History of cosmology and the cosmological constant
Yannick Fritschij
July 12, 2012
Master’s thesis Theoretical Physics, research part of the Communication variant University of Amsterdam
Supervised by prof. dr. A.J. Kox
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 the structure of the universe. However, at the start of the twentieth century, the study of cosmology evolved in a dramatic way. Where people were always used to observe the universe first, and then construct a theory based on the observations, Albert Einstein decided to turn things around. He started with his theory on gravitation, the well-known theory 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 Lemaˆıtre and others showed the in-adequacies in Einstein’s model, after which in the 1920’s models of expanding universes were developed. The prediction that the universe is actually expanding was proven to be right in the 1930’s, starting with Edwin Hubble’s 1929 results on the velocities of galaxies and nebulae. This discovery also marked the end of a short but exciting period in which cosmology 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 evidence for its existence. Einstein needed it to create his universe model, and when this model was 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; in the past decades, Einstein’s constant has been brought back and removed many times by many 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 central question why it has not disappeared after the expansion of the universe was proven, which contradicted the original reason for its existence. The role of the constant in cosmological theories from 1917 until now will be discussed, with special attention for the 1917-1930 period in which cosmology was as mentioned a purely theoretic discipline. The first part of the thesis will give an overview of the most important cosmological models from this period, starting with the Einstein Universe. But before that, in order to explain the basic idea behind this way of studying cosmology, a brief summary of Newtonian cosmology and Einstein’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 gravity was already in 1687 accurately described by Isaac Newton in his Philosophiae Naturalis Principia 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)
correct by Albert Einstein, is still viewed as the standard method to make calculations on gravity. Newton’s theory is based on two principles that have played an important role in physics ever since.
The first principle Newton used was the equality between the inertial mass from his second law of physics
F = mia (1)
and the gravitational mass
F = mgg (2)
where g depends on position and other masses. This makes the acceleration due to gravity equal for every object:
a = mg mi
g = g (3)
Although this equality could not be proven with absolute certainty, there was enough experimental evidence for Newton to assume that gravitational mass and inertial mass had to be the same.
The second part of Newton’s law of gravitation states that the gravitational force between 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 from observational evidence. From this, Newton acquired the basic equations that describes a system of particles interacting gravitationally:
mN d2xN dt2 = G X 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 had great success over the centuries; it resulted for instance in the prediction of the existence and position of Neptune. There were however also some problems that could not be solved with Newtonian gravity, for instance the observed precession of the perihelia of Mercury. Besides observational issues, the model also had a theoretical difficulty: it presented gravity as a force that could act at a distance. The movement of an object is guided by the mass and position of all the other objects in the universe, but this interaction happens instantly, without any explanation how the object “knows” about the position and mass of all the other objects. These problems were eventually solved in 1916, when Albert Einstein completely altered Newton’s description of gravity with his General Theory of Relativity.
The remainder of this section is based on Weinberg (1972), chapter 1 For instance by Ismael Bullialdus around 1640
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 of gravitation. It combines Newton’s theory of gravitation with Einstein’s theory of Special Relativity (1905). In this section, the basics of General Relativity will be explained using the underlying principles together with some straightforward calculations. The reader should be familiar with the fundamental aspects of Special Relativity as well as the basic properties of tensor calculation.
The Principle of Equivalence
A major principle which Einstein used starting his theory of General Relativity is the Principle of Equivalence of Gravitation and Inertia. It has originated from the probable equality of gravitational and inertial mass, which was already indicated by experimental results in earlier times. The previous section showed how Newton in his theory on gravity therefore assumed these masses to be equal. Einstein went one step further: he stated this equality as a postulate and used it in the following gedankenexperiment: an observer in a freely falling elevator could never tell whether the elevator is moving in a gravitational field or not, because gravitational fields always make both the observer and the elevator move with the same acceleration, so that the laws of nature will remain the same inside the elevator. This means that any experiment done by the observer will produce the same results as in the absence of gravitation. Therefore, a gravitational field can always be canceled by choosing a “freely falling” coordinate system. An important remark is that this statement only holds in relatively small regions of space and time. In large regions of space, the gravitational field can be detected by the observer in the elevator; for example by dropping two objects in the elevator, which would approach each other as the elevator falls toward the center of the earth. A time-dependent gravitational field can also be detected, if the observer does experiments over a relatively large amount of time. The Principle of Equivalence therefore states that at every space-time point in an arbitrary gravitational 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 same form 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 is only influenced by gravitational forces. According to this principle, there is a freely falling coordinate 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.
Cartesian coordinate system, namely
d2ξα
dτ2 = 0 (6)
with dτ the proper time known from Special relativity:
dτ2 = −ds2 = −ηαβdξαdξβ (7)
In any other coordinate system xµ, the equations of motion become
0 = d dτ δξα δxµ dxµ dτ = δξ α δxµ d2xµ dτ2 + δ2ξα δxµδxν dxµ dτ dxν dτ (8)
Multiplying this by δxλ/δξα gives
0 = d 2xλ dτ2 + Γ λ µν dxµ dτ dxν dτ (9)
with Γλµν the affine connection, defined by
Γλµν ≡ δx
λ
δξα
δ2ξα
δxµδxν (10)
Eq. 9 is known as the geodesic equation, because it can geometrically be described by saying that a particle in free fall through a gravitational field will move on the shortest path in space-time. “Length” is hereby measured by the proper time τ . Such paths are called geodesics. The proper time (Eq. 7) in these arbitrary coordinates becomes
dτ2 = −ηαβ δξα δxµdx µδξβ δxνdx ν = −gµνdxµdxν (11)
with gµν the metric tensor, defined by
gµν ≡
δξα δxµ
δξβ
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 Γλ
µν and the metric tensor gµν are both essential parameters
in General Relativity. Together they determine the gravitational field in any coordinate system. They are also related to each other, as can be shown by differentiating gµν with
respect to xλ and using Eq. 10 twice, which leads to
δgµν δxλ = Γ ρ λµgρν+ Γ ρ λνgρµ (13)
Adding to this the same equation with λ and µ interchanged and subtracting the equation with λ and ν interchanged gives
δgµν δxλ + δgλν δxµ − δgµλ δxν = Γ ρ λµgρν+ Γ ρ λνgρµ+ Γ ρ µλgρν + Γρµνgρλ− Γρνµgρλ− Γ ρ νλgρµ = 2gρνΓρλµ (14)
Multiplying this with gνσ finally gives
Γσλµ = 12gνσ δgµν δxλ + δgλν δxµ − δgµλ δxν (15)
The Principle of General Covariance
The previous section showed how, because of the Principle of Equivalence, equations that describe gravitational effects can be derived by using coordinate transformations. This method 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 using another principle, the Principle of General Covariance. It states that a physical equation holds in any gravitational field if two conditions are met: the equation needs to hold in the absence of gravitation and the equation has to be generally covariant, which means that it preserves its form under a general coordinate transformation. So in order to find physical equations which describe gravitational fields, it is necessary to rewrite the equations known from Special Relativity in generally covariant form. It is therefore convenient to make use of tensors, which behave relatively simple under coordinate transformations.
Curvature
In order to describe the gravitational field in generally covariant form it is necessary to look for tensors that can be formed from the metric tensor gµν and its derivatives. Using only
the metric tensor and its first derivatives will not lead to new tensors, because according to the Principle of Equivalence, there is always a coordinate system in which the first derivatives of gµν vanish. The next possibility is to construct a tensor from the metric
tensor and its first and second derivatives. To do this, it is logical to start with something which is made of the metric and its derivatives, namely the affine connection Γλ
µν, which
has the following transformation law: Γλµν = δx λ δx0τ δx0ρ δxµ δx0σ δxνΓ 0τ ρσ+ δxλ δx0τ δ2x0τ δxµδxν (16)
Differentiating this equation with respect to xκ and doing some algebra gives
0 = δx 0τ δxλ δΓλµν δxκ − δΓλµκ δxν + Γ η µνΓ λ κη− Γ η µκΓ λ νη ! − δx 0ρ δxµ δx0σ δxν δx0η δxκ δΓ0τ ρσ δx0η − δΓ0τρη δx0σ − Γ 0τ λσΓ 0λ ηρ+ Γ 0τ ληΓ 0λ σρ (17)
This can be written as a standard tensor transformation: R0τρησ = δx 0τ δxλ δxµ δx0ρ δxν δx0σ δxκ δx0ηR λ µνκ (18) with Rλ
µνκ the Riemann-Christoffel curvature tensor, defined by
Rλµνκ ≡ δΓ λ µν δxκ − δΓλ µκ δxν + Γ η µνΓλκη− ΓηµκΓλνη (19)
It can be proven that the Riemann-Christoffel curvature tensor is the only tensor that can be constructed from the metric tensor and its first and second derivatives, which is linear in the second derivatives. The fully covariant form of Rλ
µνκ can be obtained by lowering
one 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 twice
gives 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 carry energy and momentum themselves, and therefore have to be described by nonlinear partial differential equations. However, the case may be simplified by starting with weak static fields with nonrelativistic matter, where the Newtonian potential can be used. This leads to 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 the Principle of Equivalence, this leads to the educated guess that the equations for a general gravitational field take the form
Gµν = −8πGTµν (24)
where Gµν is a tensor that is formed by a linear combination of the metric and its first
and second derivatives. As mentioned in the previous section, the only tensors that can be formed from the curvature tensor Rλ
µνκ are the Ricci tensor Rµν and the curvature scalar
R. Therefore Gµν takes the form
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, the
value 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 hand side describes the energy and momentum inside this space-time. So for a region that contains a certain amount of matter (Tµν), the equations can be solved, which would lead
to a description of the curvature of space-time (Rµν, R and gµν) inside that region. In
cosmology, the space-time region one has to deal with is the universe as a whole, which obviously is not an easy task. However, only a year after the publication of General Relativity, Einstein came up with a model for the universe now known as the Einstein Universe, which shall be described in the next section.
3
The Einstein Universe
Shortly after publishing the Einstein Equations in his famous article on General Relativity in 1916, Einstein tried to use these equations in order to describe gravity on cosmological scale. This was in fact the first attempt to build a cosmological model only from mathemat-ical equations, without using astronommathemat-ical observations. The first results were published in 1917, in an article that starts with an outline of the problems Newtonian Theory gives when 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, without any 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
large extent solved with the introduction of gravitational fields, which led to Poisson’s equation for gravity:
∇2φ = 4πKρ (30)
In order to apply this equation in Newtonian gravity, one condition must be added: at spatial infinity, the potential has to reach a constant value. This means that the density of matter becomes zero at infinity. This can be explained by assuming that the gravitational field of matter, on a large scale, possesses spherical symmetry, with r the distance from the center. Poisson’s equation shows that if the potential φ becomes constant at infinity, the mean density ρ must decrease toward zero faster than 1/r2. So in the Newtonian Theory
the universe contains no matter at infinite regions, even though it may possess an infinite amount of matter.
The constant value of the gravitational potential at infinity leads to some difficulties concerning the motion of heavenly bodies. For if the potential is finite, a heavenly body will be able to move into infinity if its kinetic energy is great enough. By statistical mechanics, this should happen from time to time. It is however as mentioned impossible in the Newtonian universe that matter exists at infinite regions. One may try to solve this 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 finite regions. The potential at infinity obviously has to be smaller than in these regions, which makes it impossible to have a very high value. And if stellar systems are viewed as a gas in thermal equilibrium, Boltzmann’s law of distribution makes it even impossible for the Newtonian universe to exist. This law namely states that a finite difference of potential between 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 make sense 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
λ ρ0 (32)
with ρ0 the (constant) density. This solution corresponds to an infinite universe with both
density 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 low value of λφ, takes the form of Eq. 30. This (physically not justified) adjustment of Poisson’s equation solves the problems with Newtonian gravity. A similar method can be used in General Relativity.
The infinite universe in General Relativity
To determine the boundary conditions at spatial infinity in General Relativity, Einstein started with one assumption about inertia: that there can be no inertia relatively to “space”,
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. This condition can be formulated mathematically in the following way, beginning with the ex-pression for energy and momentum in General Relativity:
m√−g gµα
dxα
ds (33)
where g is defined as the determinant of the metric tensor gµν. The first three components
of 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 spatially isotropic, 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 also
be 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 we
obtain m√A B dx1 dx4 , m√A B dx2 dx4 , m√A B dx3 dx4 , m√B (36)
where the first three components give the (positive) momentum, and the fourth component gives 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 remain true as well, so the rest mass can only become zero if A goes to zero and B increases to infinity. This means that the potential energy m√B becomes infinite as well, against which arguments were already given in the previous section. Furthermore, concerning stellar velocities the other components of the energy-momentum tensor Tµν all have to be
very small compared to T44, which is impossible with these boundary conditions.
At this moment, Einstein was forced to give up the idea of a spatially infinite universe with boundary conditions. While De Sitter went on with the infinite universe, only without boundary conditions, Einstein took a different path: he started to investigate the possibility of 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, Einstein again started with a few assumptions. First of all, he claimed that on a large scale matter is uniformly distributed, and that its density of distribution is independent of locality and 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 T44, so that there is a coordinate system
in which all the matter is permanently at rest. In this system, T44 is equal to the mean density ρ and all the other components of Tµν are zero.
A particle in a static gravitational field can only remain at rest when g44is independent
of 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 uniform distribution of matter, the curvature must be constant. A finite space with constant curvature has to be a spherical space. Starting with a Euclidean space and removing one coordinate, the space-components of the metric of such a spherical space prove to be
gµν = − δµν+ xµxν R2− (x2 1+ x22+ x23) (39) where both µ and ν differ from 4, δµν is the Kronecker delta and R is the constant radius
of curvature of the whole spherical space.
Now that the components of the metric are found, it is possible to insert them in the Einstein field equations, which can be written in the following way:
Gµν = −8πG(Tµν− 12gµνT ) (40) Gµν = − δΓαµν δxα + ΓβµαΓανβ+ δ 2log√−g δxµδxν − Γα µν δ log√−g δxα (41) Because of the isotropy of the space, it is sufficient to perform the calculations for only one point, for example a point with coordinates x1 = x2 = x3 = x4 = 0. Inserting this point
in Eq. 40 and 41, together with the given values of the energy-momentum tensor Tµν and
the 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 is obviously not a satisfactory solution for a finite, matter-filled universe.
In order to find a satisfactory solution for the spatially finite universe, Einstein made a slight modification to his field equations, which is analogous to the extension of Poisson’s equation given by Eq. 31:
Gµν − λgµν = −8πG(Tµν− 12gµνT ) (43)
where λ is famously known as the cosmological constant, which is sufficiently small to be negligible in small regions such as the solar system. Inserting the given values of Tµν and
gµν in the extended field equations gives
λ = 8πGρ
2 =
1
which shows that in this solution, the cosmological constant defines both the density and the radius of curvature of the universe.
This model of the universe became known as the Einstein Universe: a finite space with variable curvature, but on a large scale approximately a spherical space with finite mass, density and radius of curvature, which all depend on the cosmological constant. This constant 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 based on the relatively small velocities of stars. The cosmological constant will be discussed in more detail in section 9.
The Einstein Universe received much attention among other physicists. The first major reaction came within two months from Willem de Sitter, who presented his own universe model, 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 physicist and mathematician Willem de Sitter (1872-1934) reacted with an article on the same subject. De Sitter had a lot of correspondence with Einstein, sometimes resulting in quite heavy disputes about each other’s findings. This time De Sitter also searched for a model describing a spatially finite universe, using the same field equations including the cosmological constant (Eq. 43). The solutions Einstein found did however not satisfy De Sitter, 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 known matter, calling it “world matter”. De Sitter found a solution that did not involve this world matter, by starting with one different assumption.
As mentioned in the previous section, Einstein assumed his universe to be spherical and spatially finite. The time component of the metric had to remain constant, which led to a metric with values
gij = −δij −
xixj
R2− Σx2 i
g44 = 1 (45)
where the indices i and j both differ from 4. De Sitter went even further by assuming not only 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− Σx2 µ
(46) where the indices µ and ν can have the values 1-4. By running the same calculations as Einstein did, but with this slightly different metric, De Sitter solved the Einstein equations
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 always caused by (possibly very distant) matter. However, De Sitter’s solution contains no matter at 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 with experimental evidence. This is caused by the fact that for describing phenomena in our neighbourhood, both metrics reduce to the Minkowski metric. De Sitter therefore brings up, in his own words, “metaphysical or philosophical considerations”, which will not be further discussed. The De Sitter Universe and the Einstein Universe were both slowly conquered 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 universe had 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 44 and 47) did not depend on any other coordinate. Friedman made the assumption that R could be time-dependent, which led to various new universe models. His article starts in the same way as Einstein’s and De Sitter’s articles, namely by stating a few assumptions that 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 of light, which leads to an energy-momentum tensor which has zero value in every component except T44, which has the value of c2ρg44.
The two assumptions of the second class are new. The first one indicates that the space has a constant positive curvature, which may be time-dependent. This means that the space coordinates x1, x2, x3 may depend on x4. The other assumption of the second class
states that time is orthogonal to space, which causes the metric’s components g14, g24, g34
to vanish. Friedman mentions explicitly that “no physical or philosophical reasons can be given for this second assumption; it serves exclusively to simplify the calculations”.
These assumptions lead to a metric that can be written as
ds2 = R2(dx21+ sin2x1dx22+ sin2x1sin2x2 dx23) + M2dx24 (48)
where R is a function of x4 and M can depend on all four coordinates. By plugging some
values of this metric into the Einstein equations, using the assumptions mentioned above, after some algebra the following relations can be found:
R0(x4) ∂M ∂x1 = R0(x4) ∂M ∂x2 = R0(x4) ∂M ∂x3 = 0 (49)
This means that there are two possibilities: R does not depend on x4 or M does not
depend on x1, x2, x3. Both of these options can be tested by calculating the Einstein
equations again. Following the first option leads to two possible universes: the already known stationary Einstein and De Sitter universes. The second option gives the following relations: R02 R2 + 2RR00 R2 + c2 R2 − λ = 0 (50) 3R02 R2 + 3c2 R2 − λ = κc 2ρ (51) with R0 = dR dx4 and R00= d 2R dx2 4
. The first equation can be directly integrated, which gives
RR02+ c2R − 1 3λR
3
= 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 following
integral equation: t = 1 c Z R a r x A − x + 3cλ2x3 dx + B (54)
where A, B and a are constants. When this integral is solved, the relation between R and t is obtained. Using Eq. 51 and Eq. 52, A can be expressed in terms of the total mass of space ¯M :
A = κ ¯M
6π2 (55)
In the same way, the density of matter ρ can be determined to be: ρ = 3A
Solutions
Depending on the value of the cosmological constant λ and the constant A, Eq. 54 provides several solutions. The nature of these solutions depends on the sign of the square root in the integral of Eq. 54. Because we are dealing with positive curvature, x cannot be smaller than zero. So the square root can only change its sign around the points where the denominator is zero. These points can be found by calculating the positive (because x > 0) roots of the following equation:
yx3− x + A = 0 (57)
where y = λ
3c2. 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 can easily be found by differentiating:
x = 3A
2 , y = 4
27A2 (58)
For negative λ, it can be seen in figure 1 that Eq. 57 has one positive root that lies in the interval (0, A) and is an increasing function of λ and A. If λ lies between zero and the maximum of the curves (9A4c22, from Eq. 58), the equation has two positive roots; the first
an increasing function of λ and A in the interval (A,3A2 ), the second a decreasing function of λ and A in the interval (3A2 , ∞). If λ is greater than the maximum of the curves 9A4c22,
the equation obviously has no positive roots.
From this, Friedman discerns three types of solutions. The first type deals with the situation that λ > 49Ac22, so Eq. 57 has no positive roots. This causes the square root in
Eq. 54 to be always positive. This makes that the radius of curvature R can be written as an increasing function of t. Therefore from some time in the past, R must have grown from zero to the present value. This time, called the time of growth, is given by:
t0 = 1 c Z R0 0 r x A − x + 3cλ2x3 dx (59)
where t0 is the time since the creation of the world (time of growth) and R0 is the radius
of curvature at the present time (or at any other designated time). Friedman describes this solution as a “monotonic world of the first kind”. The time of growth increases when R0 increases and decreases when A (i.e. ¯M ) or λ increases. It will always be finite when
A > 23R0, but otherwise it can reach infinite values as well, depending on λ.
The second type of solution is obtained when 0 < λ < 49Ac22. In order to keep the
square root in Eq. 54 non-imaginary, R must have a certain positive initial value x00, which depends on λ and A, and must be smaller than R0. The time since the creation of the
world then becomes:
t0 = 1 c Z R0 x00 r x A − x + 3cλ2x3 dx (60)
Friedman calls this a “monotonic world of the second kind”. It has the same properties as the first one, except that the radius of curvature in this world did not start at zero in the past, but at a certain positive value.
The third and final type of solution occurs when λ has negative value. R then becomes a periodic function of t, given by
tπ = 2 c Z x0 0 r x A − x + 3cλ2x3 dx (61)
where tπ is called the world period and x0 is a function that depends on both λ and A and
is greater than R0. This universe Friedman calls the “periodic world”; the period increases
when λ increases and is able to reach infinite values.
Friedman ends his article by stating that it is impossible to determine on observational evidence which type of solution describes our universe.
Negative curvature
Two years after this article, Friedman wrote another article on cosmology. This time he considered the possibility of a world with constant negative curvature of space. Again he distinguished between stationary and time-dependent solutions. With similar reasoning as in the previous article, Friedman shows that stationary solutions with negative curvature only exist with zero or negative density. Therefore an Einstein Universe with negative curvature is not possible, but a De Sitter Universe, which has zero density, is possible. The non-stationary universe model with negative curvature leads to equations very similar to the solutions with positive curvature. The density of matter is again provided by Eq. 56 and 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 the finiteness of the universe, because the space can still be positively curved (and therefore finite) or negatively curved (and infinite). To draw any conclusions about finiteness, one therefore needs “supplementary assumptions”.
6
Lemaˆıtre’s Solutions
Friedman’s articles did not receive much attention until the 1930’s, which perhaps can be explained by the rather complicated mathematics they involved. Therefore it happened that in 1927 the Belgian priest and astronomer Georges Lemaˆıtre (1874 - 1966) indepen-dently developed a theory very similar to Friedman’s. In his article Lemaˆıtre starts with a brief comparison of the Einstein and De Sitter universes, after which he concludes that both models have their advantages and disadvantages, and that an intermediate solution needs to be found. The key to this solution is again the introduction of a variable radius of curvature.
Lemaˆıtre finds differential equations for the radius of curvature that are nearly the same as Friedman’s (Eq. 50): 3R02 R2 + 3 R2 = λ + κρ (62) 2R00 R + R02 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, Lemaˆıtre follows a path that physically differs from Friedman, using conservation 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) Lemaˆıtre (1927)
identical to Friedman’s integral (Eq. 54): t =
Z
dR r λR2 3 − 1 + α 3R + β R2 (64)where α and β are constants defined in the following way: α = κM
π2 , β = κpR 4
(65) Friedman’s equation is retrieved by putting the pressure term β equal to zero. If α is made zero as well, De Sitter’s solution is found, while Einstein’s universe can be retrieved by making R constant and β zero.
Lemaˆıtre ends his article with three conclusions. Firstly, the mass of the universe is given by the Einstein’s relation:
√ λ = 2π 2 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 section
5). This meant that the universe actually started with a certain size a finite number of years ago. Lemaˆıtre received a lot of criticism on this view, mainly due to his priesthood; although Lemaˆıtre 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 his cosmological theory.
The third conclusion is about “an apparent Doppler effect” that this expansion causes. The initial radius of curvature R0can 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 Lemaˆıtre, unlike Friedman, makes room for observational evidence. He is also the first one to note that a large part of the universe can never be seen, because of the Doppler effect, which makes that all the light coming from it will become infra-red light. Lemaˆıtre ends his article with a suggestion about the cause of the expansion: he states that pressure of radiation may be responsible for start of the expansion of the universe.
Robertson
Again independently, Howard Percy Robertson (1903 - 1961) found a universe model in 1928 that strongly resembled the ones invented by Friedman and Lemaˆıtre. In 1929 Robert-son wrote an article in which he showed a more critical view on the starting assumptions
than his predecessors. He introduced the now often used terms of homogeneity and isotropy of 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 described the Doppler effect in a non-stationary universe.
7
Eddington’s Solutions
The expanding universe models of Friedman, Lemaˆıtre and Robertson finally received at-tention in 1930, when Arthur Eddington (1882 - 1944) wrote an article based on Lemaˆıtre’s solutions. Eddington was besides his work as a physicist also known as a populariser and translator of scientific articles; for instance, he already had brought Einstein’s articles on Relativity to the English-speaking world. Now Eddington used his reputation again in order to draw attention to the existing expanding universe models together with his own additions.
His starting point was to examine whether the Einstein Universe is stable or not. Eddington uses Lemaˆıtre’s solutions to show that this is not the case, after which he investigates 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:
3d
2R
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 the density smaller than λ/4π, ddt2R2 will be positive, which makes the universe expand. Due to
this expansion, the density will become smaller, which makes ddt2R2 again larger, so that the
universe will expand even faster. A similar thing happens when ρ > λ/4π; the universe will then start to contract and keep contracting. Eddington suggests that the universe might have started as an Einstein universe, but that a slight disturbance has lead to an expanding universe, a conclusion he bases on the “observed scattering apart of the spiral nebulae”.
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)
Eddington continues his article by calculating the rate of expansion of the universe. He sets the pressure equal to zero and uses the results from the Einstein solution (Eq 44):
2
πMe= Re = 1 √
λ (69)
where Meand Reare respectively the mass and radius of curvature of the Einstein universe.
Using Lemaˆıtre’s equations (Eq 62 and 63), the expression for the rate of expansion proves to be dR dt = r 1 3R 2λ − 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 means
that 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
=r M3
Me
(71) So when the radius of curvature gets near this value, the expansion will slow down to 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, namely
when R = Re. This means that in the neighbourhood of Re, dRdt becomes infinitely
small, 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 option because of the “philosophical satisfaction” it gives. He prefers a universe that has evolved “infinitely slowly from a primitive uniform distribution in unstable equilibrium”. Both other solutions lack, according to Eddington, a “natural starting point”. At this point he clearly distinguishes himself from Lemaˆıtre’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. He concludes 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 was strongly supported by Hubble’s paper on the recession of galaxies.
8
Hubble’s Discovery
With the improvement of astronomical measurements in the late 1920’s, experimental evidence became crucial in order to determine what type of universe we are living in. While the Einstein and De Sitter universes were purely theoretical models, both unable to describe the real universe, the expanding models presented many solutions that could describe the real universe. The experimental key that had to determine whether the universe expands or not was given by the Doppler effect (see section 6). By measuring the apparent velocity of as many extra-galactic nebulae as possible, the evidence for an expanding universe grew rapidly.
In 1929, the American astronomer Edwin Hubble (1889 - 1953) was the first to note an apparent linear correlation between the distance and speed of recession of extra-galactic nebulae. Because of the large uncertainty in the measurements of both distance and radial velocity at that time, and perhaps also because he was unaware of the supporting evidence provided by Friedman and Lemaˆıtre in previous years, Hubble remained careful in his conclusions, stating that “it is thought premature to discuss in detail the obvious consequences of the present results”. Nevertheless, Hubble’s article marked the beginning of a new period in cosmology, in which astronomical observations and theoretical research became 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 period in which Einstein’s theory of General Relativity was used to make theoretical models that describe the universe. The following sections will discuss a remarkable phenomenon that emerged from this period: the cosmological constant.
9
The Cosmological Constant
Although the universe models discussed in the previous sections are in many ways different from each other, they have one remarkable thing in common: they all make use of Einstein’s cosmological constant. This constant is always explained to have its origins in Einstein’s search for a static solution. When astronomical observations in the 1930’s gave more and more evidence for an expanding universe, Einstein was the first to reject his own constant. According to the Russian physicist George Gamow, student of Friedman, he even called it the “biggest blunder” of his life. The expanding universe models of Friedman, Lemaˆıtre 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 in a slightly different way. The cosmological constant therefore has to be more than just a constant brought in to create a static universe model. In this section, the role of Einstein’s constant 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.
the Einstein Universe where it was invented. After this, the appearance of the constant in later models will be discussed, as well as the interpretations that were attributed to the λ-term. The section ends with the reappearance of the cosmological constant in modern theories.
λ in the Einstein Universe
As shown in section 3, the cosmological constant was introduced in order to make a model for a static, finite and matter-filled universe. Einstein first tried to make this model without the constant, but the solutions he found were not satisfactory. At that point, Einstein made his 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 radius of curvature of the universe:
λ = 8πGρ
2 =
1
R2 (73)
This shows that the cosmological constant was not actually introduced in order to keep the universe from expanding, but that it was necessary from the start to create a static but also finite and matter-filled universe. Einstein emphasizes that a static solution is needed due to the relatively small velocities of stars. The universe has to be finite in order to avoid boundary 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 inertia is caused by matter only; this was an important subject in the debate between Einstein and De Sitter, which will be discussed in the next section.
Einstein immediately stated that the introduction of the cosmological constant was “not justified by our actual knowledge of gravitation”. He also seems to be reluctant to give a 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 considering λ as a constant which is peculiar to a law of nature or as a constant of integration”. The first sign of Einstein being actually unhappy about his own constant is visible in his 1919 article on atomic structure; in this article Einstein mentions that the introduction of the constant 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 cosmological constant depends on the radius of curvature in a different way:
λ = 3
R2 (74)
Einstein (1917)
Einstein to Besso (1918), in Einstein (1997), doc. 604 Einstein (1919)
This shows that also in De Sitter’s vacuum solution, the cosmological constant is needed in order to prevent the universe from blowing up to infinite values. This probably explains why De Sitter adapts λ without any hesitation.
He does however mention the constant in a postscript of the article, stating that “the introduction of this constant can only be avoided by abandoning the postulate of the relativity 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. The physical 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 the postulate. 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 makes
no mention of matter, so that it is satisfied in the De Sitter solution. However, without the cosmological constant even the mathematical postulate would not be satisfied, because the 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 Sitter debate about their universe models. However, regarding the cosmological constant, they come to the same conclusion: that the constant is needed in order to make a universe model that satisfies the postulate of inertia. Both gentlemen also agree that the constant heavily affects the beauty of the Einstein equations; De Sitter even states that it “detracts from the symmetry and elegance of Einstein’s original theory, one of whose chief attractions was that 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 perhaps expect him to remove the cosmological constant right away, because it was introduced in order 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 discuss the 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 Lemaˆıtre’s independent 1927 article on the expanding universe, the cosmological constant 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 the universe (Eq. 66), which sets λ automatically greater than zero. In later models however, where this relation did not occur anymore, Lemaˆıtre still remained a supporter of the cosmological 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 Einstein equations: “on general philosophical grounds there can be little doubt that this form of the
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 the cosmological 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 is no 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 other lengths 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 Eddington uses as “explanation of the law of gravity”.
The second main reason Eddington gives in favour of the constant is that he sees it as the cause of the expansion of the universe. Even though it is possible in his expanding universe model to set λ equal to zero, Eddington believes that there must be a certain force that directly causes the expansion of the universe. Eddington does not make any comment on the actual nature of this force, all he states is that the strength of this force is determined by the cosmological constant. This idea of the constant being the cause of expansion 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 the 1930’s, Einstein became more and more convinced that the introduction of the cosmological constant was a mistake. As mentioned before, Einstein was unhappy with the way the constant affected the simplicity of his equations. The main reason for him to keep it was that in the Einstein Universe, it was needed to allow for a finite and static distribution of matter. Already in 1923 Einstein states in a letter to the mathematician Hermann Weyl that “if there is no quasi-static world, then away with the cosmological term”. With the De Sitter solution for a universe without matter (see section 4) and especially Eddington’s proof that the Einstein Universe is unstable (see section 7), the original reasons for the introduction of the constant became a lot weaker. Einstein finally discarded the term in 1931 after the expansion of the universe was sufficiently proven. In a short article on cosmology he states that his prior assumptions about the universe being static were proven to be wrong, after which Einstein shows that in an expanding universe model λ can easily be set equal to zero, without contradicting astronomical observations.
Most physicists followed Einstein in his rejection of the constant, but there were also some important defenders of λ. Eddington remained faithful to the philosophical grounds mentioned 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. 286 Einstein (1931)
the cosmological term. As Einstein rejected the constant in order to regain simplicity in the equations, the American physicist Richard Tolman maintained the term for more or less the opposite reason; in a letter from 1931 to Einstein, he wrote that the introduction of λ provided “the most general possible expression of the second order”. Setting the constant equal to zero would therefore be “arbitrary and not necessarily correct”.
Lemaˆıtre also remained faithful to the cosmological constant, even when it was not strictly 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 age problem had to do with the estimated age of the universe, which was measured by the Hubble parameter that determined the speed of recession of galaxies. This estimated age was smaller than the age of some stars, known by theories of stellar evolution, and even smaller than the (by radioactivity) estimated age of the earth. An implemented outward force in the universe, provided by the cosmological constant, enlarges the estimated age of the universe and would therefore solve the age problem. However, in the next decade it became clear that the age problem was not caused by errors in the theory, but by errors in the measurement of the Hubble parameter. This parameter became reduced by a factor 5 to 10, which was enough to solve the age problem at that time.
The second astronomical problem Lemaˆıtre tried to solve with the help of the constant had to do with the formation of galaxies and nebulae. It was hard to explain the structure of galaxies when one only considered gravitational attraction. Lemaˆıtre therefore again used the outward force presented by the cosmological constant to explain the density per-turbations that Lemaˆıtre 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 constant a physical meaning; he only brought it in his equations in order to create a universe that 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 Lemaˆıtre, did not attribute a physical meaning to the constant as well. They both viewed the constant as an outward force that caused the universe to expand, without actually explaining the nature of this force. When Einstein and others removed the term from their equations, Eddington and Lemaˆıtre both tried to defend λ by looking for a physical phenomenon that would explain the existence of the cosmological constant. Before returning to the chronological overview, some of these interpretations will now be discussed.
As we have seen in section 7, Eddington states in his 1930 article that the cosmological constant was necessary to exist on “philosophical grounds”. In 1931 he looked for an analogy in quantum mechanics in order to explain the existence of the constant, using the
Tolman to Einstein(1931), quoted in Earman (2001), p. 197 Earman (2001)
exclusion principle and interchange of energy. This idea did not receive much attention among cosmologists.
In 1939 Eddington presented an other argument that has been used in different ways until the present day. He states that the cosmological term corresponds to the “absolute energy in a standard zero condition”. In this way, the constant fixes a zero “from which energy, 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 the universe”. Eddington thinks that it is impossible to empty the universe, but also that it “would be absurd to define our reckoning of energy by reference to a fictitious process which conflicts with the most important property of matter, namely its conservation”. Therefore, zero energy (possible) should not correspond to zero matter (impossible), so the cosmological constant is needed in order to define a new starting point for measuring energy.
It must be mentioned that Eddington’s credibility among scientists was considerably damaged at that time, due to his search for a “fundamental theory” that combined quantum theory, relativity, cosmology and gravitation. Eddington based this theory mainly on the numerical value of fundamental constants. His rigorous adjustment of the fine structure constant from 1/136 to 1/137 even gave him the nickname “Arthur Adding-one”. This probably explains why his ideas on the interpretation of the cosmological constant were not elaborated by other physicists at that time.
Lemaˆıtre 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 from a zero-level which can be chosen arbitrarily”. A theory on gravitational mass, with mass being a form of energy, therefore has to contain such an arbitrary constant. Without the cosmological constant, Lemaˆıtre states, physicists would count energy from a conventional level 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 vacuum was not noticed in earlier times, because a slight adjustment of the Einstein equations shows it immediately:
Gµν = −κTµν + λgµν (75)
By bringing λ to the right side of the equation, it has changed from a space parameter to an energy parameter, that adjusts the energy tensor, thereby defining the energy level of the vacuum. Lemaˆıtre 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 success in quantum mechanics.
Eddington (1931) Eddington (1939) Crease and Mann (1986) Lemaˆıtre (1949)
The return of the cosmological constant
Despite the effort from Eddington and Lemaˆıtre to keep the cosmological constant, the term was from the start of the 1940’s discarded by most physicists. It has however returned in many ways since then, and in the present days astronomical observations lead physicists to believe that the constant may be necessary after all. In this section the main cosmological theories involving λ will be discussed.
The first reoccurrence of the constant took place in the late 1940’s, with the rise of steady state cosmology, which was at that time rivalling the Big Bang theory. The steady state model can be seen as a model that solves the Einstein equations when these are written 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 cosmic microwave background radiation in 1965, which was predicted by Big Bang theory, steady state 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 not need one either. There were however some problems with the theory, especially the horizon problem: 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 standard Big 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 was proposed in 1981 by Alan Guth, who introduced the concept of inflation: in a very short period right after the Big Bang (10−35 sec. to 10−33 sec), the speed of expansion of the universe was much higher than it has been after that period.
The inflationary model provides a great solution to the horizon problem, but in general it does need a positive cosmological constant in order to agree with the current estimates of the amount of matter in the universe. This can be shown by looking to the most simple version of the model, which contains two dimensionless density parameters:
ΩM = 8πκρ 3H2 , Ω
λ
= λ
3H2 (77)
The current value of the total density parameter, which is obtained by adding the two parameters 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 around
0.3. So either the amount of matter in the universe is much greater than estimated, or λ must be greater than zero again. The first option is not improbable, many physical theories contain dark matter, matter that is not visible but needed in order to explain certain gravitational effects. However, dark matter only does not solve the problem, because it would contradict observations made in the 1990’s.
In 2011, Saul Perlmutter, Adam Riess and Brian Schmidt won the Nobel Prize for these observations in the 1990’s. They used supernovae in order to determine the deceleration paramater q, defined by
q = −RR
00
R02 (78)
where R is the radius of curvature of the universe. The sign of the deceleration parameter, which only depends on R00 because R and R0 are always positive, determines whether the expansion of the universe is speeding up or slowing down. When q is positive, the expansion slows down and when q is negative, the expansion accelerates. When Eq. 78 is combined with Eq. 62 and 63, q can be written in terms of the density parameters (Eq. 77):
q = 1 2Ω M 1 + 3p ρ − Ωλ (79)
Assuming that the density and pressure do not have negative values, the deceleration parameter at the present time (with p/ρ << 1) becomes
q0 ≈ 1 2Ω M 0 − Ω λ 0 (80)
The total density parameter should as mentioned be almost equal to unity:
Ωtot0 = ΩM0 + Ωλ0 ≈ 1 (81) which leads to q0 ≈ 1 2− 3 2Ω λ 0 (82)
This shows that when Ωλ
0 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 strong evidence for an accelerating expansion, with q0 < 0. This means that ΩM0 cannot be
greater than 23, 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 is therefore 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 most important 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 causes an outward pressure in the same way as the cosmological constant does. New astronomical observations will have to determine whether quintessence is the theory of the future, or that 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 order to keep his universe static, although it is often explained in this way. Einstein needed
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 was no satisfactory solution to his equations. Einstein might have created the misunderstanding about the cosmological constant himself, with his 1923 statement that if the universe is expanding, “then away with the cosmical term”. Furthermore, Einstein kept the term after De 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 that Einstein 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 is also visible in the early expanding universe models of Friedman, Lemaˆıtre and Eddington. In Lemaˆıtre’s first model it was even necessary, because he related it to the total mass of the universe. Friedman and Eddington did not particularly need the constant, but they kept it anyway; Friedman probably because it gave his model a broad range of possible solutions, Eddington because of vague “philosophical grounds”. Eddington and Lemaˆıtre remained supporters of the constant for many years, even though they had difficulties with the interpretation. Lemaˆıtre used the term to solve observational problems, which were solved later in other ways. Eddington tried to interpret the constant as the “standard unit of length” or the “cause of the expansion”; both arguments do not actually solve but rather 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 idea developed more in quantum mechanics than in cosmology.
So for many decades it seemed right that Einstein threw away his own constant. But with the discovery of a deceleration of the expansion of the universe in the 1990’s, the cosmological constant made a strong comeback. And although Eddington and Lemaˆıtre did not succeed in interpreting the constant, they may have been on the right track with their suggestions on the energy of the vacuum, a subject that is now of great interest to many physicists. But there are also other theories rivalling the cosmological constant, involving forms of dark energy. The question remains whether the cosmological constant has a real physical meaning, or that it is just an addition that proves that a theory is not entirely correct.
References
Crease, R. P. and Mann, C. C. (1986). The Second Creation: Makers of the Revolution in Twentieth-Century Physics. New York: Macmillan.
De Sitter, W. (1917). On the relativity of inertia. Remarks concerning Einstein’s lat-est hypothesis. Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 19:1217–1225.
