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Reconstruction and exact solutions for cosmological perturbations from a generalized gravity theory

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Abstract

Scalar and tensor cosmological perturbations during an inflationary universe scenario in the context of the a generalized gravity theory are studied. This analyze is carried out considering an ansatz on the variables associated to scalar and tensor perturbation (\(z_s\) and \(z_t\)) in the Jordan frame. In this context, we analyze two different Ansatze for the ratio \(z_s/z_t\), and we study in great detail the analytical and exact solutions for the cosmological perturbations together with the corresponding reconstruction of the background variables. Recent observational data from the Planck 2018 results are employed to constrain the parameters of each of the models.

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Notes

  1. Here the caret means quantity measured in Einstein frame

  2. Indeed, starting from Eqs. (11) and (17) and one finds [69] \(\frac{z_{s,t}^{\prime \prime }}{z_{s,t}}={\mathcal {H}}^{2}\lbrace (1+\delta _{s,t})(2-\epsilon +\delta _{s,t})+\frac{\delta _{s,t}^{\prime }}{{\mathcal {H}}})\rbrace \), where \(\delta _{s,t} =\frac{{\dot{Q}}_{s}}{2{\mathcal {H}} Q}\equiv \delta _{s}\) for scalar perturbations and \(\delta _{s,t}= \frac{{\dot{F}}}{2{\mathcal {H}} F}\equiv \delta _{t}\) for tensor one. In the slow roll approximation, \(\epsilon \) and \(\delta _{s,t}\) are supposed to be constant during inflation stage, hence \(\frac{z_{s,t}^{\prime \prime }}{z_{s,t}}=\frac{\gamma _{s,t}}{\eta ^{2}}\) with the constant \(\gamma _{s,t} =\frac{(1+\delta _{s,t})(2-\epsilon +\delta _{s,t})}{(1-\epsilon )^{2}}\). In contrast, from Eqs. (11), (21) and (22), we have \(Q_{s}\propto F\), hence \(\delta _{s,t}= \frac{q}{{\mathcal {H}}\eta }-1\) and, as usual, \(\epsilon =1-\frac{{\mathcal {H}}^{\prime }}{{\mathcal {H}}^2}\). Thus, we have \(\frac{z_{s,t}^{\prime \prime }}{z_{s,t}}=\frac{q(q-1)}{\eta ^2}\). As we can see, there is no need to assume slow roll approximation.

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Acknowledgements

J.A.B. was supported by COMISION NACIONAL DE CIENCIAS Y TECNOLOGIA through FONDECYT Grant N\(_0\) 11170083. C.G. was supported by Proyecto DIUDA REGULAR N\(_{0}\) 22347. R.H. was supported by Proyecto VRIEA-PUCV N\(_{0}\) 039.309/2018.

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  1. Departamento de Matemáticas, Facultad de Ingeniería, Universidad de Atacama, Avenida Copayapu 485, Copiapó, Chile

    José Antonio Belinchón

  2. Departamento de Física, Universidad de Atacama, Avenida Copayapu 485, Copiapó, Chile

    Carlos González

  3. Instituto de Física, Pontificia Universidad Católica de Valparaíso, Avenida Brasil 2950, Casilla 4059, Valparaiso, Chile

    Ramón Herrera

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Correspondence to Carlos González.

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Belinchón, J.A., González, C. & Herrera, R. Reconstruction and exact solutions for cosmological perturbations from a generalized gravity theory. Gen Relativ Gravit 52, 35 (2020). https://doi.org/10.1007/s10714-020-02685-z

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