Figure 1. The Thermal History of the The figure (adapted courtesy of Mike Turner) shows important epochs in the history of the universe according to stan- dard Big Bang cosmology. It also shows and the the of and con- stituting the universe during each epoch. The history is laid out on logarithmic Background scales to emphasize the early part. The scales are , thermal kT, universal a(t) (note that a(t) ϭ 1/(z ϩ 1) where z is the ), Salman Habib and Raymond J. Laflamme ␳(t), and . Temperature, ther- mal energy, redshift, and (or the equivalent mass density) all de- crease with time, whereas the scale of

the universe a(t) ϵ R(t)/R(t0) increases. ig Bang cosmology implies a cer- (that is, massless particles). The The temperature plotted is the tempera- Btain predictable course for the ther- during this time is highly spec- ture of the radiation, which until recombi- mal history of the universe. Here we ulative. It is assumed that thermal en- nation is the temperature of the baryonic review that history, emphasizing the ergies were initially at the scale matter ( and ) as well. two classic pieces of evidence that sup- (1028 eV), where the exotic and poorly The formulae relating these variables are port it: the relative abundances of the understood phenomenon of shown beneath the figure. They follow elements produced by primordial could have played a crucial from conservation of energy and from the and the existence, spec- role. Many and as- assumptions that matter and radiation trum, and fantastic of the cos- trophysicists believe that a bit later, as were initially in thermal equilibrium and mic microwave background. We will the temperature cooled to 1026 , that the universe has expanded adiabati- also discuss the modifications that there was a brief of inflation. cally since the hot Big Bang. The rela- might be caused by the presence of During the inflationary phase, the size tionship between ␳(t) and a(t) changes . of the universe grew exponentially by a depending on whether the universe is ra- Figure 1 outlines the calendar of factor of at least 1028. Less speculative diation-dominated or matter-dominated, events, or major epochs, in the history. is the prediction that at a temperature that is, on whether most of the energy in It can be divided into two main periods: around 1012 kelvins, a - the universe is in the form of radiation the radiation-dominated period, which transition occurred, when all the (, , and other particles lasted for approximately ten thousand and combined to form protons moving at speeds near c) or most of the (1011 ), and the matter- and neutrons. The particles of cold energy is in the form of matter. The tem- dominated period from ten thousand dark matter, if they exist, would also perature and scale-factor axes are lined years after the Big Bang to the . have been created sometime after the up with respect to each other by the ob-

During the first fraction of a inflationary phase. servation that at present (t ϭ t0) the tem- after the Big Bang, the constituents of perature of the cosmic background radia- the universe undoubtedly included the The radiation-dominated . tion is approximately 3 kelvins and that Ϫ2 particles of the of parti- From about 10 second to the present, by definition a(t0) ϵ 1. The time and den- cle physics: , quarks, and the standard Big Bang cosmology presents sity axes are lined up with the size axis gauge that mediate interactions an almost universally accepted picture, by using a specific value for the Hubble among them, all moving at velocities so which is borne out by the COBE data constant and the assumption that the di- close to the velocity of light that from and other recent observations. By that mensionless density parameter ⍀ is equal the point of view of thermodynamics, time the universe consisted of particles to 1. Then one can place any event on they behaved as particles of radiation that were stable (or long-lived com- all three axes if one knows its position on

82 Los Alamos Number 22 1994

Big Bang Cosmology and the Microwave Background

chronfig.adb 7/26/94

Cold dark matter Recombination: Quark- Primordial begins to neutral form; nucleo- form microwave background transition synthesis structure decouples Radiation 25 20 15 10 5 temperature 10 10 10 10 10 1 (kelvins)

24 21 18 15 12 9 6 3 -3 10 eV 10 eV 10 eV 10 eV 10 eV 10 eV 10 eV 10 eV 1eV 10 eV Thermal energy Kinetic energy High-energy Nuclear binding Atomic binding of a sprinter cosmic rays energy energy

Scale factor 10-25 10-20 10-15 10-10 10-5 1 a(t)

1065 1049 1033 1017 10 10-15 10-30 Density ρ(t) (g/cm3) Nuclear Water Air 1 /cm3 matter

1 103 106 109 Years

10-36 10-30 10-24 10-18 10-12 10-6 1 106 1012 1018 Seconds

Galaxy forms NOW forms Constituents of the Universe background νe νµ ντ νν Leptons ( e -)( µ -)( τ -) e ± 2 kelvins and quarks u c t + + 3 + 3 ++ 3 n, p H ,D , H , He H, D, He, ( d )( s )( h ) 4 ++ 7 ++ 4 7 He , Li + He, Li Gluons Gauge Light nuclei Neutral atoms bosons ± W , Z -to-baryon ratio ~ 3 × 109 THE BIG BANG Microwave background Photons γ 3 kelvins

Radiation-Dominated Matter-Dominated Era Era a(t ) ∝ t 1/2 a(t ) ∝ t 2/3 ρ(t ) ∝ (a(t ))-4 ∝ t -2 ρ(t ) ∝ (a(t ))-3 ∝ t -2 T ∝ (a(t ))-1 ∝ 1 + z T ∝ (a(t ))-1 ∝ 1 + z one axis (usually temperature, sometimes redshift). Note that the scale-factor axis does not show the expansion due to inflation. In the lower portion of the figure, the line for neutrinos (and antineutrinos) turns into a wiggly line to indicate that at that time the neu- trinos decoupled from matter and photons and subsequently expanded freely. The occurred when the universe became cool enough and dilute enough that neutrinos no longer interacted significantly with matter, at a temperature around 1010 kelvins. Similarly, at a temperature of about 3000 kelvins, the line for photons turns into a wiggly line to indicate their decoupling from mat- ter due to recombination, the combining of the with the positively charged nuclei to form neutral atoms. The subsequent free expansion of photons produced the cosmic microwave background radiation now observed.

Number 22 1994 Los Alamos Science 83 Big Bang Cosmology and the Microwave Background

pared to the then): proportionally, the radiation continued baryons that were neutrons, a fraction electrons, neutrinos, protons and neu- to have a -body spectrum, or that decreased with time as neutrons trons, and photons, as well as the yet- Planck distribution, a fact that explains were converted to protons through the unidentified constituents of dark matter. why the black-body spectrum of the weak interactions. Calculations predict Unstable particles such as mu and cosmic background radiation is not a that the number of neutrons at the time leptons, , and the gauge bosons surprise. However, the temperature of of primordial nucleosynthesis is approx- of the weak had disappeared the radiation, which is proportional to imately 12 percent of the total number through decay to stable particles. Fig- the average energy per quantum or in- of baryons and that almost all those ure 1 outlines the changes through versely proportional to the average neutrons ended up in -4 nuclei, time of the constituents of matter. wavelength, decreased inversely with the most tightly bound of the light- At about t ϭ 10Ϫ2 second, the uni- the scale factor, T ϰ (a(t))Ϫ1, or in pro- element nuclei. In other words, Big verse was a “primeval fireball” consist- portion to the redshift (T ϰz). The en- Bang cosmology predicts that the cos- ing of baryonic matter and radiation in ergy density of the radiation therefore mic abundance by weight, or mass frac- thermal equilibrium (and perhaps dark decreased due to both the stretching of tion, of helium-4 nuclei relative to hy- matter). Moreover, most of the energy the wavelengths of the quanta and the drogen nuclei (protons) is about 24 was in the form of radiation. That fol- dilution of their number density; thus percent, in agreement with observation. Ϫ4 lows from simply extrapolating the pre- ෆ␳radiation decreased rapidly, as (a(t)) . The second determinant of relative sent density of matter and radiation Equation 3 in the caption of Figure 3 abundances produced by primordial nu- back through the presumed adiabatic in the main text is solved to relate the cleosynthesis is the ratio of the number expansion of the universe. The energy scale factor to time. In particular, when of photons to the number of baryons (or density therefore obeyed the formula radiation was the major contributor to equivalently the photon per for black-body radiation, that is, for ra- the energy density, a(t) increased as baryon). That ratio does not change diation in thermal equilibrium with a t1/2, and so the temperature of the uni- with time and therefore, as we will see totally absorbing body. Thus the ener- verse decreased as tϪ1/2. Thus during below, places a constraint on the bary- gy density u of each relativistic species the radiation-dominated era, every de- on density today. It also controlled the present—whether photons, massless crease by a factor of 10 in temperature rates of various competing nuclear reac- neutrinos, or even electrons and corresponds to an increase by a factor tions at the time of nucleosynthesis. moving at nearly the speed of of 100 in time since the Big Bang. As For example, nucleosynthesis could not light—obeyed the Stefan-Boltzmann the primeval fireball of matter and radi- really begin until the density of ener- law: ation expanded and cooled in thermal getic photons became low enough that equilibrium, the decrease in temperature deuterons ( nuclei) formed by 2 4 uradiation ϭ ෆ␳radiation c ϭ ␴ T , was like a cosmic clock, ticking off the fusion of neutrons and protons stages in the physics of the universe. ( ϩ → deuteron ϩ pho- where T is temperature and ␴ is the ton) were unlikely to be blasted apart analogue of the Stefan-Boltzmann con- Primordial synthesis of the light by the reverse reaction. That condition stant. elements. In the first few minutes after depends on the temperature, which Black-body radiation (even with a the Big Bang, the one major event that must be about 109 kelvins, but also de- small admixture of equilibrated matter) directly left observable traces was the pends strongly on the ratio of photons behaves as a dissipationless fluid, so the primordial synthesis of the light ele- to baryons. Once the deuteron abun- radiation-dominated universe expanded ments through nuclear fusion reactions. dance increased, nuclear reactions in- adiabatically, and the total entropy was In particular most of the helium-4 ob- volving one or two deuterons could conserved. As mentioned in the main served today in was almost as- build on one another to form primordial article in the discussion of the redshift, suredly synthesized then. abundances of all of the adiabatic expansion “stretched” the As the temperature fell to 109 kel- and helium: deuterium, tritium, heli- wavelengths of photons and all other vins (t Ϸ 102 seconds), conditions um-3, and helium-4 as well as very massless (or effectively massless) were right for nucleosynthesis to begin. small amounts of -7. species in proportion to the universal Two quantities controlled the relative Primordial nucleosynthesis manufac- scale factor a(t), or ␭(t) ϭ ␭0a(t). abundances of light elements that re- tured all those nuclei in abundances Since all wavelengths were stretched sulted. The first is the fraction of the consistent with current data provided

84 Los Alamos Science Number 22 1994 Big Bang Cosmology and the Microwave Background

the photon-to-baryon ratio was between creased more rapidly than that of the ter, which by definition does not partic- 2.5 ϫ 109 and 3.6 ϫ 109. The agree- matter, at some point the energy density ipate in electromagnetic interactions, ment between nucleosynthesis calcula- of the matter became larger, and from began to develop density inhomo- tions and observations is considered con- then on the universe has been matter- geneities when the universe became vincing evidence that in the beginning dominated. The temperature (or time, matter-dominated, provided the dark- the universe was hot, radiation-dominat- teq) at which the switch from a radia- matter temperature, or average thermal ed, and in thermal equilibrium—that the tion-dominated to a matter-dominated energy, was low enough and the masses hot Big Bang did indeed occur! universe occurred depends on the total of the dark matter particles were high Further, since the ratio of photons to matter density in the universe—baryons enough that the particles were moving baryons during primordial nucleosyn- plus any species of dark matter that at nonrelativistic speeds. (Otherwise, thesis was preserved to the present, it might be out there. If baryons make up small-scale fluctuations would have can be combined with the known num- all the matter and the total matter densi- been washed out by the free streaming ber density of photons in the cosmic ty today is 2.5 ϫ 10Ϫ31 gram/centime- of dark-matter particles.) That is, the 3 microwave background to predict the ter (that is, ⍀baryon ϭ 0.05), then the dark matter must have been “cold” for baryon density in the universe today. switchover occurred at a temperature of structure to have begun to form on all 6 The bounds on the ratio yield bounds about 1000 kelvins (teq Ϸ 10 years), scales at teq, when the energy in matter on the baryon density of (2.5 Ϯ 0.5) ϫ whereas if ⍀ ϭ 1 and baryons make up was equal to the energy in radiation. 10Ϫ31 gram/centimeter3, and bounds on only 5 percent of the matter, as as- the baryonic contribution to ⍀ of 0.01 sumed by the standard CDM model, Recombination. When the universe Ϫ2 Ϫ2 h < ⍀baryon < 0.015 h . If we as- then the universe became matter-domi- cooled to about 3000 kelvins, the situa- sume that h ϭ 1/2, then ⍀baryon Ϸ 0.05, nated when the cosmic scale was a fac- tion changed dramatically. That tem- which is much larger than ⍀luminous. tor of 20 smaller or the temperature perature is far enough below the ioniza- This prediction for ⍀baryon places a was a factor of 20 higher, at about tion temperature of helium and 4 constraint on all models of structure 20,000 kelvins (teq Ϸ 10 years). Fig- hydrogen atoms that almost all the elec- formation. ure 1 was constructed with the assump- trons combine with the helium and hy- tion that ⍀ ϭ 1. drogen nuclei to form neutral atoms. The transition to a matter-domi- The beginning of the matter-domi- Cosmologists call that event “recombi- nated universe. Following nucleosyn- nated era marks the time when the nation” (a somewhat misleading term thesis, the cosmic soup consisted of mutual gravitational attraction of the since the electrons and nuclei were photons in thermal equilibrium with hy- matter became stronger than the gravi- combining for the first time). After re- drogen and helium nuclei and a number tational pull of the background sea of combination, charged particles were of electrons, left over from - radiation. Consequently matter could, rare, and so the of photons , sufficient to bal- in , begin to clump together by matter was also rare. Thus the time ance the charge of the baryonic matter. under the influence of gravity. Howev- of recombination, trecomb, marks the The radiation-dominated universe con- er, the baryons (primarily in the form end of the tight coupling between bary- tinued to expand adiabatically, and en- of hydrogen and helium nuclei) and the onic matter and photons; thermal equi- tropy conservation continued to ensure electrons are electrically charged and librium between matter and radiation is that the temperature decreased inversely therefore remained coupled to the pho- no longer maintained, and the universe with the size of the universe, T ϰ tons through electromagnetic interac- is said to be transparent to radiation Ϫ1 (a(t)) . Thus ෆ␳radiation, the equivalent tions, primarily the scattering of elec- since photons can travel through the mass density of the radiation, continued trons and photons (Thomson scatter- universe with little chance of being Ϫ4 to decrease as (a(t)) , whereas ෆ␳matter, ing). Thus, their clumping was imped- scattered. which was dominated by the matter’s ed by radiation that was much Once the ceased rest mass rather than its kinetic energy, stronger than gravitational in the to have an effect on baryonic matter, decreased inversely as the volume, or nearly uniform mass distribution—the that matter too could begin to feel the as (a(t))Ϫ3. photon-to-baryon ratio is at least a bil- pull of gravity. In fact, if dark matter lion to one—so density inhomogenei- were not present, recombination would The matter-dominated era. Since ties could not grow in baryonic matter. have to mark the beginning of the the energy density of the radiation de- In contrast, non-baryonic dark mat- growth of density fluctuations due to

Number 22 1994 Los Alamos Science 85 Big Bang Cosmology and the Microwave Background

astrofig5.adb 7/26/94 gravitational forces. However, as stated 10-14 in the introduction to the main article, recombination apparently occurred too COBE data late (and the density of baryonic matter 10-15 was too low) for the growth of gravita- Theoretical black-body tional instabilities to explain the ob- ) spectrum at 2.736 kelvins served large-scale structure. Thus the -1 Hz majority of cosmologists believe dark -1 10-16 matter must have led that development. sr -1 sec

The discovery of the cosmic back- -2 ground radiation. According to Big 10-17 Bang cosmology, the radiation that ex- isted at the time of recombination was (ergs cm thermal and consequently had a black- body spectrum at a temperature of Intensity per unit wavelength interval 10-18 about 3000 kelvins; in other words, its energy density as a function of wave- length had the Planck distribution. At recombination, the universe became 102 101 100 10-1 10-2 transparent to radiation, so the photons Wavelength (centimeters) survived unchanged except for their redshift due to cosmic expansion. As Figure 2. The Planck Distribution and the Cosmic Background Radiation the wavelength of each photon in- The figure shows the Planck distribution, or the energy density per unit wavelength of creased in proportion to the scale factor radiation in thermal equilibrium with a perfectly absorbing (black) body, at a tempera- a(t), the form of the spectrum was pre- ture of 2.736 kelvins. The red data points represent measurements by instruments served; that is, the radiation now has a other than COBE of the energy density per unit wavelength for the cosmic background black-body spectrum but at a tempera- radiation. The precision of most of the data points is so high that error bars would be ture lower by a factor a(t0)/a(trecomb) no bigger than the symbols representing the points. The red part of the curve repre- than the temperature at the time of re- sents the measurements made by the COBE satellite; its error bars and deviations combination. That spectrum is shown from the Planck distribution are too small to appear. Thus the background radiation in Figure 2. Indeed, Big Bang cosmol- appears to have a nearly perfect “black-body” spectrum, indicating that matter and ra- ogy predicts that low-temperature diation were in thermal equilibrium at the time of recombination, as predicted by stan- black-body radiation should now be ob- dard Big Bang cosmology. (Figure adapted by courtesy of D. T. Wilkinson.) servable from all empty parts of the sky—this remnant of recombination equipment, the cosmic background radi- black-body spectrum, then the inferred forms a cosmic background. ation was detected inadvertently by temperature of the spectrum would be The existence of a cosmic remnant Arno Penzias and Robert W. Wilson at between 2.5 and 4.5 kelvins. Peebles, left over from the hot Big Bang was Bell Laboratories. Penzias and Wilson Dicke, Wilkinson, and Roll in a sepa- first predicted by , R. A. were investigating the internal noise rate paper suggested that this highly un- Alpher, and R. C. Herman in 1948 on of an antenna designed to de- usual radiation was the predicted rem- the basis of their analysis of the condi- tect radio waves emitted by our own nant of the Big Bang. tions needed for primordial nucleosyn- . Quite unexpectedly they dis- Since 1965 have mea- thesis. In the early sixties R. H. Dicke covered extra noise at a wavelength of sured the intensity of the cosmic back- and P. J. E. Peebles took this notion se- 7.35 centimeters whose intensity was ground radiation at many wavelengths. riously enough to plan experiments to independent of direction. The intensity In 1989 the COBE satellite was detect its presence. Ironically, in 1965, also did not vary with the time of day launched specifically to study various while Dicke, Peebles, D. T. Wilkinson, or the time of . If that radiation aspects of the radiation. The COBRA and P. G. Roll were building their was assumed to be one point of a rocket-based experiments and one of

86 Los Alamos Science Number 22 1994 Big Bang Cosmology and the Microwave Background

the instruments on COBE confirmed to tion were seen at all angular separations consistent with the CDM model, but as very high precision that the spectrum of larger than about 10 degrees of arc (the described in the section “Cold Dark the cosmic background is indeed a resolution of the relevant instrument on Matter, Large Scales, and COBE” in black-body spectrum. COBE made a COBE). the main text, they also fix the value of particularly precise measurement of the Ten degrees of arc corresponds to the only undetermined parameter of the temperature: 2.736 kelvins plus or hundreds of megaparsecs now and to model. minus a few millikelvins. A black- hundreds of kiloparsecs at the time of body spectrum at that temperature, recombination, a distance larger than shown in Figure 2, has a peak at the the Hubble radius, RH ϵ c/H(t), at that Further Reading wavelength of 2 millimeters, which lies time. Thus the fluctuations measured E. Bertschinger. 1994. Cosmic structure forma- in the microwave region of the electro- by COBE had wavelengths larger than tion. Physica D, in press. magnetic spectrum. The ratio of the the Hubble radius (or the horizon) present temperature of the cosmic mi- when they were imprinted on the cos- H. P. Gush, M. Halpern, and E. H. Wishnow. 1990. Rocket measurement of the cosmic- crowave background to the photon tem- mic background radiation. They there- background-radiation mm-wave spectrum. Letters 65: 537Ð540. perature at recombination, T(t0)/T(trec), fore were apparently not connected Ϫ3 is about 10 , which means that the causally at the time of the imprinting. J. C. Mather et al. 1994. Measurement of the background radiation has been redshift- In that case the near of the cosmic microwave background spectrum by the ed by a factor of about 1000 since the background radiation is very difficult to COBE FIRAS instrument. The Astrophysical time of recombination. understand. This paradox is an aspect Journal 420: 439Ð444. of the “” pointed out P. J. E. Peebles. 1993. of Physical Anisotropy of the cosmic back- by Misner in the 1960s. The problem Cosmology. Princeton University Press. ground radiation and implications is solved by postulating a period of in- Michael S. Turner. 1987. A cosmologist’s tour for models of large-scale structure. flation, or exponential expansion of the through the new particle zoo (candy shop?). In about the isotropy of the very early universe. In inflationary sce- Dark Matter in the Universe, edited by J. Kor- mendy and G. R. Knapp. Reidel. cosmic background radiation is crucial narios, regions that now appear causally to studies of be- disconnected were in fact causally con- Terry P. Walker, , David N. cause it allows us to infer the size of nected prior to inflation. The point of Schramm, Keith A. Olive, and Ho-Shik Kang. 1991. Primordial nucleosynthesis redux. The density fluctuations at the recombina- this discussion is that the fact that the Astrophysical Journal 376: 51Ð69. tion time, when the cosmic background scale of the observed temperature fluc- . 1972. Gravitation and Cos- radiation decoupled from baryonic mat- tuations is larger than the Hubble radius mology: Principles and Applications of the Gen- ter. If the mass distribution at the time at trecomb implies that these large-scale eral of Relativity. John and Sons. of decoupling had been totally uniform, fluctuations could not have been affect- Steven Weinberg. 1977. The First Three Min- the cosmic background radiation would ed by small-scale physics after inflation utes: A Modern View of the Origin of the Uni- be isotropic, or the same in all direc- and must indeed carry information verse. . tions. However, in 1992 another instru- about primordial fluctuations. ment on COBE detected inhomo- Since the shape of the CDM spec- Salman Habib performed his undergraduate geneities; the temperature of the trum of fluctuations at the time of re- work at the Indian Institute of Technology, background radiation varies depending combination can be calculated from the Delhi, and his graduate work at the University of on the direction the radiation comes assumptions of the CDM model, the Maryland, College Park. He now has a joint postdoctoral fellowship in the Theoretical Astro- from. The temperature differences are observed amplitudes determine the nor- physics and the Elementary Particles and Field quite small: ␦T/T Ϸ 6 ϫ 10Ϫ6. Those malization (amplitude) of the spectrum Theory groups at the Laboratory. His differences are believed to reflect densi- on all scales. Had the measured fluctu- interests are quantum field theory, statistical me- chanics, nonlinear dynamics, cosmology, and ty inhomogeneities present at the time ations been much smaller in amplitude . of recombination; photons emanating than 6 ϫ 10Ϫ6, not only the CDM sce- Raymond J. Laflamme received his bachelor’s from a high-density region must have nario, but also the idea that gravity is degree from the Université Laval in his native lost energy in “climbing out” of the re- the decisive force responsible for struc- Canada and his Ph.D. from Cambridge Universi- gion and thus suffered a redshift over ture formation, would have been in ty under the direction of . He is now an Oppenheimer fellow in the Theoretical and above the cosmic redshift. Fluctua- jeopardy. Not only are the COBE tem- Astrophysics Group, studying cosmology, the tions in the cosmic background radia- perature-fluctuation measurements thus early universe, and .

Number 22 1994 Los Alamos Science 87