that justify processes during the inflationary epoch that result in a huge
non-vanishing positive values for the field $U$ in the RD epoch. It is
interesting to note that a behavior of the type $w_{Eff}\rightarrow-1$ is
remarkably for a model that wants to be competitive with the $\Lambda$CDM.
Indeed, in such models, and in the $\Lambda$CDM, the so called Coincidence
Problem [33] is less severe (if not a problem at all, depending on the
personal perspective), since at some point of its history, independently of
the initial conditions, the Universe always passes through a phase in which
the matter and DE densities are of the same order and then DE starts to
dominate, which in the standard model in terms of cosmic time accounts for at
least the last 3.5 billion years. On the other hand, in the nonlocal models
considered here we have to deal with a different sort of coincidence. Indeed,
the Hubble function reaches a minimum when the nonlocal fields cosmological
density starts to dominate, and the occurrence of this is roughly today. This
occurrence looks to us coincidental at least as the one present in
$\Lambda$CDM.

##  Chapter 5 Personal Contribution:
Ricci Inverse Gravity

> I don’t think there is a final theory of anything. It’s theories (turtles)
> all the way down.
>
> Jim Peebles

In Ref. [6] we proposed a novel class of modified gravity theories based on
the inverse of the Ricci tensor $R_{\mu\nu}$, which we call the anticurvature
tensor $A_{\mu\nu}$. Taking the trace of the latter we obtain the
anticurvature scalar $A$, which can then be used to construct a new type of
Lagrangian densities. It is interesting to note that with the anticurvature
scalar is very simple to write down Lagrangian densities terms with the same
dimension as $R$, like for example $A^{-1}$ or $R^{2}A$, and thus without
introducing new dimensional constants.

###  1 Field equations

The anticurvature tensor $A_{\mu\nu}$ is defined as the inverse of the Ricci
tensor:

$A^{\mu\rho}R_{\rho\nu}=\delta^{\mu}_{\nu}\;,$ (1)

from which, taking the trace, we obtain the anticurvature scalar
$A=A^{\mu\nu}g_{\mu\nu}$. Note that it is possible to write the inverse of any
matrix in terms of its adjugate, i.e. in terms of the matrix itself and the
Levi Civita symbols. In particular, for the anticurvature tensor we have:

$A^{\mu\nu}=4\frac{R_{\kappa\pi}R_{\lambda\rho}R_{\xi\sigma}\varepsilon^{\mu\kappa\lambda\xi}\varepsilon^{\nu\pi\rho\sigma}}{R_{\alpha\zeta}R_{\beta\eta}R_{\gamma\theta}R_{\delta\iota}\varepsilon^{\alpha\beta\gamma\delta}\varepsilon^{\zeta\eta\theta\iota}}\;.$
(2)

From the latter equation we can appreciate that a theory based on the
anticurvature scalar is actually a strongly nonlinear, higher order theory of
gravity.

The field equations for a general Lagrangian $f(R,A)$ are:

$\displaystyle f_{R}R^{\mu\nu}-f_{A}A^{\mu\nu}$
$\displaystyle-\frac{1}{2}fg^{\mu\nu}+g^{\rho\mu}\nabla_{\alpha}\nabla_{\rho}f_{A}A^{\alpha}_{\sigma}A^{\nu\sigma}-\frac{1}{2}\nabla^{2}(f_{A}A_{\sigma}^{\mu}A^{\nu\sigma})$
$\displaystyle-\frac{1}{2}g^{\mu\nu}\nabla_{\alpha}\nabla_{\beta}(f_{A}A_{\sigma}^{\alpha}A^{\beta\sigma})-\nabla^{\mu}\nabla^{\nu}f_{R}+g^{\mu\nu}\nabla^{2}f_{R}=T^{\mu\nu}\;,$
(3)

where $f_{A,R}$ indicate derivation with respect to the Ricci or anticurvature
scalars, see Appendix 10 for a detailed derivation of the above field
equations.111A code that evaluates the equations of motion for any $f(R,A)$ in
a given metric is made publicly available here

It is well known that one can recast an $f(R)$ theory in the form of a scalar-
tensor theory in the Einstein frame introducing a scalar field non minimally
coupled to gravity. Usually this is done by defining a scalar field
$\phi=df/dR$ and performing a Legendre transformation of the function $f$.
Such an approach, however, fails here, because $A$ is not a one-to-one
function of $R$ and therefore $df(A)/dR$ is in general not invertible.

###  2 Cosmology

We are interested in understand which kind of behavior could arise from Eq.
(3) for cosmological implications. Since we are mainly interested in Dark
Energy phenomenology, let us begin with a de Sitter ansatz for the metric
$g_{\mu\nu}$. Under this assumption $R_{\mu\nu}=Rg_{\mu\nu}/4$, and it is
straightforward to compute $A_{\mu\nu}$ from the inverse metric. In this case
all the terms with derivatives in Eq. (3) vanish, and taking the trace one has
in vacuum:

$\displaystyle f_{R}R-f_{A}A-2f=0\;.$ (4)

Since on de Sitter background $R=12H^{2}$ and $A=4/(3H^{2})$, the latter
equation can be easily solved for any $f(R,A)$ model to check whether one gets
non-trivial (i.e. $H\not=0$) solutions that could replace a cosmological
constant. For instance, if $f=R-\alpha A$ (where $\alpha$ is a constant with
dimensions $H_{0}^{4}$) then we see that $H=(\alpha/3)^{1/4}=const$.

Having seen that it is in principle possible to have de Sitter solutions in
this model, let us investigate the behavior in a flat FLRW background. In this
case we have that the Ricci and anticurvature scalars can be written:

$\displaystyle R=6\left(\dot{H}+2H^{2}\right)\;,$ (5) $\displaystyle
A=\frac{2(6+5\xi)}{3H^{2}(1+\xi)(3+\xi)}\;,$ (6)

where $\xi=\dot{H}/H^{2}$. It is straightforward to realize from Eq. (6) that
the anticurvature scalar become singular in the following cases: $H\rightarrow
0$, $\xi\rightarrow-1$, or $\xi\rightarrow-3$. The fact that for $H\rightarrow
0$ the Lagrangian is ill-defined implies, as expected, the lack of a Minkowski
solution for Lagrangians containing positive powers of the anticurvature
scalar $A^{n}$. On the other hand, this does not apply for Lagrangian
containing negative powers $A^{-n}$, which are instead singular for
$\xi\rightarrow-6/5$. Thus, just by looking at the shape of the anticurvature
scalar in FLRW background we are able to formulate the following no-go theorem
for cosmology:

######  Theorem 2 (FLRW no-go)

If the cosmic evolution passes through any one of these values of $\xi$:
$\xi=-1$, $\xi=-3$ or $\xi=-6/5$, either $A$ or $A^{-1}$, or any of their
powers, develops a singularity. If during the evolution $A$ passes through
both 0 and $\pm\infty$, then any term in the Lagrangian that contains $A^{n}$,
for $n$ positive or negative, will blow up. This behavior will reflect into
equations of motion that also contain a singularity at the same cosmic epochs.

Observations [20, 21, 27, 19] tell us that the Universe evolved from a
decelerated phase with $w_{eff}\approx 0$ (so $\xi\approx-1.5$) into an
accelerated phase $w_{eff}\approx-0.7$ (so $\xi\approx-0.45$). Therefore, the
cosmic expansion had to pass, at redshifts around unity, through both $\xi=-1$
and $\xi=-6/5$. This demonstrates that $A$ and $A^{-1}$ will both be singular
at some epoch between deceleration and acceleration. Consequently, any
Lagrangian that contains additive terms proportional to $A^{n}$ (e.g. the two
simplest scale-free models, $f(R,A)=R+\alpha A^{-1}$ and $f(R,A)=R+\alpha
R^{2}A$, with $\alpha$ a dimensionless constant) are ruled out as Dark Energy
models. Notice also that $R=6H^{2}(\xi+2)$ so no power of $R$ can cure the
singularity. In order to see some concrete realizations of the no-go theorem
in the following we will briefly illustrate the cosmological behavior for the
Lagrangians $f=R+\alpha A^{-1}$ and $f=R+\alpha R^{2}A$.

###  3 Lagrangian $R+\alpha A^{-1}$

In this case equations (3) for a FLRW background reduce to:

$\rho_{t}=3\alpha
H^{2}\frac{(\xi+3)^{2}(5\xi+6)-18\xi^{\prime}}{4(5\xi+6)^{3}}+3H^{2}\;,$ (7)

which is the modified Friedmann equations, and to:

$\displaystyle w_{t}\rho_{t}=$ $\displaystyle-\frac{\alpha
H^{2}\left[(5\xi+6)\left((\xi+3)^{2}(2\xi+3)(5\xi+6)-18\xi^{\prime\prime}\right)+270\left(\xi^{\prime}\right)^{2}-54(\xi+2)(5\xi+6)\xi^{\prime}\right]}{4(5\xi+6)^{4}}$
$\displaystyle-2H^{2}\xi-3H^{2}\;,$ (8)

which is the $(i,i)$ equation. Note that, as follows from the No-go theorem,
the singularity at $\xi=-6/5$ appears in both the equations. We have used the
subscript $t$ in $\rho_{t}$ to indicate the total matter, which of course
satisfy the continuity equation $\rho_{t}^{\prime}=-3(1+w_{t})\rho_{t}$. From
Eq. (7) we can easily define the energy density associated to the
anticurvature scalar:

$\Omega_{A}\equiv-\alpha\frac{(\xi+3)^{2}(5\xi+6)-18\xi^{\prime}}{4(5\xi+6)^{3}}\;.$
(9)

We will now consider two different cases, considering pressureless matter only
and then adding a cosmological constant

####  1 Evolution with Dust

Assuming $w_{t}=0$ we find for Eqs. (7) and (3) the following critical points:

$\displaystyle\Omega_{m}=1+\frac{\alpha}{4}\;,\qquad\xi=-\frac{3}{2}\;,$ (10)
$\displaystyle\Omega_{m}=0\,,\quad\xi_{\pm}=\frac{3(-40-\alpha\pm
6\sqrt{-\alpha})}{100+\alpha}\;.$ (11)

The first critical point corresponds to a matter dominated Universe in which
$\Omega_{A}$ behaves as matter. In particular, if $\alpha=-4$, it corresponds
to an empty Universe wich behave as it was filled with dust. The critical
points of Eq. (11) instead admit solutions only for certain values of
$\alpha$, see Fig. 1 for a graphical representation of $\xi_{+}$. We see that
for every $\alpha<0$ there are two real solutions, one above, the other below
$\xi=-6/5$, or equivalently $w_{eff}=-0.2$.

Figure 1: The two real branches of the function $w_{eff}(\alpha)=-2\xi/3-1$
from Eq. (11)

The properties of these solutions seem very interesting for cosmological
implications. For instance, for $\alpha\approx-8$, the two solutions
correspond to the observed present accelerated value $w_{eff}\approx-0.67$ and
to an expansion quite close to a matter dominated era, $w_{eff}\approx 0.06$.
Analogously, if $\alpha=-4$, one has $w_{eff}=0$, i.e. an exact matter era
evolution without matter, in which the $A$ energy density acts as a form of
Dark Matter. The other solution, $\xi_{+}$, corresponds to $w_{eff}=-0.5$,
i.e. an accelerated solution still marginally compatible with observations. In
Fig. 2 we see the behavior of the Hubble parameter for the particular case
$\alpha-4$. A cosmic evolution that moves from one such solution to the other
would be indeed an intriguing possibility, replacing both Dark Matter and Dark
Energy with the anticurvature tensor without any new scale nor fine-tuned
parameters. However, as a consequence of the no-go theorem, this cannot occur.

Figure 2:  Numerical solutions $\xi(a)$ of Eq. (11) in case of $\Omega_{m}\neq
0$, $\Omega_{\Lambda}=0$ with $w=0$ and $\alpha=-4$. The solutions $\xi=-3/2$
and $\xi=-3/4$ are confirmed to be attractors. The divide at $\xi=-6/5$ is
also evident. The red dashed line is the $\Lambda$CDM behaviour.

Through a stability analysis of the linearized dynamical system we find that
the critical point $\xi_{-}$ is a stable attractor only for $-4\leq\alpha\leq
0$. The critical point $\xi_{+}$ is a stable attractor for $\alpha\leq 0$,
while the linear analysis alone cannot assess the stability of the point
$\xi=-3/2$. These findings are supported by the numerical investigation shown
in Fig. 2, so that the cosmic evolution will end up either at $\xi_{+}$ or
$\xi_{-}$, depending on whether the initial $w_{eff}$ is above or below the
singularity at $w_{eff}=-0.2$. The crucial point is that no trajectory can
cross the $w_{eff}=-0.2$ ridge; consequently, as anticipated on general
grounds, the cosmic expansion cannot move from a decelerated phase around
$w_{eff}=0$ to an accelerated one around $w_{eff}\approx-0.7$.

####  2 Evolution with Dust and a cosmological constant

If a cosmological constant is present we can combine Eqs. (7) and (3) and
obtain:

$\xi^{\prime\prime}=\frac{6(5\xi+6)^{4}\Omega_{m}+\xi(5\xi+6)^{2}\left(9(\alpha+16)+(\alpha+100)\xi^{2}+6(\alpha+40)\xi\right)+135\alpha\left(\xi^{\prime}\right)^{2}-27\alpha\left(5\xi^{2}+11\xi+6\right)\xi^{\prime}}{9\alpha(5\xi+6)}\;,$
(12)

which must be solved together with the continuity equation:

$\Omega_{m}^{\prime}=-(3+2\xi)\Omega_{m}\;.$ (13)

In this case the phase space is more complicated, $\xi_{-}$ is now always
unstable and $\xi_{+}$ is a stable attractor for $\alpha<-16$. The critical
point $\xi=-3/2$ is always unstable, while we found a new critical point
$\xi=0$, which is a stable de Sitter attractor when $-16\leq\alpha\leq 0$.
However, the bottom line is the same, as can be immediately gleaned from Fig.
3, so the model is ruled out as a candidate for Dark Energy even when a
cosmological constant is added, regardless of the value of $\alpha$.

Figure 3:  Numerical solutions $\xi(a)$ of Eq. (12) with matter and
cosmological constant, for $\alpha=-4$. The upper curves converge toward the
de Sitter attractor at $\xi=0$. The lower curves converge towards the divide
line at $\xi=-6/5$, which is now also an attractor. The red dashed line is the
$\Lambda$CDM behavior.

###  4 Circumventing the no-go theorem

Motivated by the interesting phenomenology which could be described within the
anticurvature scalar, we will try now to address some possible escape routes
from the no-go theorem.

####  1 Modifying the geometry

To begin with, we should consider possible modifications of the background
geometry that allow for a different evolution. If we relax the assumption of a
spatially flat Universe the anticurvature scalar become:

$A=\frac{2(6+5\xi+3\Omega_{k})}{3H^{2}(1+\xi)(3+\xi+6\Omega_{k})}\;,$ (14)

where we can see that the appearance of the energy density associated with the
curvature shifts the singularity $\xi=-3$ for positive powers of $A$, while
shifts the singularity $\xi=-6/5$ of $A^{-1}$. Thus, in principle the presence
of curvature is able to shift the singularity of the Lagrangians outside the
range required by the observations. However, since observations suggest that
$\Omega_{k}\sim 0$, this possibility is very unlikely.

Another option is to take into account the contribution from spatial
anisotropies. For illustrative purposes, let us consider as an example a
Bianchi I geometry:

$ds^{2}=-dt^{2}+a(t)^{2}\left(e^{2\beta_{x}(t)}dx^{2}+e^{2\beta_{y}(t)}dy^{2}+e^{2\beta_{z}(t)}dz^{2}\right)\;,$
(15)

where we have defined the averaged scale factor

$a(t)=\sqrt[3]{a_{x}(t)a_{y}(t)a_{z}(t)}\;,$ (16)

so that $a_{i}(t)=a(t)e^{\beta_{i}}$, and the $\beta_{i}$ satisfies
$\sum_{i}\beta_{i}=0$. For the sake of simplicity, let us specialize to the
case $\beta_{x}=-\beta_{z}\equiv\beta$ and $\beta_{y}=0$. In this case the
anticurvature scalar $A$ reads:

$A=\frac{1}{H^{2}}\left[\frac{4\xi+6+\frac{(\beta^{\prime})^{2}}{2}}{3\left(3+\xi\right)\left(1+\xi+\frac{(\beta^{\prime})^{2}}{6}\right)}+\frac{2(3+\xi)}{\left(3+\xi\right)^{2}-\frac{1}{4}\left(\beta^{\prime\prime}+\beta^{\prime}(3+\xi)\right)^{2}}\right]\;,$
(17)

which for $\beta^{\prime}=0$ reduces to the FLRW case. As we can see, the
singularity $\xi=-1$ is shifted by the anisotropic term
$(\beta^{\prime})^{2}/6$. Note that also the singularity appearing in
$A^{-1}$, $\xi=-6/5$, is in general shifted. For example, if
$\beta^{\prime\prime}$ is negligible, we have that $A^{-1}$ is singular for

$\xi=\frac{24-\frac{\beta^{\prime 4}}{4}}{2\beta^{\prime
2}-20}\approx-\frac{6}{5}(1-\frac{(\beta^{\prime})^{2}}{10})\;,$ (18)

(the last approximate equality being valid for $\beta^{\prime}\ll 1$) which
recovers the FLRW case for $\beta^{\prime}=0$, while being regular in
$\xi=-6/5$ unless $\beta^{\prime 2}=48/5$, i.e. the two roots of Eq. (18) for
$\xi=-6/5$. This shows that relaxing the assumption of spatial isotropy the
singularities occurring in the anticurvature scalar and its inverse can be
arbitrarily shifted, but not removed. It is clear however that one needs
$\beta^{\prime}$ of order unity to move the singularity outside the
observational range, which on the other hand is not likely to be compatible
with experimental data.

To illustrate that, let us naively estimate $\beta$ from the evidence of
anisotropic expansion claimed recently in [162], emerged from X-ray
observations of galaxy clusters. Here the authors find that the highest and
the lowest values observed for the universe expansion rate are $H_{max}\sim
75\;$ km/s/Mpc and $H_{min}\sim 66\;$ km/s/Mpc. Assuming that the averaged
Hubble factor is $H\sim 70\;$ km/s/Mpc, we obtain

$\beta^{\prime}\sim 0.06\;,$ (19)

which shows that generally $\beta^{\prime}$ is constrained from the
observations to be too small to shift the singularities of $A$ outside the
observational range.

####  2 Non Polynomial Lagrangians

Another possibility is to consider Lagrangian densities which are not singular
when $A$ or $A^{-1}$ diverge. Still considering scale-free Lagrangians for
simplicity, we can choose for example the scalar densities $R+\alpha
R\exp[-\beta(RA)^{2}]$ or $R/(1+\alpha RA)$. In the former case, for example,
the Friedmann equation around the critical points, i.e. assuming
$\xi^{\prime}=\xi^{\prime\prime}=0$, becomes:

$3H^{2}\left(1-\frac{\alpha\mathcal{P}_{5}(\xi,\beta)}{(\xi+1)^{3}(\xi+3)^{2}}e^{-\frac{16\beta(\xi+2)^{2}(5\xi+6,)^{2}}{(\xi+1)^{2}(\xi+3)^{2}}}\right)=\rho_{m}\;,$
(20)

where $\mathcal{P}_{5}(\xi,\beta)$ is a polynomial of order five in $\xi$ and
linear in $\beta$. It is straightforward to realise that the above equation is
regular on the poles of the denominator due to the presence of the exponential
factor.

Another option is to include, as in Gauss Bonnet gravity for the Ricci tensor,
scalar combinations of higher order in the anticurvature tensor, like
$A^{\mu\nu}A_{\mu\nu}$. In FLRW background the latter looks as follows:

$A^{\mu\nu}A_{\mu\nu}=\frac{4}{9H^{4}}\frac{7\xi^{2}+15\xi+9}{(\xi+1)^{2}(\xi+3)^{2}}\;.$
(21)

We see that it still contains the singularities at $\xi=-3$ and $\xi=-1$, but
remarkably it never vanishes, and thus $(A^{\mu\nu}A_{\mu\nu})^{-1}$ can be
used to build Lagrangians which are free of this kind of singularities.

###  5 Summary and Outlooks

We have shown that it is difficult to describe Dark Energy using polynomial
Lagrangians of the anticurvature scalar because of the no-go theorem. On the
other hand, we found that an interesting phenomenology arise already for the
simplest choices of $f(R,A)$, and thus Lagrangians that escape the no-go
theorem are particularly promising for cosmological model building.

It is important to realize that in this framework we are introducing higher
order derivative terms in the Lagrangian, see Eq. (2), and then we expect that
instabilities will generally occur unless we consider degenerate Lagrangians.
The above stability issues and the no-go theorem should be taken into account
when a particular form of $f(R,A)$ is specified, which is the task we address
for future works.

##  Chapter 6 Personal Contribution:
Strong Lensing for testing Gravity and Cosmology

> Simplicity is the touchstone in finding new physical laws… If it’s elegant,
> then it’s a rough rule of thumb: you’re on the right track
>
> Kip Thorne

Lensing effects provide fertile ground for testing gravitational theories from
more than a century. In particular, it is well known that weak gravitational
lensing by Large Scale Structures can provide useful insights on the nature of
Dark Energy. In this chapter we will discuss instead the potential of strong
gravitational lensing in achieving a similar task. After a brief review of the
main equations governing this phenomenon, we will introduce two novel drift
effects proposed by us in Ref. [7], and discuss the possibility of using them
for testing violations of the Equivalence Principle and Dark Energy models
[8].

###  1 Overview of Gravitational Lensing

It is a well known results of geometrical optics that light rays passing
through a medium will generally be refracted, as encoded in the Snell’s law.
Considering the gravitational field in the empty space around its source as a
sort of ”medium”, it is a reasonable expectation that light rays traveling
through it will be deflected. It is slightly uncomfortable to give a
meaningful explanation of this effect within Newtonian gravity because photons
have no rest mass, and thus should be blind to the gravitational interaction.
On the other hand, it is a standard approach to study the motion of bodies in
a gravitational potential by mean of test particles, i.e. particles whose mass
is small enough to ignore their backreaction on the gravitational potential.
Thus, considering photons as test particles we expect already in Newtonian
gravity the deflection of light rays close to a massive body. This result was
indeed obtained by Soldner111A translation from german of the original article
is available here more than one century before Einstein’s theory of general
relativity. In GR, instead, the interpretation of the gravitational potential
in the empty space as a sort of medium is straightforward, and is logical to
conclude that trajectories of massless particles will in general be bent
because of the curvature’s gradient of the spacetime.

An effective treatment of the above phenomenon resembles the standard approach
of geometrical optics. The source of the gravitational field that deflects
light rays is then called lens, and the overall effect gravitational lensing.
We distinguish between the deflection caused by an extended, approximately
continuous distribution of sources and the one caused by a single, massive
object. The former is which is generally dubbed weak lensing, and causes
distortions on the shape of the background objects, see Ref. [163] for a
review of weak lensing and its cosmological applications. When the lens is
instead composed by a single massive object along the line of sight between
the observer and the source we are instead in a regime of Strong Lensing,
which will be our main subject during the rest of this chapter.

####  1 Strong Lensing by a point mass

To begin with, let us consider gravitational lensing by point masses. A fairly
standard configuration is given in Fig. 1, where $\theta_{0}$ is the apparent
angular position of the source as seen by the observer, and $\theta_{S}$
indicate the actual position of the source that would be observed in case of
no lensing. The above quantities are related by the lens equation in the thin
lens approximation [164]:

$\theta_{E}^{2}\equiv\theta_{0}\left(\theta_{0}-\theta_{S}\right)=\frac{4G_{N}M}{\mathcal{D}_{L}}\frac{\mathcal{D}_{LS}}{\mathcal{D}_{S}}=4G_{N}M\left(1+z_{L}\right)\left(\frac{1}{\chi_{L}}-\frac{1}{\chi_{S}}\right)\;,$
(1)

where the $\mathcal{D}$’s are the angular diameter distances:

$\mathcal{D}_{i}\equiv
a_{i}\chi_{i}=\frac{1}{1+z_{i}}\int_{0}^{z_{i}}\frac{dz^{\prime}}{H(z^{\prime})}\;,$
(2)

in which the subscript $i=L,S$ refers to the lens or to the source, while
$\mathcal{D}_{LS}$ is given by:

$\mathcal{D}_{LS}\equiv
a_{S}(\chi_{S}-\chi_{L})=\frac{1}{1+z_{S}}\int_{z_{L}}^{z_{S}}\frac{dz^{\prime}}{H(z^{\prime})}\;.$
(3)

In the above equations $z_{L,S}$ indicates the redshifts and $\chi_{L,S}$ the
comoving distances of the lens and the source respectively. Using the above
definitions it is possible to rewrite the lensing equation as:

$\theta{E}^{2}=4G_{N}M\left(1+z_{L}\right)\left(\frac{1}{\chi_{L}}-\frac{1}{\chi_{S}}\right)\;.$
(4)

Being quadratic in $\theta_{0}$, the lens equation (1) has the following two
roots:

$\theta_{0}=\frac{\theta_{S}}{2}\pm\sqrt{\frac{\theta_{S}^{2}}{4}+\theta_{E}^{2}}\;,$
(5)

which implies that, because of the lens, the original image of the source is
split into two. Notice that $\theta_{S}$ is time-independent because of the
cosmological principle. In other words, observer, lens and source form a
triangle whose sides increase due to the Hubble flow, but whose angles remain
unchanged, and therefore $d\theta_{S}/dt_{0}=0$.

Figure 1: Scheme for strong gravitational lensing induced by a point mass lens
L. $\theta_{0}$ is the apparent angle at which the source is located,
$\theta_{S}$ the true one. The deflection angle is $\delta=4GM/b$, where $b$
is the impact parameter, i.e. the distance on the lens plane between the
incoming light ray and the lens itself. Image taken from Ref. [7]

####  2 Strong Lensing for an extended Lens

In the thin lens approximation the lens equation (1) for a general mass
distribution is :

$\left(\bm{\beta}-\bm{\alpha}\right)=\nabla_{\bm{\theta}}\psi\left(\bm{\beta}\right)\;,$
(6)

where $\bm{\beta}=(\beta_{1},\beta_{2})$ and
$\bm{\alpha}=(\alpha_{1},\alpha_{2})$ are the position in the sky of the image
and the source respectively, and $\nabla_{\theta}$ is the two-dimensional
angular gradient.222From now on we will restrict the use of the $\theta_{i}$
notation for point mass lenses, while the $\alpha$ and $\beta$ notation for a
general lens profile The quantity $\psi(\bm{\beta})$ appearing in Eq. (6) is
the lensing potential and is defined as:

$\psi(\bm{\beta})\equiv\frac{2}{c^{2}}\frac{\mathcal{D}_{LS}}{\mathcal{D}_{L}\mathcal{D}_{S}}\int_{\bm{\beta}}d\lambda\;\Phi\;,$
(7)

where $\Phi$ is the standard Newtonian gravitational potential and the
integral is taken along the path of the light ray, which depends on
$\bm{\beta}$ and is parametrized by $\lambda$. Taking the divergence of Eq.
(6), as long as the extent of the lens is small compared to cosmological
distances, we can use the Poisson equation to relate the Laplacian of the
lensing potential to the mass distribution of the lens:

$\nabla^{2}_{\bm{\theta}}\psi\left(\bm{\beta}\right)=\frac{8\pi
G_{N}}{c^{2}}\frac{\mathcal{D}_{L}\mathcal{D}_{LS}}{\mathcal{D}_{S}}\Sigma(\bm{\beta})\;,$
(8)

where we have defined the surface mass density:

$\Sigma(\bm{\beta})\equiv\int_{\bm{\beta}}d\lambda\;\rho\;,$ (9)

in which appears the mass distribution of the lens $\rho$. For a detailed
derivation and an explanation on the assumptions behind Eqs. (6), (7), (8) see
for example Ref. [163].

###  2 Strong Lensing observables and Cosmology

As we saw in the previous section, in a strongly lensed system are present
several images of the same source. Thus, of course, a first important
observable is the angular separation between the various images. The entity of
this variation is typically of the order of few arcseconds. For example, using
data from Ref. [165], the quasar QSO0957 + 561 at redshift $z_{S}=1.41$ lensed
by a cluster at $z_{L}=0.31$ displays two images separated by 6.1”.

On the other hand, even if coming from the same source, these several images
at a given moment of time are not necessarily identical. Indeed, the optical
path of the photons of each image is in general different, and thus some of
the photons will require more time to reach the observer. This effect is
called Time Delay, which we indicate with $\Delta$, and could be divided into
two different contributions [164]:

$\Delta=\Delta_{geo}+\Delta_{pot}\;,$ (10)

where $\Delta_{geo}$ is the geometrical Time Delay and $\Delta_{pot}$ is the
potential Time Delay. The former is due to the bending of the trajectory of
the photon, whereas the latter is due to the motion into the lens
gravitational field. For a single image the geometric $\Delta t_{geo}$ induced
by a point mass lens is given by:

$\Delta
t_{geo}=\frac{(1+z_{L})^{2}(4GM)^{2}}{2\theta_{0}^{2}}\left(\frac{1}{\chi_{L}}-\frac{1}{\chi_{S}}\right)\;,$
(11)

from which we obtain $\Delta_{geo}=\Delta t_{geo}(\theta_{+})-\Delta
t_{geo}(\theta_{-})$, where $\theta_{\pm}$ are the two roots of Eq. (1). The
potential Time Delay between the two images due to a point mass in the thin
lens approximation is given by:

$\Delta_{pot}=\Delta t_{pot}(\theta_{+})-\Delta
t_{pot}(\theta_{-})=2GM(1+z_{L})\ln\frac{\theta_{-}}{\theta_{+}}\;.$ (12)

For an extended lens profile the Time Delay between two images $\Delta_{ij}$
is given by, see Ref. [166] :

$\Delta_{ij}=\frac{D_{\Delta_{t}}}{c}\left(\frac{\left(\bm{\beta}_{i}-\bm{\alpha}\right)^{2}}{2}-\frac{\left(\bm{\beta_{j}}-\bm{\alpha}\right)^{2}}{2}+\psi\left(\bm{\beta_{j}}\right)-\psi\left(\bm{\beta_{i}}\right)\right)\;,$
(13)

where it was defined the Time Delay distance:

$D_{\Delta_{t}}\equiv\left(1+z_{L}\right)\frac{\mathcal{D}_{L}\mathcal{D}_{S}}{\mathcal{D}_{LS}}\;.$
(14)

It is straighforward to separate in the right hand side of Eq. (13) the
contributions from the geometrical and the potential Time Delay. Indeed, the
first two terms inside the brackets are given by the differences between the
apparent and the true position of the source, and are thus of geometrical
nature. The potential Time Delay is instead given by the difference between
the lensing potential at the two apparent positions.

From Eq. (13) we can already understand the importance of precise Time Delay
measurements for cosmological implications. Indeed, if one is able to know the
position of the source $\bm{\alpha}$, measures with enough precision the
position of the images $\bm{\beta}_{i}$, and is able to infer the position of
the lens and its lensing potential, all the cosmological information is
contained in the Time Delay distance (14) through the angular diameter
distances (2). Thus, from Time Delay measurements (if a reliable description
of the lensing profile is given), one is able to reconstruct the Hubble factor
without assuming any particular cosmological model. This is precisely the goal
of the H0LiCOW collaboration
333https://shsuyu.github.io/H0LiCOW/site/index.html, which within the
COSMOGRAIL 444http://www.cosmograil.org program employed Time Delays
measurements collected over the last decade to constrain the value of the
cosmological parameter $H_{0}$ to a few percents level [167, 168, 169], with
competitive precision with respect to other cosmological probes. Moreover,
Time Delay measurements can also be used to put constraints on the Post-
Newtonian parameter $\gamma_{PPN}$, as discussed in Refs. [170, 171, 172,
173]. Furthermore, with optimistic assumptions on the surveys, in the next
years the precision of observations will be enough to provide a smoking gun
for Dark Energy [174].

In the next sections we will discuss two new observables proposed by us in
Ref. [7]: the Time Delay drift and the angular drift, which could be used to
reconstruct $H(z)$ and also to constrain violations of the Equivalence
Principle [8].

###  3 Redshift drift of Gravitational Lensing

The redshift drift is the time variation of the redshift of a source due to
the Hubble flow, see Ref. [164]. In an expanding universe described by the
FLRW metric, one straightforwardly obtains the result that a photon is
redshifted, and the redshift is given in terms of the scale factor as follows:

$1+z=\frac{a_{0}}{a_{e}}\;,$ (15)

where $a_{0}$ is the scale factor evaluated at present time (which is the time
of observation) and $a_{e}$ is the scale factor evaluated at the emission
time.

The derivative of the redshift with respect to the observation time $t_{0}$ is
the following:

$\frac{dz}{dt_{0}}=\frac{1}{a_{e}}\frac{da_{0}}{dt_{0}}-\frac{a_{0}}{a_{e}^{2}}\frac{da_{e}}{dt_{0}}=\frac{1}{a_{e}}\frac{da_{0}}{dt_{0}}-\frac{a_{0}}{a_{e}^{2}}\frac{da_{e}}{dt_{e}}\frac{dt_{e}}{dt_{0}}\;.$
(16)

It is not difficult to show that:

$\frac{dt_{e}}{dt_{0}}=\frac{a_{e}}{a_{0}}\;,$ (17)

and therefore, the redshift drift formula (16) becomes:

$\frac{dz}{dt_{0}}=\frac{1}{a_{e}}\frac{da_{0}}{dt_{0}}-\frac{1}{a_{e}}\frac{da_{e}}{dt_{e}}=(1+z)H_{0}-H(z)\;,$
(18)

where the Hubble constant $H_{0}$ and the Hubble parameter $H(z)$ have
appeared, so that measuring $dz/dt_{0}$ would allow to study the evolution of
the Hubble factor.

Since the lens equation contains the angular diameter distances, we realize in
Ref. [7] that the apparent positions of the source will acquire a time
dependence due to the Hubble flow. Thus, applying Eq. (18) to the lens
equation we predict the following angular drift and the Time Delay drift for a
point mass:

$2\frac{\dot{\theta}_{E}}{\theta}=H_{0}-\frac{H(z_{L})}{1+z_{L}}\;,\qquad\frac{\dot{\Delta}}{\Delta}=K\left(H_{0}-\frac{H(z_{L})}{1+z_{L}}\right)\;,$
(19)

where a dot indicates derivation with respect to the observer time
$\dot{}=d/dt_{0}$, and $K$ is a factor of order unity given by:

$K=\frac{\ln\frac{-\theta_{-}}{\theta_{+}}+\frac{\theta_{-}+\theta_{+}}{\theta_{-}-\theta_{+}}}{\ln\frac{-\theta_{-}}{\theta_{+}}+\frac{\theta_{+}^{2}-\theta_{-}^{2}}{\theta_{+}\theta_{-}}}\;.$
(20)

The above equations tell us that the entity of these drifts is of order
$H_{0}$. Since $H_{0}\sim 10^{-18}$ s-1, we conclude that the angular drift is
of order of $10^{-10}$ arc seconds per year, which with the current precision
of observation, $\sim 0.1$ arcseconds, would require $\sim 10^{9}$ yr to be
detected. The Time Delay drift is instad of order $\sim 10^{-3}$ arc seconds
per year, which would require $10^{8}$ years to accumulate a drift detectable
by current experiments, with sensitivity of order of days.

###  4 Test of the Equivalence Principle using Strong Lensing Time Delay

As we saw in the previous section, drift effects from Strong Lensing in GR are
too small to be detected with current experiments, being of order $\sim
H_{0}$. On the other hand, the situation could change if we move to different
theories of gravity, where the presence of a modified Poisson equation will
change the time dependence of the lensing potential. From the phenomenological
point of view, many alternative theories of gravity modify the strength of the
gravitational interaction by inducing an effective gravitational coupling
$G_{eff}$, replacing the Newton constant. For example, we already saw that in
VAAS gravity the time dependence of $G_{eff}$ is given by Eq. (43). In $f(R)$
theories of gravity instead we have $G_{eff}\sim 4G_{N}/3f_{R}$, so that
$G_{eff}$ acquires a time dependence from the term $(df/dt)\dot{R}^{-1}$.
However, in the expression for the deflection angle, we have a degeneracy
between the mass of the lens and the gravitational coupling $G$.555From now on
we will drop the subscript on $G_{eff}$ to unburden the notation. Thus, it
would not be possible in principle to distinguish between a stronger coupling
or a heavier mass.

On the other hand, the situation is different if we consider drift effects. In
Ref. [8] we studied the changes induced on the Time Delay drift by a time
dependent $G$, and analyzed for illustrative purposes the entity of the
constraints on $\dot{G}/G$ which can be extrapolated with the precision of
current experiments.

For the case of a point mass, Eqs. (19) become:

$2\frac{\dot{\theta}_{E}}{\theta_{E}}=H_{0}-\frac{H(z_{L})}{1+z_{L}}+\frac{\dot{G}}{G}\;,$
(21)
$\frac{\dot{\Delta}}{\Delta}=K\left(H_{0}-\frac{H(z_{L})}{1+z_{L}}+\frac{\dot{G}}{G}\right)\;,$
(22)

where it is understood that $G$ is the gravitational coupling at the redshift
of the lens. The above equations show explicitly that drift effects in the
context of Strong Lensing, contrary to what happens with spectroscopic
measurements, are sensitive to variations of the gravitational coupling. Since
a signal of order $H_{0}-H_{L}/(1+z_{L})$, assuming a realistic cosmological
evolution, is beyond the sensitivity of current observations, we can convert
the bounds on the drifts in upper bounds on the variation of $G$.

Since Time Delay measurements are generally more precise than angular ones we
need to obtain an expression for the time derivative of Eq. (13) for an
extended lens profile.

#### Time Delay drift for extended lenses

To begin with let us compute the time derivative of the lensing potential:

$\dot{\psi}(\bm{\beta})=\frac{2}{c^{2}}\frac{\mathcal{D}_{LS}}{\mathcal{D}_{L}\mathcal{D}_{S}}\left(\frac{1}{\left(1+z_{L}\right)}\frac{dz_{L}}{dt}\int_{\bm{\beta}}d\lambda\;\Phi+\frac{d}{dt}\left(\int_{\bm{\beta}}d\lambda\;\Phi\right)\right)\;.$
(23)

The time dependence of the second term in the right hand side of the latter
equation comes from the change in time of the light ray path due to a
variation of $\bm{\beta}$ and from the Newtonian potential. The former is
difficult to compute exactly because it is in general difficult to describe
precisely the curve identified by the light ray path. On the other hand, it is
reasonable to assume that the induced variation of the curve is small and does
not contribute to the support of the integral in Eq. (7). In particular, this
is true if we evaluate the above integral within the Born approximation, i.e.
along the unperturbed light path, as it is customary in Strong Lensing
applications where $\Phi/c^{2}\ll 1$ [163]. Within this assumption we can
interchange the operation of integration and time differentiation obtaining:

$\dot{\psi}(\bm{\beta})=\psi(\bm{\beta})\left(H_{0}-\frac{H(z_{L})}{1+z_{L}}\right)+\psi(\bm{\beta})\frac{\int_{\bm{\beta}}d\lambda\;\dot{\Phi}}{\int_{\bm{\beta}}d\lambda\;\Phi}\;.$
(24)

The above equation can be further simplified if we consider a static
distribution of matter. Indeed, in this case the only time dependence of the
Newtonian potential is through the effective gravitational coupling $G$, so
that we have:

$\dot{\psi}(\bm{\beta})=\psi(\bm{\beta})\left(\frac{\dot{G}}{G}+H_{0}-\frac{H(z_{L})}{1+z_{L}}\right)\;.$
(25)

In concrete Time Delay measurements we have to take into account corrections
due to the presence of mass distributed along the line of sight. This is done
by introducing a parameter called external convergence, $\kappa_{ext}$, and
defining the real Time Delay distance as:

$D_{\Delta t}^{real}\equiv\frac{D_{\Delta t}}{1-\kappa_{ext}}\;,$ (26)

see for example Ref. [166]. If we assume that the external convergence has a
time dependence this would be inherited by the Time Delay distance (14).

Combining the latter and Eq. (25) we can finally write down the Time Delay
drift for an extended lens with a static distribution of mass:

$\frac{\dot{\Delta}_{ij}}{\Delta_{ij}}=\left(\frac{\dot{G}}{G}+H_{0}-\frac{H(z_{L})}{1+z_{L}}\right)\left[1+\frac{D_{\Delta
t}\left(\bm{\beta}_{i}-\bm{\alpha}\right)^{2}}{2c\Delta_{ij}}-\frac{D_{\Delta
t}\left(\bm{\beta_{j}}-\bm{\alpha}\right)^{2}}{2c\Delta_{ij}}\right]-\frac{\dot{\kappa}_{ext}}{1-\kappa_{ext}}\;.$
(27)

#### Estimating the variation of the Time Delay from current data

We now illustrate how the data can put constraints on the variation of the
Time Delay. To this end, we use the package PyCS3 from the COSMOGRAIL program
666Available here., see Refs. [175, 176]. The data used are the simulated
light curves used in [176], produced in the context of the blind Time Delay
measurement competition named Time Delay Challenge 1 (TDC1) [177], and from
the quasar DES J0408-5354 [178]. We split the total time of observations in
two equal time periods. Each period consists of 658 days for the trial curves,
and of 93 days for DES J0408-5354. The Time Delay between each image is then
calculated for each period. The Time Delay estimates are shown in App. 11 and
summarized in Table 1. From it we can readily estimate the relative variation
(27) as:

$\frac{\dot{\Delta}_{ij}}{\Delta_{ij}}=\frac{\Delta_{ij}\left(t+\delta
t\right)-\Delta_{ij}(t)}{\delta
t\Delta_{ij}(t)}=\frac{\Delta_{ij}^{I+II}-\Delta_{ij}^{I}}{\Delta_{ij}^{I}\delta
t}\;,$ (28)

We display the results in Table 2.

| $\Delta_{AB}$ | $\Delta_{AC}$ | $\Delta_{AD}$ | $\Delta_{BC}$ | $\Delta_{BD}$ | $\Delta_{CD}$
---|---|---|---|---|---|---
Trial I | $-4.9^{+4.3}_{-5.5}$ | $-18.0^{+6.0}_{-8.0}$ | $-0.7^{+9.6}_{-10.3}$ | $-13.5^{+7.6}_{-7.6}$ | $+4.3^{+9.8}_{-11.5}$ | $+17.5^{+10.7}_{-12.1}$
Trial I+II | $-4.5^{+2.4}_{-2.0}$ | $-21.3^{+1.4}_{-1.7}$ | $-29.9^{+9.2}_{-7.0}$ | $-16.6^{+1.8}_{-3.4}$ | $-25.2^{+7.5}_{-6.4}$ | $-8.2^{+8.3}_{-6.7}$
DES 0408 WFI I | $-106.5^{+15.9}_{-14.1}$ | $-110.6^{+50.4}_{-21.7}$ | $-142.2^{+34.3}_{-18.3}$ | $-2.4^{+38.8}_{-19.9}$ | $-37.1^{+25.7}_{-16.2}$ | $-31.7^{+19.8}_{-30.3}$
DES 0408 WFI I+II | $-112.6^{+6.6}_{-2.2}$ | $-117.2^{+5.9}_{-7.6}$ | $-153.2^{+11.9}_{-9.5}$ | $-7.1^{+8.6}_{-8.6}$ | $-40.5^{+11.3}_{-9.3}$ | $-35.6^{+14.1}_{-10.7}$
Table 1: Time Delay $\Delta$ between four images from a simulated quasar and the DES J0408-5354 quasar [178]. Each image is labeled from $A$ to $D$. The numbers I and I+II indicate that $\Delta$ was measured over the first half of the period of observations, or over the whole period, respectively. All values are given in days. | $|\dot{\Delta}/\Delta|_{AB}$ | $|\dot{\Delta}/\Delta|_{AC}$ | $|\dot{\Delta}/\Delta|_{AD}$ | $|\dot{\Delta}/\Delta|_{BC}$ | $|\dot{\Delta}/\Delta|_{BD}$ | $|\dot{\Delta}/\Delta|_{CD}$
---|---|---|---|---|---|---
Trial ($\times 10^{-5}$) | X | $27.9\pm 12.6$ | X | $34.9\pm 20.9$ | X | X
DES 0408 WFI ($\times 10^{-5}$) | $61.6\pm 9.9$ | $64.2\pm 29.6$ | $83.2\pm 21.1$ | X | $98.5\pm 73.6$ | X
Table 2: Estimated absolute Time Delay variation for the simulated quasar and
DES J0408-5354. All values are given in $day^{-1}$. The X’s represent values
with uncertainty bigger than the central value, and are thus omitted.

#### Estimated constraint on $\dot{G}/G$

Through Eq. (27) it is possible to relate constraints on the relative time
variation of $\Delta_{ij}$ to upper bounds on the variation of $\dot{G}/G$.
The external convergence time dependence is difficult to evaluate. We expect
it to depend explicitly on $\dot{G}/G$, similarly to the potential generated
by the lens, with the two contributions having the same sign, but we will
assume that it is negligible with respect to the precision of current
observations. We want to stress however, as we discussed in the previous
sections, that it is in principle possible to disentangle such contribution by
considering differences of Time Delays drifts of multiple images. As we
already saw the drift due to the Hubble flow is of order of $H_{0}\sim
10^{-18}$ s-1, so we will neglect it as well. Another effect that could be of
relevance is the time variation of the Time Delay due to the peculiar velocity
of the lens galaxy and its transverse motion. On the other hand, according to
Refs. [179, 180], this contribution is estimated to be of the order of a few
seconds per year, so that the effect, even though it is bigger than the
cosmological one due to the Hubble flow, is still not appreciable with current
precision. Under these assumptions, and roughly estimating the term inside the
square bracket of Eq. (27) to be of order 1, the values of Table 2 can be
directly converted into upper bounds on the time variation of the effective
gravitational coupling. The results are reported in Table 3.

| $|\dot{G}/G|_{AB}$ | $|\dot{G}/G|_{AC}$ | $|\dot{G}/G|_{AD}$ | $|\dot{G}/G|_{BC}$ | $|\dot{G}/G|_{BD}$ | $|\dot{G}/G|_{CD}$
---|---|---|---|---|---|---
Trial ($\times 10^{-1}$) | X | $1.0\pm 0.5$ | X | $1.3\pm 0.8$ | X | X
DES 0408 WFI ($\times 10^{-1}$) | $2.2\pm 0.4$ | $2.3\pm 1$ | $3.0\pm 0.8$ | X | $3.6\pm 3$ | X
Table 3: Upper bounds on the absolute value of $\dot{G}/G$ in $yr^{-1}$ from
the simulated quasar and DES J0408-5354. The X’s represent values with
uncertainty bigger than the central value, and are thus omitted.

###  5 Summary and discussion

We have introduced two novel observables in the contest of Strong Lensing,
i.e. the Time Delay drift and the angular drift. We have shown that measuring
these effects one is able to estimate $H(z)$, and thus reconstruct the
cosmological evolution and the effective equation of state parameter of the
Universe $w_{eff}$. We have shown that in a modified gravity framework where
the effective gravitational coupling become time dependent, those drifts earn
a contribution $\dot{G}/G$, and thus their measurements could be used to
constraint both violations the Equivalence Principle and DE models.
Unfortunately, the current precision of the observation is not enough to
detect these drifts so that the constraints we obtain are currently not very
competitive. On the other hand, they could improve already in the near future
with upcoming data, see for example Ref. [181], or simply increasing the
observational time. Moreover, as discussed in Refs. [179, 180, 182], if in the
future strongly lensed repeating FRB will be detected, they will provide Time
Delay measurements of such extremely high precision, nominally of the order of
seconds, that even redshift drift effects due to the Hubble expansion will be
appreciable. In this scenario, the impact for cosmological implications of the
drifts we proposed in this chapter is very promising, and we keep high
expectations for the future.

##  Chapter 7 Final Considerations

> I managed to get a quick PhD — though when I got it I knew almost nothing
> about physics. But I did learn one big thing: that no one knows everything,
> and you don’t have to.
>
> Steven Weinberg

### Summary

Research on DE is a tricky business. In principle, one should just try to
answer to the question of what is causing the accelerated expansion of the
Universe, but in practice this is just the first domino tile of a long chain.
The Cosmological Constant $\Lambda$ seems to be the most reasonable and
simplest candidate of Dark Energy, in particular because together with the CDM
paradigm is able to safely match a number of different observations. However,
with the outstanding precision reached in the last years in observational
cosmology, the concordance model seems to be not so concordant, with the most
uncomfortable question being what is the value of the Hubble parameter now.
Curiously, this seems to be an evergreen question in cosmology since from its
birth, and it is inspiring the tenacity shown by the scientific community
through the last hundred years in looking for a satisfactory answer. We tried
to give an overview of the $\Lambda$CDM model in chapter 1 of this work; our
main goal was to highlight the pillars on which it is built on and the kind of
questions that it is able to answer. Of course our presentation is by no means
exhaustive or complete by itself, but we hope it is able at least to address a
satisfactory number of references for the curious reader.

DE is so interesting because it could be a window towards new physics or,
depending on one’s personal perspective, it is already a manifestation of it.
The community working on this topic has undoubtedly shown a fair amount of
creativity in the last two decades, both in boosting or trying to contain the
proliferation of possible candidates of DE. In chapter 2 we tried to give a
reasonable classification of the kind of models that were proposed in the last
years based on the Lovelock theorem. Keeping into account that a proper
treatment of the topic would have required more than a textbook, for each
class of models we presented some examples in an attempt of transmitting the
flavor and the potential of these proposals.

The ultimate goal of this thesis is to present the results of our research
from the last four years. Most of it was oriented on models of DE which does
not introduce in the playground new degrees of freedom, but instead attempt to
ascribe the accelerated expansion of the Universe to geometrical modifications
of the gravitational interaction.

An interesting phenomenology arises if in such geometrical description we
relax the assumption of locality for the field equations. Ultimately, the main
motivation for considering these kinds of models is that quantum mechanics
seems to be nonlocal at fundamental level, and such could also be a quantum
description of gravity. In chapter 3 we review the theoretical grounds and the
mathematical formalism of nonlocal models of gravity based on the inverse
d’Alembertian of the Ricci scalar $\Box^{-1}R$.

In chapter 4 we present the results of our research on nonlocal gravity
models, in particular [4, 5]:

  * •

We found that the Late-times asymptotic equation of state has a common
behavior in those models which contain explicitly a term $\Box^{-1}R$ in the
Lagrangian. In particular, even if the Hubble factor does not approach a
constant value, we found that $\dot{H}/H^{2}\rightarrow 0$ because
$H\rightarrow\infty$ due to the divergence of $\Box^{-1}R$. We also show that
this does not happen for the DW model if the distortion function
$f(\Box^{-1})R$ is chosen in such a way that $f(\infty)\neq\infty$.

  * •

We studied VAAS gravity within a dynamical system approach and found no stable
critical points. On the other hand, a qualitative study of its field equations
show that it is indeed possible to produce at background level an evolution
history compatible with $\Lambda$CDM from the radiation dominated epoch until
today.

  * •

Still in VAAS gravity, we have shown that on local scales the effects of the
nonlocal modifications is encoded in a slip parameter and in a modification of
the effective gravitational coupling $\eta,Y\neq 1$.

  * •

We checked consistency of the model with LLR constraints, which are not
passed. However, we found that the field equations admit spherically symmetric
static solutions which, if realized and stable, would satisfy trivially LLR
constraints and for which one of the gravitational potential closely resemble
the solution one would obtain in standard GR for the Schwarzschild-deSitter
solution.

In chapter 5 we discuss a geometrical theory of gravity containing higher
order derivative terms which we proposed in Ref. [6]. The theory is based on
the introduction of the anticurvature scalar $A$, which is the trace of the
inverse of the Ricci tensor $A_{\mu\nu}=R^{-1}_{\mu\nu}$. We derive the
general field equations for an $f(R,A)$ theory and assess their potential for
cosmological implications. We found that an interesting phenomenology arise
already for the simplest choices of $f$, but also a no-go theorem which claims
that polynomial Lagrangians are not able to reproduce a viable cosmological
history. Finally, we present some choices of $f$ that evades the no-go theorem
and thus are worth further investigation.

Finally, in chapter 6 we present the results of our works [7, 8] about the
potential of Strong Lensing observables to study cosmology and DE. In
particular, we introduce two new quantities, the angular drift and the Time
Delay drift, which are a manifestation of the Hubble flow in a lensed system.
These drifts, if measured, would allow for a reconstruction of $H(z)$ at the
redshift of the lens, and thus potentially very helpful for estimating the
effective equation of state parameter of the Universe $w_{eff}$. We also show
that in a modified theory of gravity with a time dependent gravitational
coupling, a similar effect arise, and we can use Strong Lensing Time Delay
observations to constrain violations of the Equivalence Principle. The entity
of these drifts is unfortunately beyond the sensitivity of current experiment,
and results in very weak constraints on $\dot{G}/G$. However, we also discuss
some possible future perspectives that would make them appealing.

### Outlooks

Most of the nonlocal models of gravity presented in the literature seems to be
ruled out by LLR constraints. On the other hand they are motivated by
fundamental physics, and as we saw in Ref.[5] in most cases they provide a
useful mechanism to trigger an accelerated expansion of the Universe only at
Late-times. For these reasons we are currently trying to understand if they
can be used, together with a Cosmological Constant $\Lambda$, to alleviate or
solve the $H_{0}$ tension by changing the rate of expansion of the universe
today without affecting CMB measurements or violating LLR constraints.
Regarding VAAS, a perturbative analysis beyond the small scales regime is
still lacking, and is crucial for properly assess the appealing of the model
with respect to $\Lambda$CDM. We address this analysis for future works.
Inverse Ricci gravity offers a completely new class of modified gravity
theories, and as we saw it can introduce a very interesting phenomenology. On
the other hand, containing higher order derivatives, one expect that it is in
general unstable unless the Lagrangian is degenerate. We address to future
works a research of which kind of $f(A,R)$ will result in a degenerate
Lagrangian capable of escaping the no-go theorem. Finally, the increasing
precision of Strong Lensing measurements motivate a deeper investigation of
the cosmological information we can extract from them. Moreover, being
equivalent to an interferometer with arms of astrophysical size, a strongly
lensed system could be sensitive to the passage of a gravitational wave. This
could be very interesting for multi-messenger astronomy, and we address to
future works a quantitative estimation of the GW impact on Time Delay
measurements to assess the potential of this idea.

##  Chapter 8 Critical points in VAAS gravity

We want to study the critical points of the system (1) satisfying the
constraint (2).

#### Critical points at finite distance

Consider first the case where $2\alpha V=1$, which is special because it
eliminates $\dot{H}$ from the field equations. From the Klein Gordon equations
we see that $V=1/(2\alpha)=$ constant only if $m^{2}=0$. The latter is a
parameter of the theory so it is not necessarily vanishing and thus, unless
$m^{2}=0$, $V$ cannot be constant. With $V=1/(2\alpha)=$ constant, then
$\dot{V}=Y=0$ and thus from Eq. (2) we get $\rho=0$. We are just left with the
Klein Gordon equation for $U$ while $H$ is completely arbitrary, since we have
lost the equations ruling its dynamics. Since providing a suitable $H$ is one
of the objectives of the model, we do not consider this possibility anymore.

Now we consider $2\alpha V\neq 1$. The right hand sides of the last three
equations of the dynamical system (1) vanish when $X=Y=0$ and $m^{2}=0$.
Again, the latter is a parameter of the theory so it is not necessarily
vanishing. Thus if $m^{2}\neq 0$ we can already conclude that there are no
critical points at finite distance.

Let us consider now the subclass of theories for which $m^{2}=0$, i.e. with
$R_{f}=0$. Demanding the vanishing of the right hand sides of the first three
equations of system (1) and from Eq. (2) we have, taking into account
$X=Y=m^{2}=0$:

$\displaystyle 3H^{2}(1-2\alpha V)=\frac{\rho-P}{2M_{Pl}^{2}}\;,$ (1)
$\displaystyle-3H(\rho+P)=0\;,$ (2) $\displaystyle H^{2}(1-2\alpha
V)=\frac{P-\rho}{2M_{Pl}^{2}}\;,$ (3) $\displaystyle 3H^{2}(1-2\alpha
V)=\frac{\rho}{M_{Pl}^{2}}\;.$ (4)

Now, from the second equation above we either have that $H=0$ or $P=-\rho$,
i.e. a vacuum energy equation of state is required. If our fluid model has not
such equation of state, then the only possibility is $H=0$ and thus
$\rho=P=0$. This critical point represents Minkowski space. Note that $U$ and
$V$ may assume whatever constant value, except $V=1/(2\alpha)$.

On the other hand, let us assume that indeed the fluid content satisfies a
vacuum energy equation of state, i.e. $P=-\rho$. In this case, the above
system becomes:

$\displaystyle 3H^{2}(1-2\alpha V)=\frac{\rho}{M_{Pl}^{2}}\;,$ (5)
$\displaystyle H^{2}(1-2\alpha V)=-\frac{\rho}{M_{Pl}^{2}}\;.$ (6)

Summing the two equations we arrive at:

$4H^{2}(1-2\alpha V)=0\;.$ (7)

Since $2\alpha V\neq 1$, we have again $H=0$ and thus the same Minkowski
critical point as before. There is a caveat here. When $m^{2}=0$ the evolution
of $U$ is disentangled from the one of $H$. Therefore, we can reduce our
dynamical system (1) to:

$\displaystyle\dot{H}$ $\displaystyle=$ $\displaystyle\frac{1}{1-2\alpha
V}\left[\frac{\rho-P}{2M_{Pl}^{2}}+2\alpha HY\right]-3H^{2}\;,$ (8a)
$\displaystyle\dot{\rho}$ $\displaystyle=$ $\displaystyle-3H(\rho+P)\;,$ (8b)
$\displaystyle\dot{Y}$ $\displaystyle=$ $\displaystyle-3HY\;,$ (8c)
$\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle Y\;.$ (8d)

From this system is not difficult to see that there is a critical point

$P=-\rho=-M^{2}_{Pl}\left(1-2\alpha V\right)3H^{2}=\mbox{ constant}\;,$ (9)

which represents a de Sitter phase. With $H>0$ constant, the equation for $U$
becomes:

$\ddot{U}+3H\dot{U}=-12H^{2}\;,$ (10)

which has the special, non-constant solution $U=-4Ht$.

#### Critical points at infinite distance

In order to investigate the critical points at infinity, we build the Poincaré
hyperspherein the variable space plus one dimension. The equation of the
sphere is the following:

$\displaystyle h^{2}+r^{2}+p^{2}+x^{2}+y^{2}+u^{2}+v^{2}+z^{2}=1\;,$ (11)

where:

$\displaystyle H\equiv\frac{h}{z}\;,\quad\rho\equiv\frac{r}{z}\;,\quad
P\equiv\frac{p}{z}\;,\quad X\equiv\frac{x}{z}\;,\quad
Y\equiv\frac{y}{z}\;,\quad U\equiv\frac{u}{z}\;\quad\textrm{and}\quad
V\equiv\frac{v}{z}\;.$ (12)

System (1) thus becomes:

$\displaystyle z\dot{h}$ $\displaystyle=$ $\displaystyle
A(1-h^{2})-h\left(rB+xC+yD+uE+vF\right)\;,$ (13a) $\displaystyle z\dot{r}$
$\displaystyle=$ $\displaystyle B(1-r^{2})-r\left(hA+xC+yD+uE+vF\right)\;,$
(13b) $\displaystyle z\dot{x}$ $\displaystyle=$ $\displaystyle
C(1-x^{2})-x\left(hA+rB+yD+uE+vF\right)\;,$ (13c) $\displaystyle z\dot{y}$
$\displaystyle=$ $\displaystyle D(1-y^{2})-y\left(hA+rB+xC+uE+vF\right)\;,$
(13d) $\displaystyle z\dot{u}$ $\displaystyle=$ $\displaystyle
E(1-u^{2})-u\left(hA+rB+xC+yD+vF\right)\;,$ (13e) $\displaystyle z\dot{v}$
$\displaystyle=$ $\displaystyle F(1-v^{2})-v\left(hA+rB+xC+yD+uE\right)\;,$
(13f) $\displaystyle z\dot{z}$ $\displaystyle=$
$\displaystyle-\left(hA+rB+xC+yD+uE+vF\right)\;,$ (13g)

where the equation for $\dot{z}$ is obtained from Eq. (11) and the terms $A$,
$B$, $C$, $D$, $E$ and $F$ are defined as follows:

$\displaystyle A$ $\displaystyle\equiv$ $\displaystyle\frac{z}{z-2\alpha
v}\left[\frac{z(r-p)}{2M_{Pl}^{2}}+\frac{m^{2}z}{2}\left(z-u\right)+2\alpha
hy\right]-3h^{2}\;,$ (14a) $\displaystyle B$ $\displaystyle\equiv$
$\displaystyle-3h(r+p)\;,$ (14b) $\displaystyle C$ $\displaystyle\equiv$
$\displaystyle-3hx-6h^{2}-\frac{6z}{z-2\alpha
v}\left[\frac{z(r-p)}{2M_{Pl}^{2}}+\frac{m^{2}z}{2}\left(z-u\right)+2\alpha
hy\right]\;,$ (14c) $\displaystyle D$ $\displaystyle\equiv$
$\displaystyle\frac{m^{2}z^{2}}{2\alpha}-3hy\;,$ (14d) $\displaystyle E$
$\displaystyle\equiv$ $\displaystyle zx\;,$ (14e) $\displaystyle F$
$\displaystyle\equiv$ $\displaystyle zy\;.$ (14f)

The critical points of the above system corresponding to $z=0$ are critical
points at infinity. In order to find them, let us first define a new time
parameter such that
$\frac{z}{3}\frac{d}{dt}\equiv\frac{d}{d\tau}\equiv\;^{\prime}$ and the
functions:

$\displaystyle G\equiv h\left(h^{2}+\gamma r^{2}+x^{2}+y^{2}+2hx-1\right)\;,$
(15) $\displaystyle\tilde{G}\equiv h\left(h^{2}+\gamma
r^{2}+x^{2}+y^{2}+2hx-\gamma\right)\;,$ (16)

where we have assumed a barotropic equation of state linking pressure to
density:

$P=(\gamma-1)\rho\;.$ (17)

The dynamical system (13) for $z=0$ can thus be written as:

$\displaystyle h^{\prime}$ $\displaystyle=$ $\displaystyle hG\;,$ (18a)
$\displaystyle r^{\prime}$ $\displaystyle=$ $\displaystyle r\tilde{G}\;,$
(18b) $\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle xG-2h^{2}\;,$
(18c) $\displaystyle y^{\prime}$ $\displaystyle=$ $\displaystyle yG\;,$ (18d)
$\displaystyle u^{\prime}$ $\displaystyle=$ $\displaystyle
u\left(G+h\right)\;,$ (18e) $\displaystyle v^{\prime}$ $\displaystyle=$
$\displaystyle v\left(G+h\right)\;.$ (18f)

It is important to emphasise that the solutions of the above system must be
compatible with the Friedmann equation, which in terms of the variables on the
Poincaré sphere provides:

$\displaystyle\left(1-2\alpha\frac{v}{z}\right)\frac{h^{2}}{z^{2}}+\frac{m^{2}u}{6z}-\frac{\alpha
y}{3z^{2}}(6h+x)-\frac{r}{3zM_{Pl}^{2}}=0\;.$ (19)

Multiplying the above equation for $z^{3}$ and then considering $z=0$, we
obtain:

$\displaystyle\alpha vh^{2}=0.$ (20)

So that at infinity Friedmann equation imposes that at least one among
$\alpha$, $v$, $h$ is vanishing. It is easy to see that when $h=0$ the
function $G$ vanishes identically, so we have a critical hyperplane in the
variables space corresponding to Minkowski spacetime.

We will discuss later the stability of this critical hypersurface, and focus
now on the only other interesting case,111We do not consider $\alpha=0$ since
it simply turns off the nonlocal interacting term. $v=0$. In this case,
Friedmann equation becomes:

$h^{2}-\alpha y\left(2h+\frac{x}{3}\right)=0\;.$ (21)

Since $h\neq 0$, then from Eqs. (18a) and (18d) we have that:

$y=h+K\;,$ (22)

where $K$ is an integration constant. From Eq. (21) we then have:

$x=\frac{3h^{2}}{\alpha y}-6h=\frac{3h^{2}}{\alpha(h+K)}-6h\;.$ (23)

This result allows us to rewrite Eq. (18c) as follows:

$x^{\prime}=G\left[\frac{3h^{2}}{\alpha\left(h+K\right)}-6h\right]-2h^{2}\;,$
(24)

and we see that it is impossible to have both (18a) and (24) vanishing without
$G=h=0$, and so there are no critical points, but $h=0$, at infinite distance.

#### Linearisation and stability of the critical point

In order to calculate the stability of the critical points at $h=0$, let us
linearise system (18) around the critical point $h_{0}=0$, i.e,
$h=h_{0}+\epsilon$, $r=r_{0}+\eta$, $x=x_{0}+\chi$, $y=y_{0}+\varphi$,
$u=u_{0}+\lambda$ and $v=v_{0}+\sigma$. Consequently, the linearised dynamical
system is given by:

$\displaystyle\epsilon^{\prime}$ $\displaystyle=$ $\displaystyle 0\;,$ (25a)
$\displaystyle\eta^{\prime}$ $\displaystyle=$ $\displaystyle r_{0}\left(\gamma
r_{0}^{2}+x_{0}^{2}+y_{0}^{2}-\gamma\right)\epsilon\;,$ (25b)
$\displaystyle\chi^{\prime}$ $\displaystyle=$ $\displaystyle x_{0}\left(\gamma
r_{0}^{2}+x_{0}^{2}+y_{0}^{2}-1\right)\epsilon\;,$ (25c)
$\displaystyle\varphi^{\prime}$ $\displaystyle=$ $\displaystyle
y_{0}\left(\gamma r_{0}^{2}+x_{0}^{2}+y_{0}^{2}-1\right)\epsilon\;,$ (25d)
$\displaystyle\sigma^{\prime}$ $\displaystyle=$ $\displaystyle
u_{0}\left(\gamma r_{0}^{2}+x_{0}^{2}+y_{0}^{2}\right)\epsilon\;,$ (25e)
$\displaystyle\lambda^{\prime}$ $\displaystyle=$ $\displaystyle
v_{0}\left(\gamma r_{0}^{2}+x_{0}^{2}+y_{0}^{2}\right)\epsilon\;.$ (25f)

Usually the stability of the critical point is studied by means of the
Jacobian matrix, but unfortunately it is degenerate in our case. However, we
easily recognise that in the above system of equations the perturbation
$\epsilon$ is constrained to be a constant by (25a). Note also that the
combinations $\left(\gamma r_{0}^{2}+x_{0}^{2}+y_{0}^{2}-1\right)$ and
$\left(\gamma r_{0}^{2}+x_{0}^{2}+y_{0}^{2}-\gamma\right)$ are constant as
long as we assume a time-independent equation of state. Thus all the
perturbations with the exception of $\epsilon$, which is constant, go linearly
with time showing an unstable behavior.

##  Chapter 9 Qualitative dynamic of nonlocal models

#### Qualitative behavior of $U$

The Klein Gordon equation for $U$ in FLRW backround, defining $X\equiv
U^{\prime}$ and using $N=\log a$ as time coordinate is:

$X^{\prime}+\left(3+\xi\right)X+6\left(2+\xi\right)=0\;.$ (1)

The formal solution for $X^{\prime}$ is:

$X(N)=C_{1}e^{-F(N)}-6e^{-F(N)}\int^{N}_{N_{i}}d\bar{N}e^{F(\bar{N})}[2+\xi(\bar{N})]\;,$
(2)

which choosing vanishing initial conditions give:

$X(N)=-6e^{-F(N)}\int^{N}_{N_{i}}d\bar{N}e^{F(\bar{N})}[2+\xi(\bar{N})]\;.$
(3)

It is straightforward to show that $0\leq X\leq 6$, indeed rewrite the
solution for $X$ as follows:

$X(N)=-6e^{-F(N)}\int^{N}_{N_{i}}d\bar{N}e^{F(\bar{N})}[3+\xi(\bar{N})]+6e^{-F(N)}\int^{N}_{N_{i}}d\bar{N}e^{F(\bar{N})}\;.$
(4)

The first integral can be cast as:

$X(N)=-6e^{-F(N)}\int^{N}_{N_{i}}d\bar{N}\frac{d(e^{F(\bar{N})})}{d\bar{N}}+6e^{-F(N)}\int^{N}_{N_{i}}d\bar{N}e^{F(\bar{N})}\;,$
(5)

and thus:

$X(N)=-6+6e^{-F(N)}+6e^{-F(N)}\int^{N}_{N_{i}}d\bar{N}e^{F(\bar{N})}\;.$ (6)

Being the second and third terms on the right hand side strictly positive, we
have then that $X(N)>-6$.

#### Qualitative behavior of VAAS

During the radiation-dominated epoch one has $\xi=-2$ and thus:

$X=0\;,$ (7)

This implies that $U$ is a constant, and this constant must be zero, because
of our initial condition. On the other hand,

$Y(N)=-e^{-(N-N_{i})}\int^{N}_{N_{i}}d\bar{N}e^{(\bar{N}-N_{i})}\frac{m^{2}}{H_{0}^{2}\Omega_{r0}e^{-4\bar{N}}}=-\frac{m^{2}}{5H_{0}^{2}\Omega_{r0}}e^{-N}\left(e^{5N}-e^{5N_{i}}\right)\;,$
(8)

from which:

$\tilde{V}=-\frac{m^{2}}{20H_{0}^{2}\Omega_{r0}}e^{4N}-\frac{m^{2}}{5H_{0}^{2}\Omega_{r0}}e^{5N_{i}-N}+C_{3}\;.$
(9)

Since $\tilde{V}(N_{i})=1$, we then have:

$\tilde{V}=-\frac{m^{2}}{20H_{0}^{2}\Omega_{r0}}e^{4N}-\frac{m^{2}}{5H_{0}^{2}\Omega_{r0}}e^{5N_{i}-N}+\frac{m^{2}}{4H_{0}^{2}\Omega_{r0}}e^{4N_{i}}+1\;.$
(10)

This is very small for $N$ large and negative, so one can basically take
$\tilde{V}=1$.

During the matter-dominated epoch we have $\xi=-3/2$ and thus:

$X=-3e^{-3N/2}\int^{N}_{\tilde{N}_{i}}d\bar{N}e^{3\bar{N}/2}=-2+2e^{3\tilde{N}_{i}/2}\;,$
(11)

where $\tilde{N}_{i}$ is some new initial value, chosen in the matter-
dominated epoch. Being this large and negative, we neglect the exponential
with respect to $-2$ and so a linear solution for $U$ follows:

$U=C_{1}-2N\;,$ (12)

with $C_{1}$ integration constant. For $Y$ we have a solution similar to the
one we found for the radiation-dominated case. We simplify it a little bit,
writing:

$\tilde{V}=1-\frac{m^{2}}{12H_{0}^{2}\Omega_{m0}}e^{3N}\;.$ (13)

We have neglected all the exponentials contributions containing the initial
e-folds number because, even in the matter-dominated epoch, it is very small.
It is only at late times that $\tilde{V}$ starts to grow different from one.

In vacuum, i.e. when matter and radiation dilute, the first Friedmann equation
is:

$3\tilde{V}=-\frac{m^{2}U}{2H^{2}}-\frac{Y}{2}(6+X)+\frac{\rho}{M_{Pl}^{2}H^{2}}\;.$
(14)

Since $U<0$, $Y<0$, $X>-6$ and of course $\rho>0$, we can conclude that
$\tilde{V}>0$. On the other hand, $Y<0$ tells us that $\tilde{V}$ always
decreases. So, in order for $\tilde{V}$ to decrease from one to zero, without
becoming negative, we need that $m^{2}/H^{2}\geq 1$ only for a limited
interval of e-folds. This, in particular, means that $H$ cannot tend to zero
in the far future, for large $N$, but it must increase in order to guarantee
that $m^{2}/H^{2}\ll 1$. On the basis of this argument, we can conclude that
at late-times, when the matter is completely diluted:

$3\tilde{V}\sim-\frac{m^{2}U}{2H^{2}}\;,$ (15)

this however provided that $X$ does not diverge, otherwise we cannot neglect
the product $XY$ in general.

Combining the two Friedmann equations, we have that:

$\xi=-3+\frac{1}{\tilde{V}}\left[-Y+\frac{m^{2}(1-U)}{2H^{2}}+\frac{\rho-P}{2M_{Pl}^{2}H^{2}}\right]\;.$
(16)

For non-exotic fluids one has $\rho-P>0$ and therefore we see that definitely
$\xi>-3$. At late times, according to our previous discussion, we have that:

$\xi\sim-3-\frac{1}{\tilde{V}}\frac{m^{2}U}{2H^{2}}\sim 0\;.$ (17)

Hence, the effective equation of state always tends to $-1$. We can check now
directly that $X$ does not diverge from its solution, computed with $\xi=0$.
One obtains:

$X=-4\;,$ (18)

i.e. a $U$ growing indefinitely negative, say:

$U=C_{1}-4N\;.$ (19)

With this solution, we are left with two equations:

$\displaystyle Y+3\tilde{V}=-\frac{m^{2}}{2H^{2}}U\;,$ (20) $\displaystyle
Y^{\prime}+3Y=-\frac{m^{2}}{H^{2}}\;.$ (21)

The derivative of the left hand side of the first equation is equal to the
left hand side of the second equation. Hence, one finds that:

$\xi=-\frac{3}{C_{1}-4N}=-\frac{3}{U}\;,$ (22)

and we have the solution for $H$:

$H=C_{2}|C_{1}-4N|^{3/4}=3|U|^{3/4}\;.$ (23)

The solution for $H^{\prime}$ is instead:

$H^{\prime}=-\frac{3C_{2}}{C_{1}-4N}|C_{1}-4N|^{3/4}\;.$ (24)

####  $\Box^{-2}R$ model

This model was proposed in Ref. [144] and is motivated from studies of
nonperturbative lattice quantum gravity. The Lagrangian is:

$\mathcal{L}=\mathcal{L}_{EH}-\frac{M^{4}}{6}\frac{1}{\Box^{2}}R\;,$ (25)

and is localized introducing the auxiliary fields:

$\displaystyle\Box U=-R\;,$ (26) $\displaystyle\Box S=-U\;,$ (27)
$\displaystyle\Box Q=-1\;,$ (28) $\displaystyle\Box L=-Q\;.$ (29)

The field equations are:

$\displaystyle
h^{2}=\frac{\gamma}{4}\left[V+WU+h^{2}\left(6Z+6Z^{\prime}-U^{\prime}Z^{\prime}-V^{\prime}W^{\prime}\right)\right]+\Omega_{R}^{0}e^{-4N}+\Omega_{M}^{0}e^{-3N}\;,$
(30) $\displaystyle\xi=\frac{1}{2\left(1-\frac{3}{2}\gamma
Z\right)}\left[\frac{-4\Omega_{R}^{0}e^{-4N}-3\Omega_{M}^{0}e^{-3N}}{h^{2}}+\frac{3}{2}\gamma\left(\frac{W}{h^{2}}-4Z^{\prime}+U^{\prime}Z^{\prime}+V^{\prime}W^{\prime}\right)\right]\;,$
(31) $\displaystyle
U^{\prime\prime}+\left(3+\xi\right)U=6\left(2+\xi\right)\;,$ (32)
$\displaystyle
V^{\prime\prime}+\left(3+\xi\right)V^{\prime}=\frac{U}{h^{2}}\;,$ (33)
$\displaystyle
W^{\prime\prime}+\left(3+\xi\right)W^{\prime}=\frac{1}{h^{2}}\;,$ (34)
$\displaystyle
Z^{\prime\prime}+\left(3+\xi\right)Z^{\prime}=\frac{W}{h^{2}}\;.$ (35)

defining $\tilde{Z}=1-\frac{3}{2}\gamma Z$ we can rewrite the late-times
Friedmann equations (30) (31), when matter is completely diluted, as:

$\displaystyle\tilde{Z}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{4h^{2}}\left(UW+V\right)-\tilde{Z}^{\prime}\left(1-\frac{U^{\prime}}{6}\right)-\frac{\gamma
W^{\prime}V^{\prime}}{4}\;,$ (36) $\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{1}{2\tilde{Z}}\left[\frac{3\gamma
W}{2h^{2}}+4\tilde{Z}^{\prime}-\tilde{Z}^{\prime}U^{\prime}+\frac{3\gamma
V^{\prime}W^{\prime}}{2}\right]\;,$ (37)

while the KG equation (35) for $\tilde{Z}$ is:

$\tilde{Z}^{\prime\prime}+\left(3+\xi\right)\tilde{Z}^{\prime}=-\frac{3\gamma
W}{2h^{2}}\;,$ (38)

whose formal solution for $\tilde{Z}^{\prime}$ is given by:

$\tilde{Z}^{\prime}=-3\gamma
e^{-F(N)}\int_{N_{i}}^{N}d\bar{N}e^{F(\bar{N})}\frac{W}{2h^{2}}\;.$ (39)

We write for convenience also the formal solutions for $V^{\prime}$ and
$W^{\prime}$:

$\displaystyle
V^{\prime}=e^{-F(N)}\int_{N_{i}}^{N}d\bar{N}e^{F(\bar{N})}\frac{U}{h^{2}}\;,$
(40) $\displaystyle
W^{\prime}=e^{-F(N)}\int_{N_{i}}^{N}d\bar{N}e^{F(\bar{N})}\frac{1}{h^{2}}\;.$
(41)

Since we chose initial conditions $W_{i}=0$ and since $W^{\prime}>0$ we can
conclude that $W>0$. This implies that $\tilde{Z}^{\prime}<0$; on the other
hand Eqs. (5) imply $V^{\prime}>0$, $0<U^{\prime}<6$ and $U>0$. In order to
understand the behavior of $\tilde{Z}$ let us define the function $T$:

$T\equiv\frac{UW+V}{h^{2}}-W^{\prime}V^{\prime}\;,$ (42)

taking its time derivative and using Eqs. (34) and (33) we are able to set up
a differential equation for $T$:

$T^{\prime}+2\xi X=\frac{U^{\prime}W}{h^{2}}+6V^{\prime}W^{\prime}\;.$ (43)

The formal solution of Eq. (43) is given by:

$T\left(N\right)=\frac{1}{h^{2}(N)}\int_{N_{i}}^{N}d\bar{N}\left(U^{\prime}W+6h^{2}V^{\prime}W^{\prime}\right)-C_{X}h^{2}(N)\;,$
(44)

where $C_{X}$ is an integration constant. Since $X(N_{i})=0$ we can conclude
from Eq. (44) that $T(N)>0$. This in turns implies that the right hand side of
Eq. (36) it’s always positive, and we can conclude that, asymptotically,
$\tilde{Z}>0$.

Since $\tilde{Z}$ is positive definite, we must have asymptotically
$\tilde{Z}^{\prime}\rightarrow 0$. It is straightforward to realize from Eq.
(39) that this is possible only if $h^{2}$ is a monotonic growing function
that grows faster than $W/2$. On the other hand, $W$ is also a monotonic
growing function since $W^{\prime}>0$. In particular, since $h^{2}$ grows
faster than $W$, it also grows faster then a constant, and so we conclude from
Eq. (41) that $W^{\prime}\rightarrow 0$. Using the latter in Eq. (36) we are
left with:

$\tilde{Z}\sim\frac{\gamma}{4h^{2}}\left(V+WU\right)\;.$ (45)

Using the above result in Eq. (37) we finally obtain:

$\xi\sim\frac{3}{\frac{V}{W}+U}\sim 0\;,$ (46)

then once again we have $w_{eff}\rightarrow-1$.

##  Chapter 10 Field equations in Inverse Ricci gravity

To begin with let us consider the following action:

Consider first the basic Action

$S=\int\sqrt{-g}d^{4}x(R+\alpha A)\;,$ (1)

where the anticurvature scalar $A$ is the trace of $A^{\mu\nu}$

$A^{\mu\nu}=R_{\mu\nu}^{-1}\;.$ (2)

By differentiating Eq. (1), we see that

$\delta A^{\mu\tau}=-A^{\mu\nu}(\delta R_{\nu\sigma})A{}^{\sigma\tau}\;.$ (3)

We have then

$\displaystyle\delta S$ $\displaystyle=\int
d^{4}x(A\delta\sqrt{-g}+\sqrt{-g}A^{\mu\nu}\delta
g_{\mu\nu}+\sqrt{-g}g_{\mu\nu}\delta A^{\mu\nu})$ $\displaystyle=\int
d^{4}x\sqrt{-g}(\frac{1}{2}Ag^{\mu\nu}\delta g_{\mu\nu}+A^{\mu\nu}\delta
g_{\mu\nu}+g_{\mu\nu}\delta A^{\mu\nu})\;,$ (4)

and since

$\delta
R_{\alpha\beta}=\nabla_{\rho}\delta\Gamma_{\beta\alpha}^{\rho}-\nabla_{\beta}\delta\Gamma_{\rho\alpha}^{\rho}\;,$
(5)

we obtain

$\displaystyle\delta A^{\mu\nu}$
$\displaystyle=-A^{\mu\alpha}(\nabla_{\rho}\delta\Gamma_{\beta\alpha}^{\rho}-\nabla_{\beta}\delta\Gamma_{\rho\alpha}^{\rho})A^{\beta\nu}$
$\displaystyle=-\frac{1}{2}A^{\mu\alpha}(g^{\rho\lambda}\nabla_{\rho}(\nabla_{\alpha}\delta
g_{\beta\lambda}+\nabla_{\beta}\delta g_{\lambda\alpha}-\nabla_{\lambda}\delta
g_{\alpha\beta})-g^{\rho\lambda}\nabla_{\beta}(\nabla_{\alpha}\delta
g_{\rho\lambda}+\nabla_{\rho}\delta g_{\lambda\alpha}-\nabla_{\lambda}\delta
g_{\alpha\rho}))A^{\beta\nu}$
$\displaystyle=-\frac{1}{2}A^{\mu\alpha}g^{\rho\lambda}(\nabla_{\rho}\nabla_{\alpha}\delta
g_{\beta\lambda}-\nabla_{\rho}\nabla_{\lambda}\delta
g_{\alpha\beta}-\nabla_{\beta}\nabla_{\alpha}\delta
g_{\rho\lambda}+\nabla_{\beta}\nabla_{\lambda}\delta
g_{\alpha\rho}+\left[\nabla_{\beta},\nabla_{\rho}\right]\delta
g_{\lambda\alpha})A^{\beta\nu}\;.$ (6)

Using integration by parts, this becomes

$\displaystyle g_{\mu\nu}\delta A^{\mu\nu}$
$\displaystyle=-\frac{1}{2}g_{\mu\nu}g^{\rho\lambda}(\delta
g_{\beta\lambda}\nabla_{\alpha}\nabla_{\rho}(A^{\mu\alpha}A^{\beta\nu})-\delta
g_{\alpha\beta}\nabla_{\lambda}\nabla_{\rho}(A^{\mu\alpha}A^{\beta\nu})-\delta
g_{\rho\lambda}\nabla_{\alpha}\nabla_{\beta}(A^{\mu\alpha}A^{\beta\nu})+$
$\displaystyle+\delta
g_{\alpha\rho}\nabla_{\lambda}\nabla_{\beta}(A^{\mu\alpha}A^{\beta\nu}))+\delta
g_{\alpha\rho}\nabla_{\lambda}\nabla_{\beta}(A^{\mu\alpha}A^{\beta\nu}))$
$\displaystyle=\frac{1}{2}\delta
g_{\iota\kappa}(-2g^{\rho\iota}\nabla_{\alpha}\nabla_{\rho}A^{\mu\alpha}A_{\mu}^{\kappa}+\nabla^{2}(A^{\mu\iota}A_{\mu}^{\kappa})+g^{\iota\kappa}\nabla_{\alpha}\nabla_{\beta}(A^{\mu\alpha}A_{\mu}^{\beta}))\;.$
(7)

So finally the variation is

$\displaystyle\delta
g_{\mu\nu}(\frac{1}{2}Ag^{\mu\nu}+A^{\mu\nu}+\frac{1}{2}(-2g^{\rho\mu}\nabla_{\alpha}\nabla_{\rho}A^{\sigma\alpha}A_{\sigma}^{\nu}+\nabla^{2}(A^{\sigma\mu}A_{\sigma}^{\nu})+g^{\mu\nu}\nabla_{\alpha}\nabla_{\beta}(A^{\sigma\alpha}A_{\sigma}^{\beta})))\;.$
(8)

Together with the variation of the standard Hilbert-Einstein Lagrangian

$\displaystyle\delta g^{\mu\nu}(-\frac{1}{2}Rg_{\mu\nu}+R_{\mu\nu})$
$\displaystyle=-\delta g_{\mu\nu}(-\frac{1}{2}Rg^{\mu\nu}+R^{\mu\nu})\;,$ (9)

we obtain finally the equations for the Action (1)

$\displaystyle R^{\mu\nu}-\frac{1}{2}Rg^{\mu\nu}-\alpha
A^{\mu\nu}-\frac{1}{2}\alpha
Ag^{\mu\nu}+\frac{\alpha}{2}\left(2g^{\rho\mu}\nabla_{\alpha}\nabla_{\rho}A_{\sigma}^{\alpha}A^{\nu\sigma}-\nabla^{2}A_{\sigma}^{\mu}A^{\nu\sigma}-g^{\mu\nu}\nabla_{\alpha}\nabla_{\rho}A_{\sigma}^{\alpha}A^{\rho\sigma}\right)$
$\displaystyle=T^{\mu\nu}\;,$ (10)

where we used the fact that
$A^{\alpha}_{\sigma}A^{\nu\sigma}=A^{\alpha\tau}g_{\tau\sigma}A^{\sigma\nu}=A^{\alpha\tau}A_{\tau}^{\nu}=A^{\alpha\sigma}A_{\sigma}^{\nu}=A_{\sigma}^{\nu}A^{\alpha\sigma}$
and we employed units in which $8\pi G=1$. It can be show that the left-hand
side of Eq. (10) is divergenceless, as it should be in order to satisfy the
Bianchi identities.

The extension to any Lagrangian $f(R,A)$ is quite straightforward:

$\delta S=\int d^{4}x\sqrt{-g}(-\frac{1}{2}f(R,A)g_{\mu\nu}\delta
g^{\mu\nu}+f_{A}A^{\mu\nu}\delta g_{\mu\nu}+f_{A}g_{\mu\nu}\delta
A^{\mu\nu}+f_{R}R_{\mu\nu}\delta g^{\mu\nu}+f_{R}g^{\mu\nu}\delta
R_{\mu\nu})\;,$ (11)

where $f_{R}=\partial f/\partial R$ and $f_{A}=\partial f/\partial A$. Then we
have

$\displaystyle f_{R}R^{\mu\nu}-f_{A}A^{\mu\nu}$
$\displaystyle-\frac{1}{2}fg^{\mu\nu}+g^{\rho\mu}\nabla_{\alpha}\nabla_{\rho}f_{A}A^{\alpha}_{\sigma}A^{\nu\sigma}-\frac{1}{2}\nabla^{2}(f_{A}A_{\sigma}^{\mu}A^{\nu\sigma})$
$\displaystyle-\frac{1}{2}g^{\mu\nu}\nabla_{\alpha}\nabla_{\beta}(f_{A}A_{\sigma}^{\alpha}A^{\beta\sigma})-\nabla^{\mu}\nabla^{\nu}f_{R}+g^{\mu\nu}\nabla^{2}f_{R}=T^{\mu\nu}\;.$
(12)

##  Chapter 11 Estimating Time delay Uncertainties with PyCS3

In this appendix, we display the time delays and their uncertainties obtained
with PyCS3. Uncertainties are obtained by simulating light curves close to the
data, and randomizing the true time delay applied to each curve, see Sec.
(3.2) of Ref. [181] for more details. The final marginalization is done
performing a hybrid approach between the “free-knot spline” and the
“regression difference” estimators as explained in Sec. (3.3) of Ref. [181].
The parameter $\tau_{thresh}=0$ indicates the marginalization is done over the
two estimators. In each figure, the top panels show the final time delay
estimates marginalizing over the two estimators. The middle figure shows the
residuals for the spline fit to the data. The top row of the bottom panels
show the distribution of data residuals for mock curves (in gray) and data (in
colors), whereas the bottom panels show their normalization over the number of
runs $z_{r}$. The time delay estimates for the simulated curves over the whole
period of observations ($\approx 1.316$ days) is shown in Fig. 1 while Fig. 2
is over half the total period. Similarly, Fig. 3 ($\approx 189$ days) and Fig.
4 show time delays for the object DES J0408-5354.

Figure 1: Time delay estimates for the simulated quasar over the full
observation period.

Figure 2: Time delay estimates for the simulated quasar over half of the
observation period.

Figure 3: Time delay estimates for the quasar DES J04078-5354 over the full
observation period.

Figure 4: Time delay estimates for the quasar DES J04078-5354 over half of the
observation period.

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