Big-Bang, Evolution of the Universe, and Mystery ofΛ

Lectio Magistralis in Occasion of the Opening Ceremony of the Accademia Gioenia CLXXXV Year

20 February 2009

Lucio PaternoDepartment of Physics and Astronomy, Astrophysics Section, University of Catania,

Citta Universitaria, Via S. Sofia 78, 95123 Catania, Italy (e-mail: lpaterno@ct.astro.it)

Abstract - The modern cosmology was born in 1915 when Albert Einstein established the theory of General Relativ-ity and discovered, with surprise and disappointment, that the equations were consistent with an universe expanding orcontracting, contrarily to the dominant Newtonian ideas and the prevalent theology of those times. These ideas inducedEinstein to modify his equations by introducing a repulsive term, the cosmological constantΛ, for balancing the attrac-tive action of gravity and rendering the universe static. However in 1929 Edwin Hubble discovered that all the observedgalaxies showed a recessional motion whose intensity was proportional to distance so establishing the famous law of theexpansion of the universe. In the light of this discovery, Einstein declared that the introduction ofΛ was the biggestmistake of his career. Other fundamental discoveries starting from the sixties, such as the cosmic microwave background(CMB), the counts of radio-galaxies and the theory of primordial nucleosynthesis, clearly demonstrated that the universeowed its origin to a primordial explosion about 14 billion years ago, the Big Bang, and is in a continuous expansion.More recently space missions have detected small temperature fluctuations in the CMB that permitted us to infer the flat,Euclidean, geometry of the universe and by consequence a necessary inflationary epoch in the primordial universe duringwhich its size increased by 1050 times in 10−33 s. The present universe is constituted by 4% of ordinary baryonic matter,23% of non-baryonic dark matter and 73% of a not well understood form of dark energy that produces an acceleratedexpansion of the universe. In this scenario the cosmological constant, rejected by Einstein, plays a fundamental role inthe dynamics of the universe.Key words: Cosmology – Big Bang – Primordial Universe – General Relativity – Cosmological Constant.

Riassunto -La cosmologia moderna ebbe inizio nel 1915 quando Albert Eistein enuncio la teoria della Relativita Generalee scoprı con sorpresa e disappunto che le equazioni predicevano un universo in espansione o contrazione, contrariamentealle idee Newtoniane dominanti e la prevalente teologia di quei tempi. Queste idee indussero Einstein a modificare lesue equazioni introducendo un termine repulsivo, la costante cosmologicaΛ, per controbilanciare l’azione attrattiva dellagravita rendendo cosı l’universo statico. Ma nel 1929 Edwin Hubble scoprı che tutte le galassie mostravano un moto direcessione proporzionale alla loro distanza, stabilendo la celebre legge dell’espansione dell’universo. Alla luce di questascoperta Einstein dichiaro che l’introduzione diΛ fu il pi u grave errore della sua carriera. Altre scoperte fondamentalia partire dagli anni sessanta, come la radiazione cosmica di microonde (CMB), i conteggi delle radiogalassie e la teoriadella nucleosintesi primordiale, hanno chiaramente dimostrato che l’universo si origino da un’esplosione primordiale, ilBig Bang, avvenuta circa 14 miliardi di anni addietro ede in continua espansione. Piu recentemente alcune missionispaziali hanno rivelato piccole fluttuazioni di temperatura nella CMB che hanno permesso di determinare la geometriapiatta, Euclidea, dell’universo con la conseguenza che dovette esserci un’epoca inflazionaria nell’universo primordiale incui le sue dimensioni aumentarono di 1050 volte in 10−33 s. L’universo attualee costituito dal 4% di materia barionicaordinaria, dal 23% di materia oscura non barionica e dal 73% di una forma non ben conosciuta di energia oscura cheproduce un’espansione accelerata. In questo scenario la costante cosmologica, ripudiata da Einstein, gioca un ruolo fon-damentale nella dinamica dell’universo.Parole chiave: Cosmologia – Big Bang – Universo Primordiale – Relativita Generale – Costante Cosmologica.

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 2

1 Introduction

The question about the origin and the end of the uni-verse in which we live was posed since the time of thehuman appearance on our planet. However until whenthe modern science and technology did not develop,any conjecture about the nature and evolution of theuniverse remained merely an act of faith.

The progresses of Astrophysics in the last fortyyears, owing to the advent of new ground-based andspace technologies, were constellated by making-epochdiscoveries comparable to the fundamental ones of thePhysics in the first fifty years of the XX century. Thesediscoveries enabled the astronomers to acquire a deepknowledge of the universe, though not definitive. Thespace missions, besides the accurate analysisin situof the components of the solar system with the con-sequent knowledge of its formation, opened the uni-verse to the optical observations extended until its re-mote limits and to the observations in electromagneticradiation spectral regions precluded to ground-basedobservations, such as the ultraviolet, infrared, X andγ bands. A new, extremelycoloureduniverse openedto the observations that allowed the deep knowledgeof a series of phenomena not visible before, such asthe birth and formation of stars in the interstellar dustyclouds, the structure and dynamics of galaxies and clus-ters of galaxies, the Quasar, very powerful energy emit-ters whose engine is constituted by a black hole billiontimes more massive than our Sun, theγ-ray bursts, themost intense energy sources of the universe which arelocated at its boundaries and are produced by the ex-plosions of the primordial supernovae stars, called hy-pernovae.

The modern Cosmology was born in 1915 whenAlbert Einstein established the theory of General Rel-ativity and discovered with some surprise and disap-pointment that equations were consistent with an ex-panding or contracting universe. At that time the ideasthat permeate the cosmology were dominated by thetheory of Newton based on the assumption of a staticand immutable universe as a consequence of the uni-form distribution of matter in all the space in such away that gravity could not act in denser regions socausing collapses with the formation of large cluster

of matter, not observed with the old technologies. TheNewtonian ideas and the prevalent theology of thattime induced Einstein to doubt of his revolutionary re-sults and led him to modify his elegant equations by in-troducing a repulsive term, the cosmological constantΛ, for balancing the attractive action of gravitationalfield and rendering the universe static.

The use ofΛ was rejected by Einstein as his ca-reer biggest mistake when Edwin Hubble establishedin 1929 his famous law on galaxy recession demon-strating that the universe is in expansion, so that heposedΛ = 0 in the successive cosmological mod-els. After Hubble, other basic discoveries, such as thecosmic microwave background (CMB), the primordialnucleosynthesis theory and the counts of radio-galaxies,posed firm grounds on the idea that the universe wasborn from an initial explosion, the Big Bang, and isstill expanding at the rate established by the Hubblelaw.

Several experiments demonstrated the validity ofthe theory of Relativity, Special and General, and themost important consequences of this theory for Cos-mology depend on the finite velocity of the light andthe impossibility of treating space and time as separateentities. These consequences imply that farther onelooks into the space farther one looks into the time andthat the space-time geometry is modelled by the pres-ence of mass and energy. Therefore to study the ori-gins of the universe one should observe the phenomenaoccurring at its farthest edge, located at an estimateddistance of about 14 billion light-years, where the firstphoton escaped from the radiation impenetrable BigBang fire-ball.

More recently, three new space experiments,COBE (1991), BOOMERANG (2000), and WMAP(2003), discovered thermal micro-fluctuations of theorder of 10µK in the CMB, interpreted as the imprintsof the past history of the universe. These fluctuationsallowed the astrophysicists to infer the large scale ge-ometrical properties of the universe that appears to beflat (Euclidean geometry). However, withΛ = 0, theuniverse density matter, both baryonic and dark, es-timated through the dynamics of clusters of galaxiesappears to be largely sub-critical indicating a non-flatgeometry but an open type one. Therefore we face

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 3

two contradictory measurements that lead to two dif-ferent space geometries. This inconsistence can onlybe removed if we re-establish the role of the rejectedcosmological constantΛ in the Einstein equations.

2 The Einstein Cosmic Equation

The final result of the theory of General Relativity, de-veloped by Einstein in 1915, is the field equation thatdescribes the space-time curvature as a function of thedensity of matter and energy. The original tensorialform of the equation that includes the cosmologicalconstantΛ is the following:

Rµν − 12

gµνR+ Λgµν =8πGc4

Tµν (1)

whereRµν is the Ricci’s curvature tensor,gµν is thespace-time metric tensor,Rthe scalar curvature namelythe trace ofRµν, Tµν is the tensor matter-energy,c is thelight-speed in the vacuum, andG is the universal grav-itation constant. On the hypothesis of large scale ho-mogeneity and isotropy of the universe, the so-calledcosmological principle, the tensorial Eq. (1) can bereduced to the following more domestic ordinary dif-ferential equation:

R2

R2c2+

k

R2− Λ

3=

8πG3c2

(%r + %m) (2)

whereR(t) is the scale factor, namely the radius of theuniverse as a function of timet, k is the scalar curva-ture (k = 0 flat Euclidean universe,k > 0 close uni-verse,k < 0 open universe),%r and%m are the radia-tion energy and matter densities respectively, and thedot represents the time derivative. Equation (2) stillcontainsΛ as a term for balancing the attractive ac-tion of gravity owing to the presence of radiation andmatter. The terms%r and%m are the following:

%r =4σT4

c3; %m =

3M4πR3

(3)

whereσ is the Stefan-Boltzmann constant of the radi-ation theory,T is the temperature of the field of radi-ation in thermodynamic equilibrium, and M the totalmass of the universe.

As already outlined, the validity of the theory ofRelativity, Special and General, has been proven by

several experiments, the first and most famous of whichwas the determination of the deflection of the star-lightrays, caused by the presence of the Sun’s mass, ob-served by Sir Arthur Eddington during the total so-lar eclipse of May 29th, 1919 (1). Other more re-cent validity proves are the explanation of the advance-ment of the Mercury’s perihelion, the increase of thedecay-time of elementary particles accelerated to ap-proach the light-speed, the gravitational lensing effectobserved when a galaxy or a cluster of galaxies inter-play between a distant source and the observer, greatlyenhancing the brightness of the distant source, the grav-itational redshift of spectral lines that has recently beenobserved in the proximity of very massive black holes,and the delay of clocks near a gravitational field withrespect to those far from it. All these effects were pre-dicted by the Einstein’s theory for bodies moving atspeeds comparable with the light-speed and in pres-ence of strong gravitational fields.

3 The Hubble Law

Immediately after the Einstein’s theory was published,Edwin Hubble,the mariner of nebulae, initiated a sys-tematic study of the so-called nebulae, that appearedas diffuse objects to the vision of the telescopes ofthat time and were argument of dispute among the as-tronomers who did not yet establish whether they werelocated in our galaxy or outside, essentially becausethe scientists did not have the possibility of measuringgreat distances. However, the existence of very brightpulsating stars called Cepheids was known since someyears before Hubble started in his survey. As discov-ered by Leavitt and Pickering in 1912 (2), these starshave the peculiarity that their pulsation period is re-lated to their absolute magnitudeM, whose relation-ship is described by the following formula:

M = −2.81 logP− 1.43 (4)

whereP is the pulsation period measured in days. Theknowledge of the absolute magnitude, connected to theintrinsic emitting power of the star, is the necessaryingredient for determining its distance once the appar-ent magnitudem, connected to the radiation fluxΦ atground through the Pogson law [m = −2.5 log(Φ/Φ0)

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 4

with Φ0 a standard reference flux1], has been mea-sured:

m− M = 5 logd − 5 (5)

whered is the distance measured in pc (parsec, 1 pc=

3.26 light-years) and the differencem−M is called dis-tance modulus. Thus the Cepheids constituted the firststandard candles for measuring the distances in theuniverse to an upper limit of about 100 million light-years, well outside our galaxy whose size is about 0.1million light-years. The period-magnitude relation-ship [Eq. (4)] was calibrated with galactic Cepheidsfor which the distances could be measured with differ-ent methods.

Hubble first discovered one of these Cepheids inthe near, great galaxy Andromeda (M31) and estab-lished its distance in some 1 million light-years, abouttwo times shorter than the modern determinations butcertainly outside our galaxy. Along a period of some5 years Hubble observed 18 galaxies measuring theiroptical spectra and distances. He discovered that allthe galaxies were characterized by a recessional mo-tion, determined by the Doppler redshift of spectrallines, whose speed was proportional to the distance.In 1929 he announced during a meeting of the Ameri-can Academy of Sciences his law of the expansion ofthe universe (3):

Vr = H0d (6)

whereVr (km s−1) is the recessional velocity, the so-called Hubble flux,d (Mpc, Megaparsec) the distance,andH0 (km s−1 Mpc−1) the Hubble constant. In Fig. 1is shown the original diagram of Hubble where in thehorizontal axis is plotted the distance in pc and in thevertical one the recessional velocity of the observedgalaxies in kms−1. The slope of the interpolating linerepresents the Hubble constantH0 defined in Eq. (6);the value ofH0 resulting from Hubble’s observationswas 500 km s−1 Mpc−1. The inverseH−1

0 of the Hubbleconstant represents a time that defines the age of theuniverse in virtue of Eq. (6). The value ofH0 foundby Hubble reflected on an age of the universe of about

1The magnitudes decrease with increasing radiation flux hav-ing also negative values, so that fainter is the star higher is itsmagnitude value.

Figure 1: The original Hubble’s diagram showing thevelocity-distance relationship for extra-galacticnebu-lae presented in 1929 at a meeting of the AmericanAcademy of Sciences.

2 billion years. Today we know that the measurementsof Hubble were affected by severe errors due to the oldobserving technologies that led to greatly overestimateH0 and by consequence to greatly underestimate theage of the universe.

After Hubble’s times until to the very recent onesthe precise determination ofH0 assumed a fundamen-tal role for the development of cosmological models.

4 The 4 Cornerstones of Big Bang

The original Hubble law was based on the observa-tions of galaxies where the brightest Cepheids couldbe visible with the telescopes of that time, whose ap-parent magnitude limits were aboutm = 20. Since thebrightest Cepheids have absolute magnitudes of the or-der ofM = −7 (P ' 100 days), then through Eq. (5) itis deduced that the distances probed by Hubble couldnot be larger than 2 million pc or 6.5 million light-years, as is evident from Fig. 1. In modern times withthe advent in 1992 of the Hubble Space Telescope theexploration of the universe could be extended to theobservations of objects withm = 28 corresponding forCepheids to distances of about 100 million light-years.However, much more powerful standard candles arethe supernovae of Ia type, carbon-oxygen white-dwarfstars in a binary system with a red-giant star as com-

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 5

panion that feeds matter onto the white-dwarf until toovercome its critical mass of about 1.4 solar masses,so causing a destructive explosion owing to the nu-clear carbon detonation. This explosion produces afixed maximum brightness equivalent to an object ofM = −19, 6× 104 times brighter than the brightestCepheids, so permitting the exploration of the universeuntil its most remote space limits and by consequencethe exploration of its origin.

Owing to the construction of new generation largetelescopes with effective collecting areas up to 200 m2,20 times larger than those used at the Hubble time, amultitude of supernovae Ia has been discovered in var-ious galaxies located from the vicinity of our galaxyto the most remote ones, so verifying the validity ofthe Hubble law over large distances up to the bound-aries of the universe. These new precise determina-tions pose firm grounds on the validity of the Hub-ble law describing the expansion of the universe. Themost recent value of the Hubble constant, derived formthe results of an international project, isH0 = 72 ±4 kms−1 Mpc−1 (4) that implies an age of the universeof 13.6± 1 billion years. In Fig. 2 is shown the Hub-ble diagram as derived from the combined observa-tions of several objects located at different distancesd, expressed in Mpc, as functions of their redshiftzor recessional velocity; in factz = ∆λ/λ representsthe Doppler shift∆λ of spectral lines with respect totheir rest wavelengthλ and is connected with the re-cessional velocityVr through the following relativisticformula:

z =

(c + Vr

c− Vr

)1/2

− 1 (7)

The Hubble law constitutes the first cornerstone onwhich the theory of Big Bang is based, as the initialbig explosion caused the expansion of the universe.

The second cornerstone is the casual discovery in1965 of the cosmic microwave background by Pen-zias and Wilson (5) who were conducting experimentson the propagation of microwaves for radio transmis-sions through the Earth’s atmosphere. They measuredin their devices a persistent background noise indepen-dently of the orientation of their antennas with the con-sequence that the noise could not be produced by some

Figure 2: The modern Hubble diagram as obtainedform the results of an international project (4).

local disturb, but it should have been produced by asignal coming from the space. They proved that thesignal had the spectral characteristics of a black-bodyemission at the temperature of about 3 K (' −270C)and that the microwave emission was isotropic (6).

Three modern space experiments, COBE in 1991(7), BOOMERANG in 2000 (8), WMAP in 2003 (9),confirmed the perfect black-body nature of the CMBemission corresponding to a temperatureT = 2.728±0.004 K as shown in Fig. 3. The CMB emission at lowtemperature is interpreted as the ashes of the primor-dial explosion that occurred at very high temperatures.

The third cornerstone is based on the illuminat-ing article on the origin of the elements published byHoyle et al. in 1956 (10). The authors describe the pri-mordial nucleosynthesis of the elements subsequent tothe Big Bang when the universe was hot enough tofuse the hydrogen into deuterium (D), helium-3 (3He)and helium-4 (4He) and then lithium-7 (7Li), but couldnot form by fusion elements heavier than lithium be-cause the universe became cooler as it expanded. Themodern observations indicate that the present cosmicabundances of4He and D are about 25% and 1% bymass respectively with traces of7Li, the same amountestimated theoretically by the authors. The heavier el-ements were produced inside the hot central regions ofthe primordial massive stars, as described in the fun-

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 6

Figure 3: The CMB spectrum as obtained from themodern space experiments.

damental article by Burbidge et al. in 1957 (11), andthen polluted into the space as consequence of their ex-plosion as supernovae. Therefore the new generationstars, as our Sun and by consequence the solar system,formed in an environment rich of heavy elements. InFig. 4 is shown the behaviour of the creation of the pri-mordial elements that happened in the first 3 min afterthe Big Bang as deduced from modern calculations byYang et al. (12) whose results are similar to those ob-tained by Hoyle et al. in 1956.

The fourth cornerstone supporting the theory ofBig Bang relies with the counts of radio-galaxies that,owing to the deep penetration of the radio-waves intothe Earth’s atmosphere, constituted the easiest observ-able objects in the far universe, when at those times(1973) space observations were precluded. If the uni-verse was infinite and uniform, since the number ofthe objectsN increases with the cube of distanced andtheir radio-fluxS decreases with the square ofd, thenthe following relationship linkingN to S should bevalid:

logN = −32

logS (8)

Equation (5) expresses the famous logN – logS rela-tionship and indicates that as the radio-flux decreasesfor the farthest galaxies the number of the observedobjects should increase with the 3/2 power, contrarilyto what appears from an inspection of Fig. 5 where

Figure 4: The behaviour of the creation of the primor-dial elements4He (top curve), D,3He and7Li that hap-pened within the first 3 min after the Big Bang.

are reported the first results of the counts of radio-galaxies by Katgert et al. in 1973 (13). It is easily seenthat below a unity of flux (a special measure for radio-astronomical measurements) the number of the objectsdecreases dramatically, indicating that the universe isfinite and non uniform as one could expect form thetheory of Big Bang.

5 The Matter in the Universe

In the light of the Hubble law of the expansion of theuniverse described by Eq. (6), the quantityR/R intoEq. (2) represents the rate of the expansion of the uni-verse, namely the Hubble constantH0. From Eq. (2)for Λ = 0 the necessary condition for having a flatuniverse (k = 0) (Einstein – De Sitter universe) is thefollowing:

%c = %r + %m =3H2

0

8πG(9)

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 7

Figure 5: The first log N – log S diagram of radio-galaxies showing the sharp decrease of the number ofobjects below the unity radio-flux. This diagram wasobtained by Katgert et al. in 1973 (13) and is thedemonstration that the universe is not infinite but itpresently reached a limit to its expansion.

For H0 = 72 kms−1 Mpc−1, as the modern measure-ments indicate, the total density of the universe shouldbe %c = 9.72× 10−30 g cm−3, where%c is the criticaldensity for a flat universe. In the expanding universethe decreasing of%r ∝ R−4 is faster than the decreas-ing of %m ∝ R−3, whereR is the radius of the universe.The present value of%r can be estimated by the tem-perature of CMB through Eq. (3) and it results to be4.63× 10−34 g cm−3 negligible with respect to the crit-ical density. Therefore%m should be of the order of%c itself. In the primordial very hot universe radiationdominated while presently the universe is dominatedby matter, the equilibrium between radiation and mat-ter being reached some 10, 000 years after Big Bang.On expressing the matter density in terms of the criti-cal density, we define the parameterΩm = %m/%c; withthis definition, neglecting%r , from Eqs. (2) and (9) weobtain:

k =R2H2

0

c2(Ωm− 1) (10)

Equation (10), obtained forΛ = 0, links the geometricproperties of the universe,k, to the amount of den-sity of matter with respect to the critical one,Ωm. Ifthe density of matter in the universe exceeds the crit-ical one (Ωm > 1) thenk > 0 and the universe isclosed; if the density is just critical (Ωm = 1) thenk = 0 and the universe is flat; if the density is sub-critical (Ωm < 1) thenk < 0 and the universe is open,as shown in Fig. 6. The expansion and the fate of

Figure 6: The various geometries of the universe de-pend on the value of the density of matterΩ0 = Ωm

the universe are therefore determined by the value ofΩm as schematically illustrated in Fig. 7 (Friedmann-Robertson-Walker universe). ForΩm > 1 (curve la-belledBound) the universe reaches a maximum expan-sion then, since the density exceeds the critical one,the gravity produces a subsequent contraction until toreach the initial conditions of Big Bang (Big Crunch);for Ωm = 1 (curve labelledMarginally Bound) the uni-verse expands smoothly, since the gravity just balancethe initial acceleration, and it will reach an infinite ra-dius in an infinite time; forΩm < 1 (curve labelledUnbound) the expansion is more rapid.

The value ofΩm can be estimated through the de-termination of the ratio mass – luminosity, M/L , ofthe clusters of galaxies. For the normal stars this ratiois of the order of unity indicating that the luminosity

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 8

Figure 7: The history and the fate of the expandinguniverse as a function of the density of matterΩm forΛ = 0; in the vertical axis is plotted the size of theuniverse and in the horizontal one the time after theBig Bang.

is only produced by the ordinary baryonic matter con-stituted by atoms. However when the astrophysicistsinitiated the study of galaxies for determining theirmasses they faced a new extraordinary problem; theyfound that the ratios M/L for galaxies were a factor often greater than those determined for stars, namely thata different form of non luminous matter should havebeen present in galaxies. The galaxies are constitutedby hundred billions of stars and therefore is unconceiv-able that the estimate of their masses is based on thecounts of stars. The estimate is based on the measure-ment of the rotational velocity as a function of the dis-tance from the centre by means of radio-astronomicaltechniques, and from the balance between gravitationaland centrifugal forces is possible to infer the total massof a galaxy. For a multitude of galaxies observed thetotal mass necessary for accounting the behaviour ofrotation resulted always about ten times larger thanthat constituted by the luminous stars, interstellar gasand dusts, including the estimate of the mass contri-bution of white dwarf, neutron stars and black holes.This excess of mass is attributed to a form of non-baryonic matter, the dark matter, whose nature is notyet well understood and whose effects manifest them-selves only through the gravitational interaction; themost accepted nature of the dark matter is presently

constituted by WIMPS (Weakly Interacting MassiveParticles), perhaps microscopic black holes not yetevaporated that formed in the primordial universe.

However in order to infer the total amount of massin the universe the observation of single galaxies doesnot constitute an efficient method. The study of clus-ters of galaxies, groups of galaxies formed by somethousand galaxies gravitationally interacting, throughthe measurement of their motions and the applicationof the virial theorem, that links gravitational to the ki-netic energy, permits to deduce the total mass presentin the clusters. The ratio M/L for all the clusters ob-served is in average a factor of about one hundred largerthan the luminous mass contained in the clusters, thatindicates that the universe is dominated by the non-baryonic dark matter. From the observations of a mul-titude of clusters of galaxies and other statistical con-siderations, it is deduced that the matter density ofthe universe is%m ' 2.6 × 10−30 g cm−3 < %c, thatmeansΩm = 0.27< 1, with a distribution of 23% non-baryonic dark matter and 4% ordinary baryonic mat-ter, half of which luminous. These data indicate that,by assumingΛ = 0, the universe should be open andits evolution described by theUnboundcurve of Fig.7.

6 CMB Temperature Anisotropy

Temperature anisotropies, or fluctuations, or inhomo-geneities, in the CMB were predicted since 1972 byZel’dovich (17) but it was not before 1992 that theywere detected by the differential radiometer instrumentonboard the satellite COBE (18). Subsequently, theexistence of these temperature anisotropies was con-firmed by other two more precise space experiments,the stratospheric balloon BOOMERANG that was firstlaunched from South Pole in 1998 (19), and WMAP,a satellite launched in 2001 (14), that is capable ofexploring all the sky differently from BOOMERANGthat could explore only a portion of it owing to theshort time of the flight (about 20 days). The results ofWMAP fully confirmed the validity of those ofBOOMERANG (Fig. 8).

The power spectrum of CMB anisotropies, namelythe angular size distribution of CMB anisotropies,

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 9

Figure 8: The all-sky image of the CMB temperatureanisotropy as obtained by the satellite WMAP in 2003(14).

marks important signatures on the structure of the uni-verse. Figure 9 shows the power spectrum as recon-structed by various measurements (15), (16) where thesquare of the normalized temperature fluctuations(µK2) is plotted versus the multipole moment`, orspherical harmonic degree, that is connected to the an-gular size of temperature inhomogeneities. On the dataare overimposed the curves that represent the predic-tions of three current cosmological models whose be-haviour is quite similar, that means that the anisotropydata describe fundamental properties of the universeindependently of models. Each of the three peaks inFig. 9 represents an important feature of the universe.

The structure of CMB anisotropies is principallydetermined by two effects, acoustic oscillations anddiffusion damping. The acoustic oscillations arise be-cause of the competition between photons and baryonsin the early universe plasma. The radiation pressure ofphotons tends to erase anisotropies, whereas the grav-itational attraction of the baryons makes them to col-lapse to form dense haloes. These two effects com-pete to create acoustic oscillations which reflect in thecharacteristic peak structure of CMB. The peaks corre-spond to resonances in which photons decouple frommatter when a particular oscillation mode is enhancedat its maximum amplitude. The first highest peak atthe multipole moment = 220, corresponding to anangular scale of 1, determines the curvature of theuniverse, namelyk; the second peak at` ' 500 deter-mines the amount of baryonic matter, while the third,located at ' 850, is connected with the dark mat-

Figure 9: The power spectrum of CMB temperatureanisotropy in terms of multipole moment or angu-lar scale as obtained by WMAP in 2006 (15) andBOOMERANG in 2005 (16). The peak at the mul-tipole 220 corresponds to an angular size of 1.

ter density. These anisotropies that originated in thesmall, early universe are now reproduced in cosmicscales giving rise to the CMB inhomogeneities andthe non uniform, quasi-filamentary distribution of theclusters of galaxies as shown in Fig. 10 that repre-sents the cone-diagram where is plotted the locationof the galaxies in clusters as a function of their re-cessional velocity, or distance, observed in a portionof sky about 2 wide. Since the distribution has theappearance of fingers this diagram is also called theGod’s fingers. The size of inhomogeneities as deter-mined by the main peak in Fig. 9 is related to the pathof the light-rays in the space-time; if their size is largerthan 1 the universe is closed, if it smaller than 1 theuniverse is open, while if the size of inhomogeneitiesis just 1 the universe is flat, or Euclidean, as illus-trated in Fig. 11. The heights of the two secondarypeaks in the CMB anisotropy power spectrum are con-sistent with the amount of baryonic and dark matteras determined from the analysis of motions in galaxyclusters. As we shall see the flatness of the universe

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 10

Figure 10: The cone diagram of distribution of galax-ies in clusters where is shown their location as a func-tion of recessional velocity, or distance, relative to aportion of sky about 2 wide.

Figure 11: The size of CMB anisotropies and the cur-vature of the universe.

and the imprints in CMB of the primordial quantumfluctuations depend on an abrupt expansion (inflation)of the order of a factor 1050 in 10−33 s that happened10−35 s after the Big Bang.

7 The Primordial Universe

The composition of the universe during its first fewminutes of existence was profoundly different fromthat of the present universe. In nature exist four funda-mental forces, gravity, electromagnetism, strong andweak nuclear forces. The first two are long rangeforces, though the gravity only, the weakest of the 4forces, extends its effects toward the infinite, while theelectric and magnetic forces vanish over large volumes

because of the parity of positive and negative chargesand the unavoidable existence of a north magnetic polefor each south magnetic pole. The gravity is attractivefor all the ordinary matter determining the behaviourof the orbits of planets, evolution of the stars, galax-ies, cluster of galaxies, and all the other large scalephenomena in the universe. The second two are veryshort range forces which act inside the atomic nucleiat distances of about 10−13 cm; the strong force keepstogether protons and neutron in atomic nuclei againstthe repulsive electromagnetic force of protons, whilethe weak force governs the radioactive decays as thetransformation of neutrons into protons or wheneveroccurs a change of type of quarks inside the protonand neutrons themselves.

In the period from the Big Bang to 10−43 s afterit, called the quantum gravity era or the Planck era,all four forces were unified, equally strong and be-haved in the same way as an unique force; in this pe-riod the universe was incredibly smallR < 10−33 cmand hotT > 1032 K. In Fig. 12 is shown the evo-lution of the four forces as a function of time; in thefigure is also indicated the temperature with the cor-responding energy in GeV (109 eV) as given by theexpressionε (GeV) ' 10−13 T(K). At Planck time

Figure 12: The 4 forces of nature were unified at thequantum gravity or Planck era, then as the universeexpanded and became cooler they decoupled.

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 11

(t = 10−43 s; R = 10−33 cm; T = 1032 K) the grav-ity first decoupled from the other forces becoming theweakest of them. The relationship between gravityand the other three forces is still uncertain becausethe theory of superstrings (Supersymmetric Theory ofStrings), that relies on the vibrational properties of aleast 10-dimensional space-time to create the funda-mental forces and the elementary particles, includingthe graviton or quantum of gravity, is not yet well es-tablished, though promising. At Planck time the re-maining three forces were unified and described as oneby the GUT (Grand Unified Theory); after the Plancktime the universe initiated its expansion at the initialrate, so creating the space-time in which we presentlylive, and the temperature dropped to 1027 K in 10−35 scausing the decoupling of the strong force from theelectromagnetic and weak ones, which remained uni-fied as the electroweak force. This decoupling is con-nected with the abrupt transition from an unstablephase of the universe, called false vacuum, to the sta-ble phase of the true vacuum; this transition happenedin the time interval between 10−35 s to 10−33 s and theradius of the universe experienced a dramatic, suddenexpansion (inflation) increasing by a factor of 1050, asillustrated in Fig. 13 where the modern inflationarytheory is compared with the old standard theory thatpredicted an expansion at the initial rate. In quantumfield theory a false vacuum is a metastable sector ofspace that corresponds to a local minimum of energybut not to the lowest one; therefore there is the sud-den phase transition (instanton effect) via the quantumtunnel effect to the lowest energy state owing to quan-tum fluctuations or creation and annihilation of highenergy virtual particles.

As pre-announced in Section 6 the inflation is re-sponsible for the large scale isotropy and homogeneityof the present universe; in fact if the universe had con-tinued to expand at the rate calculated for the Plancktime, today we should see different parts of the uni-verse with very significant temperature differences.The quantum fluctuations that occurred on microscopicscales at the pre-inflationary epoch were stretched byinflation to scales comparable with those of clustersof galaxies, namely that the tiny quantum fluctuationsultimately became the seeds for the formation of the

Figure 13: The evolution of the radius of the universepredicted by the modern inflationary theory comparedwith the old standard theory that predicts an expansionat the initial rate.

large scale structures, of the order of 0.3 billion light-years, that presently we see in the universe as illus-trated in Fig. 10, with the consequence that isotropyand homogeneity are concepts that involve scales largerthan 1 billion light-years.

The end of the inflationary epoch heralded the firstformation of elementary particles, quarks, electrons,neutrinos and their antiparticles all immersed in a bathof energetic photons at temperatures larger than 1015 K,and the universe resumed the relatively smooth ini-tial expansion rate. When the temperature droppedto 1015 K after 10−12 s the electroweak force separatedinto the electromagnetic and weak nuclear forces andthe four forces started to behave separated as today.The next significant event is calculated to have oc-curred after aµs after Big Bang when the temperaturewas 1013 K too cool for the quarks could survive asseparate particles, and consequently they were forcedto be confined in protons and neutrons and their an-tiparticles. In the first second the universe was so hot(T ≥ 1010 K) that photons were very energeticγ raysand could create pairs of particles and antiparticles con-tinuously annihilating and reproducing. However, ifall particles had annihilated their antiparticles in theearly universe no matter would be left at all, but we see

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 12

that universe is constituted by matter with a negligibleamount of antimatter observed; since particles and an-tiparticles are created or destroyed in pairs, one shouldexpect that they are found in equal numbers in the uni-verse today. The fact that the nature of our universeprefers matter is attributed to the breaking of symme-try (Charge-Parity violation) probably due to the rup-ture of the electroweak force, related to the Higss bo-son, theGod’s particle, that governs the masses of ele-mentary particles. If only a particle survived by chanceamong a billion of particles and antiparticles whichmutually annihilated, this is sufficient to explain thepresent amount of matter and absence of antimatter.

When the universe was at the temperature of 1010 K,that happened about 1 s after Big Bang, with decreas-ing density it became transparent to neutrinos whichwere free to fill the space in an enormous number,about 1 billion per proton. In the successive three min-utes first was formed the stable deuterium nucleus andthen were produced the primordial nuclei of heliumand lithium at temperatures of about 107 − 108 K. Asoutlined in Section 5, in the early universe the radia-tion density was much larger than the matter density,but with expansion the radiation density decreasedfaster than matter density, the equality having beenreached about 10,000 years after Big Bang at a tem-perature of about 105 K. The present universe is amplydominated by matter, the radiation density being aboutfour orders of magnitude lower, but the number of themicrowave, cool photons per cubic meter exceeds byabout a factor of 1 billion the present number of hy-drogen atoms per cubic meter.

At temperatures of about 105 K and densities ofabout 10−15 g cm−3 matter and radiation were still inthermodynamic equilibrium and the electrons were en-ergetic enough to prevent the formation of hydrogenand helium atoms while the ultraviolet photons un-derwent frequent collisions with free electrons, beingscattered in all directions, and ionized the few atomsformed, being adsorbed and reemitted in all directions,the radiation being trapped in a completely opaqueplasma fireball.

This situation persisted until when the tempera-ture dropped to about 104 K and the era of recombi-nation started to take place in which the electrons re-

combined with hydrogen and helium nuclei to form thefirst atoms. However it was about 380, 000 years laterthat the universe became transparent to radiation andthe first photon could escape from the fireball reach-ing us, the surface of the universe from which the firstphoton escaped being calledlast scattering surface.

The CMB (see Fig. 8) contains the most ancientphotons we expect ever to be able to observe and isa ghostly relic of the earlier dazzling splendor of theuniverse.

8 The Reinstatement ofΛ and the Endof the Universe

The power spectrum of CMB anisotropies clearlyshows that the universe is flat, namely that the curva-ture parameter should bek = 0, and that the densitymatter, baryonic plus dark, as deduced from both clus-ters of galaxies and CMB anisotropy power spectrum,is sub-critical and givesΩm = 0.27. These two resultscannot be reconciled on the basis of Eq. (10) obtainedfor Λ = 0, for which the universe should be open withk < 0.

The results become mutually consistent only if wereintroduce the cosmological constantΛ as originallypredicted by Einstein. Thus from Eq. (2) we obtainthe following equation:

k =R2H2

0

c2(Ωm + ΩΛ − 1) (11)

whereΩΛ = c2Λ/3H20 represents a sort of dark en-

ergy whose nature is obscure. From Eq. (11) it is ev-ident that the reintroduction ofΛ requires thatΩm +

ΩΛ = 1 for a flat universe with the consequence thatΩΛ = 0.73 with Λ a positive constant. The expectedvalue ofΩΛ from the the quantum field theory is 10120,some 120 orders of magnitude greater than that de-duced from cosmological observations, the most dis-astrous disagreement never found between theory andobservation. On the other hand if the value ofΩΛ

was the one predicted by the theory, the expansion ofthe universe would have been so fast that no stars andgalaxies could be formed by the gravitational force ac-tion.

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 13

This disagreement constitutes one of the biggestmysteries of the present physics: on one hand the fieldtheory that account for the existence of the various el-ementary particles and on the other hand the needs ofcosmology for accounting the observations.

The nature ofΛ is still matter of debate even if it isnow almost commonly accepted that it acts as a sort ofanti-gravitational energy, the dark energy of the vac-uum, which determines an accelerate expansion of theuniverse once radiation and matter densities diminishtheir attracting forces as the expansion proceeds.

This idea is corroborated by the fact that the super-novae observed at the edge of the universe (z > 0.5)appear fainter than what predicted by the Hubble law,namely that they are farther than what expected on thebasis of the normal Hubble expansion and thereforethey experience an accelerated expansion. Figure 14shows a magnitude-redshift diagram as obtained fromthe observations in the framework of the SupernovaCosmology Project (20), where the difference of thedistance modulus∆(m − M) with respect to a refer-ence one is plotted versus the redshiftz for a num-ber of supernovae of Ia type. The horizontal refer-

Figure 14: The magnitude-redshift diagram for super-novae of Ia type; the∆(m− M) = 0 line indicates anuniverse of Einstein-De Sitter type withΛ = 0 andΩm = 1, while the bottom curve a close, decelerat-ing universe and the upper curve, where the supernovadata are displayed, an open, accelerating universe withdark energy. Data were carried out in the frameworkof Supernova Cosmology Project (20).

ence line that intersects the zero of∆(m− M) corre-

sponds to the Einstein-De Sitter universe withΛ = 0andΩm = 1 (k = 0) expanding at the present Hubblerate as calculated from Eqs. 5, 6, and 7. The bottomcurve describes the case of an universe withΛ = 0andΩm > 1 (k > 0), namely a close, decelerating uni-verse for which the supernova magnitudes should belower than those described by the previous situation .The upper curve shows an accelerating universe withdark energy (ΩΛ = 0.73, Ωm = 0.27 ), and all theobserved points tend to lie on this curve indicating a∆(m − M) ' 0.5 for z ' 1, consistent with objectsdisplayed some 20% farther in the universe than theyshould do if dark energy did not act.

Therefore all the modern measurements from thecosmic microwave background and mass – luminosityratio of the clusters of galaxies to the faint supernovaeat high redshifts seem to indicate that the present uni-verse is made by a small quantity of baryonic matter(4% half of which luminous), by dark matter (23%)and by a large quantity of dark energy (73%), and thereintroduction ofΛ into the Einstein’s cosmic equa-tion is a necessary ingredient for explaining the presentcosmological data. The evolution of an universe withΩm = 0.3 andΩΛ = 0.7, values close to the onesmeasured, is illustrated by the top curve of Fig. 15that is similar to Fig. 7 in which are plotted onlythe curves forΩΛ = 0. Detailed calculations indicatethat the end of such an universe will happen 35 billionyears after the Big Bang (about 20 billion years fromnow) with a Big Rip, exactly the contrary of the BigCrunch, caused by the continuous, accelerated expan-sion which first separate the galaxies from their clus-ters and the constituens from the single galaxies, thenbreak up the stellar systems and the single stars, thenthe electrons from their atoms and finally reduce thenuclei in single elementary particles that become sodistant that they cannotseeeach other so leaving theuniverseeternally frozenandempty.

9 Concluding Remarks

All the data collected from precise measurements ofcosmological paramenters concorde in indicating thatthe universe was born by an initial explosion, the BigBang, some 13 – 14 billion years ago, that it expe-

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 14

Figure 15: The history and the fate of the expandinguniverse as a function of the density of matterΩm forΩΛ = 0 (the three lower curves) andΩΛ = 0.7 (the topcurve) as the modern data indicate; in the vertical axisis plotted the size of the universe and in the horizontalone the time after the Big Bang.

rienced an abrupt expansion in an earliest epoch thatproduced the present flatness and homogeneity and thatthe density matter is sub-critical. The modern mea-suremens also indicate that the cosmological constantΛ, initially rejected by Einstein, plays a fundamen-tal role for explaining both the space flatness and thefainter brightness of the far supernovae, and all thedata predict an universe expanding at an accelerate rateand a cool end (Big Rip).

However some questions are still matter of debateon the nature of dark matter and dark energy. Thoughthere are some space experiments for investigating thetrue nature of dark matter no conclusions have beenreached until now leaving the answer about its com-position to the only theoretical conjectures. Its pres-ence seems to be acquired with certainty because itsgravitational effects on the large mass distribution ofgalaxies, unless to admit that the universal gravitationtheory fails at very large distances as some scientists,who claim for a modified gravity, are substaining.

As previously outlined in Section 8, the dark en-ergy, that reflects in a positive but small value ofΩΛ,owe its anti-gravity action to the false vacuum energyin the earliest epochs of the universe. This implies an

equation of statep = w%, where% is the false vacuumdensity, with a negativew; the theory indicates that for−1 ≤ w < 0 the universe would expand at a rate higherthan that predicted in the absence of dark energy butat a finite velocity, whilst forw < −1 the expansioncaused by the dark energy should reach an infinite ve-locity so determining the Big Rip. Unfortunately theattempts to measurew did not give firm results untilnow, and this remains an open question of the moderncosmology.

Another point of discussion, as already seen in Sec-tion 8, is the fact that the quantum theory of fields, onwhich is based the description of the earliest instantsof the universe, predicts a natural value ofΩΛ in anabsurd disagreement with the needs of cosmologicalobservations, being 120 orders of magnitude larger.Nevertheless the cosmological observations require aprecise value ofΩΛ, infinitely smaller than the natu-ral value, that implies that the parameters of the natureare regulated in a so extremely fine way that it seemsinnatural.

A different approach claims for a hypothetical fifthforce, the quintessence, whose nature is of the sametype as the dark energy withw < −1/3 but variablewith the cosmic time in such a way that the expan-sion of the universe is self-adjusting at the differentepochs never reaching the situation for the Big Riptaking place. Recent crossed determinations based onCMB and supernovae measurements indicate thatw −1 (21).

All the above mentioned arguments are the openquestions of cosmology whose study is presently infull expansion with the perspective of obtaining im-portant results in a near future.

10 Acknowledgements

I would like to gratefully acknowledge the Presidentand Council of the Accademia Gioenia for the kind in-vitation to present this lecture in occasion of the open-ing ceremony of the CLXXXV Academic Year.

Lucio Paterno: Big Bang, Evolution of the Universe, and Mystery ofΛ 15

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