I Do 4-divergences affect the eqs of motion for nth order perturbed fields?

[JuanC97]

Intuitively, I'd say that adding a 4-divergence to the Lagrangian should not affect the eqs of motion since the integral of that 4-divergence (of a vector that vanishes at ∞) can be rewritten as a surface term equal to zero, but...

In some theories, the addition of a term that is equal to zero (within a given background and field configuration) gives rise to no modifications in the 0-th order equations but the equations of motion for n-th order perturbed quantities (with n≥1) do change. e.g: A term like $$ \tilde{G}^{\mu\nu}_a A_\mu^a A^\rho_b S_{\nu\rho}^b \;\;\text{with flat FLRW and}\;\; (A_\mu^a=\phi\delta_\mu^a) $$ is equal to zero but it contributes to the eqs of motion of first order perturbed quantities [https://arxiv.org/pdf/1907.07961.pdf, eq1].

I notice, though, that in this example the 'zero' that is being added to the Lagrangian might not be zero within a different configuration, so, maybe that's the reason why the equations of motion are not modified at 0-th order but the 1st order ones are. I wonder if this would be the case if we added a 4-divergence instead but, I'm not sure. I have a vibe that variating and then imposing a configuration is not the same as imposing a configuration and then variating, thus, I don't know wether from the fact that (the integral of a 4-divergence is equal to zero) follows that its 2th order variations in the action lead to zero contributions to the 1st order eqs of motion or not. What do you think?. Any help is welcomed.

 

[JuanC97]

Ok guys, I've already answer my own question:
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Yeah, the equations of motion for higher order perturbed fields remain unchanged, I checked this using XAct for Mathematica. I suppose that the difference with the case of adding a background-dependent-zero is that in that case any perturbation from the background configuration leads to a non zero contribution to the equations of motion, but in the case of adding a divergence, the zero that one is adding is a background-independent-zero and that way its perturbations should not give any contributions.