TFPT scientific source excerpt
Original repository path: documents/tfpt_2_standard_model/sections/010-section-the-explicit-mass-gap-closes-dynamics-and-cluste.tex
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Inclusive original lines: 388-408
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000388 | fully closed at the level of the covariant Einstein-side equation. With
000389 | \[
000390 |   f(R)=R+\frac{R^2}{6M^2},\qquad f_R=1+\frac{R}{3M^2},\qquad
000391 |   \boxed{\,M=\Mbar\,\cthree^{7/2}=3.06\times10^{13}\ \mathrm{GeV}\,}
000392 | \]
000393 | (the scalaron mass, since $m_{\mathrm{scal}}^2=1/(3f_{RR})=M^2$), the field equation is the $f(R)$ system
000394 | \[
000395 |   f_R R_{\mu\nu}-\tfrac12 f\,g_{\mu\nu}+\bigl(g_{\mu\nu}\Box-\nabla_\mu\nabla_\nu\bigr)f_R
000396 |   =\Mbar^{-2}T_{\mu\nu}+O\!\left(\tfrac{R^3}{M^4}\right),
000397 | \]
000398 | with $4$D trace $f_R R-2f+3\Box f_R=\Mbar^{-2}T$. At the observational pivot $N_\star=57$ this gives
000399 | \[
000400 |   n_s=1-\tfrac2{N_\star}=0.96491,\quad r=\tfrac{12}{N_\star^2}=0.00369,\quad
000401 |   A_s=\tfrac{N_\star^2}{24\pi^2}\cthree^7=2.17\times10^{-9},
000402 | \]
000403 | inside the Planck window. \textbf{Typing:} the $R+R^2$ covariant equation is \emph{closed} \mE/\mC\ in its
000404 | regime; the field equation itself is parameter-free (\vref{v358}/\vref{v359}), so the deepest open item \mO\
000405 | is the \emph{measure} --- the \emph{full} metric-sector path-integral measure beyond FRW
000406 | ($G_{\mathrm{metric}}$, status source \texttt{tfpt\_4\_frontier}). \veri{v28\_gravity\_fR.py}
000407 | \end{keybox}
000408 |