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Legendre_Suppressed.nb
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Legendre_Suppressed.nb
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(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
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Notebook[{
Cell["Growth Index In F(R) Gravity.", "Title",
CellChangeTimes->{{3.825439480468724*^9, 3.8254395376088495`*^9},
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Cell["\<\
Enhancing/Dumping Gravitational Waves Using Viable \
F(R) Models Of Gravity\
\>", "Chapter",
CellChangeTimes->{{3.8254395534702497`*^9, 3.8254395776493025`*^9},
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Cell[TextData[{
"Hereafter, MP is the Planck mass, ms is a mass scale related to ",
Cell[BoxData[
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"\[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, \[Rho] is \
the sum of dust and radiation density, R0 is the real/proper Ricci scalar \
which is replaced in the f(R) function after calculations (for instance \
derivatives) have been carried, 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 solution 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. "
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Cell["\<\
In this notebook we are mainly interested in the effective equation for state \
EoS of the universe, denotes as weff[z]. Expected value is near 0, ideally it \
should be below 1/3 in order to be accurate and rises towards such value. \
Moreover we focus also on aMdz parameter, the sign of which suggests \
enhancement or dumping of primordial gravitational waves. The present model \
is capable of yielding interesting results as the effective EoS resides near \
0 for a while and afterwards starts increasing with a slow-rate converging on \
1/3. Inserting an auxiliary mass scale here denotes as R00 manages also to \
produce a finite but quite large ratio between the first 2 derivatives of the \
f(R). \
\>", "Text",
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\[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.
\
\>", "Text",
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Cell["1. Designation Of Model.", "Section",
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Suppose we have a similar f(R) model as in arXiv:1912.13128 and \
arXiv:2003.06671 such that early-late time is unified. In order to solve \
numerically the equations of motion in the area [-0.9,1000] 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 have second order derivatives for \
Hubble. Initial conditions for statefinder yH are taken from observations so \
they should not alter. In contrast, other parameters of the model, along with \
the initial conditions of the scalar field (since it is not observed) can be \
freely altered.\
\>", "Text",
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Apart from defining the f(R) 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 as such \
differential equation shall be solved numerically in the aforementioned area \
for z. \
\>", "Text",
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"Note that changing the mass scale MP results in the appearance of \
singularities. I can reach recombination (z=3000) by making use of the Planck \
scale however higher scales can reach even further z without problems. Denote \
",
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", minimum finite value of the ratio F\[CloseCurlyQuote]\[CloseCurlyQuote]/F\
\[CloseCurlyQuote]. Think of it as a mass scale."
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Note: If I add another constant/scale, say R+ mass^2 in the input of the \
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zero and no numerical difference is observed, but then the ratio is finite \
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if I insert (R+scale^2) both in the Legendre and in front! The largest scale \
viable I observed that results in the smallest finite ratio is 10^(-52). The \
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