Can Chameleon Field be identified with Quintessence ?

We analyse the Friedmann-Einstein equations for the Universe evolution with the expansion parameter $a$ dependent on time only in the Einstein gravity with the chameleon field, changing its mass in dependence of a density of its environment. We show that the chameleon field can be identified with quintessence (a canonical scalar field responsible for the late-time acceleration of the Universe expansion and dark energy dynamics) if and only if the radiation rho_r(a) and matter rho_m(a) (dark and baryon matter) densities in the Universe evolution differ from their standard dependence on the expansion parameter a, i.e. rho_r(a) ~ a^{-4} and rho_m ~ a^{-3}, respectively, and these deviations are caused by the conformal factor, relating the Einstein and Jordan frames and defining the interaction of the chameleon field with its environment.


I. INTRODUCTION
The chameleon field, changing its mass in dependence of a density of its environment [1, 2], has been invented to avoid the problem of the equivalence principle violation [3]. Nowadays it is accepted that the chameleon field, identified with quintessence [4,5], i.e. a canonical scalar field, can be useful for an explanation of the late-time acceleration of the Universe expansion [6][7][8][9] (see, for example, [10]). In addition the chameleon field may shed light on a dark energy dynamics [11]- [16]. In terrestrial laboratories [17]- [24] chameleon-matter interactions have been investigated in terms of ultracold and cold neutrons through some effective low-energy chameleon-neutron potentials [25]- [27] and in terms of cold atoms in the atom interferometry [28][29][30][31]. However, recently there has been shown by Wang et al. [32] and Khoury [33] that the conformal factor, relating the Einstein and Jordan frames and defining the chameleon-matter interactions, is essentially constant over the last Hubble time. According to Wang et al. [32] and Khoury [33], this implies a negligible influence of the chameleon field on the late-time acceleration of the Universe expansion. To some extent this should also imply that the chameleon field cannot be identified with quintessence, responsible for a late-time acceleration of the Universe expansion [4].
Thus, the main aim of this paper is to investigate the properties of the conformal factor, i.e. its influence on the evolution of the Universe, and the conditions for the identification of the chameleon field with quintessence [4,5]. As has been shown by Khoury and Weltman [1], the chameleon field can gravitationally couple to a matter or to a matter density of its environment through the conformal factor, relating the Einstein and Jordan frames. A dependence of the chameleon field mass on a matter density of its environment through the conformal factor plays an important role for fulfilment of the equivalence principle [1, 2]. By analysing the Einstein equations for the flat Universe in the spacetime with the Friedmann metric, dependent on the expansion parameter a [34], we show that conservation of a total energy-momentum tensor of the system, including the chameleon field, radiation and matter (dark and baryon matter), demands the conformal factor to be equal to unity if and only if the dependence of the radiation ρ r (a) and matter ρ m (a) densities on the expansion parameter a does not deviate from their standard form ρ r (a) ∼ a −4 and ρ m (a) ∼ a −3 , respectively [34]. The equality of the conformal factor to unity suppresses any connection of a canonical scalar field with a matter density of its environment. In other words, this suppresses the existence of the chameleon field or makes impossible an identification of the chameleon field with quintessence. The same result we obtain by analysing the first order differential Friedmann-Einstein equation, relatingȧ 2 /a 2 to the chameleon field, radiation and matter densities, and the second order differential Friedmann-Einstein equation, relatingä/a to the chameleon field, radiation and matter densities and their pressures, whereȧ andä are the first and second time derivatives of the expansion parameter. We show that the Friedmann-Einstein equation forȧ 2 /a 2 is the first integral of the Friedmann-Einstein equation forä/a if and only if the total energy-momentum of the system, including the chameleon field, radiation and matter, is locally conserved. This means that i) if the radiation and matter densities obey their standard dependence on the expansion parameter ρ r (a) ∼ a −4 and ρ m (a) ∼ a −3 the conformal factor is equal to unity and the chameleon field cannot be identified with quintessence, and ii) if the dependence of the radiation and matter densities deviate from their standard behaviour ρ r (a) ∼ a −4 and ρ m (a) ∼ a −3 the conformal factor is not equal to unity and defines interactions of the chameleon field with its environment. In this case the chameleon field can be identified with quintessence. As a consequence of such an identification the chameleon field can be responsible for the late-time acceleration of the Universe expansion and dark energy dynamics and in addition for fulfilment of the equivalence principle.
The paper is organized as follows. In section II we derive the Einstein equations in the Einstein gravity with chameleon and matter fields. In section III in the flat Friedmann spacetime with the standard Friedmann metric g µν , i.e. g 00 = 1, g 0j = 0 and g ij = a 2 (t) η ij and η ij = −δ ij , we show that the Einstein equations reduce to the Friedmann-Einstein equations of the Universe evolution with the chameleon field, radiation and matter (dark and baryon) densities. Since the Einstein tensor G µν = R µν − 1 2 g µν R, where R µν and R are the Ricci tensor and scalar curvature, respectively, obey the Bianchi identity G µν ;µ = 0, where G µν ;µ is the covariant divergence [34], the total energy-momentum tensor of the system, including the chameleon field, radiation and matter (dark and baryon) should be locally conserved. We find that local conservation of the total energy-momentum tensor imposes two evolution equations for the radiation and matter densities, where the dependence of which on the expansion parameter a is corrected by the conformal factor in comparison with the standard dependence ρ r (a) ∼ a −4 and ρ a ∼ a −3 , respectively [34]. We show that the Friedmann-Einstein equation forȧ 2 /a 2 is the first integral of the Friedmann-Einstein equation forä/a if and only if the total energy momentum of the system, including the chameleon field, radiation and matter, is locally conserved. In other words if the radiation and matter densities acquire corrections, caused by the conformal factor relating the Einstein and Jordan frames and defining interactions of the chameleon field with its environment, the chameleon field can be identified with quintessence, responsible for the late-time acceleration of the Universe expansion and dark energy dynamics. The later can be described through the potential of the self-interaction of the chameleon field. In case of the standard dependence of the radiation and matter densities on the expansion parameter ρ r (a) ∼ a −4 and ρ m ∼ a −3 [34] local conservation of the total energy-momentum tensor of the chameleon field, radiation and matter demands the conformal factor to be equal to unity. This prohibits the identification of the chameleon field with quintessence and suppresses any interaction of quintessence with radiation and matter densities. In the Conclusion we discuss i) the obtained results and ii) our results in comparison with the results, obtained in Ref. [10] and in the Scalar-Tensor (ST) gravitational theories.

II. EINSTEIN'S EQUATIONS IN THE EINSTEIN GRAVITY WITH CHAMELEON AND MATTER FIELDS
The Einstein-Hilbert action of the Einstein gravity with the chameleon field coupled to a matter we take in the form where M Pl = 1/ √ 8πG N = 2.435×10 27 eV is the reduced Planck mass and G N is the Newtonian gravitational constant [35], R is the Ricci scalar curvature, expressed in terms of the Christoffel symbols { α µν } [34], L[φ] is the Lagrangian of the chameleon field where V (φ) is the potential of the chameleon field self-interaction. The matter fields as well as the radiation are described by the Lagrangian L m [g µν ]. The interaction of the matter fields and radiation with the chameleon field runs through the metric tensorg µν in the Jordan frame [1, 2, 36], which is conformally related to the Einstein-frame metric tensor g µν byg µν = f 2 g µν (org µν = f −2 g µν ) and √ −g = f 4 √ −g with f = e βφ/M Pl , where β is the chameleonmatter coupling constant [1,2]. The factor f = e βφ/M Pl can be interpreted also as a conformal coupling to matter fields and radiation [36] (see also [1, 2] and [37]).
Varying the action Eq.(1) with respect to metric tensor δg µν (see, for example, [34]) we arrive at the Einstein equations, modified by the contribution of the chameleon field where R µν is the Ricci tensor [34],T µν are the matter (with radiation) and chameleon energy-momentum tensors, respectively, determined bỹ The factor f 2 appears in front ofT where we have used that νµ . Then, the quantitiesρ,p andũ µ in the Jordan frame are related to the quantities ρ, p and u µ in the Einstein frame as [36] This givesT where T µν is the total energy-momentum tensor equal to Below we analyse the Einstein equations Eq.(8) in the Cold-Dark-Matter (CDM) model [35] in the Friedmann flat spacetime with the line element [34] where g 00 (x) = 1 and g ij (x) = a(t) η ij with η ij = −δ ij . Then, a(t) is the expansion parameter of the Universe evolution [34]. The Christoffel symbols { α µν }, the components of the Ricci tensor R µν and the scalar curvature R are equal to [34] where η iℓ η ℓj = δ i j andȧ andä are first and second derivatives with respect to time.

III. FRIEDMANN-EINSTEIN EQUATIONS OF THE UNIVERSE EVOLUTION
In the Friedmann spacetime the Einstein equations Eq.(8) define the following equations of the Universe evolution, which are usually called Friedmann's equations (or the Friedmann-Einstein equations) [34], where ρ r and ρ m are the radiation and matter densities. The scalar field φ couples to radiation and matter densities through the conformal factor f = e βφ/M Pl . Then, the radiation density ρ r and pressure p r are related by the equation of state p r = ρ r /3 [34]. For the description of matter we use the Cold Dark Matter (CDM) model with the pressureless dark and baryon matter [35]. The scalar field density ρ φ and pressure p φ are equal to Varying the action Eq.(1) with respect to the scalar field φ and its derivative one gets the equation of motion for the scalar field [37]. In the Friedmann spacetime it reads where V eff (φ) is the effective potential given by The contribution of the radiation density comes into the effective potential in the form (ρ r − 3p r )(f − 1). Because of the equation of state p r = ρ r /3 such a contribution vanishes. Thus, through the interaction with matter density ρ m the scalar field can acquire a non-vanishing mass if the effective potential V eff (φ) obeys the constraints i.e. the effective potential V eff (φ) possesses a minimum at φ = φ min . An important role for a dependence of a chameleon field mass on a density of an environment plays the conformal factor f and its deviation from unity. Below we analyse the conditions at which the conformal factor f can deviate from unity. This should allow i) to identify the chameleon field with quintessence and ii) to argue an importance of the chameleon field for the Universe evolution and dark energy dynamics.
A. Bianchi identity, conservation of total energy-momentum tensor and conformal factor Using Eq.(11) and taking into account that in the Friedmann flat spacetime the non-vanishing components of the Einstein tensor G µν = R µν − 1 2 g µν R are equal to one may show that the Einstein tensor G µν obeys the Bianchi identity [34] where G µν ;µ is a covariant divergence. As a result, the covariant divergence of the total energy-momentum tensor T µν ;µ should also vanish Because of only time-dependence Eq.(20) takes the form where we have taken into account Eq. (11). Using the non-vanishing components of the total energy momentum tensor we transcribe Eq.(21) into the form Since Eq. (15) can be rewritten as follows Eq.(23) is given by where we have used the equation of state p r = ρ r /3 [34]. Because of independence of radiation and matter densities Eq.(25) can be splitted into evolution equations of the radiation and matter densities For the standard dependence of the radiation and matter densities on the expansion parameter a(t) [34] where a 0 , H 0 = 1.437(26) × 10 −33 eV, Ω r and Ω m are the expansion parameter, the Hubble rate and relative radiation and matter densities at our time t 0 = 1/H 0 [35], the equations for the radiation and matter densities Eq. (26) are satisfied identically for f = 1. Thus, if the radiation and matter densities depend on the expansion parameter a as ρ r (a) ∼ a −4 and ρ m (a) ∼ a −3 , local conservation of the total energy-momentum in the Universe can be fulfilled if and only if the conformal factor f , relating the Einstein and Jordan frames and defining the chameleon-matter interactions, is equal to unity, i.e. f = 1. However, in this case there is no influence of the chameleon field on the evolution of the radiation and matter densities and a dependence of the chameleon field mass on a density of its environment. Assuming that the chameleon field can in turn make an influence on an evolution of the radiation and matter densities we obtain the following solutions to Eq.(26) where ρ r0 = 3M 2 Pl H 2 0 Ω r and ρ m0 = 3M Any observation of the corrections to the radiation and matter densities may in principle confirm an existence of the chameleon field and a correctness of an identification of the chameleon field with quintessence. Nevertheless, we have to emphasize that the contribution of the conformal factor the the radiation and matter density at our time is not practically observable. It is seen from the solutions Eq.(28) that the conformal factor affects the evolution of the radiation and matter densities during the radiation-and matter-dominated eras. However, to be more consistent with the solutions for the radiation and matter densities one has to solve Eq. (26) and Eq.(15) as a system of integro-differential equations. Of course, this system can be solved by using perturbation theory keeping the contribution of the conformal factor to leading order in the chameleon field expansion. It is well-known that the Friedmann-Einstein differential equation forȧ 2 /a 2 should be the first integral of the Friedmann-Einstein differential equation forä/a [34]. In order to find the conditions for which Eq.(12) is the first integral of Eq.(13), when the chameleon field is identified with quintessence, we propose to rewrite Eq.(12) as followṡ where ρ ch is the chameleon field density, given by Eq. (14) with the replacement V (φ) → V eff (φ), and to find ρ ch as a function of the expansion parameter a. This can be done transcribing Eq.(15) into the form where we have denoted V eff (φ) = U eff (a), assuming that φ is a function of a, i.e. φ = ϕ(a). As a function of the expansion parameter a the effective potential U eff (a) is given by where U (a) = V (φ) = V (ϕ(a)) and f (a) = e βϕ(a)/M Pl . The solution to Eq.(32) is equal to where the term C φ /a 6 corresponds to the contribution of the kinetic term of a massless scalar field [38]. The integration constant C φ we define as follows C φ = 3M 2 Pl H 2 0 Ω φ a 6 0 , where Ω φ is the integration constant, having the meaning of a relative density of a massless scalar field at our time t 0 = 1/H 0 . As a result, Eq.(31) takes the forṁ Further it is convenient to rewrite Eq.
where we have used Eq. (31). Since the second derivativeä of the expansion parameter a with respect to time can be given byä The solution to Eq.(38) amounts tȯ where C is the integration constant. Dividing both sides of Eq.(39) by a 2 we arrive at the equatioṅ where we have set C φ = 3M 2 Pl C = 3M 2 Pl H 2 0 Ω φ . Thus, Eq.  (42) and that ρ r (a)f (a) = ρ r0 f (a 0 )a 4 0 /a 4 (see Eq. (28)), we transcribe the right-hand-side (r.h.s.) of Eq.(41) into the forṁ This proves that Eq. (12) is the first integral of Eq.(13) if the total energy-momentum is locally conserved. An existence of a non-trivial conformal factor f (a) = 1 or, correspondingly, an existence of the chameleon field and a possible identification of the chameleon field with quintessence can be confirmed or rejected by observations of the corrections Eq. (30) to the radiation and matter densities. In case of the standard dependence of the radiation and matter densities ρ r (a) ∼ a −4 and ρ m (a) ∼ a −3 the conformal factor should be equal to unity. This suppresses the existence of the chameleon field as a scalar field with mass dependent on a density of its ambient matter [1, 2].

IV. CONCLUSION
We have analyse the conditions, at which one can identify the chameleon field with quintessence, a canonical scalar field responsible for the late-time acceleration of the Universe expansion and dark energy dynamics, and the role of the conformal factor, relating the Einstein and Jordan frames and describing interactions if the chameleon field with its ambient matter, in the evolution of the Universe. We have found that local conservation of the total energy-momentum of the system, including the chameleon field, radiation and matter (dark and baryon matter), leads to the equations of the evolution of the radiation and matter densities, corrected by the conformal factor. We have shown that these equations of the radiation and matter density evolution play an important role for the proof that the Friedmann-Einstein equation forȧ 2 /a 2 is the first integral for the Friedmann-Einstein equation forä/a. Hence, one may argue that if the radiative and matter densities as functions of the expansion parameter a have a standard behaviour ρ r (a) ∼ a −4 and ρ m (a) ∼ a −3 [34] the conformal factor, relating the Einstein and Jordan frames and defining the chameleon-mater and chameleon-radiation couplings, should be equal to unity. This suppresses any chameleon-matter and chameleon-radiation interaction and an identification of the chameleon field with quintessence. We would like to note that the corrections to the radiation and matter densities, coming from the conformal factor, should be noticeable in the radiation-and matter-dominated eras. In the dark energy-dominated era or in the latetime acceleration of the Universe expansion, where the expansion parameter is equal to a 0 , the contributions of the conformal factor to the radiation and matter densities in comparison with the standard values ρ r (a 0 ) = 3M 2 Pl H 2 0 Ω r and ρ m (a 0 ) = 3M 2 Pl H 2 0 Ω m are not practically observable during the Hubble time. In our analysis the cosmological constant, proportional to the dark energy density ρ Λ , can be introduced in terms of the constant part of the potential of the self-interaction of quintessence (or the chameleon field) with the scale parameter Λ = 4 3M 2 PL H 2 0 Ω Λ = 2. 24(2) meV [1], calculated for the relative dark energy density Ω Λ = 0.685 +0.017 −0.016 [35]. The φ-dependent part of the potential of the self-interaction of quintessence (or the chameleon field) V(φ) is arbitrary to some extent, i.e. model-dependent, and demands a special analysis similar to that carried out in [4,10,39], which goes beyond the scope of this paper. We are planning to perform it in terms of U(a) = V(φ) = V(ϕ(a)) in our forthcoming publication.
Thus, we may argue that the conformal factor can be in principle practically constant during the Hubble time, as has been pointed out by Wang et al. [32] and Khoury [33], but such a behaviour of the conformal factor does not mean that the chameleon field, when identified with quintessence, plays no role for the late-time acceleration of the Universe expansion and, correspondingly, for dark energy dynamics.
The influence of the interactions of quintessence with dark matter on dark energy dynamics and traces of such interactions in the Cosmic Microwave Background (CMB) have been investigated in some models of coupled quintessence in the papers [40][41][42][43]. Our analysis of the interaction of the chameleon field (quintessence), carried out in the modelindependent way apart from the CDM model of the dark and baryon matter, which is also accepted in [40][41][42][43], agrees well with the results, obtained in [40][41][42][43], concerning the existence of the traces of dark energy dynamics in the matter density distribution during the Universe evolution. Now let us discuss the results, obtained in this paper, in comparison with those, given in [10]. First of all we would like to emphasize that we solve different problems. Indeed, in present paper we have analysed the role of the conformal factor, relating the Einstein and Jordan frames and describing the couplings of the chameleon field to its ambient matter, in the evolution of the Universe. As we have discussed above, we have shown that quintessence can have the properties of the chameleon field and can couple to ambient matter and radiation through the conformal factor if and only if the radiation and matter (cold dark and baryon matter) densities deviate from their standard dependence on the expansion parameter a and such deviations are defined by the conformal factor. We have also found that the equations of the evolution of the radiation and matter densities(ρ r f ) = −4H(ρ r f ) andρ m = −3Hρ m f , where H =ȧ/a (see Eq. (26), are important for the proof that the Friedmann-Einstein equation forȧ 2 /a 2 is the first integral of the Friedmann-Einstein equation forä/a (see Eq. (42) and discussion below).
In turn, in [10] the identification of the chameleon field with quintessence has been accepted from the very beginning. As has been pointed out by Brax et al. [10], there has been shown "that the chameleon scalar field can drive the current phase of cosmic acceleration for a large class of scalar potentials that are also consistent with local test of gravity. This provides explicit realization of a quintessence model where the quintessence scalar field couples directly to baryons and dark matter with gravitational strength." Unfortunately, the role of the conformal factor, describing such a coupling, has not been investigated in [10]. Since the evolution of the radiation and matter densities are described in [10] by the equationsρ r = −4Hρ r andρ m = −3Hρ m , these densities have a standard behaviour as functions of the expansion parameter a, i.e. ρ r (a) ∼ a −4 and ρ m (a) ∼ a −3 . This is unlike our analysis of the radiation and matter density evolution. The deviations of the matter density from its standard dependence ρ m (a) ∼ a −3 are described in [10] by perturbations δ c = δ(ρ m f )/(ρ m f ) within the cosmological perturbation theory in the synchroneous gauge [44][45][46]. As a result, the perturbations δ c are described by the second order differential equation with respect to conformal time [34]. However, these perturbations have no relation to δρ m (a), calculated in this paper (see Eq. (30). Thus, one may assert that the results, which have been obtained in this paper, and the problem, which has been solved here, are fully new and do not repeat the results and problems, obtained and analysed in [10].
In [45] there has been investigated i) the effect of the time evolution of extra dimensions on CMB anisotropies and large-scale structure formation and ii) the impact of scalar fields in a low-energy effective description of a general class of brane world models on the temperature an isotropy power spectrum, iii) the effect of these fields on the polarization anisotropy spectra and iv) "the growth of large-scale structure, showing that future CMB observations will constrain theories of the Universe involving extra dimensions even further". As a toy model the authors considered two branes with matters, distributed on them. As has been found, the matter densities on the branes obey the evolution equations, the solutions of which imply that ρ m (a)a 3 = const that is similar to our results Eq. (28). However, it is not related to our analysis of i) the identification of the chameleon field with quintessence and of ii) self-consistency of the Friedmann-Einstein equations.
As regards the scalar-tensor (ST) theories of gravitation [47]- [52], which take a beginning from the well-known Jordan-Fierz-Brans-Dicke (JFBD) gravitational theories [53][54][55], they are alternative to the Einstein theory of gravitation. The ST gravitational theories start with the action in the Jordan frame [52] where G is the bare gravitational constant, i.e. the gravitational constant in the absence of scalar interactions,g µν and Φ are the metric and the scalar field, which are gravitational field variables,R is the Ricci scalar curvature in the Jordan frame, the functions F (Φ), Z(Φ) and U (Φ) are arbitrary functions of the scalar field Φ). Then, S m [Ψ m ,g µν ] is the matter field Ψ m action, where the matter field couples directly to the metricg µν only, so that weak equivalence principle is preserved by construction [52]. Because of substantial complexity of the gravitational field equations in the Jordan frame the ST gravitational theories define in the Einstein frame through the conformal transformatioñ g µν = F −1 (Φ) g µν = A 2 (φ) g µν , where A(φ) if the conformal factor and a metric g µν and a scalar field φ are new gravitational field variables in the Einstein frame. After rescaling the gravitational action of the ST gravitational theory in the Einstein frame can be given in the following form where R is the scalar curvature, defined by the metric g µν [34], and the Lagrangian L[φ] of the scalar field φ is given by Eq.
(2). According to [47]- [52], the Einstein equations of the ST gravitational theory, defined by the action Eq.(45), read where the energy-momentum tensors T µν = (ρ + p)u µ u ν − p g µν , respectively. In turn, the equation of motion for the scalar field φ is where H =ȧ/a, α(φ) = ∂ℓnA(φ)/∂φ and T (m) µν g µν = ρ − 3p. Then, the matter and radiation densities satisfy the evolution equations [47]- [52] In the CDM model with p m = 0 and p r = ρ r /3 the evolution equations Eq.(48) take the forṁ The solutions to these equations as functions of the expansion parameter a look as follows where ρ m0 = 3M 2 Pl H 2 0 Ω m and ρ r0 = 3M 2 Pl H 2 0 Ω r and φ 0 is the scalar field at a(t 0 ) = a 0 . Firstly, in the ST gravitational theories the radiation density does not deviate from its standard dependence on the expansion parameter ρ r (a) ∼ a −4 . This is unlike our result (see Eq. (28)). Secondly, the deviation of the matter (dark and baryon matter) density from its standard dependence ρ m (a) ∼ a −3 is caused by the conformal factor A(φ). However, such a deviation i) differs from that obtained in our analysis (see Eq. (28)) and ii) has no relation to the problem, which we are solving in this paper, namely, can the chameleon field be identified with quintessence or not? The analysis of self-consistency of the Friedmann-Einstein equations forȧ 2 /a 2 andä/a in the ST gravitational theories has not been investigated in [47]- [52] and goes beyond the scope of the present paper. So one may conclude that the results, obtained in the present paper, confirm the results, obtained in the ST gravitational theories concerning a dependence on the conformal factor of the deviation of the matter density from its standard behaviour ρ m (a) ∼ a −3 . Of course, the dependence on the conformal factor of the matter density, obtained in the present paper, differs from that in the ST gravitational theories and is related to the problem -can the chameleon field be identified with quintessence of not ? -the solution of which goes beyond the scope of the ST gravitational theories.
Finally we would like to emphasize that a specific analysis of a dynamics of the chameleon field such as chameleon screening mechanism and formation of a fifth force, for example, in the Galaxy and the Solar system is related to a special choice of the potential of the self-interaction of the chameleon field [10,56] and the conformal factor as a functional of the chameleon field. Such an analysis has been carried out by Brax et al. [10] and Jain et al. [56]. The repetition of such an analysis goes beyond the scope of this paper. Moreover, since we do not specify the potential of the self-interaction of the chameleon field and the conformal factor, the results, obtained in this paper, do not contradict the results, obtained by Brax et al. [10] and Jain et al. [56]. As we have found (see the solution Eq.(28)), the contributions of the conformal factor affect the evolution of the matter and radiation densities during the radiationand matter-dominated eras, but such contributions do not practically observable at our time. This agrees well with the constraints on the deviations of the radiation and matter densities from their values at our time to a few parts per million [56], which can be be obtained from the constraints on the fifth force caused by the chameleon field in the Galaxy and the Solar system.