# Exact and Slow-Roll Solutions for Exponential Power-Law Inflation Connected with Modified Gravity and Observational Constraints

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Exact and Approximate Solutions for EPL Inflation Based on Einstein Gravity

#### 2.1. The Exact and Approximate Solutions for Exponential Power-Law Inflation

#### 2.2. The Dynamics of the Universe’s Expansion

## 3. The Exact and Approximate Solutions for EPL Inflation from Conformal Connection with $\mathit{f}\left(\mathit{R}\right)$-Gravity

## 4. The Exponential Power-Law Inflation Based on Chiral Cosmological Models

## 5. The Exponential Power-Law Inflation in GR-Like Cosmological Models Based on Generalised Scalar-Tensor Gravity

## 6. The Correspondence to the Observational Constraints

## 7. EPL Inflation Based on Scalar-tensor Gravity with Quadratic Connection $\mathit{H}\propto \sqrt{\mathit{F}}$

#### Reconstruction of STG Parameters from Physical Potentials

- Higgs potential$$\begin{array}{ccc}& & {V}_{ph}\left(\varphi \right)={V}_{H}\left(\varphi \right)=\frac{{\lambda}_{H}}{4}{\left({\varphi}^{2}-{v}^{2}\right)}^{2},\hfill \end{array}$$$$\begin{array}{ccc}& & \varphi \left(t\right)=\pm {\left({v}^{2}+\frac{1}{{\beta}_{H}^{2}{t}^{2}}\right)}^{1/2},\hfill \end{array}$$$$\begin{array}{ccc}& & {\beta}_{H}={\left(\frac{{\lambda}_{H}}{4{C}_{4}}\right)}^{1/4},\hfill \end{array}$$$$\begin{array}{ccc}& & U\left(\varphi \right)=\frac{{\beta}_{H}^{2}}{{\varphi}^{2}}{\left({\varphi}^{2}-{v}^{2}\right)}^{3},\hfill \end{array}$$
- Higgs–Starobinsky potential$$\begin{array}{ccc}& & {V}_{ph}\left(\varphi \right)={V}_{0}{\left(1-{e}^{-\sqrt{\frac{2}{3}}\varphi}\right)}^{2},\hfill \end{array}$$$$\begin{array}{ccc}& & \varphi \left(t\right)=-\sqrt{6}ln\left|1-\frac{1}{{\beta}_{HS}^{2}{t}^{2}}\right|,\hfill \end{array}$$$$\begin{array}{ccc}& & {\beta}_{HS}={\left({V}_{0}/{C}_{4}\right)}^{1/4},\hfill \end{array}$$$$\begin{array}{ccc}& & U\left(\varphi \right)=6{\beta}_{HS}^{2}{e}^{-\sqrt{\frac{2}{3}}\varphi}{\left(1-{e}^{\sqrt{\frac{2}{3}}\varphi}\right)}^{3}.\hfill \end{array}$$
- Coleman–Weinberg potential$$\begin{array}{ccc}& & {V}_{ph}\left(\varphi \right)={V}_{CW}\left(\varphi \right)=\alpha {\varphi}^{4}\left(ln\left(\frac{\varphi}{{v}_{\varphi}}\right)-\frac{1}{4}\right)+\frac{\alpha}{4}{v}_{\varphi}^{4},\hfill \end{array}$$$$\begin{array}{ccc}& & \varphi \left(t\right)={v}_{\varphi}{\left[\frac{1-{\left({v}_{\varphi}{\beta}_{CW}t\right)}^{-4}}{W\left({e}^{-1}[1-{\left({v}_{\varphi}{\beta}_{CW}t\right)}^{-4}]\right)}\right]}^{1/4},\hfill \end{array}$$$$\begin{array}{ccc}& & {\beta}_{CW}={\left(\alpha /{C}_{4}\right)}^{1/4},\hfill \end{array}$$$$\begin{array}{ccc}& & U\left(\varphi \right)=\frac{{\beta}_{CW}^{2}{\left(\frac{4}{\alpha}{V}_{CW}\right)}^{5/2}}{{\left[W\left(\frac{4}{e\alpha}{V}_{CW}-\frac{{v}_{\varphi}^{4}}{e}\right)+1\right]}^{2}}{\left[\frac{W\left(\frac{4}{e\alpha}{V}_{CW}-\frac{{v}_{\varphi}^{4}}{e}\right)}{\frac{4}{\alpha}{V}_{CW}-{v}_{\varphi}^{4}}\right]}^{3/2},\hfill \end{array}$$

## 8. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Starobinsky, A.A. A New Type of Isotropic Cosmological Models Without Singularity. Phys. Lett. B
**1980**, 91, 99–102. [Google Scholar] [CrossRef] - Guth, A.H. The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Phys. Rev. D
**1981**, 23, 139–148. [Google Scholar] [CrossRef] [Green Version] - Linde, A. A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems. Phys. Lett. B
**1982**, 108, 389–393. [Google Scholar] [CrossRef] - Linde, A.D. Particle physics and inflationary cosmology. Contemp. Concepts Phys.
**1990**, 5, 1–362. [Google Scholar] - Zhuravlev, V.M.; Chervon, S.V.; Shchigolev, V.K. New classes of exact solutions in inflationary cosmology. J. Exp. Theor. Phys.
**1998**, 87, 223–228. [Google Scholar] [CrossRef] - Liddle, A.R.; Parsons, P.; Barrow, J.D. Formalizing the slow roll approximation in inflation. Phys. Rev. D
**1994**, 50, 7222–7232. [Google Scholar] [CrossRef] [PubMed] - Fomin, I.V. The models of cosmological inflation in the context of kinetic approximation. J. Phys. Conf. Ser.
**2016**, 731, 012004. [Google Scholar] [CrossRef] [Green Version] - Fomin, I.V. Cosmological inflation models in the kinetic approximation. Theor. Math. Phys.
**2017**, 191, 781–791. [Google Scholar] [CrossRef] - Ivanov, G.G. Friedmann cosmological models with a nonlinear scalar field. In Gravitation and Theory of Relativity; Kazan University Publishing House: Kazan, Russia, 1981; pp. 54–60. [Google Scholar]
- Salopek, D.S.; Bond, J.R. Stochastic inflation and nonlinear gravity. Phys. Rev. D
**1991**, 43, 1005–1031. [Google Scholar] [CrossRef] - Chervon, S.V.; Fomin, I.V.; Barrow, J.D. The method of generating functions in exact scalar field inflationary cosmology. Eur. Phys. J. C
**2018**, 4, 301. [Google Scholar] [CrossRef] [Green Version] - Fomin, I.V.; Chervon, S.V. Exact and Approximate Solutions in the Friedmann Cosmology. Russ. Phys. J.
**2017**, 60, 427–440. [Google Scholar] [CrossRef] - Fomin, I.V. Generalized Exact Solutions in the Friedmann Cosmology. Russ. Phys. J.
**2018**, 61, 843–851. [Google Scholar] [CrossRef] - Fomin, I.V.; Chervon, S.V.; Maharaj, S.D. A new look at the Schrödinger equation in exact scalar field cosmology. Int. J. Geom. Meth. Mod. Phys.
**2018**, 16, 1950022. [Google Scholar] [CrossRef] - Chervon, S.; Fomin, I.; Yurov, V.; Yurov, A. Scalar Field Cosmology; World Scientific: Singapore, 2019; p. 263. ISBN 9789811205071. [Google Scholar]
- Perlmutter, S.; Supernova Cosmology Project Collaboration. Measurements of Ω and Λ from 42 high redshift supernovae. Astrophys. J.
**1999**, 517, 565. [Google Scholar] [CrossRef] - Riess, A.G.; Supernova Search Team. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astrophys. J.
**1998**, 116, 1009. [Google Scholar] [CrossRef] [Green Version] - Sahni, V.; Starobinsky, A.A. The Case for a positive cosmological Lambda term. Int. J. Mod. Phys. D
**2000**, 9, 373. [Google Scholar] [CrossRef] - Peebles, P. The Cosmological Constant and Dark Energy. Ev. Mod. Phys.
**2003**, 75, 559. [Google Scholar] [CrossRef] [Green Version] - Padmanabhan, T. Cosmological constant: The Weight of the vacuum. Phys. Rept.
**2003**, 380, 235. [Google Scholar] [CrossRef] [Green Version] - Sahni, V.; Starobinsky, A.A. Reconstructing Dark Energy. Int. J. Mod. Phys. D
**2006**, 15, 2105. [Google Scholar] [CrossRef] - Rubakov, V.A. Relaxation of the cosmological constant at inflation? Phys. Rev. D
**2000**, 61, 061501. [Google Scholar] [CrossRef] [Green Version] - Steinhardt, P.J.; Turok, N. Why the cosmological constant is small and positive. Science
**2006**, 312, 1180. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tsujikawa, S. Quintessence: A Review. Class. Quant. Grav.
**2013**, 30, 1180. [Google Scholar] [CrossRef] [Green Version] - Durrive, J.B.; Ooba, J.; Ichiki, K.; Sugiyama, N. Updated observational constraints on quintessence dark energy models. Phys. Rev. D
**2018**, 97, 043503. [Google Scholar] [CrossRef] [Green Version] - Armendariz-Picon, C.; Mukhanov, V.F.; Steinhardt, P.J. Essentials of k essence. Phys. Rev. D
**2001**, 63, 103510. [Google Scholar] [CrossRef] - Chiba, T. Tracking K-essence. Phys. Rev. D
**2002**, 66, 063514. [Google Scholar] [CrossRef] - Sazhin, M.V.; Sazhina, O.S. The scale factor in a Universe with dark energy. Astron. Rep.
**2016**, 60, 425. [Google Scholar] [CrossRef] [Green Version] - Chervon, S.V. Chiral Cosmological Models: Dark Sector Fields Description. Quant. Matt.
**2013**, 2, 71. [Google Scholar] [CrossRef] - Chervon, S.V.; Fomin, I.V.; Pozdeeva, E.O.; Sami, M.; Vernov, S.Y. Superpotential method for chiral cosmological models connected with modified gravity. Phys. Rev. D
**2019**, 100, 063522. [Google Scholar] [CrossRef] [Green Version] - Abbyazov, R.R.; Chervon, S.V. Interaction of chiral fields of the dark sector with cold dark matter. Grav. Cosmol.
**2012**, 18, 262. [Google Scholar] [CrossRef] - Abbyazov, R.R.; Chervon, S.V.; Muller, V. σCDM coupled to radiation: Dark energy and Universe acceleration. Mod. Phys. Lett. A
**2015**, 30, 1550114. [Google Scholar] [CrossRef] [Green Version] - Fomin, I.V. The chiral cosmological models with two components. J. Phys. Conf. Ser.
**2017**, 918, 012009. [Google Scholar] [CrossRef] [Green Version] - Fomin, I.V. Two-Field Cosmological Models with a Second Accelerated Expansion of the Universe. Mosc. Univ. Phys. Bull.
**2019**, 73, 696. [Google Scholar] [CrossRef] - Starobinsky, A.A. Disappearing cosmological constant in f(R) gravity. JETP Lett.
**2015**, 86, 157. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Modified non-local-F(R) gravity as the key for the inflation and dark energy. Phys. Lett. B
**2008**, 659, 821. [Google Scholar] [CrossRef] [Green Version] - Clifton, T.; Ferreira, P.G.; Padilla, A.; Skordis, C. Modified Gravity and Cosmology. Phys. Rept.
**2012**, 513, 1. [Google Scholar] [CrossRef] [Green Version] - Rinaldi, M.; Cognola, G.; Vanzo, L.; Zerbini, S. Reconstructing the inflationary f(R) from observations. JCAP
**2014**, 1408, 015. [Google Scholar] [CrossRef] [Green Version] - Motohashi, H.; Starobinsky, A.A. f(R) constant-roll inflation. Eur. Phys. J. C
**2017**, 77, 538. [Google Scholar] [CrossRef] [Green Version] - Aldabergenov, Y.; Ishikawa, R.; Ketov, S.V.; Kruglov, S.I. Beyond Starobinsky inflation. Phys. Rev. D
**2018**, 98, 083511. [Google Scholar] [CrossRef] [Green Version] - Vernov, S.Y.; Ivanov, V.R.; Pozdeeva, E.O. Superpotential method for F(R) cosmological models. Phys. Part. Nucl.
**2020**, 51, 744–749. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration. Phys. Rev. D
**2003**, 68, 123512. [Google Scholar] [CrossRef] [Green Version] - Elizalde, E.; Nojiri, S.; Odintsov, S.D. Late-time cosmology in (phantom) scalar-tensor theory: Dark energy and the cosmic speed-up. Phys. Rev. D
**2004**, 70, 043539. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; De Laurentis, M.; Nojiri, S.; Odintsov, S.D. Evolution of gravitons in accelerating cosmologies: The case of extended gravity. Phys. Rev. D
**2017**, 95, 083524. [Google Scholar] [CrossRef] [Green Version] - Horndeski, G.W. Second-order scalar-tensor field equations in a four-dimensional space. Int. J. Theor. Phys.
**1974**, 10, 363. [Google Scholar] [CrossRef] - De Felice, A.; Tsujikawa, S. Inflationary non-Gaussianities in the most general second-order scalar-tensor theories. Phys. Rev. D
**2011**, 84, 083504. [Google Scholar] [CrossRef] [Green Version] - Starobinsky, A.A.; Sushkov, S.V.; Volkov, M.S. The screening Horndeski cosmologies. J. Cosmol. Astropart. Phys.
**2016**, 1606, 007. [Google Scholar] [CrossRef] - Fujii, Y.; Maeda, K. The Scalar-tensor Theory of Gravitation; Cambridge University Press: Cambridge, UK, 2003; p. 260. ISBN 9780521037525. [Google Scholar] [CrossRef]
- Faraoni, V. Cosmology in Scalar-Tensor Gravity. Fundam. Theor. Phys.
**2004**, 139, 267. [Google Scholar] [CrossRef] - DeFelice, A.; Tsujikawa, S.; Elliston, J.; Tavakol, R. Chaotic inflation in modified gravitational theories. J. Cosmol. Astropart. Phys.
**2011**, 1108, 021. [Google Scholar] [CrossRef] - DeFelice, A.; Tsujikawa, S. Conditions for the cosmological viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models. J. Cosmol. Astropart. Phys.
**2012**, 1202, 007. [Google Scholar] [CrossRef] - Fomin, I.; Chervon, S. Inflation with explicit parametric connection between general relativity and scalar-tensor gravity. Mod. Phys. Lett. A
**2018**, 33, 1850161. [Google Scholar] [CrossRef] [Green Version] - Kanti, P.; Rizos, J.; Tamvakis, K. Singularity free cosmological solutions in quadratic gravity. Phys. Rev. D
**1999**, 59, 083512. [Google Scholar] [CrossRef] [Green Version] - Guo, Z.K.; Schwarz, D.J. Slow-roll inflation with a Gauss-Bonnet correction. Phys. Rev. D
**2010**, 81, 123520. [Google Scholar] [CrossRef] [Green Version] - Van de Bruck, C.; Longden, C. Higgs Inflation with a Gauss-Bonnet term in the Jordan Frame. Phys. Rev. D
**2016**, 93, 063519. [Google Scholar] [CrossRef] [Green Version] - Hikmawan, G.; Soda, J.; Suroso, A.; Zen, F.P. Comment on “Gauss-Bonnet inflation”. Phys. Rev. D
**2016**, 93, 068301. [Google Scholar] [CrossRef] [Green Version] - Koh, S.; Lee, B.H.; Tumurtushaa, G. Reconstruction of the Scalar Field Potential in Inflationary Models with a Gauss-Bonnet term. Phys. Rev. D
**2017**, 95, 123509. [Google Scholar] [CrossRef] [Green Version] - Fomin, I.V.; Chervon, S.V. Exact inflation in Einstein-Gauss-Bonnet gravity. Grav. Cosmol.
**2017**, 23, 367. [Google Scholar] [CrossRef] [Green Version] - Fomin, I.V.; Morozov, A.N. The high-frequency gravitational waves in exact inflationary models with Gauss-Bonnet term. J. Phys. Conf. Ser.
**2017**, 798, 012088. [Google Scholar] [CrossRef] - Fomin, I.V.; Chervon, S.V. A new approach to exact solutions construction in scalar cosmology with a Gauss-Bonnet term. Mod. Phys. Lett. A
**2017**, 32, 1750129. [Google Scholar] [CrossRef] [Green Version] - Odintsov, S.D.; Oikonomou, V.K. Viable Inflation in Scalar-Gauss-Bonnet Gravity and Reconstruction from Observational Indices. Phys. Rev. D
**2018**, 98, 044039. [Google Scholar] [CrossRef] [Green Version] - Pozdeeva, E.O. Generalization of cosmological attractor approach to Einstein—Gauss—Bonnet gravity. Eur. Phys. J. C
**2020**, 80, 612. [Google Scholar] [CrossRef] - Lovelock, D. The Einstein tensor and its generalizations. J. Math. Phys.
**1971**, 12, 498. [Google Scholar] [CrossRef] - Ketov, S.V.; Watanabe, N. The f(R) Gravity Function of the Linde Quintessence. Phys. Lett. B
**2014**, 741, 242–245. [Google Scholar] [CrossRef] [Green Version] - Whitt, B. Fourth Order Gravity as General Relativity Plus Matter. Phys. Lett. B
**1984**, 145, 176–178. [Google Scholar] [CrossRef] - Maeda, K.I. Inflation as a Transient Attractor in R
^{2}Cosmology. Phys. Rev. D**1988**, 37, 858–862. [Google Scholar] [CrossRef] [PubMed] - Bezrukov, F.L.; Shaposhnikov, M. The Standard Model Higgs boson as the inflaton. Phys. Lett. B
**2008**, 659, 703–706. [Google Scholar] [CrossRef] [Green Version] - Bezrukov, F.L.; Magnin, A.; Shaposhnikov, M. Standard Model Higgs boson mass from inflation. Phys. Lett. B
**2009**, 675, 88–92. [Google Scholar] [CrossRef] [Green Version] - Gorbunov, D.S.; Panin, A.G. Are R
^{2}- and Higgs-inflations really unlikely? Phys. Lett. B**2015**, 743, 79–81. [Google Scholar] [CrossRef] - Fomin, I.V.; Chervon, S.V.; Tsyganov, A.V. Generalized scalar-tensor theory of gravity reconstruction from physical potentials of a scalar field. Eur. Phys. J. C
**2020**, 80, 350. [Google Scholar] [CrossRef] - Mishra, S.S.; Sahni, V.; Toporensky, A.V. Initial conditions for Inflation in an FRW Universe. Phys. Rev. D
**2018**, 98, 083538. [Google Scholar] [CrossRef] [Green Version] - Arapoglu, S.; Cikintoglu, S.; Eksi, K.Y. Relativistic stars in Starobinsky gravity with the matched asymptotic expansions method. Phys. Rev. D
**2017**, 96, 084040. [Google Scholar] [CrossRef] [Green Version] - Giudice, G.F.; Lee, H.M. Starobinsky-like inflation from induced gravity. Phys. Lett. B
**2014**, 733, 58–62. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; Nojiri, S.; Odintsov, S.D. The role of energy conditions in f(R) cosmology. Phys. Lett. B
**2018**, 781, 99. [Google Scholar] [CrossRef] - Channuie, P. Deformed Starobinsky model in gravity’s rainbow. Eur. Phys. J. C
**2019**, 79, 508. [Google Scholar] [CrossRef] - Sebastiani, L.; Cognola, G.; Myrzakulov, R.; Odintsov, S.D.; Zerbini, S. Nearly Starobinsky inflation from modified gravity. Phys. Rev. D
**2014**, 89, 023518. [Google Scholar] [CrossRef] [Green Version] - Moraes, P.; Sahoo, P.K.; Ribeiro, G.; Correa, R. A Cosmological Scenario from the Starobinsky Model within the f(R,T) Formalism. Adv. Astron. D
**2019**, 2019, 8574798. [Google Scholar] [CrossRef] [Green Version] - Odintsov, S.D.; Oikonomou, V.K.; Sebastiani, L. Unification of Constant-roll Inflation and Dark Energy with Logarithmic R
^{2}-corrected and Exponential F(R) Gravity. Nucl. Phys. B**2017**, 923, 608–632. [Google Scholar] [CrossRef] - Ketov, S.V. On the equivalence of Starobinsky and Higgs inflationary models in gravity and supergravity. J. Phys. A
**2020**, 53, 084001. [Google Scholar] [CrossRef] [Green Version] - Naruko, A.; Yoshida, D.; Mukohyama, S. Gravitational scalar-tensor theory. Class. Quant. Grav.
**2016**, 33, 09LT01. [Google Scholar] [CrossRef] [Green Version] - Saridakis, E.N.; Tsoukalas, M. Cosmology in new gravitational scalar-tensor theories. Phys. Rev. D
**2016**, 93, 124032. [Google Scholar] [CrossRef] [Green Version] - Chervon, S.V.; Fomin, I.V.; Mayorova, T.I. Chiral Cosmological Model of f(R) Gravity with a Kinetic Curvature Scalar. Grav. Cosmol.
**2019**, 25, 205. [Google Scholar] [CrossRef] - Chervon, S.V.; Fomin, I.V.; Mayorova, T.I.; Khapaeva, A.V. Cosmological parameters of f(R) gravity with kinetic scalar curvature. J. Phys. Conf. Ser.
**2020**, 1557, 012016. [Google Scholar] [CrossRef] - Garcia-Bellido, J.; Wands, D. Metric perturbations in two field inflation. Phys. Rev. D
**1996**, 53, 5437. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gordon, C.; Wands, D.; Bassett, B.A.; Maartens, R. Adiabatic and entropy perturbations from inflation. Phys. Rev. D
**2000**, 63, 023506. [Google Scholar] [CrossRef] [Green Version] - Starobinsky, A.A.; Tsujikawa, S.; Yokoyama, J. Cosmological perturbations from multifield inflation in generalized Einstein theories. Nucl. Phys. B
**2001**, 610, 383. [Google Scholar] [CrossRef] [Green Version] - Wands, D.; Bartolo, N.; Matarrese, S.; Riotto, A. An Observational test of two-field inflation. Phys. Rev. D
**2002**, 66, 043520. [Google Scholar] [CrossRef] [Green Version] - Byrnes, C.T.; Choi, K.Y. Review of local non-Gaussianity from multi-field inflation. Adv. Astron.
**2010**, 2010, 724525. [Google Scholar] [CrossRef] [Green Version] - Bartolo, N.; Komatsu, E.; Matarrese, S.; Riotto, A. Non-Gaussianity from inflation: Theory and observations. Phys. Rept.
**2004**, 402, 103. [Google Scholar] [CrossRef] [Green Version] - Ade, P.; Planck Collaboration. Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys.
**2016**, 594, A13. [Google Scholar] [CrossRef] [Green Version] - Aghanim, N.; Planck Collaboration. Planck 2018 results. VI. Cosmological parameters. arXiv
**2020**, arXiv:1807.06209. [Google Scholar] - Fomin, I.V.; Chervon, S.V. Reconstruction of general relativistic cosmological solutions in modified gravity theories. Phys. Rev. D
**2019**, 100, 023511. [Google Scholar] [CrossRef] [Green Version] - Pozdeeva, E.O.; Sami, M.; Toporensky, A.V.; Vernov, S.Y. Stability analysis of de Sitter solutions in models with the Gauss-Bonnet term. Phys. Rev. D
**2019**, 100, 083527. [Google Scholar] [CrossRef] [Green Version] - Pozdeeva, E.O.; Gangopadhyay, M.R.; Sami, M.; Toporensky, A.V.; Vernov, S.Y. Inflation with a quartic potential in the framework of Einstein-Gauss-Bonnet gravity. Phys. Rev. D
**2020**, 102, 043525. [Google Scholar] [CrossRef] - Fomin, I.V.; Chervon, S.V. The exact solutions in verified cosmological models based on generalized scalar-tensor gravity. Eur. Phys. J. C
**2018**, 78, 918. [Google Scholar] [CrossRef] [Green Version] - Fomin, I.V.; Chervon, S.V. Non-minimal coupling influence on the deviation from de Sitter cosmological expansion. J. Phys. Conf. Ser.
**2020**, 1557, 012020. [Google Scholar] [CrossRef] - Martin, J.; Ringeval, C.; Vennin, V. Encyclopedia Inflationaris. Phys. Dark Univ.
**2014**, 5–6, 75. [Google Scholar] [CrossRef] [Green Version] - Grøn, Ø. Predictions of Spectral Parameters by Several Inflationary Universe Models in Light of the Planck Results. Universe
**2018**, 4, 715. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The dependence $r=r\left({n}_{S}\right)$ for different values of the constant ${\mu}_{2}=1/2,\sqrt{2/3},2$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fomin, I.; Chervon, S.
Exact and Slow-Roll Solutions for Exponential Power-Law Inflation Connected with Modified Gravity and Observational Constraints. *Universe* **2020**, *6*, 199.
https://doi.org/10.3390/universe6110199

**AMA Style**

Fomin I, Chervon S.
Exact and Slow-Roll Solutions for Exponential Power-Law Inflation Connected with Modified Gravity and Observational Constraints. *Universe*. 2020; 6(11):199.
https://doi.org/10.3390/universe6110199

**Chicago/Turabian Style**

Fomin, Igor, and Sergey Chervon.
2020. "Exact and Slow-Roll Solutions for Exponential Power-Law Inflation Connected with Modified Gravity and Observational Constraints" *Universe* 6, no. 11: 199.
https://doi.org/10.3390/universe6110199