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Home , Cold dark matter, Dark matter halo, The Quantum Vacuum

Quantum vacuum and dark

Dragan Slavkov Hajdukovic1 PH Division CERN CH-1211 Geneva 23 [email protected] 1On leave from Cetinje, Montenegro

Abstract Recently, the gravitational polarization of the quantum vacuum was proposed as alternative to the paradigm. In the present paper we consider four benchmark measurements: the universality of the central surface density of dark matter haloes, the cored dark matter haloes in dwarf spheroidal , the non-existence of dark disks in spiral galaxies and distribution of dark matter after collision of clusters of galaxies (the Bullet cluster is a famous example). Only some of these phenomena (but not all of them) can (in principle) be explained by the dark matter and the theories of modified . However, we argue that the framework of the gravitational polarization of the quantum vacuum allows the understanding of the totality of these phenomena.

(modification of the fundamental law of gravity 1. Introduction and the assumption that in addition to and Contemporary has two cornerstones: there are still unknown fundamental and the of named dark particles) have been Physics. General Relativity is our best studied by thousands of scientists, but a solution theory of gravitation. The Standard Model is a is still not at hand. collection of Quantum Field Theories; according Recently (Hajdukovic, 2011; but see also the to the Standard Model, everything in the first appearance of the idea in Hajdukovic, 2007 is made from six quarks and six and Hajdukovic, 2008)) a third way, without leptons (and their ) which interact invoking dark matter and without invoking the through exchange of gauge ( for modification of the fundamental law of gravity, electromagnetic interactions, W ± and Z 0 for has been proposed. In simple words, according weak interactions and eight for strong to the , all baryonic interactions). matter in the Universe is immersed in quantum The problem is that our best physics is vacuum; popularly speaking a “sea” of short apparently insufficient to explain a series of living - pairs (like major phenomena discovered in Astrophysics - pairs with the lifetime of about and Cosmology. One of the unexplained 10−22 s , or -antineutrino pairs with a phenomena is that the gravitational field in the lifetime of about 10−15 s which is a record Universe is much stronger than it should be lifetime in the quantum vacuum). It is difficult to according to our theory of gravity and the believe that quantum vacuum does not interact existing amount of the baryonic matter (i.e. the gravitationally with the baryonic matter matter composed from the Standard Model immersed in it. In spite of it, the quantum particles). This phenomenon is considered as a vacuum is ignored in astrophysics and strong hint that at least one of cornerstones cosmology; not because we are not aware of its (General Relativity and Standard Model) must importance but because no one has any idea be significantly modified. Both approaches what the gravitational properties of the quantum

1 vacuum are. In absence of any knowledge, as a (a) Central surface density starting point, we have conjectured that particles There is strong evidence (Donato et al. 2009) and antiparticles have the gravitational charge of that the central surface density µ0D ≡ r0 ρ0 of opposite sign. An immediate consequence is the galaxy dark matter haloes (where r and ρ are existence of the gravitational dipoles; a virtual 0 0 pair is a gravitational dipole (in the same way as the halo core radius and central density) is nearly a virtual electron-positron pair is an electric constant and independent of galaxy luminosity. dipole), that allows the gravitational polarization The measured value (Donato et al. 2009) is of the quantum vacuum. The initial study about 140 solar masses per square parsec

(Hajdukovic, 2011) has revealed the surprising −80 M Sun kg µ ≡ r ρ =140 − = 0.29 (1) possibility that the gravitational polarization of 0D 0 0 30 pc 2 m 2 the quantum vacuum can produce phenomena The universality of the dark matter surface usually attributed to dark matter. In the present density at the core radius is a mystery for the paper we focus on four benchmark phenomena particle dark matter but can be explained within established by observations: (a) the universality the MOND phenomenology (Milgrom, 2009). of the central surface density of galaxy dark As we will see, the gravitational polarization of matter haloes (Donato et al. 2009), (b) the cored the quantum vacuum obviously leads to a dark matter haloes in dwarf spheroidal galaxies relation producing the numerical result (1). (Walker and Penarrubia, 2011), (c) the non- existence of dark disks in spiral galaxies (Moni (b) Dwarf spheroidal galaxies Bidin et al. 2010) and (d) the distribution of Dwarf spheroidal galaxies, with a typical dark matter after collisions of clusters of diameter of about 1000 light years, are the galaxies (the Bullet cluster (Clove et al. 2006) smallest galaxies observed in the Universe. For a being a famous example). In section 2 we give a number of reasons they are considered as an brief review of these four phenomena and point important “laboratory” for the study of dark to the known fact that only some of them (but matter distribution at the centres of galaxies. not all of them) can in principle be explained by Recently, Walker and Penarrubia (2011) have the dark matter and the modified theories of accomplished the first direct measurements that gravity. In section 3 we consider the same reveal how densely dark matter is packed toward phenomena in the framework of the gravitational the centres of two nearby dwarf galaxies (Fornax polarization of the quantum vacuum and argue and Sculptor) that orbit the as that it is the framework in which the totality of satellites. these phenomena can be understood. Section 4 is The measured slope devoted to discussion. ∆log M Γ ≡ (2) ∆logr 2. Four important measurements is Γ ≈ 2.61 and Γ ≈ 2.95 respectively for Let us give a brief review of four observed Fornax and Sculptor galaxy. The values of Γ in phenomena which have become benchmark for the range 2 < Γ < 3 , are consistent with cored different theories. Both, the dark matter halos of an approximately constant model and MOND fail to explain the totality of density over the central few hundred parsecs, these phenomena. The dark matter theory has what contradicts the cusp distribution ( Γ < 2 ) more problems at small scales, while modified predicted by the current cold dark matter theory. gravity (we take MOND as leading example) has Hence, Walker and Penarrubia have provided significant problems at large scales. the first direct evidence that the cold dark matter

2 paradigm cannot account for the phenomenology it behaves in the same way as the collisionless of dark matter at small scales. part of the baryonic matter. (c) Dark disks 3. Gravitational polarization of the Everyone knows that our Galaxy is immersed quantum vacuum in a halo of dark matter (a real one if we trust the 3.1 Basic ideas cold dark matter theory or a phantom halo Let us assume that particles and antiparticles according to theories of modified gravity like have the gravitational charge of the opposite MOND). It is less known that in addition to the sign. Consequently, a virtual particle-antiparticle halo, our galaxy should have a dark matter disk, pair may be considered as a gravitational dipole which is thicker than the visible galactic disk. with the gravitational dipole moment The presence of a real dark disk is a natural     expectation of the cold dark matter model (Read p = md; p < (3) c et al. 2008) while the presence of a phantom disk  (Milgrom, 2001) is a prediction of MOND Here, by definition, the vector d is directed from theory. The observations suggest (Moni Bidin et the antiparticle to the particle, and presents the al. 2010) that at this point both theories are distance between them. The inequality in (3) wrong; apparently, dark matter disk does not follows from the fact that the distance between exist. As we will show in Section 3, the non- virtual particle and antiparticle must be smaller existence of dark matter disk is a natural than the reduced Compton wavelength consequence of the gravitational polarization of  m =  mc (for larger separations a virtual pair  the quantum vacuum. becomes real). Hence, p should be a fraction (d) The Bullet cluster of  c . The observations of the Bullet cluster show the If the quantum vacuum “contains” the virtual distribution of the baryonic and dark matter after gravitational dipoles, the gravitational field of a collision of two clusters of galaxies. body immersed in the quantum vacuum, should During the collision, the galaxies within the produce , characterized with two clusters passed by each other without  interactions (because of the large distances a gravitational polarization density Pg (i.e. the between them), while the interacting clouds of gravitational dipole moment per unit volume). X-ray emitting plasma have been slowed by ram In the quantum field theory, a virtual particle- pressure. Hence, two clouds of plasma are now antiparticle pair (i.e. a gravitational dipole) located between the two separated clusters. The 3 occupies the volume λm , where λm is the (non- key point is that the distribution of dark matter reduced) Compton wavelength. As argued in (determined by the gravitational lensing) is previous papers (Hajdukovic 2010, Hajdukovic centred on clusters, while the dominant part of 2011) the (as the simplest -antiquark baryonic matter is in clouds of plasma. Such a pairs) dominate the quantum vacuum and common “destiny” of dark matter and stellar λ should be identified with the Compton components of clusters can’t be explained by m

modified gravity where dark matter should be wavelength λπ of a . Hence, the number centred on the dominant part of the baryonic density of the virtual gravitational dipoles has a matter (i.e. on clouds of plasma). However, in constant value the framework of the cold dark matter theory, 1 ∝ (4) dark matter is collisionless and it is natural that N0 3 λπ

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According to equations (3) and (4), if all dipoles As previously suggested (Hajdukovic, 2011), are aligned in the same direction, the dark matter density may be interpreted as the  density of the gravitational polarization charges. gravitational polarization density Pg has the  ρ = −∇⋅ maximal magnitude dm Pg (9)  A  If we assume the spherical symmetry, (9) may ≡ = (5) Pg Pg max 3 be reduced to λπ c 1 d 2 where A <1, should be a dimensionless ρ (r) = (r P (r)) (10) dm 2 dr g constant of order of unity. This may happen only  r in a sufficiently strong gravitational field with with Pg (r) ≡ Pg (r) .

magnitude g , larger than a critical value g cr . Let us note that from the purely mathematical point of view there are three interesting The critical field g cr should have the same possibilities: P (r) is directly proportional order of magnitude (Hajdukovic, 2011) as the g gravitational acceleration produced by a pion at to r , Pg (r) = const and Pg (r) is inversely the distance of its own Compton wavelength proportional to r . In these particular cases, the Gmπ −10 2 equation (10) leads respectively to the constant = = × g cr B 2 2.1B 10 m / s (6) λπ volume density, constant surface density and where B is a dimensionless constant of order of constant radial density of dark matter, i.e. dM unity. The numerical value of g cr is surprisingly dm Pg (r) ∝ r ⇒ = C1 (11) close to the fundamental acceleration dV dM dm a0 conjectured by MOND; in fact g cr = a0 P (r) = const ⇒ = C (12) g dS 2 implies B ≈ 0.58 ≈1 3 and we will adopt this 1 dM dm P (r) ∝ ⇒ = C (13) value for B in numerical calculations. The fact g r dr 3 that a universal critical gravitational field gcr where C1, C2 and C3 are some constants. Let us appears in our theory is only a superficial note that we continue to use the words dark similarity with MOND; in our approach there is matter, while it is not more the dark matter of no modification of the fundamental law of unknown , but the effect of the rearrangement of the virtual gravitational gravity for g < g cr . charges in the quantum vacuum. The equations (5) and (6), together with the The mathematical possibilities (11), (12) and proportionality (13) can approximate the real physical 1 situations. Before showing it, let us remember Pg max = g cr (7) 4πG that in electrodynamics, the polarization density lead to 2A = B , i.e. is a function of the electric field (in some cases a 1 1 linear function and in some cases a non-linear A ≈ 0.29 ≈ ; B ≈ 0.58 ≈ (8) 2 3 3 function). In the case of a hypothetical π gravitational polarization, the polarization Let us note that 1 4 G plays the role of the gravitational vacuum permittivity, analogous to density should be a function of the strength of the gravitational field. the vacuum permittivityε in electrodynamics). 0 The first important phenomenon is saturation; in a gravitational field stronger than the critical one, the magnitude of the polarization density

4 has the constant value determined by the situation may be approximated with (11), but equation (5). This physical situation is later we will use a more general dependence of mathematically described by (12). The examples the form r x with x ≤1being a positive number. of a region with saturation are: the central part of In principle, every baryonic body (a , a our galaxy, the central part of a globular cluster , a complex system as a galaxy, or even a and a relatively large region around a star (for single particle such as an electron) can cause instance, according to (6), our Sun produces gravitational polarization, i.e. the rearrangement saturation in a region larger than the solar of the virtual gravitational charges in the system). By the way, let us note that the surrounding quantum vacuum. This is possible saturation in the central region is not a universal only in the region in which the gravitational field property of all galaxies; for instance in the of the body is stronger than the gravitational central part of a , the field produced by other sources, what puts a gravitational field is not sufficiently strong to natural limit to the spatial extent of the dark produce saturation. matter halo of a body. If the distance of a body If the gravitational field is weaker than the from other bodies increases, the size of its dark critical one, the polarization density should matter halo should increase as well, leading to a increase when the field increases and decrease greater quotient of dark matter and baryonic when the field decreases; consequently the matter. The equation (14) obtained in the equation (12) cannot be used. previous paper (Hajdukovic, 2011) supports this Outside of a distribution of the baryonic matter, intuitive picture. the gravitational field decreases with distance; As an example of the baryonic distribution for instance it is the case outside of the saturated without spherical symmetry, let us consider a region in our galaxy. This case, corresponding to planar-like distribution (like for instance a thin (13), was already studied (Hajdukovic, 2011), galactic disk). From the mathematical point of leading to the main result: view, the simplest case is an infinite plane with a

dM dm (r) B constant baryonic surface mass densityσ b (this = mπ M b (14) dr λπ is the gravitational version of an infinite plane describing a outside of a with constant electric charge density, what is an spherically symmetric distribution of the exercise known to every student of physics). The  baryonic mass M b ; a result that mimics well gravitational field g produced by the plane is the observed galactic dark matter halo at perpendicular to the plane, oriented towards the relatively large distances from the center of the plane and has a constant magnitude which can galaxy. be determined by a trivial application of the Inside a distribution of the baryonic matter, the Gauss’s flux theorem gravitational field may increase with the distance  g = 2πGσ b (15) from the center; the simplest example is the  In a constant gravitational field g the gravitational field of the Earth or the Sun, which  increases from the centre to the surface and gravitational polarization density Pg should be a decreases after that. In the particular case of constant vector and its divergence (i.e. the right- baryonic distribution with a constant volume hand side of the equation (9)) is zero. Hence, density, the gravitational field inside the while the vacuum around the considered plane is distribution is directly proportional to the polarized, dark matter density is zero. distance r from the center. This physical Consequently, close to a large plane or between

5 two large planes, there is no significant value of A determined in (8)) a numerical value gravitational field caused by the gravitational in the excellent agreement with the polarization of the quantum vacuum. By the measurements. Alternatively we may consider way, it leads to the conclusion that the baryonic the measurement (1) as the experimental galactic disk of our galaxy can’t be accompanied determination of the constant A in equations (5) by a thicker dark matter galactic disk, what and (16). contradicts the common prediction of the cold Let us forget for the moment how we have dark matter theory (Read et al. 2008) and obtained the result (16). Even if considered in MOND (Milgrom, 2009) . Recent studies (Bidin isolation, as an ad hoc formula, it is astonishing et al., 2010) show that there is no evidence for a that a universal quantity as (1) can be expressed dark matter disk within 4 kpc from the galactic through universal constants and mass of a quark- plane, which apparently confirm our prediction. antiquark pair (what is roughly a pion). The above considerations suggest that we may According to (16) the mass of dark matter live in a Universe with a variable quotient of the enclosed inside a sphere with radius r is baryonic and dark matter. To see it, let us 2  r  imagine, that a spherical distribution of baryonic =   M dm (r) Bmπ   (17) matter is somehow “deformed” to a planar-like  λπ  distribution. In these two cases, a distinct while the acceleration produced by the dark observer would measure the same quantities of matter has a constant value equal to the critical baryonic matter, but different quantities of dark acceleration. matter! GM dm (r) gdm (r) = 2 ≡ gcr (18) 3.2 Gravitational field stronger than the r critical value So, in the region g > g cr , the total acceleration Let us turn back to the case of spherical at distance r is the sum of the acceleration symmetry. In general, there are two regions g b (r) caused by the baryonic matter (and outside a distribution of the baryonic matter; the described by the Newton law), and a very small, region with g ≥ g cr and the region with g < g cr . constant acceleration (18) caused by the dark matter and oriented towards the center of the The region with g ≥ g cr is the easiest for the study; we have the estimate (5) for the maximal spherical symmetry. In the region g < g cr , magnitude of the gravitational polarization g dm (r) is not more a constant, but depends density and we can use it in the equation (10), on r what can be wrongly interpreted as a without need for a detailed understanding of the modification of the Newton law (a mistake quantum vacuum, what is the major problem in included as cornerstone of the MOND the case g < g cr . It is evident that the phenomenology). mathematical case (12) corresponds to the By the way, the additional constant sunward physical case when the gravitational field is acceleration (18) should exist in the Solar sufficiently strong to produce saturation. From system and affect the orbital motions of the (5) and (10) it is easy to obtain the relation Solar system’s bodies, but in order to detect it,

2A  A mπ we must know orbits with higher accuracy ρ = = ≡ (16) r dm (r) 2Pg max 3 2 (which may be not so far into the future; Page et λπ c π λπ al. 2009) which explains the observed universality (1) of the central surface density and gives (using the

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3.3 Gravitational field weaker than the critical that Pg (r) is also proportional to r , what value corresponds to the mathematical case (11), For bodies like a star or our planet, the describing a cored dark matter halo. gravitational field becomes stronger than the However, at this point the problem is that we critical one in less than one meter from the do not know the properties of the quantum center of the body. Hence, the gravitational field vacuum and in particular we do not know if around a star has an inner region with g > g cr , for g < g cr , the magnitude of polarization < and an outer region with g g cr . The region density grows with the acceleration in a linear or

g > g cr should be called the region of saturation non-linear manner. because the polarization density has a maximal To be more general, let us assume a non-linear magnitude. The same should be true for a growth of the polarization density x Galaxy with a supermassive in the Pg (r) = Kr (21) center. For instance, the where K is a constant and x ≤1 a positive in the centre of the Milky Way assures condition number. Using this form for Pg (r) in the basic g > g at distances of more than 100 light cr equation (10) and after that using the obtained years (without counting other baryonic matter in result to calculate the slope (2), leads the central region). to Γ = 2 + x , i.e. 2 < Γ ≤ 3 , as observed for The other possibility is the existence of a large dwarf spheroidal galaxies (Walker and < central region with g g cr . It is possible if there Panarrubia, 2011) is a sufficiently low baryonic mass density in the 3.4 The Bullet cluster central part of a galaxy. Because of the mathematical complexity, the Let us consider a sphere filled with the numerical simulations are inevitable and crucial baryonic matter of the volume density in our present day studies of dark matter. A ρ b (r) which depends only on the distance r simulation of the Bullet cluster (and some other from the centre. The gravitational acceleration problems) in the framework of the gravitational produced by the baryonic matter is polarization of the quantum vacuum is an urgent r task. However it is easy to see that the observed 4πG 2 = ρ g b (r) 2 r b (r)dr (19) separation of dark matter and the dominant part r ∫ 0 of the baryonic matter is not a surprise. It is obvious that an analogous relation exists for The key question is why there is no significant the acceleration g dm (r) produced by the dark presence of dark matter between the clouds of matter. In the particular case of an the X-ray emitting plasma. First, while the three approximately constant baryonic volume dimensional form of clouds is not known, during density ρ b (r) ≡ ρ b , the equation (19) leads to the collision the clouds were not only slowed but the direct proportionality between acceleration flattened as well. And, as we have argued above, around a flattened distribution of the baryonic g (r) and the radial distance r , i.e. b matter, the additional field caused by the 4πGρ b gravitational polarization is not significant. The g (r) = r (20) b 3 second important factor is that the distance According to (20), the assumption of the direct between clouds is relatively small. When two

proportionality between Pg (r) and g(r) means baryonic masses are close enough, they compete to orient the same dipoles in different directions,

7 what changes the gravitational polarization the scale of the whole Universe? The answer density and its divergence. Hence, while without may be yes. Let us use in the equation (17) the the appropriate simulations a detailed picture is radius of the , which is impossible, the absence of dark matter in the estimated to be about 14 billion parsecs i.e. region of clouds has nothing unusual. ≈ 4.3×1026 m . According to (17) the 4. Discussion corresponding dark matter in the Universe is The initial paper (Hajdukovic, 2011) has about 3.4×1053 kg or 1.7×1023 solar masses. If revealed an intriguingly simple rule: find the our estimate of the current ratio of the baryonic geometrical mean of the mass of a pion and the and dark matter in the Universe is correct, the baryonic mass of a galaxy and divide it with the baryonic mass of the visible universe should be Compton wavelength of the pion; what you get 3×1022 solar masses. These numbers are close is very close to the observed radial dark matter to the already existing estimations (see for density in a galaxy (see the equation (14)). It instance Roos, 2003). Hence, everything looks was the first indication that what we call dark as if equation (17) is valid for the Universe as a matter may be the result of the gravitational whole. But if so, the ratio of the dark matter and polarization of the quantum vacuum. the baryonic matter in the universe should grow In the present paper we have revealed the with time (what was already pointed in the additional indications; the most striking one is section 3.1, from a different point of view). the result of equation (16), a universal property At the end, let us note that the gravitational of galaxies (1) can be expressed through the properties of would be tested in universal constants and mass of pion what is CERN before the end of the current decade simply astonishing. There is one point here (Kellerbauer et al. 2008) and that the recent which deserves particular attention. The Planck theoretical considerations give some support to  constant , so crucial in quantum theory, but the gravitational repulsion between matter and absent from our theory of gravitation, appears in antimatter (Villata, 2011). both equations (14) and (16) concerned with the large scale gravitational phenomena. All this Note added to the proof suggests that the gravitational polarization of the The same day when this paper was accepted for quantum vacuum may be a serious alternative to publication, Wilson et al (2011) have reported the dark matter paradigm. the observation of the conversion of virtual Let us clarify that our theory is not a support to (from the quantum vacuum) into real MOND. Yes, there is a critical gravitational photons. In addition to significant indirect field; in a field stronger than the critical one evidence accumulated in the past decades, this is there is saturation (i.e. the maximal gravitational the first direct evidence for the existence of polarization density), but there is no violation of vacuum fluctuations. A great support to the point the fundamental law of gravity. The fact that of view that the physics of the 21st Century may MOND correctly guessed the existence of a well be the physics of the quantum vacuum critical field is the reason for its partial success, which, as revealed in our paper, may explain the but (in our opinion which may be wrong) the phenomena usually attributed to the mysterious success is limited because of the dark matter. misunderstanding of the physical origin of this critical field. Let us end with one intriguing question. Are the result (16) and its consequence (17) valid at

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