# Unveiling the Dynamics of the Universe

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## Abstract

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## 1. Introduction

## 2. Scalar-tensor Theories

#### 2.1. General Formalism

#### 2.2. Conformal Picture

#### 2.3. Cosmological Dynamics

- $x=0,\phantom{\rule{0.277778em}{0ex}}y=1,\phantom{\rule{0.277778em}{0ex}}{\phi}_{0}\phantom{\rule{1.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}{\partial}_{\phi}\mathrm{V}({\phi}_{0})=0$. This case corresponds to de Sitter solutions, dominated by the scalar field, and arise at non-vanishing maxima or minima of the potential V. Notice that ${\partial}_{\phi}m$ may be different from 0. These solutions are attractors at minima of V and repellors at maxima of V.
- $x=y=0,\phantom{\rule{0.277778em}{0ex}}{\phi}_{0}\phantom{\rule{1.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}{\partial}_{\phi}\mathrm{m}=0$. The second case corresponds to matter dominated solutions with $V=0$ at maxima or minima of m. Note however that the system (15)–(18) is singular when $V=0$, translating the fact that the variables in use are not regular. In the original variables, this second class of solutions exists provided that V has also an extremum, or is identically zero.

- ${x}_{\infty}^{BM}=-W/3$, which lies on the invariant line ${x}^{2}+{y}^{2}=1$, provided $\left|W\right|<3$. These solutions (BM) correspond to those found in [114].
- ${x}_{\infty}^{S}=\frac{3\gamma /2}{Z-W},{({x}_{\infty}^{S})}^{2}+{({y}_{\infty}^{S})}^{2}=\frac{3{x}_{\infty}^{S}+Z}{Z-W}$, that lies in the interior of the phase space domain, provided that$$0<{({x}_{\infty}^{S})}^{2}+{({y}_{\infty}^{S})}^{2}=\frac{9\gamma /2+Z(Z-W)}{{(Z-W)}^{2}}<1.$$

#### 2.4. Assisted Quintessence

#### 2.4.1. General Equations

#### 2.4.2. Scalar Field Dominated Solution

#### 2.4.3. Scaling Solution

#### 2.4.4. Scalar Potential Independent Solutions

- First, we have the kinetic dominated solutions where all the energy density for the dark sector is due to the kinetic energy of the scalar fields. In this instance, we have ${w}_{\mathrm{eff}}={\Omega}_{\varphi}=1$.
- As a second possibility, we have a contribution from the kinetic energy and the matter energy densities for the dark sector. In this case we have ${w}_{\mathrm{eff}}={\Omega}_{\varphi}=\frac{2}{3}{\sum}_{i}{C}_{i\alpha}^{2}$. This will be true for any matter species α since the sum is constrained to yield the same for all matter species.
- Finally, as a third possibility we have a matter dominated scenario. This has been investigated previously in the literature for the case of a single field with two matter components [128,130,131,132]. If we extend this to more than one field, we will see that there is a consistency condition imposed on the values of the couplings to allow a flat direction in the field space, otherwise this solution will be non existent. This is by its nature a transient solution since the matter contribution will eventually decay away and the scaling solution or the scalar field dominated solution will take over. However, this can be an interesting scenario for model building since it allows the dark matter sector to have a significant contribution before the acceleration of the Universe.

## 3. Horndeski Theories and Self-tuning

#### 3.1. Dynamical Screening

- The field equation evaluated at the critical point has to be trivially satisfied leading the value of the field free to screen. This implies that the minisuperspace Lagrangian density at the critical point takes the form$$\sum _{i=0..3}{Z}_{i}\left({a}_{\mathrm{on}},\phantom{\rule{0.166667em}{0ex}}\varphi ,\phantom{\rule{0.166667em}{0ex}}\dot{\varphi}\right){H}_{\mathrm{on}}^{i}=c({a}_{\mathrm{on}})+\frac{1}{{a}_{\mathrm{on}}^{3}}\dot{\zeta}\left({a}_{\mathrm{on}},\phantom{\rule{0.166667em}{0ex}}\varphi \right).$$
- The modified Freedman equation evaluated once screening has taken place has to depend on $\dot{\varphi}$ to absorb possible discontinuities of the cosmological constant appearing on the r. h. s. of this equation. Thus, taking into account Equations (43) and (44) (and its $\dot{\varphi}$-derivative), it leads to$$\sum _{i=1..3}i\phantom{\rule{0.166667em}{0ex}}{Z}_{i,\dot{\varphi}}\left({a}_{\mathrm{on}},\phantom{\rule{0.166667em}{0ex}}\varphi ,\phantom{\rule{0.166667em}{0ex}}\dot{\varphi}\right){H}_{\mathrm{on}}^{i}\ne 0.$$
- The full scalar equation of motion must depend on $\ddot{a}$ to allow for a non-trivial cosmological dynamics before screening. This leads to a condition equivalent to (45) if ${H}_{\mathrm{on}}\ne 0$.