# Interpolation Formulas for Asymptotically Safe Cosmology

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Interpolation Formulas

## 3. Model Universe

#### 3.1. Scheme A

#### 3.2. Scheme B

## 4. Analytical Solution

#### 4.1. Scheme A

#### 4.1.1. Time Interval $0<t\le {t}_{G}$

#### 4.1.2. Time Interval ${t}_{G}\le t\le {t}_{\mathsf{\Lambda}}$

#### 4.1.3. Evolution for $t\ge {t}_{\mathsf{\Lambda}}$

#### 4.2. Scheme B

#### 4.2.1. Time Interval $0<t\le {t}_{G}$

#### 4.2.2. Time Interval ${t}_{G}\le t\le {t}_{\mathsf{\Lambda}}$

#### 4.2.3. Evolution for $t\ge {t}_{\mathsf{\Lambda}}$

#### 4.3. Typical Time Scales

## 5. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Hindmarsh, M.; Litim, D.; Rahmede, C. Asymptotically Safe Cosmology. JCAP
**2011**, 7, 019. [Google Scholar] [CrossRef] [Green Version] - Bonanno, A.; Saueressig, F. Asymptotically safe cosmology—A status report. Comptes Rendus Phys.
**2017**, 18, 254. [Google Scholar] [CrossRef] - Mandal, R.; Gangopadhyay, S.; Lahir, A. Cosmology with modified continuity equation in asymptotically safe gravity. Eur. Phys. J. Plus
**2022**, 137, 1110. [Google Scholar] [CrossRef] - Tye, S.-H.H.; Xu, J. Comment on Asymptotically Safe Inflation. Phys. Rev. D
**2010**, 82, 127302. [Google Scholar] [CrossRef] [Green Version] - Weinberg, S. Asymptotically Safe Inflation. Phys. Rev. D
**2010**, 81, 083535. [Google Scholar] [CrossRef] [Green Version] - Biemans, J.; Platania, A.; Saueressig, F. Quantum gravity on foliated spacetime - asymptotically safe and sound. Phys. Rev. D
**2017**, 95, 086013. [Google Scholar] [CrossRef] [Green Version] - Weinberg, S. Critical Phenomena for Field Theorists. In Understanding of Fundamental Constituents of Matter; Zichichi, A., Ed.; Plenum Press: New York, NY, USA, 1977. [Google Scholar]
- Reuter, M.; Saueressig, F. Quantum Einstein Gravity. New J. Phys.
**2012**, 14, 055022. [Google Scholar] [CrossRef] - Reuter, M.; Saueressig, F. Quantum Gravity and the Functional Renormalization Group: The Road towards Asymptotic Safety; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar]
- Bonanno, A.; Eichhorn, A.; Gies, H.; Pawlowski, J.M.; Percacci, R.; Reuter, M.; Saueressig, F.; Vacca, G.P. Critical reflections on asymptotically safe gravity. Front. Phys.
**2020**, 8, 269. [Google Scholar] [CrossRef] - Reuter, M. Nonperturbative Evolution Equation for Quantum Gravity. Phys. Rev. D
**1998**, 57, 971. [Google Scholar] [CrossRef] [Green Version] - Lauscher, O.; Reuter, M. Ultraviolet Fixed Point and Generalized Flow Equation of Quantum Gravity. Phys. Rev. D
**2001**, 65, 025013. [Google Scholar] [CrossRef] [Green Version] - Lauscher, O.; Reuter, M. Is Quantum Einstein Gravity Nonperturbatively Renormalizable? Class. Quantum Gravity
**2002**, 19, 483. [Google Scholar] [CrossRef] - Bonanno, A.; Reuter, M. Proper time flow equation for gravity. JHEP
**2005**, 02, 035. [Google Scholar] [CrossRef] [Green Version] - Litim, D.F. Fixed points of quantum gravity. Phys. Rev. Lett.
**2004**, 92, 201301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Reuter, M.; Saueressig, F. Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity. arXiv
**2007**, arXiv:0708.1317. [Google Scholar] - Codello, A.; Percacci, R.; Rahmede, C. Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation. Ann. Phys.
**2009**, 324, 414. [Google Scholar] [CrossRef] [Green Version] - Christiansen, N.; Litim, D.F.; Pawlowski, J.M.; Rodigast, A. Fixed points and infrared completion of quantum gravity. Phys. Lett. B
**2014**, 728, 114. [Google Scholar] [CrossRef] [Green Version] - Donà, P.; Eichhorn, A.; Percacci, R. Matter matters in asymptotically safe quantum gravity. Phys. Rev. D
**2014**, 89, 084035. [Google Scholar] [CrossRef] [Green Version] - Dupuis, N.; Canet, L.; Eichhorn, A.; Metzner, W.; Pawlowski, J.M.; Tissier, M.; Wschebor, N. The nonperturbative functional renormalization group and its applications. Phys. Rept.
**2021**, 910, 1. [Google Scholar] [CrossRef] - Trivedi, O.; Khlopov, M. Singularity formation in asymptotically safe cosmology with inhomogeneous equation of state. JCAP
**2022**, 11, 007. [Google Scholar] [CrossRef] - Gubitosi, G.; Ripken, C.; Saueressig, F. Scales and hierarchies in asymptotically safe quantum gravity: A review. Found. Phys.
**2019**, 49, 972. [Google Scholar] [CrossRef] [Green Version] - Manrique, E.; Rechenberger, S.; Saueressig, F. Asymptotically Safe Lorentzian Gravity. Phys. Rev. Lett.
**2011**, 106, 251302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pawlowski, J.M.; Strodthoff, N. Real time correlation functions and the functional renormalization group. Phys. Rev. D
**2015**, 92, 094009. [Google Scholar] [CrossRef] [Green Version] - Nagy, S.; Sailer, K.; Steib, I. Renormalization of Lorentzian conformally reduced gravity. Class. Quantum Gravity
**2019**, 36, 155004. [Google Scholar] [CrossRef] - Knorr, B.; Schiffer, M. Non-Perturbative Propagators in Quantum Gravity. Universe
**2021**, 7, 216. [Google Scholar] [CrossRef] - Platania, A. Causality, unitarity and stability in quantum gravity: A non-perturbative perspective. J. High Energy Phys.
**2022**, 2022, 167. [Google Scholar] [CrossRef] - Weinberg, S. Cosmology; Oxford University Press: Oxford, UK, 2008. [Google Scholar]
- Bonanno, A.; Reuter, M. Cosmology of the Planck era from a renormalization group for quantum gravity. Phys. Rev. D
**2002**, 65, 043508. [Google Scholar] [CrossRef] [Green Version] - Bonanno, A.; Reuter, M. Cosmology with selfadjusting vacuum energy density from a renormalization group fixed point. Phys. Lett. B
**2002**, 527, 9. [Google Scholar] [CrossRef] [Green Version] - Babić, A.; Guberina, B.; Horvat, R.; Stefancić, H. Renormalization-group running cosmologies—A scale-setting procedure. Phys.Rev. D
**2005**, 71, 124041. [Google Scholar] [CrossRef] [Green Version] - Copeland, E.J.; Liddle, A.R.; Wands, D. Exponential potentials and cosmological scaling solutions. Phys. Rev. D
**1998**, 57, 4686. [Google Scholar] [CrossRef] [Green Version] - Copeland, E.J.; Sami, M.; Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys. D
**2006**, 15, 1753. [Google Scholar] [CrossRef] [Green Version] - Mukhanov, V. Physical Foundations of Cosmology; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products; Jeffrey, A., Zwillinger, D., Eds.; Academic Press: New York, NY, USA, 2007. [Google Scholar]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nagy, S.; Sailer, K.
Interpolation Formulas for Asymptotically Safe Cosmology. *Universe* **2023**, *9*, 184.
https://doi.org/10.3390/universe9040184

**AMA Style**

Nagy S, Sailer K.
Interpolation Formulas for Asymptotically Safe Cosmology. *Universe*. 2023; 9(4):184.
https://doi.org/10.3390/universe9040184

**Chicago/Turabian Style**

Nagy, Sandor, and Kornel Sailer.
2023. "Interpolation Formulas for Asymptotically Safe Cosmology" *Universe* 9, no. 4: 184.
https://doi.org/10.3390/universe9040184