Present-day theory of gravity relies upon the presence of an omni-present energetic background field. The existence of this background field is required to explain the accelerated expansion of the universe, known since 1998, [1]. This cosmological background field has been defined on the basis of Einstein’s Cosmological Constant [2]. It is also known as “dark energy”. The unavoidable conclusion is that there is not such a thing as “empty space”, but that space is filled with an energetic fluidum. This conclusion has given rise to the idea of conceiving the vacuum as an entropic medium filled with energetic constituents, in this article to be annotated as darks. As long as these darks are not subject to any directional energetic influence, their motions remain fully chaotic. In that state the vacuum is fully symmetric, because its state before and after a time interval of “closed eyes” with an arbitrary translation or rotation of the observer, is just the same [3]. In [4,5,6,7,8] it has been argued that if the cosmological background field would consist of energetic uniformly distributed polarisable vacuum particles, the dark energy would give an explanation for the dark matter problem as well, because vacuum polarization would evoke a gravitational equivalent of the well-known Debije effect [9]. With the difference, though, that the central force from a gravitational nucleus is enhanced just opposite to the suppression of the Coulomb force from an electrically charged nucleus in an ionized plasma. It means that the awareness of the Cosmological Constant implies a symmetry break. This is the issue that will be discussed in this article.

The modeling of the omni-present background energy by energetic vacuum particles, requires a model for its elementary constituent (the dark). This element must be a source of energy, and must be force feeling as well. In those aspects it resembles an electron, which is ultimately the source of electromagnetic energy, and which is sensitive to the fields spread by other electrons. However, whereas the dark in the cosmological background field must be polarisable under the gravitational potential, an electron is non-polarisable under an electric potential. The electric dipole moment of an electron is zero, while a dark should have a non-zero gravitational dipole moment. Recently, the author of this article found a Dirac particle, of a particular type that, unlike the electron type, possesses a dipole moment that is polarisable in a scalar potential field [10,11]. It is my aim to show in this article that a particle of this kind matches with the dark. The dark shares this property with the quark [12].

The cosmological and gravity view to be developed in this article relies, next to the awareness of the darks, on a particular interpretation of the $\mathsf{\Lambda }$ parameter in Einstein’s Field Equation. Different from the common perception that Einstein’s $\mathsf{\Lambda }$ is a constant of nature, usually identified as the Cosmological Constant, it is in the author’s view a covariant integration constant that may have different values depending on the scope of a cosmological system under consideration. This awareness is based upon Einstein’s note in his 1916 article in which he equated an integration constant as zero (see footnote on p.804 in ref. [13]). More on this in the next paragraph. Because it may depend on other attributes but just time-space coordinates, such as mass content, for instance, it may have different values at the level of solar systems, galaxies and the universe. Only at the latter level, it is justified to identify the $\mathsf{\Lambda }$ as the Cosmological Constant indeed. At that level, by the way, the cosmological system is in a state of maximum symmetry and maximum entropy. The viability of this view will be proven by a calculation of Milgrom’s empirical acceleration constant of dark matter [14].

In the article, first of all the need will be revealed for accepting a fluidal energetic vacuum with the profile just described. This will be done in a hierarchic approach. It is instructive to distinguish three levels in this. The first level is the galaxy level. That part contains an analysis of the dark matter problem on the basis of the role of Einstein’s $\mathsf{\Lambda }$in his Field Equation. It will be shown that this results into a modification of Newton’s gravity law that qualitatively fits with Milgrom’s empirical one. This is possible by conceiving the galaxy as a baryonic kernel that executes a central force. The second level is the cosmological level. In this level the universe is conceived as a uniform distribution of such baryonic kernels. This will enable deriving a testable quantitative result of Milgrom’s acceleration constant. It has to be emphasized that the analysis of the two cosmological levels, i.e. galaxy and universe, does not require a microscopic identification of the vacuum energy. Accepting the role Einstein’s $\mathsf{\Lambda }$in his Field Equation is adequate here. The third level is the other extreme: the quantum level. At that level a quantum interpretation is given for the dark energy fluid as represented by Einstein’s $\mathsf{\Lambda }$. The purpose here is twofold. The first is to connect the$\mathsf{\Lambda }\mathrm{CDM}$model [15] with entropic gravity and quantum gravity. The second purpose is to strengthen the analysis made in the first and second level by showing that calculating Milgrom’s acceleration constant in a (quantum) entropic way results in an identical expression as obtained in the non- (quantum) entropic way.

In a conclusive discussion paragraph a reflection will be given on the symmetry of the universe by summarizing the analytical results from the three levels (galaxy, universe, quantum level).

As just explained, an analysis of galaxies is the first thing to be done. To do so, Einstein’s Field Equation is invoked. The Field Equation reads as,

in which $T_{\mu \nu }$is the stress-energy function, which describes the energy and the momenta of the source(s) and in which $R_{\mu \nu }$and $R$ are respectively the so-called Ricci tensor and the Ricci scalar. These can be calculated if the metric tensor components $g_{\mu \nu }$are known [13,16,17]. The $\mathsf{\Lambda }$ term is missing in Einstein’s paper of 1916, in spite of his awareness that he equated an integration constant as zero (see footnote on p.804 in ref. [13]). Later, in 1917, Einstein added this quantity as a covariant integration constant for allowing vacuum solutions of his Field Equation [18,19]. As noted in the introduction, It is usually presently taken for granted that this Lambda is a Cosmological Constant that can be regarded as a constant of nature. In fact, however, it is just a constant in the sense that its value does not depend on space-time coordinates. It may depend on attributes, like for instance mass. Hence, the Lambda may have at the galaxy level a different value from the Cosmological Constant at the level of the universe. Einstein’s Field Equation is highly non-linear. As is well known, it can be linearized under suitable constraints and approximations, eventually even up to Newton’s law of gravity. Let us summarize the linearization and let us face the problem of a non zero value for $\mathsf{\Lambda }$ in this.

In the case that a particle under consideration is subject to a central force only, the space-time condition shows a spherical symmetric isotropy. This allows to read the metric elements $g_{ij}$from a simple line element that can be written as

In which $q_0=\mathrm{i}ct$ and $\mathrm{i}=\sqrt{-1}$.

It means that the number of metric elements $g_{ij}$reduce to a few, and that only two of them are time and radial dependent.

Note: The author of this article has a preference for the “Hawking metric” (+,+,+,+) for$(\mathrm{i}ct,x,y,z)$, like, for instance also used by Dyson [20] and Perkins [21]. By handling time as an imaginary quantity instead of a real one, the ugly minus sign in the metric (-,+,+,+) disappears owing to the obtained full symmetry between the temporal domain and the spatial one.

Before discussing the impact of $\mathsf{\Lambda }$, it is instructive to summarize Schwarzschild’s solution of Einstein’s equation for a central pointlike source with mass $M$in empty space and $\mathsf{\Lambda }=0$, in which the metric components appear being subject to the simple relationship

Solving Einstein’s equation under adoption of a massive source with pointlike distribution $T_{00}=M{c}^2{\delta }^3(r)$, results in a wave equation with the format [22],

in which $G$is the gravitational constant and $U(t)$is Heaviside’s step function.

Its stationary solution under the weak field limit

is the well-known Newtonian potential,

The wave equation (4) reduces to,

With inclusion of the constant $\mathsf{\Lambda }$, the wave equation is modified into (see Appendix),

If $T_{tt}$were a pointlike source $T_{tt}=M{c}^2{\delta }^3(r)U(t)$, the static solution of this equation would be provided by the Schwarzschild-de Sitter metric, also known as Kottler metric, [23,24], given by

The viability of (9) readily follows by insertion into (8) and subsequent evaluation. Obviously, we meet a problem here, because we cannot separate a weak field $\mathsf{\Phi }$ (= gravitational potential) from the metric, because we cannot a priori identify an $r-$domain that justifies the adoption of the constraint (5). However, given the fact that a viable wave function can be obtained for $\mathsf{\Lambda }=$ 0, one might expect that it must be possible to obtain a valid wave equation for a weak field $\mathsf{\Phi }$ showing a gradual move from $\mathsf{\Lambda }=$ 0 to $\mathsf{\Lambda }\ne$ 0. Inspection of (9) reveals that if

the Kottler metric reduces to the Schwarzschild metric. It may therefore be expected that the Einstein equation can be linearized in a spatial range between an upper limit that is set by the Kottler metric and the lower limit $r>>R_S$that is set by the Schwarzschild metric. As elaborated in the Appendix, the linearization between the two limits results into a modification of the $\mathsf{\Lambda }=0$ wave equation (7), to a $\mathsf{\Lambda }\ne 0$ wave equation with the format

in which ${\lambda }^2=2\mathsf{\Lambda }$ and in which $\frac{2\mathsf{\Phi }}{c^2}=h_{t\varphi }(\ne -h_{r\varphi })$.

Under static conditions, (11) reduces to

It is instructive to interpret this linearized result in terms of the non-linearized expression (8). Whereas an empty space with $\mathsf{\Lambda }=$ 0 corresponds with virtual sources $T_{\mu \mu }=$0, the vacuum with $\mathsf{\Lambda }\ne$ 0 is a fluidal space with virtual sources $T_{\mu \mu }=-p\mathsf{\Lambda }$, with$g_{\mu \mu }=(1,1,r^2{\sin }^2\vartheta ,r^2)$, in which $p=c^4/8\pi G$ [25,26,27]. (Owing to the Hawking metric, $p$ is equal for all diagonal elements). This particular stress-energy tensor with equal diagonal elements corresponds with the one for a perfect fluid in thermodynamic equilibrium [28]. Inserting a massive source in this fluid will curve the vacuum to $g_{\mu \mu }=(g_{tt},g_{rr},r^2{\sin }^2\vartheta ,r^2)$. Hence, inclusion of the $\mathsf{\Lambda }$ implies that, under absence of massive sources, Einstein’s equation can be satisfied if empty space is given up and is replaced by a space that behaves as a molecular fluidum in thermodynamic equilibrium. If, under bias of a uniformly distributed background energy, a massive pointlike source is inserted into this fluidum, deriving a meaningful wave equation is possible. The perception that in this thermodynamic equilibrium space-time is still uncurved and that curving is the result of a disturbance on it makes the format of the wave equation shown by (11) different from the formats reviewed in [29,30]. It corresponds with what is presently known as “emergent gravity”, in which the disturbance of the entropy on the thermodynamic equilibrium is seen as the origin of gravity [31,32].

As noted before, and well known of course, Poisson’s equation and its modification is the static state of a wave equation. From the perspective of classic field theory, a wave equation, can be conceived as the result of an equation of motion derived under application of the action principle from a Lagrangian density $\mathrm{L}$of a scalar field with the generic format in which $U(\mathsf{\Phi })$is the potential energy of the field and where$\rho \mathsf{\Phi }$is the source term. Comparing various fields of energy, we have,

$U(\mathsf{\Phi })=0$ for electromagnetism.

for the nuclear forces [33].

The non-trivial solutions of wave functions) in homogeneous format derived from (14), for the first case and the third case, are respectively,

The first case applies to electromagnetism (for ${\mathsf{\Phi }}_0=Q\lambda /4\pi {\epsilon }_0$) and to Newtonian gravity (for ${\mathsf{\Phi }}_0=MG\lambda$). The third case applies to Proca’s generalization of the Maxwellian field [34]. The latter one reduces to the first case if $\lambda \to 0$, while keeping ${\mathsf{\Phi }}_0/\lambda$ constant. Generically, it represents a field with a format that corresponds with the potential as in the case of a shielded electric field (Debije [9]), as well as with Yukawa’s proposal [33], to explain the short range of the nuclear force.

Let us, after this side-step, proceed on (12). It can be readily verified that this equation can be satisfied by,

Note that the goniometric shape of this solution is a consequence of the plus sign in front of ${\lambda }^2$. It has to emphasized once more that this expression holds under the classical weak field constraint and the presence of a central source of energy that evokes this field as the tiny variation in the generic spherical metric. The shape (17) will reveal some interesting features. In accordance with the concepts of classical field theory, the field strength can be established as the spatial derivative of the potential $\mathsf{\Phi }$. We may identify this field strength as a cosmological gravitational acceleration $g$. Let us compare this acceleration with the Newtonian one $g_N$.

Hence, from (17),

Not surprisingly, the gravitational acceleration is affected by the Einstein’s $\mathsf{\Lambda }={\lambda }^2/2$. If $\mathsf{\Lambda }=0$, the gravitational acceleration equals the Newtonian one $g_N$. Under a positive value of Lambda, the gravitational acceleration has a different spatial behavior. This is illustrated in Figure 1, which shows the ratio $g/g_N$as a function of the normalized spatial quantity $\lambda r$. Up to the value $\lambda r=3\pi /4$, $g/g_N$ rises monotonously up to the value $g/g_N=$ 3.33. This figure shows that, for relative small values of $r$, the cosmological acceleration behaves similarly as the Newtonian one. Its relative strength over the Newtonian one increases significantly for large values of $r$, although it drops below the Newtonian one at $\lambda r>$ 3.45. Up to slightly below $\lambda r=3\pi /4$, this is, as will be shown, a similar behavior as heuristically implemented in MOND. The effective range is determined by the parameter $\lambda$. It might therefore well be that the cosmological gravity force manifests itself only at cosmological scale. Let us consider its consequence.

Newtonian laws prescribe that the transverse velocity$v_{\varphi }(r)$of a cosmic object revolving in a circular orbit with radius $r$ in a gravity field is determined by

in which $M(r)$is the amount of enclosed mass. This relationship is often denoted as Kepler’s third law. Curiously, like first announced by Vera Rubin [35] in 1975, the velocity curve of cosmic objects in a galaxy, such as, for instance, the Milky Way, appears being almost flat. It is tempting to believe that this can be due to a particular spectral distribution of the spectral density to compose $M(r)$. This, however, cannot be true, because $M(r)$builds up to a constant value of the overall mass. And Kepler’s law states in fact that a flat mass curve $M(r)$is not compatible with a flat velocity curve. Figure 2 illustrates the problem. It is one of the two: either the gravitational acceleration at cosmological distances is larger than the Newtonian one, or dark matter, affecting the mass distribution is responsible. Cosmological gravity as expressed by (18) may give the clue. Its effective range is determined by the parameter $\lambda$. It might therefore well be that the cosmological gravity force manifests itself only at cosmological scale. Figure 3 shows that under influence of this force, the rotation curves in the galaxy are subject to a boost. This cosmological gravity shows another intriguing phenomenon. Like shown in Figure 1 and Figure 4, at the very far cosmological distance, the attraction of gravity is inversed into repulsion [36,37,38]. This repulsion shows up at the very far end of the spatial range. It prevents the clustering of the fluidal space, thereby eliminating the major argument against the fluidal space approach.

Further exploration of this phenomenon is a subject outside the scope of this article. It has to be noted that the solution (19) is not unique. There are more solutions possible by modifying the magnitude of $\sin \lambda r$ over $\cos \lambda r$. I have simply chosen here for the symmetrical solution. Cosmological observations would be required to obtain more insight in this.

Whether this theoretically derived modification of the Newtonian gravity indeed explains the excessive orbital speeds of stars in a galaxy, such as formulated in Milgrom’s empirical law in MOND, is dependent on the numerical value of Einstein’s $\mathsf{\Lambda }$. If Milgrom’s empirical theory and the one developed in this paragraph are both true, it must be possible to relate Einstein’s $\mathsf{\Lambda }$with Milgrom’s acceleration constant $a_0$, like will be done next.

MOND is a heuristic approach based on a modification of the gravitational acceleration$g$such that

In which $\mu (x)$is an interpolation function, $g_N(=MG/r^2)$ the Newtonian gravitational acceleration and in which $a_0$ is an empirical constant acceleration. The format of the interpolation function is not known, but the objectives of MOND are met by a simple function like [14,39]

If $g/a_0<<1$, such as happens for large $r$, (20) reduces to

Under this condition, the gravitational acceleration decreases as $r^{-1}$instead of $r^{-2}$. As a result, the orbital velocity curves as a function of $r$show up as flat curves.

Algebraic evaluation of (20) and (21) results into,

This expression allows a comparison with (18).

As illustrated in Figure 5, a pretty good fit is obtained between (18) and (23) in the range up to $\lambda r=3\pi /4$(where the theoretical curve starts decaying), if

Observations on various galaxies have shown that $a_0$can be regarded as a galaxy-independent constant with a value about $a_0\approx$1.25 x 10^-10 m/s² [40].

The implication of (24) is, that $a_0$is a second gravitational constant next to $G$. The two constants determine the range$\lambda$of the gravitational force in solar systems and galaxy systems as ${\lambda }^2\approx 2a_0/5MG$, in which $M$is the enclosed mass in those systems. Whereas this second gravitational quantity $a_0$is an invariable constant, this is apparently not true for the Einsteinean parameter $\mathsf{\Lambda }$.

This result shows that Milgom’s empirical law and the theory as developed in this article are intimately related. Figure 6 shows the difference between the curves for the orbital velocity of stars in galaxies according to MOND as compared to those as predicted by the theory as developed in this article. It has to be emphasized here that establishing the fit between the two curves by setting $k=$2.5 is only meant to incorporate Milgrom’s acceleration constant $a_0$as an unknown parameter into the theory. This implies that no limitation on whatsoever is imposed, nor that the generality of the analysis is affected. From Figure 5 it is shown that beyond $r\lambda =3\pi /4$the developed theory deviates from MOND and Figure 4 shows that beyond $r\lambda =$3.66 the gravitational attraction changes into a repulsion. From this perspective, the latter phenomenon would even put a natural limit to the size of a galaxy.

Let us consider these ranges for the Milky Way. As long as $\lambda r<3\pi /4$we have

This implies a spatial coincidence range between MOND and the theory developed so far, up to a galaxy radius $R_M$to the amount of

where $R_S=2MG/c^2$ is the Schwarzschild radius and $L=c{t}_H$ is the Hubble scale ($t_H=$13.8 Gyear). Because from calculation $a_L\approx$6.9 x 10^-10 m/s² and because $a_0\approx$1.25 x 10^{- 10} m/s² from MOND’s assessment to most galaxies (if not all), we have from (26) for the Milky Way with Schwarzschild radius $R_S\approx$0.2 lightyear,

This is well beyond the radius of the Milky Way, which amounts to 180.000 - 200.000 lightyear. The coincidence range between MOND and this theory (up to $r\lambda =3\pi /4)$is well within the spatial validity range $r\lambda <6$ due to the weak field limit constraint and the linearization approximation such as derived in the Appendix.

As noted before, apart from this upper limit for the range of validity, there is a lower limit as well. This has to do with the weak field limit constraint that we have imposed to derive a single parameter wave equation from Einstein’s Field Equation. The value of this lower limit has been derived in the Appendix as

For the Milky Way ($R_S\approx$0.2 lightyear; $R_M\approx$458000 lightyear) this lower limit amounts to

Considering that our solar system is at about 26.000 lightyear from the center, it will be clear that the modified Newtonian gravitation law (18) holds for the Milky Way. Because many other galaxies are similar to the Milky Way, it is quite probable that this new theory solves the anomaly problem of the stellar rotation problem of most, if not all, galaxies.

From this result it may be concluded that that Milgrom’s acceleration constant and Einstein’s $\mathsf{\Lambda }$are closely related indeed. From (25) and (11), we have

It also means that Einstein’s $\mathsf{\Lambda }$is not a constant of nature, but instead, like noted before, a covariant integration constant that, while being independent of space-time coordinates, may be dependent on attributes of any cosmological system that is subject to Einstein’s Field Equation. If the system is a spherical one, such as solar systems or galaxies, the value of Einstein’s $\mathsf{\Lambda }$depends on the baryonic mass content of the system under consideration.

So far, we have considered a spherical gravitational system under influence of a central gravitational force, such as applies to solar systems and galaxies. But what about the universe? For any observer in the cosmos, the universe is a sphere with distributed matter. Let us model the universe as a sphere in the cosmos with radius $L$and distributed gravitational energy. We have discussed before that the vacuum is fluidal space with virtual sources $T_{\mu \mu }=-p\mathsf{\Lambda }$, in which $p=c^4/8\pi G$ . Denoting the gravitational background energy density as $\rho$, we have

We have concluded before that $\mathsf{\Lambda }$is related with some baryonic mass $M_B$, such that

The distributed energy is a gradually developed mixture of the energy from fluidal matter as meant by (31) and the energy from baryonic matter$M_B$ as meant by (32). From these expressions it can be concluded that the total gravitation energy $M_G{c}^2$in a sphere with radius $L$can be expressed as

Let $\mathsf{\Delta }{M}_G$the difference between the gravitational matter in a sphere $L+\mathsf{\Delta }L$ and the gravitational matter in a sphere $L$. It follows readily that

in which the baryonic matter is expressed as a dimensionless fraction ${\mathsf{\Omega }}_B$of the gravitational matter,

Note: In terms of the Lambda-CDM nomenclature, the baryonic share is expressed as ${\mathsf{\Omega }}_B$in the relationship

In which ${\mathsf{\Omega }}_m,{\mathsf{\Omega }}_{\mathsf{\Lambda }},{\mathsf{\Omega }}_B,{\mathsf{\Omega }}_D$, respectively, are the relative matter density, the relative dark energy matter density, the relative baryonic matter density and the relative dark matter density [17]. While the matter distribution between the matter density ${\mathsf{\Omega }}_m$(= 0.259) and dark energy density ${\mathsf{\Omega }}_{\mathsf{\Lambda }}$(= 0.741) is largely understood as a consequence from the Friedmann equations [40] that evolve from Einstein’s Field Equation under the Friedmann-Lemairtre-Robertson-Walker (FLRW) metric [17], the distribution between the baryonic matter density ${\mathsf{\Omega }}_B$(= 0.0486) and dark matter density ${\mathsf{\Omega }}_D$(= 0.210) is empirically established from observation. The quoted values are those as established by the Planck Collaboration [15].

Eq. (34) can now be integrated as

Hence, the gravitational energy density$\rho c^2$in the sphere with radius $L$ is given by

Because the visible universe is a sphere from which light cannot escape, its radius equals the Schwarzschild radius [41],

in which $M_U$ is the total gravitational mass of the universe and in which$\rho$is the overall matter density of the universe. Hence, from (38) and (39),

Identifying $L$ like before as the Hubble scale $L=c{t}_H$ ($t_H=$13.8 Gyear), we $a_L\approx$6.9 x 10^-10 m/s² and inserting the empirical $\mathsf{\Lambda }\mathrm{CDM}$value ${\mathsf{\Omega }}_B$= 0.0486 the into this expression gives the well known value $a_0\approx$1.25 x 10^-10 m/s² for Milgrom’s acceleration constant. It is a result that relates the baryonic content of the universe with Milgrom’s constant by a rather simple expression. It makes the theory developed so far testable by experimental observation, such as required for its viability.

In spite of now having obtained by theory testable numerical values for dark matter and dark energy, the true physical nature of these components has remained unclear. All we know so far is, that the universe is apparently filled with an energetic fluid that has got a mathematical abstraction in Einstein’s Field Equation in terms of virtual sources $T_{\mu \mu }=-p\mathsf{\Lambda }$, with $p=c^4/8\pi G$. The issue to be addressed next Is the question whether it is possible to give a physical profile to these constituents of the energetic background fluid. Let us conceive these constituents as vacuum particles in a state of Heisenberg unrest. Let us denote these particles as darks and let us suppose that these darks show a polarisable dipole moment in a scalar potential field. Like already discussed before, a background fluid with polarisable dipoles executes a shielding effect on a scalar potential. It may suppress its strength, like in the case of the potential field of a electrically charged particle in an ionic atomic plasma (Debije effect) or enhance is strength, like in the case of modified gravity as explained in before in this article (section 2). Let us try to a density expression for these vacuum particles (darks). To do so, let us rewrite (12) as,

and subsequently into,

${\nabla }^2\mathsf{\Phi }=-4\pi G\rho (r)$, in which

In Debije’s theory of electric dipoles [6,9,45],

The vector ${\mathrm{P}}_{\mathrm{g}}$ is the dipole density. From (58),

Assuming that in the static condition the space fluid is eventually fully polarized by the field of the pointlike source, $P_g(r)$is a constant $P_{g0}$. Hence, from (59),

Taking into account that to first order, we have from (60) and (61),

Hence, from (30), (57) and (60-62),

By assigning an elementary dipole moment${\mu }_G$ to the dark, the volume density$N/$m³ of the darks is found as,

While under adoption of Hubble’s law and the FLRW-metric the relationship between matter (${\mathsf{\Omega }}_m$) and dark energy (${\mathsf{\Omega }}_{\mathsf{\Lambda }}$), as expressed by (52-55) is a straightforward consequence of Einstein’s Field Equation, such as established in the Lambda-CDM (ΛCDM) model of the Planck collaboration group [38], the relationship between dark matter (${\mathsf{\Omega }}_D$) and baryonic matter$({\mathsf{\Omega }}_B)$ is in ΛCDM empirically assessed. In this article, I have given it a theoretical basis. A crucial element in this, is the interpretation of Einstein’s Λ in his Field Equation. Usually, this quantity is considered as a constant of nature. In the context of ΛCDM its value is established from (44) as a relationship between Λ and gravitational matter, such that

From (84) and (85),

This expression assesses a numerical value of Einstein’s Λ at the level of the universe. At this level it makes sense to indicate this value as the Cosmological Constant. Although this quantity is a measure for the gravitational energy in the universe its magnitude is very small. This small value shows about a difference of about 120 orders of magnitude with the zero-point energy suggested by quantum field energy [51]. This discrepancy is known as “the Cosmological Constant catastrophe. In the view as discussed in this article, the magnitude of the Cosmological Constant is a result from the linearization of Einstein’s Equation with a removal of the background bias. It has to be taken into account, however, that in quantum field theory such a removal is required as well. This removal is known as renormalization.

In this article, though, I have demonstrated in this article that in the application of Einstein’s Field Equation at the level of solar systems and galaxies Einstein’s Λ, while being a constant indeed in terms of independence of space-time coordinates, depends on the baryonic matter content$M_B$of the system under consideration, such that

The dependence on the baryonic matter content is obviously present at the level of the universe as well, although it remained hidden as part of the critical mass content in the considerations (85-86). It wouldn’t be correct to analyze the universe as a simple spherical system with a central gravitational force, as if it were a huge galaxy. The universe is a distributed assembly of such spherical subsystems. It is for that reason that the critical mass of the universe, has been related in (33) to distributed baryonic kernels as fractions of gravitational matter. This not only allows assigning a numerical value to the constant a₀ , but it also allows establishing it as a true cosmological invariant as a second gravitational constant next to the Newtonian $G.$  Curiously, whereas Einstein’s is not a true cosmological constant, Milgrom’s acceleration seemsto be. It has to be noted, though, that the description of the universe in the few simple parameters $a_0,G$  and is a status quo in the sense that there is no guarantee that the numerical values of the first two of these are true invariants over cosmological time, nor that Hubble’s law has been true back to the big bang.

The visible universe is a space filled with energetic vacuum particles that inherit their energy from their Heisenberg uncertainty in spatial position. It are virtually mass less ($\approx$3 10^-32 eV) gravitational quantum particles that possess a polarisable dipole moment with an eigen value $\hslash /2c.$ In free state, the particle density is N/m³  1.7 10¹⁴ particles per cubic nanometer, in which $a_0$is a true cosmological invariant, with a numerical value equal to Milgrom’s empirical acceleration constant for dark matter. If all these dipole moments were randomly orientated, the universe would show a perfect symmetry. The major part of these dipole moments ($\approx$74%) is randomly oriented indeed, because these don’t feel a polarisable influence from a baryonic cluster. This part composes the dark energy of the universe. A significant part still ($\approx$21%) of the polarisable dipole moments are polarized around baryonic clusters. This part, with its frozen symmetries, composes the dark matter, which enhances the gravitational strength of the baryonic clusters. The baryonic clusters are densely packed energetic particles that composes the observable baryonic matter (${\mathsf{\Omega }}_B\approx$5%) of the universe. This amount is related with Milgrom’s constant as ${\mathsf{\Omega }}_B=4t_H{a}_0/15c$.

It might well be that the anti-dogmatic views on Einstein’s Lambda and on the constituting elements of the energetic background energy may meet opposition. However, as shown in this article, these starting points allows a consistent derivation by theory of two important results, which so far are only established as empirical quantities. The first is a simple formula for Milgrom’s acceleration constant, derived by theory from the two starting points. It says that $a_0=(15/4){\mathsf{\Omega }}_B(c/t_H)$. In most articles on Milgrom’s constant, the analysis is restricted to spherical systems with a centric force. Those articles are subject to criticism because of that reason. In this article, I have demonstrated that the analysis can be extended to a universe with distributed matter. The second important result is the calculation by theory of the matter distribution in the universe in terms ${\mathsf{\Omega }}_B+{\mathsf{\Omega }}_D+{\mathsf{\Omega }}_{\mathsf{\Lambda }}=1$(resulting into 0.0486 + 0.210 + 0.741 = 1). The two results taken together can be viewed as a testable solution for the dark matter problem.

The main reason of including the appendix is to show the validity range for the weak field limited modification of Newton’s gravitation law, due to Einstein’s gauge constant $\mathsf{\Lambda }$. To do so properly, the derivation requires a short summary of common textbook stuff without $\mathsf{\Lambda }$, before extending it to meet the objective. This objective implies that we have to solve Einstein’s Field Equation for a spherically symmetric space-time metric that is given by the line element (2),

in which $q_0=\mathrm{i}ct$.

Note: The space-time (ict, r,$\vartheta ,\varphi$) is described on the basis of the “Hawking” metric (+,+,+,+). Once more, I would like to emphasize its merit that, by handling time as an imaginary quantity instead of a real one, the ugly minus sign in the metric (-,+,+,+) disappears owing to the obtained full symmetry between the temporal domain and the spatial one.

The components $g_{\mu \mu }$compose the metric tensor $g_{\mu \nu }$, which determine the Ricci tensor$R_{\mu \nu }$and the Ricci scalar $R$. These quantities play a decisive role in Einstein’s Field Equation, which reads as

In a space without massive sources, the Einstein Field Equation under this symmetric spherical isotropy, reduces to a simple set of equations for the elements $R_{\mu \mu }$ of the Ricci tensor,

(A3a,b,c,d)

Let us proceed by considering the Ricci scalar. It is defined generically as

In spherical symmetry the matrices contain diagonal elements only, so that (A-4) reduces to

This result can be applied to (A-3). Multiplying the first one with $g^{00}(=g^{tt})$, the second one with $g^{11}$, etc., and subsequent addition results of the terms $\mu =$1,2,3 gives,

Repeating this recipe for $g_{\mu \mu }(=1/g^{\mu \mu })$, we have for reasons of symmetry

(this result can be checked in ref. [52], under “equivalent formulations”).

Note that the subscripts and superscripts 00, 11 ,22, and 33 are, respectively, identical to $tt,rr,\vartheta \vartheta$ and $\varphi \varphi$. Applying this result to Einstein’s equation set (A-2), gives

such that after multiplication by $g_{\mu \mu }$, we have

As long as $T_{\mu \mu }$ is a pointlike centric source, like assumed in the remainder of this text, this expression is consistent with the relationship $R_{\mu \mu }-\mathsf{\Lambda }{g}_{\mu \nu }=0$ for empty space.

Let us first proceed under the conditions of the absence of massive sources ($T_{\mu \mu }=0$) and let us consider the Ricci tensor components $R_{tt}$ and $R_{rr}$under use of the results shown in Table A-1, that can be found in basic textbooks [53]. Note: $g^{\prime }$and $g^{″}$means differentiation, respectively double differentiation of $g$ into $r$; $\dot{g}$and $\ddot{g}$ means differentiation, respectively double differentiation of $g$ into $t$. Multiplying (A-3a) by $1/g_{tt}$ and (A-3b) by $1/g_{rr}$ gives,

which, under the assumption of a zero Cosmological Constant ($\mathsf{\Lambda }=$0), after subtraction and under use of the expressions in Table A-1 results into.

Hence

which can be integrated to (the Schwarzschild condition),

This, in turn, gives

Using (A-13), (A-15) and the Table A-1 values on $R_{tt}$ gives

Hence, from (A-10) and (A-16) ,

or, equivalently,

Applying well-known conditions,

$\mathsf{\Lambda }=$ 0 (already assumed) (no cosmological constant),

$g_{tt}(r,t)=1+h_{\varphi }(r,t)$, where $\left|h_{\varphi }(r,t)\right|<<1$ (the weak field limit)