astronomers’ universe · 2019. 11. 15. · [email protected] translation from the...

Astronomers’ Universe

Other titles in this series

Origins: How the Planets, Stars, Galaxies,and the Universe BeganSteve Eales

Calibrating the Cosmos: How Cosmology Explains OurBig Bang UniverseFrank Levin

The Future of the UniverseA. J. Meadows

It’s Only Rocket Science: An Introduction in Plain EnglishLucy Rogers

Rejuvenating the Sun and Avoiding OtherGlobal CatastrophesMartin Beech

The Universe Before the Big BangMaurizio Gasperini

Amedeo Balbi

The Music of the Big Bang

The Cosmic Microwave Backgroundand the New Cosmology

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Dr. Amedeo BalbiUniversità Roma, Tor VergataDipto. FisicaVia della Ricerca Scientfica, 100133 [email protected]

Translation from the Italian language edition:

La musica del Big Bang by Amedeo Balbic© Springer-Verlag Italia 2007Springer is a part of Springer Science+Business MediaAll Rights Reserved

ISBN: 978-3-540-78726-6 e-ISBN: 978-3-540-78728-0

Library of Congress Control Number: 2008931890

c© 2008 Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of thematerial is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilm or in any other way, and storage in databanks. Duplication of this publication or parts thereof is permitted only under the provisionsof the German Copyright Law of September 9, 1965, in its current version, and permissionfor use must always be obtained from Springer. Violations are liable to prosecution underthe German Copyright Law.

The use of general descriptive names, registered names, trademarks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.

Cover Image: Map and angular power spectrum of the cosmic microwave background,produced from data of the WMAP satellite (http://map.gsfc.nasa.gov/).

Printed on acid-free paper

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To my parents

Preface

Ever since its infancy, humankind has been seeking answers tosome very basic and profound questions. Did the Universe begin?If it did, how old is it, and where did it come from? What is itsshape? What is it made of? Fascinating myths and brilliant intu-itions attempting to solve such enigmas can be found all throughthe history of human thought. Every culture has its own legends,its own world creation tales, its philosophical speculations, its reli-gious beliefs. Modern science, however, cannot content itself withfanciful explanations, no matter how suggestive they are. Nowa-days, our theories about the Universe, built upon rational deduc-tion, have to survive the hard test of experiment and observation.

Cosmology, the science which studies the origin and evolu-tion of the Universe, had to overcome enormous difficulties beforeit could achieve the same level of dignity as other physical disci-plines. At first, it had no serious physical model and mathematicaltools that could be used to address the complexity of the problemsit had to face. Then, it suffered from a chronic lack of experimen-tal data, which made it almost impossible to test the theoreticalspeculations. Given this situation, answering rigorously the manyquestions on the nature of the Universe seemed nothing more thana delusion.

Today, however, things have changed. We live in the goldenage of cosmology: an exciting moment, when, for the first time, weare able to scientifically understand our Universe.

One of the greatest living cosmologists, James Peebles, oncecompared the situation of a cosmologist to that of Tantalus.According to the myth, Tantalus was punished by Zeus for hav-ing discovered the secret of ambrosia, the food which made thegods immortal. Sentenced to suffer from hunger and thirst, he wasimmersed in water, but he could not drink, because when he tried,the water receded; juicy fruits were suspended above his head, but

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when he raised his hands to grasp them, the branches rose awayfrom him.

Similarly, cosmologists can look at celestial objects as longas they wish, but they cannot “touch” them. This is one of thefacts that puts cosmology in a more difficult situation than otherfields of physics. If we want to study the properties of a material,we can analyze a certain quantity of it in a laboratory, given somecarefully controlled conditions. We can also repeat the experimentfor as long as we like. Cosmologists cannot arrange anything likethat. They just have one Universe to study, they cannot decide howto set up its conditions. Furthermore, the subject of investigationis, by its nature, very difficult to grasp. Delving into the furthestreaches of the Cosmos and observing the feeble light reaching usfrom unimaginable distances, requires sophisticated instrumentswhich have not been available to us until very recently. Muchmore than for any other science, the history of cosmology is alsoa history of the tools we use to observe the world. Our idea of theUniverse has been shaped by the extent of what we could observeof it. Starting from Galileo’s telescope, the Cosmos became largerand stranger as we were able to look into it in more detail.

As a strange sort of compensation, however, the vastity of theCosmos is also an unexpected tool that we can use to investigateits properties. The Universe is so large that it takes an enormousamount of time for light to travel through it (even at the maximumspeed allowed by the laws of physics, about 300 thousand kilome-ters per second). When we look at the Sun, we see it as it was about8 minutes ago; when we look at the closest star to the Sun, AlfaCentauri, we see it as it was about 4 years ago; the closest galaxyto our own, M31 in Andromeda, looks as it was about two and ahalf million years ago; and so on. Cosmologists measure such hugedistances in terms of light-years—the distance traveled by light inone year. One light-year corresponds to a distance of about 9,460billion kilometers.

This fact provides cosmologists with a sort of time machine.They can look at the Universe at different phases of its evolutionand reconstruct its history, much as an archaeologist can look atfossils from different epochs. Using this extraordinary opportunity,and perfecting it over the years, cosmology slowly abandoned thestatus of immature science it had at the beginning of the 20th

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century, starting a tough path which, in the last few years, has ledto its becoming one of the most advanced and successful fields ofscientific research.

We now know, for example, that the Universe expands andevolves, and that it reached its current conditions starting from amuch simpler state, when it was much smaller and denser than itis today. Pushing our physical description of the Universe to earlierand earlier times, we eventually reach a state of practically infinitetemperature and density, taking place about 14 billion years ago,which is popularly known by the name of “Big Bang”. The cos-mological model based on Big Bang is extraordinarily effective indescribing the evolution of the entire Universe—and, at the sametime, surprisingly simple. It only takes a handful of parameters tocharacterize the physical state of the Cosmos from its very begin-ning to the present. The main features of the Big Bang cosmologicalmodel are described in Chapter 1 of this book.

At some point, during their incessant research about the ori-gins of our Universe, cosmologists started asking how far in space—and then back in time—they could go with their observations. Wasit perhaps possible to look directly at the moment when the Uni-verse began? In its earliest phases, the entire Universe was extraor-dinarily hot and bright. At first, all this light could not travel along way, because of the dense fog of matter which pervaded theCosmos. But after a few hundred thousand years the Universe be-came transparent, so that light could finally stream unimpededthrough space. Today, more than 13 billion years later, a dim traceof the immense primordial glow keeps coming to us from the far-thest reaches of time and space. Although only cold cinders remainof that tremendous primordial incandescence, we can still measureits presence. It pervades the entire space, all around us. If we tuneour radio on an empty channel, about one percent of the noise wehear is made up of this cosmic signal, the most distant and ancientin the Universe. This fossil relic of the Big Bang is called the cos-mic microwave background (or CMB for short), and it is the realprotagonist of this book. By observing it, cosmologists collected anumber of important clues about the physical state of the Universein its earliest phases. Chapter 2 deals with a detailed explanationof its origin, and tells the fascinating story of its discovery.

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The primordial Cosmos was a realm of simplicity. A sort ofuniform, thick fog pervaded the entire space, and every point in theUniverse had almost identical density and temperature. But this ex-tremely uniform situation was slightly altered by the presence oftiny clumps. Matter then started collapsing around these clumps,in a slow but inexorable process. The gigantic cosmic structurewe observe in the present Cosmos—galaxies, clusters of galaxies,clusters of clusters and so on—was formed by a gradual processof aggregation around those ancient cosmic seeds. Our own exis-tence, after all, is due to those slight imperfections existing in theprimordial Universe. In Chapter 3, I will say more about the originof cosmic seeds, and will explain how we finally measured theirexistence from the imprints they left in the cosmic backgroundradiation.

Cosmic structure formation has been a tug of war betweenopposing forces. Dense regions tended to grow denser because ofself-gravity, but internal pressure opposed this growth—just as a gasresists compression—forcing matter to re-expand. A strange kindof dance then took place, a series of alternating compressions andrarefactions of the cosmic fluid. Those periodic oscillations wereidentical to the ones passing through air when sounds propagate.In other words, acoustic waves traveled across space in the earlyUniverse. By analysing the ripples these waves left imprinted in thecosmic background radiation, cosmologists can today reconstructtheir complicated overlapping, which encodes crucial informationon the physical state of the primordial Cosmos. Just as any musicalinstrument produces a characteristic spectrum of frequencies, sothe physical parameters which define the nature of our Universemanifest themselves through the specific timbre of those primor-dial acoustic waves. The hunt for this “music” of the Big Bang keptcosmologists busy for decades. Chapters 4 and 5 tell the story ofthis fascinating endeavor, which was finally successful in recentyears.

Throughout this book, then, we will see how the careful in-vestigation of the cosmic background radiation gave us the answersto many fundamental questions about our Cosmos. We now knowthat the Universe has been expanding for almost 14 billion yearsand that, perhaps, it will keep expanding forever. We know that themajestic architecture of galaxies formed over the course of billions

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of years started from tiny primordial seeds. We also know that mostof the matter in the Universe is of a completely different kind fromthe matter we are made of, and that we probably have to take intoaccount an even stranger kind of energy. But even if we can takesome pride in having made enormous progress in our understandingof the Universe, we cannot pretend to know all the answers. Ev-ery new finding generates further questions. Chapter 6 deals withsome of the unanswered problems of modern cosmology.

Acknowledgements

It is a pleasure to thank for their precious support in the mak-ing of this book Angela Lahee and Erica Kendall of Springer,Marina Forlizzi and Elisa Magistrelli of Springer-Italy, and CorradoLamberti who supervised the Italian version. It would be impos-sible to give proper credit to all my mentors and colleagues who,over the years, contributed to my knowledge of the topics coveredin this book, but I would like to thank at least Nicola Vittorio andGeorge Smoot. Finally, thanks to Mom, Dad, Andrea and Alessio.Most of all, thank you, Ilaria.

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Contents

1. The Scenery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Ancient Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3. Cosmic Seeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4. Music of the Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5. Finding Harmony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6. The Undiscovered Country . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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1. The Scenery

Naturally, we were all there,—old Qfwfq said,—whereelse could we have been? Nobody knew then thatthere could be space. Or time either: what use did wehave for time, packed in there like sardines?I say “packed like sardines,” using a literary image: inreality there wasn’t even space to pack us into. Everypoint of each of us coincided with every point of eachof the others in a single point, which was where we allwere.—Italo Calvino, All At One Point - The Cosmicomicsa

The investigation of the Cosmos by means of the scientific methodonly began in relatively recent times. The first scientifically soundideas about the nature of the Universe date back to the 16thcentury, when Nicolaus Copernicus, Tycho Brahe and JohannesKepler founded the Solar System model that we still use to inter-pret planetary motions. The idea of a motionless Earth, holdingstill at the center of the entire Universe, was replaced by a newconception—one where our planet is no different than any otherorbiting around the Sun. During the 17th century, observationsperformed by Galileo Galilei strengthened the new world-view,and Isaac Newton’s theory gave it a firmer conceptual basis. In thefollowing centuries, Newton’s notion of an absolute and unchang-ing space prevailed; later on, the prejudice of an eternal Universewas widespread among scholars. An eternal Universe eludes anyquestion about its origin and its destiny: the Universe is becauseit always was—and it will always be. Only at beginning of the20th century, a new change of perspective gave birth to moderncosmology.

Today, the scientifically accepted vision of the Universe relieson the Big Bang model. According to this model, the Universebegan its evolution at a very precise moment in the past, and itwent through radically different physical states over the course of

a Harvest Books, Harcourt Brace & Co. (USA, 1976).

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2 The Music of the Big Bang

billions of years, until it became what it currently is. The term “BigBang” was coined—as a derogatory epithet, as it happens—by FredHoyle, the great British astronomer, who was, over his entire career,one of the firmest opponents of the model. No matter how ironicwere the intentions of its author, the “Big Bang” label becameimmediately popular. It is important to emphasize, however, thatthe name conveys a misleading picture: in fact, the birth of ourUniverse has not much in common with an explosion. As we willsee in the following chapters, over the course of the past centurythe Big Bang model passed a number of observational tests, andcosmologists believe it provides the better available description ofthe structure and evolution of the Universe—at least, within thelimits of its applicability. Let us then try to understand a bit moreabout the Big Bang model.

The Great Escape

When we observe the night sky—if we are lucky enough to be faraway from city lights, perhaps on a mountain or at sea—our eyescan gaze at an astonishing view. Stars like dust, in every direction.Over the gleaming stripe that the ancients called the Milky Way,stars look so densely packed that we may not be able to single themout, unless we use a telescope. Near the end of the 18th century, as-tronomer William Herschel proved for the first time that all the starsin our sky, and our Sun of course, belong to a vast agglomerate, whoseshape can roughly be compared to that of a thick disc or a pancake.

Astronomers call such a collection of stars a galaxy. When weobserve the Milky Way, we see a larger number of stars only becausewe are looking along the denser regions of our Galaxy—along thedisc.

We now know that our Galaxy—that we still call the MilkyWay—is enormous: it has a diameter of about 100 thousand light-years, a thickness of roughly 10 thousand light-years, and it con-tains hundreds of billions of stars. We also know that the Cosmosis much, much larger than our Milky Way, and that it containshundreds of billions of galaxies similar to our own. But still atthe end of the 19th century only a few people thought that the

The Scenery 3

Universe could be any larger than our cosmic island. The MilkyWay appeared to be all that existed.

Actually, ever since Herschel’s time astronomers had observedmany objects that could not be single stars. Dim puffs of light, someof them more elongated, some rounder. These objects were callednebulae (the Latin word for “clouds”). No one was able to establishtheir distance, not to mention their real nature. For the 18th cen-tury’s astronomers, nebulae probably belonged to the Milky Way,being nothing more than clumps of interstellar matter. Germanphilosopher Immanuel Kant, who thought that the Universe hadnecessarily to be eternal and infinite, was among the first to claimthat nebulae might actually be vast aggregations of stars, lying welloutside our Milky Way.

The question about the nature of the nebulae remained unan-swered for a long time. Astronomers were divided into two factions:those who believed that nebulae were objects in our Galaxy, andthose who considered them different galaxies, spread in a Universemuch bigger than our Milky Way. The dispute was finally settledonly in 1924, by American astronomer Edwin Hubble. He madegood use of previous work by astronomer Henrietta Leavitt, whoonly a few years before had discovered a way to measure the dis-tance of a particular kind of variable stars, the Cepheids. Usingthe most powerful astronomical tool available at the time, the 2.5meters telescope of Mount Wilson, California, Hubble was able todetermine the distance to one of the most brilliant nebulae: M31in Andromeda, a nebula which is easily seen by the naked eye inoptimal conditions. Hubble established that M31 was in fact about900 thousand light-years from Earth—a much larger distance thanthe size of the Milky Way. (Today, we actually know that the dis-tance to M31 is even larger than that initially estimated by Hubble,being more than 2 million light-years away.) Given such an enor-mous distance, M31 could only be visible if it contained a hugenumber of stars, comparable to those in the Milky Way. M31 wasundoubtedly another galaxy. All of a sudden, Hubble’s discoverymade the Universe a vast place, surprisingly larger than one mighthave reasonably expected. An awful waste of space.

Such an extraordinary result, which solved a longtime con-troversy, brought Hubble immediate fame; but he did not relax.Helped by his assistant Milton Humason (who started as a janitor


at Mount Wilson, but soon became one of the most scrupulousand skilled observers of his times), Hubble went on measuring thedistance to many other nebulae, proving that they were separategalaxies1. In 1929, after they had observed and analyzed tens ofsuch galaxies, Hubble and Humason announced a new discovery—possibly even more surprising than the first one. They had beenable to determine not only the distance of many of those galaxies,but also their velocity. On average, one would expect to observeeach galaxy moving with a random velocity, with no relation tothe velocity of other galaxies. But Hubble noticed that all galaxiesseemed to be moving away from the Milky Way. A similar observa-tion had been made some years before by astronomer Vesto Slipher,but Hubble now had more and better data to confirm this finding.

Things got even stranger when Hubble decided to plot the ve-locities and distances for different galaxies on a graph (Figure 1.1).The two quantities, according to Hubble’s data, followed a roughlylinear relation. In other words, more distant galaxies seemed tomove away from the Milky Way at higher speeds—a galaxy whichwas at a distance twice as great as another galaxy, was also movingtwo times faster. This fact, which actually was not that evident inHubble’s earlier data, was later confirmed by further and more ac-curate observations in 1931. Later, this apparent runaway motionof all galaxies has been confirmed countless times with every newobservation, and is considered one of the pillars of modern cos-mology. The law which relates the velocities of galaxies to theirdistances is now called the Hubble law, to honour its discoverer.

At first sight, the Hubble law seems to force us to assign aspecial—and unpleasant—position to our own Galaxy, since allother galaxies are apparently fleeing away from us. It would looklike a suspicious anachronism, after the many centuries needed toprogressively remove Earth from the center of Cosmos, to find outthat we are indeed occupying a peculiar position in space, amidsta mysterious motion involving the entire content of the Universe.

Actually, it is possible to interpret the recession velocity ofgalaxies in an entirely different fashion. To understand this, let usbegin with a simple mental experiment. Take a long rubber band,

1 As a matter of fact, there are objects which are nebulous in nature but do belongto our own Milky Way: they have nothing to do with other galaxies, being simplyclouds of interstellar matter. We still call them nebulae.

The Scenery 5

FIGURE 1.1 The original Hubble data suggested a linear relation betweenthe velocity of galaxies and their distance from us. This law has beenconfirmed with greater precision by more recent observations. (Figureadapted from Hubble’s 1929 paper.)

make a series of knots in it at regular distances, for example 1 cm.Then extend the rubber, until its length doubles. Now, every knotwill be at a distance of 2 cm from the next one. Fine. Let us nowlook at the situation from the point of view of any one of theknots. Its nearest neighbours, which initially were at a distance of1 cm, now are 2 cm away. The next closest neighbours, however,have changed their distance from 2 cm to 4 cm, and so on. In otherwords, if we constantly extend the rubber band and assume thepoint of view of any one knot, it will look as if all the other knotsare moving away, the more distant with higher velocities. Everyknot might then be considered at the center of a recession motion,which of course is only apparent.

In the Universe, galaxies are not aligned like knots along arubber band. They are distributed in all three dimensions of space.However, with some imagination, we might imagine a situationwhich resembles that of galaxies in space. For example, we mightconsider the motion of raisins in a loaf of rising bread (Figure 1.2).As the dough expands, the raisins vary their distances in the same


FIGURE 1.2 The distance between galaxies in the expanding Universevaries in a way which is similar to raisins in a loaf of rising bread. Asthe dough expands, the separation between the raisins grows. The expan-sion rate is the same everywhere

manner as the knots on a rubber band, but in three dimensions. Inthis sense, the motion of raisins as seen from any one raisin is verysimilar to the recession motion of galaxies as seen from the MilkyWay.

At any rate, these mental exercises should only serve to con-vince you that the more sensible way to interpret the apparentrecession of the galaxies is to attribute the motion to an expan-sion of the space they are in. According to this interpretation, it isthe Universe itself which is expanding. At any given moment, thedistance between any two points in space increases. Galaxies areactually (roughly) at rest in space, while space itself carries themaway, inflating as a loaf of bread or a rubber band. If we see thingsthis way, we realize there are no special positions in space. Thereis no center to the motion, as space expands itself in exactly thesame manner everywhere.

It might be worth eliminating immediately a possible sourceof misunderstanding. The Hubble law only applies to very largedistances, comparable to the size of the entire Universe. We can-not measure the effects of the expansion on our Galaxy, on theSolar System, or on our own height. Compact objects, held to-gether by a strong gravitational influence, are not affected by theexpansion. Galaxies, for example, are so small when compared to

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the entire Universe that we may consider them as mere points inspace, carried away by the overall expansion as leaves on a river.

The discovery of the Hubble law forced cosmologists to ques-tion the idea of a static and unchanging Universe, which had dom-inated scholars’ minds for centuries. Furthermore, in order not toconsider the Milky Way as the center of a recession motion ofgalaxies, it required the introduction of a sort of “elastic” space.What could this possibly mean? Finally, the existence of the Hub-ble law seemed to imply another consequence, possibly even moreprofound. Since any distance in the Universe increases with time,galaxies had to be closer in the past. If we keep going back in time,we can extrapolate every motion until we reach a state where alldistances are null. The Hubble law then seems to imply that theUniverse began in a definite moment, separated from the presentby a finite amount of time. This is, so to speak, a first glimpse ofthe Big Bang idea.

Hubble was very cautious, however, and did not attribute anytheoretical meaning to his observations. He was a pure astronomer,and his real goal was making the most accurate observations withthe best intruments he could use. The scientific paper announcingone of the greatest discoveries of all time was simply and humblytitled “A Relation between Distance and Radial Velocity amongExtra-galactic Nebulae”. Others had to delve into Hubble’s findingsto bring theoretical cosmology toward the beginning of a new age.

Extreme Gravity

At the beginning of the 20th century, an obscure third-class clerkat the Bern patent office, Albert Einstein, revolutionized the sci-entific vision of the world by redefining such apparently obviousconcepts as space and time. Without Einstein’s ideas, it would beimpossible to answer scientifically the questions posed by moderncosmology. If Hubble’s observations provided us with a new visionof the Cosmos—a larger, stranger Cosmos—Einstein’s theory al-lowed a rigorous interpretation of such vision within a coherentframework. For the first time, the attempt to investigate the Uni-verse and its evolution as a whole could rely on a solid conceptual


basis. It would be impossible to illustrate here in detail Einstein’stheory. We can however outline its main points, those which arenecessary to comprehend our modern view of the Cosmos.

Einstein’s first revolution—that he completed in 1905, whenhe was only twenty-six years old—is known as special relativitytheory. This theory is essentially built upon two cornerstones. Thefirst one, the so-called special principle of relativity, requires thatphysical laws be the same for any observer, independently of herrelative motion at constant velocity on a straight path. Contrary toa popularly held belief, relativity theory does not state that “all isrelative”, but that physical laws are unique, and that observers haveto have certain recipes to interpret any apparent differences due totheir relative motion. Such a recipe had been available since the17th century, being first formulated by Galileo Galilei. However,Einstein had to modify it by introducing a new ingredient thatGalilei could not know, and that represents the second cornerstoneof his theory.

This ingredient, that was discovered only at the end of the 19thcentury, is the fact that light propagates in a vacuum at constantspeed—about 300 thousand kilometers per second. Normally, weestablish the velocity of motion of any object only in relation to ourown. For example, a passing train will appear in motion if we are atthe station, but it might look at rest if we are inside a car running atthe same speed and in the same direction. No such thing happenswhen we consider light. Every observer, no matter what her state ofmotion, will invariably measure the same speed of light. We wouldnot be able to “stop” light even if hypothetically we traveled atits same speed in the same direction. Moreover, light speed is aninescapable limit for the velocity of any observer.

The fact that light has a finite speed, as we already mentioned,is very important to cosmologists. They can obtain images of theUniverse in different phases of its evolution by simply looking farenough in space. However, as Einstein had to conclude with somesurprise, the constancy of light speed has much deeper implica-tions. The velocity of an object is determined by measuring thedistance traveled over a certain time interval. On the other hand,every observer, independently of her motion, always has to mea-sure the same light speed. Then, space and time have to becomeflexible concepts, losing their immutable nature, even ending up

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mixing with each other. Within the special theory of relativity,the condition of motion of different observers has to determine adiscordance in their measurements of space and time—rulers getlonger or shorter, and clocks do not register the passage of time inthe same way. These discrepancies are reconciled by introducingthe idea that space and time do not exist separately, as absolutequantities, but are partial manifestations of a single well-definedentity—spacetime.

The theory of special relativity unsettled all the certainties ofEinstein’s contemporaries. A former teacher of Einstein’s, mathe-matician Hermann Minkowski, was the first to cogently expressthe new point of view in 1908: “Henceforth space by itself, and timeby itself, are doomed to fade away into mere shadows, and only akind of union of the two will preserve an independent reality”.

But Einsten did not content himself with perturbing a world-view that had been considered unshakable for millennia. In 1915,at the end of eight years that brought him to the verge of a nervousbreakdown (“You must help me or else I’ll go crazy”, he wroteto colleague Marcel Grossmann) he completed his masterpiece,his Ninth Symphony: the general theory of relativity. It is one ofthe highest achievements of the human intellect: a beatiful andelegant theory which has been capable of predicting and correctlyinterpreting a huge number of observations. The change that thegeneral relativity theory brought to our vision of the world was sogreat that Einstein himself said that the special relativity theorywas “child’s play” in comparison.

Einstein originally took on general relativity in order to allowthe interpretation of physical phenomena by any observer, withoutrestrictions as to her motion (remember that, in special relativity,only observers moving at uniform speed along straight paths wereconsidered). But soon he realized that the theory he was develop-ing also gave a novel description of gravity. We are all familiar withgravitation, since it is the force we experience most often and di-rectly in everyday life. It keeps us anchored on the ground, it guidesthe trajectory of bullets, the fall of bodies and the orbit of celestialobjects. Before Einstein came on the scene, all these phenomenawere interpreted according to the law of universal gravitation, for-mulated by Sir Isaac Newton at the end of the 17th century. Newtonwas able to explain and unify a number of phenomena that were


considered completely unrelated at the time, such as the orbit ofplanets in the Solar System and the fall of an apple from the tree.Einstein’s ideas did not overthrow Newton’s theory, but rather ex-panded and improved it, allowing for the interpretation of extremeevents for which it was inadequate.

According to Newton’s theory, gravity required an instan-taneous—and inexplicable—action at a distance between bodies.Every body has a certain mass, governing its ability to interact bymeans of gravity. Two bodies attract with a force which is propor-tional to the product of their masses and inversely proportional tothe square of their distance. Newton could not explain the physi-cal origin of this action at a distance between bodies and, althoughuncomfortable with that, accepted it as a purely empirical fact.Einstein’s theory clarified this mystery. According to Einstein, thepresence of a mass modifies the structure of spacetime, in an evenmore radical way than special relativity. In the presence of mat-ter, spacetime gets curved. To understand this, we might imagineempty space as a large elastic sheet, kept in tension from its cor-ners, so that initially it is perfectly flat. In this analogy, the fourdimensions of spacetime—three for space and one for time—are re-placed by two space dimensions only, in order to help visualization.Now, let us imagine putting a heavy body (for example a bowlingball) at the center of the sheet. The sheet will bend and curve underthe weight of the body (Figure 1.3). If we now put a lighter body (forexample a ping-pong ball) on the sheet, it will roll down towardsthe cavity created by the larger mass, as if it were attracted by it.This example illustrates how the gravitational attraction betweenbodies can be thought of as a consequence of spacetime curvature.

FIGURE 1.3 According to Einstein’s theory of general relativity, spacetimegets warped by the presence of mass, just as an elastic sheet bends undera weight

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In Einstein’s gravitation, then, spacetime is a dynamical,everchanging entity, affected by the presence of matter; at the sametime, the motion of matter is influenced by the curvature of space-time. Or, in physicist John Wheeler’s words: “Matter tells spacehow to curve, and space tells matter how to move”.

After general relativity was completed, the time was ripe toattack the challenging task of describing how gravity shapes theentire structure of the Universe—not just the behaviour of space-time around single isolated stars, but on the largest cosmic scale.In 1917, Einstein published the paper that can be considered thefirst attempt at proposing a scientifically sound model of the Uni-verse: “Cosmological Considerations on the General Theory ofRelativity”.

Since Einstein was interested in the overall behaviour of theUniverse, he could neglect local details of matter distribution. Hethen based his cosmological model on a very simple assumption,which is now known as the cosmological principle: the Universemust appear basically the same to every observer and in any direc-tion. Certainly, locally we might see more matter in one directionrather than in the other: but, on average and on a sufficiently largescale, matter has to be uniformly distributed. Not only that, butevery point in the Universe must be absolutely equivalent to anyother point. There are no special places in the universe—just asthere is no center to galactic recession motion. In technical jargon,it is said that the Universe must be homogeneous (that is, identi-cal, on average, in every point) and isotropic (that is, identical, onaverage, in every direction).

The cosmological principle and Einstein’s general relativityare the two theoretical foundations for our modern Big Bang modelof the Universe. Based on these two cornerstones, it is possibleto correctly interpret the recession motion of galaxies observedby Hubble. Ironically, however, after radically transforming ourview of the Universe, and laying the foundation for addressing thecosmological problem with his 1917 paper, Einstein missed theopportunity of predicting an expanding Universe from his equa-tions. In this respect, he turned out to be strangely conservative.He tried to build a cosmological model for an eternal and staticUniverse—unchanging, with no beginning and no end. Einsteinclung to a widespread philosophical prejudice of his times, which


ended up influencing his cosmological model. Anyway, his obsti-nacy in looking for a static solution to his equations ultimatelyled him to introduce an idea that left profound consequences inmodern physics and cosmology. We will get back to this later, inChapter 5.

Eventually, a young Russian mathematician, Alexander Fried-mann, was the first to demonstrate that Einstein’s theory, plus thecosmological principle, implied that the Universe might be expand-ing. It is noteworthy that Friedmann derived his evolving solutionin 1922, years before Hubble’s discovery of galaxy recession. Fried-mann’s model encountered enormous resistance. Einstein himselfwas somewhat hostile to his young colleague, initially arguing thatthe model was based on wrong calculations. Later, he admitted thatthere were no mistakes, but he kept maintaining that the solutions,although correct, had no connection with reality. Only after Hub-ble’s discovery did the scientific community—including a reluctantEinstein—concede that the Universe was not static and might beevolving. Sadly, Friedmann died of typhus in 1925, never enjoyingthe belated success of his model.

Friedmann’s work is still today the backbone of the Big Bangmodel. By applying the basic idea of general relativity, namely, thatthe properties of spacetime are related to the distribution of matter,Friedmann elegantly linked the expansion of the Universe to itsmaterial content, and avoided the paradoxical conclusion that theMilky Way was at the center of the recession motion of galaxies.Let us see how.

The Weight of All Things

We all know it. When we throw a stone in the air, sooner or later itfalls back on the ground. Gravity keeps the stone tied to the Earth,and none of us is so strong to give the stone enough energy to breakthe bond. If you were Superman, however, you could easily launchthe stone at a fantastic speed, so that it would leave the Earthforever, getting lost in space. There is actually a third possibil-ity. With the incredible ability typical of any superhero, Supermanmight launch the stone at exactly the right speed to compensate

The Scenery 13

for the gravitational bond—no more, no less. In this case, the stonewill progressively slow down, but it will reach null velocity onlyafter an infinite amount of time—a behaviour which falls exactlyon the border between unlimited escape and eventual return. Theexact speed needed to attain this ideal behaviour is called escapevelocity2.

We can use the analogy of the stone trajectory within Earth’sgravitational field to understand the possible ways the Universecan expand, according to the Big Bang model. These ways are:

1. the Universe does not expand forever, but, after a certain time,it begins contracting;

2. the Universe expands forever;3. the Universe stops expanding after an infinite amount of time.

Actually, to make our analogy more accurate we should imag-ine that we could throw the stone only at a certain speed. Forexample, since Superman might be very busy saving the world, hecould lend us a catapult, built in such a way to throw the stoneat precisely the escape velocity from Earth. As long as we remainon our planet, then the stone launched by the catapult will adoptthe third behaviour, leaving Earth and stopping its motion only atinfinity. But if you now move the catapult to the Moon, which hasa much smaller mass than Earth, the same speed would be enoughfor the stone to get lost in space. In contrast, if the catapult isused within Jupiter’s gravitational field, which is much more in-tense than on Earth, the stone would not be able to leave the giantplanet, and would come back after a certain time.

With our expanding Universe, things go much in the sameway. Any portion of matter in the Universe attracts all the restof matter. The entire material content of the Universe is kept intouch by gravity. The expansion is affected by the overall influ-ence of the content of the Universe. What happens, then, is easilyunderstandable: if there is enough matter in the Universe, its over-all gravitational attraction will slow down the expansion until itstops—just as Jupiter’s pull forces the stone moving at Earth’s es-cape velocity to invert its motion. However, if the overall mass is

2 We do not need Superman to compute the value of the escape velocity fromEarth. It is about 11 kilometers per second, or about 40 thousand kilometers perhour. This is the minimum velocity required for an object to leave our planet.


not large enough, nothing will stop the Universe from expandingforever. To decide in which way our Universe is expanding, then,we must determine how much mass is contained in it. In otherwords, we have to “weigh” the Universe.

Actually, cosmologists measure the quantity of matter con-tained in the Universe in terms of its density, that is, the massdivided by the volume occupied. If the average density of the Uni-verse is larger than a certain value, called critical density, the ex-pansion will eventually stop, and the Universe will start collapsing.But if the average density is smaller than the critical density, theexpansion will go on forever (Figure 1.4).

Clearly, the exact value of the critical density depends on thevelocity at which the Universe expands. For a larger speed, a highermass will be needed to stop the expansion—just as we need a moremassive planet to bind a faster stone. The velocity at which theUniverse is currently expanding is measured in terms of a quantitynamed after the discoverer of expansion: it is called the Hubbleconstant, and represented by the symbol H0. The density of theUniverse, compared to the critical density, is represented by theGreek letter omega: Ω. A value Ω = 1 means that the Universe hasan average density exactly equal to the critical one. Values of Ω

FIGURE 1.4 In Friedmann models, the way the Universe expands dependson its average density. If it is larger than a certain critical value the Uni-verse recollapses, otherwise it expands forever

The Scenery 15

larger than 1 imply a density larger than the critical value; valuesof Ω smaller than 1 a density smaller than the critical value.

We should not think that the critical density need be huge. Ac-tually, it is roughly 10−29 grams per cubic centimeter, that is, abouta hundred billion billion billion times smaller than the density ofwater (which is 1 gram per cubic centimeter). In order to imaginesuch a small value, we could visualize six hydrogen atoms in a cubicbox one meter each side, or a gram of matter in a volume a hundredtimes larger than Earth. For comparison, in the air we breathe, andthat we consider so thin, there are roughly 1025 (1 followed by 25zeros) atoms per cubic meter. If in our experience we are used todensities much larger than the critical density it is simply becausewe live in a very crowded region of the Universe. Actually, the Cos-mos is, on average, a desolately empty place. Given the vastity ofthe Universe, however, even a very small average density can havesuch a big overall gravitational effect as to influence its destiny.

We were able to get an intuition on how the matter content ofthe Universe affects its expansion. This picture would have beeneasily understandable to a physicist at the time of Newton. Thereis nothing particularly surprising in concluding that the overallgravitational pull of the material content of the Universe can in-fluence its evolution. Why did Friedmann need general relativityto formulate his model of expanding Universe?

To start with, the only Universe we can build based onNewton’s picture is one where space and time have an absolutenature. The motion of matter takes place within the immutableframework of a fixed space, and celestial bodies are just like ballsrolling on a billiard table. To imagine that space itself is expandingwould be nonsense in this world-view. The existence of an absolutespace would lead us to interpret Hubble’s law as a real motion ofgalaxies, forcing us to consider our position as the center point ofthis motion—a clearly paradoxical situation. The dynamical space-time introduced by Einstein renders this superfluous, and is perfectto rationally interpret Hubble’s findings. But, as we are about to see,Einstein’s theory revealed even more surprising relations betweenthe overall properties of the Universe and its content—relationsthat no physicist of the 19th century could possibly imagine.


A Matter of Form

Einstein reformed our innate idea of space and time, first blur-ring the distinction between them, then altering their rigidity, byintroducing the concept that matter could curve and bend theirstructure. In his studies, he relied heavily on the ideas of math-ematician Georg Riemann, who, at the end of the 19th century,introduced a formalism which allowed him to tackle geometricalproblems on any kind of surface. We all studied at school the theo-rems of Euclidean geometry, which are valid in the familiar realmof blackboards and notebook paper—that is, on perfectly flat sur-faces. In this ordinary setting, two parallel lines never cross, thesum of the internal angles in a triangle is equal to 180 degrees,the circumference of a circle is equal to its radius multiplied by2π, and so on. But if we tried to reproduce these results not on anotepaper, but on the elastic sheet curved by the bowling ball ofour previous example, things would get very different. For example,the distances between different points would vary in a complicatedway, depending on the sheet deformation. Any geometrical formdrawn on the sheet would be altered in such a way as to lose itsfamiliar characteristics. We would suddenly realize that, for exam-ple, on a curved surface the sum of the internal angles of a triangleis not 180 degrees.

Riemann devised the mathematical tools to address just thiskind of abstract problem. Einstein went farther. He claimed that theworld itself, the spacetime we live in, is not similar to a flat pieceof paper, but is endowed with the bizarre properties of an elasticsheet, bent, curved, stretched, covered with peaks and valleys. Thekey idea of general relativity is that the geometrical propertiesof spacetime are influenced by the distribution of matter. Thisconcept must be part of any reasonable cosmological model. In fact,when we consider the Universe, we are considering all matter andall spacetime. We can then reformulate the idea as: the geometricalproperties of the Universe are influenced by the distribution of allthe matter it contains.

Trying to assess how the spacetime curves when accountingfor all the matter in the Universe would at first look like a desperateattempt. Actually, however, we are only interested in the average

The Scenery 17

modification of the spacetime curvature, not in the complicateddetails around any local distribution of matter. Furthermore, wehave to base our model on the cosmological principle, so that wecan assume that, on average, matter is evenly distributed in theUniverse. Thus, the problem simplifies considerably. In fact, itcan be shown that when general relativity is applied to describe ahomogeneous and isotropic Universe, only three kinds of geometryare possible. Before we illustrate them, let us pause for a while.

We said that spacetime is four-dimensional: there are threespace dimensions (which we usually refer to as length, width andheight) and one time dimension. Although the presence of matterinfluences all four dimensions, here we are referring to the curva-ture of the three dimensions of space. Furthermore, since none ofus can visualize a curved three-dimensional space (since we haveto imagine bending it across a fourth dimension which we cannotexperience), we will keep using analogies in which space has onlytwo dimensions (as in the case of the elastic sheet).

After these premises, here are the possible kinds of space cur-vature for the Universe (Figure 1.5):

1. the Universe can have a positive curvature: to visualize this intwo dimensions, we have to imagine the surface of a sphere;

2. the Universe can have a negative curvature: in this case, thetwo-dimensional analogy is the surface of a saddle;

3. finally, the Universe can have no curvature at all: in two dimen-sions, its geometry would be that of a perfectly flat, Euclideansurface.

The first kind of Universe has no natural boundary, althoughit has a finite extension. If we were two-dimensional beings liv-ing on the surface of a sphere—that surface would be our entire“Universe”—we would never find a sign like “Here the UniverseEnds”. Having enough time, we might travel indefinitely, on aseemingly straight path, only to discover that we eventually ar-rived back where we started—just like the characters in “Aroundthe World in 80 Days”. Positively curved universes are also calledclosed. The other two kinds of Universe are, at least in principle, in-finite. They have no natural boundaries, but as we travel throughthem we continuously find new, unexplored regions. Negatively


FIGURE 1.5 An analogy with bidimensional surfaces can help us visualizethe three possible kinds of geometry of the Universe: a plane (flat Uni-verse, null curvature), a saddle (open Universe, negative curvature) or asphere (closed Universe, positive curvature). The sum of the internal an-gles of a triangle drawn on each surface is, respectively, equal to, smallerthan or larger than 180◦

curved universes are also called open, while those with null curva-ture are simply called flat.

A flat Universe is the one we may feel more confortable with:we intuitively feel that we live in a flat space, where Euclideangeometry applies. In fact, on the small distances of our daily expe-rience, any universe would look rather flat, just as the Earth looksrather flat around us, even if we know it is not. To notice anycurvature of the Universe we have to make observations of verylarge regions of space, comparable to the dimension of the entireUniverse. For example, we might send a powerful light signal andwait to see if it shows up behind our back after many billion years.But we will see that there are actually more reasonable ways tomeasure the curvature of the Universe.

We can now get back to the idea from which we started—namely, the fact that the geometry of the Universe is determinedby its matter content. It is time to establish an important relationbetween the geometry of the Universe and what we learned beforeabout its expansion. We saw that there are three possible ways ofexpansion, depending on the value of the average density, and nowwe just learned that there are also three possible kinds of geometry.It does not take much to figure out a connection between the two

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facts. A Universe with a density larger than the critical value willbe a closed, positively curved Universe; it will be open if its densityis smaller than the critical value and flat if its density is equivalentto the critical value. This connection between the evolution ofthe Universe in time and its geometry is a direct consequence ofEinstein’s theory. Not only will a sufficiently high matter densityavoid an eternal expansion, but it will also curve the geometry ofspace so much as to close the Universe, “wrapping” it on itself.On the contrary, a low density Universe will expand forever and beinfinite in space.

What we just saw allows us to clarify something which is,usually, a strong source of confusion to those who hear about theBig Bang model for the first time. The idea that the Universe orig-inated from a great explosion is totally wrong. An explosion be-gins in a single point in space and proceeds by expelling materialoutward, starting from the initial center. The Universe does notexpand around a center, and does not expand “into” or “toward”something. It is space itself that expands. In the Big Bang model,the entire structure of spacetime originated with the Universe, andthe Universe is the whole spacetime. A good way to visualize thisis to imagine a closed, two-dimensional Universe, which will looklike an inflating balloon. For two-dimensional beings living on thesurface of the balloon, the expansion has no center: all space isexpanding exactly in the same way. The center of the sphere is notpart of the two-dimensional Universe. The better way to answerthe question: “Where did the Big Bang happen?” is “Everywhere!”.

And just before someone even thinks of asking the question,well, since time and space are tied to each other, in the Big Bangmodel time originates with the Universe as well. Unfortunatelythen (or luckily?), any question regarding things “before” or “out-side” the Universe is just meaningless, at least within the tradi-tional limits of the model.

A Fistful of Numbers

During the first half of the 20th century, science produced forthe first time in human history a coherent and logically foundedtheoretical framework to start answering questions about the birth


and the evolution of the Universe. Einstein, Hubble, Friedmannand all the other great scientists that participated in this enterprisehad slowly guided science in a previously uncharted world. TheUniverse looked different, as novel ideas were replacing the oldones.

Newborn cosmology, however, still had to face innumerableproblems. The Big Bang model still had to show itself unmistak-ably to be a valid description of reality and, as any scientificallymotivated theory, it had to do so by convincingly passing the ob-servational tests. But the really important thing was that cosmolo-gists had a model, a plausible working hypothesis to compare withreal world facts.

One of the consequences of any new scientific theory is thatof simplifying the description of phenomena that previously ap-peared to be terribly complex. For example, Newton revolutionizedthe scientific knowledge of his times by discovering that seem-ingly unrelated events—the fall of bodies on Earth and planetaryorbits—were actually explained by a single law of gravitation. Ev-ery new law or model introduces a usually small set of numericalquantities that are needed in order to predict the value of somemeasurable quantities. To some extent, the existence of only a fewfree numerical parameters in a theory is an indication of its inter-nal coherence. It shows that things are not being adjusted “ad hoc”to produce correct predictions for well-known cases, but that thetheory has a real predictive power for measurements that still haveto be performed.

In the case of Newton’s theory of gravitation, for example, theonly numerical parameter is the constant of universal gravitation,specifying the characteristic strength of attraction between anytwo masses. In Einstein’s special relativity, only the numericalvalue of the speed of light in the vacuum appears. As we tried toshow previously, the most elementary formulation of the Big Bangmodel can describe the overall evolution of the Universe in termsof only two quantities: the current expansion rate of the Universe,quantified by the Hubble constant H0, and the total density of theUniverse measured in terms of a critical density, represented by theparameter Ω. The knowledge of these two numbers would allowus to derive important conclusions on the nature of the Cosmos inwhich we live.

The Scenery 21

If we knew the exact value of the Hubble constant, for exam-ple, we might use it to extrapolate the behaviour of the Universe inthe past, and get an estimate of the time passed from the momentof the Big Bang. That is, if we knew precisely the recession speed ofa galaxy at a certain distance we might calculate the time when thesame galaxy was at a null distance from us—much in the same wayas, if we know the velocity of a train and the distance from the laststation we may compute when it left the station. Furthermore, theboldest cosmological consequence of Einstein’s theory—the con-nection between the content of the Universe, its shape and theway it expands—allows us to understand in a single shot the geo-metrical properties of the Cosmos, the nature of its evolution, andpossibly even its future destiny, by “simply” measuring its averagematter density.

The hunt for these two numbers turned out to be incrediblydifficult, and in fact kept generations of cosmologists occupiedfor a whole century. To determine the Hubble constant one hasto overcome tremendous technical difficulties, since one has todetermine simultaneously, and with great precision, the velocityand the distance of a galaxy. Both measurements are extremelyhard. Quite often, the recession speed of a galaxy is not entirelydue to the expansion of the Universe, for reasons that will becomeclearer during the rest of this book. Although measuring a recessionvelocity is relatively simple in itself, it is not always possible toexclude with certainty what other factors, which have nothingto do with the expansion, are contributing to it. To reduce thisuncertainy cosmologists have to observe the motion of very distantgalaxies, which are less likely to be perturbed by local gravitationaleffects.

If measuring velocities is a problem, determining the distanceof celestial objects is one of the most daunting tasks for a cosmolo-gist. The most commonly used technique is based on the fact thatthe energy received from a luminous source diminishes in a simpleway when observed from an increasing distance: given a certainluminosity (namely the energy effectively emitted by the source),the observed brightness (the energy we measure from a distance)decreases in an inverse proportion to the square of the distance.In other words, a light bulb will look four times dimmer when itis at twice the distance from us than another identical light bulb.


Then, if a certain class of celestial objects has a certain knownluminosity—that is, if we have what astronomers call a standardcandle—we can easily determine the distance of any object of thatclass. We can clarify the technique used by astronomers with anexample.

Suppose we are at the beginning of a long bridge. We note thaton the other side of the bridge there is a lamp of the same kind as theone next to us on this side of the bridge. We can then measure thelength of the bridge by comparing the brightness of the two lamps.Now, we also notice that on the other side of the bridge there isalso a different lamp of an unknown kind. Since we now know thelength of the bridge, we can determine the real luminosity of thatlamp. If we now find this new kind of lamp in another situation,we can use its now known luminosity to determine its distance.

This is exactly what astronomers do. They build a cosmic dis-tance ladder, determining the luminosity of a certain class of objectby comparison with objects of known luminosity, using more andmore luminous sources to observe farther and farther into the Uni-verse. The problem with this technique is that a calibration errorat some step in the ladder affects all the following steps. And if atsome point of the ladder a step is missing altogether, we cannot gofarther than that.

It is interesting to notice how the problems in distance deter-mination influenced even the original work by Hubble and Huma-son on the recession of galaxies. As we already mentioned, Hubbleused the fact that Cepheid stars are good standard candles to mea-sure the distance of galaxies. According to Hubble’s initial estimateof the relation between distance and velocity, a galaxy at 3.26 mil-lion light years should move at a velocity of about 558 kilometersper second. Unfortunately, using this estimate to extrapolate whenthe expansion started, one gets a value of only 1.8 billion years. Butalready in Hubble’s time it was known with certainty that Earthformed at least 3 billion years ago (today, we know that Earth isabout 4.5 billion years old). Then, the expanding model was facinga crisis: the Universe seemed younger than the Earth.

The situation was clarified by Walter Baade, a colleague ofHubble’s at the Mount Wilson Observatory, who discovered a flawin Hubble’s assumptions. Baade realized that there were two classesof Cepheid stars, and that Hubble interpreted his data assuming

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the wrong class, assigning a larger luminosity to the stars he wasobserving, and then overestimating the distance to the host galax-ies. Baade recalibrated the distances, and concluded that the Hub-ble constant was about twice smaller than previously estimated, sothat the age of the Universe was in fact about 3.6 billion years—amore comfortable value, although still too small if compared tothe age of the oldest stars in the Universe. Over the course of thepast decades, increasingly more precise measurements led to betterestimates of the Hubble constant and of the age of the Universe,which we now know is of the order of 14 billion years. (A popularjoke among astronomers and cosmologist is that the Hubble con-stant is not constant after all, since its value decreased by a factorof ten over a century.)

Cosmologists encountered similar difficulties when trying toestimate the average density of the Universe. The cosmic inventorycannot simply be made by counting the total number of galaxies—a gigantic yet conceivable effort. In fact, there may be more toit than meets the eye. We are in the same situation as someonewho, at night, wants to estimate the number of houses of a distantcity by counting the number of lights she sees. There is no preciserelation between these two quantities: some houses might havemany lighted windows, others not even one. This effort would bedoomed from the very beginning. Similarly, although cosmologistsassume that some relation exists between the distribution of lightand matter in the Universe, they cannot exclude that there is alarge quantity of undetectable mass—actually, as we will see lateron, it turned out that this is exactly the case.

The accurate determination of cosmological parameters, thenumbers needed to fully characterize the model of our Universe,has been one of the main activities of cosmology over the last cen-tury. This formidable challenge sharpened the wits of cosmologists,who devised more and more ingenious techniques to capture thevalue of those numbers. In the following chapters we will see howthis challenge was finally won only in the past few decades, largelyas a result of the accurate investigation of the relic afterglow of theBig Bang—the cosmic microwave background.

2. Ancient Light

And there was light.—Genesis 1:3

The cosmological model which started emerging during the firsthalf of the 20th century was initially not much more than a con-ceptual framework. Being a mathematician, Friedmann only wor-ried about the formal consistency of his equations. He found a wayto describe a Cosmos in which the spacetime structure evolvedin time, but did not care too much about the physical processeswhich might have taken place within that frame. This task wastaken on by others, who tried to build a coherent description ofthe physical status of the Universe at different epochs. It rapidlybecame clear that an expanding Universe was likely to have un-dergone an early stage when temperature, density and energy hadmuch larger values than today. Nothing we currently observe inthe Universe could have existed in the same form under such ex-treme conditions. Also, such a violent phase probably did not endwithout leaving some trace. If the Universe actually evolved froma hot and dense phase, a warm leftover of that early energy perhapswas still all around.

It just had to be found.

Some Like It Hot

Before we start our search for the fossil relic of the Big Bang, wehave to understand what happens to the content of the Universeas it expands. So far, we have just referred vaguely to “matter” assomething which is “in” the Universe, and what we mean by “mat-ter” is probably clear enough when we refer to today’s Universe:but how can we be sure that matter was always in the same state

25


in the early phases of cosmic evolution? It is time to look moreclosely at this question.

More or less at the same time as Einstein was working at hisrelativity theory and Hubble was making his observations, otherscientists were laying the foundations for a correct description ofmatter on microscopic scales. At the beginning of the 20th century,the studies of physicist Ernest Rutherford had shown that matteris made by atoms. While many different kinds of atoms exist, eachwith its peculiar chemical properties, every atom is made of thesame ingredients: a central nucleus, very heavy and with positiveelectric charge, surrounded by electrons, light particles with neg-ative electric charge. The nucleus is made of protons, positivelycharged, and neutrons, with no charge1. An atom has no electriccharge, being made of the same number of protons and electrons(the number of neutrons may vary, resulting in different isotopesof the same atom). The number of electrons in an atom establishesits chemical properties, allowing it to get bonded to other atomsto form molecules. In this way, atoms can form all the kinds ofmatter we see around us.

In the Big Bang model, the Universe does not remain identicalin time. Its physical state changes as it expands, and its contentgoes through a series of transformations at different epochs. If wehad a movie of the cosmic expansion, and could watch it in re-verse, we would see everything getting closer and closer as weproceed toward the initial frames. As if in the action of a giganticpress, the content of the entire visible Universe would appear moreand more compact, occupying a smaller and smaller volume. Justas the temperature of a gas increases as it is compressed, so theUniverse becomes hotter and hotter as it gets closer to the initialinstant. To give an idea, ten days after the Big Bang the whole visi-ble Universe was as hot as the interior of our Sun, about 10 millionCelsius degrees. A hundred-thousandth seconds after the Big Bang,the temperature was a million times higher. In such a situation,what would have been matter as we know it?

In our ordinary experience, matter is usually found in oneof three different states. A solid state, where atoms are tightly

1 Protons and neutrons are made of other elementary particles called quarks. Thisis not relevant to our discussion, though.

Ancient Light 27

packed in very rigid structures. A liquid state, where atoms areonly weakly bonded and can easily change their position. A gaseousstate, where atoms interact very weakly, and are essentially freeto move independently. As we know, matter can exist in eitherone of these states, and the choice depends, in normal conditions,essentially on temperature. If we put a solid metal in a furnaceand we increase the temperature, it will become liquid at first, andfinally, at even greater temperatures, gaseous.

There is, however, a fourth possible state for matter, which isfar less familiar to us. It is a state where even the bond betweenelectrons and nuclei gets broken, and matter becomes a mixture ofpositively charged ions and electrons. Physicists call this a plasma.A plasma is found inside neon lamps—and in flat screens, of course.

The energy needed to create a plasma out of ordinary matter isnot easily found in everyday life. But in the early Universe energieswere so high that no atom could survive in its neutral form. Theprimeval Cosmos, say, a few minutes after the Big Bang, was uni-formly filled with hot plasma, at temperatures in excess of a billiondegrees. The fact that the Universe went through such a dramaticand inhospitable phase may seem disconcerting at first, but it isactually great for cosmologists. It means that the early Universewas indeed very simple, and that it can be investigated by usingthe standard tools of modern physics.

It is tempting—and natural—to try to form some mental pic-ture of those first moments of cosmic evolution, when a devastat-ing heat shattered the very structure of matter, and the mass ofa hundred billion galaxies was packed in a mere speck. If one ofthe ingredients of your representation is a blinding flash of light,you are not very far from the description established by moderncosmology. In fact, in order to better understand the state of theUniverse in its early stages of evolution, we need to investigatemore closely the nature of light.

A Sea of Light

Light is so important for humans, lost in a mostly cold and darkUniverse, that it has nurtured the imagination of poets from every


FIGURE 2.1 The scheme of a wave with frequency f and wavelength λ,propagating at a velocity v

epoch. In the prosaic language of physicists, light is just one of thepossible manifestations of electromagnetic radiation.

Electromagnetic radiation is the way used by nature to trans-port energy in a vacuum. This transport takes place through thepropagation of waves, a common feature of physical phenomenaresulting from some oscillatory mechanism (Figure 2.1). You caneasily create a wave yourself, by handing one end of a rope to afriend and moving the other end up and down. Light waves aremade in a different way—by the vibration of electric and magneticfields—but their mathematical description is the same. Every waveis characterized by a frequency, measuring the number of oscilla-tions in a certain time interval. Frequency is measured in Hertz(represented by the Hz symbol): a frequency of 1 Hz means thatthe wave oscillates once in one second. If you know the speed ofpropagation of a wave, you can divide it by the frequency, obtainingthe wavelength, namely, the distance between two points of wavewhich are in the same oscillation phase—for example, two crests.Given a certain speed of propagation, a higher frequency wave willhave a shorter wavelength—because the extreme points of oscilla-tion will be closer. Electromagnetic waves travel in a vacuum atthe speed of light. A frequency of 1 Hz, then, corresponds to about300 thousand kilometers, while for a billion Hz the wavelengthwill be just 30 centimeters.

Our eyes are only sensitive to a small range of wavelengthsof electromagnetic radiation, between 0.0004 and 0.0007 millime-ters. Within this range, waves vibrating with different frequencies

Ancient Light 29

simply appear as differently coloured light. The highest frequencywe can observe corresponds to violet light: the lowest to red light.Between this two extremes we find the full colour spectrum. Forcomparison, FM radio waves have a wavelength of some meters,and also microwaves have larger wavelengths than visible light. Atthe other end of the electromagnetic spectrum, at much smallerwavelengths than our eyes can see, there are, for example, ultravi-olet radiation and X-rays (Figure 2.2).

An important discovery of Einstein’s (which earned him theNobel Prize in 1921) was that electromagnetic radiation is madeof quanta of energy, particles called photons. In other words, lightdoes not only have a wave-like nature, but also a corpuscular one.More intense light contains more photons. When you turn on a100 W light bulb, you are struck by a huge flux of photons—somehundreds billion billion per second.

Photons are weird in many respects. First of all, they haveno mass. Because of this, they cannot be at rest in any reference

FIGURE 2.2 The electromagnetic spectrum comprises different kinds ofradiation corresponding to different wavelengths. Visible light occupies avery restricted range


frame, and they always travel on a straight path at the speed oflight—well, this is not surprising since light is made of photons.Furthermore, every single photon has a wave-like nature, with acertain frequency and wavelength. And the energy of a photon hasnothing to do with its mass or its velocity (which are invariant) butis proportional to its frequency.

The wave-like nature of light is apparent in a phenomenoncalled the Doppler effect, which played a strong role in astronomy.It describes the frequency variation of a wave when it is emittedby a moving source (Figure 2.3). We all experience the Doppler ef-fect when we hear a higher pitch from an approaching ambulance’ssiren and a lower pitch from a receding one. The same thing hap-pens with light waves. The light of the siren gets bluer when theambulance approaches and redder when it recedes. However, whilemost of us can easily perceive the different pitch, the difference incolour is indiscernible. The fact is the frequency variation for lightwaves is related to the ratio of the source velocity with respect tothe speed of light. Then, only if the ambulance were moving at arelevant fraction of the speed of light might we notice the changein colour. For astronomical objects, by using sophisticated spectralanalyzers, it is possible to measure the Doppler shift in frequency,thus estimating the object’s velocity. (This was exactly the tech-nique used by Hubble to measure the receding speed of galaxies.)

FIGURE 2.3 The Doppler effect alters the wavelength emitted by a mov-ing source. The crests of the wave (which are produced at regular timeintervals) get denser in the direction of motion, sparser in the opposite. Astationary observer will measure a shorter wavelength (λ2) if the sourceis approaching, larger (λ1) if it is receding

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Now that we know a bit more about the nature of light, wecan get back to the issue of the state of the Universe after theBig Bang. We saw that matter was in a state of plasma, but whatabout electromagnetic radiation? It turns out that the role radiationplayed in the primordial Universe can be understood in terms ofsome quite simple physical systems.

Of Stars and Ovens

Although it may be difficult to notice it in the heavily pollutedskies of our cities, every star has a characteristic colour. Some starshave a blueish shade, like Sirius or Rigel; others, like Antares orAldebaran a reddish tint; our own Sun is yellowish. Astronomershave used this fact for a long time as a tool to measure the su-perficial temperature of stars. This is possible because the energydistribution of the photons emitted by a star (loosely speaking, thenumber of photons having a certain energy) is not random but isgoverned by a well-defined law. The form of this distribution isrelated to the superficial temperature of the star.

Strangely enough, the energy distribution of the photons emit-ted by a star is the same as the photons found in an oven at a certaintemperature, a system which is very close to what physicists call ablack body. The physics of a black body was fully understood onlyat the beginning of the 20th century, when German physicist MaxPlanck was the first to compute the correct mathematical relationdescribing black body energy distribution. Given a certain temper-ature, the electromagnetic waves of different frequencies enclosedin the oven cavity do not have the same intensity: most of the pho-tons have an energy close to a certain value. This value dependsonly on the black body temperature. For each temperature thereis one and only one possible energy distribution (Figure 2.4). Bymeasuring the energy distribution of a black body, then—or evenjust the frequency corresponding to the maximum intensity—onecan establish its temperature with great precision. When an oven isheated to 200◦C, most of the energy is emitted around wavelengthswhich are not visible to the human eye, in the infrared band. Inorder for the oven to start radiating visible light, one would have


FIGURE 2.4 The energetic distribution of photons in a black body. Eachcurve corresponds to a different temperature. For the temperatures shownin the figure the peak intensity is emitted at wavelengths correspondingto different colours of visible light

to heat it to around 500◦C, when it will begin to glow red. Stars,just like the filament of an ordinary light bulb, have a superfi-cial temperature of about 3000◦C, so that they emit most of theirenergy as visible light. This light will be a mixture of electromag-netic waves at different wavelengths, all within the visible range,and will then be mostly white, with a certain colour shade corre-sponding to the frequency of maximum emission. It is precisely bydetermining this dominant colour and comparing it with the cor-responding black body curve that we can establish the temperatureof stars.

Besides the physics of stars, black body distribution also playsa crucial role within the context of the Big Bang model, since it al-lows the description of electromagnetic radiation in the early Uni-verse. Black body is the perfect illustration of a situation whereradiation is in thermal equilibrium with matter. Let us try to elu-cidate this important concept.

Consider the room you are in. You are immersed in electro-magnetic radiation coming from different luminous sources, beingthe Sun or some artificial lights. The electromagnetic waves in the

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room behave in complicated ways. They bounce on objects and onthe room walls, diffusing among air molecules. Every object, de-pending on its characteristics, reflects or absorbs light in differentways. At the same time, each object will radiate their own electro-magnetic waves, although not as visible light—our own body is athermal source at about 36◦C, emitting infrared radiation. This sit-uation is clearly very far from equilibrium. If you were to describein detail the physical state of all the electromagnetic radiation inthe room, your task would be almost impossible.

Now, imagine a very different situation. Some metallic ob-jects are put into a furnace, and the temperature is progressivelyincreased. Initially, the situation will be very similar to that in theroom of the previous example. Every object will emit and absorbradiation in a different way. As the temperature rises, though, theobjects will start melting, losing their individuality and mixing inan undifferentiated magma. Continue increasing the temperature:the atoms will separate from each other, agitating frantically andfilling the cavity homogeneously. This situation is very close to aperfect equilibrium. It is no longer necessary to know the detailedstate of each single atom: the knowledge of the temperature in thefurnace will suffice to describe the whole system. In particular, theradiation within the furnace will follow a black body distribution.

Today, the Universe is far from a situation of thermal equilib-rium between matter and radiation. Electromagnetic sources likethe stars, with superficial temperatures of thousands of degrees,coexist with huge regions of empty and cold space. In the earlyUniverse, however, there was almost perfect thermal equilibrium.Sufficiently close to the Big Bang, there was so much energy inthe electromagnetic radiation, that matter was broken into its fun-damental constituents: a situation which resembled that of an in-candescent furnace. The primordial plasma and the photons weremixed together, forming a sort of single fluid at common tempera-ture, like milk and coffee in a cappuccino.

To understand what happens to this “cosmic cappuccino”, wehave to look back at what we know about the evolution of theUniverse. We saw that the fact that the Universe expands meansthat the distance between any two points in the Universe growswith time. What if we consider the distance between two crests ofan electromagnetic wave—its wavelength—as time goes by? This


distance, as any other distance, will be affected by the expansion,so that, according to the Big Bang model, the wavelength of anyelectromagnetic wave will increase in time. Of course, to noticethis effect one should observe the wave for a very long time, sothat the Universe size varies considerably. In astronomy jargon,this “stretch” of the wavelength of electromagnetic waves is calledredshift, because when the wavelength of visible light is increasedit moves toward the red region of the spectrum. However, the termredshift is also used to indicate the increase in wavelength of anyelectromagnetic radiation, like radio waves or X-rays.

The fact that cosmic expansion increases the wavelength ofradiation in the Universe has a very important consequence. Everysingle photon in the current Universe had to have a much shorterwavelength in the past, and therefore a much higher energy. As welook closer and closer to the Big Bang, the energy of all photonsin the Universe gets arbitrarily large. This is exactly the source ofthe enormous primordial heat. On the contrary, as the Universeexpands, the energy of photons decreases, but the distribution re-mains that of a system in thermal equilibrium—that is, a blackbody. This fact has profound implications for cosmology, whichwe will explore in detail shortly. But before that, we must addressan equally important issue, regarding the composition of the pri-mordial plasma.

Haute Cuisine

So far, we have been a bit vague when referring to the atomic nu-clei of the primordial plasma. But we know that there are manydifferent atoms in nature, each containing a different number ofprotons and neutrons in its nucleus. Some are very heavy, like lead,whose nucleus is made of 82 protons (and 125 neutrons in the mostcommon isotope); others are very light, like lithium, with only 3protons (and 4 neutrons) in its nucleus. In our Universe, the vastmajority of atoms, about 99.9%, are of only two kinds: hydrogenand helium. All other atoms have to share the remaining 0.1% ofthe pie. It is really strange, when we consider the matter we expe-rience in our everyday life. All metals, like iron, gold, copper; most

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of the atoms in the crust of our planets, like calcium and silicon;those in our own body, like carbon; those in the Earth’s atmosphere,like oxygen and nitrogen; all these kinds of atoms are but a smallpercentage of the atomic elements in the Universe. Hydrogen is byfar the most abundant element: the mass of all hydrogen atoms isabout 75% of the total atomic mass in the Universe. This is notsurprising, since hydrogen is the simplest atom one can imagine:its nucleus is simply a proton. Helium, the second most abundantelement (about 24% of the total atomic mass), has a nucleus madeof two protons and two neutrons.

One of the most important features of the Big Bang modelis being able to correctly predict the abundance of helium atomsobserved in the current Universe. According to this model, mosthelium atoms formed during the hot and dense phase which cameafter the Big Bang. The physical process which allows the formationof a helium nucleus starting from two protons and two neutronsis an example of nuclear fusion. Fusion does not happen sponta-neously. First, the strong electric repulsion between protons (bothhaving a positive charge) has to be overcome: only when protonsare sufficiently close, a different kind of interaction, the strong nu-clear force, can play a role, keeping the protons tightly packed inthe nucleus.

In order to bring protons at the right distance, extraordinaryconditions of temperature and pressure are needed. These condi-tions are normally available in stars’ interiors—and in fact it isnow well known that the heaviest atomic elements in the Universewere cooked within stellar “furnaces” over the course of billionsof years. Unfortunately, this explanation does not work for helium.In order to create all the helium atoms we observe today, stellarprocesses would have required tens of billions of years. But eventhe oldest stars in the Universe are only about ten billion yearsold. Only about 1% of all helium could have been produced in suchtime.

According to the Big Bang model, however, the entire Uni-verse was, in its earliest phases, much hotter and denser than anystar. Once its temperature dropped below a certain value—about abillion degrees, so that stable deuterium nuclei, made of one pro-ton and one neutron, could form—nuclear fusion reaction would


FIGURE 2.5 One of the processes by which helium nuclei are formed inthe early Universe. A proton and a neutron form a deuterium nucleus.Then, by adding a proton, a helium-3 nucleus (a helium isotope) is formed.Finally, helium is formed by adding a neutron

have rapidly produced helium nuclei, using all the available freeneutrons in the Universe (Figure 2.5).

In fact, based on theoretical calculations, it is estimated thathelium reached the presently measured abundance of about 24%within the first three minutes after the Big Bang. Trace amountsof other light atomic elements, like lithium, were also produced.After that time, however, although temperatures were still a fewhundred million degrees, the Universe was not dense enough tobuild heavier atomic nuclei. These had to be produced billions ofyears later, inside the stars.

The process we just outlined, namely the production of he-lium nuclei and other light elements in the first instants of theevolution of the Universe, is called primordial nucleosynthesis.Using detailed calculations, cosmologists can predict with greataccuracy the abundance of light atomic elements produced duringprimordial nucleosynthesis and compare it with the observed val-ues. The fact that theoretical predictions and observations matchalmost perfectly is a major achievement of the Big Bang model.

But there is another aspect of primordial nucleosynthesiswhich is crucial for cosmology. The exact quantity of light el-ements produced during the first three minutes is strongly af-fected by the number of protons and neutrons in the Universe.By comparing the predictions of nucleosynthesis to observations,

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cosmologists are able to literally count how many protons andneutrons exist in the Universe. Protons and neutrons have a masswhich is about a thousand times bigger than electrons. The mass ofan atom, then, is essentially the mass of its protons and neutrons—for this reason, protons and neutrons are collectively also known asbaryons, from the greek word meaning “heavy”. This means thatif we know the density of protons and neutrons in the Universe wealso know the total density of all atoms.

As we saw at the end of the last chapter, one of the main issuesin modern cosmology is the attempt to measure the average densityof the Universe. Well, the primordial nucleosynthesis calculationsand the observations of light elements abundances allows us tomeasure the density of all atoms in the Universe. The result, whichcosmologists have refined over the course of decades and is nowconsidered very reliable, is th

astronomers’ universe · 2019. 11. 15. · [email protected] translation from the...

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