The Density of the Universe and Dark Matter

Transcription

1 The Density of the Universe and Dark Matter Michal Šumbera March 12, 2018 Michal Šumbera Dark Matter March 12, / 33

2 Literature Our discussion is based on the book Michal Šumbera Dark Matter March 12, / 33

3 The Density of the Universe and Dark Matter Introduction History Reminder: Observational Parameters: Ω 0 A very brief history of the Universe Weighing the Universe Counting stars Nucleosynthesis foreshadowed Galaxy rotation curves Galaxy cluster composition Bulk motions in the Universe DM signal from CMB fluctuations The formation of structure Ultimate proof of DM: collisions of galaxy clusters What might the dark matter be? Fundamental particles Compact objects Dark matter searches Summary: Dark Matter, a mystery we can believe in... Michal Šumbera Dark Matter March 12, / 33

4 Introduction History First hints of dark matter: 1932 Oort found evidence for extra hidden matter in our galaxy Zwicky velocity dispersion of galaxies in the Coma cluster inferred a mass-to-light ratio of 400M per solar luminosity, thus exceeding the ratio in the solar neighborhood by two orders of magnitude. Consider the equation of hydrostatic equilibrium for a system with spherical symmetry dp = a(r)ρ (1) dr where p, ρ, and a are, respectively, the pressure, density, and gravitational acceleration of the gas, at radius R. For an ideal gas (p = Nk B T = k B µm p ρt ), this can be rewritten in terms of the temperature, T, and the average molecular weight µ 0.6 as: d ln ρ d ln R + d ln T d ln R = R µm p a(r) (2) T k B Michal Šumbera Dark Matter March 12, / 33

5 Introduction History T of clusters is roughly constant outside of their cores and the density profile ρ(r) of observed gas at large R roughly follows a power-law ρ(r) R α with 2 α 1.5. The mass of a cluster can be determined via application of virial theorem (2 T = Ū) to the observed distribution of radial velocities. We then find that the temperature should obey the relation: ( ) ( ) MR 1Mpc k B T ( )keV M R for the baryonic mass of a typical cluster, where M R is the mass enclosed within the radius R. The disparity between the temperature obtained using Eq.3 and the corresponding observed temperature, T 10keV, when M R is identified with the baryonic mass, suggests the existence of a substantial amount of dark matter in clusters. Michal Šumbera Dark Matter March 12, / 33 (3)

6 Reminder: Mechanical similarity Assume that potential energy U is a homogenous function of the coordinates: U(α r 1, α r 2,..., α r n ) = α k U( r 1, r 2,..., r n ) (4) For transform r a α r a and t βt v a = d r a /dt α/β v a and kinetic energy transforms as T α 2 /β 2 T. For α and β such that β = α 1 k/2 the Langrangian L = T U transforms as L α k L which leaves equations of motion unaltered. r a α r a corresponds to replacement of particle path with linear dimension l by another path with dimension l geometrically similar but differring in size (e.g. small circle large circle etc.). (4) the times of the motion between corresponding points r a α r a = r a are t /t = (l /l) 1 k/2 velocities v /v = (l /l) k/2, energies E /E = (l /l) k and angular momenta M /M = (l /l) 1+k/2. Example 1: Uniform (gravitational) field U = F r has k = 1 t /t = l /l. Example 2: k = 1, potential energy is inversely proportional to distance (Newton, Coulomb) t /t = (l /l) 3/2 (Kepler s 3 rd law). Michal Šumbera Dark Matter March 12, / 33

7 Reminder: Virial theorem Kinetic energy T = 1 2 m av 2 a is a quadratic function of velocities. Euler theorem on homogenous functions v a T / v a = 2T. 2T = v a T / v a = p a v a = d dt ( p a r a ) r a p a (5) The time-averaged value of any function f (t): 1 τ 1 τ df F (τ) F (0) f = lim f (t)dt = lim dt = lim τ τ 0 τ τ 0 dt τ τ where the last equality holds when t; F (t) <. If t, a the system occupies the region r a <, v a < p a r a < d dt ( p a r a ) = 0. Compute the time-averige of (5): = 0 (6) 2T = r a p a = r a U/ r a = ku (7) For gravitational potential k = 1 2T = U, QED. Michal Šumbera Dark Matter March 12, / 33

8 Reminder: Observational Parameters: Ω 0 For which value of H 2 = 8 3 πgρ ka 2 the Universe is flat? Flat k = 0 ρ c (t) = (3H 2 )/(8πG) (8) G = m 3 kg 1 s 2 ρ c (t 0 ) = 1.88h kgm 3 = 2.78h M /(h 1 Mpc) 3 Tiny density of matter is sufficient to halt and reverse the expansion of the Universe. ( )M M galaxy, R galaxy = Mpc the Universe cannot be far away from the critical density (within an order of magnitude or so). The Universe needs not be flat ρ c is not necessarily the true density of the Universe, but it sets a natural scale for the density: Ω(t) ρ/ρ c (9) where Ω(t) is density parameter and Ω 0 Ω(t 0 ) its present value. N.B. Ω is related to the Friedmann equation: Ω 1 = k a 2 H 2 (10) Michal Šumbera Dark Matter March 12, / 33

9 A very brief history of the Universe T GeV. Some (unknown) grand unified group, G, breaks down into the standard model gauge group, SU(3) C SU(2)L U(1)Y. Little is known about this transition, however. T 10 2 GeV. The SM gauge symmetry breaks into SU(3) C U(1)Q. This transition (EW symmetry breaking), could be the origin of baryogenesis and possibly of primordial magnetic fields. T GeV. Weakly interacting dark matter candidates with GeV-TeV scale masses freeze-out. This is true in particular for the neutralino. T 0.3GeV. The QCD phase transition occurs, which drives the confinement of quarks and gluons into hadrons. T 1MeV. Neutron freeze-out occurs. T 100keV. Nucleosynthesis: protons and neutrons fuse into light elements (D, 3 He, 4 He, Li). T 1eV. The matter density becomes equal to that of the radiation, allowing for the formation of structure to begin. T 0.4eV. Photon decoupling produces CMB. T = 2.7K 10 4 ev. Today. Michal Šumbera Dark Matter March 12, / 33

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11 Weighing the Universe How much material there is in the Universe? Crude estimate: typical galaxy weighs about M, galaxies are typically Mpc apart the Universe cannot be a long way from the critical density. But how good an estimate can be made? This is quantified by the density parameter Ω which is divided up amongst the different types i of material present in our Universe: Ω = i Ω i ρ i ρ c (11) ρ c characteristic scale for the density in the Universe is rather small; its present value is ρ c (t 0 ) = 1.88h kgm 3 = 2.78h M /(h 1 Mpc) 3 Ṇ.B. An obstacle to comparing the true density to the critical density is the factors of h, which are uncertain. Michal Šumbera Dark Matter March 12, / 33

12 Weighing the Universe N.B. various Ω i evolve with time differently, depending on the EOS of the component. A general expression for the expansion rate is: H 2 (z) H 2 0 = Ω x (1 + z) 3(1+αx ) + Ω K (1 + z) 2 + Ω M (1 + z) 3 + Ω R (1 + z) 4 (12) where M and R are labels for matter and radiation, Ω K = k 2 /a 2 0 H2 0, X refers to a generic substance with equation of state p X = α X ρ X (e.g. for the cosmological constant, α Λ = 1) and z is the redshift. Not all visible material is in the form of stars. Within clusters of galaxies there is a large amount of gas which is extremely hot and emits in the X-ray region of the spectrum. Michal Šumbera Dark Matter March 12, / 33

13 Counting stars Lets count all stars within a suitably-large region. Provided we have looked in a large enough region, we get an estimate of the overall density of material in stars. Ω stars ρ stars ρ c (13) N.B. A lot of material may reside in very low mass stars (m 0.08M ), which would be too faint to detect. Such brown dwarfs (sometimes called Jupiters) are stars with insufficient material to initiate nuclear burning. If for some reason there were a lot of objects of this kind then they could contribute substantially to the total density without being noticed, though this is not thought to be very likely on grounds of extrapolation from what we do know. Michal Šumbera Dark Matter March 12, / 33

14 Nucleosynthesis foreshadowed Very strong reason to believe that conventional material cannot contribute an entire ρ c. This is supported by the nucleosynthesis. Formation of the light elements can only match the observed element abundances if the amount of baryonic matter has a density Ω B h (14) The Hubble constant h appears as an additional uncertainty, but the constraint is certainly strong enough to insist that it is not possible to have ρ c ρ B, whether it be in the form of luminous stars or invisible brown dwarfs or gas. Adopting the Hubble Space Telescope constraints on h gives an upper limit well below 10%. A lower bound on Ω B suggests that there should be substantially more baryonic material in the Universe than just the visible stars, probably upwards of 2.5% of ρ c. Michal Šumbera Dark Matter March 12, / 33

15 Galaxy rotation curves Look at motions of various kinds of astronomical object. If the visible material is insufficient to provide the inferred gravitational force the excess gravitational attraction must be due to invisible material. Most impressive applications: A galaxy rotation curve shows the velocity of matter rotating in a spiral disk, as a function of radius from the center. The individual stars are on orbits given by Kepler s law; if a galaxy has mass M(R) within a radius R, then the balance between the centrifugal and the gravitational force demands: v 2 R = GM(R) GM(R) R 2 v(r) = (15) R N.B. The mass outside the radius R contributes no gravitational pull, due to the same theorem of Newton s we used to derive the Friedmann equation. { R if R Rgallaxy There are two possibilities: v(r) = 1 (16) R if R > R gallaxy Michal Šumbera Dark Matter March 12, / 33

16 Galaxy rotation curves Figure: The rotation curve of the spiral galaxy NGC3198. It remains roughly constant at large radii, outside the visible disk. Faster than expected orbits require a larger central force, and so they imply the existence of extra, dark, matter. Michal Šumbera Dark Matter March 12, / 33

17 Galaxy rotation curves The typical velocities at R > R gallaxy can be 3 higher than predicted from the luminous matter, implying 10 more matter than can be directly seen. This is an example of dark matter. Standard estimates suggest Ω halo 0.1 marginally consistent with equation (14) i.e. matter can be entirely baryonic. However it is probably difficult to make up all of the halo brown dwarfs. Alternative: New form of (non-baryonic) matter interacting extremely weakly with conventional matter. Such matter lacks any dissipation mechanism able to concentrate it into a disk structure resembling that of the stars. the dark matter should be in the form of a spherical halo into which the visible galactic disk and the globular clusters are embedded. Figure: A galactic disk, with a few globular clusters, embedded in a spherical halo of dark matter. Michal Šumbera Dark Matter March 12, / 33

18 Galaxy cluster composition The largest gravitationally-collapsed objects in the Universe an ideal probe of the different kinds of matter. Because of their size, they should contain a fair sample of the material in the Universe, since there is no means of segregating different types of material as all is drawn in by gravity. The visible components of a galaxy cluster are in two main parts: There are the stars within the individual galaxies, and there is diffuse hot gas seen in X-rays which has been heated up through falling into the strong gravitational potential well of the cluster. The baryon content of the latter is the greater, with about five to ten times more hot gas than stars. Michal Šumbera Dark Matter March 12, / 33

19 Galaxy cluster composition The hot gas high temperature gives it a substantial pressure p T 4, but it is confined to the galaxy cluster by gravitational attraction. However, the self-gravity of the gas alone does not provide enough attraction on its own, with the total mass of the cluster inferred to be around 10 larger than the gas mass. It is natural to assume that this extra attraction is given by dark matter. the dark matter density must be around ten times larger than the baryon density given by nucleosynthesis. Data from the Chandra X-ray satellite yeilds Ω B /Ω h 3/2 which using the nucleosynthesis constraint on Ω B given above leads to Ω 0 0.3h 1/ Michal Šumbera Dark Matter March 12, / 33

20 Bulk motions in the Universe Further dynamical evidence motions of galaxies relative to one another (i.e. the deviations from the cosmological principle). Galaxies relative motions are due to the peculiar velocities mentioned earlier. This allows one to estimate their mass under the assumption that their gravitational interaction is responsible for the motions, which are often termed bulk flows. Rather complicated analyses indicate that the total density of matter in the Universe must obey Ω This is well above the amount permitted by the nucleosynthesis constraint of equation (14). Conclusion: The Universe is largely composed of dark matter mostly of non-baryonic origin. Michal Šumbera Dark Matter March 12, / 33

21 DM signal from CMB fluctuations CMB temperature fluctuations:a comparison between COBE and WMAP. Image from The observed temperature anisotropies ( T /T 10 5 ) are expanded as T T = + +l l=2 m= l a lm Y lm (θ, ϕ) (17) Michal Šumbera Dark Matter March 12, / 33

22 DM signal from CMB fluctuations The variance of a lm is C l a lm 2 1 2l + 1 l m= l a lm 2 (18) If the temperature fluctuations are assumed to be Gaussian, as appears to be the case, all of the information contained in CMB maps is contained in the power spectrum C l = f (l). Usually plotted is l(l + 1)C l /2π. The observed power spectrum of CMB anisotropies.. Michal Šumbera Dark Matter March 12, / 33

23 DM signal from CMB fluctuations The origin of temperature fluctuations: Photons + electrons and protons are tightly coupled by Thomson scattering (elastic scattering of electromagnetic radiation by a charged particle) a photon-baryon fluid N.B. Thomson scattering linearly polarizes the CMB. Acoustic oscillations in the photon-baryon fluid: δρ/ρ T /T Sound waves (seeded by primordial fluctuations) result from the competition between attraction due to gravity and pressure restoring forces. Speed of sound photons + baryons Expansion DM (+ DE) Re-combination: e + p H + γ At T 3000K the reaction goes right, hydrogen forms and the Universe becomes transparent, photons are liberated from the plasma and travel freely for about 13.7 billion years. Acoustic waves have been imprinted on their temperature. Michal Šumbera Dark Matter March 12, / 33

24 DM signal from CMB fluctuations The methodology, for extracting information from CMB anisotropy maps, is simple, at least in principle. Starting from a cosmological model with a fixed number of parameters (usually 6 or 7), the best-fit parameters are determined from the peak of the N-dimensional likelihood surface. From the analysis of the WMAP data alone, the following values are found for the abundance of baryons and matter in the Universe: Ω B h 2 = ± 0.001, Ω M h 2 = 0.14 ± Taking into account data from CMB experiments studying smaller scales (with respect to WMAP), such asacbar and CBI, and astronomical measurements of the power spectrum from large scale structure (2dFGRS) and the Lyman α forest, the constraints become: Ω B h 2 = ± , Ω M h 2 = ± The value of Ω B h 2 thus obtained is consistent with predictions from Big Bang nucleosynthesis: < Ω B h 2 < Michal Šumbera Dark Matter March 12, / 33

25 The formation of structure: Observations vs Simulations The most widely adopted approach to the problem of large-scale structure formation involves the use of N-body simulations. The description of the evolution of structures from seed inhomogeneities is complicated by the action of many physical processes like gas dynamics, radiative cooling, photoionization, recombination and radiative transfer. Michal Šumbera Dark Matter March 12, / 33

26 Direct empirical proof of DM: collision of galaxy clusters D.Clowe et al.,the Astrophysical Journal, 648:L109-L September 10 GC consist of galaxies, intergalactic gas and DM ( 1 : 10 : 100). During collision: The gas is slowed down via friction (EM forces) plasma. Galaxies - compact objects - pass through without any friction. DM - extended object - only gravitation. Left panel: Two merging galactic clusters 1E in the visible part of the spectrum. Two concentrated regions of shining points are visible - larger (left) and smaller (right). Blue crosses mark in locations of the maximum concentration of two gaseous clouds associated with the clusters. Right panel: The same but in the X-ray region of the spectra. On both pictures green contours show the gravitation field intensity obtained using weak gravitational lensing, white contours the errors on the positions of the peaks and white bars indicate 200 kpc. Michal S umbera Dark Matter March 12, / 33

27 What might the dark matter be? Cluster of gallaxies The ultimate Copernican viewpoint: not only are we in no special place in the Universe, but we aren t even made out of the same stuff which dominates the matter density of the Universe. Although the evidence for dark matter is regarded by most as pretty much overwhelming, there is no consensus as to what form it takes. For the large-scale structure of the Universe to have evolved in the observed manner the non-baryonic matter must have negligible random motion. We therefore refer to it as cold dark matter (CDM). There are two main CDM classes: DM exists in the form of individual elementary particles. DM is some type of compact astrophysical object formed from many particles. Michal Šumbera Dark Matter March 12, / 33

28 Fundamental particles Things we know exist In the SM m ν = 0. In the Universe νabout as numerous γ. If the SM is extended to permit m ν 0.1eV, this would not affect their number density but they would have enough density to imply a closed Universe! A light ν would be a type of DM known as hot DM (v c for at least some fraction of the Universe s lifetime). In fact, HDM does not have favourable properties for structure formation and if the neutrino has such a mass it could at most contribute only part of the matter density, with some other form of dark matter also being required. Things we believe might exist Supersymmetry most trusted extension to standard particle theory associates a new companion particle to each known particle. Lightest supersymmetric particle (LSP) is stable and is an excellent CDM candidate. Depending on the model might be called the photino, or gravitino, or neutralino. They are also sometimes known as WIMPs Weakly Interacting Massive Particles. Others: axions or something completely different... Michal Šumbera Dark Matter March 12, / 33

29 Compact objects Black Holes BH formed early in the Universe s history rather than at a star s final death throes, would act like cold dark matter. BH made of baryons must form before nucleosynthesis to avoid the nucleosynthesis bound of equation (9.3). Baryons already in BH by the time of nucleosynthesis don t count as baryons, as they are not available to participate in nuclei formation. MAssive Compact Halo Object MACHOs Compact objects with masses not too far from stellar masses, and they may be baryonic or non-baryonic. Brown dwarfs are a baryonic example, but it would also be possible to conceive of non-baryonic examples. MACHOs have already been detected (!) using gravitational lensing of stars. Michal Šumbera Dark Matter March 12, / 33

30 Dark matter searches - MACHOs An illustration of the MACHO search strategy. We look from our position in the disk of the galaxy towards stars in the Large Magellanic Cloud. The line-of-sight passes through the dark matter halo, and if there are invisible compact objects there, and they pass extremely close to the line-of-sight, then gravitational lensing of the LMC star can occur. Light curves for a star in the LMC, obtained by the MACHO collaboration. The x-axis is in days with an arbitrary origin, while the y-axis shows the brightness of the star in red light and in blue light (the vertical units are a magnitude scale). The brightening only happened once, rather than periodically, and is the same in both red and blue light, whereas variable stars brighten differently at different wave- lengths. Michal Šumbera Dark Matter March 12, / 33

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32 Dark matter searches - Fundamental particles Accelerator: The LHC will search for the LSP, missing energy mass, perhaps X-section. Non Accelerator: WIMP-nucleus/nucleon scattering: nucleus/nucleon recoil deposits energy in a semiconductor (Ge) measure seasonal variation as the Earth moving around the sun sweeps through the DM halo DA(rk)MA(tter)/LIBRA (Grand Sasso), C(yogenic)D(ark)M(atter)S(earch) (Soudan), Xenon.... Axion-Photon conversion in B-fields (Primakoff) (CDMS,Pvlas) Indirect searches: e + e Excess positrons Pamela, Heat (satellites) Wimp Wimp annihilation γγ hard gammas ( > 50 GeV) Glast/Fermi (FGST) (sat). νν Amanda,Antares,IceCube (neutrino detectors) Sterile neutrinos: ν s ν a + γ X-ray background: Chandra, XMM/Newton, Integral (sats), constrain mass, mixing angle. Michal Šumbera Dark Matter March 12, / 33

33 Summary: Dark Matter, a mystery we can believe in... A window to physics beyond the standard model. A convergence between particle physics, statistical mechanics, astrophysics, cosmology. Satellites, accelerators, underground (and under-ice) detectors are the instruments for discovery. Large scale numerical simulations yield unprecedented picture of structure (and galaxy) formation. Knowledge will emerge NOT from a single approach but by a combination of space and earth based observations!!! Michal Šumbera Dark Matter March 12, / 33