- 1. Dark Matter, Dark Energy, and Unification Models Neven Bilić Ruđer Bošković Institute Zagreb
- 2. Lecture 1 1 2 3 4 5 BIG BANG History of the Early Universe - Universe Creation History of the Early Universe – Inflation Cosmological Observations and History of the Universe Empirical Grounds for Observational Cosmology 5.1 Hubble’s Law 5.2 Cosmic Microwave Background (CMB) Radiation 5.3 Big Bang Nucleosynthesis 6 Theoretical Foundation of Modern Cosmology 7 Density of Matter in Space 8 Dark Matter
- 3. Lecture 2 9 Dark Energy 9.1 Vacuum Energy – Cosmological Constant 9.2 Quintessence 9.3 Phantom quintessence 9.4 k-essence
- 4. Lecture 3 10 DE/DM Unification – Quartessence 10.1 Chaplygin gas 10.2 Problems with Nonvanishing Sound Speed 10.2.1 Spherical Model 10.3 Generalized Chaplygin Gas 10.4 Tachyon Condensate 10.5 Entropy Perturbations 10.6 Dusty Dark Energy 11 Conclusions and Outlook
- 5. 1. Big Bang What kind of explosion is the Big Bang? The Big Bang is like an explosion of a bomb that took place in a previously empty space. Wrong! The Big Bang is an eplosion of space itself which, before that, was concentrated in one point. Right! C.H. Lineweaver and T. Davis, Misconceptions About the BIG BANG, Scientific American, March 2005
- 6. 2. Early Universe - Creation The Universe begins as a quantum fluctuation of the initial vacuum The total energy of the Universe (matter energy+dark energy +gravity) = 0 Owing to the uncertainty relations DE Dt h h – Planck constant Borrowing a small amount of energy (E0 ) from the vacuum is allowed for a long time (t )! As a consequence, the lifetime of the Universe is almost infinite
- 7. 3. Early Universe - Inflation A short period of inflation follows – very rapid expansion 1025 times in 10-32 s.
- 9. The horizon problem. Observations of the cosmic microwave background radiation (CMB) show that the Universe is homogeneous and isotropic. The problem arises because the information about CMB radiation arrive from distant regions of the Universe which were not in a causal contact at the moment when radiation had been emitted – in contradiction with the observational fact that the measured temperature of radiation is equal (up to the deviations of at most about 10-5) in all directions of observation.
- 10. The flatness problem Observations of the average matter density, expansion rate and fluctuations of the CMB radiation show that the Universe is flat or with a very small curvature today. In order to achieve this, a “fine-tuning“ of the initial conditions is needed, which is rather unnatural. The answer is given by inflation:
- 11. v =5 105Hd , The initial density perturbations problem. The question is how the initial deviations from homogeneity of the density are formed having in mind that they should be about 10-5 in order to yield today’s structures (stars, galaxies, clusters). The answer is given by inflation: perturbations of density are created as quantum fluctuations of the inflaton field.
- 12. 4. Cosmological Observations and History of the Universe • • • • • • The properties we would like to find out: Spacetime geometry Age of the Universe Expansion rate Amount of ordinary (“baryonic”) matter Amount and nature of dark matter and dark energy Mechanism of structure formation • • • • • • Observations that could yield these properties: Background microwave radiation Observations of distant dalekih supernovae Large scale structure observations Concentrations of light elements Galactic rotation curves Gravitational lenses
- 13. 13 billion years Universe Today 5 billion yeras Solar System Foramation, First Supernove 1 billion years Galaxy and Star Formation 100 000 years 100 s 10-10 s 10-34 s Formation of Atoms, Decupling of Matter–Radiation Nucleosynthesis of Light Elements Synthesis of Protons and Neutrons QCD Plasma: quarks and gluons Electroweak Transition End of Inflation 10-43 s Quantum Gravity Era
- 14. After a short period of inflation the Universe continue expanding at much slower rate so that after roughly 13.5 billion years, the visible part is about 90 billion light years.
- 15. 5. Empirical Grounds for Observational Cosmology • The expansion of the Universe - Hubble’s law v = Hd • Cosmic microwave background radiation very homogeneous in all directions • Big Bang nucleosynthesis - proportions of light elements (H, D, He, Li)
- 16. Hubble’s law – Universe expansion
- 17. Hubble’s law – Universe expansion a=0.5
- 18. Hubble’s law – Universe expansion a=0.75
- 19. Hubble’s law – Universe expansion a=1
- 20. Hubble’s law – Universe expansion recession velocity a=0.5 a=0.75 a=1
- 21. CAN GALAXIES RECEDE FASTER THAN LIGHT? Of course not. Einstein’s special theory of relativity forbids that. Wrong! Sure they can. Special relativity does not apply to recession velocity. Right! RH RH - Hubble radius
- 22. CAN WE SEE GALAXIES RECEDING FASTER THAN LIGHT? Of course not. Light from those galaxies never reaches us. Wrong! Sure we can, because the expansion rate changes over time. Right!
- 23. Do objects inside the Universe expand, too? Yes. Expansion causes the universe and everything in it to grow. Wrong! Ne. No. The universe grows, but coherent objects (stars, galaxies ..) do not Right!
- 24. T = 2.723K DT = 100K DT = 200K Measuring CMB; the temperature map of the sky.
- 25. Angular (multipole) spectrum of the fluctuations of the CMB (Planck 2013)
- 26. 6. Theoretical Foundation of Modern Cosmology • General relativity – gravity G=T Matter determines the space-time geometry Geometry determines the motion of matter • Homogeneity and isotropy of space – approximate property on very large scales (~Glyrs today) • Fluctuations of matter and geometry in the early Universe cause structure formation (stars, galaxys, clusters ...)
- 27. General Relativity – Gravity 1 R g R 8 T g 2 g - metric tensor R - Riemann curvature tensor - cosmological constant T - energy – momentum tensor
- 28. Perfect Fluid Matter is described by a perfect fluid T ( p)u u pg u - fluid velocity T- energy-momentum tensor p - pressure ρ - energy density
- 29. The energy momentum conservation T ; = 0 yields, as its longitudinal part uT ; = 0 , the continuity equation 3H ( p) = 0, and, as its transverse part, the Euler equation u = 1 h p, , p where 3H = u ; ; h = g u u = u , ; u = u u ; . projector onto the three-space orthogonal to H is the local Hubble parameter. u
- 30. Cosmological principle Homogeneity and isotropy of space dr 2 2 2 ds dt a (t ) r d 2 1 kr 2 2 2 - cosmological scale the curvature constant k takes on the values 1, 0, or -1, for a closed, flat, or open universe, respectively. a (t ) – cosmological scale
- 31. Friedman equations k 8 G a 2 3 a a 4 G a ( 3 p) 3 2 T ; 0 a H a k expansion rate 0 closed 0 flat 0 open a 3(1 w) w p Various kinds of the cosmic fluids with different w • radiation • matter • vacuum pR R 3 pM 0 p w 1 3 w0 w 1 a 4 a 3 a0
- 32. 7. Density of Matter in Space The best agreement with cosmologic observations are obtained by the models with a flat space According to Einstein’s theory, a flat space universe requires critical matter density cr today cr 10-29 g/cm3 Ω= / cr ratio of the actual to the critical density For a flat space Ω=1
- 33. What does the Universe consist of? From astronomical observations: luminous matter (stars, galaxies, gas ...) lum/cr0.5% From the light element abundances and comparison with the Big Bang nucleosynthesis: barynic matter(protons, neutrons, nuclei) Bar/cr5% Total matter density fraction ΩM=M/cr 0.31 Accelerated expansion and comparison of the standard Big Bang model with observations requires that the dark energy density (vacuum energy) today ΩΛ= /cr = 0. 69%
- 34. Density fractions of various kinds of matter today with respect to the total density B B 0.05 tot DM DM 0.26 tot 0.69 tot These fractions change with time but for a spatialy flat Universe the following always holds: tot crit B DM 1
- 35. Age of the Universe Easy to calculate using the present observed fractions of matter, radiation and vacuum energy. For a spatially flat Universe from the Friedmann equation and energy consrvation we have H (a) H 0 ( M a 3 Ra 4 )1/2 H 0 h 100Gpc/s2 (14.5942Gyr)1, h 0.67 Age of the Universe is then T 1 1 da 1 da a T dt (a 3 M )1/2 aH H 0 0 0 0 T 13.78Gyr
- 36. How Large is the Observable Universe? S The universe is 14 billion years old, so the radius of the observable part is 14 billion lightyears. Wrong! Because space is expanding, the observable part of our Universe has a radius of more than 14 billion light-years. Right!
- 37. 8. Dark nature of the Universe – Dark Matter • • • • According to present recent obsrvations (Planck Satellite Mission): More than 99% of matter is not luminous Out of that less then 5% is ordinary (“baryonic”) About 26% is Dark Matter About 61% is Dark Energy (Vacuum Energy) 0.69 0.26 0.04 0.01
- 38. DARK MATTER Galactic rotation curves
- 39. Cluster 1E0657-558, Bullet Clowe et al, astro-ph/0608407 http://chandra.harvard.edu
- 42. DM contents - possible candidates Baryonic DM in the form of astrophysical objects • • • • Brown dwarfs Black holes MACHO planets Nonbaryonic DM • Sterile neutrino • Axion • SUSY stable particles: gravitino, neutralino, axino Exsperimental evidence? - great expectations from the LHC experiments at CERN
- 43. Hot DM refers to low-mass neutral particles that are still relativistic when galaxy-size masses ( 1012 M ) are first encompassed within the horizon. Hence, fluctuations on galaxy scales are wiped out. Standard examples of hot DM are neutrinos and majorons. They are still in thermal equilibrium after the QCD deconfinement transition, which took place at TQCD ≈ 150 MeV. Hot DM particles have a cosmological number density comparable with that of microwave background photons, which implies an upper bound to their mass of a few tens of eV.
- 44. Warm DM particles are just becoming nonrelativistic when galaxy-size masses enter the horizon. Warm DM particles interact much more weakly than neutrinos. They decouple (i.e., their mean free path first exceeds the horizon size) at T>>TQCD. As a consequence, their mass is expected to be roughly an order of magnitude larger, than hot DM particles. Examples of warm DM are keV sterile neutrino, axino, or gravitino in soft supersymmetry breaking scenarios.
- 45. Cold DM particles are already nonrelativistic 6 when even globular cluster masses ( 10 M ) enter the horizon. Hence, their free is of no cosmological importance. In other words, all cosmologically relevant fluctuations survive in a universe dominated by cold DM. The two main particle candidates for cold dark matter are the lowest supersymmetric weakly interacting massive particles (WIMPs) and the axion.
- 46. 9. Dark Energy Because gravity acts as an attractive force between astrophysical objects we expect that the expansion of the Universe will slowly decelerate. However, recent observations indicate that the Universe expansion began to accelerate since about 5 billion years ago. Repulsive gravity?
- 47. Lecture 2
- 48. Accelerated expansion 0 One possible explanation is the existence of a fluid with negative pressure such that p 3 0 and in the second Friedmann equation the universe acceleration a becomes positive cosmological constant vacuum energy density with equation of state p=-ρ. Its negative pressure may be responsible for accelerated expansion! New term: Dark Energy – fluid with negative presssure - generalization of the concept of vacuum energy
- 50. Time dependence of the DE density Another important property of DE is that its density does not vary with time or very weakly depends on time. In contrast , the density of ordinary matter varies rapidly because of a rapid volume expansion. The raughh picture is that in the early Universe when the density of matter exceeded the density of DE the Universe expansion was slowing down. In the course of evolution the matter density decreases and when the DE density began to dominate, the Universe began to accelerate.
- 51. Most popular models of dark energy • Cosmological constant – vacuum energy density. Energy density does not vary with time. • Quintessence – a scalar field with a canonical kinetic term. Energy density varies with time. • Phantom quintessence – a scalar field with a negative kinetic term. Energy density varies with time. • k-essence – a scalar field whose Lagrangian is a general function of kinetic energy. Energy density varies with time. • Quartessence – a model of unifying of DE and DM. Special subclass of k-essence. One of the popular models is the so-called Chaplygin gas
- 52. 9.1 Cosmological Constant – Einstein’s biggest mistake? 1917 Einstein proposes cosmological constant 1929 Hubble discovers expansion of the Universe 1934 Einstein declares the cosmological constant his biggest blunder 1998 Pearlmutter, Riess, and Schmidt find evidence for the cosmological constant or dark energy
- 53. Origin of the cosmological constant: Damn! The vacuum is not empty. Quantum fluctuations of the vacuum W+ W- e+ quark eantiquark particle antiparticle
- 54. Observations of supernovae type Ia Od 1988 Saul Perlmutter (Univ. Of California) lieder of Supernova Cosmology Project Od 1994, Brian Schmidt (Australian National University) klieder of High z Supernova Search Team In the same team Adam Riess (Space Telescope Science Institute) the first one who made the analysis
- 55. Supernovae Ia occure in binary systems made of a white dwarf and a large companion star. A low-mass white dwarf accreting matter from a nearby companion approaches the limit of 1.4 solar masses (Chandrasekar limit) and becomes unstable. Then, a thermonuclear explosion takes place ensues and an immense amount of energy is suddenly released.
- 56. In a typical galaxy, supernovae occur a few times in thousand years. In a short period a huge amount of energy is released After a few days the emission achieves a maximum with brightness comparable with the luminosity of an entire galaxy. Supernova SN 1884 in the NGC 4526 galaxy
- 57. Supernovae Ia are identified through their spectral signatures. Besides, they show a relatvely simple light profile Their spectra and light curves are amazingly uniform, indicating a common origin and a common intrinsic luminosity. This is similar to Cepheids – variable stars which serve as standard candles at relatively short distances up to about10 Mpc (~30 light years).
- 58. 1. Measuring the redshift of the spectrum λ /λ0=1+z=1/a one determines the cosmological redshift z and the scale a at the time when the observed SN appeared Cosmological redshift is similar to the Doppler redshift Doppler redshift
- 60. 2. Measuring the apparent magnitude m (logaritamic measure of the flux) and knowing the absolute magnitude M (logaritamic measure of the luminosity) one finds the so called luminosity distance DL DL Mpc 10 ( m M )/55 1Mpc=3.085 million l.years
- 61. Thus for each observed supernova we have two numbers: the redshift z and the luminosity distance DL These data are plotted on the Hubble diagram: z as a function of DL . Comparing with theoretical curves obtained from different cosmological models one infers the history of the Universe expansion.
- 62. Hubble diagram Green line: Empty Model (Ω = 0), Red : Closed Universe (ΩM = 2 ) Violet : Standard flat model (ΛCDM) with ΩM = 0.27 and ΩΛ = 0.73 Dashed violet : closed ΛCDM Black: Enstein-de Sitter Model (ΩM = 1, ΩΛ = 0, critical universe) Black dashed: Einstein - de Sitter Model with Cold DM (p=0) Blue: de Sitter Model (ΩM = 0, ΩΛ = 1) Dashed blue : Evolving Supernovae.
- 63. The observations lead to the conclusion that the Universe expansion is accelerated today. In 2011 the three astrophysicists received the Nobel prize for physics.
- 64. Transition from a decelerated to accelerated expansion It is important to look for direct evidence of an earlier, slowing phase of expansion. Such evidence would help confirm the standard cosmological model and give scientists a clue to the underlying cause of the present period of cosmic acceleration. Because telescopes look back in time as they gather light from far-off stars and galaxies, astronomers can explore the expansion history of the universe by focusing on distant objects. Observing supernovae that appeared 7 billion years ago, i.e., at the time when the universe was half the present size (a=1/2 or z=1) shows the Universe expansion was slowing down. A transition from a decelerated to accelerated expansion happened at about 4 – 5 billion years ago when the Universe siye was about a=2/3 (z=0.5) .
- 65. Alternative hypotheses Some skeptical scientists (traditional astronomers in particular) wondered, whether the teams had correctly interpreted the data from the supernovae. Was it possible that another effect besides cosmic acceleration could have caused the supernovae to appear fainter than expected? For example, dust filling intergalactic space could also make the supernovae appear dim. Or perhaps ancient supernovae were just born dimmer because the chemical composition of the universe was different from what it is today, with a smaller abundance of the heavy elements produced by nuclear reactions in stars.
- 66. Test of the alternative hypotheses Luckily, a good test of the competing hypotheses is available. If supernovae appear fainter than expected because of an astrophysical cause, such as a pervasive screen of dust, or because past supernovae were born dimmer, the dimming effects should increase with the objects’ redshift. But if the dimming is the result of a recent cosmic speedup that followed an earlier era of deceleration, supernovae from the slowdown period would appear relatively brighter. Therefore, observations of supernovae that exploded when the universe was less than two thirds of its present size could provide the evidence to show which of the hypotheses is correct. (It is possible, of course, that an unknown astrophysical phenomenon could precisely match the effects of both the speedup and slowdown, but scientists generally disfavor such artificially tuned explanation)
- 67. Test of the alternative hypotheses The Advanced Camera for Surveys, a new imaging instrument installed on the space telescope in 2002, enabled scientists to turn Hubble telescop into a supernova-hunting machine. Riess and his team found 6 supernovae 1a which exploded when the Universe was less than 1/2 its present size (z > 1, or more than 7 billion years ago). Together with SN 1997ff, these are the most distant type Ia supernovae ever discovered. The observations confirmed the existence of an early slowdown period and placed the transitional point between slowdown and speedup at about 5 billion years ago. This finding is consistent with theoretical expectations of the standard cosmological model ΛCDM
- 68. Problems with Λ 1) Fine tuning problem. The calculation of the vacuum energy density in field theory of the Standard Model of particle physics gives the value about 10120 times higher than the value of Λ obtained from observations. One possible way out is fine tuning: a rather unnatural assumption that all interactions of the standard model of particle physics somehow conspire to yield cancellation between various large contributions to the vacuum energy resulting in a small value of Λ , in agreement with observations 2) Coincidence problem. Why is this fine tuned value of Λ such that DM and DE are comparable today, leaving one to rely on anthropic arguments?
- 69. 9.2 Quintessence The coincidence problem is somewhat ameliorated in quintessence – a canonical scalar field with selfinteraction effectively providing a slow roll inflation today P.Ratra, J. Peebles PRD 37 (1988) S d 4 x L ( X , ) X g , , 1 L X V ( ) 2 2 S T ,, L g det g g Field theory description of a perfect fluid if X>0 T ( p)u u pg
- 70. where u , 1 p L X V ( ) 2 X 1 X L X V ( ) 2 A suitable choice of V(φ) yields a desired cosmology, or vice versa: from a desired equation of state p=p(ρ) one can derive the Lagrangian of the corresponding scalar field theory
- 71. 9.3 Phantom quintessence Phantom energy is a substance with negative pressure such that |p| exceeds the energy density so that the null energy condition (NEC) is violated, i.e., p+ρ<0. Phantom quintessence is a scalar field with a negative kinetic term 1 p L X V ( ) 2 1 X V ( ) 2 Obviously, for X>0 we have p+ρ<0 which demonstrates a violation of NEC! This model predicts a catastrophic end of the Universe, the so-called Big Rip - the total collapse of all bound systems.
- 72. 9.4 k-essence k-essence is a generalized quintessence which was first introduced as a model for inflation . A minimally coupled k-essence model is described by R 4 S = d x g L ( , X ) 16G where L is the most general Lagrangian, which depends on a single scalar field of dimension , and on the dimensionless quantity X g ,, For X>0 , the energy momentum tensor takes the perfect fluid form T = 2LX ,, Lg = ( p)u u p g , where LX L/X
- 73. The associated hydrodynamic quantities are p = L( , X ) = 2 XLX ( , X ) L( , X ). Examples Kinetic k-essence The Lagrangian is a function of X only. In this case p = L( X ) = 2 XLX ( X ) L( X ) To this class belong the ghost condensate L( X ) = A(1 X )2 B and the scalar Born-Infeld model L( X ) = A 1 X
- 74. Ghost condensate model N. Arkani-Hamed et al , JHEP 05 (2004) R.J. Scherrer, PRL 93 (2004) L(X) L(X) or X de-Sitter X de-Sitter
- 75. Lecture 3
- 76. 10. DE/DM Unification – Quartessence The astrophysical and cosmological observational data can be accommodated by combining baryons with conventional CDM and a simple cosmological constant providing DE. This ΛCDM model, however, faces the fine tuning and coincidence problems associated to Λ. Another interpretation of this data is that DM/DE are different manifestations of a common structure. The general class of models, in which a unification of DM and DE is achieved through a single entity, is often referred to as quartessence. Amost of the unification scenarios that have recently been suggested are basend on k-essence type of models including Ghost Condensate, various variants of the Chaplygin Gas and a model called Dusty Dark Energy.
- 77. 10.1 Chaplygin Gas An exotic fluid with an equation of state p A The first definite model for a dark matter/energy unification A. Kamenshchik, U. Moschella, V. Pasquier, PLB 511 (2001) N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002) J.C. Fabris, S.V.B. Goncalves, P.E. de Souza, GRG 34 (2002)
- 78. In a homogeneous model the conservation equation yields the Chaplygin gas density as a function of the scale factor a A B a6 where B is an integration constant.The Chaplygin gas thus interpolates between dust (ρ~a -3 ) at large redshifts and a cosmological constant (ρ~ A½) today and hence yields a correct homogeneous cosmology
- 79. Scalar Born-Infeld theory Chaplygin gas has an equivalent scalar field formulation. Considering a kinetic k-essence type of Lagrangian LBI A 1 X X g , , LBI T 2 ,, LBI g X perfect fluid T ( p)u u pg with u , X
- 80. The Chaplygin gas model is equivalent to (scalar) Born-Infeld description of a D-brane: Nambu-Goto action of a p-brane moving in a p+2 -dimensional bulk SDBI A d p 1 x ( 1) p det( g ind ) the induced metric (“pull back”) of the bulk metric Gab X a X b ind g Gab x x Xa – coordinates in the bulk, xμ – coordinates on the brane
- 81. In 4+1 dim. bulk (p=3) Choose the coordinates such that X μ =x μ, μ=0,..3, and let the fifth coordinate X 4≡ θ be normal to the brane. Then G g for G 4 0 0...3 xi G44 1 x0 Xa θ
- 82. We find a k-essence type of theory SDBI A dx 4 1 X 2 X g , , with A 1 X 2 p A 1 X 2 ; and hence p A
- 83. 10.2 Problems with Nonvanishing Sound Speed To be able to claim that a field theoretical model actually achieves unification, one must be assured that initial perturbations can evolve into a deeply nonlinear regime to form a gravitational condensate of superparticles that can play the role of CDM. The inhomogeneous Chaplygin gas based ona Zel'dovich type approximation has been proposed, N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002) and the picture has emerged that on caustics, where the density is high, the fluid behaves as cold dark matter, wherea in voids, w=p/ ρ is driven to the lower bound -1 producing acceleration as dark energy. In fact, for this issue, the usual Zel’dovich approximation has the shortcoming that the effects of finite sound speed are neglected.
- 84. In fact, all models that unify DM and DE face the problem of sound speed related to the well-known Jeans instability. A fluid with a nonzero sound speed has a characteristic scale below which the pressure effectively opposes gravity. Hence the perturbations of the scale smaller than the sonic horizon will be prevented from growing. • Soon after the Chaplygin gas was proposed as a model of unification it has been shown that the naive model does not reproduce the mass power spectrum H.B. Sandvik et al PRD 69 (2004) and the CMB D. Carturan and F. Finelli, PRD 68 (2003); L. Amendola et JCAP 07 (2003)
- 85. α<0 α>0 Power spectrum for p=-A/ρα for various α H.B. Sandvik et al, PRD 2004 α =0
- 86. CMB spectrum for p=-A/ρα for various α L. Amendola et al, JCAP 2003
- 87. The physical reason is a nonvanishing sound speed. Although the adiabatic speed of sound p A 2 cs 2 S is small until a ~ 1, the accumulated comoving size of the acoustic horizon cs ds dt H 0 1a 7 / 2 a reaches Mpc scales by redshifts of about z ~ 20, thus frustrating the structure formation at galactic and subgalactic scales. This may be easily demonstrated in a simple spherical model.
- 88. 10.2.1 Spherical Model To study the evolution of perturbations of a model with nonvanishing pressure gradients and speed of sound we will use the spherical model. For a k-essence model or any one-component type of model, the Euler equation combines with Einstein’s equations to h p, 3H 3H 2 2 4G( 3 p) = p ; 2 = = h h u( ;) H h We thus obtain an evolution equation for H . Owing to the definition of the four-velocity and orthogonality h u = 0 we may write h p, = c h , . 2 s
- 89. Hence, if cs =0 or if the pressure gradient is parallel to uμ as for dust, the right-hand side of the evolution equation and we obtain 3H 3H 2 2 4G( 3 p) = 0 wihich together with the continuity equation comprises the original spherical model E. Gaztãnaga and J.A. Lobo, ApJ. 548 (2001). However, we are not interested in dust, since generally cs ≠0 and hμν p,ν≠0 so we must generalize the spherical model to include pressure gradients. The density contrast and the deviation of the Hubble parameter from the background value are defined by = ( ) / H = H H
- 90. In the Newtonian approximation subtracting the background, eliminating δH and ∂tδH and neglecting shear we find 3 2 4 2 1 2H H (1 ) 2 2 31 a xi cs2 =0 1 xi The root of the structure formation problem is the last term, as may be understood if we solve the evolution equation at linear order
- 91. 3 2 cs2 2 H H 2 D 0 2 a with the solution (in k-space) k – comoving wave number ds –comoving size of the acoustic horizon k a 1/ 4 J 5/ 4 (ds k ) behaving asymptotically as k a for ds k 1 cos (ds k ) k for ds k 1 2 a • The perturbations whose comoving size R=1/k is larger than ds grow as δ ~ a . Once the perturbations enter the acoustic horizon, i.e., as soon as R< ds , they undergo damped oscillations.
- 92. In the case of the Chaplygin gas we have d s a 7/2/H 0 reaching Mpc scales already at redshifts of order 10-20. However, small perturbations alone are not the issue, since large density contrasts are required on galactic and cluster scales.
- 93. Nonlinear evolution of inhomogeneities As soon as δ ~ 1 the linear perturbation theory cannot be trusted. An essentially nonperturbative approach is needed in order to investigate whether a significant fraction of initial density perturbations collapses in gravitationally bound structure - the condensate. If that happens the system evolves into a two-phase structure - a mixture of CDM in the form of condensate and DE in the form of uncondensed gas.
- 94. Two phase structure – mixture of CDM in the form of condensate and DE in the form of gas gas + condensate nearly homogeneous gas at zdec 1089 collapse during expansion
- 95. To investigate the inhomogeneous Chaplygin gas we have used the so called Thomas Fermi correspondence: (to every purely kinetic k-essence there exist a corresponding canonical complex field theory) We have derived the solution for the inhomogeneous Chaplygin-gas cosmology implementing geometric version of the Zel’dovich approximation: the transformation from Lagrange to comoving-synchronous coordinates induces the spatial metric γij A B γ is the determinant of γij in comoving gauge. N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002)
- 96. The expectation to achieve a condensate formation based on the Zel’dovich approximation was naive as it is well known that Zel’dovich approximation has the shortcoming that the effects of finite sound speed are neglected. Later on, using generalizations of the spherical model which incorporate effects of the acoustic horizon we have shown that an initially perturbative Chaplygin gas evolves into a mixed system containing cold dark matter-like gravitational condensate. N.B., R. Lindebaum, G.B. Tupper, R.D. Viollier, JCAP 0411 (2003)
- 97. 3 2 4 2 1 2H H (1 ) 2 2 31 a xi The density contrast described by a nonlinear evolution equation either grows as dust or undergoes damped oscillations depending on the initial conditions in contrast to the linear evolution where all perturbations that enter the acoustic horizon undergo dumped oscillations cs2 =0 1 xi Evolution of the density contrast in the spherical model from aeq = 3 · 10-4 for the comoving wavelength 1/k = 0.34 Mpc, δk (aeq ) =0.004 (solid) δk (aeq ) =0.005 (dashed). N. B., R.J. Lindebaum, G.B. Tupper, and R.D. Viollier, JCAP 11 (2004)
- 98. Unfortunately, the estimated collapse fraction of about 1% is far too small to significantly affect structure formation. This basically decided the fate of the simple Chaplygin gas model
- 99. 10.3 Generalized Chaplygin Gas Another model was proposed in an attempt to solve the structure formation problem and has gained a wide popularity. The generalized Chaplygin gas is defined as p= A , 0 1 The additional parameter does afford greater flexibility: e.g. for small α the sound horizon d s a 2 /H 0 and thus by fine tuning α<10-5, the data can be perturbatively accommodated
- 100. Other modifications • The generalized Chaplygin gas in a modified gravity approach, reminiscent of Cardassian models 8 G 1 1 1 ) H A ) 3 2 T. Barreiro and A.A. Sen, PRD 70 (2004) • A deformation of the Chaplygin gas – Milne-Born-Infeld theory L A X bX 2 M. Novello, M. Makler, L.S. Werneck and C.A. Romero, PRD 71 (2005) • Variable Chaplygin gas p ~ an Zong-Kuan Guo, Yuan-Zhong Zhang, astro-ph/0506091, PLB (2007)
- 101. 10.4 Tachyon Condensate The failure of the simple Chaplygin gas (CG) does not exhaust all the possibilities for quartessence. The Born-Infeld Lagrangian is a special case of the string-theory inspired tachyon Lagrangian in which the constant A is replaced by a potential L = V () 1 g ,, . Tachyon models are a particular case of k-essence. It was noted in Zong-Kuan Guo, et al astro-ph/0506091, PLB (2007) that (in the FRW cosmology), the tachyon model is described by the CG equation of state in which the constant A is replaced by a n function of the cosmological scale factor p ~ a so the model was dubbed “variable Chaplygin gas”.
- 102. A preliminary analysis of a unifying model based on the tachyon type Lagrangian has been carried out in N.B., G.B. Tupper, R.D. Viollier, PRD 80 (2009) for a potential of the form V ( ) Vn 2 n n 0,1,2 n=0 gives the Dirac-Born-Infeld description of a D-brane - equivalent to the Chaplygin gas It may be shown that the model with n≠0 effectively behaves as a variable Chaplygin gas, with p ~ a 6n . The much smaller sonic horizon d s ~ a (7 23n ) H 0 enhances condensate formation by 2 orders of magnitude over the simple Chaplygin gas. Hence this type of model may salvage the quartessence scenario.
- 103. 10.5 Entropy Perturbations One way to deal with the structure formation problem, is to assume entropy perturbations such that the effective speed of sound cs vanishes. In that picture we assume that A is not a constant so we have p = cs2 A/ = 0 even if cs≠0. But in a single field model it is precisely the adiabatic speed of sound that governs the evolution. Hence, entropy perturbations require the introduction of a second field on which A depends.
- 104. 10.5 Entropy Perturbations One way to deal with the structure formation problem, is to assume entropy perturbations such that the effective speed of sound cs vanishes. This scenario has been suggested as a possible way out of the structure formation problem immanent to all DM/DE unification models. It has been noted by R. Reis et al Phys. Rev. D 69 (2004) that the root of the structure formation problem is the term ∇2δp in perturbation equations, equal to cs2∇2δ for adiabatic perturbations, and if there are entropy perturbations such that δp = 0, no difficulty arises.
- 105. Suppose that the matter Lagrangian depends on two degrees of freedom, e.g., a Born-Infeld scalar field θ and one additional scalar field φ. In this case, instead of a simple barotropic form p=p(ρ), the equation of state is parametric p=p(θ,φ) ρ=ρ(θ,φ) and involves the entropy density (entropy per particle) s=s(θ,φ) The corresponding perturbations p p p s s s
- 106. The speed of sound is the sum of two nonadiabatic terms 1 cs2 p p s s p S 0 Thus, even for a nonzero ∂p/∂ the speed of sound may vanish if the second term on the right-hand side cancels the first one. This cancellation will take place if in the course of an adiabatic expansion, the perturbation δφ grows with a in the same way as δ . In this case, it is only a matter of adjusting the initial conditions of δφ with δ to get cs=0.
- 107. Obviously, this scenario cannot work for simple or generalized Chaplygin gas models as these models are adiabatic. We have made attempts at realizing the nonadiabatic scenario in some 2 component models (hybrid Chaplygin gas and Kalb- Ramond Chaplygin gas) N. B., G.B. Tupper, and R.D. Viollier, JCAP 0510 (2005); N. B., G.B. Tupper, and R.D. Viollier, J. Phys. A40,(2007), Aside from negating the simplicity of the one-field model, our analysis has convinced us that even if δp=0 is arranged as an initial condition, it is all but impossible to maintain this condition in a realistic model for evolution.
- 108. 10.6 Dusty Dark Energy Another way to bypass the structure formation problem is to impose a constraint on pressure such that the pressure gradient is parallel to the fluid 4-velocity. The model called Dusty Dark Energy comprises two scalar fields λ and φ, λ being a Lagrange multiplier which enforces a constraint between φ and its kinetic energy term X . Starting from the action 1 S = d 4 x g K , X ) X V ), 2 where K (φ,X) is an arbitrary function of X and φ. The field λ is a Lagrange multiplier and does not have a kinetic term, while X is a standard kinetic term for φ.
- 109. The energy momentum tensor takes the usual perfect fluid form with the 4-velocity as before u = , / X The λ field equation 1 S 1 = X V ( ) = 0 g 2 imposes a constraint such that the pressure p = K ( , X ) = K ( ,2V ( )) becomes a function of φ only. Then, its gradient is proportional to φ,μ and hence parallel to the 4-velocity. In this way cs is always identically zero. In particular, cosmological perturbations reproduce the standard hydrodynamic behavior in the limit of vanishing cs . In a certain limit this model reproduces the evolution history of ΛCDM, with some potentially measurable differences.
- 110. Fate of the universe Solving the mystery of cosmic acceleration will reveal the destiny of our universe. Possible scenarios: 1. If the dark energy density ρΛ is constant or increasing with time, in 100 billion years or so all but a few hundred galaxies will be far too redshifted to be seen. 2. But if the dark energy density decreases and matter becomes dominant again, our cosmic horizon will grow, revealing more of the universe. 3. Even more extreme (and lethal) futures are possible. If dark energy density increases rather than decreases, the universe will eventually undergo a “hyper speedup” that would result in a Big Rip: tearing apart all bound systems, galaxies, solar systems, planets and atomic nuclei, in that order.
- 111. 11 Conclusions and Outlook We have discussed a just a few out of many attempts to unify DE/DM. The above mentioned examples illustrate more or less conventional trends in modern cosmology. There exist various alternative ideas such as modified theories of gravity. One of the popular ideas is the socalled brane world cosmology where our world is a fourdimensional membrane submerged in a five-dimensional space. Alternative theories explain more or less successfully a part of the phenomena related to dark energy and dark matter, but up until now there is no completely satisfactory theory which would solve all the puzzles. In any case, there remains a lot of work to be done for theoretical physicists.
- 113. Thomas-Fermi correspondence Under reasonable assumptions in the cosmological context there exist an equivalence Complex scalar field theories (canonical or phantom) Kinetic k-essence type of models
- 114. Consider L g , V (| | m ) * , 2 2 Thomas-Fermi approximation 2 e im , m , i, TF Lagrangian where X g ,, LTF 4 XY U (Y ) /m Y 2 2m 2 U (Y ) V (Y ) / m4
- 115. Equations of motion for φ and θ U X 0 Y (Yg , ); 0 We now define the potential W(X) through a Legendre transformation W ( X ) U (Y ) XY with X UY and Y WX U UY Y W WX X
- 116. correspondence Complex scalar FT L g *,, V (| |2 2 Kinetic k-essence FT m2 ) e im Eqs. of motion ( 2 g , ); 0 1 dV g ,, 2 0 2 m d | | L m4W ( X ) X g , , Eq. of motion (WX g , ); 0 Parametric eq. of state p m4W m4 (2 XWX W )
- 117. Current conservation Klein-Gordon current ig (*, * ) j , kinetic k-essence current j 2m2WX g , U(1) symmetry shift symmetry ei c
- 118. Example: Quartic potential Scalar field potential 2 V 0 m0 | |2 | |4 V 2 U (Y ) 1 1 1 Y 2 2 2 8 Kinetic k-essence 1 1 W(X ) X 2 2 2
- 119. Example: Chaplygin gas Scalar field potential | |2 m2 4 V m 2 2 | | m Scalar Born-Infeld FT 1 U (Y ) Y Y W ( X ) 2 1 X