# Dynamically Generated Inflationary ΛCDM

^{1}

^{2}

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Simple Model of Unification of Dark Energy and Dark Matter

- The first term in (10) is the standard Einstein–Hilbert action with $R\left(g\right)$ denoting the scalar curvature with respect to metric ${g}_{\mu \nu}$ in the second order (metric) formalism;
- ${\Phi}_{0}\left(A\right)$ is particular representative of a $D=4$ non-Riemannian volume-element density (6):$${\Phi}_{0}\left(A\right)=\frac{1}{3!}{\epsilon}^{\mu \nu \kappa \lambda}{\partial}_{\mu}{A}_{\nu \kappa \lambda}\phantom{\rule{0.277778em}{0ex}}.$$
- $L(\phi ,X)$ is general-coordinate invariant Lagrangian of a single scalar field $\phi \left(x\right)$:$$L(\phi ,X)=X-V\left(\phi \right),\phantom{\rule{1.em}{0ex}}X\equiv -\frac{1}{2}{g}^{\mu \nu}{\partial}_{\mu}\phi {\partial}_{\nu}\phi \phantom{\rule{0.277778em}{0ex}}.$$

**Remark**

**1.**

- A dynamically generated dark matter part given by the first term in (19), where ${p}_{\mathrm{DM}}=0$, ${\rho}_{\mathrm{DM}}={\rho}_{0}$ with ${\rho}_{0}$ as in (20), which in fact according to (21) and (22) describes a dust fluid with fluid density ${\rho}_{0}$ flowing along geodesics. Thus, we will refer to the $\phi $ scalar field by the alias “darkon.”

## 3. Inflation and Unified Dark Energy and Dark Matter

- (i) ${U}_{\mathrm{eff}}\left(u\right)$ (37) has an almost flat region for large positive u: ${U}_{\mathrm{eff}}\left(u\right)\simeq 2{\Lambda}_{0}$ for large u. This almost flat region corresponds to "early universe" inflationary evolution with energy scale $2{\Lambda}_{0}$, as will be evident from the autonomous dynamical system analysis of the cosmological dynamics in Section 4.
- (ii) ${U}_{\mathrm{eff}}\left(u\right)$ (37) has a stable minimum for a small finite value $u={u}_{*}$: $\frac{\partial {U}_{\mathrm{eff}}}{\partial u}=0\phantom{\rule{0.277778em}{0ex}}$ for $u\equiv {u}_{*}$, where:$$exp\left(\right)open="("\; close=")">-\frac{{u}_{*}}{\sqrt{3}}$$
- (iii) As it will be explicitly exhibited in the dynamical system analysis in Section 4, the region of u around the stable minimum at $u={u}_{*}$ (41) corresponds to the late-time de Sitter expansion of the universe with a slightly varied late-time Hubble parameter (dark energy dominated epoch), wherein the minimum value of the potential:$${U}_{\mathrm{eff}}\left({u}_{*}\right)=2{\Lambda}_{0}-\frac{{M}_{1}^{2}}{8{M}_{0}}\equiv 2{\Lambda}_{\mathrm{DE}}$$

## 4. Cosmological Implications

**Remark**

**2.**

- (A) Stable critical point:$${x}_{*}=0\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}{y}_{*}=0\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}{H}_{*}=\sqrt{\frac{{\Lambda}_{\mathrm{DE}}}{3}}\phantom{\rule{0.277778em}{0ex}},$$
- (B) Unstable critical point:$${x}_{**}=0\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}{y}_{**}=\sqrt{1-\frac{{\Lambda}_{\mathrm{DE}}}{{\Lambda}_{0}}}=\frac{{M}_{1}}{4\sqrt{{M}_{0}{\Lambda}_{0}}}\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}{H}_{**}=\sqrt{\frac{{\Lambda}_{0}}{3}}\phantom{\rule{0.277778em}{0ex}},$$$$\begin{array}{c}\hfill \u03f5=-\frac{\stackrel{.}{H}}{{H}^{2}}\approx {\left(\right)}^{\frac{\frac{\partial {U}_{\mathrm{eff}}}{\partial u}-\frac{1}{2\sqrt{3}}\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathrm{DM}}}{{U}_{\mathrm{eff}}+{\rho}_{\mathrm{DM}}}}2+\frac{3}{2}\frac{{\rho}_{\mathrm{DM}}}{{U}_{\mathrm{eff}}+{\rho}_{\mathrm{DM}}}\phantom{\rule{0.277778em}{0ex}},\end{array}$$$$\begin{array}{c}\hfill \eta =-\frac{\stackrel{.}{H}}{{H}^{2}}-\frac{\stackrel{..}{H}}{2H\stackrel{.}{H}}\approx -2\frac{\frac{{\partial}^{2}{U}_{\mathrm{eff}}}{\partial {u}^{2}}+\frac{1}{12}\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathrm{DM}}}{{U}_{\mathrm{eff}}+{\rho}_{\mathrm{DM}}}+\mathrm{O}\left({\rho}_{\mathrm{DM}}\right)\phantom{\rule{0.277778em}{0ex}}.\end{array}$$

## 5. Numerical Solutions

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurements of Omega and Lambda from 42 high redshift supernovae. Astrophys. J.
**1999**, 517, 565–586. [Google Scholar] [CrossRef] - Copeland, E.J.; Sami, M.; Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys.
**2006**, D15, 1753–1936. [Google Scholar] [CrossRef] [Green Version] - Novikov, E.A. Quantum Modification of General Relativity. Electron. J. Theor. Phys.
**2016**, 13, 79–90. [Google Scholar] - Benitez, F.; Gambini, R.; Lehner, L.; Liebling, S.; Pullin, J. Critical collapse of a scalar field in semiclassical loop quantum gravity. Phys. Rev. Lett.
**2020**, 124, 071301. [Google Scholar] [CrossRef] [Green Version] - Budge, L.; Campbell, J.M.; De Laurentis, G.; Keith Ellis, R.; Seth, S. The one-loop amplitude for Higgs + 4 gluons with full mass effects. arXiv
**2020**, arXiv:2002.04018. [Google Scholar] - Bell, G.; Beneke, M.; Huber, T.; Li, X.Q. Two-loop non-leptonic penguin amplitude in QCD factorization. arXiv
**2020**, arXiv:2002.03262. [Google Scholar] - Fröhlich, J.; Knowles, A.; Schlein, B.; Sohinger, V. A path-integral analysis of interacting Bose gases and loop gases. arXiv
**2020**, arXiv:2001.11714. [Google Scholar] - D’Ambrosio, F. Semi-Classical Holomorphic Transition Amplitudes in Covariant Loop Quantum Gravity. arXiv
**2020**, arXiv:2001.04651. [Google Scholar] - Novikov, E.A. Ultralight gravitons with tiny electric dipole moment are seeping from the vacuum. Mod. Phys. Lett.
**2016**, A31, 1650092. [Google Scholar] [CrossRef] [Green Version] - Dekens, W.; Stoffer, P. Low-energy effective field theory below the electroweak scale: Matching at one loop. JHEP
**2019**, 10, 197. [Google Scholar] [CrossRef] [Green Version] - Ma, C.T.; Pezzella, F. Stringy Effects at Low-Energy Limit and Double Field Theory. arXiv
**2019**, arXiv:1909.00411. [Google Scholar] - Jenkins, E.E.; Manohar, A.V.; Stoffer, P. Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching. JHEP
**2018**, 3, 16. [Google Scholar] [CrossRef] [Green Version] - Brandyshev, P.E. Cosmological solutions in low-energy effective field theory for type IIA superstrings. Grav. Cosmol.
**2017**, 23, 15–19. [Google Scholar] [CrossRef] - Gomez, C.; Jimenez, R. Cosmology from Quantum Information. arXiv
**2020**, arXiv:2002.04294. [Google Scholar] - Guth, A.H. The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Phys. Rev.
**1981**, D23, 347–356. [Google Scholar] [CrossRef] [Green Version] - Starobinsky, A.A. Spectrum of relict gravitational radiation and the early state of the universe. JETP Lett.
**1979**, 30, 682–685. [Google Scholar] - Kazanas, D. Dynamics of the Universe and Spontaneous Symmetry Breaking. Astrophys. J.
**1980**, 241, L59–L63. [Google Scholar] [CrossRef] - Starobinsky, A.A. A New Type of Isotropic Cosmological Models Without Singularity. Phys. Lett.
**1980**, 91B, 99–102. [Google Scholar] [CrossRef] - Linde, A.D. A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems. Phys. Lett.
**1982**, 108B, 389–393. [Google Scholar] [CrossRef] - Albrecht, A.; Steinhardt, P.J. Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking. Phys. Rev. Lett.
**1982**, 48, 1220–1223. [Google Scholar] [CrossRef] - Barrow, J.D.; Ottewill, A.C. The Stability of General Relativistic Cosmological Theory. J. Phys.
**1983**, A16, 2757. [Google Scholar] [CrossRef] - Blau, S.K.; Guendelman, E.I.; Guth, A.H. The Dynamics of False Vacuum Bubbles. Phys. Rev.
**1987**, D35, 1747. [Google Scholar] [CrossRef] [Green Version] - Cervantes-Cota, J.L.; Dehnen, H. Induced gravity inflation in the standard model of particle physics. Nucl. Phys.
**1995**, B442, 391–412. [Google Scholar] [CrossRef] [Green Version] - Berera, A. Warm inflation. Phys. Rev. Lett.
**1995**, 75, 3218–3221. [Google Scholar] [CrossRef] [PubMed] - Armendariz-Picon, C.; Damour, T.; Mukhanov, V.F. k - inflation. Phys. Lett.
**1999**, B458, 209–218. [Google Scholar] [CrossRef] [Green Version] - Kanti, P.; Olive, K.A. Assisted chaotic inflation in higher dimensional theories. Phys. Lett.
**1999**, B464, 192–198. [Google Scholar] [CrossRef] [Green Version] - Garriga, J.; Mukhanov, V.F. Perturbations in k-inflation. Phys. Lett.
**1999**, B458, 219–225. [Google Scholar] [CrossRef] [Green Version] - Gordon, C.; Wands, D.; Bassett, B.A.; Maartens, R. Adiabatic and entropy perturbations from inflation. Phys. Rev.
**2000**, D63, 023506. [Google Scholar] [CrossRef] [Green Version] - Bassett, B.A.; Tsujikawa, S.; Wands, D. Inflation dynamics and reheating. Rev. Mod. Phys.
**2006**, 78, 537–589. [Google Scholar] [CrossRef] [Green Version] - Chen, X.; Wang, Y. Quasi-Single Field Inflation and Non-Gaussianities. JCAP
**2010**, 1004, 27. [Google Scholar] [CrossRef] [Green Version] - Germani, C.; Kehagias, A. New Model of Inflation with Non-minimal Derivative Coupling of Standard Model Higgs Boson to Gravity. Phys. Rev. Lett.
**2010**, 105, 011302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kobayashi, T.; Yamaguchi, M.; Yokoyama, J. G-inflation: Inflation driven by the Galileon field. Phys. Rev. Lett.
**2010**, 105, 231302. [Google Scholar] [CrossRef] [Green Version] - Feng, C.J.; Li, X.Z.; Saridakis, E.N. Preventing eternality in phantom inflation. Phys. Rev.
**2010**, D82, 023526. [Google Scholar] [CrossRef] [Green Version] - Burrage, C.; de Rham, C.; Seery, D.; Tolley, A.J. Galileon inflation. JCAP
**2011**, 1101, 14. [Google Scholar] [CrossRef] - Kobayashi, T.; Yamaguchi, M.; Yokoyama, J. Generalized G-inflation: Inflation with the most general second-order field equations. Prog. Theor. Phys.
**2011**, 126, 511–529. [Google Scholar] [CrossRef] - Ohashi, J.; Tsujikawa, S. Potential-driven Galileon inflation. JCAP
**2012**, 1210, 35. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A.; Tsamparlis, M. Two scalar field cosmology: Conservation laws and exact solutions. Phys. Rev.
**2014**, D90, 043529. [Google Scholar] [CrossRef] [Green Version] - Dimakis, N.; Paliathanasis, A. Crossing the phantom divide line as an effect of quantum transitions. arXiv
**2020**, arXiv:2001.09687. [Google Scholar] - Dimakis, N.; Paliathanasis, A.; Terzis, P.A.; Christodoulakis, T. Cosmological Solutions in Multiscalar Field Theory. Eur. Phys. J.
**2019**, C79, 618. [Google Scholar] [CrossRef] [Green Version] - Benisty, D.; Guendelman, E.I. A transition between bouncing hyper-inflation to ΛCDM from diffusive scalar fields. Int. J. Mod. Phys.
**2018**, A33, 1850119. [Google Scholar] [CrossRef] - Barrow, J.D.; Paliathanasis, A. Observational Constraints on New Exact Inflationary Scalar-field Solutions. Phys. Rev.
**2016**, D94, 083518. [Google Scholar] [CrossRef] [Green Version] - Barrow, J.D.; Paliathanasis, A. Reconstructions of the dark-energy equation of state and the inflationary potential. Gen. Rel. Grav.
**2018**, 50, 82. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Olive, K.A. Inflation. Phys. Rept.
**1990**, 190, 307–403. [Google Scholar] [CrossRef] - Linde, A.D. Hybrid inflation. Phys. Rev.
**1994**, D49, 748–754. [Google Scholar] [CrossRef] [PubMed] - Liddle, A.R.; Parsons, P.; Barrow, J.D. Formalizing the slow roll approximation in inflation. Phys. Rev.
**1994**, D50, 7222–7232. [Google Scholar] [CrossRef] [PubMed] - Lidsey, J.E.; Liddle, A.R.; Kolb, E.W.; Copeland, E.J.; Barreiro, T.; Abney, M. Reconstructing the inflation potential: An overview. Rev. Mod. Phys.
**1997**, 69, 373–410. [Google Scholar] [CrossRef] - Hossain, M.W.; Myrzakulov, R.; Sami, M.; Saridakis, E.N. Variable gravity: A suitable framework for quintessential inflation. Phys. Rev.
**2014**, D90, 023512. [Google Scholar] [CrossRef] [Green Version] - Wali Hossain, M.; Myrzakulov, R.; Sami, M.; Saridakis, E.N. Unification of inflation and dark energy à la quintessential inflation. Int. J. Mod. Phys.
**2015**, D24, 1530014. [Google Scholar] [CrossRef] - Cai, Y.F.; Gong, J.O.; Pi, S.; Saridakis, E.N.; Wu, S.Y. On the possibility of blue tensor spectrum within single field inflation. Nucl. Phys.
**2015**, B900, 517–532. [Google Scholar] [CrossRef] [Green Version] - Geng, C.Q.; Hossain, M.W.; Myrzakulov, R.; Sami, M.; Saridakis, E.N. Quintessential inflation with canonical and noncanonical scalar fields and Planck 2015 results. Phys. Rev.
**2015**, D92, 023522. [Google Scholar] [CrossRef] [Green Version] - Kamali, V.; Basilakos, S.; Mehrabi, A. Tachyon warm-intermediate inflation in the light of Planck data. Eur. Phys. J.
**2016**, C76, 525. [Google Scholar] [CrossRef] - Geng, C.Q.; Lee, C.C.; Sami, M.; Saridakis, E.N.; Starobinsky, A.A. Observational constraints on successful model of quintessential Inflation. JCAP
**2017**, 1706, 11. [Google Scholar] [CrossRef] [Green Version] - Dalianis, I.; Kehagias, A.; Tringas, G. Primordial black holes from α-attractors. JCAP
**2019**, 1901, 37. [Google Scholar] [CrossRef] [Green Version] - Dalianis, I.; Tringas, G. Primordial black hole remnants as dark matter produced in thermal, matter, and runaway-quintessence postinflationary scenarios. Phys. Rev.
**2019**, D100, 083512. [Google Scholar] [CrossRef] [Green Version] - Benisty, D. Inflation from Fermions. arXiv
**2019**, arXiv:1912.11124. [Google Scholar] - Benisty, D.; Guendelman, E.I. Inflation compactification from dynamical spacetime. Phys. Rev.
**2018**, D98, 043522. [Google Scholar] [CrossRef] [Green Version] - Benisty, D.; Guendelman, E.I.; Saridakis, E.N. The Scale Factor Potential Approach to Inflation. arXiv
**2019**, arXiv:1909.01982. [Google Scholar] - Gerbino, M.; Freese, K.; Vagnozzi, S.; Lattanzi, M.; Mena, O.; Giusarma, E.; Ho, S. Impact of neutrino properties on the estimation of inflationary parameters from current and future observations. Phys. Rev.
**2017**, D95, 043512. [Google Scholar] [CrossRef] [Green Version] - Giovannini, M. Planckian hypersurfaces, inflation and bounces. arXiv
**2020**, arXiv:2001.11799. [Google Scholar] - Brahma, S.; Brandenberger, R.; Yeom, D.H. Swampland, Trans-Planckian Censorship and Fine-Tuning Problem for Inflation: Tunnelling Wavefunction to the Rescue. arXiv
**2020**, arXiv:2002.02941. [Google Scholar] - Domcke, V.; Guidetti, V.; Welling, Y.; Westphal, A. Resonant backreaction in axion inflation. arXiv
**2020**, arXiv:2002.02952. [Google Scholar] - Tenkanen, T.; Tomberg, E. Initial conditions for plateau inflation. arXiv
**2020**, arXiv:2002.02420. [Google Scholar] - Martin, J.; Papanikolaou, T.; Pinol, L.; Vennin, V. Metric preheating and radiative decay in single-field inflation. arXiv
**2020**, arXiv:2002.01820. [Google Scholar] - Cheon, K.; Lee, J. N = 2 PNGB Quintessence Dark Energy. arXiv
**2020**, arXiv:2002.01756. [Google Scholar] - Saleem, R.; Zubair, M. Inflationary solution of Hamilton Jacobi equations during weak dissipative regime. Phys. Scr.
**2020**, 95, 035214. [Google Scholar] [CrossRef] [Green Version] - Giacintucci, S.; Markevitch, M.; Johnston-Hollitt, M.; Wik, D.R.; Wang, Q.H.S.; Clarke, T.E. Discovery of a giant radio fossil in the Ophiuchus galaxy cluster. arXiv
**2020**, arXiv:2002.01291. [Google Scholar] [CrossRef] - Aalsma, L.; Shiu, G. Chaos and complementarity in de Sitter space. arXiv
**2020**, arXiv:2002.01326. [Google Scholar] - Kogut, A.; Fixsen, D.J. Calibration Method and Uncertainty for the Primordial Inflation Explorer (PIXIE). arXiv
**2020**, arXiv:2002.00976. [Google Scholar] - Arciniega, G.; Jaime, L.; Piccinelli, G. Inflationary predictions of Geometric Inflation. arXiv
**2020**, arXiv:2001.11094. [Google Scholar] - Rasheed, M.A.; Golanbari, T.; Sayar, K.; Akhtari, L.; Sheikhahmadi, H.; Mohammadi, A.; Saaidi, K. Warm Tachyon Inflation and Swampland Criteria. arXiv
**2020**, arXiv:2001.10042. [Google Scholar] - Aldabergenov, Y.; Aoki, S.; Ketov, S.V. Minimal Starobinsky supergravity coupled to dilaton-axion superfield. arXiv
**2020**, arXiv:2001.09574. [Google Scholar] - Tenkanen, T. Tracing the high energy theory of gravity: an introduction to Palatini inflation. arXiv
**2020**, arXiv:2001.10135. [Google Scholar] - Shaposhnikov, M.; Shkerin, A.; Zell, S. Standard Model Meets Gravity: Electroweak Symmetry Breaking and Inflation. arXiv
**2020**, arXiv:2001.09088. [Google Scholar] - Garcia, M.A.G.; Amin, M.A.; Green, D. Curvature Perturbations From Stochastic Particle Production During Inflation. arXiv
**2020**, arXiv:2001.09158. [Google Scholar] - Hirano, K. Inflation with very small tensor-to-scalar ratio. arXiv
**2019**, arXiv:1912.12515. [Google Scholar] - Gialamas, I.D.; Lahanas, A.B. Reheating in R
^{2}Palatini inflationary models. arXiv**2019**, arXiv:1911.11513. [Google Scholar] - Kawasaki, M.; Yamaguchi, M.; Yanagida, T. Natural chaotic inflation in supergravity. Phys. Rev. Lett.
**2000**, 85, 3572–3575. [Google Scholar] [CrossRef] [Green Version] - Bojowald, M. Inflation from quantum geometry. Phys. Rev. Lett.
**2002**, 89, 261301. [Google Scholar] [CrossRef] [Green Version] - 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.
**2003**, D68, 123512. [Google Scholar] [CrossRef] [Green Version] - Kachru, S.; Kallosh, R.; Linde, A.D.; Maldacena, J.M.; McAllister, L.P.; Trivedi, S.P. Towards inflation in string theory. JCAP
**2003**, 310, 13. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Unifying phantom inflation with late-time acceleration: Scalar phantom-non-phantom transition model and generalized holographic dark energy. Gen. Rel. Grav.
**2006**, 38, 1285–1304. [Google Scholar] [CrossRef] [Green Version] - Ferraro, R.; Fiorini, F. Modified teleparallel gravity: Inflation without inflation. Phys. Rev.
**2007**, D75, 084031. [Google Scholar] [CrossRef] [Green Version] - Cognola, G.; Elizalde, E.; Nojiri, S.; Odintsov, S.D.; Sebastiani, L.; Zerbini, S. A Class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion. Phys. Rev.
**2008**, D77, 046009. [Google Scholar] [CrossRef] [Green Version] - Cai, Y.F.; Saridakis, E.N. Inflation in Entropic Cosmology: Primordial Perturbations and non-Gaussianities. Phys. Lett.
**2011**, B697, 280–287. [Google Scholar] [CrossRef] [Green Version] - Ashtekar, A.; Sloan, D. Probability of Inflation in Loop Quantum Cosmology. Gen. Rel. Grav.
**2011**, 43, 3619–3655. [Google Scholar] [CrossRef] [Green Version] - Qiu, T.; Saridakis, E.N. Entropic Force Scenarios and Eternal Inflation. Phys. Rev.
**2012**, D85, 043504. [Google Scholar] [CrossRef] [Green Version] - Briscese, F.; Marcianò, A.; Modesto, L.; Saridakis, E.N. Inflation in (Super-)renormalizable Gravity. Phys. Rev.
**2013**, D87, 083507. [Google Scholar] [CrossRef] [Green Version] - Ellis, J.; Nanopoulos, D.V.; Olive, K.A. No-Scale Supergravity Realization of the Starobinsky Model of Inflation. Phys. Rev. Lett.
**2013**, 111, 111301. [Google Scholar] [CrossRef] [Green Version] - Basilakos, S.; Lima, J.A.S.; Sola, J. From inflation to dark energy through a dynamical Lambda: An attempt at alleviating fundamental cosmic puzzles. Int. J. Mod. Phys.
**2013**, D22, 1342008. [Google Scholar] [CrossRef] [Green Version] - Sebastiani, L.; Cognola, G.; Myrzakulov, R.; Odintsov, S.D.; Zerbini, S. Nearly Starobinsky inflation from modified gravity. Phys. Rev.
**2014**, D89, 023518. [Google Scholar] [CrossRef] [Green Version] - Baumann, D.; McAllister, L. Inflation and String Theory; Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 2015. [Google Scholar] [CrossRef] [Green Version]
- Dalianis, I.; Farakos, F. On the initial conditions for inflation with plateau potentials: the R+R
^{2}(super)gravity case. JCAP**2015**, 1507, 44. [Google Scholar] [CrossRef] [Green Version] - Kanti, P.; Gannouji, R.; Dadhich, N. Gauss-Bonnet Inflation. Phys. Rev.
**2015**, D92, 041302. [Google Scholar] [CrossRef] [Green Version] - De Laurentis, M.; Paolella, M.; Capozziello, S. Cosmological inflation in F(R,$\mathcal{G}$) gravity. Phys. Rev.
**2015**, D91, 083531. [Google Scholar] [CrossRef] [Green Version] - Basilakos, S.; Mavromatos, N.E.; Solà, J. Starobinsky-like inflation and running vacuum in the context of Supergravity. Universe
**2016**, 2, 14. [Google Scholar] [CrossRef] [Green Version] - Bonanno, A.; Platania, A. Asymptotically safe inflation from quadratic gravity. Phys. Lett.
**2015**, B750, 638–642. [Google Scholar] [CrossRef] - Koshelev, A.S.; Modesto, L.; Rachwal, L.; Starobinsky, A.A. Occurrence of exact R
^{2}inflation in non-local UV-complete gravity. JHEP**2016**, 11, 67. [Google Scholar] [CrossRef] [Green Version] - Bamba, K.; Odintsov, S.D.; Saridakis, E.N. Inflationary cosmology in unimodular F(T) gravity. Mod. Phys. Lett.
**2017**, A32, 1750114. [Google Scholar] [CrossRef] [Green Version] - Motohashi, H.; Starobinsky, A.A. f(R) constant-roll inflation. Eur. Phys. J.
**2017**, C77, 538. [Google Scholar] [CrossRef] [Green Version] - Oikonomou, V.K. Autonomous dynamical system approach for inflationary Gauss–Bonnet modified gravity. Int. J. Mod. Phys.
**2018**, D27, 1850059. [Google Scholar] [CrossRef] [Green Version] - Benisty, D.; Vasak, D.; Guendelman, E.; Struckmeier, J. Energy transfer from spacetime into matter and a bouncing inflation from covariant canonical gauge theory of gravity. Mod. Phys. Lett.
**2019**, A34, 1950164. [Google Scholar] [CrossRef] [Green Version] - Benisty, D.; Guendelman, E.I. Two scalar fields inflation from scale-invariant gravity with modified measure. Class. Quant. Grav.
**2019**, 36, 095001. [Google Scholar] [CrossRef] [Green Version] - Antoniadis, I.; Karam, A.; Lykkas, A.; Tamvakis, K. Palatini inflation in models with an R
^{2}term. JCAP**2018**, 1811, 28. [Google Scholar] [CrossRef] [Green Version] - Karam, A.; Pappas, T.; Tamvakis, K. Frame-dependence of inflationary observables in scalar-tensor gravity. PoS
**2019**, CORFU2018, 64. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D.; Saridakis, E.N. Holographic inflation. Phys. Lett.
**2019**, B797, 134829. [Google Scholar] [CrossRef] - Benisty, D.; Guendelman, E.I.; Saridakis, E.N.; Stoecker, H.; Struckmeier, J.; Vasak, D. Inflation from fermions with curvature-dependent mass. arXiv
**2019**, arXiv:1905.03731. [Google Scholar] [CrossRef] [Green Version] - Benisty, D.; Guendelman, E.; Nissimov, E.; Pacheva, S. Dynamically Generated Inflation from Non-Riemannian Volume Forms. arXiv
**2019**, arXiv:1906.06691. [Google Scholar] [CrossRef] [Green Version] - Benisty, D.; Guendelman, E.I.; Nissimov, E.; Pacheva, S. Dynamically generated inflationary two-field potential via non-Riemannian volume forms. arXiv
**2019**, arXiv:1907.07625. [Google Scholar] [CrossRef] - Kinney, W.H.; Vagnozzi, S.; Visinelli, L. The zoo plot meets the swampland: Mutual (in)consistency of single-field inflation, string conjectures, and cosmological data. Class. Quant. Grav.
**2019**, 36, 117001. [Google Scholar] [CrossRef] [Green Version] - Brustein, R.; Sherf, Y. Causality Violations in Lovelock Theories. Phys. Rev.
**2018**, D97, 084019. [Google Scholar] [CrossRef] [Green Version] - Sherf, Y. Hyperbolicity Constraints in Extended Gravity Theories. Phys. Scr.
**2019**, 94, 085005. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; De Laurentis, M.; Luongo, O. Connecting early and late universe by f(R) gravity. Int. J. Mod. Phys.
**2014**, D24, 1541002. [Google Scholar] [CrossRef] [Green Version] - Gorbunov, D.; Tokareva, A. Scale-invariance as the origin of dark radiation? Phys. Lett.
**2014**, B739, 50–55. [Google Scholar] [CrossRef] [Green Version] - Myrzakulov, R.; Odintsov, S.; Sebastiani, L. Inflationary universe from higher-derivative quantum gravity. Phys. Rev.
**2015**, D91, 083529. [Google Scholar] [CrossRef] [Green Version] - Bamba, K.; Myrzakulov, R.; Odintsov, S.D.; Sebastiani, L. Trace-anomaly driven inflation in modified gravity and the BICEP2 result. Phys. Rev.
**2014**, D90, 043505. [Google Scholar] [CrossRef] [Green Version] - Benisty, D.; Guendelman, E.I.; Vasak, D.; Struckmeier, J.; Stoecker, H. Quadratic curvature theories formulated as Covariant Canonical Gauge theories of Gravity. Phys. Rev.
**2018**, D98, 106021. [Google Scholar] [CrossRef] [Green Version] - Aashish, S.; Panda, S. Covariant quantum corrections to a scalar field model inspired by nonminimal natural inflation. arXiv
**2020**, arXiv:2001.07350. [Google Scholar] - Rashidi, N.; Nozari, K. Gauss-Bonnet Inflation after Planck2018. arXiv
**2020**, arXiv:2001.07012. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K. Geometric Inflation and Dark Energy with Axion F(R) Gravity. Phys. Rev.
**2020**, D101, 044009. [Google Scholar] [CrossRef] [Green Version] - Antoniadis, I.; Karam, A.; Lykkas, A.; Pappas, T.; Tamvakis, K. Single-field inflation in models with an R
^{2}term. In Proceedings of the 19th Hellenic School and Workshops on Elementary Particle Physics and Gravity (CORFU2019), Corfu, Greece, 31 August–25 September 2019. [Google Scholar] - Benisty, D.; Guendelman, E.I. Correspondence between the first and second order formalism by a metricity constraint. Phys. Rev.
**2018**, D98, 044023. [Google Scholar] [CrossRef] [Green Version] - Chakraborty, S.; Paul, T.; SenGupta, S. Inflation driven by Einstein-Gauss-Bonnet gravity. Phys. Rev.
**2018**, D98, 083539. [Google Scholar] [CrossRef] [Green Version] - Mukhanov, V.F.; Chibisov, G.V. Quantum Fluctuations and a Nonsingular Universe. JETP Lett.
**1981**, 33, 532–535. [Google Scholar] - Guth, A.H.; Pi, S.Y. Fluctuations in the New Inflationary Universe. Phys. Rev. Lett.
**1982**, 49, 1110–1113. [Google Scholar] [CrossRef] - Faraoni, V.; Capozziello, S. Beyond Einstein Gravity; Springer: Dordrecht, The Netherlands, 2011; Volume 170. [Google Scholar] [CrossRef] [Green Version]
- Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution. Phys. Rept.
**2017**, 692, 1–104. [Google Scholar] [CrossRef] [Green Version] - Dimitrijevic, I.; Dragovich, B.; Koshelev, A.S.; Rakic, Z.; Stankovic, J. Cosmological Solutions of a Nonlocal Square Root Gravity. Phys. Lett.
**2019**, B797, 134848. [Google Scholar] [CrossRef] - Bilic, N.; Dimitrijevic, D.D.; Djordjevic, G.S.; Milosevic, M.; Stojanovic, M. Tachyon inflation in the holographic braneworld. JCAP
**2019**, 1908, 034. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models. Phys. Rept.
**2011**, 505, 59–144. [Google Scholar] [CrossRef] [Green Version] - Berti, E.; Barausse, E.; Cardoso, V.; Gualtieri, L.; Pani, P.; Sperhake, U.; Stein, L.C.; Wex, N.; Yagi, K.; Baker, T.; et al. Testing General Relativity with Present and Future Astrophysical Observations. Class. Quant. Grav.
**2015**, 32, 243001. [Google Scholar] [CrossRef] - Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. Planck 2018 results. X. Constraints on inflation. arXiv
**2018**, arXiv:1807.06211. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Modified f(R) gravity consistent with realistic cosmology: From matter dominated epoch to dark energy universe. Phys. Rev.
**2006**, D74, 086005. [Google Scholar] [CrossRef] [Green Version] - Lozano, L.; Garcia-Compean, H. Emergent Dark Matter and Dark Energy from a Lattice Model. arXiv
**2019**, arXiv:hep-th/1912.11224. [Google Scholar] - Chamings, F.N.; Avgoustidis, A.; Copeland, E.J.; Green, A.M.; Pourtsidou, A. Understanding the suppression of structure formation from dark matter 2013 dark energy momentum coupling. arXiv
**2019**, arXiv:astro-ph.CO/1912.09858. [Google Scholar] - Liu, L.H.; Xu, W.L. The running curvaton. arXiv
**2019**, arXiv:1911.10542. [Google Scholar] - Cheng, G.; Ma, Y.; Wu, F.; Zhang, J.; Chen, X. Testing interacting dark matter and dark energy model with cosmological data. arXiv
**2019**, arXiv:1911.04520. [Google Scholar] - Cahill, K. Zero-point energies, dark matter, and dark energy. arXiv
**2019**, arXiv:1910.09953. [Google Scholar] - Bandyopadhyay, A.; Chatterjee, A. Time-dependent diffusive interactions between dark matter and dark energy in the context of k-essence cosmology. arXiv
**2019**, arXiv:1910.10423. [Google Scholar] - Kase, R.; Tsujikawa, S. Scalar-Field Dark Energy Nonminimally and Kinetically Coupled to Dark Matter. arXiv
**2019**, arXiv:1910.02699. [Google Scholar] [CrossRef] [Green Version] - Ketov, S.V. Inflation, Dark Energy and Dark Matter in Supergravity. In Proceedings of the Meeting of the Division of Particles and Fields of the American Physical Society (DPF2019), Boston, MA, USA, 29 July–2 August 2019. [Google Scholar]
- Mukhopadhyay, U.; Paul, A.; Majumdar, D. Probing Pseudo Nambu Goldstone Boson Dark Energy Models with Dark Matter—Dark Energy Interaction. arXiv
**2019**, arXiv:1909.03925. [Google Scholar] - Yang, W.; Pan, S.; Vagnozzi, S.; Di Valentino, E.; Mota, D.F.; Capozziello, S. Dawn of the dark: unified dark sectors and the EDGES Cosmic Dawn 21-cm signal. JCAP
**2019**, 1911, 44. [Google Scholar] [CrossRef] [Green Version] - Guendelman, E.I.; Kaganovich, A.B. The Principle of nongravitating vacuum energy and some of its consequences. Phys. Rev.
**1996**, D53, 7020–7025. [Google Scholar] [CrossRef] [Green Version] - Gronwald, F.; Muench, U.; Macias, A.; Hehl, F.W. Volume elements of space-time and a quartet of scalar fields. Phys. Rev.
**1998**, D58, 084021. [Google Scholar] [CrossRef] [Green Version] - Guendelman, E.I.; Kaganovich, A.B. Dynamical measure and field theory models free of the cosmological constant problem. Phys. Rev.
**1999**, D60, 065004. [Google Scholar] [CrossRef] [Green Version] - Guendelman, E.I. Scale invariance, new inflation and decaying lambda terms. Mod. Phys. Lett.
**1999**, A14, 1043–1052. [Google Scholar] [CrossRef] [Green Version] - Guendelman, E.I.; Kaganovich, A.B. Absence of the Fifth Force Problem in a Model with Spontaneously Broken Dilatation Symmetry. Ann. Phys.
**2008**, 323, 866–882. [Google Scholar] [CrossRef] [Green Version] - Guendelman, E.; Nissimov, E.; Pacheva, S.; Vasihoun, M. A New Mechanism of Dynamical Spontaneous Breaking of Supersymmetry. Bulg. J. Phys.
**2014**, 41, 123–129. [Google Scholar] - Guendelman, E.; Nissimov, E.; Pacheva, S. Vacuum structure and gravitational bags produced by metric-independent space–time volume-form dynamics. Int. J. Mod. Phys.
**2015**, A30, 1550133. [Google Scholar] [CrossRef] - Guendelman, E.; Nissimov, E.; Pacheva, S. Unified Dark Energy and Dust Dark Matter Dual to Quadratic Purely Kinetic K-Essence. Eur. Phys. J.
**2016**, C76, 90. [Google Scholar] [CrossRef] [Green Version] - Guendelman, E.; Singleton, D.; Yongram, N. A two measure model of dark energy and dark matter. JCAP
**2012**, 1211, 44. [Google Scholar] [CrossRef] [Green Version] - Guendelman, E.; Nissimov, E.; Pacheva, S. Dark Energy and Dark Matter From Hidden Symmetry of Gravity Model with a Non-Riemannian Volume Form. Eur. Phys. J.
**2015**, C75, 472. [Google Scholar] [CrossRef] [Green Version] - Guendelman, E.; Nissimov, E.; Pacheva, S. Gravity-Assisted Emergent Higgs Mechanism in the Post-Inflationary Epoch. Int. J. Mod. Phys.
**2016**, D25, 1644008. [Google Scholar] [CrossRef] - Guendelman, E.; Nissimov, E.; Pacheva, S. Modified Gravity and Inflaton Assisted Dynamical Generation of Charge Confinement and Electroweak Symmetry Breaking in Cosmology. AIP Conf. Proc.
**2019**, 2075, 090030. [Google Scholar] [CrossRef] - Guendelman, E.; Nissimov, E.; Pacheva, S. Unification of Inflation and Dark Energy from Spontaneous Breaking of Scale Invariance. In Proceedings of the 8th Mathematical Physics Meeting, Summer School and Conference on Modern Mathematical Physics, Belgrade, Serbia, 24–31 August 2014; pp. 93–103. [Google Scholar]
- Frieman, J.; Turner, M.; Huterer, D. Dark Energy and the Accelerating Universe. Ann. Rev. Astron. Astrophys.
**2008**, 46, 385–432. [Google Scholar] [CrossRef] [Green Version] - Mathews, G.J.; Kusakabe, M.; Kajino, T. Introduction to Big Bang Nucleosynthesis and Modern Cosmology. Int. J. Mod. Phys.
**2017**, E26, 1741001. [Google Scholar] [CrossRef] [Green Version] - Liddle, A. Einfuehrung in die Moderne Kosmologie; Wiley-VCH: Berlin, Germany, 2008. [Google Scholar]
- Liddle, A.R. An Introduction to Modern Cosmology; Wiley-VCH: West Sussex, UK, 2003. [Google Scholar]
- Dodelson, S. Modern Cosmology; Academic Press: Amsterdam, The Netherlands, 2003. [Google Scholar]
- Dodelson, S.; Easther, R.; Hanany, S.; McAllister, L.; Meyer, S.; Page, L.; Ade, P.; Amblard, A.; Ashoorioon, A.; Baccigalupi, C.; et al. The Origin of the Universe as Revealed Through the Polarization of the Cosmic Microwave Background. arXiv
**2009**, arXiv:0902.3796. [Google Scholar] - Baumann, D.; Cooray, A.; Dodelson, S.; Dunkley, J.; Fraisse, A.A.; Jackson, M.G.; Kogut, A.; Krauss, L.M.; Smith, K.M.; Zaldarriaga, M. CMBPol Mission Concept Study: A Mission to Map our Origins. AIP Conf. Proc.
**2009**, 1141, 3–9. [Google Scholar] [CrossRef] - Dodelson, S. Cosmic microwave background: Past, future, and present. Int. J. Mod. Phys.
**2000**, A15S1, 765–783. [Google Scholar] [CrossRef] [Green Version] - Dabrowski, M.P.; Garecki, J.; Blaschke, D.B. Conformal transformations and conformal invariance in gravitation. Annalen Phys.
**2009**, 18, 13–32. [Google Scholar] [CrossRef] [Green Version] - Angus, C.R.; Smith, M.; Sullivan, M.; Inserra, C.; Wiseman, P.; D’Andrea, C.B.; Thomas, B.P.; Nichol, R.C.; Galbany, L.; Childress, M.; et al. Superluminous Supernovae from the Dark Energy Survey. Mon. Not. R. Astron. Soc.
**2019**, 487, 2215–2241. [Google Scholar] [CrossRef] - Zhang, Y.; Yanny, B.; Palmese, A.; Gruen, D.; To, C.; Rykoff, E.S.; Leung, Y.; Collins, C.; Hilton, M.; Abbott, T.M.; et al. Dark Energy Survey Year 1 results: Detection of Intra-cluster Light at Redshift ∼0.25. Astrophys. J.
**2019**, 874, 165. [Google Scholar] [CrossRef] [Green Version] - Bahamonde, S.; Böhmer, C.G.; Carloni, S.; Copeland, E.J.; Fang, W.; Tamanini, N. Dynamical systems applied to cosmology: Dark energy and modified gravity. Phys. Rept.
**2018**, 775–777, 1–122. [Google Scholar] [CrossRef] [Green Version] - Ade, P.A.; Aghanim, N.; Armitage-Caplan, C.; Arnaud, M.; Ashdown, M.; Atrio-Barandela, F.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; et al. Planck 2013 results. XXII. Constraints on inflation. Astron. Astrophys.
**2014**, 571, A22. [Google Scholar] [CrossRef] [Green Version] - Adam, R.; Ade, P.A.; Aghanim, N.; Arnaud, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; Bartlett, J.G.; Bartolo, N.; et al. Planck intermediate results-XXX. The angular power spectrum of polarized dust emission at intermediate and high Galactic latitudes. Astron. Astrophys.
**2016**, 586, A133. [Google Scholar] [CrossRef] [Green Version] - Arkani-Hamed, N.; Hall, L.J.; Kolda, C.F.; Murayama, H. A New perspective on cosmic coincidence problems. Phys. Rev. Lett.
**2000**, 85, 4434–4437. [Google Scholar] [CrossRef] [Green Version] - Martin, J.; Ringeval, C.; Vennin, V. Encyclopædia Inflationaris. Phys. Dark Univ.
**2014**, 5–6, 75–235. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Shape of the effective potential ${U}_{\mathrm{eff}}\left(u\right)$ in the Einstein-frame (37). The physical unit for u is ${M}_{Pl}/\sqrt{2}$.

**Figure 2.**Numerical shape of the evolution of $u\left(t\right)$. The physical unit for u is ${M}_{Pl}/\sqrt{2}$.

**Figure 3.**Slow-roll parameters $\u03f5$ and $\eta $ before and around end of inflation. When $\u03f5=1$ the inflation ends.

**Figure 4.**Numerical shape of the evolution of $H\left(t\right)$. Here ${H}_{**}\equiv \sqrt{{\Lambda}_{0}/3}$ as in (58).

**Figure 5.**In the left panel—blown-up portion of the plot on Figure 2 around and after end of inflation depicting the oscillations of $u\left(t\right)$ after end of inflation. In the right panel—oscillations of $\stackrel{.}{u}\left(t\right)$ after end of inflation.

**Figure 6.**Evolution of w parameter of the equation of state with sharp growth above $w\approx -1$ for a short time interval after end of inflation—matter domination.

**Figure 7.**The scalar to tensor ratio r and the scalar spectral index ${n}_{s}$ vs. the number of e-folds for different values of the initial conditions. The sampling of the latter is done with a normal distribution ${\Lambda}_{0}=50\pm 10$, ${M}_{1}=20\pm 10$.

**Figure 8.**The relation between the scalar to tensor ratio r and the scalar spectral index ${n}_{s}$ via sampled initial conditions with a normal distribution ${\Lambda}_{0}=50\pm 10$, ${M}_{1}=20\pm 10$. All of the sampled values fall well inside the Planck data constraint (67).

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Benisty, D.; Guendelman, E.I.; Nissimov, E.; Pacheva, S.
Dynamically Generated Inflationary ΛCDM. *Symmetry* **2020**, *12*, 481.
https://doi.org/10.3390/sym12030481

**AMA Style**

Benisty D, Guendelman EI, Nissimov E, Pacheva S.
Dynamically Generated Inflationary ΛCDM. *Symmetry*. 2020; 12(3):481.
https://doi.org/10.3390/sym12030481

**Chicago/Turabian Style**

Benisty, David, Eduardo I. Guendelman, Emil Nissimov, and Svetlana Pacheva.
2020. "Dynamically Generated Inflationary ΛCDM" *Symmetry* 12, no. 3: 481.
https://doi.org/10.3390/sym12030481