Abstract
Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions.
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References
A.H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].
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 [INSPIRE].
V.F. Mukhanov and G.V. Chibisov, Quantum Fluctuations and a Nonsingular Universe, JETP Lett. 33 (1981) 532 [INSPIRE].
A.H. Guth and S.Y. Pi, Fluctuations in the New Inflationary Universe, Phys. Rev. Lett. 49 (1982) 1110 [INSPIRE].
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 [INSPIRE].
R.H. Brandenberger, Inflationary cosmology: Progress and problems, in the proceedings of the IPM School on Cosmology 1999: Large Scale Structure Formation, Tehran Islamic Republic of Iran, January 23–February 4 (1999) [hep-ph/9910410] [INSPIRE].
R. Bean, D.J.H. Chung and G. Geshnizjani, Reconstructing a general inflationary action, Phys. Rev. D 78 (2008) 023517 [arXiv:0801.0742] [INSPIRE].
J. Martin, C. Ringeval and V. Vennin, Encyclopædia Inflationaris, Phys. Dark Univ. 5–6 (2014) 75 [arXiv:1303.3787] [INSPIRE].
Planck collaboration, Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641 (2020) A10 [arXiv:1807.06211] [INSPIRE].
S. Hawking, The Occurrence of singularities in cosmology, Proc. Roy. Soc. Lond. A 294 (1966) 511 [INSPIRE].
S. Hawking, The Occurrence of singularities in cosmology. II, Proc. Roy. Soc. Lond. A 295 (1966) 490 [INSPIRE].
S. Hawking, The occurrence of singularities in cosmology. III. Causality and singularities, Proc. Roy. Soc. Lond. A 300 (1967) 187 [INSPIRE].
S.W. Hawking and R. Penrose, The Singularities of gravitational collapse and cosmology, Proc. Roy. Soc. Lond. A 314 (1970) 529 [INSPIRE].
A. Borde and A. Vilenkin, Eternal inflation and the initial singularity, Phys. Rev. Lett. 72 (1994) 3305 [gr-qc/9312022] [INSPIRE].
A. Borde and A. Vilenkin, The Impossibility of steady state inflation, in the proceedings of the 8th Nishinomiya-Yukawa Memorial Symposium: Relativistic Cosmology, Nishinomiya Japan, October 28–29 (1993), p. 111–127 [gr-qc/9403004] [INSPIRE].
A. Borde, Open and closed universes, initial singularities and inflation, Phys. Rev. D 50 (1994) 3692 [gr-qc/9403049] [INSPIRE].
A. Borde and A. Vilenkin, Singularities in inflationary cosmology: A Review, Int. J. Mod. Phys. D 5 (1996) 813 [gr-qc/9612036] [INSPIRE].
G.J. Galloway and E. Ling, Topology and singularities in cosmological spacetimes obeying the null energy condition, Commun. Math. Phys. 360 (2018) 611 [arXiv:1705.06705] [INSPIRE].
A. Borde, A.H. Guth and A. Vilenkin, Inflationary space-times are incompletein past directions, Phys. Rev. Lett. 90 (2003) 151301 [gr-qc/0110012] [INSPIRE].
J.M.M. Senovilla and D. Garfinkle, The 1965 Penrose singularity theorem, Class. Quant. Grav. 32 (2015) 124008 [arXiv:1410.5226] [INSPIRE].
M. Spradlin, A. Strominger and A. Volovich, Les Houches lectures on de Sitter space, in the proceedings of the Les Houches Summer School: Session 76: Euro Summer School on Unity of Fundamental Physics: Gravity, Gauge Theory and Strings, Les Houches France, July 30– August 31 (2001), p. 423–453 [hep-th/0110007] [INSPIRE].
A. Aguirre and S. Gratton, Steady state eternal inflation, Phys. Rev. D 65 (2002) 083507 [astro-ph/0111191] [INSPIRE].
A. Aguirre and S. Gratton, Inflation without a beginning: A Null boundary proposal, Phys. Rev. D 67 (2003) 083515 [gr-qc/0301042] [INSPIRE].
A. Aguirre, Eternal Inflation, past and future, arXiv:0712.0571 [INSPIRE].
A. Vilenkin, Arrows of time and the beginning of the universe, Phys. Rev. D 88 (2013) 043516 [arXiv:1305.3836] [INSPIRE].
A. Vilenkin and A.C. Wall, Cosmological singularity theorems and black holes, Phys. Rev. D 89 (2014) 064035 [arXiv:1312.3956] [INSPIRE].
D. Yoshida and J. Quintin, Maximal extensions and singularities in inflationary spacetimes, Class. Quant. Grav. 35 (2018) 155019 [arXiv:1803.07085] [INSPIRE].
T. Numasawa and D. Yoshida, Global Spacetime Structure of Compactified Inflationary Universe, Class. Quant. Grav. 36 (2019) 195003 [arXiv:1901.03347] [INSPIRE].
K. Nomura and D. Yoshida, Past extendibility and initial singularity in Friedmann-Lemaître-Robertson-Walker and Bianchi I spacetimes, JCAP 07 (2021) 047 [arXiv:2105.05642] [INSPIRE].
K. Nishii and D. Yoshida, String excitation by initial singularity of inflation, JHEP 10 (2021) 025 [arXiv:2105.12339] [INSPIRE].
K. Nomura and D. Yoshida, Implications of the singularity theorem for the size of a nonsingular universe, Phys. Rev. D 106 (2022) 124016 [arXiv:2206.09404] [INSPIRE].
T. Harada, T. Igata, T. Sato and B. Carr, Complete classification of Friedmann-Lemaître-Robertson-Walker solutions with linear equation of state: parallelly propagated curvature singularities for general geodesics, Class. Quant. Grav. 39 (2022) 145008 [arXiv:2110.13421] [INSPIRE].
C.J.S. Clarke, Local extensions in singular space-times, Commun. Math. Phys. 32 (1973) 205.
G.F.R. Ellis and A.R. King, Was the big bang a whimper?, Commun. Math. Phys. 38 (1974) 119 [INSPIRE].
G.F.R. Ellis and B.G. Schmidt, Singular space-times, Gen. Rel. Grav. 8 (1977) 915 [INSPIRE].
C.J.S. Clarke, Local extensions in singular space-times II, Commun. Math. Phys. 84 (1982) 329.
C.J.S. Clarke, The Analysis of space-time singularities, Cambridge Univ. Press, Cambridge, U.K. (1994) [INSPIRE].
G. Galloway and E. Ling, Some Remarks on the C0-(in)extendibility of Spacetimes, Annales Henri Poincare 18 (2017) 3427 [arXiv:1610.03008] [INSPIRE].
G.J. Galloway, E. Ling and J. Sbierski, Timelike Completeness as an Obstruction to C0-Extensions, Commun. Math. Phys. 359 (2018) 937 [arXiv:1704.00353] [INSPIRE].
J. Sbierski, The C0-inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry, J. Diff. Geom. 108 (2018) 319 [arXiv:1507.00601] [INSPIRE].
J. Sbierski, On the proof of the C0-inextendibility of the Schwarzschild spacetime, J. Phys. Conf. Ser. 968 (2018) 012012 [arXiv:1711.11380] [INSPIRE].
J. Sbierski, On holonomy singularities in general relativity and the \( {C}_{loc}^{0,1} \)-inextendibility of space-times, Duke Math. J. 171 (2022) 2881 [arXiv:2007.12049] [INSPIRE].
M. Graf and E. Ling, Maximizers in Lipschitz spacetimes are either timelike or null, Class. Quant. Grav. 35 (2018) 087001 [arXiv:1712.06504] [INSPIRE].
E. Minguzzi and S. Suhr, Some regularity results for Lorentz-Finsler spaces, Annals Global Anal. Geom. 56 (2019) 597 [arXiv:1903.00842] [INSPIRE].
D. Klein and J. Reschke, Pre-big bang geometric extensions of inflationary cosmologies, Annales Henri Poincare 19 (2018) 565 [arXiv:1604.06372] [INSPIRE].
E. Ling, Milne-like Spacetimes and their Symmetries, arXiv:1803.00174 [INSPIRE].
E. Ling, The Big Bang is a Coordinate Singularity for k = −1 Inflationary FLRW Spacetimes, Found. Phys. 50 (2020) 385 [arXiv:1810.06789] [INSPIRE].
E. Ling and E. Ling, Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes, Gen. Rel. Grav. 54 (2022) 68 [Erratum ibid. 54 (2022) 139] [arXiv:2202.04014] [INSPIRE].
E. Ling and A. Piubello, On the asymptotic assumptions for Milne-like spacetimes, Gen. Rel. Grav. 55 (2023) 53 [arXiv:2208.07786] [INSPIRE].
M. Novello and S.E.P. Bergliaffa, Bouncing Cosmologies, Phys. Rept. 463 (2008) 127 [arXiv:0802.1634] [INSPIRE].
Y.-F. Cai, Exploring Bouncing Cosmologies with Cosmological Surveys, Sci. China Phys. Mech. Astron. 57 (2014) 1414 [arXiv:1405.1369] [INSPIRE].
D. Battefeld and P. Peter, A Critical Review of Classical Bouncing Cosmologies, Phys. Rept. 571 (2015) 1 [arXiv:1406.2790] [INSPIRE].
R. Brandenberger and P. Peter, Bouncing Cosmologies: Progress and Problems, Found. Phys. 47 (2017) 797 [arXiv:1603.05834] [INSPIRE].
S.S. Boruah, H.J. Kim, M. Rouben and G. Geshnizjani, Cuscuton bounce, JCAP 08 (2018) 031 [arXiv:1802.06818] [INSPIRE].
J.D. Barrow, Varieties of expanding universe, Class. Quant. Grav. 13 (1996) 2965 [INSPIRE].
J.D. Barrow and N.J. Nunes, Dynamics of Logamediate Inflation, Phys. Rev. D 76 (2007) 043501 [arXiv:0705.4426] [INSPIRE].
S.F. Bramberger and J.-L. Lehners, Nonsingular bounces catalyzed by dark energy, Phys. Rev. D 99 (2019) 123523 [arXiv:1901.10198] [INSPIRE].
A. Anabalón, S.F. Bramberger and J.-L. Lehners, Kerr-NUT-de Sitter as an Inhomogeneous Non-Singular Bouncing Cosmology, JHEP 09 (2019) 096 [arXiv:1904.07285] [INSPIRE].
M.I. Letey et al., Quantum Initial Conditions for Curved Inflating Universes, arXiv:2211.17248 [INSPIRE].
J.E. Lesnefsky, D.A. Easson and P.C.W. Davies, Past-completeness of inflationary spacetimes, Phys. Rev. D 107 (2023) 044024 [arXiv:2207.00955] [INSPIRE].
A. Ijjas and P.J. Steinhardt, A new kind of cyclic universe, Phys. Lett. B 795 (2019) 666 [arXiv:1904.08022] [INSPIRE].
W.H. Kinney and N.K. Stein, Cyclic cosmology and geodesic completeness, JCAP 06 (2022) 011 [arXiv:2110.15380] [INSPIRE].
H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Transactions of the American Mathematical Society 36 (1934) 63.
R.T. Seeley, Extension of C∞ functions defined in a half space, Proc. Am. Math. Soc. 15 (1964) 625.
P. van Nieuwenhuizen and C.C. Wu, On Integral Relations for Invariants Constructed from Three Riemann Tensors and their Applications in Quantum Gravity, J. Math. Phys. 18 (1977) 182 [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Curvature Cubed Terms in String Theory Effective Actions, Phys. Lett. B 185 (1987) 52 [INSPIRE].
S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press (1973) [https://doi.org/10.1017/cbo9780511524646].
S.M. Carroll, Spacetime and Geometry, Cambridge University Press (2019) [https://doi.org/10.1017/9781108770385].
V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, Oxford (2005) [https://doi.org/10.1017/CBO9780511790553] [INSPIRE].
J. Sbierski, Uniqueness and non-uniqueness results for spacetime extensions, arXiv:2208.07752 [INSPIRE].
B. O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics 103, Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York, U.S.A. (1983). [ISBN: 9780125267403].
E. Ling, Aspects of C0 causal theory, Gen. Rel. Grav. 52 (2020) 57 [arXiv:1911.04438] [INSPIRE].
G.F.R. Ellis, Relativistic cosmology, Proc. Int. Sch. Phys. Fermi 47 (1971) 104 [INSPIRE].
C. Ganguly and J. Quintin, Microphysical manifestations of viscosity and consequences for anisotropies in the very early universe, Phys. Rev. D 105 (2022) 023532 [arXiv:2109.11701] [INSPIRE].
G.F.R. Ellis, R. Maartens and M.A.H. MacCallum, Relativistic Cosmology, Cambridge University Press (2012) [https://doi.org/10.1017/cbo9781139014403].
V.A. Belinsky, I.M. Khalatnikov and E.M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys. 19 (1970) 525 [INSPIRE].
C.W. Misner, Mixmaster universe, Phys. Rev. Lett. 22 (1969) 1071 [INSPIRE].
V. Belinski and M. Henneaux, The Cosmological Singularity, Cambridge Univ. Pr., Cambridge, U.K. (2017) [https://doi.org/10.1017/9781107239333] [INSPIRE].
R.M. Wald, Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant, Phys. Rev. D 28 (1983) 2118 [INSPIRE].
J.D. Barrow and J. Stein-Schabes, Inhomogeneous cosmologies with cosmological constant, Phys. Lett. A 103 (1984) 315 [INSPIRE].
Y. Kitada and K.-I. Maeda, Cosmic no hair theorem in homogeneous space-times. I. Bianchi models, Class. Quant. Grav. 10 (1993) 703 [INSPIRE].
A. Maleknejad and M.M. Sheikh-Jabbari, Revisiting Cosmic No-Hair Theorem for Inflationary Settings, Phys. Rev. D 85 (2012) 123508 [arXiv:1203.0219] [INSPIRE].
H. Andréasson and H. Ringström, Proof of the cosmic no-hair conjecture in the 𝕋3-Gowdy symmetric Einstein-Vlasov setting, J. Eur. Math. Soc. 18 (2016) 1565 [arXiv:1306.6223] [INSPIRE].
M. Kleban and L. Senatore, Inhomogeneous Anisotropic Cosmology, JCAP 10 (2016) 022 [arXiv:1602.03520] [INSPIRE].
M. Mirbabayi, Topology of Cosmological Black Holes, JCAP 05 (2020) 029 [arXiv:1810.01431] [INSPIRE].
P. Creminelli, L. Senatore and A. Vasy, Asymptotic Behavior of Cosmologies with Λ > 0 in 2+1 Dimensions, Commun. Math. Phys. 376 (2020) 1155 [arXiv:1902.00519] [INSPIRE].
P. Creminelli, O. Hershkovits, L. Senatore and A. Vasy, A de Sitter no-hair theorem for 3+1d cosmologies with isometry group forming 2-dimensional orbits, Adv. Math. 434 (2023) 109296 [arXiv:2004.10754] [INSPIRE].
J. Wang and L. Senatore, On the asymptotics of 3+1D cosmologies with bounded scalar potential and isometry group forming 2-dimensional orbits, arXiv:2111.09257 [INSPIRE].
F. Azhar and D.I. Kaiser, Flows into de Sitter space from anisotropic initial conditions: An effective field theory approach, Phys. Rev. D 107 (2023) 043506 [arXiv:2207.08355] [INSPIRE].
R. Bousso, A Covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].
R. Bousso, The Holographic principle, Rev. Mod. Phys. 74 (2002) 825 [hep-th/0203101] [INSPIRE].
R. Bousso and A. Shahbazi-Moghaddam, Singularities from Entropy, Phys. Rev. Lett. 128 (2022) 231301 [arXiv:2201.11132] [INSPIRE].
D.S. Goldwirth and T. Piran, Initial conditions for inflation, Phys. Rept. 214 (1992) 223 [INSPIRE].
R. Brandenberger, Initial conditions for inflation — A short review, Int. J. Mod. Phys. D 26 (2016) 1740002 [arXiv:1601.01918] [INSPIRE].
R. Easther, L.C. Price and J. Rasero, Inflating an Inhomogeneous Universe, JCAP 08 (2014) 041 [arXiv:1406.2869] [INSPIRE].
W.E. East, M. Kleban, A. Linde and L. Senatore, Beginning inflation in an inhomogeneous universe, JCAP 09 (2016) 010 [arXiv:1511.05143] [INSPIRE].
K. Clough et al., Robustness of Inflation to Inhomogeneous Initial Conditions, JCAP 09 (2017) 025 [arXiv:1608.04408] [INSPIRE].
K. Clough, R. Flauger and E.A. Lim, Robustness of Inflation to Large Tensor Perturbations, JCAP 05 (2018) 065 [arXiv:1712.07352] [INSPIRE].
J.C. Aurrekoetxea, K. Clough, R. Flauger and E.A. Lim, The Effects of Potential Shape on Inhomogeneous Inflation, JCAP 05 (2020) 030 [arXiv:1910.12547] [INSPIRE].
C. Joana and S. Clesse, Inhomogeneous preinflation across Hubble scales in full general relativity, Phys. Rev. D 103 (2021) 083501 [arXiv:2011.12190] [INSPIRE].
M. Corman and W.E. East, Starting inflation from inhomogeneous initial conditions with momentum, JCAP 10 (2023) 046 [arXiv:2212.04479] [INSPIRE].
D. Garfinkle, A. Ijjas and P.J. Steinhardt, Initial conditions problem in cosmological inflation revisited, Phys. Lett. B 843 (2023) 138028 [arXiv:2304.12150] [INSPIRE].
J.-L. Lehners and K.S. Stelle, A Safe Beginning for the Universe?, Phys. Rev. D 100 (2019) 083540 [arXiv:1909.01169] [INSPIRE].
C. Jonas, J.-L. Lehners and J. Quintin, Cosmological consequences of a principle of finite amplitudes, Phys. Rev. D 103 (2021) 103525 [arXiv:2102.05550] [INSPIRE].
J.D. Barrow and S. Hervik, On the evolution of universes in quadratic theories of gravity, Phys. Rev. D 74 (2006) 124017 [gr-qc/0610013] [INSPIRE].
J.D. Barrow and S. Hervik, Anisotropically inflating universes, Phys. Rev. D 73 (2006) 023007 [gr-qc/0511127] [INSPIRE].
J.D. Barrow and S. Hervik, Simple Types of Anisotropic Inflation, Phys. Rev. D 81 (2010) 023513 [arXiv:0911.3805] [INSPIRE].
J. Middleton, On The Existence Of Anisotropic Cosmological Models In Higher-Order Theories Of Gravity, Class. Quant. Grav. 27 (2010) 225013 [arXiv:1007.4669] [INSPIRE].
D. Müller, A. Ricciardone, A.A. Starobinsky and A. Toporensky, Anisotropic cosmological solutions in R + R2 gravity, Eur. Phys. J. C 78 (2018) 311 [arXiv:1710.08753] [INSPIRE].
S. Hofmann and M. Schneider, Classical versus quantum completeness, Phys. Rev. D 91 (2015) 125028 [arXiv:1504.05580] [INSPIRE].
S. Hofmann, M. Schneider and M. Urban, Quantum complete prelude to inflation, Phys. Rev. D 99 (2019) 065012 [arXiv:1901.04492] [INSPIRE].
M. Kunzinger, R. Steinbauer, M. Stojkovic and J.A. Vickers, Hawking’s singularity theorem for C1,1-metrics, Class. Quant. Grav. 32 (2015) 075012 [arXiv:1411.4689] [INSPIRE].
M. Kunzinger, R. Steinbauer and J.A. Vickers, The Penrose singularity theorem in regularity C1,1, Class. Quant. Grav. 32 (2015) 155010 [arXiv:1502.00287] [INSPIRE].
M. Graf, J.D.E. Grant, M. Kunzinger and R. Steinbauer, The Hawking-Penrose Singularity Theorem for C1,1-Lorentzian Metrics, Commun. Math. Phys. 360 (2018) 1009 [arXiv:1706.08426] [INSPIRE].
J.D.E. Grant, M. Kunzinger and C. Sämann, Inextendibility of spacetimes and Lorentzian length spaces, Annals Global Anal. Geom. 55 (2019) 133 [arXiv:1804.10423] [INSPIRE].
S.B. Alexander, M. Graf, M. Kunzinger and C. Sämann, Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems, arXiv:1909.09575 [INSPIRE].
M. Graf, Singularity theorems for C1-Lorentzian metrics, Commun. Math. Phys. 378 (2020) 1417 [arXiv:1910.13915] [INSPIRE].
M. Kunzinger, A. Ohanyan, B. Schinnerl and R. Steinbauer, The Hawking-Penrose Singularity Theorem for C1-Lorentzian Metrics, Commun. Math. Phys. 391 (2022) 1143 [arXiv:2110.09176] [INSPIRE].
R. Steinbauer, The singularity theorems of General Relativity and their low regularity extensions, arXiv:2206.05939 [https://doi.org/10.1365/s13291-022-00263-7] [INSPIRE].
F. Cavalletti and A. Mondino, A review of Lorentzian synthetic theory of timelike Ricci curvature bounds, Gen. Rel. Grav. 54 (2022) 137 [arXiv:2204.13330] [INSPIRE].
R.J. McCann, A synthetic null energy condition, arXiv:2304.14341 [INSPIRE].
J.B. Hartle and S.W. Hawking, Wave Function of the Universe, Phys. Rev. D 28 (1983) 2960 [INSPIRE].
C. Jonas, J.-L. Lehners and J. Quintin, Uses of complex metrics in cosmology, JHEP 08 (2022) 284 [arXiv:2205.15332] [INSPIRE].
D. Jia, Is singularity resolution trivial?, arXiv:2204.12304 [INSPIRE].
J. Cotler and K. Jensen, Isometric evolution in de Sitter quantum gravity, arXiv:2302.06603 [INSPIRE].
L. Boyle, K. Finn and N. Turok, The Big Bang, CPT, and neutrino dark matter, Annals Phys. 438 (2022) 168767 [arXiv:1803.08930] [INSPIRE].
L. Boyle, K. Finn and N. Turok, CPT-Symmetric Universe, Phys. Rev. Lett. 121 (2018) 251301 [arXiv:1803.08928] [INSPIRE].
L. Boyle and N. Turok, Two-Sheeted Universe, Analyticity and the Arrow of Time, arXiv:2109.06204 [INSPIRE].
L. Boyle, M. Teuscher and N. Turok, The Big Bang as a Mirror: a Solution of the Strong CP Problem, arXiv:2208.10396 [INSPIRE].
Acknowledgments
We thank Eric Woolgar for stimulating discussions in the initial stages of this project. We further thank the stimulating atmosphere at the Fields Institute throughout the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry and especially during the Low Regularity Physics and Geometry Seminar and the Workshop on Mathematical Relativity, Scalar Curvature and Synthetic Lorentzian Geometry. This publication was supported by the Fields Institute for Research in Mathematical Sciences. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the Institute. E. Ling was supported by Carlsberg Foundation CF21-0680 and Danmarks Grundforskningsfond CPH-GEOTOP-DNRF151. This research was also supported by a Discovery Grant from the Natural Science and Engineering Research Council of Canada (NSERC) and partly by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities.
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Geshnizjani, G., Ling, E. & Quintin, J. On the initial singularity and extendibility of flat quasi-de Sitter spacetimes. J. High Energ. Phys. 2023, 182 (2023). https://doi.org/10.1007/JHEP10(2023)182
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DOI: https://doi.org/10.1007/JHEP10(2023)182