Research Article | Open Access
Oscillating Cosmological Solutions in the Modified Theory of Induced Gravity
This work is the extension of author’s research, where the modified theory of induced gravity (MTIG) is proposed. In the framework of the MTIG, the mechanism of phase transitions and the description of multiphase behavior of the cosmological scenario are proposed. The theory describes two systems (stages): Einstein (ES) and “restructuring” (RS). This process resembles the phenomenon of a phase transition, where different phases (Einstein’s gravitational systems, but with different constants) pass into each other. The hypothesis that such transitions are random and lead to stochastic behavior of cosmological parameters is considered. In our model, effective gravitational and cosmological “constants” arise, which are defined by the “mean square” of the scalar fields. These parameters can be compared with observations related to the phenomenon of dark energy. The aim of the work is to solve equations of MTIG for the case of a quadratic potential and compare them with observational cosmology data. The interaction of fundamental scalar fields and matter in the form of an ideal fluid is introduced and investigated. For the case of Friedmann-Robertson-Walker space-time, numerical solutions of nonlinear MTIG equations are obtained using the qualitative theory of dynamical systems and mathematical computer programs. For the case of a linear potential, examples joining of solutions, the ES and RS stages, of the evolution of the cosmological model are given. It is shown that the values of such parameters as “Hubble parameter” and gravitational and cosmological “constants” in the RS stage contain solutions oscillating near monotonically developing averages or have stochastic behavior due to random transitions to different stages (RS or ES). Such a stochastic behavior might be at the origin of the tension between CMB measurements of the value of the Hubble parameter today and its local measurements.
This work is related to research in the field of the theory of gravity and cosmology in connection with existing problems given below.
The difference in the values of the cosmological constant obtained from astrophysical observations and predictions of the general relativity theory (GRT), taking into account the quantum effects of vacuum polarization, is known in science as the “problem of the cosmological constant” (see ). The acuity of this problem reinforces the fact that this difference is huge - .
There is a problem of “accuracy of measurement of the gravitational constant” . For example, in the International System of Units (SI), for 2008: ; the value of the gravitational constant was obtained in 2000 (Cavendish Experiment): ; in 2010, the value of was corrected: ; in 2013 a group of scientists from the International Bureau of Weights and Measures: ; in 2014, the value of the gravitational constant recommended by CODATA became as follows: In fact, is not determined even with an accuracy of the fourth decimal place.
The Cold Dark Matter model represents the current standard model in cosmology. Within this, there is a tension between the value of the Hubble constant, H0, inferred from local distance indicators (the predicted value given in article  is ), and the angular scale of fluctuations in the Cosmic Microwave Background (CMB) (as follows from  ). These two independent measurements give a discrepancy of approximately and tension with Planck+ increases to 3.7 sigma [2–4]. It also follows from the above that the measurement accuracy of is about .
In our work we present a model where, due to the oscillatory regime in the solutions of equations, the Hubble parameter also fluctuates with respect to the mean value, which is also a function of time. We investigate a mechanism associated with the nonlocal behavior of a gravitational system with scalar fields in the classical approximation.
P. A. M. Dirac put forward the principle that all large numbers are determined by the lifetime of the Universe and change with it [5, 6]. The Dirac hypothesis gave impetus to the development of a program for studying the possible dependence of physical constants on cosmological time. Dirac developed his hypothesis based on the modernization of Einstein’s theory by introducing an additional scalar field. In the 1970s he managed to shape his ideas into a conformally invariant scalar-tensor theory of gravity. Earlier, in the 1950-60s, other variants of scalar-tensor theories were created, for example, the Brans-Dicke scalar-tensor theory [7–9], based on the use of the Mach principle to explain the force of gravitational interaction as caused by the mass and size of the Universe. P. Jordan , J. Narlikar , and a number of other physicists worked in this direction. New versions of this kind of theories appear until recently, for example, modified scalar-tensor theories of gravity. The main idea of the modified theory of gravity is to eliminate the cosmological constant and the exact solution to the problem of dark matter. The theory alters the laws of the very large (on a cosmological scale) distances and lengths of time. The main idea of the modified theory of gravity is to eliminate the cosmological constant and to solve the problem of dark matter. The theory alters the very laws of large-scale (at cosmological scales) distances and lengths of time. In the MTIG, we additionally study the evolution of the gravitational constant, since it is not initially introduced into the theory but is obtained due to a fixed solution of the scalar field. Thus, MTIG is directly related to the problems of the so-called “dark energy” (DE) and “dark matter” (DM).
The theoretical challenge posed by these problems triggered many attempts to directly change Einstein’s gravity at large distances . An example of such an infrared (IR) modification is the brane-world model [13–15]. In [12–16], authors put forward the idea that if gravity is sufficiently weakened in the infrared, then vacuum energy could effectively decouple from gravity or degravitate over time. As shown in [17, 18], the current Planck data used is best suited to the model in a nonplanar . The author considers closed and open cosmological models.
Our theory is a phenomenological model used for comparison with observational data DE and DM. Within the framework of modified theory of induced gravity (MTIG), proposed in the works of [19–21], we attempted to solve the above problems based on the idea of the existence of macroscopic parameter of the theory , which generates both gravitational and cosmological “constants”: where is Planck’s constant, is speed of light, , is vacuum energy are constants of the theory, and is effective potential of the theory.
We are going to compare the experimental value of the gravitational constant with the effective “gravitational constant.”
where is the current value of the function , is the gravitational constant of Newton, the value of which is , and is a time parameter corresponding to the current value (approximately 13.8 billion years). Similarly, in accordance with astrophysical observational data, the modern value of the cosmological constant is assumed to be equal to
Functions , where , represent -dimensional Riemannian manifold described by the metric , into -dimensional flat space-time with the metric . It is convenient to leave the signature of the plane space arbitrary. For further calculations we set .
For a cosmological model (a similar model is constructed for a centrally symmetric space as well), the mechanism proposed by us reduces to the fact that the differential equations describing the evolution of the functions and have the form
where are some expressions of functions and the cosmological scale factor and their derivatives up to the second order. For , the second equation goes to the equation matching with the equation in general relativity, and the first equation disappears. Thus, there are solutions that can both match and not match with the solutions of the standard theory of gravity. Then the fundamental “constants” of theory, such as gravitational and cosmological ones, can evolve in time and also depend on coordinates. In a fairly general case, the theory describes two systems (stages): Einsteinian (ES stage), when and equation (3) disappears, and “restructuring” (RS stage), when . This process resembles the phenomenon of a phase transition, where different phases (Einstein’s gravitational systems, but with different constants) pass into each other. So far we cannot present the final mathematic mechanism of a computable description of such transitions. We can only indicate the “favorable” points at which such transitions are possible. These are the moments of time when the second derivative of the scale factor or the first derivative of equals zero. In this paper we show that the values of the observed characteristics of the gravitational field are affected not only by the values of the gravitational constants, but also, for the most part, by their derivatives.
In  to solve Problem , we considered two mechanisms for reducing the constant part of the vacuum energy . In the first variant, the value is compensated by other terms from . The reduction of two values imposes requirements on the constants of the theory (,) to the accuracy of high orders. The second mechanism for reducing the constant part of the vacuum energy reduces to the multiplicative reduction. Its principle is simple and is based on the law of conservation of energy in phase transitions corresponding to different stages of the evolution of the universe and the structure of the theory .
Note that this mechanism for reducing the vacuum energy is analogical to the mechanism for reducing the divergences in the quantum renormalization theory, despite the fact that the theory under consideration is classic. In  the influence of matter in the form of perfect fluid on the behavior of was studied. We can say that, in the cosmology based on the MTIG, the principle of the “whole” Universe (slightly analogous to Mach’s principle) is realized to some extent, which reduces to the existence of a certain parameter , which in turn depends on all material fields and generates physical “constants.”
In this theory we consider the influence of the quadratic, standard potential on the solutions of the RS stage. In our opinion, solutions containing anharmonic oscillations caused by random initial and boundary conditions are of special interest. Unlike the solutions of the Einstein equation with the asymptotics of flat space-time, the presence of a variable “cosmological term” leads to a nonlocal self-interaction of the field . Fluctuations with a complex spectrum impose monotonically varying solutions (e.g., cited in . Such behavior leads to fluctuations in the parameters relative to their mean classical values. Thus, we propose the hypothesis that the value of such parameter as the gravitational “constant” , apart from the slow evolution in the RS stage, can fluctuate near the classical value. For example, Figures 1, 2(a), and 2(b) show the normalized numerical solutions , , and , where , of equations (cited in this article).
(a) Relation of , ; is current age of the Universe. , .
(b) Relation of , ; is the current value of the Hubble constant. , .
First of all, we require the consistency of the results of the theory (after comparison with observational data) and then look for the predictive possibilities of the theory. Proceeding from this, the choice of the parameters of the theory should lead to fluctuations affecting the fourth (maximum to third) order of the solution , interpreted as the modern value of this parameter. Of course, back in time, the values of the parameters could have been substantially different and these parameters implemented qualitatively different stages of evolution (Figures 1 and 2). The graph of the scale factor shown in Figure 1 can be reconciled with the results of observational data (e.g., for calculating the Hubble constant), if the 9 percent difference obtained by measuring the CMB (corresponding to the early Universe) and local measurements is attributed to the average (for fluctuations) graph and 4.5 percent of measurement errors are attributed to the local fluctuations themselves.
Research on numerical solutions for the case of a centrally symmetric space has been made in . On the one hand, found solutions agree with the observational data; on the other hand, at far distances from the center (more than 0.01 parsecs from the Sun and more than one kiloparsec from the center of the galaxy) they can lead to a significantly different astronomical picture. We are going to inform about these decisions in the next article. Comparison of the solutions of the cosmological model and the model in the case of a centrally symmetric space leads to the fact that DM and DE can be described in a single concept.
2. Description of the Phenomenological Model MTIG
2.1. Equations of MTIG
There are arguments in favor of the theory of physical fields that it must have the property of conformal invariance, at least at the classical level, up to the time when this symmetry is not broken. The action for the membrane (-branes) admits no conformal transformations. To avoid this difficulty while remaining within the ideology of the string theory, in previous papers we proposed the following generalization of string theory .
In (4) the following notation is used: , , and is the potential depending on the fields p; is scalar curvature of manifold . The operator means covariant derivative in a manifold , where the Christoffel symbols are connected with the metric in a standard way. The variables , are arbitrary constants.
For simplicity, in this paper . characterizes all possible interactions with other fields of matter.
In the context of our paper, many modified scalar-tensor theories of gravitation can be transformed to the form (4), without taking into account the Einstein term absent in (4). Note that in [23–27], some classes of modified gravity, considered as a gravitational alternative to dark energy, were presented.
The “induced gravity” means that the initial action of the Einstein’s term is not explicitly introduced. The introduction of such term initially at first violates the conformal invariance of the theory and secondly leads to the instability of the known solutions because of the emergence of the effective gravitational “constant” At the point when it tends to zero, additional singularities arise. For example, in the case of one scalar field (), instability arises for the conformally invariant case () . In connection with what has been said, I want to point out the problem connected with the sign of (42) and (2). Unlike single scalar field (where analogy of is ), the sign of is undefined. Also, there is not enough data to compare with the proposed theory and to define all the parameters of the theory.
The consequence of these equations is the law of conservation of energy, which has the formand the equation on the field Equation (6) is an analogue of Einstein‘s equations for a macroscopic medium.
In the derivation of macroscopic equations by varying the fields , the following assumptions were made.
The induced metric (mapping) is related to the metric of the manifold by means of formula so, this model allows us to interpret the development of Universe as development of dimensional objects embedded in the multidimensional flat space-time . Self-consistency of the equations is ensured by introducing additional currents .
Due to interaction with vector fields, the equation for scalar fields acquires an additional term . Then these equations have the form The specific form of this term depends on the model. Thus, from a mathematical point of view, the solution of the inverse problem is assumed. Solving the macroscopic equation (6), we find the metric and the field . Then, solving equations (9) and (10) we find . Our approach is similar to the method of finding the unknown potentials given in [30, 31].
The condition of “embedding” (9), which was used in deriving equations, does not greatly restrict the proposed theory. Indeed, in the most general case, we can make the substitution , where are some tensor functions. If you do the calculations described above, it can be shown that the resulting equations have the same form as in (6)-(8); in these equations redefine EMT matter.
The scalar field in this theory is . Equation (6), except for the first term on the right side of the equation, is the same as a special case of Brans-Dicke.
In order to take into account the effect of vacuum polarization energy into gravity, we highlighted from EMT matter a part related to this energy, which satisfies the equation of state , where and are interpreted as the energy density and vacuum polarization pressure. Therefore, in the equations (except for (9)) to made a substitution,
The action (4) has the property of conformal invariance if , (, , where For dimension : .
Also note that (6) can be obtained by varying the action of by the metric , where is the potential.
This invariance is expressed in the fact that the equations obtained by varying the action (4) with respect to the fields and are invariant under the local Weyl scale changesfor an arbitrary function .
The condition (9) for limits the conformal invariance to scale transformation. Thus, fixing the function (in general, this parameter can be a function of coordinates) results in fixing the scale of the theory. In our notation, is a dimensionless quantity. The fields , interpreted as the coordinates of the space , have the dimension of a centimeter, which implies and . Action (4) is a dimensionless quantity ().
The authors of  investigated the possibility that probably observed additional dark radiation has an origin associated with the scale invariance.
In general case, we get the systems of “macroscopic” equations (6)-(8), “microscopic” equations (10), and constraint equations (9). The study of the complete system of equations requires the definition of the model, that is, definition of the functions . The fixed sector of the fields can play the role of Higgs scalar fields. It is proper to consider the function as the averaged field (the vacuum mean in the tree approximation) by analogy with the mechanism of spontaneous symmetry breaking (the Higgs mechanism). So the previous formulas should be understood in the following sense: , The latter can be interpreted in the sense that “geometry” is created by vacuum averages.
2.2. Different Types of Solutions
From (7) it follows that three types of solutions are possible as follows:
(I) , Note also that In this case, we obtain equations that match with the Einstein equations, with the gravitational constant and with the cosmological constant . Equations (10) can be rewritten as For the cosmological model with the EMT of perfect fluid and the potential , free fields () acquire mass , when , where is the density of energy and pressure.
(III) When separate law of conservation of matter is not necessarily fulfilled. This case is a generalization of the previous case. The law of conservation takes the form
3. Cosmological Solutions
3.1. Cosmological Vacuum Solutions: Y=const.
Let us consider the above equations under potential , for the case of a homogeneous, isotropic cosmological model.
The metric form of the manifold has the form when , respectively, for the models of open, closed, and flat types. is the metric form of a sphere, with a unit radius, expressed in spherical coordinates.
The equations for the scale factor have the form are open, closed, and flat types of spaces, respectively.
The equations for the fields take the following form: Here, eigenvalues for the three-dimensional Laplacian
Particular solutions () of these equations are found in  that satisfy the conditions of “immersion” (9). For the closed model, these solutions have the form where is immersion function of -dimensional sphere: And for the open type of space where follows from by replacing with
For the closed model manifold forms “one-sheeted hyperboloid” in a five-dimensional subspace of flat space and is described by For the case of an open model surface equation has the form
Note that when does not depend on ; and when
From the requirement (9) .
The condition for (28) leads to the relation
From (22), (31), and (32) it follows that From (34) and we getThus, if all embedding conditions of the Friedmann world into multidimensional flat space-time are met, the cosmological constant does not depend on the polarization energy of the vacuum for the model constructed by us. The result obtained can be used to investigate the case of perturbations.
The above solutions correspond to the special case without taking into account the interaction of the fields with other fields. Then, the case (34) corresponds to the minimum of the potential (Figure 3), included in (23) and (6) (in the particular case (22)).
There is a problem of defining the numerical values of the parameters of the theory. The number of essential parameters can be reduced to three. For them we use the following notation:So that
In order to generalize the case and for and to reduce the number of parameters of the theory, we will not be limited to the model described by the solutions (22)-(34). For this we adopt the following arguments.
For minimum potential energy: To agree with observational facts and taking into account (22), let us assume
Instead of selecting the ansatz (37) due to the small value of the observed cosmological constant at , for two other ansatzes were considered as well: corresponding to the minimum potential ; This case is interesting because it is possible to obtain analytical solutions of differential equations, even if the substance is present in the form of a perfect fluid.
3.2. Cosmological Solutions with Matter
In  a phenomenological model was proposed. The model of the interaction of the field and matter in the form of a perfect fluid, with the density of energy and pressure, Equations (6) and (7) take the form For convenience in computer modeling, dimensionless variables are introduced where is proper time; dot denotes the derivative by ; is some value of the field , which we associate with a constant solution , discussed above in particular (); is dimension value ; it is convenient to correspond to the modern value of the scale factor or the age of the universe. In the first case corresponds to the modern value of the scale factor and to the age of the universe. Such a scale is convenient if the required functions are expressed through the scale factor. However, we do not know the modern value of the scale factor, but we assume years. Therefore, when the desired functions are expressed in terms of time, we select for the parameter . Then correspond to the modern values of the parameters.
The last equation (taking into account the previous one) for can be rewritten as Here we have introduced the following notation:
and we also reparameterized constants taking into account their dimensions
It is interesting to compare the equations obtained from (42) and (45); in these equations we substitute , with Einstein’s equations with the same EMT, having the formwhereThe first of these equations ((42) and (48)) will match, and (43) disappears. Equivalent (at ) to (43), (45) does not match (49). We recall that in the case of Einstein’s equations the second one is a differential consequence of the first.
Thus, as already indicated in previous works, in the proposed model, the evolution of the universe contains two stages that were named as “Einstein” (ES - stage) when and “restructuring” (RS - stage) when . This process resembles the phenomenon of a phase transition, where different phases (Einstein’s gravitational systems, but with different constants) pass into each other.
From a mathematical point of view, at any time the solutions of (42) and (43), describing the ES and RS stages, can pass into each other. To describe such solutions it is necessary to join functions of the scale factor and the field and their first derivatives at the point corresponding to the moment of transition. These transitions are similar to the first-order phase transitions and apparently can be used to describe transition from the inflationary phase to the next phase :where the index “e” denotes the solutions or the corresponding ES stages and the index “r”- RS stages.
Transitions similar to the second-order phase transitions are described by the system, if the conditions (51) are supplemented by the condition of equality of second derivatives at the transition point:
A necessary condition for the existence of solution (if , then ), equations describing both ES and RS stages, is the fulfillment of the following conditions on the model parameters:
3.3. The Case without Quadratic Terms
In [20, 21] in order to obtain analytical solutions, we consider a linear approximation of , so that In this paper we want to focus on the existence of nonstandard solutions related to the branching effect of solving equations.
Equation (43) is integrated and reduced to the form where are integration constants.
Let us consider the so-called “equilibrium state” in more detail. This state is obtained by applying the conditions (53) and the additional condition on the constant : . From (53), taking into account (54), it follows . These conditions are obtained from the requirement of existence and matching solutions of (55) and (42).
After substituting these values of the parameters, besides the condition , the equations (for ) can be reduced to the formIt is of interest to study the influence of the logarithmic term in (57) and (58) on their solutions, so we left this term as some perturbation violating the solutions of the “equilibrium state.”
The solution of (57) and (58), in the case , is found by the formula The function is defined as the solution of (57) (for ). Equation (58), taking into account (59) and (57), becomes their differential consequence.
Surprisingly, the solution for the scale factor does not depend on the constant . This equation has the form where defines the cosmological constant (from which, presumably, follows). As a consequence, it follows from (60), (6), and the solution (59) that there are two “gravitational constants”: , cosmological gravitational constant, and , time-dependent function, (possibly) contributing to the gravitational interaction between massive bodies. In addition, the solution (59) is noteworthy by the fact that the transition between ES and RS stages takes place at the point when the first derivative and the second derivative of the scale factor equal zero. Thus, the transitions between the stages will be located in the vicinity of the special points () for the scale factor function. From this point of view, it is interesting to study all special points, including the equilibrium points . In the next section, we present a model of a quasistatic universe, where the scale factor fluctuates with respect to a constant value.
Joining of solution (61) with (57) and (58) at some point is performed at the equality of functions , , and their first derivatives. As for the continuity of the first derivative of a function this function is not directly related to the four-dimensional geometry, although in Section 3 we were able to interpret this parameter as the “radius” of the four-dimensional hyperboloid (embedded in the five-dimensional space-time). In the vicinity of the transition point, we require the continuity of the function , as well as its first derivative. This requirement is associated with the requirement of energy conservation. We can separately consider the question of the continuity of the second derivatives of these functions. Then we can prove the following relations at the transition point :
Let us consider the following conditions for joining of solutions at the transition point : where
In spite of the fact that the values of parameters including the value were called the “equilibrium state,” it is interesting to consider a more general case . Moreover, there is a free parameter , and we can demand that the term in (63) associated with the parameter at the critical point to equal zero. To do this, we must select .
It can be shown that transitions from the RS stage into ES stage are possible in the following form. For points : , where is defined as the solution of (57); at the point : and ; further for points :, where is defined as the solution of (61). If , then an inverse transition from ES stage into RS is possible, for example, at the point where the second derivative of the scale factor (varying by (61)) .
As an example, Figures 4 and 5 show the joining of solutions for the function describing the transition from RS into ES (Figure 4) and with the possibility of double transition (Figure 5). The first point () is defined by the moment of time , , where , ; from the point () the system can evolve along two trajectories: is transition into the ES stage or it continues to move along the same curve (remaining in RS). When the system transits and evolves according to (61), under certain initial conditions, the transition back into RS stage is possible, as shown in Figure 6. In the second transition in Figure 5, we associate it with the point , which is defined by the condition . The lifetime of the ES stage: . If we assume that the scale factor is similar to the lifetime of the universe, then is about years. Assuming that the RS stage is currently continuing, the estimate of the rate change of (for the model considered in Figure 5) is per year, and the “gravitational constant” increases with the same speed.
However, if the function is treated as a macroscopic parameter and only the function (not including its derivatives) requires the continuity, then transitions are possible at the time when and the second derivative of scale factor is not necessarily zero. From (64) follows that in the latter case .
As a result, we have possible transitions from RS stage into ES stage of the following form. For points : , is defined as the solution of (57); at the point : or and ; for points :, is defined as the solution of (61). Similarly, a solution describing the transition from ES into RS can be found.
For the solutions analysis of obtained equations we used qualitative research methods for special solutions of differential equations. From this analysis it follows that the case of the transition from ES stage into RS and back is more likely near the special points, where the first or second derivative (or both) of the scale factor equals zero or infinity.
4. Oscillating Solutions: Influence of Quadratic Terms
Let us consider (42) and (43) under the conditions (53). Recall that these conditions are the requirement for existence of common (for both stages) static solutions . In doing so, we must understand that the fulfillment of these conditions implements a “strongly” nonlinear model with potential (20). In this article, we do not claim to develop the final realistic cosmological model but want to identify the effects associated with nonlinear terms. From the conditions (53) let us express in terms of and substitute them in the equations under study that can be reduced to the formTo simplify and reduce the number of parameters, consider the following additional relations that arise from the requirement of the existence of solutions at : Also consider two different types of relations, corresponding to the ansatzes examined earlier in (37) and (38):
Let us consider some numerical solutions of the obtained equations (70) and (71), for the case of a flat space and ansatz (72). To define the unknown parameters, we use the following reasoning. If in the modern era the value of the field , then the fractions consistent with observations (similarly to ) of the effective “cosmological constant,” dust-like and ultra-relativistic matter, as follows from (70), for stage equal
By means of and we denoted the normalized Hubble constant and the deceleration parameter (with the minus sign) , calculated to the moment (). Without loss of generality we can choose . Setting , from the relations (73)-(75), we can express and substitute them in (70)-(71). The relations (73)-(75) are obtained by analogy with the model, but our model differs from , by the time dependence of the function simulating the term and the presence in the equations of the derivatives of . If , as follows from (71), there is a fraction of the energy associated with the term , which can have a larger contribution than a term interpreted as a term. Also we want to note that in this section we do not investigate the joining of solutions issues of equations describing ES and RS stages; we will study only the solutions of (71) and (70) without the multiplier . Although from the results of the previous section it follows that random transitions between the stages are possible, at the vicinity of .
Here are some numerical solutions, with the initial conditions
The equations under study are invariant with respect to time shifts . That means that the variable can also take negative values . For the initial point we will take a point with a singular solution (if it exists). As it turned out, the parameters and strongly influence the solutions. Taking into account quadratic terms, at , leads to solutions with anharmonic oscillations, where the “mean” oscillation frequency depends on the value .