An Introduction to FRW Cosmology and dark energy models
UUniversity of PatrasSchool of Natural SciencesDepartment of PhysicsDivision of Theoretical and Mathematical Physics,Astronomy and Astrophysics
An Introduction to FRW Cosmology and dark energy models
Konstantinos XenosSupervisor: Prof. Smaragda LolaThesis submitted to the University of Patrasfor the Degree of Undergraduate Studies in Physics28 09 2020 a r X i v : . [ g r- q c ] J a n bstract In this thesis we will focus on Einstein’s interpretation of gravity. We will exam-ine how the most famous equations in cosmology are derived from GR and alsosome results of cosmological significance. We will see how combining that withobservational data forces us to consider some form of dark energy or vacuum en-ergy. So we will conclude with some of the more well-known models for darkenergy and examine how the dynamics of dark energy can lead us to the so-calledcosmological inflation. 3 ontents ff el symbols . . . . . . . . . . . . . . . . . . . . . . . 142.4 The Riemman curvature tensor and the Bianchi identities . . . . . 182.5 The Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 The Einstein tensor . . . . . . . . . . . . . . . . . . . . . . . . . 22 CONTENTS hapter 1Introduction
In the year 1915 Albert Einstein had finished his 10-year attempt of constructingthe General theory of Relativity(GR). Almost at the same time the german math-ematician David Hilbert had derived the same equations of motion as Einsteinvia an axiomatic way. This would come to give space and time the impressionof a Rimannian manifold. Thus gravity would be understood as the curvature ofspacetime, a now 4-Dimensional hypersurface, due to the existence of mass.The equations that were derived gave us for the first time in history the ca-pability of studying the universe with the scientific method of mathematics andobservation, detaching cosmology from areas such as theoretical astronomy orphilosophy.In the later years this theory has been enriched with knowledge from other ar-eas of physics such as particle physics and quantum field theory, making us won-der whether a Riemannian approach is su ffi cient for giving us the bigger picture.Moreover, a geometrical approach is not close to illustrating the strong, weak, oreven the electromagnetic interactions.Nevertheless, the fundamental cosmological principles are based upon geo-metrical terms and Riemannian geometry, even just as a mathematical tool using7 CHAPTER 1. INTRODUCTION tools such as the metric or the a ffi ne connection , is needed in establishing almostall theories of gravity [1, 2] . hapter 2The Geometry of curved spacetime The basis of General Relativity is
Einstein’s Principle of Equivalence which equatesthe inertial and the gravitational mass. This came to be after a thought experimentof Einstein that connected an accelerating frame of reference with an inertial oneunder the influence of a gravitational field, where he understood that the physicallaws should remain the same for both observers in each frame. This he famouslydescribed as ”the happiest thought of his life” since it paired the gravitationalfield with the acceleration field and made up the first step for the construction ofa theory for gravity. With Einstein’s principle of equivalence we can connect anycurved path on an inertial frame to a straight line on an accelerating one, or equallyto a frame under the influence of gravity (those paths are called geodesics ). Wecan understand that the spacetime near a gravitational field is not flat since twolines that start with parallel paths don’t remain parallel. General Relativity(GR)is a theory of curved spacetime and so we need to work on a geometry of curvedsurfaces. In contrast to GR the Special theory of Relativity(SR) is a theory of flatspacetime and the space that describes this theory, known as
Minkowski space, is90
CHAPTER 2. THE GEOMETRY OF CURVED SPACETIME flat. And this holds true as two world lines that started parallel in the Minkowskispace will always remain parallel.Mathematicians at that time had already been working on theories of curvedspace. They were initially proposed by the German mathematician Bernhard Rie-mann around the year 1856, who was student of Gauss at that time. Riemann wasthe first one to work on the hypothesis that space does not need to obey Euclid’saxioms. That space is known today as
Riemannian space. It is important to statethat Riemannian space can be locally
Euclidian (flat) but is generally curved (anexample of that being a 2-dimensional sphere). The mathematical objects thatdescribe such space are called
Riemannian manifolds . In short a topological man-ifold of dimension n is a topological space that is locally like R n . In addition aRiemannian manifold is a topological manifold embedded with a metric . So inthat sense the metric describes any given manifold as we are going to see shortly . First let us look at the length element ds on a curved surface. For simplicity weare going to work on a 2-dimensional surface existing in R . Let us consider aposition vector (cid:126) r to a point P in the surface and then an infinitesimal displacement d (cid:126) r so that the new position vector of the neighboring point P (cid:48) is (cid:126) r (cid:48) = (cid:126) r + d (cid:126) r . Thelength element is going then to be: ds = d (cid:126) r · d (cid:126) r = (cid:32) ∂(cid:126) r ∂ x dx + ∂(cid:126) r ∂ y dy (cid:33) · (cid:32) ∂(cid:126) r ∂ x dx + ∂(cid:126) r ∂ y dy (cid:33) = (cid:32) ∂(cid:126) r ∂ x (cid:33) ( dx ) + (cid:32) ∂(cid:126) r ∂ x (cid:33) (cid:32) ∂(cid:126) r ∂ y (cid:33) dxdy + (cid:32) ∂(cid:126) r ∂ y (cid:33) ( dy ) = g dxdx + g dxdy + g dydy .2. THE METRIC g i j then form the matrix: [ g i j ] = g g g g = (cid:32) ∂(cid:126) r ∂ x (cid:33) ∂(cid:126) r ∂ x ∂(cid:126) r ∂ y ∂(cid:126) r ∂ x ∂(cid:126) r ∂ y (cid:32) ∂(cid:126) r ∂ y (cid:33) This is called the metric tensor and is a rank 2 tensor. One can also write downthe elements of the metric as: g i j = ∂(cid:126) r ∂ x i ∂(cid:126) r ∂ x j (2.1)Also since the spatial derivatives commute, the metric is a symmetric tensor so: g i j = g ji (2.2)The upper index indicates the contravariant form of a tensor and the lower indexindicates the covariant form. In short a contravariant vector field (rank 1 tensor)transforms by the rule ˜ a µ = ∂ ˜ x µ ∂ x ν a ν (2.3)and the covariant vector field transforms by the rule ˜ a µ = ∂ x ν ∂ ˜ x µ a ν (2.4)2 CHAPTER 2. THE GEOMETRY OF CURVED SPACETIME
The transformation occurs from the coordinate system x µ = ( x , x , ... ) to the ˜ x µ = ( ˜ x , ˜ x , ... ) and we are also using the Einstein notation or Einstein sum-mation convention where a repeated index would imply a sum (those indices arecalled dummy indices and they can be changed without altering the value of theexpression). So for a space of dimension n that would be a µ x µ ≡ n (cid:88) µ = a µ x µ = a x + a x + ... + a n x n The metric tensor being a rank 2 covariant tensor obeys the transformation rule: ˜ g i j = ∂ x κ ∂ ˜ x i ∂ x λ ∂ ˜ x j g κλ We can then write down the length element as: ds = g µν dx µ dx ν (2.5)By electing the metric g µν to be the Kronecker delta (Euclidean metric) δ µν (witchcan be easily seen that is also a rank 2 tensor), we obtain that, in a 2-dimensionalspace where x µ = ( x , y ) , the length element is: ds = δ µν dx µ dx ν = dx + dy (2.6)which is Pythagorean theorem. So one can see that the expression (2.5) is a gen-eralization of the Pythagorean theorem on a (generally) curved space. .2. THE METRIC ds = − ( cdt ) + dx + dy + dz the same expression can be derived by using the metric η µν = diag ( −
1, 1, 1, 1 ) ona coordinate system x µ = ( x , x , x , x ) = ( ct , x , y , z ) . We can now better under-stand why SR can be described as a theory of 4-dimensional spacetime by usingthe above notation. That spacetime is called Minkowski space and the correspond-ing metric that defines it
Minkowski metric . Both the Euclidean and Minkowskimetrics describe flat spaces, the di ff erance being that the Euclidean metric is Rie-mannian as it has the form g µν = diag (
1, 1, ..., 1 ) and the Minkowski metric is pseudo-Riemannian as it has the form: g µν = diag ( −
1, 1, ..., 1 ) The metric can also be used in order to rise or lower an index, as so g µν a µ = a ν So as its name suggests the General theory of Relativity is indeed a generalizationof the Special Theory of relativity as we can examine any given metric g µν otherthan η µν . In that manner we can further understand Einstein’s Principle of Equiv-alence, saying that every metric g µν is locally the Minkowski metric η µν and thatspacetime is locally Minkowski-like.4 CHAPTER 2. THE GEOMETRY OF CURVED SPACETIME ff el symbols When working on GR, Einstein realized the importance of working with tensors.As Einstein’s Equivalence Principle states, the laws of physics should be the samefor any observer in any coordinate system. Thus, expressing them in terms oftensors is indeed important since tensors are consistent to coordinate transforma-tions. So when we generalize from a flat space to a curved manifold we have todo so in a way which ensures the physical entities having a tensor-like behavior.In that manner we can see that the ordinary derivative is not behaving as atensor. Let us consider a transformation x µ → ˜ x µ = ∂ ˜ x µ ∂ x ν x ν For a scalar field φ ( x µ ) (rank 0 tensor) we can see that it transforms as ∂φ ( x µ ) ∂ x µ = ∂ µ φ ( x µ ) → ∂ ˜ µ ˜ φ ( ˜ x µ ) = ∂∂ ˜ x µ ˜ φ ( ˜ x µ )= ∂ x ν ∂ ˜ x µ ∂∂ x ν φ ( x µ )= ∂ x ν ∂ ˜ x µ ∂ ν φ ( x ν ) so in that case the e ff ect of the ordinary derivative would result in a rank 1 covari-ant tensor. However this is not the case for higher rank tensors and to prove thatwe will act the ordinary derivative on a rank 1 tensor, that being the simpler one.For example let us consider the following transformation .3. THE CHRISTOFFEL SYMBOLS ∂ ν a µ ( x ) → ∂ ˜ ν ˜ a µ ( x ) = ∂ x ρ ∂ ˜ x ν ∂ ρ ˜ a µ ( x )= ∂ x ρ ∂ ˜ x ν ∂ ρ (cid:32) ∂ x κ ∂ ˜ x µ a κ ( x ) (cid:33) = ∂ x ρ ∂ ˜ x ν ∂ ρ ∂ x κ ∂ ˜ x µ a κ ( x ) + ∂ x ρ ∂ ˜ x ν ∂ x κ ∂ ˜ x µ ∂ ρ a κ ( x ) where we can see that because of the first term existing, the quantity ∂ ν a µ ( x ) isnot a tensor.So we generalize the concept of the derivative in way that will always resultin a tensor. This is called covariant derivative and is defined as ∇ ν φ = ∂ ν φ ∇ ν a µ = ∂ ν a µ − Γ ρνµ a ρ ∇ ν a µ = ∂ ν a µ + Γ µνρ a ρ The coe ffi cients Γ αβγ are called connection coe ffi cients or Christo ff el symbols .A notation that is usually used for the ordinary and covariant derivative is thefollowing: ∂ ν a µ ≡ a µ , ν ∇ ν a µ ≡ a µ ; ν (2.7)6 CHAPTER 2. THE GEOMETRY OF CURVED SPACETIME
Given that the Christo ff el symbols transform by the law (in a way that does notrepresent a tensor) Γ αβγ → ∂ x i ∂ ˜ x β ∂ x j ∂ ˜ x γ ∂ ˜ x α ∂ x k Γ ki j + ∂ ˜ x α ∂ x i ∂ x i ∂ ˜ x α ∂ ˜ x β (2.8)we can get that the term ∇ ν a µ transforms as ∇ ν a µ → ∂ x i ∂ ˜ x ν ∂ x j ∂ ˜ x µ ∇ i a j which defines a rank 2 covariant tensor. This can be generalized to tensors ofhigher rank and the application of the covariant derivative to any tensor will in-crease its covariant rank by one.The Christo ff el symbols can be defined as ∂ e α ∂ x β = ∂ β e α = Γ γβα e γ = Γ γαβ e γ (2.9)where we assume a torsion-less manifold so that the Christo ff el symbols are sy-metric in regard to the lower indices. We can then write the following form forthe metric according to the basis vectors: g αβ = e α · e β (2.10)which derives out of (2.1). This is also a generalization of the usual connection ofthe basis vectors, the most usual being the one of a Euclidean space e α · e β = δ αβ .3. THE CHRISTOFFEL SYMBOLS e α · e β = η αβ We can now write down the partial derivative of the metric ∂ γ g αβ = ∂ γ ( e α · e β )= ( ∂ γ e α ) e β + e α ( ∂ γ e β )= Γ δαγ e δ e β + e α Γ δβγ e δ = g δβ Γ δαγ + g αδ Γ δβγ And by combining similar expressions in the following matter we get ∂ γ g αβ + ∂ β g γα − ∂ α g βγ = g αδ Γ δβγ (2.11)From that we can now get the following useful formula for the Christo ff elsymbols Γ δβγ = g αδ ( ∂ γ g αβ + ∂ β g γα − ∂ α g βγ ) (2.12)where we use the inverse metric defined as g µκ g κν = δ µν (2.13)8 CHAPTER 2. THE GEOMETRY OF CURVED SPACETIME and given that the metric is symmetric we can conclude that the inverse metric isalso symmetric.The same formula for the Christo ff el symbols can be also extracted if onenotices a very significant property of the metric ∇ ρ g µν = metric compatibility property for the covariant derivative oper-ator, and in GR it plays an important role on the development of some models, themost famous being that of the cosmological constant . We are now going to define an entity that gives a mathematical description of thecurvature of a manifold. This is the
Riemann curvature tensor and is defined assuch [ ∇ µ , ∇ ν ] a ρ = R κρνµ a κ (2.15)In short, we can understand the commutator of the covariant derivatives as themeasure of the parallel transport of a vector field (here a ) alongside two di ff erentpathways. So in the case of R κρνµ = [ ∇ µ , ∇ ν ] a ρ = ∇ µ ( ∇ ν a ρ ) − ∇ ν ( ∇ µ a ρ ) .4. THE RIEMMAN CURVATURE TENSOR AND THE BIANCHI IDENTITIES ∇ ν a ρ as a rank 2 covariant tensor and write itscovariant derivative as ∇ µ ( ∇ ν a ρ ) = ∂ µ ( ∇ ν a ρ ) − Γ λµν ( ∇ λ a ρ ) − Γ λµρ ( ∇ ν a λ ) so the above expression becomes [ ∇ µ , ∇ ν ] a ρ = ∂ µ ( ∇ ν a ρ ) − Γ λµν ( ∇ λ a ρ ) − Γ λµρ ( ∇ ν a λ ) − ∂ ν ( ∇ µ a ρ ) + Γ λνµ ( ∇ λ a ρ ) + Γ λνρ ( ∇ µ a λ )= ∂ µ ( ∂ ν a ρ − Γ κνρ a κ ) − Γ λµρ ( ∂ ν a λ − Γ κνλ a κ ) − ∂ ν ( ∂ µ a ρ − Γ κµρ a κ ) + Γ λνρ ( ∂ µ a λ − Γ κµλ a κ )= − ∂ µ ( Γ κνρ ) a κ − Γ κνρ ∂ µ a κ − Γ λµρ ∂ ν a λ + Γ λµρ Γ κνλ a κ + ∂ ν ( Γ κµρ ) a κ + Γ κµρ ∂ ν a κ + Γ λνρ ∂ µ a λ − Γ λνρ Γ κµλ a κ We can now observe that some terms cancel each other out Γ λνρ ∂ ν a λ − Γ κνρ ∂ µ a κ = Γ λµρ ∂ ν a λ − Γ κµρ ∂ µ a κ = κ , λ are dummy indices. So we end up with [ ∇ µ , ∇ ν ] a ρ = (cid:104) ∂ ν ( Γ κµρ ) − ∂ µ ( Γ κνρ ) + Γ λµρ Γ κνλ − Γ λνρ Γ κµλ (cid:105) a κ and comparing with the definition of the Riemann tensor (2.15) we get theformula R κρνµ = ∂ ν ( Γ κµρ ) − ∂ µ ( Γ κνρ ) + Γ λµρ Γ κνλ − Γ λνρ Γ κµλ (2.16)0 CHAPTER 2. THE GEOMETRY OF CURVED SPACETIME or by using the notation (2.7) R κρνµ = Γ κµρ , ν − Γ κνρ , µ + Γ λµρ Γ κνλ − Γ λνρ Γ κµλ (2.17)The Riemann tensor is a rank 4 mixed tensor of type (1,3) and we can make it arank 4 covariant tensor by applying the metric g µν : R σρνµ ≡ g σκ R κρνµ (2.18)and we can get some useful properties of the Riemann tensor using the aboveformulas R κρνµ = − R κρµν or R κρ [ νµ ] = R κρνµ + R κνµρ + R κµρν = R σρνµ + R σνµρ + R σµρν = R σρνµ = R νµσρ = − R ρσνµ = − R σρµν = R ρσνµ (2.21)and the more significant identity ∇ λ R σρνµ + ∇ ν R σρµλ + ∇ µ R σρλν ≡ R σρνµ ; λ + R σρµλ ; ν + R σρλν ; µ = R σρ [ νµ ; λ ] = .5. THE RICCI TENSOR the Bianchi identities for the Riemann tensor and they too play a significant role in establishing some important equations inGR and in cosmology. The
Ricci tensor is defined as the contraction of the Riemann curvature tensor R µν ≡ R κµκν = ∂ κ ( Γ κµν ) − ∂ ν ( Γ κκµ ) + Γ λνµ Γ κκλ − Γ λκµ Γ κνλ (2.24)We can consider other contractions for the Ricci tensor such as R κκµν , but usingthe anti-symmetry properties of the Riemman tensor we end up with R σρνµ = − R ρσνµ ⇒ g σρ R σρνµ = − g σρ R ρσνµ ⇒ R ρρνµ = − R ρρνµ ⇒ R ρρνµ = R λµνλ = − R λµλν (also trivial) and byusing a symmetry property for the Riemann tensor we can show that the Riccitensor is symmetric R µν = R νµ (2.25)This symmetry implies that for a 4-dimensional space the independent compo-nents for the Ricci tensor are 10.2 CHAPTER 2. THE GEOMETRY OF CURVED SPACETIME
Furthermore by applying the metric to the Ricci tensor we define the
Ricci scalarR ≡ R µµ = g µν R µν (2.26)and as we can see it is invariant under coordinate transformations, a property thatmakes it very useful in perceiving the magnitude of curvature for a given manifold. We can now use the Bianchi identities (2.22) to define the
Einstein tensor . Weapply the metric tensor as following g σν g ρµ ∇ λ R σρνµ + g σν g ρµ ∇ ν R σρµλ + g σν g ρµ ∇ µ R σρλν = ∇ κ g µν = ∇ λ ( g σν g ρµ R σρνµ ) + ∇ ν ( g σν g ρµ R σρµλ ) + ∇ µ ( g σν g ρµ R σρλν ) = g σν R σρνµ ≡ R νρνµ ≡ R ρµ and g ρµ R ρµ ≡ R .6. THE EINSTEIN TENSOR R σρµλ = − R ρσµλ and R σρλν = − R σρνλ so we get ∇ λ R − ∇ ν ( g σν g ρµ R ρσµλ ) − ∇ µ ( g σν g ρµ R σρνλ ) = g ρµ R ρσµλ = R σλ and g σν R σρνλ = R ρλ and then the previous expression is just ∇ λ R − ∇ ν ( g σν R σλ ) − ∇ µ ( g ρµ R ρλ ) = ( σ ↔ ρ ) , ( µ ↔ ν ) we get ∇ λ R − ∇ ν ( g σν R σλ ) = CHAPTER 2. THE GEOMETRY OF CURVED SPACETIME which is the same as ∇ ν ( δ νλ R − g σν R σλ ) = ∇ ν ( g σν R σλ − δ νλ R ) = ∇ ν ( R νλ − δ νλ R ) ≡ ( R νλ − δ νλ R ) ; ν = Einstein tensor as G µν ≡ R µν − δ µν R or G µν ≡ R µν − g µν R (2.28)And form (2.27) see that it satisfies the Bianchi identities as so ∇ µ G µν ≡ G µν ; µ = ∇ µ G µν ≡ G µν ; µ = G µν = G νµ (2.30) .6. THE EINSTEIN TENSOR CHAPTER 2. THE GEOMETRY OF CURVED SPACETIME hapter 3General Relativity
The principle of the least action or Hamilton’s principle states that the evolutionof a system will occur in a way that the action between two states is stationary forsmall variations of the variables. In mathematical terms this means that for theaction S ≡ (cid:90) t t L ( q ( t ) , ˙ q ( t ) , t ) dt (3.1)Hamilton’s principle demands the following condition: δ S δ q = L ( q ( t ) , ˙ q ( t ) , t ) is the Lagrangian function and the q(t) are the generalizedcoordinates. 278 CHAPTER 3. GENERAL RELATIVITY
We can go a step further and write the Lagrangian as so L ( q ( t ) , ˙ q ( t ) , t ) ≡ (cid:90) L ( q ( t ) , ˙ q ( t ) , t ) d n x (3.3)where the L is called Lagrangian density . Then the action can be written as S = (cid:90) L ( q ( t ) , ˙ q ( t ) , t ) d n xdt (3.4)So by using the appropriate action we can derive various equations of motionvia this variation approach. This is called the Lagrangian formalism
The action for spacetime curvature that was proposed by Hilbert in order to deriveEinstein’s field equations is known as the Hilbert-Einstein action S H − E = c π G N (cid:90) R √− gd x (3.5)or by noting a constant k = π G N c , also known as Einstein’s constant S H − E = k (cid:90) R √− gd x (3.6)(although we are not going to follow this notation, since the same symbol is usedfor the curvature of spacetime which tends to appear more often in cosmology). .3. THE EINSTEIN FIELD EQUATIONS √− g is added so that the volume element d x = dx dx dx dx is generalized to the volume element of a 4-dimensional topological manifold,where g = det ( g µν ) and can be seen as the Jacobian of a transformation from theordinary coordinate system to that of a curved spacetime. Also the ”–” derivesfrom the fact that in General Relativity we usually are occupied with pseudo-Riemannian metrics. The most simple example of a pseudo-Riemannian metric isthe Minkowski metric η µν = diag ( −
1, 1, 1, 1 ) where in this example we can seethat det ( η µν ) = − R µν R µν or even R κλµν R κλµν , in correspondence to the electromagnetic Lagrangian F µν F µν .In fact there is indeed work being done towards that direction, constructing theo-ries of generalized gravity, but we are not going to concern with any of those inthis thesis. We are now going to use the variation approach in order to derive the Einsteinfield equations by using the Hilbert-Einstein action (3.5). Since the Ricci scalar isa function of the metric it’s natural that the variation of the action is going to bein regard of the the metric. So the condition is the following: δ S H − E δ g µν = CHAPTER 3. GENERAL RELATIVITY
So by developing the term δ S H − E we get δ S H − E = δ (cid:32) c π G N (cid:90) R √− gd x (cid:33) (3.8)and for now we can ignore the constant and just work with the integral δ (cid:32)(cid:90) R √− gd x (cid:33) = (cid:90) δ ( R √− g ) d x (3.9)where we used a property of the variation (not in a strict mathematical manner) δ (cid:32)(cid:90) f (cid:33) = (cid:90) δ f (3.10)and, since its not so uncommon, we are going to treat the variation as an ordinarydi ff erential, by applying a similar product rule δ ( f g ) = g δ f + f δ g (3.11)So by expanding the Ricci scalar, while using the chain rule, we get δ ( R √− g ) = δ ( g µν R µν √− g )= R µν √− g δ g µν + g µν R µν δ √− g + √− gg µν δ R µν = R µν √− g δ g µν + R δ √− g + √− gg µν δ R µν .3. THE EINSTEIN FIELD EQUATIONS δ √− g and δ R µν . For the first one,we use the following result from linear algebra, regarding a given square matrix ad j ( A ) = det ( A ) · inv ( A ) or A − = det ( A ) C (cid:62) (3.12)where the adj(A) is the adjugate of A, meaning ad j ( A ) = C (cid:62) (3.13)if C is the cofactor matrix of A, but we are not going to go into more detail.Applying this for the metric tensor we get inv [ g µν ] = ad j [ g µν ] g = [ G µν ] (cid:62) g (3.14) G µν is not the Einstein tensor but the cofactor of the metric tensor and after mul-tiplying the above expression by [ g µν ] we get that [ g µν ] [ G µν ] (cid:62) g = I (3.15)and since the metric tensor is symmetric, its cofactor would also be symmetric,and thus equation (3.15) becomes g µν G µν g = δ νµ or g µν G µν = g (3.16)2 CHAPTER 3. GENERAL RELATIVITY and we can easily connect the above by the inverse metric if we define it as g µν ≡ G µν g (3.17)which agrees with the previous definition of then inverse metric (2.13)We can then use (3.16) to write down the cofactor of the metric in a useful waythat allows us to get an expression for the δ gG µν = ∂ g ∂ g µν (3.18)and so the variation of g would be δ g = G µν δ g µν = gg µν δ g µν (3.19)and in a similar way in regard to δ g µν it would be δ g = − gg µν δ g µν (3.20)From this we can use the chain rule to calculate δ √− g δ √− g = − √− gg δ g = − √− g ( − gg µν δ g µν )= − √− gg µν δ g µν (3.21) .3. THE EINSTEIN FIELD EQUATIONS δ ( R √− g ) = R µν √− g δ g µν + R δ √− g + √− gg µν δ R µν = R µν √− g δ g µν − √− g g µν R δ g µν + √− gg µν δ R µν = √− g (cid:32) R µν − g µν R (cid:33) δ g µν + √− gg µν δ R µν (3.22)The next thing should be to express the δ R µν term in regard to δ g µν , but we aregoing to show that this term’s contribution to the action is actually zero. Theeasiest way to accomplish that is by using an identity that connects the δ R µν withthe Christo ff el symbols. This is known as the Palatini identity δ R µν = ∇ ρ ( δ Γ ρµν ) − ∇ ν ( δ Γ ρρµ ) (3.23)The proof of it is pretty straightforward ∇ ρ ( δ Γ ρµν ) − ∇ ν ( δ Γ ρρµ ) = ∂ ρ δ Γ ρµν + Γ ρρκ δ Γ κµν − Γ κρν δ Γ ρµκ − Γ κρµ δ Γ ρκν − (cid:16) ∂ ν δ Γ ρρµ + Γ ρνκ δ Γ κρµ − Γ κνρ δ Γ ρκµ − Γ κνµ δ Γ ρρκ (cid:17) = ∂ ρ δ Γ ρµν − ∂ ν δ Γ ρρµ + δ ( Γ κνµ Γ ρρκ ) − δ ( Γ ρνκ Γ κρµ )= δ (cid:16) ∂ ρ Γ ρµν − ∂ ν Γ ρρµ + Γ κνµ Γ ρρκ − Γ ρνκ Γ κρµ (cid:17) = δ R µν CHAPTER 3. GENERAL RELATIVITY
An issue that can come up is that since the Christo ff el symbols are no tensors,then one should not be able to write down the covariant derivative in such manner.But since we take a small variation, one can ignore the non linear term that appearson (2.8) and thus the variation of the Christo ff el symbols can be treated as a tensor.Using this result we can now calculate even further g µν δ R µν = g µν (cid:16) ∇ ρ ( δ Γ ρµν ) − ∇ ν ( δ Γ ρρµ ) (cid:17) = g µν ∇ ρ ( δ Γ ρµν ) − g µρ ∇ ρ ( δ Γ ρρµ )= ∇ ρ (cid:16) g µν δ Γ ρµν − g µρ δ Γ ρρµ (cid:17) − (cid:16) δ Γ ρµν ∇ ρ ( g µν ) − δ Γ ρρµ ∇ ρ ( g µρ ) (cid:17) = ∇ ρ (cid:16) g µν δ Γ ρµν − g µρ δ Γ ρρµ (cid:17) since the second term is zero due to the metric compatibility (2.14). So the con-tribution to the action is the integral (cid:90) √− gd xg µν δ R µν = (cid:90) √− gd x ∇ ρ (cid:16) g µν δ Γ ρµν − g µρ δ Γ ρρµ (cid:17) (3.24)which is an integral over a total derivative, meaning it results to the boundaryterms. We assume however that for a realistic field those terms are going to vanish,as we go to infinity. Thus this term is going to be zero. So we end up with δ S H − E = (cid:90) √− gd x (cid:32) R µν − g µν R (cid:33) δ g µν (3.25) .3. THE EINSTEIN FIELD EQUATIONS R µν − g µν R = G µν = R µν =
0, meaning a Ricci-flat space.An obvious solution to this is the case of the Minkowski metric η µν since in thatcase Γ αµν = ∀ µ , ν , α as the partial derivatives of the metric all vanish. This so-lution leads us however to a trivial case of R κµνσ = S M = (cid:90) L M √− gd x (3.27)where L M is the corresponding Lagrangian for the contribution of matter.Then the total action is going to be S tot = S H − E + S M (3.28)and the condition from the action principle is going to be δ S tot δ g µν = CHAPTER 3. GENERAL RELATIVITY
The expression for δ S tot after ignoring the term δ R µν is going to be δ S tot = δ (cid:32) c π G (cid:90) R √− g d x + (cid:90) L M √− g d x (cid:33) = (cid:90) (cid:34) c π G √− g (cid:32) R µν − g µν R (cid:33) δ g µν + √− g δ L M + L M δ √− g (cid:35) d x And so the condition reads as follows c π G (cid:32) R µν − g µν R (cid:33) + √− g δ S M δ g µν = δ S M = (cid:90) d x (cid:32) √− g δ L M − √− gg µν L M δ g µν (cid:33) After we define the energy-momentum tensor as T µν ≡ − √− g δ S M δ g µν (3.31)the equation (3.30) yields to the Einstein field equations R µν − g µν R = π Gc T µν or G µν = kT µν k = π Gc (3.32)The energy-momentum tensor is introduced as the source term when we want toinclude any kind of sources, in this case it represents the contribution of matter.The left side of the equation refers to the curvature of spacetime while the right .3. THE EINSTEIN FIELD EQUATIONS ∇ µ T µν = ∇ µ T µν = R µν − g µν R + Λ g µν = π Gc T µν or G µν + Λ g µν = kT µν (3.34)Einstein was the first to notice this freedom that his equations provide. He pro-posed these slightly generalized equations in an attempt to strengthen his theoryof a static universe, since the previous equations lead to the conclusion that a staticuniverse can not be stable [3]. So Λ was introduced for cosmological purposesand thus it’s called the cosmological constant . This gave the universe some other8 CHAPTER 3. GENERAL RELATIVITY dynamics but was still insu ffi cient to justify a static universe and by that timethere were also astrological evidence that supported the fact that the universe isnot static, most importantly the observations of Hubble that the galaxies are mov-ing away from us supported a model of an expanding universe. This lead Einsteinto dismiss his static universe theory and the cosmological constant, famously la-beled as his ”biggest blunder”.We can connect the cosmological constant with the Lagrangian formalism by writ-ing down a corresponding action. This will be nothing more than a constant mul-tiplying the volume element integrated over the entire space. Namely S Λ = λ o (cid:90) √− g d x (3.35)So the overall action that leads to the field equations (3.34) would be S tot = S H − E + S Λ + S M (3.36)and we can easily see that this can take the form S tot = c π G (cid:90) ( R − Λ ) √− g d x + (cid:90) L M √− g d x (3.37) hapter 4The FRW Cosmology After the introduction of GR scientist were able to study the universe in a moremathematical way than ever before. This study of the evolution of the universe, aswell as the properies and the dynamics of it is known today as
Cosmology [4–7].The dynamics of the universe can be described by the Einstein field equations, butto do that we need an appropriate form for the energy-momentum tensor, which isconnected to the composition of the universe, and the metric, that is related to theRicci curvature tensor and the Ricci curvature scalar. So we are called to set up abasis of axioms that we can use in order to construct those objects. This would be the Cosmological Principle .The Cosmological Principle states that in macroscopic scales the universe canbe seen as homogeneous and isotropic . The homogeneity implies that the universeis the same everywhere, meaning that the metric that describes it should be thesame in every place of the universe and additionally the curvature too. We can al-ready see that this can not be true in smaller scales like inside a galaxy or our solarsystem, since we know that massive objects curve the spacetime around them, but390
CHAPTER 4. THE FRW COSMOLOGY on a bigger scope we believe that this holds true and most of the evidence supportthis as well. The isotropy means that the universe should look the same acrossevery direction. This suggests that there is no di ff erence in what two di ff erentobservers in di ff erent parts of the universe see. Furthermore it suggests that ourown place in the universe is not special by any means. The isotropy of the uni-verse is also heavily supported by astronomical observation, most famously theCosmic Microwave Background radiation (CMB)[8, 9]. In fact the CMB appearsas an isortopic black body radiation at a temperature of 2.7260 ± − [10]. The metric that is appropriate for a homogeneous and isotropic universe is knowas the Robertson-Walker metric [11–14] ds = − dt + a ( t ) (cid:34) dr − Kr + r ( d θ + sin θ d φ ) (cid:35) (4.1)where we have assumed that c =
1. This metric was firstly introduced by Alexan-der Friedmann on the year 1922[15, 16] after solving the Einstein equations undersome assumptions for the contents of the universe. The same metric was derivedby Howard P. Robertson and Arthur Geo ff rey Walker in the 1930’s but in a purelygeometrical approach under the assumption of a homogeneous and isotropic uni-verse. Their first approach was purely kinematic and did not predict the functiona(t). However, being influenced by Hubble’s observations, they introduced thistime-only dependent term that can explain the dynamics of the universe as pre-dicted by Hubble’s law, hence the function a(t) called the scale factor . .2. THE ROBERTSON-WALKER METRIC r ( t ) = a ( t ) r (4.2)In this relation, r(t) is the distance as measured by an observer at the time t .Thecorresponding coordinate system, which remains the same in time and does notfollow the expansion of the universe, is called physical coordinate system . Addi-tionaly r is the same distance on a coordinate system that follows the expansion(or contraction) of the universe and is called comoving coordinate system . In thiscoordinate system the distance between 2 objects always remains the same duringthe expansion (or contraction) of the universe.Taking into consideration the existence of the scale factor we can reproduceHubble’s law dr ( t ) dt = ddt (cid:16) a ( t ) r (cid:17) = ˙ a ( t ) r = ˙ a ( t ) a ( t ) r ( t ) (4.3)where we can consider the Hubble parameter to be H ≡ ˙ a ( t ) a ( t ) (4.4)2 CHAPTER 4. THE FRW COSMOLOGY so we end up with the famous expression of the Hubble’s law that describes theexpansion of the universe (cid:51) = H r (4.5)The geometrical properties of the universe are connected with the constant Kwhich is associated with the curvature of space. It takes the following 3 values: K = ⇒ space is R corresponding to a flat space.We refer to this case as a flat universe K = ⇒ space is S corresponding to a 3-sphere.We refer to this case as a closed universe K = − ⇒ space is H corresponding to a hyperbolic spatial geometry.We refer to this case as an open universeWe will later connect the curvature of the universe with the components of it. Writing down the elements of the Robertson-Walker metric we have g tt ≡ g = − g rr ≡ g = a ( t ) − Kr g θθ ≡ g = a ( t ) r g φφ ≡ g = a ( t ) r sin θ g µν = µ (cid:44) ν (4.6)and for the inverse metric we can easily get from (2.13) .3. THE FRIEDMANN EQUATIONS g tt ≡ g = − g rr ≡ g = − Kr a ( t ) g θθ ≡ g = a ( t ) r g φφ ≡ g = a ( t ) r sin θ g µν = µ (cid:44) ν (4.7)In addition the partial derivatives of the metric are ∂ µ g i j = ∀ µ , i (cid:44) j ∂ µ g = ∀ µ∂ g = ∂∂ t (cid:32) a ( t ) − Kr (cid:33) = ˙ aa − Kr ∂ g = ∂∂ r (cid:32) a ( t ) − Kr (cid:33) = Kr a ( − Kr ) ∂ g = ∂∂θ (cid:32) a ( t ) − Kr (cid:33) = ∂ g = ∂∂φ (cid:32) a ( t ) − Kr (cid:33) = ∂ g = ˙ aar ∂ g = a r ∂ g = ∂ g = ∂ g = ˙ aar sin θ∂ g = a rsin θ∂ g = a r sin θ cos θ∂ g = CHAPTER 4. THE FRW COSMOLOGY
We can now use the formula (2.12) by substituting the expressions from (4.7) and(4.8) we get in order to compute the Christo ff el symbols. Assuming the symmetryof the Christo ff el symbols in regard to the lower indices Γ αµν = Γ ανµ (4.9)we expect 40 di ff erent ones, 10 for each value of a . However in the case of theRobertson-Walker metric the non-trivial ones are way less. The computation ispretty straightforward. For example, for a non-zero one Γ = g k (cid:16) ∂ g k + ∂ g k − ∂ k g (cid:17) = g (cid:16) ∂ g + ∂ g − ∂ g (cid:17) + g (cid:16) ∂ g + ∂ g − ∂ g (cid:17) + g (cid:16) ∂ g + ∂ g − ∂ g (cid:17) + g (cid:16) ∂ g + ∂ g − ∂ g (cid:17) = g (cid:16) − ∂ g (cid:17) = ( − ) (cid:32) − ˙ aa − Kr (cid:33) = ˙ aa − Kr In a similar we compute the remaining Christo ff el symbols. We end up with:1 ) Γ = ) Γ = Γ = ) Γ = Γ = ) Γ = Γ = ) Γ = ˙ aa − Kr .3. THE FRIEDMANN EQUATIONS ) Γ = Γ = ) Γ = Γ = ) Γ = ˙ aar ) Γ = Γ = ) Γ = ˙ aar sin θ ) Γ = ) Γ = Γ = ˙ aa ) Γ = Γ = ) Γ = Γ = ) Γ = Kr − Kr ) Γ = Γ = ) Γ = Γ = ) Γ = − r ( − Kr ) ) Γ = Γ = ) Γ = − r ( − Kr ) sin θ ) Γ = ) Γ = Γ = ) Γ = Γ = ˙ aa ) Γ = Γ = ) Γ = ) Γ = Γ = r ) Γ = Γ = ) Γ = CHAPTER 4. THE FRW COSMOLOGY ) Γ = Γ = ) Γ = − sin θ cos θ ) Γ = ) Γ = Γ = ) Γ = Γ = ) Γ = Γ = ˙ aa ) Γ = ) Γ = Γ = ) Γ = Γ = r ) Γ = ) Γ = Γ = cos θ sin θ = cot θ ) Γ = ff el symbols. First, we are going to showthat R µν = ∀ µ (cid:44) ν . For that we calculate the terms Γ κµκ Γ κ κ = Γ + Γ + Γ + Γ = ˙ aa Γ κ κ = Γ + Γ + Γ + Γ = Kr − Kr + r Γ κ κ = Γ + Γ + Γ + Γ = cot θ Γ κ κ = Γ + Γ + Γ + Γ = .3. THE FRIEDMANN EQUATIONS t , the second oneof just the distance r , the third one of just θ while the last one is zero. So for thecase of µ (cid:44) ν it is actually ∂ ν Γ κµκ = ∂ Γ κµκ = ∀ µ (cid:44) ∂ Γ κµκ = ∀ µ (cid:44) ∂ Γ κµκ = ∀ µ (cid:44) ∂ Γ κµκ = ff el symbols for µ (cid:44) ν are Γ = Γ = Γ = ˙ aa Γ = Γ = r Γ = cot θ so it’s easy to see that the second partial derivative of the Christo ff el symbols thatappears on the formula of the Ricci tensor turns out to be zero as well ∂ κ Γ κµν = ∀ µ (cid:44) ν The expression for the Ricci tensor now becomes R µν = Γ λνµ Γ κκλ − Γ λκµ Γ κνλ µ (cid:44) ν (4.11)8 CHAPTER 4. THE FRW COSMOLOGY
By expanding the first term we get Γ λνµ Γ κκλ = Γ νµ Γ κκ + Γ νµ Γ κκ + Γ νµ Γ κκ + Γ νµ Γ κκ (4.12)where Γ κκ = Γ νµ = ∀ µ (cid:44) ν .So, by keeping only the non-zero Christo ff el symbols , the first term becomes Γ λνµ Γ κκλ = Γ νµ Γ κκ + Γ νµ Γ κκ = Γ Γ κκ + Γ Γ κκ + Γ Γ κκ (4.13)In a similar way we expand the second term Γ λκµ Γ κνλ = Γ κµ Γ κν + Γ κµ Γ κν + Γ κµ Γ κν + Γ κµ Γ κν = Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν + Γ µ Γ ν and for µ (cid:44) ν the only the non-zero terms are Γ λκµ Γ κνλ = Γ Γ + Γ Γ + Γ µ Γ ν (4.14) .3. THE FRIEDMANN EQUATIONS ff erentcombinations for µ and ν for µ , ν = (
1, 2, 3 ) : Γ λνµ Γ κκλ = Γ Γ κ κ = r cot θ Γ λκµ Γ κνλ = Γ Γ = r cot θ for µ = ν = (
1, 3 ) : Γ λνµ Γ κκλ = Γ Γ κκ = ˙ aa (cid:32) Kr − Kr + r (cid:33) Γ λκµ Γ κνλ = Γ Γ + Γ Γ + Γ Γ = ˙ aa (cid:32) Kr − Kr + r (cid:33) for µ = ν = : Γ λνµ Γ κκλ = Γ Γ κκ = ˙ aa cot θ Γ λκµ Γ κνλ = Γ Γ = ˙ aa cot θ Those are all the possible combinations for µ and ν so we can conclude that thosetwo expressions are indeed equal in the case of µ (cid:44) ν . Thus we can claim that forthe case of the Robertson-Walker metric R µν = ∀ µ (cid:44) ν (4.15)0 CHAPTER 4. THE FRW COSMOLOGY
This means that the only non-zero elements of the Ricci curvature tensor are only4; R , R , R , R . R = ∂ κ Γ κ − ∂ Γ κ κ + Γ λ Γ κλκ − Γ λκ Γ κ λ where Γ κ = Γ κ κ = ˙ aa Γ λκ Γ κ λ = (cid:16) Γ (cid:17) + (cid:16) Γ (cid:17) + (cid:16) Γ (cid:17) = (cid:18) ˙ aa (cid:19) + (cid:18) ˙ aa (cid:19) + (cid:18) ˙ aa (cid:19) = (cid:18) ˙ aa (cid:19) so for R we end up with R = ∂ (cid:18) ˙ aa (cid:19) − (cid:18) ˙ aa (cid:19) = − ¨ aa − ˙ a a + (cid:18) ˙ aa (cid:19) = − ¨ aa For the R element R = ∂ κ Γ κ − ∂ Γ κ κ + Γ λ Γ κλκ − Γ λκ Γ κ λ .3. THE FRIEDMANN EQUATIONS = ∂ Γ − ∂ Γ κ κ + Γ Γ κ κ − Γ κ Γ κ + ∂ Γ + Γ Γ κ κ − Γ κ Γ κ + ∂ Γ + Γ Γ κ κ − Γ κ Γ κ + ∂ Γ + Γ Γ κ κ − Γ κ Γ κ and by keeping only the non-zero terms we get R = ∂ Γ − ∂ Γ κ κ + Γ Γ κ κ − Γ κ Γ κ + ∂ Γ + Γ Γ κ κ − Γ κ Γ κ − Γ κ Γ κ − Γ κ Γ κ and further expanding whilst ignoring the non-zero terms R = ∂ Γ − ∂ Γ κ κ + Γ Γ κ κ − Γ Γ + ∂ Γ + Γ Γ κ κ − Γ Γ − Γ Γ − Γ Γ − Γ Γ CHAPTER 4. THE FRW COSMOLOGY where substituting the expressions for the Christo ff el symbols gives R = ∂ (cid:18) ˙ aa − Kr (cid:19) − ∂ (cid:32) Kr − Kr + r (cid:33) + (cid:18) ˙ aa − Kr (cid:19) (cid:18) ˙ aa (cid:19) − (cid:18) ˙ aa − Kr (cid:19) (cid:18) ˙ aa (cid:19) + ∂ (cid:18) Kr − Kr (cid:19) + (cid:18) Kr − Kr (cid:19) (cid:32) Kr − Kr + r (cid:33) − (cid:18) ˙ aa − Kr (cid:19) (cid:18) ˙ aa (cid:19) − (cid:18) Kr − Kr (cid:19) − r − r = ¨ aa − Kr + ˙ a − Kr + ˙ a − Kr + Kr − Kr (cid:32) r (cid:33) = − Kr (cid:16) ¨ aa + ˙ a + K (cid:17) = a − Kr (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) we work in a similar way for the remaining 2 elements of the Ricci tensor and weend up with R = − ¨ aa = g (cid:18) ¨ aa (cid:19) R = a − Kr (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) = g (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) R = a r (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) = g (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) R = a r sin θ (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) = g (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) (4.16) .3. THE FRIEDMANN EQUATIONS R = g µν R µν = R µµ = R + R + R + R where R = g R = g g (cid:18) ¨ aa (cid:19) = ¨ aaR = g R = g g (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) = (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) R = g R = g g (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) = (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) R = g R = g g (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) = (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) (4.17)And we end up with the following value for the Ricci scalar R = (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) (4.18)4 CHAPTER 4. THE FRW COSMOLOGY
We can also calculate the values for the Einstein tensor (2.6) G = − (cid:32) (cid:18) ˙ aa (cid:19) + Ka (cid:33) G = G = G = − (cid:32) ¨ aa + (cid:18) ˙ aa (cid:19) + Ka (cid:33) (4.19)This gives us everything we need to calculate the left side of the Einstein fieldequations (3.32). The number of the Einstein equations has also dropped downto 2, due to the symmetries that the Robertson-Walker metric provides, instead of10, given that the Einstein tensor is symmetric.The only thing that remains is to get a proper form for the energy-momentumtensor. This comes out of the assumption that the universe behaves like a perfectfluid. For a perfect fluid we know that the energy-momentum tensor takes thefollowing form T µν = ( p + ρ ) U µ U ν + pg µν (4.20)where we have assumed c =
1. The term U µ is the 4-velocity and additionally ρ , pthe density and pressure of the fluid respectively, which are functions of just time.In cosmology the density refers to the density of matter in the universe and thepressure to the relativistic particles. We usually refer to the pressure term as theradiation pressure.Since we have assumed an isotropic and homogeneous universe, the model ofa perfect fluid gives us an isotropic and homogeneous fluid as well. Thus we .3. THE FRIEDMANN EQUATIONS U µ = ( −
1, 0, 0, 0 ) The only non-zero component of the 4-velocity is the time component. That isbecause in the comoving reference system any body is at rest with the fluid, or theuniverse in this case, so the spatial components of the velocity must be zero. Wealso know that the 4-velocity satisfies the rule U µ U µ = − T µν = g µλ T λν = diag ( − ρ , p , p , p ) Now we have everything required to write down the Einstein equations for thiscase of the Robertson-Walker metric. As stated those came down to being just 2,one representing the time component and the other the spatial one. For the timecomponent we get G = π G T (4.21)6 CHAPTER 4. THE FRW COSMOLOGY which leads to the following equation (cid:18) ˙ aa (cid:19) = π G ρ − Ka (4.22)In a similar way the spatial component of the Einstein tensor leads to the equation2 ¨ aa + (cid:18) ˙ aa (cid:19) + Ka = − π G p (4.23)we can rewrite those equations using the Hubble parameter H and its timederivative ˙ H = ddt (cid:18) ˙ aa (cid:19) = ¨ aa − (cid:18) ˙ aa (cid:19) (4.24)Then we end up with H = π G ρ − Ka (4.25) ˙ H = − π G ( ρ + p ) + Ka (4.26)The above equations are called the the Friedmann equations and a universe that isdescribed by the Robertson-Walker metric and those equations is called Friedmann-Robertson-Walker universe or FRW universe in short. .3. THE FRIEDMANN EQUATIONS ¨ aa = − π G ( ρ + p ) (4.27)This equation is called the acceleration equation since it connects the expansionrate of the universe with just the components of it. The scale factor a ( t ) is inher-ently a positive value, so the sign of ¨ a depends only from the term ( ρ + p ) .We can also combine the Friedmann equation in another way by taking the partialderivative of the first (4.23) , which gives2 H ˙ H − π G ˙ ρ = K ˙ aa and then substituting the K from the second equation (4.26), which gives K = a ˙ H + π G a ( ρ + p ) Then with some simple algebra we derive the following equation ˙ ρ + H ( ρ + p ) = continuity equation as it is the equivalent of the ordinary conti-nuity equation when we apply it on the universe. To further understand why thisequation corresponds to a conservation law we are going to derive this from the8 CHAPTER 4. THE FRW COSMOLOGY conservation of the energy-momentum tensor, which is described via the Bianchiidentities (3.33) and translates to a generalized form of the conservation of energyand momentum (or mass). ∇ µ T µν = ⇒ ∂ µ T µν − Γ αµν T µα + Γ µαµ T αν = g µν = diag ( − a , a , a ) (4.29)Then the equation for ν = ∂ µ T µ = − ˙ ρ Γ αµ = g ακ ( ∂ µ g κ + ∂ g µκ − ∂ κ g µ ) Γ µαµ T α = Γ µ µ T where for µ = Γ αµ = = Γ . So for µ (cid:44) Γ µµ = g µκ ( ∂ µ g κ + ∂ g µκ − ∂ κ g µ )= g µκ ( ∂ g µκ )= ( g ∂ g + g ∂ g + g ∂ g ) .3. THE FRIEDMANN EQUATIONS = (cid:32) a ∂ (cid:16) a (cid:17)(cid:33) = (cid:32) ˙ aaa (cid:33) = ˙ aa Γ αµ = g ακ ( ∂ µ g κ + ∂ g µκ − ∂ κ g µ )= g ακ ( ∂ g µκ )= (cid:16) δ ακ a − (cid:17) ( δ µκ ˙ aa )= δ αµ ˙ aa So we end up with 0 = − ˙ ρ − Γ αµ T µα + Γ µµ T = − ˙ ρ − δ αµ (cid:18) ˙ aa (cid:19) T µα + (cid:18) ˙ aa (cid:19) T = − ˙ ρ − HT µµ + HT = ˙ ρ + H ( ρ + p ) since T = − ρ T µµ = p for µ (cid:44) CHAPTER 4. THE FRW COSMOLOGY
The Friedmann equations (4.25) and (4.26) compose a system of 2 independentequations and are the only independent equations that derive from the Einsteinfield equations. The acceleration equation (4.27) and the continuity equation(4.28) both arise from the Friedmann equations, thus they are not independent ofthem. However, for a given value of the curvature K (0,1,-1 resulting to di ff erentgeometries for the universe), the unknown variables that appear in the Friedmannequations are 3; the scale factor (and derivatives of it), the density and pressure.So in order to analytically solve them we would need to introduce a separate, in-dependent equation. This comes in the form of an equation of state , a connectionbetween the density and the pressure under the assumption that the universe to be-haves like a barotropic fluid, namely that the pressure is a function of only densityand vice versa. The simplest one is a linear expression p = w ρ (4.30)where w is a constant.So we can now get the picture of the evolution of the universe for each case ofspatial geometry for the universe (flat,open or closed). Firstly, we can rewrite theequation (4.22) as (cid:18) ˙ aa (cid:19) − π G ρ = − Ka (4.31)In that manner we can read this equation as an energy equation, justified by thefact that it derives from the ”00” components of the energy-momentum tensor.The scale factor term can be associated with the energy that causes the expansionof the energy, while the density term with that of gravitational pull. Furthermore, .4. EVOLUTION OF FRW UNIVERSE π G ρ crit = (cid:18) ˙ aa (cid:19) or ρ crit = H π G (4.32)in the same spirit we consider the density parameter as Ω ≡ ρρ crit (4.33)and by using that notation the 1st Friedmann equation (4.22) takes the form KH a = Ω − K is directly connected with thecomponents of the universe. We know that K can only take the values 0,1,-1 so,given that the denominator is inherently positive, we are left with only 3 cases Ω = ( ρ = ρ crit ) ⇒ K = ⇒ flat universe Ω > ( ρ > ρ crit ) ⇒ K = ⇒ closed universe Ω < ( ρ < ρ crit ) ⇒ K = − ⇒ open universeWe notice that a flat geometry should always remain flat. Additionally, observa-tional results [17, 18] have shown us that the current value of Ω is really close2 CHAPTER 4. THE FRW COSMOLOGY to 1 so the spatial geometry of the universe should also be really close to a flatgeometry. We will later connect this with one of the problems of the standardcosmological model; namely the flatness problem .Working in the case of K = g µν = diag ( − a , a , a ) (4.35)and the Friedmann equations are just H = π G ρ ˙ H = − π G ( ρ + p ) = − π G ρ ( + w ) by combining these we get ˙ HH = − ( + w ) ⇒ dHH = − ( + w ) dt from that we can calculate the expression for the Hubble parameter to be H = ( + w )( t − t ) , w (cid:44) − t is a constant. Given that the Hubble parameter can be expressed as H = ˙ aa , we can figure the evolution of the scale factor to be .4. EVOLUTION OF FRW UNIVERSE a ( t ) ∝ t ( + w ) (4.37)then the continuity equation (4.28) gives us the evolution for the density ρ ∝ a − ( + w ) (4.38)both the scale factor and the density are dependent on the constant w whichrepresents the composure of the components on the universe.The most typical examples are the ones were we assume the universe to be dom-inated by either non-relativistic matter (dust) or radiation, meaning relativisticparticles. In the case of dust we assume an equation of state w =
0, since non-relativistic matter should have zero pressure. In that case the scale factor anddensity evolve as a ( t ) ∝ t ρ ∝ a − A universe that evolves in that manner is called matter-dominated . In the case ofradiation the equation state can be proven to be w = (see (8.25) and (8.26) of[1] ). So in that case we get a ( t ) ∝ t ρ ∝ a − CHAPTER 4. THE FRW COSMOLOGY this corresponds to a radiation-dominated universe. These results are not coun-terintuitive. In the case of ordinary matter we expect the density to be inverselyproportional to the volume, since the mass remains always constant, and for anexpanding universe the volume is proportional to a . The same can be applied forradiation, but if we consider relativistic particles to behave as waves, then theirwavelength should also be a ff ected by the expansion of the universe by a factorequal to the scale factor a . Then since their energy is inversely proportional to thewavelength we expect the density to behave as ρ ∝ a − .If the universe started o ff with matter being dominant then the scale factor wouldevolve as a ( t ) ∝ t and the density of matter would evolve as ρ m ∝ a − ∝ t − where the density of radiation as ρ r ∝ a − ∝ t − and so matter would remaindominant. If we consider a radiation dominance in the early universe, a morepossible scenario considering that the early universe is characterized by high tem-peratures, the scale factor evolves as a ( t ) ∝ t and the corresponding densities as ρ m ∝ a − ∝ t − , ρ r ∝ a − ∝ t − . So matter should become dominant at somepoint in the evolution of the universe, a result that is also not counterintuitive aswe observe way more non-relativistic matter than radiation, in the form of galax-ies and dust clouds.Both of those cases result in a singularity for t = the Big-Bang [19].However both of those solution pose a problem which has to do with the expansionrate of the universe. From the acceleration equation (4.27) we can see that ¨ a > ρ + p <
0. So the acceleration occurs for w < − , which correspondsto a negative pressure. The latest observational evidence dictate that our universeis expanding in an accelerating way [20–25], so we are left with the hypothesis ofa ” vacuum − energy ” that came to be known as dark energy . This comes out of an .4. EVOLUTION OF FRW UNIVERSE w = − T µν ( Λ ) = λ g µν (4.39)where λ is a constant. This results to the continuity equation to be just ˙ ρ = ⇒ ρ = const .and by applying this result to the Friedmann equation we get H = − π G ρ ⇒ H = const .also since the Hubble parameter can be expressed in regard to the scale factor wecan once again calculate the evolution of the scale factor H = ˙ aa ⇒ daa = Hdt ⇒ a ( t ) ∝ e Ht a universe dominated by dark energy is called vacuum-dominated or dark energy-dominated and this result of exponential expansion is called de-Sitter universe.The same result can be obtained if we assume a cosmological constant to the Ein-stein equations.6 CHAPTER 4. THE FRW COSMOLOGY
We assume the Einstein field equations with a cosmological constant (3.34). Thenthe Friedmann equations then yield to H − π G ρ − Λ = − Ka (4.40) ¨ aa = − π G ( ρ + p ) + Λ ˙ a = and ¨ a =
0. In the case of matter-dominated universe, meaning p =
0, weget ρ = Λ π G and Λ = Ka for a matter-dominated universe ρ > Λ has to be positive as well, mean-ing K =
1, leading to a closed universe. Einstein had to abandon this idea inthe 1930’s though, as Hubble’s observations dictated an expanding universe. Fur-thermore this static solution can mathematically be proven to be unstable [27].Oddly enough the same mathematics can be used in order to explain the acceler-ating expansion of the universe, by connecting the cosmological constant with thedark energy. In a flat universe dominated by dark energy the equation of state is w = − G µν = π G (cid:16) T µν + T µν ( Λ ) (cid:17) (4.42) .5. THE COSMOLOGICAL CONSTANT MODEL λ = Λ π G and given that λ is associated with the energy density we get ρ Λ = Λ π G (4.43)the energy density appears to be constant, meaning that it is not a ff ected by theexpansion of the universe, and its pressure turns out to be negative. Furthermorefor a vacuum-dominated the Friedmann equation results in H = π G ρ Λ = Λ ⇒ a ( t ) ∝ e (cid:113) Λ t ending up with a similar exponential evolution for the expansion of the universe aswe expected for a vacuum-dominated universe. Additionally we can write downthe density parameter for the cosmological constant as Ω Λ ≡ ρ Λ ρ crit = Λ H (4.44)and so the first Friedmann equation can be written generalized as H = π G ( ρ + ρ Λ ) − Ka ⇒ Ω + Ω Λ − = Ka H (4.45)8 CHAPTER 4. THE FRW COSMOLOGY
We can now establish the standard cosmological model of Big Bang or Λ CDM .CDM comes from Cold Dark Matter (cold refers to low velocities) and Λ impliesthe existence of a cosmological constant. The standard model assumes that thetheory of General Relativity is correct and that the universe is composed of ordi-nary matter, dark matter (introduced in order to explain the rotation of galaxies)and dark energy in the form of a cosmological constant (introduced in order to ex-plain the accelerating expansion). The corresponding parameters of the standardmodel as observed by the Plank 2018 [28] are: • baryonic density parameter: Ω b = ± • cold dark matter density parameter: Ω c = ± • overall matter density parameter: Ω m = ± • dark energy density parameter: Ω Λ = ± • critical density[kg m − ]: ρ = (8.62 ± × − • Hubble constant [ km s − M pc − ]: H = ± Ω r ∼ × − .6. THE STANDARD MODEL OF COSMOLOGY a → κ a , where κ is a constant. So we can see thatthe very early universe was dominated by radiation, the energy of the particles isalso extremely high. As the universe expands and begins to cool the universe startsto become more matter-dominated, the CMB is the proof of the independence ofradiation from matter (decoupling). Today the universe has come to be dominatedby dark energy, explaining the accelerated expansion.The standard model of the Big Bang is successful at explaining our universe as weobserve it and has also made some significant predictions. Some of those are: • The expansion of our universe • The existance and spectrum of the Cosmic Microwave Background radiation (CMB) • The age of the observable universe • The ratios of the lightest atoms in the universe such as H , D , He , He and Li as well as the 3 di ff erent types (or flavors) of neutrinos.However, there are several things that cannot be explained within the standardmodel. Some of the most important problems that have arose are: • the flatness problem:The standard model does not predict a perfectly flat universe, insteadthe curvature K is a number really close to zero, but not zero. We can0 CHAPTER 4. THE FRW COSMOLOGY see from equation (4.45) that the case of K = Ω m + Ω Λ = a H ∝ t − meaning that | Ω m + Ω Λ − | ∝ t .So the case of K = • the horizon problem:Observations of the CMB show us inhomogeneities for the overalltemperature of the universe to be of a factor of 10 − , meaning that theuniverse appears to be in thermodynamic equilibrium. However, since thespeed that information can travel can not be greater than the speed of light,there is a finite distance that information can travel in order to causallyconnect two points in the universe. This distance is the horizon . We cancalculate that the horizon for the CMB corresponds to a distance of 1 o inthe sky. So the CMB should not appear to be isotropic. • the monopole problem:Grand unified theories that describe the unification of the fundamentalforces predict the existence of magnetic monopoles and other supersym-metric particles. These are created at the very early stages of the universeand should in theory dominate the present universe. However as of todaywe have been unable to observe any of those particles. .6. THE STANDARD MODEL OF COSMOLOGY • the fine tuning problem of the cosmological constant:Observational evidence shows that the vacuum energy density is compera-ble to the critical density, approximately ρ Λ (cid:39) ρ (cid:39) − GeV .However quantum field theory estimates the vacuum energy to be ρ Λ (cid:39) GeV leading to a di ff erence of 121 orders of magnitude. So ifthe cosmological constant is indeed connected to the vacuum energy thenit has to be very precisely adjusted to a really small, non-zero value, similarfine tuning problem as with the flatness of the universe.In order to address these problems we have to look for some extensions for thestandard model. The most noteworthy is the one that was proposed at the 1980’sby Alan Guth as well as Alexei Starobinsky and Andrei Linde called cosmologicalinflation .2 CHAPTER 4. THE FRW COSMOLOGY hapter 5Dark energy models
The basic idea of inflation is that at a very early stage the universe begun torapidly expand [29–37]. This rapid expansion is connected to a vacuum energy-dominated era and one of the simpler models to describe it is the cosmologicalconstant, which leads to an exponential increase for the scale factor .This immediately solves the flatness problem, since any initial value for thecurvature parameter K that is not significantly big will lead to a current value ofzero. It is believed that the inflation begun when the universe was of age 10 − sec and ended at 10 − sec. In this small amount the universe could have beenincreased in size up to roughly 10 times.Furthermore it also solves the horizon problem since the short period of timethat is required is su ffi cient to maintain thermodynamic equilibrium and at the endof the inflation the entire universe would be causally connected. And it can remainlike that until the decoupling where the CMB originates.Finally inflation solves the monopole problem as well, since after the inflationthe density of magnetic monopoles would drastically decrease and the universe734 CHAPTER 5. DARK ENERGY MODELS would become radiation-dominated, meaning that they would be extremely hardto be observed today.However inflation itself does not solve the fine tuning problem of the cosmo-logical constant. For that we have to assume that the value of the cosmologicalconstant is indeed zero and that dark energy can be explained via other mecha-nisms. In this section we will examine some other dark energy models[38–43].
Quintessence [44–48] is an attempt of describing dark energy with the help of ascalard field φ , as a scalar field can indeed lead to negative pressure. The corre-sponding action should be S φ = (cid:90) √− g (cid:32) − g αβ ∂ α φ∂ β φ − V ( φ ) (cid:33) d x (5.1)The term g αβ ∂ α φ∂ β = ( ∇ φ ) refers to the kinetic term and V ( φ ) is the potentialof the scalar field. The variation of this action leads us to an equation of motionsimilarly to the derivation of the Einstein field equations. δ S φ = − (cid:90) (cid:40) δ (cid:16) √− g (cid:17) (cid:32) g αβ ∂ α φ∂ β φ + V ( φ ) (cid:33) + √− g δ (cid:32) g αβ ∂ α φ∂ β φ + V ( φ ) (cid:33) (cid:41) d x = − (cid:90) (cid:40) δ (cid:16) √− g (cid:17) (cid:32) g αβ ∂ α φ∂ β φ + V ( φ ) (cid:33) + √− g δ (cid:16) g αβ ∂ α φ∂ β φ (cid:17) + √− g δ (cid:16) V ( φ ) (cid:17) (cid:41) d x .2. QUINTESSENCE δ (cid:16) √− g (cid:17) = − √− g g µν δ g µν δ (cid:16) V ( φ ) (cid:17) = V ( φ + δφ ) − V ( φ ) = ∂ V ∂φ δφ + O ( δφ ) (cid:39) ∂ V ∂φ δφδ (cid:16) g αβ ∂ α φ∂ β φ (cid:17) = ∂ α φ∂ β φδ g αβ + g αβ δ (cid:16) ∂ α φ∂ β φ (cid:17) and for the term δ (cid:16) ∂ α φ∂ β φ (cid:17) we use a similar product rule g αβ δ (cid:16) ∂ α φ∂ β φ (cid:17) = g αβ δ ( ∂ α φ ) ∂ β φ + g αβ δ ( ∂ β φ ) ∂ α φ = g αβ ∂ α ( δφ ) ∂ β φ + g αβ ∂ β ( δφ ) ∂ α φ = g αβ ∂ α φ∂ β δφ Furthermore we can notice that √− g g αβ ∂ α φ ∂ β δφ = ∂ β (cid:16) √− g g αβ ∂ α φ δφ (cid:17) − ∂ β (cid:16) √− g g αβ ∂ α φ (cid:17) δφ where ∂ β (cid:16) √− g g αβ ∂ α φ δφ (cid:17) leads to boundary terms after the integration, so aswe have seen before we can ignore those terms considering boundaries that go toinfinity. Thus we get the following from for the variation of the action6 CHAPTER 5. DARK ENERGY MODELS δ S φ = − (cid:90) (cid:40) − √− g g µν (cid:32) g αβ ∂ α φ∂ β φ + V ( φ ) (cid:33) δ g µν + √− g (cid:32) ∂ α φ∂ β φ (cid:33) δ g αβ (5.2) + (cid:32) − ∂ β (cid:16) √− g g αβ ∂ α φ (cid:17) + √− g ∂ V ∂φ (cid:33) δφ (cid:41) d x and from the principle of least action for the variation over the scalar field leads to δ S φ δφ = ⇒ − ∂ β (cid:16) √− g g αβ ∂ α φ (cid:17) + √− g ∂ V ∂φ = g = − a . So thederivative becomes ∂ β (cid:16) √− g g αβ ∂ α φ (cid:17) = g αβ ∂ α φ ∂ β √− g + √− g ∂ β g αβ ∂ α φ + √− g g αβ ∂ β ∂ α φ and by calculating the expressions that came up we get g αβ ∂ α φ ∂ β (cid:16) √− g (cid:17) = g ∂ φ ∂ √− g + g ii ∂ i φ ∂ i √− g (cid:16) ∂ i √− g = (cid:17) = − a ˙ a ˙ φ √− g ∂ β g αβ ∂ α φ = (cid:16) ∂ β g αβ = (cid:17) .2. QUINTESSENCE √− g g αβ ∂ β ∂ α φ = a (cid:16) − ¨ φ + a ∇ φ (cid:17) Hence the equation of motion for the scalar field takes the form ¨ φ + H ˙ φ − a ∇ φ + ∂ V ∂φ = H ˙ φ can be seen as a friction term for the momentum of the field due tothe expansion. Furthermore we can consider the field to be smooth across space,without any significant fluctuations. So comparing to the time derivative we canignore the spatial derivatives, then the above equation is just ¨ φ + H ˙ φ + ∂ V ∂φ = S tot = S H − E + S Matter + S φ the variation over the inverse metric would lead us to a similar result R µν − g µν R = π G (cid:32) T ( Matter ) µν + T ( φ ) µν (cid:33) (5.5)where we get an extra source term in the form of an energy-momentum part thatrefers to the scalar field. The definition for the energy-momentum tensor wouldalso be similar8 CHAPTER 5. DARK ENERGY MODELS T ( φ ) µν ≡ − √− g δ S φ δ g µν (5.6)and we end up with the following expression T ( φ ) µν = ∂ µ φ∂ ν φ − g µν (cid:32) g αβ ∂ α φ∂ β φ + V ( φ ) (cid:33) (5.7)from that we can calculate the corresponding energy density and pressure for thescalar field ρ φ = T = g T = ˙ φ − a (cid:16) ∇ φ (cid:17) + V ( φ ) p φ = T ii = g i j T i j = ˙ φ − a (cid:16) ∇ φ (cid:17) − V ( φ ) where we can again ignore the spatial derivatives and get a simpler form for theenergy density and pressure ρ φ = ˙ φ + V ( φ ) (5.8) p φ = ˙ φ − V ( φ ) (5.9)So the generalized Friedmann equations for the case of a flat universe are just .2. QUINTESSENCE H = π G (cid:16) ρ + ρ φ (cid:17) (5.10) ¨ aa = − π G (cid:104) ( ρ + ρ φ ) + ( p + p φ ) (cid:105) (5.11)and for a scalar field-dominated universe those equations yield to H = π G (cid:32) ˙ φ + V ( φ ) (cid:33) (5.12) ¨ aa = − π G (cid:32) ˙ φ − V ( φ ) (cid:33) (5.13)we can consider and equation of state w φ = p φ ρ φ = ˙ φ − V ( φ ) ˙ φ + V ( φ ) (5.14)where w φ is not generally constant. Given that the continuity equation becomes ˙ ρ φ + H ( ρ φ + p φ ) = ff erence being that w φ can be , generally, dependent of the scale factor ρ φ = ρ φ ( t ) e − (cid:82) ( + w φ ) daa (5.16)0 CHAPTER 5. DARK ENERGY MODELS in the quintessence model we assume the potential energy to be far greater thanthe kinetic energy of the field. In that limit the field satisfies the condition ˙ φ << V ( φ ) (5.17)so the equation of state falls into the vacuum-dominated FRW universe categoryof w φ = − ρ φ (cid:39) − V ( φ ) and p φ (cid:39) V ( φ ) (5.18)the continuity equation then dictates that the energy pressure of the scalar field,and consequently the pressure, would be constant, meaning that in that limit thepotential tends to become constant as well. Furthermore in that limit the Fried-mann equation yields to H φ (cid:39) π G V ( φ ) (5.19)thus a constant Hubble parameter, as we already know, leads to the followingexponential growth for the scale factor (de Sitter phase) a ( t ) ∝ e H φ ( t − t ) (5.20)The hypothesis that the potential energy is more potent than the kinetic energydoes not come without any basis. That is to assume a ”flat” scalar field wherewe ignore the term ¨ φ with the idea that even if this term started o ff with a high .3. K-ESSENCE H ˙ φ from the Klein-Gordon equation (5.4) would sooneror later make its value significantly smaller. In that case equation (5.4) becomes ˙ φ = − H ∂ φ V (5.21)in order to achieve an inflationary scenario the potential must be close to constant.So the term ∂ φ V has to be really small and thus we get the condition (5.17). In Quintessence model we had the requirement of the condition (5.17), meaningthat the inflation occurred due to the potential energy of a scalar field. How-ever there are inflationary models that instead, focus more so on the kinetic en-ergy named
K-inflation [49, 50], the most general form of which includes non-canonical terms. The corresponding dark energy models are called
K-essence [51 ? , 52].The Lagrangian that describes those models, in its general form, is S κ = (cid:90) √− g p ( φ , X ) d x (5.22)where the Lagrangian density p reflects purely on the pressure terms and is afunction of the field φ and the kinetic term X = − ( ∇ φ ) = − g αβ ∂ α φ ∂ β φ .We can derive the same action as in Quintessence (5.1) if we elect an energypressure for the action (5.22) of the form p ( φ , X ) = − ( ∇ φ ) − V ( φ ) (5.23)2 CHAPTER 5. DARK ENERGY MODELS so in that sense Quintessence is just a sub-case of K-essence. However K-essencemodels that have the most applications are those that focus purely on the kineticterms and ignore any potential-related contribution. These require the energy pres-sure to be of the form p ( φ , X ) = f ( φ ) p ( X ) , f ( φ ) > purely kinetic K-essence . That requirement ex-cludes any potential-oriented scenarios and consequently the Quintessence modelas it can be easily seen that the energy pressure (5.23) used in Quintessence doesnot satisfy the above condition.Considering only the contribution of kinetic terms and including non-canonicalones we get the action S X = (cid:90) √− g (cid:16) A ( φ ) X + B ( φ ) X + ... (cid:17) d x (5.25)and for small values of X we can ignore quadratic and higher order terms, so bykeeping only the terms up to the second order the Lagrangian density is just p ( φ , X ) = A ( φ ) X + B ( φ ) X (5.26)which falls under the category of K-essence if we consider a transformation forthe field φ old → φ new = (cid:90) φ old d φ (cid:114) B | A | (5.27) .3. K-ESSENCE p ( φ , X ) = f ( φ ) (cid:16) − X + X (cid:17) (5.28)the corresponding energy-momentum tensor is T µν = ∂ p ( φ , X ) ∂ X ∂ µ φ ∂ ν φ − p ( φ , X ) g µν (5.29)and so for the energy density we get ρ φ = X ∂ p ∂ X − p = f ( φ ) (cid:16) − X + X (cid:17) (5.30)the equation of state then is a function of only Xw φ = p φ ρ φ = − X − X (5.31)furthermore for the case of a flat FRW universe the Friedmann equations are H = π G f ( φ ) (cid:16) − X + X (cid:17) (5.32) ¨ aa = − π G ρ φ ( + w φ ) (5.33)so the accelerated expansion occurs for w φ < − which corresponds to values of X < . Additionally the case of a cosmological constant, meaning w φ = − X = .4 CHAPTER 5. DARK ENERGY MODELS
The case with an equation of state w < − phantom field [53–56]. This is also supported by observational data [57] and can not only giveus an inflationary scenario, but can also explain the current acceleration of theuniverse, since an energy form with equation of state w < − ff ects the universein its much later states, given the evolution equations of a flat FRW universe.In order to achieve the negative kinetic energy we take the opposite sign onthe kinetic term in the action (5.1). So we get S ph = (cid:90) √− g (cid:32) g αβ ∂ α φ∂ β φ − V ( φ ) (cid:33) d x (5.34)This leads to the following energy-density and energy-pressure values ρ ph = − ˙ φ + V ( φ ) (5.35) p ph = − ˙ φ − V ( φ ) (5.36)and to the equation of state w ph = p ph ρ ph = ˙ φ + V ( φ ) ˙ φ − V ( φ ) (5.37)which allows the case of w < − ˙ φ < V ( φ ) . .4. PHANTOM FIELD H = π G (cid:32) − ˙ φ + V ( φ ) (cid:33) (5.38) ¨ aa = − π G (cid:32) − ˙ φ − V ( φ ) (cid:33) (5.39) ¨ φ + H ˙ φ + ∂ V ∂φ = w < − the Big Rip [58], a singularity where the scale factor diverges after a finite amount of time.For that, a more generalized form of the phantom field can be introduced, oneinspired by the K-essence models, with non-canonical kinetic terms being intro-duced. This allows for the existence of a theory that predicts an energy form withnegative kinetic energy and given the equation of state (5.31) can see that thesescenarios are acceptable.6 CHAPTER 5. DARK ENERGY MODELS hapter 6Epilogue
In this thesis we have examined only but a few possible scenarios that can justifyand describe dark energy, but there is an even larger family of models that tries toaccomplish the same. The wide range of those models can be due to our lack ofunderstanding the very nature of dark energy and thus the e ff ort for such a theoryto emerge is continuous, with some believing that the answer lies with QuantumGravity [59] or
Supergravity [60–63]. However this road seems to be long andhard which makes it one of the most exciting problems in our generation.
I want to thank my supervisor Prof. Smaragda Lola for introducing me to this topicand my Co-advisor PhD Student Andreas Lymperis for helping me throughout theconstruction of this thesis, as well as the Andreas Mentzelopoulos foundation fortheir financial support. 878
CHAPTER 6. EPILOGUE ibliography [1] Sean M. Carroll. Lecture notes on general relativity. 12 1997.[2] Bernard F. Schutz.
A FIRST COURSE IN GENERAL RELATIVITY . Cam-bridge Univ. Pr., Cambridge, UK, 1985.[3] John D. Barrow, George F.R. Ellis, Roy Maartens, and Christos G. Tsagas.On the stability of the Einstein static universe.
Class. Quant. Grav. , 20:L155–L164, 2003. doi: 10.1088 / / / / Phys. Atom. Nucl. , 73:815–847,2010. doi: 10.1134 / S1063778810050091.[5] J. Plebanski and Andrzej Krasinski.
An introduction to general relativity andcosmology . 10 2006.[6] B. Ryden.
Introduction to cosmology . Cambridge University Press, 11 2016.ISBN 978-1-107-15483-4, 978-1-316-88984-8.[7] Jean-Philippe Uzan. The big-bang theory: construction, evolution and status.6 2016.[8] Ruth Durrer. The cosmic microwave background: the history of its experi-mental investigation and its significance for cosmology.
Class. Quant. Grav. ,32(12):124007, 2015. doi: 10.1088 / / / / BIBLIOGRAPHY [9] Martin J. White and J.D. Cohn. TACMB-1: The Theory of anisotropies inthe cosmic microwave background. 3 2002.[10] D.J. Fixsen. The Temperature of the Cosmic Microwave Background.
As-trophys. J. , 707:916–920, 2009. doi: 10.1088 / / / / Cosmology . 9 2008. ISBN 978-0-19-852682-7.[12] Steven Weinberg.
Gravitation and Cosmology: Principles and Applicationsof the General Theory of Relativity . John Wiley and Sons, New York, 1972.ISBN 978-0-471-92567-5, 978-0-471-92567-5.[13] Edward W. Kolb. A view of the early universe.
AIP Conf. Proc. , 478(1):3–9,1999. doi: 10.1063 / An introduction to modern cosmology . 1998.[15] A. Friedmann. On the Possibility of a world with constant negative curvatureof space.
Z. Phys. , 21:326–332, 1924. doi: 10.1007 / BF01328280.[16] A. Friedman. On the Curvature of space.
Z. Phys. , 10:377–386, 1922. doi:10.1007 / BF01332580.[17] G. Hinshaw et al. Nine-Year Wilkinson Microwave Anisotropy Probe(WMAP) Observations: Cosmological Parameter Results.
Astrophys. J.Suppl. , 208:19, 2013. doi: 10.1088 / / / / Nature , 404:955–959, 2000. doi:10.1038 / IBLIOGRAPHY
Rev. Mod. Phys. , 75:559–606, 2003. doi: 10.1103 / RevModPhys.75.559.[21] Alexei V. Filippenko and Adam G. Riess. Evidence from type Ia supernovaefor an accelerating universe.
AIP Conf. Proc. , 540(1):227–246, 2000. doi:10.1063 / Astron. J. , 116:1009–1038,1998. doi: 10.1086 / Astrophys. J. , 507:46–63, 1998. doi: 10.1086 / Ω and Λ from 42 high redshift super-novae. Astrophys. J. , 517:565–586, 1999. doi: 10.1086 / Science ,284:1481–1488, 1999. doi: 10.1126 / science.284.5419.1481.[26] Naresh Dadhich. Universality, gravity, the enigmatic Lambda and beyond. 52004.[27] A.S. Eddington. On the Instability of Einstein’s Spherical World. Mon. Not.Roy. Astron. Soc. , 90:668–678, 1930.[28] N. Aghanim et al. Planck 2018 results. VI. Cosmological parameters.
Astron.Astrophys. , 641:A6, 2020. doi: 10.1051 / / BIBLIOGRAPHY [29] Andrew R. Liddle. An Introduction to cosmological inflation. In
ICTP Sum-mer School in High-Energy Physics and Cosmology , pages 260–295, 1 1999.[30] Alan H. Guth. The Inflationary Universe: A Possible Solution to the Horizonand Flatness Problems.
Adv. Ser. Astrophys. Cosmol. , 3:139–148, 1987. doi:10.1103 / PhysRevD.23.347.[31] Alan H. Guth and S.Y. Pi. Fluctuations in the New Inflationary Universe.
Phys. Rev. Lett. , 49:1110–1113, 1982. doi: 10.1103 / PhysRevLett.49.1110.[32] Leonardo Senatore. Lectures on Inflation. In
Theoretical Advanced Study In-stitute in Elementary Particle Physics: New Frontiers in Fields and Strings ,pages 447–543, 2017. doi: 10.1142 / Adv. Ser. Astrophys. Cosmol. , 3:149–153, 1987. doi:10.1016 / Phys. Lett. B , 129:177–181, 1983. doi:10.1016 / Living Rev. Rel. , 13:3, 2010. doi: 10.12942 / lrr-2010-3.[37] Smaragda Lola, Andreas Lymperis, and Emmanuel N. Saridakis. Inflationwith non-canonical scalar fields revisited. 5 2020. IBLIOGRAPHY
Int. J. Mod. Phys. D , 15:1753–1936, 2006. doi: 10.1142 / S021827180600942X.[39] Michel Chevallier and David Polarski. Accelerating universes with scal-ing dark matter.
Int. J. Mod. Phys. D , 10:213–224, 2001. doi: 10.1142 / S0218271801000822.[40] Kazuharu Bamba, Salvatore Capozziello, Shin’ichi Nojiri, and Sergei D.Odintsov. Dark energy cosmology: the equivalent description via di ff er-ent theoretical models and cosmography tests. Astrophys. Space Sci. , 342:155–228, 2012. doi: 10.1007 / s10509-012-1181-8.[41] Pedro G. Ferreira and Michael Joyce. Cosmology with a primordial scalingfield. Phys. Rev. D , 58:023503, 1998. doi: 10.1103 / PhysRevD.58.023503.[42] Varun Sahni, Tarun Deep Saini, Alexei A. Starobinsky, and Ujjaini Alam.Statefinder: A New geometrical diagnostic of dark energy.
JETP Lett. , 77:201–206, 2003. doi: 10.1134 / Eur. Phys. J. C , 78(12):993,2018. doi: 10.1140 / epjc / s10052-018-6480-y.[44] Sean M. Carroll. Quintessence and the rest of the world. Phys. Rev. Lett. ,81:3067–3070, 1998. doi: 10.1103 / PhysRevLett.81.3067.[45] Bharat Ratra and P.J.E. Peebles. Cosmological Consequences of a RollingHomogeneous Scalar Field.
Phys. Rev. D , 37:3406, 1988. doi: 10.1103 / PhysRevD.37.3406.4
BIBLIOGRAPHY [46] R.R. Caldwell, Rahul Dave, and Paul J. Steinhardt. Cosmological imprintof an energy component with general equation of state.
Phys. Rev. Lett. , 80:1582–1585, 1998. doi: 10.1103 / PhysRevLett.80.1582.[47] A. Lymperis, L. Perivolaropoulos, and S. Lola. Sudden Future Singularitiesin Quintessence and Scalar-Tensor Quintessence Models.
Phys. Rev. D , 96(8):084024, 2017. doi: 10.1103 / PhysRevD.96.084024.[48] Andreas Lymperis. Sudden Future Singularities and their observational sig-natures in Modified Gravity.
PoS , CORFU2017:088, 2018. doi: 10.22323 / Phys. Lett. B , 458:209–218, 1999. doi: 10.1016 / S0370-2693(99)00603-6.[50] C. Armendariz-Picon, Viatcheslav F. Mukhanov, and Paul J. Steinhardt. ADynamical solution to the problem of a small cosmological constant andlate time cosmic acceleration.
Phys. Rev. Lett. , 85:4438–4441, 2000. doi:10.1103 / PhysRevLett.85.4438.[51] Takeshi Chiba, Takahiro Okabe, and Masahide Yamaguchi. Kineticallydriven quintessence.
Phys. Rev. D , 62:023511, 2000. doi: 10.1103 / PhysRevD.62.023511.[52] C. Armendariz-Picon, Viatcheslav F. Mukhanov, and Paul J. Steinhardt.Essentials of k essence.
Phys. Rev. D , 63:103510, 2001. doi: 10.1103 / PhysRevD.63.103510.[53] Sean M. Carroll, Mark Ho ff man, and Mark Trodden. Can the dark energyequation-of-state parameter w be less than − Phys. Rev. D , 68:023509,2003. doi: 10.1103 / PhysRevD.68.023509.
IBLIOGRAPHY
Phys. Lett. B , 545:23–29, 2002. doi:10.1016 / S0370-2693(02)02589-3.[55] Parampreet Singh, M. Sami, and Naresh Dadhich. Cosmological dynamicsof phantom field.
Phys. Rev. D , 68:023522, 2003. doi: 10.1103 / PhysRevD.68.023522.[56] M. Sami and Alexey Toporensky. Phantom field and the fate of universe.
Mod. Phys. Lett. A , 19:1509, 2004. doi: 10.1142 / S0217732304013921.[57] Pier Stefano Corasaniti, M. Kunz, David Parkinson, E.J. Copeland, andB.A. Bassett. The Foundations of observing dark energy dynamics with theWilkinson Microwave Anisotropy Probe.
Phys. Rev. D , 70:083006, 2004.doi: 10.1103 / PhysRevD.70.083006.[58] Robert R. Caldwell, Marc Kamionkowski, and Nevin N. Weinberg. Phantomenergy and cosmic doomsday.
Phys. Rev. Lett. , 91:071301, 2003. doi: 10.1103 / PhysRevLett.91.071301.[59] Rafael D. Sorkin. Forks in the road, on the way to quantum gravity.
Int. J.Theor. Phys. , 36:2759–2781, 1997. doi: 10.1007 / BF02435709.[60] D. Bailin and A. Love.
Supersymmetric gauge field theory and string theory .10 1994.[61] Hans Peter Nilles. Supersymmetry, Supergravity and Particle Physics.
Phys.Rept. , 110:1–162, 1984. doi: 10.1016 / Rev. Mod. Phys. ,61:1–23, 1989. doi: 10.1103 / RevModPhys.61.1.6
BIBLIOGRAPHY [63] John R. Ellis, Dimitri V. Nanopoulos, and K. Tamvakis. Grand Unificationin Simple Supergravity.
Phys. Lett. B , 121:123–129, 1983. doi: 10.1016 //