Clocking the End of Cosmic Inflation

Author(s)

Martin, Jerome, Ringeval, Christophe, Vennin, Vincent

Abstract

Making observable predictions for cosmic inflation requires determining when the wavenumbers of astrophysical interest today exited the Hubble radius during the inflationary epoch. These instants are commonly evaluated using the slow-roll approximation and measured in e-folds $\Delta N=N-N_\mathrm{end}$, in reference to the e-fold $N_\mathrm{end}$ at which inflation ended. Slow roll being necessarily violated towards the end of inflation, both the approximated trajectory and $N_\mathrm{end}$ are determined at, typically, one or two e-folds precision. Up to now, such an uncertainty has been innocuous, but this will no longer be the case with the forthcoming cosmological measurements. In this work, we introduce a new and simple analytical method, on top of the usual slow-roll approximation, that reduces uncertainties on $\Delta N$ to less than a tenth of an e-fold.

Figures

Absolute error, in {\efolds}, of the slow-roll approximated trajectory (in red) with respect to the exact value of $\Delta N(\phi)$ for various prototypical models of inflation. The blue curve shows $\Delta\Nsree(\phi) - \Delta N(\phi)$ where $\Delta\Nsree = \Nsr(\phi) - \Nsr(\phiend)$, $\phiend$ being the \emph{exact} field value at which inflation stops. The differences between the red and blue curves are the errors induced by using $\phiendsr$ instead of $\phiend$ (see text). Let us notice that the Pseudo Natural Inflationary model (lower right) is an extreme case as it has its parameters purposely chosen to be in a slow-roll violating regime (incompatible with current data).

Absolute error, in {\efolds}, of the slow-roll approximated trajectory (in red) with respect to the exact value of $\Delta N(\phi)$ for various prototypical models of inflation. The blue curve shows $\Delta\Nsree(\phi) - \Delta N(\phi)$ where $\Delta\Nsree = \Nsr(\phi) - \Nsr(\phiend)$, $\phiend$ being the \emph{exact} field value at which inflation stops. The differences between the red and blue curves are the errors induced by using $\phiendsr$ instead of $\phiend$ (see text). Let us notice that the Pseudo Natural Inflationary model (lower right) is an extreme case as it has its parameters purposely chosen to be in a slow-roll violating regime (incompatible with current data).


Absolute error, in {\efolds}, of the velocity-corrected trajectory $\Delta\Nsrvc - \Delta N$ (blue curve), of the velocity plus end-point corrected trajectories $\Delta\Nsrvcm - \Delta N$ (green curve) and $\Delta\Nsrvcp - \Delta N$ (magenta curve), with respect to the exact value of $\Delta N(\phi)$ for various prototypical models of inflation. The red curve is the error associated with the traditional slow-roll approximation, same as in \cref{fig:srtrajs}.

Absolute error, in {\efolds}, of the velocity-corrected trajectory $\Delta\Nsrvc - \Delta N$ (blue curve), of the velocity plus end-point corrected trajectories $\Delta\Nsrvcm - \Delta N$ (green curve) and $\Delta\Nsrvcp - \Delta N$ (magenta curve), with respect to the exact value of $\Delta N(\phi)$ for various prototypical models of inflation. The red curve is the error associated with the traditional slow-roll approximation, same as in \cref{fig:srtrajs}.


References
  • [1] A. A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30 (1979) 682–685.
  • [2] A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91 (1980) 99–102.
  • [3] A. H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D 23 (1981) 347–356.
  • [4] A. D. Linde, A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys. Lett. B 108 (1982) 389–393.
  • [5] A. Albrecht and P. J. Steinhardt, Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Phys. Rev. Lett. 48 (1982) 1220–1223.
  • [6] A. D. Linde, Chaotic Inflation, Phys. Lett. B 129 (1983) 177–181.
  • [7] V. F. Mukhanov and G. V. Chibisov, Quantum Fluctuations and a Nonsingular Universe, JETP Lett. 33 (1981) 532–535.
  • [8] V. F. Mukhanov and G. V. Chibisov, The Vacuum energy and large scale structure of the universe, Sov. Phys. JETP 56 (1982) 258–265.
  • [9] A. A. Starobinsky, Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys. Lett. B 117 (1982) 175–178.
  • [10] A. H. Guth and S. Y. Pi, Fluctuations in the New Inflationary Universe, Phys. Rev. Lett. 49 (1982) 1110–1113.
  • [11] S. W. Hawking, The Development of Irregularities in a Single Bubble Inflationary Universe, Phys. Lett. B 115 (1982) 295.
  • [12] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Spontaneous creation of almost scale-free density perturbations in an inflationary universe, Phys. Rev. D 28 (1983) 679–693.
  • [13] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203–333.
  • [14] J. Martin, C. Ringeval and V. Vennin, Encyclopædia Inflationaris, Phys. Dark Univ. 5-6 (2014) 75–235, [1303.3787].
  • [15] D. Baumann and L. McAllister, Inflation and String Theory. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 5, 2015, 10.1017/CBO9781316105733.
  • [16] V. Vennin, K. Koyama and D. Wands, Encyclopædia curvatonis, JCAP 11 (2015) 008, [1507.07575].
  • [17] Planck collaboration, Y. Akrami et al., Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641 (2020) A10, [1807.06211].
  • [18] D. Chowdhury, J. Martin, C. Ringeval and V. Vennin, Inflation after Planck: Judgement Day, Phys. Rev. D 100 (2019) 083537, [1902.03951].
  • [19] J. Martin, C. Ringeval and V. Vennin, Cosmic Inflation at the Crossroads, 2404.10647.
  • [20] CMB-S4 collaboration, K. N. Abazajian et al., CMB-S4 Science Book, First Edition, 1610.02743.
  • [21] Simons Observatory collaboration, P. Ade et al., The Simons Observatory: Science goals and forecasts, JCAP 02 (2019) 056, [1808.07445].
  • [22] M. Mallaby-Kay et al., The Atacama Cosmology Telescope: Summary of DR4 and DR5 Data Products and Data Access, Astrophys. J. Supp. 255 (2021) 11, [2103.03154].
  • [23] Euclid collaboration, R. Scaramella et al., Euclid space mission: a cosmological challenge for the next 15 years, IAU Symp. 306 (2014) 375–378, [1501.04908].
  • [24] Euclid collaboration, S. Ilić et al., Euclid preparation. XV. Forecasting cosmological constraints for the Euclid and CMB joint analysis, Astron. Astrophys. 657 (2022) A91, [2106.08346].
  • [25] LSST Science, LSST Project collaboration, P. A. Abell et al., LSST Science Book, Version 2.0, 0912.0201.
  • [26] LiteBIRD collaboration, E. Allys et al., Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background Polarization Survey, PTEP 2023 (2023) 042F01, [2202.02773].
  • [27] C. Ringeval, Fast Bayesian inference for slow-roll inflation, Mon. Not. Roy. Astron. Soc. 439 (Apr., 2014) 3253–3261, [1312.2347].
  • [28] J. Martin, C. Ringeval and V. Vennin, Shortcomings of New Parametrizations of Inflation, Phys. Rev. D 94 (2016) 123521, [1609.04739].
  • [29] H. Kurki-Suonio, P. Laguna and R. A. Matzner, Inhomogeneous inflation: Numerical evolution, Phys. Rev. D 48 (1993) 3611–3624, [astro-ph/9306009].
  • [30] W. E. East, M. Kleban, A. Linde and L. Senatore, Beginning inflation in an inhomogeneous universe, JCAP 09 (2016) 010, [1511.05143].
  • [31] K. Clough, R. Flauger and E. A. Lim, Robustness of Inflation to Large Tensor Perturbations, JCAP 05 (2018) 065, [1712.07352].
  • [32] J. C. Aurrekoetxea, K. Clough, R. Flauger and E. A. Lim, The Effects of Potential Shape on Inhomogeneous Inflation, JCAP 05 (2020) 030, [1910.12547].
  • [33] C. Joana and S. Clesse, Inhomogeneous preinflation across Hubble scales in full general relativity, Phys. Rev. D 103 (2021) 083501, [2011.12190].
  • [34] C. Joana, Gravitational dynamics in Higgs inflation: Preinflation and preheating with an auxiliary field, Phys. Rev. D 106 (2022) 023504, [2202.07604].
  • [35] M. Elley, J. C. Aurrekoetxea, K. Clough, R. Flauger, P. Giannadakis and E. A. Lim, Robustness of inflation to kinetic inhomogeneities, 2405.03490.
  • [36] C. Joana, Beginning inflation in conformally curved spacetimes, 2406.00811.
  • [37] A. A. Starobinsky and J. Yokoyama, Equilibrium state of a selfinteracting scalar field in the De Sitter background, Phys. Rev. D50 (1994) 6357–6368, [astro-ph/9407016].
  • [38] A. A. Starobinsky, Stochastic de Sitter (inflationary) Stage in the Early Universe, in Field Theory, Quantum Gravity and Strings (H. J. de Vega & N. Sánchez, ed.), vol. 246 of Lecture Notes in Physics, Berlin Springer Verlag, p. 107, 1986. DOI.
  • [39] V. Vennin and A. A. Starobinsky, Correlation Functions in Stochastic Inflation, Eur. Phys. J. C 75 (2015) 413, [1506.04732].
  • [40] K. Ando and V. Vennin, Power spectrum in stochastic inflation, JCAP 04 (2021) 057, [2012.02031].
  • [41] B. Blachier, P. Auclair, C. Ringeval and V. Vennin, Spatial curvature from super-Hubble cosmological fluctuations, Phys. Rev. D 108 (2023) 123510, [2302.14530].
  • [42] K. Tokeshi and V. Vennin, Why does inflation look single field to us?, 2310.16649.
  • [43] D. S. Salopek and J. R. Bond, Nonlinear evolution of long-wavelength metric fluctuations in inflationary models, Phys. Rev. D 42 (Dec, 1990) 3936–3962.
  • [44] J. A. Adams, B. Cresswell and R. Easther, Inflationary perturbations from a potential with a step, Phys. Rev. D64 (2001) 123514, [astro-ph/0102236].
  • [45] C. Ringeval, P. Brax, C. van de Bruck and A.-C. Davis, Boundary inflation and the wmap data, Phys.Rev. D73 (2006) 064035, [astro-ph/0509727].
  • [46] A. Makarov, On the accuracy of slow-roll inflation given current observational constraints, Phys. Rev. D72 (2005) 083517, [astro-ph/0506326].
  • [47] M. J. Mortonson, H. V. Peiris and R. Easther, Bayesian Analysis of Inflation: Parameter Estimation for Single Field Models, Phys.Rev. D83 (2011) 043505, [1007.4205].
  • [48] L. C. Price, H. V. Peiris, J. Frazer and R. Easther, Designing and testing inflationary models with Bayesian networks, JCAP 02 (2016) 049, [1511.00029].
  • [49] D. Seery, CppTransport: a platform to automate calculation of inflationary correlation functions, 1609.00380.
  • [50] D. Werth, L. Pinol and S. Renaux-Petel, CosmoFlow: Python Package for Cosmological Correlators, 2402.03693.
  • [51] A. Caravano, K. Inomata and S. Renaux-Petel, The Inflationary Butterfly Effect: Non-Perturbative Dynamics From Small-Scale Features, 2403.12811.
  • [52] J. Martin and C. Ringeval, Inflation after WMAP3: Confronting the Slow-Roll and Exact Power Spectra to CMB Data, JCAP 08 (2006) 009, [astro-ph/0605367].
  • [53] R. Easther and H. V. Peiris, Bayesian Analysis of Inflation II: Model Selection and Constraints on Reheating, Phys.Rev. D85 (2012) 103533, [1112.0326].
  • [54] V. F. Mukhanov, L. A. Kofman and D. Y. Pogosian, Cosmological Perturbations in the Inflationary Universe, Phys. Lett. B 193 (1987) 427–432.
  • [55] V. F. Mukhanov, Quantum Theory of Gauge Invariant Cosmological Perturbations, Sov. Phys. JETP 67 (1988) 1297–1302.
  • [56] E. D. Stewart and D. H. Lyth, A more accurate analytic calculation of the spectrum of cosmological perturbations produced during inflation, Phys. Lett. B302 (1993) 171–175, [gr-qc/9302019].
  • [57] E. D. Stewart, The Spectrum of density perturbations produced during inflation to leading order in a general slow roll approximation, Phys. Rev. D 65 (2002) 103508, [astro-ph/0110322].
  • [58] J.-O. Gong and E. D. Stewart, The Density perturbation power spectrum to second order corrections in the slow roll expansion, Phys. Lett. B 510 (2001) 1–9, [astro-ph/0101225].
  • [59] D. J. Schwarz, C. A. Terrero-Escalante and A. A. Garcia, Higher order corrections to primordial spectra from cosmological inflation, Phys. Lett. B 517 (2001) 243–249, [astro-ph/0106020].
  • [60] S. M. Leach, A. R. Liddle, J. Martin and D. J. Schwarz, Cosmological parameter estimation and the inflationary cosmology, Phys. Rev. D 66 (2002) 023515, [astro-ph/0202094].
  • [61] J. Choe, J.-O. Gong and E. D. Stewart, Second order general slow-roll power spectrum, JCAP 07 (2004) 012, [hep-ph/0405155].
  • [62] D. J. Schwarz and C. A. Terrero-Escalante, Primordial fluctuations and cosmological inflation after WMAP 1.0, JCAP 0408 (2004) 003, [hep-ph/0403129].
  • [63] P. Auclair and C. Ringeval, Slow-roll inflation at N3LO, Phys. Rev. D 106 (2022) 063512, [2205.12608].
  • [64] J. Martin and D. J. Schwarz, Wkb approximation for inflationary cosmological perturbations, Phys. Rev. D67 (2003) 083512, [astro-ph/0210090].
  • [65] S. Habib, K. Heitmann, G. Jungman and C. Molina-Paris, The inflationary perturbation spectrum, Phys. Rev. Lett. 89 (2002) 281301, [astro-ph/0208443].
  • [66] S. Habib, A. Heinen, K. Heitmann, G. Jungman and C. Molina-Paris, Characterizing inflationary perturbations: The uniform approximation, Phys. Rev. D70 (2004) 083507, [astro-ph/0406134].
  • [67] R. Casadio, F. Finelli, M. Luzzi and G. Venturi, Improved wkb analysis of cosmological perturbations, Phys. Rev. D71 (2005) 043517, [gr-qc/0410092].
  • [68] R. Easther and J. T. Giblin, The Hubble slow roll expansion for multi field inflation, Phys. Rev. D72 (2005) 103505, [astro-ph/0505033].
  • [69] F. Di Marco and F. Finelli, Slow-roll inflation for generalized two-field Lagrangians, Phys. Rev. D71 (2005) 123502, [astro-ph/0505198].
  • [70] R. Casadio, F. Finelli, M. Luzzi and G. Venturi, Higher order slow-roll predictions for inflation, Phys. Lett. B625 (2005) 1–6, [gr-qc/0506043].
  • [71] X. Chen, M.-x. Huang, S. Kachru and G. Shiu, Observational signatures and non-Gaussianities of general single field inflation, JCAP 0701 (2007) 002, [hep-th/0605045].
  • [72] T. Battefeld and R. Easther, Non-Gaussianities in Multi-field Inflation, JCAP 0703 (2007) 020, [astro-ph/0610296].
  • [73] W. H. Kinney and K. Tzirakis, Quantum modes in DBI inflation: exact solutions and constraints from vacuum selection, Phys. Rev. D77 (2008) 103517, [0712.2043].
  • [74] S. Yokoyama, T. Suyama and T. Tanaka, Primordial Non-Gaussianity in Multi-Scalar Slow-Roll Inflation, JCAP 0707 (2007) 013, [0705.3178].
  • [75] L. Lorenz, J. Martin and C. Ringeval, K-inflationary Power Spectra in the Uniform Approximation, Phys.Rev. D78 (2008) 083513, [0807.3037].
  • [76] K. Tzirakis and W. H. Kinney, Non-canonical generalizations of slow-roll inflation models, JCAP 01 (2009) 028, [0810.0270].
  • [77] N. Agarwal and R. Bean, Cosmological constraints on general, single field inflation, Phys. Rev. D79 (2009) 023503, [0809.2798].
  • [78] T. Chiba and M. Yamaguchi, Extended Slow-Roll Conditions and Primordial Fluctuations: Multiple Scalar Fields and Generalized Gravity, JCAP 0901 (2009) 019, [0810.5387].
  • [79] K. Ichikawa, T. Suyama, T. Takahashi and M. Yamaguchi, Non-Gaussianity, Spectral Index and Tensor Modes in Mixed Inflaton and Curvaton Models, Phys. Rev. D78 (2008) 023513, [0802.4138].
  • [80] D. Langlois, S. Renaux-Petel, D. A. Steer and T. Tanaka, Primordial perturbations and non-Gaussianities in DBI and general multi-field inflation, Phys.Rev. D78 (2008) 063523, [0806.0336].
  • [81] A. De Felice and S. Tsujikawa, Conditions for the cosmological viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models, JCAP 1202 (2012) 007, [1110.3878].
  • [82] J. Martin, C. Ringeval and V. Vennin, K-inflationary Power Spectra at Second Order, JCAP 06 (2013) 021, [1303.2120].
  • [83] J. Beltran Jimenez, M. Musso and C. Ringeval, Exact Mapping between Tensor and Most General Scalar Power Spectra, Phys. Rev. D88 (2013) 043524, [1303.2788].
  • [84] A. Karam, T. Pappas and K. Tamvakis, Frame-dependence of higher-order inflationary observables in scalar-tensor theories, Phys. Rev. D 96 (2017) 064036, [1707.00984].
  • [85] E. Bianchi and M. Gamonal, Primordial power spectrum at N3LO in effective theories of inflation, 2405.03157.
  • [86] J. Martin and C. Ringeval, First CMB Constraints on the Inflationary Reheating Temperature, Phys. Rev. D 82 (2010) 023511, [1004.5525].
  • [87] C. Ringeval, T. Suyama and J. Yokoyama, Magneto-reheating constraints from curvature perturbations, JCAP 09 (2013) 020, [1302.6013].
  • [88] M. Hindmarsh and O. Philipsen, WIMP dark matter and the QCD equation of state, Phys. Rev. D71 (2005) 087302, [hep-ph/0501232].
  • [89] G. Dvali and M. Redi, Phenomenology of 1032 Dark Sectors, Phys. Rev. D 80 (2009) 055001, [0905.1709].
  • [90] C. Ringeval, The exact numerical treatment of inflationary models, Lect. Notes Phys. 738 (2008) 243–273, [astro-ph/0703486].
  • [91] A. D. Linde, Hybrid inflation, Phys.Rev. D49 (1994) 748–754, [astro-ph/9307002].
  • [92] A. R. Liddle, P. Parsons and J. D. Barrow, Formalizing the slow roll approximation in inflation, Phys. Rev. D 50 (1994) 7222–7232, [astro-ph/9408015].
  • [93] V. Vennin, Horizon-Flow off-track for Inflation, Phys. Rev. D 89 (2014) 083526, [1401.2926].
  • [94] J. Ellis, M. A. G. Garcia, D. V. Nanopoulos and K. A. Olive, Calculations of Inflaton Decays and Reheating: with Applications to No-Scale Inflation Models, JCAP 07 (2015) 050, [1505.06986].
  • [95] L. Iacconi, M. Fasiello, J. Väliviita and D. Wands, Novel CMB constraints on the α parameter in alpha-attractor models, JCAP 10 (2023) 015, [2306.00918].
  • [96] V. Mukhanov, Quantum Cosmological Perturbations: Predictions and Observations, Eur. Phys. J. C 73 (2013) 2486, [1303.3925].
  • [97] I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products. 2007.