Physics Beyond the Standard Model and Dark Matter
aa r X i v : . [ h e p - ph ] A p r PHYSICS BEYOND THE STANDARD MODELAND DARK MATTER
Hitoshi Murayama
Department of Physics, University of CaliforniaBerkeley, CA 94720, USA and
Theoretical Physcis Group, Lawrence Berkeley National LaboratoryBerkeley, CA 94720, USA ontents
1. Introduction 51.1. Particle Physics and Cosmology 51.2. Next Threshold 72. Why Beyond the Standard Model 92.1. Empirical Reasons 92.2. Philosophical and Aesthetic Reasons 102.3. Positron Analogue 132.4. Hierarchy Problem 163. Examples of Physics Beyond the Standard Model 163.1. Supersymmetry 173.2. Composite Higgs 183.3. Extra Dimensions 204. Evidence for Dark Matter 245. What Dark Matter Is Not 275.1. MACHOs 275.2. Neutrinos 295.3. CHAMPs and SIMPs 306. WIMP Dark Matter 306.1. WIMP 306.2. Boltzmann Equation 326.3. Analytic Approximation 346.4. Numerical Integration 356.5. The New Minimal Standard Model 366.6. Direct Detection Experiments 386.7. Popular WIMPs 406.8. Indirect Detection Experiments 417. Dark Horse Candidates 427.1. Gravitino 427.2. Axion 437.3. Other Candidates 468. Cosmic Coincidence 469. Conclusions 48Appendix A. Gravitational Lensing 48Appendix A.1. Deflection Angle 48Appendix A.2. Amplification in Microlensing 51Appendix A.3. MACHO search 53Appendix A.4. Strong Lensing 55References 57 . Introduction I’m honored to be invited as a lecturer at a Les Houches summer school whichhas a great tradition. I remember reading many of the lectures from past summerschools when I was a graduate student and learned a lot from them. I’m alsolooking forward to have a good time in the midst of beautiful mountains, eventhough the weather doesn’t seem to be cooperating. I’m not sure if I will ever getto see Mont Blanc!I was asked to give four general lectures on physics beyond the standardmodel. This is in some sense an ill-defined assignment, because it is a vast subjectfor which we know pratically nothing about. It is vast because there are so manypossibilities and speculations, and a lot of ink and many many pages of paper hadbeen devoted to explore it. On the other hand, we know practically nothing aboutit by definition, because if we did, it should be a part of the standard model ofparticle physics already. I will therefore focus more on the motivation why weshould consider physics beyond the standard model and discuss a few candidates,and there is no way I can present all the examples exhaustively. In addition, afterreviewing the program, I’ve realized that there are no dedicated lectures on darkmatter. Since this is a topic where particle physics and cosmology (I believe)are likely to come together in the near future, it is relevant to the theme of theschool “Particle Physics and Cosmology: the Fabric of Spacetime.” Therefore Iwill emphasize this connection in some detail.Because I try to be pedagogical in lectures, I will probably discuss many pointswhich some of you already know very well. Given the wide spectrum of back-ground you have, I aim at the common denominator. Hopefully I don’t end upboring you all!
At the first sight, it seems crazy to talk about particle physics and cosmology to-gether. Cosmology is the study of the universe, where the distance scale involvedis many Gigaparsecs ∼ cm. Particle physics studies the fundamental con-stituent of matter, now reaching the distance scale of ∼ − cm. How can theyhave anything in common? 5 H. Murayama
The answer is the Big Bang. Discovery of Hubble expansion showed that thevisible universe was much smaller in the past, and the study of cosmic microwavebackground showed the universe was filled with a hot plasma made of photons,electrons, and nuclei in thermal equilibrium. It was hot . As we contemplateearlier and earlier epochs of the universe, it was correspondingly smaller andhotter.On the other hand, the study of small scales d in particle physics translates tolarge momentum due to the uncertainty principle, p ∼ ~ /d . Since large momen-tum requires relativity, it also means high energy E ∼ cp ∼ ~ c/d . Physics athigher energies is relevant for the study of higher temperatures T ∼ E/k , whichwas the state of the earlier universe.This way, Big Bang connects microscopic physics to macroscopic physics.And we have already seen two important examples of this connection.Atomic and molecular spectroscopy is based on quantum physics at the atomicdistance d ∼ − cm. This spectroscopy is central to astronomy to identify thechemical composition of faraway stars and galaxies which we never hope to getto directly and measure their redshifts to understand their motion including theexpansion of the space itself. The cosmic microwave background also originatesfrom the atomic-scale physics when the universe was as hot as T ∼ Kand hence was in the plasma state. This is the physics which we believe weunderstand from the laboratory experiments and knowledge of quantum mechan-ics and hence we expect to be able to extract interesting information about theuniverse. Ironically, cosmic microwave background also poses a “wall” becausethe universe was opaque and we cannot “see” with photons the state of the uni-verse before this point. We have to rely on other kinds of “messengers” to extractinformation about earlier epochs of the universe.The next example of the micro-macro connection concerns with nuclear phy-sics. The stars are powered by nuclear fusion, obviously a topic in nuclearphysics. This notion is now well tested by the recent fantastic development inthe study of solar neutrinos, where the core temperature of the Sun is inferredfrom the helioseismology and solar neutrinos which agree at better than a per-cent level. Nuclear physics also determines death of a star. Relatively heavy starseven end up with nuclear matter, i.e. neutron stars, where the entire star basicallybecomes a few kilometer-scale nucleus. On the other hand, when the universewas as hot as MeV (ten billion degrees Kelvin), it was too hot for protons andneutrons to be bound in nuclei. One can go through theoretical calculations onhow the protons and neutrons became bound in light nuclear species, such as deu-terium, He, He, Li, based on the laboratory measurements of nuclear fusioncross sections, as well as number of neutrino species from LEP (Large ElectronPositron collider at CERN). This process is called Big-Bang Nucleosynthesis(BBN). There is only one remaining free parameter in this calculation: cosmic hysics Beyond the Standard Model and Dark Matter baryon density. The resulting predictions can be compared to astronomical de-terminations of light element abundances by carefully selecting the sites whichare believed to be not processed by stellar evolutions. There is (in my humbleopinion) reasonable agreement between the observation and theoretical predic-tions (see, e.g. , [1]). This agreement gives us confidence that we understand thebasic history of the universe since it was as hot as MeV. D a r k M a tt e r ? b a r y o n a s y m m e t r y ? li gh t e l e m e n t s C M B D a r k E n e r gy ? − sec 1sec 300Kyr Today o b s e r v a t o r i e s r e c o n s t r u c t B i g B a n g a c c e l e r a t o r s r e c r e a t e B i g B a n g BigBang C M B B - m od e? Fig. 1. Possible messengers from early universe.
We currently do not have messengers from epochs in early universe above theMeV temperature. In other words, our understanding of early universe physicsis not tested well for
T > ∼ MeV. Yet many of the topics discussed in this schoolare possible messengers from earlier era: dark matter ( GeV?), baryon asym-metry of the universe ( GeV?), density perturbations (scalar and tensor com-ponents) from the inflationary era ( GeV?). These are the energy scales thatlaboratory measurements have not reached to reveal the full particle spectrum andtheir interactions, hence the realm of physics beyond the standard model . Under-standing of such early stages of the universe requires the development in particlephysics, while the universe as a whole may be regarded as a testing ground ofhypothesized particle physics at high energies beyond the reach of accelerators.This way, cosmology and particle physics help and require each other.
There is a strong anticipation in the community that we are just about to reveal anew threshold in physics. Let me tell you why from a historical perspective.We (physicists) do not witness crossing a new threshold very often, but eachtime it happened, it resulted in a major change in our understanding of MotherNature.
H. Murayama
Around year 1900, we crossed the threshold of atomic scale. It is impressiveto recall how much progress chemists have made without knowing the underlyingdynamics of atoms and molecules. But the empirical understanding of chemistryhad clear limitation. For example, van der Waals equation of state showed therewas the distance scale of about − cm below which the state-of-art scientificknowledge of the time could not be applied, namely the size of atoms. Oncethe technology improved to study precision spectroscopy that allowed people toprobe physics inside the atoms, a revolution followed. It took about three decadesfor quantum mechanics to be fully developed but it forever changed our under-standing of nature. The revolution went on well into the 40’s when the marriageof quantum mechanics and relativity was completed in Quantum ElectroDynam-ics.Next important threshold was crossed around 1950 when new hadron reso-nances and strange particles were discovered, crossing the threshold of the stronginteraction scale ∼ − cm. Discovery of a zoo of “elementary particles” ledto a great deal of confusion for about three decades. It eventually led to the reve-lation of non-perturbative dynamics of quantum field theory, namely confinementof quarks, dimensional transmutation, and dynamical symmetry breaking of chi-ral symmetry. More importantly, it showed a new layer in nature where quarksand gluons take over the previous description of subatomic world with protonsand neutrons. The experimental verification of this theory, Quantum ChromoDy-namics, took well into 90’s at numerous accelerators PETRA, PEP, TRISTAN,LEP, and HERA.One more force that is yet to be fully understood is the weak interaction. Itsscale was known from the time of Fermi back in 1933 when he wrote the firsttheory of nuclear beta decay. The theory contained one dimensionful constant G F ≈ (300 GeV ) − ≈ (10 − cm ) . Seven decades later, we are just aboutto reach this energy scale in accelerator experiments, at Tevatron and LHC. Wedo not really know what Nature has in store for us, but at least we’ve known allalong that this is another important energy scale in physics. If we are not misled,this is the energy scale associated with the cosmic superconductor. Just like theMeissner effect lets magnetic field penetrate into a superconductor only over afinite distance, the cosmic superconductor lets the weak force carried by W and Z bosons go over a tiny distance: a billionth of a nanometer. Right now we areonly speculating what revolution may take place at this distance scale. A newlayer of matter? New dimensions of space? Quantum dimensions? Maybe stringtheory? We just don’t know yet.Of course historical perspective does not guarantee that history repeats itselfin an equally exciting fashion. But from all what we know, there is a good reasonto think that indeed a new threshold is waiting to be discovered at the TeV energyscale, as I will discuss in the next section. Another simple fact is that crossing hysics Beyond the Standard Model and Dark Matter a new threshold is something like twice-in-a-century experience. I’m excited tothink that we are just about to witness one, a historic moment.An interesting question is what fundamental physics determines these thresh-olds. The atomic scale, that looked like a fundamental limitation in understandingback in the 19th century, did not turn out to be a fundamental scale at all. It is aderived scale from the mass of the electron and the fundamental constants, a B = ~ e m e ≈ − cm . (1.1)The strong-interaction scale is also a derived energy scale from the coupling con-stant a s = M e − π /g s ( M ) b ≈ − cm , (1.2)where g s is the strong coupling constant defined at a high-energy scale M and b is the beta function coefficient. Because of the asymptotic freedom, the strongcoupling constant is weak (what an oxymoron!) at high energies, while it be-comes infinitely strong at low energies. The scale of strong interaction is wherethe strength of the interaction blows up. In other words, the two thresholds wehave crossed so far were extremely exciting, yet they turned out to be not funda-mental! They point to yet deeper physics that determine these parameters in na-ture. Maybe the weak-interaction scale is also a derived scale from some deeperphysics at yet shorter distances.
2. Why Beyond the Standard Model
Until about ten years ago, particle physicists lamented that the standard modeldescribed every new data that came out from experiments and we didn’t havea clue what may lie beyond the standard model. Much of the discussions onphysics beyond the standard model therefore were not based on data, but ratheron theoretical arguments, primarily philosophical and aesthetic displeasure withthe standard model. It all changed the last ten years when empirical evidenceappeared that demonstrated that the standard model is incomplete: • Non-baryonic dark matter, • Dark energy, • Neutrino mass, • Nearly scale-invariant, Gaussian, and apparently acausal density perturba-tions, • Baryon asymmetry. H. Murayama
I will discuss strong evidence for non-baryonic dark matter and dark matter laterin my lectures. Density fluctuation is covered in many other lectures in thisschool by Lev Kofman, Sabino Matarrese, Yannick Mellier, Simon Prunet, andRomain Teyssier. Neutrino mass is discussed by Sergio Pastor, and baryon asym-metry by Jim Cline. The bottom line is simple: we already know that there mustbe physics beyond the standard model. However, we don’t necessarily know theenergy (or distance) scale for this new physics, nor what form it takes. One con-servative approach is to try to accommodate all of these established empiricalfacts into the standard model with minimum particle content: The New Mini-mal Standard Model [2]. I will discuss some aspects of the model later. Buttheoretical arguments suggest the true model be much bigger, richer, and moreinteresting.
What are the theoretical arguments that demand physics beyond the standardmodel? As I mentioned already, they are based on somewhat philosophical ar-guments and aesthetic desires and not exactly on firm footing. Nonetheless theyare useful and suggestive, especially because nature did solve some of the similarproblems in the past by invoking interesting mechanisms. A partial list relevantto my lectures here is • Hierarchy problem: why G F ∼ − GeV − ≪ G N ∼ − GeV − ? • Why θ QCD < ∼ − ≪ ? • Why are there three generations of particles? • Why are the quantum numbers of particles so strange, yet do anomalies cancelso non-trivially?For an expanded list of the “big questions”, see e.g. , [3].To understand what these questions are about, it is useful to remind ourselveshow the standard model works. It is a gauge theory based on the SU (3) × SU (2) × U (1) gauge group with the Lagrangian L SM = − g ′ B µν B µν − g Tr( W µν W µν ) − g s Tr( G µν G µν )+ ¯ Q i i DQ i + ¯ L i i DL i + ¯ u i i Du i + ¯ d i i Dd i + ¯ e i i De i +( Y iju ¯ Q i u j ˜ H + Y ijd ¯ Q i d j H + Y ijl ¯ L i e j H + h . c . )+( D µ H ) † ( D µ H ) − λ ( H † H ) − m H † H + θ π ǫ µνρσ Tr( G µν G ρσ ) . (2.1) hysics Beyond the Standard Model and Dark Matter It looks compact enough that it should fit on a T-shirt. Why don’t we see such aT-shirt while we see Maxwell equations a lot?The first two lines describe the gauge interactions. The covariant derivatives D = γ µ D µ in the second line are determined by the gauge quantum numbersgiven in this table: SU (3) SU (2) U (1) chirality Q +1 / left U +2 / right D − / right L − / left E − rightThis part of the Lagrangian is well tested, especially by the LEP/SLC data in the90’s. However, the quantum number assignments (especially U (1) hypercharges)appear very strange and actually hard to remember. Why this peculiar assign-ment is one of the things people don’t like about the standard model. In addition,they are subject to non-trivial anomaly cancellation conditions for SU (3) U (1) , SU (2) U (1) , gravity U (1) , U (1) , and Witten’s SU (2) anomalies. Many of usare left with the feeling that there must be a deep reason for this baroque quantumnumber assignments which had led to the idea of grand unification.The third line of the Lagrangian comes with the generation index i, j = 1 , , and is responsible for masses and mixings of quarks and masses of charged lep-tons. The quark part has been tested precisely in this decade at B -factories whilethere is a glaring omission of neutrino masses and mixings that became estab-lished since 1998. In addition, it appears unnecessary for nature to repeat el-ementary particles three times. The repetition of generations and the origin ofmass and mixing patterns remains an unexplained mystery in the standard model.The last line is completely untested. The first two terms describe the Higgsfield and its interaction to the gauge fields and itself. Having not seen the Higgsboson so far, it is far from established. The mere presence of the Higgs field posesan aesthetic problem. It is the only spinless field in the model, but it is introducedfor the purpose of doing the most important job in the model. In addition, wehave not seen any elementary spinless particle in nature! Moreover, the potential It reminds me of an anecdote from when the standard model was just about getting off the groundaround 1978. There was a convergence of the data to the standard model and people got very excitedabout it. Then Tini Veltman gave a talk asking “do you really think this is great model?” and wrotedown every single term in the Lagrangian without using a compact notation used here over pages andpages of transparencies. Unfortunately I don’t remember who told me this story. I often told my friends that I chose physics over chemistry or biology because I didn’t want tomemorize anything, but this kind of table casts serious doubt on my choice! H. Murayama needs to be chosen with m < to cause the cosmic superconductivity whichdoes not give any reason why our universe is in this state. I will discuss moreproblems about it in a few minutes. Overall, this part of the model looks veryartificial.The last term is the so-called θ -term in QCD and violates T and CP . Thevacuum angle θ is periodic under θ → θ + 2 π , and hence a “natural” value of θ is believed to be order unity. On the other hand, the most recent experimentalupper limit on the neutron electric dipole moment | d n | < . × − e cm (90%CL) [4] translates to a stringent upper limit θ < (1 . ± . × − using theformula in [5]. Why θ is so much smaller than the “natural” value is the strongCP problem, and again the standard model does not offer any explanations.Now we have more to say about the Higgs sector (the third line). Clearly it isvery important because (1) this is the only part of the Standard Model which has adimensionful parameter and hence sets the overall energy scale for the model, and(2) it has the effect of causing cosmic superconductivity without explaining itsmicroscopic mechanism. For the usual superconductors studied in the laboratory,we can use the same Lagrangian, but it is derived from the more fundamentaltheory by Bardeen, Cooper, and Schrieffer. The weak attractive force betweenelectrons by the phonon exchange causes electrons to get bound and condense.The “Higgs” field is the Cooper pair of electrons. And one can show why it hasthis particular potential. In the standard model, we do not know if Higgs field iselementary or if it is made of something else, nor what mechanism causes it tohave this potential.All the puzzles raised here (and more) cry out for a more fundamental theoryunderlying the Standard Model. What history suggests is that the fundamentaltheory lies always at shorter distances than the distance scale of the problem.For instance, the equation of state of the ideal gas was found to be a simpleconsequence of the statistical mechanics of free molecules. The van der Waalsequation, which describes the deviation from the ideal one, was the consequenceof the finite size of molecules and their interactions. Mendeleev’s periodic tableof chemical elements was understood in terms of the bound electronic states,Pauli exclusion principle and spin. The existence of varieties of nuclide was dueto the composite nature of nuclei made of protons and neutrons. The list couldgo on and on. Indeed, seeking answers at more and more fundamental level isthe heart of the physical science, namely the reductionist approach.The distance scale of the Standard Model is given by the size of the Higgsboson condensate v = 250 GeV. In natural units, it gives the distance scale of d = ~ c/v = 0 . × − cm. We therefore would like to study physics at distance This is an amazing limit. If you blow up the neutron to the size of the Earth, this limit correspondsto a possible displacement of an electron by less than ten microns. hysics Beyond the Standard Model and Dark Matter scales shorter than this eventually, and try to answer puzzles whose partial listwas given in the previous section.Then the idea must be that we imagine the Standard Model to be valid downto a distance scale shorter than d , and then new physics will appear which willtake over the Standard Model. But applying the Standard Model to a distancescale shorter than d poses a serious theoretical problem. In order to make thispoint clear, we first describe a related problem in the classical electromagnetism,and then discuss the case of the Standard Model later along the same line [9]. In the classical electromagnetism, the only dynamical degrees of freedom areelectrons, electric fields, and magnetic fields. When an electron is present in thevacuum, there is a Coulomb electric field around it, which has the energy of ∆ E Coulomb = 14 πε e r e . (2.2)Here, r e is the “size” of the electron introduced to cutoff the divergent Coulombself-energy. Since this Coulomb self-energy is there for every electron, it has tobe considered to be a part of the electron rest energy. Therefore, the mass of theelectron receives an additional contribution due to the Coulomb self-energy: ( m e c ) obs = ( m e c ) bare + ∆ E Coulomb . (2.3)Experimentally, we know that the “size” of the electron is small, r e < ∼ − cm.This implies that the self-energy ∆ E is greater than 10 GeV or so, and hencethe “bare” electron mass must be negative to obtain the observed mass of theelectron, with a fine cancellation like . − . . . (2.4)Even setting a conceptual problem with a negative mass electron aside, sucha fine-cancellation between the “bare” mass of the electron and the Coulombself-energy appears ridiculous. In order for such a cancellation to be absent, weconclude that the classical electromagnetism cannot be applied to distance scalesshorter than e / (4 πε m e c ) = 2 . × − cm. This is a long distance in thepresent-day particle physics’ standard.The resolution to this problem came from the discovery of the anti-particle ofthe electron, the positron, or in other words by doubling the degrees of freedomin the theory. The Coulomb self-energy discussed above can be depicted by a dia-gram Fig. 2 where the electron emits the Coulomb field (a virtual photon) which Do you recognize π ? H. Murayama
Fig. 2. The Coulomb self-energy of the electron.Fig. 3. The bubble diagram which shows the fluctuation of the vacuum. is absorbed later by the electron (the electron “feels” its own Coulomb field). But now that we know that the positron exists (thanks to Anderson back in 1932),and we also know that the world is quantum mechanical, one should think aboutthe fluctuation of the “vacuum” where the vacuum produces a pair of an electronand a positron out of nothing together with a photon, within the time allowedby the energy-time uncertainty principle ∆ t ∼ ~ / ∆ E ∼ ~ / (2 m e c ) (Fig. 3).This is a new phenomenon which didn’t exist in the classical electrodynamics,and modifies physics below the distance scale d ∼ c ∆ t ∼ ~ c/ (2 m e c ) = 200 × − cm. Therefore, the classical electrodynamics actually did have a finite ap-plicability only down to this distance scale, much earlier than . × − cm asexhibited by the problem of the fine cancellation above. Given this vacuum fluc-tuation process, one should also consider a process where the electron sitting inthe vacuum by chance annihilates with the positron and the photon in the vacuumfluctuation, and the electron which used to be a part of the fluctuation remainsinstead as a real electron (Fig. 4). V. Weisskopf [10] calculated this contributionto the electron self-energy, and found that it is negative and cancels the leading The diagrams Figs. 2, 4 are not Feynman diagrams, but diagrams in the old-fashioned perturba-tion theory with different T -orderings shown as separate diagrams. The Feynman diagram for theself-energy is the same as Fig. 2, but represents the sum of Figs. 2, 4 and hence the linear divergenceis already cancelled within it. That is why we normally do not hear/read about linearly divergentself-energy diagrams in the context of field theory. hysics Beyond the Standard Model and Dark Matter Fig. 4. Another contribution to the electron self-energy due to the fluctuation of the vacuum. piece in the Coulomb self-energy exactly: ∆ E pair = − πε e r e . (2.5)After the linearly divergent piece /r e is canceled, the leading contribution in the r e → limit is given by ∆ E = ∆ E Coulomb + ∆ E pair = 3 α π m e c log ~ m e cr e . (2.6)There are two important things to be said about this formula. First, the correction ∆ E is proportional to the electron mass and hence the total mass is proportionalto the “bare” mass of the electron, ( m e c ) obs = ( m e c ) bare (cid:20) α π log ~ m e cr e (cid:21) . (2.7)Therefore, we are talking about the “percentage” of the correction, rather than ahuge additive constant. Second, the correction depends only logarithmically onthe “size” of the electron. As a result, the correction is only a 9% increase inthe mass even for an electron as small as the Planck distance r e = 1 /M P l =1 . × − cm.The fact that the correction is proportional to the “bare” mass is a consequenceof a new symmetry present in the theory with the antiparticle (the positron): thechiral symmetry. In the limit of the exact chiral symmetry, the electron is mass-less and the symmetry protects the electron from acquiring a mass from self-energy corrections. The finite mass of the electron breaks the chiral symmetryexplicitly, and because the self-energy correction should vanish in the chiral sym-metric limit (zero mass electron), the correction is proportional to the electronmass. Therefore, the doubling of the degrees of freedom and the cancellationof the power divergences lead to a sensible theory of electron applicable to veryshort distance scales. An earlier paper by Weisskopf actually found two contributions to add up. After Furry pointedout a sign mistake, he published an errata with no linear divergence. I thank Howie Haber for lettingme know. H. Murayama
In the Standard Model, the Higgs potential is given by V = m | H | + λ | H | , (2.8)where v = h H i = − m / λ = (176 GeV) . Because perturbative unitarityrequires that λ < ∼ , − m is of the order of (100 GeV) . However, the masssquared parameter m of the Higgs doublet receives a quadratically divergentcontribution from its self-energy corrections. For instance, the process where theHiggs doublets splits into a pair of top quarks and come back to the Higgs bosongives the self-energy correction ∆ m = − h t π r H , (2.9)where r H is the “size” of the Higgs boson, and h t ≈ is the top quark Yukawacoupling. Based on the same argument in the previous section, this makes theStandard Model not applicable below the distance scale of − cm. This is thehierarchy problem. In other words, if we don’t solve this problem, we can’t eventalk about physics at much shorter distances without an excessive fine-tuning inparameters.It is worth pondering if the mother nature may fine-tune. Now that the cos-mological constant appears to be fine-tuned at the level of − , should we bereally worried about the fine-tuning of v /M P l ≈ − [6]? In fact, some peo-ple argued that the hierarchy exists because intelligent life cannot exist otherwise[7]. On the other hand, a different way of varying the hierarchy does seem tosupport stellar burning and life [8]. I don’t get into this debate here, but I’d liketo just point out that a different fine-tuning problem in cosmology, horizon andflatness problems, pointed to the theory of inflation, which in turn appears to beempirically supported by data. I just hope that proper solutions will be foundto both of these fine-tuning problems and we will see their manifestations at therelevant energy scale, namely TeV. You have to be an optimist to work on bigproblems.
3. Examples of Physics Beyond the Standard Model
Given various problems in the standard model discussed in the previous section,especially the hierarchy problem, many possible directions of physics beyond thestandard model have been proposed. I can review only a few of them here giventhe spacetime constraint. But I especially emphasize the aspect of the modelsthat leads to a (nearly) stable neutral particle as a good dark matter candidate. hysics Beyond the Standard Model and Dark Matter The motivation for supersymmetry is to make the Standard Model applicableto much shorter distances so that we can hope that the answers to many of thepuzzles in the Standard Model can be given by physics at shorter distance scales[11]. In order to do so, supersymmetry repeats what history did with the positron:doubling the degrees of freedom with an explicitly broken new symmetry. Thenthe top quark would have a superpartner, the stop, whose loop diagram givesanother contribution to the Higgs boson self energy ∆ m = +6 h t π r H . (3.1)The leading pieces in /r H cancel between the top and stop contributions, andone obtains the correction to be ∆ m + ∆ m = − h t π ( m t − m t ) log 1 r H m t . (3.2)One important difference from the positron case, however, is that the massof the stop, m ˜ t , is unknown. In order for the ∆ m to be of the same order ofmagnitude as the tree-level value m = − λv , we need m t to be not too farabove the electroweak scale. TeV stop mass is already a fine tuning at the level ofa percent. Similar arguments apply to masses of other superpartners that coupledirectly to the Higgs doublet. This is the so-called naturalness constraint on thesuperparticle masses (for more quantitative discussions, see papers [12]).Supersymmetry doubles the number of degrees of freedom in the standardmodel. For each fermion (quarks and leptons), you introduce a complex scalarfield (squarks and sleptons). For each gauge boson, you introduce gaugino, apartner Majorana fermion (a fermion field whose anti-particle is itself). I do notgo into technical aspect of how to write a supersymmetric quantum field theory;you should consult some review articles [13, 14].One important point related to dark matter is the proton longevity. We knowfrom experiments such as SuperKamiokande that proton is very long lived (if notimmortal). The life time for the decay mode p → e + π is longer than . × years, at least twenty-three orders of magnitude longer than the age of theuniverse! On the other hand, if you write the most general renormalizable theorywith standard model particle content consistent with supersymmetry, it allowsfor vertices such as ǫ ijk u i d j ˜ s k and eu i ˜ s ∗ i (here i, j, k are color indices). Then This is a terrible name, which was originally meant to be “scalar top” or “supersymmetric top.”Some other names are even worse: sup , sstrange , etc. If supersymmetry will be discovered at LHC,we should seriously look for better names for the superparticles, maybe after the names of rich donors. H. Murayama one can draw a Feynman diagram like one in Fig. 5. If the couplings are O (1) ,and superparticles around TeV, one finds the proton lifetime as short as τ p ∼ m s /m p ∼ − sec; a little too short! ˜ sudu u ¯ ue + Fig. 5. A possible Feynman diagram with supersymmetric particles that can lead to a too-rapid protondecay p → e + π . Because of this embarrassment, we normally introduce a Z symmetry called“ R -parity” defined by R p = ( − B + L +2 s = ( − matter R π (3.3)where s is the spin. What it does is to flip the sign of all matter fields (quarksand leptons) and perform π rotation of space at the same time. In effect, itassigns even parity to all particles in the standard model, and odd parity to theirsuperpartners. Here is a quick check. For the quarks, B = 1 / , L = 0 , and s = 1 / , and we find R p = +1 , while for squarks the difference lies in s = 0 and hence R p = − . This symmetry forbids both of the bad vertices in Fig. 5.Once the R -parity is imposed, there are no baryon- and lepton-number vi-olating interaction you can write down in a renormalizable Lagrangian with thestandard model particle content. This way, the R -parity makes sure that pro-ton is long lived. Then the lightest supersymmetric particle (LSP), with odd R -parity, cannot decay because there are no other states with the same R -paritywith smaller mass it can decay into by definition. In most models it also turns outto be electrically neutral. Then one can talk about the possibility that the LSP isthe dark matter of the universe. Another way the hierarchy problem may be solved is by making the Higgs bosonto actually have a finite size. Then the correction in Eq. (2.9) does not requiretremendous fine-tuning as long as the physical size of the Higgs boson is about An obvious objection is that imposing R -parity appears ad hoc. Fortunately there are severalways for it to emerge from a more fundamental theory. Because the R -parity is anomaly-free [15],it may come out from string theory. Or R p can arise as a subgroup of the SO (10) grand unifiedgauge group because the matter belongs to the spinor representation and Higgs to vector, and hence π rotation in the gauge group leads precisely to ( − matter . It may also be an accidental symmetrydue to other symmetries of the theory [16, 17] so that it is slightly broken and dark matter mayeventually decay. hysics Beyond the Standard Model and Dark Matter r H ≈ (TeV) − ≈ − cm. This is possible if the Higgs boson is a compositeobject made of some elementary constituents.The original idea along this line is called technicolor (see reviews [18, 19]),where a new strong gauge force binds fermions and anti-fermions much likemesons in the real QCD. Again just like in QCD, fermion anti-fermion pairhave a condensate h ¯ ψψ i 6 = 0 breaking chiral symmetry. In technicolor theo-ries, this chiral symmetry breaking is nothing but the breaking of the electroweak SU (2) × U (1) symmetry to the U (1) QED subgroup. Because the Higgs bosonis heavy and strongly interacting, it is expected to be too wide to be seen as aparticle state.It is fair to say, however, that the technicolor models suffer from various prob-lems. First of all, it is difficult to find a way of generating sufficient masses forquarks and leptons, especially the top quark, because you have to rely on higherdimension operators of type ¯ qq ¯ ψψ/ Λ . The scale Λ must be low enough to gen-erate m t , while high enough to avoid excessive flavor-changing neutral current.In addition, there is tension with precision electroweak observables. These ob-servables are precise enough that they constrain heavy particles coupled to Z -and W -bosons even though we cannot produce them directly. Because of this issue, there are various other incarnations of composite Higgsidea, which try to get a relatively light Higgs boson as a bound state [26, 27]. Oneof the realistic models is called “little Higgs” [24, 25]. Because of the difficultyof achieving Higgs compositeness at the TeV scale, we are better off puttingoff the compositeness scale to about 10 TeV to avoid various phenomenologicalconstraints. Then you must wonder if the problem with Eq. (2.9) comes back.But there is a way of protecting the scale of Higgs mass much lower than thecompositeness scale by using symmetries similar to the reason why a pion isso much lighter than a proton. If you are clever, you can arrange the structureof symmetry such that it eliminates the one-loop correction in Eq. (2.9) and thecorrection arises only at the two-loop level. Then the compositeness ∼ TeVis not a problem.Another attractive idea is to use extra dimensions to generate the Higgs fieldfrom a gauge field, called “Higgs-gauge unification” [28, 29, 30, 31, 32]. Weknow the mass of the gauge boson is forbidden by the gauge invariance. If theHiggs field is actually a gauge boson (spin one), but if it is spinning in extra di-mensions, we (as observers stuck in four dimensions) perceive it not to spin. Notonly this gives us raison d’être of (apparently) spinless degrees of freedom, italso provides protection for the Higgs mass and hence solves the hierarchy prob-lem. The best implementation of this line of thinking is probably the holographic It is curious that higher dimensional versions of technicolor models called Higgsless models[21] do much better [22]. A supersymmetric version of technicolor also does better than the originaltechnicolor [23]. H. Murayama
Higgs model in Refs. [33, 34] which involves the warped extra dimension I willbriefly discuss in the next section. It should also be said that many of the ideasmentioned here are closely related to each other [35].Similarly to the case of supersymmetry, people often introduce a Z symmetryto avoid certain phenomenological embarrassments. In little Higgs theories, tree-level exchange of new particles tend to cause tension with precision electroweakconstraints. Then the new states must be sufficiently heavy so that the hierarchyproblem is reintroduced. By imposing “ T -parity,” new particles can only appearin loops for low-energy processes and the constraints can be easily avoided [36].Then the lightest T -odd particle (LTP) becomes a candidate for dark matter. Intechnicolor models, the lightest technibaryon is stable (just like proton in QCD)and a dark matter candidate [37]. The source of the hierarchy problem is our thinking that there is physics at muchshorter distances than − cm. What if there isn’t? What if physics ends at − cm where quantum field theory stops applicable and is taken over bysomething more radical such as string theory? Normally we associate the ulti-mate limit of field theory with the Planck scale d P l ≈ − cm = G / N wherethe gravity becomes as strong as other forces and its quantum effects can nolonger be ignored. How then can the quantum gravity effects enter at a muchlarger distance scale such as − cm?One way is to contemplate “large” extra dimensions of size R [38]. Imaginethere are n extra dimensions in addition to our three-dimensional space. If youplace two test masses at a distance r much shorter than R , the field lines ofgravity spread out into n dimensions and the force decreases as the surfacearea r − ( n +2) . However, if the distance is longer than R , there is a limit to whichhow much the field lines can spread because they are squeezed within the size R . Therefore, the force decreases only as r − for r ≫ R , reproducing the usualinverse square law of gravity. It turns out that the inverse square law is tested onlydown to 44 µ m [39] (even though this is very impressive!) and extra dimensionssmaller than that are allowed experimentally.Matching two expressions for the gravitational force at r = R , we can relatedthe Newton’s constant in n + 3 dimensions to the Newton’s constant G N in threedimensions G n +3 R n +2 = G N R , (3.4)and hence G n +3 = G N R n . In the natural unit ~ = c = 1 , the mass scaleof gravity is related to the Planck scale by M n +2 R n = M P l . Even if the true hysics Beyond the Standard Model and Dark Matter energy scale of quantum gravity is at M ∼ TeV, we may find an apparent scaleof gravity to be M P l ∼ GeV. Then the required size of extra dimensions is R = cm ( n = 1)10 − cm ( n = 2)10 − cm ( n = 3)10 − cm ( n = 6) (3.5)Obviously the n = 1 case is excluded because R is even bigger than 1AU ≈ cm. The case n = 2 is just excluded by the small-scale gravity experiment,while n ≤ is completely allowed. Fig. 6. Large extra dimensions. Even though the three-brane is drawn at the ends of extra dimensions,it does not have to be.
If we don’t see the extra dimensions directly, what do they do to us? Let uslook at the case of just one extra dimension with periodic boundary condition y → y + 2 πR . Then all particles have wave functions on the coordinate y thatsatisfies ψ ( y + 2 πR ) = ψ ( y ) . They can of course depend on the usual four-dimensional space time x , too. One can expand it in Fourier modes ψ ( x, y ) = X n ψ n ( x ) e iny/R . (3.6)The momentum along the y direction is p y = − i∂ y = n/R , and the total energyof the particle is E = q ~p + p y = r ~p + (cid:16) nR (cid:17) . (3.7)Namely that you find a tower of particles of mass m n = n/R , called Kaluza–Klein states. H. Murayama
Of course, the standard model is tested down to − cm, and we have notfound Kaluza–Klein excitation of electron, etc. This is not a problem if we arestuck on a three-dimensional sheet (three- brane ) embedded inside the n + 3 di-mensional space. The branes are important objects in string theory and it is easyto get particles with gauge interactions stuck on them. The brane may be freelyfloating inside extra dimensions or may be glued at singularities ( e.g. , orbifoldfixed points). The simplest way to use large extra dimensions is to assume thatonly gravity is spread out in extra dimensions, while the standard model particlesare all on a three-brane.Cosmology with large extra dimension is an iffy subject, however. The Kaluza–Klein excitation of gravitons can be produced in early universe and the cosmol-ogy would be different from the standard Friedmann univese (see, e.g. , [40]).I will not get into this discussion here.Instead of models with large extra dimensions, models with small extra di-mensions of size R ≈ − cm ≈ TeV − are also interesting, which allowfor normal cosmology below TeV temperatures. This would also allow us stan-dard model particles to live in extra dimensions, too, because our Kaluza–Kleinexcitations have been too heavy to be produced at accelerators so far. There aremany versions of small extra dimensions.One very popular version is warped extra dimension [41]. Instead of flat met-ric in the extra dimensions, it sets up an exponential behavior. It is somethinglike Planck scale varies from GeV to TeV as you go across the 5th dimen-sion. Therefore, physics does end at TeV if you on one end of the 5th dimension,while it keeps going to GeV on the other end. The hierarchy problem maybe solved if Higgs resides on (or close to) the “TeV brane.” This set up attracteda lot of attention because the bulk is actually a slice of anti-de Sitter space whichhas nice features of preserving supersymmetry, leading to AdS/CFT correspon-dence [42], etc. It is also possible to obtain quite naturally from string theory [43].In a grand unified model from warped extra dimension, the proton longevity isan issue which is solved by a Z symmetry, and the lightest Z -charged particle(LZP) is a candidate for dark matter [44].It is also possible to have the “flat” extra dimension at the TeV scale and putall the standard model particles in the 5D bulk, called Universal Extra Dimension(UED) [45]. It is tricky to get chiral fermions in four dimensions if they areembedded in higher dimensional space. If you start out with five-dimensionalDirac equation ( iγ µ ∂ µ + γ ∂ y ) ψ ( x, y ) = 0 , (3.8) Historically, unified theories and string theory assumed R ≈ d Pl ≈ − cm. TeV-sized extradimensions are much larger than this, but I’m calling them “small” for the sake of distinction fromthe large extra dimensions. hysics Beyond the Standard Model and Dark Matter Fig. 7. Warped extra dimension. Even though the standard model particles are shown to be on theTeV brane, they may propagate in the bulk depending on the models. the Fourier-mode expansion for the mode ψ n ( x ) e − iny/R gives (cid:16) iγ µ ∂ µ − i nR γ (cid:17) ψ n ( x ) = 0 . (3.9)After a chiral rotation ψ n → ψ n e iπγ / ψ n , the second term turns into the usualmass term without γ . The problem is that here are always two eigenvalues γ = ± and you find both left- and right-handed fermions with the same quantumnumbers. Namely, you get Dirac fermion, not Weyl fermion. Then you don’t getthe standard model that distinguishes left from right. In terms of spectrum, whatis on the left of Fig. 8 is the spectrum because the Fourier modes n and − n givethe degenerate mass n/R each of them with its own Dirac fermion. Fig. 8. The spectrum of fermions in the 5D bulk. After orbifold identification in Fig. 9, the spectrumis halved and one can obtain chiral fermions in 4D.
The trick to get chiral fermions is to use an orbifold Fig. 9. Out of a circle S ( y ∈ [ − πR, πR ] ), you identify points y and − y to get a half-circle S / Z . Thereare two special points, y = 0 and πR , that are identified only with themselvescalled “fixed points.” In addition, we take the boundary condition that ψ ( y ) = − γ ψ ( − y ) . For n = 0 , we use cos ny/R for γ = − and sin ny/R for γ =+1 , without the degeneracy between n and − n . For n = 0 , only γ = − H. Murayama survives with the wave function ψ ( y ) = 1 . This way, we keep only a half of thestates as shown in Fig. 8, and we can get chiral fermions. As a consequence, wefind the system to have a Z symmetry under y → πR − y , under which modeswith even n are even and odd n odd. This Z symmetry is called KK parity andthe lightest KK state (LKP) becomes stable. At the tree-level, all first Kaluza–Klein states are degenerate m = 1 /R . Radiative corrections split their masses,and typically the first Kaluza–Klein excitation of the U (1) Y boson is the LKP[46]. Because the mass splittings are from the loop diagrams, they are small.Similarly to supersymmetry, there is a large number of new particles beyond thestandard model, namely Kaluza–Klein excitations. Its collider phenomenologyvery much resembles that of supersymmetry and it is not trivial to tell them apartat the LHC (dubbed “bosonic supersymmetry” [47]). Fig. 9. The orbifold S / Z . Points connected by the solid lines are identified.
4. Evidence for Dark Matter
Now we turn our attention to the problem of non-baryonic dark matter in theuniverse. Even though this is a sudden change in the topic, you will see soon thatit is connected to the discussions we had on physics beyond the standard model.We first review basics of observational evidence for non-baryonic dark matter,and then discuss how some of the interesting candidates are excluded. It leadsto a paradigm that dark matter consists of unknown kind of elementary particles.By a simple dimensional analysis, we find that a weakly coupled particle at theTeV-scale naturally gives the correct abundance in the current universe. We willtake a look at a simple example quite explicitly so that you can get a good feelon how it works. Then I will discuss more attractive dark matter candidates thatarise from various models of physics beyond the standard model I discussed inthe previous section. Here I’ve ignored possible complications due to brane operators and electroweak symmetrybreaking. hysics Beyond the Standard Model and Dark Matter The argument for the existence of “dark matter,” namely mass density thatis not luminous and cannot be seen in telescopes, is actually very old. Zwickyback in 1933 already reported the “missing mass” in Coma cluster of galaxies.By studying the motion of galaxies in the cluster and using the virial theorem(assuming of course that the galactic motion is virialized) he determined the massdistribution in the cluster and reported that a substantial fraction of mass is notseen. Since then, the case for dark matter has gotten stronger and stronger andmost of us regard its existence established by now. I refer to a nice review fordetails [48] written back in 1997, but I add some important updates since thereview.Arguably the most important one is the determination of cosmological param-eters by the power spectrum of CMB anisotropy. In the fit to the power-law flat Λ CDM model gives Ω M h = 0 . +0 . − . and Ω B h = 0 . +0 . − . [49].The point here is that these two numbers are different . Naively subtracting thebaryon component, and adding the errors by quadrature, I find (Ω M − Ω B ) h =0 . +0 . − . = 0 at a very high precision. This data alone says most of the mattercomponent in the universe is not atoms, something else.Another important way to determine the baryon density of the universe isbased on Big-Bang Nucleosynthesis (BBN). The baryon density is consistentwith what is obtained from the CMB power spectrum, Ω B h = 0 . +0 . − . from five best measurements of deuterium abundance [50] using hydrogen gasat high redshift (and hence believed to be primordial) back-lit by quasars. Thisagrees very well with the CMB result, even though they refer to very differentepochs: T ∼ MeV for BBN while T ∼ . eV for CMB. This agreement givesus confidence that we know Ω B very well.A novel technique to determine Ω M uses large-scale structure, namely thepower spectrum in galaxy-galaxy correlation function. As a result of the acousticoscillation in the baryon-photon fluid, the power spectrum also shows the “baryonoscillation” which was discovered only the last year [51]. Without relying on theCMB, they could determine Ω M h = 0 . ± . . Again this is consistentwith the CMB data, confirming the need for non-baryonic dark matter.I’d like to also mention a classic strong evidence for dark matter in galaxies.It comes from the study of rotation curves in spiral galaxies. The stars and gasrotate around the center of the galaxy. For example, our solar system rotates inour Milky Way galaxy at the speed of about 220 km/sec. By using Kepler’s law,the total mass M ( r ) within the radius r and the rotation speed at this radius v ( r ) are related by v ( r ) = G N M ( r ) r . (4.1)Once the galaxy runs out of stars beyond a certain r , the rotation speed is hence H. Murayama expected to decrease as v ( r ) ∝ r − / . This expectation is not supported byobservation.You can study spiral galaxies which happen to be “edge-on.” At the outskirtsof a galaxy, where you don’t find any stars, there is cold neutral hydrogen gas.It turns out you can measure the rotation speed of this cold gas. A hydrogenatom has hyperfine splitting due to the coupling of electron and proton spins,which corresponds to the famous λ = 21 cm line emission. Even though thegas is cold, it is embedded in the thermal bath of cosmic microwave backgroundwhose temperature 2.7 K is hot compared to the hyperfine excitation hc/kλ =0 . K. Therefore the hydrogen gas is populated in both hyperfine states andspontaneously emits photons of wavelength 21 cm by the M1 transition. This canbe detected by radio telescopes. Because you are looking at the galaxy edge-on,the rotation is either away or towards us, causing Doppler shifts in the 21 cm line.By measuring the amount of Doppler shifts, you can determine the rotation speed.Surprisingly, it was found that the rotation speed stays constant well beyond theregion where stars cease to exist.
Fig. 10. Rotation curve of a spiral galaxy [52].
I mentioned this classic evidence because it really shows galaxies are filledwith dark matter. This is an important point as we look for signals of dark matterin our own galaxy. It is not easy to determine how much dark matter there is,however, because eventually the hydrogen gas runs out and we do not know howfar the flat rotation curve extends. Nonetheless, it shows the galaxy to be madeup of a nearly spherical “halo” of dark matter in which the disk is embedded. If we had lived in a universe a hundred times larger, we would have lost this opportunity ofstudying dark matter content of the galaxies! hysics Beyond the Standard Model and Dark Matter
5. What Dark Matter Is Not
We don’t know what dark matter is, but we have learned quite a bit recentlywhat it is not . I have already discussed that it is not ordinary atoms (baryons).I mention a few others of the excluded possibilities.
The first candidate for dark matter that comes to mind is some kind of astronom-ical objects, namely stars or planets, which are is too dark to be seen. Peopletalked about “Jupiters,” “brown dwarfs,” etc. In some sense, that would be themost conservative hypothesis. Because dark matter is not made of ordinaryatoms, such astronomical objects cannot be ordinary stars either. But one canstill contemplate the possibility that it is some kind of exotic objects, such asblack holes. Generically, one refers to MACHOs which stand for MAssive Com-pact Halo Objects.Black holes may be formed by some violent epochs in Big Bang (primordialblack holes or PBHs) [53] (see also [54]). If the entire horizon collapses into ablack hole, which is the biggest mass one can imagine consistent with causality,for example in the course of a strongly first order phase transition, the black holemass would be M PBH ≈ M ⊙ (cid:18) T
100 MeV (cid:19) − (cid:16) g ∗ . (cid:17) − / . (5.1)Therefore, there is no causal mechanism to produce PBHs much larger than M ⊙ assuming that universe has been a normal radiation dominated universefor T < ∼ MeV to be compatible with Big-Bang Nucleosynthesis. Curiously, onefinds M PBH ≈ M ⊙ if it formed at the QCD phase transition T ≈ MeV [55].On the other hand, PBHs cannot be too small because otherwise they emit Hawk-ing radiation of temperature T = (8 πG N M PBH ) − that would be visible. Thelimit from diffuse gamma ray background implies M PBH > ∼ − M ⊙ .How do we look for such invisible objects? Interestingly, it is not impossibleusing the gravitational microlensing effects [56]. The idea is simple. You keepmonitoring millions of stars in nearby satellite galaxies such as Large Magel-lanic Cloud (LMC). Meanwhile MACHOs are zooming around in the halo of ourgalaxy at v ≈ km/s. By pure chance, one of them may pass very close alongthe line of sight towards one of the stars you are monitoring. Then the gravitywould focus light around the MACHO, effectively making the MACHO a lens.You typically don’t have a resolution to observe distortion of the image or multi-ple images, but the focusing of light makes the star appear temporarily brighter. Somehow I can’t call primordial black holes a “conservative” candidate without chuckling. H. Murayama
This is called “microlensing.” By looking for such microlensing events, you caninfer the amount of MACHOs in our galactic halo.I’ve shown calculations on the deflection angle by the gravitational lensingand the amplification in the brightness in the appendix. (Just for fun, I’ve alsoadded some discussions on the strong lensing effects.) The bottom line is thatyou may expect the microlensing event at the rate ofrate ≈ × − (cid:18) M ⊙ M MACHO (cid:19) / (5.2)towards the LMC, with the duration ofduration ≈ × sec (cid:18) M MACHO M ⊙ (cid:19) / (cid:18) √ d d kpc (cid:19) , (5.3)where d ( d ) is the distance between the MACHO and us (the lensed star).Two collaborations, the MACHO collaboration and the EROS collaboration,have looked for microlensing events. The basic conclusion is that MACHOs ofmass − –30 M ⊙ cannot make up 100% of our galactic halo (Fig. 11). See also[58, 57]. Fig. 11. Limit on the halo fraction f of MACHOs from the EROS collaboration [57]. The sphericalisothermal model of halo predicts the optical depth towards the LMC of τ = 4 . × − . For moredetails, see the paper. Even though the possibility of MACHO dark matter may not be completelyclosed, it now appears quite unlikely. The main paradigm for the dark matter ofthe universe has shifted from MACHOs to WIMPs. hysics Beyond the Standard Model and Dark Matter Having discovered neutrinos have finite mass, it is also natural to consider neu-trinos to be dark matter candidate. As a matter of fact, neutrinos are a componentof dark matter, contributing Ω ν h = P i m ν i eV . (5.4)It is an attractive possibility if the particles which we already know to exist couldserve as the required non-baryonic dark matter.However, as Sergio Pastor discussed in his lectures, neutrinos are not goodcandidates for the bulk of dark matter for several reasons. First, there is an upperlimit on neutrino mass from laboratory experiments (tritium beta decay) m < eV [59]. Combined with the smallness of mass-squared differences ∆ m ⊙ =8 × − eV and ∆ m ⊕ = 2 . × − eV , electron-volt scale neutrinos shouldbe nearly degenerate. Then the maximum contribution to the matter density is Ω ν h < (3 × / < . . This is not enough.Second, even if the laboratory upper limit on the neutrino mass turned out to benot correct, there is a famous Tremaine-Gunn argument [60]. For the neutrinosto dominate the halo of dwarf galaxies, you need to pack them so much thatyou would violate Pauli exclusion principle. To avoid this, you need to makeneutrinos quite massive > ∼ eV so that you don’t need so many of them [61].This obviously contradicts the requirement that Ω ν < .Third, neutrinos are so light that they are still moving at speed of light (HotDark Matter) at the time when the structure started to form, and erase structure atsmall scales. Detailed study of large scale structure shows such a hot componentof dark matter must be quite limited. The precise limit depends on the exactmethod of analyses. A relatively conservative limit says P i m ν i < . eV [62]while a more aggressive limit goes down to 0.17 eV [63]. Either way, neutrinoscannot saturate what is needed for non-baryonic dark matter.In fact, what we want is Cold Dark Matter, which is already non-relativisticand slowly moving at the time of matter-radiation equality T ∼ eV. Naively alight (sub-electronvolt) particle would not fit the bill.A less conservative hypothesis may be to postulate that there is a new heavyneutrino (4th generation). This is a prototype for WIMPs that will be discussedlater. It turns out, however, that the direct detection experiments and the abun-dance do not have a compatible mass range. Namely the neutrinos are too stronglycoupled to be the dark matter! H. Murayama
Even though people do not talk about it any more, it is worth recalling that darkmatter is unlikely be charged (CHAMP) [64] or strongly interacting (SIMP) [65].I simply refer to papers that limit such possibilities, from a multitude of searchmethods that include search for anomalously heavy “water” molecule in the seawater, high-energy neutrinos from the center of the Earth from annihilated SIMPsaccumulated there, collapsing neutron stars that accumulate CHAMPs, etc.
6. WIMP Dark Matter
WIMP, or Weakly Interactive Massive Particle, is the main current paradigm forexplaining dark matter of the universe. With MACHOs pretty much gone, it isindeed attractive to make a complete shift from astronomical objects as heavy as M ⊙ ≈ GeV to “heavy” elementary particles of mass ∼ GeV. I willdiscuss why this mass scale is particularly interesting.
The idea of WIMP is very simple. It is a relatively heavy elementary particle χ so that accelerator experiments so far did not have enough energy to create them,namely m χ > ∼ GeV. On the other hand, the Big Bang did once have enoughenergy to make them.Let us follow the history from when
T > ∼ m χ . WIMPs were created as muchas any other particles. Once the temperature dropped below m χ , even the uni-verse stopped creating them. If they are stable, whatever amount that was pro-duced was there, and the only way to get rid of them was to get them annihilatingeach other into more mundane particles ( e.g. , quarks, leptons, gauge bosons).However, the universe expanded and there were fewer and fewer WIMPs in agiven volume, and at some point WIMPs stopped finding each other. Then theycould not annihilate any more and hence their numbers become fixed (“freeze-out”). This way, the universe could still be left with a certain abundance ofWIMPs. This mechanism of getting dark matter is called “thermal relics.”Let us make a simple estimate of the WIMP abundance. In radiation domi-nated universe, the expansion rate is given by H = ˙ aa = g / ∗ T M P l (cid:18) π (cid:19) / , (6.1) I once got interested in the possibility that Jupiter is radiating heat more than it receives fromthe Sun because SIMPs are annihilating at its core [66]. It does not seem to explain heat from otherJovian planets, however, once empirical limits on SIMPs are taken into account. hysics Beyond the Standard Model and Dark Matter where M P l = 1 / √ πG N = 2 . × GeV is the reduced Planck scale. Forsimple estimates, we regard ( π / / = 0 . ≈ and ignore many other fac-tors of O (1) . Hence, H ≃ g / ∗ T /M P l . The entropy density is correspondingly s = g ∗ T (cid:18) π (cid:19) / ≃ g ∗ T . (6.2)Given the thermally averaged annihilation cross section h σ ann v i , and the numberdensity of WIMPs n χ , the annihilation rate of a WIMP is Γ = h σ ann v i n χ . (6.3)The annihilation stops at the “freeze-out temperature” T f when Γ ≃ H , andhence n χ ( T f ) ≃ g / ∗ T f h σ ann v i M P l . (6.4)The yield of WIMPs is defined by Y χ = n χ /s . This is a convenient quantitybecause it is conserved by the expansion of the universe as long as the expansionis adiabatic, i.e. , no new source of heat. This is due to the conservation of boththe total entropy and total number of particles and their densities both scale as /a . The estimate of the yield is Y χ ∼ g − / ∗ h σ ann v i T f M P l = g − / ∗ x f h σ ann v i m χ M P l . (6.5)Here, we defined T f = m χ /x f . We will see later from more detailed calculationsthat x f ∼ . The abundance in the current universe is calculated using the yieldand the current entropy density, divided by the current critical density, Ω χ = m χ n χ s s ρ c ∼ g − / ∗ x f h σ ann v i M P l s ρ c . (6.6)We use s = 2890 cm − and ρ c = 1 . × − h GeV cm − , where the currentHubble constant is H = 100 h km/sec/Mpc with h ≈ . . In order of obtain Ω χ h ∼ . , we need h σ ann v i ∼ g − / ∗ x f . × − GeV − Ω χ h ∼ − GeV − . (6.7)Recall a typical annihilation cross section of a particle of mass m χ by a relativelyweak interaction of electromagnetic strength ( e.g. , e + e − → γγ ) is σ ann v ∼ πα m χ . (6.8) H. Murayama
To obtain the correct abundance, what we need is m χ ∼ GeV . (6.9)This is a very interesting result. Namely, the correct abundance of thermal relicsis obtained for a particle mass just beyond the past accelerator limits and wherewe expect new particles to exist because of the considerations of electroweaksymmetry breaking and the hierarchy problem. In other words, it is exactly theright mass scale for a new particle!In the next few sections, we will firm up this naive estimate by solving theBoltzmann equations numerically. We will also study a concrete model of a newparticle for dark matter candidate and work out its annihilation cross section. Inaddition, we will see if we have a chance of “seeing” the dark matter particlein our galactic halo, or making it in future accelerator experiments. Then wewill generalize the discussions to more theoretically attractive models of physicsbeyond the standard model. You have already seen Boltzmann equation in lectures by Sabino Matarrese andI don’t repeat its derivations. We assume kinetic equilibrium, namely that eachparticle species has the Boltzmann distribution in the momentum space exceptfor the overall normalization that is given by its number density. Considering theprocess of χ χ ↔ χ χ , where χ i refers to a certain elementary particle, theBoltzmann equation for the number density n for the particle χ is a − d ( n a ) dt = h σv i n n (cid:18) n n n n − n n n n (cid:19) . (6.10)Here, σv is the cross section common for the process χ χ → χ χ and its in-verse process χ χ → χ χ assuming the time reversal invariance. The numberdensities with superscript refer to those in the thermal equilibrium.In the case of our interest, χ , are “mundane” light (relativistic) particles inthe thermal bath, and hence n , = n , . In addition, we consider the annihilation χχ ↔ ( mundane ) , and hence n = n . The Boltzmann equation simplifiesdrastically to a − dn χ a dt = h σ ann v i [( n χ ) − ( n χ ) ] . (6.11)This time we pay careful attention to all numerical factors. We use Y = n χ s , (6.12) hysics Beyond the Standard Model and Dark Matter s = g ∗ T (cid:18) π (cid:19) / , (6.13) H = 8 π G N g ∗ π T = g ∗ π T M P l , (6.14) x = m χ T . (6.15)Even though we start out at temperatures
T > m χ when χ are relativistic,eventually the temperature drops below m χ and we can use non-relativistic ap-proximations. Then the equilibrium number density can be worked out easilyas n χ = Z d p (2 π ) e − E/T (cid:18) E = m χ + ~p m χ (cid:19) = e − m χ /T (cid:18) m χ T π (cid:19) / = e − x m χ (2 πx ) / . (6.16)Therefore Y = n χ s = 1 g ∗ π (cid:16) x π (cid:17) / e − x = 0 . x / e − x . (6.17)Changing the variables from n χ to Y and t to x , the Boltzmann equationbecomes dYdx = − x s ( m χ ) H ( m χ ) h σ ann v i ( Y − Y ) . (6.18)Here, we used s ( T ) = s ( m χ ) /x and dt = − H ( T ) dTT = − m χ H ( m χ ) T dT = 1 H ( m χ ) xdx. (6.19)It is useful to work out s ( m χ ) H ( m χ ) = 2 π (cid:18) π (cid:19) / g / ∗ m χ M P l = 1 . g / ∗ m χ M P l . (6.20)Note that the annihilation cross section h σ ann v i is insensitive to the temperatureonce the particle is non-relativistic T ≪ m χ . Therefore the whole combina-tion s ( m χ ) H ( m χ ) h σ ann v i is just a dimensionless number. The only complication is This statement assumes that the annihilation is in the S -wave. If it is in the l -wave, h σ ann v i ∝ v l ∝ x − l . H. Murayama that Y has a strong dependence on x . We can further simplify the equation byintroducing the quantity y = s ( m χ ) H ( m χ ) h σ ann v i Y. (6.21)We obtain dydx = − x ( y − y ) , (6.22)with y = 0 . g − / ∗ M P l m χ h σ ann v i x / e − x . (6.23) Here is a simple analytic approximation to solve Eq. (6.22). We assume Y tracks Y for x < x f . On the other hand, we assume y ≫ y for x > x f because y drops exponentially as e − x . Of course this approximation has a discontinuity at x = x f , but the transition between these two extreme assumptions is so quickthat it turns out to be a reasonable approximation. Then we can analytically solvethe equation for x > x f and we find y ( ∞ ) − y ( x f ) = 1 x f . (6.24)Since y ( ∞ ) ≪ y ( x f ) , we obtain the simple estimate y ( ∞ ) = x f . (6.25)Given this result, we can estimate x f as the point where y ( x ) drops down ap-proximately to x f , . g − / ∗ M P l m χ h σ ann v i x / f e − x f ≈ x f , (6.26)and hence x f ≈ ln . m χ M P l h σ ann v i x / f g / ∗ ! ≈
24 + ln m χ GeV + ln h σ ann v i − GeV − −
12 ln g ∗ . (6.27) hysics Beyond the Standard Model and Dark Matter I’ve gone through numerical integration of the Boltzmann equation Eq. (6.22).Fig. 12 shows the x -evolution of y . You can see that it traces the equilibriumvalue very well early on, but after x of about 20, it starts to deviate significantlyand eventually asymptotes to a constant. This is exactly the behavior we expectedin the analytic approximation studied in the previous section. Fig. 12. Numerical solution to the Boltzmann equation Eq. (6.22) for m = 100 GeV, g ∗ = 100 , h σ ann v i = 10 − GeV − . Superimposed is the equilibrium value y . Fig. 13 shows the asymptotic values y ( ∞ ) which we call x f . I understand thisis a confusing notation, but we have to define the “freeze-out” in some way, andthe analytic estimate in the previous section suggest that the asymptotic value y ( ∞ ) is nothing but the freeze-out value x f . This is the result that enters thefinal estimate of the abundance and is hence the only number we need in theend anyway. It does not exactly agree with the estimate in the previous section,but does very well once I changed the offset in Eq. (6.27) from 24 to 20.43.Logarithmic dependence on m χ is verified beautifully. Fig. 13. x f values as a function of m χ , for g ∗ = 100 , h σ ann v i = 10 − GeV − . The dots are theresults of numerical integrations, while the solid line is just ln m χ with an offset so that x f = 20 . for m χ = 100 GeV. H. Murayama
Putting everything back together, ρ χ = m χ n χ = m χ Y s = m χ H ( m χ ) s ( m χ ) x f h σ ann v i s (6.28)As before, we use s = 2890 cm − and ρ c = 1 . × − h GeV cm − , wherethe current Hubble constant is H = 100 h km/sec/Mpc with h ≈ . . To obtain Ω M h = 0 . , we find h σ ann v i = 1 . × − GeV − , confirming the simpleestimate in Section 6.1. Now we would like to apply our calculations to a specific model, called the NewMinimal Standard Model [2]. This is the model that can account for the empiricalfacts listed in Section 2.1 with the minimal particle content if you do not pay any attention to the theoretical issues mentioned in Section 2.2. It accomplishesthis by adding only four new particles to the standard model; very minimalindeed! The dark matter in this model is a real scalar field S with an odd Z parity S → − S , and its most general renormalizable Lagrangian that should beadded to the Standard Model Eq. (2.1) is L S = 12 ∂ µ S∂ µ S − m S S − k | H | S − h S . (6.29)The scalar field S is the only field odd under Z , and hence the S boson is stable.Because of the analysis in the previous sections, we know that if m S is at theelectroweak scale, it may be a viable dark matter candidate as a thermal relic.This is a model with only three parameters, m S , k , and h , and actually the lastone is not relevant to the study of dark matter phenomenology. Therefore this isa very predictive model where one can work it out very explicitly and easily.To calculate the dark matter abundance, what we need to know is the annihila-tion cross section of the scalar boson S . This was studied first in [68] and later in[69], but the third diagram was missing. In addition, there is a theoretical boundson the size of couplings k and h so that they would stay perturbative up to highscales ( e.g. , Planck scale). The cosmic abundance is determined by m S and k inaddition to m h . Therefore on the ( k, m h ) plane, the correct cosmic abundancedetermines what m S should be. This is shown in Fig. 15. You can see that for avery wide range m S ≃ . GeV–1.8 TeV, the correct cosmic abundance can beobtained within the theoretically allowed parameter space. For heavy m S ≫ m h ,the cross section goes like k /m S and is independent of m h . This is why the m S contours are approximately straight vertically. For light m S ≪ m h , the cross The other three are the inflaton and two right-handed neutrinos. hysics Beyond the Standard Model and Dark Matter hSS ¯ ff − ikv − i m f v hSS V = W, ZV = W, Z − ikv ig V m V g µν hSS hh − ikv − i m h v SSS hh − ikv − ikv SS hh − ik Fig. 14. Feynman diagrams for the annihilation of S scalars. The final states in the first diagram canbe any of the quark or lepton pairs f ¯ f .Fig. 15. The region of the NMSM parameter space ( k ( m Z ) , m h ) that satisfies the stability andtriviality bounds, for h ( m Z ) = 0 , 1.0, and 1.2. Also the preferred values from the cosmic abundance Ω S h = 0 . are shown for various m S . Taken from [2]. section goes like k m S /m h . This is why the m S contours approximately havea fixed k/m h ratio. Note that when m S ≃ m h / , the first two diagrams can hitthe Higgs pole and the cross section can be very big even for small k . This res-onance effect is seen in Fig. 15 where m S = 75 GeV line reaches almost k = 0 for m h = 150 GeV.You may wonder why I am talking about S as light as 5.5 GeV. Shouldn’twe have seen it already in accelerator experiments? Actually, no. The onlyinteraction the S boson has is with the Higgs boson which we are yet to see.Therefore, we could not have produced the S boson unless we had produced theHiggs boson. That is why even such a light S boson does not contradict data. Inother words, it wouldn’t be easy to find this particle in accelerator experiments. H. Murayama
How do we know if dark matter is indeed in the form of WIMP candidate youlike? One thing we’d love to see is the direct detection of WIMPs. The idea isvery simple. You place a very sensitive device in a quite location. WIMPs aresupposed to be flying around in the halo of our galaxy with the typical speed of ∼ km/s ∼ − c . Because they are only very weakly interacting, they can gothrough walls, rocks, even the entire Earth with little trouble, just like neutrinos.For a mass of m χ ∼ GeV, its typical kinetic energy is E kin = m χ v ∼ keV. If the WIMP (ever) scatters off an atomic nucleus, the energy deposit isonly (at most) of this order of magnitude. It is a tiny energy deposit that is verydifficult to pick out against background from natural radioactivity (typically MeVenergies). Therefore you have to make the device very clean, and also place itdeep underground to be shielded from the cosmic-ray induced backgrounds, mostimportantly neutrons ejected from the rocks by cosmic-ray muons. One you’vedone all this, what you do is to wait to see this little “kick” in your detector.Let us do an order of magnitude estimate. The local halo density is esti-mated to be about ρ halo χ ≈ . GeV/cm . The number density of WIMPs is n χ = ρ halo χ /m χ . The flux of WIMPs is roughly vn χ ≈ − cn χ . The elasticcross section of WIMP on neutron or proton may be spin-independent or spin-dependent. In the spin-independent case, the amplitude of the WIMP-nucleuscross section goes as A (mass number) and hence the cross section on the nucleus σ A goes as A . Of course the detailed scaling is model-dependent, but in mostphenomenological analyses (and also analyses of data) we assume σ A = A σ p .Let us also assume Ge as the detector material so that A = 56 . Then theexpected event rate is R = n halo χ v m target m A σ A ≈ GeV m χ m target kg A σ p − cm . (6.30)To prepare a very sensitive device as big as 100 kg and make it very clean is abig job. You can see that your wait may be long .Now back to the New Minimal Standard Model. The scattering of the S bosonoff a proton comes from the t -channel Higgs boson exchange. The coupling ofthe Higgs boson to the nucleon is estimated by the famous argument [70] usingthe conformal anomaly. The mass of the proton is proportional to the QCD scale m p ∝ Λ QCD ( m u , m d , m s are ignored and hence this is the three-flavor scale).It is related to the Higgs expectation value through the one-loop renormalizationgroup equation as (we do not consider higher loop effects here) Λ QCD = m / c m / b m / t M e − π /g s ( M ) (6.31) hysics Beyond the Standard Model and Dark Matter where M is some high scale and each quark mass is proportional to v . Thecoupling of the Higgs to the proton is given by expanding the vacuum expectationvalue as v → v + h , and hence y pph = ∂m p ∂v = 29 m p v . (6.32)This allows us to compute the scattering cross section of the S boson and thenucleon. hp, nS − i m p v − ikv Fig. 16. Feynman diagrams for the scattering of the S -boson off a proton or neutron. The result is shown in Fig. 16 as the red solid line. The point is that the elasticscattering cross section tends to be very small. Note that a hypothetical neutrinoof the similar mass would have a cross section of σ νp ∼ G F m ν /π ∼ − cm which is much much bigger than this. This is the typical WIMP scattering crosssection. Superimposed is the limit from the CDMS-II experiment [71] and hencethe direct detection experiments are just about to reach the required sensitivity. Inother words, this simple model is completely viable, and may be tested by futureexperiments. For the resonance region m S ≃ m h / , the coupling k is very smallto keep enough abundance and hence the direct detection is very difficult.The future of this field is not only to detect WIMPs but also understand itsidentity. For this purpose, you want to combine the accelerator data and thedirect detection experiments. The direct detection experiments can measure theenergy deposit and hence the mass of the WIMP. It also measures the scatteringcross section, even though it suffers from the astrophysical uncertainty in theestimate of the local halo density. On the other hand, assuming m S < m h / ,the Higgs boson decays invisibly h → SS . Such an invisible decay of the lightHiggs boson can be looked for at the LHC using the W -fusion process. Quarksfrom both sides radiate an off-shell W -boson that “fuse” in the middle to producea Higgs boson. Because of the kick by the off-shell W -boson, the quarks acquire p T ∼ m W / in the final state and can be tagged as “forward jets.” Even thoughthe Higgs boson would not be seen, you may “discover” it by the forward jetsand missing E T [74]. The ILC can measure the mass of the Higgs preciselyeven when it decays dominantly invisibly (see, e.g. , [75]) and possibly its width.Combining it with the mass from the direct detection experiments, you can infer H. Murayama
Fig. 17. The elastic scattering cross section of Dark Matter from nucleons in NMSM, as a functionof the Dark Matter particle mass m S for m h = 150 GeV. Note that the region m S > . TeV isdisallowed by the triviality bound on k . Also shown are the experimental bounds from CDMS-II [71]and DAMA [72], as well as improved sensitivities expected in the future [73]. Taken from [2]. the coupling k and calculate its cosmic abundance. It would be a very interestingtest if it agrees with the cosmological data Ω M ≈ . . If it does, we can claima victory; we finally understand what dark matter is ! Superpartners of the photon and Z , and neutral Higgs bosons (there are two ofthem), mix among each other once SU (2) × U (1) symmetry breaks. Out of foursuch “neutralino” states, the lightest one is often the LSP and is the most pop-ular candidate for dark matter in the literature (see, e.g. , [77, 78, 79] for some ofthe recent papers). One serious problem with the supersymmetric dark matter isthat there are many parameters in the model. Even the Minimal Supersymmet-ric Standard Model (MSSM) has 105 more parameters than the standard model.It is believed that the fundamental theory determines all these parameters or atleast reduces the number drastically, and one typically ends up with five or soparameters in the study. Depending on what parameter set you pick, the phe-nomenology may be drastically different. For the popular parameter set calledCMSSM (Constrained MSSM, also called minimal supergravity or mSUGRA)with four parameters and one sign, see a recent study in [76]. I do not go into Superpartner of neutrinos is not out of the question [80]. hysics Beyond the Standard Model and Dark Matter detailed discussions about any of them. I rather mention a few generic points.First, we do get viable dark matter candidates from sub-TeV supersymmetryas desired by the hierarchy problem. This is an important point that shouldn’tbe forgotten. Second, what exactly is the mass and composition of the neu-tralino depends on details of the parameter set. The supersymmetric standardmodel may not be minimal either; an extension with additional singlet calledNext-to-Minimal Supersymmetric Standard Model (NMSSM) is also quite pop-ular. Third, sub-TeV supersymmetry can be studied in great detail at LHC and(hopefully) ILC, so that we can measure their parameters very precisely (see [81]for the early work). One can hope to correlate the accelerator and undergrounddata to fully test the nature of the dark matter [82]. Fourth, a large number ofparticles present in the model may lead to interesting effects we did not considerin the discussions above. I’ll discuss one of such effects briefly below.Universal Extra Dimension (UED) is also a popular model. I’m sure GeraldineServant will discuss dark matter in this model in her lectures because she is oneof the pioneers in this area [86]. Because the LKP is stable, it is a dark mattercandidate. Typically the first KK excitation of the U (1) Y gauge boson is theLKP, and its abundance can be reasonable. Its prospect for direct and indirectdetection experiments is also interesting. See a review article that came out afterLes Houches [87].The most striking effect of having many particle species is the coannihilation [83]. An important example in the case of supersymmetry is bino and stau. Binois the superpartner of the U (1) Y gaugino (mixture of photino and zino), and itsannihilation cross section tends to be rather small partly because it dominantlygoes through the P -wave annihilation. If, however, the mass of the stau is not toofar above bino, stau is present with the abundance suppressed only by e − ∆ m/T ( ∆ m = m ˜ τ − m ˜ B ) assuming they are in chemical equilibrium ˜ Bτ ↔ γ ˜ τ . Thereare models that suggest the mass splitting is indeed small. The cross sections ˜ B ˜ τ → γτ and ˜ τ ˜ τ ∗ → γγ, f ¯ f etc tend to be much larger, the former goingthrough the S -wave and the latter with many final states. Therefore, despite theBoltzmann suppression, these additional contributions may win over h σ ˜ B ˜ B v i .There are other cases where the mass splitting is expected to be small, such ashiggsino-like neutralinos [84]. In the UED, the LKP is quite close in its mass tothe low-lying KK states and again coannihilation is important. On the experimental side, there are other possible ways of detecting signals ofdark matter beyond the underground direct detection experiments and collidersearches. They are indirect detection experiments, namely that they try to detectannihilation products of dark matter, not the dark matter itself. For annihilation to H. Murayama occur, you need some level of accumulation of dark matter. The possible sites are:galactic center, galactic halo, and center of the Sun. The annihilation productsthat can be searched for include gamma rays from the galactic center or halo, e + from the galactic halo, radio from the galactic center, anti-protons from thegalactic halo, and neutrinos from the center of the Sun. Especially the neutrinosignal complements the direct detection experiments because the sensitivity ofthe direct detection experiments goes down as /m χ because the number densitygoes down, while the sensitivity of the neutrino signal remain more or less flatfor heavy WIMPs because the neutrino cross section rises as E ν ∝ m χ . You canlook at a recent review article [85] on indirect searches.
7. Dark Horse Candidates
Assuming R -parity conservation, superparticles decay all the way down to what-ever is the lightest with odd R -parity. We mentioned neutralino above, but an-other interesting possibility is that the LSP is the superpartner of the gravitino,namely gravitino. Since gravitino couples only gravitationally to other particles,its interaction is suppressed by /M P l . It practically removes the hope of directdetection. On the other hand, it is a possibility we have to take seriously. This isespecially so in models with gauge-mediated supersymmetry breaking [88].The abundance of light gravitinos is given by Ω / h = m / keV (7.1)if the gravitinos were thermalized. In most models, however, the gravitino isheavier and we cannot allow thermal abundance. The peculiar thing about alight gravitino is that the longitudinal (helicity ± / ) components have a muchstronger interaction m SUSY / ( m / M P l ) if m / ≪ m SUSY . Therefore theproduction cross section scales as σ ∼ m SUSY / ( m / M P l ) , and the abundancescales as Y / ∼ σT /H ( T ) ∼ m SUSY
T / ( m / M P l ) . Therefore we obtainan upper limit on the reheating temperature after the inflation [89, 90]. If thereheating temperature is right at the limit, gravitino may be dark matter.There is however another mechanism of gravitino production. The abundanceof the “LSP”is determined the usual way as a WIMP, while it eventually decaysinto the gravitino. This decay can upset the Big-Bang Nucleosynthesis [89], butit may actually be helpful for a region of parameter space to ease some tensionamong various light element abundances [91]. Note that the “LSP” (or more cor-rectly NLSP: Next-Lightest Supersymmetric Particle) may be even electrically hysics Beyond the Standard Model and Dark Matter Fig. 18. The upper bound on T max as a function of the gravitino mass from the requirement thatthe relic stable gravitinos do not overclose the Universe. Taken from [90]. It assumes h = 1 and Ω / h < and hence the actual constraint is nearly an order of magnitude more stringent than thisplot. charged or strongly coupled as it is not dark matter. When superparticles areproduced at colliders, they decay to the “LSP” inside the detector, which escapesand most likely decays into the gravitino outside the detector. If the “LSP” ischarged, it would leave a charged track with anomalously high dE/dx . It is inprinciple possible to collect the NLSP and watch them decay, and one may evenbe able to confirm the spin / nature of the gravitino [92] and its gravitationalcoupling to matter [93].If the gravitino is heavier than the LSP, its lifetime is calcuated as τ ( ˜ G → γ ˜ γ ) = 3 . × (cid:16) m / TeV (cid:17) − sec . (7.2)It tends to decay after the BBN and upsets its success. Its production cross sectionscales as σ ∼ /M P l and hence Y / ∼ T /M
P l . Depending on its mass anddecay modes, one can again obtain upper limits on the reheating temperature.The case of hadronic decay for m / ∼ TeV is the most limiting case thatrequires T RH < ∼ GeV [94], causing trouble to many baryogenesis models.
One of the puzzles about the standard model I discussed earlier is why θ < ∼ − in QCD. A very attractive solution to this problem is to promote θ to a dynamicalfield, so that when it settles to the minimum of the potential, it automatically H. Murayama makes θ effectively zero [95]. The dynamical field is called axion , and couples(after integrating out some heavy fields) as L = (cid:18) θ + af a (cid:19) ǫ µνρσ Tr( G µν G ρσ ) . (7.3)Here, f a is called axion decay constant which has a dimension of energy. Thepotential is given approximately as V ∼ m π f π (cid:20) − cos (cid:18) θ + af a (cid:19)(cid:21) , (7.4)and indeed the axion field settles to a = − θf a and the G ˜ G term vanishes. Theaxion mass is therefore m a ≈ µ eV GeV f a . (7.5)Various astrophysical limits basically require f a > ∼ GeV and hence theaxion is a very light boson (see, e.g. , [96]). Most of these limits come from thefact that the axion can carry away energy from stars and would cool them tooquickly, such as white dwarfs, red giants, and SN1987A. Models of such high f a are called “invisible axion” models because then the axion coupling is veryweak to other particles, avoids these limits, and hence is very difficult to observe.There are two popular versions, KSVZ [97, 98] and DFSZ [99, 100] models.In the early universe T ≫ GeV, the axion potential looks so flat that it cannottell where the minimum is. Therefore we expect it starts out wherever it findsitself, mostly likely not at the minimum. The likely initial misplacement is of theorder of f a . Now we would like to know what happens afterwards.Let us consider a scalar field in an expanding universe. Neglecting the spatialvariation and considering the time dependence alone, the equation of motion is ¨ φ + 3 H ˙ φ + V ′ ( φ ) = 0 . (7.6)For a quadratic potential (mass term) V ( φ ) = m φ , the equation is particu-larly simple and homogeneous, ¨ φ + 3 H ˙ φ + m φ = 0 . (7.7) This is especially true for the axion because its mass originates from the QCD instaton effectswhich are suppressed by powers of the temperature in hot thermal bath [101]. hysics Beyond the Standard Model and Dark Matter It is useful to solve this equation for a constant (time-independent) H first. ForFourier modes φ ∼ e − iωt (of course we take the real part later on), we need tosolve − ω − iH + m = 0 , (7.8)and we find ω ± = 12 h − iH ± p − H + 4 m i = (cid:26) − iH, − i m H ( H ≫ m ) m − i H, − m − i H ( H ≪ m ) (7.9)Therefore for H ≫ m (early universe), one of the solutions damps quickly φ + ∼ e − Ht , while the other is nearly stationary φ − ∼ e − ( m / H ) t . This is becauseit is “stuck” by the friction term − H ˙ φ . On the other hand the field oscillatesaround the minimum as φ = φ e − imt e − Ht/ .One can improve this analysis for time-dependent H ( t ) = 1 / t and a ( t ) ∝ t / in the radiation-dominated univese, by replacing e − iω ± t by e − i R t ω ± ( t ′ ) dt ′ (adiabatic approximation). When H ≫ m ( mt ≪ ), we find φ + = φ e − R tt H ( t ′ ) dt ′ = φ (cid:18) t t (cid:19) − / = φ (cid:16) a a (cid:17) − ,φ − = φ e − m R tt t ′ dt ′ / = φ e − m t / ≈ φ . (7.10)The second one is the solution that is stuck by the friction. On the other handwhen H ≪ m , φ ± = φ e ± imt e − R tt H ( t ′ ) dt ′ = φ e ± imt (cid:18) t t (cid:19) / = φ e ± imt (cid:16) a a (cid:17) / . The field damps as a − / , and its energy as V = m φ / ∝ a − , just likenon-relativistic matter. In fact, a coherently oscillating homogeneous field can beregarded as a Bose–Einstein condensate of the boson at zero momentum state.Therefore, the axion field can sit on the potential and does not roll down be-cause of the large friction term − H ˙ φ when H ≫ m . On the other hand for lateruniverse H ≪ m , it oscillates as a usual harmonic oscillator e ± imt and dilutesas non-relativistic matter. This is why a very light scalar field can be a candi-date for cold dark matter. Counterintuitive, but true. This way of producing colddark matter is called “misalignment production” because it is due to the initialmisalignment of the axion field relative to the potential minimum. Because theamount of misalignment is not known, we cannot predict the abundance of axion H. Murayama precisely. Assuming the misalignment of O ( f a ) , f a ≃ GeV is the preferredrange for axion dark matter.There is a serious search going on for axion dark matter in the halo of ourgalaxy. In addition to the required coupling of the axion to gluons, most modelspredict its coupling to photons a ( ~E · ~B ) of the similar order of magnitude. TheADMX experiment places a high- Q cavity in a magnetic field. When an axionenters the cavity, this coupling would allow the axion to convert to a photon,which is captured resonantly by the cavity with a high sensitivity. By changingthe resonant frequencies in steps, one can “scan” a range of axion mass. Theirlimit has just reached the KSVZ axion model [103, 104], and an upgrade toreach the DFSZ axion model using SQUID is in the works. See [105] for moreon axion microwave cavity searches. I focused on the thermal reclic of WIMPs primarily because there is an attractivecoincidence between the size of annihilation cross section we need for the correctabundance and the energy scale where we expect to see new partices from thepoints of view of electroweak symmetry breaking and hierarchy problem. Axionis not connected to any other known energy scale, yet it is well motivated fromthe strong CP problem. On the other hand, nature may not necessarily tell usa “motivation” for a particle she uses. Indeed, people have talked about manyother possible candidates for dark matter. You may want to look up a couple ofkeywords: sterile neutrinos, axinos, warm dark matter, mixed dark matter, coldand fuzzy dark matter, Q -balls, WIMPZILLAs, etc. Overall, the candidates inthis list range in their masses from − eV to eV, not to mention stillpossible MACHOs < ∼ − M ⊙ = 10 eV. Clearly, we are making progress.
8. Cosmic Coincidence
Whenever I think about what the univese is made of, including baryons, photons,neutrinos, dark matter, and dark energy, what bothers me (and many other people)is this question: why do they have energy densities within only a few orders ofmagnitude? They could have been many orders of magnitude different, but theyaren’t. This question is related to the famous “Why now?” problem. The problemis clear in Fig. 19. As we think about evolution of various energy densities overmany decades of temperatures, why do we live at this special moment when thedark matter and dark energy components become almost exactly the same? I feel They ignored theoretical uncertainty in the prediction of the axion-to-photon coupling, and theKSVZ model is not quite excluded yet [102]. hysics Beyond the Standard Model and Dark Matter like I’m back to Ptolemy from Copernicus. We are special , not in space anymore, but in time. Is that really so?
Fig. 19. The evolution of radiation, matter, and cosmological constant ( Λ ) components of the univeseas the temperature drops over many orders of magnitude. "Now" is a very special moment whenmatter and Λ are almost exactly the same, and the radiation is not that different either. In Fig. 19, the radiation component goes down as T − , while the matter T − .The cosmological constant is by definition constant T . Matter and Λ meet now .When thinking about this problem, it is always tempting but dangerous to bring“us” into the discussion. Then we will be forced to talk about conditions foremergence of intelligent lifeforms, which we don’t know very well about. In-stead, it may be better to focus on physical quantities; namely the triple coinci-dence problem that three lines with different slopes seem to more or less meet ata point. In fact, dimensional analysis based on TeV-scale WIMP suggests ρ matter ∼ (cid:18) TeV M P l (cid:19) T , (8.1)which agrees with ρ radiation ∼ T at the temperature T ∼ TeV /M P l ≈ meV =10 K; this is about now! In order for the cosmological constant to meet at thesame time, we suspect there is a deep reason ρ Λ ∼ (cid:18) TeV M P l (cid:19) . (8.2)Indeed, ρ / is observationally about 2 meV, while TeV /M P l ≈ . meV.Maybe that is why we see a coincidence [106]. H. Murayama
Actually, an exact coincidence does not leave a window for structure forma-tion, which requires matter-dominated period. Fortunately, WIMP abundance isenhanced by weakness of the annihilation cross section, which goes like /α .This enhancement of matter relative to the triple coincidence gives us a windowfor matter domination and structure formation. May be that is why we seems tobe in this triangle. But then, why is the baryon component also just a factor offive smaller than dark matter? Are they somehow related?Oh well, we know so little.
9. Conclusions
In my lectures, I tried to emphasize that we are approaching an exciting time tocross new threshold of rich physics at the TeV energy scale in the next few yearsat the LHC. At the same time, the dark matter of the universe is now established tobe not made of particles we know, requiring physics beyond the standard model.The main paradigm for dark matter now is WIMPs, TeV-scale particles producedby the Big Bang which naturally give the correct order of magnitude for its abun-dance. Even though nature may be tricking us by this coincidence, many of us(including I) think that there is indeed a new particle (or many of them) waitingto be discovered at the LHC (or ILC) that tells us something about the dark sideof the universe. If this is so, I would feel lucky to be born to this age.
Appendix A. Gravitational Lensing
Gravitational lensing is an important tool in many studies in cosmology and as-trophysics. In this appendix I introduce the deflection of light in a sphericallysymmetric gravitational field (Schwarzschild metric)
Appendix A.1. Deflection Angle
Using the Schwarzschild metric ( c = 1 ) ds = r − r S r dt − rr − r S dr − r dθ − r sin θ dφ (A. 1)where r S = 2 G N m is the Schwarzschild radius. The Hamilton–Jacobi equa-tion for light in this metric is g µν ∂S∂x µ ∂S∂x ν (A. 2) For an introduction to Hamilton–Jacobi equations, see http://hitoshi.berkeley.edu/221A/classical2.pdf . hysics Beyond the Standard Model and Dark Matter = rr − r S (cid:18) ∂S∂t (cid:19) − r − r S r (cid:18) ∂S∂r (cid:19) − r (cid:18) ∂S∂θ (cid:19) − r sin θ (cid:18) ∂S∂φ (cid:19) = 0 . We separate the variables as S ( t, r, θ, φ ) = S ( t ) + S ( r ) + S ( θ ) + S ( φ ) (A. 3)where rr − r S (cid:18) dS dt (cid:19) − r − r S r (cid:18) dS dr (cid:19) − r (cid:18) dS dθ (cid:19) − r sin θ (cid:18) dS dφ (cid:19) = 0 . Because the equation does not contain t or φ explicitly, their functions must beconstants, dS dt = − E, (A. 4) dS dφ = L z . (A. 5)We can solve them immediately as S ( t ) = − Et, (A. 6) S ( φ ) = L z φ. (A. 7)Then Eq. (A. 2) becomes rr − r S E − r − r S r (cid:18) dS dr ( r ) (cid:19) − r (cid:18) dS dθ ( θ ) (cid:19) − r sin θ L z = 0 . The θ dependence is only in the last two terms and hence (cid:18) dS dθ ( θ ) (cid:19) + 1sin θ L z = L (A. 8)is a constant which can be integrated explicitly if needed. Without a loss ofgenerality, we can choose the coordinate system such that the orbit is on the x - y plane, and hence L z = 0 . In this case, S ( φ ) = 0 and S ( θ ) = Lθ . Finally, theequation reduces to rr − r S E − r − r S r (cid:18) dS dr ( r ) (cid:19) − L r = 0 . (A. 9) H. Murayama
Therefore, S ( r ) = Z s r ( r − r S ) E − L r ( r − r S ) dr. (A. 10)Since S ( t, r, θ, φ ) = S ( r ) − Et + Lθ , S can be regarded as Legendre transform S ( r, E, L ) of the action, and hence the inverse Legendre transform gives ∂S ( r, E, L ) ∂L = − θ. (A. 11)Using the expression Eq. (A. 10), we find θ ( r ) = Z rr c Ldr p E r − L r ( r − r S ) . (A. 12)The closest approach is where the argument of the square root vanishes, E r c − L r c ( r c − r S ) = 0 . (A. 13)It is useful to verify that the m = 0 ( r S = 0 ) limit makes sense. The clos-est approach is E r c − L r c = 0 and hence r c = L/E , which is the impactparameter. The orbit Eq. (A. 12) is θ ( r ) = Z rr c Ldr √ E r − L r = Z rr c r c drr p r − r c . (A. 14)Change the variable to r = r c cosh η , and we find θ ( r ) = Z η r c sin ηdηr c cosh ηr c sinh η = Z η dη cosh η = 2 arctan tanh η . (A. 15)Hence tan θ = tanh η , and cos θ = 1 − tan θ/
21 + tan θ/ − tanh η/
21 + tanh η/ η = r c r . (A. 16)Therefore r c = r cos θ which is nothing but a straight line.To find the deflection angle, we only need to calculate the asymptotic angle θ ( r = ∞ ) . Going back to Eq. (A. 12), we need to calculate θ ( ∞ ) = Z ∞ r c Ldr p E r − L r ( r − r S ) . (A. 17) hysics Beyond the Standard Model and Dark Matter We would like to expand it up to the linear order in r S ≪ r c . If you naivelyexpand the integrand in r S , the argument of the square root in the resulting ex-pression can be negative for r = r c < L/E . To avoid this problem, we changethe variable to r = r c /x : θ ( ∞ ) = Z Lr c dx p E r c − L r c ( r c − r S x ) x . (A. 18)Using Eq. (A. 13), we write E r c and obtain θ ( ∞ ) = Z r c dx p r c (1 − x ) − r c r S (1 − x ) . (A. 19)Expanding it to the linear order in r S /r c , we find θ ( ∞ ) = Z (cid:18) √ − x + (1 + x + x ) r S x ) √ − x r c + O ( r S ) (cid:19) dx = π r S r c . (A. 20)The deflection angle is ∆ θ = π − θ ( ∞ ) = 2 r S r c = 4 G N m/r c . It is easy torecover c = 1 by looking at the dimensions, and we find ∆ θ = 4 G N m/c r c .It is also useful to know the closest approach r c to the first order in m . Weexpand r c as r c = LE + ∆ . Then Eq. (A. 13) gives L E ∆ − L E ∆ + E L r S + O ( r S ) = 0 , (A. 21)and hence r c = LE − r S O ( r S ) . (A. 22) Appendix A.2. Amplification in Microlensing
Once the deflection angle is known, it is easy to work out the amplification usingsimple geometric optics. Throughout the discussion, we keep only the first orderin very small angles. Just by looking at the geometry in Fig. 20, the deflectionangle is ∆ θ = θ + θ = r − r d + r − r d = 4 G N mr . (A. 23) H. Murayama
Here, r is the impact parameter. When r = 0 (exactly along the line of sight),the solution is simple: r ( r = 0) = R ≡ r G N m d d d + d . (A. 24)This is what is called the Einstein radius, R in Paczynski’s notation [56]. Forgeneral r , Eq. (A. 23) can be rewritten as r ( r − r ) − R = 0 , (A. 25)which is Eq. (1) in the Packzynski’s paper. It has two solutions r ± ( r ) = 12 (cid:18) r ± q r + 4 R (cid:19) . (A. 26)The solution with the positive sign is what is depicted in Fig. 20, while the solu-tion with the negative sign makes the light ray go below the lens. Fig. 20. The deflection of light due to a massive body close to the line of sight towards a star.
To figure out the amplification due to the gravitational lensing, we considerthe finite aperture of the telescope ( i.e. , the size of the mirror). We assume aninfinitesimal circular aperture. From the point of view of the star, the finite aper-ture is an image on the deflection plane of size δ , namely the plane perpendicularto the straight line from the star to the telescope where the lens is. The ver-tical aperture changes the impact parameter r to a range r ± δ (size of themirror is δ × ( d + d ) /d ). Correspondingly, the image of the telescope is at r ± ( r ± δ ) = r ± ( r ) ± δ dr ± dr . Using the solution Eq. (A. 26), we find that thevertical aperture always appears squashed (see Fig. 21), δ × (cid:12)(cid:12)(cid:12)(cid:12) drdr (cid:12)(cid:12)(cid:12)(cid:12) = δ × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ± r p r + 4 R !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = δ × p r + 4 R ± r p r + 4 R < δ. Note that this Taylor expansion is valid only when δ ≪ r . For δ ∼ r , we have to work it outmore precisely; see next section. hysics Beyond the Standard Model and Dark Matter On the other hand, the horizontal aperture is scaled as δ × rr . (A. 27)Because the amount of light that goes into the mirror is proportional to the ellip-tical aperture from the point of view of the star that emits light isotropically, themagnification is given by A ± = rr (cid:12)(cid:12)(cid:12)(cid:12) drdr (cid:12)(cid:12)(cid:12)(cid:12) = ( p r + 4 R ± r ) r p r + 4 R = 2 r + 4 R ± r p r + 4 R r p r + 4 R . The total magnificiation sums two images, A = A + + A − = r + 2 R r p r + 4 R = u + 2 u √ u + 4 (A. 28)with u = r /R . Basically, there is a significant amplification of the brightnessof the star when the lens passes through the line of sight within the Einsteinradius.
Appendix A.3. MACHO search
We estimate the frequency and duration of gravitational microlensing due to MA-CHOs in the galactic halo. The Large Magellanic Cloud is about 50kpc awayfrom us, while we are about 8.5kpc away from the galactic center. The flat ro-tation curve for the Milky Way galaxy is about 220 km/sec (see Fig. 6 in [48]).The Einstein radius for a MACHO is calculated from Eq. (A. 24), R = r G N mc d d d + d = 1 . m (cid:18) mM ⊙ (cid:19) / (cid:18) √ d d kpc (cid:19) . (A. 29)To support the rotation speed of v ∞ = 220 km/sec in the isothermal modelof halo, we need the velocity dispersion σ = v ∞ / √ . The average velocitytransverse to the line of sight is h v x + v y i = 2 σ = v ∞ . (A. 30)The time it takes a MACHO to traverse the Einstein radius is R v ∞ = 5 . × sec (cid:18) mM ⊙ (cid:19) / (cid:18) √ d d kpc (cid:19) , (A. 31) The singular behavior for r → is due to the invalid Taylor expansion in δ . This is practicallynot a concern because it is highly unlikely that a MACHO passes through with r < ∼ δ . Note thatthe true image is actually not quite elliptic but distorted in this case. H. Murayama
Fig. 21. The way the mirror of the telescope appears on the deflection plane from the point of viewof the star. For the purpose of illustration, we took R = 2 , r = 3 . about two months for m = M ⊙ and d = d = 25 kpc. A microlensing event ofduration shorter than a year can be in principle be seen. The remaining question is the frequency of such microlensing events. It is theprobability of a randomly moving MACHO coming within the Einstein radiusof a star in the LMC. We will make a crude estimate. The flat rotation curvesrequires G N M ( r ) r = v ∞ r and hence the halo density ρ ( r ) = v ∞ πG N r . The numberdensity of MACHOs, assuming they dominate the halo, is then n ( r ) = v ∞ πG N mr .Instead of dealing with the Boltzmann (Gaussian) distribution in velocities, wesimplify the problem by assuming that ~v ⊥ = v x + v y = σ . From the transversedistance r ⊥ = p x + y , only the fraction R /r ⊥ heads the right directionfor the distance σ ∆ t . Therefore the fraction of MACHOs that pass through theEinstein radius is Z σ ∆ t πr ⊥ dr ⊥ R r ⊥ = 2 πR σ ∆ t. (A. 32)We then integrate it over the depth with the number density. The distance fromthe solar system to the LMA is not the same as the distance from the galactic MACHO collaboration did even more patient scanning to look for microlensing events longerthan a year [67]. hysics Beyond the Standard Model and Dark Matter center because of the relative angle α = 82 ◦ . The solar system is away from thegalactic center by r ⊙ = 8 . kpc. Along the line of sight to the LMA with depth R , the distance from the galactic center is given by r = R + r ⊙ − Rr ⊙ cos α with α = 82 ◦ . Therefore the halo density along the line of sight is n ( r ) = v ∞ πG N m ( R + r ⊙ − Rr ⊙ cos α ) (A. 33)The number of MACHOs passing through the line of sight towards a star in theLMA within the Einstein radius is Z R LMC dRn ( r )2 πR σ = Z R LMC dR v ∞ πG N m ( R + r ⊙ − Rr ⊙ cos α ) 2 πR σ ∆ t (A. 34)Pakzynski evaluates the optical depth, but I’d rather estimate a quantity that isdirectly relevant to the experiment, namely the rate of the microlensing events.Just by taking ∆ t away, rate = Z R LMC dR v ∞ πG N m ( R + r ⊙ − Rr ⊙ cos α ) 2 πR σ = Z R LMC dR v ∞ R + r ⊙ − Rr ⊙ cos α s R ( R LMC − R ) G N mR LMC σc = v ∞ σc √ G N mR LMC Z R LMC p R ( R LMC − R ) dRR + r ⊙ − Rr ⊙ cos α . (A. 35)The integral can be evaluated numerically. For R LMC = 50 kpc, r ⊙ = 8 . kpc, α = 82 ◦ , Mathematica gives 3.05. Then with σ = v ∞ / √ , v ∞ = 220 km/sec,we find rate = 1 . × − sec − (cid:18) M ⊙ m (cid:19) / = 5 . × − year − (cid:18) M ⊙ m (cid:19) / . Therefore, if we can monitor about a million stars, we may see 5 microlensingevents for a solar mass MACHO per year, even more for ligher ones.
Appendix A.4. Strong Lensing
Even though it is not a part of this lecture, it is fun to see what happens when r < ∼ δ . This can be studied easily with a slightly tilted coordinates in Fig. 22. H. Murayama
Fig. 22. A slightly different coordinate system to work out the distortion of images.
Using this coordinate system, we can draw a circle on the plane ( x, y ) =( x , y ) + ρ (cos φ, sin φ ) , and the corresponding image on the deflector plane is (˜ x, ˜ y ) = (˜ x , ˜ y ) + ˜ ρ (cos φ, sin φ ) = d d + d ( x, y ) . The impact parameter is then r = p ˜ x + ˜ y which allows us to calculate r ± ( r ) using Eq. (A. 26) for each φ .Obviously φ is the same for the undistorted and distorted images. Fig. 23 showsa spectacular example with ( x , y ) = (1 , , d d + d = , ρ = 0 . . Because ρ ∼ r , the Taylor expansion does not work, and the image is far from an ellipse. Fig. 23. A highly distorted image due to the gravitational lensing. Yellow circle is the undistortedimage, while the two blue regions are the images distorted by the gravitational lensing.
This kind of situation is not expected to occur for something as small as themirror of a telescope, but may for something as big as a galaxy. When an imageof a galaxy is distorted by a concentration of mass in the foreground, such asa cluster of galaxies, people have seen spectacular “strong lensing” effects, asshown in Fig. 24. hysics Beyond the Standard Model and Dark Matter Fig. 24. A Hubble Space Telescope image of a gravitational lens formed by the warping of images ofobjects behind a massive concentration of dark matter. Warped images of the same blue backgroundgalaxy are seen in multiple places. The detailed analysis of lensing effects allows one to map out themass distribution in the cluster that shows a smooth dark matter contribution not seen in the opticalimage. Taken from [107].
Acknowledgements
I’d like to express my gratitute to the organizers of the school, Francis Bernardeauand Christophe Grojean. I am especially indebted to Christophe for his patiencewaiting for this contribution. I also thank Adam Brown, Edoardo Di Napoli,Alexander Sellerholm, Ethan Siegel, Daniel Sunhede, Federico Urban, and GillesVertongen for our excursion to Tête Rousse. Matt Buckley kindly read and cor-rected the original manuscript. This work was supported in part by the U.S.Department of Energy under Contract DE-AC03-76SF00098, and in part by theNational Science Foundation under grant PHY-04-57315.
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