Theoretical conditions on phenomenological
parameterizations of dark energy on
cosmological scales
THESIS
submitted in partial fulfillment of the requirements for the degree of
MASTER OFSCIENCE
in PHYSICS
Author : F.T. (Fr´e) Vink
Student ID : ...
Supervisor : Dr. A. Silvestri
2ndcorrector : Prof. dr. A. Ach ´ucarro
Theoretical conditions on
phenomenological parameterizations of
dark energy on cosmological scales
F.T. (Fr´e) Vink
Instituut-Lorentz, Leiden University P.O. Box 9506, 2300 RA Leiden, The Netherlands
July 12, 2018
Abstract
We present a method to link phenomenological parameterizations of dark energy on cosmological scales with theory. We will use the phenomenological parameterization adopted by the Planck collaboration [1], parameterizing the phenomenological functions µ
and η as functions of scale k, and time using the scale factor a. When linking the phenomenological functions with theory, we restrict ourselves to the Horndeski class of
theories [2] and neglect time derivatives, i.e. we work with the general and model independent quasi static approximation (QSA), to be able to find analytical expressions for
our phenomenological parameterizations. Despite its generality, the QSA forces the phenomenological parameterization adopted by the Planck collaboration into a scale-independent one; which motivated us to propose an alternative phenomenological parameterization using a parameterization of the phenomenological functions µ(a, k)and
Σ(a, k). This phenomenological parameterization does allow for scale dependence in µ under our conditions. The effect of imposing extra conditions on our models, e.g. imposing
the speed of gravitational waves to equal the speed of sound, is investigated by searching the parameter space of the phenomenological parameterizations for points that yield physically viable models. The viability here is evaluated by means of ghost and gradient
Contents
1 Introduction 1
2 Theory 5
2.1 Phenomenological functions 5
2.1.1 Definition of µ, η andΣ 5
2.2 Effective Field Theory of Dark Energy 7
2.2.1 γfunctions 8
3 Matching parameterizations of the phenomenological functions with theory 9
3.1 Conventions and definitions 9
3.2 Parameterization of the phenomenological functions 10
3.2.1 The Planck parameterization 10
3.3 µand η in terms of the EFT functions 11
3.3.1 Perturbed Einstein equations 11
3.3.2 Quasi-Static Approximation 12
3.3.3 Rewriting the operators to the EFT functions 12
3.4 Conditions on the parameterizations 14
3.4.1 Condition imposed by the Quasi Static Approximation 14
3.4.2 Parameterizing µ andΣ 15
3.4.3 Matching theory with the parameterization 16
4 Models 19
4.1 Definition of the considered models 19
4.1.1 Horndeski constraint 19
4.1.2 Gravitational waves constraint 20
4.1.3 Division of the different models 20
4.1.4 ConstrainingΩ 20
4.2 Case specific expressions for the EFT functions 21
4.2.1 µ–η parameterization 21
4.3 Properties of the case specific expressions of the EFT functions 31
5 Viability of the models 33
5.1 Viability conditions 33
5.2 Model viability 35
5.2.1 Choice of undetermined functions 36
5.2.2 Cosmological parameter values 36
5.2.3 Parameters priors 36
5.2.4 Free parameters 37
5.2.5 µ–η parameterization 37
5.2.6 µ–Σ parameterization 41
6 Conclusion & Discussion 47
7 Outlook 49
8 Acknowledgements 51
Chapter
1
Introduction
In this work in theoretical cosmology, we will link existing theoretical models of modified gravity to the phenomenological functions. However, before we can do so, we should first explain what the field of cosmology actually is. Cosmology can be seen as an area of physics, aimed at understanding the origin and evolution of the universe as a whole. This leads us to some very fundamental questions, e.g. which forces and materials shape the universe and how do we describe them? At the basis of the current description of the universe lies the assumption of the cosmological principle. It states that the universe is isotropic and homogeneous, and tells us that neither we, nor any point in the universe is privileged or special.
The force of gravity plays a dominant role in the evolution of the universe and is a good starting point in cosmology. We therefore turn to Albert Einstein, who in 1915 published his theory of general relativity (GR), describing gravity with a geometrical interpretation [3]. General relativity tells us that we need to choose a metric to describe space and time. One solution of general relativity is the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric [4–7]. Which is the most general metric that obeys the cosmological principle. In polar coordinates, the metric is given by:
ds2 =dt2−a2(t) dr2 1−κr2 +r 2 dθ2+sin2 θdφ2 (1.1) Here t is the proper, or cosmic time measured by a comoving observer, i.e. an observer moving along with the expansion of the universe, a(t)is the scale factor and we use natural units, hence c = 1. Spatial hypersurfaces, i.e. surfaces with t = 0, have positive, zero or negative curvature if κ = −1, 0, 1 respectively. We can see that the scale factor scales the coordinates, which leads to an expansion of the universe. a(t)therefore parameterizes the expansion of the universe. Let us now consider the the Einstein field equation:
Rµν−
1
2Rgµν+Λgµν =8πGTµν. (1.2)
Where gµν is the metric tensor, Rµν is the Ricci curvature tensor, R = gµνRµν is the Ricci
energy momentum tensor. We model matter and energy by a perfect fluid, which thus is isotropic and homogeneous. The energy momentum tensor is then given by Tνµ=diag(−ρ, p, p, p).
Where ρ is the density and p the pressure of the fluid. Using the FLRW metric, we can find a solution for the Einstein field equation [8].
˙a a 2 = 8πG 3 ρ− κ a2 + Λ 3 (1.3)
The dot indicates a derivative with respect to proper time andΛ is the cosmological con-stant. The above equation is called the Friedmann equation and gives us the evolution of the scale factor a(t). This equation will describe the background cosmology in our work.∗
From equation (1.3) we see that the expansion of the universe is influenced by the cos-mological constant. The coscos-mological constant was initially added by Einstein to make sure that the static solution to the Einstein equations would not collapse on itself. At the time, the expansion of the universe was not yet observed, which favored a static solution of the Einstein equations. After the observation of the expansion of the universe by Hubble [10], the constant was no longer required and was removed from the theory. The cosmological constant has made a big comeback and takes a prominent place in the name of the currently favored cosmological model, theΛCDM cosmology.
At recent times, we observe an exponential expansion of the universe. Solving (1.3) whenΛ is the dominant term, gives us an exponential evolution of a(t). The cosmological constant can thus be used to explain the observed accelerated expansion of the universe at recent times. To match with observations, the cosmological constant should account to roughly 70% of the energy density in the universe. We would like to be able to derive this value for the energy density from theory to check whether we understand the physics behind the expansion of the universe.
Unfortunately, this is where things get tricky. We could imagine that the energy from the cosmological constant comes from empty space itself. Due to the Heisenberg uncertainty principle, a vacuum can have energy, i.e. vacuum energy. We can compute the amount of energy we expect from vacuum energy and find that it adds up to an energy way higher then the observed value of the cosmological constant. Assuming general relativity to hold up to the Planck scale, we find a difference between the observed value and the value predicted by theory of a factor 10118[11]. Therefore, to obtain the observed value of the cosmological constant, one could define the observed cosmological constant as an effective cosmological constant.
Λe f f =Λvac+Λextra (1.4)
The observed value is then the result of the vacuum energy plus some extra true cosmolog-ical constant needed to bring down the predicted value. This however requires one to fine tune this extra term to a very high degree, which is known as the fine tuning problem.
One might wonder whether we are using the right theory to describe the universe when we encounter a mismatch between observations and predictions this severe. Perhaps, a dif-ferent theory might describe our observations equally well or even better, without having ∗If one is interested to know how to use this equation to describe the evolution of the universe during the
3
the fine tuning problem. We do not want to start from scratch, as GR does fit extremely well with solar system experiments and will likely point us in the right direction. Therefore, it might be better to modify the theory of gravity, which will give us modified gravity. On the other hand the cosmological constant might also be caused by a dynamical field in the universe. The solution to the unexplained energy density is yet unclear. We do know that we find a large amount of energy, which we do not observe; it acts as energy and it is dark. We will therefore group all the possible explanations for the observed energy, including a modification of the theory of gravity, under the name of dark energy.
When changing the theory of gravity, we want to be able to check whether our theories match with observations. We can test the theories and the cosmology they give using the large scale structure of the universe (LSS), i.e. the clustered matter around us and its interac-tion with distant light through weak lensing. The large scale structure can be well modeled by the linear evolution of perturbations in matter fields and scalar perturbations in the met-ric. The different theories will affect the evolution of perturbations of the homogeneous FLRW metric, therefore all observables linked to the perturbations will give us information on dark energy. Weak lensing, redshift space distortion and galaxy clustering are often con-sidered as large scale structure observables. The cosmic microwave background (CMB) is not, however it is also very relevant for testing dark energy models.
The perturbations evolve due to gravity, therefore changing the theory of gravity is ex-pected to change the evolution of the perturbations. Let us see how this works in the FRLW metric. We start with (1.2), and as observations like [12] tell us that the universe is very close to a flat universe, we take κ = 0. It is then also convenient to use Cartesian coordinates, which gives us:
ds2 = −dt2+a2(t)δijdxixj. (1.5)
Where x are comoving coordinates, i.e. they stay constant under the expansion of the uni-verse. We factor out the scale factor:
ds2 =a(τ)
−dτ2+δijdxixj
, (1.6)
which defines the conformal time τ as the analog of comoving spatial coordinates for time. We will now perturb this metric:
ds2 =a(τ)
−(1+2A)dτ2+2Bidxidτ+ (δij+hij)dxixj
. (1.7)
Where A, Biand hij are functions of space and time. We do a scalar-tensor-vector
decompo-sition of these functions as done in [13].
Bi =∂iB+Bˆi (1.8)
hij =2Cδij+2∂hi∂ki+2∂(iEˆj)+2 ˆEij (1.9)
Where in the equation above the first term on the right hand side is scalar, the second a vector and (if present) the third a tensor. If the notation using the subscript brackets does
not look familiar to you, do not worry about it. In this work we are only interested in scalar perturbations, so we do not have to deal with the brackets.
The metric perturbations are not uniquely defined but depend on the change of coor-dinates. A change in coordinates can induce some fictitious perturbations. We therefore prefer to use a specific combination of the perturbation variables that are invariant under coordinate transformations, called Bardeen variables [14].
Ψ ≡ A+ H(B−∂τE) +∂τ(B−∂τE), Φˆi ≡∂τEˆi−Bˆi, Eˆij, (1.10)
Φ≡ −C− H(B−∂τE) +
1 2∆
2E. (1.11)
Where ∂τ denotes a derivative with respect to conformal time and H ≡ 1adadτ is the Hubble
parameter in conformal time. We then choose to use the Newtonian gauge, given by
B=E=0. (1.12)
Note that this choice leaves us with with scalar perturbations only. The resulting metric is given by:
ds2= −(1+2Ψ)dt2+ (1−2Φ)a2dx2. (1.13) Where dx2 = δijdxixj. The Newtonian potential Φ will influence growth of structure and
peculiar velocities of galaxies, while the lensing potentialΦ+Ψ will affect the geodesics of light, thus showing up in weak lensing and the integrated Sachs-Wolfe effect in the cosmic microwave background. Modified theories will modify these potentials; we can thus test our theories using the large scale structure using observations of the observables mentioned above.
In chapters 2 and 3 we will see that we can link these potentials to theory without as-suming a particular model of gravity. We can look at the phenomenology and be agnostic about the underlying theory. This is a powerful approach as we do not know which theories to use a priory. In this work we will present a way to establish a link between these phe-nomenological approaches and theoretical models, using the Effective Field theory of Dark Energy (EFT) [15–18]. This allows us to impose conditions on theoretical models from the EFT in terms of the parameters used in observations [1]. Matching the observations with theory will give us an understanding of which models are feasible and gives insight in the restrictions imposed by parameterizing the phenomenological functions. The link between observations and theory is important and can help both fields to work together optimally. In the coming chapters, we will make sure to explain all theories, terms and statements in the paragraph above and guide you through the steps taken in the research leading to this work.
Chapter
2
Theory
In this chapter we will present the theoretical knowledge necessary to understand the steps taken in the following chapters. We will introduce the theories used in this work, but we will leave the application of the theories to derive our results for the next chapters.
2.1
Phenomenological functions
As we saw in chapter 1, we can use the large scale structure (LSS) of the universe to test modified gravity theories. We saw how the perturbed FRLW metric is characterize by the potentials Ψ and Φ. As Φ affects the growth of structure and Φ+Ψ affects the geodesics of light, we will find the effect ofΨ and Φ in LSS measurements. Using the ΛCDM model, we can compute the values for Ψ and Φ theoretically. Therefore, we can find deviations from ΛCDM when the potentials are measured. These deviations at the level of the LSS will be parameterized by the phenomenological functions, which we will show below. The phenomenological functions play an important role in the investigation of the cosmological model. They allow one to be agnostic about the underlying theories at play and describe what data tell us about the universe without restricting ourselves to certain theories. 2.1.1 Definition of µ, η andΣ
We again look at the perturbed FLRW metric in conformal Newtonian gauge, as found in (1.13).
ds2= −(1+2Ψ)dt2+ (1−2Φ)a2dx2, (2.1) We can expand Einstein’s equations to linear order in these perturbations to find coupled differential equations (that include perturbations to matter as a source) for Φ and Ψ. In general relativity, working in Fourier space, it is possible to reduce these equations into a system of two algebraic equations relating Φ, Ψ and the density perturbation of matter. Following [19], we see that by combining the 00 and 0i components of the Einstein equations,
we can find the Poisson equation
k2Φ= −4πGa2ρ∆, (2.2)
and from the i6=j component we find
k2(Φ−Ψ) =12πGa2(ρ+P)σ. (2.3)
Here k = ˆkk is the Fourier vector associated with the scale of the perturbations, G is the gravitational constant, ρ is the background matter density,∆ the comoving density contrast, P the pressure and σ the dimensionless shear perturbation∗. By combining the weak lensing shear and galaxy redshift data from surveys like e.g. EUCLID and LSST, we can test if these equations hold. A deviation from these equations would show us there is a deviation from ΛCDM and might indicate that we need a different theory of gravity, or consider a different source, i.e. dark energy. These deviations are therefore of great importance to us and we will show that we can describe them using the phenomenological functions.
Since the epoch of interest to us is the matter dominated one, as well as the subsequent period of accelerated expansion, we will drop σ. When one considers extended theories of gravity, these equations are generally modified to possibly include time derivatives too. It has been shown that two functions of time and scale are sufficient to describe the dynamics of linear scalar perturbations in any general theory of gravity (e.g. [20, 21]). These functions are typically defined through modifications of the equations above:
k2Ψ= −4πGµ(a, k)a2ρ∆, (2.4)
Φ=η(a, k)Ψ. (2.5)
µ(a, k)now describes departures fromΛCDM in the Poisson equation for the gravitational
potentialΨ. η(a, k)describes the ratio between the two gravitational potentials, which re-flects the presence of non-zero anisotropic stress for η6=1.
Alternatively, one could define another phenomenological functionΣ using the follow-ing relation:
Σ(a, k) = µ(a, k)
2 (η(a, k) +1) (2.6)
Which in terms of the potentials and densities gives us:
k2(Ψ+Φ) = −8πGΣ(a, k)a2ρ∆. (2.7)
Σ(a, k)describes modification to the lensing effects of gravity, with lensing potentialΨ+Φ [1].
When ΛCDM holds, the phenomenological functions equal 1; however, values other than unity indicate a departure fromΛCDM. In general, the phenomenological functions are functions of time and scale in beyondΛCDM models. This is a phenomenological approach,
∗The comoving density contrast is defined by∆≡
δ+3 ˙av/k, where δ≡δρ/ρ is the density contrast in the
Newtonian conformal gauge, v is the irrotational component of the peculiar velocity and ˙a is the derivative of the scale factor with respect to cosmic time
2.2 Effective Field Theory of Dark Energy 7
that compresses all our ignorance about what the extended theory of gravity could be, into two functions closely linked to the observables. They allow for an agnostic exploration of data from large scale structure, without having to specialize to a model.
While the phenomenological functions have many advantages in the exploration of data, they lack a link with the underlying theory. In other words, were we to find that data pre-ferred some value of these functions different from 1, it would be very difficult to link these to a model of gravity. However, if we restrict to the so called quasi static regime (which we will discuss later in section 3.3.2 in more detail), it is possible to obtain explicit expressions for µ, η, Σ for any given theory. Yet, which theories shall we focus on? Many alternative gravity theories exist, but which one would we like to use in our computations? Though someone might have a preference for a particular theory, it would be nice to approach this problem agnostically and systematically. This is where the Effective Field Theory of Dark Energy comes in.
2.2
Effective Field Theory of Dark Energy
With the Effective Field Theory of Dark Energy (EFT) [15–18], we are able to group together a wide range of different dark energy and modified gravity theories in one large action. This enables us to simultaneously study the effects of these theories.
When such an action is constructed, a few conditions are imposed. The theory should describe the perturbations of a cosmological background solution containing gravity, a sin-gle extra scalar field, and a matter sector which is assumed to obey the weak equivalence principle [16]. Furthermore, the theory should include all couplings from the scalar to itself and to gravity, and contain both dark energy and modified gravity models. This general class of models is the Horndeski class [2].
How would one go about creating such an action? The approach is to use the ideas of the EFT of Inflation [22] and the EFT of quintessence [23]. We will skip to the resulting action immediately, but we refer the interested reader to [16] for the intermediate steps. As presented in [19], the EFT action can be written as†:
S= Z d4xp −g m 2 0 2 [Ω(t) +1]R+Λ(t) −c(t)δg 00+ M24(t) 2 (δg 00)2 (2.8) −M¯ 3 1(t) 2 δg 00 δKµµ− ¯ M22(t) 2 (δK µ µ)2− ¯ M32(t) 2 δK i jδK j i + ˆ M2(t) 2 δg 00 δR(3) +m22(t)(gµν+nµnν) ∂µ(g00)∂ν(g00) +Sm[gµν, χi].
Here m20 = (8πG)−1is the Planck mass‡, and δg00, δKµν, δK and δR
(3) are the perturbations of the time-time component of the metric, the extrinsic curvature, the trace of the extrinsic curvature and the three dimensional spatial Ricci scalar of the constant-time hypersurfaces respectively. The Sm[gµν, χi] term is the action for all matter fields χi minimally coupled
†Note that we adopt the definition ofΩ(t)used in [24]. We therefore effectively replaceΩ→
e
Ω(t) +1 and rename eΩ(t)toΩ(t).
to the metric gµν. We restricted ourselves to 1 scalar degree of freedom in the action. The
Lagrangian found from this action is called the Horndeski Lagrangian.
We have a few unknowns left, i.e. Ω(t), Λ(t), c(t) and the functions multiplying the second order operators, i.e. the Mji terms. These are the EFT functions. Ω(t),Λ(t)and c(t) are the only EFT functions that describe the background dynamics, and are therefore called the background functions. The functions multiplying the second order operators describe the dynamics of the linear scalar perturbations.
The power of the EFT formalism lies in these functions. When we change the operators, we get a different theory. For instance, as shown in [16], whenΩ(t) = 1, a constantΛ is included, c(t) = 0 and all other operators are excluded, we find ΛCDM. Including Ω(t), Λ(t)and requiring c(t) =0 gives us f(R)gravity, and so forth. Endless combinations and conditions on the functions can be thought of, each giving us different gravity theories. We now have a systematic and unifying way of writing many gravity theories, without favoring one above the other, thus remaining agnostic about the true nature of gravity.
2.2.1 γ functions
We will follow [24] and redefine the functions multiplying the second order operators to the
γ-functions. This redefinition makes them dimensionless, which helped in the
implementa-tion in the EFTCAMB/EFTCosmoMC code presented in [24]. The dimensionless property also proves useful when constructing their expressions, as quantities holding dimensions can now only enter as ratios. This makes dimensional analysis faster and a convenient way to check for mistakes in our expressions. The γ-functions are defined as:
γ1 = M24 m2 0H0 , γ2= ¯ M13 m2 0H0 , γ3 = ¯ M22 m2 0 , (2.9) γ4 = ¯ M23 m2 0 , γ5= ˆ M2 m2 0 , γ6= m22 m2 0 . Where H0is the value of the Hubble parameter today.
Chapter
3
Matching parameterizations of the
phenomenological functions with
theory
In chapter 2 we introduced the phenomenological functions and the EFT language. The phe-nomenological functions allow us to be agnostic about the deviations fromΛCDM, while the EFT allows us to consider many different modified gravity theories at once. If we can link the two, we might be able to derive conditions on the phenomenological functions given by theory. Our goal for this chapter is thus to find an expression for the phenomenological function µ and η, and combine them with the EFT language to find the sought after link of phenomenology with theory.
3.1
Conventions and definitions
Before we get our hands dirty and dive deep into the theory of this work, we should consider some conventions. We already came across the scale factor, a, proper time t and conformal time τ, but it will be convenient to recall the relations between them below.
We defined the cosmic time as t, and the conformal time, i.e. the time measured in a comoving frame, as τ, with dτdt = a, where a is the scale factor. The cosmic Hubble parameter is defined as H = 1adadt and the conformal Hubble parameter asH = 1
a da
dτ. This gives us the
following relations: H = 1 a d dτa= 1 a dt dτ d dta= d dta=a 1 a d dta= aH(a). (3.1)
To avoid confusion between derivatives with respect to conformal or cosmic time, we will avoid using Newton’s convention for time derivatives using a dot above the quantity. In stead we will avoid time derivatives all together and rewrite our time derivatives as deriva-tives with respect to a. We will use the following rules and denote derivaderiva-tives with respect
to a with a prime∗. d dtX(a(t)) = H(a) d daX(a) = HX 0( a), (3.2) d dτX(a) =aH(a) d daX(a) =aHX 0(a). (3.3)
Where X(a(t))is some quantity that depends on the scale factor or time.
3.2
Parameterization of the phenomenological functions
With the conventions out of the way, we can continue our work and start by explaining the chapter title. The parameterizations of the phenomenological functions were not mentioned before, why do we need to parametrize in the first place? In general the phenomenological functions are expected to be functions of time and scale. When fitting them to data, unless one does some binning, it is necessary to parametrize these time and scale dependences in term of numerical parameters. Note that a parameterization will always restrict your freedom in the evolution of the quantity you consider a bit. For instance, if you choose to parameterize a discontinuous function in terms of polynomials, e.g. when using a Taylor approximation, we find that there is no exact solution without using an infinite number of terms, or parameters. We do not want to introduce an infinite number of parameters in our theory; we can therefore see that the restriction to polynomials will prevent us from describing discontinuous evolution of quantities. Whether this is a problem is a different question and is case-dependent.
When choosing a parameterization, one necessarily restricts to certain classes of models. Which models exactly? and, more importantly, does the final parametrized form of µ and
ηcorrespond to a viable physical model? I.e. for instance a model that does not develop
instabilities. This is the key question that we address in this thesis, with focus on the the parameterization recently adopted by the Planck collaboration. Using the link between the EFT formalism and the phenomenological functions, we investigate under which conditions the parametrized µ, η andΣ fitted to Planck data correspond to viable models of gravity. 3.2.1 The Planck parameterization
We consider the parameterization used by the Planck collaboration [1], where they write their parameterization as µP(a, k) =1+ f1(a) 1+c1(λH/k)2 1+ (λH/k)2 , (3.4) ηP(a, k) =1+ f2(a) 1+c2(λH/k)2 1+ (λH/k)2 . (3.5)
∗Note that inΛCDM time and scale factor are coupled, we merely use a different way of looking at time
3.3 µ and η in terms of the EFT functions 11
Here we have three parameters per phenomenological function, i.e. the constant ci, constant
λand a time dependent function fi(a)which for generality purposes will be parameterized
later rather than now. k = ˆkk is the Fourier vector associated with scales. We wrote the subscript P to indicate we are considering the parameterization used by Planck [1]. Note that the chosen form, i.e. the ratio of quadratic polynomials in k2, suggest that the parame-terization is inspired by the quasi static regime†of scalar-tensor theories with second order equations of motions, i.e. the quasi static regime of Horndeski [25]. However, no explicit as-sumption about the quasi static regime is made in the phenomenological parameterization.
3.3
µ and η in terms of the EFT functions
3.3.1 Perturbed Einstein equations
In equations 2.4, 2.5, we saw that we can find µ and η in terms of the Poisson equation for Ψ and the relation between the gravitational potentials respectively. When we find an expression for the potentials in the EFT language, we would thus be able to connect this to
µand η. We will follow [25] and start from the metric introduced in equation (2.1),
ds2= −(1+2Ψ)a2dτ2+ (1−2Φ)a2dx, (3.6) where we now changed from cosmic to conformal time, introducing an extra factor of a2in the time component of the metric. We consider a very general action defined in term of a Lagrangian density that contains an arbitrary function of geometric invariants R, Rαβαβ, Rαβγδαβγδ, ∆R, Rαβ∇
α∇βR, ... as well as any number of scalar degrees of freedom. Note that this is a
very similar to the action introduced in equation (2.8).
If we vary this action with respect to the metric tensor, we find 4 Einstein equations. The time-time and time-space component can then be combined to form the Poisson equation. In linear order in perturbations and Fourier space, this will have the following general form
ˆ
AΨ+BˆΦ+Cˆiδφi = −4πGa2ρ∆. (3.7)
The traceless space-space Einstein equation give us ˆ
DΨ+EΦˆ +Fˆiδφi =0. (3.8)
If we vary the action with respect to each scalar field φi and linearize in the perturbations,
we find
ˆ
HiΨ+KˆiΦ+ ˆLiiδφj =0. (3.9)
Here, the operators of the three equations above are linear operators that contain functions of the background, time derivatives and/or powers of k, which sounds very similar to the operators we found in the EFT action (2.8).
3.3.2 Quasi-Static Approximation
We have a system of 3 equations for 3 unknowns, i.e.Ψ, Φ and δφi. We could therefore solve
for this system to find the expressions for the unknowns. However, the time derivatives that are in the equations make life difficult by turning the system into a system of differential equations, which in general does not need to have an analytical solution. The solution to this problem is surprisingly simple; drop the derivatives. This is known as the Quasi Static Approximation (QSA) and is not as bad as the above text might imply.
Ignoring the time derivatives is justified by the two assumptions in the QSA. Namely, the time derivatives of metric perturbations are relatively small compared to the spatial deriva-tives, and we assume sub-horizons scales, i.e. k/aH 1. InΛCDM, the second assumption implies the first, as the perturbed quantities evolve on time scales comparable to the expan-sion rate. The time derivatives therefore are only comparable to the spatial derivatives for horizon sized perturbations. However, for alternative gravity theories, this might not be the case. Note that when one applies the QSA to beyondΛCDM models, the time derivatives of the perturbations with respect to their spatial gradients are also neglected.
[25] has investigated the validity of the QSA for the Horndeski models considered, which is reported in appendix III of [25]. They conclude that a conservative way to use the QSA would be to separately fit the model to a subset of data corresponding to clustering on sub-horizon scales. If a departure fromΛCDM is then seen, we would have a clear idea of the scales involved and the validity of the QSA can be judged. We therefore conclude that we can use the QSA in our work.
When using the QSA, the operators become polynomials in k. As we have 3 equations for 3 unknowns, we can solve forΨ, Φ and δφi. Using equations (2.4) and (2.5), we can now
write µ and η in terms of the operators
µ(a, k) = k 2 FˆiKˆ i−E ˆLˆ ii ˆLi i B ˆˆD−E ˆˆA +Kˆi A ˆˆFi−D ˆˆCi +Hˆi E ˆˆCi−B ˆˆFi , (3.10) η(a, k) = ˆ D ˆLii−FˆiHˆi ˆ FiKˆi−E ˆLˆ i i . (3.11)
3.3.3 Rewriting the operators to the EFT functions
As shown above, we can find µ and η in terms of the operators in the perturbed Einstein equations. We are left with the task of finding these operators, which is done by following the procedure described above in section 3.3.1. We vary the action with respect to the metric tensor, and with respect to the scalar field. However, we now vary the EFT action of equation (2.8) and compare the results with equations (3.7 – 3.9). The Newtonian gauge is used for these computations as it is a convenient gauge to work in for the effective Poisson equation and has the advantage that the metric tensor gµνis diagonal [16, 26].
3.3 µ and η in terms of the EFT functions 13
and a dependence directly and renaming the operators:
A1(a) k2 a2φ+A2(a) k2 a2Π+A3(a) k2 a2ψ= −∆ρm (3.12) B1(a)ψ+B2(a)φ+B3(a)Π=0 (3.13) C1(a) k2 a2φ+C2(a) k2 a2ψ+ C3(a) k2 a2 +M(a) 2 Π=0 (3.14)
Here we explicitly write a mass term M2and take out the k dependence from the operators. In this form, we are restricted to a Lagrangian that contains only one degree of freedom, hence restricting our theory. The perturbed equations are written in the form above by [16], which have also done the variation of the action. We can thus avoid reinventing the wheel, but we need to make sure we are using the same EFT-action before using the results of [16]. Comparison of equation (2.8) with equation (2.1) in [16] shows that we consider the same EFT functions, with the exception of us usingΩ(t) +1 where [16] usesΩ(t). We thus have to make sure that we substituteΩ(t) +1 where [16] findΩ(t)in their operators. The operators are shown in appendix A equations (A.3 – A.12).
Using the expressions for the operators, we can find expressions for µ and η in terms of the EFT functions by substituting the operator expressions in the expressions below.
µ(a, k) = B3C1k2−B2 a2M2+C3k2 4πG A1 B1 a2M2+C3k2 −B3C2k2 (3.15) −A3 B2 a2M2+C3k2 −B3C1k2 +A2k2(B2C2−B1C1) η(a, k) = B3C2k 2−B 1 a2M2+C3k2 B2(a2M2+C3k2) −B3C1k2 (3.16)
Here we dropped the a dependence of the operators for readability purposes. Note that the operators do not depend on k.
One can imagine that substitution of the operators in the equations above will result in some incredibly long equations. We will therefore first take a closer look at the equations in operator form. We rewrite the terms such that we collect the factors of k2. It now comes in handy that the operators themselves do not depend on k.
µ(a, k) = a2B2M2+ (B2C3−B3C1)k2 4πG −a2A1B1M2+a2A3B2M2 (3.17) + (A2B1C1−A3B3C1−A2B2C2+A1B3C2−A1B1C3+A3B2C3)k2 η(a, k) = −B1a 2M2+ (B 3C2−B1C3)k2 B2a2M2+ (B2C3−B3C1)k2 (3.18)
numerator of µ and the denominator of η. µ(a, k) = 1+ B2C3−B3C1 a2B 2M2 k2 4πG − a2A 1B1M2+a2A3B2M2 a2B 2M2 (3.19) +A2B1C1−A3B3C1−A2B2C2+A1B3C2−A1B1C3+A3B2C3 a2B 2M2 k2 η(a, k) = −B1a2M2 a2B 2M2 + B3C2−B1C3 a2B 2M2 k 2 1+ B2C3−B3C1 a2B 2M2 k 2 (3.20) The ’p-functions’
In the way we have written the phenomenological functions above, we can see an interesting feature. They can be written in terms of 5 time-independent functions.
µ(a, k) = 1+p3(a)k 2 p4(a) +p5(a)k2 , (3.21) η(a, k) = p1(a) +p2(a)k 2 1+p3(a)k2 . (3.22)
We can read off the expressions for the p-functions if we compare equations (3.21) and (3.22) with equations (3.19) and (3.20). As these p-functions are expressed in terms of the opera-tors, we can also express them in terms of the EFT functions using equations (A.3 – A.12)‡.
p1(a) = −B1a2M2 a2B 2M2 (3.23) p2(a) = B3C2−B1C3 a2B 2M2 (3.24) p3(a) = B2C3 −B3C1 a2B 2M2 (3.25) p4(a) =4πG −a2A1B1M2+a2A3B2M2 a2B 2M2 (3.26) p5(a) =4πG A2B1C1−A3B3C1−A2B2C2+A1B3C2−A1B1C3+A3B2C3 a2B 2M2 (3.27)
3.4
Conditions on the parameterizations
3.4.1 Condition imposed by the Quasi Static Approximation
Keeping in mind that we want to match the parameterizations introduced in (3.4) and (3.5) with the EFT language, for which we now found expression of µ and η using the p-functions, we want to rewrite the parameterizations in the form used for the p-functions as ‡We will leave the substitution of the operators in terms of the EFT functions for later, i.e. when conditions
3.4 Conditions on the parameterizations 15
well. Rewriting the equations to match this form gives§
µP(a, k) = 1+ 1+f1(a) (1+c1f1(a))(λH)2a 2k2 1 1+f1(a)c1 + a2k2 (1+c1f1(a))(λH)2 , (3.28) ηP(a, k) = 1+c2f2(a) + 1(+λfH)2(a2)a2k2 1+ (a2k2 λH)2 . (3.29)
From equations (3.21) and (3.22), we see that the numerator of µ is equal to the denominator of η, while we only imposed the Quasi Static Approximation (QSA) to arrive at this form of the phenomenological functions. However, looking at equations (3.28) and (3.29) above, we see that this is not satisfied directly in the phenomenological parameterization. We need to impose a condition on the Planck parameterization in order to satisfy the shape of the phenomenological functions dictated by the QSA, i.e.
1+ f1(a)
(1+c1f1(a))(λH)2 =
1
(λH)2 ⇒ f1(a) =0∨c1 =1. (3.30)
This condition will only affect the parameterization of µ. If we consider the effect of this condition on the original form of the parameterization given in (3.4, 3.5), we see that for
f1(a) =0 we find
µP(a, k) =1, (3.31)
and for c1 =1 we get
µP(a, k) =1+ f1(a). (3.32)
Therefore, even in the less restrictive c1 = 1 case, we completely lose the scale dependent
part of the parameterization¶. Though this is a result on its own, we might find the Planck parameterization [1] too restrictive when considering the QSA. We thus propose an alterna-tive parameterization.
3.4.2 Parameterizing µ andΣ
We propose to use the same parameterization as used in (3.4, 3.5), but using µ andΣ, defined by equations (2.5, 2.7), in stead of µ and η:
µMS(a, k) =1+ f1(a) 1+c1(λH/k)2 1+ (λH/k)2 , (3.33) ΣMS(a, k) =1+ f2(a) 1+c2(λH/k)2 1+ (λH/k)2 . (3.34)
Where we label them with the subscript MS for µ–Σ. We can then use the relation η = 2Σ/µ−1, introduced in equation (2.6), to rewrite this to a parameterization in terms of µ §Note that we also changed from cosmic H to the conformal Hubble parameterHcompared to equations
(3.4) and (3.5).
and η. Furthermore, we again use the conformal Hubble parameter and write the equations in the form used by the p-functions.
µMS(a, k) = 1+ 1+f1(a) (1+c1f1(a))(λH)2a 2k2 1 1+f1(a)c1 + a2k2 (1+c1f1(a))(λH)2 (3.35) ηMS(a, k) = 2c2f2(a)−c1f1(a)+1 1+c1f1(a) + 2 f2(a)−f1(a)+1 (1+c1f1(a))(λH)2a 2k2 1+ 1+f1(a) (1+c1f1(a))(λH)2a 2k2 (3.36)
The advantage of the µ–Σ parameterization is now apparent. The numerator of µMSis equal
to the denominator of ηMS, which is required by the QSA. We can thus work with this
pa-rameterization without having to impose conditions on the papa-rameterization from the start.
3.4.3 Matching theory with the parameterization
We are now ready to do what was promised by the chapter title. We will match the param-eterization of the phenomenological function with theory, i.e. with the EFT language. This will be done using the p-functions introduced above. The p-functions can be written both in terms of the parameterizations and the EFT operators.
P-functions in terms of the parameterization
We find the expressions for the p-functions by comparing the equations for the phenomeno-logical functions in terms of the p-functions (3.21, 3.22), to the corresponding expressions for the parameterizations, (3.28, 3.29) and (3.35, 3.36). For the Planck parameterization we then find: p1(a) =1+c2f2(a), (3.37) p2(a) =a2 1+ f2(a) (λH)2 , (3.38) p3(a) = a2 (λH)2, (3.39) p4(a) = 1 1+ f1(a)c1 , (3.40) p5(a) = a2 (1+c1f1(a))(λH)2. (3.41)
3.4 Conditions on the parameterizations 17
While for the µ–Σ parameterization we have: p1(a) = 2c2f2(a) −c1f1(a) +1 1+c1f1(a) , (3.42) p2(a) = 2 f2(a) − f1(a) +1 (1+c1f1(a))(λH)2a 2, (3.43) p3(a) = 1+ f1(a) (1+c1f1(a))(λH)2a 2, (3.44) p4(a) = 1 1+ f1(a)c1 , (3.45) p5(a) = a2 (1+c1f1(a))(λH)2. (3.46)
Note that the expressions for p4and p5are equal for the different parameterizations. This is
due to the fact that the same parameterization for µ is used in both parameterizations and p4and p5only appear in µ.
We can equate these expressions to the expression found for the p-functions in terms of the EFT operators in equations (3.23 – 3.27). The theory is then matched with the param-eterization of the phenomenological functions. To make progress, we will have to impose some conditions on either the theory or the parameterizations and see what this means for the equations. One example of this would be to set the speed of gravitational waves to the speed of light, which has been very fashionable in cosmology since the detection of the grav-itational wave event GW170817 from a binary neutron star merger, which was detected both gravitationally and electro-magnetically [27]. We will explore this in the next chapter.
Chapter
4
Models
In chapter 3 we matched the Planck parameterization [1] and our proposed parameteriza-tion to the EFT funcparameteriza-tions. In secparameteriza-tion 3.4.1 we saw that the Planck parameterizaparameteriza-tion loses the scale dependence for µ in order to be matched with the quasi static approximation. This in-hibits the evolution of µ with scale, and therefore the gravity models that can be investigated using this parameterization. We would like to further investigate the effects and restrictions of the parameterizations considered. We will distinguish between different cases by setting values of the EFT functions ourselves. We will then consider the implications of setting the EFT functions on the parameterizations by computing the p-functions and EFT functions for the different models in terms of the parameters of the phenomenological parameteriza-tions. The chapter will be concluded with a summary of the properties of the case specific expression we obtain.
4.1
Definition of the considered models
Setting the EFT functions can give us insight in the parameterization under specific physical conditions. As briefly mentioned at the end of chapter 3, we could for instance only consider models for which the speed of gravitational waves is equal to the speed of light.
4.1.1 Horndeski constraint
In using the Horndeski Lagrangian (2.8), we restrict ourselves to using Horndeski theories [2]. In the EFT language, as presented in [24], this means setting
γ3= −γ4 =2γ5 ∧ γ6 =0. (4.1)
Such a restriction comes with the penalty of being less general in the theories we consider. Horndeski theories however correspond to the most general scalar tensor theories with sec-ond order equations of motion. This is still a very broad class of models. Therefore, we do not restrict ourselves a lot in the models available for study, while it greatly simplifies equations (A.3, A.12). We will thus use this condition for the entirety of this work.
I: γ3 =0∧Ω=0 II: γ3 =0∧Ω6=0
III: γ3 6=0∧Ω=0 IV: γ3 6=0∧Ω6=0
Table 4.1: The four different models we consider by changing our choices for the EFT functions γ3 andΩ. These models can be considered for both parameterizations, giving us 8 cases to study.
4.1.2 Gravitational waves constraint
We can set the EFT functions with a physical motive by requiring the speed of gravitational waves to equal the speed of light. This is a reasonable condition as the detection of the gravitational wave event GW170817 and its electro-magnetic counter part GRB 170817A has constrained the difference between the speed of gravitational waves and the speed of light at the present moment, to be within−3×10−15and+7×10−16times the speed of light [27]. It is however still useful to consider models where the speed of light and the speed of the gravitational waves differ. The detected difference is only valid at the present time, as the detection was a low redshift measurement. A gravitational wave speed other than the speed of light is therefore not yet ruled out for all times.
We can set the speed of light equal to the speed of gravitational waves in our models by setting γ3=0 in equations (A.3, A.12). Combined with the Horndeski constraint, we have
γ3= γ4 =γ5 =γ6=0, (4.2)
which greatly simplifies our operator expressions. 4.1.3 Division of the different models
We present our choices of the EFT functions γ3andΩ in table 4.1. A combination of choices
for EFT function values will be considered as a ’case’. We thus see that we distinguish between 4 cases. Note that we do not restrict ourselves in the possible values for γ3andΩ.
By allowing them both to be 0 and non 0, all possible values for the functions are considered. In section 3.4.3 we showed how we can relate the EFT functions to the phenomenological functions by equating (3.37 – 3.41) for the µ–η parameterization and (3.42 – 3.46) for the µ–Σ parameterization, to (3.23 – 3.27). Therefore, setting the EFT functions γ3andΩ affects the
phenomenological functions. We can compute the phenomenological functions for the four different models and compare them with the two parameterizations we consider, µ–η and µ –Σ; giving us 8 different cases to study.
4.1.4 ConstrainingΩ
The effects of setting γ3are considered in section 4.1.2, but settingΩ is new. [Ω(a) +1]is a
function multiplying the Einstein-Hilbert term√−gm20R/2 in the action (2.8). This function describes a non-minimal coupling between the scalar field and the metric and in General Relativity would giveΩ(a) = 0 [16]. We will distinguish between a zero and non-zero Ω, i.e. a minimally and non-minimally coupling between the scalar field and the metric. We
4.2 Case specific expressions for the EFT functions 21
will see later that setting Ω(a) 6= 0 specifically, combined with γ3 = 0 will give us the
possibility of computing γ2analytically for our models.
4.2
Case specific expressions for the EFT functions
With the division of our models in specific cases, we can take a look at the equations for the p-functions and equate the parameter expressions to the EFT expressions. This allows us to compute the EFT functions in terms of the parameterizations. Below, we will present the expressions of the p-functions in terms of the EFT functions for the 8 cases considered, as well as the expressions for the EFT functions in terms of the parameters in our phenomeno-logical parameterizations. We will consider the µ–η and µ–Σ parameterizations separately. We start with general expressions applicable to all 4 cases within a parameterization and show the reduction of the expressions induced by the case specific conditions.
The expressions of the EFT functions in terms of the parameters of the phenomenolog-ical parameterizations allow us to study the behavior of the EFT function as a function of our phenomenological parameterizations. This is interesting to us, as the EFT functions describe the physical properties of the models we consider. The expression presented be-low will therefore albe-low us to see the effects of the parameterizations on the physics of our models and will be used to identify physically stable models in the parameter space of our phenomenological parameterization in chapter 5.
4.2.1 µ–η parameterization
In equations (3.37 – 3.41) we found the expression for the p-functions in terms of the µ–η parameterization. We will now equate them to the expressions for pi in terms of the EFT
operators, given by equations (3.23 – 3.27). p1(a) =1+c2f2(a) = −B1M2 B2M2 , (4.3) p2(a) =a21 + f2(a) (λH)2 = B3C2−B1C3 a2B 2M2 , (4.4) p3(a) = a2 (λH)2 = B2C3−B3C1 a2B 2M2 , (4.5) p4(a) = 1 1+ f1(a)c1 =4πG−A1B1M 2+A 3B2M2 B2M2 , (4.6) p5(a) = a2 (1+c1f1(a))(λH)2 (4.7) =4πG A2B1C1−A3B3C1−A2B2C2+A1B3C2−A1B1C3+A3B2C3 a2B 2M2 .
We impose the Horndeski constraint (section 4.1.1), substitute equations (A.3 – A.12) in the right hand side of the equations above and simplify the expressions using Mathematica [28].
The result for the right hand side of the equations is shown below. p1(a) = γ3(a) +Ω(a) +1 Ω(a) +1 , (4.8) p2(a) = −2H02˜c(γ3+Ω+1) − H H γ3 2aγ30+aΩ0+4Ω+4 (4.9) +a aγ30Ω0+2(Ω+1)γ30+aΩ02 +4γ32 −2aγ3(γ3+Ω+1) H0 +aH0H γ2 −aγ30−aΩ0+Ω+1 +a(γ3+Ω+1)γ20 3H(Ω+1) aH0H0 2H0˜c−a2γ2H0 −H 2H02˜c+a3H0 γ2H00+γ20H0 +2a2H02 γ3+2aΩ0 +aH2 H 0 aγ20+γ2 −4 aΩ0 aH00+ H0 −γ3H0 −2H3 γ3−6aΩ0 , p3(a) = −2H02(Ω+1)˜c−2H H 2γ3 aγ30+aΩ0+Ω+1 (4.10) +a 2aγ30Ω0+γ30 aγ30+Ω+1 +aΩ02 +γ32 −aγ3(Ω+1)H0 +aH0H(Ω+1) aγ20+γ2 3H(Ω+1) aH0H0 2H0˜c−a2γ2H0 −H 2H02˜c+a3H0 γ2H00+γ20H0 +2a2H02 γ3+2aΩ0 +aH2 H 0 aγ20+γ2 −4 aΩ0 aH00+ H0 −γ3H0 −2H3 γ3−6aΩ0 , p4(a) = (γ3(a) +Ω(a) +1)2 Ω(a) +1 , (4.11) p5(a) = − a 1 a 2(γ3+Ω+1) −2H 2 0˜c(γ3+Ω+1) (4.12) −H H γ3 2aγ30+aΩ0+4Ω+4 +a aγ30Ω0+2(Ω+1)γ30+aΩ02 +4γ32 −2aγ3(γ3+Ω+1) H0 +aH0H γ2 −aγ30−aΩ0+Ω+1 +a(γ3+Ω+1)γ20 + H0γ2+ HΩ0 (aH0γ2(Ω+1) −H 2γ3 aγ30+aΩ0+Ω+1 +a(Ω+1) 2γ30+Ω0 +2γ32 23H(Ω+1) aH0H0 a2γ2H0−2H0˜c +H 2H02˜c+a3H0 γ2H00+γ20H0 +2a2H02 γ3+2aΩ0 +aH2 4 aΩ0 aH00+ H0 −γ3H0 −H0 aγ20+γ2 +2H3 γ3−6aΩ0
Note the presence of the background function ˜c in stead of c as seen before. We use a dimen-sionless version of c defined in (A.2). Except for p1and p4, the expressions are very lengthy,
4.2 Case specific expressions for the EFT functions 23
we thus start with p1and p4.
p1(a) =1+c2f2(a) = γ3(a) +Ω(a) +1 Ω(a) +1 , (4.13) p4(a) = 1 1+ f1(a)c1 = (γ3(a) +Ω(a) +1) 2 Ω(a) +1 . (4.14)
We get a system of 2 equations with 2 unknowns, i.e. γ3 andΩ. We can thus solve for γ3
andΩ. Ω(a) = 1− (1+c1f1(a))(1+c2f2(a)) 2 (1+c1f1(a))(1+c2f2(a))2 , (4.15) γ3(a) = c2f2(a) (1+c1f1(a))(1+c2f2(a))2 . (4.16)
We can find general expressions forΩ and γ3for all 4 cases at once. Setting one of them to
0 will then impose conditions on the parameters, which will simplify the expression for the other. This will be discussed for the 4 different cases below.
When we assume aΛCDM background,His known and this sets ˜c, as it only depends onΩ andH. Therefore, we only need to find γ2 to be able to compute the evolution of our
models. To find γ2, we need to solve one of the differential equations for γ2 given by p2,
p3and p5. Here we will need the different models considered in table 4.1, as the conditions
will simplify the expressions for p2, p3and p5.
Case I
We require γ3 = Ω= 0. We substitute this in equations (4.8 – 4.12) and equate them to the
expressions for the p-functions in terms of the parameterization found in (3.37 – 3.41).
p1(a) =1+c2f2(a) =1, (4.17) p2(a) =a21 + f2(a) (λH)2 = − aγ20 +γ2 3(a2γ 2H02+a2H (γ2H00+γ02H0) − H2(aγ02+γ2)) , (4.18) p3(a) = a2 (λH)2 = − aγ20 +γ2 3(a2γ 2H02+a2H (γ2H00+γ02H0) − H2(aγ02+γ2)) , (4.19) p4(a) = 1 1+ f1(a)c1 =1, (4.20) p5(a) = a2 (1+c1f1(a))(λH)2 (4.21) = 2H (aγ2 0+ γ2) +aH0γ22 6H (−a2γ 2H02−a2H (γ2H00+γ20H0) + H2(aγ20+γ2)) .
thus equate the right hand sides of p3and p5to find: − aγ 0 2+γ2 3(a2γ 2H02+a2H (γ2H00+γ20H0) − H2(aγ02+γ2)) (4.22) = 2H (aγ2 0+ γ2) +aH0γ22 6H (−a2γ 2H02−a2H (γ2H00+γ20H0) + H2(aγ20+γ2)) (4.23) ⇒aγ02+γ2=aγ02+γ2+aH0γ22/2H (4.24) ⇒aH0γ22/2H =0 (4.25) ⇒γ2(a) =0 (4.26)
WithΩ= 0, we also find that the background function c= 0. We thus find that all the EFT functions are 0, which corresponds to aΛCDM model.
Case II
We still have γ3 =0, but now requireΩ6=0. It is then possible to write γ2as follows:
γ2= HΩ0(a) H0 1+ p4(a)p3(a)−p5(a) p2(a)p4(a)−p5(a) 1− p4(a)p3(a)−p5(a) p2(a)p4(a)−p5(a) (4.27) Which can be shown to be true by substitution of the EFT expressions for the p-functions. From equations (3.39 – 3.41), we see that p3p4−p5=0 when we look at the parameterization
expressions. The expression for γ2therefore simplifies to
γ2(a) =
HΩ0(a) H0
. (4.28)
We substitute the derivative of (4.15).
= H
H0
−c1f10(a)(1+c1f1(a)) +c1f1(a)f10(a) (1+c1f1(a))2
(4.29) Note that the expression for γ2does not depend on c2 or f2. The condition γ3 = 0 gives us
f2c2=0. (4.15) therefore simplifies to:
Ω(a) = −c1f1(a) 1+c1f1(a)
(4.30) Which is also independent of either f2 or c2. Therefore, we find that the EFT functions do
not depend on the parameterization of η directly. We do require a condition on c2f2to set
γ3(a) = 0 though. In equation (3.30), we saw that for the µ–η parameterization the QSA
imposed f1=0∨c1 =1. As f1=0 givesΩ=0 in (4.30), which is not allowed in case II, we
need to use c1=1: γ2(a) = H H0 −f10(a)(1+ f1(a)) + f1(a)f10(a) (1+ f1(a))2 , (4.31) = −H H0 f1(a) (1+ f1(a))2 . (4.32)
4.2 Case specific expressions for the EFT functions 25
Case III
We return to equations (4.15, 4.16). We still have the condition f1(a) =0∨c1 =1, however,
the condition on c2 and f2(a) is no longer required. The fraction of p-functions used in
equation (4.27) no longer reduces to an algebraic expression for γ2. We need to solve for the
differential equation given by either (4.10), (4.11) or (4.12). Ω=0 combined with equation (4.15) implies:
1− (1+c1f1(a))(1+c2f2(a))2 =0 (4.33)
From the condition f1(a) = 0∨c1 = 1, we see that we have two options to consider. For
f1(a) = 0, we now require c2f2 = 0 forΩ = 0. However, equation (4.16) would then give
us γ3 = 0. As we are focusing on γ3 6= 0 now, this is not allowed. We thus continue with
c1 =1, which gives us:
1− (1+ f1(a))(1+c2f2(a))2 =0. (4.34)
We rewrite this to a condition for f1(a).
f1(a) =1−
1
(1+c2f2(a))2. (4.35)
We get two free parameters, i.e. c2and f2(a). We now want to find our EFT functions, which
now simplify with respect to (4.15) and (4.16) to:
Ω(a) =0. (4.36)
γ3(a) =
c2f2(a)
2c22f2(a)2+4c2f2(a) +1
. (4.37)
We still need to find γ2. For Ω = 0, we can not use equation (4.27) to find γ2. We need
to use the unused p-functions to find Ω and γ3, which give differential equations for γ2.
The differential equations they give are not equally difficult. We prefer the ones without γ22 terms, as they are easier to solve. Furthermore, by looking at ratio’s of p-functions, we might eliminate some of the parameters. Looking for the parameter expressions of the p-functions, we see that taking the ratio p2/p3will eliminate the a/(λH)2term, which also simplifies our
differential equation. We thus solve:
p2(a)
p3(a)
=1+ f2 (4.38)
We expect γ2to be zero at a=0, we therefore use γ2(0) =0 as a boundary conditions when
solving the differential equation. The result for γ2is given by:
γ2(a) =e Ra 1 f2(x)(f1(x)−xc2 f0 1(x)+1)−x(f10(x)+c2(f1(x)+1)f20(x)) x(c2−1)(f1(x)+1)f2(x) dx Z a 1 A (y)dy− Z 0 1 A (y)dy (4.39)
Where A(y) = " e− Ry 1 f2(x)(f1(x)−xc2 f0 1(x)+1)−x(f10(x)+c2(f1(x)+1)f20(x)) x(c2−1)(f1(x)+1)f2(x) dx (4.40) × 2(c2−1)H02˜c(y) (f1(y) +1)3f2(y) (c2f2(y) +1)3 −H(y) 2y(c2−1)c2(f1(y) +1)2(c2f2(y) +1)H0(y)f2(y)2 +H(y) 2 f2(y) f1(y) −y f10(y) +1 2 f2(y)3− 2y f20(y) +1 f1(y)2 + 4y f20(y) + f10(y) y−2y2f20(y) +2 f1(y) −y2f10(y)2+2y f20(y) +f10(y) y−2y2f20(y) +1 f2(y)2+y(f1(y) +1)f20(y) × 2y f10(y) +y f20(y) + f1(y) y f20(y) −1 −1 f2(y) +y2(f1(y) +1)2f20(y)2 c32 +f2(y) 2 3 f1(y)2+ 6−5y f10(y) f1(y) +3y2f10(y)2 −5y f10(y) +3 f2(y)2− 6y f20(y) +2 f1(y)2 + 12y f20(y) +y f10(y) 5−8y f20(y) +4 f1(y) −5y2f10(y)2+6y f20(y) +y f10(y) 5−8y f20(y) +2 f2(y) +y(f1(y) +1) ×f20(y) 5y f10(y) +2y f20(y) +2 f1(y) y f20(y) −2 −4 c2 2 + (f1(y) +1)f10(y)f20(y)y2+ f2(y) 2 f20(y) (f1(y) +1)2 +f10(y) 4y f20(y) −3 (f1(y) +1) +4y f10(y)2 y +f2(y)2 4 f1(y)2+ 8−6y f10(y) f1(y) +6y2f10(y)2−6y f10(y) +4 c2 +y2(2 f2(y) +1)f10(y)2 y2H(y) (c 2−1)H0(f1(y) +1)3f2(y) (c2f2(y) +1)3
Note that (4.35) still needs to be substituted. This will be done in Mathematica and showing it here will not give us any more insight. We find two integral with the same integrand. The second integral, i.e. the one with boundaries 1 to 0, comes from our boundary condition. It can be seen that for a=0, we subtract the same integrals inside the brackets of (4.39), giving us γ2(a) =0 imposed by the boundary condition.
Case IV
This is the most general case we can consider in the µ–η parameterization as both the EFT functionΩ and γ3 are considered to be non zero. This means that we will use equations
(4.15) and (4.16). Note that the condition imposed by the QSA, i.e. f1(a) =0∨c1 = 1 does
4.2 Case specific expressions for the EFT functions 27 For f1(a) =0 we find: Ω(a) = 1− (1+c2f2(a)) 2 (1+c2f2(a))2 , (4.41) γ3(a) = c2f2(a) (1+c2f2(a))2 . (4.42)
We find γ2(a)through the p2/p3ratio again. We set the boundary condition γ2(0) =0, but
unfortunately, no solution is then found. The functions tends to diverge at a = 0, which is why we have to use a slightly different boundary condition. We will use γ2(q) = 0 and
solve the differential equation. We can then later decide the the value of q and determine how close to a=0 we can get while still finding a well behaved γ2. Similar to (4.39), we find
γ2= a 1 c2−1f 2(a) − c2 c2−1 Z a 1 B (x)dx− Z q 1 B (x)dx. (4.43) Where for B(x)we find
B(x) = −h2x1−1c2−2f 2(x) c2 c2−1 c 2H(x) (c2−1)x f2(x) (c2f2(x) +1) H0(x) (4.44) +H(x) −c2f2(x)2 c2 2x f20(x) +1 −3 +f2(x) c22x f20(x) x f20(x) −1 −c2 3x f20(x) +1 +2 +x f20(x) c22x f20(x) +c2 x f20(x) −2 +1 +c22f2(x)3 − (c2−1)H02˜c(x) (c2f2(x) +1)3 (c2−1)H0H(x) (c2f2(x) +1)3 For c1=1 we find: Ω(a) = 1− (1+ f1(a))(1+c2f2(a)) 2 (1+ f1(a))(1+c2f2(a))2 , (4.45) γ3(a) = c2f2(a) (1+ f1(a))(1+c2f2(a))2 . (4.46)
The expression we find for γ2is equal to the expression we found for case III with c1 =1, i.e.
(4.39), yet without the extra condition on f1given by settingΩ=0. Note that this condition
was not yet substituted in equation (4.39), we can thus use the expression for γ2 for both
cases III and IV.
4.2.2 µ–Σ parameterization
For the µ–Σ parameterization we will apply the same method as used for the µ–η parame-terization. The EFT expressions for the p-function remain the same, but as we saw in section
3.4.3, the expressions found by the parameterization change and are now given by (3.42– 3.46). We consider the same cases presented in table 4.1. However, before doing so, we can again find expressions forΩ and γ3valid for all 4 cases. Using p1and p4, we have
p1(a) = 2c2f2(a) −c1f1(a) +1 1+c1f1(a) = γ3(a) +Ω(a) +1 Ω(a) +1 , (4.47) p4(a) = 1 1+ f1(a)c1 = (γ3(a) +Ω(a) +1) 2 Ω(a) +1 . (4.48)
Which we can solve forΩ and γ3,
Ω(a) = 1+c1f1(a) − (c1f1(a) −2c2f2(a) −1) 2 (c1f1(a) −2c2f2(a) −1)2 , (4.49) γ3(a) = 2(c2f2(a) −c1f1(a)) (c1f1(a) −2c2f2(a) −1)2 . (4.50)
Depending on the cases given in table 4.1, setting the functions above to zero will give us conditions on our parameters.
Case I
Here bothΩ and γ3 are set to 0. In our discussion of the γ3 = 0 case of the µ–η
param-eterization, we found that p3(a) = p5(a), combined with Ω(a) = 0 gives us γ2 = 0. The
gravitational waves constraint, combined withΩ(a) =0 gives:
c1f1(a) =c2f2(a) =0, (4.51)
⇒c1=0∨ f1(a) =0, c2=0∨ f2(a) =0 (4.52)
For p3and p5we then find:
p3(a) = 1+ f1(a) (λH)2 a 2, (4.53) p5(a) = a2 (λH)2. (4.54)
It is thus clear that for f1 = 0, we find p3 = p5 and therefore γ2 = 0. However, for f1 6=
0∧c1=0, we need to solve a differential equation to find γ2(a).
From the differential equation
p2(a)/p3(a) = (−f1(a) +2 f2(a) +1)/(f1(a) +1) (4.55)
with boundary condition γ2(0) =0, we find:
γ2(a) =e Ra 1 −2 f2(x)+f1(x) 2x(f1(x)−f2(x))dx (4.56) × Z 1 0 e R1 y −2 f2(x)+f1(x) 2x(f1(x)−f2(x))dx 4H 2 0˜c(y) 2y2H(y)H 0 dy + Z a 1 e R1 y −2 f2(x)+f1(x) 2x(f1(x)−f2(x))dx 4H 2 0˜c(y) 2y2H(y)H 0 dy
4.2 Case specific expressions for the EFT functions 29
Which seems fine except for ˜c. For the dark energy equation of state wDE = −1∧Ω=0, we
find c(a) = ˜c(a) =0. Therefore, we have γ2 =0 for both f1 =0 and f1 6=0∧c1=0.
Case II
We still keep the gravitational waves constraint, i.e. γ3 = 0, but letΩ be non-zero. From
equation (4.50) we then see that this gives us the condition c1f1(a) =c2f2(a). This simplifies
our expression forΩ to:
Ω(a) = −c1f1(a) 1+c1f1(a)
= −c2f2(a) 1+c2f2(a)
. (4.57)
Expression (4.27) is valid forΩ0 6=0, we can thus use this expression for the µ–Σ param-eterization too. While the fraction of p-functions reduced to 0 before, we now find:
p4(a)p3(a) −p5(a)
p2(a)p4(a) −p5(a)
= (c1−1)f1(a) 2((c1+1)f1(a) −2 f2(a))
(4.58) Therefore, after substitution of equation (4.57), we find:
γ2 = HΩ0(a) H0 1± p4(a)p3(a)−p5(a) p2(a)p4(a)−p5(a) 1∓ p4(a)p3(a)−p5(a) p2(a)p4(a)−p5(a) (4.59) = H H0 −c1f10(a) (1+c1f1(a))2 1± (c1−1)f1(a) 2((c1+1)f1(a)−2 f2(a)) 1∓ (c1−1)f1(a) 2((c1+1)f1(a)−2 f2(a)) (4.60) We find that we have a model with 3 free parameters, i.e. c1, E11and E22. This makes sense,
as we start with 5 free parameters, cancel the λ parameter by using fractions of p-functions and use the condition c1f1 =c2f2, which allows us to get rid of another parameter.
Case III
Setting γ3 6=0 now gives us c1f1(a) 6=c2f2(a)through equation (4.50). From equation (4.49),
we see that settingΩ=0 gives us:
1+c1f1(a) − (c1f1(a) −2c2f2(a) −1)2 =0 (4.61)
Where we also want to make sure that the denominator of (4.49) is non-zero. If we consider the case where the denominator is zero, we find:
c1f1(a) −2c2f2−1=0 (4.62)
Combined with the numerator being zero we then get:
1+c1f1(a) − (c1f1(a) −2c2f2(a) −1)2 =0 (4.63)
⇒1+c1f1(a) =0 (4.64)
If we then substitute this result back into (4.62), we find:
c2f2(a) = −1 (4.66)
Therefore we get:
c1f1(a) =c2f2(a) (4.67)
Which is not allowed in the γ3 6= 0 case. By proof by contradiction, we conclude that the
denominatorΩ is non-zero for γ3 6=0. We can thus safely derive a condition from (4.61).
f1(a) =
4c2f2(a) +3±p8c2f2(a) +9
2c1
(4.68) We are left with the task of finding γ2, which can be found solving the differential equation
given by p2/p3: p2(a) p3(a) = 2 f2(a) − f1(a) +1 1+ f1(a) (4.69) When solving for γ2(a), we use the boundary condition γ2(0) = 0. The solution for γ2 is
given by a lengthy integral, but has the following structure:
γ2(a) =e Ra 1Q(x)dx Z 1 0 −eR1yQ(y)dxA(y)/B(y)dy+ Z a 1 −eR1yQ(y)dxA(y)/B(y)dy (4.70) Where, A(y), B(y)and Q(x)are functions that generally depend on c1, c2, f1(a), f2(a). The
integral from 0 to 1 comes from the boundary condition. These functions can be identified in Mathematica and (4.68) can be substituted for the case ofΩ=0. Without the substitution of (4.68), we find: Q(x) = f1(x) xγ03(x) +xΩ0(x) −2Ω(x) −2 +2 f2(x)(Ω(x) +1) (4.71) +x γ30(x) +Ω0(x) [x(f1(x) (γ3(x) +2Ω(x) +2) −2 f2(x)(Ω(x) +1) +γ3(x))], A(y) = H(y) 2yγ3(y)H0(y) (f1(y) (γ3(y) +2Ω(y) +2) (4.72) −2 f2(y)(Ω(y) +1) +γ3(y)) +H(y) 4 f2(y) 2γ3(y) yγ03(y) +yΩ0(y) +Ω(y) +1
+y 2yγ30(y)Ω0(y) +γ03(y) yγ03(y) +Ω(y) +1+yΩ0(y)2+γ3(y)2
−f1(y) γ3(y) 6yγ30(y) +5yΩ0(y) +8Ω(y) +8
+y 5yγ30(y)Ω0(y) +2γ30(y) yγ03(y) +2Ω(y) +2+3yΩ0(y)2 +6γ3(y)2 +3y2γ30(y)Ω0(y) +2y2γ30(y)2+y2Ω0(y)2+3yγ3(y)Ω0(y) +2yγ3(y)γ03(y) −2γ3(y)2 −2H02˜c(y) (f1(y) (γ3(y) +2Ω(y) +2) −2 f2(y)(Ω(y) +1) +γ3(y)) B(y) =H0y2H(y) (f1(y) (γ3(y) +2Ω(y) +2) −2 f2(y)(Ω(y) +1) +γ3(y)) (4.73)
Because of the length of the functions have already, we will not show the substitution ofΩ,
4.3 Properties of the case specific expressions of the EFT functions 31
Case IV
As the most general case for this parameterization, the expression forΩ and γ3in this case
are given by equation (4.49) and (4.50) respectively. Furthermore, it turns out that it is more convenient to find the general expression for γ2for bothΩ = 0 andΩ6=0 together, as the
conditions imposed byΩ do not result in nice simplifications of the parameter expressions of the p-functions. The expression for γ2 found in case III, i.e. (4.70), is therefore the same
expression as found for this case, yet without the extra condition on f1.
4.3
Properties of the case specific expressions of the EFT functions
We find many similarities in the properties of the EFT functions between the two phe-nomenological functions. This was to be expected, as p4and p5are equal for both
param-eterizations. In both the phenomenological parameterizations, the EFT functions reduce to theirΛCDM form for case I. Case II allows for fully analytical expressions of the EFT func-tions, while cases III and IV show integrals for γ2that can not be solved analytically for both
parameterizations. An important difference between the two phenomenological parameter-ization is given by the condition, f1 =0∨c1 =1 imposed by the QSA for all cases in the µ–η
parameterization, which is not present in the µ–Σ parameterization. This condition reduces the number of free parameters allowed in the µ–η parameterization with respect to the µ–Σ parameterization, which will be discussed in greater detail in section 5.2.4. Armed with our expressions for the EFT functions, we are now ready to proceed to the study of physically viable models in the parameter space of the phenomenological parameters.
