Life Evolution Universe Lecture Notes - AUC

7/27/2019 Life Evolution Universe Lecture Notes - AUC

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Life, Evolution, Universe Theme Course

The Big Bang - setting the stage for the emergence of life

Jan Pieter van der Schaar

February 11, 2013

Abstract

This set of lecture notes belongs to the first (physics) part of the Life, Evolution, Universetheme course, part of the first year curriculum at Amsterdam University College. The startingpoint is a description of the theory and observational evidence behind the hot Big Bang model.After a mostly conceptual introduction of special and general Relativity, the Friedmann equa-

tions governing the evolution of the universe as a whole are derived and some basic solutionsare discussed. Next a brief introduction of thermodynamics and quantum mechanics is given,after which a basic description of particle physics is discussed and applied in the context of theearly universe. The crucial events of nucleosynthesis and decoupling (last scattering) can thenbe explained, leading up to a discussion of the observed Cosmic Microwave Background (CMB)radiation. Explaining what has been learned from the CMB we arrive at the hypothesis of cos-mological inflation as a theoretical paradigm responsible for the initial conditions of the hot BigBang. Under the influence of gravity the small inhomogeneities in the initial conditions grewto form galaxies and stars. Star formation and energy production is then discussed, with anemphasis on the production of heavy nuclei, among which carbon, explaining the abundancesof the different elements in the universe.

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Contents

1 Introduction 3

2 Part I: Relativity and the Big Bang 42.1 The laws of Newton and the structure of space and time . . . . . . . . . . . . . . . 4

2.2 The speed of light and its space-time consequences . . . . . . . . . . . . . . . . . . 72.3 Relativistic momentum and energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Relativistic Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Big Bang Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Worksheet 1: Relativity and the Big Bang 22

4 Part II: The hot Big Bang and Quantum Physics 254.1 Pressure and the ideal gas law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Heat energy, temperature and entropy . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 A quantum universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Worksheet 2: The hot quantum universe 33

6 Part III: from Nucleosynthesis to Last Scattering 356.1 Thermal history and primordial nucleosynthesis . . . . . . . . . . . . . . . . . . . . 356.2 The cosmic microwave background . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7 Worksheet 3: Matter and radiation in the universe 42

8 Part IV: Galaxies, Stars and Heavy Elements 448.1 Initial conditions for the origin of structure . . . . . . . . . . . . . . . . . . . . . . 448.2 Stellar nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.3 The life of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9 Summary 45

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1 Introduction

To start out I found this little joke on the internet, which nicely summarizes in chronological ordera large part of what we intend to cover in this course:1

The History of the Universe in 200 Words or Less

(http://members.verizon.net/ vze3fs8i/hist/hist.html)

Quantum fluctuation. Inflation. Expansion. Strong nuclear interaction. Particle-antiparticleannihilation. Deuterium and helium production. Density perturbations. Recombination. Black-body radiation. Local contraction. Cluster formation. Reionization? Violent relaxation. Virializa-tion. Biased galaxy formation? Turbulent fragmentation. Contraction. Ionization. Compression.Opaque hydrogen. Massive star formation. Deuterium ignition. Hydrogen fusion. Hydrogen de-pletion. Core contraction. Envelope expansion. Helium fusion. Carbon, oxygen, and silicon fusion.Iron production. Implosion. Supernova explosion. Metals injection. Star formation. Supernova ex-plosions. Star formation. Condensation. Planetesimal accretion. Planetary differentiation. Crustsolidification. Volatile gas expulsion. Water condensation. Water dissociation. Ozone production.

Ultraviolet absorption. Photosynthetic unicellular organisms. Oxidation. Mutation. Natural se-lection and evolution. Respiration. Cell differentiation. Sexual reproduction. Fossilization. Landexploration. Dinosaur extinction. Mammal expansion. Glaciation. Homo sapiens manifestation.Animal domestication. Food surplus production. Civilization! Innovation. Exploration. Religion.Warring nations. Empire creation and destruction. Exploration. Colonization. Taxation withoutrepresentation. Revolution. Constitution. Election. Expansion. Industrialization. Rebellion.Emancipation Proclamation. Invention. Mass production. Urbanization. Immigration. Worldconflagration. League of Nations. Suffrage extension. Depression. World conflagration. Fissionexplosions. United Nations. Space exploration. Assassinations. Lunar excursions. Resignation.Computerization. World Trade Organization. Terrorism. Internet expansion. Reunification. Dis-solution. World-Wide Web creation. Composition. Extrapolation?

Although short and concise, obviously this list does little to increase your understanding ofwhat actually happened over the last 14 billion years. But it does appropriately remind us ofthe challenges to teach an introductory theme course like this. A full, in-depth, understandingof all these events and phenomena is clearly impossible and a careful selection has to be madethat on the one hand allows for a more in-depth discussion and on the other hand preserves thechronological ordering, connecting events dynamically. This is far from easy and any selectionof topics has its drawbacks. For the physics part of the Life, Evolution, Universe theme courseI have decided to focus attention on essentially thee themes: 1) the structure of space-time andhow that connects to the central idea of the Big Bang, 2) microscopic quantum physics and howthat is related to the creation of matter in a hot Big Bang universe, and finally, 3) star formation

and energy production and how that is related to the production of the heavier elements in ouruniverse. In all three themes a significant level of depth will be achieved such that some basiccalculations can be done that should solidify your understanding, going beyond the conceptualnarrative level that you might have encountered before. Every chapter typically contains severalexercises in support of the material that will be part of the hand-in assignments. Solutions willbe presented and discussed in the exercise classes. In addition, a separate textbook is provided tomaintain a general and conceptual birds-eye perspective, exhibiting the (cosmological) connectionsbetween the different sciences. Lets hope we have succeeded!

Please report any typos, mistakes, confusions and remarks to me during the lectures or bysending me an email using [email protected] . This version of the lecture notes wasdrafted on February 11 2013.

1For the physics part this summary more or less describes our intentions up until Supernova explosion.

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2 Part I: Relativity and the Big Bang

The universe is 13.73 0.12 billion years old2. This truly amazing fact, that the universe had asingular beginning 13.7 billion years ago serves as the starting point of this course and will typicallybe referred to as t = 0. For a long time many physicists and philosophers believed the universe wasinfinite in space and time. However, already Isaac Newton was aware of the fact that a universe

with an infinite number of stars was an unstable situation; the slightest perturbations would leadto a dynamical, collapsing universe due to the fact that gravity makes initial inhomogeneities inthe distribution of matter grow bigger and bigger.

Another obvious, but often neglected, fact was that the sky is dark at night. If the universeconsisted out of a static infinite collection of stars, at an approximately constant density and anaverage brightness, for example, equal to the Sun, it is straightforward to show that the night wouldbe as bright as the day. The decrease in surface brightness with distance of a single star is easilycompensated by the increase in the (infinite) number of stars contributing to the total surfacebrightness. This became known as Olbers paradox, after the amateur astronomer HeinrichWilhelm Olbers, who described this paradox in 1823 (although he was certainly not the first toworry about this). As we now know, the solution to this paradox is that the universe has a finite

age.To better understand and appreciate this far-reaching conclusion, it is necessary to take a step

back and ask ourselves some basic, but fundamental, questions about the nature of space and time.More than a century ago, it was Albert Einstein who overturned the basic and intuitive notions ofspace and time, which were unquestioned since Isaac Newton. Based on the remarkable propertiesof light and the principle of relativity Einstein was lead to a unified and dynamical description ofspace and time in his theory of relativity. This laid the foundations for cosmology as a discipline inphysics, by allowing physicists to study the evolution of space and time, the universe, as a whole.

2.1 The laws of Newton and the structure of space and time

What is space and what is time? Are they absolute concepts that are the same for every observeror do they depend on the details of the state (of motion) of the observer? To describe and locateevents in space and time one uses coordinates and often different observers do not use the samecoordinate systems. This could mean that results of certain measurements of these observers arerelated to each other by, what is known as, a coordinate transformation. The simplest examplewould be to consider two observers that are moving with a constant velocity with respect to eachother. In this case the measured velocities of an object under study are related by a coordinatetransformation from the frame of reference in rest to the moving frame of reference. On the otherhand, based on our experience one might expect distances (differences in space) and durations(differences in time) to stay invariant under such a coordinate transformation.

An important property or principle is that one would like the laws of physics to be independent

of the specific observer. Every observer, independent of his or her state of motion should be ableto discover and test the same laws of physics. Isaac Newton already argued that there do existspecial frames of reference, namely those tied to observers on which no force is acting, the so-calledinertial frames of reference, in which fiducial, or pseudo-, forces3 are absent.

The study of motion and dynamical changes in the state of motion, is called classical mechanicswhen dealing with relatively large objects as compared to the scale of atoms. According to Newtonforces are connected to changes in the state of motion, i.e. changes in velocity or better momentum,implying that inertial frames of reference are tied to observers that are moving with a constantvelocity. As a corollary, objects on which no force is acting are either in rest or moving at aconstant velocity v = dxdt , which is known as Newtons first law. Any changes in velocity, ormore appropriately the momentum p = m v in which the inertial mass m appears that is a

2As reported by the latest January 2010 WMAP data release.3Fiducial forces are forces that arise due to using an accelerated frame of reference, a good example being the

centrifugal force which one experiences when positioned in a rotating reference frame.

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measure of an objects resistance to a change in velocity, signal the presence of an external forcedoing work on the system. In terms of a precise formula Newtons second law defines what wemean with a force

F =d

dtp = m

d2

dt2x , (1)

where in the final equality we assumed that the inertial mass of the object under considerationis constant. Also note that we introduced little arrows over the force, momentum and positionvariables to indicate that these are vectors in three-dimensional space, implying one equation forevery direction in space, i.e. x = (x,y ,z). Importantly, in all inertial frames of reference Newtonssecond law should be unambiguous. All inertial observers will agree on whether in some process orexperiment a force is acting or not. This must mean that the mathematical form of Newtons lawin unaffected under coordinate transformations taking you from one inertial frame to another. Forexample, consider two inertial frames moving with respect to each other with velocity u along thex-axis. In that case the different coordinates would be related by the following so-called Galileantransformation

x = x

u t , t = t

y = y

z = z , (2)

and it is easy to check that indeed Newtons law is unaffected by this transformation. This isequivalent to saying that absolute velocity does not exist, only relative velocities can be determined,and forces only depend on acceleration a d2x

dt2. Clearly, velocities are not invariant under Galilean

coordinate transformations, since v = v u. A key property of the coordinate transformationwritten here is that according to it time is absolute, i.e. all observers agree on the elapsedtime once they have synchronized their clocks. The Galilean coordinate transformation realizesthe principle of relativity: two observers moving at a (constant) speed relative to each other willobserve the same laws of physics. We would like this principle to be satisfied by any physical theory,since without it a physical theory would suffer from a certain amount of subjectivity related to theparticular choice of observer.

What is important to realize is that conversely, starting from the principle of relativity, thefact that Newtons second law is invariant under Galilean coordinate transformations (2), teachesus something about the properties of space and time. Namely that different inertial observerswill agree on measured space and time differences. The reason you might not be too impressedis because this seems to be an obvious and unshakable truth. However, as we will soon discover,this supposed truth is far from obvious and in in fact wrong when you look carefully enough. Butbefore we go there, let us first remind you of some important and useful properties that followfrom the laws of Newton.

Because a force, according to Newtons second law (1), creates a change in momentum of anobject it is acting on, in the absence of (external) forces the total momentum has to be conserved

d

dtptotal =

d

dt

Ni=1

pi = 0 . (3)

This turns out to be a very useful property in many systems of interest. For instance if onecollides two (or more) objects, like particles or billiard balls, the total momentum p = p1 + p2 willbe a conserved quantity because the net force vanishes. In the case of billiard balls the combinedelectromagnetic forces of all atoms making up the billiard balls causes them to repel each otherwhen they hit and the two forces at work are exactly opposite and of the same magnitude. This

indeed better be true, since in the absence of an external forced

dtp =

d

dt[p1 + p2] = 0 F1 + F2 = 0 . (4)

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This is also known as the action equals reaction law, or Newtons third law, and followsimmediately from Newtons second law under the assumption that external forces are absent. It isimportant to realize that the vector nature of this equation implies that the two forces are oppositein direction and equal in magnitude, and that the property of momentum conservation holds foreach direction in space independently.

In Newtons laws forces are simply defined as changes in the momentum of a particular object,which does not help in understanding the true origin of these forces. As far as we know today anychange in momentum can actually be associated to one of the four fundamental forces present innature, which are gravity, electromagnetism, the weak nuclear forces responsible for radio-activedecay and the strong nuclear force which holds the atomic nucleus together. Three out of the fourforces are microscopically understood as interactions between fundamental particles that carrycertain charges (like electric charge in the electromagnetic force), i.e. that are absent when objectsare uncharged, or neutral by combining charges with different signs. Gravity is the odd oneout, in the sense that nothing can escape the influence of the gravitational force. Even thoughgravity is an extremely weak force, relatively speaking, the fact that the total energy (or mass)is an additive quantity implies that gravity is the force that is most familiar to us and the mostimportant in shaping the (large-scale) universe. Surprisingly however, it has also remained themost mysterious force from a fundamental microscopic point of view. That said, you are probablyall familiar with the following expression for the gravitational force

FG = G m1 m2|r1 r2|2 , (5)

where G is the gravitational constant and m1 and m2 are the gravitational masses of the bodiesunder consideration. This is saying that the magnitude of the gravitational force is inverselyproportional to the squared distance between the objects.

Plugging this expression (5) in Newtons second law one can then try to find the trajectoriesr1(t) and r2(t) for the two objects that solve the differential equations. Important examples

include spherical distributions of matter, for which one can show that the resultant gravitationalforce is equal to the total mass located at the center of the spherical distribution, meaning that thedetails of the spherical distribution of mass are unimportant, simplifying calculations considerably.Another related property of the gravitational force that we will make use of later on, is that thetotal gravitational force on a test-object of mass m at a radius r due to a spherically uniformdistribution of mass will only depend on the total mass within the region r. So all contributionsto the gravitational force from regions larger than r will cancel.

Besides forces, there is another quantity appearing in Newtons second law that requires a bit ofexplanation, which is the inertialmass. The inertial mass can be thought of as an ob jects resistanceto being accelerated, a constant proportionality factor that relates the acceleration to the actualforce and that measures the so-called inertia. A priori this does not have to be the same as the

objects gravitational mass, which appears in the expression for the gravitational force (5). Notethat the objects weight is again something different: its the force on the ob ject that is necessaryto counteract the gravitational attractive force on the Earths surface4. Experiments have beenunable to detect even the slightest difference between gravitational and inertial mass and theequivalence between inertial and gravitational mass is a key principle (known as the equivalenceprinciple) in Einsteins theory of gravitation, which supersedes Newtons description of gravityfor large enough masses and velocities, as we will describe in a bit more detail later on.

A final (mechanical) ingredient that we need to discuss is energy. Energy is an extremelyimportant concept throughout physics (and beyond) and we will return to it in different contexts.The most important property of it is that in the absence of forces, just like momentum, it is aconserved quantity. So how is it defined in the the context of mechanics? In mechanics energy

is a measure of the amount of work done on an object by some force. Assuming a constant4On the Earths surface the gravitational force is well approximated by a constant acceleration equal to g = 9.8

(m/s)/s.

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force F0 the work W done by a force on an object is simply W = F0 s, where s is the distanceover which the force acted. Note that the result is a number, instead of a vector. Since work isdefined as an inner product between force and distance, the only nonzero contribution stems fromthe force in the direction of the path one is considering. Equivalently, in components this readsW = F0x sx + F

0y sy + F

0z sz. When the force is not a constant, this expression is easily generalized

to the following integral definition of work

W =

F ds =

Fs ds , (6)

where in the last expression Fs indicates the force along the direction of the path under consider-ation. Using Newtons second law that relates a force to a change in momentum, the work donecan be expressed in terms of the initial and final velocities, leading to the identification of whatis known as kinetic energy Ek =

12 mv

2, representing the amount of energy that is stored in theform of motion

W =

dp

dt ds = 1

2m

v2tf

ti=

1

2m

v2f v2i

. (7)

From this expression we conclude that a force increases the amount of kinetic energy containedin an object. In the absence of forces, since the velocity is then unchanged, kinetic energy willbe a conserved quantity. In collisions, where external forces are absent, together with momentumconservation this significantly constrains the possible outcomes once the initial velocities beforethe collision are known. In the presence of what are known as conservative forces the notion ofenergy conservation remains useful by introducing a form of energy related to the force known aspotential energy. A conservative force can be defined in terms of a potential V(s) as follows

F = dds

V(s) V(s) . (8)

Using the definition of work and conservative forces the potential energy stored in a a (conservative)

and spherically symmetric force field (depending only on the radius) is easily derived as

W =

d

dsV(s) ds = V(ri) V(rf) . (9)

So the amount of work done by a conservative, spherically symmetric, force (like gravity) is simplythe difference between the potential energy at the final and initial radius V V(ri) V(rf).From the derivation of the kinetic energy of an object this should also be equal to Ek Ek(rf)Ek(ri). As a consequence

Ek(rf) + V(rf) = Ek(ri) + V(ri) , (10)

so the sum of potential and kinetic energy is a conserved quantity in all processes that involveconservative forces (explaining the terminology). All fundamental forces in nature are in factconservative forces, which greatly enhances the application of the concept of energy conservation.

This ends our summary of the laws of Newton, whose important applications will be furtherexplored in the working group sessions. Before moving on to a discussion of the special theoryof relativity, we would like to emphasize again that the principle of relativity in the context ofthe laws of Newton tells us that distance and durations are independent of the state of motion of(inertial) observers. In other words, we seem to have learned something about the properties ofspace and time by combining the principle of relativity with the laws of Newton. Let us now seewhether these properties of space and time remain intact upon further and more detailed scrutiny.

2.2 The speed of light and its space-time consequences

Since the laws of Newton are unaffected by a Galilean coordinate transformation connecting dif-ferent observers at different constant velocities, implying that velocities are relative quantities, onehas to conclude that there cannot exist an upper bound on the relative velocity. In particular

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this implies that by applying a Galilean coordinate transformation one should always be able toconsider a reference frame in which the object under consideration is at rest. Or, in other words,an observed or computed velocity should always depend on the state of motion of the observer,and is never uniquely determined. However, doing this experiment with light one finds that thespeed of light c is always 299792458 3 108 m/s, completely independent of the relative motionof the observer. What could be wrong?

Indeed, as was known at the start of the 20th century, light can be described as an electro-magnetic wave in Maxwells theory of electromagnetism. Calculating the propagation speedof electromagnetic waves always gives c, and Maxwells equations are not invariant under Galileancoordinate transformations. Instead, it turns out that Maxwells equations are invariant underwhat is known as a Lorentz transformation

x =x u t1 u2/c2 , t

=t ux/c2

1 u2/c2 (11)

y = y

z = z ,

First notice that when u/c 1 this reduces to the Galilean coordinate transformation (2). Thissuggests Newtons laws have to be adapted at large velocities to not contradict the principle ofrelativity and the constancy of the speed of light. Importantly and rather counter-intuitively,the Lorentz transformation suggests that time is no longer absolute! This means differentobservers, in different states of motion, experience their own time intervals, that do not agree ingeneral, i.e. that are related to each other by the Lorentz transformation. One can also readilycheck that a light-ray moving at the speed of light c in one frame of reference, is also moving atthe speed of light in the other frame of reference by calculating v dxdt .

Einstein realized that he should take the Lorentz transformation, which was already known atthe time, very seriously and adapt Newtons law to be in accordance with the principle of relativity(absolute velocity should have no physical meaning) and the constancy of the speed of light.So let us follow through all of the consequences of the Lorentz transformations, i.e. what does ittell us about the properties of space and time?

Let us first define what are called events; points in four-dimensional space and time, definedby four numbers, 3 to define location xi, with i = 1, 2, 3 (also represented as the vector x) andone to express the time of the event t. In a particular reference frame, using coordinates (t, xi),a specific time t = t0 labels all events in space that occur simultaneously. All events that occursimultaneously define a 3-dimensional space, a slice through 4-dimensional space-time. Before Ein-stein and his relativity theory, the notion ofsimultaneity of two events was absolute; all observersagreed on whether two events occurred simultaneously or not. From the Lorentz transformations(12) it follows that two events that were simultaneous in one frame, say (t0, x1) and (t0, x2), wherewe neglected the y and z coordinate, are not simultaneous in the primed frame of reference whichis moving relative to the unprimed reference frame. Clearly then, the Lorentz transformation isreferring to a structure of space and time where simultaneity is an observer dependent concept!

Einstein concluded that simultaneity, and therefore time itself, as an observer independentconcept had to be given up. All observers experience their own time, and equipping them withclocks the Lorentz transformation tells us how to transform from one clock to another. Thefact that simultaneity is no longer absolute also has important consequences for the measuringof lengths or distances, which is done using predefined rulers (meter sticks) at some fixed time,i.e. the distance between the simultaneous events (t0, x1) and (t0, x2), with x2 > x1, is of course(x2 x1). However, in the primed reference frame these events are not simultaneous and cantherefore not be used to measure distance. Using the Lorentz transformation, fixing t1 = t

2 to

force simultaneity in the primed reference frame, one can derive the following expression for themeasured (simultaneous) distance in the primed reference frame

L (x2 x1) =

1 u2/c2 L . (12)

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In other words, the distance is smaller in the primed reference, a famous result known as lengthcontraction. All lengths and distances are smaller for observers that move with respect to theobjects or events for which the lengths and distances were defined at rest.

Another famous and experimentally well tested consequence of special relativity is that oftime dilation. Moving clocks run slower! Again, this can be easily derived from the Lorentztransformations. Fix the time between two events in a rest frame, i.e. consider two events at thesame location x1, (t1, x1) and (t2, x1) and t2 > t1, for which the amount of elapsed time betweenthe two events is then obviously t = t2 t1. Now perform the Lorentz transformation on thesetwo events and calculate t t2 t1. The result is

t =1

1 u2/c2 t , (13)

implying that a moving clock runs slower. Be careful: in the primed frame of reference the clock ismoving and registering t, which is smaller as compared to t, so one concludes the moving clockhas registered less time! This is perhaps most obvious by considering the example of a light-clockat rest as compared to a light-clock moving at some velocity u. The light-clock consists of two

mirrors bouncing back and forth light rays vertically and one tick of the clock is defined as a light-ray going up and down once. Now give the light-clock a horizontal velocity (or better, considera moving observer with a velocity u). Using the fact that the speed of light is constant, but thedistances between the mirrors has increased because the light-clock is moving one concludes themoving clock is ticking slower as compared to the (same) light-clock at rest.

It is important to realize that the relativistic effects of length contraction and time dilation arenot due to some special (internal) features of the clocks or meter sticks considered. All lengths areshortened and all processes are slowed down from the point of view of a moving observer. If thatwere not the case, an observer at rest relative to the clocks and meter sticks under consideration,would be able to figure out that he or she is moving by comparing to other (unaffected) metersticks and clocks, which should not be possible according to the principle of relativity; velocity

is a relative concept and all observers have every right and reason to consider themselves atrest. Another important thing to stress is that these relativistic effects are tiny in every daycircumstances, they are governed by the ratio u/c, so only when the velocities under considerationreach a significant fraction of the speed of light will these effects become apparent. Nevertheless,they tell us something truly fundamental about the structure of space and time.

The theory of special relativity tells us that time and space intervals depend on the state ofthe observer. Different observers can relate their time and space measurements to each other byperforming a Lorentzian coordinate transformation. To obtain more insight into the structure ofspace-time implied by the Lorentz transformations note that the Lorentz transformation keeps aparticular combination of coordinates invariant

c2

t2

x2

= ct2

x2

. (14)

This means one can define an invariant space-time distance or proper time 2 between twoevents

c22 c2(t)2 (x)2 . (15)This can also be written infinitesimally, considering the limit of smaller and smaller time andspace intervals, i.e. c2d2 = c2dt2 dx2. In the classical Galilean or Newtonian universeof absolute space and time t and x, time and length, were both separately invariant underGalilean coordinate transformations that connect two frames of reference. In special relativity theonly invariant distance is this particular combination of time and space intervals, also known asproper time, since it reduces to the elapsed coordinate time t in the frame of reference where two

events are only separated in time (x = 0). From this expression of the invariant proper time onecan immediately read off the formula for the time dilation effect (13) by extracting a factor c2(t)2

and realizing that u xt . Note that 2 should then be interpreted as the time elapsed on the

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clock in its rest frame, and t > as the time elapsed in the moving frame (so the moving clockis running slower).

Clearly, the expression for the proper time is not positive definite. We say that two eventsare timelike separated if 2 > 0, spacelike separated if 2 < 0 and lightlike separated if2 = 0. Since the proper time is an invariant quantity under Lorentz transformations thesenotions are invariant statements relating the two events, so two events that are timelike separatedin one frame will be timelike separated in any frame of reference. Two timelike separated eventscan be connected by a worldline, meaning an observer or object always moving at a speed slowerthan the speed of light. Two lighttlke separated events can only be connected by a lightray andtwo spacelike separated events cannot be connected by a lightray or an observer or object movingslower than the speed of light, such events can not be causally connected, meaning that noinformation can be exchanged between them because it would require going faster than the speedlight.

We are lead then to a structure of space-time that is very different from what Newton and Galileihad discovered. Because Newtons law is inconsistent with the Lorentz transformation Einsteinunderstood that Newtons law had to be modified. He concluded the same about Newtons law ofgravitation. How should Newtons second law be modified to be consistent with the principle ofrelativity and the constancy of the speed of light?

To make an educated guess, we start by noting that the structure of space-time defined by theLorentz transformation, with the Lorentz invariant proper time denoting a sum of squared timeand space intervals, suggests a mathematical formulation in terms of so-called four-vectors x =(ct,xi), where the greek index runs from 0 to 3 to distinguish it from ordinary (space) vectors xi,where the latin index runs from 1 to 3. Rotations in three space dimensions, which transform thedifferent axes into each other, do of course not affect the distance between two points, which canmathematically be expressed by noting that the distance d2 = x2 + y2 + z2 =

i,j x

ixjij isinvariant under rotations, where we introduced the 3 by 3 unit matrix ij

ij = 1 0 00 1 0

0 0 1

. (16)

The idea is that a Lorentz transformation can be thought of as a rotation in 4-dimensionalspace-time, since it mixes space and time coordinates. To be concrete, in this mathematicalnotation the Lorentz transformation (12) can be written as

x =

x , (17)

where the matrix equals

=

11u2/c2 (u/c) 11u2/c2 0 0(u/c) 1

1u2/c21

1u2/c20 0

0 0 1 00 0 0 1

. (18)

With the indices suppressed this simply reads x = x, where it is understood that x and x are4-vectors and is a 4 by 4 matrix. The crucial ingredient in this mathematical notation is thatthe invariant distance should be the proper time c22 = c2t2 x2, which can be written as

xx, where we introduced the 4 by 4 matrix , equal to

=

1 0 0 0

0 1 0 00 0 1 00 0 0 1

. (19)

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The important difference with the 3 by 3 unit matrix ij is the appearance of 1 entries in allthe diagonal space components, as required to reproduce the invariant proper time. The 4 by 4matrix essentially carries all information about the structure of 4-dimensional space-time, andis known as the Minkowski metric.

If this was all a bit too formal, remember that the key point is that in special relativity thenatural objects appearing are four-vectors x, where the first component corresponds to time(which was considered separately in in the Newtonian/Galilean framework), which is intrinsicallytied to space through the Lorentz transformations. Although not absolutely necessary, we will usethis mathematical observation to make an inspired guess to what the relativistic generalization ofNewtons law should be. It will also be useful when introducing the foundations of the generaltheory of relativity, which relies on the same mathematical machinery.

2.3 Relativistic momentum and energy

Since we started with Newtons law and noted that it was invariant under Galilean coordinatetransformations, it should be clear that Newtons laws should be modified in special relativity,where instead the laws of motion should be invariant under Lorentz transformations that mix up

space and time. How should Newtons law be modified to achieve this? There are several ways toproceed, but the fastest is to make an informed guess based on the concept of momentum and themathematical machinery of 4-vectors alluded to at then end of the previous section.

To arrive at the relativistic generalization of Newtons law the informed guess is that the 3-vector of momentum should be replaced by a 4-vector of momentum, so p or pi generalizes to p,with running from 0 to 3. This also implies that the 3-vector of force F should be generalized toa 4-vector of force F. So the proposal is that the relativistic version of Newtons law should looksomething like F = ddt p

. This is however not the final answer because the time derivative is notinvariant under a Lorentz transformation. To complete the generalization we therefore introducethe time derivative with respect to the invariant proper time that was introduced In the previoussection (15), giving

F = dd

p , (20)

and to no longer put you in suspense; this is indeed the correct relativistic generalization ofNewtons law. All inertial observers, whose coordinates are connected by Lorentz transformations,will now agree on this form of the law of nature connecting forces to changes in motion. For thoseof you who wonder about how to transform this equation from one frame to another: just act withthe Lorentz transformation matrix on the 4-vectors on both sides of the equation (F and p).So both sides of the equation transform in the same way, meaning the form of the equation isindependent of the reference frame under consideration.

Of course, to connect the relativistic expression above to the starting point F = m a, we stillneed an expression connecting the generalized 4-vector of momentum to the 3-vector of momentum

p = mv. Again, lets make an informed guess based on the mathematical machinery of 4-vectors,the natural variables in a relativistic description. The proposal is

p = mdx

d, (21)

where is again the invariant proper time and x the coordinate 4-vector (ct,xi), with i runningfrom 1 to 3. By construction a Lorentz transformation acts in the same way on both sides, implyingthe relation is invariant under Lorentz transformations, as required.

If you have not followed all the details of the construction outlined above, dont worry. Inthat case you can just take the above expression for p as given and work out the consequences.For instance, starting from (21), one would like to know the relativistic expression for the spatial

components p, i.e. the pis in p = (p0, pi), with i running from 1 to 3. Using that v = dxdt , from(15) we find that

d2 = dt2

1 v2/c2 (22)11


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and one arrives at

p =mv

1 v2/c2 . (23)

From this one concludes that momentum is no longer linear in velocity v, as it was in the Newtonianframework p = mv. This makes perfect sense, because if it was linear in velocity a continuouslyacting force would at some point lead to velocities greater than the speed of light, which shouldnot be allowed in a relativistic theory.

By generalizing Newtons law to a 4-vector equation we have added a fourth component, whichis also conserved in addition to the ordinary conservation of the 3-vector of momentum p, i.e.

ddp

= 0 in the absence of forces. What could that fourth component be? From the definition of4-momentum (21), using that x = (ct,xi), we find that

p0 =mc

1 v2/c2 . (24)

Now to get an idea of the non-relativistic meaning of this expression, expand it in ( v/c)2, which isindeed small in everyday circumstances. Since 1

1v2/c2 1 + 12 v2/c2 + O(v4/c4) we obtain

p0 mc

1 +1

2v2/c2

= 1/c

mc2 +

1

2mv2

. (25)

Multiplying with c we recognize the usual kinetic energy 12 mv2 as one of the contributions, so we

are lead to the conclusion that the fourth component of the relativistic 4-vector of momentumis just the energy! Whereas in the non-relativistic Newtonian description momentum and energywere separately conserved in the absence of external forces, in the relativistic theory energyand momentum combine into a single (4-vector) object that is conserved. Just like time andspace are transformed into each other under a Lorentz transformation, entangled in the unifiedconcept of space-time, energy and momentum mix when transforming to a different inertial frameof reference. Importantly an additional term has appeared in (25), which should represent the zerovelocity limit of the energy, equal to E = mc2, apparently stating that mass is equivalent toenergy! Undoing the small velocity expansion, defining p0 E/c, we obtain the expression forthe relativistic energy

E p0c = mc2

1 v2/c2 . (26)

Finally, in the previous section it was noted that one could construct an invariant distance, theproper time, from the 4-vector x = (ct,xi), using the 4 by 4 matrix . Similarly, starting fromthe 4-vector of momentum p = (E/c,pi) one can construct an object which is invariant underLorentz transformations in exactly the same way. Calculating the object pp one finds thatthe combination

E

2

c2 p2 , (27)should be the same in any (inertial) reference frame. Plugging in the explicit expressions for themomentum p (23) and E (26) allows one to explicitly calculate this invariant quantity, which isequal to

E2

c2p2 = m2c2 . (28)

In words this says that the total energy squared minus the total momentum squared always equalsthe mass squared. Since in the rest-frame the total energy is given by E0 = mc

2, after multiplyingby c2 this can also be written as

E2 p2c2 = E20 . (29)One particularly interesting consequence of this equation (that we will use later on) is that formassless objects or particles the momentum apparently equals their energy, i.e. E =|p|c.

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This ends our introduction to special relativity, which is of course extremely brief, but it shouldbe just enough so you can hopefully appreciate a first glimpse on the theory of General Relativity,before we move on to the cosmological implications and applications of relativity.

2.4 Relativistic Gravity

Based on the observed fact that the speed of light is independent of the particular frame ofreference, combined with the principle of relativity, i.e. no (inertial) observer should be preferredor be special, Einstein concluded that Newtons law of inertia F = ddt p had to be modified. Thiswas relatively easy as compared to the next goal Einstein set for himself. Just like Newtons lawof inertia, Newtons theory of gravity did not obey the principles of special relativity and thereforeshould be modified as well. An obvious indication why Newtonian gravity can not be correct isthat it acts instantaneously. Two gravitating bodies act upon each other without any time delay,signaling that the force of gravity is communicated with an infinite velocity. This is not allowedin relativity where the speed of light is the maximum velocity at which anything, information orforces, can be exchanged. Indeed, in Maxwells theory of electromagnetism the electromagneticforce is communicated at the speed of light.

As compared to Newtons law of inertia, which was relatively easy to transform into a fullyrelativistic version, it took Einstein almost 10 years to finalize the relativistic theory of gravity.A real tour de force, but the result was absolutely stunning and until this day is consideredone of the most beautiful theories in physics. The key ingredient that set Einstein on the right(but long) track was the remarkable equivalence between inertial and gravitational mass. Theforce of gravity Fg =

GMr2

mg is proportional to the gravitational mass mg of a particular object,

whereas the opposition to a change in velocity as described by Newtons law of inertia F = mi a,is proportional to the inertial mass mi. A priori, there is no reason why these two types of massesshould be proportional or equal to each other, but nevertheless they are: mi mg. This has theimportant consequence that all objects, no matter how massive, accelerate in the same way in agravitational field, a fact that can still be very surprising to some people. For centuries this feature

was considered more or less an accident, something without an explanation. But Einstein realizedit revealed something fundamental about how gravity works.

The equivalence of inertial and gravitational mass implies that all masses behave in exactly thesame way in an external gravitational field, in contrast to for instance electromagnetism where thepresence of positive and negative charges that have no relationship to the inertial mass of an objectleads to very different behavior. For gravity it follows that a collection of different objects andobservers, all in free-fall, would have no means to detect a gravitational field in their immediateneighborhood: they are all falling in exactly the same way, no relative accelerations exist thatcould signal an external gravitational force. This also explains why you dont feel gravity infree-fall, all the different component out of which your body is made are accelerating in exactlythe same way. Relative accelerations, which would make you feel gravity, are absent. At the same

time this means that when you are not in free-fall, when you do of course feel gravity, the effect isindistinguishable from acceleration. Einstein promoted this feature of gravity to a principle, theequivalence principle, saying that the effect of a gravitational field can not be distinguished fromacceleration. Acceleration and gravity are equivalent notions. The famous (graphical) exampleusually given is that of an observer in an elevator, where the elevator serves to isolate the observerso he or she can only make observations or do experiments in the elevator. There is not a singleexperiment the observer locked up in the elevator can do to distinguish between a situation wherethe elevator is put on the surface of the Earth, i.e. in a gravitational field, and a situation wherethe elevator is accelerating upwards in space by means of an external force (say a rocket).

So instead of saying we feel gravity because there is a gravitational force acting on our body,according to the equivalence principle one would conclude that the reason we feel gravity isbecause we are simply not in a free-fall frame of reference. Now compare this to the so-calledfiducial or pseudo-forces that were briefly alluded to in section 2.1, which arise when one is

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not using an inertial frame of reference and are directly connected to the acceleration of the frameof reference. This suggests that gravity as we know it should also be considered a pseudo-force,only arising when an observer is not in a free-fall frame of reference. Indeed, since pseudo-forcesarise due to observers in non-inertial, accelerating, frames of reference they are always proportionalto the inertial mass of the observer, just like the force of gravity that we experience when we areprevented from freely falling. Conversely, in free-fall no measurement can be done in the immediateneighborhood that would establish the presence of a gravitational field.

Proposing that the equivalence principle, i.e. the equivalence between a gravitational force andacceleration, is a fundamental ingredient of gravity immediately has profound consequences. Inparticular, ever since Newton, it was unclear what was the effect of gravity on light, whichas far as one could tell were not particles of a specific mass, so one typically thought gravity didnot influence light. In one of the exercises you are asked to show that the equivalence principleimplies that light will be deflected by the gravitational force. Another consequence of theequivalence principle is that light is not only deflected by gravity but also red- or blue-shifteddepending on the the direction a lightray is traveling (up or down the gravitational potential).This gravitational time-delay effect has been experimentally verified and needs to be takeninto account when determining accurate positions on Earth using GPS satellites.

Even though the equivalence principle was an important breakthrough towards constructing arelativistic theory of gravity, some additional conceptual and technical insights were necessary. Themost important was the notion of curved space-time geometry. According to the equivalenceprinciple the gravitational generalization of the inertial (force-less) frames of special relativityare the free-fall frames of reference, where all forces are again absent. Although this assuresthat locally, meaning in the immediate neighborhood, gravitational forces can not be detected,a gravitational field that is changing from one location to another will cause small tidal effectsthat can be ascertained by the deformations on nearby freely falling trajectories. Think of a (very)large elevator with different objects inside falling towards the (center of the) Earth. The free-fallingtrajectories of the different objects in the elevator will ever so slightly move towards each other,allowing careful observers to find out whether a (position dependent) gravitational field is present.So variations in a gravitational field can be detected through tidal effects, as opposed to thegravitational field itself.

Variations in a gravitational field can cause freely falling trajectories to converge or diverge in aspace-time diagram, instead of forever remaining parallel at a fixed distance. These two features,that a gravitational field is equivalent to acceleration and that variations in a gravitational fieldlead to tidal effects producing universal converging or diverging behavior of free-fall trajectories,convinced Einstein that relativistic gravity should be understood in terms of curved space-time.The notion of a free-fall trajectory in curved space-time should be compared to the notion ofthe shortest distance between two points on curved manifolds like the two-dimensional sphere andsaddle. On a sphere two initially parallel lines of shortest distance starting at two nearby points andmoving in the same direction will intersect each other at some point (convince yourself by drawinga sphere, or getting your hands on a globe). On a two-dimensional saddle the opposite happens,two initially parallel trajectories will diverge away from each other. Only on a flat two-dimensionalsurface will two initially parallel lines remain parallel forever.

According to Einstein then, the converging or diverging nature of nearby free-fall trajectoriesshould be a consequence of the curvature of space-time! Realize that this interpretation wouldbe impossible without the equivalence principle. The equivalence principle assures that the effectsof gravity only manifest themselves in terms of variations of the gravitational field and are thesame for all objects, independent of any particular properties of the object under consideration.These features open up the possibility for a description of gravity in terms of space-time geometry,whose effects should indeed be universal. This situation is very different from electromagnetism,where the resulting accelerating motions depend on the charges of the objects under consideration.

Space-time geometry can be curved, the curvature of space-time being related to the gravita-tional field, which is caused by gravitational sources, i.e. matter, which because of E = mc2 is now

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equivalent to (any kind of) energy. To get a little bit of intuition on the curvature of space-time,let us return to the remarkable fact that clocks run differently at different heights in a gravitationalpotential. Consider two observers at different heights in a gravitational potential. Their clockswill run differently, and the curved nature of space-time can be visualized by connecting the twoobservers by a line (or a meter stick equal to their hight difference) after the same time has elapsedon either clock in a space-time diagram. Since the clocks run differently, the resulting shape is nota perfect square, it is skew, representing the curved nature of space-time. For those interested,the variable measuring lengths in a curved space-time is the metric g(x

), a fourbyfour matrixwith entrances that can depend on the coordinates x, generalizing the Minkowski metric that we encountered in the theory of special relativity corresponding to flat space-time. Thegeneralized invariant distance, or proper time, then equals

c2d2 =,

gdxdx . (30)

After many years of technical struggles, trying to learn, apply and generalize the mathematicsof curved spaces, Einstein finally succeeded to construct a relativistic theory of gravity. In thistheory the effects of gravity are completely described by the properties of space and time, i.e.their curvedness. Space and time are dynamical objects that can bend and turn, in response tothe presence of mass-energy, and at the same time instructs test-objects how to move, i.e. whatthe freely falling frames of reference are. So the Earth is not moving around the Sun because aNewtonian force is acting on it, but because the Sun is deforming space and time which causes theEarth to move around the Sun in the straightest (free-fall) line possible, somewhat analogous to alight glass marble on a rubber sheet deformed (dented) by a heavy lead marble in the middle.

Clearly the general theory of relativity, as Einstein called his relativistic version of gravity, hasimportant consequences for cosmology. With the help of the equations governing the evolutionof space-time, making some assumptions about the distribution of matter and energy, one can inprinciple calculate to see what the universe was like in the past or will be in the future. Fortunately,the full machinery of general relativity is not actually required to find the relevant cosmologicalequations. Nevertheless it is important to stress that the only sensible interpretation of the resultsthat will be derived is in terms of the evolution of curved space and time.

2.5 Big Bang Cosmology

With the promotion of space and time to dynamical entities, cosmology, i.e. the evolution ofthe universe as a whole, became a part of physics. The equations of general relativity, couplingthe distribution of energy-matter to the geometry and evolution of space-time, are non-linear andtherefore difficult to solve in general. One needs to make some assumptions on the distribution ofenergy-matter, before being able to find cosmological solutions. A first guess is to assume that thedistribution of matter and energy on the largest scales in the universe is approximately uniform, so

the universe looks the same no matter where you are. To be more precise, one says that the largescale universe should be homogeneous, or the samein every location, and isotropic, meaningthe same in every direction, in a leading approximation. This assumption was actually promotedto a principle: the cosmological principle.

First one should realize that the homogeneity and isotropy do not imply one another. Thereexist perfectly good examples of homogeneous distributions of energy or matter that are notisotropic, like for instance a homogeneous distribution of matter on a two-dimensional cylinder,where one spatial direction is infinite and the other is a circle. Similarly one can consider isotropicdistributions that are not homogeneous, like the distribution of matter around a center where thedensity of matter drops as a function of radius. However, if a distribution is isotropic around everypoint then it does imply homogeneity as well. Intuitively, the cosmological principle simply states

that there are no special locations or directions in the universe, generalizing the Copernicusprinciple that states we are not in a special place, like the center, of the solar system. Althoughinitially this was nothing more than an unsubstantiated guess, nowadays observations of the cosmic

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microwave background and the large scale distribution of (clusters of) galaxies have confirmed thatthe cosmological principle is an excellent first approximation at distance scales starting around 100million lightyears ( 1024 meter).

The cosmological principle immediately has consequences for the possible geometry of space-time on cosmological scales. Since no location in space is special, the distribution of energy-matterand correspondingly, according to the general theory of relativity, the geometry of space cannotdepend on the spatial coordinates xi. It could however depend on the time coordinate, so the mostgeneral way to parametrize distances in a universe in line with the cosmological principle is

l2 = a(t)22k , (31)

where 2k describes how to measure distances on the different homogeneous and isotropic curvedthree-dimensional spaces that one can write down. The cosmological principle implies that the cur-vature of space, if any, has to be a constant, i.e. the same everywhere in space. Only three differentconstant curvature spaces exist: the three-dimensional generalization of the sphere has constantpositive curvature (for convenience normalized to k = +1), the three-dimensional generalizationof the saddle has negative curvature (normalized to k = 1) and in between one finds flat space(k = 0), for which 0 simply becomes equal to the usual x2 + y2 + z2. The time-dependentoverall factor is known as the scale-factor, translating time-independent distances as defined bythe curved space metric 2k to physical distances in the dynamical universe. Translating thisspatial structure into the space-time measure of relativistic proper time distances one obtains theFriedmann, Robertson, Walker (FRW) metric of space-time

c22 = c2t2 a(t)22k , (32)

corresponding to the cosmological generalization of the Minkowski metric . The FRW metricserves as the starting point for doing cosmology. Using it as an ansatz, plugging it in the equationsof general relativity, one can try to find cosmological solutions for the variable of interest, the scalefactor a(t). But before doing that, one can already derive some important consequences of thisparticular structure of space-time.

First of all, note that the assumption of a homogeneous and isotropic universe excludes thepresence of a center, or any other type of special placein the universe. Anticipating the Big Bangmodel, which relies on the cosmological principle, it should be clear that the Big Bang, definingthe beginning of time, should not be thought of as occurring at a special point in space. TheBig Bang was a time, not a place! It happened everywhere, in line with the cosmologicalprinciple. Moreover, another very general conclusion follows by considering test-objects, think ofgalaxies, at fixed locations on the three-dimensional constant curvature space. Let us call thefixed distance between any two test-objects x. The coordinates xi on that space are usuallycalled co-moving coordinates to emphasize the fact that test-objects at fixed locations in these

coordinates correspond to time-dependentphysical

locations, due to the dynamical nature of spaceas expressed by the scale factor a(t) that relates the two coordinate sets. The physical distance is ofcourse r(t) = a(t)x and from this one can derive that the velocity v drdt = dadt x = 1a dadt a(t)x.By identifying the Hubble parameter H 1a dadt this can be summarized as Hubbles law

v = H r , (33)

stating that galaxies further apart move away from each other at a higher velocity, as long as dadt > 0.It was Edwin Hubble who in 1929 found the first evidence for this linear relationship betweenvelocity and distance, indicating that our universe is homogeneous, isotropic and expanding! Theparameter H indicates the rate of expansion (or contraction) and in general changes over time. Forrelatively nearby galaxies it is a good approximation to consider it equal to the Hubble parameter

now H0, which is usually called the Hubble constant in a somewhat confusing terminology.To determine actual solutions for the scale factor a(t) that tell us exactly how the universe has

been, and will be, evolving in time one needs another ingredient: how does the energy distribution

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behave as a function of the scale factor a(t)? What is needed is an equation governing the evolutionof the mass-energy distribution. How would one model the behavior of typical matter and energycomponents in the universe, assuming the spatial distribution is homogeneous and isotropic, asdemanded by the cosmological principle? One type of matter is easily described: consider aconstant density of non-interacting massive particles, with negligible average kinetic energy, alsoknown as dust. How will the density depend on time? Obviously the volume is proportional toa(t)3, so the mass- or energy-density will scale as 1/a(t)3. However, in a relativistic set-up oneshould be a bit more careful in general. In particular consider a gas of photons with a constantdensity, or any other massless particles for that matter. From the relativistic expression for energyand momentum (29) one concludes that |p| = E/c, so in this case the magnitude of the momentumdensity is proportional to the energy-density. The presence of non-negligible momentum for theindividual photons necessarily implies the presence of a pressure component, just like the pressureof a gas is due to the collective collisions, or exchange of momentum, of particles with a surface.Assuming isotropy the average size of this pressure should be equally divided between the 3 spatialdirections, giving

P=

pc

V3

=

1

3 , (34)

where is the energy density = EV3 . This is indeed well-known as the equation of state forradiation. As we will see the presence of this pressure leads to an additional factor of a(t) forthe behavior of the energy density of radiation, or any gas of approximately massless particles,which can also be understood in terms of the additional redshift of the energy giving 1/a(t)4.Even for massive particles, in the limit that the average magnitude of momentum is a lot higherthan the energy represented by the rest-mass of the particle, which can for instance be achievedby considering very high temperatures, the energy distribution will effectively behave as radiationand scale as 1/a(t)4 instead of 1/a(t)3.

In general the behavior of matter can be described by considering a homogeneous fluid with adensity and a pressure P. The equation governing the evolution of can be derived by applyingthe first law of thermodynamics, which we will discuss in more detail in the next lecture and

just mention hereE = PV + TS , (35)

that says that any change in the internal energy of a system at a temperature T in a volume V3 isgiven by the sum of the work done by the fluid(PV) and the amount of added heat (TS). Inthe second term on the right the temperature T and the entropy S, a measure of disorder, appear.For our purposes we will only be considering reversible processes where no energy is lost or gainedin the form of heat Q = TS, which would require a temperature difference to generate theflow of heat. This implies that the entropy difference S vanishes and this term can be neglectedfrom now on. As a consequence any change in the internal energy of the fluid must be due to thework done by the fluid

PV. Now write E = V3 and take the time derivative on both sides

of the equation, taking into consideration that the three-volume V3 is proportional to a(t)3 in adynamical universe. Collecting all the different terms results in the continuity equation

d

dt+ 3

1

a

da

dt( + P) = 0 , (36)

governing the behavior of the energy density of a general fluid in an evolving homogeneous andisotropic space-time. Using this equation one can check that for dust, i.e. ordinary matter withouta pressure component, this indeed leads to a density proportional to 1 /a3, as it should.

The next step is to derive the actual equation governing the evolution of the scale factor a(t),which requires the details of general relativity and is therefore beyond our scope. The final result

is called the Friedmann equation

H2 =8G

3 k c

2

a(t)2, (37)

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where G is the gravitational constant measured to equal (roughly) G = 6.7 1011 m3 kg1 s2and k is the curvature parameter that was introduced earlier to distinguish the three differenthomogeneous and isotropic spaces (k = +1, 1, 0). Before discussing solutions of a(t), let usconsider a derivation of the same equation using just Newtonian gravity! A note of caution:even though the actual equation can be derived from strictly Newtonian physics, this is a luckycoincidence having to do with the symmetry in the problem as a consequence of the cosmologicalprinciple. In no way does it imply that Newtonian gravity is on the same footing as generalrelativity and the only proper interpretation of the equation, after it has been derived, is interms of dynamical curved space-time. Nevertheless, it is an instructive exercise applying somemechanical principles that were introduced in the first lecture.

The setup is the following: consider an infinite homogeneous and isotropic distribution of matterwith a density . Pick a center, any point will do because as a consequence of the cosmologicalprinciple it should not matter. From this center draw a sphere of radius r to a thin shell with atest-object on it with mass m. What is the gravitational force on this particle m? An importantproperty of Newtonian gravity that we mentioned briefly was that only the mass in the internalsphere will contribute to the force, all contributions from outside the sphere will cancel. The totalmass inside the sphere equals M =

V3 =

4

3

r3 . The total energy of the particle is the sumof kinetic and potential energy due to Newtonian gravity (which is negative), to be precise

E =1

2m

dr

dt

2 GM m

r(t) 1

2m(kc2)x2 . (38)

The total amount of energy should be conserved, so we can define it to be some time-independentconstant, and for reasons that will become clear later on we pick the combination 12 m(kc2)x2,where k is some constant, c is the speed of light and x2 is the co-moving coordinate distance ofthe particle m. Remember that r(t) can be written as a(t) x as a consequence of the cosmologicalprinciple. We calculated the total mass M already, so plugging that in (38) and canceling thesmall mass m on both sides we obtain

dr

dt

2 8

3 G r(t)2 = (kc2)x2 . (39)

Using that r(t) = a(t)x, we see that all dependence on the co-moving coordinate x drops out in(39), as it should because the evolution should not depend on the co-moving distance x. So weobtain

da

dt

2 8

3 G a(t)2 = (kc2) , (40)

which after identifying H 1a dadt can be rewritten as the already familiar Friedmann equation

H2 =8G

3 k c2

a(t)2 . (41)

Again, the fact that this can be derived using only Newtonian physics is a coincidence and thefinal equation should really be interpreted in terms of general relativity where the constant k isrelated to the curvature of space.

In the Newtonian derivation k is related to the total energy Etot 12 m(kc2)x2 , which inaddition also had to depend on the co-moving distance x2 to guarantee consistency with thecosmological principle. This allows us to reach some interesting conclusions on the evolution ofthe universe in the three cases of interest. In the Newtonian framework positive total energy,corresponding to k < 0, implies that the negative gravitational potential energy never wins overthe positive kinetic energy contribution, which means the universe will expand forever at some

positive limiting velocity. In the opposite case, when the total energy is negative, correspondingto k > 0, something more interesting happens, because the gravitational potential energy willdominate at some point, turning around initial expansion into contraction. So the universe will

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reach a maximum size and then collapse. Finally, when the total energy vanishes (k = 0), thegravitational potential energy and the kinetic energy cancel each other exactly and the universewill expand forever with a vanishing limiting velocity.

In Einsteins general relativity theory these three different cases instead correspond to thethree constant curvature spaces that can be considered, but the conclusions on the dynamics ofthe universe remains the same. So a compact, closed, spherical universe will expand to a maximumsize and then contract to collapse at some point. An infinite saddle-like universe will expand foreverand reach a finite limiting expansion rate, whereas a flat universe will also expand forever but witha vanishing expansion rate in the infinite future.

Before deriving some actual solutions to the Friedmann equation, the important role of pressurecan be made clear by cleverly combining the Friedmann (37) and the continuity equation (36). We

would like to derive an equation for the acceleration of the expansion 1ad2adt2

. Acting on both sidesof the Friedmann equation with a time derivative we get

2HdH

dt=

8G

3

d

dt+ 2H

kc2

a2. (42)

Now use the continuity equation, replacingd

dt , and at the same time the fact thatdH

dt =1

a

d2a

dt2 H2

.Dropping an overall factor of H this results in

21

a

d2a

dt2 2H2 = 8G ( + P) + 2 kc

2

a2. (43)

Using the Friedmann equation (37) one more time to replace H2, one ends up with the final resultfor the acceleration equation

1

a

d2a

dt2= 4G

3( + 3P) , (44)

where the curvature k dependence has dropped out. From this we can draw two important conclu-sions. First, for ordinary positive densities and pressures the expansion of the universe is always

slowing down. Secondly, any positive pressure component adds to the deceleration of the expan-sion of the universe. Under the assumption that negative pressure, as opposed to negative density,can be physically realized, the only way to obtain a universe whose expansion is accelerating is byhaving the negative pressure component satisfy P< (1/3) .

So in what kind of universe are we living? In particular, is our universe positively curved,negatively curved or flat? For a long time it was hard to answer this question because a directobservational probe of the large scale spatial geometry did not exist. Instead cosmologists andastronomers tried to get a handle on the total density of matter in the universe by in essencecounting the visible masses, like stars, galaxies, gas clouds etcetera. This would be tied to thegeometry if this visible mass would correspond to all matter and energy existing in the universe.This is clear from the Friedmann equation by re-ordering it in the following way

k c2

a(t)2H2=

8G

3H2

1 . (45)

Obviously, when the density of matter or energy is smaller than some critical density definedas crit 3H28G , the curvature of space-time has to be negative, so that would mean saddle-shaped.Oppositely, when the density is bigger than the critical amount, the curvature is necessarily positiveimplying a spherical universe. Only when the density is exactly equal to the critical amount willthe universe be flat, i.e. k = 0. Note that the sign of k is fixed, meaning that the shape of theuniverse can be determined at any time, in particular at the current age of the universe. Becausethe current Hubble parameter H0 has been measured we can define the current critical density as

0crit

3H20

8G and we can ask how the visible mass density compares to this number to determine

the spatial curvature of the universe.In fact the density of visible matter is not that far off from the critical density, but it does

seem to be smaller. More accurate measurements and estimates of the density of visible matter

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have confirmed this: the total matter density of the stuff we are made off, including photons,only corresponds to about 4 percent of the current critical density. So one would be tempted toconclude that we are living in a saddle-like universe that will expand forever, and that was indeedthe general expectation until about 15 years ago.

The Cosmic Microwave Background that will be discussed later can actually be used asa direct probe of the geometry of the universe. The most recent result on the current curvature ofthe universe, as measured by the parameter k k c2a(t)2H2

0

equals

k = 0.006 0.007 , (46)

where the error corresponds to one sigma, so an 68 percent confidence level. In other words: theuniverse is flat to incredible precision! This is even more surprising when one realizes that flatsolutions of the Friedmann equation are typically unstable, implying that the slightest departurefrom a flat universe will evolve exponentially fast into a manifestly spherical or saddle-like universe.Correspondingly, if the universe is flat to high precision now, it had to have been incredibly flatin the early universe! At first sight this therefore looks like an incredible fine-tuning of the initialconditions. The density in the early universe had to have been exactly equal to the critical density.

Who ordered that? Moreover, the visible part is apparently only a small fraction of the criticaldensity, so what is it that makes up for the rest of our critical density universe?

Leaving these important questions behind for the moment, we will now look for solutionsa(t) under the assumption the universe is flat (k = 0), allowing us to drop the last term on theright in (37). The easiest way to find solutions is to assume a dominant form of matter (dust orradiation) and make an informed ansatz for the behavior of the scale factor as a function of time.Let us assume the scale factor is a power-law of time, i.e. a(t) tp. Plugging this ansatz intothe Friedmann equation one can derive the following solutions for pressureless matter (dust) andradiation

P= 0, a(t)3: p = 2/3, H(t) = 23t . P= 13 , a(t)4: p = 1/2, H(t) = 12t .

In both cases the Hubble parameter behaves as 1/t, but as compared to the first estimate of the ageof the universe (t0 = 1/H0) both results imply a younger universe equal to respectively t0 =

23H0

(matter) and t0 =1

2H0(radiation). Of course, in our universe both radiation and matter should

play a role. Under the assumption that matter was dominant for the larger part of the existenceof the universe (radiation was dominant for a brief while in the very early universe) this wouldmake the age of the universe closer to 9 billion years, which is in conflict with observations of theoldest stars. What is the resolution?

About a decade ago two teams of astronomers that had been searching for supernovas an-nounced a stunning result. The collected data on a particular type of supernova for which the

absolute brightness was known (allowing these objects to be used as standard candles) all seemedto be a bit too faint in relation to their distance as determined by redshift. The only explanationfor the data was that the universe is expanding in an accelerating fashion! This was a shock-ing conclusion, because from the acceleration equation (44) we concluded that ordinary matteror radiation can never give rise to accelerating expansion. Instead, we found that only negativepressure fluids with w < 1/3 can give rise to acceleration. The prototype case of the latter beingthe cosmological constant , corresponding to an energy density that does not decrease as theuniverse expands, necessarily implying a negative equation of state parameter w = 1 as can bereadily derived from the continuity equation (36). The latest observations, combining differentdata sets, including the Cosmic Microwave Background, conclude that a cosmological constantequal to approximately 75 percent of the current critical density can explain the data. This also

solves the age problem. In the presence of a cosmological constant that accelerates the expan-sion the age of the universe is increased, putting it back in the right ballpark of roughly 14 billionyears. If one instead relaxes the assumption of a perfect cosmological constant ( w = 1) in favor

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of a more general negative pressure fluid w < 1/3, then one typically refers to dark energy todistinguish this contribution from (presureless) dark matter.

So currently the dominant contribution to the critical energy density is provided by the cos-mological constant (or slightly more general dark energy), followed by a significant contributionof dark matter that is needed to explain galaxy rotation curves and the formation of structures inthe universe. Only about 4 percent of the critical density corresponds to the visible matter thatwe are made off and about which we know a lot. The other 96 percent is really mysterious andcan only be distinguished due to the different effects they have on the evolution of the universeand the (matter) structures that are part of it.

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3 Worksheet 1: Relativity and the Big Bang

Summary

Inertial frames of reference Principle of relativity Newtons second law F = ddt p Absolute distance and duration versus relative distance and duration Length contraction and time dilation Inertial and gravitational mass Free-fall frames of reference

Curved space-time

Homogeneous and isotropic Cosmological expansion and the scale-factor a(t) Hubble parameter H aa and Hubbles law v = H r

Friedmann equation H2 = 8G3 k c2

a(t)2

Discussion

A truck hits a small car. What force is bigger, the one on the truck or the one on the smallcar? When the same force acts for the same amount of time on two similar objects, with one

objects mass being twice as big as the other one, what will be the ratio between the finalvelocities of the two objects?

The average lifetime of an (unstable) particle is the time it takes for the particle to decay.Will this lifetime appear to us to be longer or shorter when the particle reaches velocitiesclose to the speed of light?

Since simultaneity is a relative statement, might cause and effect be interchanged for someobservers? Why not?

A cylinder is an example of a homogeneous space. Can you explain why it is not isotropic? Why is the fact that light travels further than c t in an expanding universe not in conflict

with the theory of special relativity?

What is more general about the theory of general relativity? Explain the observed redshifts of far away galaxies.

Exercise 1: Slingshot

Consider two objects colliding, with one object much heavier than the other m2 m1. We willconsider the limit m1/m2 0. A nice way to determine the velocity of the light object aftercollision makes use of changing perspectives from one reference frame to another. Assume that

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in the standard observers frame of reference the heavy object is coming in from the right withvelocity v1 and the light object is on a collision course from the left with velocity v2. Now considerthe reference frame in which the heavy object is at rest.a) Determine the velocity of the light object in the frame of reference in which the heavy object isat rest.b) Assuming no energy is lost in the collision (known as an elastic collision), what is the velocityof the light object after the collision in the frame of reference in which the heavy object is at rest?c) Returning back to the observers frame of reference, what is the velocity of the light object afterthe elastic collision?

Exercise 2: Length contractionDerive the equation for the length contraction using the Lorentz transformations. Hint: first derivea relation between t2 t1 and L x2 x1 by demanding simultaneity in the primed referenceframe, t1 = t

2. Then calculate L

x2 x1.

Exercise 3: Time dilationUsing the light-clock example that was discussed in class together with Pythagoras theorem toderive the expression (13) for the time dilation.

Exercise 4: Bending of lighta) Consider an elevator with an observer (draw it!) accelerating upwards by a constant externalforce. In the elevator a laser beam travels horizontally from the left top corner of the elevator tothe other side of the elevator. Where would the laser beam end up on the right side wall of theelevator: higher, lower or at the same height? Explain!b) Now apply the equivalence principle. To what gravitational setting should the acceleratingobserver be equivalent (draw it!). What does the equivalence principle then imply for the effect ofgravity on light?

c) Using the equivalence principle, can you calculate the amount of bending (the distance the lightdrops) in a gravitational field with a gravitational acceleration constant g = 10 (m/s)/s (roughlythe strength on Earth) when the elevator is 3 meters wide?

Exercise 5: Clocks in a gravitational fieldA direct consequence of the equivalence principle, combined with some features of special relativity,that can be readily derived is the running of clocks in a gravitational field, i.e. locations withdifferent gravitational potentials.a) Consider a rocket of height h accelerating upwards with acceleration g carrying two clocks,one in the top of the rocket and one at the bottom (again draw it!). The clock high up in therocket sends a flash of light down to the clock at the bottom every second (as read on the top

clock). Based on the fact that the rocket is accelerating upwards, is the interval between the flashesshorter, longer or equal at the bottom of the rocket? So does the clock at the top of the rocketappear to run faster, slower or equal to the clock at the bottom? Explain!b) Now apply the equivalence principle. To what gravitational setting should the acceleratingrocket be equivalent (draw it!). What does the equivalence principle then imply for the running ofclocks deeper down in a gravitational potential?c) We can actually calculate the size of the effect by using the relativistic Doppler effect that saysthat the frequency of light emitted fe by a source that is moving with a velocity v in the directionof the observer is blue- or red-shifted (depending on whether the source is moving away or comingtowards us) by the following amount

fo = fe1 + v/c1 v2/c2 . (47)

When g is the acceleration of the rocket, when a lightray is send from the top of the rocket to reach

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the bottom, what is the difference between the velocity v between emission and observation in atime t? Is it blue- or red-shifted?d) Assuming the velocity of the rocket is still far from the speed of light, can you approximatelycalculate the time difference t in terms of the height h and the speed of light c?e) Plugging the expression for v into the expression for the blue- or red-shift (47), calculate theratio between the frequency at the observer (or the rate at the observer) and the frequency atemission (rate at emission). What does this imply for the running of clocks at the top and atthe bottom of the rocket (use that frequency can be considered a rate) and consequently for therunning of clocks in a gravitational potential?

Exercise 6: The Hubble constant and the redshift parametera) Take the current rate of expansion to equal H0 = 70 km/s/Mpc. Assuming this rate was con-stant throughout the history of the universe, calculate a reasonable first estimate for the age ofthe universe (in years!).[Hint: use that 1 Mpc 3. 1022 meter.]b) Would the universe be older or younger if the Hubble parameter was higher in the past (ex-plain!)?c) The cosmological principle can also be used to derive an expression for the redshift of light(from far away galaxies) in terms of the scale factors at emission and observation. Use the factthat the wavelength in terms of the co-movingcoordinates x is independent of time to derive anexpression for z obsemem in terms of the scale factors a(tobs) and a(tem). The redshift parameterz is the typical parameter used to indicate distance in cosmology.

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4 Part II: The hot Big Bang and Quantum Physics

Our universe is expanding, so in the past it necessarily was a lot denser. Note that the universe wasnot necessarily smaller, in an absolute sense, in the past, because that hinges on the assumptionthat the universe has a finite total size, which is unknown. For all we know the universe could beinfinite in size. What we do know is that the observable part of the universe, which is finite in size

because the universe has a finite age, is surprisingly flat.All visible matter that has over time accumulated into galaxies and formed stars, due to small

initial inhomogeneities in the density, had to correspond to a dense gaseous type medium in theearly universe that was extremely isotropic and homogeneous. Interactions between the basicconstituents of this gas made sure all available (kinetic) energy in terms of random motions wasdistributed equally. This is the criterium for thermodynamical equilibrium as we will discuss insomewhat more detail in the next paragraphs and it defines a temperature: all available energy isspread evenly among all degrees of freedom, at an average of

E =1

2kBT , (48)

per degree of freedom (known as the equipartition energy), where kB is Boltzmanns constantkB = 1.38 1023 (Joules per degree Kelvin, where Joules is the standard unit of energy equal toNewton-meter). This defines the absolute temperature scale in degrees Kelvin. The associatedpressure of this hot gas leads to negative work being done (so energy is lost) as the universeexpands and a corresponding lowering of the temperature as time progresses. Reversing this logic,as one con

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Life Evolution Universe Lecture Notes - AUC