# Non-Standard Hierarchies of the Runnings of the Spectral Index in Inﬂation

## Abstract

**:**

## 1. Introduction

## 2. Predictions of Single Field Slow-Roll Inflation

## 3. Multi-Field Scenarios and Isocurvature Perturbations

## 4. Other Approaches

#### 4.1. Variable Speed-Of-Light Cosmology

#### 4.2. Cosmological Perturbations in Quantum Gravity

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Guth, A.H. The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Phys. Rev. D
**1981**, 23, 347–356. [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. B
**1982**, 108, 389–393. [Google Scholar] [CrossRef] - Escudero, M.; Ramírez, H.; Boubekeur, L.; Giusarma, E.; Mena, O. The present and future of the most favoured inflationary models after Planck 2015. J. Cosmol. Astropart. Phys.
**2016**, 2016, 020. [Google Scholar] [CrossRef] - Martin, J.; Ringeval, C.; Vennin, V. Encyclopaedia Inflationaris. Phys. Dark Universe
**2014**, 5–6, 75–235. [Google Scholar] [CrossRef] - Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; et al. Planck 2015 results. XX. Constraints on inflation. Astron. Astrophys.
**2015**, 594, A20. [Google Scholar] - Ade, P.A.R.; Aghanim, N.; Ahmed, Z.; Aikin, R.W.; Alexander, K.D.; Arnaud, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barkats, D.; et al. Joint Analysis of BICEP2/KeckArray and Planck Data. Phys. Rev. Lett.
**2015**, 114, 101301. [Google Scholar] [CrossRef] [PubMed] - Cabass, G.; Di Valentino, E.; Melchiorri, A.; Pajer, E.; Silk, J. Constraints on the running of the running of the scalar tilt from CMB anisotropies and spectral distortions. Phys. Rev. D
**2016**, 94, 023523. [Google Scholar] [CrossRef] - Cabass, G.; Melchiorri, A.; Pajer, E. μ distortions or running: A guaranteed discovery from CMB spectrometry. Phys. Rev. D
**2016**, 93, 083515. [Google Scholar] [CrossRef] - Chluba, J. Which spectral distortions does ΛCDM actually predict? Mon. Not. R Astron. Soc.
**2016**, 460, 227–239. [Google Scholar] [CrossRef] - Chluba, J.; Hamann, J.; Patil, S.P. Features and New Physical Scales in Primordial Observables: Theory and Observation. Int. J. Mod. Phys. D
**2015**, 24, 1530023. [Google Scholar] [CrossRef] - Kogut, A.; Fixsen, D.J.; Chuss, D.T.; Dotson, J.; Dwek, E.; Halpern, M.; Hinshaw, G.F.; Meyer, S.M.; Moseley, S.H.; Seiffert, M.D.; et al. The Primordial Inflation Explorer (PIXIE): A nulling polarimeter for cosmic microwave background observations. J. Cosmol. Astropart. Phys.
**2011**, 2011, 025. [Google Scholar] [CrossRef] - Di Valentino, E.; Brinckmann, T.; Gerbino, M.; Poulin, V.; Bouchet, F.R.; Lesgourgues, J.; Melchiorri, A.; Chluba, J.; Clesse, S.; Delabrouille, J.; et al. Exploring Cosmic Origins with CORE: Cosmological Parameters. arXiv, 2016; arXiv:1612.00021. [Google Scholar]
- Finelli, F.; Bucher, M.; Achúcarro, A.; Ballardini, M.; Bartolo, N.; Baumann, D.; Clesse, S.; Errard, J.; Handley, W.; Hindmarsh, M.; et al. Exploring Cosmic Origins with CORE: Inflation. arXiv, 2016; arXiv:1612.08270. [Google Scholar]
- André, P.; Baccigalupi, C.; Banday, A.; Barbosa, D.; Barreiro, B.; Bartlett, J.; Bartolo, N.; Battistelli, E.; Battye, R.; Bendo, G.; et al. PRISM (Polarized Radiation Imaging and Spectroscopy Mission): An Extended White Paper. J. Cosmol. Astropart. Phys.
**2014**, 2014, 006. [Google Scholar] [CrossRef] [PubMed] - Battye, R.A.; Davies, R.D.; Weller, J. Neutral hydrogen surveys for high redshift galaxy clusters and proto-clusters. Mon. Not. R. Astron. Soc.
**2004**, 355, 1339–1347. [Google Scholar] [CrossRef] - Maartens, R.; Abdalla, F.B.; Jarvis, M.; Santos, M.G. Overview of Cosmology with the SKA. In Proceedings of the Advancing Astrophysics with the Square Kilometre Array (AASKA14), Giardini Naxos, Italy, 9–13 June 2014.
- Amendola, L.; Appleby, S.; Bacon, D.; Baker, T.; Baldi, M.; Bartolo, N.; Blanchard, A.; Bonvin, C.; Borgani, S.; Branchini, E.; et al. Cosmology and Fundamental Physics with the Euclid Satellite. Living Rev. Relativ.
**2016**, 16, 6. [Google Scholar] [CrossRef] - Pourtsidou, A. Synergistic tests of inflation. arXiv, 2016; arXiv:1612.05138. [Google Scholar]
- Adams, J.A.; Cresswell, B.; Easther, R. Inflationary perturbations from a potential with a step. Phys. Rev. D
**2001**, 64, 123514. [Google Scholar] [CrossRef] - Ashoorioon, A.; van de Bruck, C.; Millington, P.; Vu, S. Effect of transitions in the Planck mass during inflation on primordial power spectra. Phys. Rev. D
**2014**, 90, 103515. [Google Scholar] [CrossRef] - Muñoz, J.B.; Kovetz, E.D.; Raccanelli, A.; Kamionkowski, M.; Silk, J. Towards a measurement of the spectral runnings. arXiv, 2016; arXiv:1611.05883. [Google Scholar]
- Garcia-Bellido, J.; Roest, D. Large-N running of the spectral index of inflation. Phys. Rev. D
**2014**, 89, 103527. [Google Scholar] [CrossRef] - Gariazzo, S.; Mena, O.; Ramirez, H.; Boubekeur, L. Primordial power spectrum features in phenomenological descriptions of inflation. arXiv, 2016; arXiv:1606.00842. [Google Scholar]
- Kohri, K.; Matsuda, T. Ambiguity in running spectral index with an extra light field during inflation. J. Cosmol. Astropart. Phys.
**2015**, 2015, 019. [Google Scholar] [CrossRef] - Peloso, M.; Sorbo, L.; Tasinato, G. A falsely fat curvaton with an observable running of the spectral tilt. J. Cosmol. Astropart. Phys.
**2014**, 2014, 040. [Google Scholar] [CrossRef] - Wands, D.; Bartolo, N.; Matarrese, S.; Riotto, A. Observational test of two-field inflation. Phys. Rev. D.
**2002**, 66, 043520. [Google Scholar] [CrossRef] - Ashoorioon, A.; Krause, A.; Turzynski, K. Energy Transfer in Multi Field Inflation and Cosmological Perturbations. J. Cosmol. Astropart. Phys.
**2009**, 2009, 014. [Google Scholar] [CrossRef] - Lalak, Z.; Langlois, D.; Pokorski, S.; Turzynski, K. Curvature and isocurvature perturbations in two-field inflation. J. Cosmol. Astropart. Phys.
**2007**, 2007, 014. [Google Scholar] [CrossRef] - Di Marco, F.; Finelli, F.; Brandenberger, R. Adiabatic and isocurvature perturbations for multifield generalized Einstein models. Phys. Rev. D
**2003**, 67, 063512. [Google Scholar] [CrossRef] - Di Marco, F.; Finelli, F. Slow-roll inflation for generalized two-field Lagrangians. Phys. Rev. D
**2005**, 71, 123502. [Google Scholar] [CrossRef] - Van de Bruck, C.; Robinson, M. Power Spectra beyond the Slow Roll Approximation in Theories with Non-Canonical Kinetic Terms. J. Cosmol. Astropart. Phys.
**2014**, 2014, 024. [Google Scholar] [CrossRef] - Van de Bruck, C.; Longden, C. Running of the Running and Entropy Perturbations During Inflation. Phys. Rev. D
**2016**, 94, 021301. [Google Scholar] [CrossRef] - Kaiser, D.I.; Sfakianakis, E.I. Multifield Inflation after Planck: The Case for Nonminimal Couplings. Phys. Rev. Lett.
**2014**, 112, 011302. [Google Scholar] [CrossRef] [PubMed] - Schutz, K.; Sfakianakis, E.I.; Kaiser, D.I. Multifield Inflation after Planck: Isocurvature Modes from Nonminimal Couplings. Phys. Rev. D
**2014**, 89, 064044. [Google Scholar] [CrossRef] - Moffat, J.W. Superluminary universe: A Possible solution to the initial value problem in cosmology. Int. J. Mod. Phys. D
**1993**, 2, 351–366. [Google Scholar] [CrossRef] - Afshordi, N.; Magueijo, J. Critical geometry of a thermal big bang. Phys. Rev. D
**2016**, 94, 101301. [Google Scholar] [CrossRef] - Moffat, J.W. Variable speed of light cosmology and bimetric gravity: An Alternative to standard inflation. Int. J. Mod. Phys. A
**2005**, 20, 1155–1162. [Google Scholar] [CrossRef] - Magueijo, J. New varying speed of light theories. Rep. Prog. Phys.
**2003**, 66, 2025–2068. [Google Scholar] [CrossRef] - Moffat, J.W. Variable speed of light cosmology: An Alternative to inflation. arXiv, 2002; arXiv:hep-th/0208122. [Google Scholar]
- Van de Bruck, C.; Burrage, C.; Morrice, J. Vacuum Cherenkov radiation and bremsstrahlung from disformal couplings. J. Cosmol. Astropart. Phys.
**2016**, 2016, 003. [Google Scholar] [CrossRef] [PubMed] - Salzano, V.; Dabrowski, M.P. Statistical hierarchy of varying speed of light cosmologies. arXiv, 2016; arXiv:1612.06367. [Google Scholar]
- Moffat, J.W. Bimetric Gravity, Variable Speed of Light Cosmology and Planck2013. arXiv, 2013; arXiv:1306.5470. [Google Scholar]
- Kamenshchik, A.Y.; Tronconi, A.; Venturi, G. Quantum Cosmology and the Evolution of Inflationary Spectra. Phys. Rev. D
**2016**, 94, 123524. [Google Scholar] [CrossRef] - Brizuela, D.; Kiefer, C.; Kraemer, M. Quantum-gravitational effects on gauge-invariant scalar and tensor perturbations during inflation: The de Sitter case. Phys. Rev. D
**2016**, 93, 104035. [Google Scholar] [CrossRef] - Brizuela, D.; Kiefer, C.; Kraemer, M. Quantum-gravitational effects on gauge-invariant scalar and tensor perturbations during inflation: The slow-roll approximation. Phys. Rev. D
**2016**, 94, 123527. [Google Scholar] [CrossRef] - Bojowald, M.; Calcagni, G.; Tsujikawa, S. Observational test of inflation in loop quantum cosmology. J. Cosmol. Astropart. Phys.
**2011**, 2011, 046. [Google Scholar] [CrossRef]

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Longden, C.
Non-Standard Hierarchies of the Runnings of the Spectral Index in Inﬂation. *Universe* **2017**, *3*, 17.
https://doi.org/10.3390/universe3010017

**AMA Style**

Longden C.
Non-Standard Hierarchies of the Runnings of the Spectral Index in Inﬂation. *Universe*. 2017; 3(1):17.
https://doi.org/10.3390/universe3010017

**Chicago/Turabian Style**

Longden, Chris.
2017. "Non-Standard Hierarchies of the Runnings of the Spectral Index in Inﬂation" *Universe* 3, no. 1: 17.
https://doi.org/10.3390/universe3010017