Dark matter, the Equivalence Principle and modied gravity

Adn Sus 1

Universidad de Valladolid, Spain

a r t i c l e i n f o

Article history:Received 30 October 2013Accepted 30 December 2013Available online 21 January 2014

Keywords:Dark matterBullet clusterModied gravityScientic evidence

a b s t r a c t

Dark matter (DM) is an essential ingredient of the present Standard Cosmological Model, according towhich only 5% of the mass/energy content of our universe is made of ordinary matter. In recent times, ithas been argued that certain cases of gravitational lensing represent a new type of evidence for theexistence of DM. In a recent paper, Peter Kosso attempts to substantiate that claim. His argument is that,although in such cases DM is only detected by its gravitational effects, gravitational lensing is a directconsequence of Einstein0s Equivalence Principle (EEP) and therefore the complete gravitational theory isnot needed in order to derive such lensing effects. In this paper I critically examine Kosso0s argument: Iconfront the notion of empirical evidence involved in the discussion and argue that EEP does not haveenough power by itself to sustain the claim that gravitational lensing in the Bullet Cluster constitutesevidence for the DM Hypothesis. As a consequence of this, it is necessary to examine the details ofalternative theories of gravity to decide whether certain empirical situations are indeed evidence for theexistence of DM. It may well be correct that gravitational lensing does constitute evidence for the DMHypothesisat present it is controversial whether the proposed modications of gravitation all need DMto account for the phenomenon of gravitational lensing and if so, of which kindbut this will not be adirect consequence of EEP.

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1. Introduction

The Dark Matter Hypothesis is an essential ingredient of theStandard Cosmological Model (which is sometimes referred to asCDM: it introduces cold dark matter dark energy on top ofordinary matter/energy). The model describes the dynamicalevolution of the universe as a whole (it is a solution of Einstein0sGeneral Theory of Relativity: GTR) and tries to explain many otheraspects of the universe using different physical theories. It isbeyond doubt that CDM is a very successful model. It is successfulbecause it provides explanations of many different observedphenomena, such as the redshift in the radiation coming fromdistant galaxies; the cosmic microwave background (CMB); theamount of structure observed at present; the observed propor-tions of the elements; and the velocities of dispersion of galaxiesin clusters. In many of these explanations the theory of gravitationplays a central role (at both the galactic and cosmic scales gravityis the most important interaction); that is why one can say that

GTR (or the Newtonian approximation of certain phenomena)provides the dynamical backbone of the model.

All this success, though, does not come without a cost. It hasbeen known for almost a century that in order to reproduce someastronomical observations (the rotational velocities of spiralgalaxies and velocities of dispersion of galaxies in clusters) usingour current gravitational theories (GTR or Newtonian gravity) onemust assume the existence of a considerable amount of extramass/energy that is not observed. More recently it was realisedthat in order for CDM to reproduce other observations, also inaccordance with GTR, such as the amount of galactic structure orthe anisotropies found in the CMB, the observed quantity of massis not enough. The conclusion is that, at different levels, in order tomake good predictions with our current best gravitational theory,we must assume the existence of much more matter than theamount that we directly detect through its emitted radiation. At acertain point in the history of developing this idea, this missingmass was labelled dark matter (DM).

One can invert the previous statement by saying that our bestgravitational theory together with certain empirical observationsand, more widely CDM, predicts the existence of unobservedmatter. Nonetheless, the predictions made in these two contextsare not exactly the same. In the context of CDM, the DM

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Hypothesis gains a more specic content: if at rst it was possibleto assume that DM might be ordinary dim matter, differentarguments have proved that the missing mass should consist ofnon-baryonic particles for which, after different arguments thatrule out neutrinos, there is no suitable candidate in the StandardModel (SM) of particle physics that describes all theoreticalparticles. So the prediction of the present cosmological model isthat stuff made out of a hitherto undetected kind of particle isresponsible for the discrepancy between observations and thepredictions derived from GTR, which use as the total mass thatinferred directly from observations of visible matter.

All the phenomena explained by the CDM model can be takenas evidence in favour of the DM Hypothesis. This essentiallyassumes that the current gravitational theory is correct: weobserve the alleged gravitational effects of DM on the motion ofother bodies or on the outcome of some specic process. Due tothis fact of its dependence on gravitational theory, such evidencecan be called indirect; in contrast to an ideal direct observation ofDM itself.2 Part of this evidence now comes from the phenomenonof gravitational lensing: the bending of light near massive bodiespredicted by GTR, although not uniquely by that theory. Byobserving the deviation of light rays as they pass through certainregions of space, via deformations in the images of some visiblecelestial objects, one can infer the presence of non-visible mass inthose regions and therefore of DM. This description immediatelyindicates the ambiguous nature of such evidence: the presence ofDM is inferred by assuming a given gravitational theoryin thissense it would undoubtedly be indirect evidencethanks to theeffects it produces directly on electromagnetic radiationperhapsin this sense then it is closer to being direct evidence.3 But behindthe discussion about the direct nature of such evidence, morefundamental questions lurk. Is gravitational lensing (and for thatmatter, the other alleged evidence for DM) evidence in a suf-ciently strong sense? The idea of calling gravitational lensingdirect evidence seems to include the conviction that it is rmerin some way than observing the effects of DM on, let us say,the motion of stars and galaxies. Is such a conviction justied? Isgravitational lensing of a more reliable nature than other evi-dence? And, if so, in what sense? In this paper I try to answerthese questions.

In recent times, the observation of certain gravitational lensingeffects has reinforced the condence of many cosmologists in theDM Hypothesis; they claim that such phenomena conrm beyondreasonable doubt the existence of DM. In a recent paper, PeterKosso defends the notion that it is right to consider the gravita-tional lensing observed in the Bullet Cluster as evidence for theDM Hypothesis. To do so, he neutralises a potentially lethal threatfor such a claim: that because interpreting gravitational lensingpresupposes a theory of gravity, taking it as evidence of DM wouldbe circular. I think that Kosso0s assault on this challenge leadstowards the necessary conceptual discussion, but I see his answersas incomplete in a way that may lead to confusion. He argues thatone can escape the accusation of circularity by showing that onlypart of the complete gravitational theory, a part that is common toany real alternative to it, is needed to reproduce gravitationallensing. This seems to be perfectly correct, but not that easy toachieve. In this paper I will argue that the main reason that Kossoprovides for his defence of the evidential status of gravitationallensing is not correct, which means that his conclusion is toohasty. He argues that Einstein0s Equivalence Principle (EEP) isenough to derive the phenomenon in question; I argue to the

contrary that one needs to go into the details of the gravitationaltheories to settle the question regarding the evidential status ofgravitational lensing.

The layout of the paper is the following. In Section 2, I enterinto the discussion concerning criteria for deciding when someobservation is to be taken as evidence for a hypothesis. Section 3uses the previous discussion to frame the question about whetherthe observation of gravitational lensing can be evidence for theexistence of DM. In Section 4, I confront the relation betweengravitational lensing and the Equivalence Principle that is centralto Kosso0s argument. Section 5 brings to the fore what I think isthe missing element in Kosso0s discussion: the consideration oftheories of modied gravity. Finally, in Section 6, I specify in whatlimited sense, in the light of the previous discussion, one can saythat gravitational lensing is evidence for the DM Hypothesis.

2. What is evidence?

As with any hypothesis that deals with empirical matters, theDM Hypothesis can be conrmed to a greater or lesser extentdepending on the empirical evidence put forward. Such evidencecan be divided in two main types: direct and indirect evidence.The DM Hypothesis is a conjecture about the presence of somekind of matter in certain parts of the universe. Direct evidence forthis hypothesis would involve direct detection of DM (that is, forthe modern DM Hypothesis, detection of DM particles) or at leastthe detection of products resulting from collisions of DM particles.Indirect evidence, in general, can be understood as evidencemediated by some scientic theory. In this general sense, it isnow agreed that any evidence for a scientic theory is, strictlyspeaking, indirect; but this does little more than show thelimitations of such a characterisation. The basic idea behind thenotion of direct evidence is that we arrive at the evidence in a waythat is independent of the theories that are being tested (ques-tioned) at the time; or that the theories on which it is dependentare considered basic theories involved in the process of observa-tion and, in this sense, not questionable in this context. Inparticular, for the DM Hypothesis, indirect evidence is usuallyunderstood as evidence mediated by gravitational theory.

Two observations are in order here. First, this distinctionbetween direct and indirect evidence involves an inevitably vagueand contextual element. In some cases it is arguable whether thetheories involved in arriving at a given piece of evidence are basicenough for that evidence to be considered direct. It seemsunquestionable that some of the theories involved in the detectionof subatomic particles, for instance, might not deserve such aqualication. Nonetheless, one might think that there is a naturaldivide where we can draw the line: evidence is direct, relative to agiven hypothesis, only if none of the theoretical input needed forits determination is given by theories whose empirical conrma-tion somehow partly depends on the truth of that hypothesis.The rationale behind this characterisation is that all evidence isconditional on the correctness of the theories involved in itsdetermination, and evidence can be called indirect for a hypoth-esis when those theories involved in its determination are them-selves under question in the given context, which happens whenthese theories are partly dependent on the hypothesis beingexamined. Thus, not all theories should be considered along thesame lines in order to evaluate the nature of evidence. Some ofthem are more basic in the following sense: their justication doesnot depend substantially on that evidence; in other words, theevidence in question is not considered as part of their evidentialsupport. But these considerations are still vague because, formally,any observed empirical consequence of a theory can be taken aspart of its evidential support; which means that something else,

2 As I discuss below, this distinction is not as clear as one might have originallythought.

3 At least some authors do call gravitational lensing direct evidence (Clowe,Bradac, Gonzalez, Markevitch, Randall, & Jones, 2006).

A. Sus / Studies in History and Philosophy of Modern Physics 45 (2014) 6671 67

apart from its derivability from the theory, is implied in consider-ing a piece of evidence as part of the evidential support of a theory.And that something is partly determined by what the scienticcommunity takes as evidence in relation to the collection oftheories available at a certain moment. The contextual elementseems unavoidable.

The foregoing discussion has led us to my second point: thedistinction between direct and indirect evidence is intertwinedwith the much more interesting and general question about whatmakes an empirical observation evidence, simpliciter, for a giventheory or hypothesis. At rst glance, what I say above seems toquestion the idea that such a thing as indirect evidence may beevidence at all; if the determination of an empirical observation asevidence for a given hypothesis needs us to presuppose the truthof a theory for which that observation is also an essential part ofthe empirical evidence, then it would seem that we are faced witha agrant case of self-justication and that we should not call thatobservation evidence at all. Even if we rule out such observationsas evidence for that hypothesis, it is still in principle possiblethat some empirical item be considered evidence, because even ifit presupposes the truth of some theories, the conrmation ofthose theories is not fully dependent on that evidence; eitherbecause they are independently well tested or because we onlyneed to take part of the theory as true (which is itself indepen-dently well tested) in order to derive the observational data fromthe hypothesis. Nonetheless, this still would not be direct evi-dence, because its determination depends on taking some part ofthe theories as true (so it is not completely independent of thetruth of the theory). Be that as it may, the point is that this raisesthe question of what the minimal requirements are for anobservation to be counted as evidence for a given theory or hypo-thesis.

Implicit in Kosso0s discussion of the evidential status ofgravitational lensing we nd a necessary condition for somethingto be considered evidence that is intimately related to the discus-sion above. He presents, following Vanderburgh, what they callthe DM double bind. This consists of the fact that, since DM canonly be detected by its gravitational effects, in order to claim itsdetection one needs to assume a theory of gravity; but the theoryof gravity is only corroborated insofar as one presupposes that acertain mass distribution obtains. Thus, claiming detection of DMthrough its gravitational effects would be to restrain the possibilityof challenging GTR (at least, astronomically): it would always bepossible to claim DM detection to save the astronomical/cosmo-logical observations. If we add to this the fact that the gravitationaltheory is not independently well tested at the galactic scale, wehave the DM double bind.

Nevertheless, Kosso notes that this situation need not bedisastrous; its potential will depend on whether the completegravitational theory, or only part of it, is needed to detect DM. Theargument would be something like the following: if any candidatetheory of gravity needs to hypothesise DM in order to explaincertain phenomena, then observations of those phenomena cancondently be taken as detection of DM. Even if the gravitationaltheory is part of the determination of some observations asevidence in favour of DM, insofar as what the observationscontribute is common to any candidate gravitational theory, wecan say that such observations are, in effect, evidence (since itsdetermination is independent of any particular theory) even if weare not ready to call them direct evidence. Kosso claims moreoverthat only a part of the theory that is common to any viablegravitational theory is needed to claim detection of DM and thatpart is EEP.

As I mentioned above, implicit in Kosso0s discussion there is anecessary condition for an observation to be taken as evidence for agiven hypothesis; namely, that the observed empirical fact be

independent from the hypothesis in the following sense: it can bederived independent of (without assuming) the truth of any parti-cular theory for which the truth of the hypothesis is an essential partof its empirical conrmation. This necessary condition would be metif the hypothesis were needed in order to derive the empirical factfrom absolutely any candidate theory (if it were part of any theoryfromwhich the empirical fact in question could be derived). Taken inthis sense, however, it would be an unreasonably strong conditionitseems that it must always be possible, in principle, to devisealternative theories to derive certain given empirical facts that neednot assume a given hypothesisand surely not one that is met by theDM Hypothesis. The condition of independence can be given aweaker sense, as Kosso does; that is, that the hypothesis is part ofevery available theory from which the empirical fact could bederived; in Kosso0s words and referring specically to the DMHypothesis that it is part of all currently viable theories of gravity.This means that a test to decide whether a phenomenon is evidenceor not for a given hypothesis is to check whether that phenomenon isalso a consequence of every alternative viable theory that does notcontain the hypothesis.4 This is precisely what we should do inrelation to the DM Hypothesis.

3. Gravitational lensing as evidence for the DM Hypothesis

It has been argued that the phenomenon of gravitationallensing provides new (direct) evidence for the DM Hypothesis.Kosso, in his discussion, provides an argument to defend this view.The starting point for his argument is that the prediction ofgravitational lensingthat there is deection of light, to somedegree, in the proximity of massive objectsis common to allmetric theories of gravity. In fact, Kosso argues that the occurrenceof gravitational lensing (though not the degree to which light isdeected) is a consequence, not of GTR, but of EEP. If this is thecase, observation of gravitational lensing can be taken, in accor-dance with the discussion in Section 2, as detection of matter(visible or not) in certain regions; as long as we agree that theviable alternatives to GTR are all metric theories of gravity and,therefore, that they all satisfy EEP. If we add to this the fact that insome of those regions no ordinary matter is present, then thatshould be taken as a case of DM detection, i.e., as evidence for theexistence of DM. Kosso puts it this way:

A gravitational-lensing telescope can indicate where mass iswithout presuming the accuracy of GTR, but determining howmuch mass is there requires GTR, or some alternative metrictheory of gravity. This means that lensing could be used to detectdark matter, independent of GTR, only if the dark matter isseparated from normal, luminous matter. (Kosso, 2013, p. 146)

This is claimed to be the situation with respect to observations ofthe Bullet Cluster. Contrary to what usually happens, which is thatordinary mass/energy and DM are mixed together in galaxies andclusters, observations of the Bullet Cluster can be interpreted aspresenting a separation of ordinary matter and putative DM. Theseobservations are believed to be of a situation that involves thecollision of two clusters of galaxies. In the case of such collisionsand assuming that there is DM at the level of clusters, it would beexpected for the ordinary mass/energy and the DM to separate;ordinary matter from the two clusters, in the form of intergalacticgas, would interact and be left behind near the collision site, whileDMwhich is electrically neutralwould pass straight through thecollision site. The result would be that the centre of mass of the

4 Incidentally, here we can see very clearly how the conventional elemententers into consideration.

A. Sus / Studies in History and Philosophy of Modern Physics 45 (2014) 667168

baryonic matter, inferred from the electromagnetic interactions,and the centre of mass of the DM, inferred from the effects ongravitational lensing, would not coincide. This is exactly what is infact observed in the Bullet Cluster and is the reason why it isclaimed that the Bullet Cluster provides direct evidence for DM. InKosso0s words,

the important point is that any lensing theory puts the centerof gravity of the clusterthe centers, really, since there are twoclusters separatingdisplaced from the baryonic matter. Anyviable metric theory nds dark matter in the sieve. (Kosso,2013, p. 147)

So, if Kosso0s argument is right, the observation of gravitationallensing in the Bullet Cluster is indeed evidence for the existenceof DM, as has been claimed. In order to evaluate the argument,we must address two different points: whether it is right thatgravitational lensing is a consequence of EEP; and whether allviable alternatives to GTR full EEP. I will answer the secondquestion afrmatively, but the rst negatively. Even so, it couldstill be possible that the alternatives to GTR agree in their need forpostulating the existence of DM but, as I will endeavor to show,this does not guarantee that they support what is at present calledthe DM Hypothesis.

4. Einstein0s Equivalence Principle and gravitational lensing

Our rst concern must be whether gravitational lensing can betaken to be a straightforward consequence of EEP. In a loose sense,it seems clear that this is indeed the case. As Kosso notes,following Will (1993), any theory meeting EEP is what is called ametric theory of gravity. This means, according to Kosso, that thephenomenon of gravity is described as the shaping of the metricby matter in the sense that the motion of material test bodies willbe determined to follow time-like geodesics of the metric, which isitself shaped by the presence of matter. Moreover, as light rays areforced to follow null geodesics of the metricalso a consequenceof EEPthe deection of light in the presence of matter will be aconsequence of EEP. Nonetheless, the relevant question for ourconcern is whether this is enough for the more specic statementthat gravitational lensing can indicate where mass is. To answer thisa question, we need to examine the structure of EEP in somewhatmore detail.

Will (1993) provides a classic characterisation of EEP. Hepresents it as a generalisation of what he calls, following otherauthors, the Weak Equivalence Principle (WEP): the independenceof free fall with respect to the composition of bodies. EEP addsto WEP the requirement that the outcomes of all local non-gravitational experiments are independent of the velocity andthe spacetime location of the free-falling apparatus. Will providesan argument to justify a result that is essential for our discussionthat every theory that meets EEP is what he calls a metric theory ofgravity. Such theories are constrained by the requirement to meetthree postulates: that spacetime is endowed with a metric; thattest bodies follow geodesics of the metric; and that, locally, thelaws of physics are those of Special Relativity (Will, 1993, p. 22).As noted above, a consequence of this is that light rays follow nullgeodesics of the metric, and this is at the heart of gravitationallensing. Thus far, I nd no reason to disagree with Kosso0sconclusionthat gravitational lensing is a consequence of EEPso long as the shape of the geodesics is dependent on the presenceof matter in some form.

This does not, however, mean that gravitational lensingthedeection of light in a certain regionindicates the presence ofmatter in a given region, a least not fromwhat we have said so far.EEP concerns what matter and radiation do given a certain metric,

but not how that metric is shaped by matter. The conclusion thatKosso extracts from the latter consideration, namely, that thedeection of light indicates the presence of matter in a givenregion, seems to be jumping the gun somewhat. Agreed: we knowthat some form of matter must ultimately be responsible for theeffect; but we do not know where that matter is located. At least itdoes not seem at all correct to say that the determination of thelocation of matter is a consequence of EEP; its determination willdepend on the particular metric theory of gravity that we use toperform the calculations. Thus, the relevant question is, again,whether every viable alternative to GTR produces the same resultin relation to the cases of gravitational lensing in which thereis separation of ordinary matter and putative DM. Our previousdiscussion informs us that the answer to this question cannot beyes, because every viable alternative to GTR is a metric theory ofgravity. This is simply not enough. The answer would turn out tobe afrmative if the viable alternatives do in fact coincide on thelocation of the matter responsible for the gravitational lensing, butwe need further reasons to afrm that. In particular, it seemsnecessary here to examine one of the most promising alternativesto the DM Hypothesis that provides a different explanation forobserved discrepancies with GTR: the so-called modied theoriesof gravity.

5. Modied gravity

Alternatives to the DM Hypothesis that explain away the problemof the missing mass have been around for about three decades. In1983, Milgrom proposed a modication of Newtonian physics basedon the idea of modifying the relation between force and accelerationfor weak accelerations (Milgorm, 1983). This can also be formulatedas a modication of the effect of mass on the gravitational force. Theresult is a theory that changes the effect that mass has on thegravitational motion of bodies. Milgrom0s original proposal (ModiedNewtonian Dynamics: MOND) reproduced with great success, with-out DM, some of the observations which are accounted for byNewtonian gravitation only through the introduction of DM. Inparticular, the reproduction of observed rotation curves (that repre-sent the dependence of the rotational velocity of matter in a galaxyon the distance to the centre) for spiral galaxies is spectacular. So, atthe level of galactic dynamics, with no doubt, MOND is a seriouscontender: it derives galactic dynamics without introducing DM inany form. When it comes to reproducing observations of clusterdynamics, the success of MOND is not complete; to derive theobserved phenomena one still needs 23 times more matter thanis actually visible. Nonetheless, as defenders of MOND might claim,this still could be accomplished with ordinary matter that is difcultto detect (Bekenstein, 2010, p.7).5

Nevertheless, not much reection is needed to realise that atheory such as MOND cannot, by itself, be an alternative to the DMHypothesis. As I point out in the Introduction above, the DMHypothesis is needed in CDM to reproduce different cosmologicalphenomena. This means that a serious alternative to it must berelativistic and be able to account for various observations at thecosmological scale, such as structure formation or the nature ofthe CMB. Furthermore, a relativistic context is also the naturalsetting for an explanation of gravitational lensing effects. Beyondthese reasons, even advocates of MOND would consider that thetheory is not fundamental.

In recent times, relativistic alternatives to GTR plus DM havebeen proposed. Probably the one that has won most attention is

5 I return to the nature of the DM that alternative gravitational theoriesrequire, and the relevance of this for the discussion about the evidence forDM, below.

A. Sus / Studies in History and Philosophy of Modern Physics 45 (2014) 6671 69

Bekenstein0s relativistic generalisation of MOND: known as TeVeS,because on top of the metric of GR it introduces a vector eld and ascalar eld as fundamental gravitational elds. In its standardformulation, the equations of TeVeS are written in terms of twodifferent metrics: one of them responds to the material contents ofthe universe (sometimes called the geometric metric), the otherencodes the motion of bodies and radiation in the absence of non-gravitational interactions (the geodesic metric).6 The two metricsare related by a transformation involving the scalar and vectorelds. Such a distinction between the metrics allows the theory,in principle, to reproduce the difference between the predicted(according to GTR) and the observed response of matter andradiation to the visible mass without introducing DM. The basicidea is that the effect of mass on the motion of matter andradiation is mediated by the other elds and this would explainwhy the observed kinematics differs from that predicted by New-tonian or Einsteinian dynamics.

TeVeS, with the right choice of vector eld, is able to reproduceMONDian dynamics for galaxies. It also promises the possibility ofaccounting for many of the cosmological observations that CDMsuccessfully accounts for.

Without entering into the details of the theory, I wish to stresstwo points that are essential for the current discussion. The rst isthat TeVeS meets EEP according to the characterisation of theprinciple given above. In TeVeS, the geodesic metric plays the roleof the metric in that it is responsible for universal coupling; thedifference now being how this metric responds to the mattercontent. This means that the geodesic metric determines matterand radiation kinematics just as the (unique) metric does in GTR.Formally this is reected in the Lagrangian formulation of thetheory, where the matter Lagrangian only includes the geodesicmetric and the matter elds coupled to it.

The second point is that TeVeS does not satisfy Birkhoff0stheorem (Ferreira & Starkman, 2009, p. 8). Birkhoff0s theorem issatised by GTR and can be taken as a relativistic generalisation ofthe Gauss law that operates in Newtonian gravitation. The theo-rem implies that the gravitational effect on a test body inside aspherically symmetrical mass shell vanishes; therefore, any grav-itational pull towards the centre of the shell in such a situationmust be due to a mass concentration at the centre. A consequenceof the violation of this theorem is that in cases in which thegravitational effects seem to be caused by an interior massaccording to GTR (or Newtonian gravitation), one can attributesuch effects to the environment if the right theory were MONDian.Therefore, situations such as that arising in the Bullet Cluster inwhich gravitational lensing apparently locates some mass whereno ordinary mass is detected might be interpreted as produced byordinary mass that is simply located differently.

From this we can conclude, then, that saying that the gravita-tional lensing observed in the Bullet Cluster is evidence for thepresence of DM in places where no ordinary matter can be, due tothe fact that all the alternative theories to GTR are metric theories(and therefore meet EEP), is not correct. There are metric theoriesof gravity that, in principle, might reproduce the observationswithout the postulation of DM in places where GTR needs it.

The next step would be to ask whether in effect, and not onlyin principle, TeVeS or another theory can do the job withoutpostulating DM; and this issue is controversial. The consensusseems to be that MONDian theories need at least some form of DM(in the context of clusters, this was known before consideringgravitational lensing) which might be in the form of ordinarymatter (neutrinos, hot DM, might do the job) but there is noagreement about whether non-ordinary DM is also necessary.

Ferrerira and Starkman (p. 9), for instance, say that addingneutrinos plus the extra degrees of freedom of the environmentis sufcient to reproduce the lensing phenomenon at the BulletCluster. They stress, nonetheless, that MONDian theories introducedark elds on top of the gravitational eld and that this fact can beunderstood as a betrayal of the aim of modifying gravity in orderto avoid DM. Nevertheless, as Famaey and McGough (2013) pointout, the presence of the so-called dark elds means that the darksector does not disappear in MONDian theories, but the role thatsuch elds play is very different from that played by DM in CDM;so, one should not be too quick to conrm the DM Hypothesis dueto the existence of the dark sector in MONDian theories.Bekenstein (2010), on the other hand, discusses several attemptsmade in the literature to reproduce observed lensing using TeVeS.His conclusion seems to be that TeVeS can reproduce stronglensing with neutrinos only, but it might need other forms ofDM to t situations of weak lensing. Weak gravitational lensing,then, seems problematic for modied theories of gravity. In anycase, the issue does not seem to be decided yet; and this supportsthe idea that if gravitational lensing is to be taken as evidence forDM, it can only be so in a very weak sense, which is the theme ofthe next section.

6. Conclusion: Which DM Hypothesis?

The situation so far, in accordance with the preceding discus-sion, is the following: gravitational lensing is an observed phe-nomenon that seems to indicate, if we take the gravitationaltheory accepted in the Standard Cosmological Model for granted(namely GTR) that there is collisionless (non-baryonic) DM sepa-rate from ordinary matter. In this sense it seems clear thatgravitational lensing is evidence for the DM Hypothesis. If, incontrast, we take into account some popular alternative theoriesto GTR, it seems that gravitational lensing at most indicates that inclusters there is more matter than that which can be luminouslydetected (something known in advance). It is possible, however,that this extra matter is made up of neutrinos and is locatedsomewhere other than at the centre of the lens. In the light of this,can we consider gravitational lensing to be evidence for DM; and ifso, for precisely which hypothesis concerning DM?

In order to answer these questions it is useful to distinguishtwo extreme, broadly differentiated senses of the DM Hypothesis.The rst, a general DM Hypothesis, would simply state that thereis more matter than that detected electromagnetically. The second,the one maintained in the context of the present cosmologicalmodel, would say that DM is formed of particles that are notincluded in the Standard Model; baryons and neutrinos areexcluded as DM candidates for different reasons. Kosso, togetherwith many cosmologists, defends the notion that gravitationallensing, in particular as observed in the Bullet Cluster, constitutesevidence for the DM Hypothesis, but he does not specify suf-ciently clearly for which hypothesis concerning DM he claims thisto be the case. According to his criterion for determining when anobservation is to be considered evidence for a hypothesis, gravita-tional lensing can only be evidence for the most general DMHypothesis; what all viable alternative theories have in common isthat they must assume more mass than that inferred from theemitted radiation. If this is important, it is an extremely weaksense of being evidence for the DM Hypothesis. This is so for twodifferent reasons: there is no novelty involvedit was alreadyknown that the alternative theories to GTR need to introduce extramatter to reproduce cluster dynamicsand this general hypothesisis not what provides the specic content of the DM Hypothesisthat forms part of the present Standard Cosmological Model.According to that model, DM must necessarily be cold DM: it6 See, for instance, Ferreira & Starkman (2009, p. 5).

A. Sus / Studies in History and Philosophy of Modern Physics 45 (2014) 667170

cannot consist of either baryons or neutrinos. For gravitationallensing to be strong evidence for the DM Hypothesis, it shouldtherefore be the case that MONDian theories of gravity need non-standard particles to reproduce the phenomena. But we can nowsay, albeit with certain reservations, that it is agreed that suchtheories can do the job with standard particles only: either withdark baryons or with neutrinos. The conclusion then is thatgravitational lensing is not evidence for what at present is calledthe DM Hypothesis.

It is also important to stress a second point; apart from theanswer to the question of whether gravitational lensing is evidencefor DM, Kosso0s grounds for defending his position do not seemsound. He states that the reasonwhy gravitational lensing is evidencefor DM is that all the viable alternatives to GTR are metric theories ofgravity, which means that all of them satisfy EEP. He goes on to assertthat EEP is enough to reproduce the phenomena observed in thecases of gravitational lensing, because it encodes the fact that lightbends in the presence of matter. He uses this claim to defend the ideathat GTR is not needed to reproduce the observed lensing; any viablealternative would also do so. As I discuss above, this argument fails inits crucial step: EEP is not sufcient to derive gravitational lensing ifone does not include how matter content affects the metric. Hence,being a metric theory is not enough to provide an explanation of the

phenomenon of gravitational lensing; it is only when one analysesthe details of the alternative theories that one can decide whether ornot all of them need to introduce extra matter and of which kind.Ultimately, I should point out the strong sense of Kosso0s statementmay turn out to be correctit might be the case that the proposedmodications of gravity all end up needing non-standard DMbutthis is not something that can be derived from EEP alone.

References

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A. Sus / Studies in History and Philosophy of Modern Physics 45 (2014) 6671 71

Dark matter, the Equivalence Principle and modified gravityIntroductionWhat is evidence?Gravitational lensing as evidence for the DM HypothesisEinsteinprimes Equivalence Principle and gravitational lensingModified gravityConclusion: Which DM Hypothesis?References