Expected constraints on dark energy parameters with euclid
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Expected constraints on Dark Energy Parameters with Euclid Nelson Daniel de Aguiar Lima Departamento de F´ısica e Astronomia Faculdade de Ciˆencias da Universidade do Porto July / 2012
Expected Constraints on Dark Energy Parameters with Euclid Nelson Daniel de Aguiar Lima Departamento de F´ısica e Astronomia Faculdade de Ciˆencias da Universidade do Porto July/2012
Orientador: Pedro Pereira Teixeira Viana Examinador Externo: Ismael Alexandre Borges Tereno Presidente do J´uri: Maria Augusta Oliveira Pereira dos Santos
Um particular agradecimento ao Professor Pedro Viana, pelo seu tempo e dedica¸c˜ao a este trabalho. Um especial obrigado aos meus pais e a ti, Ana.
Abstract Euclid is a future space experiment that has the objective of constraining Dark Energy with unprecedented accuracy using two surveys: a photometric survey for Weak Lensing, and a spectroscopic survey for Baryonic Acoustic Oscillations (BAO) and Galaxy Clustering. In this work, I present forecasts for the Weak Lensing and the BAO surveys’ expected constraining power over Dark Energy parameters, using the Fisher Matrix formalism. The forecasts were obtained for the parameters of two distinct Quintessence models that break the traditional quintessence slow-roll conditions, which results in a much more interesting and dynamical behavior of the respective equations of state, wQ. The first model consists of a scalar field rolling in the vicinity of a non-zero potential minimum and depends on three parameters: K2, which controls the curvature of the potential at the minimum (K2<0 implies significantly curved potentials); φi, which is the field’s initial position, and ΩQ0, the present-day value of the quintessence energy density. The second model consists of a field initially slowly rolling in a nearly flat region of the potential that later on enters a curved region with a zero minima, around which it rapidly oscillates. And so does the respective equation of state, oscillating around wQ= 0. It has two free parameters: ΩQ0, and M, which defines the point where the field enters the curved region of the potential. To calculate the forecasts, I had to establish the fiducial values for each model’s parameters. For this, I’ve determined for which values the parameters minimized χ2for the most recent data from the Supernova Cosmology Project. I have found out that, for the first model, the fiducial values are K2=−15.0, φi= 0.30 and ΩQ0= 0.74. Even more interestingly, I have concluded that, for these fiducial values, this model is preferred over the widely accepted ΛCDM model. For the second model, I have determined that M= 0.00202 and ΩQ0= 0.75 and established that this model is not preferred in detriment of the ΛCDM model. The expected constraints are shown by the two-dimensional marginalized 68% and 95% confidence regions for each fiducial model. I have concluded that the Weak Lensing survey is much more constraining than the BAO survey. This is supported by much larger marginalized 1σvalues for each parameter for the BAO survey. This corroborates some previous analysis of the constraining power of these surveys over the linear dark energy parameters wa−w0. The combined constraints obtained were: K2=−15.0±2.798, φi= 0.30±0.0278 and ΩQ0= 0.74±0.00167 for the first model; M= 0.00202± 0.000835 and ΩQ0= 0.75 ±0.00723 for the second model. 9
5.3 Difference between the observational µ(z) data and the ΛCDM’s expected µ(z) values (black crosses), and between the ΛCDM’s and this section’s quintessence model, taken with its fiducial values, µ(z) predictions (red crosses). For this analysis, h0= 0.70 and ΩΛ= 0.74. ................................... 88 5.4 Evolution of the equation of state, w, of the model of section 3.3 in a flat Universe, as a function of the scale factor a, for its fiducial values: ΩQ0= 0.75 and M= 0.00202. The equation of state starts to rapidly oscillate around 0 at a≈0.8, going from −1 to 1, reflecting the field’s oscillations around its potential’s zero minimum. . . . . . 89 5.5 Difference in percentage for the comoving distance registered by an observer today between this quintessence model and the ΛCDM with ΩΛ= 0.74. The Quintessence model takes its fiducial values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.6 Difference between the observational µ(z) data and the ΛCDM’s expected µ(z) values (black dots), and between the ΛCDM’s and this section’s quintessence model, taken with its fiducial values, µ(z) predictions (red dots). For this analysis, h0= 0.70 and ΩΛ= 0.74. ........................................ 90 5.7 Expected normalized galaxy redshift distribution for the Weak Lensing survey, obtained using Smail et al. analytical approximation. The galaxies are equally distributed into 10 redshift bins along the redshift range mentioned before for this survey. . . . 91 5.8 Expected normalized galaxy redshift distribution for the BAO survey, according to Geach et al. empirical distribution for Hα emitters, considering a limiting flux of 3×10−6. The galaxies are distributed into 14 redshift bins along the redshift range mentioned before for this survey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.9 Weak Lensing expected constraints on the dark energy parameters of the first model, obtained for the Euclid experiment. The red contour represents the 95% confidence region, while the black contour is the 68% confidence region. For each parameter, the respective one-dimensional distribution is also presented. . . . . . . . . . . . . . 93 5.10 BAO expected constraints on the dark energy parameters of the first model. The red contour represents the 95% confidence region, while the black contour is the 68% confidence region. For each parameter, the respective one-dimensional distribution isalsopresented...................................... 94 5.11 Combined Weak Lensing and BAO expected constraints on the dark energy parameters of the first model, obtained for the Euclid experiment. The red contour represents the 95% confidence region, while the black contour is the 68% confidence region. For each parameter, the respective one-dimensional distribution is also presented............................................ 95 5.12 Weak Lensing expected constraints on the dark energy parameters of the first model, obtained for the Euclid experiment. The red contour represents the 95% confidence region, while the black contour is the 68% confidence region. For each parameter, the respective one-dimensional distribution is also presented. . . . . . . . . . . . . . 96 5.13 BAO expected constraints on the dark energy parameters of the second model, obtained for the Euclid experiment. The red contour represents the 95% confidence region, while the black contour is the 68% confidence region. For each parameter, the respective one-dimensional distribution is also presented. . . . . . . . . . . . . . 96 5.14 Combined Weak Lensing and BAO expected constraints on the dark energy parameters of the second model, obtained for the Euclid experiment. The red contour represents the 95% confidence region, while the black contour is the 68% confidence region. For each parameter, the respective one-dimensional distribution is also presented............................................ 97 16
1 Introduction Our Universe is a truly fascinating entity. From the smallest planet to the largest clusters of galaxies, there is still much light to be shed on the intricacies of its composition, dynamics and even the primordium of its own existence, despite the latest technological developments in instrumentation. Recent observations have, however, contributed immensely to the building of our knowledge about it. The analysis of the Cosmic Microwave Background (CMB) radiation by the Wilkinson Microwave Anisotropies Probe (WMAP) revealed the staggering homogeneity and isotropy of the temperature of such radiation [1, 2]. This is a relic of the beginning of our Universe, and is consistent with it having started from a state of extreme density known as the Big Bang. A Big Bang followed by an inflationary epoch would have provided the necessary conditions for the homogeneity observed between regions that are not in causal contact at the present. This homogeneity at large scales is corroborated by the analysis of the large scale structures of our Universe identified by the Sloan Digital Sky Survey (SDSS) [3]. Such a beginning would also explain the observed flat geometry of our Universe, since whatever existent curvature would have been flattened by the rapid inflationary expansion (check [4] for a review on inflation). This fact is confirmed by the characteristics of the primary peak of the CMB [5, 6, 7]. Nonetheless, the most surprising revelation would come from the analysis of distant Type I Supernovae (SNIa). The observation of the light emitted by such bright objects confirmed not only that the different regions of our Universe are receding from each other (which had already been confirmed in 1929 by Edwin Hubble [8]), but are doing it in an accelerated manner [9, 10]. According to General Relativity (GR), such a dynamic Universe is only possible if, apart from its matter and radiation content, there is a negative pressure component. Such a component would be able to provide the necessary repulsive energy to overcome the gravitational attraction of the other elements. The truth is that the most recent observations [11, 12, 13, 14] are consistent with the existence of a currently dominant component with a negative equation of state (the ratio between its pressure and density), named Dark Energy. This leads to the most accepted description of our Universe, the Lambda Cold Dark Matter (ΛCDM)model. According to this model, this mysterious dark energy consists of a Cosmological Constant Λ with a constant energy density. This implies that the present day value of its equation of state is −1. Dark energy comprises almost 75% of the total density of our Universe, while the remaining 25% exists mostly in the form of non-luminous cold dark matter. Of this percentage, only 4% exists in the form of baryonic matter [15]. 17
However, the observed value for the dark energy density parameter ΩΛ0 is almost 121 orders of magnitude inferior to the theoretical predictions. Besides that, there is the coincidental problem related to the initial conditions that could have led to this particular moment in cosmic history in which we exist, where the density of dark energy is comparable to the matter density (for a review on this subject, check [16]). These problems led theorists to pursue other hypothesis for dark energy. Some of these are based on a scalar field, φ, slowly-rolling down its potential, V(φ), and constitute the Quintessence theories of dark energy [17, 18, 19]. The quintessence’s slow roll conditions (analog to the inflationary ones, but broader) are imposed so that the present day value of wQis close to −1, respecting current observations. Imposing the slow-roll conditions, the quintessence models usually present a slow and monotonic evolution for the respective equation of state as a function of the scale factor a[20, 21, 22, 23]. So far, all of these models turned out to be consistent with present observations. There have been, however, some quintessential propositions that are more radical, such as rapidly oscillating models. These are based on power-law potentials which enable the field to rapidly oscillate around its zero-valued minimum [24, 25, 26]. This leads to an equation of state that varies rapidly between 1 and −1, taking an average value related to the power of the exponential. Due to the particular behavior of these models, some fine-tuning is required in order to adjust them to the present day observations, and some have even been ruled out [27]. One of the two quintessence models studied in this work presents such rapid oscillations near to the present time. The other model that was studied lies somewhere in between the models of the two last paragraphs. It basically consists of a quintessence scalar field rolling down a potential whose minimum is not zero, but is fixed at the current value of the dark energy’s density. If certain conditions related with the value of the potential’s curvature at the minimum are met, the field oscillates around the nonzero minimum. The field’s equation of state accompanies this behavior without getting too far away from the observed value of −1. Although this model requires extremely low values for the field’s mass to be viable, it seems a reasonable and natural way of obtaining a dark energy equation of state close to −1 with an interesting dynamical behavior. Despite interesting, these models, as any others, have yet to be confirmed or ruled out by observations. Therefore, it is extremely useful to assess the power of future surveys to constrain cosmological parameters related to dark energy and eventually differentiate between a cosmological constant and a dynamical dark energy model. In the case of the present work, a simple analysis for a future experiment will be performed. 18
There are several methods for evaluating the capability of a survey for constraining cosmological parameters. For example, there is the formalism presented by the Figure of Merit Science Working Group (FoMSWG) for the Joint Dark Energy Mission (JDEM) [28]. In this work, the Fisher Matrix [29] for a certain experiment is produced and from it the principal components or eigenmodes are extracted. This method allows one to find out for which values of the redshift an experiment has the best power for constraining a certain model. For that, w(z) is assumed constant in bins of equally spaced values of z. This comes as a generalization of the more usual w(z) model given by w(z) = w0+wa(z/(1+z)) [30], which is one of the best accepted parameterizations for the dark energy evolution. In the present work, a simpler method than the one mentioned in the last paragraph is used: the Fisher matrices related to a Weak Lensing and Baryonic Accoustic Oscillations (BAO) surveys to be performed by a future experiment (the European Space Agency’s Euclid mission) were constructed for the dark energy models of interest. From those, the joint probability regions for the model’s relevant parameters are plotted. These are an estimate of the confidence one can have on a certain experiment’s capability for constraining some parameters. This work is organized as follows: Section 2 provides a detailed review on general’s relativity framework for cosmology and the dynamics of our Universe. It also presents the problems related to the Cosmological Constant that led to the need of exploring other dark energy models. This sets the path for Section 3, in which the basics of the dynamics associated to simplest quintessence models are presented. Also, the dark energy models of interest are thoroughly analyzed. Section 4 delivers the basics on the Fisher Matrix formalism as well on the mapping of the joint credible regions for constraining the cosmological parameters of interest. Lastly, Section 5 presents the results obtained using the adopted method for constraining parameters, while the Conclusion presents some final thoughts on the subject and suggestions for future works. The results presented in this work were obtained using the freely available software package iCosmo [31], which uses the programming language IDL. The routines used were a mixture of the ones on the package and others modified/created by the author of this work for the intended purposes. 19
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2 Cosmological Foundations 2.1 General Relativity The Theory of General Relativity (GR) is, without question, one of the most complex and revolutionary mathematical formulations of the 20th century, being initially understood only by a handful of people. It is the product of the ingenious mind of Albert Einstein1, who published it in 1916 in the scientific magazine Annalen der Physik as a coherent relativistic description of gravity. Achieving this was no easy task, and the laborious work of Einstein had one particular objective in mind: to obtain a field equation that would relate the distribution of the gravitational source with the corresponding relativistic gravitational field or, in other words, the development of the relativistic generalization of the well known Newtonian Equation ∇2φ= 4πGNρm,(2.1) where φaccounts for the classic gravitational potential, GNis the Newtonian gravitational constant, and ρmis the gravitational source’s mass density. In the context of Special Relativity (SR), mass is nothing more than an organized form of energy, corresponding to the rest energy of an object. This is translated in the iconic equation E=γm0c2, where γis the Lorentz factor, c is the frame-invariant light velocity and m0is the rest mass of the object. This equation is reduced to E=m0c2when the object is at rest. Under this criteria, one could make the assumption that the simplest relativistic generalization of the newtonian field equation would be replacing the classical source of the gravitational field, ρm, by the total energy density of the source, ρ. However, what one learns from special relativity is that energy and momentum come up as being equivalent, since they can be transformed into each other according to generic lorentz transformations when measured by different observers. Therefore, in GR and SR, they end up constituting a very important rank-two 2 0tensor, the energy-momentum or stress-energy tensor, T. Therefore, in order to construct a frame invariant theory that does not favor a particular frame or observer over any other, one has to consider the whole energy-momentum tensor as the source of the gravitational field in the relativistic approach to equation (2.1). 1Curiously, Einstein would receive the Nobel Prize in Physics in 1920 for its work in the description of the photoelectric effect and not for the conception of GR. 21
In an arbitrary frame, the components of T,Tαβ (where αand βrun over the values 0, 1, 2 and 3)2, can be thought of as representing the flux of αmomentum across a surface of constant coordinate xβ, with x0=ct. For future reference, a brief description of the components of the stress-energy tensor will be given. This will be useful later on, when describing the Universe and its content as a fluid3 with certain properties. So, generically, these will be: •T00 is the flux of zero momentum, or energy, across a surface of ct =constant, which is just the energy density ρ; •T0iis associated to the energy flux across a surface of constant xi, and is related to the motion of particles and, for instance, heat conduction; •Ti0represents the imomentum density, or the flux of imomentum across a surface of constant ct and, again, is associated to moving particles and also to the momentum of transmitted energy as heat, for example; •Tij is the flux of imomentum across the surface of xj=constant and represents the force between adjacent fluid elements, which may not be perpendicular to the surfaces between them, as in cases where, for example, viscosity is present in the fluid. Tij is associated to stress, since it comes in units of [force/area]. It can be shown that this tensor is symmetric [32], which means that it has 10 independent components which, when provided, describe the energy and momentum content of a fluid. Therefore, according to this, the energy-momentum tensor Tshould incorporate, in some way, the law of local conservation of energy and momentum. In special relativity, this is expressed by the equation Tαβ ,β = 0, where the lower index, separated by comma, represents a common derivative of the tensor with respect to the coordinate4xβ, and the repeated upper and lower indices imply a summation over all of their possible values. In GR, the latter equation is generalized to a covariant differentiation, as follows: Tαβ ;β≡Tαβ ,β +TανΓβνβ +TνβΓανβ = 0,(2.2) where Γαβν is a Christoffel Symbol, which is defined in terms of the most relevant tensor in GR, the metric tensor,g[see eq. (2.15)]. 2Throughout this thesis, the greek indices can take the already designated values, while the latin indices, such as i and j, will only assume the spatial coordinates associated values 1, 2 and 3. 3A fluid can be understood as being a collection of so many particles that its description can only be accomplished in terms of its average or bulk quantities, which can vary from point to point. 4This will be the notation used throughout this thesis to represent a common derivative. 22
2.1.1 The Equivalence Principle and the metric tensor The metric tensor, g, is of special importance, since it is at the core of the definition of General Relativity as a geometric theory of gravity, in the sense that any physical phenomenon related to gravity can be attributed to the underlying curved geometry of space and time. This concept comes as a consequence of the generalization of Galileo’s Inertial Equivalence Principle, sometimes referred to as the Weak Equivalence Principle. According to the latter, the inertial mass of a body, mI, is the same as the gravitational mass of the body, mg. This means that exists an equivalence between the mass of an object that responds to all of the applied forces in the body and the mass that’s associated to the gravitational force. That is best translated in the observational fact that distinct objects, when submitted to the gravitational force, experience the same acceleration. Therefore, if an observer is in a freely falling frame, everything happens as if there was an absence of gravity, since all released objects will fall with the same acceleration. That way, no gravitational effect can be detected, and the observer will not be able to tell if the aforementioned objects are falling with respect to its surroundings. This was the basis for Einstein’s Equivalence Principle (EP), also known as the Strong EP, which states that physics in a freely falling frame in a gravity field is equivalent to physics in an inertial frame where gravity is absent, therefore accelerating with ~a =−~g (~g is the classical acceleration vector associated to the existent gravitational field). This means that accelerating frames can be dealt with as if they were inertial frames, where the differential laws of physics will have the same form as they do in Special Relativity, when gravitational fields are absent [33]. However, one can not say that there exists a global inertial frame that fills all of spacetime, since any gravitational field is nonuniform. The inertial frames can only have a limited extent in space, and also time, in regions small enough so that the field’s nonuniformities can be neglected. Therefore, the best one can have in General Relativity are frames that, at a point of the manifold, are equivalent to Special Relativity’s inertial frames [32]. The Strong EP is behind some interesting observable effects, such as clocks running at different rates according to their position in a gravitational field. The geometrical interpretation of this phenomenon easily allows one to understand the defining role of the metric tensor in GR. To do so, one has to refer to the gravitational redshift effect, according to which the frequency of a light ray emitted at a point with lower gravitational potential, φem, and received at a point with higher gravitational potential, φr, decreases, even if the receiver observer is stationary with respect to the emitter. 23
Gravitational redshift is a fundamental result, since it explains why an observer outside a freely falling frame is not able to detect any frequency shift whatsoever in a light ray emitted in that frame: it exactly cancels the expected non-relativistic Doppler effect that comes up when two frames are in relative motion. It is expressed by the formula [34] νr−νem νem =−φr−φem c2,(2.3) where νrand νem refer, respectively, to the frequency of the light ray measured at the reception and emission points. This effect has a fundamental explanation that’s related to the fact that clocks do run at different rates when located at different gravitational field points, which has become known as the gravitational time dilation effect. Since frequency is proportional to the inverse of the local proper time dτ, which is just the time measured in the rest frame, equation (2.3) can be worked out to dτ1−dτ2 dτ2 =φ1−φ2 c2, where the emitter and receiver subscripts have been replaced by a number. This is a simple generalization of the redshift equation to a time dilation formula that relates the rates at which two clocks run at two generic different gravitational field points 1 and 2. If the field is static, then the latter equation can be integrated to yield τ1−τ2 τ2 =φ1−φ2 c2.(2.4) According to this equation, a clock at a higher gravitational potential will run faster then another clock at a lower gravitational potential, even if they are at rest with respect to one another. This is different from the well established timedilation effect of SR, where two observers in relative motion will see the other’s clock running at a slower, different rate. If the clocks in the gravitational field are also in relative motion, then equation (2.4) does not apply. This physical phenomenon can be given a simple geometric interpretation if one posits the existence of a nontrivial and position dependent metric, gνµ (4 x 4 matrix), that reflects the underlying structure of space and time. It embodies information about the rate at which clocks run and the distances between the points of a manifold which, without a metric, would just be a shapeless collection of points. That’s the case of Riemannian manifolds, which have a symmetric 0 2metric tensor gdefined at every point. 24
As a simple generalization of SR, one can define a frame invariant interval ds which can be written in terms of the coordinates xµof the manifold as ds2=gνµdxνdxµ,(2.5) that if null, positive or negative, defines light-like, space-like and time-like 4vectors, respectively. Special Relativity’s squared invariant interval should come as a limit of the latter equation, when gνµ =ηνµ =diag(−1,1,1,1) is verified, which is just Minkowski’s flat spacetime metric5. So, in order to re-obtain the formula stated in equation (2.4) as a result of a curved spacetime metric gνµ, one should find it to have a g00 component equal to g00 =−1 + φ(~x) c22 ,(2.6) where φ(~x) is the gravitational potential value at a point defined by a 3-dimensional spatial vector ~x. This comes about because equation (2.4) can be thought of as if comparing the proper time interval of a clock situated at a given point in space dτ(~x), where the gravitational field is φ(~x), with the coordinate time dt measured by a clock at rest at a point where the gravitational potential is null. This allows the gravitational time-dilation to be rewritten as dτ(~x) = 1 + φ(~x) c2dt. (2.7) Therefore, recalling from Special Relativity that the proper time interval is related to the invariant interval by ds2=−c2dτ2, and realizing that equation (2.5) is reduced to ds2=g00dx0dx0for ~ dx = 0, as necessary to attain the proper time measured in the rest frame, one concludes that6 dτ2=−g00dt2.(2.8) The only way for the latter equation to be equivalent to eq. (2.7) is if the metric element g00 has the form presented by equation (2.6). This simple result is of fundamental importance, since it allows one to conclude that the geometrical interpretation of a phenomenon brought up by the Equivalence Principle of gravitation is implying that gravity modifies spacetime, deviating it from SR’s Minkowski flat spacetime. This is translated by a warping of the metric gνµ, such that gνµ 6=ηνµ, which plays the role of the relativistic gravitational potential. 5In a flat spacetime, two light rays emitted parallel to each other remain parallel throughout their trajectory, no matter how far extended that may be. 6Recall that x0=ct. 25
Figura 2.2: A picture of the CMB radiation temperature distribution. Taken from [37]. Now, for our Universe to be isotropic, it means there can not be an observed privileged direction in any sense. This implies, for instance, that the mass density can be a function of radius only (in fact, as stated, it is constant, on average) and also that there can be no preferred axes for any other physical properties, such as a velocity field. In this context, the only allowed velocity field is one in which all objects are receding from each other, either in an expanding or contracting manner, with a receding velocity vector ~v, linearly proportional to the distance vector ~r between them ~v =H~r, (2.20) where His a constant of proportionality. Vesto Slipher was the first researcher to obtain evidence for an expanding Universe, by finding that the spectrum of neighboring galaxies was redshifted [38], which would only be possible, by doppler effect, if they were getting further away from us. His work was then followed up by Edwin Hubble, an american astronomer like Slipher, who presented his results in 1929 [8] and is, since then, credited for such finding. Therefore, eq. (2.20) has become known as the Hubble relation, and His referred to as the Hubble parameter. The perception that galaxies are receding from us could potentially lead to the consideration that we occupy a privileged place of observation of the Universe. However, eq. (2.20) linear12, holding for any observer on any other galaxy. Therefore, the Hubble expansion law is compatible with the Copernican Principle, according to which we do not occupy a privileged position in the Universe, and nor does any other observer. Therefore, it is reasonable to expect other observers to see an isotropic and homogeneous Universe just as we do. 12This would not be the case if, for instance, the Hubble parameter depended either on position and/or velocity 32
In an isotropic Universe, it is possible to define a cosmic time as a universal time with which all observers agree. In the picture envisaged so far, one can imagine a set of observers, all at rest with respect to the matter distribution in their immediate vicinity. Therefore, the time tmeasured by the clocks of these observers will be their proper time. And, while they’ll be receding from each other according to the natural expansion of the Universe, they can exchange light signals between them and agree to set their clocks to a standard time [33]. This is a useful notion that takes part in the cosmological extension of the Copernican Principle, establishing a proper and rather powerful framework for work in Cosmology. This is the Cosmological Principle: at a given epoch, or fixed value of the cosmic time t, the Universe is homogeneous and isotropic, presenting the same properties, except for local irregularities, from different observation points. The Cosmological Principle allows us to divide the Universe in separate regions of constant time which are homogeneous and isotropic. Therefore, any metric developed for a cosmological model should incorporate these facts. The standard metric in such conditions, with constant curvature and the desired spherical symmetry, is the Robertson-Walker metric ds2=−c2dt2+R2(t)dr2 1−kr2+r2dΩ2(2.21) where dΩ2=sin2θdφ2+dθ2is the differential solid angle (for a full derivation of this metric see [32]). The coordinates r,θand φare comoving coordinates, which remain fixed in time. This way, the dynamics of the expansion due to the Hubble flow is associated to a general scale factor R(t). Everything happens as if the elements of our Universe, like galaxies, occupy a fixed coordinate position on the surface of a balloon, as in fig. (2.3). As the balloon inflates, conserving its shape, the galaxies on the surface maintain their coordinates while the distance between them grows accordingly with the expansion at a rate that’s proportional to the increasing distance, as stated by Hubble’s law. Figura 2.3: The expanding Universe as the surface of a balloon. As the distance between the elements on its surface increases with the general expansion, so does the rate at which they grow further apart. Taken from [32]. 33
Returning to the metric, eq. (2.21), one last reference is in order. The parameter kis associated to the constant curvature of the 3-dimensional space described by such metric. It can take one of three values: k=−1,0 or 1, corresponding to three different ”Universes”. For k= 0, one has the metric of a ”flat Universe”; k= 1 corresponds to a ”closed Universe”and the Universe having k=−1 is an ”open Universe”. As will be pointed out in the next section, this parameter is related to the total matter/energy density of the Universe, in a clear demonstration that its matter/energy content is associated to its geometry. And, finally, it is sometimes useful to express the Robertson-Walker metric in terms of a time dependent dimensionless scale factor a(t) = R(t)/R0, where R0 is the value of the previously presented scale factor at the present epoch, t0, such that a(t0) = 1. In terms of this scale factor, the metric takes the form ds2=−c2dt2+R02a2(t)dr2 1−kr2+r2(dθ2+ sin2θdφ2),(2.22) which will be of much use throughout the next section. 2.3 The Friedmann Equations Now that the metric for an isotropic and homogeneous expanding Universe has been established, it is possible to solve the Einstein field equations for such a Universe. The last thing necessary is to determine the stress-energy tensor T that best describes the Universe constituents. According to the Cosmological Principle, one can define hypersurfaces of constant cosmic time, which will be isotropic, homogeneous and at rest in the comoving coordinates13. The only interaction between them is the gravitational interaction, which will determine their free fall motion, following the geodesic equations. Therefore, the energy-momentum tensor must take into account that, since the considered fluid element is at rest in the chosen reference frame, there will be no momentum density, Ti0= 0. And, due to symmetry, T0iwill also be null, which means there is not energy flowing between the fluid elements. Lastly, since the only interaction is the gravitational one, there should be no viscosity, given that the fluid elements only interact through a perpendicular force. This means that the tensor matrix should be a diagonal one, independent of the comoving frame in consideration, since no viscosity is a statement independent of the spatial axes. This is only accomplished through a multiple of the identity matrix, the only matrix that is diagonal in all frames. 13Such inertial frames that momentarily accompany a fluid element’s movement and in which they will be at rest are called Momentarily Comoving Rest Frames (MCRF). 34
Accordingly with these last considerations, the stress-energy tensor of the elements of such a fluid will be diagonal, with only four non-vanishing elements: T00, or the total energy density ρ, and Tij =pδij, where prepresents the direction independent pressure applied perpendicularly between all of the fluid’s elements, such as the fluid’s isotropy requires. This corresponds to the stress-energy tensor of a Perfect Fluid, which in the MCRF is given by Tαβ = ρc20 0 0 0p0 0 0 0 p0 0 0 0 p .(2.23) Another way of describing it, in the MCRF, is according to the formula Tαβ =ρ+p c2UαUβ+pηαβ,(2.24) where Uαis the four-velocity field, given by the expression Uα=dxα/dτ. It is normalized in such a way that UαUα=−c2, which means that, in the MCRF it takes the simple form Uα= (c,~ 0). This last expression for the stress-energy tensor is a frame-invariant one, which means it can be generalized for any given system by the form Tαβ =ρ+p c2UαUβ+pgαβ.(2.25) This implies, finally, that the form of the tensor of this perfect fluid, in GR, will be Tαβ =ρc20 0pgij .(2.26) The First Friedmann Equation corresponds to the G00 =−8πGNρ/c2component of the Einstein equation, as in eq. (2.17). This is a laborious calculation which, for the time being, will be intentionally left out. In terms of the scale factor, a(t), and the curvature parameter, k, of the Robertson-Walker metric, this equation is given by ˙a2(t) a2(t)+kc2 R02a2(t)=8πGN 3ρ, (2.27) where the dot in overscript represents a differentiation with respect to coordinate time ct, such that ˙a(t) = 1 c ∂a ∂t , and ρrefers to the total energy density from all of the components of the Universe, such as matter and radiation. 35
The Second Friedmann Equation corresponds to the Gij =−8πGNpgij/c4 equation. The derivation of Gij will not be done here. It suffices to say that it can be obtained by differentiating eq. (2.27) with respect to coordinate time and using the local conservation law of energy and momentum, eq. (2.2), which, in terms of the RW metric, is given by T0µ;µ≡˙ρc2+ 3 ˙a aρc2+p= 0.(2.28) With this, the second Friedmann equation is expressed by ¨a(t) a(t)=−4πGN c2p+1 3ρc2,(2.29) where the double dot in overscript represents a double differentiation with respect to coordinate time, such that ¨a(t) = 1 c2 ∂2a ∂t2, and pis the total pressure of the Universe, considering all of its components; ρis defined as in equation (2.27). Both of these equations can have rather simple Newtonian interpretations. This does not, in any way, contradict the necessity of applying General Relativity in Cosmology. The Cosmological Principle states that our Universe can be sliced in smaller regions that can be treated as the isotropic and homogeneous whole. Therefore, on reduced scales, the Newtonian description is valid since the gravitational interaction is very weak and a small region can be considered almost flat by the flatness theorem. The First of the Friedmann Equations is a statement of the energy balance for a central force problem. It is the sum of a kinetic term and a potential energy one, where the total energy that these terms add up to is related to the negative of the curvature parameter k. The second Friedmann equation, as its derivation implies, can be seen as the usual F=ma equation [34]. A simple rearrangement allows eq. (2.27) to be rewritten as −k=˙a(t)R0 c21−ρ ρc,(2.30) where ρcis known as the time-dependent critical density parameter: the value of total energy density for which the curvature parameter k, according to the First Friedmann equation, vanishes, and is expressed by ρc(t) = 3 8πGN ˙a2(t) a2(t).(2.31) 36
Therefore, the total density of the Universe determines its curvature, as another clear sign of the influence that mass/energy have on the geometrical structure of the Universe. Defining Ω = ρ/ρc, if Ω >1, the curvature of the Universe will be positive, k= +1, as in a closed universe. When Ω = 1, k= 0 and the universe will have a flat spatial geometry. Lastly, if Ω <1, the universe will have an open geometry since the curvature will be negative, k=−1. The total energy density plays a fundamental role on the dynamics of our Universe. It determines the type of spatial curvature it has and, by defining the sign of the curvature signature k, it is also imposing the sign of its total energy, according to the Newtonian approach to the Friedmann Equations. As a consequence, the fate of our Universe is dictated as well. This is easily perceived for a single component Universe, non-relativistic matter or radiation, which may be seen as an approximation for a multi-component Universe dominated either by one of those components. Although different, the qualitative behavior of such Universes is similar. A Universe with negative total energy, hence positively curved, is a bound and spatially closed one, therefore destined to have its expansion halted at some point due to the gravitational attraction between the fluid’s elements. If it has a positive total energy, it represents an unbound and open Universe, which should continue to expand forever until all of the fluid’s elements are causally disconnected from each other. A Universe with null total energy is a flat one, but also infinite in extent. All of these Universes, however, have in common the fact that they all begin in a state of extreme density where the scale factor goes to zero. It is a violent picture of divergent spacetime curvature and density predicted by the Big Bang Cosmology. This is depicted in fig. (2.4). Figura 2.4: Evolution of the scale factor as a function of time for the open, flat and closed Universes, according to the sign of the curvature parameter k, for a matter or radiation dominated Universe. Taken from [34]. 37
The basic equations for the time evolution of the scale factor of the Universe for such models can be seen either in [34] or [33]. Here, it will only be necessary to establish how the density and pressure of a general component of the Universe, such as matter or radiation, evolves with the scale factor, since it will be useful for future reference. This is achieved through the energy-momentum conservation condition given by eq. (2.28), which can be expressed also as ˙ρc2=−3(ρc2+p)˙a a.(2.32) This last equation can be simplified by an Equation of State that relates pressure with density. These can have rather complicated forms, but since Cosmology deals with a dilute gas, this relation can be expressed by this simple equation [34]: p=wρc2,(2.33) where wis a constant that characterizes the component of the system. For nonrelativistic matter, where pressure can be neglected when compared to the rest mass energy of the constituting particles, w= 0. For radiation, one has w= 1/3. With this last relation, eq. (2.32) can be simplified to dρ ρ=−3(1 + w)da a,(2.34) which is easily integrated, for constant w, to the general form ρ(a) = ρ0a−3(1+w),(2.35) where ρ0is the present day value that component’s density, usually obtained from observations. In a matter and/or radiation dominated Universe, the respective densities evolve as ρmatter(a) = ρm0a−3(2.36) and ρradiation(a) = ρr0a−4.(2.37) Since both the densities and pressures of these components are positive, Universes dominated by such components should all be in a decelerating expansion, perhaps as in a closed Universe, where its expansion is reversed at some point, such that a(t) returns to zero at some finite moment in time, resulting in a Big Crunch. 38
2.3.1 The accelerating Universe The basic dynamical equations of the Universe have been established in the previous section. These are the Friedmann equations. The first one, eq. (2.27), defines the rate at which the scale factor changes with time, normalized by the scale factor itself. In fact, ˙a(t)/a(t) is just the Hubble parameter at a cosmic time t,H(t), which means that one can rewrite such equation to yield H2(t) = 8πGN 3ρ−kc2 R2 0a2(t).(2.38) From equation (2.30), the present day value of the Hubble parameter can be extracted, defining that H0= ˙a(t0), since a(t0) = 1, resulting in kc2 R2 0 =H2 0(Ω0−1),(2.39) where Ω0is the present day value of the fractional density, ρ0/ρc. With this information, one can replace kin eq. (2.38) to give H(a) = H0 X i Ωi(a) + 1−Ω0 a2!1/2 ,(2.40) where the sum runs over all of the possible components of the Universe and their fractional densities, Ωi(a) = ρi(a)/ρc, as a function of the scale factor a. From now on, the time dependence will be implicit in a. This last equation is implying that the Hubble parameter will remain unchanged in time if our Universe is an empty one. This could already be concluded from the analysis of the Second Friedmann equation and observed in fig. (2.4), given that the acceleration of such a Universe would be null. However, if some components with positive density/pressure exists, then the expansion will be a decelerated one, and the Hubble parameter diminishes with time. The neglected fact is that, in reality, our Universe could be accelerating or even non-accelerating due to the existence of negative density/pressure components. An accelerating Universe is possible, theoretically, if there exists some form of negative pressure component that surpasses the gravitational attractive effect of matter/radiation in the Second Friedmann equation. To make this clear, eq. (2.29) can be rewritten as ¨a a=−4πGN 3ρ(1 + 3weff ),(2.41) 39
where, since both ρand prefer to total energy density and pressure of the Universe, an effective equation of state has been defined as weff =p/ρc2. Therefore, this last equation will result in a positive acceleration, ¨a > 0, if weff <−1 3,(2.42) which means that, in some way, a negative pressure component must exist in order to sustain the positive contributions of matter and radiation. But, no matter how intriguing such theoretical considerations may be, it is observation that dictates the designed models for our Universe. Therefore, it is necessary some kind of experiment that observes if the Hubble parameter is changing in time. This can be achieved trough the plotting of Hubble curves, that measure the recession velocity of astronomical objects as a function of their distance. If some curvature is obtained for such a relation, then one can conclude that the Universe is either accelerating or decelerating. The Hubble relation can be expressed in terms of redshift, z, a dimensionless quantity that measures the deviation in wavelength, ∆λ, between the wavelength of the emitted radiation λem and the received radiation’s λr z≡λr−λem λem .(2.43) Such a deviation may arise due to Doppler effect, as already mentioned in section 2.1.1 and, for non relativistic motion, it can be stated as z≈v c,(2.44) where vis the relative velocity between the observers. This means that the Hubble relation can be rewritten as z=H cr, (2.45) or, equivalently, as r=zc H.(2.46) 40
Thus, an accelerating/decelerating expansion means that the expansion rate was smaller/bigger in the past. This implicates that, in order for an object to reach a certain recession velocity, it should be located further away/closer than expected. Or, in terms of the observed light, it should be measured dimmer/brighter than expected. For the Hubble curve, this implies an upward/downward curvature relatively to the straight curve expected for an empty or non-accelerating Universe. Bright objects much needed for distant observations do exist: supernovas, more concretely, type Ia supernovae (SN Ia)14. Circa 1998, two distinct groups analyzed approximately 50 SN Ia at high redshifts comprehended between 0.4−0.7 [9, 10]15. An example of the relation obtained between the logarithmic luminosity distance and redshift for those SN Ia is illustrated in fig. (2.5). Figura 2.5: First evidence for an accelerating Universe. Taken from [9]. 14A type Ia supernovae is a powerful thermonuclear explosion induced by gravitational contraction and consequent heating of the core of a white dwarf, which then collapses into a neutron star. 15Three of the multiple authors of the referred works, Saul Perlmutter, Brian P. Schmidt and Adam G. Riess were awarded the conjoint Nobel Prize in Physics in 2011 for the first evidence of an accelerating expanding Universe 41
48
3 Dynamical Dark Energy The compelling evidence that our Universe is experiencing an accelerated expansion has led theorists to propose alternative solutions that could explain such phenomenon. However, they have yet to be undoubtedly confirmed, or excluded, by observations. These range from the introduction of modifications to the geometrical left-hand side of Einstein’s field equation to the possible existence of a dynamical form of dark energy [45]. Either way, there must definitely be some new Physics that lies beyond the grasp of General Relativity and the standard model of particle physics. The first set of hypothesis arise from the assumption that the current framework of cosmology, General Relativity, may not be entirely sufficient to explain the undergoing accelerated expansion. Therefore, these became known as alternative gravity theories, which incorporate higher-order curvature terms on the metric side of the field equation, as in f(R) theories [46], or consider higher dimensions, as in braneworld models [47]. It should be noted that precision GR tests severely limit deviations from GR itself. However, most of these are local tests and their results may not necessarily apply on truly cosmological scales. The other set of solutions assumes that the framework established by General Relativity is correct. Standing from this point of view, the current accelerated expansion of the Universe is the result of a negative pressure component that takes part of the stress-energy tensor on the right hand side of Einstein’s field equation. Data reunited from the most recent observations indicate that almost 75% of the total energy density is in the form of this mysterious component, called dark energy. The equation of state of such a component, wDE in natural units, is given by wDE =pDE/ρDE,(3.1) where pDE and ρDE represent, respectively, the dark energy pressure and energy density. Its present value is currently constrained to be approximately wDE ≈ −1 [15], with an accuracy around 10% [12, 13, 14]. The simplest possible solution to explain the observed dark energy is, as already mentioned, the introduction of a Cosmological Constant in the field equations, according to which ρDE remains constant throughout cosmic evolution, with w=−1. This basically implies that vacuum ”weights”something and, therefore, should contribute to the total energy density. This is the picture envisaged by the standard model of cosmology, the ΛCDM model, which is the model that best fits current observations and is favored by statistical methods of model analysis, such as the bayesian criteria [48]. 49
The introduction of a cosmological constant is, fundamentally, a phenomenological description of dark energy. It suffers from a severe fine-tuning problem: the currently observed fractional dark energy density ΩDE is approximately 121 orders of magnitude inferior to the expected value extracted from quantum field theory calculations. Therefore, it is only natural for theorists to search for other solutions for dark energy, most of which consider it to be a dynamical entity and allow its equation of state to deviate from the cosmological constant value of −1. 3.1 Quintessence Present observations point out that the equation of state of the mysterious and dominant dark energy component should have a value of approximately −1. However, these observations are restricted to a small window in cosmic time and, therefore, don’t give a clear insight about how the evolution of this equation of state could have been like. So, there is no reason why one shouldn’t consider it to have been dynamical in the past, evolving with time until reaching the present ”cosmological-constant-like”value of −1. Models that allow a dynamical evolution of the dark energy’s equation of state with time are, for instance, Quintessence models. Quintessence is, in the broadest sense, a dynamical, slowly evolving energy component with negative pressure. It should account for the missing energy density between the matter density and the critical density of our Universe, considering it flat. One of the most studied possibilities for Quintessence is to consider that its contribution corresponds to the energy of a canonical scalar field, φ, slowly evolving down its potential, V(φ). A scalar field is the simplest type of field one can consider to account for this mysterious energy component, and is physically motivated by particle physics, including string theory, in which they arise naturally [17]. Quintessence as a rolling scalar field was first introduced by Peebles and Ratra as being a smooth and homogeneous component minimally coupled to matter. They also presented the motivation for choosing certain potentials in which it should evolve in order to become a late-time dominant energy component in cosmic history, thereby driving the Universe into an accelerated expansion [18]. These models have been intensively studied as a strong hypothesis for dark energy ([19, 49, 50, 51] and references therein), and still are. Particularly, in [49], is argued that quintessence should be a heterogeneous time-dependent energy component, because the definition of a smooth component ignores its response to the inhomogeneities of the cosmic fluid. Also, its fluctuations should leave a distinguishable imprint on the CMB anisotropy spectrum, possibly useful for discriminating quintessence from the cosmological constant in future observations. 50
Subsequent authors, in [50] and [51], have continued the analysis of quintessence models, however noticing the intrinsic fine-tuning demanded by these models in order to obtain a correct and suitable cosmology. Therefore, these authors introduced the notion of tracker quintessence, as models for which little fine-tuning is required in the initial conditions in order to obtain a common evolutionary track for the field. In these models, generally, the dark energy’s equation of state follows the dominant component’s one (either radiation or matter) for much of the cosmic evolution. The density of the quintessence field takes over only in the present epoch, when its density becomes comparable to the matter density, driving the Universe into an accelerated expansion and behaving as a cosmological constant with an equation of state approaching −1. Therefore, these tracker models may represent a possible solution to the coincidence problem. Following this line of work, there are models known as k-essence which have been proposed and also constitute a good attempt for tackling these problems [52, 53]. Other interesting models explore more and complex interactions between the dark energy quintessence field and matter [54, 55], being known as interacting quintessence models. 3.2 Slow-roll Quintessence In the following section, the basics of the dynamics of a slowly rolling quintessence field are presented. Throughout it, natural units of c=~= 8πG = 1 will be assumed, as will the metric signature (−,+,+,+). The action of a quintessence scalar field, φ, minimally coupled to gravity, is given by [17, 33] S=Zd4x√−g−1 4R+Lm+LQ,(3.2) where gis the determinant of the metric tensor, Ris the Ricci scalar, Lmand LQ are the matter and quintessence Lagrangian densities. In the simplest possible realization of a quintessence scalar field minimally coupled to gravity and ordinary matter, the lagrangian density of the field is given by [17] LQ=−1 2(δµφ) (δµφ)−V(φ),(3.3) where the first term is the field’s canonical kinetic term and the second is the potential energy of the field. The minimal coupling to gravity and ordinary matter is common to many quintessence models, because it avoids the dragging of matter perturbations by the field, which would prevent them from collapsing even before the field begins to dominate [18]. This makes the detection of quintessence extremely hard through its interaction with Standard Model particles, as opposed to suggested interacting quintessence models [55]. 51
The canonical kinetic term indicates that the field is expected to roll in the potential away from local unstable maxima and towards the stable extremum minima of the potential. However, there are models that consider the possibility of the field evolving with non-canonical kinetic terms, whose field lagrangian’s are built from non-linear terms of φand X=−(1/2) (δµφ) (δµφ). Examples of such models are the already mentioned k-essence models and also phantom energy models [56, 57]. In the latter, the field’s lagrangian has a negative kinetic term, and the field is expected to roll towards the potential stable maxima, contrary to the canonical case. These models produce an equation of state wDE <−1, which is not completely ruled out by observations. This violates the null energy condition, since the energy density of the field grows with time, rather then decaying [45]. Under such conditions, the fate of our Universe is to end in a Big Rip singularity, since the scale factor and expansion rate diverge in a finite time, ripping apart all known matter [58]. The stress-energy momentum tensor of the field can be found by variational principles, varying the action in terms of the metric tensor gµν, such that [17] Tµν(Q)=−2 √−g δSQ δgµν ,(3.4) where SQis the quintessence scalar field action, given by SQ=Z√−gLQd4x=Z√−g−1 2gµν(δµφ)(δνφ)−V(φ)d4x. (3.5) Knowing that δ√−g=−(1/2)√−ggµνδgµν and δgµν =gµαgνβδgαβ, the variation of the field’s action results in δSQ=Z√−g−1 2(δµφ)(δνφ)−1 2gµνLQδgµνd4x, (3.6) allowing to conclude that the field’s energy-momentum tensor is Tµν(Q)= (δµφ)(δνφ)−gµν 1 2gαβ(δαφ)(δβφ) + V(φ).(3.7) In a flat, isotropic and homogeneous Universe, the Robertson-Walker metric, eq. (2.22), can be written as ds2=−dt2+a2(t)dx2+dy2+dz2. From this, the quintessence’s field energy density, ρQ, and pressure, pQ, assuming the field depends on time alone, are determined by ρQ=−T0 0=1 2˙ φ2+V(φ), pQ=Ti i=1 2˙ φ2−V(φ).(3.8) 52
From these last equations, the quintessence’s field equation of state can be put simply in the form wQ=pQ ρQ = 1 2˙ φ2−V(φ) 1 2˙ φ2+V(φ).(3.9) This equation immediately reveals that, in order to have a present-day value for the quintessence field equation of state close to −1, the condition ˙ φ2/2<< V(φ) has to be verified, which means that the field’s kinetic energy has to be substantially smaller to its potential energy. That is why, generically, quintessence is associated to a slowly rolling scalar field evolving in its potential. This can be achieved through the usual slow-roll conditions, similar to the inflationary conditions for accelerated expansion [20]: 1 V dV dφ 2 << 1,(3.10) and 1 V d2V dφ2<< 1.(3.11) These conditions imply that, in order to satisfy observations today, the potentials in which the field evolves must be nearly flat. This is in agreement with the results of reference [59]. Its authors used a Monte Carlo simulation to derive the class of potentials that yield a value of wQ≈ −1 today, and found out that these should either be nearly flat or have a very sharp curvature potential change in the present epoch. It is worth noticing that, in the case of the latter set of potentials, a second coincidental question is raised: this is concerned with the reasons and initial conditions that have led the field to enter such a special region of the potential at the present epoch, particularly when the dark energy energy density began to dominate over the matter density [20]. The slow-roll conditions are extracted from the field’s equation of motion, ¨ φ+ 3H˙ φ+dV dφ = 0,(3.12) which can be derived from the zeroth component of the covariant energy-momentum conservation law, T0µ;µ= 0, using the Robertson-Walker metric. H= ˙a/a is the Hubble parameter, given by eq. (2.40). Equation (3.12) states that the scalar quintessence field, φ, is expected to roll down its potential, with its motion being damped by a term proportional to the Hubble parameter, H. 53
The inflationary slow-roll conditions allow the equation of motion of the field to be simplified by neglecting the acceleration term ¨ φ, yielding simply 3H˙ φ= −dV/dφ, which is the inflationary’s equation of motion. In the inflationary regime, the cosmic expansion is dominated purely by the scalar field. Therefore, the Hubble parameter is given by the expression H2≈ρQ/3 (recall eq. (2.53). It is, then, solely determined by the potential of the field, which is approximately the field’s energy density. Therefore, under these slow-roll conditions, the acceleration term, ¨ φ, can be shown to be smaller than the friction term, 3H˙ φ[4]. However, the inflationary slow-roll conditions do not fully apply to the quintessence counterpart. This is because, for quintessence, it is not required that the acceleration of the field, ¨ φ, to be smaller than the friction term, 3H˙ φ, since the Hubble parameter’s H2= 1/3 [ρrad +ρmatter +ρQ]evolution in a flat Universe is determined by the matter/radiation energy density along with the quintessence’s field energy density. Inflation only accounts for the scalar field’s energy density, which is required to dominate over all other forms of energy during the inflationary expansion [21, 22]. Even though inflationary slow-roll conditions need not apply to quintessence, this does not mean the potential where the field evolves can not obey the slow-roll equations (3.10) and (3.11). In reference [20], a Thawing quintessence model was considered in which the potential where the field rolls is nearly flat, satisfying the slow-roll equations for the field’s initial value φ=φ0. The authors have obtained a general result for the evolution of the quintessence’s equation of state, wQ, with the scale factor, a, for such models that does not differ much from the present day value −1. It consists on a single expression that depends only on the dark energy’s fractional energy density and equation of state present-day values, ΩQ0 and w0, respectively. Slow-roll thawing models offer a natural way of obtaining a dark energy equation of state that does not deviate much from the present-day value imposed by observations of wDE ≈ −1. In these models, the field is initially almost frozen at a value φ=φ0by the damping Hubble parameter, H, resulting in a value for wQ≈ −1. Then, as the Hubble parameter decreases with cosmic evolution, the field is released from its initial position and rolls down the potential towards a zero minimum, slowly increasing its kinetic energy. Accordingly, the equation of state also increases with time to values grater than −1, but not much [23, 60]. Similarly, there are the Freezing models, where the field is already rolling down the potential towards its minimum and wQ>−1, prior to the onset of acceleration. However, the field’s evolution is slowed down and critically damped by the Hubble’s friction term when its density starts to dominate, and wQapproaches asymptotically −1[20, 23]. Both models pose, therefore, an experimental challenge when trying to distinguish them from the cosmological constant. 54
3.2.1 Quintessence field rolling near a potential minimum In the previous section it was well established that, in order for a minimally coupled quintessence model to reproduce the present-day value of −1 observed for the dark energy’s equation of state, the field has to be evolving towards the minimum of its potential in a slow manner, such that the field’s kinetic energy is much less than its potential energy. This implies that the potential has, somehow, to obey a set of slow-roll conditions in order for this to be verified. However, the usual inflationary slow-roll conditions need not apply to quintessence. Recently, in [60], it has been shown that, although the common slow-roll conditions given by equations (3.10) and (3.11) are sufficient to obtain a equation of state that is close to −1 today, they are not necessary. In the referred article, the author derived a more general set of conditions on the potential assuming that the equation of state is always close to −1 throughout the evolution of the field. What’s important to understand is that, in fact, the slow-roll conditions are more flexible than those of inflation, which has led to models that relax such conditions. One of such models envisages a scenario where a quintessence field is rolling in the vicinity of a minimum of the potential [61]. This model was inspired in another one [62], which considers a field very close to an unstable potential maximum rolling away from it. In that way, the validity of the slow-roll condition given by equation (3.10) alone was guaranteed, while the condition imposed by equation (3.11) was somewhat relaxed. This model introduces a new degree of freedom to the dark energy equation of state, wQ, besides those already introduced by the more general result of [20]. So, wQdepends not only on the dark energy’s equation of state and energy density present-day values, but also on a parameter that consists on the value of the curvature of the potential, (1/V )(d2V/dφ2), at the extremum. And, as expected, when the limit (1/V )(d2V/dφ2)→0 is considered, one recovers the result of [20], since both slow-roll equations are respected. The model of reference [61], the main focus of this section, considers a canonical scalar field, φ, minimally coupled to ordinary matter and gravity evolving very close to a stable extremum of the potential, V(φ), therefore a minima. The model can also be extended to a phantom field rolling near a potential maximum, which will not be done here. It is convenient, before proceeding, to recall the equation of motion of the scalar field derived under the Robertson-Walker isotropic metric ¨ φ+ 3H˙ φ+V0(φ) = 0,(3.13) where, as before, the dots are derivatives with respect to coordinate time x0 (x0=t, in natural units). The prime corresponds to a derivative with respect to the field φ, such that V0=dV/dφ, and His the Hubble parameter. 55
Following reference [62], it is possible to get rid of the ˙ φterm in equation (3.13) by performing the change of variables φ(t) = u(t) a(t)3/2,(3.14) which results in the following ¨u−3 2"¨a a+1 2˙a a2#u+a3/2V0(u/a3/2) = 0.(3.15) This equation can be simplified by applying the Friedmann equations for the evolution of the scale factor a(t), equations (2.27) and (2.29). In natural units, for a flat Universe (hence k= 0), the first is given by H2≡˙a a2 =ρ 3,(3.16) where ρis the Universe’s total energy density, and the density of each component evolves with the scale factor according with equation (2.35). The second Friedmann equation, in natural units, then becomes ¨a a=−1 6(ρ+ 3p),(3.17) where pis the Universe’s total pressure. Each component’s pressure contribution relates to its energy density through the respective equation of state. Using these last two equations, equation (3.15) acquires the form ¨u+3 4p u +a3/2V0(u/a3/2) = 0.(3.18) Subsequently, it is assumed a Universe of pressureless matter and a quintessence scalar field that plays the role of dark energy. To realistically mimic the observed behavior for the dark energy, and due to the nature of the model assumed, such a quintessence component should have an equation of state wQ≈ −1 throughout the cosmic evolution. Therefore, by equation (2.35), it should have an approximately constant energy density ρQ=ρQ0, where ρQ0is the present-day value observed for the dark energy’s density. In these conditions, the total pressure of the Universe can be approximated by p=pQ=−ρQ0. Then, equation (3.18) takes the form ¨u−3 4ρQ0u+a3/2V0(u/a3/2)=0.(3.19) 56
Since the point of interest of this model is to obtain the dynamics of a scalar field rolling near a stable extremum of the potential (namely a minima for quintessence) located at φ=φ?, it is useful to perform an expansion of the potential near this extremum. A simple Taylor-series expansion up to the quadratic order gives: V(φ) = V(φ?) + 1 2V00(φ?)(φ−φ?)2+O(φ−φ?)3,(3.20) where the linear term in φis absent due to the fact that V0(φ?) is null at the extremum. Substituting this expression into equation (3.19) (and imposing that V(φ?) = ρQ0)18, one obtains the final differential equation for the field’s evolution [61] ¨u−3 4V(φ?)−V00(φ?)u= 0,(3.21) Its solution, defining the constant kto be given by k≡p(3/4)V(φ?)−V00(φ?),(3.22) is simply u=Asinh(kt) + Bcosh(kt),(3.23) where Aand Bare constants. With the assumption of a dark energy, with a nearly constant energy density, and pressureless matter dominated Universe, it is reasonable to assume that the evolution of the scale factor is well approximated by its value in the ΛCDM model [63] a(t) = 1−ΩQ0 ΩQ01/3 sinh2/3(t/tΛ),(3.24) where ΩQ0is the present-day value of the fractional quintessence’s density, ΩQ= ρQ/ρc, and a= 1 at the present time t=t0. On the other hand, tΛis defined as tΛ=2 p3ρQ0 =2 p3V(φ?).(3.25) 18This comes by as a good approximation, given that the field’s kinetic energy should be much less than its potential energy. Therefore, the value of the potential extremum close to which the field rolls should account for almost all of the field’s energy density 57
Figura 3.4: Evolution of the quintessence’s field equation of state, wQ(a), according to this model’s analytical equations [dashed red lines] (3.43) [for K2= 0] and (3.40) [for K2=−10 and K2=−20], against its exact evolution for the exponential potential (solid lines), for the indicated values of K2. The value of φiis the same for the three cases. The most interesting case is when K2<0. This needs a large V00/V , implying a sharp curvature of the potential at the extremum, a minimum for quintessence. As figure (3.2) tells, the field seems to be frozen to its initial value in the first moments of cosmic evolution. However, due to the potential’s curvature, the Hubble’s friction term in equation (3.12) can be overcome and the field is able to start rolling down the potential, and eventually begins oscillating around its minimum. Therefore, one should expect to see a similar behavior for the respective equation of state wQ(a), as shown by figures (3.3) to (3.5). Figura 3.5: Evolution of the quintessence’s field equation of state, wQ(a), according to this model’s analytical equations [dashed red lines] (3.43) [for K2= 0] and (3.40) [for K2=−10 and K2=−20] against its exact evolution for the PNGB potential (solid lines), for the indicated values of K2. The value of φiis the same for the three cases. 64
The analytical solution for the evolution of the equation of state for this last case is given by equation (3.40) and is depicted in figures (3.3) to (3.5) against the exact numerical solutions for, respectively, the quadratic, exponential and PNGB potentials. The values for K2are K2=−10 and K2=−20, along with the limiting, monotonically evolving case of K2= 0. ΩQ0was taken to be equal to 0.74. The value of φidetermines the numerical value of w0, which decreases with φi. This was chosen so that w0≈ −1, which was then inserted in the analytical equation. The agreement is very good, and corroborates the usefulness of this model for producing a general expression for wQ(a) for a diverse set of potentials. As one can see in figures (3.3) to (3.5), the equation of state for the curved regime K2<0 presents some form of oscillation, as would be expected due to the field’s oscillation around the extremum. This oscillation is, of course, more pronounced for a steeper potential, therefore with a greater absolute value of K2. However, it never deviates a lot from −1. And, as predictable, the agreement of the analytical solution to the exact one is smaller as the steepness of the potential increases, since a steeper potential decreases the accuracy of the slow-roll condition imposed by eq. (3.10). Summing up, this model consists of a quintessence scalar field rolling near a potential minimum. One of the usual inflationary slow-roll conditions, that of eq. (3.11), is relaxed, allowing for curved potentials. For this model, one can obtain an analytical solution for the evolution of the equation of state wQ(a) depending on three parameters: ΩQ0,w0and K2. The most interesting case comes up when K, which is a measure of the potential’s curvature at its extremum, is imaginary. In this regime, the field is able to overcome the Hubble friction term that damps its evolution and is capable of oscillating around the potential’s minimum, which reflects in the field’s equation of state. This particular regime is the case analyzed in this work. However, although wQ(a) oscillates, it never deviates that much from the observed present-day value of −1, meaning that its evolution is still potential dominated. This is because, unlike other oscillating models [24, 25, 26, 65], the potential’s minimum is different from zero and equal to the dark energy’s present-day energy density ρQ0. If V(φ?) = 0, then the equation of state would oscillate between ±1. The model considered here seems to lie somewhere between the lines of the thawing models of [20, 23, 60] and the rapidly oscillating models of [24, 25, 26]. It provides a reasonable way of obtaining a dynamical evolution for the equation of state that does not differ much from −1. And, despite the analytical form for wQ(a) depends on a set of three parameters, it is usefully applicable to a wide range of potentials. Nonetheless, it has a downside. In order for the field to be evolving today, its mass should be of the order of the Hubble parameter today, which comes as a very unnatural small value [17, 18, 20, 49]. 65
3.3 Rapidly oscillating Quintessence In this section, the second model of oscillating quintessence studied in this work is presented. Unlike the previous model, which consisted of a scalar field rolling in the vicinity of a non-zero minimum potential, this one considers a scalar quintessence field rapidly oscillating around a zero-potential minimum near the present. This type of model has been studied for a while now [24, 25, 65], and broke the traditional lineage of quintessence models that tried to conservatively obey the slow-roll conditions necessary to have a dark energy equation of state close to −1 today and throughout cosmic evolution. These studies have shown that with a fairly simple power-law potential for the evolution of the field, V(φ) = k|φ|n,(3.48) it is possible to obtain a model in which, if certain conditions are met, the field is initially slow-rolling through the potential (maintaining an almost constant value) to, later on, roll towards its zero minimum and rapidly oscillate around it. kis a constant and 0 ≤n < 1 is the potential’s power-law exponential. This seems very reasonable and, most importantly, these oscillating models were shown to have a constant dark energy equation of state wDE (averaged over a period of oscillation T) that depends solely on the value of n[24, 25, 65] wDE =n−2 n+ 2.(3.49) Therefore, according to the last equation, it is perfectly possible to have an oscillating quintessence model that, after entering the oscillatory regime, has an equation of state sufficiently close to −1 such that it does not contradict current observations. This usually means that nmust be much smaller than 1. Although such a potential seems odd, it may arise from Lagrangians with a φ2potential and a non-canonical kinetic term [66]. There’s an important paper by Johnson and Kamionkowski [27] that analyzes the suitability of these oscillating models for driving the present-day cosmic acceleration. Based on a simple argument they conclude that the dynamical and gravitational instabilities arising from the presence of an anharmonic potential term in these models renders them unsuitable for the growth of matter inhomogeneities and, therefore, for driving cosmic acceleration. This is why both [24, 25] present conditions and restrictions over the potential’s initial conditions that are necessary for the field to start oscillating close to today. Ultimately, the late-time oscillation regime resists as the only acceptable scenario for these models. 66
The rapidly oscillating model under scrutiny in this section is slightly different than the one presented previously, and is described in full detail in [26]. The focus of this paper is not exactly the rapidly oscillatory phase that leads the present day acceleration of our Universe. Its main interest is the dynamical instability arising from the anharmonic terms in the potential and its consequences on the field’s fluctuations and on the gravitational potential. Despite that, this article provides the necessary initial and fine-tuning conditions for the oscillations to start close to today in such a way that the observable expansion history is not much affected. Besides, the authors of this article also identify some possible observational signatures that could be useful for future dark energy surveys, making this model discernible from other dark energy models. According to the model of [26], motivated by monodromy and supergravity models, the field is supposed to evolve in a potential of the form V(φ) = m2M2 2"(φ/M)2 1+(φ/M)2(1−α)#,(3.50) where 0 ≤α < 1. The parameter mdetermines the curvature of the potential, d2V/dφ2, at its minimum. According to the potential’s formulation, one can envisage a scenario where the field is initially slowly rolling through the potential towards its minimum, falling later onto it and entering the oscillatory regime. The constant parameter Mwill determine where the shape of the potential changes according to the evolving value of φ. This scenario can be accomplished if one takes the initial value of the field, φi, to be much larger than M, since the potential has a quadratic minimum for φ << M and a shallower than quadratic form for φ >> M, as follows V(φ)≈m2M2/2(φ/M)2αφ >> M, m2/2φ2φ << M. As has been well established in the previous sections, current observations are consistent with a form of dark energy with a constant equation of state, close to the value −1, throughout the cosmological history. In order for this quintessence model to be in agreement with the observed expansion history, while allowing for a rapid growth of structure with a few oscillations close to the present time, or a≈1, there is the necessity to impose certain conditions on the field’s initial evolution and the potential parameters. This is, once again, a classic fine-tuning problem that appears in many other dark energy models, even in the simplest model of all, the ΛCDM model. 67
In order for the potential to produce oscillations near the present time, there is the requirement that 102H0.M,m.10−2mP l (where MPl is the Planck mass which, in the adopted planckian units, is equal to 1) and α << 1 [26]. However, there are several more constraints that one can derive from the homogeneous equation for the evolution of the quintessence field, which is given by the familiar equation ¨ φ+ 3H˙ φ+V0(φ) = 0 (3.51) where the Hubble expansion parameter, H(with the contribution of a quintessence field and pressureless matter only), as a function of the scale factor a, in a flat Universe, as has already been stated before, is given by H2(a) = 1 3"˙ φ2 2+V(φ)#+H2 0 Ω0m a3.(3.52) In these last equations, the dot refers to a derivative with respect to cosmic time t, and the prime to a derivative with respect to the field φ; Ω0mand H0are the present-day values of the fractional matter density and the Hubble parameter, respectively. Following this model’s derivation in [26], imposing that the field starts oscillating close to today and, up until that moment, presents a behavior similar to that of a cosmological constant with an equation of state wDE =−1, implies that the field is initially slowly rolling ( ˙ φ2/2<< V (φ)) through the potential. This is better achieved if the initial value of the field, φi, during the matter dominated epoch is larger than M, and if αis much smaller than 1, which greatly reduces the slope of the potential. In order for the quintessence field to have an energy density ρQ, from the beginning of cosmic evolution, close to that predicted by the observational favorite ΛCDM model, ρΛ, it is necessary that V(φi)∼ρΛ⇔(mM)2φi M2α ∼6 (1 −Ω0m) (mP lH0)2.(3.53) Assuming that the field slow rolls during most of the matter dominated era allows one to obtain an approximate solution for its evolution from eq. (3.51). To do so, one should be convinced that, during this epoch, ¨ φis approximately zero. And, the Hubble parameter reduces to the matter term, proportional to a−3. With these considerations, taking φi>> M, one has φ−φi M∼−αa3(m/H0)2(φi/M)2α−1, α 6= 0 −a3(m/H0)2(M/φi)3, α = 0.(3.54) 68
Despite the fact that the last equation is not applicable during the expected oscillatory regime, it can be used to extract an estimative for the value of awhen the field starts to oscillate, aosc. This can be obtained by setting φ= 0 in equation (3.54) [26] a3 ∗∼((H0/m)2(φi/M)2(1−α)α−1, α 6= 0 (H0/m)2(φi/M)4, α = 0.(3.55) The true value of aosc for which the field starts to oscillate actually differs of a∗, but both of them are monotonically related. Combining the last equation with equation (3.53), one can obtain two expressions for (φi/M) and (m/H0) in terms of mPl,M,αand a∗only: φi M∼(α1/2a3/2 ∗(mPl/M), α 6= 0 (mPl/M)1/2a3/4 ∗, α = 0,(3.56) and, m H0∼a−3α/2 ∗α−α/2(mPl/M)1−α, α 6= 0 (mPl/M), α = 0.(3.57) Since αis directly related to the asymptotic slope of the potential, it should determine the time the field takes to transition from the slow-roll evolution to the oscillation around the zero minimum of the potential. In order to see this, one can determine the approximate value that the field takes when slow-roll ends, φes, by considering the condition ˙ φ2∼V(φ) and using equation (3.51) with ¨ φ≈0 to extract ˙ φ[26]. This yields φes ∼αmPl, α 6= 0 M(mPl/M)1/3, α = 0.(3.58) Using the last result, one can determine the approximate time this transition takes to occur by expressing ˙ φas aH(dφ/da)≈aH(φosc −φes)/∆ain equation (3.51), yielding ∆a∼α1−α, α 6= 0 (M/mPl)2/3, α = 0.(3.59) φosc is taken to be 0, and designates the value of φthat marks the beginning of the oscillatory regime. The latter result is obtained when considering the transition taking place close to today (a≈1). 69
So, as one can see in the last equation, the larger αis, the longer it takes for oscillations to start. Therefore, in order to produce oscillations close to the present time and still have consistency with the ΛCDM expansion history, αshould be much smaller than 1 for the case of α6= 0, as has been stated before. The case considered in this work will only be that of α= 0, and one should also guarantee that M << mPl is verified. Throughout the rest of this work, α= 0 will be fixed. The initial value of the quintessence field will be assumed to be larger than M. At φ≈Mthe potential changes its shape, making the transition from the slow-roll zone to its minimum, where the field starts to oscillate. So, φi>> M contributes to the start of oscillations close to today. The only free parameter of this model will be M. The other two parameters will be fixed by two constant ratios, such that m/H0= 1130.6 and φi/M = 23.7 [26]. The initial value of the field’s velocity ˙ φiis determined by equation (3.54). This construction assures that the value of H0produced by this model agrees very closely with that of the ΛCDM model for a value of M= 0.00200 [26]; not only that, but the field will be slowly-rolling until φ≈M, starting to rapidly oscillate around a≈0.8, which is close to the present. Figure (3.6) shows displays the evolution of the dark energy equation of state for M= 0.00200 and M= 0.00250. Figura 3.6: Evolution of the quintessence equation of state wQ=w, in a flat Universe, for the rapidly oscillating quintessence model with M= 0.00200 (solid black line) and M= 0.00250 (red dashed line). The present-day dark energy density is taken to be ΩDE = ΩΛ= 0.74. 70
From fig. (3.6) one can see how the different moments in the evolution of the quintessence field are reflected in the behavior of the respective equation of state. For φ >> M, which corresponds to the initial moments of cosmic evolution, given that φiis set to be much larger than M, the evolution of the field is practically static, with almost no change whatsoever in its value and the potential’s. During this phase, the slow-roll conditions are respected, and the evolution of the field is dominated by the Hubble friction term in equation (3.51). Consequently, the quintessence equation of state, w, presents an approximately constant ’cosmological constant’ value of −1. However, although slowly, the field is indeed rolling down its potential towards its zero minimum. As time passes, the field’s value steadily decreases until a point at which φ≈M. As already referred, this is the point in which the shape of the potential changes from the almost constant/zero curvature slow-roll region to its quadratic minima. As the slope of the potential increases, so does the field’s kinetic energy. This reflects on the equation of state, whose value slowly increases, setting itself apart from the −1 value of the slow-roll epoch associated to the Cosmological Constant. The higher Mis, the later the field enters this region, as one can see in fig. (3.6), with the equation of state for M= 0.00250 deviating from −1 a bit later than for the case of M= 0.00200. Figura 3.7: Evolution of the Quintessence field φfor the rapidly oscillating quintessence model with M= 0.00200 (solid black line) and M= 0.00250 (red dashed line). The horizontal dashed black and solid red lines mark where φ= 0.00200 and φ= 0.00250, respectively. When the field φcrosses this values, this means that the it entered a region of the potential where the slope is steeper and will roll down increasingly faster towards the zero minimum of the potential where it starts to oscillate. 71
Increasingly faster, the field approaches the zero minimum of the potential and starts oscillating in a rapid manner around it. Therefore, when the field crosses the potential minimum, it has maximum kinetic energy and, consequently, the equation of state goes to w= 1. When the field reaches the maximum ’height’ on the potential it possibly can, it has zero kinetic energy and, therefore, the value of the equation of state will be w=−1. Between these moments, the equation of state obviously crosses zero, where the kinetic energy of the field is equal to its potential energy. From the moment oscillations kick in, the field’s behavior becomes similar to that of pressureless matter, with its equation of state taking a constant value of w= 0 (as one can check in equation (3.49) with n = 2), when averaged over an oscillation period [24, 25, 26, 65]. Figure (3.7) plots the evolution of the quintessence field for both M= 0.00200 and M= 0.00250, and corroborates the last paragraphs. As the field enters the oscillatory regime, it behaves like pressureless matter with a constant time-averaged equation of state w= 0. The field rapidly oscillates around the minimum because, close to the present, the potential’s curvature value mis much larger than the Hubble parameter’s friction term. Nonetheless, it is expected that the amplitude of the field’s oscillations of the decreases with time, which becomes evident in figure (3.7) for aclose to 1. The field’s energy density now evolves with a−3, recalling equation (2.35). So, if the energy density of the field is equal to the sum of its kinetic and potential energies, then the amplitude of the field should decrease with a−3/2, given that its potential energy is proportional to φ2[26]. It was shown in [25] that, for a generic power-law potential, the ratio between the frequency of the oscillations, ν, and the Hubble parameter, H, is an increasing function of time. This happens for both a matter or a quintessence field dominated Universe as long as the field’s oscillations amplitude decreases with time, as is the case studied here. What this means is that the field does not ”rest”at the minimum, but approaches it asymptotically, oscillating around it with an ever increasing frequency. And, as this happens, the quintessence field’s equation of state takes a time averaged value equal to zero, which means that the field’s energy density will decrease with a−3from thereon. 72
4 Constraining Dark Energy 4.1 Present Constraints Although there are many different quintessence models that can possibly explain the observed dark energy, it is imperative to understand if they are viable by checking if the present-day data available from different experiments is compatible with those models. It is also important to evaluate the ability that future astronomical experiments will have in constraining the dark energy cosmological parameters, such as the present-day value of its equation of state, w0, or the present-day value of the dark energy’s fractional energy density, ΩDE0. According to the most recent compilation on Cosmological Parameters (see Cosmological Parameters 2010 [15]), the results of recent experiments are more consistent with the cosmological constant case [w=−1], although they are not particularly sensitive to the evolution of the equation of state in the past. The aim of future experiments is precisely to constrain such evolution and eventually observe, for higher redshifts, a deviation of wfrom the cosmological constant value of −1. Figura 4.1: Representation of the 68.3%, 95.4% and 99.7% confidence contour plots for wand Ωmatter, assuming a flat Universe. Taken from [12] Figure (4.1) is a likelihood plot of the confidence regions for the dark energy equation of state, w, and the matter fractional energy density, Ωm, assuming a flat Universe [12]. This plot results from individual data from a CMB, baryonic acoustic oscillations (BAO) and a SNa experiment. The combined constraints (grey region) are in particular consistent with that w=−1. Such analysis concluded that w=−0.96 ±0.06 ±0.06 and Ωmatter = 0.274 ±0.016 ±0.012, taking statistical and systematic errors into consideration. Similar results were found in [13]. 73
iCosmo is an interactive software package (available at http://www.icosmo.org) used for the computation of cosmological quantities for the low-redshift universe, accepting as input any values of the cosmological parameters. It has the particularly important feature of being able to derive observed quantities, such as the cosmic shear, for an existing or upcoming probe. Cosmic shear is the statistical measurement of shear (γ1, γ2), and uses the two-point function of the shear field as an observable, Cij(l), where lis the order of the multipole moment [31]. From that, it is also able to determine the associated uncertainties ∆Cij(l), and compute the expected constraints by calculating the respective Fisher matrices through equation (4.5). For the weak lensing case specifically, it produces the constraints associated to weak lensing tomography [71]. For that, firstly, it divides galaxies into separate redshift bins (usually 10) and then computes the two-point correlation function of the shear field. This way, the shear patterns at each slice are studied, looking not only for cross-correlation in each of them, but also for correlations between the different slices. After that, it determines the associated fisher matrices, which are used to plot the joint credible regions. For weak lensing, the Fisher matrices can be calculated through the following equation [31, 72, 73] Flens αβ =X l 2l+ 1 2fSurvey sky ∂Cij(l) ∂pα[Cl]−1 jk ∂Ckm(l) ∂pβ,(4.18) where the partial derivatives are taken with respect to the model’s parameters pα; fEuclid sky is the fraction of the sky covered by the analyzed survey and [Cl]−1 ij is the covariance matrix for a given lfor the i−jbin pair [72, 73], written as [Cl]−1 ij =Cij(l) + γ2 int Ng δij2 ,(4.19) where γint is the galaxy-intrinsic shear rms in one component [73]; δij is the Kronecker symbol and Ngis the number density of galaxies in the i−th bin. For the weak lensing photometric survey, the fraction of the sky expectedly covered by Euclid is 1/2, corresponding to an area of 20.000 deg2. It should able to measure the shape of over 2 billion galaxies at a median redshift of 1.0, covering a redshift range of approximately 0 < z < 2. For calculation purposes, the number of galaxies per arcmin2assumed is 30, although Euclid has the optimistic goal of achieving 40. The galaxy distribution is expected to be close to that given by the analytical expression of Smail et al. [74], with parameters α= 2 and β= 1.5 [72, 73]. The galaxies are then equally distributed in 10 redshift bins. The galaxyintrinsic shear rms is taken to be γint = 0.22 [71], and the photometric redshifts for these galaxies should reach a precision of σz(1 + z)=0.05, with the goal of achieving a value of 0.03 [68, 72, 73]. 80
4.3.2 Baryonic Acoustic Oscillation Basics In the early stages of our Universe, before the re-combination era, its composition consisted of a dense and hot plasma where photons and baryons not only intensely interacted, but they were actually coupled through Thomson Scattering. Despite this seemingly harmonious existence, this primordial plasma struggled between the possibility of a collapse due to its self gravitational attraction and the ripening repulsion of the outward electromagnetic and kinetic pressure. These two competing effects led to the formation of acoustic waves in the photon-baryon fluid. These are still imprinted in the Cosmic Microwave Background (CMB) radiation until today, where they have frozen right after the primordial plasma cooled down and the decoupling of photons and baryons took place at z≈1100. This is also visible in the distribution of matter/galaxy which, in Fourier space or, equivalently, in the matter power spectrum P(k), corresponds to a series of acoustic peaks, named Baryonic Acoustic Oscillations (BAOs) [31, 75, 76]. These oscillations have a characteristic scale, the sound horizon at recombination s, which can be accurately measured by present CMB anisotropies observation surveys, such as the WMAP [2]. That renders the acoustic oscillations as a standard ruler for cosmology, with a great potential for constraining dark energy parameters [76]. This scale is present both in the transverse (y) and radial (y0) directions, such that [77] y=r(z) s(4.20) and y0=c H(z)s,(4.21) which means that the BAOs can be used to measure the co-moving distance, r(z), and the Hubble parameter H(z) at a redshift slice at z, in units of the sound horizon s. It is, therefore, very useful to have a way to estimate the accuracy (or, equivalently, the error) with which the BAO scale can be measured. This is necessary for estimating the power of future experiments for constraining cosmological parameters. The iCosmo software makes us of the analytical fitting formulae calculated in [77] to determine such accuracy. These were derived from several simulations that depended on the survey’s volume V, central redshift and average number density of galaxies. 81
The iCosmo software uses equations (6)-(9) from [77] to determine the accuracy with which a given spectroscopic survey can determine the BAO characteristic scale and evaluate if such experiment has the precision to properly resolve the acoustic features. As pointed out in [77], its conservative approach (disregarding, for instance, the contribution from the non-linear scales) leads to smaller accuracies when compared to other results that use full Fisher matrix simulations [75, 76, 77]. Not only that, the analytical fitting formulae was obtained for a ΛCDM cosmology, even though it should remain as a good approximation for other cosmological models [77]. From an experimental point of view, given a certain survey configuration, one is able to extract the radial and transverse BAOs components yand y0. Using the fitting formulae from [77], one can also determine the associated relative errors x= ∆y/y and x0= ∆y0/y0using the iCosmo software. These fractional errors are assumed uncorrelated between each other and between the redshift bins. This results in a diagonal data covariance matrix. Afterwards, iCosmo determines the associated Fisher Matrices Fαβ as given by [78]. Fαβ =X i 1 y(zi)2x2 i ∂y(zi) ∂pα ∂y(zi) ∂pβ+X i 1 y0(zi)2x02 i ∂y0(zi) ∂pα ∂y0(zi) ∂pβ,(4.22) where the partial derivatives are taken with respect to the parameter’s (pα) fiducial value, and the sums run over the observational bins. Euclid’s BAO survey will be a spectroscopic survey, cover an area of 20.000 deg2[68, 79]. It should observe galaxies at a median redshift of 1.1, in a redshift range of approximately 0.7< z < 2.1 [68]. The redshift distribution of galaxies is expected to follow the empirical predictions of Hα emitters with a limiting flux of 3 ×10−6erg s−1cm−2[68, 80, 81]. The number of galaxies per arcmin2is expected to be of 1.2, and these should be distributed in 14 redshift bins with a 0.1 spacing between them [80]. The spectroscopic survey should measure the galaxies’ redshifts with a precision of σz(1 + z) = 0.001 [68, 79, 80]. 4.4 What about Supernovae? In section 2.3.1 it was mentioned the importance that Supernovas, namely those of type SNIa, had in establishing that the Universe is currently undergoing an accelerated phase of expansion. These bright, distant objects have become very useful to astronomers, giving them the chance to grasp and measure astronomical distances in a precise manner. Although these objects aren’t a focus of the Euclid mission, they were important for the objectives of this work. 82
Supernovas have been given the epithet of Standard Candles, since their intrinsic luminosity can be deduced from other properties and their apparent magnitude is directly related to the distance at which they are observed. The most common procedure in determining the distance to a stellar object of any kind assumes that one has a rigorous method for evaluating its intrinsic, true luminosity, which should then be compared with the observed light flux of such object. The measured flux of light of an object is related to its intrinsic luminosity, only reduced by the squared distance as given by f=L 4πd2 L ,(4.23) where Lis the intrinsic luminosity of the object and dLdefines the luminosity distance. This distance differs from the proper distance dpto the object by dL(z) = dp(1 + z).(4.24) This means that, in fact, equation (4.23) takes the form f=L/4πd2 p(1 + z)2. The proper distance dpcan be calculated from the Robertson −Walker metric (eq. 2.22) by considering a light ray (ds2= 0) that connects a light source located at a certain comoving distance from an observer along a radial path (dΩ= 0), which results in the equation dp(z) = Zz 0 cdz0 H(z0).(4.25) In a static Universe, the luminosity distance equals the proper distance to the light source, dp=dL. However, in an expanding Universe, the observed light flux f, being proportional to the energy transfer per unit of time, should be reduced by a factor of (1+z)2[34]. This is a consequence of the wavelength lengthening of the emitted light. This ends up reflecting in the light’s frequency wand, consequently, in its energy, which is directly proportional to w. The ratio between the emitted energy/frequency wem to the observed one w0is given by wem w0 =λ0 λem =1 a= 1 + z, (4.26) which is eq. (2.43) re-written. Decreasing the frequency, reciprocally, the time interval must be increased by δt0=δtem(1+z), leading to a conjugated reduction of the energy transfer rate w0/δt0by (1 + z)2, implied by eq. (4.23). 83
In practice, as seen in figure (2.5), astronomers actually plot of the logarithmic luminosity distance m−Mvs. redshift. Mis called the absolute magnitude, and is a logarithmic scale for the intrinsic luminosity of a star; mis the apparent magnitude and, similarly, constitutes a logarithmic scale of the measured flux of a star. The difference between mand Mresults in an expression related to the luminosity distance by µ(z)≡m−M= 5 log10 dL 10.(4.27) This equation allows a direct confrontation between experiment and theory. Some of the most recent data was gathered by the Supernova Cosmology Project (SCP), which compiles µ(z) for a large number of Supernovae at different z values. These values can be compared to the expected ones using equation (4.27) for the respective redshifts at which the supernovas are located. In order to do so, one has to calculate the expected dL(z) using eq. (4.24). Therefore, µ(z) depends explicitly on the adopted cosmological model through H(z), enabling one to test both of the dark energy models studied in this work against observations. This allowed the maximization of the likelihood given by equation (4.2) and the determination of the true value of the models’ parameters pi 0that best fitted the available data. These constituted the ’fiducial’ values for each of the studied dark energy models. To do so, one needs only to minimize χ2. This was done with the generalization of eq. (4.1) that considers that the errors associated to each datum can be correlated. Therefore, not only the errors could have different variances, but these can also be correlated. Using a matrix formulation, eq. (4.1) is written as [67] χ2=X i,j (fi(p)−yi)E−1ij (fj(p)−yj),(4.28) where E−1ij is the inverse of the data covariance matrix. For Supernovae, such matrix can be obtained from the SCP website. As will be explained in the next section, χ2was calculated for all of the dark energy models’ free parameters, which were allowed to vary. From that, one obtained a matrix of χ2values from which the minimum was determined. The corresponding free parameters’ values were then taken to be the fiducial values of the corresponding dark energy model. 84
5 Results 5.1 The Fiducial Models In the first part of this work, according to what was described in section 4.3.2, I have used the data gathered by the Supernova Cosmology Project (SCP) in order to find the values of the parameters of the two dark energy models under analysis that best fitted the observational data. To achieve this, I had to find for which values the parameters of these two models maximized the likelihood function given by equation (4.2), thereby minimizing the quantity χ2as given by equation (4.1). These values, alongside with other important cosmological quantities that will also be specified, constitute the fiducial values of the respective dark energy model. For Supernovae, χ2takes the particular form given by equation (4.28). The observed quantity, yi, is the logarithmic luminosity distance, µ(z), for supernovae located at a redshift z. According to equation (4.27), µ(z) is related to the luminosity distance, dL(in parsecs), at the redshift of measurement. Therefore, with the theoretical expression for dL(z), equation (4.24), one can calculate the expected values of µ(z) for the quintessence model’s studied in this work, which will be fi(p) in eq. (4.28). Once dL(z) depends on the Hubble parameter, H, it will also depend on the models parameters, p, which influence the evolution of the quintessence energy density. The data covariance matrix, [E]ij, was also obtained from the SCP. I chose to use the covariance matrix that included systematic uncertainties. This is a more precise and realistic approach, as it also assumes that the uncertainties for the different elements of data are correlated. 5.1.1 The First Fiducial Model Here, I present the fiducial values for the parameters of the dark energy model of section 3.2.1. These results were obtained for the exact numerical evolution of the model, using the quadratic potential, and not for the parametrization obtained in that section for the dark energy quintessence equation of state wQ. Considering a flat Universe composed mainly of pressureless matter and a quintessence scalar field, the exact evolution of the field has three free parameters: the present-day value of dark energy fractional density ΩQ0, the initial value of the field, φi, and the value of K2, which is directly related to the potential’s curvature. The field is assumed to be static at the beginning of cosmic evolution, which means that its initial velocity is taken to be zero. This means the equation of state’s initial value will be −1. The present-day value of the Hubble parameter was taken to be h0= 0.70 Km s−1Mpc−1(H0= 100 ×h0) [15, 14]. 85
For this analysis, I have allowed each of the three parameters to vary between 10 different values. For K2specifically, I have only considered K2≤0, which is the main case of interest for this work due to the associated oscillatory behavior of the quintessence’s field. The list of values for all the parameters is listed below: •φi= [0.0,0.1,0.20,0.30, ..., 0.80,0.90]; •K2= [0.0,−5.0,−10.0,−15.0,−20.0, ..., −40.0,−45.0]; •ΩQ0= [0.70,0.71,0.72,0.73,0.74, ..., 0.78,0.79]. The values of these three parameters that minimized χ2take part of the fiducial values of this model. These are presented in the following table, alongside the matter’s present-day fractional energy density, Ωm0= 1 −ΩQ0in a flat Universe and the present-day value of the Hubble parameter, h0. All other fiducial cosmological quantities necessary for computation take the default values defined on the iCosmo software package, and are also presented. First Fiducial Model K2−15.0h00.70 φi0.30 n1.0 ΩQ00.74 τ0.09 Ωm00.26 σ80.8 Tabela 1: The Fiducial Values for the first Dark Energy model under analysis. This result deserves some comments. The model under scrutiny in this section is that of a quintessence scalar field rolling near a potential minimum. Given its characteristics (refer to section 3.2.1), one can see that if the value of φiand the field’s initial velocity is null, then the field will remain immobile at the minimum, independently of the value of the curvature at the potential’s minimum, or K2 . This means that the dark energy’s equation of state will not deviate from −1, being indistinguishable from that associated to the Cosmological Constant case, as discussed in section 2.3.2. Therefore, this simple analysis shows that this fiducial dynamical model yields a better fit to the observational supernovae data than the cosmological constant case. The best fit happens for a value of K2=−15.0 and φi= 0.30 and, as fig. (5.1) shows, the associated equation of state, w(a), presents some dynamical behavior: it starts to deviate from −1 at a≈0.05 and steadily increases its value until a maximum of w(a)≈ −0.84, when a≈0.5. It then decreases back to w(a) = −1, a value that is reached at a≈0.9. From that point on, the equation of state exhibits an attenuated increasing tendency that seems to persist up to the present, registering a present-day value of w0≈ −0.99. 86
Figura 5.1: Evolution of the equation of state wof the dark energy model of section 3.2.1 in a flat Universe, as a function of the scale factor a, for its fiducial values: ΩQ0= 0.74; K2=−15.0, and φi= 0.30. Figure (5.2) compares the comoving distance dp(a) (eq. (4.25)) between the ΛCDM model and this section’s model. The difference is positive in the distant past given the fact that φi6= 0, contributing to a larger dark energy density in the distant past than that verified for the ΛCDM. From then on, the divergence increases until a≈0.3, due to this model’s larger Hubble parameter in the distant past. The difference between both models’ predictions then decreases, as the field’s energy density, now with an equation of state larger than −1, decreases with a. Also, due to the damping action of the Hubble factor on the field’s motion, the equation of state tends to −1. Therefore, the difference between the predictions diminishes and seems to stabilize close to a value of 0.5 %. These results indicate that this dark energy model, for this fiducial values, can not be ruled out by observations, and may be preferred by them. Figura 5.2: Difference in percentage for the comoving distance registered by an observer today between this quintessence model, with its fiducial values, and the ΛCDM with ΩΛ= 0.74. 87
Figura 5.3: Difference between the observational µ(z) data and the ΛCDM’s expected µ(z) values (black crosses), and between the ΛCDM’s and this section’s quintessence model, taken with its fiducial values, µ(z) predictions (red crosses). For this analysis, h0= 0.70 and ΩΛ= 0.74. Figure (5.3) plots the difference between the ΛCDM’s expected µ(z) values and this quintessence model’s µ(z) predictions, taken with its fiducial values. This shows that this quintessence model’s predicted values are extremely close to the cosmological constant ones and, therefore, are also approximate to the observed data values. This helps to accept the determined preference of this quintessence model by the observational data according to the χ2analysis. 5.1.2 The Second Fiducial Model This section presents the fiducial values for the dark energy quintessence model of section 3.3. These values were obtained for the exact numerical evolution of its equation of state, wQ. In a flat Universe composed of pressureless matter, with a present-day energy density Ωm0, and a dark energy component, this model has two free parameters: the quintessence’s energy density present-day value ΩQ0and the parameter, M, which determines where the field’s potential changes its shape. The initial velocity of the field was assumed to be zero. The present-day value of the Hubble parameter was considered to be h0= 0.70 Km s−1Mpc−1. For this model, I have allowed the free parameters to vary between 20 different values and determined χ2for every combination of them. The list below shows the allowed values for both parameters •M= [0.001800,0.001820,0.001840,0.001860, ..., 0.002160,0.002180]; •ΩQ0= [0.60,0.61,0.62,0.63,0.64, ..., 0.78,0.79]. 88
For this particular case, I have restricted the Mvalues as shown due to the fact that the derivation of this quintessence model was done so that the characteristic rapidly oscillatory behavior of the associated equation of state, w, for M= 0.00200, starts around a= 0.8, as figure (3.6) shows. This guarantees that, for this value of Mand a dark energy present-day density of ΩQ0≈0.75, the cosmic evolution history predicted by this model does not deviate much from that of the ΛCDM model, when comparing the predictions of both models for the comoving history registered by an observer today. If oscillations started much sooner than a≈0.8, this would give rise to high deviations from the observed expansion history [26]. Therefore, I have allowed Mto vary narrowly around M= 0.00200 to avoid this problem. The fiducial values determined for this model according to the observational supernovae data are presented on the table below. Again, all other cosmological quantities necessary for computations take the default values of the iCosmo software package. Second Fiducial Model M0.00202 n1.0 ΩQ00.75 τ0.09 Ωm00.25 σ80.8 h00.70 Tabela 2: The Fiducial Values for the second Dark Energy model under analysis. Figura 5.4: Evolution of the equation of state, w, of the model of section 3.3 in a flat Universe, as a function of the scale factor a, for its fiducial values: ΩQ0= 0.75 and M= 0.00202. The equation of state starts to rapidly oscillate around 0 at a≈0.8, going from −1 to 1, reflecting the field’s oscillations around its potential’s zero minimum. 89
Figura 5.12: Weak Lensing expected constraints on the dark energy parameters of the first model, obtained for the Euclid experiment. The red contour represents the 95% confidence region, while the black contour is the 68% confidence region. For each parameter, the respective one-dimensional distribution is also presented. Figura 5.13: BAO expected constraints on the dark energy parameters of the second model, obtained for the Euclid experiment. The red contour represents the 95% confidence region, while the black contour is the 68% confidence region. For each parameter, the respective one-dimensional distribution is also presented. 96
Figura 5.14: Combined Weak Lensing and BAO expected constraints on the dark energy parameters of the second model, obtained for the Euclid experiment. The red contour represents the 95% confidence region, while the black contour is the 68% confidence region. For each parameter, the respective one-dimensional distribution is also presented. Figure (5.14) presents the expected constraints that Euclid will impose on these model’s parameters from the combination of its Weak Lensing and BAO surveys. These are, as expected, very similar to the constraints expected from the Weak Lensing survey. The overall improvement over those constraints is extremely subtle, and is more easily noticeable on the 1σmarginalized errors of each parameter that are presented in table (4). Weak Lensing BAO Combined ΩQ0.00814 0.0551 0.00723 M0.000983 0.0143 0.000835 Tabela 4: 1σmarginalized values for the second model’s parameters for Euclid’s Weak Lensing and BAO surveys, as well for the combination of both surveys. 97
98
6 Conclusion and Final Remarks This work focused on one of the most discussed subjects in contemporary cosmology: the accelerated expansion of our Universe, as demonstrated by recent observations of distant type Ia Supernovae and the respective distance-luminosity relations. More specifically, it concentrated on one of the hypothetical possibilities that could be responsible for promoting such acceleration: Dark Energy. In particular, this work explored two distinct Dark Energy models and presented forecasts for the constraints that a future space experiment, Euclid, will be able to impose on such models. The mostly accepted model to explain this accelerated expansion is the ΛCDM model. According to it, our Universe is currently composed by Cold Dark Matter and dominated by a Cosmological Constant as the mysterious Dark Energy component. Basically, this is a component with constant energy density, ρΛ, and negative pressure, pΛ, such that its equation of state, w=pΛ/ρΛ, is equal to −1 and, therefore, is responsible for the present-day acceleration of our Universe. However, there is a severe fine-tuning problem associated to this model on top of a significant difference between the observed energy density, ρΛ, and the estimated value from quantum-field calculations, with the latter being 121 orders of magnitude larger. This calls for new Physics and, amongst the various hypothesis, there are the Quintessence Dark Energy models. In Quintessence models, the Dark Energy component that permeates our Universe is associated to a canonical scalar field, φ, slowly evolving in its potential, V(φ), according to equation (3.12). Most Quintessence models, in order to present an equation of state close to −1 near to the present, impose slow-roll conditions similar to those of inflation: the potential in which the field rolls should be nearly flat. However, recent articles [60, 61, 62] show that those conditions need not necessarily apply to Quintessence, and present a different approach to this problem, relaxing one of those conditions. The first model analyzed in this work does precisely that, and allows the scalar field to roll on a potential that may present some significant curvature on its nonzero minimum. Its derivation was presented with detail in section 3.2.1, based on reference [61]. According to this model, the field is initially frozen in some position φito later start rolling down the potential towards its minimum as the Hubble friction term is overcome. The more curved the potential is, which one can control trough the K2parameter, the more dynamical this evolution becomes. As the field rolls around the minimum, one can see this behavior reflected on the respective equation of state, wQ(a), as the latter gets larger (although never that much) than its initial −1 value, according to the field’s velocity and potential value. 99
Therefore, this model provides a way of having a moderately oscillatory Dark Energy equation of state, for negative K2values, that does not deviate much from −1 and is accepted by present-day observations, as shown in [61]. This model also yields an analytical approximate solution to the field’s equation of state that is very close to the numerical results and which can be usefully applied to a wide range of potentials with non-zero minima. As for the second model, presented in section 3.3 according to reference [26], it loosely breaks the slowly-roll pattern usually persistent in most quintessence models. The field initially evolves slowly in an almost flat region of the potential until it reaches a point, controlled by the free parameter M, where the potential completely changes its shape. The field then enters a region where the curvature is very prominent and rolls rapidly towards the zero-minimum of the potential, around which it starts to rapidly oscillate with decreasing amplitude due to the ever-present Hubble friction term. As the field oscillates, its equation of state, wQ, which seems not to deviate from −1 until the beginning of oscillations, also oscillates between ±1, presenting an averaged constant value of wQ= 0. Therefore, the field behaves like pressureless dark matter. Even though this contradicts the necessity for having a negative pressure Dark Energy component that drives the acceleration of our Universe, this model (with some fine-tuning involved) predicts an evolution history that is very close to the ΛCDM, and thus it can not be ruled out. With these two models in hand, the first step in this work towards obtaining constraints for both of them was to determine plausible fiducial values for each of their free parameters. In order to do so, I have realized a simple χ2analysis in which the predictions of these models for the distance modulus, µ(z), are compared to the observed values coming from the most recent Supernova data from the Supernova Cosmology Project. This was done for multiple combinations of each models’ parameters and the one that minimized χ2was chosen as the fiducial combination of values for the respective model. The first relevant result comes with the fiducial values for the first model. This model is, in fact, indistinguishable from the Cosmological Constant for φi= 0, and this case was included in the analysis done. However, what was found in this work is actually that the χ2minimization was obtained for φi= 0.30, K2=−15.0 and a Quintessence present-day energy density of ΩQ0= 0.74. This means that a dynamical source of Dark Energy is preferred to the Cosmological Constant for the Supernova data used in this work. The equation of state for these fiducial values presents some evolution despite not getting much larger than −1. Particularly close to the present, wQ≈ −1. Therefore, the cosmic history predicted by this model does not differ much from the ΛCDM0sexpectation. 100
For the second model, the fiducial values that were obtained were M= 0.00202 and ΩQ0= 0.75, almost identical to the fiducial values of the article where the model was presented [26]. This result, therefore, corroborates the analysis of that work, where it was argued that the equation of state, wQ, should start to oscillate close to a= 0.8, which happens for Mclose to 0.00200. This guarantees that the evolution history predicted by this model does not deviate much from the ΛCDM0sprediction and, therefore, can not be ruled out by current observations. Even though this model can not be directly related to the Cosmological Constant, I have compared the χ2value obtained for this fiducial model with the ΛCDM0s for ΩΛ= 0.74, and the latter is smaller. Therefore, this model is not preferred against the ΛCDM by the Supernova Cosmological Project observational data, contrary to the first model. Then, following the most recent Euclid information, namely the respective Red Book [68], I have constructed two surveys corresponding to Euclid’s Weak Lensing and Baryonic Acoustic Oscillations (BAO) surveys (refer to section 5.2). These were then inserted in the public software iCosmo [31] to obtain forecasts for this future space experiment. This gives an idea of the constraining power that each of those surveys could have on these fiducial models, and many others, like the usual w0−waDark Energy parametrization (equation 4.16): in fact, one of Euclid’s main objectives is to improve the constraints on these two parameters. Analyzing the results of section 5.3 one can conclude that Euclid’s Weak Lensing survey has undoubtedly more constraining power than the BAO’s survey on the Dark Energy parameters of the two models studied in this work. The 1σ(68%) and the 2σ(95%) confidence regions for both models’ parameters are extraordinarily more localized around the fiducial values for Weak Lensing. Also, the marginalized uncertainties, which are the best that one could possibly obtain according to the Cramer −Reo theorem [28, 29], are much smaller on the Weak Lensing case. The combined constraints of the two surveys, obviously, don’t present much improvement over the Weak Lensing constraints. This discrepancy might seem puzzling, as the available literature does not seem to clearly indicate such a difference between the constraining power of both surveys. However, observing the forecasts for the w0−waparameters obtained for Euclid in figure (3) of reference [82], one sees that the constraints imposed by BAO are substantially looser than the Weak Lensing’s. However, the constraints are significantly better when considering the full power spectrum, P(k), which, as mentioned in section 2.2.2 of Euclid’s Red Book [68], improves Euclid’s Figure of Merit (which is a quantitative way of evaluating the constraining power of a future experiment). 101
The last paragraph, combined with the more conservative approach used in the calculation of the BAO measurement accuracy by the iCosmo software, justifies the results obtained in this work. They indicate that a more complete analysis, involving the full power spectrum, is necessary in order to fully evaluate Euclid’s constraining capabilities. Nonetheless, Euclid’s Weak Lensing results alone are very promising for constraining Dark Energy parameters. In the future, it would also be interesting to do an analysis similar to that presented on the Dark Energy Joint Mission performed by the Figure of Merit Science Working Group [28], in which the Dark Energy equation of state, w, is separated into a number of piecewise constant parameters corresponding to increasing values of redshift. This is called the Principal Component analysis and would allow one to verify at which redshift the Euclid experiment would be able to better constrain the Quintessence models of this work. I believe this would work particularly better for the model of section 3.2.1, since the binning of the equation of state into sufficiently small spaced intervals would still reflect the dynamical behavior of the equation of state, whereas for the second model this spacing would have to be extremely (perhaps impractically) small in order to include the extreme variations of wbetween ±1. 102
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