Gravitation, cosmology and dark matter

In this work, we present the fundamentals of the general theory of relativity, the standard cosmology and a description of current and interesting research topics about our current Universe, in addition the emergence of the dark matter problem has been raised, trying to outline possible dark matter candidates. This assumption seems realistic, however the analysis, from the point of view of standard cosmology, which is also studied in this work, turns out to be more complex than if dark matter were considered with a single species of elementary particle

the same reality, the notion of curvature of space-time and the principle of general covariance; this principle argues that the laws of physics behave in the same way in all coordinate systems.
Variational principles [5]  but in the case of the general relativity; since it is a eld theory [6].
The last decade has seen an explosive increase in the amount and precision of data obtained from cosmological observations. The number of techniques for data analysis and verication has proliferated in recent years.
Theoretical cosmologists have not yet nished interpreting this amount of data. Likewise, the wonderful ideas we have for the beginning of the Universe have not yet been connected with concrete models of elementary particles. One of our hopes, in addition to providing an overview of the possible candidates that may constitute the subject, is to give an updated overview of the theoretical developments necessary to make this connection.
Understanding the nature of dark matter (DM) is one of the major problems of contemporary cosmology, astrophysics and particle physics. Since 1930s, when the problem of missing matter rst arose in the context of galaxy rotation curves, despite major advances in studies of the universe and the role of dark matter in its formation, evolution and present behaviour, we are still unable to nd out how dark matter incorporates into the particle physics framework.

Numerous theories beyond the Standard Model (SM)
[711] with sensible dark matter candidates have been proposed. Among them are: supersymmetric particles, additional scalars in the extensions of the SM Higgs sector, axions, dark matter in technicolour or extra dimensions theories, to mention only the most intensely stud-ied. Nonetheless no traces of new physics were found, neither at the LHC, nor at other particle colliders, while scalar particle, which ts the Standard Model predictions for Higgs boson was discovered. Furthermore experimental eorts leading to direct detection of dark matter particles are still inconclusive.
Our intention is to oer a brief introduction to general theory of relativity and standard cosmology [12] [13] [14]; having emphasis on the relativistic theory of gravitation, as well as a description of current and interesting research topics about our current Universe respect to dark matter. In section 2, we present the general relativity, as well as a consistent derivation of the eld equations for relativistic gravitation. In section 3, we show the foundations of standard cosmology. In section 4, we present a description of dark matter. We will use units in which c = 1.

General relativity 2.1 Principles of General Relativity
We are ready to face our task, that is, to extend ourselves to a theory that incorporates gravitation. In this paper, we will undertake a detailed consideration of the physical principles that guided Einstein in his investigation of the general theory. Indeed, we could adopt the same attitude with general relativity. Furthermore, if we discover limitations in the theory, then there are more salvage changes by investigating the physical basis of the theory rather than playing with the mathematics. It is perhaps signicant that Einstein spent much of his life trying to unify general relativity and electromagnetism [2,3,3,1527,2937,4056] by various mathematical tricks, but without success 1 .
There are ve principles that, explicitly or implicitly, The next postulate is that it is impossible to talk about motion or geometry in an empty universe, so there would be no correspondence in an empty universe.
M2. If there is no matter then there is no geometry.
The nal postulate refers to a universe that contains a body, so since there is nothing interacting with it, it does not possess any inertial properties. M3. A body in an empty universe does not possess inertial properties.

Equivalence principle
Galileo Galilei did some experiments that were revolutionary for his time. Galileo found from his experiments two concepts that are cornerstones in the theory of gravitation. First, Galileo showed that the rate at which an object falls under gravity does not depend on its weight.
Second, Galileo measured the rate at which objects fall and found that their acceleration is constant, that is, independent of time. In other words, the rate of fall of an object in free fall, that is, an object that only falls under the inuence of gravity, is independent of its mass.
In contrast to the electromagnetic eld, the gravitational eld shows an extremely curious property; which we will present below. The bodies that move under the action of the gravitational eld, experience an acceleration that does not depend in the least on the material or the physical state of the body.
In classical mechanics, the ratio of the masses of two bodies are dened in two ways that dier fundamentally from each other. The rst of these ways of dening it; it is what makes said ratio equal to the inverse of the ratio of the accelerations that the same force imparts to moving objects, that is, inertial mass. The second way to dene the mass ratio is to make it equal to the ratio of the forces acting on them in the same gravitational eld, that is, active gravitational mass. The equality of both masses, dened in such a radically dierent way, is a fact conrmed by experimental physics with great precision and accuracy, that is, the facts are conrmed by the Eötvos experiment [76]. Classical mechanics does not justify this equality of masses.
According to the Eötvos experiment, which supports the strength of the equality between gravitational and inertial mass, it turns out that from Newton's equation of motion we deduce that acceleration is independent of the nature of the body. Now let K0 be an inertial frame. Material masses that are far enough from each other and from other masses are devoid of acceleration with respect to K0. Let us also refer these masses to a coordinate reference system K1 endowed with uniform acceleration with respect to K0. With respect to K1 all the masses have accelerations equal and parallel to each other, in other words, that with respect to said system, all the masses behave exactly as if a gravitational eld were present and K1 were not accelerated. We can consider this gravitational eld as real. We call the physical equivalence between the two reference systems K0 and K1 the "principle of equivalence" [77].

Principle of minimal gravitational coupling
The principles we have discussed so far do not show us how to obtain the eld equations of the systems in General Relativity when the corresponding equations are known in special relativity. The principle of minimal gravitational coupling is a simple principle that essentially says that we do not add unnecessary terms to make the transition from the special to the general theory.

General covariance principle
The general covariance principle ensures the validity of any physical law in a gravitational eld if: 1. The mathematical equation that represents the physical law is valid in the absence of gravitation, that is, the metric tensor for curved spaces gµν tends to the Minkowski tensor ηµν and the ane connection vanishes.

The physical equation is generally covariant, that
is, it preserves its form under any coordinate transformation.
The general covariance principle is not a principle of invariance but information about the eects of gravitation.

The principle of correspondence
The correspondence principle argues that some new theory, with its range of validity, must be consistent with some acceptable theories prior to the new one at some limiting value. General relativity agrees on the one hand with special relativity in the absence of gravitation and on the other hand is consistent with the Newtonian theory of gravitation in the limit of weak gravitational elds and in the limit of low velocities (compared to the velocity of the light). This gives rise to the principle of correspondence.
As we know, special relativity is consistent with classical Newtonian mechanics at low speeds (compared to the speed of light), and the Newtonian theory of gravitation is consistent with classical mechanics in the absence of gravitation.
2.2 Some aspects of the metric tensor g µν The covariant metric tensor In the invariant expression for the square of the line element, the part represented by the dx µ is that a contravariant vector could be changed to a covariant one. Besides, gµν = gνµ, it follows from the expression for the square of the line element, that gµν is a covariant tensor of second rank. We will call it the metric tensor. In what follows, we derive the properties of this tensor.

The contravariant metric tensor
We take the cofactor of each of the elements gµν and divide by its determinant g = |gµν |, thus we calculate the inverse matrix for the metric tensor, thus having a contravariant tensor. As a consequence, we are only interested in non-singular symmetric metric tensors, such tensors have an inverse given by the equation where the kronecker symbol δ ν µ denotes

Scalar volume
We look rst for the transformation law of the determinant g = |gµν |. The transformation law for the metric tensor is given by Therefore, by the rule for the multiplication of determinants, we nd On the other hand, the law of transformation of the volume element dτ = dx1dx2dx3dx4, is in accordance with Jacobi's theorem dτ = ∂x σ ∂x µ dτ.

Instead of
√ g, we introduce the quantity √ −g, which is always real since only Pseudo-Riemannian metrics will be considered.
Forming new tensors by using the metric tensor The internal, external and mixed multiplication of a tensor by means of the metric tensor generates tensors of dierent character and rank. We can now use gµν and g µν to raise and lower indices, that is,

Christoel symbols
Consider a particle that moves freely under the inuence of purely gravitational forces. According to the equivalence principle, there is a coordinate system in free fall in which the particle's equation of motion is that of a straight line in space-time, i.e., where dτ denotes the proper time interval Suppose that now, we use another coordinate system x µ (a Cartesian system at rest with respect to the laboratory, or a curvilinear system, or an accelerating one, or rotating, or otherwise). The coordinates ξ α are functions of the x µ , that is, which by multiplying it by ∂x λ ∂ξ α and using the relation ∂ξ α ∂x µ ∂x λ ∂ξ α = δ λ µ , we obtain as result where Γ λ µν is the ane connection dened by The proper time can also be expressed in an arbitrary where the metric tensor is dened by Relationship between gµν and Γ λ µν We have just seen that the eld that determines the gravitational eld acting on a free particle in a non-inertial frame is Γ λ µν (2.7) and that the proper time interval between two innitesimally separated events is determined by gµν . Next we will see that gµν is also the gravitational potential, that is, that its derivatives determine the eld Γ λ µν . Dierentiating equation (2.10) with respect to x λ , we get: and with the help of (2.10), we can write ∂gµν ∂x λ = Γ ρ λµ gρν + Γ ρ λν gρµ. (2.11) If we add to equation (2.11) the result of exchanging µ and ν in it and subtract the same equation exchanging ν with λ, we have: since Γ ρ µν and gµν are symmetric under the exchange of µ with ν.
Dening the matrix g νσ as the inverse of gνσ, that is, g νσ gρν = δ σ ρ ; and multiplying the above equation by g νσ : As a comment, the relationship between Γ λ µν and gµν has two important consequences: (1) the equation of motion of a particle in free fall automatically maintains the shape of the proper interval, dτ , and (2) the law of motion of bodies in free fall can be stated as a variational principle.

Covariant dierentiation
Let A µ be a contravariant vector whose components are given with respect to the coordinate system xν . Let P1 and P2 also be two innitely close points on the continuum. For the innitesimal region surrounding the point P1 there exists a coordinate system of xν for which the manifold is Euclidean. Let us imagine a vector drawn at the point P2 using the local system of xν with the same coordinates (parallel vector passing through P2).
This parallel vector is therefore determined by the vector through P1 and the oset. This operation is called parallel displacement of the vector A µ from P1 to the innitely close point P2. If we make the vector dierence between the vector A µ at the point P2 and the vector obtained by the parallel displacement from P1 to P2 we obtain a vector that can be considered as a dierential.
If A ν are the coordinates of the vector at P1, and A ν +δA ν the coordinates of the vector shifted to P2 along the interval dx ν , the δA ν (innitesimal quantity indicating how much the vector eld under consideration has shifted) do not cancel out in this case. Consequently, we can write the equation used to express parallel transport [78]: If we consider the invariant of the vector A ν , that is, the norm of A ν , dened as gµν A µ A ν , this quantity is an invariant, it cannot vary in a parallel displacement.
In a general manifold, the intuitive concept of parallelism fails. If we transport a vector from one point to another through two dierent curves, we will obtain two dierent vectors. However, if we transport a vector from one point to some other and the resulting vector is independent of the path, then, for the usual concept of parallelism, space-time must have zero curvature, that is, the Riemann tensor-Christoel is null (R λ µνκ = 0).
We can obtain the law of parallel displacement of the covariant vector Bµ by stipulating that the parallel displacement will be carried out in such a way that the scalar Φ = A µ Bµ remains invariant. We then get δBµ = Γ α µσ Bαdx σ .
(2.14) Note that if the connection Γ λ µν is zero in the covariant derivative of some tensor, then we have the ordinary partial derivative as the equivalent of the covariant derivative.
From equation (2.18), it follows that In addition to gµσg νσ = δ ν µ , it is found by dierentiation that and interchanging µ with ν we can write we can make the sum of these two equations obtaining the equation Substituting the right-hand side of equation ( (2. 25) Isometries in the metric tensor The metric tensor gµν invariant under the transformation So a metric tensor that ceases to be invariant is called an isometry. Since gµν is a covariant tensor, we can write the transformation law Then, using equation (2.26), x λ → x λ will be an isometry if In general, condition (2.27) is very complicated, but it can be signicantly simplied if we consider the special case of an innitesimal coordinate transformation where ε is small and arbitrary and X a is a vector eld.
Working to rst order in ε and subtracting gµν (x) from each side, it follows that the quantity in square brackets vanishes 0 ∼ = ε X ρ ∂gµν ∂x ρ + gµρ This quantity is simply the Lie derivative of gµν with respect to X, that is, using the Lie derivative: We can now replace the ordinary derivatives by covariant derivatives in some expression for a Lie derivative LX gµν = X ρ ∇ρgµν + gµρ∇ν X ρ + gνρ∇µX ρ , and consequently, using equations (2.23), the condition for an innitesimal isometry is LX gµν = ∇ν Xµ + ∇µXν = 0.

The RiemannChristoel tensor
We can make use of equation (2.8), to have the resulting equation (2.30) this equation is called transformation of the ane connection.
We can isolate the inhomogeneous term, that is, we multiply equation (2.30) by ∂x ξ ∂x λ to obtain (2.31) We dierentiate equation (2.31) with respect to x κ , to arrive at the equation then we take into account equation (2.31), to rewrite the preceding equation in the form: then after rearranging terms, we get If in this equation we interchange ν with κ and subtract the result from the original equation, all the products of the ane connections cancel, thus we nd the tensor equation: we can write this equation as a tensor transformation rule as follows: On the other hand, we dene the equation [85] ∇σ∇ν Aµ − ∇ν ∇σAµ = [∇σ, ∇ν ] Aµ = R ρ µνσ Aρ, is called the Riemann-Christoel tensor [86] [87].
From the tensor character of ∇σ∇ν Aµ − ∇ν ∇σAµ together with the fact that Aρ is an arbitrary vector and the transformation rule µνσ is a tensor (the Riemann-Christoel tensor). The mathematical signicance of this tensor is as follows: If the manifold is of such a nature that there is a coordinate system with reference for which the components of gµν are constant, then all the components of R ρ µνσ cancel out. If we change to a new coordinate system instead of the original one, the referred components of gµν will not be constant, but due to their tensor nature, the transformed components of R ρ µνσ would disappear in the new system.
A necessary and sucient condition for a manifold to be at is that the Riemann tensor be zero, as is the case with the special theory of relativity (Minkowski space).
Contracting equation (2.33) with respect to the indices σ and ρ we obtain the second rank covariant tensor which is often called the Ricci tensor.
From the Ricci tensor we can contract again to form the so-called scalar of curvature R = g µν Rµν . Therefore, it can be thought that the gravitational eld in the absence of matter, that is, in a vacuum, requires that the Ricci tensor Rµκ = 0 be derived from the Riemann tensor R λ µνκ , if and only if R λ µνκ = 0. Therefore, eld equations of gravitation can be written as Field equations for momentum and energy in the absence of matter In this section, we will show the equations corresponding to the laws of momentum and energy. For this we start from a Lagrangian density function dened by Let us propose the Lagrangian density LG as a function of the term √ −gg µν and the quantity ∂σ ( First of all, let us take the covariant derivative of the quantity √ −gg µν , that is, the tensor equation then we multiply the resulting equation by Γ ρ µν , consequently, we get: We can contract equation (2.39) with respect to ρ and ν, to arrive at: Usually; to nd the second equation that will help us to nd the eld equations, we dierentiate equation (2.41) with respect to √ −gg αβ and consequently we have: The third equation is found by dierentiating equation (2.40) with respect to ∂η √ −gg αβ , to then do algebraic manipulations and get: Finally, to enter the treatment of the relativistic equations of gravity in a vacuum, let us dierentiate equation (2.41) with respect to ∂η √ −gg αβ . Therefore, we have the result Going back to equation (2.38), therefore, this leads us to compute the partial derivatives then substitute equations (2.43) and (2.44) into the resulting derivative of the Lagrangian density, and after making the necessary reductions we nd; the desired partial derivative: Carrying out the variation of the Lagrangian density, we then integrate by parts the second term resulting from the variation and use the divergence theorem Ω ∂ρF ρ dτ = ∂Ω FdSρ, so we nd; the Euler derivative dened by: then we take fact that the second and third terms of the resulting equation reduce to ∂L G ∂xρ , therefore the previous equation becomes or (the appearance of the −2κ factor will appear after), Note that t σ ρ is not a tensor. Equation (2.48) expresses the law of conservation of momentum and energy of gravitation.
We will express equation (2.36) in a third form, which will be used particularly to nd the general form of the relativistic equations of the gravitational eld, that is, with matter.
We can multiply (2.36) by then we transform the rst term of the resulting equation by Leibniz's rule of dierentiation, then by equation (2.44) and realizing appropriate index manipulations we nd: If we take into account equation (2.50) and replace the second term of the preceding equation and if we also take into account the contraction of equation (2.50) where we have dened the expression But if we consider the solar system, the total mass of the system and its total gravitational action as well, will depend on the total energy of the system. Therefore, it is necessary to introduce the sum t σ µ + T σ µ of the components of the gravitational eld and of the matter.
Consequently, instead of equations (2.52) we write the eld equations If we dierentiate the rst term of equation (2.53), (2.51), we arrive at: In order for us to express the eld equations in covariant form, we need to multiply the above eld equations by gνκ and using equation ( since all other terms are rst and second order in Γ λ µν .
Cyclically permuting ν, σ and λ, we get Contracting ρ with ν: where equation (2.23) has been used, that is, ∇ λ g ρν = 0. Contracting again µ with σ, we have: which can be rewritten as then contracting ν with ρ and with the help of equation It has already been shown that the divergence of the rst term vanishes, therefore, the principle of conservation of energy of matter can be expressed by equation It can be seen that both operations lead to an identity. where LG = R √ −g is the scalar Lagrangian density and g is the determinant of the metric tensor gµν . Making small variations δgµν in the metric tensor gµν and keeping the tensor gµν and its rst derivatives constant on the boundary, in eect, we can nd that δS = 0 for δgµν gives Einstein's equations in the absence of matter [92].

Indirect derivation of the eld equations
From the Lagrangian density we would think of this, a function dependent on the tensor gµν , in addition to its rst and second derivatives; namely, LG = LG gµν , ∂gµν ∂x ρ , To substantiate this statement. In the rst place, we make use of equation (2.27) Secondly, if we carry out the variation of equation ( which is equivalent to the equation and from this equation and from the Euler-Lagrange equations it is possible to write Einstein's law of gravity in a vacuum; that is, the equation where we have used the equation By virtue of the stationary gravitational action principle, we obtain the eld equations which are equivalent to equation (2.61) and the contracted Bianchi identities: as the constraints of the eld equations.
Next we do a variation with respect to Γ σ µν , so Integrating by parts and discarding the divergence term by the usual argument, we get δS = δ ν ρ ∇σg µσ − ∇ρg µν δΓ ρ µν d 4 x.
The variations δΓ ρ µν are arbitrary, but the symmetry in µ and ν, and therefore only the symmetrical part of the expression in square brackets vanishes, i.e.
then it follows that Γ ρ µν is necessarily the metric connection In Palatini's proposal, the variation with respect to the metric leads to the eld equations, and the variation with respect to Γ ρ µν reveals that the an connection is necessarily the metric connection (see appendix A).

Complete eld equations
So far, we have been working with the eld equations in a vacuum. To obtain the complete eld equations, we assume that there are other elds present alongside the gravitational eld, which can be described by an appropriate Lagrangian density LM (the Lagrangian density of matter). The action integral is then   The FLRW metric can be determined from the geometry of homogeneous spaces as shown below.
Let us consider the transformations in a fourdimensional Euclidean space with the coordinates (x, y, z, w) w = a cos ψ, x = a sin ψ cos θ, y = a sin ψ sin θ cos φ, z = a sin ψ sin θ sin φ. The innitesimal distance in this system is given by:   The line element for a homogeneous hypersurface in a hyperbolic space is expressed by the equation  which also has a line element given by dσ = dx 2 + dy 2 + dz 2 . If we proceed in a similar way as in the past occasions with the help of equations (3.7), the line element for the spherical surface is dσ 2 = a 2 (t) dr 2 + r 2 dθ 2 + sin 2 θdφ 2 . (3.8) Instead of the four coordinates for which the spatial isotropy of the universe is most evident, we will now choose dierent coordinates that are more convenient from the point of view of physical interpretation.
In such a coordinate system the metric is of the form: where i, k = 1, 2, 3.
If we take the radial transformations, sin ψ = r in equation

Friedmann Equations
Suppose now that the Universe is lled with an ideal uid; frictionless adiabatic uid, ie uid characterized by the fact that in a local coordinate system of a uid element there is only one isotropic pressure. Therefore, the energy-momentum tensor according to the theory of general relativity can be represented by: (3.14) A direct consequence of equations (3.13) and (3.14) is the continuity equation. If we dierentiate equation   The Hubble parameter refers to the speed with which most distant galaxies are receding from us through Hubble's law [99] v = Hd, (3.17) where v is the speed and tells us how an object moves away or approaches and d is the distance between the observer and the distant galaxy that moves away.
There is another physically feasible way to nd equation (3.15). This way would be through the application of the equation of conservation of momentum and energy for matter in the relativistic theory of gravitation. The conservation of energy is expressed in General Relativity by nullifying the divergence of the energy-momentum tensor, that is, by the equation  , (3.22) where a0 is the current scale factor, unless ω = −1 , in which case we get a (t) ∝ exp (Ht). It is important to note that the matter and global radiation in at Universes start with a = 0, this is a singularity, known as   Unless ω is close to −1, it is often convenient to approximate equation (3.24) to the quantity: This is why the quantity is known as the Hubble time.
If we consider the previous analysis and use equations

Cosmological parameters
The best known cosmological parameter is the Hubble parameter, dened in equation (3.16), whose value today is called the Hubble constant H0 = 1 The Hubble constant is given by H0 = 100h (km/seg/Mpc) .  where this amount is approximately equivalent to 6 protons per cubic meter.
In terms of the energy density of the Universe and the Hubble parameter it is possible to dene the density parameter by the mathematical equation For a closed Universe (k = +1) we have that Ω > 1.
Flat Universe (k = 0), the density parameter is equal to zero, that is, Ω = 0.
Cosmological observations allow us to estimate the different density parameters as follows: Baryonic matter (matter made up of electrons, neutrons and protons): ΩB ≈ 0.04 [106] [107].
An important consequence of the density parameters is the consistency relationship between the cosmological parameters [108]: ΩB + ΩDM + ΩDE = 1.
The uniform expansion corresponds to q = 0 and requires a cancellation between matter and vacuum energy.
For matter we have q > 0, otherwise, for the vacuum energy domain, q < 0. According to the current density parameter, the presence of radiation is negligible, but in the past radiation was dominant. At present, the total energy content is dominated by dark energy, similar to a cosmological constant, and therefore the expansion of the Universe at present is accelerating (see gure 1).

Geometry and dark energy
In recent years it has become clear that the dominant element of energy density in the Universe today is neither dust nor radiation, but rather dark energy [111]. This So the solutions are where they have encountered the case where k = 0, above. It is clear that, in the limit as t → ∞, all solutions expand exponentially, regardless of the spatial curvature.
In fact, these solutions are exactly the same space -the de (3.34) Once again, this solution does not cover all Anti-Sitter spaces (see [117]).

Dark Matter
In the previous section we have studied the tools needed to analyze the kinematics and dynamics of homogeneous and isotropic cosmologies in General Relativity. In this part, we refer to the real Universe in which we live, and we will discuss the properties, evidence and other topics about dark matter.

Matter: Ordinary and Dark
For "ordinary matter" is meant anything made of atoms and their components (protons, neutrons, and electrons), which would include all of the stars, planets, gas, and dust in the universe, immediately visible or otherwise.
Usually such matter in question is called "baryonic mat- ter", where "baryons" include protons, neutrons, and particles (strongly interacting particles with a conserved quantum number known as "baryon number").
Ordinary baryonic matter turns out not to be enough to account for the mass density. Our current best estimate for the baryon density [118] is  The commonly adopted cosmological model is the  Evidence for dark matter extends from the scale of galaxies to the Universe as a whole. From various observations and theoretical models, the essential properties of dark matter are inferred, namely, that it is electrically neutral (thus not luminous), massive, non-baryonic, non-relativistic (cold), and weakly interacts with ordinary matter. Furthermore, dark matter must be stable or decay with a lifetime much greater than the age of the Universe. These requirements cannot be met by any known particle. Many reasonable candidates beyond the Standard Model were proposed. To become one of them, a particle must pass a ten-point test 3 [120].
The "missing matter" problem was rst noted by Franck Zwicky in 1933 [121]. He was looking at the Coma cluster and, using the viral theorem, found that it contains an insucient amount of luminous matter to explain the distribution of galaxies' velocities. Similar conclusions can be reached by examining the rotation curves of individual galaxies. Measurements show (gure 2) that the rotation curves of galaxies at great distances remain at, while Newton's gravitational law gives the following expression for the circular velocity v (r): where the mass of matter at radius r is given by  5. Stellar evolution.
6. Support for self-interaction restrictions.

Can it be veried experimentally?
ible and X-ray spectrum and weakly interacting dark halos detected by gravitational lensing, which did not slow down during the collision and were separated. of the baryons. This point of view is also supported by N-body simulations like the Millennium project [126] or Illustris [127].
We can derive information about the role of dark matter in the early Universe from relics, the most important of which is the cosmic microwave background (CMB) radiation. It originates at the time of recombination, when electrons were bonded to protons to form neutral atoms.
From that moment on, photons were able to ow freely through the Universe, which is now observed as radiation with a perfect blackbody spectrum of temperature  place. Due to these properties, one can separate the distribution of energy from the signal [129]. An interesting case is the two-photon annihilation considered in [139].
Due to conservation of energy and non-relativistic motion, the particle will annihilate into photons with energy nearly equal to its rest mass Eγ = mχ. Also, the motion where A = 0, 1, 2, 3, 4. The fth dimension y is compacted (it must be in a "compacted" form since we don't usually see manifestations of this dimension in our 4D universe) into a circle of radius R called the radius of compactication. At a scale M R, the eect of this extra dimension is not apparent. Since compactication is on a circle, y is periodic such that y −→ y + 2πR (Φ (x, y) = Φ (x, y + 2πR)).
(with φ * (x) = φ−n (x)). So from (4.8) and (4.9) we get (4.10) The action is given by dyL . (4.11) Integrating over the extra dimension (4.10) takes the Lorentz invariance in 5D, we have the relation The number k is conserved. This could give us a light and stable Kaluza-Klein particle (LKP).

The scalar singlet as dark matter
This model was rst proposed by Silveira and Zee in [148] and explored in more detail in [149] [150]. The most general form for the potential of the scalar sector adding a real scalar singlet to the Standard Model is as follows: (4.14) and the Lagrangian density of the model is given by After electroweak symmetry breaking the scalar mass terms can be written as  The term δ2H † HS 2 shows us the interaction between the two scalar elds and the physical Higgs eld and λ = δ2v/2 is the coupling between the two scalars and the Higgs. With this coupling it is possible to calculate the scattering cross section σ of the scalar S of a nucleon and annihilation cross section Γ of S, where the two scalars annihilate via the Higgs to a fermion-antifermion pair f f or to W + W − ,Z Z or hh. We require σ to calculate direct detection range and Γ for relic density calculations.

Axions
The existence of the axion was rst postulated to solve the CP problem of strong interactions [156] [157]. These particles are pseudo Goldstone bosons associated with the spontaneous breaking of the Peccei-Quinn (PQ) U (1) symmetry, which occurs on a scale fa [158] [159]. Axions are mostly produced by non-thermal mechanisms, even though they are extremely light and their nature is non-relativistic. For this reason they are candidates to contribute to cold dark matter. One of the production mechanisms is through coherent oscillations of the axion eld. This mechanism produces the following relic axion density [160] Ωah 2 = Ca fa 10 12 GeV 1.175 where Ca is a constant between 0.5 and 10 and θi ∼ O (1) is the initial angle of the oscillations. In addition to nonthermal mechanisms, axions can also be produced thermally [161].

Hot dark matter
The most popular candidates for hot dark matter are neutrinos [162]. Neutrinos can be produced or destroyed in the early Universe by the reaction γ + γ ν + ν e + + e − , (4.38) in thermal equilibrium, neutrinos also interacted with matter through the reactions ν + n e − + p, (4.39) ν + p e + + p.
where GF is the Fermi constant, which satises GF / ( c) 3 = 1.16 × 10 −5 GeV and Eν is the energy of the neutrino. At very high temperatures, the density number nν ∼ T 3 so the annihilation rate is given by: The expansion rate of the universe at high temperatures can be estimated from the Hubble parameter H =ȧ/a, where a is the cosmological scale factor, as: where, g (T ) is the eective number of degrees of freedom of the spin in thermal equilibrium. Therefore, the rate of annihilation is greater than the rate of expansion in thermal equilibrium.
Neutrinos with a mass greater than 1 MeV will begin to annihilate before decoupling, being in equilibrium, so their density will be exponentially suppressed. where Y = nν /nγ is the density of ν s relative to the photon density, which is currently 411 photons per cm 3 . In a universe in adiabatic expansion Yν = 3/11. This suppression is a result of the e + e − annihilation that occurs after neutrino decoupling. The matter component is required to be bounded by  The calculation of the relic density for heavy neutrinos is determined by the cooling of the neutrino annihilations, which occurs at T mν , after the annihilations have begun to reduce the neutrino density. The annihilation range is given by [164]: where it is assumed that annihilation is dominated by νν −→ f f via Z boson exchange and that σν ∼ When the annihilation rate becomes slower than the expansion rate of the universe, the relative abundance of neutrinos becomes xed.
Based on the invisible lepton width of the Z bo-

MACHOs
The baryons that make up dark matter could be found in dierent forms. The most plausible candidates for baryonic dark matter could be Jupiter-type planets or brown dwarfs (which are stars with masses less than 0.08M ).
These objects are called massive compact halo objects (MACHOs). Their pressure is not enough for them to support the combustion of hydrogen, so their only source of light radiation is the gravitational energy that they lose during their slow combustion. If a MALE passes in front of a distant star, it will act as a gravitational lens.

However, MALES have already been detected between
Earth and the Large Magellanic Cloud [167]. Recent results from the MACHO Collaboration indicate the existence of many of these objects with masses around 0.5 solar masses, which may correspond to 20% of the section of the halo made up of dark matter [168].

Asymmetric dark matter
The models of Asymmetric dark matter (ADM) [169] are based on the hypothesis that the current dark matter abundance has the same origin as ordinary matter, an asymmetry between particle and antiparticle density. The inelastic dark matter model assumes that a dark excited state χ * exists together with a dark matter χ with a mass δ. The inelastic scattering of dark matter by the nucleus can be expressed as Nχ ←− Nχ * . The dark matter particle χ scatters inelastically from the nucleus of mass mN , satisfying δ < β 2 mχmN 2 (mχ + mN ) . We have studied from dierent perspectives the cosmological parameters related to the ordinary matter, dark matter and dark energy of the Universe.
There is compelling evidence for the existence of dark matter. Although our understanding of its nature and distribution is still incomplete, many independent observations suggest that about 30% of the total energy density of the Universe is made of some sort of non-baryonic dark matter.
The dark matter problem is not only relevant to astrophysicists but also to the particle and high-energy physics community. In fact, some of the best dark matter candidates come from possible extensions of the Stan- Appendix A: Tranformation of the ane connection From equation (2.8), passing to any other system x µ , we will nd that: The rst term on the right is what would be expected if Γ λ µν were a tensor, however, the second term indicates that Γ λ µν it is not a tensioner. On the other hand, from the transformation law for the metric tensor g µν = gρσ ∂x ρ ∂x µ ∂x σ ∂x ν , note that: