Dark Matter as a Result of the Multiverse and an Improved Prediction Algorithm for Galaxy Rotation Curves and Cluster Velocity Dispersions

1. Introduction

The hypothesis of dark matter is a way to explain why among other galaxies seem not to obey Newton’s law of gravity. As well, dark matter is needed to explain the statistical distribution of ‘cold’ and ‘hot’ spots in the background radiation, that would still need the existence of (much) dark matter vs. baryonic matter to be understandable in terms of Big Bang nucleosynthesis, as well as matters like gravitational lensing. In this article it will be argued dark matter is a physical entity arising from the multiverse hypothesis.

Nevertheless, there exist several alternative approaches to account for the additional gravity it yields, like Modified Newtonian Dynamics (MOND) [1,2], Bekenstein’s TensorVectorScalar gravity (TeVeS) [3] or Covariant Emergent Gravity (CEG) [4]. But, they assume dark matter does not really exist, and so leave the other matters mentioned up here, and the gravity in galaxy clusters, see Banik [2], unresolved. Besides that, they do not give a natural explanation for the concepts and additional fields they introduce.

Therefore, in the article in hand the existence of dark matter is the starting point and it is argued it is the effect of a superposition of all the 165 possible 3-dimensional universes in an 11-dimensional space, in the present state with 10 spatial dimensions, of which zero to two dimensions overlap with our universe. This number however is not fixed by this proposal, but it will be shown this number yields a good match with the observations in galaxies and clusters.

Dark matter in first approximation appears like a set of wire-masses, without any modification to Newton’s law of gravity, because the dark matter is stretched in one or more directions. So, the hypothesis retains the standard gravitational field equations, but attributes the additional approximate $1/r$ contribution to a different projected mass distribution rather than to a modification of the gravitational law itself.

Compactification and superposition are distinct ingredients of the proposal: compactification specifies the geometry within each state, whereas superposition is separately postulated to yield the set of 165 geometrically distinct three-dimensional embeddings.

The universes with non-overlapping dimensions are assumed to be interwoven with our universe at the level of strings, inspired by intersecting brane-words concepts. This fine fabric is postulated to make the other universes have effect throughout our entire universe. These dimensions are compactified at the GUT-scale, which however is larger than the Planck-length, the assumed size of all the dimensions at the start of the Big Bang, Hawking [7,8]. As a result, the effective distances gravity from another universe has to act over, are stretched and thus being much longer. So, this hypothesis may bring the smallest and large scales together. The less dimensions of a superposed universe overlap with ours, in the more directions the effect of dark matter is stretched out and thus weakened, so the universes with two overlapping dimensions, i.e., where dark matter appears stretched in one direction, dominate gravity in galaxies and clusters.

As a result, so as to describe its effect on gravity in General Relativity (GR), the cylindrically symmetric solution from Levi-Civita [5] as described by Santos [6] might well be used as a first approximation, in linearized or numerical GR. With dark matter as a first approximation appearing like a linear mass density.

It will be argued that this all logically follows from Hawking’s Cosmology [7,8].

It solves the problem that Lelli and Mistele [9] showed, that the common way to project dark matter NFW-halos around galaxies cannot be valid. The alternative naturally assumes dark matter is distributed like the visible matter in a galaxy, in a wire-mass shape, consistent with [5] and [6]. As such, it does not need ad-hoc assumptions about the nature of dark matter, like the dark particles being collisionless and not losing energy through radiation, that must lead to a halo-shape in galaxies. So, the hypothesis gives a natural explanation of dark matter, why it is undetectable and why its gravity shows flat rotation curves at large distances from the galaxy center and Newtonian behaviour in its vicinity, and as a result even allows elliptical orbits of stars near the center. The term naturalness is extensively discussed by Hossenfelder [10] (p. 57). In short it means that a theory is without fine-tuned constants.

In this paper the Spitzer Space Telescope satellite data of 175 galaxies, SPARC, as processed and reported by Lelli et al. [11] and Starkman et al. [12] are used to assess several predictions that follow from this theory. The mass-to-light ratio has been used as the only fitting parameter to fit the baryonic rotation velocity, and hence the baryonic gravitational acceleration, in each galaxy to the observed values near the center of the galaxies. After that, the hypothesis in hand is used to predict the additional gravitational acceleration at all radii without any further fitting and to compare the predictions with the observed values.

After a brief introduction of Big Bang theory in chapter 2.1 and Hawking’s cosmology in chapter 2.2 and some other indispensable literature about quantum systems in chapter 2.3, MOND and TeVeS will be discussed in chapter 2.4. This forms the fundament for the proposal presented in chapter 3.

In chapter 3 the proposal will stepwise be derived in a logical manner from Hawking’s cosmology and String theory.

In chapter 4 this will all be worked out. The hypothesis for dark matter will be elaborated and its consequences and behaviour will be explored.

In chapter 5, some curve fitting, some testable predictions and some speculations are proposed and support for them presented, one using the work of Levi-Civita, and in chapter 5.3 an improved alternative to MOND for the prediction of rotation velocities is presented.

In chapter 5.4 using Monte Carlo simulations of galaxy clusters, it will be shown it gives a much improved prediction for dispersion velocities in a wide range of NGC and Abell clusters. In chapter 5.5 it is argued that this can explain the rapid development of large galaxies in the early universe. The rest of chapter 5 gives three other predictions.

In chapter 6 the conclusions and suggestions for further work are presented.

2. Hawking’s Cosmology and Superposition State of Universe, MOND and TeVeS

In this chapter the fundament for the proposal of chapter 3 will be laid, by giving an overview of existing theories that contain vital building elements.

2.1. Big Bang Theory

The line of thought of the universe as a quantum system is an elaboration of Hartle & Hawking [7]. The universe, according to the Big Bang theory, comes from an infinitesimal small point in which only elementary particles existed in the form of a plasma, with an extremely high temperature [13] (pp. 127-136) and as a result was in a quantum state, see chapter 4.

The originally extremely high temperature is still visible and measurable in the so-called background radiation. Its properties are direct evidence that the universe originated from a hot Big Bang stage. The Big Bang theory is also a logical extrapolation of the expansion of the universe that we observe, among other things due to the redshift of the spectrum of the radiation of stars, but also of the history/evolution of stars and galaxies as visible through our telescopes.

In addition, the non-uniform distribution of stellar objects as quasars over the different redshifts proves the universe is not static.

Moreover, Big Bang theory can quantitatively explain many phenomena, such as the distribution over the various elements of the mass in the universe, the cosmic composition, based on nuclear physics.

The fact that it is dark at night also proves that the universe cannot be infinitely large and infinitely old, because then the entire sky would be filled with light from stars. So, our universe indeed has a beginning. Moreover, the Big Bang theory forms a well-cohesive whole with astronomy and the rest of physics.

2.2. Hawking’s Cosmology and String Theory

Somewhere at the beginning, our universe has been in a quantum state, because that’s where one ends upon extrapolating the expansion of the universe back to the very smallest starting point, [7,8] have derived solutions to the wave function of the universe as proposed by Everett [14] and further elaborated by DeWitt [15]. As derived and explained by Hartle & Hawking [7] these solutions must satisfy the Wheeler-DeWitt equation.

Hartle & Hawking [7] show the Wheeler-DeWitt equation has the following form:

$$\mathrm{Ĥ}\left(x\right)\mathrm{ }|\mathrm{ѱ}\mathrm{ }⟩=0$$

(1)

where $|\mathrm{ѱ}⟩$ is the wave function of the universe [7]. The Hamiltonian, in this case derived from General Relativity [7], describes the total energy of a system and $Ĥ$ is the Hamiltonian constraint operator [16] (p. 27). The so-called Hamiltonian constraint described by (1) follows from the total energy of the universe being zero, gravitational energy cancelling out the mass energy. Hawking’s & Hartle’s solutions of this equation describe a universe that has no beginning, the Hartle-Hawking state, [7]. Hawking [8] explains this in simpler terms as well: time must have been indeterminate there on the smallest scale in that quantum state, because of the extreme gravitational warpage of space-time at that moment, [8] (p. 172). The time t=0 therefore is not precisely defined and at these scales time reduces to a fourth spatial dimension. This is fundamental to the theory in hand and will be further elaborated in chapter 4.2.

So, the universe has no exact measurable beginning. Hawking calls this the ‘no-boundary-condition’, [8] (pp. 172-173). It makes it impossible to trace the development of our universe from the beginning to this time in a deterministic ‘bottom-top’ way and, hence, there is a need for a statistical ‘top-down cosmology’, considering all possible alternative histories of the universe.

He states that as a consequence of this, at the very beginning time acted as a fourth spatial dimension, “In the early universe-when the universe was small enough to be governed by both general relativity and quantum theory, there were effectively four dimensions of space and none of time”, [8] (p. 172) This is the starting point of the proposal of this paper. String theory, however, suggests as much as eleven spatial dimensions.

The quantum aspects of the Big Bang become clearer when considering so-called ‘double-slit’ experiments, with a light beam split in two that are directed at a wall with two narrow slits. Especially the variant where only one photon is fired at a time. The same interference patterns then arise as with continuous beams of photons, so the probability waves of single photons interfere with themselves, as it were. One photon behaves as if it passed through both slits. That can only happen if the photon itself follows all possible alternative paths simultaneously, as it were like a split probability wave. So, the behaviour of the single photon can be seen as a superposition of all possible alternative paths it follows, so alternative histories, [8] (p.104). The superposition causes wave interference and that determines the paths the photon follows in the experiment.

Hawking’s and other’s point about the probabilities is that a quantum experiment will only have a certain outcome when it is performed. The Big Bang can be regarded as such an experiment [8] p. 179), where the universe in the quantum state may have had a statistical probability distribution of many ‘alternative histories’, following the interpretation of Feynman. Maybe 10⁵⁰⁰ ones as String theory and the more general M-theory suggest, [8] (pp. 152 and 181). At page 77 Hawking states that “the universe does not have a single existence or history, but rather every possible version of the universe exists simultaneously in what is called a quantum superposition”.

This does not a-priori imply we still are in a real a state of superposition between all, or part of these alternative histories now, but this paper will argue that this is indeed the case with our universe for a specific part of these histories.

The parameters and hence the quantum state of our universe are known now. Our universe has known single values for the fundamental parameters and constants. Of all the ‘alternative histories’, ours is the one that has come true. The experiment has been performed; we know the outcome. This is only possible when there is an observer to the experiment, [8] (pp. 107 and 179). This where the idea of an ‘observed universe’ of Hawking and others like Wheeler comes from. The assumption is that man or other sentient beings can perform this role of external observer, as Hawking and Wheeler argue, based upon the ‘delayed-choice’ experiment by Wheeler, see Zeilinger’s overview [17]. That shows that the moment of time where the observer enters the history is not relevant [8] p. 106-107), which is consistent with Zeilinger’s interpretation and conclusions in [17] and the citation in chapter 5.5.

The ideas in Hawking’s cosmology [7] are consistent with certain approaches to quantum gravity, such as String theory. The idea from Hawking that space and time are quantum phenomena are central themes in quantum gravity research. The Feynman path integrals that Hawking uses, are an important computational method in quantum gravity, especially in the context of Euclidean quantum gravity. This is explained by himself in [7]. This work provides an early foundation for later developments in quantum cosmology, like String theory, and the idea that gravity itself is a quantum phenomenon. From String theory, the assumption in the article in hand of more dimensions in a multiverse has been taken over, which thus is consistent with taking Hawking’s cosmology as a starting point for the theory of the article in hand. But, there yet is no direct evidence for String theory. Future experiments in gravitational waves, particle physics and cosmology could provide clues. If the LHC or future accelerators find supersymmetry, it would be a boost for String theory and hence the assumption of the article in hand.

But, String theory does shed light on the behaviour of black holes. Strominger and Vafa [18] in 1996 showed that in String theory the entropy of certain extremal black holes exactly matches the Bekenstein-Hawking formula [18]. They calculated the number of microscopic states of D-branes and found a perfect match.

Maldacena [19] in 1999 introduced the AdS/CFT correspondence, a duality rooted in String theory, that couples gravity in a (D+1)-dimensional anti-de Sitter space to a D-dimensional conformal field theory. This idea has major implications for the so-called black hole information paradox. The black hole information paradox is the problem that, according to Hawking’s calculations, information appears to be lost when a black hole evaporates due to Hawking radiation, which violates the laws of quantum mechanics that require that information is always conserved. Maldacena’s duality suggests that information is not lost but remains encoded in the dual theory.

Mathur [20] in 2003 proposed that black holes do not contain a singularity, but instead consist of a complex collection of string states, or a fuzzball. This potentially solves the information paradox, because information can be stored in the quantum structure of the fuzzball instead of being destroyed in a singularity. This all is support for the String theory and hence for the assumptions it makes.

In the meanwhile, it is fruitful to explore the potential for a natural explanation of what dark matter is, as is done in the article in hand.

2.3. How a Superposition State Can Have Classical Effects

Quantum superposition can be forced by a beam-splitter like in the famous ‘double-slit’ experiment discussed up here. It can be forced as well by a dedicated device like in a Qubit or in an MRI-scanner. A tensor-interaction like in the deuteron may as well yield a superposition state. The latter will be discussed into more depth in the sequel, since it might be very relevant to the behaviour of our universe. So, quantum effects can affect classical effects through a variety of mechanisms, with microscopic quantum phenomena affecting macroscopic classical phenomena.

A deuteron Is a bare proton and a neutron, glued together, without electrons. It forms a vital step in the fusion of helium, and thus of the existence of stars. The binding force between the neutron and the proton is the sum of the resulting forces of the superposition of two quantum spin states, see Bethe [21]. So, this force would not be strong enough if the deuteron were in just one of those states. It is evident this has a huge classical impact.

Another example is superconductivity. Superconductivity occurs when electrons behave as a collective quantum mechanical entity, completely eliminating electrical resistance. This has applications in powerful magnets and lossless current transport, Ginzburg et al. [22]. In the early stages of the universe, quantum fluctuations caused variations in the density of matter, which later evolved into the large-scale structures of the universe such as galaxies and clusters, Guth [23]. Chemical reactions, enzymatic processes and even biological phenomena such as photosynthesis are influenced by quantum mechanical principles, which has macroscopic consequences, McFadden et al. [24].

This all is essential to the following part of this paper: as with the single photon in the double-slit or with the deuteron, our universe could still be in a superposition of multiple histories. The result should be able to interfere with itself very well like the single photon, and forces like gravity might add up like in the deuteron. It will be argued why for electro-magnetism this cannot have a measurable impact.

These possibilities for classical effects lead to a testable hypothesis of the nature of dark matter. But firstly, MOND and TeVeS theories are briefly visited, because they give a useful mathematical description of the gravitational effects of dark matter of which some starting points are used in the article in hand.

2.4. Introduction to MOND and TeVeS Theories

Modified Newtonian Dynamics (MOND) is an empirical alternative to the hypothesis of dark matter to explain why galaxies and open clusters seem not to obey Newton’s law of gravity, see Kroupa [25]. It is explored in this chapter and among other described by Schilling [26].

First published in 1983 by Milgrom [1] and extensively assessed by Banik [2], the aim was to explain why the observed velocities of stars in galaxies are larger than expected based on Newtonian gravity.

An example of the so-called ‘rotation curves’ discussed down here, is shown below in Figure 1. It shows the rotation velocities as a function of radius from the center of a galaxy, as well as the logarithmic brightness curve, which is a good measure for radial mass distribution. It comes from Lelli et al. and Starkman [11,12]. It is one of the 175 galaxies of the SPARC database (NGC6503). The black dots are observed velocities, to be called V_obs in the sequel. They are higher than the velocities calculated from gravitational attracting force according to Newton’s law of gravity. Taking this equal to the centrifugal force, results in the theoretically expected velocity, called V_bar, the blue line.

Further examples are shown in chapter 5. If at large radii both the total observed gravity and the opposing centrifugal force decrease linearly with radius, the observed velocities, V_obs, can remain constant over a long range of radii as is shown by Lelli et al. and Starkman [11,12] for many of the other galaxies with Spitzer photometry. See Annex 2 for all the rotation curves.

Milgrom noted that instead of assuming dark matter to solve this, the discrepancy might be resolved if the gravitational force experienced by a star in the outer regions of a galaxy would vary inversely linearly with radius R (as opposed to the inverse square of the radius, as in Newton’s law of gravity). MOND has been fitted empirically such that it differs from Newton’s laws at extremely small accelerations that are characteristic of the outer regions of galaxies with formula (2). The transition would occur below an acceleration of a_m = 1.2 x 10⁻¹⁰ m²/s, Milgrom’s constant. The area with lower gravitational acceleration is called the MOND regime.

The theory needs an interpolation algorithm for the acceleration beneath a_m. The interpolation depends on the variable µ ($x$) with $x={\displaystyle \frac{g}{a_m}}$, so the predicted total acceleration over Milgrom’s constant, as follows:

$$\mathrm{\mu }\left(x\right)={\displaystyle \frac{x}{\sqrt{1+\mathrm{ }{x}^2}}}$$

(2)

and the Newtonian acceleration g_N is related to the resulting total predicted acceleration a through:

$$g_N=\mathrm{ }\mathrm{\mu }\mathrm{ }\left(x\right)g\mathrm{ }$$

(3)

Since this equation needs to be solved iteratively when it is used to predict the total acceleration from the Newtonian, and since this interpolation formula allows for inversion, it can be rewritten as follows:

$$g=g_N\mathrm{ }{\left({\displaystyle \frac{1}{2}}+\mathrm{ }{\displaystyle \frac{1}{2}}\sqrt{1+{\left(2\mathrm{ }{\displaystyle \frac{a_m}{g_N}}\right)}^2}\right)}^{{\displaystyle \frac{1}{2}}}$$

(4)

See for example Platschorre [27]. With back and forth calculating a series of values for $a_0$and $a$it can easily be shown that this formula works correctly. It will be used in chapter 5.3.

So, MOND is designed such that it always yields flat rotation curves, but in the sequel it will be shown that very often, in the SPARC data, such flat rotation curves do not appear.

However, this MOND theory of gravity does improve the calculations on the velocities of stars but does not explain the observed deviations from Newtonian mechanics. Bekenstein even states it is not a theory at all, but only a recipe [3]. Furthermore, as Bekenstein [3] mentions it does not specify how to calculate gravitational lensing by galaxies and clusters of galaxies, and it violates conservation of momentum.

Early attempts to generalise MOND by making a relativistic version of it were relativistic AQUAL [3] (p. 22) and Phase Coupled Gravity (PCG) [3] (p. 22). Fascinating is that PCG yields a description of total gravity as a sum of two competing fields, one with quadratic decay of gravity with distance x and one that decays linearly [3] (equation (17) at p. 7).

It is interesting to note that Covariant Emergent Gravity (CEG) [4] as well yields a sum of two competing fields, as Platschorre [27] shows. Zhou et al. [28] found this as well upon applying a conformal gravity approach.

But both AQUAL and PCG attempts had problems like waves propagating faster than light and incorrect light deflection [3]. Bekenstein provided a theory, that accounts for this, TeVeS, which has been formulated in terms of GR, but with additional scalar and vector fields [3] and which in the Newtonian limit gives the same results as MOND. However, TeVeS still does not give an explanation for the source of the additional scalar and vector fields that make it deviate from GR.

The paper in hand, presents a hypothesis that provides a natural explanation for the physical existence of dark matter, provides a valid solution in GR and that does not need any interpolation algorithm and that significantly can improve the predictions for all galaxies in the SPARC database. This hypothesis is presented in chapter 3 and worked out in chapter 4.

3. A Hypothesis on the Physical Existence and the Nature of Dark Matter

The thoughts leading to the hypothesis can be logically summarized as follows, and will be elaborated in the next chapter. The first five steps come from Hawking himself, as elaborated in [7,8].

• 1. Hawking’s cosmology is a logical combination of two well proven theories, quantum mechanics and Big Bang theory, and thus, it is a good description of the earliest stages of our universe.
• 2. Our universe results from a Big Bang that was in a quantum superposition state at its start, that can be interpreted as 10⁵⁰⁰ alternative histories in an 11-dimensional space, using the Feynman interpretation of quantum mechanics and String-theory.
• 3. The realization of our universe from the 10⁵⁰⁰ alternative histories cannot have occurred without a sentient observer.
• 4. Our universe has been realized.
• 5. At least one sentient observer exists, which can have come into being in the universe following the conclusion of Wheeler’s delayed choice experiments.
• 6. Since it is not economical to consider 10⁵⁰⁰ a fine-tuned number, aimed at creating exactly one universe with sentient being, there still remains a superposition state of more than one alternative histories of the universe. This makes it a multiverse, each universe with sentient beings. This multiverse might still exists by means of a state of superposition, which must not necessarily be disturbed by de-coherence, since nothing exists outside the multiverse and the different histories only concern the accessible dimensions in a history, which are not disturbed by any chaotical processes in a universe. That is why, we here adopt, as a working premise rather than a result derived from standard quantum mechanics, that a coherent sector of these alternative geometrical states can remain gravitationally relevant at late times.
• 7. The other universes in superposition can follow a history comparable with ours that leads to sentient beings, but do not necessarily share all our spatial dimensions in the 11-dimensional space, but do have nearly exactly the same constants of nature. From the delayed choice experiment it follows they all have the same causal status.
• 8. The superposed gravity of all 165 potential universes in an 11-dimensional multiverse with 10 spatial dimensions acts together just like the binding force in a deuteron and as a result the gravitational accelerations and potentials caused by baryonic matter in these universes should be added.
• 9. Since there are more ways to yield partly overlapping universes in an 11-dimensional space than fully overlapping, the odds are that there exist multiple universes that share zero or only one or two dimensions with our universe.
• 10. Gravity acting in our universe resulting from the mass in another one, if it is tightly interwoven with our universe at the smallest scales, appears stretched like a wire-mass because the third dimension is compactified to a GUT-scale that, however, is much larger than the Planck-length. This leads to a linear decrease of the gravitational acceleration as a function of distance from such a stretched mass and hence to a logarithmic potential.
• 11. The existence of 165 universes that share zero, one or two dimensions with our universe in a state of superposition, forms a natural explanation of what dark matter is and together with the previous step to and explanation for the flat rotation curves at large distances from the centre of galaxies as well as the high velocity dispersions in galaxy clusters.

An argument like this is as strong as its premises. Therefore, the word proof or evidence is avoided here and it is called an argument. In the sequel, this path of thinking will be further worked out and the premises explained.

4. Elaboration of the hypothesis on the Physical Existence and the Nature of Dark Matter

Firstly, in chapter 4.1, the underlying natural explanation for this concept will be presented after the basic assumptions from Hawking’s cosmology and the String theory that was developed from it, will be explored and then the consequences for the gravitational potential will be worked out.

4.1. Exploring the Logical Consequences of Hawkings’s Cosmology and String Theory

In the sequel, the case is argued for a natural explanation for the flat rotation curves explained in chapter 2, starting from this cosmology, from String theory that is founded in it and from the other elaborations made about superposition in chapter 2. This will not only merge to a natural explanation but yield an improvement of the rotation curves compared to MOND too, in the form of a simple physical model.

The crucial observation of the paper in hand is that if the moment of time where the observer enters the history is not relevant, as discussed in chapter 2.2, this would give all possible observers the same causal status.

Now, it is of paramount importance to realize that there is no natural relationship between the numbers 1, for one universe, and 10⁵⁰⁰ for the number of possibilities, mentioned in chapter 2. Arguing that this number can only lead to one single universe with sentient beings, has created a fine-tuned number, which is not the most economical of explanations, since it would require more explanations itself.

If a universe in which man originated is a realization of 10⁵⁰⁰ possibilities, it is irrational to assume that not at least one more history of the universe, with sentient beings who can also act as observers, has been realized. Who was first or last does not play a role in this, as the ‘delayed choice’ experiments show. We are then in a multiverse, which state of real superposition, as defined by [8] (p. 77), would result necessarily from the existence of multiple observers. The superposition has then been maintained in the way presented earlier in this essay. But then, the real superposition must necessarily exist if man is the needed observer of our universe as Hawking states it. The states will be able to interact with our universe by adding up certain effects, as in the deuteron or the double-slit experiments.

The additional gravity attributed to dark matter can be such an effect. The constants of nature in those universes will have nearly exactly the same value as ours, since the existence of sentient beings does not allow very different values, as explained by [8] (chapter 7 p. 203 in particular) and by Rees in Just Six Numbers [29].

When one would argue that universes can never get in a superposition state, one ends in a ‘reductio ad absurdum’. There must necessary be a real superposition, but there cannot be one…

But, the values of some of the forces or energies in our universe, like gravity or the cosmological constant, or the mass might be explained as the sum of contributions from different quantum states or histories of the universe if it still would be in real superposition. Then their value should match the sum of two or more allowed values conforming to their probability distribution, as defined by for instance Weinberg regarding the cosmological constant, see Hossenfelder [10] (p. 155). Cosmic forces would then act on the sum of all mass in this superposed universe. The necessary existence of sentient beings in more than one universe, will then be the mechanism that maintains part of the original superposition. Because the ‘delayed-choice’ experiment by Wheeler shows that the moment of time where the observer enters the history is not relevant [8] (p. 106), the observers in the parallel universes possess exactly the same causal status, so they must necessarily all act as observers then. That might be the natural and necessary cause of such a maintained superposition state.

This is a logical way of creating a multiverse from one Big Bang that results inevitably from Hawking’s cosmology if 10⁵⁰⁰ is not a fine-tuned number, as that was presented in chapter 3. The result should interfere with itself very well, as the single photon in a double-slit experiment and yield a sum of binding forces (each with their own amplitude) like in the deuteron.

We adopt, as said, a conjectural premise rather than a result derived from standard quantum mechanics, that a coherent sector of these alternative geometrical states can remain gravitationally relevant at late times. For, there is nothing outside our universe that could disturb the superposition state, it might remain in that state forever, so without de-coherence effects disturbing it. Since a universe has one history as defined by Feynman, so a common, shared, set of values of nature’s constants, its size is not a reason to disturb it either. And that is why Hawking and others [7,14] can speak of the wave function of the universe in the first place.

Equation 1 can then be rewritten as follows with, for instance, four of the eleven dimensions:

$$\mathrm{Ĥ}\left(x\right)\mathrm{ }\left({\mathrm{ }a}_{xyz}\left|{\mathrm{ѱ}}_{xyz}\mathrm{ }⟩+\mathrm{ }{a}_{wxy}\right|{\mathrm{ѱ}}_{wxy}\mathrm{ }⟩+a_{wxz}\left|{\mathrm{ѱ}}_{wxz}\mathrm{ }⟩+a_{wyz}\right|{\mathrm{ѱ}}_{wyz}\mathrm{ }⟩\mathrm{ }\right)=0$$

(1*)

In a sense (1*) says the total energy of the multiverse is zero.

4.2. Geometrical Consequences of Compactified Dimensions

Now, the thought steps of the previous section will be worked out so as to as to study the consequences for the gravitational potential.

The proposed natural explanation for dark matter comes from the line of thought set in motion with Hawkings’s cosmology and M-theory as well as String theory. The present proposal assumes a superposition of 165 geometrically distinct 3-dimensional universes embedded in an 11-dimensional space with ten spatial dimensions. Compactification determines the geometry of each universe, whereas superposition is a separate assumption concerning their joint quantum state.

The idea is that one or more dimensions of our universe locally are compactified, i.e., curled up at the smallest scale R in other universes in the fine fabric that is shaped by the universes in 11-dimensional space, and vice versa.

Please note, it has been assumed here each of the eight remaining dimensions could right after the Big Bang become the time dimension, following the reasoning of Hawking cited in chapter 2.2. So, in effect all eleven dimensions can potentially act as a spatial dimension in one or more of the possible universes. This leaves 11* 10 * 9 = 990 possibilities for 3-dimensional universes, of which groups of six only have a different order of the same three dimensions and hence are mutually undistinguishable. This yields 990 / 6 = 165 different 3-dimensional universes. In line with this reasoning it has been assumed the distribution of time and the seven compactified dimensions over the eight remaining dimensions is irrelevant to the discussion.

This idea of a fine fabric is consistent with String theory, see Aldazabal [30] and Lüst [31] with their theory of intersecting brane-worlds in particular. At the very start of the Big Bang, the diameter of our 3-dimensional universe, then amounting to the Planck-length Lp, would exactly match that of all other universes, being Lp too. Thus, they could all be represented by the same particle. This could be Lemaitre’s primeval atom Lemaître [32]. And this was in a state of superposition. Combination calculus shows the other 164 possible universes are distributed as follows, see Table 1.

So, according to M-theory, our 3-dimensional universe does have eight other dimensions that can be compactified, but of which one is time. This does not mean the save spatial ones have zero size, but have a thickness equal to the GUT-scale, which cannot be smaller than the Planck-length L_p, see for example Hossenfelder [33] and Spallucci and Fontanini [34] (Section 3).

As said, at the very beginning of the Big Bang, ours and the superposed universes, a multiverse in a sense, would then have a perfect geometrical and physical match, without any internal contradiction. All would be perfectly overlapping and form a very fine fabric, since they all exist in the same underlying 11-dimensional space the dimensions w, x, y and z as depicted in Figure 2 (simplified to four dimensions). All the constants of nature could be the same or differ only very slightly and just the distribution of the dimensions that are not curled up, would differ. Then, the thickness of the other as seen from our universe would remain Lp or, as a result of the energy decrease, increase to the GUT-scale, R. All universes fill the higher dimensional multiverse each in the same way, but mutually orthogonal. For example, like in the above, imagining four dimensions instead of eleven, the dimensions will be distributed over the universes could be as follows: xyz, wxy, wxz and wyz. This is elaborated in M-theory by describing the multiverse as 3-dimensional so-called branes in an 11-dimensional bulk, see Liu [35] and Randall et al. [36].

The dimensions of a higher-dimensional space should not be viewed as fixed Cartesian axes. More fundamentally, they represent independent degrees of freedom of the underlying manifold. Together with the induced metric, these degrees of freedom determine how distances and angles are experienced within a given embedded universe.

Consequently, lower-dimensional universes embedded in a higher-dimensional bulk need not appear as straight orthogonal lines in a Cartesian picture. Depending on their embedding, they may be curved, twisted, or a pair of them even schematically represented by intertwined structures such as a double helix.

All the universes could fill the entire higher dimensional multiverse in exactly the same manner as branes with a thickness R as a fine fabric, but with one dimension orthogonal to one of the others, as depicted in Figure 2, with one dimension ignored for simplicity. Here it is assumed two 2D universes share the x- dimensions, but the blue one has the y- and the red one the z-dimension curled up. They are orthogonal to each other at the location of every single string. See for example Bergshoeff and Riccioni [37]. These strings are not objects that exist in space, but that they define space itself. This is vital to the discussion in the sequel. This fine fabric is postulated to make sure the other universes can have effects everywhere in our universe and vice versa. In brane-world scenarios, however, they do not necessarily need to be a fine fabric, but could just as well be planes intersecting in one place or not at all, but those scenarios do not explain the effects studied here and are thus ignored in the paper in hand.

In the multiverse, galaxies from different 3-dimensional universes might be part of one larger structure in 10-dimensional space, as sketched in Figure 2, analogue to the 4-dimensional representation of the bookshelves towards the end of the motion picture Interstellar at 2 h: 16 m.

This is, because galaxies in different superposed universes will attract each other by the multi-dimensional gravity, like in our universe but acting with a larger resolution, i.e., acting over the compactification radius R, see Arkani-Hamed [38] and Maartens [39]. Hence the objects tend to overlap in 10-dimensional space (11 including time), called the ‘bulk’ in M-theory, and they will appear to us as dark matter. This is only possible throughout our entire universe if the branes fill the entire bulk like a fine fabric, so if the branes are intertwined at the scale of every single string, which are stretched and rolled up in the compactified dimension and protrude from the brane and hence give it a certain thickness. These one-dimensional strings themselves then have an effective thickness equal to the Planck-length because of quantum effects. This can be considered as the smallest resolution of space in our universe, see Hossenfelder [33] and Spallucci and Fontanini 2005 [34].

It is vital to this reasoning, that distances are measured directly from one mass to another and that the gravitational effects from one brane to another are remediated locally by gravitons emitted by those masses and not in another way, like by deformations of a brane that themselves at any location will deform other intertwined branes, because such a mechanism would generally be expected to lead to deviations from the inverse-square law. After all, gravity would then leak to the other branes everywhere and hence the force would decrease faster than as the inverse square of the distance between the masses.

Here it is important to recall that mathematically, an assumed linear gravity field, following an inversed linear law and hence with a logarithmic potential as seems visible in the rotation curves of galaxies, could only occur as a result of a line-mass, so mass distributed as a wire. Now, it is proposed here, these apparent line-masses are an emergent effect of the galaxies in the 24 universes that share two overlapping dimensions with ours, see Table 1, that only appear as stretched, because distances towards their mass are stretched in one direction as seen from our vantage point. That is the starting point for the line of thought to be pursued in the sequel.

The way higher dimensional gravity is projected on planes of lower dimensions is described by the Gauss-Codazzi equations, see Maartens [39], and is compatible with the idea of ‘brane-world shadow matter’ as described by Liu [35], where gravity in the fourth dimension is also acting over the distance R. But, the proposal of the paper in hand tries to give an natural explanation of how this has come into existence and how this works out in galaxies and galaxy clusters. Besides this, the difference with Liu [35] is that in that paper it was assumed the two different universes considered shared the same three dimensions, which did not explain why dark matter yields a more or less logarithmic potential. And given the reasoning in chapter 3 it is more logical and probable they will share less than 3 dimensions.

4.3. GUT-scale and Apparent Wire-Masses

Gravity from objects located at the intersections of these universes would appear in our universe as coming from a stretched projection. This is determined by the way a 3-dimensional universe, or brane in M-theory, is compactified to a larger size than the Planck-length, the resolution of space in our universe, see Croon et al. [39]. In this manner, a galaxy from another universe can appear as stretched out in our universe, see Figure 3. This is consistent with the description of Spallucci and Fontanini [34]. This is because the rolling up everywhere will happen at a GUT-scale of multiple times the Planck-length L_p, the size the original primeval atom possessed in this fourth dimension at the start_. This size is assumed in line with Hawking [7], since it is the smallest scale our theories allow. In Figure 2 it can be seen that as a result for gravity in the xy-universe to act from one line to another in the z-direction, it must work from one rolled up string to another and hence feel a much longer distance than when it would just directly act along a line in the z-direction, as it might do in the other corresponding universe.

The size of the compactified dimension is increased by 2$\mathsf{\pi }$ x GUT-scale/L_p since the start of the Big Bang. This determines how these dimensions are compactified and how, all distances in the inaccessible dimensions thus appear stretched in our universe, because they are measured over a longer, but compactified, i.e., rolled up length scale like it is elaborated in Spallucci and Fontanini [34].

In Croon et al. [39] the said GUT-scale is 1200 times the Planck-length L_p, so to assume a factor 22 (as is needed to match the data that are to be calculated in chapters 5.3 and 5.4.1) an increased energy may be needed, which is not in contradiction with current physics on beforehand.

Here, it is vital to note that if in one wxz universe a galaxy or cluster has coordinates x and w, it will have those in a wxy universe too. In this way, a galaxy disk can appear at the same location but in a stretched out way, so forming a straight wire at the location wx. The radial distribution of the wire-like mass will closely reflect that of the baryonic mass in a galaxy and it will have a finite length. This is mostly effective, when the stretching occurs perpendicular to the disc plane, since then it will remain a concentrated mass, i.e., a thickened disc.

But, since the orientations of the galaxies relative to the stretched directions or axes can vary, it is not just the galaxy discs that always appear thickened; in others of the 24 relevant cases they could be stretched out in the disc plane, leading to a much lower apparent mass density. Or just the distances from a vantage point in our universe to a certain mass in a superposed universe can be stretched as well, depending on which dimension is stretched in a certain universe, leading to a much lower mutual gravitational attraction. This means that not all 24 cases will lead to significant gravitational acceleration, but only two or three of them. In clusters the same effect will effectively reduce the effect of the dark mass in an equivalent way, but since clusters are spherical and not disc shaped, the reduction will be less. Therefore, a set of different possible angles (13) will be studied in the sequel in galaxies and a large number of galaxies (2400) in the simulations of clusters.

That galaxies and galaxy clusters in one universe will appear at the locations of those in other universes can simply be explained by gravity itself and by the way the density fluctuations in the Big Bang appeared as a result of standing waves from baryonic acoustic oscillations, BAO’s Schilling [26]. These standing waves appeared in all superposed universes and may have had a perfect geometrical match. The mass in the cross sections through galaxies, clusters and filaments will be held together in the expanding 3-dimensional universe by gravity just like in our universe. So gravity will hold galaxies in our universe and in the others together as well, since gravity works in all dimensions, Liu [35], Arkani-Hamed [38].

This as well suggests that there will be strong correlation between the mass and rotation velocity of the baryonic cloud and the dark matter clouds at the location of a developing galaxy disk, since larger amounts of baryonic mass will capture larger amounts of dark matter, which is in line with the findings of conventional cosmology with this respect, as for example is illustrated by the existence of the so-called spin-parameter, Mo et al. [40] which relates their radii tightly together. As well, the very existence of the Tully-Fisher relation confirms this, as will be explained in chapter 5.1.

So, our 3-dimensional universe would from the start be totally keep filled with 165 additional gravity fields caused by the baryonic matter the superposed universes. Effectively, an amount of 24 linear fields as a first approximation, because the 24 universes that share two dimensions with ours are dominating, since they appear stretched in only one direction and hence yield most gravity. Some of these appear as wire-masses in our universe, depending on how the galaxy discs are orientated with respect to the stretched directions.

4.4. Effect on Rotation Curves

But what will the apparent wire-like masses in the plane of rotation, so in the plane of the galaxy disks, or more generally, the stretching of the distances between masses, do with the rotation velocity pattern? There must occur velocity variations, because the orbiting stars can cross the axes defined by the stretched dimensions, the gravitational force towards the center of the galaxy being maximal then. For, when an orbiting star crosses an axis, the distance towards the center is not stretched in some of the 24 scenarios of Table 1, i.e. in the scenarios with only one dimension stretched. But, they will orbit further and move towards the other axis. When they are right between the two axes, at a 45° angle with the axes, the gravitational force will have its minimum, since then the distance towards the center is stretched in more scenarios. It can be shown the distance towards the center will slightly increase then and hence the velocity will decrease, because some kinetic energy will become potential energy. So, it will vary along the orbit. This will become visible in Table 3 in the sequel, upon comparing runs 1 and 2.

This can cause periodic orbits that can be slightly non-Keplerian, but that can still be periodic and closed Gutiérrez [41]. The combination of all scenarios from Table 1 will cause a combined pattern with many different possible orbits in the case of single test masses, but it is logical that the viscosity of the system, caused by collisions and magnetism Begelman & Rees 2021 pp 66 and 67 [42], will force the system in again closed and periodic orbits. In the meanwhile, the variation of the velocity along the orbit will cause a limited variation in mass density along the orbit. This all may contribute to the causes of the apparent boxiness of many brightness distributions in disks and bulges and the disturbances or noise that cause the development of bars in galaxies, but, the net effect will in the first place be strongly dampened by the gravitational attraction coming from the baryonic mass and from the other apparent line-masses, the ones perpendicular to the rotation plane, that as well tend to force the orbits in a circular shape.

Now, to place this in the right perspective within other physics, it is essential to note that this all does not apply to electromagnetic waves for an obvious reason: a thin intersecting plane with the thickness of the Planck-length or GUT-scale would act as an infinitely narrow polarization filter, through which only an infinitesimal fraction of the wave could propagate. An electromagnetic wave will need three dimensions to propagate, since the electric field and the magnetic field have orthogonal polarizations. Such a wave cannot exist in the 0-, 1- or 2-dimensions that overlap with other universes. That as well is part of the explanation why dark matter is undetectable, see for example Darling [43].

In chapter 5 it will be shown this proposal indeed works and leads to a good description of the contribution of dark matter to gravity in galaxies and the effects of, for example, the orientations of galaxies relative to the axes of the universe and the supposed dark matter halos, see Han [44], or filaments . To get there, firstly for all 175 galaxies in the SPARC database as reported by Lelli et al. [11] and Starkman et al. [12], the contributions of the visible ‘baryonic’ matter distribution to the gravitational acceleration and from the invisible gas have been recalculated.

The resulting matter distribution has been derived from the brightness profiles and HI gas concentrations as reported by Lelli et al. [11] and Starkman et al. [12] and then compared with their results. This has done so as to be sure that the author has performed the conversion from brightness to mass distribution correctly, for gas, disk and bulges. In chapter 5.2 this is all explained in depth.

5. Testable Predictions, Curve Fitting and some Speculations

In the sequel, some curve fitting and predictions, some of which of a more speculative nature, that follow from the hypothesis and support for them will be presented.

5.1. GR, Tully-Fisher Relationship and Milgrom’s Constant

Gravity coming from the superposed universe in first approximation appears like a line mass. Accordingly, the summed Newtonian and logarithmic potentials are used here as a weak-field phenomenological ansatz; demonstrating that they arise from a complete higher-dimensional solution of Einstein’s equations remains an open task.

This can give a weak-field solution of Einstein’s field equations in linearized GR, with some well explained natural source of the apparent logarithmic potential caused by dark matter, as proposed in chapter 4. The hypothesis in hand must give a natural explanation for this added gravitational potential and in the specific case of galaxy rotation, i.e., in the non-relativistic limit of GR, lead to the well-established Tully-Fisher relation $V^4\propto GM$. In this section it will be shown that the cylindrically symmetric solution of Levi-Civita [5] as worked out by Santos et al. [6] is a working solution for galaxies and flat rotation curves and it will be shown it is indeed consistent with the linear gravity hypothesis.

The Schwarzschild metric is a solution of Einstein’s field equations for an empty space with only a point-mass M, like the galactic center. The Schwarzschild line element is as follows:

$$d{s}^2=-\left(1-{\displaystyle \frac{2GM}{R}}\right)d{t}^2+{\displaystyle \frac{d{R}^2}{1-\frac{2GM}{R}}}+R^2\left(d{\mathsf{\theta }}^2+{sin}^2\mathsf{\theta }\mathrm{d}{\mathsf{\phi }}^2\right)$$

(5)

Which in the Newtonian approximation becomes $g_{00}$ = 1 + 2 $\mathsf{\Phi }$. This leads to the Newtonian potential that is proportional to 1/R. But in 1919 Levi-Civita found the solution for a cylindrical vacuum spacetime, which has the following form:

$${d{s}^2}_{LC}=r^{4\sigma }d{t}^2-r^{4\sigma \left(2\sigma -1\right)}\left(d{r}^2+d{z}^2\right)-{\displaystyle \frac{1}{a}}\left(r^{2\left(1-2\sigma \right)}\right)d{\mathsf{\phi }}^2$$

(6)

Santos et al. show that in the Newtonian limit, this metric yields the logarithmic potential:

$$\mathsf{\Phi }\left(R\right)=2\sigma \mathit{ln}\left(r\right)$$

(7)

Here r is the radial coordinate, i.e., the distance to the axis of the cylinder. The constant a in (6) must have the value unity to be consistent with the Minkowski flat space when σ = 0 [6] (p. 6). Besides this, it does not appear in the gravitational potential that results from (10), as can be seen up here, so it is put to unity in the sequel. Now, since the theory in hand does not in any way modify this description of gravity, but only states that the dark mass should be projected at larger distances, because of the stretching of the rolled up dimensions, this metric can still be used without modifications or added terms.

Modelling the total mass acting in a galaxy as a visible point mass together with a set of dark matter line-masses would yield the sum of both a linear and a logarithmic potential. Now, the hypothesis in hand for a wire-mass leads to $\sigma =\mathrm{G}{M}^{\text{'}}$, in which G is the gravitational and M’ the linear mass density, i.e., mass over thickness of the galaxy disc, in line with Santos et al. [6]. So, the line element to model dark matter in galaxies becomes:

$${d{s}^2}_{DM}=r^{2G{M}^{\text{'}}}d{t}^2-r^{2\left(G{M}^{\text{'}}\right)\left(G{M}^{\text{'}}-1\right)}\left(d{r}^2+d{z}^2\right)-\left(r^{2\left(1-G{M}^{\text{'}}\right)}\right)d{\mathsf{\phi }}^2$$

(8)

This is utterly consistent with the conclusions of Santos et al., who conclude that $\sigma$ must be the Newtonian mass per unit length as produces by an infinitely long line-mass, which as well Levi-Civita himself already concluded [6] (p. 4 and p. 11).

These two metrics (5) and (8) cannot be added straightforwardly in GR, since GR is strongly non-linear. However, in very weak fields, GR can be treated as linear, using linearized GR, see [3] (p. 18) and [26] (p. 200). This is derived by treating the metric as split in two components: $g_{\alpha \beta }={\eta }_{\alpha \beta }+h_{\alpha \beta }$, i.e., the Minkowski flat space and an added small deviation, so with $\left|h_{\alpha \beta }\right|\ll 1$. The same can be done with the line element, denoting the deviation as h simply. Comparing Milgrom’s constant with, for example, Earths gravitational acceleration makes clear this is a valid approach in galaxies. And in chapter 5.3 it will be shown in the Newtonian limit it gives an improved prediction method for rotation velocities, compared with MOND and TeVeS. So, upon studying flat rotation curves, these metrics may be added to one another to yield a valid solution of GR. As a result, the Newtonian potential and the logarithmic one (12) that followed from (11) will add up. This combination of metrics yields a good way to model dark matter n GR in weak fields, without the need to modify GR. In stronger fields, the point-mass and line-mass can only be combined in a numerical manner.

Now, M’ being the linear mass density, i.e., mass over thickness of the galaxy disc, directly leads to the Tully-Fisher relationship $V^4\propto GM$, since, as further explained in chapter 5.2, the thickness d of the galaxy disc controls this mass density. The larger d for a certain galaxy mass, the lower the linear mass density. In the next section d will be taken proportional to the vertical disk scale length h_z. It can be shown with dedicated literature, for example de Kregel et al. [45] that this, together with the disk scale length h_r, is proportional to the flat rotation velocity squared, i.e.,${d\sim h}_z\propto V^2$. The summarizing SPARC data-table Table1.mrt [11,12] clearly confirms this. So, d can be written as $d=\overline{h_z}/\overline{V^2}*V^2$. Since the logarithmic potential leads to $V^2\propto G{M}^{\text{'}}=GM/d$, so with d appearing in the denominator of the right-hand-side of equation (8) in chapter 4.4, the latter two proportionalities simply lead to the Tully-Fisher relationship $V^4\propto GM$.

But, the above is about visible baryonic disks, while the dark matter is described by a wire-like mass, existing in superposed universes. However, as argued in chapter 4.2 there will be strong correlation between the mass and rotation velocity of the baryonic mass and the dark matter, since larger amounts of baryonic mass will capture larger amounts of dark matter. And, this simply is in line with the findings of conventional cosmology with this respect, as for example is illustrated by the existence of the spin-parameter by Mo et al. [40] which relates the baryonic disk and halo radii tightly together. So, it can be assumed that the above applies as well to the length and hence the linear mass density of the wires.

Comparing this with the alternative formulation following from MOND, $V^4\propto a_{m}GM$, shows Milgrom’s constant has a deeper relationship with 1/d and hence with the vertical disk scale length h_z. It takes the place of the average ratio of velocity and scale height, so $\overline{V^2}/\overline{h_z}$, which is indeed an acceleration scale. An increasing rotation velocity tends to increase h_z, but there must be a counteracting force too, which is the gravitational attraction of the mass in the disk towards the disk plane. This is easily seen when one considers the height h_z a ball reaches when thrown upwards in the Earth’s gravitational field. Kinetic versus potential energy determines the height the ball reaches. This height is proportional to the square of the velocity V divided by the acceleration due to gravity g, i.e.,.${d\sim h}_z\propto V^2/g$, which yields, $V^4\propto gGM$, with $g=\overline{V^2}/\overline{h_z}$. Comparing this with the expression that followed from MOND, Milgrom’s constant takes the place of the average Newtonian gravitational acceleration g towards the disk plane of a large set of galaxies. Since dark matter in the article in hand is baryonic mass in superposed universes, i.e., universes with an alternative history, it in the deepest sense is the average g over a large set of ‘dark’ galaxies intersecting with ours. So, MOND’s way of describing the effect of dark matter has a link with the linear mass density of the ‘wires’ too, by controlling d through the acceleration that is expressed by a_m. However, since M’ appears in equation (8) in the article in hand it is not predicted that this acceleration appears as a constant in galaxies in our universe. These findings will be further applied in the next section, upon applying the SPARC data [11,12].

5.2. Galaxy Rotation Curves

The second finding is based upon curve fitting. It is that the additional acceleration is the result of a set of additional dark galaxies that appear stretched in one or more directions, at least in the in the Newtonian limit. The effect will be proportional to the amount of dark matter in a galaxy, which will vary between different galaxies. The curve fitting that supports this, has been done for all 175 galaxies from the SPARC database measured with the Spitzer Space Telescope [11,12].

The core assumption, as mentioned, is that the distribution of dark matter closely resembles the that of the visible matter, since they attract each other through gravity. This assumption is consistent with the findings of Lelli and Mistele [9] mentioned in the introduction. They inferred the gravitational potential around isolated galaxies from weak gravitational lensing with the said SPARC data. With these data, they showed circular velocity curves that remain flat for hundreds of kpc, greatly extending the classic result from 21 cm observations. Indeed, they state there is no clear hint of a decline out to 1 Mpc, well beyond the expected virial radii of dark matter halos. This means the common way to project dark matter halos around galaxies cannot be valid. The hypothesis in the paper in hand clearly does not have this problem.

To assess the validity of this constant for linear gravity, firstly for all 175 galaxies the contributions of the visible baryonic matter distribution to the gravitational acceleration and from the invisible gas have been recalculated from the brightness profiles and HI-gas concentrations as reported by Starkman et al. [12]. This has been done in a numerical manner by dividing the discs and bulges in a series of small patches and by evaluating all mutual attractions through gravity. This has been expressed in the form of velocity contributions, as the SPARC team did too. For the visible disk contribution, it is called V_disk and for the HI gas V_gas. This done in order to verify and show that the author has interpreted the brightness profiles from the visible disk, from the bulges and the gas mass distributions correctly.

Figure 4 shows the contributions to gravitational acceleration as a function of radial distance. It is plotted for the galaxies NGC 6503 and NGC 6674, the first of which with a bulge contribution, called V_bulge. The ‘recalc’ subscripts refer to the values as calculated by the author, The ‘SPARC’ indications refer to the values as reported by Starkman [12] at the website, in the file MaximumDisk_Mass_Models_mrt.txt. This is the file produced by [12]. It contains disk brightness profiles as well as observed rotation velocities, V_obs and bulge brightness profiles as well as the theoretical velocities as calculated by the SPARC team with Newton’s law of gravity. It also contains error estimates, except for V_bulge and for the HI gas V_gas. As a consequence, for those variables, no error bars will be shown in the graphs down here in Figure 4 and in the Annexes.

The squared theoretical velocities can be added and then result in the total Newtonian or baryonic gravitational acceleration, which can as well be expressed as a velocity contribution V_bar. But the contribution of V_disk and V_buls depend on the mass-light-ratio Y_ml, as follows:

$$V_{bar}=\sqrt{Y_{ml}\left(V_{disk}^2+V_{bul}^2\right)+\left|V_{gas}\right|V_{gas}}$$

(9)

V_gas in particular can have a significant negative contribution from gas outside the observed radius. Therefore, it is multiplied with its absolute value here to maintain the correct sign.

The mass-light-ratio, Y_ml is assumed 1 at this stage and will later act as the single fitting parameter. The contributions are calculated from the brightness profiles under the assumption that in thin disks the latter directly represent a distribution of the mass density. In the bulges this is not true; here brightness represents a cumulative mass density distribution since all observations of brightness run through the entire bulge and each layer adds brightness to the inward layers. So, it must be converted to a distributive mass distribution first, by subsequently subtracting the brightness contributions from larger radii at each observed radius, the part between two radii considered as a slice of a sphere. A complication with this is that the integration path length through each slice of the bulge is dependent on the radius observed. For example, at the most inner radius the brightness contribution from the outmost slice is much smaller than at the second outmost radius, since there one looks a long way perpendicularly through the outmost slice.

The HI-gas densities have been retrieved from the reported total HI mass and from the reported HI-radius by fitting the reported V_gas to the equation from Martinsson [46], see equation (10). Three of the parameters that were fixed by Martinsson have been replaced by fitted parameters a, b and ${\sum }_{HI}.$ The latter is fitted to match the total reported HI mass of the galaxy. Multivariate regression has been used to find the optimal values in:

$${\sum }_{R}HI\left(R\right)={\sum }_{\mathit{HI}}{e}^{-{\left({\displaystyle \frac{R-a{R}_{\mathit{HI}}}{0.36R_{\mathit{HI}}}}\right)}^b}$$

(10)

Following Lelli et al. [11], the total gas mass has been multiplied by a factor of 1.33 to account for helium gas as well. Following Patra [47] the vertical scale length of the HI-gas is modelled three times bigger than h_z, because this reference among others mentions values up to 1 kpc which is three times bigger than mean value of the SPARC data set. Both scale lengths have been modelled explicitly, which means that for both the disk and the bulge a 3-dimensional model has been used.

Since, as mentioned, the goal of the calculations in the above merely is to verify and show that the author has interpreted the brightness profiles from the visible disk, from the bulges and the gas mass distributions correctly and not to obtain an improved mass-model, for some galaxies interpolations and extrapolations of the brightness profiles have been made to come closer to the SPARC graphs.

Then gravitational acceleration for each particle at each radius and each angle of its orbit and each distance from the central line of the disk can be calculated by summing up masses in each part of the galaxy disk and bulge with:

$$g_c\left(R\right)=G{\sum }_i{\sum }_{\mathrm{\varnothing }}{\sum }_{\beta }{\displaystyle \frac{m_i}{X_{i,\mathrm{\varnothing },\beta }^2}}[\mathrm{m}/{\mathrm{s}}^2]$$

(11)

G is Newtons constant of gravity. Mass outside the orbit of each particle as far as it is not at the side of the center of rotation as seen from the particle has a negative sign since it has a negative contribution to the centrifugal force. X is the distance between two masses.

Gravitational acceleration as observed in each galaxy, is calculated from the observed velocities V_obs i.e., from the centrifugal force. This V_obs is plotted in Figure 5, as well as the baryonic contribution to the acceleration expressed as V_bar, see its definition in formula (9). The lines V_mond , V_bekst and V_recalc will be discussed in chapter 5.3.

The mass-to-light ratio has been used as the only fitting parameter to fit the baryonic rotation velocity, and hence the baryonic gravitational acceleration in each galaxy to the observed values near the core of the galaxies. After that, the hypothesis in hand is used to predict the additional gravitational acceleration at all radii without any further fitting.

Striking feature of the current scenario is that it does predict flat rotation curves where they appear in the observations, see Figure 5 (a) but as well linearly decreasing rotation curves, contrary to MOND, see Figure 5 (b)! MOND is designed such that it always yields flat rotation curves, but they do not always appear.

The Newtonian gravitational accelerations, expressed by V_bar, are now calculated with a fitted mass-to-light ratio, Y_ml. It has for each galaxy simply be fitted such that V_bar < 0.95 V_obs, at all radii, so following the maximal disk hypothesis, in line with the findings of Lelli [11]. This value has been determined by testing a wide range from 0.75 up to 0.99 and yields the best match with the observations in the current scenario. And, this value of the ratio of 0.95 gives the best overall predictive performance of MOND for the 175 galaxies as well, after again testing a range of values. However, it is hard to distinguish significant differences between values of this ratio in the range from 0.9 to 0.99, with only a 5 % variation of the errors when comparing with the observations. But, for another reason, a relatively high value is to be preferred, namely because of the findings of Lelli and Mistele [9], which revealed flat or more or less linearly decreasing rotation curves extending up to huge distances from the galaxy center. Therefore the fitting ratio that tends to give the flattest rotation curves, i.e. a high value of the ratio, is to be preferred.

This fitting assumes that the contribution from the Newtonian gravity never can be larger than the observed value, with some margin at all radii, so assuming there always is some contribution of dark matter at the smallest radii where Newtonian gravity will dominate too.

The error bars have been computed from the error estimates provided by the SPARC team, which concern eV_disk, eV_bar, eV_obs and the error of the disk surface brightness eSB_disk. It has been assumed that deviations occurring in the measurements of the surface brightness at each radius are independent from each other and that those measurements are independent from the measurements of the rotation velocities and from the calculated velocities. Furthermore, the contributions of eSB_disk. at specific radii have been weighted with the inverse of the squared distance $X_{i,\varnothing ,\beta }^2$ of each mass m_i as defined in formula (11) to the observed point. $X_{i,\varnothing ,\beta }^2$. The errors in the variables computed in equation (11) have then been combined at each radius R after Ku [48] (pp. 265-269).

Now, for the scenario as predicted by the theory in hand, the gravitational contributions from dark matter, i.e., space in other universes appearing stretched to us, will be modelled in exactly the same manner, but using a loop over all 165 possible scenarios as summed up in Table 1.

Firstly in Table 2 a numerical example is given for a simple situation of two unity masses at a unity distance. Newton's constant of gravity has as well been put to one. The stretched axes per scenario from Table 1 are listed in the first column.

The distances in this example are taken such that an angle between the nearest axis and the line between two points amounts to 33 °, which can be shown is the average value of this angle in a sphere.

The second columns shows the number of scenarios from the sum of 165 that correspond to the stretching direction.

This shows how for an intermediate angle between the axes and the line between two points the gravitational action from the masses in the superposed universes increase the total acceleration. The effect from universes that share two dimensions with ours and hence are stretched in one direction, see Table 1, were the largest if there would be as many of them as ones that share only one dimension. The latter have the largest absolute contribution because of their large number (N = 28 x 3 = 84).

Now, the gravitational contributions from dark matter, i.e., space in other universes appearing stretched to us, has been modelled in exactly the same manner, for all 175 galaxies in the sample for all 165 scenarios. Gravitational acceleration at the central line through the plane of rotation has been recalculated over the full three dimensions of the disks, the bulges and the gas clouds, comparable with the procedure in equation (11), so over all other particles i at radii within the observed radius and outside, for all azimuths. This has been done in a exactly the same numerical manner, so with a limited resolution of patches at different radial distances and for 24 azimuth angles and assuming a vertical scale length h_z as calculated from the disk scale length ratio h_r/h_z, defined after Kruijt [49] (p. 11) and Sparke and Gallagher [50] (p. 202). The latter reference states at that page that typically the disk is about 10% as thick as it is wide, so h_r/h_z ≈ 10. But, De Grijs [51] has found it depends on the Hubbel-type of the galaxy. This has been implemented in the paper in hand. As said, based upon [47] for the gas the common factor three to increase h_z has been used. This yields a vertical mass distribution that then has been stretched in one or more directions according to Table 1.

It should, however, be noted that in the disk and gas cloud a certain amount of ‘viscosity’ because of magnetism occurs, see Begelman & Rees [42] (pp. 66 and 67), This yields an exchange of angular momentum over the disks cross dimension. As a result, still one value of V_bar and V_obs can still be attributed to or measured at each radius in the galaxy.

Then for each galaxy, the ratio of the linear mass density of dark matter in each superposed galaxy to the baryonic one in our universe is determined. It is denoted ML_ratio. Since the theory does not predict the baryonic and dark masses should be the same, the amount of matter in the galaxies in the superposed universes can vary according to their history, since there is no fundamental reason why it should be exactly distributed as in our universe. The same applies to the vertical scale length h_z, of the dark galaxies compared to the galaxies in our universe. The mass ratio and the ratios of h_z together determine the linear mass density in the dark galaxies, which is the true factor that determines the gravitational acceleration caused by an apparent wire-mass. Therefore both could vary. To avoid huge numerical effort, the choice has been made to vary the mass proportional to this ratio in all cases to test the constraint they give, as explained in the sequel, since some preliminary tests showed this gives the best rotation curve fits.

The values of ML_ratio for the case the dark and visible galaxies are outlined with the coordinate axes of the multiverse have been plotted in Figure 6. The average over all tilting angles relative to the axes, which can be randomly divided, taken over all 175 galaxies and all angles should equal unity. This makes it a constraint, and not a free parameter of the proposed scenario. The values show a certain bandwidth, which indicates that the amount of dark matter or the vertical scale height can vary between different galaxies and have a different proportion to the visible matter. After all, the values of ML_ratio represent the effect of the matter in the superposed universes as observed in our galaxies. Therefore a range of tilting angles and a range of observation points in the rotation planes have been simulated, for all 175 galaxies. They are shown in Table 3.

Table 3. Values of ML_ratio for all possible angles.

Run	Observation angle in rotation plane [°]	Galaxy angles x/z axes [°]	MLratio
1	0	0/0	0.29
2	45	0/0	0.48
3	0	0/45	1.0
4	45	0/45	1.6
5	90	0/45	0.32
6	0	35/45	0.85
7	45	35/45	1.6
8	90	35/45	1.4
9	135	35/45	1.6
10	0	14/30	0.7
11	45	14/30	1.9
12	90	14/30	1.2
13	135	14/30	1.2

Table 3. Values of ML_ratio for all possible angles.

Run	Observation angle in rotation plane [°]	Galaxy angles x/z axes [°]	MLratio
1	0	0/0	0.29
2	45	0/0	0.48
3	0	0/45	1.0
4	45	0/45	1.6
5	90	0/45	0.32
6	0	35/45	0.85
7	45	35/45	1.6
8	90	35/45	1.4
9	135	35/45	1.6
10	0	14/30	0.7
11	45	14/30	1.9
12	90	14/30	1.2
13	135	14/30	1.2

When runs 1, 2 and 4 are taken twice, for reasons of symmetry and so as to give all the four scenario’s the same weight, the average value equals unity, which should be, since there is no reason to assume the average mass of the 175 should differ from the average value of their counterparts in the 164 other universes. So this makes ML_ratio being unity, on average, a constraint.

Then it is interesting to know how much each of the scenarios of Table 1 contribute to the additional gravitational acceleration in galaxies. This has been worked out in Table 4.

This shows that the dark universes with one stretched dimension are dominating in this scenario, which confirms the earlier statement that dark matter in first approximation can be modelled as a wire mass. This is caused by the fact that galaxy disks are so thin, amplifying the contribution from scenarios where the dimension in the h_z-direction are stretched, since then even the stretched distances remain small.

5.3. Prediction Model for Gravity by Dark Matter in Galaxies

With this natural explanation and the equations (10) to (12) a prediction model for the total acceleration, g_linear + g_bar (as defined in equation (9)) can be made that is as accurate as MOND and TeVeS.

And, in galaxies (like in clusters), the predictive power of the present scenario is indeed comparable, as will be shown in the sequel. The proof of this prediction comes from the same 175 measurements with SPARC. To this end, The observed Newtonian gravitational accelerations, corrected with MOND or TeVeS using equation (4) are divided by the observed gravitational acceleration V_obs²/R. The values for the inclination of the 175 galaxies and the distances to ourselves have been taken from the SPARC database without varying them.

This differs from the approach of Lelli [11] which varied these two parameters within the reported error margins, so as to find the best correspondence of MOND with the observations, based upon the assumption that a_m is a constant. The future will show whether with smaller error margins in these parameters that will still hold. Since as mentioned in chapters 4 and 5.1 in the theory in hand ML_ratio can vary, this approach is obsolete here.

So, in the article in hand, the inclination and distance are not varied, but just the reported values have been used, just to compare the predictions as they are with V_obs. Optimisation of ML_ratio has been done such that the r.m.s. error value of the deviations of g_linear + g_bar from g_obs over all radii where g is smaller than a_m is minimised. Following Ku [47] (p. 269) the error estimate has been calculated as the r.m.s. value of the 175 error estimates of ML_ratio over $\sqrt{175}$ (175 the number of SPARC galaxies). This gives an error estimate of ± 0.02 [m³ kg⁻¹ s⁻²]. But, because the results are depending on the ratio chosen for the determination of Y_ml, this seems to be a too flattering value.

The rotation velocities predicted with the linear model presented in this paper and the observed accelerations, (g_innear + g_bar) / g_obs have been compared for all 175 galaxies. The predictions lie as close to the observed values as with MOND and TeVeS, when the square root of the deviations of g_MOND and (g_linear + g_bar) compared to g_obs are added for all radii of all 175 galaxies. To get here, the function µ(y) from TeVeS had to be solved in an iterative manner at each radius.

This was based upon fitting the mass-to-light ratio Y_ml based upon the maximal disk hypothesis. This approach is used in the 175 plots in Annex 2 and in Figure 5 and Figure 7. The overall performance of MOND is slightly better than that of TeVeS. With dedicated values for the baryonic to dark mass ratio per galaxy, so consistent with the theory as outlined in chapter 4, used in the entire range of radii in a galaxy, the accuracy is comparable with MOND.This is based upon the velocities V_gas and V_disk as calculated by the SPARC team from the detailed density distributions they measured, resulting in V_bar. After that, the Newtonian gravitational accelerations as calculated by that team were modified with MOND as well as Bekenstein’s TeVeS.

In Annex 2 all the 175 rotation curves with the predictions are depicted. Some show that the predictions with the multiverse hypothesis, so linear gravity in first approximation, V_lin, reproduce more details of the observed rotation curves too, see for example Figure 5 in the previous section and Figure 7 down here for two additional ones. In the legends, TeVeS is indicated as V_bekst in Figure 7.

They have been made for the situation that the dark and visible galaxies are outlined with the axis system as described in chapters 4.2 and 4.3. This appears to give the best match for galaxies with clearly flat rotation curves, which, by the way, might give a clue to why some rotation curves are more flat than others in the first place. This is an assumption that needs more research. Striking feature of Figure 7 (a), again, is that the current scenario does predict flat rotation curves where they appear in the observations, see Figure 5 (a) but as well linearly decreasing rotation curves, contrary to MOND, see Figure 5 (b).

The error-bars in the graphs have again been calculated from the error margins as reported by the SPARC team with the assumption that the different quantities are independent.

But, what causes the improvements? The central point is that MOND and TeVeS do modify the gravitational acceleration g acting on a mass. It can easily be seen that when the mutual interaction of a small and a large mass is considered, this violates the conservation of momentum, Bekenstein [3]. The present predictions avoid this, by calculating the mutual acceleration for all separate masses, with linear dependence from the mutual distance. And as a result, the interpolation function of MOND and TeVeS, see equation (2) in chapter 2, has become obsolete now, since all mutual interactions of mases in a galaxy are treated separately and in a consistent manner, with conservation of momentum. And this has a large effect, because, given the MOND parameter µ is order of 0.1 to 0.5 in the flat rotation part of the 175 SPARC galaxies (so x in equation (2) adopt values in the range of 0,1 to 0,6). So, in the intermediate MOND regime This makes clear this MOND interpolation function is dominating the MOND predictions. Almost all the SPARC observation points are in this intermediate, and hence not in the deep MOND regime. As a result, the interpolation function, acting on g on a mass is determining.

The question is what this brings in terms of improving calculation methods for galaxies or simulation models for the evolution of galaxies. The application of this present calculation scheme, alternative to MOND, would take the following steps for a given radius R in a galaxy:

• Calculate the Newtonian gravitational acceleration at R, from the baryonic mass distribution with equations (10) and (11).
• From the same baryonic mass distribution, already available from step 1), calculate the additional linear gravitational acceleration by stretching the distance the gravitational force acts by the said factor of approximately 21, in one, two or three directions conforming to all 164 scenarios summed up in Table 1.
• Add the Newtonian gravitational acceleration to the 164 linear gravitational accelerations and compute the rotation velocity at R.

As said, the theory in hand states the dimensions of all the other universes are stretched, according to the scenarios in Table 1. As well, the baryonic and dark galaxies attract each other and are assumed to have the same orientations because of gravity, but together they can be tilted in any direction with respect to the stretches dimensions, i.e., with respect to the axes. When a wide range of possible orientations (i.e., a range of 0 to 45° to the nearest axis) and a range of points in the 175 galaxies are simulated a mean value of the amount of stretching can be determined. The mean factor amounts to F = 21 ± 1. In clusters it will show a value of 22, so the predicted GUT-scale will amount to approximately 22/2π = 3.5 x L_p. This approach works at all radii, without the need for a distinction between two regimes, and without an interpolation scheme between the two regimes, as with MOND. Simulations of clusters, discussed in the next section will increase the fitted value of F somewhat, and show an overall value of 22. Because the results are depending on the ratio chosen for the determination of Y_ml, and it is hard to distinguish between values of this ratio in the range from 0.9 to 0.99, the resulting values for F would suggest the said error interval of ± 1.

5.4. Cluster Velocity Dispersions

A prediction is the gravitational effects of dark matter as described by the theory in hand appear in galaxy clusters as well. The theory is able to significantly improve the predictions of the velocity dispersions in galaxy clusters compared with MOND and gives a valid explanation for the way dark matter acts in clusters.

The clusters show velocity dispersions that depend on the gravitational potential and obey the Virial theorem Milgrom [52]. Milgrom used data from NGC clusters only, but Tian, McGaugh et al. [53] analyzed the much larger Abell clusters as well. The data of both sources have been combined in the graph depicted in the sequel in Figure 9.

The paper in hand proposes an approach, in which the real existence of dark matter of which the gravitational attraction acts over one or more stretched dimensions, is the starting point.

Again, it is assumed the number of superpositions amounts to N = 165. For the conventional approach based upon Newtonian gravity, the assumed ratio of the sum of baryonic and dark matter to baryonic matter only amounts to N = 6. The later ratio has been used in the validation of the Monte Carlo simulations of clusters that will be presented in the sequel.

The Newtonian, and the MOND approach, as well as the scenario proposed in this paper will be elaborated and used to predict the dispersion velocities from the mass M and the diameter R_h of the clusters.

The Virial theorem states that if a spherical distribution of objects of equal mass is stable and self-gravitating (such as a galaxy cluster), the total gravitational potential energy, (U, of the objects is equal to minus two times the total kinetic energy, T:

$$2T+U=0$$

(12)

With U being the sum of all the mutual gravitational potentials caused by the gravitational forces acting on test masses and the vectors to the mass centers, as follows:

$$U=G{\sum }_{ij}{m}_i{m}_j/\left(\right|r_i-r_j\left|\right)$$

(13)

For simple geometrical reasons it is commonly assumed the ratio between the squared velocity dispersion and the squared line-of-sight velocity, the latter being the actually measured quantity, amounts to 3. Then, equation (14) for Newtonian gravity typically results from dynamic N-body simulations or static Monte Carlo simulations for the line-of-sight velocity dispersions σ², after Navarro et al. [54], Binney and Tremaine [55], Lokas et al. [56], Carlberg et al. [57] Kravtsov et al. [58] the Planck Collaboration [59], Wolf et al. [60], Evrard et al. [61] :

$${\sigma }^2={\displaystyle \frac{0.3G{M}_{200}}{R_{200}}}$$

(14)

In [53] as well a factor 0.3 in this equation (J = 3) is shown to give a close fit with the observations. With the radius R₂₀₀ = ½ R_h, the latter being the cluster diameter referred to in [52] and [53]. As shown in Figure 9 this equation is in good correspondence with the said observational data, when as in conventional cosmology it is assumed the total amounts of matter amounts to approximately six times the baryonic matter. This factor N = 6 has been applied to the data in the graph in Figure 9 down here to yield the Newtonian velocity dispersions including the effect of dark matter.

But, Milgrom has applied the Virial theorem to the MOND equation (5) for spherical clusters in (Milgrom 2018). Milgrom’s formula for the dispersion velocity in a cluster is:

$${\sigma }^2=\sqrt{{\displaystyle \frac{G{M}_{200}{a}_m}{9}}}$$

(15)

Contrary to Milgrom’s findings for smaller and nearer by clusters only, Milgrom [52], the full range including the Abell clusters reveals the MOND data are significantly lower than the observations in the Abell clusters and the corresponding curve in Figure 9 does not have the right slope.

Now, in line with the references cited in the above for this paper Monte Carlo simulations have been performed with Matlab^® to firstly reproduce equation (14) for reasons of validation. This has been done using an NFW-mass profile, see Navarro et al.l [54] with a typical expected value for the concentration parameter of c = 3 for large clusters and to 10 for smaller ones, after Bullock et al. [62] and Groener et al. [63], which then is varied statistically using a log-normal distribution around this value. This parameter c is defined as R₂₀₀/r_s, see equation (16).

$$\rho \left(r\right)={\displaystyle \frac{{\rho }_s}{\left(\frac{r}{r_s}\right){(1+\frac{r}{r_s})}^2}}$$

(16)

Firstly, this has been rewritten into a normalized cumulative radial mass distribution. Then random normalized radii have been pulled and accepted/rejected, each data point representing a galaxy. The accepting/rejecting has been based upon the cumulative mass distribution, after assigning each pulled galaxy a mass as discussed in the sequel. This has been continued up to the point the total cluster mass M₂₀₀ was reached.

In cosmological N-body or Monte Carlo simulations, the concentration parameter c is log-normally distributed for a given mass, with a scatter in log c of order 0.15–0.25, see Comerford & Natarajan [64]. This supports the use of a log-normal Monte Carlo calculations with σ ∼ 0.2–0.3 for variations in c in NFW haloes e.g., Bullock et al. [62]. Here σ ∼ 0.2 has been applied using the ‘Randn’ function, to pull random figures from a standard normal distribution. The said value of c is then taken as the mean expected value of the distribution (, P 51971 c = 100, UGC 10045 c = 16, NGC 5005 _c = 12, NGC, 5353: c = 10, Abell 3526: c = 6 and Abell 2142: c = 3) and the value that has been pulled has the value e^{(c +σRandn(N,1))}, with N the number of pulled values, which equals unity in each separate run.

The adopted distribution of the mass of individual galaxies is as well dependent on the radial position r in a cluster, with larger galaxies prevailing near the cluster center. The corresponding radial dependence of the mass density is modelled as a saturated power law, motivated by dynamical friction and merger-driven mass segregation (e.g., Munari et al. [65], De Lucia & Blaizot [66]; Niederste-Ostholt [67] van der Burg et al. [68] Annunziatella M. et al. [69] (showing Schechter functions describing the SMF that are much steeper in the outskirts than in the center; an effect that can be approximated with equation (17))). The adopted approximation is as follows, with r_ref = 0.3 * R₂₀₀.:

$$M\left(r\right)=M_{core}{(1+{\left({\displaystyle \frac{r}{r_{ref}}}\right)}^2)}^{-1}+M_{field}$$

(17)

This has been done up to the radius R₂₀₀ using the following galaxy masses: M_field = 3 *10¹⁰ M_sun and M_core = 4*10¹⁰ M_sun so as to match equation (14) and the observations exactly. Using the same Monte Carlo approach, the masses M of each galaxy have varied analogue as the halo concentration parameter c as described up here, following Bahé et al. [70]. Based upon this reference a value of σ ∼ 0.2 has been applied here as well. These values have been kept constant for the two largest clusters that have been simulated, Abell 3526 and Abell 2142, since the mass per galaxy does not vary significantly with cluster mass, see Lin et al. [71] and has been decreased for the smallest two again using [71].

The velocity dispersions occurring in the simulation of each cluster have been determined for al 2400 galaxies and then the average value has been taken to make Figure 9.

After, as discussed up here, the galaxies were distributed over the radii so as to match with equation (16), the galaxies were assigned random angles to distribute them randomly in the cluster. This has been done by pulling random directions in a unit cube and by accepting/rejecting on the criterium that a galaxy should fall within the unit sphere. This avoids creating an uneven distribution with more galaxies at the poles of a cluster than at the equator.

This has been simulated a dozen times for 6 clusters, i.e., the largest in the said data set (Abell 2142, NGC 5353 and Abell 3526) and three smaller ones (NGC 5005, P51971 and UGC 10045) to validate the model with equation (14), before moving to a modified simulation conforming the theory in hand. The latter entails stretching the dimensions according to the 164 scenarios of Table 1 in Section 4.2, but furthermore with the same settings as the said validation simulation. This stretching, with say a factor F, of one or more dimensions makes the gravitational force decrease faster, proportional to F². The amount of work a moving object has to perform is the integral of the force times the distance covered by a test mass, the distance covered being measured in our universe, so non-stretched. This simply means the gravitational potential contribution from another universe with a stretched dimension decreases with F² too. This means the isotropy of the velocity dispersions in a spherical cluster is broken when the contributions from other, stretched, universes are considered. For one stretched dimension, the integral, the mutual distance $|r_i-r_j|$ expressed in terms of x,y and z- distances, and assuming movement in the x-direction is modified as follows:

$$U=\int {\displaystyle \frac{G{m}_i{m}_j}{x^2+y^2+z^2}}dx={\displaystyle \frac{G{m}_i{m}_j}{\sqrt{x^2+y^2+z^2}}}$$

(18)

The sum of this over all pairs of test masses is just equation (13). Upon stretching one dimension it will become:

$$U=\int {\displaystyle \frac{G{m}_i{m}_j}{{\left(Fx\right)}^2+y^2+z^2}}dx={\displaystyle \frac{G{m}_i{m}_j}{F\sqrt{{\left(Fx\right)}^2+y^2+z^2}}}$$

(18*)

The integral for movement in the y (and likewise in the z direction) however will become:

$$U=\int {\displaystyle \frac{G{m}_i{m}_j}{{\left(Fx\right)}^2+y^2+z^2}}dy={\displaystyle \frac{G{m}_i{m}_j}{\sqrt{{\left(Fx\right)}^2+y^2+z^2}}}$$

(18**)

When two dimensions are stretched, for example both x and y, the potential for movement in the y-direction will as well be conforming to equation (18*) and when three dimensions are stretched all three will have this shape.

Firstly, this has been worked out in Table 5 upon computing the gravitational potential U for two masses with each unity mass and at unity mutual distance. Newton's constant of gravity has as well been put to one just as in chapter 5.2. The stretched axes per scenario from Table 1 are again listed in the first column.

The contribution to the potential of a particle moving in the x-direction is called ‘Px’ etc in the second column.

The number of scenarios N that have a certain stretched axis and yield a contribution Pc is given in the second column, i.e. 1/9 of the number listed in Table 1. They appear three times each, since the contribution to the potential is calculated for movement in three directions for each stretching scenario. So, each contribution is weighted with a factor 1/9 (1/3 in the case of three stretched dimensions in the last three rows) to get the proper sum of 165 and the proper sum of the potential U.

One mass is in the origin and the other at a position near the x-axis with an angle of 17° from the x-axis. The resulting distances along the axes can be seen in the first row. The effective distance still amounts to R₂₀₀ = 1.

Because of the stretching, the spherical symmetry is broken and all the stretched distances have to be computed for each scenario from Table 1. They are called 'Effx', 'Effy' and 'Effz' and except in the first row (with the Newtonian contribution) are thus modified. The Newtonian potential, 'Pot.' in the last column, as a result equals 1.

The sum of all the potentials 'Pot.' from baryonic and dark matter, so U amounts to 6 approximately, which is in line with the theory as will be explained in the sequel and with Figure 9.

Striking feature here is that the contributions to the potential are very low when the corresponding axis is stretched (see row 2 for example). On the other hand, when, for example, the distances along the y- and z-axis are stretched, the contribution Px will be largest (row 17). This as well happens when only the distances along the y- or z-axis are stretched (row 5, row 8), but here the number N is lower, which dampens the effect.

The angle of 17° adopted in Table 5 was just taken arbitrarily in this example. When the angle is zero, this ratio will be approximately 20, a value which decreases rapidly when the angle becomes larger than zero, and at the maximum deviation from all the axes, i.e. an angle 55° the minimum value will be approximately 4. In a cluster, with many galaxies, these angles of the lines between any pair of two galaxies, will be randomly distributed. These indeed have been modelled in such a random way, in the sequel.

Secondly, the Monte Carlo simulations have been performed. Upon employing equations (18**), the assumed superposition of 165 universes with stretching of space in one, two or three directions as described Table 1 has been modelled with the further unchanged Monte Carlo model.

The resulting gravitational potential varies with the radial distance from the galaxies to the center of the cluster and at R₂₀₀ it converges to zero, see Figure 8 for an example.

This exactly yields the same result as equation (14) and thus matches the observations well, provided for the factor F a value of F = 22 is adopted. The r.m.s. value of this factor F over a dozen runs for each simulated cluster amounts to 3.4. This cannot be reduced by taking into account that six clusters have been simulated, since these simulations cannot be considered as independent. So this gives an error estimate of ± 3.4.

Figure 9. Velocity dispersions in NGC and Abell clusters as function of cluster mass.

Figure 9. Velocity dispersions in NGC and Abell clusters as function of cluster mass.

Figure 9 shows equations (18 and 18* and 18**) give a better match with the than MOND and perform comparable with the conventional Newtonian approach.

The factor F = 22 ± 3.4 derived from these cluster velocity dispersions matches quite well with the factor derived from the 175 galaxies discussed in chapter 5.3, which amounted to F = 21 ± 1. To summarize, the description of dark matter as a result of potentially 165 different superposed universes in 11-dimensional space is fundamental to describe both flat rotation curves in galaxies and the cluster velocity dispersions in a consistent manner.

Then, like it was done with the galaxies, it is interesting to know how much each of the scenarios of Table 1 contribute to the additional gravitational potential in clusters. This has been worked out in Table 6.

This distribution happens to be comparable to that in galaxies, but with a slightly larger contribution from the scenarios that share less dimensions with our universe. This has to do with the galaxies being thin, disc shaped, which is still gives relatively much gravity when the vertical direction is stretched, since then the mass is still relatively concentrated.

5.5. Structure Formation

The current scenario can strongly effect structure formation in the young universe.

For, at very long mutual distances, gravity from the 165 superposed universes will start to act as 165 point sources. This, at very long distances, when the cloud has begun to flatten, can have a considerably larger total effect than N = 6 Newtonian potentials assumed in conventional cosmology in the shape of a spherical NFW-halo. This is, because on average in 2/3 of the 24 cases with one stretched dimension from Table 1, this long distance between two masses is not further stretched, but only directions perpendicular to it. Likewise this applies to 1/3 of the 84 cases with two stretched dimensions on average. But, first simulations with a model of a gas cloud, based upon the model made for the clusters described in chapter 5.4, reveal this only works in the current scenario when the flattened cloud and hence the first galaxies appearing shortly after the Big Bang are in line with one or more of the axes of the universe as discussed in the paper in hand, see chapter 5.2 Table 3. These axes might correspond with the filaments. This is a prediction that can be falsified.

Another preliminary simulation on the total gravitational force towards the crossing point of three filaments filled with gas, shows the current scenario might largely increase the forces pulling the gas from the filaments to the crossing point, where the first galaxies in the young universe developed.

In these cases, this larger gravitational potential at long distances can have considerable impact on the development of galaxies from rotating clouds of gas and dust. Recently a paper published in Nature by Labbé et al. [72] confirmed this. The James Webb telescope showed there is indeed a very rapid development of large galaxies already at 600 million years after the Big Bang, Labbé et al. [72], see Figure 10.

This rapid development of large galaxies is much sooner than the current theories predict.

The said much larger gravitational potential in the current scenario can greatly accelerate the contraction of gas clouds and the development of stars and galaxies, as discussed by Sanders [73] and McGaugh [74] and Kroupa [75] and Banik [2] and hence can give a good explanation for this rapid development.

5.6. Dark Matter Undetectable

Dark matter is undetectable, since it cannot interact with visible matter, except by gravity, which deforms space and time in 11 dimensions. In the current scenario this becomes logical. In chapter 4 it is argued that electromagnetic waves cannot propagate in and through the 0-, 1- or 2-dimensionality of the overlap with other universes in the superposition state.

But, in the current scenario, there is another fundamental reason why dark matter is undetectable. That is orthogonality, see Griffiths & Schroeter [16] (p. 98 and 151) in terms of quantum mechanics. It follows from the definition of a superposition state, namely as a linear combination of independent quantum states, i.e., orthogonal states. They are per definition not accessible to each other. For the superposition of 3-dimensional universes in a 4-dimensioanl space with two overlapping dimensions, the meaning of this is evident, because one of the dimensions is orthogonal to that of the other universes. But it is in all cases necessary to maintain a superposed state. For instance, Zeilinger [76] states about the double-slit experiment: “The superposition of amplitudes .. is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear”.

That is why we can never perform any measurement on the properties of dark matter. This information must be and remain absent for the superposition at the earliest stage of the Big Bang to have been possible at all.

5.6. Orientations of Galaxies Relative to the Structure of the Universe

The following is more speculative of nature, but as mentioned in chapter 4, Han [44] shows that the Milky Way’s assumed stellar halo is tilted with respect to the disk plane, suggesting that at least some component of the dark matter halo may also be tilted, see Figure 11. The origin of this misalignment of the assumed dark halo, of approximately 25°, can be explained by the gravity hypothesis in the paper in hand, since the orientations of the stretched dimensions of another superposition state overlapping with ours, can deviate from the orientation of a galaxy and the effects per directions can be anisotropic if the amounts of matter in overlapping galaxies differ according to their history. In the study in hand it was revealed that some rotation curves, namely the most flat, show the best correspondence with observations for the situation that the dark and visible galaxies are outlined with the axis system as described in chapters 4.2 and 4.3. This appears to give the best match for galaxies with clearly flat rotation curves, which might give a clue to why some rotation curves are more flat than others in the first place.

Thus, the idea is that the orientations of the supposed halos of different galaxies and in particular the orientations of the galaxies with the flattest rotation curves might reveal a deep underlying structure in the universe, i.e., will display the directions of the stretched dimensions of other universes overlapping with our universe. In other words, the orientations of the assumed dark halos will, when compared with each other on a large scale, display something of the underlying coordinate system of our multiverse.

5.7. GUT-Scale

A prediction is that the fitted value of the stretch ratio F ≈ 22 found in the paper in hand will yield a good estimation of the GUT-scale. Assuming it indeed represents the compactification radius of the other dimensions, so the ratio of string length over thickness in our observed universe, i.e., 2$\pi$ x GUT-scale/L_p.

This means that from both the galaxy rotation curves and cluster velocity dispersions a consistent order of magnitude value of GUT-scale = 22 x L_{p /}2$\pi$ = 3.5 x L_p can be deduced. So, of the GUT-scale will ever be measured, it may give insight in the amount of universes in our multiverse.

6. Conclusions and Suggestions for Further Work

In this article it is argued dark matter has a deep link with the multiverse hypothesis. The hypothesis of dark matter is a way to explain why among other galaxies seem not to obey Newton’s law of gravity. As well, dark matter is needed to explain the statistical distribution of ‘cold’ and ‘hot’ spots in the background radiation, that would still need the existence of (much) dark matter vs. baryonic matter to be understandable in terms of Big Bang nucleosynthesis, as well as matters like gravitational lensing and gravity in galaxy clusters. Alternative approaches like MOND and TeVeS work well to describe the flat rotation curves in galaxies as such, but do not give a natural explanation for the concepts and additional fields they introduce.

But, it should be insisted, the existence of dark matter is taken as a starting point in this article. A natural explanation for the physical existence and for the nature of dark matter is presented based upon Hawking’s cosmology and String theory that has its foundations in it.

The hypothesis in hand does not introduce any modifications to Newton’s law of gravity, nor new physical parameters or concepts, nor new fields. It does not need ad-hoc assumptions about the nature of dark matter, like the dark particles being collisionless and not losing energy through radiation, that must lead to an NFW-halo around galaxies. Yet, it explains the observations.

The conclusion is that the universe consists of 165 3-dimensional universes or branes (i.e. 11*10*9/6 different versions, with Hawking assuming that at the very start of the Big Bang time cannot be distinguished from spatial dimensions), existing as states of a superposed 11-dimensional multiverse or bulk, which each have zero up to two overlapping dimensions with the observed universe. For there is nothing outside it that could disturb the superposition state, and nothing inside it, since the states only concern which dimensions are compactified or not, it might be in that state forever, without de-coherence effects ending it. This is why this superposition is a working premise in the current article and, besides that, it is why Hawking and others can speak of the wave function of the universe in the first place.

The assumed superposition leads to the existence of additional baryonic matter, but in superposed universes and hence ‘dark’, with gravity that attracts matter in other superposed universes over stretched distances because of the compactification of the other seven dimensions. It is assumed the universes are interwoven at the smallest scale as a fine fabric, so as to make the other universes have effect throughout our entire universe. The 0-, 1- or 2-dimensionality of the overlap with other universes explains why electromagnetic waves cannot propagate through them. And this, together with the orthogonality of superposition states, gives a natural explanation for dark matter particles being undetectable.

Gravity from dark matter and visible matter is well interpreted as the sum of two gravitational accelerations. Dark matter in first approximation appears like a line mass, however without any modification to Newton’s law of gravity, because of the stretching of dark matter. Therefore the Levi-Civita metric and adding this to the Schwarzschild metric for a baryonic point mass, in first approximation gives an insightful solution of the Einstein field equations in the weak fields that occur in the galaxies studied that directly leads to the Tully-Fisher relation. The dimensions of strings in String-theory cause this stretching, because the other dimensions at the level of strings are interwoven with our universe as a fine fabric as described by intersecting brane-world scenarios and is rolled up at the GUT-scale, which however is much larger than the Planck-length. All distances gravity has to cover through the fabric into another universe are thus stretched in that other universe. This explains why the effect of dark matter is negligible near the center of a galaxy, even allowing elliptical orbits of the stars there, but dominates at large radii. So, this hypothesis may bring the smallest and large scales together. Dark matter does in this hypothesis still appear in the center of galaxies in our universe, since dark matter naturally is attracted by the visible matter in the galaxies.

From the values of the calculated baryonic and the observed velocities in galaxies in the SPARC data, an average value for the stretching of one up to three of the dimensions in superposed universes is deduced: F = 21 ± 1. This is not a fine-tuned number as meant in Hossenfelder [10], but an empirical value that represents the average effect of the matter in the superposed universes as observed in our galaxies. The amount of matter in the galaxies in the superposed universes can vary according to their history, which becomes visible in the values of the optimal baryonic to dark mass ratio that vary from galaxy to galaxy.

A value of F = 22 ± 3.4 seems to give a much improved prediction of dispersion velocities in galaxy clusters.

So, the present galaxy and cluster analyses empirically constrain the effective stretch factor to $F\approx 22$; this value should therefore be regarded as a calibrated model parameter until it, in some other way, can be derived independently from the compactification geometry.

The mass-to-light ratio has been used as the only fitting parameter to fit the baryonic rotation velocity, and hence the baryonic gravitational acceleration in each galaxy to the observed values near the core of the galaxies. After that, the above-mentioned value for ML_ratio is used to predict the additional gravitational acceleration at all radii without any further fitting. Applying this to predict rotation velocities from the baryonic matter distribution in a galaxy, upon using the mass density in the plane of rotation, will yield predictions that are as close to the observations as MOND or Bekenstein’s work, TeVeS.

The description of dark matter as a result of stretched dimensions is fundamental to describe both flat rotation curves in galaxies and the cluster velocity dispersions in a consistent manner. In galaxy clusters, the resulting improvement of the predictions of the velocity dispersions is even much more than in galaxies.From both the galaxy rotation curves and cluster velocity dispersions a consistent order of magnitude value of GUT-scale = 22 x L_{p /}2$\pi$ = 3.5 x L_p has been deduced. So, of the GUT-scale will ever be measured, it will give insight in the amount of universes in our multiverse.

But the 11 dimensions are taken from M-theory and are not prescribed by the current scenario. If in the future alternative theories with a few more dimensions would appear to give a better description of gravity, this might be helpful, since the more possible 3-dimension universes there are, the higher the fitted stretch factor F will appear to be and as a result, the flatter the rotation curves will become in the current scenario.

Further investigation of the shapes and orientations of dark matter halos, and the assumption made in the current scenario that the galaxies with the most flat rotation curves are more aligned with the coordinate axes of the universe, is needed. The stretched mass may have comparable orientation as the ones of the halos currently calculated by many researchers, but still more concentrated at the center of galaxies. This avoids the fundamental problems with the current view of halos surrounding galaxies as recently reported by Mistele and Lelli based upon the SPARC data.

Using the work of Levi-Civita and Santos it is shown a consistent relativistic formulation of the hypothesis can be constructed in linearized, for weak fields, or numerical GR. But in much stronger fields linearised calculations in GR will break down and numerical approaches are needed.

More future work is to implement the linear gravity approach in existing simulation software for the evolution of filaments and galaxies to verify whether that will yield better agreement with the observed trends, in particular the rapid evolution of large galaxies in the early universe as well as to study how the current scenario can be applied to gravitational lensing, including dark matter as a source.

The author looks forward to receiving responses to this hypothesis from the field.