History of cosmology and the cosmological constant

Yannick Fritschij

July 12, 2012

Master’s thesis Theoretical Physics, research part of the Communication variantUniversity of AmsterdamSupervised by prof. dr. A.J. KoxInstitute for Theoretical Physics Amsterdam, History of Physics program

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Introduction

Cosmology is a discipline that has been studied since the beginning of human civilisation.Already in ancient times people looked up at the sky at night and wondered about thestructure of the universe. However, at the start of the twentieth century, the study ofcosmology evolved in a dramatic way. Where people were always used to observe theuniverse first, and then construct a theory based on the observations, Albert Einsteindecided to turn things around. He started with his theory on gravitation, the well-knowntheory of General Relativity, and tried to implement this on the universe as a whole,without using any astronomical data. This resulted in a model that described a static,finite and unbounded matter-filled universe.

Other physicists soon followed Einstein with theoretical models of the universe. Im-portant contributions by Willem de Sitter, Georges Lemaıtre and others showed the in-adequacies in Einstein’s model, after which in the 1920’s models of expanding universeswere developed. The prediction that the universe is actually expanding was proven to beright in the 1930’s, starting with Edwin Hubble’s 1929 results on the velocities of galaxiesand nebulae. This discovery also marked the end of a short but exciting period in whichcosmology had become a purely theoretic study.

A typical phenomenon that has been produced in this period is the so-called cosmolog-ical constant. This constant was invented by Einstein without any observational evidencefor its existence. Einstein needed it to create his universe model, and when this modelwas proven to be wrong he discarded it almost immediately. One would expect the cos-mological constant to have disappeared forever from that time, but the opposite is true; inthe past decades, Einstein’s constant has been brought back and removed many times bymany great physicists, and even in the present time it is still subject of debate.

The second part of this thesis will deal with the cosmological constant, with as centralquestion why it has not disappeared after the expansion of the universe was proven, whichcontradicted the original reason for its existence. The role of the constant in cosmologicaltheories from 1917 until now will be discussed, with special attention for the 1917-1930period in which cosmology was as mentioned a purely theoretic discipline. The first partof the thesis will give an overview of the most important cosmological models from thisperiod, starting with the Einstein Universe. But before that, in order to explain the basicidea behind this way of studying cosmology, a brief summary of Newtonian cosmology andEinstein’s theory of General relativity will be presented.

1 Newtonian Cosmology

In cosmology, all motion is determined by the force of gravity. This force of gravitywas already in 1687 accurately described by Isaac Newton in his Philosophiae NaturalisPrincipia Mathematica. His theory on gravity, although it was proven to be not entirely

The illustration on the front page is an adaption of a cartoon published in the Algemeen Handelsblad(July 9, 1930)

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correct by Albert Einstein, is still viewed as the standard method to make calculations ongravity. Newton’s theory is based on two principles that have played an important role inphysics ever since.

The first principle Newton used was the equality between the inertial mass from hissecond law of physics

F = mia (1)

and the gravitational massF = mgg (2)

where g depends on position and other masses. This makes the acceleration due to gravityequal for every object:

a =

(mg

mi

)g = g (3)

Although this equality could not be proven with absolute certainty, there was enoughexperimental evidence for Newton to assume that gravitational mass and inertial mass hadto be the same.

The second part of Newton’s law of gravitation states that the gravitational forcebetween two objects decreases as the inverse square of the distance between these objects.This was already suggested in earlier times, but Newton was the first to deduce it fromobservational evidence. From this, Newton acquired the basic equations that describes asystem of particles interacting gravitationally:

mNd2xNdt2

= G∑M

mNmM(xM − xN)

|xM − xN |3(4)

where mN is the mass of the Nth particle and xN is its position at time t. For two objects,this can be written as:

Fg =Gm1m2

r2(5)

with Fg the force of gravity and r the distance between the two objects.Using Kepler’s laws on the elliptical motion of planets around the sun, Newton made

a model that described the motion of all the planets in our solar system. This model hadgreat success over the centuries; it resulted for instance in the prediction of the existenceand position of Neptune. There were however also some problems that could not be solvedwith Newtonian gravity, for instance the observed precession of the perihelia of Mercury.Besides observational issues, the model also had a theoretical difficulty: it presented gravityas a force that could act at a distance. The movement of an object is guided by themass and position of all the other objects in the universe, but this interaction happensinstantly, without any explanation how the object “knows” about the position and mass ofall the other objects. These problems were eventually solved in 1916, when Albert Einsteincompletely altered Newton’s description of gravity with his General Theory of Relativity.

The remainder of this section is based on Weinberg (1972), chapter 1For instance by Ismael Bullialdus around 1640

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2 Einstein’s Theory of General Relativity

Einstein’s theory of General Relativity belongs to the most important theories in physics.Since its publication in 1916 it has become the standard theory for describing effects ofgravitation. It combines Newton’s theory of gravitation with Einstein’s theory of SpecialRelativity (1905). In this section, the basics of General Relativity will be explained usingthe underlying principles together with some straightforward calculations. The readershould be familiar with the fundamental aspects of Special Relativity as well as the basicproperties of tensor calculation.

The Principle of Equivalence

A major principle which Einstein used starting his theory of General Relativity is thePrinciple of Equivalence of Gravitation and Inertia. It has originated from the probableequality of gravitational and inertial mass, which was already indicated by experimentalresults in earlier times. The previous section showed how Newton in his theory on gravitytherefore assumed these masses to be equal. Einstein went one step further: he stated thisequality as a postulate and used it in the following gedankenexperiment: an observer ina freely falling elevator could never tell whether the elevator is moving in a gravitationalfield or not, because gravitational fields always make both the observer and the elevatormove with the same acceleration, so that the laws of nature will remain the same insidethe elevator. This means that any experiment done by the observer will produce the sameresults as in the absence of gravitation. Therefore, a gravitational field can always becanceled by choosing a “freely falling” coordinate system. An important remark is thatthis statement only holds in relatively small regions of space and time. In large regions ofspace, the gravitational field can be detected by the observer in the elevator; for exampleby dropping two objects in the elevator, which would approach each other as the elevatorfalls toward the center of the earth. A time-dependent gravitational field can also bedetected, if the observer does experiments over a relatively large amount of time. ThePrinciple of Equivalence therefore states that at every space-time point in an arbitrarygravitational field it is possible to choose a “locally inertial coordinate system” such that,within a sufficiently small region of the point in question, the laws of nature take the sameform as in unaccelerated Cartesian coordinate systems in the absence of gravitation.

Freely falling particle

The Principle of Equivalence can be used to describe the movement of a particle which isonly influenced by gravitational forces. According to this principle, there is a freely fallingcoordinate system ξα in which the equations of motion are the same as in an unaccelerated

Based on Weinberg (1972), chapters 3-7“Unaccelerated Cartesian coordinate systems” are, roughly speaking, the coordinate systems described

by Special Relativity.

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Cartesian coordinate system, namely

d2ξα

dτ 2= 0 (6)

with dτ the proper time known from Special relativity:

dτ 2 = −ds2 = −ηαβdξαdξβ (7)

In any other coordinate system xµ, the equations of motion become

0 =d

dτ

(δξα

δxµdxµ

dτ

)=

δξα

δxµd2xµ

dτ 2+

δ2ξα

δxµδxνdxµ

dτ

dxν

dτ(8)

Multiplying this by δxλ/δξα gives

0 =d2xλ

dτ 2+ Γλµν

dxµ

dτ

dxν

dτ(9)

with Γλµν the affine connection, defined by

Γλµν ≡δxλ

δξαδ2ξα

δxµδxν(10)

Eq. 9 is known as the geodesic equation, because it can geometrically be described bysaying that a particle in free fall through a gravitational field will move on the shortestpath in space-time. “Length” is hereby measured by the proper time τ . Such paths arecalled geodesics. The proper time (Eq. 7) in these arbitrary coordinates becomes

dτ 2 = −ηαβδξα

δxµdxµ

δξβ

δxνdxν

= −gµνdxµdxν (11)

with gµν the metric tensor, defined by

gµν ≡δξα

δxµδξβ

δxνηαβ (12)

which shows that the metric tensor gµν is actually a generalization (to arbitrary coordi-nates) of the Minkowski metric ηµν .

The affine connection Γλµν and the metric tensor gµν are both essential parametersin General Relativity. Together they determine the gravitational field in any coordinatesystem. They are also related to each other, as can be shown by differentiating gµν withrespect to xλ and using Eq. 10 twice, which leads to

δgµνδxλ

= Γρλµgρν + Γρλνgρµ (13)

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Adding to this the same equation with λ and µ interchanged and subtracting the equationwith λ and ν interchanged gives

δgµνδxλ

+δgλνδxµ− δgµλ

δxν= Γρλµgρν + Γρλνgρµ + Γρµλgρν + Γρµνgρλ − Γρνµgρλ − Γρνλgρµ

= 2gρνΓρλµ (14)

Multiplying this with gνσ finally gives

Γσλµ = 12gνσ(δgµνδxλ

+δgλνδxµ− δgµλ

δxν

)(15)

The Principle of General Covariance

The previous section showed how, because of the Principle of Equivalence, equations thatdescribe gravitational effects can be derived by using coordinate transformations. Thismethod could also be used to derive all sorts of physical equations in gravitational fields,but it would lead to very complicated calculations. The problem can be simplified by usinganother principle, the Principle of General Covariance. It states that a physical equationholds in any gravitational field if two conditions are met: the equation needs to hold in theabsence of gravitation and the equation has to be generally covariant, which means that itpreserves its form under a general coordinate transformation. So in order to find physicalequations which describe gravitational fields, it is necessary to rewrite the equations knownfrom Special Relativity in generally covariant form. It is therefore convenient to make useof tensors, which behave relatively simple under coordinate transformations.

Curvature

In order to describe the gravitational field in generally covariant form it is necessary to lookfor tensors that can be formed from the metric tensor gµν and its derivatives. Using onlythe metric tensor and its first derivatives will not lead to new tensors, because accordingto the Principle of Equivalence, there is always a coordinate system in which the firstderivatives of gµν vanish. The next possibility is to construct a tensor from the metrictensor and its first and second derivatives. To do this, it is logical to start with somethingwhich is made of the metric and its derivatives, namely the affine connection Γλµν , whichhas the following transformation law:

Γλµν =δxλ

δx′τδx′ρ

δxµδx′σ

δxνΓ′τρσ +

δxλ

δx′τδ2x′τ

δxµδxν(16)

Differentiating this equation with respect to xκ and doing some algebra gives

0 =δx′τ

δxλ

(δΓλµνδxκ

−δΓλµκδxν

+ ΓηµνΓλκη − ΓηµκΓ

λνη

)

− δx′ρ

δxµδx′σ

δxνδx′η

δxκ

(δΓ′τρσδx′η

−δΓ′τρηδx′σ

− Γ′τλσΓ′ληρ + Γ′τληΓ′λσρ

)(17)

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This can be written as a standard tensor transformation:

R′τρησ =δx′τ

δxλδxµ

δx′ρδxν

δx′σδxκ

δx′ηRλ

µνκ (18)

with Rλµνκ the Riemann-Christoffel curvature tensor, defined by

Rλµνκ ≡

δΓλµνδxκ

−δΓλµκδxν

+ ΓηµνΓλκη − ΓηµκΓ

λνη (19)

It can be proven that the Riemann-Christoffel curvature tensor is the only tensor that canbe constructed from the metric tensor and its first and second derivatives, which is linearin the second derivatives. The fully covariant form of Rλ

µνκ can be obtained by loweringone index:

Rλµνκ ≡ gλσRσµνκ (20)

This can be contracted to give the Ricci tensor,

Rµκ ≡ gλνRλµνκ (21)

which is the only second-rank tensor that can be formed from Rλµνκ. Contracting twicegives the curvature scalar,

R ≡ gλνgµκRλµνκ (22)

which is also the only scalar that can be constructed from Rλµνκ.

The Einstein Field Equations

The field equations for gravitation are not easy to derive, because gravitational fields carryenergy and momentum themselves, and therefore have to be described by nonlinear partialdifferential equations. However, the case may be simplified by starting with weak staticfields with nonrelativistic matter, where the Newtonian potential can be used. This leadsto the following equation:

∇2g00 = −8πGT00 (23)

where G is Newton’s constant and T00 is a component of the energy-momentum tensor,which describes the density and flux of energy and momentum in space-time. Using thePrinciple of Equivalence, this leads to the educated guess that the equations for a generalgravitational field take the form

Gµν = −8πGTµν (24)

where Gµν is a tensor that is formed by a linear combination of the metric and its firstand second derivatives. As mentioned in the previous section, the only tensors that can beformed from the curvature tensor Rλ

µνκ are the Ricci tensor Rµν and the curvature scalarR. Therefore Gµν takes the form

Gµν = C1Rµν + C2gµνR (25)

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where C1 and C2 are constants. Using the fact that Tµν is symmetric and conserved (so Gµν

as well), and that in the nonrelativistic limit Gµν must reduce to ∇2g00 as in Eq. 23, thevalue of the constants can be determined. This finally leads to the Einstein field equations :

Rµν − 12gµνR = −8πGTµν (26)

An alternative version can be obtained by contracting this equation

R = 8πGT (27)

and using this in 26, which gives

Rµν = −8πG(Tµν − 12gµνT ) (28)

This form shows that the Einstein field equations in vacuum reduce to

Rµν = 0 (29)

The Einstein field equations are the final result of the Theory of General Relativity.The left-hand side of Eq. 26 describes the curvature of space-time, while the right handside describes the energy and momentum inside this space-time. So for a region thatcontains a certain amount of matter (Tµν), the equations can be solved, which would leadto a description of the curvature of space-time (Rµν , R and gµν) inside that region. Incosmology, the space-time region one has to deal with is the universe as a whole, whichobviously is not an easy task. However, only a year after the publication of GeneralRelativity, Einstein came up with a model for the universe now known as the EinsteinUniverse, which shall be described in the next section.

3 The Einstein Universe

Shortly after publishing the Einstein Equations in his famous article on General Relativityin 1916, Einstein tried to use these equations in order to describe gravity on cosmologicalscale. This was in fact the first attempt to build a cosmological model only from mathemat-ical equations, without using astronomical observations. The first results were publishedin 1917, in an article that starts with an outline of the problems Newtonian Theory giveswhen it is applied in cosmology.

The infinite universe in Newtonian Theory

The first problem Newton’s theory of gravity contains is that it involves action at a distance(see section 1); it assumes that gravity instantly acts from one object on another, withoutany mediator of the interaction. This makes the force of gravity move at infinite speed,which is impossible according to Einstein’s theory of Relativity. This problem was to a

Einstein (1917)

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large extent solved with the introduction of gravitational fields, which led to Poisson’sequation for gravity :

∇2φ = 4πKρ (30)

In order to apply this equation in Newtonian gravity, one condition must be added: atspatial infinity, the potential has to reach a constant value. This means that the density ofmatter becomes zero at infinity. This can be explained by assuming that the gravitationalfield of matter, on a large scale, possesses spherical symmetry, with r the distance from thecenter. Poisson’s equation shows that if the potential φ becomes constant at infinity, themean density ρ must decrease toward zero faster than 1/r2. So in the Newtonian Theorythe universe contains no matter at infinite regions, even though it may possess an infiniteamount of matter.

The constant value of the gravitational potential at infinity leads to some difficultiesconcerning the motion of heavenly bodies. For if the potential is finite, a heavenly bodywill be able to move into infinity if its kinetic energy is great enough. By statisticalmechanics, this should happen from time to time. It is however as mentioned impossiblein the Newtonian universe that matter exists at infinite regions. One may try to solvethis problem by assuming that the gravitational potential at infinity has a very high value,but this assumption is quickly rejected by the observed values of the potential in finiteregions. The potential at infinity obviously has to be smaller than in these regions, whichmakes it impossible to have a very high value. And if stellar systems are viewed as a gasin thermal equilibrium, Boltzmann’s law of distribution makes it even impossible for theNewtonian universe to exist. This law namely states that a finite difference of potentialbetween the center and spatial infinity corresponds to a finite ratio of densities. Therefore,if the density vanishes at infinity, it should also vanish at the center, which does not makesense at all.

A small adjustment of Poisson’s equation can be made in order to solve these difficulties:

∇2φ− λφ = 4πκρ (31)

where λ denotes a universal constant. This equation has the solution

φ = −4πκ

λρ0 (32)

with ρ0 the (constant) density. This solution corresponds to an infinite universe with bothdensity and potential having a constant value in all regions. In local, non-uniform systems,a second non-constant potential φ will appear, so that Eq. 31, due to the relatively lowvalue of λφ, takes the form of Eq. 30. This (physically not justified) adjustment of Poisson’sequation solves the problems with Newtonian gravity. A similar method can be used inGeneral Relativity.

The infinite universe in General Relativity

To determine the boundary conditions at spatial infinity in General Relativity, Einsteinstarted with one assumption about inertia: that there can be no inertia relatively to “space”,

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but only an inertia of masses relatively to one another. This means that if a mass is suf-ficiently distant from all other masses in the universe, its inertia should be zero. Thiscondition can be formulated mathematically in the following way, beginning with the ex-pression for energy and momentum in General Relativity:

m√−g gµα

dxαds

(33)

where g is defined as the determinant of the metric tensor gµν . The first three componentsof this expression give the negative momentum and the fourth component gives the energy.Choosing a system of coordinates where the gravitational field at every point is spatiallyisotropic, the space-time interval becomes

ds2 = −A(dx21 + dx22 + dx23) +Bdx24 (34)

where x1,2,3 are space-coordinates and x4 is the time-coordinate. The coordinates can alsobe chosen in such a way that √

−g =√A3B = 1 (35)

For small velocities, dx4 is much greater than the other components, so using Eq. 33 weobtain

mA√B

dx1dx4

,mA√B

dx2dx4

,mA√B

dx3dx4

,m√B (36)

where the first three components give the (positive) momentum, and the fourth componentgives the energy. The assumption about inertia leads to the view that the expression m A√

B,

which is the rest mass in these equations, has to become zero at infinity. Eq. 35 must remaintrue as well, so the rest mass can only become zero if A goes to zero and B increases toinfinity. This means that the potential energy m

√B becomes infinite as well, against

which arguments were already given in the previous section. Furthermore, concerningstellar velocities the other components of the energy-momentum tensor T µν all have to bevery small compared to T 44, which is impossible with these boundary conditions.

At this moment, Einstein was forced to give up the idea of a spatially infinite universewith boundary conditions. While De Sitter went on with the infinite universe, only withoutboundary conditions, Einstein took a different path: he started to investigate the possibilityof a universe that is spatially finite, so that there is no need for boundary conditions.

The spatially finite universe

In order to be able to determine the metric tensor in a spatially finite universe, Einsteinagain started with a few assumptions. First of all, he claimed that on a large scale matteris uniformly distributed, and that its density of distribution is independent of localityand varies extremely slowly over time. In this way, the universe can be regarded as static.Then he stated, just as in the previous section, that all the other components of the energy-momentum tensor are very small compared to T 44, so that there is a coordinate system

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in which all the matter is permanently at rest. In this system, T 44 is equal to the meandensity ρ and all the other components of T µν are zero.

A particle in a static gravitational field can only remain at rest when g44 is independentof locality. Because it should also be time-independent, it is possible to state that

g44 = 1 (37)

The other time-components of the metric are, as always in a static situation, equal to zero:

g14 = g24 = g34 = 0 (38)

What remains is to determine the space-components of the metric. Because of the uniformdistribution of matter, the curvature must be constant. A finite space with constantcurvature has to be a spherical space. Starting with a Euclidean space and removing onecoordinate, the space-components of the metric of such a spherical space prove to be

gµν = −(δµν +

xµxνR2 − (x21 + x22 + x23)

)(39)

where both µ and ν differ from 4, δµν is the Kronecker delta and R is the constant radiusof curvature of the whole spherical space.

Now that the components of the metric are found, it is possible to insert them in theEinstein field equations, which can be written in the following way:

Gµν = −8πG(Tµν − 12gµνT ) (40)

Gµν = −δΓαµνδxα

+ ΓβµαΓανβ +δ2 log

√−g

δxµδxν− Γαµν

δ log√−g

δxα(41)

Because of the isotropy of the space, it is sufficient to perform the calculations for only onepoint, for example a point with coordinates x1 = x2 = x3 = x4 = 0. Inserting this pointin Eq. 40 and 41, together with the given values of the energy-momentum tensor Tµν andthe metric tensor gµν , gives

−8πGρ

2= − 2

R2= 0 (42)

which means that the density should be zero and the radius of curvature infinite. This isobviously not a satisfactory solution for a finite, matter-filled universe.

In order to find a satisfactory solution for the spatially finite universe, Einstein made aslight modification to his field equations, which is analogous to the extension of Poisson’sequation given by Eq. 31:

Gµν − λgµν = −8πG(Tµν − 12gµνT ) (43)

where λ is famously known as the cosmological constant, which is sufficiently small to benegligible in small regions such as the solar system. Inserting the given values of Tµν andgµν in the extended field equations gives

λ =8πGρ

2=

1

R2(44)

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which shows that in this solution, the cosmological constant defines both the density andthe radius of curvature of the universe.

This model of the universe became known as the Einstein Universe: a finite spacewith variable curvature, but on a large scale approximately a spherical space with finitemass, density and radius of curvature, which all depend on the cosmological constant. Thisconstant was, according to Einstein, not justified by the present knowledge of gravitation,but necessary to make a quasi-static distribution of matter possible, a requirement basedon the relatively small velocities of stars. The cosmological constant will be discussed inmore detail in section 9.

The Einstein Universe received much attention among other physicists. The first majorreaction came within two months from Willem de Sitter, who presented his own universemodel, which shall be described in the following section.

4 The De Sitter Universe

Less than two months after Einstein had published his article on cosmology, Dutch physicistand mathematician Willem de Sitter (1872-1934) reacted with an article on the samesubject. De Sitter had a lot of correspondence with Einstein, sometimes resulting inquite heavy disputes about each other’s findings. This time De Sitter also searched for amodel describing a spatially finite universe, using the same field equations including thecosmological constant (Eq. 43). The solutions Einstein found did however not satisfy DeSitter, because of the large amount of matter that has to exist according to these solutions.De Sitter even distinguished this matter from the relatively very small amount of knownmatter, calling it “world matter”. De Sitter found a solution that did not involve thisworld matter, by starting with one different assumption.

As mentioned in the previous section, Einstein assumed his universe to be sphericaland spatially finite. The time component of the metric had to remain constant, which ledto a metric with values

gij = −δij −xixj

R2 − Σx2ig44 = 1 (45)

where the indices i and j both differ from 4. De Sitter went even further by assuming notonly space to be finite, but time as well: he proposed a four -dimensional finite space-time.This gives the metric the values

gµν = −δµν −xµxν

R2 − Σx2µ(46)

where the indices µ and ν can have the values 1-4. By running the same calculations asEinstein did, but with this slightly different metric, De Sitter solved the Einstein equations

De Sitter (1917)

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with cosmological constants. The results were as follows:

λ =3

R2

ρ = 0 (47)

This was a rather shocking result, because Einstein assumed that inertia is alwayscaused by (possibly very distant) matter. However, De Sitter’s solution contains no matterat all. De Sitter tries to compare both models in the remainder of the article, but imme-diately brings up the problem that it is impossible to distinguish between the models withexperimental evidence. This is caused by the fact that for describing phenomena in ourneighbourhood, both metrics reduce to the Minkowski metric. De Sitter therefore bringsup, in his own words, “metaphysical or philosophical considerations”, which will not befurther discussed. The De Sitter Universe and the Einstein Universe were both slowlyconquered in the next decade, when expanding universe models were presented.

5 Friedman’s Universe

Einstein and De Sitter both assumed in their models without hesitation that the universehad to be concerned as static. In 1922, the Russian physicist Alexander Friedman (1888 -1925) was the first to use Einstein’s equations to create a model of an expanding universe.In the Einstein and De Sitter universes, the radius of curvature R (e.g. in equation 44and 47) did not depend on any other coordinate. Friedman made the assumption that Rcould be time-dependent, which led to various new universe models. His article starts inthe same way as Einstein’s and De Sitter’s articles, namely by stating a few assumptionsthat simplify the Einstein equations.

Assumptions

Friedman divides his assumptions into two classes. The first class contains the same as-sumptions as Einstein and De Sitter made: the use of the Einstein equations with cosmo-logical constant (Eq. 43) and the matter being at relative rest, compared to the speed oflight, which leads to an energy-momentum tensor which has zero value in every componentexcept T44, which has the value of c2ρg44.

The two assumptions of the second class are new. The first one indicates that the spacehas a constant positive curvature, which may be time-dependent. This means that thespace coordinates x1, x2, x3 may depend on x4. The other assumption of the second classstates that time is orthogonal to space, which causes the metric’s components g14, g24, g34to vanish. Friedman mentions explicitly that “no physical or philosophical reasons can begiven for this second assumption; it serves exclusively to simplify the calculations”.

Friedman (1922)

13

These assumptions lead to a metric that can be written as

ds2 = R2(dx21 + sin2x1 dx22 + sin2x1 sin2x2 dx

23) +M2dx24 (48)

where R is a function of x4 and M can depend on all four coordinates. By plugging somevalues of this metric into the Einstein equations, using the assumptions mentioned above,after some algebra the following relations can be found:

R′(x4)∂M

∂x1= R′(x4)

∂M

∂x2= R′(x4)

∂M

∂x3= 0 (49)

This means that there are two possibilities: R does not depend on x4 or M does notdepend on x1, x2, x3. Both of these options can be tested by calculating the Einsteinequations again. Following the first option leads to two possible universes: the alreadyknown stationary Einstein and De Sitter universes. The second option gives the followingrelations:

R′2

R2+

2RR′′

R2+c2

R2− λ = 0 (50)

3R′2

R2+

3c2

R2− λ = κc2ρ (51)

with R′ =dR

dx4and R′′ =

d2R

dx24. The first equation can be directly integrated, which gives

RR′2 + c2R− 1

3λR3 = A (52)

with A a constant of integration. This can be written as

1

c2

(dR

dt

)2

=A−R + λ

3c2R3

R(53)

with the time component x4 written as t. This can be transformed into the followingintegral equation:

t =1

c

∫ R

a

√x

A− x+ λ3c2x3dx+B (54)

where A, B and a are constants. When this integral is solved, the relation between R andt is obtained. Using Eq. 51 and Eq. 52, A can be expressed in terms of the total mass ofspace M :

A =κM

6π2(55)

In the same way, the density of matter ρ can be determined to be:

ρ =3A

κR3(56)

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Solutions

Depending on the value of the cosmological constant λ and the constant A, Eq. 54 providesseveral solutions. The nature of these solutions depends on the sign of the square root in theintegral of Eq. 54. Because we are dealing with positive curvature, x cannot be smaller thanzero. So the square root can only change its sign around the points where the denominatoris zero. These points can be found by calculating the positive (because x > 0) roots of thefollowing equation:

yx3 − x+ A = 0 (57)

where y = λ3c2

. This can be treated as a family of functions of y and x with parameter A.Figure 1 shows this family of curves in the x,y-plane. The maximums of the curves caneasily be found by differentiating:

x =3A

2, y =

4

27A2(58)

Figure 1: Solutions to Eq. 57

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For negative λ, it can be seen in figure 1 that Eq. 57 has one positive root that lies in theinterval (0, A) and is an increasing function of λ and A. If λ lies between zero and themaximum of the curves ( 4c2

9A2 , from Eq. 58), the equation has two positive roots; the firstan increasing function of λ and A in the interval (A, 3A

2), the second a decreasing function

of λ and A in the interval (3A2,∞). If λ is greater than the maximum of the curves 4c2

9A2 ,the equation obviously has no positive roots.

From this, Friedman discerns three types of solutions. The first type deals with thesituation that λ > 4

9c2

A2 , so Eq. 57 has no positive roots. This causes the square root inEq. 54 to be always positive. This makes that the radius of curvature R can be writtenas an increasing function of t. Therefore from some time in the past, R must have grownfrom zero to the present value. This time, called the time of growth, is given by:

t′ =1

c

∫ R0

0

√x

A− x+ λ3c2x3dx (59)

where t′ is the time since the creation of the world (time of growth) and R0 is the radiusof curvature at the present time (or at any other designated time). Friedman describesthis solution as a “monotonic world of the first kind”. The time of growth increases whenR0 increases and decreases when A (i.e. M) or λ increases. It will always be finite whenA > 2

3R0, but otherwise it can reach infinite values as well, depending on λ.

The second type of solution is obtained when 0 < λ < 49c2

A2 . In order to keep thesquare root in Eq. 54 non-imaginary, R must have a certain positive initial value x′0, whichdepends on λ and A, and must be smaller than R0. The time since the creation of theworld then becomes:

t′ =1

c

∫ R0

x′0

√x

A− x+ λ3c2x3dx (60)

Friedman calls this a “monotonic world of the second kind”. It has the same properties asthe first one, except that the radius of curvature in this world did not start at zero in thepast, but at a certain positive value.

The third and final type of solution occurs when λ has negative value. R then becomesa periodic function of t, given by

tπ =2

c

∫ x0

0

√x

A− x+ λ3c2x3dx (61)

where tπ is called the world period and x0 is a function that depends on both λ and A andis greater than R0. This universe Friedman calls the “periodic world”; the period increaseswhen λ increases and is able to reach infinite values.

Friedman ends his article by stating that it is impossible to determine on observationalevidence which type of solution describes our universe.

16

Negative curvature

Two years after this article, Friedman wrote another article on cosmology. This time heconsidered the possibility of a world with constant negative curvature of space. Again hedistinguished between stationary and time-dependent solutions. With similar reasoning asin the previous article, Friedman shows that stationary solutions with negative curvatureonly exist with zero or negative density. Therefore an Einstein Universe with negativecurvature is not possible, but a De Sitter Universe, which has zero density, is possible. Thenon-stationary universe model with negative curvature leads to equations very similar tothe solutions with positive curvature. The density of matter is again provided by Eq. 56and therefore again positive.

Friedman discusses at the end of the article the physical meaning of these results.He claims that knowledge of the world equations do not lead to any certainty about thefiniteness of the universe, because the space can still be positively curved (and thereforefinite) or negatively curved (and infinite). To draw any conclusions about finiteness, onetherefore needs “supplementary assumptions”.

6 Lemaıtre’s Solutions

Friedman’s articles did not receive much attention until the 1930’s, which perhaps can beexplained by the rather complicated mathematics they involved. Therefore it happenedthat in 1927 the Belgian priest and astronomer Georges Lemaıtre (1874 - 1966) indepen-dently developed a theory very similar to Friedman’s. In his article Lemaıtre starts witha brief comparison of the Einstein and De Sitter universes, after which he concludes thatboth models have their advantages and disadvantages, and that an intermediate solutionneeds to be found. The key to this solution is again the introduction of a variable radiusof curvature.

Lemaıtre finds differential equations for the radius of curvature that are nearly the sameas Friedman’s (Eq. 50):

3R′2

R2+

3

R2= λ+ κρ (62)

2R′′

R+R′2

R2+

1

R2= λ− κp (63)

The only difference is the appearance of the pressure p, which is zero in Friedman’s so-lutions. From here, Lemaıtre follows a path that physically differs from Friedman, usingconservation of energy and momentum and the assumption that the total mass in the uni-verse is constant. This leads however to a solution for the time-component that is almost

Friedman (1924)North (1990)Lemaıtre (1927)

17

identical to Friedman’s integral (Eq. 54):

t =

∫dR√

λR2

3− 1 +

α

3R+

β

R2

(64)

where α and β are constants defined in the following way:

α =κM

π2, β = κpR4 (65)

Friedman’s equation is retrieved by putting the pressure term β equal to zero. If α is madezero as well, De Sitter’s solution is found, while Einstein’s universe can be retrieved bymaking R constant and β zero.

Lemaıtre ends his article with three conclusions. Firstly, the mass of the universe isgiven by the Einstein’s relation:

√λ =

2π2

κM=

1

R0

(66)

His second conclusion is that the universe expands from a certain radius of curva-ture R0 to infinity, like Friedman’s “monotonic world of the second kind” (see section5). This meant that the universe actually started with a certain size a finite number ofyears ago. Lemaıtre received a lot of criticism on this view, mainly due to his priesthood;although Lemaıtre always claimed that he kept religion and science separated, many sci-entists believed that he deliberately tried to involve the concept of divine creation into hiscosmological theory.

The third conclusion is about “an apparent Doppler effect” that this expansion causes.The initial radius of curvature R0 can therefore be calculated by an approximating formula:

R0 =rc

v√

3(67)

where r is the distance to a certain star or nebula and v is the speed at which it recedes,which can be calculated by measuring the frequency of the light that it emits.

With the third conclusion Lemaıtre, unlike Friedman, makes room for observationalevidence. He is also the first one to note that a large part of the universe can never beseen, because of the Doppler effect, which makes that all the light coming from it willbecome infra-red light. Lemaıtre ends his article with a suggestion about the cause ofthe expansion: he states that pressure of radiation may be responsible for start of theexpansion of the universe.

Robertson

Again independently, Howard Percy Robertson (1903 - 1961) found a universe model in1928 that strongly resembled the ones invented by Friedman and Lemaıtre. In 1929 Robert-son wrote an article in which he showed a more critical view on the starting assumptions

Robertson (1928)

18

than his predecessors. He introduced the now often used terms of homogeneity and isotropyof space to justify his assumptions. After this, he found that the Einstein and De Sitter uni-verses are the only stationary solutions of the Einstein equations. Robertson also describedthe Doppler effect in a non-stationary universe.

7 Eddington’s Solutions

The expanding universe models of Friedman, Lemaıtre and Robertson finally received at-tention in 1930, when Arthur Eddington (1882 - 1944) wrote an article based on Lemaıtre’ssolutions. Eddington was besides his work as a physicist also known as a populariser andtranslator of scientific articles; for instance, he already had brought Einstein’s articles onRelativity to the English-speaking world. Now Eddington used his reputation again inorder to draw attention to the existing expanding universe models together with his ownadditions.

His starting point was to examine whether the Einstein Universe is stable or not.Eddington uses Lemaıtre’s solutions to show that this is not the case, after which heinvestigates several properties of the expanding universe.

Instability of Einstein’s universe

To show that the Einstein universe is unstable, Eddington starts with the following differ-ential equation for the curvature radius:

3d2R

dt2= R(λ− 4πρ) (68)

In equilibrium the left hand side of the equation is zero, which gives Einstein’s solution,with ρ = λ/4π (c.f. Eq. 44). If however there is a slight disturbance which makes thedensity smaller than λ/4π, d2R

dt2will be positive, which makes the universe expand. Due to

this expansion, the density will become smaller, which makes d2Rdt2

again larger, so that theuniverse will expand even faster. A similar thing happens when ρ > λ/4π; the universewill then start to contract and keep contracting. Eddington suggests that the universemight have started as an Einstein universe, but that a slight disturbance has lead to anexpanding universe, a conclusion he bases on the “observed scattering apart of the spiralnebulae”.

Robertson (1929)Eddington (1930)Kragh and Smith (2003)Essentially the same results were found by De Sitter, published in two articles: De Sitter (1930) and

De Sitter (1931b)The gravitational constant G is set equal to 1 in Eddington’s notation, which will be followed here.

19

Eddington continues his article by calculating the rate of expansion of the universe. Hesets the pressure equal to zero and uses the results from the Einstein solution (Eq 44):

2

πMe = Re =

1√λ

(69)

where Me and Re are respectively the mass and radius of curvature of the Einstein universe.Using Lemaıtre’s equations (Eq 62 and 63), the expression for the rate of expansion provesto be

dR

dt=

√1

3R2λ− 1 +

4M

3πR(70)

This form of the square root makes room for three different scenarios:

1. M > Me: the square root does not vanish for any positive value of R, which meansthat the universe in this case is able to expand from very small to very large radius.By differentiating Eq. 70, the rate of expansion proofs to be minimal when

R

Re

= 3

√M

Me

(71)

So when the radius of curvature gets near this value, the expansion will slow downto its minimum and then become faster again.

2. M < Me: in this case, the square root is imaginary between two values R1 and R2.This means that the universe either expands to radius R1 and then contracts again,or that it contracts to radius R2 and then expands again.

3. M = Me: now there is only one value for which the square root vanishes, namelywhen R = Re. This means that in the neighbourhood of Re,

dRdt

becomes infinitelysmall, so that the universe can remain near this value for an infinite time.

Eddington, who was also known as a philosopher of science, favours the third optionbecause of the “philosophical satisfaction” it gives. He prefers a universe that has evolved“infinitely slowly from a primitive uniform distribution in unstable equilibrium”. Bothother solutions lack, according to Eddington, a “natural starting point”. At this point heclearly distinguishes himself from Lemaıtre’s views, whose model contains such an “unnat-ural starting point” (see section 6).

In the remainder of the article, Eddington makes some numerical estimates and com-ments on observational phenomena, such as the recession of the galactic nebulae. Heconcludes that this observed recession, together with the instability of the Einstein uni-verse, makes it very likely that the universe is indeed expanding. This conclusion wasstrongly supported by Hubble’s paper on the recession of galaxies.

20

8 Hubble’s Discovery

With the improvement of astronomical measurements in the late 1920’s, experimentalevidence became crucial in order to determine what type of universe we are living in. Whilethe Einstein and De Sitter universes were purely theoretical models, both unable to describethe real universe, the expanding models presented many solutions that could describe thereal universe. The experimental key that had to determine whether the universe expandsor not was given by the Doppler effect (see section 6). By measuring the apparent velocityof as many extra-galactic nebulae as possible, the evidence for an expanding universe grewrapidly.

In 1929, the American astronomer Edwin Hubble (1889 - 1953) was the first to note anapparent linear correlation between the distance and speed of recession of extra-galacticnebulae. Because of the large uncertainty in the measurements of both distance andradial velocity at that time, and perhaps also because he was unaware of the supportingevidence provided by Friedman and Lemaıtre in previous years, Hubble remained carefulin his conclusions, stating that “it is thought premature to discuss in detail the obviousconsequences of the present results”. Nevertheless, Hubble’s article marked the beginningof a new period in cosmology, in which astronomical observations and theoretical researchbecame equally important in order to describe the universe in a single model.

This also ends the historical overview on cosmology from 1917 until 1930, the periodin which Einstein’s theory of General Relativity was used to make theoretical models thatdescribe the universe. The following sections will discuss a remarkable phenomenon thatemerged from this period: the cosmological constant.

9 The Cosmological Constant

Although the universe models discussed in the previous sections are in many ways differentfrom each other, they have one remarkable thing in common: they all make use of Einstein’scosmological constant. This constant is always explained to have its origins in Einstein’ssearch for a static solution. When astronomical observations in the 1930’s gave moreand more evidence for an expanding universe, Einstein was the first to reject his ownconstant. According to the Russian physicist George Gamow, student of Friedman, he evencalled it the “biggest blunder” of his life. The expanding universe models of Friedman,Lemaıtre and Eddington do however, as we have seen, keep the cosmological constant.And even in modern cosmological theories, Einstein’s constant seems to have returned ina slightly different way. The cosmological constant therefore has to be more than just aconstant brought in to create a static universe model. In this section, the role of Einstein’sconstant in the universe models that we have discussed will be investigated, starting with

Hubble (1929)Gamow (1958). Einstein is not directly quoted here, which makes it uncertain whether Einstein used

this exact phrase or that it is Gamow’s interpretation.

21

the Einstein Universe where it was invented. After this, the appearance of the constant inlater models will be discussed, as well as the interpretations that were attributed to theλ-term. The section ends with the reappearance of the cosmological constant in moderntheories.

λ in the Einstein Universe

As shown in section 3, the cosmological constant was introduced in order to make a modelfor a static, finite and matter-filled universe. Einstein first tried to make this model withoutthe constant, but the solutions he found were not satisfactory. At that point, Einstein madehis famous adjustment of the Einstein equations:

Gµν − λgµν = −8πG(Tµν − 12gµνT ) (72)

with λ as the symbol for the cosmological constant that depends on the density and radiusof curvature of the universe:

λ =8πGρ

2=

1

R2(73)

This shows that the cosmological constant was not actually introduced in order to keep theuniverse from expanding, but that it was necessary from the start to create a static butalso finite and matter-filled universe. Einstein emphasizes that a static solution is neededdue to the relatively small velocities of stars. The universe has to be finite in order to avoidboundary conditions, which Einstein believes to be “contrary to the spirit of relativity”.And finally, the universe needs to be matter-filled in order to obey the principle that inertiais caused by matter only; this was an important subject in the debate between Einsteinand De Sitter, which will be discussed in the next section.

Einstein immediately stated that the introduction of the cosmological constant was “notjustified by our actual knowledge of gravitation”. He also seems to be reluctant to givea physical interpretation of the constant; in a letter to his friend Michele Besso, a Swiss-Italian engineer, in 1918 he writes that “there is no essential difference between consideringλ as a constant which is peculiar to a law of nature or as a constant of integration”. Thefirst sign of Einstein being actually unhappy about his own constant is visible in his 1919article on atomic structure; in this article Einstein mentions that the introduction of theconstant is “gravely detrimental to the formal beauty of the theory”.

λ in the De Sitter Universe

In the solutions found by De Sitter (section 4), the density is zero and the cosmologicalconstant depends on the radius of curvature in a different way:

λ =3

R2(74)

Einstein (1917)Einstein to Besso (1918), in Einstein (1997), doc. 604Einstein (1919)

22

This shows that also in De Sitter’s vacuum solution, the cosmological constant is neededin order to prevent the universe from blowing up to infinite values. This probably explainswhy De Sitter adapts λ without any hesitation.

He does however mention the constant in a postscript of the article, stating that “theintroduction of this constant can only be avoided by abandoning the postulate of therelativity of inertia altogether”. This postulate was first used by Einstein (see section 3),who stated that there is no inertia relatively to space, but only relatively to matter. Thephysical interpretation of this statement is that without matter, there is no inertia possible.This clearly contradicts De Sitter’s vacuum solution, which is why he slightly adjusted thepostulate. De Sitter introduced the mathematical postulate of the relativity of inertia,which states that gµν should be invariant at infinity. This version of the postulate makesno mention of matter, so that it is satisfied in the De Sitter solution. However, withoutthe cosmological constant even the mathematical postulate would not be satisfied, becausethe universe would then blow up, which makes gµν non-invariant at infinity. In this case,according to De Sitter, inertia would remain unexplained.

The postulate of the relativity of inertia plays a central role in the Einstein - De Sitterdebate about their universe models. However, regarding the cosmological constant, theycome to the same conclusion: that the constant is needed in order to make a universe modelthat satisfies the postulate of inertia. Both gentlemen also agree that the constant heavilyaffects the beauty of the Einstein equations; De Sitter even states that it “detracts fromthe symmetry and elegance of Einstein’s original theory, one of whose chief attractions wasthat it explained so much without introducing any new hypothesis or empirical constant”.

λ in the early expanding universe models

When Friedman created a model for an expanding universe in 1922, one would perhapsexpect him to remove the cosmological constant right away, because it was introduced inorder to find solutions for the completely different static situation. It is therefore remark-able that Friedman not only keeps λ in his equations, but that he does not even discussthe constant in his entire article. In Friedman’s solutions, λ is allowed to be equal to zero,but Friedman never makes any remark in favour of or against this possibility.

In Lemaıtre’s independent 1927 article on the expanding universe, the cosmologicalconstant is also kept without any hesitation. In this case the constant is also still necessary,because of the relation between λ and the total mass and starting radius of curvature of theuniverse (Eq. 66), which sets λ automatically greater than zero. In later models however,where this relation did not occur anymore, Lemaıtre still remained a supporter of thecosmological constant.

An other supporter of the constant was the philosopher Eddington, who in the in-troduction of his 1930 article immediately made a statement about the renewed Einsteinequations: “on general philosophical grounds there can be little doubt that this form of the

De Sitter (1917)

23

equations is correct rather than his earlier form Gµν = 0”. These “philosophical grounds”are explained in more detail in two books Eddington wrote about cosmology.

According to Eddington, there are two fundamental reasons for the existence of thecosmological constant. The first has to do with the universe being finite and unbounded.This means that even empty space, because it is finite, has a certain radius of curvature.This radius of curvature is defined by the cosmological constant. Furthermore, there isno such thing as absolute length; length can only be measured relative to something else.Eddington therefore views λ as a “natural unit of length”, in terms of which all the otherlengths can be expressed. The “standard metre” at some point in space then becomes al-ways the same fraction of the radius of curvature at that point, a statement that Eddingtonuses as “explanation of the law of gravity”.

The second main reason Eddington gives in favour of the constant is that he sees it asthe cause of the expansion of the universe. Even though it is possible in his expandinguniverse model to set λ equal to zero, Eddington believes that there must be a certainforce that directly causes the expansion of the universe. Eddington does not make anycomment on the actual nature of this force, all he states is that the strength of this forceis determined by the cosmological constant. This idea of the constant being the cause ofexpansion was at the same time supported by De Sitter.

λ after 1930

When the evidence of the Hubble expansion of the universe started to grow rapidly in the1930’s, Einstein became more and more convinced that the introduction of the cosmologicalconstant was a mistake. As mentioned before, Einstein was unhappy with the way theconstant affected the simplicity of his equations. The main reason for him to keep it wasthat in the Einstein Universe, it was needed to allow for a finite and static distribution ofmatter. Already in 1923 Einstein states in a letter to the mathematician Hermann Weylthat “if there is no quasi-static world, then away with the cosmological term”. With theDe Sitter solution for a universe without matter (see section 4) and especially Eddington’sproof that the Einstein Universe is unstable (see section 7), the original reasons for theintroduction of the constant became a lot weaker. Einstein finally discarded the termin 1931 after the expansion of the universe was sufficiently proven. In a short article oncosmology he states that his prior assumptions about the universe being static were provento be wrong, after which Einstein shows that in an expanding universe model λ can easilybe set equal to zero, without contradicting astronomical observations.

Most physicists followed Einstein in his rejection of the constant, but there were alsosome important defenders of λ. Eddington remained faithful to the philosophical groundsmentioned in the previous section. But there were also new arguments given in favour of

Eddington (1930)Eddington (1929) and Eddington (1933)De Sitter (1931a)Einstein to Weyl (1923), quoted in Pais (1982), p. 286Einstein (1931)

24

the cosmological term. As Einstein rejected the constant in order to regain simplicity in theequations, the American physicist Richard Tolman maintained the term for more or lessthe opposite reason; in a letter from 1931 to Einstein, he wrote that the introduction of λprovided “the most general possible expression of the second order”. Setting the constantequal to zero would therefore be “arbitrary and not necessarily correct”.

Lemaıtre also remained faithful to the cosmological constant, even when it was notstrictly necessary anymore in his own model. He used the constant to solve two astro-nomical problems: the age problem and the problem of structure formation. The ageproblem had to do with the estimated age of the universe, which was measured by theHubble parameter that determined the speed of recession of galaxies. This estimated agewas smaller than the age of some stars, known by theories of stellar evolution, and evensmaller than the (by radioactivity) estimated age of the earth. An implemented outwardforce in the universe, provided by the cosmological constant, enlarges the estimated age ofthe universe and would therefore solve the age problem. However, in the next decade itbecame clear that the age problem was not caused by errors in the theory, but by errorsin the measurement of the Hubble parameter. This parameter became reduced by a factor5 to 10, which was enough to solve the age problem at that time.

The second astronomical problem Lemaıtre tried to solve with the help of the constanthad to do with the formation of galaxies and nebulae. It was hard to explain the structureof galaxies when one only considered gravitational attraction. Lemaıtre therefore againused the outward force presented by the cosmological constant to explain the density per-turbations that Lemaıtre thought were needed to create the existing galaxies and nebulae.This idea was supported by many physicists, and was not rejected until the late 1960’s.

Interpretations of λ

It has been mentioned before that Einstein did not try to give the cosmological constanta physical meaning; he only brought it in his equations in order to create a universethat obeyed his starting assumptions. When these assumptions were proven to be wrong,Einstein and many other physicists did not hesitate to remove the “unphysical” constant.For a long time, the most important defenders of λ, Eddington and Lemaıtre, did notattribute a physical meaning to the constant as well. They both viewed the constant as anoutward force that caused the universe to expand, without actually explaining the natureof this force. When Einstein and others removed the term from their equations, Eddingtonand Lemaıtre both tried to defend λ by looking for a physical phenomenon that wouldexplain the existence of the cosmological constant. Before returning to the chronologicaloverview, some of these interpretations will now be discussed.

As we have seen in section 7, Eddington states in his 1930 article that the cosmologicalconstant was necessary to exist on “philosophical grounds”. In 1931 he looked for ananalogy in quantum mechanics in order to explain the existence of the constant, using the

Tolman to Einstein(1931), quoted in Earman (2001), p. 197Earman (2001)Eddington (1930)

25

exclusion principle and interchange of energy. This idea did not receive much attentionamong cosmologists.

In 1939 Eddington presented an other argument that has been used in different waysuntil the present day. He states that the cosmological term corresponds to the “absoluteenergy in a standard zero condition”. In this way, the constant fixes a zero “from whichenergy, momentum and stress are measured”. This point cannot be actually zero because“the zero condition must correspond to a possible rearrangement of the matter of theuniverse”. Eddington thinks that it is impossible to empty the universe, but also thatit “would be absurd to define our reckoning of energy by reference to a fictitious processwhich conflicts with the most important property of matter, namely its conservation”.Therefore, zero energy (possible) should not correspond to zero matter (impossible), sothe cosmological constant is needed in order to define a new starting point for measuringenergy.

It must be mentioned that Eddington’s credibility among scientists was considerablydamaged at that time, due to his search for a “fundamental theory” that combined quantumtheory, relativity, cosmology and gravitation. Eddington based this theory mainly on thenumerical value of fundamental constants. His rigorous adjustment of the fine structureconstant from 1/136 to 1/137 even gave him the nickname “Arthur Adding-one”. Thisprobably explains why his ideas on the interpretation of the cosmological constant werenot elaborated by other physicists at that time.

Lemaıtre presented similar reasons in favour of the constant many years later in 1949.He states that “energy essentially contains an arbitrary constant; it can be counted froma zero-level which can be chosen arbitrarily”. A theory on gravitational mass, with massbeing a form of energy, therefore has to contain such an arbitrary constant. Without thecosmological constant, Lemaıtre states, physicists would count energy from a conventionallevel that is “more fundamental than any other they could have chosen just as well”.

It is remarkable that this connection between the constant and the energy of the vacuumwas not noticed in earlier times, because a slight adjustment of the Einstein equations showsit immediately:

Gµν = −κTµν + λgµν (75)

By bringing λ to the right side of the equation, it has changed from a space parameter toan energy parameter, that adjusts the energy tensor, thereby defining the energy level ofthe vacuum. Lemaıtre and De Sitter started to investigate this possibility in the 1930’s,but they remained close to cosmology, while the vacuum energy theories had more successin quantum mechanics.

Eddington (1931)Eddington (1939)Crease and Mann (1986)Lemaıtre (1949)Kragh (2011)

26

The return of the cosmological constant

Despite the effort from Eddington and Lemaıtre to keep the cosmological constant, the termwas from the start of the 1940’s discarded by most physicists. It has however returned inmany ways since then, and in the present days astronomical observations lead physicists tobelieve that the constant may be necessary after all. In this section the main cosmologicaltheories involving λ will be discussed.

The first reoccurrence of the constant took place in the late 1940’s, with the rise ofsteady state cosmology, which was at that time rivalling the Big Bang theory. The steadystate model can be seen as a model that solves the Einstein equations when these arewritten in the following way:

Gµν + Cµν = 8πκTµν (76)

where Cµν is a tensor that is responsible for the continuous creation of matter. Its prop-erties resemble that of a positive cosmological constant. With the discovery of cosmicmicrowave background radiation in 1965, which was predicted by Big Bang theory, steadystate cosmology was soon discarded and the constant disappeared again.

Big Bang theory did not contain a cosmological constant, and for a long time it did notneed one either. There were however some problems with the theory, especially the horizonproblem: the cosmic microwave background radiation looks the same in every direction,so all parts of the universe must have a common origin. But according to the standardBig Bang model, certain parts of the universe have never been able to contact each other,because the distance between them is too large. A solution to the horizon problem wasproposed in 1981 by Alan Guth, who introduced the concept of inflation: in a very shortperiod right after the Big Bang (10−35 sec. to 10−33 sec), the speed of expansion of theuniverse was much higher than it has been after that period.

The inflationary model provides a great solution to the horizon problem, but in generalit does need a positive cosmological constant in order to agree with the current estimatesof the amount of matter in the universe. This can be shown by looking to the most simpleversion of the model, which contains two dimensionless density parameters:

ΩM =8πκρ

3H2, Ωλ =

λ

3H2(77)

The current value of the total density parameter, which is obtained by adding the twoparameters above, has to be very close to unity in order to solve the horizon problem.Current estimates of the amount of matter in the universe however give ΩM a value around0.3. So either the amount of matter in the universe is much greater than estimated, or λmust be greater than zero again. The first option is not improbable, many physical theoriescontain dark matter, matter that is not visible but needed in order to explain certaingravitational effects. However, dark matter only does not solve the problem, because itwould contradict observations made in the 1990’s.

Based on Earman (2001) and Kragh (2011)

27

In 2011, Saul Perlmutter, Adam Riess and Brian Schmidt won the Nobel Prize for theseobservations in the 1990’s. They used supernovae in order to determine the decelerationparamater q, defined by

q =−RR′′

R′2(78)

where R is the radius of curvature of the universe. The sign of the deceleration parameter,which only depends on R′′ because R and R′ are always positive, determines whether theexpansion of the universe is speeding up or slowing down. When q is positive, the expansionslows down and when q is negative, the expansion accelerates. When Eq. 78 is combinedwith Eq. 62 and 63, q can be written in terms of the density parameters (Eq. 77):

q =1

2ΩM

(1 +

3p

ρ

)− Ωλ (79)

Assuming that the density and pressure do not have negative values, the decelerationparameter at the present time (with p/ρ << 1) becomes

q0 ≈1

2ΩM

0 − Ωλ0 (80)

The total density parameter should as mentioned be almost equal to unity:

Ωtot0 = ΩM

0 + Ωλ0 ≈ 1 (81)

which leads to

q0 ≈1

2− 3

2Ωλ

0 (82)

This shows that when Ωλ0 is negative, zero or smaller than 1

3, q0 is positive and the expansion

of the universe slows down. However, the observations on supernovae presented strongevidence for an accelerating expansion, with q0 < 0. This means that ΩM

0 cannot begreater than 2

3, so dark matter is only able to solve part of the problem; too much matter

will cause the expansion to slow down. In this model, a positive cosmological constant istherefore needed to make everything right.

This obviously made room for a definitive comeback of the cosmological constant. How-ever, there are also some rivalling theories without the cosmological term. The mostimportant theory without λ makes use of a hypothetical form of dark energy, called“quintessence”. This dark energy is given by a special type of dark matter that causesan outward pressure in the same way as the cosmological constant does. New astronomicalobservations will have to determine whether quintessence is the theory of the future, orthat the cosmological constant once again will play a prominent role in physics.

Conclusions

It has become clear that the cosmological constant was not just added by Einstein in orderto keep his universe static, although it is often explained in this way. Einstein needed

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the constant to create a universe that was not only static, but also finite and matter-filled.These were the basic assumptions Einstein started with, and without the constant there wasno satisfactory solution to his equations. Einstein might have created the misunderstandingabout the cosmological constant himself, with his 1923 statement that if the universe isexpanding, “then away with the cosmical term”. Furthermore, Einstein kept the term afterDe Sitter’s vacuum solution and Eddington’s proof that the Einstein Universe is unstable,but quickly discarded it when it became clear that the universe expands. This shows thatEinstein over the years related the constant more and more to the assumption on staticness,while it essentially played a bigger role in his equations.

That the cosmological constant does more than keeping the Einstein Universe static isalso visible in the early expanding universe models of Friedman, Lemaıtre and Eddington.In Lemaıtre’s first model it was even necessary, because he related it to the total mass ofthe universe. Friedman and Eddington did not particularly need the constant, but theykept it anyway; Friedman probably because it gave his model a broad range of possiblesolutions, Eddington because of vague “philosophical grounds”. Eddington and Lemaıtreremained supporters of the constant for many years, even though they had difficulties withthe interpretation. Lemaıtre used the term to solve observational problems, which weresolved later in other ways. Eddington tried to interpret the constant as the “standardunit of length” or the “cause of the expansion”; both arguments do not actually solve butrather move the problem. In the 1940’s they finally started to find better interpretations,by relating the cosmological constant to the energy of the vacuum. However, this ideadeveloped more in quantum mechanics than in cosmology.

So for many decades it seemed right that Einstein threw away his own constant. Butwith the discovery of a deceleration of the expansion of the universe in the 1990’s, thecosmological constant made a strong comeback. And although Eddington and Lemaıtredid not succeed in interpreting the constant, they may have been on the right track withtheir suggestions on the energy of the vacuum, a subject that is now of great interestto many physicists. But there are also other theories rivalling the cosmological constant,involving forms of dark energy. The question remains whether the cosmological constanthas a real physical meaning, or that it is just an addition that proves that a theory is notentirely correct.

References

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De Sitter, W. (1930). The expanding universe. Discussion of Lemaıtre’s solution of theequations of the inertial field. Bulletin of the Astronomical Institutes of the Netherlands,5:211–218.

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