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The same process may therefore occur when negative masses are included. To test this hypothesis, simulations are setup within a cube-shaped region of volume 3. Particles are located within this arbitrarily-sized region: 25 positive mass and a further 25 negative mass particles — with the x , y , and z coordinates of every particle being drawn from a uniform distribution. The total mass within the cube is 0. The initial velocities of all particles were equal to zero. The simulation is scaled such that each side of the box has a length of Mpc.

The simulation runs over This provides a coarse mass resolution of 1.

As the simulation progresses, structure including filaments and voids clearly begin to be formed. Similarly to Sect. The initial particle distribution has formed a complex network that comprises filaments, voids, and rich clusters. From these early simulations, it is unclear which is the predominant effect: the additional pressure from the negative masses being attracted towards positive masses leading to more rapid structure formation than can occur in a positive mass only universe, or the mutual repulsion between negative masses tending to counteract this and leading to slower structure formation.

Whichever the case, the presence of negative mass particles leads to modification of the relative spatial distribution between filaments and voids 6. N-body simulations showing the formation of large-scale structure from an initially motionless, uniform, particle distribution of 25 positive masses in yellow and 25 negative masses in purple. Both the initial top and the final bottom time-steps are shown.

One-dimensional simulations of particles with a negative gravitational mass have also been reported in a recent study Manfredi et al. More sophisticated N-body simulations with larger numbers of particles and more sophisticated initial conditions will be able to compare the resulting filaments and voids from these simulations with the observed large-scale structure in our Universe.

Nevertheless, we have obtained a key result — that structure formation is possible in a universe with negative mass. This has previously been perceived as a problem for theories of negative masses e. Bonnor , partly due to the conventional maxim that massive particles cannot accelerate to c , and primarily due to the general reasoning that we do not observe such high-speed particles.

Both components of this argument are however of dubious merit. Firstly, the theory of positive—negative mass particle pairs provides clear rules that govern such interactions. The mechanics of these interactions are governed by the usual physical laws: the conservation of energy and momentum remain fundamental, and hence it is unclear why we should object to this potentially physical law of nature on grounds of aversion alone. Secondly, and more importantly, observations provide evidence for significant numbers of ultra-high-energy cosmic rays which are known to be extragalactic in origin, although the mechanism of their production remains a mystery Pierre Auger Collaboration From this perspective, runaway motion is not a challenge for negative mass models, but is rather a useful observational constraint.

The idea that all negative masses in a universe should form gravitational dipoles and accelerate to high energies is not supported by the simulations presented here which have a limited number of particles , in which no runaway particles can be identified. While runaway motion is a legitimate physical facet of negative mass particle interactions, the simulations indicate that this behaviour is only common for idealised particle pairs and occurs more rarely as a bulk behaviour within a negative mass fluid. This is likely as the particles in such a fluid are subject to numerous counteracting forces from the surrounding medium.

One can assume that some amount of runaway particles must still exist, although these would likely be highly scattered by Brownian motion e. Landis However, no runaway particles were detected in these simulations and the computational results remain unchanged. One can conclude that runaway motion must be sufficiently rare within a bulk fluid that the effect does not occur with any regularity in a simulation of 50 particles.

Simulations with higher numbers of particles of the order of millions will be able to place numerical constraints on the runaway particle rate, in order to provide direct observational comparisons with ultra-high-energy cosmic ray detection rates. Furthermore, this model suggests that these negative masses can flatten the rotation curve of a galaxy.

An Introduction to Dark Matter

This suggests that negative masses could possibly be responsible for dark matter and dark energy. While this paper is primarily focussed on the theoretical and simulated consequences of such exotic matter, for the sake of completeness I now review the literature and consider any potential for compatibility between the toy model and contemporary cosmological observations.

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There is strong observational evidence from high-redshift supernovae that the expansion of the Universe is accelerating due to a positive cosmological constant Riess et al. However, inspection of these results reveals that the observations themselves may demonstrate initial evidence for a negative mass dominated Universe. In both of these seminal works, the very reasonable assumption was made that all matter in the Universe has positive mass.

For the Bayesian fits in Perlmutter et al. In other words, a probability of zero was assigned to a negative mass cosmology.

Dark Matter

In Riess et al. The entire supernovae analysis was re-run in this former work, in order to further test this point. This would be evidence for a negative mass dominated cosmology. To assume that all mass in the Universe is positive is highly reasonable, as there has never been a pre-detection of such exotic material. However, as I have shown in this paper, negative mass density may not be unphysical.

In fact, one can argue that its presence can be inferred from cosmological and galaxy rotation data, and it may possibly be able to provide an explanation for dark matter and dark energy. This further demonstrates that this is not a systematic that can be alleviated by better data, but rather a conceptual challenge with the data analysis. As shown in Sect. However, a cosmological constant is simply one form that can be taken by the negative energy density. Although beyond the scope of this current paper, future works will be able to reanalyse the supernovae data and obtain updated measurements of the cosmological parameters when matter creation is fully included within the Bayesian analysis.

One of the major outcomes from measurements of the CMB has been locating the precise position of the first acoustic peak. Spergel et al.

However, as shown in Sect. For a large universe in which the local geometry appears to be flat, it would therefore imply an especially low value for the magnitude of the cosmological constant — otherwise the universe would have recollapsed before reaching such a size. This is of course a possibility, but could also be perceived by a sceptical reader as a hand-waving way to allow any geometry for the Universe — regardless of the observational data. Naively, in the open spatial geometry of the negative mass dominated cosmology, the position of the first CMB peak would be expected to be located at a considerably smaller angle.

I now extensively follow that earlier work. The angular position of the first CMB peak is defined by the angle under which the sound horizon is seen at recombination, which is given by. By definition, the sound horizon is the distance that acoustic waves can propagate in a primordial plasma, which is typically assumed to only consist of positive masses.

Accounting for the expanding universe, the distance of the sound horizon is given by. However, in a negative mass cosmology, sound waves would be generated at the interfaces between positive and negative mass dominated regions. I note that inflation is itself not required in this particular cosmology, which does not have an age or horizon problem see Sect. An extended exposition can be found in BLC The aforementioned paper shows that acoustic waves would propagate in the plasma while positive and negative masses are in contact.

The expression for the angular position of the first CMB peak is then given by. Within a factor of approximately two, a negative mass universe can therefore predict the location of the first CMB peak and hence may be consistent with CMB observations. However, this model assumes that there is repulsion between positive and negative masses that leads to a subsequent gravitational decoupling, whereas the negative masses proposed in this paper obey the weak equivalence principle.

In the latter case, sound can be continuously generated up until the present epoch. As this model contains numerous approximations, there appears to be the potential for a negative mass universe to be fully consistent with observations of the CMB. It is quite surprising in a negative mass cosmology with negative spatial curvature, that the first CMB acoustic peak can naturally emerge at the correct scale — a simple back of the envelope approximation is already within a factor of two and consistent with the typically interpreted flat spatial geometry.

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In an effort to identify any plausible mechanism that due to the nature of negative masses may allow for further adjustment of the CMB peaks, we can also speculate about various other compressive and expansive effects in the early Universe. These overdensities would gravitationally attract matter, while heat from photon—baryon interactions would seek thermal equilibrium and create an outward pressure. Counteracting gravity and pressure thereby give rise to oscillations analogous to sound waves. In a negative mass dominated universe, it is apparent that the sound generation mechanism would be modified.

There would be two opposing effects: repulsive pressure from negative masses within an overdensity would tend to erase anisotropies, while conversely negative mass haloes would tend to surround positive mass baryons and increase both the effective gravitational attraction and the subsequent collapse of these overdensities. Matter creation would also exert a further influence. While these additional effects could modify the physics of sound generation in the early Universe and the predicted anisotropies in the CMB, the precise effects would depend upon the particle physics of the negative masses themselves, which is beyond the scope of this current paper.

In these regions, the positive and negative masses would tend to interact. These interactions could lead to runaway motion and possible annihilation of positive—negative mass particle pairs, which would further affect the precise mechanism of sound generation.

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However, such effects are not considered here. Further consideration of these additional plausible physical effects, together with theoretical constraints for the second and third CMB peaks, can form a robust test to either validate or rule out the cosmic presence of negative mass. In summary, while CMB modelling has provided an exceptional fit to observational data, the parameterisation of a model is only ever as good as the selected model itself.

Nevertheless, I again emphasise that a reanalysis of the CMB is not essential for this purpose, as the Universe could simply be large with minimal local curvature. Allen et al. Such observations make a critical assumption — that clusters are standard buckets that contain a representative mix of the constituent components of the cosmos. Within the presented toy model, galaxy clusters therefore do not represent standard buckets.

Although not standard buckets, one could still anticipate that at least some observations of a few galaxies or galaxy clusters may have found hints of a negative mass. For example, Chandra observations of the merging cluster Abell found hints of a negative mass and therefore did not plot those data Kempner et al.

Perhaps these findings can be trivially explained by mundane observational biases and systematics. Nevertheless, given that we have identified other possible evidence for the influence of negative masses on other spatial scales, the repeated observation of negative mass in clusters appears to be yet another piece of evidence that allows us to infer the plausible existence of this exotic material.

It is not immediately clear whether the relative ratios of 4. Claiming a cosmology with negative spatial curvature and negative cosmological constant would seem to be a heretical, renegade, and insane point of view. While this is true, the interpretation of these observations has been derived using the critical assumption that all mass in the Universe is positive. As scientists, we aim to be motivated purely by the scientific evidence alone and endeavour to remain entirely uninfluenced by confirmation bias. We have thus allowed ourselves to indulge in this unconventional thought experiment.

One can ask whether this negative mass cosmology could possibly be our cosmology.

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  6. In Sects. I do not claim an all-encompassing or rigorous proof of a negative mass cosmology, but simply highlight that the toy model raises numerous interesting questions. Future work will be able to further test the compatibility with additional cosmological observations. There are several outstanding theoretical challenges for a theory such as the one presented in this paper. I here provide some brief speculation as to the possibilities and future theoretical considerations.

    It may be possible to directly validate this theory via the direct capture and detection of a negative mass particle. Particles undergoing runaway motion would be highly scattered due to Brownian motion see Sect. At face-value, this is consistent with the origin of ultra-high-energy cosmic rays, and could lead to particles with energies above the Greisen—Zatsepin—Kuzmin GZK limit, such as the so-called Oh My God particle.

    Although this paper only considers particles with identical inertial and gravitational mass, there are also a number of other negative mass models in which the inertial or gravitational mass alone may have an inverted sign. These models have recently been presented in Manfredi et al. Experiments underway at CERN are expected to soon provide verification or refutation of these alternative negative mass models. It is possible that multiple forms of negative mass may possibly exist, and observational constraints will play an important role in testing these various scenarios.

    It seems that the proposed negative mass fluid can be modelled as either matter or vacuum energy. It has previously been proposed that space-time arises as a form of large-scale condensate of more fundamental objects, that are typically of an unknown nature e. One could therefore speculate that the negative masses could be interpretable as a quantised form of energy associated with space-time itself. The introduction of negative masses to the vacuum can also potentially provide a solution to the cosmological constant problem.

    The predicted vacuum energy can be a factor of 10 larger than the observed value e. Hobson et al. By invoking negative masses, the vacuum energy density can now take on essentially any value depending upon the precise cancellation of positive and negative energy states.

    If the negative oscillator modes exactly balance the positive modes, then. In theories of quantum gravity, gravitation is mediated by the graviton — a massless, spin-2, boson. This means that any pair of negative masses would attract, and not repel as suggested in this theory. There appear to be two options: either it is possible that the graviton could be modelled as a bound state of a positive and a negative mass, in a theory of composite gravity or some other mechanism which provides a modification of graviton properties.

    Alternatively, this could also indicate that the proposed theory cannot be modelled by real, physical, particles, but rather by the presence of effective negative masses within a superseding theory. Electrically-charged negative masses may tend to coalesce into highly charged clumps, eventually reaching a critical mass at which all other masses would be gravitationally repelled. This has previously been briefly described Landis One suggestion is that negative mass particles are always electrically neutral and remain in a diffuse form.

    No attempt has been made to reconcile the presented theory with the standard model of particle physics. Can a viable Higgs mechanism allow for a negative mass? Is there a way to introduce negative masses into the standard model that could allow for the combination of fundamental forces at high energies, in a grand unified theory? Would supersymmetry be required? Is it possible that a negative mass particle travelling backwards in time may be measured as having a positive mass?

    These questions would be interesting future avenues that could be explored further by the particle physics community. I have considered the introduction of negative masses and matter creation to cosmology, both via a theoretical approach and via computational simulations. Neither negative masses nor matter creation are new ideas. When considered individually, neither idea can explain modern astrophysical observations. This paper has reinvoked these two previous concepts and combined them together.

    Commonly presumed issues with negative masses include incompatibility with general relativity however, this was shown to be compatible in e. Bondi , and the vacuum instability which is not a bug, but rather a feature of the proposed theory, see Sect. By reintroducing the creation term into general relativity, but only for negative masses, it is possible to construct a toy model that has the potential to possibly explain both dark energy and dark matter within a simple and unified theoretical framework.

    These hypothesised negative masses can push against positive mass galaxies and galaxy clusters, thereby modifying their dynamics. As an illustrative concept, empty space-time would behave almost like popcorn — with more negative masses continuously popping into existence. From an astrophysical perspective, this cosmological theory surprisingly has some successes in describing observations. While such a proposal is a renegade and heretical one, it has been suggested that negative values for these parameters may possibly be consistent with cosmological observations, which have critically always made the reasonable assumption that mass can only be positive.

    The theory, simulations, and observations suggest that this particular cosmology has the following properties:. Negative masses can be intrinsically attracted towards regions of positive mass, thereby leading to an increase in density that manifests itself as a dark matter halo that extends out to several galactic radii.

    Due to mutual self-repulsion between negative masses, dark matter haloes formed from negative masses are not cuspy, and could thereby possibly provide a resolution of the cuspy-halo problem. The rotation curves of galaxies can be flattened by the negative masses in the surrounding dark matter halo, however the curve is also predicted to increase linearly in the outermost regions of galaxies.

    This may be consistent with previous observational findings, which have found that most rotation curves are rising slowly even at the farthest measured point e. Structure formation appears to be able to take place in a positive and negative mass universe, leading to the conventional suite of filaments, voids, rich clusters, and field galaxies. Supernovae observations of a positive cosmological constant made the reasonable critical assumption that all mass is positive.

    Upon relaxing this assumption, the supernovae data of Riess et al. It appears that the first acoustic peak in the CMB could naturally emerge at the correct scale in a negative mass cosmology. Several additional physical effects on the CMB need to be fully considered, and determining the effects that negative masses have on the second, third, and higher order CMB peaks can enable a robust test to either validate or rule out the presence of negative mass in the Universe.

    In this cosmology, negative masses are distributed throughout all of space-time, so that galaxy clusters cannot represent standard buckets. A number of galaxy cluster observations appear to have inferred the presence of negative mass in cluster environments. The introduction of negative masses can lead to an Anti-de Sitter space. Negative masses are predicted to produce a vacuum instability, which would suggest the vacuum itself is undergoing a slow and stable decay. In this cosmology, the universe would be taking on an increasingly negative energy state due to the continuous creation of negative masses.

    While such a vacuum instability is normally considered to be a theoretical insufficiency of negative masses, in this particular case it is not a bug, but rather a feature of the proposed cosmology. This accumulation of evidence could possibly indicate that while we cannot currently directly detect negative masses, we may have been able to infer the presence of these negative masses via their gravitational effects. These effects would seem bizarre, peculiar, and unfamiliar to us, as we reside in a positive mass dominated region of space.

    As the interactions between positive and negative masses are mediated by gravitation, the effects are typically fundamentally related to the physical scale — generally requiring a sufficiently large accumulation of positive mass in order for negative masses to influence the dynamics of a physical system.

    One aspect that is particularly preposterous is the concept of runaway motion, but as quantum mechanics has shown, many absurd concepts constitute real, testable, and repeatable facets of nature. There appears to be the potential and scope for this concept to be fully tested in order to make complete comparisons with observational data. A number of testable predictions have been made, including using cutting-edge telescopes such as the SKA, constraining the CMB acoustic peaks, and attempting direct detection from ultra-high-energy cosmic rays.

    Meanwhile, laboratory tests may be able to confirm whether antimatter could possibly be responsible for these gravitational effects — although it would seem that a far more exotic material would likely be required. In addition, future state-of-the-art N-body simulations on GPUs with larger numbers of particles, that allow for the creation of negative masses, will help to provide a refined comparison with observations.

    I here emphasise that several well-accepted theories can be modelled using non-real or effective negative masses. Air bubbles in water can be modelled as having a negative effective mass Brennen The Casimir effect can be modelled using a region of negative energy density Morris et al. Hawking radiation can be modelled using virtual negative mass particles that fall into the black hole Hawking Even Bose—Einstein condensates have observable regions with negative effective mass Khamehchi et al.

    While the results in this paper appear to be consistent with vacuum states that have negative energy density, it is possible that these findings may imply a superseding theory that in some limit can be modelled by negative masses. In this way, the toy model could possibly be compatible with our own Universe, which may still satisfy the weak energy condition. I suggest that a negative mass Universe is also a beautiful one.

    It naturally implies a symmetry, in which all physical systems are polarised into positive and negative states. A polarised cosmology that contains both positive and negative masses can literally bring balance to the Universe.

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