I am looking for decent books or articles covering Lovelock and Horndeski second order extensions to the Einstein field equations as I have a need to understand how and why its employed in line intensity mappings for a calibration project I am on. So far I've isolated few of the terms but have encountered others I still need to isolate for its functionality.

I am currently studying the following but could use other recommended literature

https://www.ugr.es/~bjanssen/text/Tesis-JoseAlbertoOrejuela.pdf

What I'm trying to resolve is how it fits into the line power spectrum

\[P_{Cluster}(k,z)=b^2(z)I^2(z)P_m(k,z)\]

where I is the mean intensity.

shot noise given by

\[P_{obs}(k,z)=P_{clust}(k,z)+p_{shot}(z)+P_N\]

with Fourier mode at scale K

\[N_m(k)=\frac{k^2\Delta kV_s}{4\pi^2}\]

and variance

\[\sigma^2(k,z)=\frac{P^2_{obs}(k,z)}{n_m k}\]

with anistropic matter power spectrum

linear plane parallel approximation

\[P_{obs}(k,\mu,z)=[b,(z)^2 I(z)^2+f(z)^2I(z)^2\mu^2]^2P_m(k)\]

the above being a more generalized format than the one being employed but its far easier to relate to.

Edited 17 hours ago17 hr by Mordred

so far I have determined that it ties into the monopole and quadrupole moments via Legendre Polynomials

\[P_I(k)\frac{2l+1}{2}\int_{-1}^1\mathcal{L}_I{\mu}P(k,\mu)d\mu\]

\[P_o(k)(1+2/3\beta(bl)^2+1/5(bl^2\beta^2)P_m(k)\]

\[P_2(k)=(4/3(bl)^2\beta+4/7(bI)^2\beta^2)P_m(k)\]

Edited 16 hours ago16 hr by Mordred

I will have to wait till Monday to confirm but it looks like the Horndeskii relations were being employed to pull time elapsed data from the spectrographic filtration software and have nothing to do with the software calibration itself.