K-Essence Lagrangians of Polytropic and Logotropic Unified Dark Matter and Dark Energy Models
Abstract
:1. Introduction
2. Nonrelativistic Theory
2.1. The Gross-Pitaevskii Equation
2.2. The Madelung Transformation
2.3. Lagrangian of a Quantum Barotropic Gas
2.4. Lagrangian of a Classical Barotropic Gas
2.5. Reduced Lagrangian
2.6. Polytropic Gas
2.7. Isothermal Gas
2.8. Chaplygin Gas
2.9. Standard BEC
2.10. DM Superfluid
2.11. Logotropic Gas
2.12. Summary
3. Relativistic Theory
3.1. Klein-Gordon Equation
3.2. The de Broglie Transformation
3.3. TF Approximation
3.4. Reduced Lagrangian
3.5. Nonrelativistic Limit
3.6. Enthalpy
4. General Equation of State
4.1. Equation of State of Type I
4.1.1. Determination of , and
4.1.2. Determination of , and
4.1.3. Lagrangian
4.2. Equation of State of Type II
4.2.1. Determination of and
4.2.2. Determination of , and
4.2.3. Lagrangian
4.3. Equation of State of Type III
4.3.1. Determination of and
4.3.2. Determination of , and
4.3.3. Lagrangian
5. Polytropes
5.1. Polytropic Equation of State of Type I
5.1.1. Determination of , and
5.1.2. Determination of , and
5.1.3. Lagrangian
5.2. Polytropic Equation of State of Type II
5.2.1. Determination of and
5.2.2. Determination of , and
5.2.3. Lagrangian
5.3. Polytropic Equation of State of Type III
5.3.1. Determination of and
5.3.2. Determination of , and
5.3.3. Lagrangian
6. Logotropes
6.1. Logotropic Equation of State of Type I
6.1.1. Determination of , and
6.1.2. Determination of , and
6.1.3. Lagrangian
6.2. Logotropic Equation of State of Type II
6.2.1. Determination of and
6.2.2. Determination of , and
6.2.3. Lagrangian
6.3. Logotropic Equation of State of Type III
6.3.1. Determination of and
6.3.2. Determination of , and
6.3.3. Lagrangian
7. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. General Identities for a Nonrelativistic Cold Gas
Appendix B. K-Essence Lagrangian of a Real SF
Appendix B.1. General Results
Appendix B.2. Canonical SF
Appendix B.3. Tachyonic SF
Appendix B.4. Nonrelativistic Limit
Appendix B.5. Cosmological Evolution
Appendix C. Equation of State of Type I
Appendix C.1. Friedmann Equations
Appendix C.2. Canonical SF
Appendix C.3. Tachyonic SF
Appendix D. Equation of State of Type II
Appendix D.1. General Results
Appendix D.2. Cosmology
Appendix D.3. Two-Fluid Model
Appendix E. Equation of State of Type III
Appendix E.1. General Results
Appendix E.2. TF Approximation
Appendix F. Analogies and Differences between u and V
Appendix G. Energy Conservation Equation for a SF
Appendix G.1. Complex SF
Appendix G.2. Real Canonical SF
Appendix G.3. Real Tachyonic SF
Appendix H. Some Studies Devoted to Polytropic and Logotropic Equations of State of Type I, II and III
Appendix I. Conservation Laws for a Nonrelativistic SF
Appendix I.1. Conservation Laws in Terms of Hydrodynamic Variables
Appendix I.2. Conservation Laws in Terms of the Wave Function
Appendix I.3. Conservation Laws from the Lagrangian Expressed in Terms of the Wave Function
Appendix I.4. Conservation Laws from the Lagrangian Expressed in Terms of Hydrodynamic Variables
Appendix J. Conservation Laws for a Relativistic Complex SF
Appendix J.1. Conservation Laws from the Lagrangian Expressed in Terms of the Wave Function
Appendix J.2. Conservation Laws from the Lagrangian Expressed in Terms of Hydrodynamic Variables
1 | |
2 | SFs have been used in a variety of inflationary models [34] to describe the transition from the exponential (de Sitter) expansion of the early universe to a decelerated expansion (radiation era). It was therefore natural to try to understand the present acceleration of the universe, which has an exponential behaviour too, in terms of SFs [35,36]. However, one has to deal now with the opposite situation, i.e., describing the transition from a decelerated expansion (matter era) to an exponential (de Sitter) expansion. |
3 | |
4 | To explain the accelerated expansion taking place today, the universe must be dominated by a fluid of negative pressure violating the strong energy condition, i.e., . For a fluid with a linear equation of state , where is a constant, we need to have an acceleration. But, in that case, , yielding instabilities at small scales [18]. This is the usual problem for a fluid description of domain walls () and cosmic strings (). Quintessence models with a standard kinetic term do not have this problem because the speed of sound is equal to the speed of light [51]. K-essence models [37,38,39] with a nonstandard kinetic term are different in this respect [41], but they still have a positive squared speed of sound (note that can exceed the speed of light in that case). |
5 | Negative pressures arise in different domains of physics such as exchange forces in atoms, stripe states in the quantum Hall effect, Bose-Einstein condensates with an attractive self-interaction etc. |
6 | A d-brane is a d-dimensional extended object. For example, a ()-brane is a membrane. d-branes arise in string theory for the following reason. Just as the action of a relativistic point particle is proportional to the world line it follows, the action of a relativistic string is proportional to the area of the sheet that it traces by traveling through spacetime. The close connection between a relativistic membrane [()-brane] in three spatial dimensions and planar fluid mechanics was known to J. Goldstone (unpublished) and developed by Hoppe and Bordemann [66,70] (the Chaplygin equation of state appears explicitly in [66] and the Born-Infeld Lagrangian associated with the action of a membrane appears explicitly in [70]). These results were generalized to arbitrary d-branes by Jackiw and Polychronakos [58]. The same Lagrangian appears as the leading term in Sundrum’s [75] effective field theory approach to large extra dimensions. The Born-Infeld Lagrangian can be viewed as a k-essence Lagrangian involving a nonstandard kinetic term. The Chaplygin equation of state is obtained from the stress-energy tensor derived from this Lagrangian. Therefore, the Chaplygin gas is the hydrodynamical description of a SF with the Born-Infeld Lagrangian. A more general k-essence model is the string theory inspired tachyon Lagrangian with a potential [76,77,78,79,80,81,82,83,84,85,86,87]. It can be shown that every tachyon condensate model can be interpreted as a brane moving in a bulk [88,89]. The Born-Infeld Lagrangian is recovered when is replaced by a constant [81]. |
7 | This generalization was mentioned by Kamenshchik et al. [50], Bilic et al. [71] and Gorini et al. [94], and was specifically worked out by Bento et al. [95]. Equation (7) can be viewed as a polytropic equation of state with a polytropic index and a polytropic constant . A further generalization of the GCG model has been proposed. It has an equation of state
|
8 | If a solution to these problems cannot be provided, this would appear as an evidence for an independent origin of DM and DE (i.e., they are two distinct substances) [103]. |
9 | This procedure is not well-defined mathematically because it yields an infinite additional constant . This constant disappears if we take the gradient of the pressure as in [118]. However, in general, an infinite constant term remains. Therefore, the above procedure simply suggests a connection between the polytropic and logotropic equations of state, but this connection is rather subtle. |
10 | |
11 | We could also consider the case of self-gravitating BECs. In that case, one has to introduce a mean field gravitational potential in the GP equation which is produced by the particles themselves through a Poisson equation (see Ref. [126] for more details). |
12 | We note that the potential is defined from the pressure up to a term of the form , where A is a constant. If we add a term in the potential V, we do not change the pressure. On the other hand, if we add a constant term C in the potential V, this adds a term in the pressure. However, for nonrelativistic systems, this constant term has no observable effect since only the gradient of the pressure matters. |
13 | Similarly, the invariance of the Lagrangian with respect to spatial translation implies the conservation of linear momentum. In relativity theory, these apparently separate conservation laws are aspects of a single conservation law, that of the energy-momentum tensor (see Section 3). |
14 | The variable is related to the phase (angle) of the SF by . |
15 | In our framework, the limit corresponds to the TF approximation where the quantum potential can be neglected. |
16 | The condition is necessary for local stablity since . When the system is subjected to a potential , the function F determines the relation between the density and the potential at equilibrium (see Section 3.4 of [136]). |
17 | |
18 | We stress that is not the rest-mass density (see below). It is only in the nonrelativistic regime that coincides with the rest-mass density . |
19 | The variable is related to the phase (angle) of the SF by . In the nonrelativistic limit, we shall denote the variable by . |
20 | The pseudo quadrivelocity does not satisfy so it is not guaranteed to be always timelike. Nevertheless, can be introduced as a convenient notation. |
21 | It differs from the pseudo quadrivelocity introduced in Equation (136). |
22 | We note that
|
23 | We note that the potential is defined from the pressure up to a term of the form , where A is a constant. If we add a term in the potential V, we do not change the pressure P but we introduce a term in the energy density. On the other hand, if we add a constant term C in the potential V (cosmological constant), this adds a term in the pressure and a term in the energy density. |
24 | We cannot directly take the limit in the KG equation. This is why we have to average over the oscillations. Alternatively, we can directly take the limit in the hydrodynamic equations associated with the KG equation. This is equivalent to the WKB method. |
25 | |
26 | We note that the expression of for an equation of state of type III coincides with the expression of for an equation of state of type II of the same functional form provided that we make the replacements and (see Appendix F). |
27 | The true equation of state of a relativistic BEC is given by Equation (331) or (346) corresponding to a polytrope of type III with index [149,152,153,154,155,156]. However, in order to have a unified terminology throughout the paper, we shall always associate the polytropic index to a BEC even if this association is not quite correct for models of type I and II in the relativistic regime (see Appendix H). As explained in Appendix F, all the models coincide in the nonrelativistic limit. |
28 | The linear equation of state (227) with corresponds to the ultrarelativistic limit of the equation of state (286) associated with a polytrope of type II with index . Indeed, for a polytrope , Equation (286) yields in the ultrarelativistic limit. The index corresponds to (radiation) and the index corresponds to (stiff matter). |
29 | The linear equation of state (227) with corresponds to the ultrarelativistic limit of the equation of state (340) associated with a polytrope of type III with index . Indeed, for a polytrope , Equation (340) yields in the ultrarelativistic limit. The index corresponds to (radiation) and the index corresponds to (stiff matter). |
30 | |
31 | Note that V represents here the total potential including the rest mass term. For brevity, we write V instead of . |
32 | K-essence Lagrangians were initially introduced to describe inflation (k-inflation) [40,41]. They were later used to described dark energy [37,38,39]. K-essence Lagrangians can also be obtained from a canonical complex SF in the TF limit [71,89]. In that case, the real SF corresponds to the action (phase) of the complex SF (see Section 3.4). |
33 | Note that the more general Lagrangian (A23) does not conserve the charge (or the boson number). |
34 | |
35 | In a homogeneous universe, |
36 | The case of a phantom universe is treated in [99]. |
37 | |
38 | We have taken so that the charge of the SF coincides with the boson number. |
39 | For a spatially homogeneous SF, it is shown in Ref. [149] that the TF approximation is equivalent to the fast oscillation approximation . |
References
- Planck Collaboration; Ade, P.A.R.; Aghanim, N.; Armitage-Caplan, C.; Arnaud, M.; Ashdown, M.; Atrio-Barandela, F.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; et al. Planck 2013 results. XVI. Cosmological parameters. Astron. Astrophys. 2014, 571, A16. [Google Scholar]
- Planck Collaboration; Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; Bartlett, J.G.; et al. Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. 2016, 594, A13. [Google Scholar]
- Sahni, V.; Starobinsky, A. The Case for a Positive Cosmological Λ-Term. Int. J. Mod. Phys. D 2000, 9, 373. [Google Scholar] [CrossRef]
- Carroll, S.M. The Cosmological Constant. Living Rev. Relativ. 2001, 4, 1. [Google Scholar] [CrossRef]
- Guth, A.H. Inflationary universe: A possible solution to the horizon and flatness problems. Phys. Rev. D 1981, 23, 347. [Google Scholar] [CrossRef]
- Weinberg, S. The cosmological constant problem. Rev. Mod. Phys. 1989, 61, 1. [Google Scholar] [CrossRef]
- Padmanabhan, T. Cosmological constant-the weight of the vacuum. Phys. Rep. 2003, 380, 235. [Google Scholar] [CrossRef]
- Steinhardt, P. Critical Problems in Physics; Fitch, V.L., Marlow, D.R., Eds.; Princeton University Press: Princeton, NJ, USA, 1997. [Google Scholar]
- Zlatev, I.; Wang, L.; Steinhardt, P.J. Quintessence, Cosmic Coincidence, and the Cosmological Constant. Phys. Rev. Lett. 1999, 82, 896. [Google Scholar] [CrossRef]
- Moore, B.; Quinn, T.; Governato, F.; Stadel, J.; Lake, G. Cold collapse and the core catastrophe. MNRAS 1999, 310, 1147. [Google Scholar] [CrossRef]
- Kauffmann, G.; White, S.D.M.; Guiderdoni, B. The formation and evolution of galaxies within merging dark matter haloes. Mon. Not. R. Astron. Soc. 1993, 264, 201. [Google Scholar] [CrossRef]
- Klypin, A.; Kravtsov, A.V.; Valenzuela, O. Where Are the Missing Galactic Satellites? Astrophys. J. 1999, 522, 82. [Google Scholar] [CrossRef]
- Kamionkowski, M.; Liddle, A.R. The Dearth of Halo Dwarf Galaxies: Is There Power on Short Scales? Phys. Rev. Lett. 2000, 84, 4525. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Boylan-Kolchin, M.; Bullock, J.S.; Kaplinghat, M. Too big to fail? The puzzling darkness of massive Milky Way subhaloes. MNRAS 2011, 415, L40. [Google Scholar] [CrossRef]
- Bullock, J.S.; Boylan-Kolchin, M. Small-Scale Challenges to the ΛCDM Paradigm. Ann. Rev. Astron. Astrophys. 2017, 55, 343. [Google Scholar] [CrossRef]
- Vilenkin, A.; Shellard, E.P.S. Cosmic String and Topological Defects; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Kamionkowski, M.; Toumbas, N. A Low-Density Closed Universe. Phys. Rev. Lett. 1996, 77, 587. [Google Scholar] [CrossRef]
- Fabris, J.C.; Martin, J. Amplification of density perturbations in fluids with negative pressure. Phys. Rev. D 1997, 55, 5205. [Google Scholar] [CrossRef]
- Fabris, J.C.; de Borba Gonçalves, S.V. Perturbations in a stringlike fluid. Phys. Rev. D 1997, 56, 6128. [Google Scholar] [CrossRef]
- Spergel, D.; Pen, U.L. Cosmology in a String-Dominated Universe. Astrophys. J. 1997, 491, L67. [Google Scholar] [CrossRef]
- Turner, M.S.; White, M. CDM models with a smooth component. Phys. Rev. D 1997, 56, R4439. [Google Scholar] [CrossRef]
- Silveira, V.; Waga, I. Cosmological properties of a class of Λ decaying cosmologies. Phys. Rev. D 1997, 56, 4625. [Google Scholar] [CrossRef]
- Chiba, T.; Sugiyama, N.; Nakamura, T. Cosmology with x-matter. Mon. Not. R. Astron. Soc. 1997, 289, L5. [Google Scholar] [CrossRef]
- Caldwell, R.R.; Dave, R.; Steinhardt, P.J. Cosmological Imprint of an Energy Component with General Equation of State. Phys. Rev. Lett. 1998, 80, 1582. [Google Scholar] [CrossRef] [Green Version]
- Peebles, P.J.E.; Ratra, B. Cosmology with a Time-Variable Cosmological “Constant”. Astrophys. J. 1988, 325, L17. [Google Scholar] [CrossRef]
- Ratra, B.; Peebles, P.J.E. Cosmological consequences of a rolling homogeneous scalar field. Phys. Rev. D 1988, 37, 3406. [Google Scholar] [CrossRef] [PubMed]
- Wetterich, C. Cosmology and the fate of dilatation symmetry. Nucl. Phys. B 1988, 302, 668. [Google Scholar] [CrossRef]
- Wetterich, C. An asymptotically vanishing time-dependent cosmological "constant". Astron. Astrophys. 1995, 301, 321. [Google Scholar]
- Frieman, J.; Hill, C.; Stebbins, A.; Waga, I. Cosmology with Ultralight Pseudo Nambu-Goldstone Bosons. Phys. Rev. Lett. 1995, 75, 2077. [Google Scholar] [CrossRef]
- Coble, K.; Dodelson, S.; Frieman, J.A. Dynamical Λ models of structure formation. Phys. Rev. D 1997, 55, 1851. [Google Scholar] [CrossRef]
- Ferreira, P.G.; Joyce, M. Structure Formation with a Self-Tuning Scalar Field. Phys. Rev. Lett. 1997, 79, 4740. [Google Scholar] [CrossRef]
- Copeland, E.J.; Liddle, A.R.; Wands, D. Exponential potentials and cosmological scaling solutions. Phys. Rev. D 1998, 57, 4686. [Google Scholar] [CrossRef]
- Ferreira, P.G.; Joyce, M. Cosmology with a primordial scaling field. Phys. Rev. D 1998, 58, 023503. [Google Scholar] [CrossRef]
- Linde, A. Particle Physics and Inflationary Cosmology; Harwood: Chur, Switzerland, 1990. [Google Scholar]
- Starobinsky, A. How to determine an effective potential for a variable cosmological term. JETP Lett. 1998, 68, 757. [Google Scholar] [CrossRef]
- Saini, T.D.; Raychaudhury, S.; Sahni, V.; Starobinsky, A. Reconstructing the Cosmic Equation of State from Supernova Distances. Phys. Rev. Lett. 2000, 85, 1162. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Chiba, T.; Okabe, T.; Yamaguchi, M. Kinetically driven quintessence. Phys. Rev. D 2000, 62, 023511. [Google Scholar] [CrossRef]
- Armendariz-Picon, C.; Mukhanov, V.; Steinhardt, P.J. Dynamical Solution to the Problem of a Small Cosmological Constant and Late-Time Cosmic Acceleration. Phys. Rev. Lett. 2000, 85, 4438. [Google Scholar] [CrossRef]
- Armendariz-Picon, C.; Mukhanov, V.; Steinhardt, P.J. Essentials of k-essence. Phys. Rev. D 2001, 63, 103510. [Google Scholar] [CrossRef]
- Armendariz-Picon, C.; Damour, T.; Mukhanov, V. k-Inflation. Phys. Lett. B 1999, 458, 209. [Google Scholar] [CrossRef]
- Garriga, J.; Mukhanov, V. Perturbations in k-inflation. Phys. Lett. B 1999, 458, 219. [Google Scholar] [CrossRef]
- Caldwell, R.R. A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state. Phys. Lett. B 2002, 545, 23. [Google Scholar] [CrossRef]
- Caldwell, R.R.; Kamionkowski, M.; Weinberg, N. Phantom Energy: Dark Energy with w < −1 Causes a Cosmic Doomsday. Phys. Rev. Lett. 2003, 91, 071301. [Google Scholar]
- Bronstein, M.P. On the expanding universe. Phys. Z. Sowjetunion 1933, 3, 73. [Google Scholar]
- Bertolami, O. Brans-Dicke Cosmology with a Scalar Field Dependent Cosmological Term. Fortschr. Phys. 1986, 34, 829. [Google Scholar]
- Bertolami, O. Time-dependent cosmological term. Nuovo Cimento 1986, 93, 36. [Google Scholar] [CrossRef]
- Özer, M.; Taha, M.O. A model of the universe free of cosmological problems. Nucl. Phys. B 1987, 287, 776. [Google Scholar] [CrossRef]
- Brax, P.; Martin, J. Quintessence and supergravity. Phys. Lett. B 1999, 468, 40. [Google Scholar] [CrossRef]
- Steinhardt, P.J.; Wang, L.; Zlatev, I. Cosmological tracking solutions. Phys. Rev. D 1999, 59, 123504. [Google Scholar] [CrossRef] [Green Version]
- Kamenshchik, A.; Moschella, U.; Pasquier, V. An alternative to quintessence. Phys. Lett. B 2001, 511, 265. [Google Scholar] [CrossRef]
- Grishchuk, L.P. Density perturbations of quantum-mechanical origin and anisotropy of the microwave background. Phys. Rev. D 1994, 50, 7154. [Google Scholar] [CrossRef]
- Makler, M.; Oliveira, S.Q.; Waga, I. Constraints on the generalized Chaplygin gas from supernovae observations. Phys. Lett. B 2003, 555, 1. [Google Scholar] [CrossRef]
- Chaplygin, S. Gas jets. Sci. Mem. Moscow Univ. Math. Phys. 1904, 21, 1. [Google Scholar]
- Tsien, H.S. Two-Dimensional Subsonic Flow of Compressible Fluids. J. Aeron. Sci. 1939, 6, 399. [Google Scholar] [CrossRef]
- Von Kármán, T. Compressibility Effects in Aerodynamics. J. Aeron. Sci. 1941, 8, 337. [Google Scholar] [CrossRef]
- Stanyukovich, K. Unsteady Motion of Continuous Media; Pergamon: Oxford, UK, 1960. [Google Scholar]
- Bazeia, D.; Jackiw, R. Nonlinear Realization of a Dynamical Poincaré Symmetry by a Field-Dependent Diffeomorphism. Ann. Phys. 1998, 270, 246. [Google Scholar] [CrossRef]
- Jackiw, R.; Polychronakos, A.P. Fluid Dynamical Profiles and Constants of Motion from d-Branes. Commun. Math. Phys. 1999, 207, 107. [Google Scholar] [CrossRef]
- Jackiw, R. Lectures on Fluid Dynamics. A Particle Theorist’s View of Supersymmetric, Non-Abelian, Noncommutative Fluid Mechanics and d-Branes; Springer: New York, NY, USA, 2002. [Google Scholar]
- Hassaïne, M.; Horváthy, P.A. Field-Dependent Symmetries of a Non-relativistic Fluid Model. Ann. Phys. 2000, 282, 218. [Google Scholar] [CrossRef]
- Hoppe, J. Supermembranes in 4 Dimensions. arXiv 1993, arXiv:hep-th/9311059. [Google Scholar]
- Bazeia, D. Galileo invariant system and the motion of relativistic d-branes. Phys. Rev. D 1999, 59, 085007. [Google Scholar] [CrossRef]
- Jackiw, R.; Polychronakos, A.P. Supersymmetric fluid mechanics. Phys. Rev. D 2000, 62, 085019. [Google Scholar] [CrossRef] [Green Version]
- Bergner, Y.; Jackiw, R. Integrable supersymmetric fluid mechanics from superstrings. Phys. Lett. A 2000, 284, 146. [Google Scholar] [CrossRef]
- Hassaïne, M. Supersymmetric Chaplygin gas. Phys. Lett. A 2001, 290, 157. [Google Scholar] [CrossRef]
- Bordemann, M.; Hoppe, J. The dynamics of relativistic membranes. Reduction to 2-dimensional fluid dynamics. Phys. Lett. B 1993, 317, 315. [Google Scholar] [CrossRef]
- Jevicki, A. Light-front partons and dimensional reduction in relativistic field theory. Phys. Rev. D 1998, 57, R5955. [Google Scholar] [CrossRef]
- Ogawa, N. Remark on the classical solution of the Chaplygin gas as d-branes. Phys. Rev. D 2000, 62, 085023. [Google Scholar] [CrossRef]
- Born, M.; Infeld, L. Foundations of the New Field Theory. Proc. R. Soc. A 1934, 144, 425. [Google Scholar] [CrossRef]
- Hoppe, J. Some classical solutions of relativistic membrane equations in 4 space-time dimensions. Phys. Lett. B 1994, 329, 10. [Google Scholar] [CrossRef]
- Bilic, N.; Tupper, G.B.; Viollier, R.D. Unification of dark matter and dark energy: The inhomogeneous Chaplygin gas. Phys. Lett. B 2002, 535, 17. [Google Scholar] [CrossRef]
- Polchinski, J. String Theory; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Rubakov, V.A. Large and infinite extra dimensions. Phys. Usp. 2001, 44, 871. [Google Scholar] [CrossRef]
- Nambu, Y. Lectures on the Copenhagen Summer Symposium 1970. unpublished.
- Sundrum, R. Effective field theory for a three-brane universe. Phys. Rev. D 1999, 59, 085009. [Google Scholar] [CrossRef] [Green Version]
- Garousi, M.R. Tachyon couplings on non-BPS D-branes and Dirac-Born-Infeld action. Nucl. Phys. B 2000, 584, 284. [Google Scholar] [CrossRef]
- Sen, A. Rolling Tachyon. J. High Energy Phys. 2002, 04, 048. [Google Scholar] [CrossRef]
- Sen, A. Tachyon Matter. J. High Energy Phys. 2002, 07, 065. [Google Scholar] [CrossRef]
- Sen, A. Field Theory of Tachyon Matter. Mod. Phys. Lett. A 2002, 17, 1797. [Google Scholar] [CrossRef]
- Gibbons, G. Cosmological evolution of the rolling tachyon. Phys. Lett. B 2002, 537, 1. [Google Scholar] [CrossRef]
- Frolov, A.; Kofman, L.; Starobinsky, A. Prospects and problems of tachyon matter cosmology. Phys. Lett. B 2002, 545, 8. [Google Scholar] [CrossRef]
- Felder, G.; Kofman, L.; Starobinsky, A. Caustics in Tachyon Matter and Other Born-Infeld Scalars. JHEP 2002, 09, 026. [Google Scholar] [CrossRef]
- Padmanabhan, T. Accelerated expansion of the universe driven by tachyonic matter. Phys. Rev. D 2002, 66, 021301. [Google Scholar] [CrossRef]
- Padmanabhan, T.; Choudhury, T.R. Can the clustered dark matter and the smooth dark energy arise from the same scalar field? Phys. Rev. D 2002, 66, 081301. [Google Scholar] [CrossRef]
- Bagla, J.S.; Jassal, H.K.; Padmanabhan, T. Cosmology with tachyon field as dark energy. Phys. Rev. D 2003, 67, 063504. [Google Scholar] [CrossRef] [Green Version]
- Gorini, V.; Kamenshchik, A.; Moschella, U.; Pasquier, V. Tachyons, scalar fields, and cosmology. Phys. Rev. D 2004, 69, 123512. [Google Scholar] [CrossRef]
- Feinstein, A. Power-law inflation from the rolling tachyon. Phys. Rev. D 2002, 66, 063511. [Google Scholar] [CrossRef]
- Bilic, N.; Tupper, G.B.; Viollier, R.D. Chaplygin gas cosmology-unification of dark matter and dark energy. J. Phys. A 2007, 40, 6877. [Google Scholar] [CrossRef]
- Bilic, N.; Tupper, G.B.; Viollier, R.D. Cosmological tachyon condensation. Phys. Rev. D 2009, 80, 023515. [Google Scholar] [CrossRef]
- Randall, L.; Sundrum, R. An Alternative to Compactification. Phys. Rev. Lett. 1999, 83, 4690. [Google Scholar] [CrossRef]
- Banados, M.; Henneaux, M.; Teitelboim, C.; Zanelli, J. Geometry of the 2+1 black hole. Phys. Rev. D 1993, 48, 1506. [Google Scholar] [CrossRef]
- Hawking, S.; Page, D. Thermodynamics of black holes in anti-de Sitter space. Commun. Math. Phys. 1983, 87, 577. [Google Scholar] [CrossRef]
- Kamenshchik, A.; Moschella, U.; Pasquier, V. Chaplygin-like gas and branes in black hole bulks. Phys. Lett. B 2000, 487, 7. [Google Scholar] [CrossRef]
- Gorini, V.; Kamenshchik, A.; Moschella, U. Can the Chaplygin gas be a plausible model for dark energy? Phys. Rev. D 2003, 67, 063509. [Google Scholar] [CrossRef]
- Bento, M.C.; Bertolami, O.; Sen, A.A. Generalized Chaplygin gas, accelerated expansion, and dark-energy-matter unification. Phys. Rev. D 2002, 66, 043507. [Google Scholar] [CrossRef] [Green Version]
- Benaoum, H.B. Accelerated Universe from Modified Chaplygin Gas and Tachyonic Fluid. arXiv 2002, arXiv:hep-th/0205140. [Google Scholar]
- Chavanis, P.H. Models of universe with a polytropic equation of state: I. The early universe. Eur. Phys. J. Plus 2014, 129, 38. [Google Scholar] [CrossRef]
- Chavanis, P.H. Models of universe with a polytropic equation of state: II. The late universe. Eur. Phys. J. Plus 2014, 129, 222. [Google Scholar] [CrossRef]
- Chavanis, P.H. Models of universe with a polytropic equation of state: III. The phantom universe. arXiv 2012, arXiv:1208.1185. [Google Scholar]
- Bilic, N.; Lindebaum, R.J.; Tupper, G.B.; Viollier, R.D. Nonlinear evolution of dark matter and dark energy in the Chaplygin-gas cosmology. J. Cosmol. Astropart. Phys. 2004, 11, 008. [Google Scholar] [CrossRef]
- Neves, R.; Vas, C. Brane dynamics, the polytropic gas and conformal bulk fields. Phys. Lett. B 2003, 568, 153. [Google Scholar] [CrossRef]
- Avelino, P.P.; Beca, L.M.G.; de Carvalho, J.P.M.; Martins, C.J.A.P. The ΛCDM limit of the generalized Chaplygin gas scenario. J. Cosmol. Astropart. Phys. 2003, 9, 2. [Google Scholar] [CrossRef]
- Sandvik, H.B.; Tegmark, M.; Zaldarriaga, M.; Waga, I. The end of unified dark matter? Phys. Rev. D 2004, 69, 123524. [Google Scholar] [CrossRef]
- Fabris, J.C.; Gonçalves, S.V.B.; de Sá Ribeiro, R. Generalized Chaplygin Gas with α = 0 and the ΛCDM Cosmological Model. Gen. Relat. Grav. 2004, 36, 211. [Google Scholar] [CrossRef]
- Bento, M.C.; Bertolami, O.; Sen, A.A. WMAP constraints on the generalized Chaplygin gas model. Phys. Lett. B 2003, 575, 172. [Google Scholar] [CrossRef]
- Bento, M.C.; Bertolami, O.; Sen, A.A. Generalized Chaplygin gas and cosmic microwave background radiation constraints. Phys. Rev. D 2003, 67, 063003. [Google Scholar] [CrossRef]
- Bento, M.C.; Bertolami, O.; Sen, A.A. Generalized Chaplygin Gas Model: Dark Energy-Dark Matter Unification and CMBR Constraints. Gen. Relat. Grav. 2003, 35, 2063. [Google Scholar] [CrossRef]
- Carturan, D.; Finelli, F. Cosmological effects of a class of fluid dark energy models. Phys. Rev. D 2003, 68, 103501. [Google Scholar] [CrossRef]
- Amendola, L.; Finelli, F.; Burigana, C.; Carturan, D. WMAP and the generalized Chaplygin gas. JCAP 2003, 7, 5. [Google Scholar] [CrossRef]
- Fabris, J.C.; Gonçalves, S.V.B.; de Souza, P.E. Fitting the Supernova Type Ia Data with the Chaplygin Gas. arXiv 2002, arXiv:astro-ph/0207430. [Google Scholar]
- Dev, A.; Alcaniz, J.S.; Jain, D. Cosmological consequences of a Chaplygin gas dark energy. Phys. Rev. D 2003, 67, 023515. [Google Scholar] [CrossRef]
- Alcaniz, J.S.; Jain, D.; Dev, A. High-redshift objects and the generalized Chaplygin gas. Phys. Rev. D 2003, 67, 043514. [Google Scholar] [CrossRef]
- Colistete, R.; Fabris, J.C.; Gonçalves, S.V.B.; de Souza, P.E. Bayesian Analysis of the Chaplygin Gas and Cosmological Constant Models Using the SNe ia Data. Int. J. Mod. Phys. D 2004, 13, 669. [Google Scholar] [CrossRef]
- Silva, P.T.; Bertolami, O. Expected Constraints on the Generalized Chaplygin Equation of State from Future Supernova Experiments and Gravitational Lensing Statistics. Astrophys. J. 2003, 599, 829. [Google Scholar]
- Bean, R.; Doré, O. Are Chaplygin gases serious contenders for the dark energy? Phys. Rev. D 2003, 68, 023515. [Google Scholar] [CrossRef]
- Chavanis, P.H. New logotropic model based on a complex scalar field with a logarithmic potential. Phys. Rev. D 2022, 106, 063525. [Google Scholar] [CrossRef]
- Abdullah, A.; El-Zant, A.A.; Ellithi, A. The growth of fluctuations in Chaplygin gas cosmologies: A nonlinear Jeans scale for unified dark matter. arXiv 2021, arXiv:2108.03260. [Google Scholar]
- Chavanis, P.H. Is the Universe logotropic? Eur. Phys. J. Plus 2015, 130, 130. [Google Scholar] [CrossRef]
- Chavanis, P.H. The Logotropic Dark Fluid as a unification of dark matter and dark energy. Phys. Lett. B 2016, 758, 59. [Google Scholar] [CrossRef]
- Chavanis, P.H.; Kumar, S. Comparison between the Logotropic and ΛCDM models at the cosmological scale. J. Cosmol Astropart. Phys. 2017, 5, 18. [Google Scholar] [CrossRef] [Green Version]
- Chavanis, P.H. New predictions from the logotropic model. Phys. Dark Univ. 2019, 24, 100271. [Google Scholar] [CrossRef]
- Chavanis, P.H. Predictions from the logotropic model: The universal surface density of dark matter halos and the present proportions of dark matter and dark energy. Phys. Dark Univ. 2022, 37, 101098. [Google Scholar] [CrossRef]
- Donato, F.; Gentile, G.; Salucci, P.; Frigerio Martins, C.; Wilkinson, M.I.; Gilmore, G.; Grebel, E.K.; Koch, A.; Wyse, R. A constant dark matter halo surface density in galaxies. Mon. Not. R. Astron. Soc. 2009, 397, 1169. [Google Scholar] [CrossRef]
- Ferreira, V.M.C.; Avelino, P.P. New limit on logotropic unified dark energy models. Phys. Lett. B 2017, 770, 213. [Google Scholar] [CrossRef]
- Chavanis, P.H. Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, Toulouse, France. in preparation.
- Chavanis, P.H. Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions. I. Analytical results. Phys. Rev. D 2011, 84, 043531. [Google Scholar] [CrossRef]
- Gross, E.P. Classical theory of boson wave fields. Ann. Phys. 1958, 4, 57. [Google Scholar] [CrossRef]
- Gross, E.P. Structure of a quantized vortex in boson systems. Nuovo Cimento 1961, 20, 454. [Google Scholar] [CrossRef]
- Gross, E.P. Hydrodynamics of a Superfluid Condensate. J. Math. Phys. 1963, 4, 195. [Google Scholar] [CrossRef]
- Pitaevskii, L.P. Vortex Lines in an Imperfect Bose Gas. Sov. Phys. JETP 1961, 13, 451. [Google Scholar]
- Suárez, A.; Chavanis, P.H. Hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit. J. Phys. Conf. Ser. 2015, 654, 012008. [Google Scholar] [CrossRef]
- Chavanis, P.H.; Matos, T. Covariant theory of Bose-Einstein condensates in curved spacetimes with electromagnetic interactions: The hydrodynamic approach. Eur. Phys. J. Plus 2017, 132, 30. [Google Scholar] [CrossRef] [Green Version]
- Chavanis, P.H. Phase transitions between dilute and dense axion stars. Phys. Rev. D 2018, 98, 023009. [Google Scholar] [CrossRef]
- Chavanis, P.H. Quantum tunneling rate of dilute axion stars close to the maximum mass. Phys. Rev. D 2020, 102, 083531. [Google Scholar] [CrossRef]
- Madelung, E. Quantentheorie in hydrodynamischer Form. Zeit. F. Phys. 1927, 40, 322. [Google Scholar] [CrossRef]
- Chavanis, P.H. Dissipative self-gravitating Bose-Einstein condensates with arbitrary nonlinearity as a model of dark matter halos. Eur. Phys. J. Plus 2017, 132, 248. [Google Scholar] [CrossRef]
- Eckart, C. The Electrodynamics of Material Media. Phys. Rev. 1938, 54, 920. [Google Scholar] [CrossRef]
- Schakel, A. Effective Field Theory of Ideal-Fluid Hydrodynamics. Mod. Phys. Lett. B 1996, 10, 999. [Google Scholar] [CrossRef]
- Chandrasekhar, S. An Introduction to the Theory of Stellar Structure; University of Chicago Press: Chicago, IL, USA, 1939. [Google Scholar]
- Pethick, C.J.; Smith, H. Bose-Einstein Condensation in Dilute Gases; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Ferreira, E.G.M. Ultra-light dark matter. Astron. Astrophys. Rev. 2021, 29, 7. [Google Scholar] [CrossRef]
- Bekenstein, J.; Milgrom, M. Does the missing mass problem signal the breakdown of Newtonian gravity? Astrophys. J. 1984, 286, 7. [Google Scholar] [CrossRef]
- Chavanis, P.H.; Sire, C. Logotropic distributions. Physica A 2007, 375, 140. [Google Scholar] [CrossRef]
- Noether, E. Invariante Variationsprobleme. Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse 1918, 40, 235–237. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Fluid Mechanics; Pergamon Press: London, UK, 1959. [Google Scholar]
- De Broglie, L. La mécanique ondulatoire et la structure atomique de la matière et du rayonnement. J. Phys. 1927, 8, 225. [Google Scholar] [CrossRef]
- De Broglie, L. Sur le rôle des ondes continues Ψ en mécanique ondulatoire. Compt. Rend. Acad. Sci. 1927, 185, 380. [Google Scholar]
- De Broglie, L. Corpuscules et ondes Ψ. Compt. Rend. Acad. Sci. 1927, 185, 1118. [Google Scholar]
- Suárez, A.; Chavanis, P.H. Cosmological evolution of a complex scalar field with repulsive or attractive self-interaction. Phys. Rev. D 2017, 95, 063515. [Google Scholar] [CrossRef]
- Suárez, A.; Chavanis, P.H. Hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit: General formalism and perturbations analysis. Phys. Rev. D 2015, 92, 023510. [Google Scholar] [CrossRef]
- Tooper, R.F. General Relativistic Polytropic Fluid Spheres. Astrophys. J. 1964, 140, 434. [Google Scholar] [CrossRef]
- Colpi, M.; Shapiro, S.L.; Wasserman, I. Boson stars: Gravitational equilibria of self-interacting scalar fields. Phys. Rev. Lett. 1986, 57, 2485. [Google Scholar] [CrossRef] [PubMed]
- Li, B.; Rindler-Daller, T.; Shapiro, P.R. Cosmological constraints on Bose-Einstein-condensed scalar field dark matter. Phys. Rev. D 2014, 89, 083536. [Google Scholar] [CrossRef]
- Chavanis, P.H. Partially relativistic self-gravitating Bose-Einstein condensates with a stiff equation of state. Eur. Phys. J. Plus 2015, 130, 181. [Google Scholar] [CrossRef]
- Chavanis, P.H.; Harko, T. Bose-Einstein condensate general relativistic stars. Phys. Rev. D 2012, 86, 064011. [Google Scholar] [CrossRef]
- Chavanis, P.H. Cosmological models based on a complex scalar field with a power-law potential associated with a polytropic equation of state. Phys. Rev. D 2022, 106, 043502. [Google Scholar] [CrossRef]
- Chandrasekhar, S. A limiting case of relativistic equilibrium. In General Relativity; Papers in Honour of Synge, J.L., O’ Raifeartaigh, L., Eds.; Oxford University Press: Oxford, UK, 1972. [Google Scholar]
- Yabushita, S. Pulsational instability of isothermal gas spheres within the frame-work of general relativity. Mon. Not. R. Astron. Soc. 1973, 165, 17. [Google Scholar] [CrossRef]
- Yabushita, S. On the analogy between neutron star models and isothermal gas spheres and their general relativistic instability. Mon. Not. R. Astron. Soc. 1974, 167, 95. [Google Scholar] [CrossRef] [Green Version]
- Chavanis, P.H. Gravitational instability of finite isothermal spheres in general relativity. Analogy with neutron stars. Astron. Astrophys. 2002, 381, 709. [Google Scholar] [CrossRef]
- Chavanis, P.H. Relativistic stars with a linear equation of state: Analogy with classical isothermal spheres and black holes. Astron. Astrophys. 2008, 483, 673. [Google Scholar] [CrossRef]
- Tooper, R.F. Adiabatic Fluid Spheres in General Relativity. Astrophys. J. 1965, 142, 1541. [Google Scholar] [CrossRef]
- Chavanis, P.H. Cosmology with a stiff matter era. Phys. Rev. D 2015, 92, 103004. [Google Scholar] [CrossRef]
- Banerjee, R.; Ghosh, S.; Kulkarni, S. Remarks on the generalized Chaplygin gas. Phys. Rev. D 2007, 75, 025008. [Google Scholar] [CrossRef]
- Chavanis, P.H. Laboratoire de Physique Théorique, Université de Toulouse, CNRS, Toulouse, France. in preparation.
- Chavanis, P.H. A simple model of universe describing the early inflation and the late accelerated expansion in a symmetric manner. AIP Conf. Proc. 2013, 1548, 75. [Google Scholar]
- Chavanis, P.H. A Cosmological model based on a quadratic equation of state unifying vacuum energy, radiation, and dark energy. J. Gravity 2013, 2013, 682451. [Google Scholar] [CrossRef]
- Chavanis, P.H. A Cosmological Model Describing the Early Inflation, the Intermediate Decelerating Expansion, and the Late Accelerating Expansion of the Universe by a Quadratic Equation of State. Universe 2015, 1, 357. [Google Scholar] [CrossRef]
- Chavanis, P.H. A simple model of universe with a polytropic equation of state. J. Phys. Conf. Ser. 2018, 1030, 012009. [Google Scholar] [CrossRef]
- Chavanis, P.H. A Real Scalar Field Unifying the Early Inflation and the Late Accelerating Expansion of the Universe through a Quadratic Equation of State: The Vacuumon. Universe 2022, 8, 92. [Google Scholar] [CrossRef]
- Chavanis, P.H.; Lemou, M.; Méhats, F. Models of dark matter halos based on statistical mechanics: The classical King model. Phys. Rev. D 2015, 91, 063531. [Google Scholar] [CrossRef] [Green Version]
- Chavanis, P.H.; Lemou, M.; Méhats, F. Models of dark matter halos based on statistical mechanics: The fermionic King model. Phys. Rev. D 2015, 92, 123527. [Google Scholar] [CrossRef]
- Chavanis, P.H. Predictive model of BEC dark matter halos with a solitonic core and an isothermal atmosphere. Phys. Rev. D 2019, 100, 083022. [Google Scholar] [CrossRef]
- Kaup, D.J. Klein-Gordon Geon. Phys. Rev. 1968, 172, 1331. [Google Scholar] [CrossRef]
- Ruffini, R.; Bonazzola, S. Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State. Phys. Rev. 1969, 187, 1767. [Google Scholar] [CrossRef]
- Braaten, E.; Zhang, H. The physics of axion stars. Rev. Mod. Phys. 2019, 91, 041002. [Google Scholar] [CrossRef]
- Hui, L.; Ostriker, J.; Tremaine, S.; Witten, E. Ultralight scalars as cosmological dark matter. Phys. Rev. D 2017, 95, 043541. [Google Scholar] [CrossRef]
- Chavanis, P.H. Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas. Eur. Phys. J. Plus 2020, 135, 290. [Google Scholar] [CrossRef]
- Mukhanov, V.; Vikman, A. Enhancing the tensor-to-scalar ratio in simple inflation. J. Cosmol. Astropart. Phys. 2006, 2, 4. [Google Scholar] [CrossRef]
- Babichev, E. Global topological k-defects. Phys. Rev. D 2006, 74, 085004. [Google Scholar] [CrossRef]
- Fang, W.; Lu, H.Q.; Huang, Z.G. Cosmologies with a general non-canonical scalar field. Class. Quantum Grav. 2007, 24, 3799. [Google Scholar] [CrossRef] [Green Version]
- Mukhanov, V. Physical Foundations of Cosmology; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Joyce, M. Electroweak baryogenesis and the expansion rate of the Universe. Phys. Rev. D 1997, 55, 1875. [Google Scholar] [CrossRef]
- Gorini, V.; Kamenshchik, A.; Moschella, U.; Pasquier, V.; Starobinsky, A. Stability properties of some perfect fluid cosmological models. Phys. Rev. D 2005, 72, 103518. [Google Scholar] [CrossRef]
- Chimento, L.P. Extended tachyon field, Chaplygin gas, and solvable k-essence cosmologies. Phys. Rev. D 2004, 69, 123517. [Google Scholar] [CrossRef]
- Scherrer, R.J. Purely Kinetic k Essence as Unified Dark Matter. Phys. Rev. Lett. 2004, 93, 011301. [Google Scholar] [CrossRef]
- Ellis, G.F.R.; Madsen, M.S. Exact scalar field cosmologies. Class. Quantum Grav. 1991, 8, 667. [Google Scholar] [CrossRef]
- Copeland, E.J.; Sami, M.; Tsujikawa, S. Dynamics of Dark Energy. Int. J. Mod. Phys. D 2006, 15, 1753. [Google Scholar] [CrossRef]
- Bamba, K.; Capozziello, S.; Nojiri, S.; Odintsov, S.D. Dark energy cosmology: The equivalent description via different theoretical models and cosmography tests. Astrophys. Space Sci. 2012, 342, 155. [Google Scholar] [CrossRef]
- Halliwell, J.J. Scalar fields in cosmology with an exponential potential. Phys. Lett. B 1987, 185, 341. [Google Scholar] [CrossRef]
- Lucchin, F.; Matarrese, S. Power-law inflation. Phys. Rev. D 1985, 32, 1316. [Google Scholar] [CrossRef]
- Arbey, A.; Lesgourgues, J.; Salati, P. Cosmological constraints on quintessential halos. Phys. Rev. D 2002, 65, 083514. [Google Scholar] [CrossRef] [Green Version]
- Gu, J.-A.; Hwang, W.-Y.P. Can the quintessence be a complex scalar field? Phys. Lett. B 2001, 517, 1. [Google Scholar] [CrossRef]
- Kasuya, S. Difficulty of a spinning complex scalar field to be dark energy. Phys. Lett. B 2001, 515, 121. [Google Scholar] [CrossRef]
- Boyle, L.A.; Caldwell, R.R.; Kamionkowski, M. Spintessence! New models for dark matter and dark energy. Phys. Lett. B 2002, 545, 17. [Google Scholar] [CrossRef]
- Lane, H.J. On the Theoretical Temperature of the Sun; under the Hypothesis of a Gaseous Mass maintaining its Volume by its Internal Heat, and depending on the Laws of Gases as known to Terrestrial Experiment. Am. J. Sci. 1870, 50, 57. [Google Scholar] [CrossRef]
- Zöllner, E. Über Die Stabilität Kosmischer Massen; Universitätsbibliothek Leipzig: Leipzig, Germany, 1871. [Google Scholar]
- Emden, R. Gaskugeln; B.G. Teubne: Leipzig, Germany, 1907. [Google Scholar]
- McLaughlin, D.; Pudritz, R. A Model for the Internal Structure of Molecular Cloud Cores. Astrophys. J. 1996, 469, 194. [Google Scholar] [CrossRef]
- Latifah, S.; Sulaksono, A.; Mart, T. Bosons star at finite temperature. Phys. Rev. D 2014, 90, 127501. [Google Scholar] [CrossRef] [Green Version]
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Chavanis, P.-H. K-Essence Lagrangians of Polytropic and Logotropic Unified Dark Matter and Dark Energy Models. Astronomy 2022, 1, 126-221. https://doi.org/10.3390/astronomy1030011
Chavanis P-H. K-Essence Lagrangians of Polytropic and Logotropic Unified Dark Matter and Dark Energy Models. Astronomy. 2022; 1(3):126-221. https://doi.org/10.3390/astronomy1030011
Chicago/Turabian StyleChavanis, Pierre-Henri. 2022. "K-Essence Lagrangians of Polytropic and Logotropic Unified Dark Matter and Dark Energy Models" Astronomy 1, no. 3: 126-221. https://doi.org/10.3390/astronomy1030011
APA StyleChavanis, P. -H. (2022). K-Essence Lagrangians of Polytropic and Logotropic Unified Dark Matter and Dark Energy Models. Astronomy, 1(3), 126-221. https://doi.org/10.3390/astronomy1030011