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Exact solutions and constraints on the dark energy model in FRW Universe

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Abstract

The inflationary epoch and the late time acceleration of the expansion rate of universe can be explained by assuming a gravitationally coupled scalar field. In this article, we propose a new method of finding exact solutions in the background of flat Friedmann–Robertson–Walker (FRW) cosmological models by considering both scalar field and matter where the scalar field potential is a function of the scale factor. Our method provides analytical expressions for equation of state parameter of scalar field, deceleration parameter and Hubble parameter. This method can be applied to various other forms of scalar field potential, to the early radiation dominated epoch and very early scalar field dominated inflationary dynamics. Since the method produces exact analytical expression for H(a) (i.e., H(z) as well), we then constrain the model with currents data sets, which includes-Baryon Acoustic Oscillations, Hubble parameter data and Type 1a Supernova data (Pantheon Dataset). As an extension of the method, we also consider the inverse problem of reconstructing scalar field potential energy by assuming any general analytical expression of scalar field equation of state parameter as a function of scale factor.

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Notes

  1. In this article, we use super or subscript 0 to represent the values of the dynamical variables at present time \(t_0\) with the convention \(a_0 = 1\), so that present day redshift \(z_0 = 0\).

  2. We note that with the aforementioned normalization choice both V and \({{\dot{\phi}^2}}\) henceforth become dimensionless variables.

  3. Solutions corresponding to \(H>0\) correspond to expanding cosmological models and \(H<0\) corresponds to collapsing models.

  4. Apart from the usual argument that the classical Friedmann equation must only be valid up to some initial scale factor well above the Planck length scale, the problem of divergence of \({{\dot{\phi}^2}}\) can also be bypassed if we assume that the scalar field theory valid up to some initial scale factor \(a_i\) corresponding to \({{\dot{\phi}^2}}(a_i) = \dot\phi^2_i\), a finite value.

  5. \(H>0\) correspond to expanding cosmological models and \(H<0\) corresponds to collapsing models.

  6. In this case, we have assumed that \(\omega _\phi (a) \ne 1\) or equivalently, \(V(a) \ne 0\) for the domain of interest of a. The solution V(a) = 0 when \(\omega _\phi (a) = 1\) can be easily obtained from Equation (13).

  7. In fact, the CPL equation of state parameterized by, \( \omega _\phi (z) = \omega _a + \omega _b({z}/({1+z}))\), where z denotes the redshift. Using \(1/a = 1+z\) and a calculation with some redefinition of variables, one can show this is equivalent to Equation (37).

  8. The further incorporation of the early inflationary era will be followed in a future article.

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Acknowledgements

The authors acknowledges financial support from Ministry of Human Resource and Development, Government of India via Institute fellowship at IISER Bhopal.

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, 462 066, India

    Albin Joseph & Rajib Saha

Authors
  1. Albin Joseph
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  2. Rajib Saha
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Correspondence to Rajib Saha.

Appendix

Appendix

To derive Equation (34) using Equation (32) we first obtain

$$\begin{aligned} \frac{d{{\dot{\phi}^2}}}{da}&= 2\left[ \frac{dV}{da} \left( \frac{1+\omega _\phi }{1-\omega _\phi }\right) \right. \\&\quad \left. +\,V\left\{ \frac{1}{1-\omega _\phi }+\frac{1+\omega _\phi }{(1-\omega _\phi )^2}\right\} \frac{d\omega _\phi }{da}\right] , \end{aligned}$$
(54)

which can be simplified as

$$ \frac{d{{\dot{\phi}^2}}}{da} = \frac{2V}{(1-\omega _\phi )^2}\frac{d\omega _\phi }{da} + 2\frac{dV}{da} \left( \frac{1+\omega _\phi }{1-\omega _\phi }\right) . $$
(55)

Using Equation (55) in Equation (9) and rearranging terms we obtain

$$\begin{aligned}&\frac{dV}{da}\left[ 1 + \frac{1+\omega _\phi }{1-\omega _\phi }\right] \\&\quad + V\left[ \frac{6}{a}\frac{(1+\omega _\phi )}{(1-\omega _\phi )} + \frac{1}{( 1 - \omega _\phi )^2}\frac{d\omega _\phi }{da} \right] = 0, \end{aligned}$$
(56)

which can be easily simplified into the form of Equation (34).

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Joseph, A., Saha, R. Exact solutions and constraints on the dark energy model in FRW Universe. J Astrophys Astron 42, 111 (2021). https://doi.org/10.1007/s12036-021-09776-6

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  • DOI: https://doi.org/10.1007/s12036-021-09776-6

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