Review Article | Open Access
Boris E. Meierovich, "Macroscopic Theory of Dark Sector", Journal of Gravity, vol. 2014, Article ID 586958, 23 pages, 2014. https://doi.org/10.1155/2014/586958
Macroscopic Theory of Dark Sector
A simple Lagrangian with squared covariant divergence of a vector field as a kinetic term turned out to be an adequate tool for macroscopic description of the dark sector. The zero-mass field acts as the dark energy. Its energy-momentum tensor is a simple additive to the cosmological constant. Massive fields describe two different forms of dark matter. The space-like massive vector field is attractive. It is responsible for the observed plateau in galaxy rotation curves. The time-like massive field displays repulsive elasticity. In balance with dark energy and ordinary matter it provides a four-parametric diversity of regular solutions of the Einstein equations describing different possible cosmological and oscillating nonsingular scenarios of evolution of the Universe. In particular, the singular big bang turns into a regular inflation-like transition from contraction to expansion with the accelerated expansion at late times. The fine-tuned Friedman-Robertson-Walker singular solution is a particular limiting case at the lower boundary of existence of regular oscillating solutions in the absence of vector fields. The simplicity of the general covariant expression for the energy-momentum tensor allows displaying the main properties of the dark sector analytically. Although the physical nature of dark sector is still unknown, the macroscopic theory can help analyze the role of dark matter in astrophysical phenomena without resorting to artificial model assumptions.
Currently there are two most intriguing long standing problems in astrophysics pointing to the existence of so called “hidden sector,” containing “dark energy” and “dark matter.” So far their interaction with the ordinary matter (baryons and leptons) is observed only via gravitation.
The first problem, named “galaxy rotation curves,” appeared in 1924, after Oort discovered the galactic halo, a group of stars orbiting the Milky Way outside the main disk . In 1933, Zwicky  postulated “missing mass” to account for the orbital velocities of galaxies in clusters.
At first glance, these two problems have little to do with one another. The accelerated expansion of the Universe indicates the existence of a hidden mechanism of repulsion , while the plateau of the galaxy rotation curves is the result of additional attraction caused by the dark matter . Macroscopic approach to the dark sector problems, based on the analysis of vector fields in general relativity, provides an appropriate universal tool for the theoretical description of both these phenomena. The space-like massive vector field is attractive. It is responsible for the observed plateau in galaxy rotation curves. The time-like massive vector field displays repulsive elasticity. In the scale of the whole Universe it is the source of accelerated expansion. Naturally, the previous solutions of the Einstein equations, describing the expansion of the Universe filled with the mutually attracting matter only, inevitably contained a singularity. Inclusion of the repulsive dark matter into consideration allows the existence of nonsingular solutions describing various possible regular scenarios of evolution of the Universe.
This paper contains the macroscopic theory of dark sector, based on the analysis of vector fields in general relativity. The step by step derivations are accompanied by the references to the benchmark achievements of the predecessors. The main attention is paid to clarify the validity of basing assumptions.
Vector fields are used to describe quantum particles of the ordinary matter . A zero-mass particle—photon—is a quantum of electromagnetic field obeying Maxwell equations. Massive bosons obey Proca equations [8–11]. The relation of spinors and vectors (, page 88) facilitates establishing Dirac equations for fermions.
Field equations for quantum particles are easily established in accordance with the properties of their free motion in plane geometry. If necessary (for the secondary quantization, for instance), the Lagrangian of a particle is then constructed in such a way that the field equations minimize the functional of action. This approach is convenient for description of already known particles. However, it does not help to describe the unknown substance of the dark sector.
In general relativity, the standard approach starting from a general form of the Lagrangian of a vector field is capable to describe not only the already known particles. It is reasonable to start from a general form of the Lagrangian of a vector field in general relativity and derive the vector field equations and the energy-momentum tensor. Excluding the terms associated with the ordinary matter, one can separate the Lagrangian having a chance to describe the dark sector. The separation of the Lagrangian of the dark sector is necessary, especially if the ordinary matter in the Universe is considered as a continuous medium with the macroscopic energy-momentum tensor (56). Otherwise the ordinary matter would be taken into account twice: as a medium with the energy-momentum tensor (56) and as quantum particles described by the vector field.
It turns out that the most simple Lagrangian of a vector field (58) (with the squared covariant divergence as a kinetic term) allows to describe the main observed manifestations of the dark sector completely within the frames of the minimal general relativity. In this case, the massless field corresponds to the dark energy, the massive space-like field is responsible for a plateau in the galaxy rotation curves, and the massive time-like vector field displays the repulsive elasticity. The competition of repulsive dark matter and attractive ordinary matter leads to a variety of possible regular scenarios of evolution of the Universe.
According to the NASA “sliced cake” diagram  (see Figure 1) today there is only 4.6% of the ordinary matter among the stuff of the Universe. All other 95% is the unknown substance, referred to as the dark matter and dark energy. For this reason it is natural to start with the analysis of the properties of dark sector and add the ordinary matter into consideration after the role of dark sector is clarified.
The main properties of vector fields in general relativity are recalled in Section 2 in order to clarify the specifics of the approach based on the principle of regularity. The features of the Proca equations allow separating the terms in the Lagrangian, which are not connected with the ordinary matter. It turns out that a simple Lagrangian (with the squared covariant divergence as a kinetic term) is a proper tool for macroscopic description of main observed properties of the dark sector. The necessary conditions of regularity for a space-like and a time-like vectors are different. Nevertheless, the field equations and the energy-momentum tensor have the same covariant form for both kinds of vector fields. The simplicity of equations allows getting analytical solutions in the most interesting cases. The galaxy rotation curves, driven by the space-like vector fields, are derived in Section 3. Various uniform and isotropic scenarios of evolution of the Universe under the joint influence of the zero-mass vector field, the time-like massive vector field, and the ordinary matter are analyzed in Section 4. The current situation with the dark sector is summarized in Section 5. Some major astrophysical problems are specified, where the macroscopic theory can be applied, helping to avoid the unnecessary model assumptions. Unraveling the still unknown physical nature of the dark energy and dark matter remains the most pressing issue.
2. Vector Fields in General Relativity
2.1. Lagrangian of a Vector Field
Within the frames of minimal general relativity (field equations no higher than of the second order), the Lagrangian of a vector field is a scalar consisting of bilinear combinations of the covariant derivatives and a scalar potential . A bilinear combination of the covariant derivatives is a 4-index tensor A general form of the scalar , formed via contractions of , is where , , and are arbitrary constants. Therefore, a general form of the Lagrangian of a vector field , contains three kinetic terms with the arbitrary coefficients , , and . Applying the least action principle, in view of it is convenient to consider the Lagrangian (3) as a function of , , , and as independent variables.
2.1.1. Bumblebee Models
Strictly speaking, (3) is not yet the most general form of the Lagrangian of a vector field. Scalars can be made out of not only via contractions, but also by convolutions with participation of , like , and . In principle, the number of independent constants can exceed 3.
The direction of a vector is specified, and solutions can be less symmetric than the initial Lagrangian. In this case the symmetry of a system is considered as spontaneously broken. Spontaneous breaking of Loretz symmetry by a vector field , having a nonzero expectation value in vacuum, is a subject of so called “bumblebee theories;” see [13–16] and references therein. Within the frames of field equations no higher than of the second order, the most general form of the action is Here is the Riemann tensor of curvature and , and are arbitrary tensors formed out of the vector and the metric tensor . In practice people restrict themselves by simplified models. For instance, Seifert  considered recently the particular case , , where and and ars constants. The potential is taken to have a minimum at some nonzero value of its argument . In this model, a perturbation of a Lorentz-violating vector field can be interpreted as a photon field.
Why “bumblebee”? Perhaps, there was something looking as strange as the ability of the insect bumblebee to fly successfully despite being sometimes questioned on theoretical grounds . By the way, a possibility of existence of macroscopic objects, moving at the speed of light and having zero mass due to gravitational mass-defect, had been mentioned by Andreev back in 1973 [19, 20].
While a vector remains small compared to its vacuum expectation value at a minimum of , the Lagrangian (3) is sufficient. However, in case of a small mass (108) the longitudinal massive field ; see (110). If in the process of compression the field comes close to its vacuum expectation value, then a phase transition with spontaneous symmetry breaking occurs. The bumblebee approach looks promising for future consideration of a symmetry breaking phase transition in the state of maximum compression.
2.1.2. Specificity of Curved Space-Time
The third kinetic term in (3) can be transformed via differentiation by parts to Here is a total differential. It does not change the integral of action, and one can use the equivalent Lagrangian instead of (3).
In general relativity the second covariant derivative of a vector is not invariant against the replacement of the order of differentiation: is the Riemann tensor of curvature. Hence where is the Ricci tensor. In a curved space-time , and the term affects the integral of action. From the point of view of general relativity all three kinetic terms in (3) are equally important. If we adhere the view that in the quantum physics each elementary particle is a quantum of some field and vice versa each field corresponds to its own quantum particle , then, in principle, any linear combination of the three kinetic terms in the Lagrangian (3) could be associated with some sort of matter.
In the plane geometry , the term drops out, a covariant derivative reduces to the ordinary one , and there are only two arbitrary constants (, and ) in the Lagrangian:
It is convenient to classify the vector fields according to their properties of invariance and symmetry.
The sign of the scalar is invariant against the arbitrary transformations of coordinates. Therefore, if there is no interaction other than via gravitation, there can be three different independent vector fields with , , and . If , then in general relativity one can choose a reference frame where either when (space-like vector) or if (time-like vector). is a separate case. The field equations for an ordinary massive particle are easily derived basing on the statement that in the plain space-time there is a reference frame where the particle is at rest . Hence, the ordinary massive particles are described by the space-like fields. For the zero-mass particles (such as photons) there is no reference system, where they are at rest. The photons are associated with the massless vector field . The time-like vector fields can not be associated with the ordinary massive particles because in the case there is no frame where . Nevertheless, one can not deny the existence of some substance corresponding to a time-like field. From the general relativity view point all three kinetic terms in the Lagrangian (3) are equally important, as well as any of the three types of vector fields could describe some sort of matter.
The covariant derivative can be presented as a sum of a symmetric and an antisymmetric parts: In view of the scalar (2) can be presented in the form The last term with antisymmetric derivatives is identical to electromagnetism. It becomes clear in common notations , . The bilinear combination of the antisymmetric derivatives is the same as in electrodynamics. In view of the symmetry of Christoffel symbols , and the scalar does not depend on the derivatives of the metric tensor. On the contrary, the two first terms in (12) with symmetric covariant derivatives contain not only the components of the metric tensor , but also the derivatives . The difference between the two terms with symmetric tensors is caused by the curvature of space-time.
2.2. Regularity in General Relativity
In the notations the Lagrangian is
In plane space-time it is necessary to require . Otherwise for a space-like vector the action could not have a minimum as required by the least action principle. If is negative, then it is possible to make the action negative with an arbitrarily large absolute value via fairly rapid change of with time (within the considered time interval); see , page 98.
In the regular solutions of the Einstein equations all invariants of the Riemann curvature tensor are finite. Hence, the invariants of the Ricci tensor are finite too. By virtue of Einstein equations the requirement of regularity automatically excludes a possibility to achieve an infinite value for all the invariants of the energy-momentum tensor . In general relativity the distribution/motion of matter and the curvature of space-time are mutually balanced. Practically, there is no need to require in advance. Necessary restrictions, if any, on the signs of the constants , , and arise as a consequence of the condition of regularity.
The requirement that all the invariants of the Riemann curvature tensor are finite is a necessary condition of regularity in general relativity.
2.3. Vector Field Equations
The vector field obeys the Eiler-Lagrange equations In terms of , , and the variational derivative is For we have and are considered as independent variables in the Lagrangian (3). In a locally geodesic reference system (where the Christoffel symbols together with the derivatives are zeros) and . Here The vector field equations (16), having a covariant form in a locally geodesic system, remain the same in all other reference frames.
In terms of , and
There are two independent terms with the symmetric tensor in (21) and one with the antisymmetric tensor . The physical origin of the two symmetric terms is connected with the curvature of space-time. It becomes clear if we set . Then , , and (21) reduces to The two left terms differ by the order of differentiation. In accordance with (9) the difference between the two terms in (22) exists only in the curved space-time: In the flat space-time the Ricci tensor , and in case there is no physical difference between the two left terms in (22).
If the vector field is weak, so that the second and higher derivatives of the potential can be neglected, then (20) are linear, and the principle of superposition takes place, as it should be in the case of free (noninteracting) fields.
2.3.1. Proca Equations
In the particular case , the field equations (20) reduce to the Proca equations [8–11] in the case of no sources: Usually is referred to as the “mass” of a field. Proca equations are used to describe a free massive spin-1 particle. In the case of a massless field (25) reduce to the Maxwell equations.
For any tensor the scalar is symmetric with respect to the lower indexes. Renaming the blind indexes, in view of the symmetric properties of the Riemann and Ricci tensors we have The scalar because in accordance with (27) , while for the antisymmetric tensor . Thus, it follows from the Proca equations (25) that in the particular case , the covariant divergence of the vector is zero: In electrodynamics is referred to as Lorentz gauge. The fact that does not mean that the Proca equations are gauge invariant. (29) is the consequence of the particular choice , .
If still , but , we have The Lorentz gauge restriction (29) would automatically exclude the case from the consideration. Applying to (30), the term ~ drops out in view of (28), and, instead of the Lorentz condition (29), the scalar obeys the Klein-Gordon equation In plain space-time “the additional condition (29) excludes the part of belonging to spin 0” , page 72. In general relativity the condition (29) excludes also longitudinal vectors, existing in curved space-time. As it turns out, these longitudinal vector fields are just suitable for the description of dark matter and energy.
In the most simple case the field equations (24) reduce to if (massless field) and to if (massive field).
The covariant divergence of a zero-mass field remains constant through the whole space-time: The fact that “the gauge-fixing term exactly behaves as a cosmological constant throughout the history of the Universe, irrespective of the background evolution” had been mentioned by Beltran Jimenez and Antonio . The massive field has a potential: it is a gradient of the scalar obeying (31).
2.3.2. Einstein-Aether Models
Vector field considered as the gradient of a scalar field , , was used in a scalar variant of the Einstein-aether theory; see a recent paper by Haghani et al. , providing a brief comprehensive review of the topic. Einsten-aether theories [25, 26] consider phase transitions with spontaneous violation of Lorentz symmetry by a vector field whose nonzero vacuum expectation value plays the role of the order parameter. “Einstein-aether” is a kind of bumblebee models oriented mainly on time-like vector fields. It is not clear how to associate a time-like vector field with a massive quantum particle of ordinary matter, because there is no reference frame where such a particle could be at rest. The word “aether” in a title reflects the situation that a time-like vector field should correspond to something different from the ordinary matter.
A tipical action in the Einsten-aether theory [27–30] is a particular case of (5) with and In  the potential is taken as , where is a Langrange-Eiler multiplier. The sign being chosen to enforce the vector field to be time-like. Another commonly used example for the potential is a smooth quadratic function .
If the symmetry breaking vector is chosen as a gradient of a scalar, , then it is possible to consider the potential as a function of the scalar only, . In this case a Lagrangian contains not only and , but also second derivatives of . From my point of view, a vector field approach is more convenient than the scalar one. The equations become more simple, while their solutions are more general. Though, in the particular case , the vector is a gradient of a scalar (see (33)), this scalar is a covariant divergence . The Lagrangian contains only and .
In the state of broken symmetry there are 4 constants in (35) instead of 3 constants in (3). If a vector field is small compared to its vacuum expectation value and , then and the second and higher derivatives of can be omitted. While the field is small, the symmetry remains unbroken, and there is no need in the fourth term in (35). On the contrary, the full scale bumblebee models, including the Einsten-aether theory, provide a solid basis for analyzing phase transitions with spontaneous symmetry breaking in a strongly compressed state.
2.3.3. Nongauge Longitudinal Vector Field in Plane Geometry
In the plain centrally symmetric metric , and (31) for a static longitudinal vector field, depending only on the distance from the center, is The center is a singular point of the spherical coordinate system. Like a pole on the globe, the singularity of a coordinate system has nothing to do with physical properties of matter. The finite, regular at , solution of (37) (blue curve in Figure 2) exists if Restriction (39) is a necessary condition of regularity for a space-like vector field.
According to (33) the nonzero component of the vector is : Radial dependences of and are shown in Figure 2. The field is space-like and longitudinal: it is directed along and depends upon the same coordinate .
In plain geometry , and in the case of a longitudinal time-like vector field the Klein-Gordon equation (31) is simply Its solution is finite if and are constants of integration.
In plain space-time conditions of regularity determine the sign of the ratio , which appears different for space-like and time-like fields. In plain space-time the specific sign of is not restricted by the requirement of regularity. However, there is no reason why the sign of should be different for space-like and time-like vector fields. Actually it turns out that the sign of , is restricted by the requirement of regularity in a curved space-time see Section 4.2 below. Then, as it follows from (39), (43), and (44), the regular solutions exist, if is positive for a space-like field and negative for a time-like one.
2.4. Energy-Momentum Tensor
Using the identity the energy-momentum tensor can be expressed as It differs from (94.4) in , where the Lagrangian is a function of and . The form (46) is more convenient when and are considered as independent variables in the Lagrangian (3). In view of in a locally geodesic system (where ) the energy-momentum tensor can be written as follows: It is worth mentioning that should be set to zero after the variational differentiations and are done. In terms of symmetric and antisymmetric tensors (11) we find The tensor (50) is presented in a symmetric form against the indexes .
2.4.1. Checking the Zero of the Covariant Divergence
Using the vector field equations (21), can be presented as The coefficients , and are arbitrary constants. However, it does not mean that the three vectors , , and in (53) are zeros separately. These vectors are reduced to a similar form (see  for details): The covariant divergence of the energy-momentum tensor (53) with , , and , given by (54) is evidently zero due to the vector field equations (21).
The covariant field equations (20) and the energy-momentum tensor (51) describe the behavior of vector fields in the background of any arbitrary given metric . If the back reaction of the field on the curvature of space-time is essential, then the metric obeys the Einstein equations with (51) added to . Here and are the cosmological and gravitational constants, respectively. With account of back reaction the field equations (20) are not independent. They follow from the Einstein equations (55) with (51) due to the Bianchi identities.
2.4.2. Ordinary Matter and Dark Sector
The ordinary matter enters the Einstein equations via the well known energy-momentum tensor of macroscopic objects. Consider The energy , pressure , and, generally speaking, temperature of the ordinary matter obey the equation of state. So far there is no evidence of any direct interaction between dark and ordinary matter other than via gravitation. The gravitational interaction is described by Einstein equations (55) with The general expression (51) for the energy-momentum tensor of a vector field describes (but is not limited to) the vector particles, which are also the ordinary matter. In order to describe the dark sector via vector fields it is reasonable to separate from (51) the part which does not relate to the ordinary matter.
The Proca equations (25) are associated with a spin-1 particle—a quantum of the ordinary matter. The particular choice of the parameters ensures the Lorentz condition (29), which allows avoiding “the difficulties of negative contribution to the energy” . However, in curved space-time the energy is not a scalar, and its sign is not invariant against the arbitrary coordinate transformations. As a result of the Lorentz gauge restriction the terms drop out from the Lagrangian (3), from the vector field equations (20) and from the energy-momentum tensor (51).
To avoid the double contribution of particles of the ordinary matter, it is reasonable to consider and set . As simple a Lagrangian as possible turns out to be an adequate tool for macroscopic description of the dark sector. Accordingly, (20) and the energy-momentum tensor (51) reduce to
Gravity is currently considered the basic interaction of cosmic objects in the scale of galaxies and larger. The three types of vector fields (massless, massive space-like, and massive time-like) can be of different physical nature.
The energy-momentum tensor (60) of a zero-mass vector field reduces to is the constant value of the potential of the massless field. acts in the Einstein equations (55) as a simple addition to the cosmological constant, changing to is the constant divergence of the zero-mass vector field (34).
It is convenient to consider the values of the potentials of massive fields as already included into : so that the power series of the potentials of massive fields start with : In the case of weak vector fields the second and higher derivatives of the potentials can be neglected, and the energy-momentum tensor of a massive field is
In general, it could be necessary to consider both independent vectors— for a space-like and for a time-like—massive fields with different potentials and . As far as the energy-momentum tensor of a massless field (dark energy) is included into (62), the remaining energy-momentum tensor of massive fields is the sum of two tensors corresponding to and , respectively.
In the scale of a galaxy (~10 kpc) the space-like vector field dominates. It is responsible for the plateau in galaxy rotation curves; see Section 3. The time-like field dominates at the scales much larger than the distance between the galaxies, where the Universe can be considered as uniform and isotropic. The time-like field displays the repulsive elasticity. Together with the zero-mass vector field (dark energy) and the ordinary matter it gives rise to a variety of possible regular scenarios of evolution of the Universe and rules out the problem of fine tuning; see Section 4. In particular, the singular Big Bang turns into a regular inflation-like bounce with accelerated expansion at late times.
It would be interesting to trace how the additional attraction of the space-like dark matter, dominating in the galaxy scale, transforms into the elastic repulsion of the time-like dark matter, dominating in the scale of the Universe. Both types of massive fields and are supposed to be active in the intermediate region, so the energy-momentum tensor of the dark sector should be the sum (66).
The study of the structure of the Universe in the intermediate range (Mpc to hundred Mpcs) had been initiated in the pioneering papers by Zel’dovich . Continuous research by his followers shows that dark energy and dark matter significantly affect the structural dynamics of galaxies and clusters in this range; see a review by Gurbatov et al. . Utilizing the energy-momentum tensor ((66), (60)) in the analysis of the large scale structure of the Universe would allow avoiding unnecessary model assumptions.
3. Galaxy Rotation Curves
The description of the dark sector via vector fields allows deriving the galaxy rotation curves directly from the first principles within the minimal Einstein’s general relativity .
The velocity of a star, orbiting around the center of a galaxy and satisfying the balance between the centrifugal and centripetal accelerations, should decrease with the radius of its orbit as at . However, the numerous observed dependences , named galaxy rotation curves, practically remain constant at far periphery of a galaxy. An example is presented in Figure 3. It was a fundamental problem, because the general relativity reduces to Newton’s theory in the limit of nonrelativistic velocities and weak gravitation.
3.1. Benchmarks in History
The “galaxy rotation curves” problem appeared after Oort discovered the galactic halo, a group of stars orbiting the Milky Way outside the main disk . In 1933, Zwicky  postulated “missing mass” to account for the orbital velocities of galaxies in clusters. Persistent investigations by Rubin and colleagues [35, 36] in seventies practically dispelled the skepticism about the existence of dark matter on the periphery of the galaxies.
Among numerous attempts to solve the problem of galaxy rotation curves the most discussed one is the empirical explanation named MOND (modified Newtonian dynamics), proposed by Milgrom back in 1983 [37, 38]. For a relativistic justification of MOND Bekenstein , Sanders , and Brownstein and Moffat [41–43] introduce additional scalar, vector, or tensor fields. Though these (and many others) empirical improvements of MOND are able to fit a large number of samples for about a hundred galaxies, the concern still remains. So far we had neither self-consistent description of the dark sector as a whole nor direct derivation of MOND from the first principles within Einstein’s general relativity. The survey  by Famaey and McGaugh and a recent review article  by Bernabei et al. reflect the current state of research and contain the comprehensive lists of references.
3.2. Rotation Curve Driven by a Massive Vector Field
Applying general relativity to the galaxy rotation problem it is reasonable to consider a static centrally symmetric metric with two functions and depending on only one coordinate—circular radius . Real distribution of stars and planets in a galaxy is neither static nor centrally symmetric. However this simplification facilitates analyzing the problem and allows displaying the main results analytically. If a galaxy is concentrated around a supermassive black hole, the deviation from the central symmetry caused by the peripheral stars is small.
In the “dust matter” approximation and the only nonzero component of the energy-momentum tensor (56) is . Whatever the distribution of the ordinary matter is, the covariant divergence is automatically zero. In the dust matter approximation the curving of space-time by ordinary matter is taken into account, but the back reaction of the gravitational field on the distribution of matter is ignored. If the energy is considered as a given function.
In the space-time with metric (67) the energy-momentum tensor (65) of a weak space-like longitudinal vector field is In the scale of galaxies the role of expansion of the Universe as a whole is negligible, and one can omit (62) in the Einstein equations. In the dust matter approximation the Einstein equations are See , page 382 for the derivation of the left-hand sides. The prime stands for , except . Among the four equations (59), ((70)–(72)) for the unknowns , and any three are independent.
Extracting (71) from (70) we get a relation With account of (68) and (73) the vector field equation (59) takes the form Equations (73) and (74) are derived with no assumptions concerning the strength of the gravitational field.
Omitting the second and higher derivatives of the potential , in the dust matter approximation , we get from (70) the following expression for : Here is the squared mass of the vector field. The sign minus is in conjunction with the necessary condition of regularity (39) for a space-like vector field.
In a static centrally symmetric gravitational field determines the centripetal acceleration of a particle (, page 323). Without dark matter (75) gives Newton’s attractive potential far from the center: The first term in the r.h.s. of (75) appears due to the dark matter.
The constant in the Lagrangian (58) is considered the same for any vector, be it space-like or time-like, massive or massless. Regularity condition for a space-like vector (39) does not determine the sign of . The negative sign (107) is dictated by the self-consistent requirement of regularity for massless and massive time-like vectors acting together; see Section 4.2 below. In view of (39) and (107) the requirement of regularity for a space-like vector field is satisfied if is positive: The first term with square brackets in (75) with is positive, and both terms in the r.h.s. are of the same sign, so the presence of dark matter increases the attraction to the center.
The curvature of space-time caused by a galaxy is small. In the linear approximation the influence of dark and ordinary matter can be separated from one another. For (75) reduces to where the first term does not contain . However, the contribution of dark matter comes from both additives. The vector field equation (74) and the Einstein equation (70) at are simplified: The boundary conditions for these equations, are determined by the requirement of regularity in the center. Here , .
The term in (79) reflects the interaction of dark and ordinary matter via gravitation. If the curvature of space-time caused by the ordinary matter is small, this term is negligible compared to . The nonlinear term is small compared to at , but at , despite being small, it decreases a little bit quicker than . Neglecting both nonlinear terms in square brackets, the field equation (79) reduces to the one in the plane space-time. obeys the Klein-Gordon equation (37). The regular solution is ((38), (40)): where .
The balance of the centripetal and centrifugal accelerations determines the velocity of a rotating object as a function of the radius of its orbit: is the velocity of light. Far from the center decreases as , while . The dependence (85) turns at from a linear to a plateau (86) with damping oscillations. The plateau appears entirely due to the vector field. At the same time the vector field contributes to as well. Substituting (82) into (80) we get a regular at solution for : where the last term in (87) gives Newton’s potential. Substituting (87) into (85) we finally have Without dark matter (88) would give Newton’s at . In the presence of dark matter the velocity of rotation tends to at with damping oscillations. If the contribution of stars and planets to the total mass of a galaxy is small compared to the mass of the black hole in the center, then outside the black hole is the Schwarzschild radius of a black hole. Outside a black hole , and the condition is fulfilled for all galaxies. Far outside the Schwarzschild radius the velocity of rotation around a black hole is The deviation from Newton’s law due to the dark matter takes place at . At the curve of rotation around a black hole is a universal function. In dimensionless units there are no parameters; see Figure 4.
While the contribution of dark matter to the rotation curve is described by the universal function shown in Figure 4, the distribution of stars and planets, circulating around a black hole, differs from one galaxy to another. The rotation curves of different galaxies look different. However, the deviation from Newton’s on the periphery of a galaxy is their common feature. Dark matter manifests itself most clearly in the periphery of galaxies. Therefore, in order to compare with (90), among the numerous available rotation curves, it is natural to choose those having the stars outside the main disk.
Fitting the rotation curves of two such galaxies via the universal function (90) is presented in Figure 5. These two spiral galaxies are located in the Ursa Major cluster (UMa). Their numbers are from “The New General Catalogue of Nebulae and Clusters of Stars” (abbreviated as NGC). It is a catalogue of deep-sky objects in astronomy compiled by Dreyer in 1888 , as a new version of John Herschel’s Catalogue of Nebulae and Clusters of Stars.
Agreement with the oscillations is hardly accidental. Actually, damping oscillations of a rotation curve in the far periphery of a galaxy can be considered as a “signature of dark matter.” I would strongly recommend this observational test as confirming the existence of dark matter, along with its adequate description by a longitudinal nongauge vector field.
The fact of small deviations from the universal curve indicates that the main contribution comes from the dark matter. It is in agreement with the modern concept that there is only some 5% of ordinary matter in the Universe, while the amount of dark matter is about five times as much; see Figure 1. The fitting in Figure 5 also testifies that the deviation from the central symmetry by a disk of circulating stars and planets is small. It confirms the existence of a heavy object, like a black hole, in the center of a galaxy. In general, the observed deviations of rotation curves from the universal curve could clarify the average distribution of the ordinary matter within other galaxies.
The contribution of dark matter to a rotation curve (90) is expressed via two observable parameters: the limiting plateau value and mass . They allow restoring the value of the parameter at in the boundary conditions (81): (86). As far as there is no evidence of any direct interaction of dark and ordinary matter, the origin of specific values and of a particular galaxy depends on what happens in the center. The values and differ from one galaxy to another. It looks like for each galaxy these values are driven by a heavy object (may be a black hole, may be a neutron star) located in the center and, by the way, supporting the central symmetry of the gravitational field.
Dark matter, described by a vector field with the Lagrangian (58), actually justifies the empirical Milgrom’s hypothesis of MOND—the modified Newton’s dynamics . Newton’s dynamics really get modified by the vector field so that the rotation curve flattens out at the far periphery of a galaxy. This is because the perturbation of the gravitational field due to a massive longitudinal vector field decreases slower than the perturbation caused by the ordinary matter. The empirical Milgrom’s hypothesis of MOND was a real breakthrough in the 80s. Naturally, basing only on the intuitive arguments, it was scarcely possible to guess that the transition to a plateau is accompanied by damping oscillations.
However, the question of the physical origin of dark matter remains open. In other words, what makes different from zero? Solutions of the linearized Einstein equations do not answer this question. Within the approximation of weak fields and remain free parameters. The energy is an arbitrary function in the dust matter approximation, so the parameters and of dark matter are in no way connected with the ordinary matter.
According to the empirical MOND prediction the limiting plateau value is connected with the mass of a galaxy: Milgrom postulates the existence of a very small acceleration , and that at the violation of Newton’s law takes place . To answer the question “what the empirical MOND relation (91) should be replaced by?” one has to find out the reason why the divergence is not zero at . The gravitational field of a collapsing black hole is neither static nor weak. In the close vicinity of a black hole the velocities of circulating stars and planets are relativistic. It is necessary to get a self-consistent solution of the nonlinear Einstein equations. At strong interaction via gravitation should affect the dynamic balance of the ordinary and dark matter. For nonlinear equations the requirement of regularity can impose an additional restriction on the parameters and in the boundary conditions (81). It is very likely that it will fix the connection between and , providing the dependence of the limiting plateau value on the mass of a galaxy.
Actually it is a revision of equilibrium  and collapse  of supermassive bodies with the dark matter taken into account. The ordinary matter could be still considered as a degenerate relativistic Fermi gas (see , problem 3 in the end of the paragraph 61, page 207). The dark matter should be included into Einstein equations via the energy-momentum tensor (66). Considering a collapsing system, there is no reason to ignore a time-like vector. It is possible that the repulsive ability of a time-like vector field can dynamically balance the collapse. In this case there will be a regular solution of Einstein’s equations describing the internal structure of a black hole without a singularity in the center. This is a worthy task for future.
One can trace two main trends in the literature in trying to unravel the puzzle of a plateau in the rotation curves of galaxies—to “improve” general relativity and to compose a mixture of fields able to fit the observations without a “mysterious dark matter.”
For instance, Sanders argues that “…the correct theory may well be one in which MOND reflects the influence of cosmology on local particle dynamics and arises only in a cosmological setting” and concludes: “It goes without saying that this theory is not general relativity, because in the context of general relativity local particle dynamics is immune to the influence of cosmology” . Obviously, I do not share Sanders’ conclusion. I have presented above the complete derivation from the Einstein equations ((70)–(72)) to the galaxy rotation curve (85). The formulae ((88), (90)) are derived completely within the Einstein’s theory. For the time being, there is no need in any modifications of the general relativity to explain the observable plateau in galaxy rotation curves.
An example of opposing fields and dark matter is the Moffat’s attempt of applying a mixture of scalar, vector, and tensor fields in order “to explain the flat rotation curves of galaxies and cluster lensing without postulating exotic dark matter” [42, 43]. It is a question of terminology. In quantum physics there always is a quantum particle corresponding to the field describing a material substance. From my point of view, the fields are convenient mathematical instruments that we utilize to describe the physical phenomena, no matter how we name them.
According to observations the period of oscillations (see Figure 5) is around 15 kpc. If in quantum mechanics it is the de Broglie wavelength , then the rest energy of a quantum particle, corresponding to the space-like vector field, should be eV. The lightest particles as candidates for the cosmological nonbaryonic dark matter are discussed by Khlopov  in connection with spontaneously symmetry breaking in phase transitions in the early Universe.
The theory predicts the oscillating features with no baryonic counterparts in the rotation curves of the outer regions of galaxies. As this would be the main observational signature of existence of the dark matter, I persistently recommend this observational test.
The “unprecedented constraints on the stellar and dark matter mass distribution within our Milky Way” are reported by Bovy and Rix . Continuous progress in the accuracy of observations would be able to provide the values of the main parameters and for Milky Way and other galaxies and clusters.
Having the energy-momentum tensor (60), it is worth considering a possible role of dark matter in the “pioneer anomaly” in the scale of the solar system. It appeared that a very small unexpected force caused an approximately constant additional acceleration of m/s2 directed towards the Sun for both spacecraft Pioneer 10 and Pioneer 11 [52, 53]. It is interesting to trace the dependence (78) along the two spacecraft hyperbolic orbits at distances between 20–70 astronomical units ( cm) from the Sun. Manifestation of dark matter on the periphery of the solar system would be a great surprise!
4. Regular Cosmology
From the standpoint of general relativity the matter curves the space-time, giving rise to mutual attraction between the bodies. However, according to modern observations, the Universe is expanding as a whole, despite the gravitational attraction between material objects. The expanding solution of Einstein’s equations due to the cosmological constant belongs to De Sitter [54, 55]. The expanding solutions of the Einstein equations without the cosmological constant (Friedman , Robertson [57, 58], and Walker  (FRW)) inevitably contained the singularity. The unknown origin of expansion of the Universe, containing only mutually attracting objects, was supposed to be hidden within the singularity. For a long time the singularity was considered a general property of the Universe. The singular point, referred to as “Big Bang,” is still widely recognized as the “date of birth” of the Universe.
Discovery of the accelerated expansion of the Universe [3, 4] shows that the source of acceleration continues to exist for a long time after the Big Bang. Naturally, the fact of accelerated expansion gave rise to the assumption that the physical vacuum is not just the absence of the ordinary matter. The existence of dark energy and dark matter, as the unknown source of the Universe’s expansion, is widely discussed in modern literature and Internet [60, 61].
The macroscopic approach to the theory of evolution of the Universe driven by vector fields plays the central role among numerous attempts to guess the riddle of accelerated expansion. It allows avoiding unnecessary model assumptions (like “”, quintessence, phantom-like cosmologies,…, see a review ) and remains in the classical frames of Einstein’s general relativity. Utilization of vector fields in general relativity shows undoubtable advantages in comparison with scalar fields and with multiplets of scalar fields. The equations appear to be more simple, while their solutions are more general. The solutions have additional parametric freedom, allowing forgetting the fine-tuning problem . However, starting from the pioneer paper by Dolgov , people considered mostly gauge vector fields [65–69] in applications to the dark sector. The Lorentz gauge restriction allowed avoiding the difficulty of negative contribution to the energy. But at the same time it does not allow utilizing all the advantages of vector fields. In general relativity (in curved space-time) the energy is not a scalar, and its sign is not invariant against the arbitrary coordinate transformations. Considering the vector fields in general relativity, it is worth rejecting the gauge restriction, using instead a more weak condition of regularity. Step by step, people are now getting rid of the Lorentz gauge restrictions [70–73].
Today it is generally accepted that among the stuff of the Universe only 4.5% is the ordinary matter; see Figure 1 . It is reasonable to analyze the role of vector fields in cosmology step by step. First step—dark energy only (zero-mass field): about 72%. Second step is adding massive vector field (dark matter, about 23%) into consideration. The final step is to include the ordinary matter, after the main role of vector fields is clarified.
According to observations the Universe expands, and its large scale structure remains homogeneous and isotropic. Consider the space-time with the metric depending on only one time-like coordinate (In the derivations below the units with , and are used.). The metric tensor is diagonal. The uniform and isotropic expansion is characterized by the single metric function , and the rate of expansion is . The Ricci tensor is also diagonal:
4.1. Massless Field as Dark Energy
The energy-momentum tensor (61) of a massless field acts in the Einstein equations (55) as a simple addition to the cosmological constant (62): The contribution of the zero-mass field to the curvature of space-time remains constant in the process of the Universe evolution. The fact that “the gauge-fixing term exactly behaves as a cosmological constant throughout the history of the Universe, irrespective of the background evolution” had been mentioned by Beltran Jimenez and Antonio .
The metric function is the self-consistent regular solution of the Einstein equations (94), provided that is a linear function; is a constant of integration. The metric (92) with the metric function (95) is called de Sitter (or anti-de Sitter, depending on the sign definition of the Ricci tensor). It describes either expansion (sign +) or contraction (sign −) of the Universe at a constant rate; see Figure 6.
In the case of sign + the rate of expansion is called Hubble constant.
The zero-mass vector field determines the constant rate of expansion. Available today properties of the so called dark energy (presently unknown form of matter providing the major contribution to the uniform isotropic expansion of the Universe) can be described macroscopically by the zero-mass vector field with a simple Lagrangian As long as the physical nature of vacuum is not known, the “geometrical” origin of the cosmological constant and the “material” contribution to (62) by the zero-mass vector field can not be separated from one another. The combined action of the massless field and/or the cosmological constant is described by the single parameter—Hubble constant (97).
4.2. Massless Field + Massive Field
Over the scales much larger than the distances between the galaxies the Universe is uniform and isotropic. In the scale of the whole Universe the massive vector field is time-like and longitudinal: the only nonzero component is , and it depends upon the time coordinate .
The energy-momentum tensor (65) for the longitudinal massive time-like field is The massless field enters the Einstein equations only via the cosmological constant (62). remains constant throughout the whole history of the Universe. The massive field is described by the function , which enters Einstein equations (100)-(101) directly . In accordance with the necessary condition of regularity for a time-like massive vector field (43) , and the squared mass of the field is . The applicability of (100)-(101) is restricted by the condition that the field is small, so that the second and higher derivatives of the potential can be ignored.
4.2.1. Asymptotic Behavior at Large
As it is confirmed below, set (100)-(101) has regular solutions with the rate changing from in the past to in future. Far back in the past and in the late future the temporal evolution of the massive field is described by (102) with . Its solution is a linear combination of two functions, vanishing at . The functions are monotonic if or oscillating with a decreasing magnitude if . If is small, the field decreases very slowly: In the limit the term with disappears as , while the term with becomes indistinguishable from the massless field, which remains constant during the whole process of evolution. In dimensional units the ratio is .
4.2.2. Regular Bounce
Extracting (100) from (101), we have Without massive field, that is, if , the second derivative , and we return to the de Sitter metric with the metric function (95) describing the two isolated solutions—compression and expansion—at a constant rate (97).
The second order set of Einstein equations ((100), (101)) for the unknowns and looks more complicated than the equivalent set ((102), (105)). At the same time the set (102), (105) is of the third order. Hence, it has extra solutions. So, working with the set (102), (105), it is necessary to eliminate extra solutions that are not the solutions of the Einstein equations ((100), (101)).
The time coordinate is a cyclic variable, and it is convenient to set the origin at a moment when . Initial conditions for the Einstein equations contain only . The derivative is strictly fixed by the solutions of (100), (101). As for the set (102), (105), the value in the initial conditions is a free parameter, independent of . The connection between and , eliminating extra solutions, follows from (100) at : The l.h.s. of (106) is positive. The initial conditions (106) are self-consistent if both and are of the same sign. According to the requirement of regularity of the de Sitter metric (96) is negative. Hence is negative too: Then (105) is positive, . We conclude that the massive time-like vector field makes the rate of evolution a monotonically growing function from in the past to in future. The Universe contracts at , and expands at . is the moment of maximum compression.
One of the two constants and at remains arbitrary within the initial conditions (106). The Einstein equations ((100), (101)) are invariant. In the case the field is a symmetric function, and if it is an antisymmetric one. In both cases is antisymmetric. If both constants and at are not zeroes, a regular solution still exists, but there is no symmetry with respect to . The scale factor decreases with time while , reaches its minimum, and grows when becomes positive.
In the case of a small mass, (in dimensional units ) the compression-to-expansion transition is described by the analytical solution for the symmetric configuration as follows: The rate of evolution (109) is shown in Figure 7. With no ordinary matter the time interval of transition is of the order of Hubble time ~. At it does not depend on the mass of the massive field. The scale factor is The metric function enters the Einstein equations ((100), (101)) only via the derivatives of but not directly. Without ordinary matter is a free parameter, and is defined up to an arbitrary constant factor . The acceleration is positive: