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Trippe Missing Mass and Modified Gravity ·

Yet another indication toward a lack of observed mass is pro- and dust distributed between the stars. Accordingly, it is tempt- vided by cosmology. Modern cosmological observations imply ing to identify the missing mass with additional ISM, and in- a certain fixed value for the density of matter/energy within the deed ISM distributed throughout galaxy clusters largely ex- universe. A combination of theoretical predictions and the dy- plains the enormous mass discrepancies first reported by [1]. namical observations discussed above1 leads to the conclusion However, else than at the time of Zwicky, modern astronomical that only about 16% of the matter present in the universe is ac- observations cover the entire electromagnetic spectrum from tually luminous – on cosmological scales, we observe a mass radio to γ-rays and are able to trace the characteristic signa- discrepancy Mdyn/M0 6 [7, 8]. tures of interstellar matter. Atomic gas – mostly hydrogen – is Evidently, the missing≈ mass problem is a substantial chal- identified via radio observationsof the H I 21-cmline. Hot, ion- lenge not only for astronomy but for various fields of physics. ized gas is traced via its thermal X-ray emission. Molecular gas In the following, I introduce the two main concepts proposed and dust show emission lines at radio and infrared wavelengths for solving the missing mass problem, non-baryonic dark mat- and are sources of a characteristic wavelength-dependent ab- ter on the one hand and modified laws of gravity on the other sorption of light emitted by background sources (interstellar hand. I present and discuss the increasing body of observa- extinction). And, finally, interstellar dust is an emitter of black- tional evidence in favor of modified gravity that has been ac- body radiation peaking in the mid to far infrared depending on cumulated especially within the last decade. Eventually, I ar- the dust temperature (for an exhaustive review of interstellar gue that these recent observations have the potential to initiate matter, see, e.g., [10]). Modern observations exclude the possi- a paradigm shift in astrophysics from theories based on dark bility that substantial amounts of diffuse interstellar matter have matter toward theories based on modifications of the laws of been missed within or around galaxies (cf. also [6]). The situa- gravity on astronomical scales. tion is somewhat more complex for the case of galaxy clusters: here the observational limits are less well established and could 2. DarkMatter lead to corrections of M0 by factors on the order of two [11].

The most evident approach to the missing mass problem is 2.2. MACHOs the assumption that our observations are incomplete and that In the 1980s, the absence of sufficient amounts of diffuse in- the mass discrepancy arises from additional matter components terstellar matter led to the idea that the missing mass might be simply not yet observed. A first clue toward the distribution of stored in small compact bodies – dubbed massive compact halo this additionaldark mass is providedby the rotation of galaxies: objects, or MACHOs – like interstellar planets or stellar mass the rotation curves behave as if galaxieswere surroundedby ha- black holes distributed within and around galaxies. Such ob- los of matter extending well beyond the visible components of jects are extremely difficult to detect as both the emission and the galaxies. As one can estimate from (2) easily, υ const c the absorption of light by them would be marginal. However, indicates halos with density profiles ρ(r) ∝ r 2. Technically,≈ − the compactness of those bodies makes them suitable as grav- a simple parameterization of a dark matter halo is achieved by itational lenses [12]: a MACHO located within or around the a mass density profile ρ(r)= ρ (r /r)2, with ρ being a scal- 0 0 0 Milky Way focuses the light of a backgroundstar toward an ob- ing factor of the dimension of a mass density and r denoting 0 server on Earth, leading to a substantial (by factors 2 10) a characteristic radius.2 From this follows that any dark mat- and characteristic increase of the observed brightness≈ of− the ter halo model for a given galaxy requires (at least) three free star over a time of a few hundred days. Accordingly, simul- parameters: the scaling parameters ρ and r plus the galaxy’s 0 0 taneous long-term monitoring of large numbers of background mass-to-light ratio ϒ which is needed to estimate the luminous stars should unveil the presence of MACHOs. However, sev- mass of a galaxy from its brightness. This ratio is a function eral studies observing the Magellanic Clouds – satellite galax- of the composition of the stellar population that makes up the ies of the Milky Way that provide millions of background stars galaxy.3 – as well as the Galactic spiral arms found only few gravita- Historically, the concept of dark matter evolved in three main tional lensing events. In conclusion, MACHOs contribute only steps. few per cent of the missing mass of the Milky Way at most [13, 14, 15, 16]. 2.1. Interstellar matter A notable (crudely 10–50% depending on the actual galaxy) 2.3. WIMPs fraction of the total – luminous– mass of galaxiesis contributed With interstellar matter – diffuse or compact – largely be- not by stars but by the diffuse interstellar medium (ISM), gas ing ruled out, the search for dark matter candidates eventu- 1And ignoring here an additional component, dark energy, an electromagnet- ally reached the realm of particle physics by postulating the ically dark fluid generating a negative pressure that has been postulated as universal presence of non-baryonic weakly interacting massive explanation for the accelerated expansion of the universe. particles, or WIMPs for short. Massive numerical simulations 2 Such a functional form of the density profile follows naturally from the as- (e.g., [17]) show that consistency with cosmic structure forma- sumption that a dark matter halo is a self-gravitating isothermal ensemble of particles in equilibrium; cf. § 4.3.3(b) of [2]. Modern studies use more so- tion requires dark matter to be cold, meaning that dark matter phisticated dark matter density profiles, especially the Navarro–Frenk–White particles move at non-relativistic speeds (this excludes massive profile [9]. All those profiles require at least as many parameters – i.e., three neutrinos as WIMP candidates – they would be hot dark matter – as the simple powerlaw profile given here. moving at relativistic speeds). 3The mass-to-light ratio is usually quoted in units of solar masses per solar luminosity, M /L ; for the inner regions of the Milky Way, ϒ 3M /L Theoretically, the most “popular” source of cold dark mat- (e.g., [2]). ⊙ ⊙ ≈ ⊙ ⊙ ter particles is the supersymmetric sector (SUSY) of particle S. Trippe Missing Mass and Modified Gravity 3 · physics [18, 19]. Supersymmetric particles are supposed to be A more sophisticated proposal was provided in 1983 by more massive than standard model (SM) particles. Further- Mordehai Milgrom [27, 28, 29]. At the time of Milgrom’s more, a conserved quantum number – the R-parity – ensures work, the missing mass problem in galaxies was a fairly recent that SUSY particles may decay into lighter SUSY particles but discovery, and the data base was still sparse. Observational not into SM particles. Accordingly, one may tentatively iden- features known to Milgrom were the asymptotically flat rota- tify the WIMP with the lightest supersymmetric particle; usu- tion curves of disk galaxies and the Tully–Fisher relation [30] 4 ally, this is supposed to be either a gravitino or a neutralino. which suggested that M0 ∝ υc for disk galaxies (the Tully– However, despite massive experimental efforts, observational Fisher relation will be discussed in detail in 4.2). This led evidence for a supersymmetric sector has not been found yet Milgrom to conjecture a modified law of gravity§ 6 related to a [20]. characteristic acceleration (or gravitational field strength) such that 3. Modified Gravity gN = µ(x)g (3) “ It is worth remembering that all of the discussion [on dark with x = g/a , a being a constant of the dimension of an matter] so far has been based on the premise that New- M M acceleration – today known as Milgrom’s constant – and µ(x) tonian gravity and general relativity are correct on large µ scales. In fact, there is little or no direct evidence that being a transition function with the asymptotic behavior (x)= 1 for x 1 and µ(x)= x for x 1. Obviously, the first limiting conventional theories of gravity are correct on scales much ≫ ≪ larger than a parsec or so. ” case correspondsto the usual Newtonian dynamics. The second limiting case however results in — [21], § 10.4.3, p. 635 4 Despite the overwhelming success of Einstein’s General The- υc = GM0 aM = const (4) ory of Relativity (GR) [22], modified laws of gravity can be, for g aM. By construction, (4) assumes ordered circular mo- and have been, considered as solutions of the missing mass tion of≪ stars (or any test mass), corresponding to dynamically problem. As indicated by the quote from [21] given above, cold stellar systems7 like disk galaxies. For stellar systems those proposals are motivated by the fact that GR is experi- dominated by random motions, i.e., dynamically hot systems mentally well-established in the regime of small spatial scales like elliptical galaxies and galaxy clusters, one finds [31, 32] – and strong gravitational fields (e.g., [23]) but cannot be tested again for g aM – the relation directly on spatial scales much larger than, and for gravitational ≪ fields much weaker than within, the solar system. Accordingly, 4 σ 4 = GM a = const (5) modifications of Newton’s law of gravity on spatial scales much 9 0 M larger than the scale of the solar system cannot be ruled out a with σ being the (three-dimensional) velocity dispersion. The priori. proposed modified law of gravity (3) and the resulting relations The most obvious modification one can make is a change between luminous mass and velocity (4, 5) have been summa- of the scaling of gravitational acceleration g with radial dis- 4 rized under the term Modified Newtonian Dynamics, or MOND tance r for a point mass with luminous mass M0. In stan- 2 for short. dard gravity, g = gN = GM0/r , with gN denoting the standard The MOND laws provide a variety of explicit predictions (P) Newtonian gravitational field strength. For a test particle or- that put them in contrast to dynamics based on dark matter and biting M0 with circular speed υc, the centripetal acceleration is 2 2 that can be tested observationally: ac = υc /r g, resulting immediately in υc = GM0/r. We ≡ P1. Like for any modified law of gravity, the source of the now assume a modified law of gravity g = η(r/r0)gN with r0 gravitational field can only be the luminous source mass M0. being a constant characteristic length, η(r/r0)= 1 for r r0, ≪ This implies a one-to-one correspondence between the spatial and η(r/r0)= r/r0 for r r0. The first limiting case corresponds to the usual Newtonian≫ law of gravity; the second limit- structure of the gravitational field and the spatial distribution of mass. This is sharply distinct from the assumption of gravity ing case however results in g = GM0/(rr0) and, consequently, 2 being partially caused by dark matter: in this case, luminous υc = GM0/r0 = const – we have found a way to create constant circular speeds in the outer regions of disk galaxies. Our and dark mass components may be spatially separate, resulting in a gravitational field not following the distribution of the simple law of gravity is easily falsified however: we postulate luminous mass. here a modification at a characteristic length scale r0, meaning that the mass discrepancy Mdyn/M0 for galaxies should corre- P2. According to (3), MOND requires a universal scaling of late with the distance from the galactic center – which is ruled the mass discrepancy Mdyn/M0 (i.e., g/gN) with acceleration, µ out by observations [24].5 Notable examples for those modified with the scaling law being the function (g/aM). This scaling laws of gravity are given by [25, 26]. law comes with only one free parameter, Milgrom’s constant aM, which is a constant of nature. 4For simplicity, I only quote absolute values of positions, velocities, and accelerations. side the orbit via the same functional relation (cf. also § 4.3). Similar, general 5This is actually a non-trivial statement because for force laws deviating from parameterizations also exist for the distribution of mass in elliptical galaxies 2 a r− scaling, Gauss’ theorem is invalid (cf. § 2.1 of [2]). This implies that [2]. a test mass is influenced also by the mass distribution outside its orbit. For- 6Milgrom actually formulated his relation initially as a modified law of inertia tunately, nature is on our side: the most important probes of stellar dynamics but abandoned this interpretation quickly. are disk galaxies. To good approximation, all disk galaxies can be parameter- 7This nomenclature stems from analogy to kinetic gas theory: a system is ized by exponential surface mass density profiles. This means that for all disk “cold” (“hot”) if the random velocities of the constituent particles are much galaxies the galaxy mass outside a given orbit is determined by the mass in- lower (much higher) than any ordered, streaming velocity within the system. 4 S. Trippe Missing Mass and Modified Gravity ·

P3. In the limit g aM, stellar speeds necessarily scale like 4 4 ≪ υc ∝ M0 and σ ∝ M0 for ordered circular and random motions, respectively. Those scaling relations between velocity and luminous mass are not expected for dark matter based dynamics where the masses and spatial distributions of the luminous and dark components are, a priori, independent.

4. Observational Evidence

4.1. Search for Dark Matter Particles By definition, dark matter particles are supposed to interact with electromagnetic radiation either extremely weakly or not at all. Obviously, this makes a direct detection very difficult and requires the use of non-electromagnetic signatures as trac- ers of WIMPs [33]. The last two decades have seen several large-scale experiments searching for characteristic recoil sig- nals arising from the scattering of WIMPs at atomic nuclei. Those experiments are based in underground laboratories in order to minimize the non-WIMP background and use either crystals cooled to cryogenic temperatures or liquid noble gases, especially xenon, as detector materials (e.g., [34, 35, 36]). Ifan ensemble of WIMPs were to orbit the Galactic center along the solar orbit, the terrestrial motion around the sun should change the relative velocity of Earth and the WIMP ensemble during the year, leading to a characteristic annual modulation of the signal rate. In addition to direct detection methods, signatures of the decay or annihilation of WIMP particles have been searched for in cosmic γ [37], neutrino [38], and antipro- Fig. 1: The baryonic Tully–Fisher relation (in logarithmic representa- tion): luminous galaxy mass M0 (in units of solar mass M ) as func- ton [39] radiation. To date, none of the multiple experiments ⊙ tion of asymptotic rotation speed υc. Circles denote observational val- and observations has returned a detection of dark matter parti- ues for 92 galaxies [44, 45]; the data span four orders of magnitude cle candidates (see [40, 41] for very recent null results). 4 in mass. The straight line shows the relation M0 = υc /(1.3GaM) for 10 2 aM = 1.15 10 m s ; the factor 1.3 arises from geometry. Error × − − 4.2. Mass–velocity scaling relations bars have been omitted for clarity; the intrinsic scatter of the relation is consistent with zero. In 1977, [30] reported the discovery of an empirical correlation – the Tully–Fisher relation – between the luminosity L of disk galaxies and their (asymptotic) circular speed υc such that L ∝ Already one year before the discovery of the Tully–Fisher 4 υc . The luminosity of a galaxy is a proxy for the combined relation, in 1976, [46] reported observational evidence for a mass of its stars, M⋆; deriving this mass requires knowledge of strikingly similar scaling law in elliptical galaxies. Elliptical the galaxy’s mass-to-light ratio ϒ. As it turns out [42, 43, 44], galaxies are dominated by random stellar motions with velocity the actual, underlying fundamental relation is found by taking dispersion σ; [46] found a relation between the velocity disper- into account the total luminous (baryonic) mass of galaxies, sions and the luminosities of galaxies like L ∝ σ 4 – a relation including stars and the diffuse interstellar matter. This results known today as the Faber–Jackson relation. In analogy to the 4 in a baryonic Tully–Fisher relation M0 ∝ υc holding over at procedure for the baryonic Tully–Fisher relation, one can as- 4 least four orders of magnitude in galactic mass – see Fig. 1 for sume an underlying relation M0 ∝ σ and attempt a comparison an illustration. Comparison of the data with the scaling law (4) to (5). Taking into account that σ is actually the velocity dis- leads to the relation persion integrated along the line of sight through an elliptical galaxy, it turns out that (5) indeed provides a good description 4 υc of the kinematics of elliptical galaxies for g aM [47]. M0 = (6) ≪ ∝ 1.3Ga In addition to the kinematics of galaxies, scaling laws M0 M 4 4 υc and/or M0 ∝ σ have been found for where the (approximate) geometry factor 1.3 arises from the difference between the distributions of mass in idealized ho- (i) the motion of galaxies in galaxy clusters [47]; mogeneous spheres and realistic disk galaxies (cf. (2) and ∝ σ 2 2.6.1(b) of [2]). As illustrated in Fig. 1, data and model are in (ii) the temperature T of the diffuse intra-cluster medium § 10 2 in galaxy clusters, i.e., M ∝ T 2 ∝ σ 4 [48]; excellent agreement for aM 1.15 10 m s . Taking into 0 ≈ × − − account the measurement errors, the intrinsic scatter of the data (iii) dwarf galaxies orbiting the Milky Way [49]; about the theoretical line is consistent with zero [44, 45] – in other words, (6) provides a complete description of the large- (iv) stellar motions in the outer regions (where g < aM) of scale kinematics of disk galaxies. globular star clusters [50]; and S. Trippe Missing Mass and Modified Gravity 5 ·

Ignoring the constants, this relation predicts g ∝ Σ0. In case of modified dynamics, we start off from the relation 4 υc = GM0aM. In analogy to the procedureapplied to the New- tonian case we can (i) divide both sides by r2, (ii) use the defi- 2 nition of Σ0, (iii) use g = ac = υc /r, and (iv) divide both sides 2 by aM . Eventually, we find

g Gπ Σ 1/2 = 0 . (8) aM aM

Evidently, modified dynamics predicts g ∝ √Σ0 . In Fig. 2, I compare the predicted surface density– acceleration (SDA) relations (7, 8) with observational values obtained by [52] for 71 disk galaxies. For each galaxy, the radial surface density profile was (as usual in galactic astronomy) approximated as an exponential disk, i.e., Σ0(r)= Σ0(0)exp( r/rd), with a “disk scale length” rd as free parameter. Rotation− velocities were consistently measured at r ≈ 2.2rd, surface densities were derived from the luminous galactic mass enclosed within r . 2.2rd [53]; this procedure ensures Fig. 2: The empirical relation between baryonic surface mass den- that data from different galaxies can be compared in a straight- 10 2 sity Σ and scaled gravitational acceleration g/a . Data points with forward manner. I assume here aM = 1.15 10− m s− as 0 M × error bars indicate measurements from a sample of 71 disk galax- suggested by the baryonic Tully–Fisher relation (Fig. 1). ies [52]. The continuous grey line corresponds to (8) with aM = The SDA observations provide three key results. Firstly, the 10 2 1.15 10− m s− ; this line is purely theoretical and not a fit to the baryonic surface mass densities of disk galaxies span over three × data. The dotted line denotes the relation (7) expected from Newtonian orders of magnitude. This immediately falsifies the hypothesis 10 2 dynamics (assuming again aM = 1.15 10 m s ). × − − that the Tully–Fisher relation arises from Newtonian dynamics combined with (approximately) constant surface densities. Secondly, the very existence of an empirical SDA relation rules (v) wide binary stars (where g . a for M 1M and r & M ⋆ out the possibility that disk galaxy dynamics is dominated by 7000 astronomical units) [51]. ≈ ⊙ dark matter; if this were the case, we would not expect a corre- In total, the scaling laws (4) and (5) are consistent with the lation between baryonic surface density and acceleration at all. kinematics of stellar systems spanning about eight orders of Thirdly, the SDA data follow the theoretical line provided by magnitude in size and 14 orders of magnitude in mass (see also MOND8 and not the line expected from Newtonian dynamics. [52] for an exhaustive review); this comprises all gravitation- ally bound stellar systems beyond the scale of planetary sys- 4.4. Mass Discrepancy–Acceleration Relation tems. 4 However, at least in case of disk galaxies a M0 ∝ υc scaling Scaling relations between the luminous mass of a stellar system law can be found not only from (4) but – a priori – also from and the velocities of and accelerations experienced by its con- standard Newtonian dynamics. Starting from the usual Newto- stituents, as discussed in 4.2 and 4.3, probe the asymptotic 4 2 §§ nian relation υc = (GM0/r) , we can introduce a (baryonic) dynamics in the limit of weak gravitational fields. Deeper in- 2 surface mass density Σ0 = M0/(πr ) which averages over all sights require an analysis of the transitional regime from New- matter at galactocentric radii from 0 to r. Combining the two tonian to modified dynamics. As pointed out first by [24] about 4 expressions leads to υc ∝ M0 Σ0; this implies a Tully–Fisher- a decade ago, this regime can be tested by observations of stel- like relation if all disk galaxies have approximately the same lar velocities in disk galaxies. A luminous mass M0 is derived surface mass density. Accordingly, we need to take a close look from imaging a galaxy and summing up the masses of stars and at the surface density of disk galaxies. interstellar matter; a dynamical mass Mdyn is calculated from stellar velocities (derived from spectroscopic measurements) 4.3. Surface Density–Acceleration Relation via (2). The empirical mass discrepancy (MD) Mdyn/M0 can be analyzed as function of galactocentric radius r, angular fre- 2 Assuming the universal validity of either the Newtonian scaling quency, and Newtonian acceleration gN = GM0/r of stars or- law (2) or the modified law (4) leads to important corollaries biting the galactic center. Empirically, the MD is uncorrelated regarding the relation between gravitational acceleration g and with radius or angular frequency and strongly anti-correlated 2 baryonic surface mass density Σ0 = M0/(πr ) of disk galaxies. with acceleration; this behavior is illustrated in Fig. 3. Evi- 2 In case of Newtonian dynamics, υc = GM0/r. Taking this dently, the mass discrepancy is a function of gravitational field 2 2 relation and (i) dividing both sides by r, (ii) using g = ac = strength g = υc /r = GMdyn/r = gN(Mdyn/M0). Notably, the 2 υc /r, (iii) using the definition of Σ0, and (iv) expressing g in mass discrepancy–acceleration (MDA) relation is universal be- units of aM results in cause all disk galaxies studied – 60 galaxies spanning about

g Gπ Σ 8The agreement with the MOND line can only be approximate because the = 0 . (7) condition g aM is not fulﬁlled for the majority of the data. aM aM ≪ 6 S. Trippe Missing Mass and Modiﬁed Gravity ·

feature in the luminosity profile there is a corresponding feature in the rotation curve and vice versa”. A more recent and more detailed study [58] based on a sample of 43 disk galaxies finds that “the observed rotation curves of disk galaxies [...] can be fitted remarkably well simply by scaling up the contri- butions of the stellar and H I disks” – in other words, at any galactocentric radius the dynamical mass is completely determined by the enclosed luminous mass. Quantitatively, the best description of galactic rotation curves is achieved by (9) with 10 2 aM = (1.2 0.3) 10− m s− [59]. Such a behavior is self- evident in± the context× of modified gravity models (where the luminous mass is the dynamical mass when taking into account the modified law of gravity); it is extremely difficult to understand when starting off from the assumption of a separate dark mass distribution around a galaxy.

4.6. Dwarf Galaxies Fig. 3: The mass discrepancy–acceleration relation: mass discrep- The modern textbook picture of galaxies [60, 61] usually as- ancy Mdyn/M0 as function of Newtonian gravitational acceleration sumes that most (crudely 50–90%) of the total mass of any 2 gN = GM0/r . Please note the logarithmic–linear axis scales. Grey galaxy (regardless of the type of galaxy) is stored in a spherical circles denote observational values, in total 735 measurements from 60 dark matter halo that extends well beyond the luminous com- galaxies [24, 52, 54]. The black continuous curve denotes the function 10 2 ponents of the galaxy. A key prediction of this picture is the (9) for aM = 1.06 10 m s . × − − formation of a large number of small “dwarf” satellite galaxies – due to “condensation”of luminousmatter in local dark matter concentrations within the dark halo – distributed isotropically two orders of magnitude in size in case of the data set shown in around the main galaxy. This is actually not observed: the Fig. 3 – follow the same empirical curve [24, 52, 54]. number of satellite galaxies observed around galaxies is sub- The observed MDA relation naturally suggests a description stantially – by more than one order of magnitude– smaller than using the model (3); by construction, µ x g g M M ( )= N/ = 0/ dyn expected (a result known as the missing satellite problem). Fur- (i.e., the inverse of the mass discrepancy). The transition func- thermore, the distribution of these satellites is not isotropic; the tion µ x does not follow from theory; a priori, various – em- ( ) satellites of the Milky Way and a few other galaxies are located pirically motivated – choices for the functional form of µ x ( ) in planes that coincide with the planes of past galaxy–galaxy are available [28, 55]. A good description of the MDA data is collisions – which indicates that all these satellites are “tidal provided [56] by the choice dwarf galaxies” formed via tidal interactions between galaxies. x g For a recent exhaustive review, see [62] and references therein. µ(x)= with x = ; (9) 1 + x aM 4.7. Colliding Galaxy Clusters a possible physical motivation for this form of µ(x) will be discussed briefly in 5. As illustrated in Fig. 3, model and data On the one hand, modified laws of gravity demand that the lu- § 10 2 are in very good agreement for aM = 1.06 10 m s ; Inote minous matter of any stellar system traces the total dynami- × − − that here the best-fit value for aM depends on the model chosen cal mass. On the other hand, dynamical models based on dark for µ(x). Within a combined, largely systematic relative un- matter do not comprise this restriction; the spatial distribution certainty of about 10%, the value for aM found from use of (9) of the total dynamical mass does not necessarily follow the lu- is identical to the one derived independently from the baryonic minous matter. Accordingly, a decision in favor of dark mat- Tully–Fisher relation (cf. Fig. 1). ter based models could be enforced by observations of astronomical objects where dynamical mass and luminous mass are 4.5. Local Coupling of Mass and Kinematics spatially separate. A key test is provided by clusters of galaxies that are colliding or, more appropriately speaking, passing In 4.2 and 4.3, I discussed global scaling relations between through each other. Due to their very small effective cross sec- the§§ luminous mass and the rotation speed of disk galaxies – tion, the individual galaxies are de facto collisionless and just “global” in the sense that the scaling relations discussed there pass each other without interaction. The same can be assumed are obtained from averaging over entire galaxies. An impor- for the (inter)galactic dark matter halos – if those actually exist. tant amendment to this approach is provided by taking a close The intergalactic cluster gas – that comprises the overwhelming look at the corresponding local scaling relations obtained from fraction of the total luminous mass of a galaxy cluster – how- spatially resolving the luminous mass distributions and kine- ever experiences collisional ram pressure and lags behind the matics of individual galaxies. Observations of baryonic mass galaxies and the dark matter distributions. Observationally, the profiles – luminous mass as function of galactocentric radius distribution of the luminous mass is – essentially – traced by – and rotation curves – rotation speed as function of galacto- the X-ray emission from the intracluster gas; the distribution of centric radius – on a galaxy-by-galaxy base unveil a relation the gravitating dynamical mass is derived from the gravitational that has since become known as Renzo’s rule [57]: “For any lensing of background sources. S. Trippe Missing Mass and Modified Gravity 7 ·

2 The first system of colliding clusters that has been studied in out that the presence of dark matter halos with ρ ∝ r− can detail is the “Bullet Cluster” 1E0657-56 [63]. Comparison of explain the asymptotic flattening of galactic rotation curves the distributions of intracluster gas and dynamical mass shows (υc(r) const at large r) – but nothing else. Indeed, none of a clear spatial separation – most of the dynamical mass indeed the relevant≈ empirical scaling relations of galactic kinematics appears to be stored in a non-luminous mass component, and – notably the Tully–Fisher and Faber–Jackson relations ( 4.2), accordingly, this result was interpreted as a triumph of the dark the surface density–acceleration ( 4.3) and mass discrepancy–§ matter paradigm over its competitors. Ironically, more recent acceleration ( 4.4) relations, and§ Renzo’s rule ( 4.5) – follow studies of the kinematics of the Bullet Cluster [64] have turned from dark matter§ models a priori. Explaining§ the baryonic around this interpretation completely: the relative velocities of Tully–Fisher relation requires a “fine tuning” of dark and lu- the colliding galaxy clusters are far too high ( 3000km/s com- minous matter such that for all galaxies the total surface mass ≈ pared to .1800km/s expected theoretically) to be understood density Σ = Σ0 + ΣDM (with ΣDM being the dark matter sur- by dark matter driven dynamics. Using results from cosmolog- face density) is approximately the same despite the fact that Σ0 ical numerical simulations for reference, [64] conclude that the varies over three orders of magnitude. Arguably the only real- probability for the standard dark matter paradigm being correct istic way to achieve Σ const is the assumption that ΣDM Σ0 9 ≈ ≫ is less than 10− . At this point it is important to note that the for all galaxies plus ΣDM const. However, this ansatz is con- initial interpretation of the mass distributions within the Bullet tradicted by the empirical≈ SDA relation which demonstrates Cluster is based on the presumption that (essentially) the entire that galactic dynamics is controlled by the luminous matter, im- baryonic intracluster medium in galaxy clusters is observable plying ΣDM Σ0. To date, the necessary fine tunings between as X-ray luminous gas. This is probably not the case: analyses (i) dark and≪ luminous surface densities, as well as (ii) between of the baryonic mass content of galaxy clusters conclude that dark and luminous mass profiles within galaxies (Renzo’s rule) crudely half of the total baryonic matter seems to have been – also referred to as a dual “conspiracy between luminous and missed by observations as yet [11, 65]. Furthermore, a sec- dark matter” [42] – are unexplained. ond system of colliding galaxy clusters has become available II. Successes of modified dynamics. The three predictions for analysis: the “Train Wreck Cluster” A520 [66]. In A520, P1–3 provided by Modified Newtonian Dynamics ( 3) can be observations find that the spatial distributions of dynamical and compared to observations in a straightforward manner.§ Predic- luminous mass coincide [67]. tion P1 states that luminous and dynamical mass are identical when applying an appropriate scaling. This is observed in indi- 5. Discussion vidual galaxies as Renzo’s rule ( 4.5) as well as in at least one galaxy cluster, A520 ( 4.7). Prediction§ P2 demands a univer- Ever since its discovery, the missing mass problem has been a sal scaling of mass discrepancies§ according to (3) and, indeed, challenge to our understanding of the physics of the cosmos. observations find a universal mass discrepancy–accelerationre- Historically, the standard approach has been the postulate of lation of disk galaxies ( 4.4). Prediction P3 states that, for § cold dark matter ( 2; cf. also [5, 68]), and, initially, for a very g aM, stellar velocities and luminous masses are related ac- good reason: else§ than a modified law of gravity, the assump- cording≪ to (4, 5). Comparisons to observations show that P3 tion of dark matter does not per se require “new physics” – as naturally provides for the baryonic Tully–Fisher and Faber– long as it can be reconciled with the standard model of particle Jackson relations on all scales ( 4.2) as well as for the surface physics. As I outlined in 2.3, Occam’s razor is no longer in density–acceleration relation of§ disk galaxies ( 4.3). Remark- favor of dark matter: at least§ since the failed searches for MA- ably, the scaling laws (4, 5) are consistent with§ the kinematics CHOs in the 2000s, models based on dark matter inevitably re- of stellar systems spanning eight orders of magnitude in size quire physics beyond the standard model of particle physics – and 14 orders of magnitude in mass, thus covering all gravi- both dark matter and modified gravity now require new physics. tationally bound stellar systems beyond the scale of planetary Else than usually, and implicitly, assumed (or hoped for), the systems. I emphasize that the statements P1–3 are indeed pre- two approaches are not mutually exclusive; this is important dictions: even though Milgrom constructed (3) based on his for the discussion of structure on cosmological scales. knowledge of flat rotation curves and the Tully–Fisher relation, A review of the observational evidence provided in 4 leads most of the relevant observations, notably the baryonic Tully– to conclusions that fall into either of two categories: § Fisher and Faber–Jackson relations, Renzo’s rule, and the SDA I. Failures of the dark matter paradigm. The most evident and MDA relations, have been achieved only as late as two problem of the dark matter paradigm is the fact that a parti- decades after Milgrom’s proposal. Last but not least, MOND cle with the required physical properties is not known; to date, is not only physically more successful than dark-matter driven none of the multiple direct and indirect searches for dark matter dynamics but also technically simpler: modeling the dynamics particles has returned any viable candidate ( 4.1). In order to of any given galaxy with MOND requires one free parameter, § the mass-to-light ratio ϒ (cf. especially 4.2, 4.4). When fit- understand the kinematics of galaxies it is necessary to assume §§ that galaxies are embedded in extended, more or less spheri- ting a dark matter halo to a given galaxy, one needs at least 2 three parameters: ϒ and (at least) the scaling parameters ρ and cal, dark matter halos with mass density profiles ρ ∝ r− over 0 r0 (cf. 2). a sufficient range of galactocentric radii r. Those models imply § the presence of large numbers of isotropically distributed dwarf Regarding the combined evidence, it is rather obvious that satellite galaxies – which are not observed ( 4.6) – as well as the missing mass problem is treated much better by a modified certain limits on the collision velocities of§ galaxy clusters – law of gravity than by postulating dark matter. This being said, which are in disagreement with observations ( 4.7). it is also clear that MOND is incomplete: it is a law of gravity Taking a more careful look at galactic kinematics,§ it turns but not yet a theory of gravity – meaning it does not provide 8 S. Trippe Missing Mass and Modified Gravity · the underlying physical mechanism a priori. A key question to curves. Once this is achieved, the only free parameter remain- be addressed is the physical role of Milgrom’s constant; empiri- ing for the description of any given galaxy is the galactic mass- 10 2 cally ( 4), aM = (1.1 0.1) 10 m s , with the error being to-light ratio ϒ [84, 85]. § ± × − − largely systematic. Coincidentally, aM = cH0/2π within er- 1 1 Conformal gravity follows from the insight that (i) the Einstein rors, with c being the speed of light and H0 70kms Mpc ≈ − − equations are not the only possible field equations of GR, and being Hubble’s constant [69]; this might indicate an as yet un- (ii) the Einstein–Hilbert action is not the only possible action of covered connection between galactic dynamics and cosmology. GR (cf. also the case of f (R) gravity). Accordingly, it is pos- Over the last decades, there have been multiple approaches to- sible to derive a theory of gravitation that keeps the coordinate ward theories of gravity beyond Einstein’s GR, and the follow- invariance and equivalence principle structure of general rela- ing is a brief, incomplete overview over six important exam- tivity but adds a local conformal invariance in which the action ples: is invariant under a specific local transformation of the metric. Tensor–vector–scalar theory (TeVeS) was derived by Jacob This results in additional universal gravitational potential terms Bekenstein as a relativistic gravitation theory of MOND [70, of linear and higher orders. The modified gravitational poten- 71]. In TeVeS, gravitation is mediated by three dynamical grav- tial is supposed to mimic the effect of dark matter in galaxies itational fields: the Einstein metric tensor gµν , a timelike 4- [86, 87, 88]. µν vector field Uα obeying g UµUν = 1, and a scalar field φ − Any theory of modified gravity necessarily needs to com- (using here the usual Einsteinian index conventions). By con- prise the – empirically well established – MOND laws as well struction, TeVeS is consistent with general relativity and the as non-dynamicalsignatures of gravity, notably (i) gravitational relevant limits, Newtonian dynamics (for g aM) and MOND ≫ lensing and (ii) gravitational waves. Studies assuming TeVeS (for g aM). ≪ as the underlying theory of gravity find good agreement with f (R) gravity results from theories aimed at generalizations of observations of gravitational lensing [89, 90, 91]. An example the Lagrangian Ł in the Einstein–Hilbert action of general rel- for a theory naturally including gravitational waves is provided ativity (see [72, 73, 74] and references therein for reviews). by massive gravity; as already pointed out by [76], in massive In standard GR, Ł ∝ R with R being the Ricci scalar. In a gravity the GR equations of gravitational waves are recovered generalized formulation, this expression becomes Ł ∝ f (R) in the limiting case of vanishing graviton mass. with f (R) being a general function of R; a simple example The discussion of the missing mass problem and modified n is f (R)= R with n being a real number. f (R) theories have dynamics has implications for, and has to be consistent with, been motivated mostly by cosmology, especially the cosmic in- cosmology. Modern cosmological models are constrained best flation and dark energy problems, but have also found applica- by observations of the angular power spectrum of the cosmic 9 tion to the missing mass problem (e.g., [75]). microwave background(CMB) radiation [8, 92, 93]; the current Massive gravity follows from the assumption (motivated by concordance cosmology is the Λ Cold Dark Matter (ΛCDM) quantum field theories) that gravitation is mediated by virtual model, with Λ denoting the cosmological constant [7]. Us- particles, gravitons, with non-zero mass. This possibility was ing a total of six fit parameters, the ΛCDM model indeed pro- first pointed out by Fierz and Pauli in 1939 [76] and has been vides a very good description of the CMB power spectrum [8]. studied extensively especially in view of potential applications This success comes at a high prize however: it requires the to cosmology (see [77, 78] for reviews). Massive gravity pro- assumption that only 5% of the mass/energy content of the uni- vides an intuitive approach to the missing mass problem: as- verse are provided by “ordinary” luminous matter; 27% are suming a luminous mass M0 and no graviton–graviton inter- provided by non-baryonic cold dark matter, and the remaining actions, the emitted gravitons form a halo with mass density 68% by dark energy, a dark fluid generating negative gravita- 2 profile ρ(r)= β M0 r− with r being the radial distance and β tional pressure and thus causing an accelerated expansion of the 2 Λ being a scaling factor. Using β = aM/(4πυc ), with υc being universe. Whereas the CDM model is technically successful, the circular speed of a test mass orbiting M0, and integrating the fact that it requires 95% of the mass/energy content to be over r leads to (9) [56, 79, 80]. provided by exotic dark components makes it physically du- 10 Λ Non-local gravity is based on the perception that the postulate bious. More importantly, the CDM model is not the only of locality in the special theory of relativity is only approx- possible description of the CMB power spectrum: the data can imately correct for realistic accelerated observers. This sug- be explained equally well by assuming Modified Newtonian gests the construction of a non-local general theory of relativ- Dynamics plus leptonic, hot dark matter – distributed over the ity wherein the gravitational field is local but satisfies non-local spatial scales of galaxy clusters and larger – composed of neu- integro-differential equations. In the Newtonian limit one finds trinos with masses on the order of few electron volts [95]. Fur- a modified Poisson equation and from this a gravitational point thermore, and very recently, new tools have been developed for mass potential which comprises an extra term that mimics an testing the impact of modified laws of gravity on the large-scale additional “dark” mass component [81, 82, 83]. structure of cosmic matter via numerical simulations. First results indicate that the observational signatures of a combina- Scalar–tensor–vector gravity, also referred to as Modified tion of f (R) gravity and massive neutrinos are indistinguish- Gravity (MOG), modifies Einsteinian (tensor) GR by adding one massive vector field and two scalar fields as mediators of 10Quoting the more drastic wording by [94]: “According to Planck [the CMB gravitation. The theory contains two free parameters that can space observatory], the universe consists of 4.9% ordinary matter, 26.8% be constrained empirically by comparison to galaxy rotation mysterious dark matter whose gravity holds the galaxies together, and 68.3% weird, space-stretching dark energy.” In essence: when assuming a ΛCDM 9Technically, already the introduction of a cosmological constant Λ into Ein- cosmology, 95% of the mass/energy content of the universe are “mysterious” stein’s field equations leads to an f (R) gravity with f (R)= R 2Λ [73]. and/or “weird”. − S. Trippe Missing Mass and Modified Gravity 9 · able from those of ΛCDM models [96, 97]. I note however galaxies. To date, dark matter particles have not been detected that these results do not yet imply the discovery of a cosmol- despite massive experimental efforts. Furthermore, the pres- ogy based on modified gravity; such a cosmology needs to be ence of dark matter halos is incompatible with distributions of derived from first principles out of a modified theory of gravity dwarf galaxies around “mother” galaxies as well as the kine- – which has not been achieved yet. matics of colliding galaxy clusters. A complication arises for the dynamics of X-ray bright On the other hand, all observations are explained naturally groups and clusters of galaxies: even when assuming MOND, in the frame of Modified Newtonian Dynamics (MOND) which the amount of luminous mass observed is too small – by fac- assumes a characteristic re-scaling of Newtonian gravitational tors on the order of two – to explain the dynamics [98, 52]. acceleration as function of acceleration (field strength). Most of This discrepancy is especially pronounced in the central re- the aforementioned kinematic scaling relations were predicted gions of those systems. A consistent dynamical description re- successfully by MOND as early as about two decades before quires (i) a modified theory of gravity that leads to scaling laws they were actually observed. MOND is also found to be consis- more complex than (3) and (5) on scales of groups of galaxies tent with cosmological observations, most notably the angular (i.e., hundreds of kiloparsecs); (ii) cold baryonic dark matter power spectrum of the cosmic microwave background radia- (CBDM) presumably in the form of cold (few Kelvin) com- tion. pact gas clouds [99]; or (iii) the presence of massive (preferred Regarding the combined evidence, it becomes more and masses being about 11eV) sterile neutrinos [100]. more obvious that the solution for the missing mass problem is Summarizing over the multiple lines of evidence – from stel- to be found in a modified theory of gravity that comprises the lar dynamics to cosmology – it is difficult to resist the im- MOND laws. Despite multiple attempts on theories of gravity pression that a solution of the missing mass problem requires beyond Einsteinian general relativity, a consistent picture has a modified law of gravity; the standard dark matter postulate not yet emerged and is left as work still to be done. seems to be more and more disfavoredby observations. The lat- ter statement is specifically aimed at “naive” dynamical models Acknowledgments: This work made use of the galactic dy- based solely on non-baryonic cold dark matter; certain com- namics data base provided by STACY S. MCGAUGH at Case binations of modified gravity and standard-model dark matter Western Reserve University, Cleveland (Ohio),11 and of the – especially massive neutrinos and/or CBDM – are in good software package DPUSER developed and maintained by 12 agreement with cosmological observations and might actually THOMAS OTT at MPE Garching. 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