Erf_Swampland.nb

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Rescaled f(R) Canonical Scalar Field Inflation In The Presence Of A \
Gau\[Beta]-Bonnet Term. \
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1. Designation of scalar coupling function-Solution of continuity equation\
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We start by designating the scalar functions of the model. For simplicity, \
the Gau\[Beta]-Bonnet scalar coupling function is assume to be the error \
function such that the ratio of the first two derivatives of the scalar \
functions, which plays an important role in the constrained Gau\[Beta]-Bonnet \
model as shown in arXiv:2003.13724, is simplified greatly. Now the form of \
the scalar potential is extracted from the continuity equation of the scalar \
field by treating it as a first order differential equation with respect to \
the scalar potential. The equations of motion are extracted by assuming the \
slow-roll condition is valid. Slow-roll indices along with observed indices \
are taken from arXiv:0412126.\
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Every parameter that ends with 0 or 1 is an auxiliary free parameter that \
needs to be designated later or is used as an intermediate state. With \[Xi] \
and V the Gau\[Beta]-Bonnet scalar coupling and the scalar potential are \
denoted while H is Hubble's parameter and H2 the squared Hubble. Each \
parameter ending with dot implies time derivative for simplicity and E0, Qa, \
Qe, Qt are auxiliary function that depend solely on  the scalar field. \
Finally, indices \[CurlyEpsilon]1 through \[CurlyEpsilon]4 are the slow-roll \
indices of the model and ns, nt, r indicate the observed indices.\
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The proper functions depending on the potential are specified here\
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We know specify the auxiliary parameters that are needed in order to simplify \
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Here, the observables which shall be computed during the first horizon \
crossing are specified. The sound wave velocity is included not only due to \
the tensor to scalar ratio but to also analyse whether the model is free of \
ghost instabilities due to the Gau\[Beta]-Bonnet scalar coupling functions \
which is assumed to be dynamical and not constant. \
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To study the inflationary era, one needs to insert as an input the value of \
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The numerical value of the slow-roll indices in general suffices in order to \
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