a;ç"'1fi}t-ís{:\{ f:¡¡j if.ifli*åJi.i {:lt{}t'¿'it-} t''t"t"l"7l{:'i ¡"¡'¿yj CHAPTER 6 4,ZzE*9ï.a{:c?,89 BIG BANG COSMOLOGY - QUESTIoN s.11 THE EVOTVING UNIVERSE Anumberofimportanteventsinthehistoryofcosmologyhavebeenmentionedin starling with the irri, compile a chronological listing of these events, puUfi"*ion"r.,upt"r. of Einìtein's theory of general relativity in 1916' #""8 ãsæäræs#acsäåæäE In Chapter 5 we saw how general relativity can be used to construct models of the QUESTION s.12 Universe. These models describe how the scale factor varies in a Universe that is models and List the assumptions that underpin the Friedmann-Robertson-walker filled with a smooth distribution of matter and radiation, but say very little about the the Friedmann equation. properties and behaviour ofthese components. So, for instance, they do not account for the consequences of microscopic processes such as the interactions QUESTION 5.13 between the parlicles that make up the matter within the Universe. Describesomeofthepossibleconsequencesofpositivecurvatureinathree- However, such interactions play an impofiant role in cosmology. One example dimensional space' in the context of the FRW models' that we shall see later in this chapter is that cosmological theories can offer an explanation for the observation that most of the stars in the Universe have a QUESTIoN 5.14 composition that is approximately 75o/ohyûogen and 25Yohelium (by mass). A that the behaviour described fundamental aspect of the process that forms helium is that it involves reactions The detailed argument given in Section 5.4.1 showed in a FRW model' between nuclei and parlicles at an early stage in the history of the Universe. Thus by Hubble's law is an e-xpected consequence of expansion was only approximately physical processes on small scales must be taken into consideration in order to However, in Section s.+.i itwas stated that this argument ofthe expansion' Carefully develop an understanding of the evolution of the Universe. This chapter starts by true because it ignored the acceleration or deceleration the key step at which acceleration examining the conditions under which parlicles interacted in the early Universe reread the argument in section 5.4.1 and identify (Section 6.2).The main focus, however, is to follow a chronological sequence is ignored. from very early times in the history of the Universe (Section 6.3) through the formation of the first nuclei (Section 6.4) and the first neutral atoms (Section 6.5) QUËSTION s.1s to the stage at which gravitational clustering gives rise to the large-scale structure parameters mentioned List the values that have been assigned to all the observational that we observe in the present-day Universe (Section 6.6). inSection5.4.Whereaquantityisexpressedinmorethanoneunitsystem,confirm At first sight, it might seem impossible to use the models in Chapter 5 to make any the equivalence of all the given values' predictions about the small-scale behaviour of matter. Cosmological models describe how the scale factor varies with time, but at the present time any change in the scale factor ceftainly does not have any effect on, for instance, the atoms that make up your body. However one of the major assumptions made in Chapter 5 was that the matter in the Universe is smoothly distributed. This assumption of a uniform distribution of matter is a key to linking the large-scale dynamical behaviour of the Universe to small-scale effects.

To see why this is so, consider a volume of the Universe that, at some particular time, is bounded by an imaginary cube, as shown in Figure 6.1. Let us further suppose that we want to follow the evolution of the matter within this cube at all times using some particular Friedmann-Robertson-Walker model with scale factor R(r). To do this, the edges of the cube must follow the expansion (or contraction) of the model universe.

I Each edge of the cube has an associated length /. How must the length of each edge change with time if the cube is to follow the expansion or contraction of the model universe?

I The length of each edge of the cube must be proportional to the scale factor, i.e.l æ R(t). 6 BIC BANE COSMOLOCY * THE EVOLVING UNIVËRsE ,AN INTRÕDUETION TO 6ALAXIES AND EOSMOLOGY

is proportional n(t) x R(l) x R(r) volume /:) of the cube at any time ¡ 1o 6.2 The thermal history at the Universe Thus the 1: Although by the changó in volumé shown in Figure 6'l' time ll : (Ã(l))3.This is iffustíaie¿ A key physical parameter in particle interactions is temperature. It may seem to a volume of any shape that we have chosen to discuss a cubic volume here' a volume make no sense to talk of the 'temperature' of the Universe. The temperature of qái contraction) of such a model universe will have follows the expansion matter seems to range from a few degrees above absolute zero within giant volume is called a co-moving volume' i æ (R(t))3. Such a molecular clouds, up to temperatures of over 107 K that are found in extreme l*-/(rr)-l BecausethevolumeZofaco-movingregion,suchasthecube,changesasthe astrophysical environments. However, we will see shorlly that there is a cosmic is constant, the density of matter the malter within the Universe, but scale factor changes, but the mass ø within it temperature; it does not refer to temperature of p^, also varies with scale within the co-movirrgìotu*., which we denote by to the radiation that pervades the Universe. This radiation, which is observable time tz (CMB), plays impofiant factor. In fact, today as the cosmic microwave background an extremely role in modern cosmology. V/e shall discuss several aspects of the CMB in this, and M1* (6.1) in the following chapter, but in this section we shall consider how the existence of P^=T t*1ry this background radiation in an expanding Universe allows us to determine how j temperature has changed over cosmic history. The starting point for this discussion l physical parameter in determining how Now, the density of matter is an imporlant is to consider the cosmic microwave background in a liule more detail. scale' For example' the interactions between particles progress on a microscopic with one another increases as utwhich molecules in u ,ample of gas collide 6.2"1 The tennp€rature of the background radÊatÊor¡ ,it" is related to gas is increaseà. Thui, the large-scale behaviour Q)-1 the density ofthe In Section 5.2.2 you saw that the most significant contribution to the radiation small-scale effects. content of the Universe is the cosmic microwave background radiation. The 6.1 An imaginarY cubic Figure spectral flux density of the CMB peaks at a wavelength of about I mm - this is (with sides of length I that ThisstillmaynotappeartobeagleathelpinunderstandingtherealUniverseas at uolu-. since we know that the matter in the universe illustrated in the spectrum of the background radiation that is shown in Figure 6.2. of a to u modJ, evolves with the exPansion we saw in Chapter 4 "pp"r"¿ "o.mól,ogicaldoes ãot have a uniform density. Specifically, The form of this spectrum is highly significant: it is, to a very good approximation, model universe such that the mass the present time of matter is homogeneous only when we c_onsider scales a black-body spectrum implying it can be associated with a particular temperature. within the volume is constant' AnY that the distribution - Mpc: if we look at the Universe on smaller scales' we see volume that behaves in this waY, Ërrut",. than about 200 the average density of the matter in the universe whatever its shaPe, is called a íarge density variations. ihus 7 lri co-moving volume. wouldseemtobeaquantityoflimitedpracticaluse.However,weshallseelater past the matter in the that there is good evidence'that at timei in the distant it is at present even on universe wai much more smoothly distributed than - does telate to the small- x Ir small scales. At such times, the average density 6 I relatively ! 5.0 Irl scale behaviour of matter. p rl d I that is described by Equation 6'1 The relationship between density and scale factor s 4.0 in Chapter 5' The majority b ,il holds true for any of the cosmoiogical models described gtú ofthesemodelsarecharacterizedbyascalefactorR:0attime/:0'Aswas * notedinSection5.3'5theearlyexpansionphaseofanysuchmodelisreferredto strongly favours a model of the Ë as the big bang. cwreÀt"orfnologi"ul evidence Universethatwentthroughabig-bangphase-andfortheremainderofthis consequence that .fruf,., we shall only coÃider b1g bang models. An immediate bang model, early stages in the can be noted from Equation 6.1, i-s thaiin a big historyoftheuniverse(when,R(l)wasverysmall)arecharacterizedbyhigh speaking, the_mathematical densities. you may also irave noticed that, sìrictly relationshipp^æ' 'lR3impliesaninfinitedensitywhenR:0.Weshallreturnto Figure 6.2 The spectrum of the cosmic microwave background. (Note that this spectrum later' consider the'signifrcance of such infinite quantities shows the spectral flux density F¿.) I Finally,weshouldmakeabriefnoteaboutterminology:muchofthediscussionof i the Universe' A convention that this chapter 1".r.r, to ti*e, in the history of Ç."ific by I (so' The characteristic temperature T indicated by any given black-body spectrum is iÄe chapter is that ,h: ug" tf th. Uniu.rse is àenoted we adopt throughout (tr'¡) is a Universe was 1 second old)' The related to the wavelength Âp,ut at which the spectral flux density for instance, / : I s denotes ihe time at whicñ the to as 16' So, /0 represents maximum. According to Wien's displacement law, time now, at which we observe the Universe, is referred the current value of the age of the Universe' 2.90 x l0-3 (ip.*/m) = glK) (6.2)

257 256 EIC AN ¡N]"RODUCTION TO CALAXILS AND COSMÕLÕCY 6 BANú CÛsMOLûçY - THE F,VÕLVINC UNIV¡:¡i5E

Calculate the characteristic temperature of the cosmic microwave background r If a microwave background photon currently has L : I mm, what wavelength radiation. would it have had when the scale factor was 1000 times smaller than its present-day value? In which part of the electromagnetic spectrum does this Equation 6.2 ü Rearranging wavelength 1ie? 2.90 x l0-3 Using Equation 6.3, with values of 1 (T/K\ = + tl ),0: mm and R(t)lR(tù: l/1000 gives (Âo.u¡ /m) h:10-3 m/I000: l0-6m and using LpeakÉ 1 mm : I x 10-3 m So when the scale factor was 1000 times smaller than at present, photons that are at 2.90 x 10-3 currently the peak of the cosmic microwave background had a (T -' " '-= \- lK\'-l = = 2.90 wavelength of 10-6 m, which lies in the infrared part of the spectrum. (l x l0_l)

So, to one significant figure, the temperature of the cosmic microwave Thus, at high redshift, the wavelengths of photons in background radiation is 3 K. the cosmic background radiation would have been much shofter than at present, and consequently interactions between photons and matter would have been much more likely. However, before discussing Detailed spectral measurements have been used to determine the temperature of this interaction, we need to consider the form of the red-shifted spectrum the cosmiõ microwave background to a high degree of accuracy, with a value of in a little more detail. T : 2.725 + 0.002 K being widely accepted. An important feature of the black-body spectrum is that if the photons that make up such a spectrum The fact that the CMB follows a black-body spectrum is, at first sight, puzzling' are all red-shifted by the same amount, then it will remain a black- body spectrum. Photons Black-body spectra are formed when photons are continually absorbed and that are currently at the wavelength at which the spectrum has peak, re-emitted by matter. However, matter in the nearby Universe is transparent to a will always be at the peak, but the wavelength of that peak will change. The way in cosmic microwave background photons. Thus there is no interaction between which this wavelengfh, hpeak, changes with scale factor is given by Equation 6.3. R(10) matter and photons, and so nearby matter could not give rise to the observed and Ls are the current values of R(r) and  respectively, and so black-bodyìpectrum. So, if the CMB did form by the interaction of radiation and can be considered as constants in Equation 6.3. Thus Equation 6.3 can be written as this have occurred? To answer this question we have to matter - how could Lp"uu æ R(t) (6.4) consider the effect of the expansion of the Universe on the photons of the cosmic microwave background. However, the temperature of a black-body spectrum is related to the wavelength of the peak of emission by Wien's displacement law (Equation 6.2) which can be 6.2.2 The evoh¡tÊ¿¡m of the tererperature of backgrCIund rearranged and expressed as radÍati¡rn 7 A clue to the origin of the microwave background lies in an effect that was Tn Lp"uk (6.s) introduced in Chapter 5 - the cosmological red-shift of photons. In Section 5.4.1 that the effect of the expansion of the Universe on a single photon was to you saw using the relationship between hpeayand the scale factor (Equation 6.4) gives in.r"ur. its wavelength. The relationship between the wavelenglh )'s of a photon that is observed now (when the scale factor has the value R(rs)) and the wavelength ), that the photon had when the scale factor was rR(r) is I R(r) (6.6) L R(/) (6.3) R(ro) ho The temperature of the cosmic background radiation aI any time is inversely proportional to the scale factor at that time. Thus, when the scale factor was smaller than it is at present, the wavelengths of photons lhaL are now seen in the cosmic microwave background were all correspondingly smaller. In fact, the background radiation that is now observed as This relationship is important because, in principle, it allows us to calculate the the cosmic microwaye background, would, when the scale factor was much temperature of the background radiation at any given epoch for any cosmological smaller, have had a peak in another part of the electromagnetic spectrum. For this model. Remember from Chapter 5 that different cosmological models provide reason we shall use the term cosmic background radiation to denote this radiation just different relationships for the scale factor À as a function of time (see, for example, at" any time in cosmic history. The cosmic microwave background is the Figure 5.23). observable form of the cosmic background radiation at the present time.

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scale factor varies with time, The question asks for a sketch Even if we do not know the exact \vay in which the of how z varies with time. The implication of this is than is at that the curve R Equation 6.6 shows that if the scale factor was once much smaller it that shows z(r) does not have to be exact, but thai it should show background radiation at that time would have the most important features of how the Lemaître prãsent, then the temperature of the temperature varies with time. A way of model doing this is to been much higher than it is at present' consider a few times (labell.d A, B, c and D) on the corresponding curve of R(r) as shown in Figure 6.3a. At each time, we shaíl use Equation 6.6 to at two times (t1 and t2) deduce how 7 is il Use Equation 6.6 to express the ratio of the temperature behaving and use this information to help us draw a sketch of r(t) The deductions that in terms of the scale factor at those two times. can be made about z at these times aie shown in Table 6.1. e Equation 6.6 can be exPressed as A C D t (a) Table 6.1 The behaviour of the scale factor R at various constant times indicated on Figure 6.3a 7'(tl = and the inferred behaviour of the temperature T althose times. R(/) T Behaviour Time ofR at this time Behaviour of T at this time So for times /1 and t2 we can write A ,R: O T:1/R: æ constant constant B R has T(r) = and T(t") = increased to some value Zmust decrease to some value R(r,) R(rz) and now does not vary much and also only change slowly withtime withtime respectively. Dividing the first of these equations by the second gives A B C D C (b) R has a value that is slightly higher Zmust have a value that is slightly R(tz) T(tt) than that at B lower than that at B Figure 6.3 (a) Scale factor, T(rz) R(¡r ) and À is (b) temperature D increasing to very high values Zmust decrease to very low values as functions of time for the Lemaître cosmological model. A, B, C and D are times that is by the we can now use the deductions about the way There is, howeveq a problem in applying Equation 6.6 which highlighted in which z varies at these times to are referred draw to in Table 6. l. following question. a sketch. starting with time A, we cleariy cannot plot an infinite temperature at A, so we simply show z as having a very high varueãs we approach / - 0. At time B, I What is the predicted temperature of the Universe if the scale factor has a value we simply choose a finite value of temperature, and note that the temperature of zero? at time c is slightly lower than at B. Finally, at time D, the temperature decreases to very low values. These points are shown ån Figure 6.3b, andthe final A Since Z *.llR(t), if ,R : 0, the predicted temperature would be infinite! stage is to draw a smooth curve through these points to complete the sketch.

A prediction of an infinite value of any physical quantity is treated with great suspicion by most physicists. Rather than taking this infinite value at face value, it QUESTTON 6.1 is assumed that our understanding of physical processes is incomplete. The limits For all the Friedmann-Robertson-walker at which our knowledge of physical laws break down will be discussed briefly in models with k: 0, shown in Figure 5.23, Section 6.3 and taken up again in Chapter 8. However, for the present discussion, use Equation 6.6 to draw a corresponding curve T(t) that shows approximately how the important point is that at times when the scale factor was very small, the the temperature of the cosmic background radiation would våry with time. temperature would have been very high.

EXAMPLE 6.1 The answer to Question 6.1 illustrates the point that in any big bang model (i.e. one that has R : 0 at For the Lemaître cosmological model (in which,'t > ,'1s) and k : +1, use Figure 5.23 r: 0) the temperature oflhe cosmic background radiation would have been very high and Equation 6.6, to sketch a conesponding curve T(t) that shows approximately in its early stages. Such a scenario is often referred to as the i hot big bang. how the temperature of the cosmic background radiation varies with time. i' The change in wavelength of the cosmic microwave background has a profound i soLuTtoN effect on the way t in which photons interact with matter. Al times when the temperature of the cosmic background radiation The curve that shows how the scale factor R varies with time in the Lemaître was very high, the typical photon energy would have been greater than the model is shown in Figure 5.23 and is reproduced here as Figure 6'3a. ionization energy of the hydrog"n ãto-. cosmological Under these conditions, the baryonic matter in the Universe would-have been in the In order to draw a sketch of how the temperature Z varies with time, we need to form of a plasma. make use of the relationship between the temperature and the scale factor. This relationship is given by Equation 6.6, T æ llR(t).

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of a plasma and that of The opacityof a medium is a I What is the qualitative difference between the opacity The energy density of the cosmic microwave background is the total energy of all of the extent to which the measure an un-ionized gas? the microwave background photons per cubic metre of space. Note that thl energy medium is opaque to radiation. density (like the mass density) is defined per iJ A plasma tends to have a much higher opacity than an un-ionized gas, i.e. cubic metre, i.e. for a physical volume of space that is not co-moving. In an expanding un-ionized gases tend to be much more transparent than plasmas' universe, the energy áensity can therefore be expected to decrease as the universe expands.

r what are the SI units in which energy The reason for the dramatic difference in opacity between a plasma and an density should be expressed? presence free electrons in the plasma. Photons interact with un-ionized gas is the of J Since the SI unit of energy is the joule, and the unit of volume is m3, the sI unit a plasma primarily by scattering from the free electrons in the plasma (Figure 6.4 - of energy density is Jm-3. this process is called Thomson scattering after the discoverer of the electron J. J. Thomson). The degree of interaction between photons and electrons in a plasma can be very high, and this offers a clue as to the origin of the near perfect The current energy density ofthe cosmic background radiation can be found from forming a black-body spectrum of the background radiation. The conditions for measurements of the CMB and has a value of ur,s x 5 x 10-1a J m-3. (ø.,s is a black-body spectrum are that there must be many collisions between the material shorlhand way of writing ur(ts) - the value of u, at the present time.) that makes up a thermal source and the photons that are radiated by it. so an The energy density of matter ü^,scànbe found from density of matter interpretation of the black-body spectrum of the cosmic microwave background is in a straightforward way as the following question illustrates. that it was formed at a time when the Universe consisted of a hot plasma, and so between the photons and the free electrons. As the there were many collisions QUESTTON 6.2 Universe has expanded, the wavelengths ofthe photons have increased, and the black-body spectrum has shifted to longer wavelengths. Consequently, the The current average mass densily of all matter, both luminous and dark, is estimated N temperature associated with this black-body spectrum has dropped with the to be about pm,o 3 x 10-27 kgm-:. By using the equivalence between energy ð and expansion of the Universe. mass rîx given by E: mcz, calculate the current average energ/ densily due to matter.

BEFORE AFTER photon The answer to Question 6.2 shows that at the present time, the energy density of matter is ar.,s : J ; lQ-loJm-3. Thus at the present time, the energy density áue to matter exceeds the energy density in the cosmic microwave background by a factor Figure 6.4 The interaction of several thousand. between photons and free o Finally, we consider the energy density due to the cosmological constant. In electrons in a plasma. Chapter 5, it was noted that the cosmological constant,4 has an associated density -{ø*a* p¡: &"'P""3 æwabïap?¡"âqasa €}€ **eæY€êY c€a:s¡sË€-Ë*s åax ÊEce åireåwers€3 Ac2lSnG rc3) So far, we have only considered one physical properly of the cosmic background r Give an expression for the energ)t density (ø¿) of the vacuum in radiation - its temperature. However, to establish whether this background radiation terms of ,zl. plays an important role in the evolution of the Universe, it is necessary to consider iJ The energy density of the vacuum is obtained by multiplying Equatio n 6.7 by c2, iÍs energy density and how this quantity varies with scale factor' u¡: Ptc2 Recall that in Chapter 5, the behaviour of cosmological models was shown to and so depend on the density of matter p^, and on the cosmological constant,'l' In the radiation models that we considered there, we simply assumed that electromagnetic u¡: AcalSnG (6.g) was a minor constituent of the Universe. However, we now want to question this assumption - so we must compare the imporlance of these three components: electromagnetic radiation. The physical matter, cosmological constant, and As was noted in Chapter 5, the energy that may be associated with the cosmological components within a parameter that determines the importance of any one of these constant is often referred to as dark energy. consequently the quantity u¡canbe model is the energy density, i.e. the energy per unit volume due to that cosmological interpreted as the energy density of dark energy. The nature orlnis dark energy is a the energy density of any one component far exceeds that of the component. If mystery, but recent observations imply that u¡has a value of about 9 x 10-10 J m-3. component that will have the dominant effect on the other two, then it is this So, rather surprisingly, dark energy makes the dominant contribution to the total dynamical behaviour of the Universe. energy density of the Universe at the present time. So, let us now consider the current energy densities due to radiation u. mattet u^ and the cosmological constant ø1.

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where h is the Planck constant. The It might seem then that the cosmic background radiation is an insignificant frequency f and wavelength  of electromagnetic radiation of the total energy density of the Universe. However, this was not are always related byf : cü",,o *" also say compionent "un always the case. To see wfry, it is necessary to compare the way in which the three c hc energy densities u' u^and u^ change with scale factor' 'pn --_ ). (6.12) dark We starl with the simplest case of the three, which is the energy density of the can see But as the universe expands (i.e. as R(l) increases) energy. By inspecting the terms on the right-hand side of Equation 6'8 we the wavelength of a photon will (c, and A). also increase (Figure 6.5). The photon wavelength that iúis .r"rgy densìty depends only on values of physical constants G is proportionãl to the scale factor u¡ Thus, this density does not change with scale factor. As noted above, L æ R(r) "nãigy (6.13) currently has a value ofabout 9 x 10-10Jm-3, and this value has been constant throughôut the history of the Universe (with perhaps one brief, but important' that we shàll discuss later). However, the faclThat ez,1is constant and L(t) "*""piion not mean that it has always been the most important factor in relatively large, does time I, scale factor:À(lr) determining how the Universe evolves. Next, let,s consider the how the energy density due to matter changes with scale 1(tz) (normal) density of matter' X'igure 6.5 As the Universe expands (from scale factor. This is found from the time t, scale factor:À(tz) factor rR(/1) to R(t)), the wavelength of any photon down will increase I For a cosmological model in which matter is uniformly distributed, write in proportion to the scale factor. an equation thãt describes how the densify of matter changes with scale factor' Hence for each photon in the cosmic background radiation i-l We have already seen that the density of matter p.(f) varies according to 1 Equation 6.1 spr, * R(Ð (6.14) (6.1) ,^ç,¡*fi Now, the energy density of radiation ur(t) is given at any time t by

u,(t): n(t) x ern?) (6.1s)

The energy density of matter is related to the density of matter by u^: p^c2 , buf c Tt follows that is a constant, so we can write

I u, (t) oc_ (6. u^(t)æÃ; (6.e) R(t)4 16)

comparing this with Equation 6.9 shows that the energy density of radiation can be analysed by taking a The behaviour of the energy density of radiqtion behaves differently from the energy density of matter. specifically, the energy the way in which the density of similar approach to that taken when we examined density of radiation is inversely proporlional ro thefourth power ôf tne scatð-factor, is to consider the number of matter .fráng"r with scale factor. The first step whereas the energy density of matter is inversely proportional to the third power of photons per cubic metre, i.e. the number density of photons n(r). Assuming that the scale factor. cosmic bãckground photons are neither created nor destroyed during the relevant Figure 6.6 shows how all three energy densities (u,+, part of the expansion we can expect that u^, and u) vary as a function ofscale factor. At the present, the energy densities ofthe dark energy (9 x 10-lor--:; and of matter (3 x l0-loJm-3) are far in excess of the energy n(ù n (6.10) density =)-R(/)', of radiation (5 x 10-t+ J 6-3). As we look back to earlier times howevËr, when the value of the scale factor was smaller, we can see that both the energy way to So the number density of cosmic background photons behaves in a similar densities of radiation and of matter were greater in the past than at present. when is a difference' the density of matter. But what about the enefgy density? Here there the value of R(r)/R(re) was less than about 0.1, both the energy density of matter The energy of a photon of frequency/is given by and ofradiation exceeded the energy density ofthe dark energy. tpn: hf (6'11)

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Figure 6.6 The energy densities of 10 10 6,3,4 Å radiaüi'n'Ê-dnurnÊå*æ*ed matter (blue line) and radiation (red ãÊ?{Þ{åeE u,Ë Ëg?q} åjsãÊv*¡rs* line) as a function of scale factor. we have just seen that in the early universe, R the dominant energy density is that due At a tirne whenA(l)/R(r6) r 10 a the to the radiation within the universe. The Friedmann equation tn"ut *u, described in energy densities of matter and l0 Chapter 5 (Box 5'4) can be solved for such conditions and the way in which the radiation were equal. Prior to this scale factor varies with time for such a moder is shown rn nigure å.2. one time, the energy density of important feature of such a model is that the scale factor varie-s in 0 the following way: radiation exceeded that of matter - I during this era the dynamical >\ R(r) æ ¡rrz evolution of the Universe was eÍa rcJ7) -2 determined by its radiation content. o 0 After this time, the energy density 0 bo Because H matter the energy densify of radiation of matter was greater, so it was the o is dominant for times when R(r)/R(/') < l0-4, () all cosmological models which start : matter in the Universe that 0 -6 at t 0 with À(0) : 0, willf through a phase Figure 6.7 The evolution of the that is well described by this radiation-dominated controlled its dynamical evolution. model. Thus rrre are in the rather scale factor with time in a remarkable position that regardless The behaviour ofthe energy of which type of cosmoroÀi"ui model best cosmological model in which the describes the universe at the present, density due to the cosmological l0-10 cosmological constant we can te reasonably c.înfident that we dominant contribution to the know how the scale constant is also shown (purple line) factor vaiied with time in the first few tens of thousands energy density arises from the years of - this does not vary with redshift ofthe big bang. radiation within the Universe and is exceeded by the energy (i.e. during the radiation-dominated 10-1 However, the temperature of the background radiation densities in matter and radiation at l0-6 l0-5 l0-4 l0-3 l0-2 10-1 I varies with scale factor era). according to T(t) æ. llR(t) (Equation ø.ø¡. early times. scale factor À(t) / R(t o) tt follows that during the radiation- dominated era the temperature of the background radiation varies with time according to Fufthermore, as shown in Figure 6.6,the energy density of radiation has declined T(t) æ ¡ -uz more rapidly than the energy density of matter. Indeed there was a time when the (6. l8) value of R(r)/R(re) was such that these two energy densities were equal. This This describes how temperature changes with time in an appears have occurred when R(r)/R1¡o) ¡v a. plausible expanding universe where to l0 For most cosmological the energy density of radiation is the áominant component. of course, to use models this corresponds to a time when the age of the Universe was a few times Equation 6.18 to predict the temperature, it is n.."r.ury to know the constant 104 years. As we look back to even earlier times, when R(¡)/Rfto) was even smaller, proportionality of between 7 un¿ ¡-1t2.In fact, this can be derived we see that the energy density of radiation exceeded that of matter this period from the Friedmann - of equation and Equation 6.1g becomes the history of the Universe is called the radiation-dominated era. points The key of this discussion so far can be summarized as follows: (T/K) x 1.5 x 10ro x Qts¡_trz (6.19) I At the present time, the energy density due to radiation is much lower than the energy density of matter or the energy density of dark energy. (Note that z is measured in kelvin and t in seconds.) Equation 6.19 is an 2 As the Universe expanded, the energy density of radiation decreased more approximate relationship. you As will see later, other physical processes can change rapidly than the energy density of matter. The energy density of dark energy the temperature of the radiation in the real Universe during the radiation-dominated has remained constant with time. phase ofits expansion. 3 At times when the scale factor was less than about lO-a of its current value, the what is the energy density of radiation would have exceeded the energy density due to t temperature when the age of the universe is one second? matter. At this time, the energy density of dark energy would have been r By substituting a : varue of ¡ 1 s in Equation 6. l9 the temperature is negligible in comparison to the energy densities of radiation or matter. 1.5 x 1010K.

As you will shorlly see, the existence of an early radiation-dominated era has a profound effect on the dynamical evolution of the Universe. Thus the temperature of the universe in the first few seconds of the big bang was At this point you may be wondering how is it that the energy density of matter and higher than the highest temperatures that are found in the cores of the most radiation change in different ways? The numbers of photons and parlicles within a massive stars (where temperatures may reach about r0eK). This immediately co-moving volume remain constant, so this cannot be the origin of the difference. suggests that nuclear reactions may have occurred in any matter that was present The answer lies in the factthat the energies of photons change with the expansion at this time. we will look into such processes in more deiail in Section 6.4, but in of the Universe, whereas the masses (and hence energies) of particles such as the next section we will discuss .u.n tno." extreme conditions: we will consider the protons and electrons and of any cold-dark matter particles remain constant. processes that occurred when the universe was less than I second old. 2{tr}