Multiverse Model: External Universe(s) as Source of Dark Energy

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1. Introduction

The standard model of cosmology, ΛCDM (Lambda Cold Dark Matter), is able to explain the status of our Universe starting from the Big Bang via formation of elements, stars and galaxies until present-day cosmological observations. ΛCDM is based on Einstein’s General Relativiy (GR) and the field equations derived from GR:

${G}_{\mu \nu}+\Lambda {g}_{\mu \nu}=k{T}_{\mu \nu}$

with *G _{μν}* = Einstein tensor, Λ = cosmological constant,

Six parameters are minimally necessary for the ΛCDM model, age of the Universe, scalar spectral index, curvature fluctuation amplitude, reionization optical depth, baryon matter density, Dark Energy (DE) density, and CDM density. Despite huge efforts, the latter two parameters still cannot be explained physically at present. Nevertheless, the ΛCDM model has been and still is the most efficient tool to cover cosmological observations. The characterization and check of feasibility of a physical explanation of DE is the objective of this paper.

DE has negative pressure and contributes to *T _{μν}*, the stress-energy tensor, leading to accelerated expansion of the Universe. DE is acting like repulsive gravity constituting approx. 68% of the mass-energy density of the Universe [5]. The remaining 32% are CDM (28%) and normal, baryonic matter (4%).

The focus of this paper is to check the feasibility for the following physical explanation of DE: warping of spacetime in our Universe due to a source or sources located outside of our Universe. The basis for this hypothesis is the existence of one or more other universes in addition to our Universe (Multiverse model). The concept of multiple universes or Multiverses has controversially been discussed by many physicists, dividing them into two groups, the believers such as for example Tegmark [6], Riess [7], and Hawking [8] and the skeptics, for example, Penrose [9] and Mukhanov [10].

As a check of feasibility of the proposed hypothesis, the kinetic energy of our Universe due to the gravitational pull by one or more external universes has been calculated and from the resulting energy density, the two parameters, accelerated expansion and Λ, have been obtained using Einstein’s field equation. Both, acceleration and Λ, are in agreement with cosmological observations.

2. Calculations

In order to simplify the calculations, it is assumed that all universes have a zero curvature, *i.e.* they are flat, follow the same laws of physics and are similar in size (radius *r* = 4.40 × 10^{26} m) and mass (*m* [baryonic + dark matter] = 1.01 × 10^{54} kg). A three-dimensional coordinate system, *x*, *y*, *z*, is used with its origin U_{0} (0, 0, 0) at the center of our Universe. Accordingly, any other universes have *x*-, *y*-, and *z*-coordinates, which are equal to or larger than twice the radius of the Universe, *r*. In the following, all coordinates are provided as multiples of *r*.

The net gravitational pull, *F*, of external universes is calculated according to Newton’s law,
$F=G{m}_{1}{m}_{2}/{R}_{i}^{2}$ with *G* = 6.674 × 10^{−11} m^{3}/kg/s^{2}, *m*_{1} = *m*_{2} = 1.45 ×10^{53} kg (baryonic mass) + 8.7 × 10^{53} kg (Cold Dark Matter [CDM] mass) = 1.01 × 10^{54} kg (total mass). The mass of CDM is obtained from its density of 2.24 × 10^{−27} kg/m^{3}, as reported by Carmeli [11], and the volume of our Universe (3.57 × 10^{80} m^{3}). *R _{i}* is the vector pointing from the center of our Universe to the center(s) of the external universe(s).

The overall gravitational pull of more than one external universe, *F*, is obtained by additive vector calculation after determination of the individual *x*-, *y*-, and *z*-components, *F _{i}*, for the individual pull of each external universe.

The total gravitational pull, *F _{i}*, of any external universe is obtained from the individual components

The acceleration of our Universe induced by gravity of external universes *U _{i}*, is calculated from

The kinetic energy *E _{kin}* of our Universe due to the gravitational pull of the external universe(s) is calculated according to

${G}_{\mu \nu}+\Lambda {g}_{\mu \nu}=k{T}_{\mu \nu}$

With *G _{μν}* = Einstein tensor, Λ = cosmological constant,

${G}_{\mu \nu}+\Lambda {g}_{\mu \nu}=8\pi G{T}_{\mu \nu}$,

and, finally to

$\Lambda =8\pi G{\rho}_{\lambda}$,

with *ρ _{λ}* = density of DE, for Λ with the dimension s

and to

$\Lambda =4\pi G{\rho}_{\lambda}/{c}^{2}$,

with *c* = speed of light, for Λ with the dimension m^{−2}.

3. Results

The force, *F*, on our Universe by the gravitational pull of one external universe U (2, 0, 0) at a location twice the radius *r* from the center of our Universe, *i.e.* directly adjacent, is 8.88 × 10^{43} N (Figure 1, Table 1) resulting in a net acceleration of our Universe of 8.75 × 10^{−11} m/s^{2} and a net velocity of 35,865 km/s. As a consequence, after 13 billion years the Universe has moved 1.47 × 10^{25} m or 0.03 r from its original position in the direction of U_{1} and the latter has moved the same distance in the direction of our Universe.

The kinetic energy, *E _{kin}*, of our Universe due to the gravitational pull by one external universe is 6.53 × 10

If the external universe is located further away, e.g. at position U (10, 0, 0), we get *F* = 3.55 × 10^{42} N, *a* = 3.50 × 10^{−12} m/s^{2}, and *v* = 1434 km/s, for U (100, 0, 0) we have *F* = 3.55 × 10^{40} N and *v* = 14.3 km/s and for *U* (1000, 0, 0) we obtain *F* = 3.55 × 10^{38} N, *a* = 3.50 × 10^{−16} m/s^{2}, and *v* = 0.14 km/s. The dependence of *F*, *a*, *ρ*_{Λ}, and Λ on the location of an external universe at position U (*n*, 0, 0) with *n* = 2 - 10 is illustrated in Figure 2. The density of DE, *ρ _{λ}*, is

Figure 1. Four models with external universes exerting gravitational pull on our Universe. Top: Two-dimensional examples. Bottom: Three-dimensional versions. The coordinates of the universes are multiples of the radius, *r*. In the two-dimensional graphs, the red arrows represent the direction and the value of the gravitational force *F*. The lengths of the arrows are relative to the gravitational force exerted by the gravitational pull of external universes. In the 3-dimensional graphs, only the relative lengths of the arrows are provided, not their directions. The absolute values of *F* and the corresponding angles Φ and *θ* are listed in Table 1. The ranking of the absolute values of *F* is *F*_{4} > *F*_{1} > *F*_{2} > *F*_{3}.

Table 1. Comparison of the parameters obtained for four different Mulitverse models with one to four external universes.

0, 0) *ρ _{λ}*= 2.62 × 10

On the other hand, if the mass of the external universe is increased, this also increases *F*, *a*, *ρ*_{Λ}, and Λ, as illustrated in Figure 2 for acceleration and Λ.

In the model with two external universes at positions U_{1} (3, 0, 0) and U_{2} (−4, 4, 0), we obtain a gravitational pull of *F* = 3.26 × 10^{43} N, a net acceleration of *a* = 3.21 × 10^{−11} m/s^{2}, and a net velocity of *v *= 13,157 km/s. The DE density is 2.74 × 10^{−30} kg/m^{3} and Λ = 4.59 × 10^{−39} s^{−2} or 2.56 × 10^{−56} m^{−2}.

In the two three-dimensional graphs, the respective values are *F* = 2.30 × 10^{43} N and *F* = 9.86 × 10^{43} N, *a *= 2.27 × 10^{−11} m/s^{2} and *a* = 9.72 × 10^{−11} m/s^{2}, and *v* = 9292 km/s and *v* = 39,846 km/s, respectively. For*ρ _{λ}* we obtain 1.37 × 10

4. Discussion

Although the number of theories as modifications or alternatives to the ΛCDM model is constantly increasing, the introduction of the Cosmological Constant Λ by Einstein in 1917 [1] [2] was the first and still remains the most effective approach to keep the General Relativity equation for large cosmological scales on track. A physical explanation for Λ led to the introduction of a scalar field [12], without, however, being able to explain what kind of scalar field this could be. The model was modified later on in various ways [13], including the incorporation of additional dimensions [14] and numerous other approaches without

Figure 2. Top Left: Acceleration in m/s^{2} of our Universe due to the external pull of one external universe vs. the positions U (*n*, 0, 0) with *n* = 2 - 10. The coordinates of U (*n*, 0, 0) are multiples of the radius, *r*. Top Right: Density of Dark Energy, *ρ*_{Λ}, in kg/m^{3} as a function of the distance *r* (radius of the universe) of one external universe located at U (*n*, 0, 0) with *n *= 2-10. The y-axis is on a logarithmic scale. 2^{nd} Row Left: Cosmological constant, Λ, in s^{−2} depending on the location of one external universe at position U (*n*, 0, 0) with*n* = 2 - 10. The y-axis is on a logarithmic scale. 2^{nd} Row Right: Cosmological constant, Λ, in m^{−2} depending on the location of one external universe at position U (*n*, 0, 0) with *n* = 2 - 10. The y-axis is on a logarithmic scale. Bottom Left: Acceleration *a* in m/s^{2} depending on the relative mass *n* of one external universe at position U (2, 0, 0) with *n* = 1 - 20 multiples of the mass of the external universe. Bottom Right: Cosmological constant, Λ, in s^{−2} depending on the relative mass *n* of one external universe at position U (2, 0, 0) with *n* = 1 - 20 multiples of the mass of the external universe.

being able to replace the ΛCDM model.

Other models that have widely been discussed are those based on extended gravity, for example, interferometric detection of gravitational waves as a definitive test of GR, as proposed by Corda [15], or in general, extensive evaluation of all aspects of gravitational waves leading to gravitational physics and astronomy, as described by several groups [16] - [21].

The objective of the present paper is based on the idea that repulsive gravity, which is considered as the driving force of DE, might easily be substituted by “normal”, attractive gravity if the source of this force is placed outside of our Universe. This proposal requires the presence of a Multiverse, which means the existence of more than one universe. Using three-dimensional vector calculation and Newton’s Law on Gravity, the gravitational pull of external universes on our Universe was calculated for one, two, three, and four external universes. The total mass taken into account for a universe included baryonic matter and Dark Matter. For calculating the latter mass, DM density data published by Carmeli [11] were used. A recent paper by Kusenko [22] proposes CDM as black holes in a Multiverse.

The acceleration achieved due to the gravitational pull by external universes is 2.3 to 9.7 × 10^{−11} m/s^{2} for all four models. This is in agreement with *a* ~1 km^{2}/s^{2}pc ≈ 3 × 10^{−11} m/s^{2} as reported by Walker [23].

From the acceleration we can calculate the kinetic energy, *E _{kin}*, induced by the gravitational pull and the density of

The data provided in the present paper for *E _{kin}* and Λ have to be considered a first and rough check of the feasibility of this proposal and need further modeling. For example, the kinetic energies provided above do not take into consideration that the universes are approaching each other and therefore the distance

5. Conclusion

The data obtained in this paper indicate that the accelerated expansion of our Universe can be explained by defining DE as the warping of spacetime by a source outside of our Universe. The kinetic energy caused by the gravitational pull by one or more external universes provides a direct measure for the warping of spacetime from outside. The concept is based on the assumption that we are living in a Multiverse and that the external universes are identical to ours in terms of physical laws and constants, at least regarding Newton’s Law on Gravity.

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