form being similar to that of the ideal fluid by making use of the followingprocedure. First, we define the analogue of the specific enthalpy of the quintessence ”fluid” µ q = q − g αβ Ψ α Ψ β , (166)30 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 and the e ff ective four-velocity, v α , of the ”fluid” µ q v α = − Ψ α . (167)The four-velocity v α is normalized to g αβ v α v β = −
1. Therefore, the scalar field enthalpy µ q can be expressed in termsof the partial derivative from the scalar field µ q = v α Ψ α . (168)We introduce the analogue of the rest mass density ρ q of the quintessence ”fluid” by identification, ρ q = µ q . (169)As a consequence of the above definitions, the energy density, ǫ q and pressure p q of the quintessence ”fluid” can beintroduced as follows ǫ q ≡ − g αβ Ψ α Ψ β + W ( Ψ ) = ρ q µ q + W ( Ψ ) , (170) p q ≡ − g αβ Ψ α Ψ β − W ( Ψ ) = ρ q µ q − W ( Ψ ) . (171)We notice that relation µ q = ǫ q + p q ρ q , (172)between the specific enthalpy µ q , density ρ q , pressure p q and the energy density, ǫ q , of the scalar field ”fluid” formallyholds on the same form (147) as in the case of the barotropic ideal fluid.After substituting the above-given definitions of various “thermodynamic” quantities into equation (165), it for-mally reduces to the stress-energy tensor of an ideal “fluid” T q αβ = (cid:16) ǫ q + p q (cid:17) v α v β + p q g αβ . (173)It is worth emphasizing that the analogy between the stress-energy tensor (173) of the scalar field ”fluid” with thatof the barotropic ideal fluid (162) 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 ( Ψ ) in the energy density ǫ q , andpressure p q of the scalar field. The dark energy in the form of quintessence is physically di ff erent from dark matter inthe form of the ideal fluid! In particular, the “speed of sound” of the quintessence “fluid” is always equal to the speedof light c independently of the equation of state of the quintessence, p q = w q ǫ q , where parameter w q = ρ q µ q − W ( Ψ )12 ρ q µ q + W ( Ψ ) , (174)and it can take the values in the range from − + W [84]. The Lagrangian L P of the baryonic matter represents a bare perturbation of the cosmological manifold. It entersthe total Lagrangian (79) and can be chosen in accordance with the specific problem we want to solve. We keep itunspecified as long as the theory permit. We assume that the baryonic matter of the bare perturbation is described bydynamic fields Θ B which geometric nature depends on the type of the baryonic matter. In what follows, we shall omitindex B from the baryonic fields to simplify notations as it does not lead to confusion. Metrical stress-energy tensor ofthe baryonic matter, T αβ , has been defined in terms of the variational derivative in (117). Tensor T αβ is a source of thebare gravitational perturbation of the background manifold which generates the small-scale structures in the universe.A particularly familiar form of the stress-energy tensor of the baryonic matter is given by that of the ideal fluid T αβ = ( ǫ + p ) u α u β + pg αβ , (175)31 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where ǫ , p are the energy density and pressure of the fluid comprising the bare perturbation, and u α is its four-velocitynormalized to g αβ u α u β = −
1. It is worth emphasizing that the four-velocity u α of the baryonic matter has a peculiarcomponent and di ff ers from the velocity of the Hubble flow (see below). Notice that the stress-energy tensor T µν ofthe baryonic matter defined in (114), is not fully identical with T µν . We have derived relation between the two tensorsin (121) and (122). All geometric objects on background cosmological manifold ¯ M will be denoted with a bar over the object. TheFLRW metric on the background manifold is given in (67). It is convenient to introduce global isotropic coordinates X α = ( X , X i ) by changing the cosmic time T to the conformal time η ≡ X via di ff erential equation dT = a ( η ) d η ,where a cosmological scale factor a ( η ) ≡ R [ T ( η )]. The FLRW metric tensor in the isotropic coordinates reads [4]¯ g µν = a ( η ) f µν (176)where f µν = ( − , f i j ), and f i j = + kr ! − δ i j , (177)depends on the curvature of the spatial hypersurfaces, k = {− , , + } . In case, k =
0, the metric f µν is reduced to theMinkowski metric, η µν so that the physical metric g µν is confromally-flat. In fact, FLRW metric g µν is conformally-flatin any case but the conformal factor is not reduced to a ( η ) but is given a more complicated function of time and space.This question is discussed in more detail in an excellent article by M. Ibison [99]. Congruence of world lines offreely-falling particles which have constant spatial coordinates makes up the Hubble flow. Four-velocity of each sucha particle in the isotropic coordinates is ¯ U α = dX α / dT = ( a − , , , ffi neconnection, etc.) when expressed in the isotropic coordinates, depend only on time X = η but do not depend onspatial coordinates X i . Nonetheless, we can, and will, use arbitrary coordinates x α = ( x , x i ) on the manifold whichare connected to the isotropic coordinates X α by di ff eomorphism x α = x α ( X β ). Partial coordinate derivative of abackground geometric object, ¯ F = ¯ F ( η ), in the arbitrary coordinates is given by¯ F ,α = − ¯ F ′ a ¯ u α = − ˙¯ F ¯ u α , (178)where ¯ u α is four-velocity of the Hubble flow in the arbitrary coordinates, ¯ F ′ = d ¯ F / d η , and ˙¯ F ≡ d ¯ F / dT . Equation(178) applied to the scale factor, yields a ,α = − ˙ a ¯ u α = −H ¯ u α , and the partial derivative from the conformal Hubbleparameter H ,α = − ˙ H ¯ u α . These expressions for the partial derivatives are very useful in calculations.Einstein’s field equations on the background cosmological manifold with FLRW metric are given by (73)-(75).After substitution FLRW metric to these equations they yield two Friedmann equations describing the temporal evo-lution of the scale factor a , H = π ǫ − ka , (179)2 ˙ H + H = − π ¯ p − ka (180)where ¯ ǫ and ¯ p are the e ff ective energy density, ¯ ǫ = ¯ ǫ m + ¯ ǫ q , and pressure, ¯ p = ¯ p m + ¯ p q , of the background dark matterand dark energy. A consequence of the Friedmann equations (179), (180) is equation˙ H = − π (¯ ǫ + ¯ p ) + ka , (181)that relates the time derivative of the Hubble parameter to the sum of the overall energy density and pressure of darkmatter and dark energy ¯ ǫ + ¯ p = ¯ ρ m ¯ µ m + ¯ ρ q ¯ µ q . (182)32 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 Equation of continuity for the rest mass density ¯ ρ m of the background dark matter is given by (72) where we haveto make a replacement, L M → L m and ¯ Φ A → ¯ Φ for the background value of the Clebsch potential. The equationreads ( ¯ ρ m ¯ u α ) | α = , (183)that is equivalent to ¯ ρ m | α − H ¯ ρ m ¯ u α = . (184)The background equation of the conservation of the energy density ǫ m of dark matter is derived from its definition(146), the law of conservation of thermal energy (148), and the continuity equation (184). It yields,¯ ǫ m | α − H (¯ ǫ m + ¯ p m ) ¯ u α = . (185)Background equation for the evolution of the dark energy is also given by the Euler-Lagrange equation (72) afterreplacements L M → L q and ¯ Φ A → ¯ Ψ . It reads ¯ g αβ ¯ Ψ | αβ − ∂ ¯ W ∂ ¯ Ψ = . (186)After making use of definition of the background specific enthalpy of the scalar field ¯ µ q ≡ ¯ u α ¯ Ψ | α , an equality ¯ µ q = ¯ ρ q ,and definition (170) of the specific energy ¯ ǫ q of the scalar field, equation (186) can be recast to¯ ǫ q | α − H (cid:16) ¯ ǫ q + ¯ p q (cid:17) ¯ u α = , (187)that is completely similar to the hydrodynamic equation (185) of conservation of the energy density of dark matter.Because of this similarity, the second Friedmann equation (180) is not really independent, and can be derived directlyfrom the first Friedmann equation (179) by taking a time derivative and applying the energy conservation equations(185) and (187) to simplify the result.Equation of continuity for the density of dark energy, ¯ ρ q , is obtained by di ff erentiating definition (169) of ¯ ρ q , andmaking use of (186). It yields (cid:16) ¯ ρ q ¯ u α (cid:17) | α = − ∂ ¯ W ∂ ¯ Ψ , (188)or, equivalently, ¯ ρ q | α − H ¯ ρ q ¯ u α = ∂ ¯ W ∂ ¯ Ψ ¯ u α , (189)which shows that the density ¯ ρ q is not conserved. This fact again points out that the similarity of the scalar field andan ideal fluid is not complete. Dark energy is not thermodynamically equivalent to dark matter. Only if the scalar field¯ Ψ is potential-free, the quintessence can be treated as an ideal fluid. We should emphasize that non-conservation ofthe density ¯ ρ q does not violate any physical law since (189) is simply another way of writing the evolution equation(186) for dark energy. In the present paper, FLRW background manifold is defined by the metric ¯ g αβ which dynamics is governed by thetwo scalar fields - the Clebsch potential ¯ Φ of dark matter and the scalar field ¯ Ψ of dark energy. We assume that thebackground metric and the fields are perturbed intrinsically (the primordial perturbations) and extrinsically (the bareperturbations) by the presence of baryonic matter described by the field Θ . The perturbed metric and the matter fieldscan be split in their background values and the corresponding perturbations, g αβ = ¯ g αβ + h αβ , Φ = ¯ Φ + φ ,
Ψ = ¯ Ψ + ψ . (190)These equations are exact. Equation for the perturbation of the metric tensor g αβ = ¯ g αβ + κ αβ (191)33 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 will be also treated as exact. Corresponding perturbation of the contravariant component of the metric is not indepen-dent and is determined from the isomorphism g αγ g γβ = ¯ g αγ ¯ g γβ = δ βα , yielding g αβ = ¯ g αβ − κ αβ + κ αγ κ γβ + . . . , (192)where the ellipses denote terms of the higher order.We consider perturbation of the metric - κ αβ , that of the potential of dark matter - φ , and that of the potential ofdark energy - ψ as weak with respect to their corresponding background values ¯ g αβ , ¯ Φ , and ¯ Ψ , which dynamics isgoverned by equations that have been explained in section 5.4. Perturbations κ αβ , φ , and ψ have the same order ofmagnitude as Θ . Calculations also prompt us to single out √− ¯ g from h αβ , and operate with a variable l αβ ≡ h αβ √− ¯ g . (193)Tensor indices of the metric tensor perturbations, l αβ , h αβ , etc., are raised and lowered with the help of the backgroundmetric, for example, l αβ ≡ ¯ g αµ ¯ g βν l µν . The field variable l αβ relates to the perturbation κ αβ of the metric tensor asfollows l αβ = − κ αβ +
12 ¯ g αβ κ + κ γα κ βγ − κ αβ κ −
14 ¯ g αβ κ µν κ µν − κ ! + . . . , (194)where κ ≡ κ σσ = ¯ g ρσ κ ρσ , and ellipses denote terms of the higher orders in κ αβ .Perturbations of four-velocities, w α and v α , entering definitions of the stress-energy tensors (162), (173), are fullydetermined by the perturbations of the metric and the potentials of dark matter and dark energy. Indeed, according todefinitions (156) and (167) the four-velocities are defined by the following equations w α = − Φ α µ m , v α = − Ψ α µ q . (195)where µ m and µ q are given by (157) and (166) respectively. We define perturbations δ w α and δ v α of the covariantcomponents of the four-velocities as follows w α = ¯ u α + δ w α , v α = ¯ u α + δ v α , (196)where the unperturbed values of the four-velocities coincide and are equal to the four-velocity of the Hubble flow dueto the requirement of the homogeneity and isotropy of the background FLRW spacetime, that is ¯ w α = ¯ v α = ¯ u α . Hence,we have ¯ u α = − ¯ Φ α ¯ µ m , ¯ u α = − ¯ Ψ α ¯ µ q . (197)Making use of (195) and (197) in the left side of definitions (195), and expanding its right side by making useof expansions (191) and (192), yield relation between the four-velocity perturbations and the perturbations of thedynamic field variables δ w α = − µ m ¯ P βα φ β − q ¯ u α , δ v α = − µ q ¯ P βα ψ β − q ¯ u α , (198)where φ β ≡ φ | β , ψ β ≡ ψ | β , ¯ P αβ = ≡ ¯ g αβ + ¯ u α ¯ u β is a projector tensor onto the hypersurface orthogonal to the Hubbleflow, and q ≡ − ¯ u α ¯ u β κ αβ = ¯ u α ¯ u β l αβ + l , (199)is a scalar-type projection of the metric tensor perturbation on the Hubble flow ( l ≡ ¯ g αβ l αβ ). Equations (198) are validin linear approximations. Higher-order corrections can be obtained by the same procedure by keeping more terms inthe expansions. 34 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 Field equations for metric tensor perturbation are given by the Euler-Lagrange equations (103) where F G µν is givenby exact expression (110), and operator F M µν is a linear superposition of two pieces F M µν = F m µν + F q µν , (200)corresponding to dark matter (index ’m’) and dark energy (index ’q’). These pieces are defined in accordance with(106) that is F m µν ≡ − π √− ¯ g δδ ¯ g µν h ρσ δ ¯ L m δ ¯ g ρσ + φ δ ¯ L m δ ¯ Φ ! , (201) F q µν ≡ − π √− ¯ g δδ ¯ g µν h ρσ δ ¯ L q δ ¯ g ρσ + ψ δ ¯ L q δ ¯ Ψ ! . (202)Calculation of variational derivatives from various functions entering L m and L q is straightforward and follows fromtheir definitions, the chain rule (58), and a set of variational derivatives from thermodynamic quantities given inAppendix A.2.Making use of the Lagrangian’s definitions (159) and (163) taken on the background manifold and calculatingvariational derivatives in (201), (202), we obtain F m µν = − π ( ¯ p m − ¯ ǫ m ) l µν + π ¯ ρ m (cid:16) ¯ u µ φ ν + ¯ u ν φ µ − ¯ g µν ¯ u α φ α (cid:17) (203) + π ¯ ρ m − c c ! ¯ u α φ α −
12 ¯ µ m q ! ¯ u µ ¯ u ν , F q µν = − π (cid:16) p q − ǫ q (cid:17) l µν + π ¯ ρ q (cid:16) ¯ u µ ψ ν + ¯ u ν ψ µ − ¯ g µν ¯ u α ψ α (cid:17) + π ¯ g µν ∂ ¯ W ∂ ¯ Ψ ψ , (204)where ¯ ρ q = µ q ≡ ˙¯ Ψ / a in accordance with definition (169) projected on the background manifold. The dark energypotential function, ¯ W = ¯ W ( ¯ Ψ ), is arbitrary. We emphasize that expressions (203), (204) are exact .Substituting (203), (204) along with (110) to the left side of (103) yields the field equations for gravitationalperturbations l αβ in a covariant form [53, Eq. 161] l µν | α | α + ¯ g µν A α | α − A ( µ | ν ) − R α ( µ l ν ) α − R µαβν l αβ + (cid:16) F m µν + F q µν (cid:17) = π Λ µν , (205)where A α ≡ l αβ | β is the gauge vector function. This form of the field equation for gravitational perturbation l αβ ofthe background FLRW manifold is exact , gauge-invariant and covariant. The left side of (205) contains only linearterms while all quadratic, cubic, etc. perturbations are included in its right side to Λ µν which also contains the stress-energy tensor T µν of the baryonic matter (the bare perturbation). The linear operator in the left side of (205) is rathercomplicated but it can be significantly simplified by choosing a gauge condition imposed on the variable A α ≡ l αβ | β inthe following form [53] A α = − Hl αβ ¯ u β + π (cid:16) ¯ ρ m φ + ¯ ρ q ψ (cid:17) ¯ u α . (206)This gauge condition is analogous to the de Donder (harmonic) gauge condition that is frequently used in the approxi-mation schemes of solving Einstein’s equations in asymptotically flat spacetime [14, 18]. Equation (206) extrapolatesthe harmonic gauge condition to the realm of cosmological FLRW spacetime.This gauge condition (206) cancels a large number of terms in the field equations (205) and allows us to decouplethe field equations for di ff erent components of the metric tensor perturbation, l αβ . Picking up the isotropic coordinates35 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 of the Hubble observers we bring the gravity field equations to the following form [53] (cid:3) q + Hq , + kq − π − c c ! ¯ ρ m ¯ µ m q = π ( Λ + Λ kk ) − π a ¯ ρ m − c c ! φ − (207a)16 π a ∂ ¯ W ∂ ¯ Ψ ψ + π aH (cid:16) ¯ ρ m φ + ¯ ρ q ψ (cid:17) , (cid:3) l i + Hl i , + kl i = π Λ i , (207b) (cid:3) l < i j > + Hl < i j >, + (cid:16) ˙ H − k (cid:17) l < i j > = π Λ < i j > , (207c) (cid:3) l + Hl , + (cid:16) ˙ H + k (cid:17) l = π Λ kk . (207d)where we denoted the wave operators (cid:3) q ≡ f µν g ; µν and (cid:3) l µν ≡ ¯ f αβ l µν ; αβ . Other notations in (207) are φ ≡ φ , , q ≡ ( l + l kk ) / l ≡ l kk = l + l + l , l < i j > = l i j − (1 / δ i j l , and the same index notations are applied to the e ff ectivestress-energy tensor Λ kk = Λ + Λ + Λ , Λ < i j > = Λ i j − (1 / δ i j Λ kk .Solution of the linearised gravitational field equation (205) (and (207)) consists of two parts - a general solution, l H µν , of homogeneous equation (205) with the source Λ µν =
0, and a particular solution, l P µν of inhomogeneous equation(205) with the source Λ µν ,
0. They form a linear superposition l µν = l H µν + l P µν , (208)which is crucial for understanding the physical e ff ects of cosmological perturbations. The homogeneous solution, l H µν , is not trivial but associated with the primordial cosmological perturbations originating at the Big Bang [4, 5].This perturbation dominates on the horizon and super-horizon scales, and its gauge-invariant scalar part (which exactdefinition and equations are given in [53, section 7]) evolves over time to form the large-scale structure of the universegoverned by dark matter. The tensor part of the homogeneous solution represents relic gravitational waves. Theparticular solution, l P µν , represents gravitational perturbations produced by the small-scale structures in the universeconsisting of baryonic matter. We notice that l H µν and l P µν correspond to the long wavelength and short wavelengthperturbations introduced by Green and Wald [45], and denoted in their paper as γ ( L ) µν and h ( S ) µν respectively (see [45,eq. 70]). In what follows, we operate with a single value of the perturbation l µν without substituting the explicitdecomposition (208) into subsequent formulas as it was not a primary goal of the present paper. Decomposition (208)is required for discussion the problem of averaging, back-reaction and precise definition of the Newtonian limit incosmology [45, 46]. Evolution of dark matter perturbation is described by the perturbation φ of the Clebsch potential. Equation for φ is derived from a general equation (130) is, in case of dark matter, reads F m Φ = π Σ m . (209)where all terms can now be explicitly written down because the Lagrangian of dark matter is fully determined by(159). The linear di ff erential operator F m Φ is derived from (133) which is split in two independent parts for dark matterand dark energy. The dark matter part reads F m Φ ≡ − π √− ¯ g δδ ¯ Φ h ρσ ¯ T m ρσ − h ¯ T m + p − ¯ g φ ¯ I m ! , (210)where ¯ I m ≡ ρ m ¯ u α ) | α . (211)The source density in the right side of (209) represent contribution of non-linear perturbations Σ m ≡ √− ¯ g δ L dyn δ ¯ Φ , (212)36 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 and we shall calculate it explicitly in section 7.2.2. Notice that because of the non-linearity the source term, Σ m ,depends not only on the dark matter variables but on the dynamic variables describing the gravitational and darkenergy perturbations as well.Calculation of the variational derivative in (210) requires taking the variational derivative from various thermo-dynamic quantities like the energy density ǫ m , pressure p m , etc. with respect to the background value of the Clebschpotential ¯ Φ . All of them are functions of the specific enthalpy µ m which, in its own turn, depends only on the deriva-tives Φ α of the potential. As an example, let us consider the density of the ideal fluid ¯ ρ m = ¯ ρ ( ¯ µ m ). We have δρ m δ ¯ Φ = − ∂∂ x α ∂ρ m ∂ ¯ Φ α = − ∂∂ x α ∂ρ m ∂ ¯ µ m ∂µ m ∂ ¯ Φ α ! , (213)where the partial derivative of the density with respect to the specific enthalpy is calculated with the help of (155) bymaking use of equation of state of the ideal fluid, and the partial derivative ∂µ m ∂ ¯ Φ α = ¯ u α , (214)as follows from the definition of µ m . The same procedure is applied for calculation of the variational derivative fromother thermodynamic quantities. The variational derivative from the Hubble four-velocity ¯ u α is calculated from therelation, ¯ µ m ¯ u α = − ¯ Φ α , between the specific enthalpy, four-velocity and the gradient of the Clebsch potential. Allvariational derivatives that enter the calculation are given in Appendix A.3.1 of the present paper. Finally, we come tothe following result, F m Φ ≡ π Y α | α , (215)where the vector field Y α ≡ ¯ ρ m ¯ µ m φ α − ¯ ρ m l αβ ¯ u β + − c c ! ¯ ρ m ¯ µ m ¯ u β φ β −
12 ¯ ρ m q ! ¯ u α . (216)It shows that in the linear approximation of the dynamic perturbation theory, where Σ m =
0, the current √− ¯ gY α isconserved.Taking covariant derivative in (215) brings about the field equations for φφ αα − µ m A α ¯ u α − µ m H (4 q − l ) + − c c ! ¯ u α ¯ u β φ αβ −
12 ¯ µ m ¯ u α q α ! (217) − H ¯ µ m ∂ ln c ∂ ¯ µ m ¯ u α φ α −
12 ¯ µ m q ! = Σ m , where φ αα ≡ φ | α | α , q α ≡ q | α , and the very last term accounts for the fact that the speed of sound is not constantin inhomogeneous medium - the e ff ect which is important for a more adequate treatment of precise cosmologicalobservations. Indeed, the speed of sound, c s , relates to other thermodynamic quantities by equation of state makingthe speed of sound a function of the specific enthalpy, c s = c s ( ¯ µ m ). Covariant derivative from the speed of sound is c s | α = ( ∂ c s /∂ ¯ µ m ) ¯ µ m | α , where the covariant derivative ¯ µ m | α = ( ∂ ¯ µ m /∂ ¯ ρ m ) ¯ ρ m | α and, according to equation of continuity,¯ ρ m | α = H ¯ ρ m ¯ u α . It yields − c c ! | α = H ¯ µ m ∂ ln c ∂ ¯ µ m ¯ u α , (218)that explains how the last term in (217) originates from (215), (216).After imposing the gauge condition (206), the covariant equation (217) is reduced to φ αα + π ¯ µ m (cid:16) ¯ ρ m φ + ¯ ρ q ψ (cid:17) − µ m H q + − c c ! ¯ u α ¯ u β φ αβ −
12 ¯ µ m ¯ u α q α ! (219) − H ¯ µ m ∂ ln c ∂ ¯ µ m ¯ u β φ β −
12 ¯ µ m q ! = Σ m , . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 which is linearly coupled to the dynamic perturbation, ψ , of dark energy besides of coupling with gravitational fieldperturbation l αβ . Equation (219) is to be solved by iterations starting from equating the right side of it, Σ m =
0, andsolving for φ , which is used then for calculation of Σ m , and solving (219) again, and so on. Since equation (219) islinearly coupled with ψ , we will need equation for the dark energy perturbation.We should underline that the field equation (219) for the perturbations of the dark matter is nothing else but thecovariant form of equation for the sound waves propagating through the substance of the background dark matter, ¯ M ,with the speed of sound c s . Indeed, in the isotropic coordinates x α = ( η, x i ), the operator φ αα + − c c ! ¯ u α ¯ u β φ αβ = a − c ∂ φ∂η + f i j ∂ φ∂ x i ∂ x j ! , (220)which is a wave operator describing propagation of the perturbations of dark matter with the speed of sound. We alsonotice that the wave equation (219) is homogeneous in linearised order of approximation in which the source Σ m canbe neglected because it is quadratic with respect to perturbations. It means that solution of the linearised equation(219) corresponds only to the primordial excitations of the sound waves in dark matter. Baryonic matter (stress-energytensor of the bare perturbation) cannot produce any direct perturbation of the background distribution of dark matterin the linearised approximation. Calculation of the field equation for dark energy perturbation, ψ , follows the similar path like we did in the previoussubsection 5.6.2. The field equations follow from (130), and they are F q Ψ = π Σ q , (221)where F q Ψ and Σ q are determined by the variational derivatives from the Lagrangian of dark energy (163) and thedynamic Lagrangian (97) respectively. The linear operator F q Ψ is calculated by substituting the Lagrangian (163) into(133) which yields F q Ψ ≡ − π √− ¯ g δδ ¯ Ψ h ρσ ¯ T q ρσ − h ¯ T q + p − ¯ g ψ ¯ I q ! , (222)where ¯ I q ≡ "(cid:16) ¯ ρ q ¯ u α (cid:17) | α + ∂ ¯ W ∂ ¯ Ψ . (223)The source density Σ q ≡ √− ¯ g δ L dyn δ ¯ Ψ , (224)and we shall calculate it explicitly in section 7.2.3.According to equation (163), the Lagrangian density of the scalar field L q depends on both the field Ψ and its firstderivative, Ψ α . For this reason, unlike the operator F m , the di ff erential operator F q is not reduced to the covariantdivergence from a vector field as the partial derivative of the Lagrangian L q with respect to Ψ does not vanish. Wehave F q Ψ ≡ π Z α | α − l ∂ ¯ W ∂ ¯ Ψ − ψ ∂ ¯ W ∂ ¯ Ψ ! (225)where l ≡ ¯ g αβ l αβ , and vector field Z α ≡ ψ α − ¯ ρ q l αβ ¯ u β , (226)where we have used equation ¯ Ψ α = − ¯ u β ¯ Ψ β ¯ u α = − ¯ ρ q ¯ u α . The current Z α is not conserved unlike Y α for the dark matter.Taking covariant derivative in (225) and making use of the gauge condition (206) yield the field equations for ψψ αα + π ¯ µ m (cid:16) ¯ ρ m φ + ¯ ρ q ψ (cid:17) − µ q H + ∂ ¯ W ∂ ¯ Ψ ! q − ∂ ¯ W ∂ ¯ Ψ ψ = Σ q , (227)38 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where ψ αα ≡ ψ | α | α , and we have use the equality ¯ ρ q = ¯ µ q . Equation (227) is to be solved by iterations starting fromequating the right side of it, Σ q =
0, and solving for ψ , which is used then for calculation of Σ q , and solving (227)again, and so on. This procedure is going on along by simultaneously solving (219) for φ . Of course, before solvingequation (227) we have to specify the structure of the scalar potential ¯ W .Like in the case of dark matter, the wave equation (227) is homogeneous in linearised order of approximation inwhich the source Σ q can be neglected because it is quadratic with respect to perturbations. Hence, solution of thelinearised equation (227) corresponds only to the primordial excitations of the scalar field in dark energy. Baryonicmatter (stress-energy tensor of the bare perturbation) cannot produce any direct perturbation of dark energy in thelinearised approximation.
6. Stress-Energy Tensor of the Dynamic Field Perturbations
In order to solve the field equations (205) for the gravitational perturbations in the quadratic and higher orderapproximations, we have to know the e ff ective stress-energy tensor Λ µν entering the right side of these equations. Thee ff ective stress-energy tensor Λ µν is defined as a variational derivative (104) taken from the e ff ective Lagrangian (100).According to (114), it consists of two parts – the stress-energy tensor of matter of the baryonic (bare) perturbation T µν , and the stress-energy tensor of the dynamic field perturbations, T µν . We shall keep tensor T µν unspecified as longas theory permits and focus on calculation of T µν which also consists of two parts, t µν and τ µν , according to (123).Tensor t µν is the stress-energy tensor of gravitational field perturbations (124). Tensor τ µν is the stress-energy tensororiginating from the coupling of the background dark matter and dark energy perturbations to the gravitational fieldperturbations. General formula for calculating τ µν is given in (125). In case of FLRW universe gowerved by darkmatter and dark energy, tensor τ µν is linearly split in two counterparts τ αβ = τ m αβ + τ q αβ , (228)where τ m αβ and τ q αβ describe contributions of dark matter and dark energy respectively. In this section we calculateall the components of the e ff ective stress-energy tensor in the quadratic approximation. Higher-order terms will bepublished somewhere else. Stress-energy tensor of gravitational perturbations, t µν , has a universal and unique presentation on any pseudo-Riemannian manifold because it originates from a pure geometric part of the perturbed Hilbert Lagrangian. We begincalculation of t µν from its definition which is given by (124) in the form of variational derivative from the followingscalar density F G ≡ h ρσ F G ρσ − (1 / h F G , (229)where the tensor F G ρσ is given by (110), F G = ¯ g ρσ F G ρσ , h ρσ = √− ¯ gl ρσ , h ≡ ¯ g ρσ h ρσ . We put together all terms entering F G , and, then, employ the Leibniz rule to single out the total divergence from the products of two functions - themetric tensor perturbation and its second derivative. It results in F G = h ρσ | λ l λρ | σ − h ρσ | λ l ρσ | λ + h | λ l | λ + div , (230)where l ≡ ¯ g µν l µν , and div denote the terms which form a total divergence that vanishes upon taking a variationalderivative and, hence, can be discarded. For this reason, we drop it o ff from further calculation. Next step is to applythe covariant definition (45) of variational derivative to (230) in definition (124) of t αβ which can be written as follows16 π t αβ = √− ¯ g ¯ g αµ ¯ g βν δ F G δ ¯ g µν , (231)that conforms with the lower (subscript) position of indices of the metric tensor entering in the denominator of defini-tion (45) of variational derivative. 39 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 It is worthwhile to remind the reader that perturbation h ρσ is an independent variable which has been used inderivation of (124). It means that the partial derivative ∂ h αβ ∂ ¯ g µν = . (232)On the other hand, the covariant components of the gravitational perturbation, h αβ = ¯ g ακ ¯ g βλ h κλ , contain explicitly thebackground metric tensor and, hence, cannot be considered as independent from it. Therefore, we have for the partialderivatives ∂ h αβ ∂ ¯ g µν = ∂ ¯ g ακ ∂ ¯ g µν ¯ g βλ h κλ + ∂ ¯ g βλ ∂ ¯ g µν ¯ g ακ h κλ = δ ( µα h ν ) β + δ ( µβ h ν ) α . (233)Let us consider now a functional dependence of the covariant derivative h αβ | λ on the metric tensor. We noticethat the perturbation h αβ is a tensor density of weight −
1. Therefore, its covariant derivative has one more term ascompared with that of a tensor of a second rank. More specifically, h αβ | κ = h αβ,κ + ¯ Γ ασκ h σβ + ¯ Γ βσκ h σα − ¯ Γ σσκ h αβ . (234)It reveals that the derivative h αβ | κ depends merely on the Christo ff el symbols and is independent of the metric tensor¯ g αβ . Hence, the partial derivative ∂ h αβ | λ ∂ ¯ g µν = . (235)It agrees with our postulate that the metric tensor and the Christo ff el symbols are true independent variables along withthe tensor density h αβ and its covariant derivative h αβ | λ . Equation (230) given in terms of the independent variables,reads (with the divergence term discarded) F G = ¯ g κλ ¯ g βρ √− ¯ g ¯ g αλ h ρσ | κ h αβ | σ −
12 ¯ g ασ h ρσ | κ h αβ | λ +
14 ¯ g ασ h ασ | κ h βρ | λ ! , (236)where we have discarded the total divergence.Variational derivative in the form of (45) taken from (236) engages partial derivatives with respect to the back-ground metric tensor, ¯ g µν , and those with respect to the background Christo ff el symbols, ¯ Γ αβγ . The partial derivativewith respect to the metric tensor yields1 √− ¯ g ∂ F G ∂ ¯ g µν = −
12 ¯ g µν l ρσ | λ l λρ | σ − l ρσ | λ l ρσ | λ + l | λ l | λ ! (237) − l µσ | ρ l νσ | ρ + l µσ | ρ l νρ | σ + l ρσ | µ l ρσ | ν + l µν | σ l | σ − l | µ l | ν , where we have used (235). The partial derivative with respect to the Christo ff el symbols taken from h ρσ | κ is calculatedfrom its presentation in the form of (234) with the help of (51). It gives ∂ h ρσ | κ ∂ ¯ Γ αλγ = δ ρα δ ( λκ h γ ) σ + δ σα δ ( λκ h γ ) ρ − δ ( λα δ γ ) κ h ρσ . (238)After making use of this formula, the partial derivative of F G with respect to the Christo ff el symbols results in1 √− ¯ g ∂ F G ∂ ¯ Γ αµγ = l ργ l µ ( ρ | α ) + l ρµ l γ ( ρ | α ) − l ρα | ( µ l γ ) ρ (239) − δ ( µα l γ ) ρ | σ l ρσ + l α ( γ l | µ ) + l ρσ l ρσ | ( µ δ γ ) α − δ ( µα l γ ) l . . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 It allows us to calculate the linear combination of the partial derivatives entering definition of variational derivative(45), namely, −
12 1 √− ¯ g ¯ g σν ∂ F G ∂ ¯ Γ σµγ + ¯ g σµ ∂ F G ∂ ¯ Γ σνγ − ¯ g σγ ∂ F G ∂ ¯ Γ σµν ! = l ρ ( µ l ν ) ρ | γ − l γρ | ( µ l ν ) ρ − l ργ l µν | ρ − l µν l | γ (240) + ¯ g µν l ρσ l γρ | σ − l ρσ l ρσ | γ + ll | γ ! . After making use of (237) and (240) in expression (231) for variational derivative defined by the rule (45), the stress-energy tensor of the gravitational field perturbations takes on the following form16 π t µν = −
12 ¯ g µν l ρσ | γ l γρ | σ − l ρσ | γ l ρσ | γ + l | γ l | γ ! (241) − l µσ | ρ l νσ | ρ + l µσ | ρ l νρ | σ + l ρσ | µ l ρσ | ν + l µν | σ l | σ − l | µ l | ν + ¯ g µν l ρσ l γρ | σ − l ρσ l ρσ | γ + ll | γ ! | γ + l ρ ( µ l ν ) ρ | γ − l νρ l γρ | µ − l µρ l γρ | ν − l γρ l µν | ρ − l µν l | γ ! | γ . It apparently depends on the second derivatives of the gravitational perturbation, l µν which is a consequence of ourcovariant field-theoretical approach for description of perturbations of gravitational field [85, 100]. Most of alterna-tive approaches to construct the stress-energy tensor of gravitational field perturbations without second derivativesunavoidably make it non-covariant that is coordinate-dependent. For this reason such “tensors” of gravitational fieldperturbations are commonly-known as pseudo-tensors [101]. Babak and Grishchuk [102, 103] proposed an interestingmethod to constructing a tensor of gravitational field perturbations which does not include the second derivatives ofthe field perturbations. The method requires an introduction of an additional (Lagrange multiplier) term to the grav-itational field Lagrangian which is proportional to the Riemann tensor of the background manifold. This procedurehas been worked out in [102, 103] for the case of Minkowski-flat background. Further research should be conductedto extend it to the case of an arbitrary curved background manifold.Significant number of the second derivatives in expression (241) can be eliminated on-shell by making use of thecovariant field equation (205). To this end we write down the terms with the second covariant derivatives in (241) andexpress the commutator of the second-order derivatives from the metric tensor perturbation in terms of the Riemanntensor, l αρ | σβ = l αρ | βσ − l γρ ¯ R αγσβ + l αγ ¯ R γρσβ . (242)A useful consequence of this equation is a contraction with respect to index α which gives l αρ | σα = A ρ | σ + l αρ ¯ R σα + l αγ ¯ R γρσα , (243)where A α ≡ l αβ | β , and A α = ¯ g αβ A β . Straightforward but tedious rearrangement of the second-order derivatives fromthe metric tensor perturbations with the help of (242), (243) allows us to put (241) into the following form16 π t µν = l µρ | σ l ν ( ρ | σ ) − l ρσ | µ l νρ | σ − l ρσ | ν l µρ | σ + l ρσ | µ l ρσ | ν − l | µ l | ν − l ρσ l µν | ρσ (244) +
12 ¯ g µν l ρσ | γ l γρ | σ − l ρσ | γ l ρσ | γ + l | γ l | γ ! + l ρ ( µ A ν ) | ρ − l µν A ρ | ρ − l µν | ρ A ρ + π " l ρ ( µ Θ ν ) ρ − l µν Θ − ¯ g µν l ρσ Θ ρσ − l Θ ! + l ρµ l σν ¯ R ρσ + l αβ l ρ ( µ ¯ R ν ) αβρ where Θ αβ = T αβ − π (cid:16) F m αβ + F q αβ (cid:17) , (245)41 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 and the spur, Θ ≡ ¯ g αβ Θ αβ .As we can see, most of the second-order derivatives from the metric tensor perturbations have vanished. Theremaining second-order derivatives remain as the very last term in the first line of (244)) and in the terms whichdepend on the gauge function A α in the second line of (244). On-shell expression (244) of the tensor of gravitationalfield perturbations also depends on the Riemann (curvature) tensor of the background manifold. Had the backgroundmanifold ¯ M been flat such terms would not be present. The last but not least notice is that t µν includes on-shellcoupling of the gravitational field perturbations with the perturbations of the background matter as well as with thebare perturbations. These terms are proportional to the terms with Θ αβ which come from F G αβ because on shell, F G αβ = Θ αβ , due to the field equations (103).We expressed the stress-energy tensor of gravitational field perturbations t µν in terms of the variable l αβ . It isinstructive to reformulate it in terms of the perturbation of the metric tensor, κ αβ , defined in (191) and related to l αβ according to (194). Calculations are done in two steps. First, we replace all l αβ in (241) with κ αβ and retain only linearterms in (194). Second step is to replace l αβ in the linear operator (110) with κ αβ by taking into account quadraticterms in expansion (194). All quadratic terms with respect to κ αβ are combined together to produce the stress-energytensor of gravitational field perturbations expressed in terms of the dynamic variable κ αβ . This tensor coincides (up tothe sign convention) with that given in the textbook by S. Weinberg [56, equation 7.6.15] which also depends on thesecond-order derivatives from the metric tensor perturbations. The advantage of our perturbation scheme as comparedwith S. Weinberg’s book [56] is that we have worked out an iterative procedure of calculation of the field perturbationsat any order of approximation. In particular, we can derive an exact analytic form of the gravitational stress-energytensor t µν which reads [85] t µν = π δ ρµ δ σν −
12 ¯ g µν ¯ g ρσ ! (cid:16) G αρβ G βσα − G αρσ G βαβ (cid:17) (246) + π " h µν ¯ g ρβ G ααβ −
12 ¯ g µν h αβ G ραβ − h ρ ( µ G αν ) α + h ρβ ¯ g α ( µ G αν ) β + h β ( µ G ρν ) β − h β ( µ ¯ g ν ) α ¯ g ρσ G αβσ | ρ , where G αβγ ≡ Γ αβγ − ¯ Γ αβγ is the di ff erence between the Christo ff el symbols of the perturbed, M , and the background,¯ M , manifolds G αβγ = g αρ (cid:16) κ ρβ | γ + κ ργ | β − κ βγ | ρ (cid:17) , (247)where κ µν ≡ g µν − ¯ g µν . We emphasize that the geometric object G αβγ is a tensor with respect to coordinate transforma-tion on the background manifold since it represents the di ff erence between the two Christo ff el symbols [104]. It doesnot mean, of course, that we employ a bi-metric theory of gravity being di ff erent from general theory of relativity. Thebackground metric ¯ g µν is simply the lowest (unperturbed) state of the gravitational field which dynamical propertiesare described by the full metric g µν . Since both the background metric ¯ g µν , its perturbation κ µν , and the object G αβγ are tensors, t µν is a stress-energy tensor of the gravitational field perturbations. It defines energy, a linear momentum,and other physical characteristics of the perturbations at each point of the background spacetime [105]. Expansion of(246) in Taylor series with respect to perturbations and leaving only quadratic terms yields (241) The part of the stress-energy tensor describing the dark matter perturbation is given in (228) by τ m µν that, accordingto (125), is calculated as a variational derivative16 πτ m αβ = √− ¯ g ¯ g αµ ¯ g βν δ F m δ ¯ g µν , (248)from the Lagrangian density given by F m ≡ h ρσ F m ρσ − h F m + p − ¯ g φ F m Φ , (249)42 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where the individual terms entering the right side of (249) are taken from (203) and (215) respectively, and F m ≡ ¯ g αβ F m αβ . We single out the total divergence in (249), discard it, and brings (249) to the following form F m ≡ h ρσ F m ρσ − h F m − π p − ¯ g φ α Y α , (250)where the total divergence has been dropped o ff , φ α ≡ φ | α , and the current Y α is given in (216). After reducing similarterms equation (250) takes on the following form F m ≡ − π p − ¯ g ¯ ρ m ¯ µ m φ α φ α − ρ m l αβ ¯ u α φ β ! − π p − ¯ g ( ¯ p m − ¯ ǫ m ) l αβ l αβ − l ! (251) − π p − ¯ g ¯ ρ m ¯ µ m − c c ! " (¯ u α φ α ) −
32 ¯ µ m q ¯ u α φ α + µ q . where again we have used notation φ α ≡ φ | α and l ≡ ¯ g αβ l αβ .Taking variational derivative in (248) is rather straightforward but tedious procedure. Because the Lagrangian F m depends neither on the Christo ff el symbols nor on the curvature tensor, the variational derivative (248) is reduced toa partial derivative with respect to the metric tensor δ F m /δ g µν = ∂ F m /∂ g µν . Calculation of the partial derivative isdone with the help of the chain rule and equations in Appendix A.1. It yields the stress-energy tensor of dark matter τ m µν = ¯ ρ m µ m φ µ φ ν − ¯ ρ m µ m " φ α φ α + − c c ! ( ¯ u α φ α ) ¯ g µν (252) − ¯ ρ m µ m − c c ! "
32 ¯ µ m q − u α φ α ! ¯ u ( µ φ ν ) −
34 ¯ µ m ¯ u α φ α l µν + ¯ ρ m µ m − c c ! φ α φ α − µ m l αβ ¯ u α φ β +
32 ¯ µ m l ¯ u α φ α ! ¯ u µ ¯ u ν + ¯ ρ m µ m " − c c ! − c c ! − ¯ ρ m ¯ µ m c c ∂ ln c ∂ ¯ p m ( ¯ u α φ α ) −
32 ¯ µ m q ¯ u α φ α + µ q ¯ u µ ¯ u ν −
18 ¯ ρ m ¯ µ m − c c ! " q l µν − q ¯ g µν + l αβ l αβ − l + q l ! ¯ u µ ¯ u ν −
12 ( ¯ p m − ¯ ǫ m ) " l αµ l να − ll µν − l αβ l αβ − l ! ¯ g µν , where we have used thermodynamic relation (147) to make a replacement ¯ ǫ m + ¯ p m = ¯ ρ m ¯ µ m .As we have modelled dark matter by the ideal fluid, equation (252) represents the stress-energy tensor of soundwaves propagating on the background cosmological manifold. This tensor depends on the speed of sound, c s , whichenters denominators in some terms of (252). It may cause an impression that in case of dust, when c s →
0, the tensor τ m µν is divergent. This impression is not true as the numerators of the corresponding terms also approach to zero withthe same rate as the denominator. It leaves τ m µν well-defined even in case of a model of dark matter consisting ofnon-interacting dust particles. The part of the stress-energy tensor describing the dark energy perturbation is given in (228) by τ q µν that, accordingto (125), is calculated as a variational derivative16 πτ q αβ = √− ¯ g ¯ g αµ ¯ g βν δ F q δ ¯ g µν , (253)from the Lagrangian density given by F q ≡ h ρσ F q ρσ − h F q + p − ¯ g ψ F q Ψ , (254)43 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where the individual terms entering the right side of (254) are taken from (204) and (225) respectively, and F q ≡ ¯ g αβ F q αβ . We single out the total divergence and bring (254) to the following form F q ≡ h ρσ F q ρσ − h F q − π p − ¯ g ψ α Z α − π p − ¯ g ψ l ∂ ¯ W ∂ ¯ Ψ + ψ ∂ ¯ W ∂ ¯ Ψ ! , (255)where the total divergence has been dropped o ff . More explicitly, F q ≡ − π p − ¯ g " ψ α ψ α + l ψ ∂ ¯ W ∂ ¯ Ψ + ψ ∂ ¯ W ∂ ¯ Ψ − µ q l αβ ¯ u α ψ β + π p − ¯ g ¯ W ( ¯ Ψ ) l αβ l αβ − l ! , (256)where ψ α ≡ ψ | α . Taking variational derivative from the left side of (256) with respect to ¯ g µν , we obtain the stress-energy tensor of dark energy perturbation τ q µν = ψ µ ψ ν − ψ α ψ α + ψ ∂ ¯ W ∂ ¯ Ψ ! ¯ g µν − l µν ψ ∂ ¯ W ∂ ¯ Ψ + ¯ W ( ¯ Ψ ) " l αµ l να − ll µν − l αβ l αβ − l ! ¯ g µν . (257)This tensor depends on the potential ¯ W ( ¯ Ψ ) of the scalar field and on its first and second derivatives. The potential hasbeen kept arbitrary which makes expression (257) rather general and applicable to discussion of a wide spectrum ofphysical situations.
7. Post-Friedmanian Equations of Motion in Cosmology
In this section we shall derive equations of motion of the baryonic matter in the universe governed by dark matterand dark energy. Baryonic matter falls freely in the gravitational field produced by dark matter and dark energyprimordial perturbations which are responsible for the formation of the large scale structure in the universe [4, 5].Since luminous matter is made of baryons, its astronomical observations traces the gravitational potential of darkmatter and helps us to identify where it confines and clumps to clusters. We shall also take into account the self-gravitational interaction of the baryonic matter, thus, extending the post-Newtonian treatment of equations of motionin asymptotically-flat spacetime [14, 82] to the realm of cosmology where FLRW background metric is not flat.
Let us consider a background spacetime manifold, ¯ M , with the e ff ective Lagrangian L e ff = L e ff (cid:16) ¯ g µν , ¯ Γ αβγ ; ¯ Φ A , ¯ Φ α ; Θ B , Θ B α ; h µν , h µν | a ; φ A , φ A α (cid:17) , (258)depending on a set of the independent dynamic variables and their conjugated counterparts which are covariant deriva-tives on the background manifold. We have proved in section 4.8 that the e ff ective Lagrangian L e ff is gauge-invariant on shell modulo a total divergence. The gauge invariance of L e ff suggests that its Lie derivative along an arbitraryvector field, ξ α , must be also nil modulo a total divergence: £ ξ L e ff = ∂ α U α , where U α is a vector field. Because atotal divergence added to the Lagrangian do not a ff ect the field equations we drop it out of the subsequent equations.We compute the Lie derivative of the e ff ective Lagrangian by making use of (64) that reduce calculation of the Liederivative to that of variational derivatives modulo a total divergence. After dropping o ff the divergence, we have£ ξ L e ff = δ L e ff δ ¯ g αβ £ ξ ¯ g αβ + δ L e ff δ ¯ Φ A £ ξ ¯ Φ A + δ L e ff δ h µν £ ξ h µν + δ L e ff δφ A £ ξ φ A + δ L e ff δ Θ B £ ξ Θ B . (259)Field equations (101), (128), (134) describing evolution of the dynamic field perturbations h µν , φ A , θ B on the back-ground manifold exterminate the last three terms in the right side of (259). The first term in the right side of (259) canbe written down as follows δ L e ff δ ¯ g αβ £ ξ ¯ g αβ = − p − ¯ g Λ αβ ξ α | β , (260)44 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where we have used definitions (104) and equation for the Lie derivative of the background metric£ ξ ¯ g αβ = − ξ α | β − ξ β | α , (261)In order to develop a second term in the right side of (259), we have to know the Lie derivative of the field, ¯ Φ A ,which depends on its geometric properties. In a particular case of a tensor density ¯ Φ A ≡ (cid:16) ¯ Φ A (cid:17) µ ...µ p ν ...ν q of weight m , theLie derivative is given by (61) that can be written symbolically as follows£ ξ ¯ Φ A = ξ α ¯ Φ A α + ¯ K A αβ ξ α | β , (262)where ¯ Φ A α ≡ (cid:16) ¯ Φ A (cid:17) µ ...µ p ν ...ν q | α , ¯ K A αβ = ¯ K A σα ¯ g σβ , and¯ K A σα ≡ m δ σα (cid:16) ¯ Φ A (cid:17) µ ...µ p ν ...ν q (263) − δ µ α (cid:16) ¯ Φ A (cid:17) σµ ...µ p ν ...ν q − . . . − δ µ p α (cid:16) ¯ Φ A (cid:17) µ ...µ p − σν ...ν q + δ σν (cid:16) ¯ Φ A (cid:17) µ ...µ p αν ...ν q + . . . + δ σν q (cid:16) ¯ Φ A (cid:17) µ ...µ p ν ...ν q − α . Making use of definition (132) and (262) we can present the second term in the right side of (259) in the followingform δ L e ff δ ¯ Φ A £ ξ ¯ Φ A = p − ¯ g Σ M A (cid:16) ξ α ¯ Φ A α + ¯ K A αβ ξ α | β (cid:17) . (264)Substituting (260), (264) to the right side of (259) results in£ ξ L e ff = p − ¯ g Σ M A ¯ Φ A α ξ α + p − ¯ g − Λ αβ + Σ M A ¯ K A αβ ! ξ α | β . (265)Applying the Leibniz rule to change the order of di ff erentiation in the terms depending on ξ α | β , we can recast (265) tothe following form £ ξ L e ff = p − ¯ g " Σ M A ¯ Φ A α + Λ αβ | β − (cid:16) Σ M A ¯ K A αβ (cid:17) | β ξ α + p − ¯ gW β | β , (266)where the vector field W β ≡ − Λ αβ + Σ M A ¯ K A αβ ! ξ α . (267)The last term in (266) is reduced to the total divergence of a vector density p − ¯ gW β | β = ∂ β (cid:16) p − ¯ gW β (cid:17) , (268)where W β = ¯ g αβ W α . The Lie derivative (266) of the e ff ective Lagrangian vanishes modulo the divergence of the vectorfield U β ≡ √− ¯ gW β if, and only if, the combination of terms enclosed to the square brackets in (266) is nil. It yieldsthe equations of motion of matter Λ αβ | β = − Σ M A ¯ Φ A α + (cid:16) Σ M A ¯ K A αβ (cid:17) | β . (269)It should be compared with the law of conservation of matter in flat background spacetime, Λ αβ,β =
0, with the rightside equal to zero [75]. The presence of the background matter fields ¯ Φ A on the curved background manifold makesthe right side of (269) di ff erent from zero. This result was established in [85].Equation (269) can be interpreted as the integrability condition of the gravitational field equation (103). Taking acovariant derivative from both sides of the field equation (103) and applying the equations of motion (269) yields (cid:16) F G αβ + F M αβ (cid:17) | β = − π (cid:20) Σ M A ¯ Φ A α − (cid:16) Σ M A ¯ K A αβ (cid:17) | β (cid:21) . (270)In the linear approximation, when all quadratic and higher-order terms with respect to the perturbations are discarded( Σ M A → F G αβ + F M αβ ) | β =
0. It agrees with the assumption that the stress-energy tensor ofthe bare perturbation is conserved in the linearised perturbative order, T αβ | β =
0. Now we are set to start calculatingequations of motion of matter of the baryonic matter in FLRW universe goiverned by the dark matter and dark energywhich we consider in the next few subsections. 45 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 The dark matter and dark energy components of matter that governs the temporal evolution of the universe aremodelled by two scalar fields Φ ≡ Φ and Φ ≡ Ψ . For scalar fields the tensor ¯ K A αβ ≡ T µν | ν + t µν | ν + τ m µν | ν + τ q µν | ν = (cid:16) ¯ µ m Σ m + ¯ µ q Σ q (cid:17) ¯ u µ , (271)where we have used equation (197) for expressing the gradients of the scalar fields ¯ Φ and ¯ Ψ in terms of the backgroundfour-velocity ¯ u α as well as equations (114), (123), (228) defining the e ff ective stress-energy tensor Λ αβ . Equation (271)is a covariant equation of motion of the baryonic matter described by the stress-energy tensor T µν , in the presence ofdynamic perturbations of gravitational field, dark matter and dark energy. In case of asymptotically flat spacetime theright side of (271) would vanish while in the left side of (271) only the first two terms would remain among which thestress-energy tensor of gravitational field, t µν , would be made of the perturbations of gravitational field caused by thebaryonic matter itself.In FLRW universe with dark matter and dark energy, more terms appear in equations of motion (271) whichshould be properly treated. Our goal is to calculate the explicit form of Σ m and Σ q as well as the covariant divergencesof stress-energy tensors entering (271). We split the process of calculation in three parts. First, we calculate thedivergence, t µν | ν , of the stress-energy tensor of gravitational field, then, we proceed to calculation of the divergence, τ m µν | ν , of the stress-energy tensor of dark matter, and that τ q µν | ν of dark energy. It becomes clear in the course of thecalculations, that a large group of terms making up Σ m and Σ q can be represented in the form of a covariant divergence.Such terms are combined with τ m µν | ν and τ q µν | ν respectively to reduce the number of similar terms. We give more detaileddescription in the text which follows. Covariant divergence from the stress-energy tensor of gravitational field, t µν , is derived by means of direct cal-culation from its definition (244). In the process of calculation we can simplify a significant number of terms byemploying the commutation relations (242), (243) for second-order covariant derivatives along with a rule for thethird order derivative l λµ | ρσν = l λµ | ρνσ − l γµ | ρ ¯ R λγσν + l λγ | ρ ¯ R γµσν + l λµ | γ ¯ R γρσν , (272)which allows us (after one more commutation of the covariant derivative in l λµ | ρνσ ) to derive l νµ | ρσν = A µ | ρσ + l αµ | σ ¯ R αρ + l αµ | ρ ¯ R ασ + l αβ | σ ¯ R βµρα + (273) l αβ | ρ ¯ R βµσα + l αµ | β ¯ R βρσα + l αµ ¯ R αρ | σ + l αβ ¯ R βµρα | σ . A significant number of similar terms is cancelled out, and after a multi-page analytic calculation we obtain a fairlysimple result, t µν | ν = l ρν Θ ρµ − l µν Θ ! | ν − l ρσ Θ ρσ | µ − l Θ | µ ! , (274)where a tensor Θ αβ was defined in (245), and Θ = ¯ g αβ Θ αβ . After taking the covariant divergence, it is convenient tosplit the right side of (274)algebraically in three parts t µν | ν = l ρν T ρµ − l µν T ! | ν − l ρσ T ρσ | µ − l T | µ ! (275) − π l ρν F m ρµ − l µν F m ! | ν + π l ρσ F m ρσ | µ − lF m | µ ! − π l ρν F q ρµ − l µν F q ! | ν + π l ρσ F q ρσ | µ − lF q | µ ! , . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where F m αβ and F q αβ have been given in (203) and (204) respectively, and F m ≡ ¯ g αβ F m αβ , F q ≡ ¯ g αβ F q αβ . The first linein the right side of this equation describes the coupling of gravitational perturbation with the stress-energy tensor T µν of the baryonic matter, and the second and the third lines outline the contribution of dark matter (index ‘m’) and darkenergy (index ‘q’). The source density Σ m for dark matter dynamic perturbations in quadratic and higher orders, is defined by equation(212) where we shall take into account in the dynamic Lagrangian only terms of the second order, L dyn = L , (276)and keep in L only dark matter variables. By a simple inspection, we find out that Σ m depends only on the derivatives,¯ Φ α , of the scalar field ¯ Φ and, thus, can be written in the form of a covariant divergence Σ m = J m ν | ν , (277)where J m ν = π √− ¯ g δ F m δ ¯ Φ ν , (278)is a second order (quadratic) correction to the conserved dark matter current Y µ given in (216).The current J m ν can be algebraically split in two components - one being parallel to the Hubble velocity, ¯ u α , andanother one being orthogonal to it, J m α = ¯ ρ m ¯ u α j + ¯ ρ m ¯ P αβ j β , (279)where P αβ ≡ δ βα + ¯ u α ¯ u β . The corresponding projections, which appear in (279), are given by the following expressions, j =
12 ¯ µ − c c ! " φ α φ α + − c c ! (¯ u α φ α ) +
32 ¯ µ m c c " l αβ ¯ u α φ β + − c c ! (¯ u α φ α ) q (280) − − c c ! " + c c ! q + l αβ l αβ − l −
12 ¯ µ m c c ∂ ln c ∂ ¯ µ m " ( ¯ u α φ α ) −
32 ¯ µ m q (¯ u α φ α ) +
12 ¯ µ q , j β = µ − c c ! ( ¯ u α φ α ) φ β −
32 ¯ µ m l βα φ α −
32 ¯ µ m − c c ! " ( ¯ u α φ α ) l βγ ¯ u γ + q φ β + − c c ! q l βα ¯ u α . (281)Direct calculation of the covariant divergence from J m α in (277) entangles a lot of algebraic operations whichnumber can be significantly reduced by making use of the following procedure. First of all, we notice that the term,¯ µ m Σ m ¯ u µ , in the right side of (271) can be replaced on shell with ¯ µ m J m | νν ¯ u µ due to (277). Then, we use the chain ruleand derivatives ¯ µ m | ν = ∂ ¯ µ m ∂ ¯ ρ m ¯ ρ m | ν = c c H ¯ µ m ¯ u ν , (282)¯ u µ | ν = H ¯ P µν . (283)in order to transform ¯ µ m Σ m ¯ u µ = (cid:16) ¯ µ m J m ν ¯ u µ (cid:17) | ν + c c H ¯ ρ m ¯ µ m j ¯ u µ − H ¯ ρ m ¯ µ m j ν ¯ P µν , (284)where we have used (279). 47 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 Now, we combine the total divergence in the right side of (284) with the divergence of the stress-energy tensor ofdark matter in the left side of (271). In doing so, we notice the following equations16 π p − ¯ g τ m µν = ¯ g µρ ¯ g νσ ∂ F m ∂ ¯ g ρσ + ∂ F m ∂ ¯ µ m ∂ ¯ µ m ∂ ¯ g ρσ ! , (285)16 π p − ¯ gJ m ν = ∂ F m ∂ ¯ Φ ν + ∂ F m ∂ ¯ µ m ∂ ¯ µ m ∂ ¯ Φ ν , (286)which are just more explicit form of the definitions (248) and (278) of the corresponding quantities expressed as thevariational derivatives with respect to the metric tensor and the derivative of the scalar field respectively. Accountingfor the variational derivatives (A17), (A40) we obtain τ m µν −
12 ¯ µ m J m ν ¯ u µ = π √− ¯ g ¯ g µρ ¯ g νσ ∂ F m ∂ ¯ g ρσ − ∂ F m ∂ ¯ Φ ν ¯ u µ ! . (287)This equation elucidates that we do not need to directly calculate a large number of terms depending on the partialderivatives with respect to the specific enthalpy ¯ µ m when calculating the covariant divergence in the left side ofequations of motion (271). It saves us from doing a lot of redundant algebraic operations.It is also reasonable to combine (287) with the dark matter term representing a total divergence in the second lineof (275) and denote X µν ≡ τ m µν −
12 ¯ µ m J m ν ¯ u µ − π l ρν F m ρµ − l µν F m ! . (288)Notice that tensor X µν is not symmetric with respect to its indices. Making use of (201), (252), (279) in the right sideof (288), and reducing similar terms, we obtain a rather short expression X µν ≡ ¯ ρ m µ m φ µ φ ν − φ α φ α ¯ g µν ! + ¯ ρ m µ m − c c ! " ( ¯ u α φ α ) φ µ ¯ u ν −
12 ( ¯ u α φ α ) ¯ g µν (289) − ¯ ρ m φ µ l νρ ¯ u ρ +
14 ¯ u µ l νρ φ ρ ! − ¯ ρ m − c c ! " q φ µ ¯ u ν + ( ¯ u α φ α ) l µν +
14 ¯ u µ l νρ ¯ u ρ ! + " ¯ ρ m ¯ µ m − c c ! q + ( ¯ p m − ¯ ǫ m ) l ρσ l ργ − l ! ¯ g µν . Let us denote the density of the force caused by dark matter on the motion of the baryonic matter by f m µ . Aftergrouping together all terms in (271), belonging to the dark matter sector, the force density is defined by the followingexpression f m µ ≡ − X µν | ν − π l ρσ F m ρσ | µ − lF m | µ ! + c c H ¯ ρ m ¯ µ m j ¯ u µ − H ¯ ρ m ¯ µ m j ν ¯ P µν , (290)where the second term in the right side was taken from (275), and the last two terms – from (284). We can split theforce density, f m µ , in two orthogonal components f m µ = a m ¯ u µ + a m ν ¯ P νµ , (291)48 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where a m ≡ − ¯ u ν f m ν and a m µ ≡ ¯ P µν f m ν . We have, more explicitly, a m =
14 ¯ ρ m h l αβ φ αβ + A α φ α − u α φ α ) (cid:16) ¯ u β A β (cid:17)i (292) +
18 ¯ ρ m − c c ! h ¯ u α ¯ u β l βγ φ αγ + q ¯ u α ¯ u β φ αβ − (¯ u α φ α ) (cid:16) ¯ u β q β (cid:17) + ( ¯ u α φ α ) (cid:16) ¯ u β A β (cid:17)i + ρ m H (¯ u α φ α ) q − l ! +
18 ¯ ρ m H − c c ! h l αβ ¯ u α φ β + ( ¯ u α φ α ) (3 q − l ) i +
38 ¯ ρ m ¯ µ m H ∂ ln c ∂ ¯ µ m q − l ! (¯ u α φ α ) , a m µ =
12 ¯ ρ m ( ¯ u α A α ) φ µ (293) +
18 ¯ ρ m − c c ! h l µα ¯ u β φ αβ − q ¯ u α φ µα + (¯ u α q α ) φ µ + (¯ u α φ α ) A µ i − ρ m H q − l ! φ µ +
12 ¯ ρ m H − c c ! " (¯ u α φ α ) l µβ ¯ u β − q φ µ + l µα φ α +
38 ¯ ρ m ¯ µ m H ∂ ln c ∂ ¯ µ m h (¯ u α φ α ) l µα ¯ u α − q φ µ i , where φ αβ ≡ φ | αβ . Our next goal is to calculate the force density exerted by dark energy on the motion of the baryonicmatter in the universe. The procedure of calculation of the dark energy force density is similar to that described in the previous subsection(7.2.2). The dark energy source , Σ q , which is defined in (224) as a variational derivative from the dynamic Lagrangian L dyn , depends not only on the derivatives of the scalar field ¯ Ψ but on the field itself through the field potential W = W ( Φ ). We take into account in L dyn only the quadratic terms with respect to the dynamic perturbations which yield Σ q = π √− ¯ g − ∂ F q ∂ ¯ Ψ + ∂ F q ∂ ¯ Ψ ν ! | ν , (294)where the Lagrangian density F q is given in (255). After taking the variational derivatives in (294), we obtain Σ q = ψ ∂ ¯ W ∂ ¯ Ψ + l ψ ∂ ¯ W ∂ ¯ Ψ − ∂ ¯ W ∂ ¯ Ψ l αβ l αβ − l ! + (cid:16) l αβ | γ ψ a ¯ u β ¯ u γ + l αβ ψ αγ ¯ u β ¯ u γ + Hl αβ ¯ u α ψ β (cid:17) . (295)The force density exerted by dark energy on the motion of the baryonic matter is combined from all terms in (271)which depend on dark energy components, f q µ ≡ − τ q µν | ν + π l ρν F q ρµ − l µν F q ! | ν − π l ρσ F q ρσ | µ − lF q | µ ! +
12 ¯ µ q Σ q ¯ u µ , (296)where the second and third terms standing in the irght side of this definition come from the third line of (275). Inorder to calculate the right side of (296) we use equation (257) for τ q µν , equation (204) for F q µν , and equation (295) for49 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 Σ q . After long but straightforward calculation and reduction of many similar terms, we get f q µ = ¯ ρ q ¯ u µ l αβ ψ αβ + l αβ | γ ψ α ¯ u β ¯ u γ + l αβ ψ αγ ¯ u β ¯ u γ + Hl αβ ¯ u α ψ β ! (297) + ( A α ψ α ) ¯ u µ +
12 ( A α ¯ u α ) ψ µ + ¯ ρ q H q − l ! ψ µ − ∂ ¯ W ∂ ¯ Ψ (cid:16) ψ A µ + l µν ψ ν − q ψ µ (cid:17) +
14 ¯ ρ q ψ ∂ ¯ W ∂ ¯ Ψ l µν ¯ u ν − l ¯ u µ ! , where we denoted ψ αβ ≡ ψ | αβ , and ¯ ρ q = µ q . After making use of the results of the presiding section, equations of motion (271) of the baryonic matter take onthe following form T µν | ν + l ρν T ρµ | ν − T ρν | µ ! − l µν − l δ µν ! T | ν + A ρ T ρµ − A µ T = f m µ + f q µ . (298)The left side of this equation can be brought to a more conventional form of a covarinat derivative with respect to thefull metric, if we use relation (122) between the stress-energy tensor of the bare perturbation T µν given in (117) and T µν defined in (115).Let us take a covariant divergence of T µν with respect to the full metric g µν that is ∇ ν T µν ≡ g νρ ∇ ν T ρµ where the ∇ ν denotes a covariant derivative with respect to the full metric, and we rise and lowered indices with the help of the fullmetric. Covariant derivatives from the stress-energy tensor T µν are calculated with the help of ∇ α T µν = T µν | α − G βαµ T νβ − G βαν T µβ , (299)where G βαµ is the Christo ff el symbol being associated with the full metric. It is rather straightforward to prove thatthey have the following exact form, G βαµ =
12 ¯ g βγ (cid:16) κ γα | µ + κ γµ | α − κ αµ | γ (cid:17) . (300)In the linear approximation with respect to the l µν equation (300) reads G βαµ = −
12 ¯ g βγ (cid:16) l γα | µ + l γµ | α − l αµ | γ (cid:17) + (cid:16) δ βα l | µ + δ βµ l | α − ¯ g αµ l | β (cid:17) . (301)Two contracted values of the Christo ff el symbols are G α ≡ G βαβ = l | α , ¯ g αβ G γαβ = − l γβ | β = − A γ , (302)Making use of these notations and definitions, and doing a direct calculation results in ∇ ν T µν = T µν | ν + l ρν T ρµ | ν − T ρν | µ ! − l µν − l δ µν ! T | ν + A ρ T ρµ − A µ T . (303)It elucidates that equation of motion (298) has the following form ∇ ν T µν = f m µ + f q µ . (304)Had the background spacetime been flat, the right side of (304) would vanish yielding the conventional law of con-servation, ∇ ν T µν =
0. However, in cosmology the spacetime manifold is given by the perturbed FLRW metric. Theperturbations interact with themselves causing an e ff ective force f µ = f m µ + f q µ which disturbs microscopic motion ofthe baryonic matter and “violates” the law of conservation of its stress-energy tensor.50 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 The important case of the baryonic matter is a perfect fluid with the stress-energy tensor (175), T µν = ( ǫ + p ) u µ u ν + pg µν , (305)where ǫ and p are the energy density and pressure of the fluid, u µ is the four-velocity of the fluid element, and g µν isa full (contravariant) metric. The pressure, p = p ( µ ), and the energy density, ǫ = ǫ ( µ ), are functions of the specificenthalpy, µ , of the fluid defined by µ = q − g µν Θ µ Θ ν , (306)where Θ µ ≡ ∂ µ Θ , and Θ is the Clebsch potential of the baryonic fluid. The baryonic mass density ρ is defined bythermodynamic equation ρµ = ǫ + p . (307)Substituting (305) to the left side of (304)and projecting this equation on the four-velocity u α of the baryonic fluid,we get the post-Friedmannian law of conservation of energy u µ ∇ µ ǫ + ( ǫ + p ) ∇ µ u µ = − u µ (cid:16) f m µ + f q µ (cid:17) , (308)and the post-Friedmannian Euler equation( ǫ + p ) u ν ∇ ν u µ = ( g µν + u µ u ν ) (cid:16) −∇ ν p + f m ν + f q ν (cid:17) . (309)One more equation is obtained by direct variation of the Lagrangian of the baryonic matter with respect to the Clebschpotential, leading to the law of conservation of baryonic mass density ρ ∇ µ ( ρ u µ ) = , (310)which is an exact relation.
8. Discussion
The present paper employs a new gauge-invariant approach to the theory of cosmological perturbations. Thisapproach utilizes the dynamic field theory on curved geometric manifolds introduced by Bruce DeWitt [87], andrepresents a systematic development of the iterative scheme for deriving a decoupled system of field equations for theperturbations of the metric tensor and material fields considered as dynamic variables on background FLRW manifold.We also demonstrate how to formulate the covariant equations of motion for the perturbations of the material variableslike density, pressure, velocity of matter, etc., on the expanding spacetime of FLRW universe.The original motivation for the development of the dynamic field theory of the gauge-invariant perturbations incosmology was the task of generalization of the post-Minkowskian (PMA) and post-Newtonian (PNA) approximationschemes used in experimental gravitational physics for testing general relativity in the solar system, binary pulsars,other localized astronomical systems like the Milky Way [14, 86, 106–108], and in gravitational wave astronomy[15, 16, 18, 41, 88, 109] for studying the process of generation, propagation, and emission of gravitational wavesby the isolated system comprised of massive bodies. Standard PMA and PNA schemes assume that the backgroundspacetime is asymptotically flat which does not correspond to cosmological observations clearly indicating that thebackground spacetime is described by the curved FLRW metric. Therefore, the standard PMA and PNA schemesare totally missing cosmological e ff ects which can become important in discussion of certain experimental situations[71, 110].Earlier existing perturbation frameworks in cosmology developed by Lifshitz [54, 55], Bardeen [57], Mukhanov et al [3, 111], Ellis et al [62, 63, 112] made use a principle of separation of the metric tensor perturbations in scalar,vector, and tensor harmonics but it does not comply with the theoretical foundation of experimental gravitationalphysics in asymptotically-flat spacetime [14, 106]. For this reason, we do not use the scalar-vector-tensor decomposi-tion of the metric tensor but operate directly with the components of the metric tensor perturbations and scalar fieldsfor which we derive the gauge-invariant equations as described in sections 4.6 and 4.7. Moreover, those perturbativeapproaches of previous researchers did not clearly separate the gravitational e ff ects of small-scale and large-scale51 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 inhomogenities of matter so that it remained fuzzy how to split the matter and gravitational field of an astronomi-cal N-body system, which is an external perturbation of the background geometry, from the matter and gravitationalfield caused by the primordial perturbations of the background matter including cosmological gravitational waves.Recently, Green and Wald [45, 46] have developed a new framework for separation of the gravitational e ff ects of thesmall and large-scale inhomogeneities in cosmology based on generalized Burnetts shortwave approximation [113].Green-Wald’s framework resembles the similar approach developed by Futamase [32, 50], but it has considerablywider applicability.We compared the Lagrangian-based framework of the present paper with that of Green and Wald and noticedthat both frameworks are based on somewhat similar assumptions. In particular, both approaches admit that there isa background spacetime metric, ¯ g αβ , which is kept arbitrary for development of general formalism (until section 5in this paper). Both approaches assume that the metric tensor perturbations, h αβ = g αβ − ¯ g αβ , are small so that theperturbative series are conjectured to be convergent, but no restrictions like, h αβ,µν ≪ h αβ,µ ≪ h αβ , are imposed onthe first and second derivatives of the metric tensor perturbations. Both Green and Wald, and we, allow the matterperturbations to have a high-density contrasts, δρ/ ¯ ρ ≫ ff erent physicalorigin depending on the situation under discussion. Further comparative analysis of the results of the present paperand those of Green and Wald [45, 46] revealed the following:1. We consistently rely upon the perturbative approach to develop the dynamic field theory of cosmological per-turbations and never assume, for example, that quadratic products of the first derivatives h αβ,µ are of the sameorder as the curvature of the background metric. Thus, we do not include explicitly to our scheme the case ofgeneration of the background metric ¯ g αβ by the small-scale perturbations of the metric and / or its derivatives, viashort-wave averaging of Einstein’s equations like Green and Wald [45, 46] did. We do admit the back-reactionof the metric perturbations (both small and large scale) on the background metric but it can produce in ourapproach only small pertubative corrections to the expansion rate of the universe. In this sense the short-waveapproximation approach in cosmology developed by Green and Wald [45, 46] seems to have wider application,at least in the geometric sector of the theory.2. We assume that the background metric, ¯ g αβ , obeys Friedmann’s equations exactly (see section 5.4) while Greenand Wald derived the di ff erential equations governing the evolution of the background metric by making useof the short-wave approximation of Burnett [113]. It means that in our approach the dynamic evolution ofthe background metric is driven exclusively by the background value of the stress-energy tensor of the back-ground dark matter and dark energy while the e ff ective stress-energy tensors of the gravitational and matterperturbations (see section 6) do not contribute to the background value of the metric of FLRW manifold.3. Green and Wald [45, 46] separate the cosmological perturbations of the background matter and gravitationalfield in short wavelength (index ( S )) and long wavelength (index ( L )) perturbations which are treated di ff erentlyby making use of additional assumptions and / or limitations on the mathematical behaviour of the perturbationsdepending on the expansion parameter λ (see section III in [45]). The analogue of the short wavelength pertur-bations in our approach are the bare perturbations. They are described by the particular solutions of the fieldequations (205) while the long wavelength perturbations of the background metric tensor are given by theirhomogeneous solution. The bare perturbations correspond to the gravitational field in the Newtonian limit ofN-body problem in cosmology and are caused by the bare stress-energy tensor of baryonic matter making upstars, galaxies and their clusters.4. The present paper makes use of a systematic dynamic-field approach based on the Noether’s variational prin-ciple to disentangle the background quantities from the perturbations in the iterative sense, and to derive thefield equations for the perturbations at each iteration by taking variational derivatives. This makes our approachfully algorithmic and the process of calculation of the variational derivatives can be written down as a recursivecomputer program. In principle, we can calculate the field equations for perturbations of any order starting fromthe background Lagrangian while Green-Wald’s approach [45, 46] is less algorithmically formalized and getsmore and more laborious as one goes to higher approximations.5. Green and Wald [45, 46] directly operated with the Einstein tensor, G αβ = R αβ − (1 / g αβ R , to decompose itinto the background and perturbative parts and to derive the field equations for the perturbations. However, they52 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 did not pay su ffi cient attention to the structure of the perturbation of the stress-energy tensor which is clearlyshown in their equations (see, for example, T (1) ab in [45, eq. 87]) but remains unspecified. The perturbation of thebackground stress-energy-tensor of matter contains the linear-in-metric-tensor perturbation terms which shouldbe included to the left side of the field equations for the metric tensor perturbations. Our approach carefullytreats this problem of extracting the linear-in-metric tensor perturbations terms to formulate the linear operatorof the field equations for the perturbations of the dynamic variables (see how the left side of the linearised fieldequation (205) is defined).6. At last, but not least, we notice the di ff erence in the choice of the gauge condition (206) used in the presentpaper and that in [45] (see discussion at the end of section III in [45] and [45, eq. 91]). No doubt, that the gaugecondition is, in a sense, a matter of taste of a researcher serving to one or another particular task. The advantageof our gauge condition (206) is that it allows to decouple the field equations for the metric perturbations intime domain and put them into the form being very similar to that implemented in the canonical PMA andPNA approaches used for testing general relativity. It allows to compare the results of the cosmological tests ofgeneral relativity to those performed in the solar system much more easier. Furthermore, our gauge condition(206) reduces the field equations for the dynamic variables to the Bessel-type wave equations which have well-defined retarded Green functions and can be solved in terms of the retarded integrals [114–116].Our field-theoretical approach to cosmological perturbations can be extended to incorporate more general physicalsituations. One of the main advantages of our formalism is the method of treatment of the perfect fluid as a dynamicfield that makes its e ff ects to be very similar to those produced by a scalar field [117]. It is straightforward to includein our approach more realistic fluids with entropy, viscosity, anisotropic stresses, etc. The Lagrangian for such fluidshave been discussed in a number of papers [91, 98, 118, 119], and is well-established. It is also possible to incorporateto our formalism additional vector and tensor fields which may be important for researchers doing quantum gravityand / or looking for violations of general relativity on cosmological scales [120–122].Finally, it would be highly desirable to apply our dynamic field theory approach to perform calculations of grav-itational radiation emitted by isolated astronomical sources (like binary stars) with taking into account various cos-mological e ff ects. This will yield a key to precise measurement of cosmological parameters with gravitational wavedetectors. So far, this problem was considered only under assumption that spacetime is asymptotically flat (see review[18] and references therein), thus, severely limiting the domain of possible fundamental applications of gravitationalwave astronomy to cosmology. Acknowledgements
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A.1. Variational derivative from the Hilbert Lagrangian
The goal of this section is to prove relation (74) being valid on the background manifold ¯ M . We shall omit the barover the background geometric objects as it does not bring about confusion. We notice that the Hilbert Lagrangiandensity, L G = − (16 π ) − √− gR , di ff ers from the Einstein Lagrangian density L E = − (16 π ) − √− gL by a total derivativethat is a consequence of (16). Due to relation (32) the Lagrangian derivatives from L G and L E coincides δ L G δ g µν = δ L E δ g µν , (A1)thus, pointing out that we can safely operate with the Einstein Lagrangian density L E . Because of (54), we have δ L E δ g µν = √− g A ρσµν δ L E δ g ρσ , (A2)which suggests that calculation of the variational derivative with respect to the metric tensor is su ffi cient.Calculation of the variational derivative δ L E /δ g ρσ demands the partial derivatives of the contravariant metric andChristo ff el symbols with respect to g µν . The partial derivatives of the metric are calculated with the help of (38),(49). The Christo ff el symbols are given in terms of the partial derivatives from covariant metric tensor, g αβ,γ whichare not conjugated with the dynamic variable g αβ . Thus, calculation of the partial derivative with respect to g µν fromthe Christo ff el symbols demands its transformation to the form where the conjugated variables g αβ,γ are used instead.This form of the Christo ff el symbols is Γ αβγ = (cid:16) g ρκ,σ g ασ g ρβ g κγ − g ασ,β g γσ − g ασ,γ g βσ (cid:17) . (A3)Taking the partial derivative of (A3) with respect to the contravariant metric yields ∂ Γ αβγ ∂ g µν = − g ασ n Γ [ σβ ]( µ g ν ) γ + Γ [ σγ ]( µ g ν ) β + Γ ( βγ )( µ g ν ) σ o , (A4)and ∂ Y α ∂ g µν = − Γ ( µν ) α , (A5)where we have used (3). Contracting (A4), (A5) with the Christo ff el symbols and the metric tensor results in g σγ ∂ Γ αβγ ∂ g µν Γ βσα = − Γ αβµ Γ βνα , (A6) g σγ ∂ Y β ∂ g µν Γ βσγ = − Γ ( µν ) α Γ α , (A7) g σγ ∂ Γ βσγ ∂ g µν Y β = Γ ( µν ) α Y α − Γ αµν Y α − Y µ Y ν . (A8)Partial derivatives of the Christo ff el symbols with respect to the metric derivatives are calculated from (A3) withthe help of (39). We get ∂ Γ αβγ ∂ g µν,ρ = h g ρα g β ( µ g ν ) γ − δ ργ δ α ( µ g ν ) β − δ ρβ δ α ( µ g ν ) γ i , (A9) ∂ Y β ∂ g µν,ρ = − g µν δ ρβ . (A10)58 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 Contracting (A9), (A10) with the Christo ff el symbols and the metric tensor results in g σγ ∂ Γ αβγ ∂ g µν,ρ Γ βσα = − Γ ρµν , (A11) g σγ ∂ Y β ∂ g µν,ρ Γ βσγ = − g µν Γ ρ , (A12) g σγ ∂ Γ βσγ ∂ g µν,ρ Y β = g µν Y ρ − δ ρ ( µ Y ν ) . (A13)Explicit expression for the variational derivative of the Einstein Lagrangian is − π δ L E δ g µν = ∂ √− g ∂ g µν g σγ + √− g ∂ g σγ ∂ g µν ! (cid:16) Γ αβγ Γ βσα − Y β Γ βσγ (cid:17) (A14) + √− gg σγ ∂ Γ αβγ ∂ g µν Γ βσα − ∂ Y β ∂ g µν Γ βσγ − ∂ Γ βσγ ∂ g µν Y β ! − ∂∂ x ρ " √− gg σγ ∂ Γ αβγ ∂ g µν,ρ Γ βσα − ∂ Y β ∂ g µν,ρ Γ βσγ − ∂ Γ βσγ ∂ g µν,ρ Y β ! . Replacing the partial derivatives in (A14) with the corresponding right sides of equations (38), (49), (A11)–(A13) andtaking the partial derivative with respect to spatial coordinates, yields − π δ L E δ g µν = √− g R µν − g µν R ! , (A15)where we have used expressions (12), (14) for the Ricci tensor and Ricci scalar respectively. Substituting equation(A15) to (A2) yields δ L E δ g µν = − π R µν . (A16) A.2. Variational derivatives of dynamic variables with respect to the metric tensorA.2.1. Variational derivatives of dark matter variables
The primary thermodynamic variable of dark matter is µ m defined in (157). Variational derivative from µ m iscalculated directly from its definition and yields δ ¯ µ m δ ¯ g µν =
12 ¯ µ m ¯ u µ ¯ u ν . (A17)Variational derivative of pressure ¯ p m is obtained from thermodynamic relation (151a) by making use of the chaindi ff erentiation rule along with (A17), that is δ ¯ p m δ ¯ g µν =
12 ¯ ρ m ¯ µ m ¯ u µ ¯ u ν . (A18)Variational derivative of the rest mass and energy density are obtained by making use of (A17) along with equationof state that allows us to express partial derivatives of ρ m and ǫ m in terms of the variational derivative for µ m . Morespecifically, δ ¯ ρ m δ ¯ g µν = c c ¯ ρ m ¯ u µ ¯ u ν , (A19) δ ¯ ǫ m δ ¯ g µν = c c ¯ ρ m ¯ µ m ¯ u µ ¯ u ν , (A20)59 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 where the speed of sound appears explicitly. Variational derivatives from products and / or ratios of the thermodynamicquantities are calculated my applying the chain rule of di ff erentiation and the above equations, δ ( ¯ ρ m ¯ µ m ) δ ¯ g µν = + c c ! ¯ ρ m ¯ µ m ¯ u µ ¯ u ν , (A21) δδ ¯ g µν ¯ ρ m ¯ µ m ! = − − c c ! ¯ ρ m ¯ µ m ¯ u µ ¯ u ν , (A22) δ ( ¯ p m − ¯ ǫ m ) δ ¯ g µν = − c c ! ¯ ρ m ¯ µ m ¯ u µ ¯ u ν . (A23) A.2.2. Variational derivatives of dark energy variables
The primary thermodynamic variable of dark energy is ¯ µ q defined in (157). Variational derivative from ¯ µ q iscalculated directly from its definition, δ ¯ µ q δ ¯ g µν =
12 ¯ µ q ¯ u µ ¯ u ν . (A24)Variational derivative of the mass density ¯ ρ q of the dark energy “fluid” follows directly from ¯ ρ q = ¯ µ q , and reads δ ¯ ρ q δ ¯ g µν =
12 ¯ ρ q ¯ u µ ¯ u ν . (A25)Variational derivative of pressure ¯ p q is obtained from definition (171) along with (A24), which yields δ ¯ p q δ ¯ g µν =
12 ¯ ρ q ¯ µ q ¯ u µ ¯ u ν . (A26)Variational derivative of energy density ¯ ǫ q is obtained by making use of (A24) along with (170). More specifically, δ ¯ ǫ q δ ¯ g µν =
12 ¯ ρ q ¯ µ q ¯ u µ ¯ u ν . (A27)Variational derivatives from products and ratios of other quantities are calculated my making use of the chain rule ofdi ff erentiation and the above equations δ (cid:16) ¯ ρ q ¯ µ q (cid:17) δ ¯ g µν = ¯ ρ q ¯ µ q ¯ u µ ¯ u ν , (A28) δδ ¯ g µν ¯ ρ q ¯ µ q ! = , (A29) δ (cid:16) ¯ p q − ¯ ǫ q (cid:17) δ ¯ g µν = . (A30) A.2.3. Variational derivatives of four-velocity of the Hubble flow
Variational derivatives from four-velocity of the fluid are derived from the definition (197) of the four-velocitygiven in terms of the potential ¯ Φ or ¯ Ψ which are independent dynamic variables that do not depend on the metrictensor. Taking variational derivative from (197) and making use either (A17) or (A24) we obtain δ ¯ u α δ ¯ g µν = −
12 ¯ u α ¯ u µ ¯ u ν , (A31) δ ¯ u α δ ¯ g µν = −
12 ¯ u α ¯ u µ ¯ u ν − ¯ g α ( µ ¯ u ν ) , (A32) δ (¯ u α φ α ) δ ¯ g µν = − φ ( µ ¯ u ν ) −
12 ¯ u µ ¯ u ν (¯ u α φ α ) , (A33)where equation (A33) accounts for the fact that φ α is an independent variable that does not depend on the metrictensor. 60 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 A.2.4. Variational derivatives of the metric tensor perturbations
Variational derivatives from the metric tensor perturbations l αβ are determined by taking into account that l αβ = h αβ / √ ¯ g and h αβ is an independent dynamic variable which does not depend on the metric tensor. Therefore, itsvariational derivative is nil, and we have δ l αβ δ ¯ g µν = δδ ¯ g µν h αβ √ ¯ g ! = h αβ δδ ¯ g µν √ ¯ g ! = − l αβ ¯ g µν . (A34)Other variational derivatives are derived by making use of tensor operations of rising and lowering indices with thehelp of ¯ g αβ and applying from (A34). It gives δ l αβ δ ¯ g µν = − l αβ ¯ g µν + l α ( µ δ ν ) β , (A35) δ l δ ¯ g µν = l µν − l ¯ g µν , (A36) δ q δ ¯ g µν = − q ¯ u µ ¯ u ν +
12 ¯ g µν ! +
12 ( l µν + l ¯ u µ ¯ u ν ) , (A37) δδ ¯ g µν l αβ l αβ − l ! = l α ( µ l ν ) α − ll µν − ¯ g µν l αβ l αβ − l ! . (A38) A.3. Variational derivatives with respect to matter variablesA.3.1. Variational derivatives of dark matter variables
The dark matter variables do not depend on the Clebsch potential ¯ Φ directly but merely on its first derivatives ¯ Φ α .Therefore, any variational derivative of dark matter variable, say, Q = Q ( ¯ Φ α ), is reduced to a total divergence δ Q δ ¯ Φ = − ∂∂ x α ∂ Q ∂ ¯ Φ α . (A39)We present a short summary of the partial derivatives with respect to ¯ Φ α . ∂ ¯ µ m ∂ ¯ Φ α = ¯ u α , (A40) ∂ ¯ p m ∂ ¯ Φ α = ¯ ρ m ¯ u α , (A41) ∂ ¯ ρ m ∂ ¯ Φ α = c c ¯ ρ m ¯ µ m ¯ u α , (A42) ∂ ¯ ǫ m ∂ ¯ Φ α = c c ¯ ρ m ¯ u α , (A43) ∂ ( ¯ ρ m ¯ µ m ) ∂ ¯ Φ α = + c c ! ¯ ρ m ¯ u α , (A44) ∂∂ ¯ Φ α ¯ ρ m ¯ µ m ! = − − c c ! ¯ ρ m ¯ µ ¯ u α , (A45) ∂ ( ¯ p m − ¯ ǫ m ) ∂ ¯ Φ α = + − c c ! ¯ ρ m ¯ u α . (A46)Partial derivatives of four velocity ∂ ¯ u α ∂ ¯ Φ β = − ¯ P αβ ¯ µ m , ∂ ¯ u α ∂ ¯ Φ β = − ¯ P αβ ¯ µ m . (A47)61 . M. Kopeikin and A. N. Petrov / Annals of Physics 00 (2018) 1–62 It allows us to deduce, for example, ∂ (¯ u α φ α ) ∂ ¯ Φ β = − µ m ¯ P αβ φ β , (A48) ∂ q ∂ ¯ Φ α = − µ m ¯ P αµ l µν ¯ u ν . (A49) A.3.2. Variational derivatives of dark energy variables
The dark energy variables depend on both the scalar potential ¯ Ψ and its first derivative ¯ Ψ α in the most genericsituation. This is because there is a potential of the scalar field W ( ¯ Φ ) that is absent in case of the dark matter.Therefore, variational derivative of the dark energy variable, say, ⊣ = A ( ¯ Ψ , ¯ Ψ α ), is δ A δ ¯ Ψ = ∂ A ∂ ¯ Ψ − ∂∂ x α ∂ A ∂ ¯ Ψ α . (A50)Partial derivatives ∂ A /∂ ¯ Ψ = ( ∂ A /∂ W )( ∂ W /∂ ¯ Ψ , and their particular form depends on the shape of the potential W . Asfor the patial derivatives with respect to the derivatives of the field, they can be calulated explicitly for each variable,and we present a short summary of these partial derivatives below. More specifically, ∂ ¯ µ q ∂ ¯ Ψ α = ¯ u α , (A51) ∂ ¯ p q ∂ ¯ Ψ α = ¯ ρ q ¯ u α , (A52) ∂ ¯ ρ q ∂ ¯ Ψ α = ¯ u α , (A53) ∂ ¯ ǫ q ∂ ¯ Ψ α = ¯ ρ q ¯ u α , (A54) ∂ (cid:16) ¯ ρ q ¯ µ q (cid:17) ∂ ¯ Ψ α = ρ q ¯ u α , (A55) ∂∂ ¯ Ψ α ¯ ρ q ¯ µ q ! = , (A56) ∂ (cid:16) ¯ p q − ¯ ǫ q (cid:17) ∂ ¯ Ψ α = . (A57)Partial derivatives of four velocity ∂ ¯ u α ∂ ¯ Ψ β = − ¯ P αβ ¯ µ q , ∂ ¯ u α ∂ ¯ Ψ β = − ¯ P αβ ¯ µ q . (A58)It allows us to deduce, for example, ∂ (¯ u α ψ α ) ∂ ¯ Ψ β = − µ q ¯ P αβ ψ β , (A59) ∂ q ∂ ¯ Ψ α = − µ q ¯ P αµ l µν ¯ u ν ..