WIXLIB

9 6;: and 9 6;: are the pressure and energy density of the fluid at a given time in the expansion, andwhere is the comoving four-velocity, satisfying .Let us now write the equations of motion of such a fluid in an expanding universe. According togeneral relativity, these equations can be deduced from the Einstein equations (1), where we substitute the FRW metric (2) and the perfect fluid tensor (7). The 2component of the Einstein equationsconstitutes the so-called Friedmann equation;!#( 7 ' .377v .(8)44.(where I have treated the cosmological constant as a different component from matter. In fact, it canbe associated with the vacuum energy of quantum field theory, although we still do not understand whyshould it have such a small value (120 orders of magnitude below that predicted by quantum theory), if itis non-zero. This constitutes today one of the most fundamental problems of physics, let alone cosmology. ), a direct consequence of the general covariance of theThe conservationof energy ( theory (), can be written in terms of the FRW metric and the perfect fluid tensor (7) as,1, 6 c7 E') ,1, 67 E B3(9) k' 3where the energy density and pressure can be split into its matter and radiation components, 5 H 3 4 5 ' 5 , with corresponding equations of state,w . Together, the Friedmannand the energy-conservation equation give the evolution equation for the scale factor,7 !#( ' 4 '397:(10)44tI will now make a few useful definitions. We can write the Hubble parameter today v in units of100 km s Mpc & , in terms of which one can estimate the order of magnitude for the present size andage of the universe,t B 8 & ?3 & " ¡Q 1 & v(11) t4 B 3 & B ¡Q (12)v * 4Q *t & # § v (13) * for decades, and only in the last fewThe parameter has been measured to be in the range ª *G « . I will discuss those recent measurements in theyears has it been found to lie within 10% ofnext Section. One can also define a critical density , that which in the absence of a cosmological constant wouldcorrespond to a flat universe,4 t v . * E. .p # w "!N(14)E * 3 & p # w 9 & ¡² :(15)pEE * g is a solar mass unit. The critical density corresponds to approximately 4where protons per cubic meter, certainly a very dilute fluid! In terms of the critical density it is possible to define the ratios ³B µw , for matter, radiation, cosmological constant and even curvature, today,;!#% "!#% ¶³³(16)4 t4 tv .v .( *7 t t(17)³Q·³B 4 tv . v .112

We can evaluate today the radiation component ³ , correspondingto relativistic particles,from the &¹ º »¼ ¾½ ¿ B¹ º » \ E* Ep\ E9x w «: w 9pà :, whichdensity of microwave background photons, ¹ º » * ÁÀ gives ³ À . . Three massless neutrinos contribute an even smaller amount. Therefore,we can safely neglect the contribution of relativistic particles to the total density of the universe today,which is dominated either by non-relativistic particles (baryons, dark matter or massive neutrinos) or bya cosmological constant, and write the rate of expansion vª. in terms of its value today,7 \t7 Et7 t7 ' ''*.t\E77v . 9 :v . ³³³²·³B 7(18).An interesting consequence of these redefinitions is that I can now write the Friedmann equation today,7 7 t, as a cosmic sum rule, Ä'' 3(19)³³Q·³B today. That is, in the context of a FRW universe, the total fraction of matterwhere we have neglected ³density, cosmological constant and spatial curvature today must add up to one. For instance, if we measureone of the three components, say the spatial curvature, we can deduce the sum of the other two. Makinguse of the cosmic sum rule today, we can write the matter and cosmological constant as a function of the7 tK scale factor ()tÅÇ ÉÆ È;!#% 7³3ÅÇ &ÆKÈ Ê77 ' 87 '7 E 87(20)³ 9 :4 v . 9 :³ 9:³Q· 9:tÅÇ ÉÆ È(7 E *7³ ·ÅÇ &ÆKÈ Ê7 ' {7 '7 E 877³Q· 9 :(21)4 ³ 9:³Q· 9:v . 9 :7Ë} This implies that for sufficiently early times,, all matter-dominated FRW universes can be de 3 5). On the other hand, the vacuum energyscribed by Einstein-de Sitter (EdS) models ( ³ ³ ·will always dominate in the future.Another relationship which becomes very useful is that of the cosmological deceleration parameterzt, in terms of the matter and cosmological constant components of the universe, see Eq. (10),7 z t 37(22)³Q· ³tv .zt which is independent of the spatial curvature. Uniform expansion corresponds toand requires a Ì ³²· . It represents spatial sections that are expanding at a fixed rate, its scaleprecise cancellation: ³factor growing by the same amount in equally-spaced time intervals. Accelerated expansion correspondszpt ³Q· : spatial sections expand at an increasing rate, their scaleto and comes about whenever ³z tÎÍ factor growing at a greater speed with each time interval. Decelerated expansion corresponds to Í ³Q· : spatial sections expand at a decreasing rate, their scale factor growingand occurs whenever ³at a smaller speed with each time interval.today,2.1.3Mechanical analogyIt is enlightening to work with a mechanical analogy of the Friedmann equation. Let us rewrite Eq. (8) as( 7 AZ?xÐ]Ñ;A Ð 37 . µÏ7(23) . G\E E ½ 7is the equivalent of mass for the whole volume of the universe. Equation (23) canwhere 'ÔÓfor a test particle of unit mass in the centralbe understood as the energy conservation law Òpotential 'Ó 39 Z:(24)  . 5Note that in the limit Õ Ö8d the radiation component starts dominating, see Eq. (18), but we still recover the EdS model.113

corresponding to a Newtonian potential plus a harmonic oscillator potential with a negative spring con ( 4( w . Note that, in the absence of a cosmological constant (), a critical universe, definedstant Âas the borderline between indefinite expansion and recollapse, corresponds, through the Friedmann equa tions of motion, precisely with a flat universe (). In that case, and only in that case, a spatially ¾ ) corresponds to an eternally expanding universe, and a spatially closed universeopen universe ( Ø' ) to a recollapsing universe in the future. Such a well known (textbook) correspondence is( incorrect when ³Q·8Ù : spatially open universes may recollapse while closed universes can expand forever. One can see in Fig. 1 a range of possible evolutions of the scale factor, for various pairs of values 3of 9 ³³Q·²: .3.03.0B2.5Û2.0AC2.5Þ2.0No 01.0K0.5LM0.5F0.00.0-3-2-10123-3-2-1Fig. 1: Evolution of the scale parameter with respect to time for different values of matter density and cosmological parameter.ºThe horizontal axis represents æ¼j ç èpéëêXì êíèxî , while the vertical axis is ïðj Õ ñeÕ è in each case. The values of ( òò ó )&for different plots are: A (1,0), B (0.1,0), C (1.5,0), D (3,0), E (0.1,0.9), F (0,1), G (3,.1), H (3,1), I (.1,.5), J (.5, ì²b ),,K (1.1,2.707), L (1,2.59), M (0.1,1.5), N (0.1,2.5). From Ref. [7].One can show that, for ³Q·ôÙ 7 t 7 Í , for which õ 9 0²:0wv .E 'õ 9 0²:0 ³0õ ö 9 0²:40 . ³ v, a critical universe ( v) corresponds to those pointsÍ 79 : and õ ö 9 0 : vanish, while õ ö ö 9 0²:, . ³Q Ä' 0 ³B '³Q· Q3 (25)00114 4 ³B w ³Í 3(26)

õ ö ö 9 0²:G 0 ³ ' ³Q ' ³B ³ B Í Í0 0Using the cosmic sum rule (19), we can write the solutions as ³Q· ? @ A E E Ñ ? @ A 9 ³ ³ & : ùø³³³B ø 4 w ³ Ìú Ìû **(27)(28)Í 7ð ýü(), and ³B , while the secondoneThe first solution corresponds to the critical point 0 þø ÿ \ Eø 4 9³³Q w ³ , and ³Q : w ³Q. , for. Expanding around ³, we find ³Q·to 0 û . Ì 3. These critical solutions are asymptotic to the Einstein-de Sitter model ( ³), see³³Q·Fig. 2.3.0ce ounB2.52.01.5ΩΛgatinelerccAing celerat eDExpansion ecollapseC l RosedO �ò ó î . º The line ò ó jËb#ìkò corresponds to a flat universe, ò 8j d , separating open fromclosed universes. The line ò&óKj)òñ corresponds to uniform expansion, ;èQjªd , separating accelerating from deceleratingFig. 2: Parameter spaceéÁòº universes. The dashed line corresponds to critical universes, separating eternal expansion from recollapse in the future. Finally,the dotted line corresponds to ê2.1.4è ç è j , beyond which the universe has a bounce.Thermodynamical analogyIt is also enlightening to find an analogy between the energy conservation Eq. (9) and the second law ofThermodynamics, ') Ó 3, ,5 ,(29) ÓÓ 7 Ewhere is the total energy of the closed system andis its physical volume. Equation (9) ), corresponding to a fluidimplies that the expansion of the universe is adiabatic or isoentropic ( , ª Q in thermal equilibrium at a temperature T. For a barotropic fluid, satisfying the equation of state,we can write the energy density evolution as, 7 E , 7 E 4 7 E *99 :9:v:1, 61, 6(30)For relativistic particles in thermal equilibrium, the trace of the energy-momentum tensor vanishes (be 5 4 4ww . In that case, the energy density ofcause of conformal invariance) and thusradiation in thermal equilibrium can be written as [8]115

! . \ 34 Z o1nxq n ;q \' r! s#" n ;q \(31)3(32)where is the number of relativistic degrees of freedom, coming from both bosons and fermions. Using ª K'ð 9:p, w , and thereforethe equilibrium expressions for the pressure and density, we can write , Î )' Ó 9 Î') 'Ó &ù Î') Ó ,'AZ?xÐ]*:9 Ï % (33), , 9:: ,.H7 E ' Ó 9That is, up to an additive constant, the entropy per comoving volume is : w , which isconserved. The entropy per comoving volume is dominated by the contribution of relativistic particles,so that, to very good approximation, !7] EAS?xÐ Ñ"A Ð 3. µÏ :(34) ( 9 ù«EE '* (35) ( o]npqh n ;q n*; q)r s " A consequence of Eq. (34) is that, during the adiabatic expansion of the universe, the scale factor grows7, inversely proportional to the temperature of the universe,w . Therefore, the observational factthat the universe is expanding today implies that in the past the universe must have been much hotter anddenser, and that in the future it will become much colder and dilute. Since the ratio of scale factors canube described in terms of the redshift parameter , see Eq. (5), we can find the temperature of the universeat an earlier epoch by t '8u *9:(36)Such a relation has been spectacularly confirmed with observations of absorption spectra from quasarsat large distances, which showed that, indeed, the temperature of the radiation background scaled withredshift in the way predicted by the hot Big Bang model.2.2 Brief thermal history of the universeIn this Section, I will briefly summarize the thermal history of the universe, from the Planck era to thepresent. As we go back in time, the universe becomes hotter and hotter and thus the amount of energyavailable for particle interactions increases. As a consequence, the nature of interactions goes from thosedescribed at low energy by long range gravitational and electromagnetic physics, to atomic physics, nuclear physics, all the way to high energy physics at the electroweak scale, gran unification (perhaps), andfinally quantum gravity. The last two are still uncertain since we do not have any experimental evidencefor those ultra high energy phenomena, and perhaps Nature has followed a different path. 6The way we know about the high energy interactions of matter is via particle accelerators, which areunravelling the details of those fundamental interactions as we increase in energy. However, one shouldbear in mind that the physical conditions that take place in our high energy colliders are very differentfrom those that occurred in the early universe. These machines could never reproduce the conditions ofdensity and pressure in the rapidly expanding thermal plasma of the early universe. Nevertheless, thoseexperiments are crucial in understanding the nature and rate of the local fundamental interactions available at those energies. What interests cosmologists is the statistical and thermal properties that such a6See the recent theoretical developments on large extra dimensions and quantum gravity at the TeV [9].116

plasma should have, and the role that causal horizons play in the final outcome of the early universe expansion. For instance, of crucial importance is the time at which certain particles decoupled from theplasma, i.e. when their interactions were not quick enough compared with the expansion of the universe,and they were left out of equilibrium with the plasma.One can trace the evolution of the universe from its origin till today. There is still some speculation about the physics that took place in the universe above the energy scales probed by present colliders. Nevertheless, the overall layout presented here is a plausible and hopefully testable proposal. Accordingto the best accepted view, the universe must have originated at the Planck era ( GeV, \pEs) from a quantum gravity fluctuation. Needless to say, we don’t have any experimental evidence for such a statement: Quantum gravity phenomena are still in the realm of physical speculation. However, it is plausible that a primordial era of cosmological inflation originated then. Its consequences willbe discussed below.Soon after, the universe may have reached the Grand Unified Theories (GUT) era E( - GeV, À s). Quantum fluctuations of the inflaton field most probably left their imprint then astiny perturbations in an otherwise very homogenous patch of the universe. At the end of inflation, thehuge energy density of the inflaton field was converted into particles, which soon thermalized and became the origin of the hot Big Bang as we know it. Such a process is called reheating of the universe.Since then, the universe became radiation dominated. It is probable (although by no means certain) thatthe asymmetry between matter and antimatter originated at the same time as the rest of the energy of theuniverse, from the decay of the inflaton. This process is known under the name of baryogenesis sincebaryons (mostly quarks at that time) must have originated then, from the leftovers of their annihilationwith antibaryons. It is a matter of speculationt whether baryogenesis could have occurred at energies as low as the electroweak scale (100 GeV, & s). Note that although particle physics experiments havereached energies as high as 100 GeV, we still do not have observational evidence that the universe actually went through the EW phase transition. If confirmed, baryogenesis would constitute another ‘window’into the early universe. As the universe cooled down, it may have gone through the quark-gluon phase transition ( . MeV, À s), when baryons (mainly protons and neutrons) formed from their constituentquarks.The furthest window we have on the early universe at the moment is that of primordial nucleosyn * MeV, 1 s – 3 min), when protons and neutrons were cold enough that bound systemsthesis (could form, giving rise to the lightest elements, soon after neutrino decoupling: It is the realm of nuclearphysics. The observed relative abundances of light elements are in agreement with the predictions of thehot Big Bang theory. Immediately afterwards, electron-positron annihilation occurs (0.5 MeV, 1 min) and all their energy goes into photons. Much later, at about (1 eV,À yr), matter and radiation have4 equal energy densities. Soon after, electrons become bound to nuclei to form atoms (0.3 eV,À yr),in a process known as recombination: It is the realm of atomic physics. Immediately after, photons decouple from the plasma, travelling freely since then. Those are the photons we observe as the cosmic Gyr), the small inhomogeneities generated during inflamicrowave background. Much later (tion have grown, via gravitational collapse, to become galaxies, clusters of galaxies, and superclusters,characterizing the epoch of structure formation. It is the realm of long range gravitational physics, perhaps dominated by a vacuum energy in the form of a cosmological constant. Finally (3K, 13 Gyr), theSun, the Earth, and biological life originated from previous generations of stars, and from a primordialsoup of organic compounds, respectively.I will now review some of the more robust features of the Hot Big Bang theory of which we haveprecise observational evidence.2.2.1Primordial nucleosynthesis and light element abundanceIn this subsection I will briefly review Big Bang nucleosynthesis and give the present observational constraints on the amount of baryons in the universe. In 1920 Eddington suggested that the sun might derive its energy from the fusion of hydrogen into helium. The detailed reactions by which stars burn hy117

drogen were first laid out by Hans Bethe in 1939. Soon afterwards, in 1946, George Gamow realizedthat similar processes might have occurred also in the hot and dense early universe and gave rise to thefirst light elements [4]. These processes could take place when the universe had a temperature of around /.10 * MeV, which is about100 times the temperature in the core of the Sun, while the densityE .20 ½E t ¿ .2\ 0 isg cm , about the same density as the core of the Sun. Note, however, that although both processes are driven by identical thermonuclear reactions, the physical conditions in starand Big Bang nucleosynthesis are very different. In the former, gravitational collapse heats up the core ofthe star and reactions last for billions of years (except in supernova explosions, which last a few minutesand creates all the heavier elements beyond iron), while in the latter the universe expansion cools the hotand dense plasma in just a few minutes. Nevertheless, Gamow reasoned that, although the early period ofcosmic expansion was much shorter than the lifetime of a star, there was a large number of free neutronsat that time, so that the lighter elements could be built up quickly by succesive neutron captures, starting' 5È 4 7' 6. The abundances of the light elements would then be correlated withwith the reaction 3their neutron capture cross sections, in rough agreement with observations [6, 10].Fig. 3: The relative abundance of light elements to Hidrogen. Note the large range of scales involved. From Ref. [10].Nowadays, Big Bang nucleosynthesis (BBN) codes compute a chain of around 30 coupled nuclearreactions, to produce all the light elements up to beryllium-7. 7

space-time,but with some knowledge of quantum fieldtheory and the standard model of particle physics. 2. INTRODUCTION TO BIG BANG COSMOLOGY Our present understanding of the universe is based upon the successful hot Big Bang theory, which ex-plains its evolution from the firstfraction of a