f(R)_Thermal_Matter_Era.nb

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Phase Space of f(R) Gravity In The Presence Of Thermal Effects.\
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In the following, functions f denotes the differential equations that we are \
interested in. For fi, subscript i which runs from 1 to 4 suggests the \
differential equation for variable xi, for instance f1=dx1/dN with N being \
the e-folding number. Parameter constraint marks the Friedmann constraint, or \
in other words the temporal component of the field equations that must also \
be satisfied by the fixed points of the phase space, thus if a fixed point \
does not satisfy the relation Constraint=1, it is discarded. Upon finding the \
fixed points, parameter m is designated a value so as to make the system \
truly autonomous. For details of the nature of m, see arXiv 2009.12897. There \
exist 4 total values of interest for m, namely m=0 which marks (quasi) \
de-Sitter expansion, m=-9/2 which is used in this code and describes the \
matter dominated era, m=-8 which describes the radiation dominated era and \
finally m=-9 which refers to stiff matter. For each fixed point separately, \
the eigenvalues of the Jacobian are extracted in order to ascertain the type \
of stability of the respective fixed point. Afterwards, we introduce \
dependence of the dynamical variables on the e-folding number N, here denotes \
as t for simplicity, and by solving the corresponding nonlinear system of \
differential equations numerically in a given area, we plot the trajectories \
on certain subspaces of the phase space.\
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This code was devised in order to study the behavior of an f(R) gravity \
assuming that a thermal term is present in Friedmann\[CloseCurlyQuote]s \
equation. References to such equation can be found in arXiv:2009.12897. There \
exists 6 variables in total however for simplicity we shall limit our work to \
only 4. So, as a first step, one must construct the system of nonlinear \
differential equations. The aforementioned problem is described by the \
following equations.\
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The above system has unknown for the time being fixed points, however some of \
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the Friedmann equation. In consequence, the constraint below suggests that \
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As a first step, the fixed points are identified. Naturally, a fixed point is \
derived as a root of the previous equations. For simplicity, we study the \
matter dominated era of the universe, which corresponds to a value of m equal \
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In order to ascertain whether a fixed point is stable or not, the \
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linearized system near a fixed point and provided that the eigenvalues of the \
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system is topologically equivalent to that of the nonlinear system. The \
difference between the two pictures resides in the actual trajectories of the \
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We now proceed with the phase space analysis. Random trajectories are chosen \
for the 4 variables. For this procedure, the system of differential equations \
is solved numerically for the choice of such initial conditions. Here, t is \
an auxiliary variable,  not to be confused with time, that described the \
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