Dark Matter in Galaxies and Clusters: Unraveling Cosmic Mysteries

1. Cosmology • Dark matter in galaxies and clusters • Newtonian cosmology • The Big Bang model • Relativistic cosmology • The primordial Universe • The cosmic microwave background

2. Dark matter in galaxies and clusters Rotation curves of spiral galaxies Assuming circular orbits and spherical symmetry with M(r) = total mass inside the orbit In external regions (where there is very little visible matter), one would expect: M(r)≈ ct→ v ~ r –1/2 However, in our Galaxy and other spirals, one measures v ~ ct in these external regions → one assumes that mass continues to increase despite the fact that nothing can be seen

3. Dark matter in galaxies and clusters - 2 Dark matter halo → one postulates the existence of a spherical halo of dark matter Rotation curve: Conservation of mass: → in external regions: (= singular isothermal sphere) To avoid the central singularity, one assumes: (must also be truncated at large distance to avoid an infinite mass)

4. Dark matter in galaxies and clusters - 3 Dark matter halo Other formulae have been suggested for the dark matter distribution From numerical simulations, Navarro, Frank and White (1996) propose the following equation: It is generally considered that dark matter constitutes 80 to 95% of the mass of our Galaxy … and similar values in other spiral galaxies

5. Dark matter in galaxies and clusters - 4 Dark matter in spiral galaxies • Estimated from rotation curves – by assuming the galaxies are circular (ellipticity ↔ inclination) – by subtracting visible matter – by assuming total mass is spherically distributed → in the outskirts, v≈ ct, interpreted as indicating the presence of a dark matter halo Rotation curves of spiral galaxies

6. Dark matter in galaxies and clusters - 5 Dark matter in elliptical galaxies (out ?) 2003: Romanowsky et al. (Science301, 1696) measure the velocity dispersion of planetary nebulae in 3 elliptical galaxies → find a decrease compatible with the absence of dark matter in these galaxies → strange if ellipticals are the result of spirals merging…

7. Dark matter in galaxies and clusters - 6 Dark matter in elliptical galaxies (in ?) 2005: Dekel et al. (Nature437, 707) From simulations of elliptical galaxies formation by mergers of spirals with dark matter halos find that the stars observed far away from the center have highly eccentric orbits → their velocities are not oriented at random: the further away they are from the center in the plane of the sky, the higher the transverse component with respect to the radial component, which is the one measured → phew!... Results are compatible with the presence of dark matter

8. Dark matter in galaxies and clusters - 7 Dark matter in elliptical galaxies (episode 3) Most gravitational lensing galaxies are ellipticals → measurement of their total inside the Einstein radius from the lensing effect, in particular if ≈ Einstein ring → one gets a measurement of M(rE)H0 M(rE): mass inside the Einstein radius → M determined with similar accuracy as H0 (~10%)

9. Dark matter in galaxies and clusters - 8 Dark matter in elliptical galaxies (episode 3) Results for 15 gravitational mirages with elliptical lenses 1´´

10. Dark matter in galaxies and clusters - 9 Dark matter in elliptical galaxies (out) M/L ratio inside the Einstein ring as a function of RE/R1/2 (Einstein radius / effective radius) medium blue: visible matter alone light blue: with baryonic dark matter distributed as visible matter beige: with dark matter halo similar to spiral galaxies

11. Dark matter in galaxies and clusters – 10 Dark matter in clusters 1933: Zwicky measures radial velocities of galaxies in the Coma cluster He obtains a dispersion σ(vrad) = 977 km/s With such velocities, in order for the galaxies to remain gravitationally bound to the cluster, one needs Mtot ~ 3 1015M This is much higer than visible mass: Mvis ~ 1013M Galactic dark matter halos are not massive enough to fill the gap → there must be additional matter, distributed in between the galaxies Fritz Zwicky

12. Dark matter in galaxies and clusters – 11 Hot gas in clusters Observations in X-rays → discovery of very hot gas (~ 108 K) in galaxy clusters Mgas (Coma) ~ 3 1013M → this hot gas + `orphan´ stars are not enough to account for the total mass → dark matter also in clusters Chandra and HST images of two clusters

13. Dark matter in galaxies and clusters – 12 Detection of dark matter in clusters • Concentrated clusters (Σ >> Σcr) → background galaxies are imaged as giant arcs → allow to constrain the total mass distribution → distribution of dark matter • Less concentrated clusters → slight distortions of background galaxies → statistical analysis → similar results Part of the Abell 1699 cluster (HST)

14. Dark matter in galaxies and clusters – 13 Detection of dark matter in clusters Image of two clusters that just collided: the `bullet cluster´ In the optical (HST) + X-rays (pink, shock) + total mass (≈ dark matter) (blue) Hot gas is separated from the total mass distribution → presented as evidence in favor of WIMP dark matter (collisionless particles)

15. Dark matter in galaxies and clusters – 14 Detection of dark matter in clusters Image of two other clusters in collision: Abell 520 in the optical (HST) + X-rays (green) + dark matter (blue) orange: smoothed version of visible matter → incompatible with the bullet cluster

16. Newtonian cosmology The Olbers paradox: Why is the sky dark at night? • If Universe infinite and homogeneous → all lines of sight should encounter the surface of a star → the night sky should be as luminous as the surface of the Sun Olbers suggested absorption of light by dust as an explanation Principle of energy conservation: → Olbers solution does not work as dust would heat until emitting as much flux as the stars Modern solution: • The finite age of the Universe implies that light could not travel over infinite distances

17. Newtonian cosmology - 2 The cosmological principle To be able, from observations the local Universe, to test models representing the Universe as a whole, one must make the hypothesis that our neighborhood is representative of the whole Universe → one assumes that any sufficiently large portion of the Universe is representative of the Universe as a whole This is the cosmological principle It thus postulates that the Universe is homogeneous and isotropic if one considers sufficiently large scales N.B. In practice, sufficiently large => ~100 Mpc

18. Newtonian cosmology - 3 Model the Universe • From the cosmological principle, one can attempt to model the Universe • As gravity dominates on large scales, generalrelativity is the branch of science which should allow a: − quantitative − complete understanding of the Universe and its evolution (apart from the very tiny first fraction of a second, when quantum effects dominate) • However, we can develop our intuition by considering the expansion of the Universe from a newtonian point of view

19. Newtonian cosmology - 4 Simple `pressureless´ model • Universe filled with pressureless matter (`cosmic dust´) – of density ρ(t) (one assumes uniformly scattered matter) – does not take into account photons and neutrinos (since pressure in that case) • Let us consider a spherical shell of mass m and radius r(t) – expanding with the Universe at a speed: – such that it always contains the same particles

20. Newtonian cosmology - 5 • Total energy: with Mr = mass inside the sphere: With t0 = present time, one introduces the variable = r(t0) = present radius of the sphere, and parameter k such that: [k] = L–2

21. Newtonian cosmology - 6 (1) Parameter k determines the fate of the Universe: • k > 0 (E < 0): expansion will stop and reverse (`closed´Universe) • k < 0 (E > 0): expansion will go on forever (`open´Universe) • k = 0 (E = 0): v → 0 when t → ∞ (`flat´ Universe) !!! In this newtonian frame, spacetime is flat and the words `closed´, `open´ and `flat´ only refer to the dynamics of expansion → will be generalized in the frame of general relativity

22. Newtonian cosmology - 7 • Cosmological principle → same expansion rate everywhere r(t) = `distance´ coordinate = comoving coordinate (attached to expanding space and matter) R(t) = dimensionless scale factor such that R(t0) = 1 • Redshift:

23. Newtonian cosmology - 8 • However, for us, Robs = R(t0) = 1 → the scale factor at emission R and the redshift z are related by: • Mr = constant → ρ r3 = constant • mean density of the Universe as a function of redshift:

24. Newtonian cosmology - 9 Evolution of the pressureless model → determination of R(t) • Hubble law: with

25. Newtonian cosmology - 10 = Friedmann equation, as obtained in general relativity If flat Universe (k = 0, each shell expands at exactly its escape speed), the corresponding density is called the critical density ρc(t) Its present value is: → ρc,0 ≈ 10−26 kg/m3≈ 6 protons/m3

26. Newtonian cosmology - 11 • The estimated baryonic matter density (in particular from: − WMAP/Planck observations of the CMB − primordial nucleosynthesis models) amounts to: ρb,0 ≈ 0.04 ρc,0 (17) The remaining part would be dark matter • Density parameter: Present value:

27. Newtonian cosmology - 12 • The M/L ratio and the corresponding density parameter Ω0 are shown for various objects, as a function of their characteristic size • Increase with size, up to a maximum Ω0 < 0.3 • The analysis of Planck data gives: (with h= 0.68, assuming k = 0) – for total matter (baryonic + dark): – for baryonic matter:

28. Newtonian cosmology - 13 • Ω0 > 1 → k > 0 (closed Universe) • Ω0 < 1 → k < 0 (open Universe) (Universe with matter only) • Ω0 = 1 → k = 0 (flat Universe)

29. Newtonian cosmology - 14 • at very early times (z → ∞): H→ ∞ • if z → ∞ then Ω → 1, whatever Ω0: → the Universe was basically flat during the first stages of its expansion

30. Newtonian cosmology - 15 Expansion of a flat pressureless Universe with tH = 1/H0 = Hubble time

31. Newtonian cosmology - 16 Expansion of a pressureless Universe (general case) • If Ω0≠ 1, the solution is a bit more complicated • Evolution of R(t) adjusted on present time data (t = 0) • Note the similarity of the curves for t < 0 • This model is not valid at the very first stages (but negligible influence on the age of the Universe)

32. Newtonian cosmology - 17 Age of a pressureless Universe • flat Universe: • General case: more complicated expression, which simplifies at high redshift:

33. Newtonian cosmology - 18 Taking pressure into account • Non relativistic particles (v << c) : ρ = density of matter • Relativistic particles (v = c) : ρ = equivalent mass density = energy density / c2 • Spherical comoving shell of radius r, density ρ, pressure P, temperature T: 1st law of thermodynamics: U = internal energy, W = work and dQ = 0 (no heat flux since the temperature is the same everywhere in the Universe)

34. Newtonian cosmology - 19 Let u be the internal energy per unit volume: with (4): r = R → `fluid equation´:

35. Newtonian cosmology - 20 – multiply (12) by R – derive with respect to t – use (36) to replace – use (12) to replace – kc2 N.B.: the effect of P (like ρ) is to slow expansion down (pressure is acting outside as well as inside of the sphere, this is the equivalent mass density that matters)

36. Newtonian cosmology - 21 (36) + (37) = 2 equations for 3 unknowns R, ρ, P To get a solution, one must add the equationof state, which links P and ρ – for matter: wm = 0 – for radiation: wrad = 1/3 For pressureless matter, one gets (7) back

37. Newtonian cosmology - 22 Deceleration parameter q(t) = dimensionless quantity that describes the acceleration (or rather the deceleration) of expansion: For a pressureless Universe:

38. The Big Bang model • 1946: Gamov suggests that all chemical elements were synthesized during a very hot and densephase of the early Universe → Alpher, Bethe & Gamov paper (ApJ, 1st April 1948) (= αβγ) • 1948: Alpher and Herman realize that there were some mistakes in the computations → practically no nucleosynthesis beyond 4He • Moreover, at that time, the Hubble constant indicated an age of the Universe ~109 years, shorter than the age of the Earth, estimated at several billion years by radioactivity, as soon as 1928

39. The Big Bang model - 2 Steady state cosmology These different problems + the uncomfortable idea of a beginning of the Universe led Gold, Bondi and Hoyle to propose an alternative to the Gamov model → steady state cosmological model (Gold, Bondi & Hoyle, 1948) Based on the perfect cosmological principle: the Universe appears the same everywhere and at every time However, galaxies move away from each other → continuous creation of matter to keep ρ constant (~ 1 H atom per m3 per billion years) → tH does not give the order of magnitude for the age of the Universe, but for the creation of matter

40. The Big Bang model - 3 Radiation in the Big Bang model • One of the main ideas of the αβγ paper was that the young Universe was very hot and filled with radiation close to thermodynamic equilibrium (since photon mean free path very short → opaque medium) • 1948: Alpher and Herman predict that the radiation should have cooled down to a present value ~ 5 K With wrad = 1/3 and urad = aT4 − a factor R3 is due to the volume increase of the Universe − a factor R is due to the stretching of the photons wavelength

41. The Big Bang model - 4 • To produce the observed quantities of 4He and 3He, one needs: T ~ 109 K and ρb ~ 10−2 kg/m3 value similar to the original one estimated by Alpher and Herman, who had predicted the existence of the cosmic microwave background (CMB) as early as 1948

42. The Big Bang model - 5 Discovery of the cosmic microwave background 1964: Penzias and Wilson want to measure the radio emission of the Milky Way → they discover an isotropic, non seasonal radiation → cannot come from the atmosphere, nor from the Milky Way Brought in contact with cosmologists Dicke and Peebles, who had `rediscovered´ the results of Alpher and Herman and predicted T(CMB) ≈ 10 K → they realize that Penzias and Wilson just discovered the `relic´ radiation from the Big Bang, with T(CMB) ≈ 3 K → hard blow on steady state theory Robert Wilson and Arno Penzias

43. The Big Bang model - 6 Recent measurements of the CMB • 1991: the COBE satellite measures with very high accuracy the CMB spectrum → remarkable agreement with a black body spectrum at 2.725 ± 0.002 K • 2001-2010: the WMAP satellite measures the CMB with a higher angular resolution (to study the tiny anisotropies) • 2009-2013: the Planck satellite measures the CMB with even higher angular resolution CMB spectrum (COBE)

44. The Big Bang model - 7 Dipole anisotropy of the CMB An observer in motion with respect to the expanding Universe sees a Doppler shift of the CMB depending on direction → this translates into an apparent variation of the CMB T° as a function of the angle θ between the directions of motion and observation → the Sun has a peculiar velocity of 371 km/s → the local group has a peculiar velocity ~600 km/s in the direction of Hydra constellation Dipole anisotropy of the CMB (COBE)

45. The Big Bang model - 8 2-components Universe model • Presently, radiation has a negligible effect on the Universe expansion • However, this was not the case during the first epochs → for these epochs, we need a 2-components model: − matter (non relativistic, v << c) − photons and neutrinos (relativistic, v ~ c) • For photons : With g = number of degrees of freedom (grad = 2 since 2 spin states)

46. The Big Bang model - 9 The radiation era → for relativistic particles: for matter (baryons + dark) : • Presently, matter dominates: • However, going back in time, ρrel increases faster than ρm → during the 1st epochs, radiation (= all relativistic particles) dominated • The transition happened when ρrel ≈ ρm, which correponds to: R≈ 3 × 10−4, z ≈ 3000 and, from (43): T ≈ 9000 K

47. The Big Bang model - 10 • One calculates that the transition between radiation and matter era happened when the Universe was: t ~ 55000 years old • When radiation was fully dominant (ρrel >> ρm), by integrating (44) with k = 0 (since the Universe was basically flat at that epoch), one gets: and: Where g* is the effective number of degrees of freedom for radiation in the broad sense (all relativistic particles)

48. The Big Bang model - 11 Primordial nucleosynthesis (1) t ~ 10−4 s; T ~ 1012 K • the Universe contains a `soup´ of photons (γ), electrons (e−), positrons (e+), electronic and muonic neutrinos and their antiparticles , plus a small fraction of protons and neutrons (~5 for 1010 photons), constantly transformed into one another by the reactions:

49. The Big Bang model - 12 • the nn/np ratio is given by Boltzmann law: • the energy (~100 MeV) is so high compared to the mass difference between protons and neutrons: that the reactions are in equilibrium and the ratio nn/np≈ 0.985

50. The Big Bang model - 13 (2) t ~ 1 s; T ~ 1010 K • The rates of reactions (50) to (52) strongly decrease as: − the mean energy of neutrinos becomes too low for them to take part in these reactions − soon after, the mean energy of photons, kT, goes under 1 MeV, threshold for the creation of electron-positron pairs by the reaction: γ→ e− + e+ → huge decrease of the number of e−and e+ • At that time, the ratio nn / np given by (53) amounts to 0.223 • When T = 1010 K, 4He becomes stable but stays out of reach since 2H remains unstable as long as T > 109 K → neutrons remain free and the ratio nn / np continues to decrease through the neutron desintegration reaction