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Late_Time_Scalar_Tensor_Model.nb
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Notebook[{
Cell["Late-Time Phenomenology For A General Scalar-Tensor Model.", "Title",
CellChangeTimes->{{3.825439480468724*^9, 3.8254395376088495`*^9},
3.8254456165835333`*^9},ExpressionUUID->"d7ab949b-cfc1-4129-a464-\
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Cell[CellGroupData[{
Cell["\<\
Canonical Scalar Field Non-Minimally Coupled To Ricci Scalar And Gauss-Bonnet \
Topological Invariant.\
\>", "Chapter",
CellChangeTimes->{{3.8254395534702497`*^9, 3.8254395776493025`*^9},
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Hereafter, MP is the Planck mass, ms is a mass scale related to \[Kappa]^2\
\[Rho]0 with \[Rho]0 being the current density of dust, \[Chi] is the current \
density of radiation over dust, M is a different Mass scale, each \
function/parameter ending with \[OpenCurlyDoubleQuote]dot\
\[CloseCurlyDoubleQuote] implies derivative with respect to cosmic time t \
hence it is rewritten appropriately as a derivative of redshift z, X is the \
kinetic term of the scalar field while V is the scalar potential, \[Rho] is \
the sum of dust and radiation density, R0 is the real/proper Ricci scalar \
which is replaced in the f(R,\[CapitalPhi]) function after calculations (for \
instance derivatives) have been carried, similar for \[CapitalPhi] and H2 \
stands for Hubble squared, it is easy to remember. YH is the function that we \
find as solution to the equations of motion instead of Hubble however yH is \
the proper statefinder we focus on, which is assigned the function satisfying \
the equations of motion, similar for the scalar field. Moreover, functions \
with the number 1 are similarly functions of the proper yH and \[CurlyPhi] \
solutions and facilitate us into extracting results for the statefinders. \
Referring to statefinders, q is as usual the deceleration parameter, j is the \
jerk and s is the snap, all of them connected with derivates of Hubble, \
however must be written with respect to redshift. Om is an extra statefinder \
for which mainly the current value matters, as it must coincide or actually \
be quite close to the current value of the matter density parameter \
\[CapitalOmega]m. Finally, functions ending with DE are related to dark \
energy, meaning all the extra geometric terms appearing in the equations of \
motion. In particular, \[Omega]DE stands for the Equations of state (EoS) \
parameter whereas \[CapitalOmega]DE as before signifies the Dark Energy \
density parameter.
\[CapitalLambda]CDM Values: Roughly speaking, q near -0.535, j near unity, \
snap near 0, Om near 0.315, \[Omega]DE near -1 (does not have to be exactly \
-1, it can also vary infinitesimally as shown in this particular model) and \
\[CapitalOmega]DE near 0.685 such stat Om+\[CapitalOmega]DE near unity \
(Friedman constraint). More details+accurate values can be found in arXiv: \
1807.06209.
Fun fact, f(R) gravity is known for producing dark energy oscillations for \
large values of redshift, exactly as shown in section 3. Higher derivatives \
of Hubble (or for consistency yH) imply faster oscillations as shown in the \
jerk and snap. Adding a simple canonical kinetic term does not nullify them. \
The non-minimal coupling here was neglected along with the scalar potential \
and the Gauss-Bonnet scalar coupling function \[Xi](\[CurlyPhi]) for \
simplicity, the user however is capable of adding even more scalar terms, for \
instance a k-essence term or a noncanonical kinetic term (perhaps a \
Born-Infeld term). In particular, for this f(R) model, given that R^(1/100) \
becomes dominant as z approaches zero (since R->0 as well) the \
\[OpenCurlyDoubleQuote]singularity\[CloseCurlyDoubleQuote] which appears in \
the equations of motion is too difficult to overcome so a plethora of models, \
and perhaps with different free parameters, may yield the same results. As an \
example, see the following two arXiv papers.\
\>", "Text",
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Suppose we have a similar f(R) model as in arXiv:1912.13128 and \
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scalar field with an arbitrary and not necessarily quadratic scalar potential \
is also present. In order to solve numerically the equations of motion in the \
area [-0.9,10] for redshift, one needs to specify the initial conditions, \
here denoted with 0 in the end and D in front for derivatives, given that we \
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participating in the equations of motion. Initial conditions for statefinder \
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parameters of the model, along with the initial conditions of the scalar \
field (since it is not observed) can be freely altered. Pay attention to the \
dimensionality of each free parameter, in particular [f(R,\[CurlyPhi])]=ev^2, \
[\[CurlyPhi]]=ev, [yH]=dimensionless and so on.\
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1.2. Specifying The Scalar-Tensor Model And The Equations Of Motion.\
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Apart from defining the f(R,\[CurlyPhi]) function, it is also useful to \
rewrite time derivatives as derivatives with respect to redshift z. Also, \
instead of Hubble\[CloseCurlyQuote]s parameter, we focus on a new statefinder \
function, denoted as YH in this framework in order to facilitate our study by \
simplifying a bit the equations of motion which were rendered intricate due \
to the variable change a(t)->z. The main focus lies with input eq1 and eq2 as \
these differential equations shall be solved numerically in the \
aforementioned area for z. All functions are defined as g[x] instead of \
g[x_]: since in the latter case both the time response and the \
\[OpenCurlyDoubleQuote]accuracy\[CloseCurlyDoubleQuote] decreases, as the \
code will always summon the bare function and if it contains the solution of \
the equations of motion it will be unable to run properly, or for the same \
result one would need to add a lot more intermediate functions/steps. Either \
way, such definition works.\
\>", "Text",
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