Functors of Actions

Abstract

In this document, we introduce a novel formalism for any field theory and apply it to the effective field theories of large-scale structure. The new formalism is based on functors of actions composing those theories. This new formalism predicts the actionic fields. We discuss our findings in a cosmological gravitology framework. We present these results with a cosmological inference approach and give guidelines on how we can choose the best candidate between those models with some latest understanding of model selection using Bayesian inference.

Appendix A: The Higgs Matter Field

As [26, 31, 32] have shown, the matter fields have one main component, the Higgs field, which gives mass to the other fields of the standard model of particle physics. In particular the action for the Higgs field is the following:

$$\begin{aligned} S_\mathrm{m, Higgs} = c^4 \int d^4x \sqrt{\eta } {\mathcal {L}}_\mathrm{Higgs}\left[ \eta _{\mu \nu }, \overrightarrow{\phi }(\eta ),A_{\mu } \right] \end{aligned}$$

(A1)

where $\eta$ is the determinant of the Minkowski metric, $\eta _{\mu \nu }$, which is taken as, $-+++$, $\overrightarrow {\phi }(\eta )$ is a vector of real scalar fields and $A_{\mu }$ is a real vector field used for the interactions. The Lagrangian density is composed as:

$$\begin{aligned} {\mathcal {L}}_\mathrm{Higgs}\left[ \eta _{\mu \nu },\overrightarrow {\phi }(\eta ),A_{\mu } \right] = - \frac{1}{2} \left( \nabla \phi _1 \right) ^2 - \frac{1}{2} \left( \nabla \phi _2 \right) ^2 - V\left( \phi _1^2 + \phi _2^2 \right) - \frac{1}{4} F_{\mu \nu }F^{\mu \nu } \end{aligned}$$

(A2)

where $\phi _i\equiv \phi _i(\eta _{\mu \nu }),\ i = 1,2$ are the two real scalar fields which interact with the $A_{\mu }$ field and $\left( \nabla \phi _i \right) ^2 \equiv \nabla _{\mu }\phi _i \nabla ^{\mu } \phi _i \equiv \eta _{\mu \nu } \nabla ^{\mu }\phi _i \nabla ^{\nu } \phi _i$ . Note that this $\nabla ^{\mu }$ is different than the $\nabla ^{\mu }$ in Sect. 2.1. Here this $\nabla ^{\mu }$ is defined as:

$$\begin{aligned} \nabla _{\mu } \phi _1&= \partial _{\mu } \phi _1 - e A_{\mu } \phi _2 \end{aligned}$$

(A3)

$$\begin{aligned} \nabla _{\mu } \phi _2&= \partial _{\mu } \phi _2 - e A_{\mu } \phi _1 \end{aligned}$$

(A4)

$$\begin{aligned} F_{\mu \nu }&= \partial _{\mu }A_{\nu } - \partial _{\nu }A_{\mu } \end{aligned}$$

(A5)

where e is a dimensionless coupling constant. Note that ${\mathcal {L}}_\mathrm{Higgs}\left[ \eta _{\mu \nu },\overrightarrow{\mathbf{\phi}}(\eta ),A_{\mu } \right]$ is invariant under simultaneous gauge transformation of the first kind on $\phi _1\pm i \phi _2$ and of the second kind on $A_{\mu }$ . In the case where $V'(\phi _0^2) = 0$ and $V'' ( \phi ^2_0 ) > 0$, where $\phi _0$ is the ground state of either $\phi _i$, then spontaneous breakdown of U(1) symmetry occurs. Note that we present a generic description of the Higgs field, which can be applied to more specific interactions which constitute the standard model.