AnnoAccademico2013/2014 ManuelColavincenzo Dott.LuigiTedesco Laureando : Relatore : NonLinearElectrodynamicsEﬀectsofAnisotropicExpansionofTheUniverse UniversitàdegliStudidiBari"AldoMoro"CorsodiLaureainFisica

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Università degli Studi di Bari "Aldo Moro"

Corso di Laurea in Fisica

TESI DI LAUREA MAGISTRALE

Non Linear Electrodynamics

Effects of Anisotropic Expansion

of The Universe

Laureando:

Manuel Colavincenzo

Relatore:

Dott. Luigi Tedesco

Anno Accademico 2013/2014

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1

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2

Your future hasn’t been written yet.

No one’s has.

Your future is whatever you make it, so make it a good one.

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3

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4

Ai miei genitori e a mia Sorella.

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List of Figures

1.1 Flat space-time . . . 3

1.2 Curved space-time . . . 3

2.1 Observer 1 looks at different directions and sees the same sky . . . 15

2.2 Observer 1,2 and 3 “see” pretty much the same sky . . . 16

2.3 Sky for the observer 2 is different from the one of observer 1 . . . . 16

2.4 Trend of density radiation and density matter. The intersection represents the equilibrium era . . . 21

2.5 Photon travelling to the origin. . . 22

3.1 Artististic impression of a “cosmic foam” Universe. . . 31

3.2 Cosmic string . . . 32

3.3 Magnetic field data from the Whirlpool Galaxy, M51 . . . 32

4.1 ^Equantum energy divided by the energy E of a single photon. . . 55

4.2 Quasars (QUASi-stellAR objects) lie near the edge of the observable Universe. This gallery of quasar portraits from the Hub- ble Space Telescope offers a look at their local neighborhoods: the quasars themselves appear as the bright star-like objects with diffraction spikes. The images in the center and right hand columns reveal quasars associated with disrupted colliding and merging galaxies which should provide plenty of debris to feed a hungry black hole. 56 5.1 Dark Energy is responsible for the accelerated expansion of the Universe . . . 59

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Mathematical Notations

In this thesis we use the index 0 for temporal coordinate and the indices 1,2,3 for spatial coordinates. For the metric gµν of space-time we adopt the following signature:

g_µν = diag(+, −, −, −).

The mathematical conventions for geometrical objects are the following:

Riemann Tensor:

R_µνα^β = ∂_µΓ_να^β+ Γ_µρ^βΓ_να^ρ− (µ ↔ ν);

Ricci tensor:

R_να = R_µνα^µ; covariant derivative:

∇_µV^α = ∂_µV^α+ Γ_µβ^αV^β and

∇_µV_α = ∂_µV_α− Γ_µα^βV_β.

The symbol indicates usual d’Alambert operator in the Minkowski space, that is

≡ η^µν∂_µ∂_ν = 1 c²

∂²

∂t² − ∇²

where η is the Minkowski metric and ∇² = δ^ij∂_i∂_j, the Laplacian operator in the Euclidean three-dimensional space. We use latin lower-case letters i, j, k . . . to indicate spatial indexes 1,2,3; the greek lower-case letters µ, ν, α . . . to indicate space-temporal indexes 0,1,2,3. The indices in round

iv

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Mathematical Notation v or square brackets, respectively, satisfy the simmetry or antisimmetry properties defined by the rule:

T_(αβ)≡ 1

2(T_αβ+ T βα) and

T_[αβ]≡ 1

2(T_αβ− T_βα).

It’s possible to extend the process of symmetrization or antisymmetrization, with n ≥ 2 (n number of indices), taking into account all the possible per- mutations and dividing for the number of permutation n!. If an object has more than two indices, and the indeces to be symmetrize or antysimmetrize are not neighboring, that indices will be separated from the other by with a vertial bar. For example:

T_(µ|α|ν) ≡ 1

2(T_µβν+ T_ναµ) and

T_[µ|α|ν] ≡ 1

2(T_µβν− T_ναµ).

At last, the totally antisimmetric tensor (Levi-Civita symbol) in the Minkowski space, ^µναβ, is defined with the following convention:

⁰¹²³ = +1 , µναβ = −^µναβ.

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Introduction

As an object of speculation and philosophical thought, often integrated with religious and mytical ideas, cosmology is as old as humankind. The practice of this kind of cosmology, may even be said to be so throughly integrated with basic mental characteristic of the human race that it defines us as being human [1]. For this reason, cosmology is, first of all, free speculation on the world and Universe, rappresenting the natural interest of human being in natural phenomena. This type of cosmology is released by any mathematical and physical bond; using the words of H. Kragh, this is cosmo-mythology [2].

The period of cosmo-mythology, from the ancient Egyptians to the birth of scientific cosmology, was very important for it heavily contribuited to the development of scientific, philosophical and religious thought. Only at the baginning of 20th century cosmology will turn from cosmo-mythologies into scientific cosmology.

Starting with Albert Einstein and his theory of general relativity, sci- entists begin to insert free speculation in a mathematical framework that automatically imposes boundary to the speculation itself. An other impor- tan improvement involves the possibility to compare theoretical models with the experimentals data; in fact with the development of technology it become possible to measure more and more precisely weak and elusive phenomena, as CMB(Cosmic Microwave Background) among the most rappresentative ones. This is, in my opinion, the foundamental step that allowed cosmology to become a real science.

A clue of the evolution of cosmology into a real science is the formula- tion of the Standard Model of cosmology; as an other branch of physics, for example particle physics with standard model which describes strong, electromagnetic and weak interactions and all the elementary particle involved, also cosmology needs a conceptual rappresentation of the real word, capable to explain its functioning.

Around 1920s Aleksandr Friedmann and, indipendently, Georges Lemaître, established that, at the biginning, the Universe was extremely hot and dense.

Furthermore, on very large scale, the Universe nowadays is omogeneus and

vi

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Introduction vii

isotropic (Cosmological Principle), on average. A few years later, Edwin Hubble will prove the expansion of the Universe, thanks to observation of redshift of galaxy.

The standard model, as one can easily imagine, isn’t perfect, and around 1980s the idea that the Universe may have undergone a period of inflation, during which its expansion rate accelerated and any initial inhomogeneities were smoothed out, was put forward. Inflation provides a model which can, at least in theory, explain how such homogeneity might have arisen and it does not require the introduction of the Cosmological Principle. The idea of inflation guarantees the presence of small fluctuations in the cosmological density, needed to produce the gravitational instability at the base of the origin of galaxies and other complex structures.

This developments of cosmology are not the most recent, in fact, in this thesis we discuss the so called alternative cosmology, in particular, the anisotropic cosmology. The standard cosmology treats the small density variation like perturbation of Friedmann standard model and therefore the results are in- herently approximate. The small variations are treated perturbatively because of the cosmological principle of isotropy and homogenity, but the Uni- verse could be neither homogeneous or isotropic and one should solve Ein- stein’s equations exactly. Only for a few anisotropic model is possible to find an exact solution. [3]

The structure of the thesis is the following:

In the first two chapters I will analyze the transition from special relativity to the general relativity, with the introduction of curved space-time, and the Standard cosmological model.

In Chapter 3 we discuss the anisotropic model and in general about cosmology beyond standard model of the Universe.

In chapter 4 we study the electrodynamic in curved space and in particular in FLRW and Bianchi I Universe.

In chapter 5 we discuss about the generalized Maxwell’s equations for anisotropic model of the Universe and in particular we use the examples of Lagrangian in order to test the theory.

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Chapter 1 Curved Space-Time

1.1 Theory of General Relativity

Transistion from special relativity(SR) to general relativity(GR) [4] is necessary to study the theory of gravitation in a relativistic framework. The first matter is the identification of the type of the physical quantity to describe, in a relativistic fashion, the gravitational theory. The Newtonian’s theory is not relativistic because it gives gravitational information about two static masses, but, in analogy with electromagnetism, one may presume that the behavior of two masses in motion is different.

The first idea, in analogy with Coulomb force, is to build a vectorial theory with a gravitational potential φ and a vector potential ~A. If we follow the vectorial idea, the result would be a repulsive force for “charges” of same sign and a attractive one for “charges” of opposite sign. This is the general result of vectorial theory. The presence of two opposite forces leads up to drop the vectorial theory because, the gravitational force is only actractive.

The second possibility is a scalar theory, in this case Newtonian’s potential is equivalent to a Lorentz scalar. This second assumption is experimentally incoherent because it’s possibile to predict the precession of the perihelion of Mercury, but the numerical value disagreed with the experimental data.

Once we have eliminated all the alternatives, the right one remains: the gravitational theory must be described by a tensorial theory. The tensorial language allows to describe the gravitation from a geometrial point of view too.SR is based on two foundamental principles:

• the laws of physic are the same in all inertial frames of reference 1

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Chapter 1. Curved Space-Time 2

• the speed of electromagnetic signal, in vacuum, is the same (in modu- lus) in all inertial frames of reference.

GR requires a generalization of the first point:

• the laws of physic are the same in all frames of reference

The difference between the two principles is that GR allows non inertial frame, too. A direct consequence of this idea is the so-colled principle of general covariance. This is the spirit of general relativity; the flat Minkowski space-time will be replaced with a more generic geometry.

We consider a generic coordinate transformation:

x^µ → x^0µ =⇒ dx^µ= ∂x^µ

∂x^0νdx^0ν (1.1)

with _∂x^∂x^0ν^µ = (J⁻¹)^µ_ν Jacobian.

The line element in the new coordinates will be:

ds² = η_µνdx^µdx^ν = η_µν∂x^µ

∂x^0α

∂x^µ

∂x^0βdx^0αdx^0β. (1.2) At this point it’s possible to define a new generic metric:

g_αβ = η_µν∂x^µ

∂x^0α

∂x^µ

∂x^0βdx^0αdx^0β = g_αβ(x⁰). (1.3) The new element of line, after coordinate transformation, will be:

ds² = g_αβ(x⁰)dx^0αdx^0β. (1.4) From eq. (1.3) and eq. (1.4) it’s possible to realize that, in contrast to the Euclidean geometry (fig.1.1), this new geometry changes locally from point to point (fig. 1.2). The geometry used in the description of GR is the Rie- mann geometry.

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Figure 1.1: Flat space-time

Figure 1.2: Curved space-time It is useful to consider two work hypothesis:

1. ds² is a quadratic homogeneous form in the coordinates differential: ds² = g_µνdx^µdx^ν

2. ds² is invariant for a general coordinates transformation eq. (1.1): ds² = ds⁰².

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1.2 Equivalence Principle

One of the most important principles is the so called "Principle of equivalence" and its earliest form known at the as the Weak equivalence Principle, that states that "inertial mass" and "gravitational mass" of any object are the same. On the other hand, the equivalence principle says that in small enough regions of space-time, the laws of physics reduce to those of special relativity: it is impossible to detect the existence of the gravitational field".

This principle assures that it is always possible to cancel the gravitational field, locally, exerting a force equal and opposite to the gravitational force.

It is possible to apply this principle only to the gravitational interactions, because gravity acts on all bodies universally, e.g. the electromagnetic field acts in a different manner on two opposite charges, therefore, even locally, it’s impossible to eliminate electromagnetic field because we should apply two different opposite forces on the two charges. Using the famous example of the free fall elevator, a test body in the elavetor will fluctuate in the air without measure any gravitational field. The geometry of space-time is affected by the equivalence, but locally, it’s should be possible to restore the Minkowski geometry. At this point, the first work hypothesis eq. (1) intervenes, because thanks to quadratic homogeneous form, the Riemann’s geometry shrinks to the flat Minkowski space-time. Ultimately, the two work hypotheses are not the most general hypothesis, but they are indispensable physical requests.

Equivalance Principle ⇐⇒ ∃ x^µ→ x^0µ : gµν(x0) = ηµν

The transformation x^µ→ x^0µ always exists.

1.3 Minimal coupling

General Relativity requires general coordinates transformations, hence the equations of physical systems must be covariant for general transformation of coordinates. Moreover, equivalence principle must always hold true, therefore the equations of special relativity holds true in the setting of general relativity, for an opportune choice of inertial card and in a space-time region that is “small” enough.

This properties are satisfied adopting the minimal coupling prcedure, which

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Chapter 1. Curved Space-Time 5 consists of 3 steps:

• metric tensor g replaces metric tensor η; this means that scalar product will be performed with the new coordinates dipendent tensor

η → g

• the usual derivative operator ∂µ, must be replaced with the operator of covariant derivative ∇µ. This step is necessary because the transition to general transformations of coordinates implies that the differential of a vector doesn’t change oneself like a vector, so it is necessary to restore the invariance with the introduction of the affine connection.

∂_µ→ ∇_µ

• it’s necessary to work with the usual tensor, but general transformation brings to the definition of tensorial density, a geometrical object more general than usual tensors that dipends on the transformation determinant. One should multiplay the tensorial density of weight w by (−g^w/2) to keep the covariance.

density of weight w → (−g^w/2) ∗ (density of weight w).

1.4 Presence of Gravitational Field and Curva-

ture Tensor

How is it possible to distinguish, without ambiguity in the choice of the coordinates, the presence of the gravitational field?

To answer to this question, we start from the equivalence principle: it’s always possible to eliminate, locally, the gravitational field; the equivalance principle is not valid when we consider two test bodies; using again the example of the elavetor, initially, the behavior of the two bodies in free fall will be the same as in absence of gravity, but after a while the two body will acquire a relative acceleration and their trajectories will cross. This effect is due to the spherical symmetry of the Earth’s gravitation field. Then:

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in presence of gravitational field, it is impossible to eliminate the acceleration between two distinct points, no matter how close they are, in the same instant of time.

Between two distinct points that move along two distinct geodetics, there is always be a relativity acceleration that bends the trajectory. This phe- nomenon can be used to mark out the presence of the gravitational field:

the acceleration between two points on two different geodetics, dipends only by the distorsion of the trajectories by the gravitational field and distinguish, without ambiguity, the presence of the field itself.

Two geodetics with a relativity acceleration ¨η^µ have the following equations:

¨

x^µ+ Γ_αβ^µ(x) ˙x^α˙x^β = 0 (1.5a)

¨

x^µ+ ¨η^µ+ Γ_αβ^µ(x)( ˙x^α+ ¨η^α)( ˙x^β+ ¨η^β) = 0 (1.5b)

We assume that the relative motion of the second geodetic in respect to the first, is very small so that x^µ= x^µ+ δx^µ = x^µ+ η^µ; using this expression in the second equation of eq. (1.5), and expanding Γ we have:

¨

x^µ+ ¨η^µ+ Γ_αβ^µ(x) ˙x^α˙x^β + 2.Γ_αβ^µ(x) ˙x^αη˙^β + η^ν∂_νΓ_αβ^µ(x) ˙x^α˙x^β = 0 (1.6)

Therefore, by taking advantage of the first of eq. (1.5) we have:

¨

η^µ+ 2Γ_αβ^µ(x) ˙x^αη˙^β + η^ν∂νΓ_αβ^µ(x) ˙x^α˙x^β = 0. (1.7) This last equation states the geodetic’s deviation. As we can see it’s impossible to eliminate, locally, the effect of the gravitational field due to the presence of the derivative of the connection.

It’s useful to express eq. (1.7) in an explicit covariant form, because only in this way it’s possible to make explicit the curvature tensor.

The explicit covariant form is obtained using the covariant derivative instead of the usual derivative:

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Dη^µ

dτ = ˙η^µ+ Γ_αβ^µ˙x^αη^β (1.8)

D²η^µ dτ² = d

dτ

˙

η^µ+ Γ_αβ^µ˙x^αη^β

+ Γ_λσ^µ˙x^λDη^σ dτ =

¨

η^µ+ Γ_αβ^µ(¨x^αη^β + ˙x^αη˙^β) + ˙x^ν∂_ν(Γ_αβ^µ) ˙x^αη^β + Γ_λσ^µ˙x^λ( ˙η^σ+ Γ_αβ^σ˙x^αη^β) (1.9)

Inserting eq. (1.7) and the first eq. (1.5) equation in eq. (1.9) we have:

− ˙x^α˙x^βη^ν[∂_νΓ_αβ^µ+ Γ_ρν^µΓ_αβ^ρ− ∂_αΓ_νβ^µ− Γ_αρ^µΓ_νβ^ρ]. (1.10) Finally the deviation equation in explicit covariant form is

D²η^µ

dτ² = −R_ναβ^µ˙x^α˙x^βη^ν (1.11) where

R_ναβ^µ= ∂_νΓ_αβ^µ+ Γ_ρν^µΓ_αβ^ρ− ∂_αΓ_νβ^µ− Γ_αρ^µΓ_νβ^ρ (1.12) is the curvature tensor of Riemann of rank four.

The curvature tensor is the "object" that allows to distinguish a flat variety from a curved one.

The vanishing of the Riemann’s tensor is a necessary and sufficient condition for the existence of a coordinates transformation that reduces the metric to the Minkowski one in every point.

If R = 0 the metric is an accelerated "parametrization" of the Minkowski space then there isn’t any type of deviation between the geodetics.

At this point we can answer to the question of this section: it’s possible to identify the physical effects of the gravitational interaction, with the curvature and so with the geometric distorsion of the space-time. The next step is the description of the source of the curvature and the evolution of the curvature through time.

The main properties of the Riemann tensor are:

• Rµναβ = R_[µν]αβ

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• Rµναβ = R_[µν][αβ] → riduces the component from 4⁴ to 36

• R_[µνα]^β = 0 I Bianchi identity, reduces the component to 20

• ∇[λR_[µν]α^β II Bianchi identity

• [∇α, ∇_β]A^µ = (∇_α∇_β− ∇_β∇_α)A^µ= R_αβν^µA^ν.

The last property is the most important because, generally speaking the com- mutator of the covariant derivative is non null. The gravity is the reason; if R = 0 the gravitational field is null in fact the covariant derivative reduces to the ordinary derivative and [∂µ, ∂_ν] = 0.

1.5 Maxwell’s equation in curved space

The starting point is the description of the coupling of electromagnetic field to the geometry. For special relativity, the electromagnetic field si described by a tensor:

F_µν = ∂_µA_ν − ∂_νA_µ (1.13) with Fµν = −F_νµ and Aµ potential. With minimal coupling electromagnetic tensor in Riemann curved space-time becomes:

F˜_µν = ∇_µA_ν − ∇_νA_µ. (1.14)

It’s useful to underline that we consider a simmetrical and metrical compat- ible connection (Christoffel connection), therefore:

Γ_µν^α = 1

2g^αρ(∂_µg_ρν + ∂_νg_µρ− ∂_ρg_µν) = Γ_νµ^α. (1.15) We have:

F˜_µν = F_µν. (1.16)

From eq. (1.16) it’s possible to infer that the coupling of electromagnetic

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Chapter 1. Curved Space-Time 9 field with geometry is the same of Minkowski geometry one. It’s important to specify that the invariance of electromagnetic field excludes the form of connection and it lies in the structure of potential Aµ.

The invariance of Fµν leads to two important outcomes:

1. the invariance of Maxwell’s equations that describes the divergence of magnetic field and the rotational of electric field

∇_[αF_µν]= 0 = ∂_[αF_µν] (1.17) 2. Fµν is invariant for gauge transformation Aµ → Aµ+ ∂µf.

Thanks to the point 2, the conservation of electrical charge, indipendently of the geometry, is ensured. From the variation of action

S = − Z

Ω

d⁴x[ 1

16πF_µνF^µν +1 c

J˜^µA_µ] (1.18) we obtain the relation of charge conservation in a curved geometry

∂µ(√

−g ˜J^µ) = 0. (1.19)

Minimal coupling doesn’t leave unchanged the Maxwell equations which describe the divergence of electric field and rotational of magnetic field, that are the equations of the dynamic of the system. Starting from the action eq.

(1.18), the Eulero-Lagrange equations are:

∂_µ∂(√

−gL(A, ∂A))

∂(∂_µA_ν) = ∂(√

−gL(A, ∂A))

∂A_ν , (1.20)

therefore

∂_µ(√

−g F^µν) = 4π c

√−g ˜J^ν. (1.21)

1.6 Einstein’s equations

The Einstein’s equations represent the equation for a relativistic gravitational field. The starting point is the action, in this case the gravitational action:

S_g = Z

Ω

d⁴x√

gL_g(g, ∂g, ...). (1.22)

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The expression eq. (1.22) represents the most general action for a gravitational field. Einstein’s action is the simple choice

S_E = − Z

Ω

d⁴x√

−gR

χ (1.23)

the minus is due to the metric convenction. χ comes from dimensional reasons and to controll the intensity of the coupling between matter and geometry.

R is the curvature scalar (see mathematical notation iv ).

The action that represents the source is S_m =

Z

Ω

d⁴x√

−gL_m(ψ, ∇ψ, g). (1.24)

Total action is the sum of eq. (1.23) and eq. (1.24)

S = SE + Sm. (1.25)

Actually, it’s necessary an additional contribution to the action eq. (1.25), in fact

R ∼ ∂Γ + Γ² Γ ∼ ∂g therefore

R ∼ (∂g)²+ ∂²g. (1.26)

This is a problem because Lagrangian will include second order metric derivative and, therefore, there will be a boundary term e.g.

Z

Ω

d⁴x ∂²g = Z

∂Ω

dSµ ∂g (1.27)

and it is not possible to remove this contribution using the condition δg|_∂Ω= 0. The problem is solved by the introduction of a boundary action

S_{Y GH} = Z

∂Ω

dS_µ√

−gV^µ (1.28)

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so that δgS_{Y GH} ∼ (−)R

∂Ω∂g.

Finally the correct total action is the sum of eq. (1.25) and eq. (1.28)

S = S_E+ S_m+ S_{Y GH}. (1.29)

At this point it is necessary to use the principle of minimal action

δgS = δgSE+ δgSY GH+ δgSm. (1.30) The variation is at fixed x and ψ. The first contribution of eq. (1.30) is

δ_gS_E = −δ_g Z

Ω

d⁴x√

−gR

χ = −1 χ

Z

Ω

d⁴xδ_g(R√

−g). (1.31) Using the definition of scalar curvature R = g^µνR_µν, the integral becomes

−1 χ

Z

Ω

d⁴x√

−g[R_µνδg^µν+ g^µνδ(R_µν) − R

2g_µνδg^µν]. (1.32) The second term in the eq. (1.32) generates the boundery contribution, therefore:

δ_gS_E+δ_gS_{Y GH} = −1 χ

Z

Ω

d⁴x√

−g(R_µν−1

2g_µνR)δg^µν = −1 χ

Z

Ω

d⁴x√

−gG_µνδg^µν (1.33) with Gµν = R^µν− ¹₂g_µνR.

The last term in the eq. (1.30) is:

δ_gS_m = Z

Ω

d⁴x[∂√

−gL

∂g^µν δg^µν+∂(√

−gL)

∂(∂αg^µν)δ(∂_αg^µν) + . . . ]. (1.34) The dots represent high order derivatives of g. Using Gauss theorem in eq.

(1.34) and using the boundary condition δg|∂Ω = 0, it’s straightforward to obtain:

δ_gS_m = 1 2

Z

Ω

d⁴x√

−gT_µνδg^µν (1.35)

with

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1 2

√−gT_µν = δ√

−gL

δg^µν . (1.36)

In our case

δ√

−gL

δg^µν = ∂√

−gL

∂g^µν − ∂_α(∂L√

−g

∂(∂_αg^µν)). (1.37) Therefore, from eq. (1.33) and eq. (1.35)

δ_g(S_E+ S_m+ S_{Y GH}) = − Z

Ω

d⁴x1

χG_µνδg^µν = −1 2

Z

d⁴x√

−gT_µνδg^µν = 0.

(1.38) The ipervolume of the two integrals is the same and generic, therefore it’s possible to match the arguments of the two integrals. The Einstein’s equations are:

Gµν = χ

2Tµν. (1.39)

Eq. (1.39) is a set of 10 indipendent equations, because the tensors Tµν

and Rµν are symmetric. As we can see Einstein’s equations represent the relation between energy and geometry, in particular how energy curved the space-time. The energy-momentum tensor Tµν in the eq. (1.39) rapresents the conserved current for local transformations, e.g.:

x^µ → x^0µ = x^µ+ ξ^µ(x). (1.40) The variation of eq. (1.24) respect to the ψ (field changes due to transformation eq. (1.40)) and g, leads to a generalized conservation’s law:

∇^µTµν = 0. (1.41)

The conservation of the energy-momentum tensor is consistent with the Bianchi identity eq. (3.6) that forces the null covariant divergence of the

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Chapter 1. Curved Space-Time 13 left members of the Einstein’s equation eq. (1.39). The condition eq. (1.41) is a constraint on the equation, in fact it imposes four conditions on the ten components of the Einstein’s equations and it leaves only six indipendent components. The resolution of the equations allows to define only 6 of the 10 component of the metric tensor. It’s easy to see that the six equations describe the dynamic of the system because this equations include the second order time derivative while the other four represent the constraints on the initial conditions and they include only first order time derivative so they don’t impose conditions on the dynamic.

This four free components of the metric are along the covariance of the theory which requires the freedom to change frame of reference and to impose 4 gauge condition on the metric .

1.6.1 Einstein’s equations Properties

• Gµν satisfies the Bianchi identity

∇_νG^µν = 0. (1.42)

• It’s possible to rewrite the Ricci tensor in term of energy-momentum tensor

R_µν = χ

2(T_µν− 1

2g_µνT ). (1.43)

• It’s possible to choose a more simple gravitational action then eq. (1.23) S_E = −

Z

Ω

d⁴x√

−g(R

χ + Λ). (1.44)

One can understand Λ as the zero order of curvature. The presence of Λ leads to a modification of the Einstein’s equations, that become

G_µν = χ

2(T_µν+ g_µνΛ). (1.45) Λis the so-called Cosmological constant and it allows to explain the acceleration of the Universe.

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Chapter 2 Cosmological Standard Model

2.1 Cosmological Principle

Cosmological principle [5] assures the isotropy (fig.2.1) and the homo- geneneity(fig.2.2) of the Universe on large scale; numerically speaking, on large scale this means a cube of side 15 Mpc (Mega Parsec). It is very important to underline the meaning of isotropy and homogeneity.

• ISOTROPY

The Universe is the same in every direction

Figure 2.1: Observer 1 looks at different directions and sees the same sky

15

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Chapter 2. Cosmological Standard Model 16

• HOMOGENEITY

The Universe is the same from every point in the space-time

Figure 2.2: Observer 1,2 and 3 “see” pretty much the same sky

Iff the Universe is isotropic arround every point then is also homogeneous.

It’s easy to understand this observation by means of an other picture

Figure 2.3: Sky for the observer 2 is different from the one of observer 1 If the Universe was isotropic, the only way it could not be homogeneous, is if there was a sort of ring(fig.2.3), in such a way that looks the same in every direction, but if that was the case, for an observer at point 2 the Uni- verse wouldn’t be not isotropic anymore.

The introduction of cosmological principle at the beginning of 20th century, allows to guess on the structure of the Universe, at least in first approxi- mation, without solving exactly Einstein’s equations for an arbitrary distri-

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Chapter 2. Cosmological Standard Model 17 bution of matter and without any type of observational justification. Ein- stein himself was attracted by this idea because Mach’s Principle states that which the law of physics are determinated by the distribution of matter on large scales. Einstein thought that the only way for cosmology to become a theoretical science is to assume a certain semplicity for the global structure of the Universe.

One can approach to the cosmological principle either in a philosophical way or in a physical way. Philosophically speaking, if the laws of physics change drammatically from place to place, then it could be very difficult for us to understand the mechanisms of the Universe. This idea leads up to the so-called perfect cosmological principle which permits a steady-state Universe, that is a isotropic, homogeneous and the same at all times Universe. This model was abandoned after the discovery of the CMB and the helium abundance, which is better explained with the Big bang theory.

The physics approach is more pragmatic, infact we assume the isotropy and the homogeneity of the Universe to be valid, because it agrees with observations, at least on large scale. There are a lot of physical problems, like the cosmological horizon problem, of which we talk in the following section, that undermines the semplicity of the cosmological principle, but the policy professed is steel the empirical one, so that the other theories always start from the hypothesis of validity of the cosmological priciple.

As we will see in the next chapter if we relaxe the hypothesis of isotropy the calculation might be more complex, but the results could help to understand other important evidences as the CMB anomalies.

Geometry for an isotropic and homogeneous space-time doesn’t depend on the origin of the reference frame nor on the spatial orientation of axes. The line element for this type of space-time is:

ds² = c²dt²b²(t) − a²(t)[|d~x|²+ (~x · d~x)k

1 − k|~x|²]. (2.1) eq. (2.1) represents a maximally symmetric variety with constant spatial curvature k, and it’s valid for a comovent observer, that is an observer at rest with the geometry.

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2.2 FLRW metric

Isotropy and homogeneity concern the spatial part of the metric, therefore it’s always possible to do a transformation of the time coordinate, so that homogeneity and isotropy remain valid. The transformation used in this work, called synchronous gauge, is the following:

t → t⁰ : g⁰⁰(t⁰) = 1. (2.2) Under transformation eq. (2.2), the eq. (2.1) becomes

ds² = c²dt²− a²(t)[|d~x|²+ (~x · d~x)k

1 − k|~x|²] (2.3) with t cosmic time.

The standard cosmology is based upon the maximally spatially symmetric Friedmann-Lamaitre-Robertson-Walker line element, in polar coordinates eq.

(2.1), with a(t) is the cosmic-scale factor.

The convention of standard model is to use eq. (2.1) with cosmic time and polar coordinate, so that eq. (2.3) becomes:

ds² = c²dt²− a²(t)h dr²

1 − kr² + r²(dθ² + sin²θdφ²)i

. (2.4)

It is very interesting to note that this form of the line element was origi- nally introduced for the sake of mathematical simplicity, but we know that it is well justified at early times or today on large scales(»10Mpc). The r, θ, φ coordinates are referred to a comoving system, that is to say a particle at rest in these coordinates remains at rest.

The definition of FLRW metric, makes possible to write the hypothesis of cosmological model:

• Einstein’s equations describe the interactions on large scale

• The dominant form of matter and energy are arranged in isotropic and homogeneous manner, therefore the geometry of space-time is properly described by FLRW metric

• The sources of the cosmic gravitational field are described by a perfect barotropic fluid, with two main component, matter and radiation

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• The radiation, at cosmic level, is in termodynamic equilibrium.

The resolution of Einstein’s equations eq. (1.39) leads to the equation of evolution for the scale factor a(t). Using the FLRW metric eq. (2.4) and a stress-energy tensor consistent with the hypothesis of perfect fluid, in the comovent system

T_µ^ν = (ρ + p)u_µu^ν − pδ_µ^ν (2.5) it’s easy to find







¨ a

a = −^χ₆(ρ + 3p)

¨ a

a+ 2H²+ 2_a^k2 = ^χ₂(ρ − p).

(2.6)

The combination of the two equations of eq. (2.6) allows us to write the famous

Friedmann equation

H²+ k a² = χ

3ρ. (2.7)

The conservation of stress-energy tensor eq. (1.41) leads to continuity equation

˙

ρ + 3H(ρ + p) = 0. (2.8)

The unknown quantities are a(t), ρ(t) and p(t), then we need to consider another equation besides eq. (2.7) and eq. (2.8). Therefore the system is well determined. The third equation comes from the hypothesis of barotropic fluid, for which is the barotropic equation is valid

p = γρ (2.9)

with γ constant. The cosmic fluid is a mixture of barotropic fluid

ρtot =X

n

ρn, ptot =X

n

pn (2.10)

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where n indicates the type of component such as radiation and matter.

If every component evolves indipendently, it’s possible to write the continuity equation for every component.

The final system is











H²+_a^k2 = ^χ₃ρ_n

˙

ρ_n+ 3H(ρ_n+ p_n) = 0 p_n= γ_nρ_n

(2.11)

Using barotropic relation in the continuity equation, it’s possible to find, very easily, the density in function of scal factor

ρn = ρ0(a

a₀)^−3(1+γⁿ⁾. (2.12)

Cosmological model predicts two dominant components:

• Radiation: all the relativistic “object” → γ_r = ¹₃

• Matter: all the non-relativistic “object” → γ_m = 0. In this way we abtain:

ρ_r = ρ₀(a₀

a)⁴ ρ_m = ρ₀(a₀

a )³. (2.13)

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Figure 2.4: Trend of density radiation and density matter. The intersection represents the equilibrium era

In figure eq. (2.4) we show two cosmological phases, one before the equilibrium, Radiation phase, and one after equilibrium, Matter phase. At present days we are in the matter phase: ^ρ_ρ^m_r ∼ 10⁴.

Using the results of eq. (2.13) into Friedmann equation eq. (2.7), it’s easy to obtain the evolution of scale factor through the time, with k = 0

a(t) ∼ t¹² γrad = 1

3 (2.14)

a(t) ∼ t²³ γmat= 0. (2.15)

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2.3 Horizons

Generally it’s possible to identify two types of horizons

• of Particle: it’s around every line of Universe and it devides the observed space-time from the one not yet observed.

• of Event: it devides the portion of space-time that we can observe from the portion that we never can observe.

2.3.1 Particle Horizon

We consider a photon travelling along a null radial, dθ = 0 = dφ geodetic, ds² = 0. The radial condition implies dΩ = dθ²+ sin²θdφ² = 0. The FLRW metric eq. (2.4) becomes:

dt

a(t) = − dr

√1 − kr² (2.16)

where the minus indicates a photon that travels to the origin of the frame of reference. We suppose the signal beams at time t1 and receved at time t0, with t1 < t₀ (fig.2.5)

Figure 2.5: Photon travelling to the origin.

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Integrating eq. (2.16) on the trajectory we have Z t0

t1

dt a(t) =

Z r1

0

√ dr

1 − kr². (2.17)

Now it’s useful to define the proper distance:

the spatial distance between two points of space-time with null time separation.

From eq. (2.4) with dt = 0, and remembering that dΩ = 0, we have

dl² = a²(t) dr²

1 − kr². (2.18)

Therefore the proper distance from the emission point evaluated at t = t0

(fig. 2.5) is given by

d(t₀) = a(t₀) Z r1

0

√ dr

1 − kr². (2.19)

Comparing eq. (2.19) with eq. (2.17) we have the proper distance covered by the photon

d(t₀) = a(t₀) Z t0

t1

dt

a(t). (2.20)

It’s possible to vary t1 that represents the emission time. For our interest we consider an inferior limit:

t₁ → t_{M in} maximal possible extension in the past (the signal is sent from the past).

In other words this means that it doesn’t exist any moment previous tM in. Now it’s possible to define the particle horizon for an observer in the origin at the time t0:

the paricle horizon is the spheric surface, centered in the origin, of proper radius

d_p(t₀) = a(t₀) Z t0

tmin

dt

a(t). (2.21)

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2.4 Problems

The cosmological standard model agrees with the observation for what concern the homogeneity and the isotropy of the space on large scale, it predicts the “age of the Universe” along the lower limit of 12Gyr and expacially it predicts the expansion and the resulting cooling of the Universe. The expansion with a FLRW geometry, makes possible to understand the redshift of cosmic source; finally this model predicts the presence of cosmic background radiation. Dispite that, the cosmological model doesn’t answer to other important questions.

The principle issues of cosmological model are:

1. missing matter problem 2. acceleration of the Universe 3. cosmological constant problem 4. flatness problem

5. horizon problem.

The problem of missing matter leads to the introduction of dark matter;

according to this theory a huge part of matter isn’t “visible” and should be affected only by gravitational effect.

Looking at eq. (2.14) and eq. (2.15) both in radiation and matter phase, we deduce that the Universe is decelerating, infact:

a(t) ∼ t^α =⇒ ¨a(t) ∼ α(α − 1)t^α−2 (2.22)

therefore

¨ a

a = α(α − 1)

t² . (2.23)

Both for α = ¹₂ and α = ²₃, from eq. (2.23), ¨a < 0 but observations indicate a Universe in acceleration; to solve this problem we introduce the so-called dark energy. Dark energy should be another cosmic fluid, like radiation and matter, nowdays dominant, with negative pressure; in this way ¨a > 0.

Problem (4) and (5) can be solved assuming a period of, so-called- inflac- tion- during which the Universe was expanded more quickly than eq. (2.14) and eq. (2.15).

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Chapter 2. Cosmological Standard Model 25 Dark energy, dark matter and theory of inflaction are modifications of cos- mologial model, so the metric is the FLRW one, therefore is preserved the isotropy and homogeneity of space. However, previsions on the behavior of fossil radiation don’t look good on large scale. An anisotropic model of Uni- verse might solve some of this problems.

2.4.1 Horizon Problem

It’s useful to examine the problem of the horizon because it explains the sense of the inflaction and it’s useful to understand the result of the following chapter.

According to the standard model there is a particle horizon eq. (2.21):

dp(t) = a(t) Z t

0

dt⁰ 1

a(t⁰). (2.24)

From eq. (2.14) and eq. (2.15) we know that

a ∼ t^α α < 1 (2.25)

therefore

d_p(t) ∼ t^α Z t

0

dt⁰(t⁰)^−α ∼ t^α+1−α = t. (2.26) We know that the Hubble parameter H = _a^˙a then H ∼ ¹_t, therefore

d_p(t) ∼ 1

H. (2.27)

Now we consider the following ratio:

horizon radius

observable space radius ≡ r(t). (2.28) Nowdays the portion of observable space is inside the horizon radius, so that all the points are casually connected, but in the past epoch the situation was different.

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Chapter 2. Cosmological Standard Model 26 We look at r(t) and its tendency back in time:

r(t) ∼ H⁻¹

a = 1

Ha ∼ 1

˙a ∼ t^1−α →

t→00 (2.29)

this means that back in time the Hubble horizon decreases with respect to the region of observable space; there was an instant, e.g. tf, in which the present observable region of space became wider then the Hubble radius and so the point that was included in this region of space wasn’t casually connected becouse d > dp. This is a problem, because, if we assume the existence of non casually connected point in a certain epoch, is then impossible an homogeneous thermalization of every point in space.

The solution of this problem is the inflaction, a period of extremely fast expansion of scale factor. We analize the situation step by step from a certain initial instant ti to the starting of inflation and then to the end of inflaction and the beginning of standard model.

• t = ti: the observable space is inside the horizon;

• ti < t < t_f inflationary phase: the scale factor and then the observable space, was expanded quickly then Hubble radius so that the observable region was larger then the horizon;

• tf < t < t0: the scale factor slow down its expansion trending up to the standard model t^α

In conclusion for t = t0 the observable region is inside the horizon; for t = tf, even if the observable region is larger then the horizon, the points with d > dp

come from a portion of space that at t = ti was inside the horizon.

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Chapter 3 An Alternative Cosmology

3.1 Anisotropic Model

The experimental data show that the CMB’s temperature is isotropic, up to small perturbations. On this base the Universe, in standard model is described by FLRW metric. The problems are the small perturbations, infact according to Ehlers, Geren and Sachs (EGS) theorem

if the CMB temperature were exactly isotropic about every point in space- time, then the Universe would have to be exactly an FLRW model.

CMB is not exactly isotropic, as one can deducts from the experiments, then a correct description of our Universe could require a non isotropic metric [6].

This is only one of the many evidences in favour of the broken isotropy. We have two very important observational evidences showing that we don’t have exact isotropy. Both evidences may be caused by an anisotropic phase during the evolution of the Universe; in other terms the existence of anomalies in CMB suggests the presence of anomalous plane-mirroring symmetry on large scales [7].

The same anomalous features in seven years WMAP data and Plank data seems to suggest that our Universe might be non-isotropic.

The first evidence is the presence of small anisotropy deviations as regards the isotropy of the CMB, infact we have small anisotropies with 10⁻⁵ ampli- tude.

The second evidence is connected with the presence of large anomalies [8].

These anomalies can be considered in 4 parts:

• the alignment of quadrupole and octupole moments [9], [10],[11], [12]

28

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Chapter 3. Alternative Cosmology 29

• the large scale asymmetry [13], [14]

• the very strange cold spot [15]

• the low quadrupole moment of the CMB that it is very important because it may indicate an ellipsoidal anisotropic evolution of the Universe [16], [17], [18] and [19].

This is a consequence of the fact that the low quadrupole moment is sup- pressed at large scale. The common cosmological model is not be able to explain this. On the other hand a lot of studies based on anisotropy try to explain the initial phase of inflation, the local spherical voids and a non-trivial spherical topology [20], [21], [22] and [23].

3.2 The Bianchi models

Bianchi models are particular non-isotropic and homogeneous models. From a mathematical point of view, this means that there is a sort of symmetry that connect the Universe for two observers, in such a way that both of them see the same cosmic history. It’s possible to classify this simmetry to form the Bianchi types.

The Bianchi classification is based on the construction of spacelike hyper- surfaces upon which it’s possible to define at least three indipendent vector fields, ξi, that satisfy the constrain

∇_µξ_ν+ ∇_νξ_µ = 0, (3.1)

the so called “killing’s equation”, where ξ are the killing vectors that satisfy the following commutation relations:

[ξ_i, ξ_j] = ξ_iξ_j − ξ_jξ_i = C_ij^kξ_k (3.2)

C_ij^k are called “structure constants”.

The killing vectors are important because allow to define a transformation,

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Chapter 3. Alternative Cosmology 30 called (isometry) that leaves invariant the metric gµν eq. (3.3)

δg = g^0µν(x) − g^µν(x) = −ξ^ρ∂_ρg^µν+ g^µβ∂_βξ^ν + g^νβ∂_βξ^ν = 0. (3.3) The knowledge of the isometries of a metric tensor, means the knowledge of the killing vectors. If we know the isometry, it is possible to consider the reference frame in the most efficient way.

We choose a particular decomposition, so that the structure constants are C_ij^l= _ijkn^kl+ δ_j^la_i− δ^l_ia_j (3.4) with ijk the total antisimmetric tensor and δ_i^j the Kronecker delta. n^ij has diagonal form, diag(n1, n₂, n₃), and vector ai = (a, 0, 0)with constant a. It’s possible to normalize a and ni to ±1 or 0. If (a n2 n₃) = 0then n2 = n₃ = ±1 and a = p|h|, where h is a parameter used in the classification. The Bianchi types are fixed by the value of n1 and a. Following [3] we classify the Bianchi types in terms of the numbers of functions needed to specify the solution in vacuum (r) or in the precence of a perfect fluid (s) (tab. 3.1).

Bianchi type Group Dimension p Vacuum r Fluid s

I 0 1 2

II 3 2 5

III 5 3 7

VI0 6 4 8

VII0 6 4 8

VIII 6 4 8

IX 6 4 8

IV 5 3 7

V 3 1 5

VIh 6 4 8

VIIh 6 4 8

VIh=¹₉ 6 4 7

Table 3.1: The Bianchi types

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Chapter 3. Alternative Cosmology 31

3.3 The Bianchi I Universe

Bianchi I metric is very similar to FLRW metric eq.(2.4). The generic form, in cartesian coordinates, is the following:

ds² = dt²− a²(t)dx²− b²(t)dy²− c²(t)dz² (3.5) (we assume c = 1).

As we can see, eq.(3.5) shows three different time dipendent scale factors a, b and c, therefore homogeneity is preserved, but the expansion is different in the three spatial directions so the isotropy is lost.

In this work I will use a particular Bianchi I metric with planar symmetry in the xy-plane:

ds² = dt² − a²(t)(dx²− dy²) − b²(t)dz² (3.6) where a(t) and b(t) are the two scale factors.

A planar simmetry of this type, could be induced by different mechanisms as

• cosmic domain wall: two-dimensional membranes that form when a discrete symmetry is broken at a phase transition. These walls resemble the walls of a closed-cell foam, dividing the Universe into cells (fig.3.1)

Figure 3.1: Artististic impression of a “cosmic foam” Universe.

• cosmic string: one-dimensinal lines that are formed when an axial or cylindrical symmetry is broken, which may have formed in the very

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Chapter 3. Alternative Cosmology 32 early Universe and may be responsible for the formation of large-scale structure observed in the Universe today (fig.3.2);

Figure 3.2: Cosmic string

• cosmic magnetic field: acts at every cosmic scale

– at stellar level enables a protostar to unload angular momentum – at galactic level, accretion disks around stellar-sized black holes

create jets that inject hot ionized material into the interstellar medium – while central supermassive black holes may create jets that inject such material into the intergalactic medium fig.(3.3).

Figure 3.3: Magnetic field data from the Whirlpool Galaxy, M51

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Chapter 3. Alternative Cosmology 33 These three types of mechanisms are described by three energy-momentum tensor of the following form:

• domain wall

(T_w)^µ_ν = ρ_w diag(1, 1, 1, 0) (3.7)

• cosmic string

(T_s)^µ_ν = ρ_s diag(1, 0, 0, 1) (3.8)

• uniform magnetic field

(T_B)^µ_ν = ρ_B diag(1, −1, −1, 1) (3.9)

3.3.1 Einstein’s equations in Bianchi I Universe

From the metric eq.(3.6) the metric tensor is:

g_µν = diag(1, −a², −a², −b²) (3.10)

g^µν = diag(1, −a⁻², −a⁻², −b⁻²). (3.11) The controvariant metric tensor is written in such a way that g^µαg_αν = δ^µ_ν. The Einstein’s equations in presence of source of gravitation field are:

R_µν− 1

2g_µνR = 8πGT_µν (3.12)

where R ≡ Rµν^µν is scalar of curvature, G gravitational constant,

T_µν stress-energy tensor.

The derivation of Einstein’s equation, requires the computation of Christoffel connection (see page iv) defined by:

Γ_αβ^µ= 1

2g^µρ(∂_αg_ρβ+ ∂_βg_µρ− ∂_ρg_αβ). (3.13)

AnnoAccademico2013/2014 ManuelColavincenzo Dott.LuigiTedesco Laureando : Relatore : NonLinearElectrodynamicsEﬀectsofAnisotropicExpansionofTheUniverse UniversitàdegliStudidiBari"AldoMoro"CorsodiLaureainFisica

Università degli Studi di Bari "Aldo Moro"

Corso di Laurea in Fisica

Non Linear Electrodynamics

Effects of Anisotropic Expansion

of The Universe

Laureando:

Manuel Colavincenzo

Relatore:

Dott. Luigi Tedesco

Anno Accademico 2013/2014

Contents

List of Figures

Mathematical Notations

Introduction

Chapter 1

Curved Space-Time

1.1 Theory of General Relativity

1.2 Equivalence Principle

1.3 Minimal coupling

1.4 Presence of Gravitational Field and Curva-

ture Tensor

1.5 Maxwell’s equation in curved space

1.6 Einstein’s equations

1.6.1 Einstein’s equations Properties

Chapter 2

Cosmological Standard Model

2.1 Cosmological Principle

2.2 FLRW metric

2.3 Horizons

2.3.1 Particle Horizon

2.4 Problems

2.4.1 Horizon Problem

Chapter 3

An Alternative Cosmology

3.1 Anisotropic Model

3.2 The Bianchi models

3.3 The Bianchi I Universe

3.3.1 Einstein’s equations in Bianchi I Universe