einstein’s general theory of relativity, on its own, predictedthat space-time began at the big bang singularity and wouldcome to an end either at the big crunch singularity (if thewhole universe recollapsed), or at a singularity inside a blackhole (if a local region, such as a star, were to collapse). anymatter that fell into the hole would be destroyed at thesingularity, and only the gravitational effect of its mass wouldcontinue to be felt outside. on the other hand, when quantumeffects were taken into account, it seemed that the mass orenergy of the matter would eventually be returned to the restof the universe, and that the black hole, along with anysingularity inside it, would evaporate away and finally disappear.

could quantum mechanics have an equally dramatic effect onthe big bang and big crunch singularities? what really happensduring the very early or late stages of the universe, whengravitational fields are so strong that quantum effects cannot beignored? does the universe in fact have a beginning or anend? and if so, what are they like?

throughout the 1970s i had been mainly studying blackholes, but in 1981 my interest in questions about the origin andfate of the universe was reawakened when i attended aconference on cosmology organized by the jesuits in thevatican. the catholic church had made a bad mistake withgalileo when it tried to lay down the law on a question ofscience, declaring that the sun went round the earth. now,centuries later, it had decided to invite a number of experts toadvise it on cosmology. at the end of the conference theparticipants were granted an audience with the pope. he toldus that it was all right to study the evolution of the universeafter the big bang, but we should not inquire into the big bangitself because that was the moment of creation and thereforethe work of god. i was glad then that he did not know thesubject of the talk i had just given at the conference - thepossibility that space-time was finite but had no boundary,which means that it had no beginning, no moment of creation.

i had no desire to share the fate of galileo, with whom i feela strong sense of identity, partly because of the coincidence ofhaving been born exactly 300 years after his death!

in order to explain the ideas that i and other people havehad about how quantum mechanics may affect the origin andfate of the universe, it is necessary first to understand thegenerally accepted history of the universe, according to what isknown as the “hot big bang model.” this assumes that theuniverse is described by a friedmann model, right back to thebig bang. in such models one finds that as the universeexpands, any matter or radiation in it gets cooler. (when theuniverse doubles in size, its temperature falls by half.) sincetemperature is simply a measure of the average energy - orspeed - of the particles, this cooling of the universe would havea major effect on the matter in it. at very high temperatures,particles would be moving around so fast that they couldescape any attraction toward each other due to nuclear orelectromagnetic forces, but as they cooled off one would expectparticles that attract each other to start to clump together.

moreover, even the types of particles that exist in the universewould depend on the temperature. at high enoughtemperatures, particles have so much energy that wheneverthey collide many different particle/antiparticle pairs would beproduced - and although some of these particles wouldannihilate on hitting antiparticles, they would be produced morerap-idly than they could annihilate. at lower temperatures,however, when colliding particles have less energy,particle/antiparticle pairs would be produced less quickly - andannihilation would become faster than production.

at the big bang itself the universe is thought to have hadzero size, and so to have been infinitely hot. but as theuniverse expanded, the temperature of the radiation decreased.

one second after the big bang, it would have fallen to aboutten thousand million degrees. this is about a thousand timesthe temperature at the center of the sun, but temperatures ashigh as this are reached in h-bomb explosions. at this time theuniverse would have contained mostly photons, electrons, andneutrinos (extremely light particles that are affected only by theweak force and gravity) and their antiparticles, together withsome protons and neutrons. as the universe continued toexpand and the temperature to drop, the rate at whichelectron/antielectron pairs were being produced in collisionswould have fallen below the rate at which they were beingdestroyed by annihilation. so most of the electrons andantielectrons would have annihilated with each other to producemore photons, leaving only a few electrons left over. theneutrinos and antineutrinos, however, would not haveannihilated with each other, because these particles interact withthemselves and with other particles only very weakly. so theyshould still be around today. if we could observe them, itwould provide a good test of this picture of a very hot earlystage of the universe. unfortunately, their energies nowadayswould be too low for us to observe them directly. however, ifneutrinos are not massless, but have a small mass of theirown, as suggested by some recent experiments, we might beable to detect them indirectly: they could be a form of “darkmatter,” like that mentioned earlier, with sufficient gravitationalattraction to stop the expansion of the universe and cause it tocollapse again.

about one hundred seconds after the big bang, thetemperature would have fallen to one thousand million degrees,the temperature inside the hottest stars. at this temperatureprotons and neutrons would no longer have sufficient energy toescape the attraction of the strong nuclear force, and wouldhave started to combine together to produce the nuclei ofatoms of deuterium (heavy hydrogen), which contain oneproton and one neutron. the deuterium nuclei would then havecombined with more protons and neutrons to make heliumnuclei, which contain two protons and two neutrons, and alsosmall amounts of a couple of heavier elements, lithium andberyllium. one can calculate that in the hot big bang modelabout a quarter of the protons and neutrons would have beenconverted into helium nuclei, along with a small amount ofheavy hydrogen and other elements. the remaining neutronswould have decayed into protons, which are the nuclei ofordinary hydrogen atoms.

this picture of a hot early stage of the universe was firstput forward by the scientist george gamow in a famous paperwritten in 1948 with a student of his, ralph alpher. gamowhad quite a sense of humor - he persuaded the nuclearscientist hans bethe to add his name to the paper to makethe list of authors “alpher, bethe, gamow,” like the first threeletters of the greek alphabet, alpha, beta, gamma: particularlyappropriate for a paper on the beginning of the universe! inthis paper they made the remarkable prediction that radiation(in the form of photons) from the very hot early stages of theuniverse should still be around today, but with its temperaturereduced to only a few degrees above absolute zero (-273?c). itwas this radiation that penzias and wilson found in 1965. atthe time that alpher, bethe, and gamow wrote their paper, notmuch was known about the nuclear reactions of protons andneutrons. predictions made for the proportions of variouselements in the early universe were therefore rather inaccurate,but these calculations have been repeated in the light of betterknowledge and now agree very well with what we observe. itis, moreover, very difficult to explain in any other way whythere should be so much helium in the universe. we aretherefore fairly confident that we have the right picture, at leastback to about one second after the big bang.

within only a few hours of the big bang, the production ofhelium and other elements would have stopped. and after that,for the next million years or so, the universe would have justcontinued expanding, without anything much happening.

eventually, once the temperature had dropped to a fewthousand degrees, and electrons and nuclei no longer hadenough energy to overcome the electromagnetic attractionbetween them, they would have started combining to formatoms. the universe as a whole would have continuedexpanding and cooling, but in regions that were slightly denserthan average, the expansion would have been slowed down bythe extra gravitational attraction. this would eventually stopexpansion in some regions and cause them to start torecollapse. as they were collapsing, the gravitational pull ofmatter outside these regions might start them rotating slightly.

as the collapsing region got smaller, it would spin faster - justas skaters spinning on ice spin faster as they draw in theirarms. eventually, when the region got small enough, it wouldbe spinning fast enough to balance the attraction of gravity,and in this way disklike rotating galaxies were born. otherregions, which did not happen to pick up a rotation, wouldbecome oval-shaped objects called elliptical galaxies. in these,the region would stop collapsing because individual parts of thegalaxy would be orbiting stably round its center, but the galaxywould have no overall rotation.

as time went on, the hydrogen and helium gas in thegalaxies would break up into smaller clouds that would collapseunder their own gravity. as these contracted, and the atomswithin them collided with one another, the temperature of thegas would increase, until eventually it became hot enough tostart nuclear fusion reactions. these would convert thehydrogen into more helium, and the heat given off would raisethe pressure, and so stop the clouds from contracting anyfurther. they would remain stable in this state for a long timeas stars like our sun, burning hydrogen into helium andradiating the resulting energy as heat and light. more massivestars would need to be hotter to balance their strongergravitational attraction, making the nuclear fusion reactionsproceed so much more rapidly that they would use up theirhydrogen in as little as a hundred million years. they wouldthen contract slightly, and as they heated up further, wouldstart to convert helium into heavier elements like carbon oroxygen. this, however, would not release much more energy,so a crisis would occur, as was described in the chapter onblack holes. what happens next is not completely clear, but itseems likely that the central regions of the star would collapseto a very dense state, such as a neutron star or black hole.

the outer regions of the star may sometimes get blown off ina tremendous explosion called a supernova, which wouldoutshine all the other stars in its galaxy. some of the heavierelements produced near the end of the star’s life would beflung back into the gas in the galaxy, and would provide someof the raw material for the next generation of stars. our ownsun contains about 2 percent of these heavier elements,because it is a second- or third-generation star, formed somefive thousand million years ago out of a cloud of rotating gascontaining the debris of earlier supernovas. most of the gas inthat cloud went to form the sun or got blown away, but asmall amount of the heavier elements collected together to formthe bodies that now orbit the sun as planets like the earth.

the earth was initially very hot and without an atmosphere.

in the course of time it cooled and acquired an atmospherefrom the emission of gases from the rocks. this earlyatmosphere was not one in which we could have survived. itcontained no oxygen, but a lot of other gases that arepoisonous to us, such as hydrogen sulfide (the gas that givesrotten eggs their smell). there are, however, other primitiveforms of life that can flourish under such conditions. it isthought that they developed in the oceans, possibly as a resultof chance combinations of atoms into large structures, calledmacromolecules, which were capable of assembling other atomsin the ocean into similar structures. they would thus havereproduced themselves and multiplied. in some cases therewould be errors in the reproduction. mostly these errors wouldhave been such that the new macromolecule could notreproduce itself and eventually would have been destroyed.

however, a few of the errors would have produced newmacromolecules that were even better at reproducingthemselves. they would have therefore had an advantage andwould have tended to replace the original macromolecules. inthis way a process of evolution was started that led to thedevelopment of more and more complicated, self-reproducingorganisms. the first primitive forms of life consumed variousmaterials, including hydrogen sulfide, and released oxygen. thisgradually changed the atmosphere to the composition that ithas today, and allowed the development of higher forms of lifesuch as fish, reptiles, mammals, and ultimately the human race.

this picture of a universe that started off very hot andcooled as it expanded is in agreement with all the observationalevidence that we have today. nevertheless, it leaves a numberof important questions unanswered:

1. why was the early universe so hot?

2. why is the universe so uniform on a large scale? whydoes it look the same at all points of space and in alldirections? in particular, why is the temperature of themicrowave back-ground radiation so nearly the same when welook in different directions? it is a bit like asking a number ofstudents an exam question. if they all give exactly the sameanswer, you can be pretty sure they have communicated witheach other. yet, in the model described above, there would nothave been time since the big bang for light to get from onedistant region to another, even though the regions were closetogether in the early universe. according to the theory ofrelativity, if light cannot get from one region to another, noother information can. so there would be no way in whichdifferent regions in the early universe could have come to havethe same temperature as each other, unless for someunexplained reason they happened to start out with the sametemperature.

3. why did the universe start out with so nearly the criticalrate of expansion that separates models that recollapse fromthose that go on expanding forever, that even now, tenthousand million years later, it is still expanding at nearly thecritical rate? if the rate of expansion one second after the bigbang had been smaller by even one part in a hundredthousand million million, the universe would have recollapsedbefore it ever reached its present size.

4. despite the fact that the universe is so uniform andhomogeneous on a large scale, it contains local irregularities,such as stars and galaxies. these are thought to havedeveloped from small differences in the density of the earlyuniverse from one region to another. what was the origin ofthese density fluctuations?

the general theory of relativity, on its own, cannot explainthese features or answer these questions because of itsprediction that the universe started off with infinite density atthe big bang singularity. at the singularity, general relativity andall other physical laws would break down: one couldn’t predictwhat would come out of the singularity. as explained before,this means that one might as well cut the big bang, and anyevents before it, out of the theory, because they can have noeffect on what we observe. space-time would have a boundary- a beginning at the big bang.

science seems to have uncovered a set of laws that, withinthe limits set by the uncertainty principle, tell us how theuniverse will develop with time, if we know its state at any onetime. these laws may have originally been decreed by god, butit appears that he has since left the universe to evolveaccording to them and does not now intervene in it. but howdid he choose the initial state or configuration of the universe?

what were the “boundary conditions” at the beginning of time?

one possible answer is to say that god chose the initialconfiguration of the universe for reasons that we cannot hopeto understand. this would certainly have been within the powerof an omnipotent being, but if he had started it off in such anincomprehensible way, why did he choose to let it evolveaccording to laws that we could understand? the whole historyof science has been the gradual realization that events do nothappen in an arbitrary manner, but that they reflect a certainunderlying order, which may or may not be divinely inspired. itwould be only natural to suppose that this order should applynot only to the laws, but also to the conditions at theboundary of space-time that specify the initial state of theuniverse. there may be a large number of models of theuniverse with different initial conditions that all obey the laws.

there ought to be some principle that picks out one initialstate, and hence one model, to represent our universe.

one such possibility is what are called chaotic boundaryconditions. these implicitly assume either that the universe isspatially infinite or that there are infinitely many universes.

under chaotic boundary conditions, the probability of findingany particular region of space in any given configuration justafter the big bang is the same, in some sense, as theprobability of finding it in any other configuration: the initialstate of the universe is chosen purely randomly. this wouldmean that the early universe would have probably been verychaotic and irregular because there are many more chaotic anddisordered configurations for the universe than there aresmooth and ordered ones. (if each configuration is equallyprobable, it is likely that the universe started out in a chaoticand disordered state, simply because there are so many moreof them.) it is difficult to see how such chaotic initial conditionscould have given rise to a universe that is so smooth andregular on a large scale as ours is today. one would also haveexpected the density fluctuations in such a model to have ledto the formation of many more primordial black holes than theupper limit that has been set by observations of the gammaray background.

if the universe is indeed spatially infinite, or if there areinfinitely many universes, there would probably be some largeregions somewhere that started out in a smooth and uniformmanner. it is a bit like the well-known horde of monkeyshammering away on typewriters - most of what they write willbe garbage, but very occasionally by pure chance they will typeout one of shakespeare’s sonnets. similarly, in the case of theuniverse, could it be that we are living in a region that justhappens by chance to be smooth and uniform? at first sightthis might seem very improbable, because such smooth regionswould be heavily outnumbered by chaotic and irregular regions.

however, suppose that only in the smooth regions weregalaxies and stars formed and were conditions right for thedevelopment of complicated self-replicating organisms likeourselves who were capable of asking the question: why is theuniverse so smooth.? this is an example of the application ofwhat is known as the anthropic principle, which can beparaphrased as “we see the universe the way it is because weexist.”

there are two versions of the anthropic principle, the weakand the strong. the weak anthropic principle states that in auniverse that is large or infinite in space and/or time, theconditions necessary for the development of intelligent life willbe met only in certain regions that are limited in space andtime. the intelligent beings in these regions should therefore notbe surprised if they observe that their locality in the universesatisfies the conditions that are necessary for their existence. itis a bit like a rich person living in a wealthy neighborhood notseeing any poverty.

one example of the use of the weak anthropic principle is to“explain” why the big bang occurred about ten thousand millionyears ago - it takes about that long for intelligent beings toevolve. as explained above, an early generation of stars firsthad to form. these stars converted some of the originalhydrogen and helium into elements like carbon and oxygen, outof which we are made. the stars then exploded as supernovas,and their debris went to form other stars and planets, amongthem those of our solar system, which is about five thousandmillion years old. the first one or two thousand million yearsof the earth’s existence were too hot for the development ofanything complicated. the remaining three thousand millionyears or so have been taken up by the slow process ofbiological evolution, which has led from the simplest organismsto beings who are capable of measuring time back to the bigbang.

few people would quarrel with the validity or utility of theweak anthropic principle. some, however, go much further andpropose a strong version of the principle. according to thistheory, there are either many different universes or manydifferent regions of a single universe, each with its own initialconfiguration and, perhaps, with its own set of laws of science.

in most of these universes the conditions would not be rightfor the development of complicated organisms; only in the fewuniverses that are like ours would intelligent beings develop andask the question, “why is the universe the way we see it?”

the answer is then simple: if it had been different, we wouldnot be here!

the laws of science, as we know them at present, containmany fundamental numbers, like the size of the electric chargeof the electron and the ratio of the masses of the proton andthe electron. we cannot, at the moment at least, predict thevalues of these numbers from theory - we have to find themby observation. it may be that one day we shall discover acomplete unified theory that predicts them all, but it is alsopossible that some or all of them vary from universe touniverse or within a single universe. the remarkable fact is thatthe values of these numbers seem to have been very finelyadjusted to make possible the development of life. for example,if the electric charge of the electron had been only slightlydifferent, stars either would have been unable to burnhydrogen and helium, or else they would not have exploded. ofcourse, there might be other forms of intelligent life, notdreamed of even by writers of science fiction, that did notrequire the light of a star like the sun or the heavier chemicalelements that are made in stars and are flung back into spacewhen the stars explode. nevertheless, it seems clear that thereare relatively few ranges of values for the numbers that wouldallow the development of any form of intelligent life. most setsof values would give rise to universes that, although they mightbe very beautiful, would contain no one able to wonder at thatbeauty. one can take this either as evidence of a divinepurpose in creation and the choice of the laws of science oras support for the strong anthropic principle.

there are a number of objections that one can raise to thestrong anthropic principle as an explanation of the observedstate of the universe. first, in what sense can all these differentuniverses be said to exist? if they are really separate from eachother, what happens in another universe can have noobservable consequences in our own universe. we shouldtherefore use the principle of economy and cut them out of thetheory. if, on the other hand, they are just different regions ofa single universe, the laws of science would have to be thesame in each region, because otherwise one could not movecontinuously from one region to another. in this case the onlydifference between the regions would be their initialconfigurations and so the strong anthropic principle wouldreduce to the weak one.

a second objection to the strong anthropic principle is that itruns against the tide of the whole history of science. we havedeveloped from the geocentric cosmologies of ptolemy and hisforebears, through the heliocentric cosmology of copernicus andgalileo, to the modern picture in which the earth is amedium-sized planet orbiting around an average star in theouter suburbs of an ordinary spiral galaxy, which is itself onlyone of about a million million galaxies in the observableuniverse. yet the strong anthropic principle would claim thatthis whole vast construction exists simply for our sake. this isvery hard to believe. our solar system is certainly aprerequisite for our existence, hand one might extend this tothe whole of our galaxy to allow for an earlier generation ofstars that created the heavier elements. but there does notseem to be any need for all those other galaxies, nor for theuniverse to be so uniform and similar in every direction on thelarge scale.

one would feel happier about the anthropic principle, at leastin its weak version, if one could show that quite a number ofdifferent initial configurations for the universe would haveevolved to produce a universe like the one we observe. if thisis the case, a universe that developed from some sort ofrandom initial conditions should contain a number of regionsthat are smooth and uniform and are suitable for the evolutionof intelligent life. on the other hand, if the initial state of theuniverse had to be chosen extremely carefully to lead tosomething like what we see around us, the universe would beunlikely to contain any region in which life would appear. inthe hot big bang model described above, there was not enoughtime in the early universe for heat to have flowed from oneregion to another. this means that the initial state of theuniverse would have to have had exactly the same temperatureeverywhere in order to account for the fact that the microwaveback-ground has the same temperature in every direction welook. the initial rate of expansion also would have had to bechosen very precisely for the rate of expansion still to be soclose to the critical rate needed to avoid recollapse. this meansthat the initial state of the universe must have been verycarefully chosen indeed if the hot big bang model was correctright back to the beginning of time. it would be very difficult toexplain why the universe should have begun in just this way,except as the act of a god who intended to create beings likeus.

in an attempt to find a model of the universe in whichmany different initial configurations could have evolved tosomething like the present universe, a scientist at themassachusetts institute of technology, alan guth, suggested thatthe early universe might have gone through a period of veryrapid expansion. this expansion is said to be “inflationary,”

meaning that the universe at one time expanded at anincreasing rate rather than the decreasing rate that it doestoday. according to guth, the radius of the universe increasedby a million million million million million (1 with thirty zerosafter it) times in only a tiny fraction of a second.

guth suggested that the universe started out from the bigbang in a very hot, but rather chaotic, state. these hightemperatures would have meant that the particles in theuniverse would be moving very fast and would have highenergies. as we discussed earlier, one would expect that atsuch high temperatures the strong and weak nuclear forcesand the electromagnetic force would all be unified into a singleforce. as the universe expanded, it would cool, and particleenergies would go down. eventually there would be what iscalled a phase transition and the symmetry between the forceswould be broken: the strong force would become different fromthe weak and electromagnetic forces. one common example ofa phase transition is the freezing of water when you cool itdown. liquid water is symmetrical, the same at every point andin every direction. however, when ice crystals form, they willhave definite positions and will be lined up in some direction.

this breaks water’s symmetry.

in the case of water, if one is careful, one can “supercool”

it: that is, one can reduce the temperature below the freezingpoint (o?c) without ice forming. guth suggested that theuniverse might behave in a similar way: the temperature mightdrop below the critical value without the symmetry between theforces being broken. if this happened, the universe would be inan unstable state, with more energy than if the symmetry hadbeen broken. this special extra energy can be shown to havean antigravitational effect: it would have acted just like thecosmological constant that einstein introduced into generalrelativity when he was trying to construct a static model of theuniverse. since the universe would already be expanding just asin the hot big bang model, the repulsive effect of thiscosmological constant would therefore have made the universeexpand at an ever-increasing rate. even in regions where therewere more matter particles than average, the gravitationalattraction of the matter would have been outweighed by therepulsion of the effective cosmological constant. thus theseregions would also expand in an accelerating inflationarymanner. as they expanded and the matter particles got fartherapart, one would be left with an expanding universe thatcontained hardly any particles and was still in the supercooledstate. any irregularities in the universe would simply have beensmoothed out by the expansion, as the wrinkles in a balloonare smoothed away when you blow it up. thus the presentsmooth and uniform state of the universe could have evolvedfrom many different non-uniform initial states.

in such a universe, in which the expansion was acceleratedby a cosmological constant rather than slowed down by thegravitational attraction of matter, there would be enough timefor light to travel from one region to another in the earlyuniverse. this could provide a solution to the problem, raisedearlier, of why different regions in the early universe have thesame properties. moreover, the rate of expansion of theuniverse would automatically become very close to the criticalrate determined by the energy density of the universe. thiscould then explain why the rate of expansion is still so close tothe critical rate, without having to assume that the initial rate ofexpansion of the universe was very carefully chosen.

the idea of inflation could also explain why there is so muchmatter in the universe. there are something like ten millionmillion million million million million million million million millionmillion million million million (1 with eighty zeros after it)particles in the region of the universe that we can observe.

where did they all come from? the answer is that, in quantumtheory, particles can be created out of energy in the form ofparticle/antiparticle pairs. but that just raises the question ofwhere the energy came from. the answer is that the totalenergy of the universe is exactly zero. the matter in theuniverse is made out of positive energy. however, the matter isall attracting itself by gravity. two pieces of matter that areclose to each other have less energy than the same two piecesa long way apart, because you have to expend energy toseparate them against the gravitational force that is pulling themtogether. thus, in a sense, the gravitational field has negativeenergy. in the case of a universe that is approximately uniformin space, one can show that this negative gravitational energyexactly cancels the positive energy represented by the matter.

so the total energy of the universe is zero.

now twice zero is also zero. thus the universe can doublethe amount of positive matter energy and also double thenegative gravitational energy without violation of the conservationof energy. this does not happen in the normal expansion ofthe universe in which the matter energy density goes down asthe universe gets bigger. it does happen, however, in theinflationary expansion because the energy density of thesupercooled state remains constant while the universe expands:

when the universe doubles in size, the positive matter energyand the negative gravitational energy both double, so the totalenergy remains zero. during the inflationary phase, the universeincreases its size by a very large amount. thus the totalamount of energy available to make particles becomes verylarge. as guth has remarked, “it is said that there’s no suchthing as a free lunch. but the universe is the ultimate freelunch.”

the universe is not expanding in an inflationary way today.

thus there has to be some mechanism that would eliminate thevery large effective cosmological constant and so change therate of expansion from an accelerated one to one that isslowed down by gravity, as we have today. in the inflationaryexpansion one might expect that eventually the symmetrybetween the forces would be broken, just as super-cooled wateralways freezes in the end. the extra energy of the unbrokensymmetry state would then be released and would reheat theuniverse to a temperature just below the critical temperaturefor symmetry between the forces. the universe would then goon to expand and cool just like the hot big bang model, butthere would now be an explanation of why the universe wasexpanding at exactly the critical rate and why different regionshad the same temperature.

in guth’s original proposal the phase transition was supposedto occur suddenly, rather like the appearance of ice crystals invery cold water. the idea was that “bubbles” of the new phaseof broken symmetry would have formed in the old phase, likebubbles of steam surrounded by boiling water. the bubbleswere supposed to expand and meet up with each other untilthe whole universe was in the new phase. the trouble was, asi and several other people pointed out, that the universe wasexpanding so fast that even if the bubbles grew at the speedof light, they would be moving away from each other and socould not join up. the universe would be left in a verynon-uniform state, with some regions still having symmetrybetween the different forces. such a model of the universewould not correspond to what we see.

in october 1981, i went to moscow for a conference onquantum gravity. after the conference i gave a seminar on theinflationary model and its problems at the sternbergastronomical institute. before this, i had got someone else togive my lectures for me, because most people could notunderstand my voice. but there was not time to prepare thisseminar, so i gave it myself, with one of my graduate studentsrepeating my words. it worked well, and gave me much morecontact with my audience. in the audience was a youngrussian, andrei linde, from the lebedev institute in moscow.

he said that the difficulty with the bubbles not joining up couldbe avoided if the bubbles were so big that our region of theuniverse is all contained inside a single bubble. in order for thisto work, the change from symmetry to broken symmetry musthave taken place very slowly inside the bubble, but this is quitepossible according to grand unified theories. linde’s idea of aslow breaking of symmetry was very good, but i later realizedthat his bubbles would have to have been bigger than the sizeof the universe at the time! i showed that instead thesymmetry would have broken everywhere at the same time,rather than just inside bubbles. this would lead to a uniformuniverse, as we observe. i was very excited by this idea anddiscussed it with one of my students, ian moss. as a friend oflinde’s, i was rather embarrassed, however, when i was latersent his paper by a scientific journal and asked whether it wassuitable for publication. i replied that there was this flaw aboutthe bubbles being bigger than the universe, but that the basicidea of a slow breaking of symmetry was very good. irecommended that the paper ? published as it was because itwould take linde several months to correct it, since anythinghe sent to the west would have to be passed by sovietcensorship, which was neither very skillful nor very quick withscientific papers. instead, i wrote a short paper with ian mossin the same journal in which we pointed out this problem withthe bubble and showed how it could be resolved.

the day after i got back from moscow i set out forphiladelphia, where i was due to receive a medal from thefranklin institute. my secretary, judy fella, had used her notinconsiderable charm to persuade british airways to give herselfand me free seats on a concorde as a publicity venture.

however, i .was held up on my way to the airport by heavyrain and i missed the plane. nevertheless, i got to philadelphiain the end and received my medal. i was then asked to give aseminar on the inflationary universe at drexel university inphiladelphia. i gave the same seminar about the problems ofthe inflationary universe, just as in moscow.

a very similar idea to linde’s was put forth independently afew months later by paul steinhardt and andreas albrecht ofthe university of pennsylvania. they are now given joint creditwith linde for what is called “the new inflationary model,”

based on the idea of a slow breaking of symmetry. (the oldinflationary model was guth’s original suggestion of fastsymmetry breaking with the formation of bubbles.)the new inflationary model was a good attempt to explainwhy the universe is the way it is. however, i and several otherpeople showed that, at least in its original form, it predictedmuch greater variations in the temperature of the microwavebackground radiation than are observed. later work has alsocast doubt on whether there could be a phase transition in thevery early universe of the kind required. in my personalopinion, the new inflationary model is now dead as a scientifictheory, although a lot of people do not seem to have heard ofits demise and are still writing papers as if it were viable. abetter model, called the chaotic inflationary model, was putforward by linde in 1983. in this there is no phase transitionor supercooling. instead, there is a spin 0 field, which, becauseof quantum fluctuations, would have large values in someregions of the early universe. the energy of the field in thoseregions would behave like a cosmological constant. it wouldhave a repulsive gravitational effect, and thus make thoseregions expand in an inflationary manner. as they expanded,the energy of the field in them would slowly decrease until theinflationary expansion changed to an expansion like that in thehot big bang model. one of these regions would become whatwe now see as the observable universe. this model has all theadvantages of the earlier inflationary models, but it does notdepend on a dubious phase transition, and it can moreovergive a reasonable size for the fluctuations in the temperature ofthe microwave background that agrees with observation.

this work on inflationary models showed that the presentstate of the universe could have arisen from quite a largenumber of different initial configurations. this is important,because it shows that the initial state of the part of theuniverse that we inhabit did not have to be chosen with greatcare. so we may, if we wish, use the weak anthropic principleto explain why the universe looks the way it does now. itcannot be the case, however, that every initial configurationwould have led to a universe like the one we observe. one canshow this by considering a very different state for the universeat the present time, say, a very lumpy and irregular one. onecould use the laws of science to evolve the universe back intime to determine its configuration at earlier times. according tothe singularity theorems of classical general relativity, therewould still have been a big bang singularity. if you evolve sucha universe forward in time according to the laws of science,you will end up with the lumpy and irregular state you startedwith. thus there must have been initial configurations thatwould not have given rise to a universe like the one we seetoday. so even the inflationary model does not tell us why theinitial configuration was not such as to produce something verydifferent from what we observe. must we turn to the anthropicprinciple for an explanation? was it all just a lucky chance?

that would seem a counsel of despair, a negation of all ourhopes of understanding the underlying order of the universe.

in order to predict how the universe should have started off,one needs laws that hold at the beginning of time. if theclassical theory of general relativity was correct, the singularitytheorems that roger penrose and i proved show that thebeginning of time would have been a point of infinite densityand infinite curvature of space-time. all the known laws ofscience would break down at such a point. one might supposethat there were new laws that held at singularities, but it wouldbe very difficult even to formulate such laws at such badlybehaved points, and we would have no guide from observationsas to what those laws might be. however, what the singularitytheorems really indicate is that the gravitational field becomes sostrong that quantum gravitational effects become important:

classical theory is no longer a good description of the universe.

so one has to use a quantum theory of gravity to discuss thevery early stages of the universe. as we shall see, it is possiblein the quantum theory for the ordinary laws of science to holdeverywhere, including at the beginning of time: it is notnecessary to postulate new laws for singularities, because thereneed not be any singularities in the quantum theory.

we don’t yet have a complete and consistent theory thatcombines quantum mechanics and gravity. however, we arefairly certain of some features that such a unified theory shouldhave. one is that it should incorporate feynman’s proposal toformulate quantum theory in terms of a sum over histories. inthis approach, a particle does not have just a single history, asit would in a classical theory. instead, it is supposed to followevery possible path in space-time, and with each of thesehistories there are associated a couple of numbers, onerepresent-ing the size of a wave and the other representing itsposition in the cycle (its phase). the probability that theparticle, say, passes through some particular point is found byadding up the waves associated with every possible history thatpasses through that point. when one actually tries to performthese sums, however, one runs into severe technical problems.

the only way around these is the following peculiarprescription: one must add up the waves for particle historiesthat are not in the “real” time that you and i experience buttake place in what is called imaginary time. imaginary time maysound like science fiction but it is in fact a well-definedmathematical concept. if we take any ordinary (or “real”)number and multiply it by itself, the result is a positivenumber. (for example, 2 times 2 is 4, but so is - 2 times -2.) there are, however, special numbers (called imaginarynumbers) that give negative numbers when multiplied bythemselves. (the one called i, when multiplied by itself, gives -1, 2i multiplied by itself gives - 4, and so on.)one can picture real and imaginary numbers in the followingway: the real numbers can be represented by a line goingfrom left to right, with zero in the middle, negative numberslike - 1, - 2, etc. on the left, and positive numbers, 1, 2, etc.

on the right. then imaginary numbers are represented by aline going up and down the page, with i, 2i, etc. above themiddle, and - i, - 2i, etc. below. thus imaginary numbers arein a sense numbers at right angles to ordinary real numbers.

to avoid the technical difficulties with feynman’s sum overhistories, one must use imaginary time. that is to say, for thepurposes of the calculation one must measure time usingimaginary numbers, rather than real ones. this has aninteresting effect on space-time: the distinction between timeand space disappears completely. a space-time in which eventshave imaginary values of the time coordinate is said to beeuclidean, after the ancient greek euclid, who founded thestudy of the geometry of two-dimensional surfaces. what wenow call euclidean space-time is very similar except that it hasfour dimensions instead of two. in euclidean space-time there isno difference between the time direction and directions inspace. on the other hand, in real space-time, in which eventsare labeled by ordinary, real values of the time coordinate, it iseasy to tell the difference - the time direction at all points lieswithin the light cone, and space directions lie outside. in anycase, as far as everyday quantum mechanics is concerned, wemay regard our use of imaginary time and euclideanspace-time as merely a mathematical device (or trick) tocalculate answers about real space-time.

a second feature that we believe must be part of anyultimate theory is einstein’s idea that the gravitational field isrepresented by curved space-time: particles try to follow thenearest thing to a straight path in a curved space, but becausespace-time is not flat their paths appear to be bent, as if by agravitational field. when we apply feynman’s sum over historiesto einstein’s view of gravity, the analogue of the history of aparticle is now a complete curved space-time that representsthe history of the whole universe. to avoid the technicaldifficulties in actually performing the sum over histories, thesecurved space-times must be taken to be euclidean. that is,time is imaginary and is indistinguishable from directions inspace. to calculate the probability of finding a real space-timewith some certain property, such as looking the same at everypoint and in every direction, one adds up the waves associatedwith all the histories that have that property.

in the classical theory of general relativity, there are manydifferent possible curved space-times, each corresponding to adifferent initial state of the universe. if we knew the initial stateof our universe, we would know its entire history. similarly, inthe quantum theory of gravity, there are many differentpossible quantum states for the universe. again, if we knewhow the euclidean curved space-times in the sum over historiesbehaved at early times, we would know the quantum state ofthe universe.

in the classical theory of gravity, which is based on realspace-time, there are only two possible ways the universe canbehave: either it has existed for an infinite time, or else it hada beginning at a singularity at some finite time in the past. inthe quantum theory of gravity, on the other hand, a thirdpossibility arises. because one is using euclidean space-times, inwhich the time direction is on the same footing as directions inspace, it is possible for space-time to be finite in extent and yetto have no singularities that formed a boundary or edge.

space-time would be like the surface of the earth, only withtwo more dimensions. the surface of the earth is finite inextent but it doesn’t have a boundary or edge: if you sail offinto the sunset, you don’t fall off the edge or run into asingularity. (i know, because i have been round the world!)if euclidean space-time stretches back to infinite imaginarytime, or else starts at a singularity in imaginary time, we havethe same problem as in the classical theory of specifying theinitial state of the universe: god may know how the universebegan, but we cannot give any particular reason for thinking itbegan one way rather than another. on the other hand, thequantum theory of gravity has opened up a new possibility, inwhich there would be no boundary to space-time and so therewould be no need to specify the behavior at the boundary.

there would be no singularities at which the laws of sciencebroke down, and no edge of space-time at which one wouldhave to appeal to god or some new law to set the boundaryconditions for space-time. one could say: “the boundarycondition of the universe is that it has no boundary.” theuniverse would be completely self-contained and not affected byanything outside itself. it would neither be created nordestroyed, it would just be.

it was at the conference in the vatican mentioned earlierthat i first put forward the suggestion that maybe time andspace together formed a surface that was finite in size but didnot have any boundary or edge. my paper was rathermathematical, however, so its implications for the role of god inthe creation of the universe were not generally recognized atthe time (just as well for me). at the time of the vaticanconference, i did not know how to use the “no boundary” ideato make predictions about the universe. however, i spent thefollowing sum-mer at the university of california, santa barbara.

there a friend and colleague of mine, jim hartle, worked outwith me what conditions the universe must satisfy if space-timehad no boundary. when i returned to cambridge, i continuedthis work with two of my research students, julian luttrel andjonathan halliwell.

i’d like to emphasize that this idea that time and spaceshould be finite “without boundary” is just a proposal: it cannotbe deduced from some other principle. like any other scientifictheory, it may initially be put forward for aesthetic ormetaphysical reasons, but the real test is whether it makespredictions that agree with observation. this, how-ever, isdifficult to determine in the case of quantum gravity, for tworeasons. first, as will be explained in chapter 11, we are notyet sure exactly which theory successfully combines generalrelativity and quantum mechanics, though we know quite a lotabout the form such a theory must have. second, any modelthat described the whole universe in detail would be much toocomplicated mathematically for us to be able to calculate exactpredictions. one therefore has to make simplifying assumptionsand approximations - and even then, the problem of extractingpredictions remains a formidable one.

each history in the sum over histories will describe not onlythe space-time but everything in it as well, including anycomplicated organisms like human beings who can observe thehistory of the universe. this may provide another justificationfor the anthropic principle, for if all the histories are possible,then so long as we exist in one of the histories, we may usethe anthropic principle to explain why the universe is found tobe the way it is. exactly what meaning can be attached to theother histories, in which we do not exist, is not clear. this viewof a quantum theory of gravity would be much moresatisfactory, however, if one could show that, using the sumover histories, our universe is not just one of the possiblehistories but one of the most probable ones. to do this, wemust perform the sum over histories for all possible euclideanspace-times that have no boundary.

under the “no boundary” proposal one learns that thechance of the universe being found to be following most of thepossible histories is negligible, but there is a particular family ofhistories that are much more probable than the others. thesehistories may be pictured as being like the surface of the earth,with the distance from the north pole representing imaginarytime and the size of a circle of constant distance from thenorth pole representing the spatial size of the universe. theuniverse starts at the north pole as a single point. as onemoves south, the circles of latitude at constant distance fromthe north pole get bigger, corresponding to the universeexpanding with imaginary time (fig. 8.1). the universe wouldreach a maximum size at the equator and would contract withincreasing imaginary time to a single point at the south pole.

ever though the universe would have zero size at the northand south poles, these points would not be singularities, anymore than the north aid south poles on the earth aresingular. the laws of science will hold at them, just as they doat the north and south poles on the earth.

the history of the universe in real time, however, would lookvery different. at about ten or twenty thousand million yearsago, it would have a minimum size, which was equal to themaximum radius of the history in imaginary time. at later realtimes, the universe would expand like the chaotic inflationarymodel proposed by linde (but one would not now have toassume that the universe was created somehow in the rightsort of state). the universe would expand to a very large size(fig. 8.1) and eventually it would collapse again into what lookslike a singularity in real time. thus, in a sense, we are still alldoomed, even if we keep away from black holes. only if wecould picture the universe in terms of imaginary time wouldthere be no singularities.

if the universe really is in such a quantum state, therewould be no singularities in the history of the universe inimaginary time. it might seem therefore that my more recentwork had completely undone the results of my earlier work onsingularities. but, as indicated above, the real importance of thesingularity theorems was that they showed that the gravitationalfield must become so strong that quantum gravitational effectscould not be ignored. this in turn led to the idea that theuniverse could be finite in imaginary time but withoutboundaries or singularities. when one goes back to the realtime in which we live, however, there will still appear to besingularities. the poor astronaut who falls into a black hole willstill come to a sticky end; only if he lived in imaginary timewould he encounter no singularities.

this might suggest that the so-called imaginary time is reallythe real time, and that what we call real time is just a figmentof our imaginations. in real time, the universe has a beginningand an end at singularities that form a boundary to space-timeand at which the laws of science break down. but in imaginarytime, there are no singularities or boundaries. so maybe whatwe call imaginary time is really more basic, and what we callreal is just an idea that we invent to help us describe what wethink the universe is like. but according to the approach idescribed in chapter 1, a scientific theory is just a mathematicalmodel we make to describe our observations: it exists only inour minds. so it is meaningless to ask: which is real, “real” or“imaginary” time? it is simply a matter of which is the moreuseful description.

one can also use the sum over histories, along with the noboundary proposal, to find which properties of the universe arelikely to occur together. for example, one can calculate theprobability that the universe is expanding at nearly the samerate in all different directions at a time when the density of theuniverse has its present value. in the simplified models thathave been examined so far, this probability turns out to behigh; that is, the proposed no boundary condition leads to theprediction that it is extremely probable that the present rate ofexpansion of the universe is almost the same in each direction.

this is consistent with the observations of the microwavebackground radiation, which show that it has almost exactly thesame intensity in any direction. if the universe were expandingfaster in some directions than in others, the intensity of theradiation in those directions would be reduced by an additionalred shift.

further predictions of the no boundary condition arecurrently being worked out. a particularly interesting problem isthe size of the small departures from uniform density in theearly universe that caused the formation first of the galaxies,then of stars, and finally of us. the uncertainty principle impliesthat the early universe cannot have been completely uniformbecause there must have been some uncertainties orfluctuations in the positions and velocities of the particles. usingthe no boundary condition, we find that the universe must infact have started off with just the minimum possiblenon-uniformity allowed by the uncertainty principle. theuniverse would have then undergone a period of rapidexpansion, as in the inflationary models. during this period, theinitial non-uniformities would have been amplified until theywere big enough to explain the origin of the structures weobserve around us. in 1992 the cosmic background explorersatellite (cobe) first detected very slight variations in theintensity of the microwave background with direction. the waythese non-uniformities depend on direction seems to agree withthe predictions of the inflationary model and the no boundaryproposal. thus the no boundary proposal is a good scientifictheory in the sense of karl popper: it could have been falsifiedby observations but instead its predictions have been confirmed.

in an expanding universe in which the density of matter variedslightly from place to place, gravity would have caused thedenser regions to slow down their expansion and startcontracting. this would lead to the formation of galaxies, stars,and eventually even insignificant creatures like ourselves. thusall the complicated structures that we see in the universe mightbe explained by the no boundary condition for the universetogether with the uncertainty principle of quantum mechanics.

the idea that space and time may form a closed surfacewithout boundary also has profound implications for the role ofgod in the affairs of the universe. with the success of scientifictheories in describing events, most people have come to believethat god allows the universe to evolve according to a set oflaws and does not intervene in the universe to break theselaws. however, the laws do not tell us what the universeshould have looked like when it started - it would still be upto god to wind up the clockwork and choose how to start itoff. so long as the universe had a beginning, we could supposeit had a creator. but if the universe is really completelyself-contained, having no boundary or edge, it would haveneither beginning nor end: it would simply be. what place,then, for a creator?