“Dark energy” in the Local Void M. VillataarXiv:1201.3810v1 [astro-ph.CO] 18 Jan 2012 Abstract The unexpected discovery of the accelerated 1 Introduction cosmic expansion in 1998 has ﬁlled the Universe with the embarrassing presence of an unidentiﬁed “dark en- Since the end of the last century, observations of high- ergy”, or cosmological constant, devoid of any phys- redshift type Ia supernovae have unexpectedly shown ical meaning. While this standard cosmology seems that the cosmic expansion is currently in an accel- to work well at the global level, improved knowledge eration phase (e.g. Riess et al. 1998; Perlmutter et al. of the kinematics and other properties of our extra- 1999), whose physical cause is unknown. Formally, this galactic neighborhood indicates the need for a better acceleration is ascribed to an additional term having theory. We investigate whether the recently suggested a negative pressure in the expansion equations, which repulsive-gravity scenario can account for some of the represents about 75% of the total energy density of the features that are unexplained by the standard model. Universe, in the simplest case corresponding to a cos- Through simple dynamical considerations, we ﬁnd that mological constant, perhaps associated to the energy the Local Void could host an amount of antimatter of the quantum vacuum. Besides this standard cos- (∼ 5×1015 M⊙ ) roughly equivalent to the mass of a typ- mology of the ΛCDM model, various alternatives have ical supercluster, thus restoring the matter-antimatter been proposed to explain the cosmic speed-up, invok- symmetry. The antigravity ﬁeld produced by this “dark ing scalar ﬁelds or modiﬁcations of general relativity, repulsor” can explain the anomalous motion of the Lo- such as extensions to extra dimensions or higher-order cal Sheet away from the Local Void, as well as several curvature terms (e.g. Amendola 2000; Dvali et al. 2000; other properties of nearby galaxies that seem to require Carroll et al. 2004; Capozziello et al. 2005). In a vari- void evacuation and structure formation much faster ant of these alternative theories, Villata (2011) proposes than expected from the standard model. At the global to extend general relativity to antimatter, intended as cosmological level, gravitational repulsion from anti- CPT-transformed matter, whose immediate result is matter hidden in voids can provide more than enough the prediction of a gravitational repulsion between mat- potential energy to drive both the cosmic expansion and ter and antimatter, which, with antimatter hidden in its acceleration, with no need for an initial “explosion” cosmic voids, could explain the accelerated expansion. and dark energy. Moreover, the discrete distribution of these dark repulsors, in contrast to the uniformly per- Knowledge of peculiar motions and spatial distribu- meating dark energy, can also explain dark ﬂows and tion of galaxies allows us to explore gravitational inter- other recently observed excessive inhomogeneities and actions within and among clusters. This is particularly anisotropies of the Universe. true for our extragalactic neighborhood, where dis- tances can be measured with higher precision. Through Keywords Gravitation — Cosmology: theory — Dark a study performed on a database of about 1800 galaxies energy — Large-scale structure of Universe within 3000 km s−1, Tully et al. (2008) ﬁnd that the pe- culiar velocity of the Local Sheet decomposes into three M. Villata dominant, almost orthogonal components. One of them INAF, Osservatorio Astronomico di Torino, Via Osservatorio 20, (of 185 km s−1) is ascribed to the gravitational pull of I-10025 Pino Torinese (TO), Italy the Virgo Cluster and its surroundings, while a second, e-mail: firstname.lastname@example.org larger component (of 455 km s−1) seems to be due to an attraction on larger scales nearly in the direction of the
2Centaurus Cluster. The third component (259 km s−1) Ni 2004; Hajdukovic 2010, 2011c,b; see also Cabboletis not directed toward anything prominent, but points 2010), or that it is even self-repulsive (e.g. Noyes 2008;away from the Local Void. Some evacuation of voids is Benoit-L´vy & Chardin 2012). eexpected in standard cosmology, due to the density con- In a recent paper, Villata (2011) argued that theretrast that pushes matter outward, but the high speeds is no need to change the sign of the gravitational massof galaxies at the edge of the Local Void indicate that it of antimatter (which would represent a violation of theshould be very large and very empty, more than usually weak equivalence principle) to get repulsion betweenexpected. Consequently, alternative models have been matter and antimatter; but he showed that antigravityconsidered. For instance, Dutta & Maor (2007) argue appears as a prediction of general relativity, once it isthat dark energy with a varying equation of state could assumed that this theory is CPT invariant and that,have higher density where mass density is lower, so that consequently, matter is transformed into antimatter byrepulsion from voids would be stronger. these three joint operations (charge conjugation, parity, Moreover, as discussed by Peebles & Nusser (2010), and time reversal).many other properties of galaxies in our neighborhood Besides representing the universal symmetry of ﬁeldare not explained by the standard model. Besides the theories, the CPT transformation is consistent with theobserved extreme emptiness of the Local Void (∼ 10−5 Feynman-Stueckelberg interpretation of antiparticles asprobability), the presence of large, luminous galaxies particles traveling backwards in time, since this viewin the adjacent low-density regions is in contrast with explains the need of applying T (and P) in additionexpectations (< 1% probability). These and other ob- to C to describe the behavior of antimatter. In otherserved features seem to indicate the need for a better words, a physical system (or a component of a physi-theory, entailing a mechanism by which matter is more cal system) traveling in the opposite time direction israpidly repelled by voids and gathered into galaxies and observed from our time direction through a CPT trans-structures. formation, and this would be the reason why physical In this paper we want to investigate whether the re- laws are CPT symmetric, since they must describe thepulsive gravity scenario proposed by Villata (2011) as a behavior of physical systems as observed from eitherpotential explanation for the cosmic acceleration could time directions, i.e. when they appear to be composedalso help solve some of these problems. A brief intro- of either matter or antimatter, or both.duction to antimatter gravity is given in Sect. 2, while In general relativity, the equation of motion for ain Sect. 3 this view is applied to the dynamics of our matter test particle in a matter-generated gravitationalextragalactic surroundings. Discussion and conclusions ﬁeld iscan be found in Sect. 4. d2 xλ dxµ dxν 2 = −Γλ µν . (1) dτ dτ dτ2 Antimatter gravity If we CPT-transform all the four elements in Eq. (1), we get an identical equation describing the motion ofThe existence of antigravity, i.e. gravitational repulsion an antimatter test particle in an antimatter-generatedbetween matter and antimatter, has been debated at gravity ﬁeld, since all the four changes of sign cancel onelength during the last several decades, without reach- another. Thus, this CPT symmetry ensures the sameing any ﬁrm conclusion. After strong opposition dur- self-attractive gravitational behavior for both mattering the second half of last century, the idea of anti- and antimatter. But, if we transform only one of thegravity has experienced a period of renewed interest two components, either the ﬁeld Γλ or the particle µνbecause of the discovery of the accelerated expansion (represented by the remaining three elements), we getof the Universe in 1998. In particular, against anti- a change of sign that converts the original gravitationalgravity in the Fifties–Sixties were proposed some the- attraction into repulsion, so that matter and antimatteroretical arguments that seemed to rule out its exis- repel each other.tence (Morrison 1958; Schiﬀ 1958, 1959; Good 1961), The equation for a massless particle (e.g. a photon)but which were later criticized and questioned (e.g. is formally equal to Eq. (1), except for the parameterNieto & Goldman 1991; Chardin & Rax 1992; Chardin τ , which can no longer be taken as the proper time,1993, 1997; Hajdukovic 2011a), and then lost much being dτ = 0, but it will be another parameter describ-of their eﬀectiveness. At the same time, several au- ing the world line. Therefore, a (retarded) photon willthors have pursued the idea of antigravity, either assum- be repelled by an antimatter gravity ﬁeld, as well asing that antimatter has negative gravitational mass, a CPT-transformed photon, i.e. an advanced photon,and thus is self-attractive (e.g. Noyes & Starson 1991;
3will be repelled by matter1 . This may provide a test which can be decomposed into three quasi-orthogonalfor the theory of antigravity: the presence of antimat- components (Tully et al. 2008). One of them is dueter in cosmic voids suggested by Villata (2011, 2012) to the well-known attraction towards the Virgo Clus-might be revealed by its gravitational eﬀect on the ra- ter and its dense surroundings (e.g. Tonry & Davisdiation coming from background sources, in a sort of 1981; Aaronson et al. 1982; Hoﬀman & Salpeter 1982;antigravitational lensing. Tully & Shaya 1984; Tonry et al. 2000). The sec- In the Newtonian approximation, the law of gravity ond, largest component has a less clear origin: it isextended to antimatter becomes F (r) = ∓GmM/r2 , ascribed to gravitational pull on scales larger thanwhere the minus sign refers to the gravitational self- 3000 km s−1 nearly in the direction of the Centau-attraction of both matter and antimatter, while the rus Cluster, but the identiﬁcation and relative impor-plus sign indicates the gravitational repulsion between tance of the “great attractors” is still debated (e.g.matter and antimatter. Shaya 1984; Lynden-Bell et al. 1988; Scaramella et al. The change of sign in the CPT-transformed (i.e. 1989; Raychaudhury 1989; Kocevski & Ebeling 2006;antimatter-generated) gravitational ﬁeld comes from Erdo˘du et al. 2006). gthe inversion of the derivatives of the metric tensor The last component corresponds to what is his-in the expression of Γλ . Thus, while the metric ten- µν torically known as the “local velocity anomaly” (e.g.sor, i.e. the potential, does not change sign and re- Faber & Burstein 1988; Tully 1988; Tully et al. 1992).mains the same for matter and antimatter, the sign It is large, 259 km s−1, and is anomalous because is notchange appears only in the ﬁeld. This means that in directed towards any important structure. It can notthe above generalized Newton law, the plus sign indi- be due to a simple gravitational pull, otherwise also thecating repulsion does not formally come from an in- Leo Spur galaxies, which would be in between, shouldverted potential, but from its inverted gradient. In- be aﬀected by it, while they appear to be at rest withdeed, the potential φ is usually deﬁned as −∇φ = g, respect to this motion. Thus, the cause of this compo-so that, being the attractive ﬁeld g(r) = −GM/r2 , nent must be on the other side, where the Local Sheetthe potential is φ(r) = −GM/r, and, with the same borders the Local Void. Moreover, this velocity vec-potential, the repulsive ﬁeld comes from its inverted tor points just in the opposite direction with respect togradient: g (r) = GM/r2 . This formally correct view ¯ an approximate center of the void, and is orthogonal tois actually not intuitive, so that it is preferable to re- the disk-like structure of the Local Sheet. Whatever theverse the potential and to get the ﬁeld according to the cause of the anomalous motion, it must aﬀect the Localusual −∇φ deﬁnition. The repulsive potential is thus Sheet more than the rest of our neighborhood (which¯φ(r) = GM/r, and the gravitational picture of the cos- is indeed more distant from the void), and also seemsmic alternance of matter and antimatter suggested in to have the property to squash the closest structures,Villata (2011) can be represented as an alternation of besides giving them high velocities.potential wells and hills. Of course, from the antimat- These and several other features (see Sects. 1 andter point of view this situation would be seen upside 4), such as the extreme emptiness of the Local Void,down. do not ﬁnd a satisfactory explanation within the stan- dard model, and seem to require the intervention of a repulsive force. In particular, the ﬂattened shape of3 Dynamics of the Local Sheet the Local Sheet looks just like the signature of a repul- sive gravitational ﬁeld. Indeed, while the eﬀect of tidalOur peculiar motion with respect to the general ex- forces in an attractive gravity ﬁeld is to stretch struc-pansion of the Universe can be inferred by the ob- tures along the radial direction, since the side closer toserved dipole anisotropy in the cosmic microwave back- the ﬁeld source is more attracted than the side fartherground (e.g. Fixsen et al. 1996). This motion has well- away, the tidal eﬀect of a repulsive ﬁeld is to squashknown local contributions, including the orbital veloc- them, the near side being more pushed than the farity of the Sun in our Galaxy and the attraction of one: dF/dr = −2GmM/r3 .the latter towards M31. Once these components are Under some simple assumptions in linear perturba-taken into account, it is found that the Local Sheet tion theory, the peculiar velocity ﬁeld Vpec of an idealhas still a very large peculiar velocity of 631 km s−1, pressureless ﬂuid is related to the gravitational acceler- ation g and the expansion time t by the simple formula1 As a consequence, the energy of a retarded-advanced photon (see e.g. Peebles 1993; Peebles et al. 2011) Vpec = g t.pair in a gravitational ﬁeld would be conserved, thus invalidating We can check whether this equation can represent a rea-the Morrison (1958) argument against antigravity. sonable approximation for the peculiar velocity of the
4Local Sheet with respect to the gravitational attraction the total energy per unit mass, given by V 2 /2 + GM/dby the Virgo Cluster. Rewriting it as (where V = H0 d + Vpec is the total velocity with re- spect to the void center) at the Local Sheet and Leo- Vpec M Mpc2 t = 4.4 × 10−12 , (2) Spur/Virgo distances. In the intermediate case seen km s −1 M⊙ d2 Gyr above, with MLV = 6.6 × 1015 M⊙ , dLS = 25 Mpc LSwith t = t0 = 13.75 Gyr (Jarosik et al. 2011), d = and Vpec = 634 km s−1 for the Local Sheet, dLeo =17 Mpc and Vpec = 185 km s−1 (Tully et al. 2008), we 32.5 Mpc and Vpec = 375 km s−1 for Leo-Spur/Virgo2, Leoget for the mass of the Virgo Cluster and surroundings and adopting H0 = 74 km s−1 Mpc−1 from Tully et al.MVir = 8.8 × 1014 M⊙ . This value is in good agree- (2008), we obtain 4.2×106 and 4.7×106 km2 s−2 for thement with previous estimates (e.g. Tully & Shaya 1984; respective total speciﬁc energies. We then ﬁnd a smallMohayaee & Tully 2005), so that we may be conﬁdent diﬀerence of about 10%, which could also be due toin Eq. (2), at least at the scale of our supercluster. the various uncertainties aﬀecting both measures and Assuming that Eq. (2) holds also for repulsive accel- model assumptions, or could come from slightly diﬀer-eration, we can try a rough estimate of the mass of anti- ent initial conditions for the diﬀerently located systems.matter (see Sect. 2), possibly located around the center This approximate energy conservation at diﬀerent dis-of the Local Void, needed to push the Local Sheet at tances from the antigravity center conﬁrms that thethe observed velocity of 259 km s−1. The expected dis- dynamics in the radial direction is actually dominatedtance of the Local Sheet from the center of the void may by this repulsive interaction, and supports our scenario.be assumed in the range 25 ± 5 Mpc (e.g. Tully et al. If we go back in time conserving mechanical energy,2008), which gives MLV = 2.7+1.2 × 1015 M⊙ . How- −1.0 we ﬁnd V = 0 at d = 6.7 Mpc for the Local Sheet, andever, it would be strange that this repulsive interaction at d = 6.0 Mpc for Leo Spur, whereas one could expectaﬀected only the Local Sheet, leaving the rest of our much smaller distances, approaching a singularity. Thisneighborhood unaﬀected and at rest with respect to might mean that our data are too imprecise to allow athe Local Void. Indeed, what is measured as a peculiar similar extrapolation in the past, or that our simplisticvelocity of 259 km s−1 is actually the peculiar velocity dynamical assumption of a dominant, point-like-sourceof the Local Sheet relatively to the rest of the Local ﬁeld can no longer hold at these earlier stages and alsoSupercluster, i.e. mainly to the Virgo Cluster, and in surrounding ﬁelds should be taken into account, or thatparticular to the Leo Spur galaxies, which are both far- some of the energy has been somehow dissipated, or ather than the Local Sheet from the approximate cen- combination of all these things.ter of the Local Void by about 7–8 Mpc. Thus, for In any case, what we are ﬁnding throughout this sec-a more signiﬁcant estimate of the mass in the Local tion is that the repulsive gravity ﬁeld can account notVoid, we take the diﬀerence of Eq. (2) between the dis- only for peculiar (anomalous) velocities and accelera-tance of the Local Sheet (again assumed as 25 ± 5 Mpc) tion of the Hubble ﬂow, but even for the Hubble ﬂowand the Leo-Spur/Virgo radial distance from the void itself. There seems to be more than enough energy forcenter (correspondingly estimated as 32.5 ± 4.5 Mpc), driving all the Universe expansion, stored in the poten-which yields MLV = 6.6+4.6 × 1015 M⊙ , i.e. signiﬁcantly −3.1 tial energy, when, going back in time, matter get closerlarger than the former result, but of the same order of and closer to the peaks of the antimatter potential hills.magnitude. These estimates come from the simplistic No need for initial (artiﬁcial) velocities provided by anassumption that the various components are aﬀected, explosive “Big Bang”, whose energy would not havein the radial direction from the void center, only by the any identiﬁed origin. No need for unknown dark en-dominant repulsive interaction with an antimatter mass ergy to drive the cosmic acceleration, or dark energy isconcentrated close to the assumed void center, and dis- nothing else than the potential energy of gravitationalregarding other possible contributions in the same di- repulsion.rection. Thus, we are not claiming that these valuesmust be considered as extremely realistic, and we cannot even choose between the two results, but we can 4 Discussion and conclusionssay that an antimatter mass of ∼ 5 × 1015 M⊙ , withina factor of 2, can easily account for the local velocity There remains the question of why antimatter in voidsanomaly. This value is comfortably comparable to the should not be visible. It seems that something “dark”mass of a medium-size supercluster, which means that must necessarily exist: dark matter, dark energy, andwe would have found the location for antimatter in amatter-antimatter symmetric Universe. 2 V LS Leo Another argument in favor of the above picture and pecand Vpec are calculated from Eq. (2), and have the correctmass estimate is the following one. We can calculate diﬀerence of 259 km s−1 between them.
5now “dark repulsors”. On the other hand, were it oth- of the expanding Universe4 , but at the cost of intro-erwise, we would have understood everything a long ducing ad hoc elements that do not have a physicaltime ago. explanation, like the initial “explosion” causing the cos- Like matter, antimatter is self-attractive, so we can mic expansion and the dark energy that should accel-expect that it forms anti-galaxies and anti-stars, which erate it. As we have seen, antigravity can explain bothwould emit electromagnetic radiation, we should can the expansion and its acceleration, and this theory alsodetect. However, as pointed out by Villata (2012, has the virtue of solving the long-standing problemsee also Sect. 2), antimatter, if emitting, should emit of matter-antimatter asymmetry. It might seem thatadvanced radiation, which can be undetectable for at one is replacing something invisible (dark energy) withleast two reasons. First, it could be intrinsically unde- something else equally invisible (dark repulsors). Thetectable, at least with the usual detectors and observing diﬀerence is that the presence of the latter is theoreti-procedures3 . Then, even if it were somehow detectable, cally expected if not required, whilst the presence of thewe know from Sect. 2 that advanced photons are re- former is not requested, unexpected, and embarrassing.pelled by matter, so that the chance of detecting the Moreover, some recent observational ﬁndings are un-few that possibly succeeded in reaching us on top of the dermining the standard model (e.g. Walker & Pe˜ arrubia npotential hill of our Galaxy is further greatly reduced. 2011 on unexpectedly uniform distribution of dark mat-The possibility remains that antimatter emits also re- ter in dwarf galaxies5 ). In particular, the existencetarded radiation (as well as matter might emit advanced of “dark ﬂows”, i.e. large-scale, peculiar bulk motionsradiation). In this case we should see antimatter stars, of galaxy clusters (e.g. Kashlinsky et al. 2010 and ref-galaxies and clusters ﬁlling the voids, but well sepa- erences therein), of excess clustering on large scalesrated from matter ones by gravitational repulsion. Yet, (Thomas et al. 2011), of anisotropy in the expansionas already said, antimatter would be matter traveling acceleration (e.g. Cai & Tuo 2012) seem to indicate thebackwards in time, so that, taking this concept to the need for a distribution of acceleration sources less ho-extreme, we would observe its cosmic evolution in re- mogeneous than the uniformly permeating dark energy,verse, and its current evolutionary stage could be very thus favoring the discrete distribution in cosmic voids.diﬀerent from ours. For example, if antimatter were Let us return to our extragalactic neighborhood and“created” in a distant future, now it might be collapsed its anomalies partly mentioned in the Introduction andinto supermassive black holes, or be in some other non- described in detail in Peebles & Nusser (2010). Allemitting form, before deﬁnitively collapsing, 14 billion those observed properties, such as the extreme empti-years ago. Perhaps to be “reborn” as matter, and re- ness of the Local Void, the unexpected presence of largetrace in the opposite direction the time already passed galaxies at its edge, the existence of pure disk galaxies,through, but well separated from the “anti-itself” by and the insensitivity of galaxy properties to environ-gravitational repulsion. A further step would lead us ments, indicate the existence of a mechanism that canto suggest a Universe “recurring over time”, in a cos- produce a fast evacuation of voids and rapid assemblymic loop, but such a speculative rumination is beyond of matter into structures, which the standard theorythe scope of this paper. based on attractive-only gravity can not provide. Re- In any case, there seem to be more reasons for anti- pulsive gravity can be the cause of fast evacuation and,matter invisibility than for visibility, so that we are not also through the squashing tidal eﬀect, can help a lotsurprised not to see anything in cosmic voids. Another to rapidly gather matter into galaxies and structures,point is the diﬀerent, quasi-spherical symmetry of voids especially at the borders of voids.with respect to the squashed/ﬁlamentary shape of mat- In conclusion, repulsive gravity between matter andter structures. This would support the view of diﬀerent antimatter, which is a natural outcome of general rela-evolutionary stages, where antimatter black holes close tivity extended to CPT-transformed systems, is an ex-to the void centers produce spherical ﬁelds, and the cellent candidate to explain not only the cosmic expan-more uniformly distributed matter can only adapt in sion and its acceleration, but also the anomalous kine-the potential valleys among the quasi-circular potential matics and several other properties of our extragalactichills. surroundings, as well as other observed features of the Passing several cosmological tests, the ΛCDM stan-dard model seems to provide a fairly good description 4 However,in a recent paper, Benoit-L´vy & Chardin (2012) have e shown that most of these observational constraints can also be3 Actually there is some confusion about advanced radiation in the explained by their quite diﬀerent cosmology, based on the as-literature; for discussions around the issue of advanced radiation sumption of a matter-antimatter symmetric Universe.in general, see e.g. Davies (1975), Cramer (1980), and references 5 See Hajdukovic (2011c) for an alternative interpretation of darktherein. matter based on antigravity.
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