Abstract
Massive scalar fields provide excellent dark matter candidates, whose dynamics are often explored analytically and numerically using nonrelativistic Schrödinger-Poisson (SP) equations in a cosmological context. In this paper, starting from the nonlinear and fully relativistic Klein-Gordon-Einstein (KGE) equations in an expanding universe, we provide a systematic framework for deriving the SP equations, as well as relativistic corrections to them, by integrating out ‘fast modes’ and including nonlinear metric and matter contributions. We provide explicit equations for the leading-order relativistic corrections, which provide insight into deviations from the SP equations as the system approaches the relativistic regime. Upon including the leading-order corrections, our equations are applicable beyond the domain of validity of the SP system, and are simpler to use than the full KGE case in some contexts. As a concrete application, we calculate the mass-radius relationship of solitons in scalar dark matter and accurately capture the deviations of this relationship from the SP system towards the KGE one.
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References
F. Wilczek, Problem of Strong P and T Invariance in the Presence of Instantons, Phys. Rev. Lett. 40 (1978) 279 [INSPIRE].
R.D. Peccei and H.R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38 (1977) 1440 [INSPIRE].
J. Preskill, M.B. Wise and F. Wilczek, Cosmology of the Invisible Axion, Phys. Lett. B 120 (1983) 127 [INSPIRE].
L.F. Abbott and P. Sikivie, A Cosmological Bound on the Invisible Axion, Phys. Lett. B 120 (1983) 133 [INSPIRE].
M. Dine and W. Fischler, The Not So Harmless Axion, Phys. Lett. B 120 (1983) 137 [INSPIRE].
T. Matos, F.S. Guzman and L.A. Urena-Lopez, Scalar field as dark matter in the universe, Class. Quant. Grav. 17 (2000) 1707 [astro-ph/9908152] [INSPIRE].
A. Ringwald, Axions and Axion-Like Particles, in 49th Rencontres de Moriond on Electroweak Interactions and Unified Theories, pp. 223–230 (2014) [arXiv:1407.0546] [INSPIRE].
W. Hu, R. Barkana and A. Gruzinov, Cold and fuzzy dark matter, Phys. Rev. Lett. 85 (2000) 1158 [astro-ph/0003365] [INSPIRE].
L. Hui, J.P. Ostriker, S. Tremaine and E. Witten, Ultralight scalars as cosmological dark matter, Phys. Rev. D 95 (2017) 043541 [arXiv:1610.08297] [INSPIRE].
A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell, String Axiverse, Phys. Rev. D 81 (2010) 123530 [arXiv:0905.4720] [INSPIRE].
A.H. Guth, M.P. Hertzberg and C. Prescod-WEinstein, Do Dark Matter Axions Form a Condensate with Long-Range Correlation?, Phys. Rev. D 92 (2015) 103513 [arXiv:1412.5930] [INSPIRE].
M.P. Hertzberg, Quantum and Classical Behavior in Interacting Bosonic Systems, JCAP 11 (2016) 037 [arXiv:1609.01342] [INSPIRE].
E.W. Kolb and I.I. Tkachev, Nonlinear axion dynamics and formation of cosmological pseudosolitons, Phys. Rev. D 49 (1994) 5040 [astro-ph/9311037] [INSPIRE].
H.-Y. Schive, T. Chiueh and T. Broadhurst, Cosmic Structure as the Quantum Interference of a Coherent Dark Wave, Nature Phys. 10 (2014) 496 [arXiv:1406.6586] [INSPIRE].
D.G. Levkov, A.G. Panin and I.I. Tkachev, Gravitational Bose-Einstein condensation in the kinetic regime, Phys. Rev. Lett. 121 (2018) 151301 [arXiv:1804.05857] [INSPIRE].
M.A. Amin and P. Mocz, Formation, gravitational clustering, and interactions of nonrelativistic solitons in an expanding universe, Phys. Rev. D 100 (2019) 063507 [arXiv:1902.07261] [INSPIRE].
L. Hui, A. Joyce, M.J. Landry and X. Li, Vortices and waves in light dark matter, JCAP 01 (2021) 011 [arXiv:2004.01188] [INSPIRE].
L.A. Ureña-López and A.X. Gonzalez-Morales, Towards accurate cosmological predictions for rapidly oscillating scalar fields as dark matter, JCAP 07 (2016) 048 [arXiv:1511.08195] [INSPIRE].
R. Hlozek, D. Grin, D.J.E. Marsh and P.G. Ferreira, A search for ultralight axions using precision cosmological data, Phys. Rev. D 91 (2015) 103512 [arXiv:1410.2896] [INSPIRE].
D.J.E. Marsh, Axion Cosmology, Phys. Rept. 643 (2016) 1 [arXiv:1510.07633] [INSPIRE].
E.G.M. Ferreira, Ultra-Light Dark Matter, arXiv:2005.03254 [INSPIRE].
L. Hui, Wave Dark Matter, arXiv:2101.11735 [INSPIRE].
M.H. Namjoo, A.H. Guth and D.I. Kaiser, Relativistic Corrections to Nonrelativistic Effective Field Theories, Phys. Rev. D 98 (2018) 016011 [arXiv:1712.00445] [INSPIRE].
B. Salehian, M.H. Namjoo and D.I. Kaiser, Effective theories for a nonrelativistic field in an expanding universe: Induced self-interaction, pressure, sound speed, and viscosity, JHEP 07 (2020) 059 [arXiv:2005.05388] [INSPIRE].
C. Cheung, P. Creminelli, A.L. Fitzpatrick, J. Kaplan and L. Senatore, The Effective Field Theory of Inflation, JHEP 03 (2008) 014 [arXiv:0709.0293] [INSPIRE].
J.J.M. Carrasco, M.P. Hertzberg and L. Senatore, The Effective Field Theory of Cosmological Large Scale Structures, JHEP 09 (2012) 082 [arXiv:1206.2926] [INSPIRE].
J. Eby, K. Mukaida, M. Takimoto, L.C.R. Wijewardhana and M. Yamada, Classical nonrelativistic effective field theory and the role of gravitational interactions, Phys. Rev. D 99 (2019) 123503 [arXiv:1807.09795] [INSPIRE].
E. Braaten, A. Mohapatra and H. Zhang, Classical Nonrelativistic Effective Field Theories for a Real Scalar Field, Phys. Rev. D 98 (2018) 096012 [arXiv:1806.01898] [INSPIRE].
J. Adamek, D. Daverio, R. Durrer and M. Kunz, General Relativistic N-body simulations in the weak field limit, Phys. Rev. D 88 (2013) 103527 [arXiv:1308.6524] [INSPIRE].
J. Adamek, D. Daverio, R. Durrer and M. Kunz, General relativity and cosmic structure formation, Nature Phys. 12 (2016) 346 [arXiv:1509.01699] [INSPIRE].
P. Mocz et al., First star-forming structures in fuzzy cosmic filaments, Phys. Rev. Lett. 123 (2019) 141301 [arXiv:1910.01653] [INSPIRE].
P. Mocz et al., Galaxy formation with BECDM — II. Cosmic filaments and first galaxies, Mon. Not. Roy. Astron. Soc. 494 (2020) 2027 [arXiv:1911.05746] [INSPIRE].
N. Musoke, S. Hotchkiss and R. Easther, Lighting the Dark: Evolution of the Postinflationary Universe, Phys. Rev. Lett. 124 (2020) 061301 [arXiv:1909.11678] [INSPIRE].
B. Schwabe, J.C. Niemeyer and J.F. Engels, Simulations of solitonic core mergers in ultralight axion dark matter cosmologies, Phys. Rev. D 94 (2016) 043513 [arXiv:1606.05151] [INSPIRE].
N. Glennon and C. Prescod-WEinstein, Using PySiUltraLight to Model Scalar Dark Matter with Self-Interactions, arXiv:2011.09510 [INSPIRE].
M.P. Hertzberg, Y. Li and E.D. Schiappacasse, Merger of Dark Matter Axion Clumps and Resonant Photon Emission, JCAP 07 (2020) 067 [arXiv:2005.02405] [INSPIRE].
M.A. Amin and Z.-G. Mou, Electromagnetic Bursts from Mergers of Oscillons in Axion-like Fields, JCAP 02 (2021) 024 [arXiv:2009.11337] [INSPIRE].
L. Lancaster, C. Giovanetti, P. Mocz, Y. Kahn, M. Lisanti and D.N. Spergel, Dynamical Friction in a Fuzzy Dark Matter Universe, JCAP 01 (2020) 001 [arXiv:1909.06381] [INSPIRE].
B. Bar-Or, J.-B. Fouvry and S. Tremaine, Relaxation in a Fuzzy Dark Matter Halo, Astrophys. J. 871 (2019) 28 [arXiv:1809.07673] [INSPIRE].
P. Mocz et al., Galaxy formation with BECDM — I. Turbulence and relaxation of idealized haloes, Mon. Not. Roy. Astron. Soc. 471 (2017) 4559 [arXiv:1705.05845] [INSPIRE].
X. Du, C. Behrens and J.C. Niemeyer, Substructure of fuzzy dark matter haloes, Mon. Not. Roy. Astron. Soc. 465 (2017) 941 [arXiv:1608.02575] [INSPIRE].
S. May and V. Springel, Structure formation in large-volume cosmological simulations of fuzzy dark matter: Impact of the non-linear dynamics, arXiv:2101.01828 [INSPIRE].
K. Kirkpatrick, A.E. Mirasola and C. Prescod-WEinstein, Relaxation times for Bose-Einstein condensation in axion miniclusters, Phys. Rev. D 102 (2020) 103012 [arXiv:2007.07438] [INSPIRE].
J. Veltmaat and J.C. Niemeyer, Cosmological particle-in-cell simulations with ultralight axion dark matter, Phys. Rev. D 94 (2016) 123523 [arXiv:1608.00802] [INSPIRE].
F. Edwards, E. Kendall, S. Hotchkiss and R. Easther, PyUltraLight: A Pseudo-Spectral Solver for Ultralight Dark Matter Dynamics, JCAP 10 (2018) 027 [arXiv:1807.04037] [INSPIRE].
A. Khmelnitsky and V. Rubakov, Pulsar timing signal from ultralight scalar dark matter, JCAP 02 (2014) 019 [arXiv:1309.5888] [INSPIRE].
M. Bošković, F. Duque, M.C. Ferreira, F.S. Miguel and V. Cardoso, Motion in time-periodic backgrounds with applications to ultralight dark matter haloes at galactic centers, Phys. Rev. D 98 (2018) 024037 [arXiv:1806.07331] [INSPIRE].
D. Blas, D. López Nacir and S. Sibiryakov, Secular effects of ultralight dark matter on binary pulsars, Phys. Rev. D 101 (2020) 063016 [arXiv:1910.08544] [INSPIRE].
S.-J. Sin, Late time cosmological phase transition and galactic halo as Bose liquid, Phys. Rev. D 50 (1994) 3650 [hep-ph/9205208] [INSPIRE].
G. Fodor, P. Forgacs, Z. Horvath and M. Mezei, Computation of the radiation amplitude of oscillons, Phys. Rev. D 79 (2009) 065002 [arXiv:0812.1919] [INSPIRE].
G. Fodor, A review on radiation of oscillons and oscillatons, Ph.D. Thesis, Wigner RCP, Budapest (2019) [arXiv:1911.03340] [INSPIRE].
H.-Y. Zhang, M.A. Amin, E.J. Copeland, P.M. Saffin and K.D. Lozanov, Classical Decay Rates of Oscillons, JCAP 07 (2020) 055 [arXiv:2004.01202] [INSPIRE].
I.L. Bogolyubsky and V.G. Makhankov, Lifetime of Pulsating Solitons in Some Classical Models, Pisma Zh. Eksp. Teor. Fiz. 24 (1976) 15.
M. Gleiser, Pseudostable bubbles, Phys. Rev. D 49 (1994) 2978 [hep-ph/9308279] [INSPIRE].
E.J. Copeland, M. Gleiser and H.R. Muller, Oscillons: Resonant configurations during bubble collapse, Phys. Rev. D 52 (1995) 1920 [hep-ph/9503217] [INSPIRE].
S. Kasuya, M. Kawasaki and F. Takahashi, I-balls, Phys. Lett. B 559 (2003) 99 [hep-ph/0209358] [INSPIRE].
M.A. Amin and D. Shirokoff, Flat-top oscillons in an expanding universe, Phys. Rev. D 81 (2010) 085045 [arXiv:1002.3380] [INSPIRE].
M.A. Amin, K-oscillons: Oscillons with noncanonical kinetic terms, Phys. Rev. D 87 (2013) 123505 [arXiv:1303.1102] [INSPIRE].
E. Seidel and W.M. Suen, Oscillating soliton stars, Phys. Rev. Lett. 66 (1991) 1659 [INSPIRE].
L. Visinelli, S. Baum, J. Redondo, K. Freese and F. Wilczek, Dilute and dense axion stars, Phys. Lett. B 777 (2018) 64 [arXiv:1710.08910] [INSPIRE].
P.-H. Chavanis, Phase transitions between dilute and dense axion stars, Phys. Rev. D 98 (2018) 023009 [arXiv:1710.06268] [INSPIRE].
J. Eby, M. Leembruggen, L. Street, P. Suranyi and L.C.R. Wijewardhana, Global view of QCD axion stars, Phys. Rev. D 100 (2019) 063002 [arXiv:1905.00981] [INSPIRE].
M.A. Amin, Inflaton fragmentation: Emergence of pseudo-stable inflaton lumps (oscillons) after inflation, arXiv:1006.3075 [INSPIRE].
M.A. Amin, R. Easther, H. Finkel, R. Flauger and M.P. Hertzberg, Oscillons After Inflation, Phys. Rev. Lett. 108 (2012) 241302 [arXiv:1106.3335] [INSPIRE].
M. Gleiser, N. Graham and N. Stamatopoulos, Generation of Coherent Structures After Cosmic Inflation, Phys. Rev. D 83 (2011) 096010 [arXiv:1103.1911] [INSPIRE].
P. Grandclement, G. Fodor and P. Forgacs, Numerical simulation of oscillatons: extracting the radiating tail, Phys. Rev. D 84 (2011) 065037 [arXiv:1107.2791] [INSPIRE].
K.D. Lozanov and M.A. Amin, Self-resonance after inflation: oscillons, transients and radiation domination, Phys. Rev. D 97 (2018) 023533 [arXiv:1710.06851] [INSPIRE].
J.-P. Hong, M. Kawasaki and M. Yamazaki, Oscillons from Pure Natural Inflation, Phys. Rev. D 98 (2018) 043531 [arXiv:1711.10496] [INSPIRE].
T. Ikeda, C.-M. Yoo and V. Cardoso, Self-gravitating oscillons and new critical behavior, Phys. Rev. D 96 (2017) 064047 [arXiv:1708.01344] [INSPIRE].
J.R. Bond, J. Braden and L. Mersini-Houghton, Cosmic bubble and domain wall instabilities III: The role of oscillons in three-dimensional bubble collisions, JCAP 09 (2015) 004 [arXiv:1505.02162] [INSPIRE].
S. Antusch, F. Cefala, S. Krippendorf, F. Muia, S. Orani and F. Quevedo, Oscillons from String Moduli, JHEP 01 (2018) 083 [arXiv:1708.08922] [INSPIRE].
A. Arvanitaki, S. Dimopoulos, M. Galanis, L. Lehner, J.O. Thompson and K. Van Tilburg, Large-misalignment mechanism for the formation of compact axion structures: Signatures from the QCD axion to fuzzy dark matter, Phys. Rev. D 101 (2020) 083014 [arXiv:1909.11665] [INSPIRE].
P. Brax, J.A.R. Cembranos and P. Valageas, Nonrelativistic formation of scalar clumps as a candidate for dark matter, Phys. Rev. D 102 (2020) 083012 [arXiv:2007.04638] [INSPIRE].
M. Kawasaki, W. Nakano, H. Nakatsuka and E. Sonomoto, Oscillons of Axion-Like Particle: Mass distribution and power spectrum, JCAP 01 (2021) 061 [arXiv:2010.09311] [INSPIRE].
E. Seidel and W.-M. Suen, Formation of solitonic stars through gravitational cooling, Phys. Rev. Lett. 72 (1994) 2516 [gr-qc/9309015] [INSPIRE].
L.A. Urena-Lopez, Oscillatons revisited, Class. Quant. Grav. 19 (2002) 2617 [gr-qc/0104093] [INSPIRE].
M. Alcubierre, R. Becerril, S.F. Guzman, T. Matos, D. Núñez and L.A. Urena-Lopez, Numerical studies of Φ2 oscillatons, Class. Quant. Grav. 20 (2003) 2883 [gr-qc/0301105] [INSPIRE].
H.-Y. Zhang, Gravitational effects on oscillon lifetimes, JCAP 03 (2021) 102 [arXiv:2011.11720] [INSPIRE].
R.L. Arnowitt, S. Deser and C.W. Misner, The Dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].
E. Gourgoulhon, 3+1 formalism and bases of numerical relativity, gr-qc/0703035 [INSPIRE].
P.-H. Chavanis, Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: I. Analytical results, Phys. Rev. D 84 (2011) 043531 [arXiv:1103.2050] [INSPIRE].
P.H. Chavanis and L. Delfini, Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: II. Numerical results, Phys. Rev. D 84 (2011) 043532 [arXiv:1103.2054] [INSPIRE].
D. Croon, J. Fan and C. Sun, Boson Star from Repulsive Light Scalars and Gravitational Waves, JCAP 04 (2019) 008 [arXiv:1810.01420] [INSPIRE].
M.P. Hertzberg, Quantum Radiation of Oscillons, Phys. Rev. D 82 (2010) 045022 [arXiv:1003.3459] [INSPIRE].
K. Mukaida, M. Takimoto and M. Yamada, On Longevity of I-ball/Oscillon, JHEP 03 (2017) 122 [arXiv:1612.07750] [INSPIRE].
F.S. Guzman and L.A. Urena-Lopez, Newtonian collapse of scalar field dark matter, Phys. Rev. D 68 (2003) 024023 [astro-ph/0303440] [INSPIRE].
G. Kane, K. Sinha and S. Watson, Cosmological Moduli and the Post-Inflationary Universe: A Critical Review, Int. J. Mod. Phys. D 24 (2015) 1530022 [arXiv:1502.07746] [INSPIRE].
K.D. Lozanov and M.A. Amin, Gravitational perturbations from oscillons and transients after inflation, Phys. Rev. D 99 (2019) 123504 [arXiv:1902.06736] [INSPIRE].
J.T. Giblin and A.J. Tishue, Preheating in Full General Relativity, Phys. Rev. D 100 (2019) 063543 [arXiv:1907.10601] [INSPIRE].
M. Khlopov, B.A. Malomed and I.B. Zeldovich, Gravitational instability of scalar fields and formation of primordial black holes, Mon. Not. Roy. Astron. Soc. 215 (1985) 575 [INSPIRE].
J.Y. Widdicombe, T. Helfer, D.J.E. Marsh and E.A. Lim, Formation of Relativistic Axion Stars, JCAP 10 (2018) 005 [arXiv:1806.09367] [INSPIRE].
Z. Nazari, M. Cicoli, K. Clough and F. Muia, Oscillon collapse to black holes, JCAP 05 (2021) 027 [arXiv:2010.05933] [INSPIRE].
T. Helfer, E.A. Lim, M.A.G. Garcia and M.A. Amin, Gravitational Wave Emission from Collisions of Compact Scalar Solitons, Phys. Rev. D 99 (2019) 044046 [arXiv:1802.06733] [INSPIRE].
E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys. 40 (1927) 322.
A. Suárez and P.-H. Chavanis, Hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit: General formalism and perturbations analysis, Phys. Rev. D 92 (2015) 023510 [arXiv:1503.07437] [INSPIRE].
J. Cookmeyer, D. Grin and T.L. Smith, How sound are our ultralight axion approximations?, Phys. Rev. D 101 (2020) 023501 [arXiv:1909.11094] [INSPIRE].
P.W. Graham, J. Mardon and S. Rajendran, Vector Dark Matter from Inflationary Fluctuations, Phys. Rev. D 93 (2016) 103520 [arXiv:1504.02102] [INSPIRE].
P. Adshead and K.D. Lozanov, Self-gravitating Vector Dark Matter, Phys. Rev. D 103 (2021) 103501 [arXiv:2101.07265] [INSPIRE].
P.G. Ferreira, Cosmological Tests of Gravity, Ann. Rev. Astron. Astrophys. 57 (2019) 335 [arXiv:1902.10503] [INSPIRE].
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Salehian, B., Zhang, HY., Amin, M.A. et al. Beyond Schrödinger-Poisson: nonrelativistic effective field theory for scalar dark matter. J. High Energ. Phys. 2021, 50 (2021). https://doi.org/10.1007/JHEP09(2021)050
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DOI: https://doi.org/10.1007/JHEP09(2021)050