Effective theories for a nonrelativistic field in an expanding universe: Induced self-interaction, pressure, sound speed
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Springer
Received: May 26, 2020 Accepted: June 5, 2020 Published: July 9, 2020
Borna Salehian,a Mohammad Hossein Namjooa and David I. Kaiserb a
School of Astronomy, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran b Department of Phyics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: A massive, nonrelativistic scalar field in an expanding spacetime is usually approximated by a pressureless perfect fluid, which leads to the standard conclusion that such a field can play the role of cold dark matter. In this paper, we systematically study these approximations, incorporating subleading corrections. We provide two equivalent effective descriptions of the system, each of which offers its own advantages and insights: (i) A nonrelativistic effective field theory (EFT) with which we show that the relativistic corrections induce an effective self-interaction for the nonrelativistic field. As a byproduct, our EFT also allows one to construct the exact solution, including oscillatory behavior, which is often difficult to achieve from the exact equations. (ii) An effective (imperfect) fluid description, with which we demonstrate that, for a perturbed Friedmann-LemaˆıtreRobertson-Walker (FLRW) universe: (a) The pressure is small but nonzero (and positive), even for a free theory with no tree-level self-interactions. (b) The sound speed of small fluctuations is also nonzero (and positive), reproducing already known leading-order results, correcting a subdominant term, and identifying a new contribution that had been omitted in previous analyses. (c) The fluctuations experience a negative effective bulk viscosity. The positive sound speed and the negative bulk viscosity act in favor of and against the growth of overdensities, respectively. The net effect may be considered a smoking gun for ultra-light dark matter. Keywords: Classical Theories of Gravity, Effective Field Theories ArXiv ePrint: 2005.05388
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)059
JHEP07(2020)059
Effective theories for a nonrelativistic field in an expanding universe: Induced self-interaction, pressure, sound speed, and viscosity
Contents 1 Introduction
1
2 A suitable field redefinition 2.1 FLRW background
4 8
effective field theory in the nonrelativistic limit Smearing and mode expansion Effective field theory for the background Effective field theory for fluctuations
9 10 12 16
4 Effective fluid description
20
5 Comparison with numerical solutions
23
6 Summary and outlook
26
A The Hamiltonian
27
B Higher-order terms for fluctuations
31
C Viscous fluid
34
D EFT for the self-interacting field
36
E Analyses in a multicomponent universe
38
F Nonlocal field redefinition and the sound speed at arbitrary scale
40
G Gauge transformation G.1 Time-averaged comoving gauge
43 45
1
Introduction
Ultra-light particles have been considered a promising candidate for dark matter [1
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