Generalised Proca theories in teleparallel gravity

Abstract

Generalised Proca theories of gravity represent an interesting class of vector–tensor theories where only three propagating degrees of freedom are present. In this work, we propose a new teleparallel gravity analog to Proca theories where the generalised Proca framework is extended due to the lower-order nature of torsion-based gravity. We develop a new action contribution and explore the example of the Friedmann equations in this regime. We find that teleparallel Proca theories offer the possibility of a much larger class of models in which do have an impact on background cosmology.

Appendix: Teleparallel Proca scalars

In this appendix, we will expand all the generators from Table 1 in all their possible index configurations. We will denote the generator or groups of generators with brackets like in the example after Table 1, . Note that the following sets of scalars are the full list of possible independent scalars.

1.1 Torsion vector component $v_{\mu }$

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

(A10)

(A11)

(A12)

1.2 Torsion axial component $a_{\mu }$

(A13)

$$\begin{aligned}&\left\{ \epsilon aAFF,\epsilon aA\tilde{F}\tilde{F}\right\} \nonumber \\&I_{14}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \mu \nu \gamma }F{}^{\alpha }{}_{\mu }F{}_{\nu \gamma },\end{aligned}$$

(A14)

$$\begin{aligned}&I_{15}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \mu \nu \gamma }F{}^{\beta }{}_{\mu }F{}_{\nu \gamma }, \end{aligned}$$

(A15)

$$\begin{aligned}&I_{16}:=A_{\alpha }a{}^{\alpha }\epsilon {}^{\gamma \mu \nu \beta }F{}_{\gamma \mu }F{}_{\nu \beta }, \end{aligned}$$

(A16)

$$\begin{aligned}&\left\{ \epsilon aAFFF,\epsilon aA\tilde{F}\tilde{F}F\right\} \nonumber \\&I_{17}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \beta \mu \nu }F{}_{\gamma \rho }F{}^{\gamma }{}_{\mu }F{}^{\rho }{}_{\nu },\end{aligned}$$

(A17)

$$\begin{aligned}&I_{18}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\mu \rho \gamma \nu }F{}^{\alpha }{}_{\mu }F{}^{\beta }{}_{\rho }F{}_{\gamma \nu },\end{aligned}$$

(A18)

$$\begin{aligned}&I_{19}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \rho \gamma \nu }F{}^{\alpha }{}_{\mu }F{}_{\gamma \nu }F{}^{\mu }{}_{\rho },\end{aligned}$$

(A19)

$$\begin{aligned}&I_{20}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \rho \gamma \nu }F{}^{\beta }{}_{\mu }F{}_{\gamma \nu }F{}^{\mu }{}_{\rho },\end{aligned}$$

(A20)

$$\begin{aligned}&I_{21}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\nu \rho \gamma \mu }F{}^{\alpha \beta }F{}_{\gamma \mu }F{}_{\nu \rho }, \end{aligned}$$

(A21)

$$\begin{aligned}&I_{22}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \beta \nu \rho }F{}_{\nu \rho }F{}^{2}, \end{aligned}$$

(A22)

$$\begin{aligned}&\left\{ \epsilon aAFFFF,\epsilon aA\tilde{F}\tilde{F}\tilde{F}\tilde{F},\epsilon aA\tilde{F}\tilde{F}FF\right\} \nonumber \\&I_{23}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \mu \rho \sigma }F{}^{\alpha }{}_{\mu }F{}^{\gamma }{}_{\sigma }F{}_{\nu \gamma }F{}^{\nu }{}_{\rho }, \end{aligned}$$

(A23)

$$\begin{aligned}&I_{24}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \mu \rho \sigma }F{}^{\beta }{}_{\mu }F{}^{\gamma }{}_{\sigma }F{}_{\nu \gamma }F{}^{\nu }{}_{\rho }, \end{aligned}$$

(A24)

$$\begin{aligned}&I_{25}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\rho \gamma \nu \sigma }F{}^{\alpha }{}_{\mu }F{}^{\beta }{}_{\rho }F{}^{\mu }{}_{\gamma }F{}_{\nu \sigma },\end{aligned}$$

(A25)

$$\begin{aligned}&I_{26}:=A_{\alpha }a{}^{\alpha }\epsilon {}^{\mu \beta \nu \sigma }F{}_{\gamma \rho }F{}^{\gamma }{}_{\mu }F{}_{\nu \sigma }F{}^{\rho }{}_{\beta },\end{aligned}$$

(A26)

$$\begin{aligned}&I_{27}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\mu \gamma \nu \sigma }F{}^{\alpha }{}_{\mu }F{}^{\beta }{}_{\rho }F{}_{\nu \sigma }F{}^{\rho }{}_{\gamma },\end{aligned}$$

(A27)

$$\begin{aligned}&I_{28}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \gamma \nu \sigma }F{}^{\alpha }{}_{\mu }F{}^{\mu }{}_{\rho }F{}_{\nu \sigma }F{}^{\rho }{}_{\gamma }, \end{aligned}$$

(A28)

$$\begin{aligned}&I_{29}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \gamma \nu \sigma }F{}^{\beta }{}_{\mu }F{}^{\mu }{}_{\rho }F{}_{\nu \sigma }F{}^{\rho }{}_{\gamma },\end{aligned}$$

(A29)

$$\begin{aligned}&I_{30}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\sigma \gamma \nu \rho }F{}^{\alpha }{}_{\mu }F{}^{\beta \mu }F{}_{\nu \rho }F{}_{\sigma \gamma }, \end{aligned}$$

(A30)

$$\begin{aligned}&I_{31}:=A_{\alpha }a{}^{\alpha }\epsilon {}^{\nu \rho \sigma \beta }F{}_{\nu \rho }F{}_{\sigma \beta }F{}^{2}, \end{aligned}$$

(A31)

$$\begin{aligned}&I_{32}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\beta \rho \sigma \gamma }F{}^{\alpha }{}_{\rho }F{}_{\sigma \gamma }F{}^{2}, \end{aligned}$$

(A32)

$$\begin{aligned}&I_{33}:=A_{\alpha }a{}_{\beta }\epsilon {}^{\alpha \rho \sigma \gamma }F{}^{\beta }{}_{\rho }F{}_{\sigma \gamma }F{}^{2}, \end{aligned}$$

(A33)

1.3 Purely tensorial component $t_{\alpha \beta \gamma }$

(A34)

(A35)

(A36)

(A37)

$$\begin{aligned}&\left\{ tAFF,tA\widetilde{F}\widetilde{F}\right\} \nonumber \\&I_{38}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }t{}^{\alpha \mu \gamma }, \end{aligned}$$

(A38)

$$\begin{aligned}&I_{39}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \mu }t{}^{\beta \gamma \mu },\end{aligned}$$

(A39)

$$\begin{aligned}&I_{40}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }t{}^{\gamma \mu \alpha },\end{aligned}$$

(A40)

$$\begin{aligned}&\left\{ tAFFF,tA\widetilde{F}\widetilde{F}F\right\} \nonumber \\&I_{41}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }F{}^{\mu }{}_{\nu }t{}^{\alpha \gamma \nu },\end{aligned}$$

(A41)

$$\begin{aligned}&I_{42}:=A_{\alpha }F{}_{\beta \gamma }F{}^{2}t{}^{\alpha \beta \gamma }, \end{aligned}$$

(A42)

$$\begin{aligned}&I_{43}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \nu }F{}^{\gamma }{}_{\mu }t{}^{\beta \nu \mu },\end{aligned}$$

(A43)

$$\begin{aligned}&I_{44}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}^{\beta }{}_{\gamma }F{}_{\mu \nu }t{}^{\gamma \mu \nu }, \end{aligned}$$

(A44)

$$\begin{aligned}&I_{45}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \nu }F{}^{\gamma }{}_{\mu }t{}^{\mu \nu \beta },\end{aligned}$$

(A45)

$$\begin{aligned}&\left\{ tAFFFF,tA\widetilde{F}\widetilde{F}\widetilde{F}\widetilde{F},tA\widetilde{F}\widetilde{F}FF,tAFFFF\right\} \nonumber \\&I_{46}:=A_{\alpha }F{}_{\beta }{}^{\mu }F{}^{\beta }{}_{\gamma }F{}_{\mu \nu }F{}^{\nu }{}_{\rho }t{}^{\alpha \rho \gamma },\end{aligned}$$

(A46)

$$\begin{aligned}&I_{47}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }F{}^{2}t{}^{\alpha \mu \gamma },\end{aligned}$$

(A47)

$$\begin{aligned}&I_{48}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \nu }F{}^{\gamma }{}_{\mu }F{}^{\nu }{}_{\rho }t{}^{\beta \mu \rho },\end{aligned}$$

(A48)

$$\begin{aligned}&I_{49}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}_{\gamma \mu }F{}^{2}t{}^{\beta \gamma \mu }, \end{aligned}$$

(A49)

$$\begin{aligned}&I_{50}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}^{\beta }{}_{\gamma }F{}_{\mu \rho }F{}^{\mu }{}_{\nu }t{}^{\gamma \rho \nu },\end{aligned}$$

(A50)

$$\begin{aligned}&I_{51}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}^{\beta }{}_{\gamma }F{}^{\gamma }{}_{\mu }F{}_{\nu \rho }t{}^{\mu \nu \rho },\end{aligned}$$

(A51)

$$\begin{aligned}&I_{52}:=A_{\alpha }F{}_{\beta }{}^{\mu }F{}^{\beta }{}_{\gamma }F{}_{\mu \nu }F{}^{\nu }{}_{\rho }t{}^{\gamma \rho \alpha }, \end{aligned}$$

(A52)

$$\begin{aligned}&I_{53}:=A_{\alpha }F{}_{\beta \mu }F{}^{\beta }{}_{\gamma }F{}^{2}t{}^{\gamma \mu \alpha },\end{aligned}$$

(A53)

$$\begin{aligned}&I_{54}:=A_{\alpha }F{}^{\alpha }{}_{\beta }F{}^{\beta }{}_{\gamma }F{}_{\mu \rho }F{}^{\mu }{}_{\nu }t{}^{\nu \rho \gamma }. \end{aligned}$$

(A54)