For example, one could imagine spatial pockets of negative mass with particularly high or low creation rates. This would give rise to an inhomogeneous and anisotropic distribution of expansion speeds across the sky that varies around individual voids in the galaxy distribution, and which could therefore be tested using observational data. This can be done by looking at the measured galaxy distribution using upcoming radio telescopes such as the Square Kilometre Array SKA and its pathfinders and precursors in combination with future optical and infrared surveys such as Euclid and LSST e.
Maartens et al. The continuous-creation of negative masses implies that the universe would be taking on an increasingly negative energy state. This could be interpreted as the vacuum instability being a real, physical, phenomena. In other words, rather than positive mass matter collapsing into an infinitely negative energy state, this cosmology results in the continuous creation of potentially real or virtual particles with negative energy.
Such speculations can be considered more rigorously in future works. Let us now consider the case of negative masses that are being constantly created. These negative masses gravitationally repel each other — thereby pushing apart structures in the universe. I showed in Sect. In fact, these continuously-created negative masses appear to resemble dark energy.
This degeneracy is present in the supernovae observations of an accelerating Universe see Sect. I am arguing that these negative masses are created at such a rate that they retain — on large scales — a constant density. This is the standard Friedmann equation given in Eq. In conclusion, I have shown that constantly-created negative masses are a natural explanation for the cosmological constant. The physical properties of negative masses that were undergoing constant creation would be rather similar to the accepted and inferred properties of the cosmological constant.
As these negative masses can take the form of a cosmological constant, one can deduce that they are a property of the vacuum rather than non-relativistic matter in the conventional sense. In summary, the field equations are modified to. If I consider the positive mass matter as negligible, we have the Friedmann equation.
The solution for the scale-factor is given by. This AdS universe undergoes a cycle of expansion and contraction with a timescale of. This is a counterintuitive result, as although the negative masses are gravitationally repelling one another, the cosmological effect appears to be for the negative energy associated with negative masses to cause the universe to recollapse.
The solution describes an open universe which expands from a Big Bang, reaches a maxima, and then recontracts to a Big Crunch. This appears to be an elegant outcome — by introducing negative masses that undergo continuous creation, one obtains a cyclic cosmology. Even with the addition of positive mass matter, a universe with a negative cosmological constant would eventually recollapse due to this extra attractive force. The continuous negative mass matter creation would cause objects in this universe to be gravitationally accelerated apart, but only on a local level.
In late-times, this cyclic cosmology with AdS space collapses due to its negative energy. One could also attempt to calculate the age of the universe in this cosmology, which can be conveniently estimated from Eq. While this appears to be compatible with observations of our own Universe, there are several notable caveats that could either extend or shorten the calculated age.
The existence of AdS space implies that string theory may possibly be directly applicable to our Universe. I note that this cyclic universe does not violate the positive energy theorem of Witten , which approximately states that it is not possible to construct an object out of ordinary matter with positive local energy density , that has a total energy which is negative.
Another interesting result is that the theory directly predicts a time-variable Hubble parameter, which may be consistent with recent cosmological measurements Efstathiou ; Riess et al. Such scenarios can be explored further in future works. I now consider the localised properties of positive masses that are immersed in a negative mass fluid. Such a scenario will clearly have an effect on the dynamics of positive masses and the subsequent evolution of a positive mass system.
This thereby allows us to consider the implications for the dynamics of galaxies and similar structures. For a stable circular orbit, the gravitational force which acts inwardly towards the orbital centre is equal to the centripetal force which is related to the component of the velocity acting tangentially to the orbital path. This provides the simple equation. By rearranging Eq. Nevertheless, very few galaxies show any evidence for such a Keplerian rotation curve.
I now construct an alternative model for the galaxy rotation scenario, that also includes a cosmological constant. In this case, the same setup exists as in the standard galaxy rotation scenario.
The positive mass test particle is now immersed in a negative mass fluid that, as shown in Sect. We can therefore examine the Newtonian limit of the field equations in the case of a non-zero cosmological constant. We can define. We can therefore modify Eq. However, for non-zero values of the cosmological constant, the rotation curve is modified.
For a negative cosmological constant, the rotation curve clearly increases linearly towards the outskirts of a galaxy, such that. This appears consistent with observational results, and previous studies have found that most rotation curves are rising slowly even at the farthest measured point Rubin et al. The data again appear to be consistent with a negative cosmological constant.
I emphasise that the rotation curves are being affected by the local negative mass density, which can coalesce into halo-like structures see Sect. Hence, more generally the solution for the rotation curve is given by. Predicted circular velocity as a function of radius, for a galaxy of similar size and mass to the Milky Way and that is influenced by a cosmological constant.
The displayed rotation curves are for increasing magnitudes of a positive in blue and a negative in red cosmological constant. Whereas a positive cosmological constant steepens the decline of a rotation curve, a negative cosmological constant flattens the rotation curve, causing a steady increase at larger galactic radii. Solid body rotation in the centre of the galaxy is not shown. Although the majority of observed rotation curves are largely flat, there is some observational evidence that rotation curves can continue to rise out to large galactocentric radii e.
Rubin et al. However, there are also reasons why the theoretical rotation curves may generally appear to rise more rapidly than is often seen in observations. The presented toy model is just a simple case, which demonstrates the change to a rotation curve enacted by a negative mass halo. There are several simplifying assumptions that have been made: i the positive mass within a galaxy is not concentrated at a central point and has a radially-dependent mass profile, ii a typical galaxy has separate bulge, halo, and disk components which are not modelled here, and iii the local negative mass density may be asymmetric and itself has a radial mass distribution.
In order to fit observational rotation curve data, these additional factors would all need to be modelled in further detail. This would then be able to either validate or rule out the proposed cosmology and the existence of a negative cosmological constant.
However, we consider this fitting process to be beyond the scope of this current paper. One of the most effective manners of testing a physical hypothesis related to particle interactions is via N-body simulations. Most modern N-body software packages do not support exotic and rarely-studied phenomena such as negative masses. The code is parallelised using DASK in order to make use of the multiple processing cores available in most modern machines.
The N-body code used here evaluates the particle positions and velocities at each timestep by using direct methods, thereby avoiding the introduction of any approximations and maintaining the highest accuracy — albeit at the cost of substantial computing time, of the order of O N 2 per timestep. The primary motivation for this computational perspective is not focussed on performance, but rather on providing easily-understandable, open-source software.
The simulations presented here are therefore necessarily primitive in comparison with the state-of-the-art e. Springel et al. Unless stated otherwise, all of the simulations use a total of 50 particles. Due to computational reasons, no matter creation was included in the current simulations 4. The simulations are presented in Sects. Furthermore, due to the mutually-repulsive nature of negative masses, the formed halo is not cuspy.
This provides a resolution of the cuspy-halo problem e. An initially uniform distribution of both positive and negative masses leads to the conventional filamentary and void-like structures observed in standard large-scale structure simulations.
Moreover, the positive mass component of these simulations naturally becomes surrounded by negative mass material — leading to ubiquitous dark matter haloes for every astrophysical object. In the first set of simulations, a positive mass galaxy is located at the centre of the initial particle distribution. This positive mass galaxy is initialised with spherical-symmetry and following the conventional Hernquist model, with a scale radius equal to 1.
The Hernquist model is used to set the initial particle positions and velocities of the positive masses. The velocities consist of a radial component and are also built hot, with a velocity dispersion provided by a small Gaussian component with a standard deviation of 0.
The initial positive mass particle distribution is located at the centre of a cube of negative masses with volume 3. The initial conditions of these negative masses are set to be uniformly distributed in position and with zero initial velocity.
The simulation is scaled such that the positive mass galaxy has similar properties to the Milky Way, with a characteristic radius of 15 kpc and a mass of 5. The simulation consequently runs over Each side of the box of negative masses has a length of 3 Mpc. Assuming spherical particles, each of the 45 negative masses therefore has an initial descriptive radius equivalent to 52 kpc between each particle.
The resulting particle distribution from this simulation is shown in Fig. The full animated video from this simulation is available online. The negative masses at the outskirts of the cube are mutually-repelled by other surrounding negative masses and the cube begins to expand in volume, as discussed in Sect.
Meanwhile, the negative masses within the central portion of the cube are attracted towards the positive mass galaxy. From their initially zero velocities, the negative mass particles are slushed to-and-fro from either side of the positive mass galaxy. Eventually, the negative mass particles reach dynamic equilibrium in a halo that surrounds the positive mass galaxy and which extends out to several galactic radii.
The negative mass particles have naturally formed a dark matter halo. N-body simulations showing the formation of a non-cuspy dark matter halo from an initially motionless particle distribution of 45 negative masses in purple , that surround a Hernquist-model galaxy of positive masses in yellow.
Both the initial top panel and the final bottom panel time-steps are shown. An animated video from this simulation is available online. Observations of galaxies typically indicate an approximately constant dark matter density in the inner parts of galaxies, while conventional cosmological simulations of positive mass dark matter indicate a steep power-law-like behaviour. This is known as the core—cusp problem or the cuspy halo problem and is currently unsolved de Blok The negative mass halo that has formed in the simulations presented here can be clearly seen in the simulations to have a flat central dark matter distribution.
The typically assumed positive mass particles that are used in conventional simulations are gravitationally attractive and thereby accumulate into a sharp cusp. However, negative mass particles are self-interacting and gravitationally repulsive — thereby yielding a flat inner density profile. To demonstrate this, the extracted density profile from the simulations is shown in Fig. It is worth noting that the precise form of the density profile will be further modified by matter creation.
Nevertheless, the magnitude of the simulated negative mass density profile is shown and compared to both the cuspy Navarro—Frenk—White NFW profile derived from standard N-body simulations with positive mass dark matter e. Navarro et al. The Burkert profile is clearly a much better representation of the simulated dark matter halo than the NFW profile.
A negative mass cosmology can therefore provide a solution to the cuspy halo problem. This appears to be the only cosmological theory within the scientific literature that can explain and predict the distribution of dark matter in galaxies from first principles.
Magnitude of the density profile as a function of radius from the galactic centre, as extracted from the N-body simulations. The density profiles shown are: i as empirically determined for a negative mass cosmology from the N-body simulations in blue , ii for a NFW profile in dark red , and iii for an observationally motivated Burkert profile in light red.
The negative mass density profile is calculated in equally-spaced radial bins, with the measurements indicated by data points and overplotted with a moving average to guide the eye.
The simulated dark matter halo is non-cuspy and best described by the Burkert profile. Beyond the radius of the negative mass halo, the radial profile of the diffuse negative mass background becomes visible. The sharp cut-off to the density profile at the edge of the halo may be further modified by matter creation.
Negative masses can therefore simply reproduce the key features of observed dark matter density profiles in real galaxies. As the halo consists of negative mass particles, this will screen the positive mass galaxy from long-range gravitational interactions. This is similar to the screening effects that are seen in electrical plasmas with positive and negative charges, except this is a gravitational plasma with positive and negative masses.
The negative mass sheath surrounding a typical galaxy effectively begins to shield the positive mass from external gravitational effects. The rotation curves of galaxies have been shown to remain essentially flat out to several tens of kpc. We can therefore attempt to measure the effect that negative masses have on the rotation curves of galaxies.
I use similar initial conditions as in Sect. However, in order to reliably measure the rotation curve, the positive mass galaxy is setup as a kinematically cold system with no velocity dispersion and only a circular, orbital, velocity component. The rotation curve was first measured from a simulation with an initial particle distribution that consisted solely of positive masses in a Hernquist model galaxy.
The resulting rotation curve is indicated by the black line in Fig. The rotation curve for the positive mass galaxy clearly follows a Keplerian curve, with solid body rotation within the scale radius of the galaxy 5 , followed by a steady decline.
The rotation curve was then also measured from another simulation with an identical particle distribution for the positive masses, but now also surrounded by 45 initially uniformly distributed negative masses. The resulting rotation curve is indicated by the red line in Fig. The rotation curve for this positive mass galaxy with a negative mass halo also exhibits solid body rotation within the scale radius of the galaxy, but then appears to slowly increase, remaining essentially flat out to several galactic radii albeit with a slight positive incline.
I emphasise that the only difference between these two simulations is that one contains only positive mass matter, whereas the other contains both positive mass matter and a negative mass halo.
The negative masses have flattened the rotation curve of the galaxy. Sep 28, Recommended for you. First observation of an inhomogeneous electron charge distribution on an atom 14 hours ago. Nuclear radiation used to transmit digital data wirelessly Nov 10, Load comments Let us know if there is a problem with our content.
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