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Deposited in DRO: 01 May 2014 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Zhao, H. and Li, B. (2010) 'Dark uid : a uni ed framework for modi ed Newtonian dynamics, dark matter, and dark energy.', Astrophysical journal., 712 (1). pp. 130-141. Further information on publisher's website: http://dx.doi.org/10.1088/0004-637X/712/1/130
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Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk The Astrophysical Journal, 712:130–141, 2010 March 20 doi:10.1088/0004-637X/712/1/130 C 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
DARK FLUID: A UNIFIED FRAMEWORK FOR MODIFIED NEWTONIAN DYNAMICS, DARK MATTER, AND DARK ENERGY
HongSheng Zhao1 and Baojiu Li2 1 Scottish University Physics Alliance, University of St. Andrews, KY16 9SS, UK; [email protected] 2 Department of Applied Math and Theoretical Physics, Cambridge University, CB3 0WA, UK; [email protected] Received 2008 April 7; accepted 2009 December 7; published 2010 February 25
ABSTRACT Empirical theories of dark matter (DM) like modified Newtonian dynamics (MOND) gravity and of dark energy (DE) like f (R) gravity were motivated by astronomical data. But could these theories be branches rooted from a more general and hence generic framework? Here we propose a very generic Lagrangian of such a framework based on simple dimensional analysis and covariant symmetry requirements, and explore various outcomes in a top–down fashion. The desired effects of quintessence plus cold DM particle fields or MOND- like scalar field(s) are shown to be largely achievable by one vector field only. Our framework preserves the covariant formulation of general relativity, but allows the expanding physical metric to be bent by a single new species of dark fluid flowing in spacetime. Its non-uniform stress tensor and current vector are simple functions of a vector field with variable norm, not coupled with the baryonic fluid and the four-vector potential of the photon fluid. The dark fluid framework generically branches into a continuous spectrum of theories with DE and DM effects, including the f (R) gravity, tensor–vector–scalar-like theories, Einstein–Aether, and νΛ theories as limiting cases. When the vector field degenerates into a pure scalar field, we obtain the physics for quintessence. Choices of parameters can be made to pass Big Bang nucleosynthesis, parameterized post- Newtonian, and causality constraints. In this broad setting we emphasize the non-constant dynamical field behind the cosmological constant effect, and highlight plausible corrections beyond the classical MOND predictions. Key words: cosmology: theory – dark matter – galaxies: kinematics and dynamics – gravitation Online-only material: color figures
1. INTRODUCTION anomaly. Rather in the same vein as how symmetry motivated GR, these theories meet the astronomical data only a posteri, Gravity, the earliest and the weakest of the known forces, has e.g.,Lietal.(2008) showed a vector field in the gravity sec- never been very settled. The beauty of covariant symmetry mo- tor could not be excluded by the accurate cosmic microwave tivated Einstein to supersede Newton’s paradigm with general background (CMB) data. relativity (GR), but (inadequate) empirical evidence motivated Nevertheless, the above order is not the only way to dis- Einstein to introduce first and then withdraw the cosmologi- cover theories. The puzzling blackbody radiation spectrum and cal constant, a concept defying quantum physics understanding Balmer’s curious empirical formula for hydrogen lines are even in modern day. While making generally no more than a fac- among the odd pieces of classical physics that led to the full tor of 2 corrections to Newton’s theory, GR and its equivalence formulation of quantum mechanics. This bottom–up approach is principles insist on covariant symmetries in spacetime. There often gradual, the arrival of the final theory taking several gener- is also no frame to measure locally any absolute direction of ations of formulations (e.g., from Planck’s model for blackbody gravitational acceleration for a free-falling observer. However, radiation and Bohr’s model for a hydrogen atom to Heisenberg’s symmetry can be spontaneously broken if there are dynamical matrices-based formulation in general) with different levels of interactions or couplings of fields, a well-known mechanism in mathematical rigor and sophistication. several branches of physics, especially the Higgs mechanism in Historically, Milgrom’s modified Newtonian dynamics particle physics that gives a mass to a particle. Many attempts (MOND) was invented without any packaging by covariant the- have been made to break the strong equivalence principle by ories of gravity, just as the concept of dark matter (DM) was adding new fields (degrees of freedom) in the gravity sector, invented by Zwicky without packaging first with supersymmetry- which essentially means the gravitational “constant” G may be like particle field theories. MOND is a model motivated to a new dynamical degree of freedom governed by other fields explain the curious uniform rules (or facts) underlying rota- coupled to the metric. The best known is the Brans–Dicke the- tion curves of many spiral galaxies, as Balmer’s formula and ory (Brans & Dicke 1961). The lesser known is that a vector its generalizations suggesting strongly a fundamental rule for field of a non-zero absolute value in vacuum can also be coupled all atomic lines. Since the rule is empirical and bare (without to gravity, to give absolute directions (Will 1993). It has long covariance), it waits to be enshrouded by a theory preserving been suggested that Lorentz symmetry can be broken locally in basic symmetries to predict any logical corrections to situations the quantum gravity and string theory context (Kostelecky & where the empirical rule must fail slightly, e.g., by a factor of 2 in Samuel 1989) to yield a vector field of a non-zero expectation some gravitational lenses made by elliptical galaxies and clus- value (e.g., pointing toward the direction of time) in the vacuum. ters of galaxies. The tensor–vector–scalar (TeVeS) framework The most successful attempt so far is the Einstein–Aether the- of Bekenstein (2004), building up earlier constructions (e.g., ory of Jacobson & Mattingly (2001). A common theme of these Bekenstein 1988; Bekenstein & Sanders 1994) and especially theories is that they are not invented for certain observational the introduction of a vector field (Sanders 1997), makes the first
130 No. 1, 2010 DARK FLUID: A UNIFIED FRAMEWORK FOR MOND, DM, AND DE 131 step to the integration of the MOND formula with fundamental scalar potential or interaction or the mass term:4 physics. A time-like vector field is shown to be the necessary 2 2 c ingredient of a MOND gravity. However, the vector field does Λint(ϕ,K4) = Λ0F (ϕ )+ϕ K4,K4 ≡∇Æc∇Æ , (2) not completely replace the role of the CDM particles (Zuntz −10 −1 2 et al. 2008) or neutrinos (Skordis et al. 2006), even though it is where Λ0 ≡ (1.2 × 10 ms ) and K4 are important capable of driving some structure growth (Dodelson & Liguori for creating the MOND effect, and K4 = 0 for a uniform a 2006). Yet the original aim of TeVeS was limited, e.g., it is nei- cosmology; the notation ∇ ≡ Æ ∇a is a derivative parallel a ac c ther a model for dark energy (DE) nor for structure formation. to the local time direction, and ∇ = g ∇c,Æa = gacÆ .The Orthogonally many literatures considered theories of modified term Λint takes on several roles: it gives an effective potential or gravity such as the f (R) gravity (Chiba 2003) and scalar infla- mass for the scalar field, and allows the mass or sound speed to tion theory as ad hoc fixes of the cosmological constant problem vary with the interaction of ϕ and Æ; to achieve the cold dark and the horizon problem, respectively, without aiming to address matter (CDM) effect during structure formation would require −2 −2 outstanding questions on galaxy rotation curves. Recently, Zhao ϕ Λint → Λ0ϕ F → cst at high redshift when ϕ →∞, i.e., (2007) and Halle et al. (2008), building on the work of Zlosnik a nearly constant scale-free mass. To achieve the cosmological et al. (2007), showed that these outstanding problems of DM constant effect would require that Λint → Λ0F → cst in the and DE can find at least one common solution simultaneously future when ϕ → 0. in the framework of a massive vector field, which is called a νΛ Other interaction terms are collected in model. In these theories, there is “one field which rules them all 2 and in the darkness bind them.” These models share as inspi- L/ = e0ϕ R +[c1∇aÆc + c3∇cÆa b a c rations models like Chapligin gas (e.g., Bento et al. 2002) and − (e1Æagbc + e2Æbgac) ∇ ϕ]∇ Æ , (3) other unified models (e.g., Peebles & Vilenkin 1999), but use the Tully–Fisher relations explicitly as the basis of the models. 1 − c2 The goal of the paper is to build a framework of new kinds a 2 ϕ 2 a b ∗ L = c (∇ Æ ) − (∇ϕ) +(g Æ Æ − 1)L , of models. We shall show that MOND, f (R) gravity, Einstein– Æ 2 a N 2 ab Aether theory, and DE theories can be integrated into a common (4) covariant framework with a Lagrangian depending on a unit 3 vector and a dynamical scalar field. MOND would become where L∗ is the Lagrangian multiplier. a specific choice of the potential of the scalar field. Having In an effective sense, we can pack our Æa and ϕ fields as such a framework allows one to explore the consequences of one field: our Lagrangian can be cast in terms of a vector field modified gravity systematically. It can be meaningless to even of dynamics norm Za ≡ Æaϕ where one can pack up the 3+1 differentiate DE and modified gravity. Modified gravity contains degrees of freedom into a vector of 4 degrees of freedom in the extra fields, which can be treated as a DE field. same style as in Halle et al. (2008). The implications of such a One of the goals of the paper is to show covariant corrections vector field with a norm λ ≡ ϕ2 is discussed in Appendix A, to the MOND formulae for a time-dependent system. There can where we illustrate how a TeVeS Lagrangian L(Za) = L(ϕÆa) also be corrections at very small spatial scale. These are the reduces to various special cases, TeVeS,BSTV,Einstein–Aether, generic consequences of Lagrange equations. f (K), f (R) (e.g., Carroll & Lim 2004;Lim2005; Sanders 2005; The outline is as follows. We propose our general Lagrangian Kanno & Soda 2006). in Section 2, and choose a subset of models with MOND and For a first study, we shall set DE effects. We discuss the properties of our dark fluid in the cases of Hubble expansion (Section 3), and for static galaxies L/ = 0(5) (Section 4), where we give the modified Poisson equation and the equation for the Hubble expansion. We discuss corrections hereafter by setting the coefficients c1,c3,e0,e1,e2 (which are to MOND in Section 5. We summarize the properties of the generally functions of ϕ) to zero. Specifically, we choose to set dark fluid in Section 6. In Appendices A–E we give the full e0(ϕ) = 0toeliminatef (R)-gravity-like coupling and set the Lagrangian, full equations of motion (EOMs) of the theory, functions e1(ϕ) = e2(ϕ) = 0 to eliminate kinetic cross-coupling an estimate of the damping frequency of the dark fluid, and a like ∇Æ∇ϕ, and c1 = c3 = 0 to eliminate the spin-1 mode choice for the fluid’s potential function, and some background gravitational wave. This is done partly to simplify the dynamics, information of the 3+1 decomposition formulation. partly because any complicated couplings can be dangerous theoretically (see Appendix B); these coupling terms serve as 2. A SIMPLE LAGRANGIAN FOR THE DARK FLUID an effective potential for the scalar ϕ, and this dynamically changing potential does not appear to have a well-defined lower We propose a Lagrangian L containing a scalar ϕ and a time- a bound to safeguard the causal structure. It is tempting to set like unit vector Æ , and the metric tensor gab as = = − 2 c2 0 1 cϕ for simplicity; however, we will not do so yet since these terms in L could be desirable for a theory to be c2 Æ L = R − ϕ (∇ ϕ∇cϕ)+2Λ + L + L/, (1) healthy. N 2 c int Æ 2.1. Parameters for MOND-inspired Subset of Dark 2 where N, cϕ are various coupling constants or functions of Fluid Theories λ = ϕ2, R is the Ricci scalar, and Λ takes the role of the int A theory is√ built by minimizing the action S = −1 4 −(16πG) dx −g(Lm + L) once the Lagrangian density 3 A possible origin for the dark fluid is a field coupled to neutrinos of various flavors; this field allows transitions of neutrinos of various energy, mass, flavor, 4 Λ = 2 Λ = K4 Other functions int f (y)(ϕ +1) 0 with y 2 for some and helicity, and there are many possibilities within theories of neutrino mass (1+ϕ )Λ0 (Zhao 2008a). function f can also create MOND, DM, and cosmological constant. 132 ZHAO & LI Vol. 712
30 1
25 0.8
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F 15 mu
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0 -4 -2 0 2 4 0 0 2 4 6 8 10 phi x (a) (b) Figure 1. Panel (a): F (ϕ2)vs.ϕ for k = 3, 3/2, 2 (top to bottom); note√ the symmetric potential with the global minimum F = 0at√ϕ = 0, and the plateau at λ = 1. 2 Panel (b) shows our function μ ≡ 1 − ϕ as function of x ≡|∇Φ|/ Λ0 for the models with k = 3, 3/2, 2(toptodown),adopting Λ0 → a0. (A color version of this figure is available in the online journal.)
L is specified. Our Lagrangian L has three dynamical fields: the λ ≡ ϕ2 and F = 9(ϕ2/3 − 1)3 +9= 9(ϕ2 − 3ϕ4/3 +3ϕ2/3). scalar field λ (or equivalently ϕ), the Æther field Æa, and the An important asymptotic property of the functions F and metric field gab. Now minimizing the action S with respect to d F (a prime always means a derivative dλ)isthat variations of the three fields will lead to the scalar field EOM, d Æther field EOM, and the modified energy–momentum tensor. F ≡ F ∼ (1 − λ−1/3)2, if λ ≡ ϕ2 → 1, (7) Variations with the non-dynamical Lagrange multiplier L∗ gives dλ the unit-vector constraint for the Æ field. The general results are ∝ (λ − 2)−1/2, if λ ≡ ϕ2 → +2 ∼ 10−7, (8) more tedious and are presented elsewhere (Halle et al. 2008). To illustrate the physics and stay away from models with obvious ∼ 0ifλ ≡ ϕ2 →∞. (9) pathological dynamics, we consider only a specific choice of functions to select a MOND-related subset of dark fluid the- We shall show that this property describes a non-uniform ories. Starting from the Lagrangian given in Equation (1), we (DE) fluid which gives the MOND-like (DM) effects in select models with L/ = 0 and select the functions in Lϕ and LÆ galaxies, and Newtonian-like effects in the solar system. as follows. 2. We set the scalar field sound speed as a finite positive The dimensionless function F (λ) has the meaning of the constant, i.e., 2 potential of the scalar field λ ≡ ϕ . The coupling constants ∞ >cϕ > 0, (10) 2 cϕ and c2 are of order unity, and will be shown to be related to and we shall treat N as constants too. In general, theories the sound speeds of the dark fluid. with finite positive propagation speeds have a finite causal- 1. Our choice for the dark fluid potential (shown in Figure 1) ity cone, hence are well behaved (Bruneton & Esposito- with a constant energy scale Λ is Farese 2007). The opening angle of the cone of propagation 0 can be wider than that of photons. 3Λ 3Λ 3. We set the equivalent part of Jacobson’s unit vector field 2 k 0 k 0 1/k 3 Λ F (ϕ ) = + (|λ±| − 1) ,k= 3, (6) Lagrangian to be c K + c K + c K + c K = 0 × K + 0 3 3 1 1 3 3 4 4 2 2 1 0 × K3 +2λK4 + c2K2, where, e.g., c2K2 corresponds to 2 the second term in L . This choice c = c = 0inthe where we shall adopt λ± ≈ λ = ϕ essentially up to a small Æ 1 3 correction.5 The reason to choose k = 3 is such that we Lagrangian kills spin-1 mode waves of the vector field, 2 and guarantees that the normal gravitational wave in the make the potential Λ0F → 9Λ0ϕ to be harmonic at large 6 tensor mode will propagate with the normal speed (of light). ϕ to mimic CDM. In galaxies we can neglect ;soλ± ≈ This might not be necessary, but it simplifies the analysis of parameterized post-Newtonian (PPN) parameters in the ϕ2+2 5 More rigorously we make a small correction, e.g., λ± = or 1+2 solar system and the sound speed of the vector field. ϕ2±ϕ−24 − λ± = with 2 ∼ O(10 7) being a small positive constant, which One can use the analysis by Foster & Jacobson (2006)to 1−4 forms a barrier preventing the theory from entering unphysical regime in the predict the sound speed square of the spin-0 mode of Aether − − − solar system (cf. Appendix C). Note our choice of the function 2 ≡ (c2+c1+c3) (2 c4 c1) = (1 λ)c2 2 3 1 3/k 2/k 1/k fluctuation, c0 (c +c )(1−c −c ) (2+c +3c +c ) (2+3c )λ . Λ F (ϕ ) ≈ k Λ [ λ − λ + λ ] has the zero point Λ F | 2 = 0, 4 1 1 3 1 2 3 2 0 0 3 0 λ=ϕ =0 7 2 i.e., it is designed to give no cosmological constant like effect in the solar Among the many choices of c2 to guarantee that c0 and c2 3 system. At the other extreme Λ0F (1) = Λ0k /3, hence creating a constant 7 For example, a simple choice for c is c = 2 1−λ . This guarantees that potential at the cosmological scale. 2 2 3N2 1+λ 6 = − 2 Other choices are also interesting: a model with k 3/2 would represent c2 = (1 λ) 0; a large N can guarantee that the effect of the c K term is Λ → 9 Λ 4 0 3N2(1+λ)λ 2 2 relativistic DM at large ϕ with 0F 8 0ϕ . A model with k>3 could 2 | | 2K2 ∼ 2H drive inflation with an appropriate N. A model with k = 2 is the simplest with small since c2K2 2 2 , although it is unclear if it is desirable that Λ 3N N Λ = 8 0 | |− 3 = Λ 1 | |3 −| |2 | | 2 = = 2 0F 3 [( ϕ 1) +1] 8 0( 3 ϕ ϕ + ϕ ). c0 0atλ 1, and c0 can be small in the solar system. No. 1, 2010 DARK FLUID: A UNIFIED FRAMEWORK FOR MOND, DM, AND DE 133
1 2
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w1 0 w2 0.5 -2 -1 0 1 2 3 4 log10(lambda) 0 -2 -1 0 1 2 3 4 -0.5 log10(lambda) -0.5
-1 -1 (a) (b) 2 − 2 Figure 2. Panel (a): equation of state estimator w = ϕ F F (ϕ ) vs. log (λ)fork = 3, 3/2, 2 (bottom to top); note that w becomes negative for λ<10. Panel (b) 1 ϕ2F +F (ϕ2) 10 1 2 − 2 shows the sound of speed estimator w = δ[ϕ F F (ϕ )] . Note that k = 3 has very small sound speed a large λ, but there is a singularity near λ ∼ 1/3. 2 δ[ϕ2F +F (ϕ2)] (A color version of this figure is available in the online journal.)
are smooth functions for all values of λ, the simplest choice as the masses of the scalar and vector fields, hence our model is such that describes effectively an unstable slowly decaying dark particle. To be more explicit, consider the approximation α → 0 and c 1 1 − 2 = → 2 = Λ F ∼ 9ϕ2 for large ϕ in our model with k = 3. The scalar c0 . (11) 0 2 3λ 3λ EOM becomes a damped harmonic equation, c2 The important thing is that 0 is a finite real positive number ϕ¨ +3H ϕ˙ =−18N 2Λ ϕ, (14) 2 = 1 0 given by c0 3λ > 0. Interestingly, in the solar system, where the scalar field λ is expected to settle to a very small which gives the approximate solution − equilibrium value λ ∼ 2 ∼ O(10 7) for our choice of the 2 2 2 penalizing scalar field potential F (λ), the spin-0 mode of 9Λ0ϕ ≈ 3Ω0H cos (Nt 18Λ0), the vector field propagates with a finite superluminal speed, 1 = 2 −1/2 ϕ˙2 ≈ 3Ω H 2 sin2(Nt 18Λ ), (15) c0 (3 ) 1, avoiding the Cherenkov radiations 2 0 0 constraint in the solar system. All PPN parameters are 2N expect to be equal to that of GR in the solar system as −1 = 3 which is rigorous in the matter-dominated era when H 2 t, well; the PPN parameters α1 =−8λ, α2 = (3λ − 1)λ are where Ω0 ∼ O(1) is a constant, determined by the initial non-zero, but can be made small enough to fit current limits condition of ϕ. Taking the average over the rapid oscillations, | | −4 | | × −7 α1 < 10 and α2 < 4 10 (Foster & Jacobson 2006). we have Ω The general EOMs and Einstein equations are given in 1 2 3 0 2 (F − 9)Λ0 ≈ ϕ˙ ≈ H (16) Appendix A. In the following section, we will apply these to 2N 2 2 simple configurations: uniform cosmology and static galaxies. at high redshift. A numerical example of the oscillating scalar field is given in Figure 3. 3. BACKGROUND COSMOLOGY The modified 00 term of the Einstein equation becomes (see Consider background cosmology in the FRW flat metric, Equation (C8)) 2 = ¯ ¯ ds2 = dt2 − a2(t)(dx2 + dy2 + dz2). (12) 3H 8πG[ρ + ρDS] , (17) First, the scalar field follows an EOM exactly as a quintessence 1 1 (see Equation (C2)), ρ¯ ≡ ϕ˙2 + Λ F − α4πGρ¯ (18) DS α 2N 2 0 8πG 1+ 2 3α ˙ =− Λ 2 2 ϕ¨ +3H ϕ 0F + H (2N ϕ), (13) where ρ¯DS and ρ¯ are the (background) energy densities of the 2 dark sector (DM plus DE) and the baryon-radiation fluid, respec- → ϕ H =˙a/a tively. At high redshift when F λ we get a DM-like effect, so tracks the Hubble rate . Here we introduce an 1 2 2 α(λ) ˙ Λ − ≡ ≡ and [ 2N 2 ϕ + 0ϕ ] behaves like a CDM of the Broglie fre- auxiliary function b(λ)orα(λ) defined by b(λ) 1 2 2 c1+c3+3c2 =− −1 =−1 quency 2N Λ0. The DM-like effect is sub-dominant at very 2 λ for our choice of coefficients c2 3λ , where = 2 high redshift, e.g., radiation era or Big Bang nucleosynthesis λ ϕ . The vector field EOM gives the Lagrange multiplier →∞ → − 2 (BBN), where λ , hence α 0. Therefore, the BBN con- ∗ = 1 cϕ ˙2 ∗ 2Λ L ∂t (αH)+ N 2 ϕ . Since L and N 0F can be thought straint is automatically satisfied because the Hubble expansion 134 ZHAO & LI Vol. 712 4. STATIC GALAXY LIMIT To work out perturbations in static galaxies, remember that in the Newtonian gauge we have only two scalar mode perturbation potentials, Φ and Ψ, which appear in the perturbed metric: 2 = Φ 2 − 2 Ψ 2 2 2 ds (1 + 2 )dt a0 (1 + 2 )(dx + dy + dz ), (22)
where we will let a0 = 1. As a first study we assume no Hubble expansion. b The vector field EOM in static systems fixes Æa = gabÆ = (1 + Φ, 0, 0, 0), so the vector field tracks the metric exactly without any freedom in static galaxies. The 00 component of the Einstein equation becomes a Poisson equation (see Equation (C8)), −(2Ψ),ii = 8πG(ρ + ρDM + bρDE), (23) 2 2 i=x,y,z Figure 3. Contributions to the Hubble expansion ln(H /H0 ) from three components as a function of the scale factor ln a: baryonic matter (green straight line with an a−3 fall-off), radiation (yellow straight line with an a−4 fall-off), and 2 −3 dark fluid’s potential term a0 F/3 (the oscillating curve with a nearly a fall- i=x,y,z off of its amplitude for the epoch between 1 H0t 1/N). For illustration ρDM ≡ [2λΦ,i ],i , (24) purposes, we assume parameters Ωb = 10 Ωr = 0.03, and N = 1000, an initial 8πG condition φ = 1.8N at ln a =−10. Note the dark fluid is sub-dominate in the ≡ radiation era, but dominates the baryonic fluid by a factor of 2 in the matter where we use the notation F,i ∂i F , and the dummy index era, and starts to flat out to a constant amplitude at the present epoch, similar implies covariant or contra-variant derivatives with respect to to the cosmological constant effect. More realistic models with a larger N are x,y,z. We use the approximation that the DE part 8πGbρDE = challenging to integrate numerically (appropriate for studying inflation and DM ϕ˙2 Λ effects in galaxies), and the results are sensitive to the initial condition of the 2N 2 + 0F is a negligible source compared to 8πGρ from the scalar field. Nevertheless, the dark fluid seems to have the desired properties of baryons, and that in between that of a uniform cosmological constant and that of a CDM. −Ψ = Φ (25) (A color version of this figure is available in the online journal.) from the spatial cross term of the Einstein equation. The above result is essentially a Poisson equation where the vector field creates an effective DM-like source term ρDM. Rearrange the rate at BBN is equal to that of a radiation-only universe. We also terms; the same equation becomes the MOND Poisson equation: expect the early structure formation to resemble CDM models. Equivalently, the Einstein equation can be written as ∇·[(1 − λ)∇Φ] = 4πGρ, λ ≡ ϕ2. (26) To see that 1 − λ can be identified with the MOND μM a¨ 2 − 2 + H = 8πG(p¯ + p¯DS) (19) function, first we define a value of the scalar field ϕ such that a M |∇Φ|2 | ≡ F λ=ϕ2 . (27) 1 1 2 M Λ p¯DS ≡ ϕ˙ − Λ0F +2α Hϕϕ˙ − α4πGp¯ , 0 8πG 1+ α 2N 2 2 We find that the scalar field EOM is given as (see Equation (C2)) (20) 2 2 2 2 −c ∇ ϕ =−[Λ0F −|∇Φ| ](2N ϕ), (28) where the pressure of the baryon-radiation fluid p¯ = 0inthe ϕ matter-dominated era. Applying the harmonic approximation where we neglect all time-dependent terms. This equation is 1 ˙2 ∼ Λ − 2N 2 ϕ 0[F F (1)] we find the effective pressure of the similar to the equation of the Yukawa potential with a screening dark sector 2 ∇2 − 2 = length of cϕ/ω,(cϕ ω )ϕ 0. In the simplest case, we 9Λ0 2 2 −¯ ∼ adopt c → 0 to kill the Laplacian term ∇ ≡ = ∂i ∂i , pDS α . (21) ϕ i x,y,z 8πG 1+ 2 and we find that the equation for the scalar field becomes
This behaves like a DE with a characteristic scale Λ0, which is 2 3 −1 −1/k 2 2 k Λ λ k (λ − 1) ≈|∇Φ| (29) of the same order of magnitude (smaller by a factor of a few)8 as 0 ∼ × −10 −2 2 the observed cosmological constant (8 10 ms ) .Note for our choice of F (λ), where we neglect the term 4ϕ2 for a that in writing the above equations we have implicitly assumed lighter expression. For k = 3, the equation can then be easily that the effective DM and effective DE components couple to solved as9 each other. This can been seen by checking that neither DM ˙¯ ¯ ¯ = x −3 nor DE satisfies the conservation law ρ +3H(ρ + p) 0, but ≡ 2 → 2 ≈ ¯ λ ϕ ϕM 1+ . (30) their sum ρDS does. Figure 2 show two crude estimators for the 3 = √|∇Φ| x Λ equation of state parameter and possible problematic regions. 0
9 The model with k = 2 gives a MOND μ = 1 − λ 8 n3 1/3 2n − However, an interesting choice for Λint = Λ0(1 + θ)+(1+θ ) 4 3 x2 x 1/3 1/3 3 = 1+ + . (1 − nθ )K4,forθ ≡ (λ − 1) , especially n = 6 to produce the right 16 4 = √|∇Φ| 3Λ ∼ Λ x Λ amount of cosmological constant n 0/3 72 0. 0 No. 1, 2010 DARK FLUID: A UNIFIED FRAMEWORK FOR MOND, DM, AND DE 135
To see if we recover the properties of MOND function μM is perhaps longer). In the process of damping, there will − 2 (see 1 ϕM vs. x shown in Figure 1), we rewrite the solution of be a correction to the MOND μM function by the q term, − 2 = − 2 the scalar field as heuristically, 1 ϕ 1 ϕM ,if ≡ |∇√ Φ| 2 x, where x Λ 1 |∇Φ|2 1 − ϕ ≡ μ ≈ − 0 q M M 2 x 3 x → + , (35) 1 − − , where x 1. 2 3 Λ0 2N Λ0 (31) ≡ 2 − 2 ∇2 − 2 where q ∂t cϕ +(1 cϕ)η∂t is an operator. In tidally This is exactly the physics of MOND, if acting systems, the value for ϕ will oscillate between its pre- merging value and its equilibrium value. Λ → 0 a0 (32) Models with a small N would not give MOND, e.g., if N = 1–10, πν−1 ∼ (100 − 10) Gyr, then the universe would is identified with the MOND acceleration scale a0.Inthe be too young dynamically to have a precise MOND effect in solar system or strong gravity regime, the modification factor galaxies because ϕ would not have enough time to respond to − −3 ∼ 1 (x/3) 1 to the Newtonian Poisson equation is small the formation of galaxies. Rather ϕ would lag behind and might and reduces sharply. In weak gravity, applying the spherical remain close to its cosmological average, approximation around a dwarf galaxy of mass mb,wehave |∇Φ|2 = −2 2 = ∇Φ /a0 Gmbr , and the rotation curve Vcir(r) r . ϕ ∼¯ϕ, (36) The big success of MOND in dwarf spiral galaxies is to explain 4 = ∼ −10 −2 their Tully–Fisher relation Vcir(r)/(Gmb) a0 10 ms which would mean a boost of the gravity of the baryon by a −10 −2 2 2 −1 if Λ0 ∼ (1×10 ms ) , which is of the order of magnitude of constant factor (1 −¯ϕ ) everywhere. the observed amplitude of “the cosmological constant” effect. In the intermediate regime, our μM resembles the “standard” 6. GENERIC PROPERTIES OF DARK FLUID μ = √ x function of MOND, so it will fit rotation curves of 1+x2 It is still uncertain whether the time-dependent correction and galaxies very well. a possible diffusion term are enough to help MOND to explain the Bullet clusters (Angus et al. 2007; Angus & McGaugh 2008). 5. TEMPORAL AND SPATIAL CORRECTIONS TO However, it seems robust that the dark fluid—described by the MOND: OSCILLATIONS AND DIFFUSIONS field Æaϕ(λ)—is generally out of phase from the baryonic fluid. When considering merging systems like galaxy clusters, The dynamics of galaxies inside the dark fluid√ are governed by time-dependent terms λ¨ ∼ λ˙ 2 ∼ O(ω2) are important, where five variables, two for the scalar field ϕ = λ, the metric or − = | | g00 1 ω O( k σ ) is the inverse of the timescale to cross a system of potential Φ =−Ψ = = Æ0 − 1, and three for the vector − 2 | | 1 10 size k by stars of velocity dispersion σ in unit of the speed field perturbation part Æi = Æi − ui . These evolve according of light. There can also be diffusion on small scale due to a to the five coupled Equations (C2), (C5), and (C9) in the absence 2 2 pressure-like term ∇ λ =−|k| λ. of the Hubble expansion. The proof is given in Appendix C;the The scalar field EOM becomes (see Equation (C2)) following is a summary of the equations: 2 2 2 2 2 2 2 2 2 2 ∂ − c ∇ + 1 − c η∂ ϕ =−(ϕ − ϕ ) ν , (33) ∂ − c ∂ − 1 − c (∂i æi )∂t (∂ æ − ∂ Φ) t ϕ ϕ t M t ϕ i ϕ ϕ − t i i ϕ 2 2N Λ0 Λ0 F −F 2 ≡ 2 Λ M ∼ 2Λ 2 = 2 where ν 2N ϕ 0 − 4N 0F ϕ , and F F (ϕ ), ϕ ϕM M + F ϕ = 0, (37) ≡ 2 = ∇Φ 2 Λ FM F (ϕM ) ( ) / 0 by definition of ϕM .Wealso introduce η−1 as a damping timescale due to coupling of ϕ with Φ − the Æ vector field (cf. Equation (C5) in the Appendix for the − Φ (∂i æi +3∂t )( c2) − 2 ∇2 = 2 | |2 ∂t [λ(∂t æi ∂i )] + ∂i EOM of the vector field). The diffusion term cϕ cϕ k 2 2 = 2 is non-zero unless cϕ 0. The scalar field ϕ then follows 1 − c + ϕ ∂ ϕ∂ ϕ = 0, (38) the equation of a damped harmonic oscillator with a damping 2 t i − 2 ∼−2 2N rate (1 cϕ)η, a slightly nonlinear restoring force ν ϕ, 2 2 2 and an external force ∼ ν ϕM ∼|∇Φ| (2N ϕM ). Assuming 2 bary that the correction due to Hubble expansion O(3H +2Hη)b ∂i [λ∂t æi +(1− λ)∂i Φ] = +4πGρ , (39) is negligible for a very small b , the scalar field ϕ eventually approaches the MOND-like static solution ϕM , thanks to the where the time-derivative terms are all moved to the left-hand damping term with a timescale η−1, which kills any history √side, the summation over dummy index i = x,y,z is implicit, Λ ∼ −10 −2 ∼ dependence. Rapid oscillations will likely keep the fluid’s time- 0√ 10 ms , cϕ 1, and N are constants, and we used averaged property close to MOND-like solution as well. = =− =−2 = ϕ λ, η ∂i æi , c2 3λ , and c4 2λ. We estimate the oscillation timescale These equations, when supplemented by the continuity equa- 8 tion and the three momentum equations for the normal baryonic ϕ −1 −1 −1/2 10 ∼ ν = (2N) (Λ0F λ) | = 2 ∼ × 300 yr, λ ϕM 10 ϕ¨ N It is not necessary to compute Æa = ua +æa (cf. Equation (B13)), or to = Φ (34) evolve the auxiliary unit vector field ua (1 + ,u1,u2,u3) with the a∇ = =− Φ 9 ∼ = geodesics of test particles u aui Ai ∂i explicitly. All 3+1 √which is about 10 yr if N 10. Here we assume ϕM = − − formulated physical quantities are expressed in terms of æa (0, æ1, æ2, æ3), = = ∼ 8 2 a ˙ λ O(1) F for systems of mild gravity ( 10 cm s , and the local volume expansion rate θ =∇au = 3H +3Ψ and the local e.g., clusters; for systems of stronger gravity, the timescale acceleration Ai. 136 ZHAO & LI Vol. 712
bary fluid described by ρ and vi with a certain equation of state H.S.Z. acknowledges Martin Feix and Gil Esposito-Farse for P bary(ρbary), completely specify the dynamics. discussions on PPN and causality issues. The dark fluid has two types of deviations from MOND in general. APPENDIX A 1. The dark fluid has a generic scalar field vector oscillation MOTIVATING A GENERIC LAGRANGIAN RELATING − − and damping timescales ∼ O(ν 1) ∼ O(η 1) ∼ O(N) VARIOUS THEORIES WITH ONE FIELD times the orbital√ crossing time roughly unless the system We first motivate a generic Lagrangian behind our specific is hotter than c/ N or the forces are in resonance. A very a fast damping would mean an almost instantaneous relation Lagrangian L(ϕ,Æ ) for the dark fluid. The idea is mainly to − 2 relate various theories together with only one field. We start between gravity and the scalar field 1 ϕ ,astheμM in the a classical Bekenstein & Milgrom (1984) modified gravity from any vector field of 4 degrees of freedom, denoted by Z .It interpretation of MOND. A slow damping would mean has generally a variable or dynamic norm a history-dependent relation, reminiscent of Milgrom’s ≡ 2 ≡| a b| modified inertia interpretation of MOND: the dark fluid λ ϕ gabZ Z , (A1) adds a dynamically varying inertia around the baryons that it surrounds. A possible test could be in galaxies with where we adopt the convention of Einstein summation of identical upper and lower indices. The field λ is an auxiliary rotating bar(s), where there could be a phase lag between a the bar and the effective DM (Debattista & Sellwood 1998). scalar field characterizing the norm of the vector field Z , hence This has intriguing consequences to the bar’s pattern speed is not an independent dynamical freedom (we will return to this point below). A generic coupling of the vector field Za because of nontrivial corrections to the MOND pictures a b of dynamical friction (Ciotti & Binney 2004;Nipotietal. with the spacetime is through the contractions among the Z Z 2008; Tiret & Combes 2008); the properties of the dark tensor, the gab metric tensor, and the Ricci tensor Rab. Hence the most generic theory √ of the vector should have an action fluid are in between that of real particle dark halo and that −1 4 S =−(16πG) dx −g(Lm + L), where the Lagrangian naively expected from MOND. L 2. The dark fluid has a pressure, controlled by a propagation density containing a term m due to matter, and the term c speed ϕ, where the speed of light is unity here, and the L = Λ ab − a b −2 ∇a ∇ ··· 2 ∼ 2 ∼ 2 0F (λ)+[g f1 Z Z ϕ f2]Rab + f3 λ aλ + , dark fluid can be made cold by cϕ 0, or hot by cϕ 1, 2 (A2) or superluminal by cϕ 1. The ϕ would no longer be a function of the local gravity at r (as in MOND), rather it where F (λ) and fi are dimensionless functions of λ, and the is a weighted average of a volume of all points r1 by a Λ ν|r1−r| typical energy density set by a constant 0, which is the only Yukawa-type screening function exp(− ), where ab cϕ dimensional scale in the dark fluid. Note that R = g Rab is the Ricci scalar, and has the dimension of Λ . 8 0 −1 10 The above Lagrangian is generic enough, and many DE Screening length = cϕν ∼ cϕ × 300 lt yr. (40) N models can be derived from it. For example, the terms Λ0(λ − 1/(1+n) n 1) + f1R with f1 = λ can lead to an R + const/R (Li Note this spatial correction to MOND can exist even in et al. 2008) gravity (as could be checked by solving λ from c2 = static systems; even a small pressure term with ϕ 0 the EOM of λ and then substituting back into the original might smooth out MOND effects on small-scale structures a Lagrangian); and the terms f1R + f3∇ λ∇aλ with f1 = λ and (wide binaries, star clusters, dwarf galaxies), where the f =−cstλ−1 would lead to the Brans–Dicke theory of gravity. wave number |k|2 is much bigger than in galaxy clusters. 3 ∼ The usual quintessence theories can be recovered from the terms The screening length can be set at 100 pc for either an Λ ∇a ∇ 8 5 −1 4 0F (λ)+R + λ aλ with a suitable potential F (λ). N ∼ 10 , cϕ ∼ 3×10 km s hot dark fluid or an N ∼ 10 −1 The essential dynamics of a cosmological vector field is de- and cϕ ∼ 30 km s cold dark fluid. This scale, 100 pc, is a −2 a b scribed by the term f2ϕ Z Z Rab, which can be moulded into scale dividing dense star clusters and fluffy dwarf galaxies. very different but equivalent forms, e.g., L = Kac∇ Zb∇ Zd + Observational DM effects are only seen in the universe on bd a c f R Λ F ϕ2 f ϕ2∇aϕ∇ ϕ ··· 11 Kac scales larger than 100 pc. It has been challenging for MOND 1 +2 0 ( )+4 3 a + , where the tensor bd is a function of ϕ (Ferreira et al. 2007; Halle et al. 2008). If to explain this observed scale (Zhao 2005; Sanchez-Salcedo a & Hernandez 2007;Baumgardtetal.2005). we require that the field Z has a unit norm guaranteed by a Lagrange multiplier, we would recover Einstein–Aether theory In conclusion, we find a framework of dark fluid theories (Jacobson & Mattingly 2001) and its generalizations (Zlosnik where MOND corresponds to a special choice of potentials et al. 2007;Lietal.2008). or mass for the vector field. The dark fluid can run cold or On the other hand, the generality of the Lagrangian can also a b −2 hot depending on the sound speed cϕ (which could even be a create problems, especially a coupling like Z Z Rabϕ f2, running function of the vector field). These theories degenerate which can dangerously widen the cone of causality. Care must be into scalar field theories for DE effects in the Hubble expansion. taken in selecting the coefficients f1,f2,f3, etc. as the package It is possible√ to create an exact w =−1 DE effect. The scale is simply too broad to be healthy as a whole. It is necessary = Λ a0 0 in MOND in equilibrium spiral galaxies derives its to eliminate, by hand, certain mathematically allowed terms to Λ physics from the amplitude of the DE 0. MOND or DM effects pick a safe subset of the theory. To facilitate the selection, let us are hence indications of a non-uniform DE fluid described decompose the four dynamical degrees of freedom in the vector generally by a vector field ϕÆa. For non-equilibrium systems like the Bullet clusters or galaxies with satellites, the properties 11 b = −∇ ∇ c ∇ ∇ c Note that Z Rab√ ( a cZ + c aZ ) by definition√ of the Ricci tensor, of the dark fluid do not follow exactly the usual expectations of 4 − a b −2 → 4 − ac∇ b∇ d so the term dx gZ Z Rabϕ f2(ϕ) dx gKbd aZ cZ , MOND or cold/hot DM, but (not so surprisingly) in between. where a full divergence term is dropped after turning into a surface integral. No. 1, 2010 DARK FLUID: A UNIFIED FRAMEWORK FOR MOND, DM, AND DE 137
a field Z into a scalar degree and three dynamical degrees of a K =∇Æ (B4) unit-norm vector field Æa, 3 c a a a c Z K =∇Æ ∇ Æ , (B5) Æa ≡ . (A3) 4 c a ac c ϕ where ∇ = g ∇c,Æa = gacÆ . We use the following shorthand notations d = 1/N 2, d = (1 − c2 )/N 2, V = 2λ F , These three degrees of freedom for the unit-norm vector field 1 2 ϕ 0 = = = 2 = = Æa could be identified by a similar analysis as that of Graesser d3 e1d4 e2, c4 2ϕ , c14 c1 + c4, d34 d3 + d4, and = ˙ = 2 et al. (2005). This decomposition makes sense unless the norm α c1 +3c2 + c3, ϕ ∂t ϕ.Theterme0ϕ R in the Lagrangian of Za is zero, i.e., ϕ2 = 0, at which point the choice of Æa is not density is the scalar–tensor term which we do not use in this unique. If ϕ2 =|ZaZ | is always positive, then the vector field work, and the terms in field equations due to this can be found a = Æa will be space-like/time-like if Za is space-like/time-like. A easily in the literature, so here we neglect it by setting e0 0. space-like vector field picks out a preferred spatial direction, The stress–energy tensor is which could violate various constraints, so it is safer to limit ϕ Æ 8πGTab = 8πGT +8πGT , (B6) our Za (hence Æa) to be time-like. Because we do not desire ab ab Za to change from time-like to light-like (ϕ2 = 0) in the solar where system, care must be taken to the choice of the potential F, f , 1 3 8πGTϕ = g V +(d − d ) ∇ ϕ∇ ϕ − g ∇ ϕ∇cϕ etc. to prevent ϕ from reaching zero (see Appendix D about the ab ab 1 2 a b 2 ab c function F). c d3 c Finally, we add a cautionary note: + ∇c d3Æ (Æa∇bϕ +Æb∇aϕ) − ÆaÆb∇ ϕ L a 2 1. Any Lagrangian of the form (Z ) could be expressed − as L(Æa,ϕ) and vice versa; one only needs to substitute d4 d3 c d + (∇d ϕ)[(Æ ∇cÆ )(gab − ÆaÆb) b a a b 2 ϕ = |Z Zb| and Æ = Z / |Z Zb|. The expressions d ∇ c can be very compact in one and very lengthy in another. It +Æ ( cÆ )ÆaÆb] seems safe to speak of Za as a unifying field at least in the d − d d +ÆcÆd 2 3 ∇ ϕ∇ ϕ − 3 ∇ ∇ ϕ Æ Æ effective sense if not fundamentally meaningful. 2 c d 2 c d a b 2. A vector field Za of a dynamical norm would be time-like − if it is actually related to a unit-norm time-like field Æa c d d2 + d4 d4 +Æ Æ ∇cϕ∇d ϕ + ∇c∇d ϕ by a metric redefinition. The redefined metric can absorb 2 2 the norm (see Section 4.2 of Zhao 2008b); hence, what × (gab − ÆaÆb), (B7) appears as a unit vector in one metric can have a dynamical norm in another metric. For the same reason, Zlosnik et al. 1 (2007) found a lengthy Lagrangian when combining the Æ = K ∇ cd 8πGTab gab +[Æd cJ ]ÆaÆb unit vector and scalar of TeVeS into one time-like non-unit 2 vector. Another possibility is for Za to be a complex unit ∇ c − c − c + c Æ(aJb) Æ J(ab) Æ(aJ b) vector related to neutrino mass (cf. Zhao 2008b). c c + c1[∇ Æa∇cÆb −∇aÆ ∇bÆc] A detailed discussion on various interpretations of Za is + c ÆcÆd [∇ Æ ∇ Æ − Æ Æ ∇ Æ ∇ Æe], beyond this paper. While it is conceptually satisfactory to relate 4 c a d b a b c e d a all dynamics of a Lagrangian to one vector field Z , a sensible where we have defined Lagrangian is more readily constructed in a compact way using J a ≡ Kab ∇ Æd , K ≡ Kab ∇ Æc∇ Æd = c K the Æa unit vector field of 3 degrees of freedom and the ϕ field c cd b cd a b j j a j of 1 degree of freedom instead of Z of 4 degrees of freedom. (B9) We shall restrict to the Æa and ϕ notations only in the main text Kab = c gabg + c δaδb + c δaδb + c ÆaÆbg . (B10) of the paper. cd 1 cd 2 c d 3 d c 4 cd The⎡ scalar EOM is⎤ APPENDIX B c d − ⎣V + j K ⎦ = gab 1 ∇ ϕ∇ ϕ + d ∇ ∇ ϕ EOM AND STRESS TENSOR FOR THE GENERAL 2 j 2 a b 1 a b LAGRANGIAN j
d3 a b d4 a b ab a b The most general Lagrangian that we consider (Equation (1)) + ∇b(Æ ∇aÆ )+ ∇a(Æ ∇bÆ ) − (g − Æ Æ ) can be cast to the form 2 2 d L = 2 2 a − ∗ × 2 ∇ ϕ∇ ϕ + d ∇ ∇ ϕ + d ∇ ∇ (ÆaÆb). (B11) (1 + e0ϕ )R + V (ϕ )+(Æ Æa 1)L + cj Kj 2 a b 2 a b 2 a b j=1,4 The vector EOM is +[−d gab + d (gab − ÆaÆb)]∇ ϕ∇ ϕ 1 2 a b − ∗ =−∇ b ∇ b∇ c b a c L Æa bJ a + c4 aÆc(Æ bÆ ) − (d3Æagbc + d4Æbgac) ∇ ϕ∇ Æ , (B1) d + d ∗ + 3 4 − d (Æb∇ ϕ)∇ ϕ where R is the Ricci scalar, and L is the Lagrange multiplier (a 2 2 b a kind of potential), where − d3 d4 b b a c + [(∇bÆ )∇aϕ − (∇aÆ )∇bϕ] K1 =∇aÆc∇ Æ (B2) 2 d3 + d4 b + Æ ∇b∇aϕ, (B12) a 2 2 K2 = (∇aÆ ) (B3) 138 ZHAO & LI Vol. 712 ∗ where the dependence on the Lagrangian multiplier L can be ϕÆ d1 2 α 2 ab − a b 8πGp = ϕ˙ − V + (θ + η) eliminated by multiplying both sides by (g Æ Æ ) and 2 6 contract over index a. Alternatively, L∗ can be computed by a α(θ + η) d4 multiplying both sides by Æ and contract over the index a. + ∂t − ∂t ϕ˙ , (B19) In the following, we list the field equations in the weak field 3 2 limit, applicable to galaxies in an expanding background metric. In galaxies, the metric is nearly flat but the overdensity of matter can be much bigger than unity. We use the 3+1 formulation (see d1 α 8πGρϕÆ = ∇ˆ a(c B )+ ϕ˙2 + V − (θ + η)2 Appendix E). We write 14 a 2 6 = d3 d3 d4 Æa ua +æa, (B13) + ϕη˙ − ∇ˆ a ∇ˆ ϕ + ϕ˙(θ + η). (B20) 2 2 a 2 where ua is the four-velocity of an observer on geodesic, and like Æa is a unit-norm time-like vector field. Since the background One can verify that these stress-tensor components satisfy ˆ a universe is observed to be homogeneous and isotropic to high the conservation equations, ρ˙ +(ρ + p)θ + ∇ qa = 0 and precision, the difference vector field, æ , between them should 4 ˆ ˆ b a q˙a + θqa +(ρ +p)Aa −∇ap +∇ πab = 0, where the superscript be of first order in perturbation. The relations 3 ϕÆ is omitted. a a a The ∇ϕ∇Æ coupling leads to terms like O(d ∇2ϕ)inthe ÆaÆ = uau = 1 → u æa = 0 (B14) 3 density ρϕÆ, which do not appear to have a lower bound, hence which means that the field æa is perpendicular to the four- could lead to unbound Hamiltonian if d3 and d4 were not set velocity of the observer (see Li et al. 2008). to zero. Also a non-zero c13 term clearly creates anisotropic As discussed in Halle et al. (2008), it is highly nontrivial to stress, and hence excites the spin-1 mode. The ϕ2R term could keep track of the necessary orders of the equation to be valid both lead to a Hamiltonian density O(∇2ϕ), which can be unbound for static highly nonlinear galaxies in a static empty universe and when ϕ is big. Undesirable modified gravity effects in the solar a b for linear perturbations around a uniformly expanding universe. system could also be created by the term Æ ∇aÆ ∇bϕ ∼ Here we treat the zeroth order being a static galaxy in a nonlinear O(1)∂i g00∂i ϕ, where ∂i stands for spatial derivatives and can be b a overdensity. Any temporal changing terms or terms involving very big on small scales. The term Æ ∇aÆ ∇bϕ ∼ O(H )∂t ϕ Hubble expansion are at least first-order perturbations. Besides can have big influence for structure formation with little effects the first order, we have kept some second-order terms whenever on the solar system, and hence might be desirable. For above the first order can become zero in a static galaxy. For a lighter considerations, we set d3 = d4 = c1 = c3 = 0 = f , in which ˆ a notation, we also define intermediate variables η ≡ ∇ æa, case the equations are greatly simplified. θ ˆ ≡ ˙ ≡ ∇ Ba Aa + æa + 3 æa, Qab c13(σab + aæb ). The scalar EOM is APPENDIX C EQUATIONS AND DAMPING RATES OF 2 ˆ 2 d3 ˆ a d1∂ +(d1 − d2)∇ ϕ + (d1θ + d2η) ϕ˙ =− ∇ Ba SCALAR–VECTOR FIELDS IN GALAXIES t 2 d4 ∂ Here we give some relations between our 3 + 1 variables − [(θ˙ + η˙)+(θ + η)2] − V + c K , (B15) 2 ∂ϕ i i and the conventional gravitational potentials in the Newtonian gauge: =−c14 a α 2 ··· where ci Ki 2 B Ba + 6 (H + η) + , which includes second-order terms, non-negligible for a nearly static galaxy. ds2 = (1 + 2Φ)dt2 − a2(t)(1 + 2Ψ)(dx2 + dy2 + dz2),a(t) = 1. We will not go into complete expressions for the second-order (C1) terms, which are beyond this paper but have been discussed in In the following, we shall neglect the Hubble expansion, and Halle et al. (2008). limit the discussion to our choice of parameters with d3 = d4 = The vector EOM is 0, and c13 = α − 3c2 = 0. The metric has neither spin-2 nor spin-1 mode, and has purely spin-0 mode due to variations of 2θ ˆ α(θ + η) αθ ˆ b Φ =−Ψ ∂t + (c14Ba) + ∇a − æa∂t =−∇ Qab Newtonian potentials (t,x,y,z) (t,x,y,z). We neglect 3 3 3 any vorticity generated by the scalar field and non-relativistic d + d d θ d θ baryons. + ∂ 3 4 (∇ˆ ϕ −˙ϕæ ) − d ϕ˙ − 3 + 4 t 2 a a 2 6 2 The scalar EOM reduces to Equation (33): ˆ × (∇aϕ −˙ϕæa). (B16) 2 − 2 ∇2 − 2 = −Λ ∇Φ −˙ 2 2 ∂t cϕ + 1 cϕ η∂t ϕ [ 0F +( æ) ](2N ϕ), The stress–tensor components due to the Æ and ϕ fields are (C2) given by ϕÆ which has the form of a damped harmonic oscillator with the 8πGπ =−Q˙ − θQ , (B17) ab ab ab damping rate η determined by
≡ ∇ˆ j =− =− α(θ + η) η æj ∂j æj ∂j ∂j Y, (C3) ϕÆ = ˙∇ˆ ∇ˆ 8πGqa d1ϕ aϕ + a 3 with an implicit summation of the dummy index j = x,y,z.In d4 ˆ ˆ b the following, we consider mainly the evolution of the vector + (θ − ∂t ) ∇aϕ + ∇ Qab, (B18) 2 field perturbation æj = ∂j Y 1 in the compressional spin-0 mode with a potential Y. No. 1, 2010 DARK FLUID: A UNIFIED FRAMEWORK FOR MOND, DM, AND DE 139
That c13 = 0 means that two Newtonian potentials are related This equation reduces to the form of MOND, by Ψ =−Φ and in the vector field EOM and heat flux we can bary −2 2 drop terms such as Qab.Thetermsofσab, ab in the constraint ∂i [(1 − λ)∂i Φ] = 4πGρ + O(N |k| ), (C10) Equation (E5) can be dropped if we assume negligible (non- relativistic) vorticity and shear due to the baryon heat flux, so in steady-state systems without the Hubble expansion, in which 2 case we can set æ˙ i = 0. The scalar Equation (C2)forλ = ϕ = = Λ = 2 recovers MOND√ only if cϕ η 0 and 0 a , in which case 2θ ∇Φ = = −1/3 − − 2A˙ = 2∂ Φ˙ =−2∂ Ψ˙ =−∂ ( )/a0 F (λ) 3(λ 1) for our choice F (λ) with i i i i 3 a negligible (Equation (6)). However, we shall show in the following the importance of corrections due to η in the scalar bary 1 = 8πGq + ϕ∂˙ iϕ + ∂i [c2(η + θ)] (C4) field equation, and æ˙ in the Poisson equation. i N 2 i
a a C.1. Approximate Solutions gives the relation of the variables θ =∇au and Ai = u ∇aui in the 3+1 formalism with the gravitational potentials in the The evolution equations can be summarized as follows: conformal Newtonian gauge; for the last equation, Equation (E5) 1 √ and Equation (B18) are combined, with the total heat flux √ 2− 2 ∇2− − 2 ∂t cϕ (1 cϕ)(∂i æi )∂t λ ϕÆ bary 2 = (2N Λ0 λ) qi qi + qi , where the baryon heat flux contribution bary 2 = bary bary (∂i Φ − ∂t æi ) qi ρ vi for a baryon density ρ and velocity vi . =−F + , (C11) Dropping higher order terms, the vector field EOM Λ0 (Equation (B16)) becomes 2 c ∂i λ∂t λ ∂ (λ∂ æ )+(1− λ)∂ ∂ Φ − ∂ λ∂ Φ = ϕ +4πGρbaryv , − 2 t t i t i t i 2 i 1 cϕ 8N λ ∂t [2λBi ] =−∂i [c2 (η + θ)] − ϕ∂˙ i ϕ, (C5) N 2 (C12) for the index i = x,y,z, where Bi =−∂i Φ + æ˙ i , and we ϕ −˙ϕæ [ ,i i ] ∼ ∂i ϕ | | |˙|| | − Φ =− bary approximated N 2 N 2 because æi 1, and ϕ ϕ,i ∂i [(1 λ)∂i ] ∂i [λ∂t æi ] +4πGρ . (C13) ϕÆ inside the causal horizon. Using the expression for 8πGqi in Equation (C4), the EOM becomes There is also a constraint on Equation (C4): ∂t ϕ∂j ϕ bary ∂j [(2 + 3c2)∂t Φ + c2∂i æi ] − = +8πGρ vj , (C14) c2 N 2 [2λ(æ¨ − ∂ Φ˙ )+2λ˙æ˙ ] =−[2∂ Φ˙ − 8πGρbaryv ]+ ϕ ϕ∂˙ ϕ. i i i i i N 2 i which is not independent from the combination of the above (C6) equations and the continuity equation of the baryon fluid; here The equation can be simplified as =−2 c2 3λ for our choice. The above equations are hard to solve in general. Neverthe- ˙ ˙ − ˙ Φ − Φ˙ = 2[λæ¨ i + λæi λ∂i +(1 λ)∂i ] Si , less, Equation (C4) can be heuristically integrated to a simpler 2 c ∂i λ relation: S ≡ ϕ λ˙ +8πGρbaryv . (C7) i 4N 2λ i θ c − Φ˙ = Ψ˙ = ∼− 2 η − Φ˙ ϕ bary, (C15) 3 (2 + 3c2) = √Thus, the vector perturbation æi and the scalar field φ λ evolve as two coupled damped harmonic oscillators with where Φ˙ ϕ bary is a fudged contribution to Φ˙ due to the evolution ˙ damping rates λ and η, respectively. The source term on the of the scalar field and the movement of baryons. We can then λ ˙ right-hand side of the EOM of the vector would be zero, Si = 0, express Φ,i in terms of −η,i = æj,ji. and the EOM can be cast 2 2 = for a very cold dark fluid cϕ/N 0 and in the vacuum where to the form ρbary = 0. λ˙ j 2 − 2 = ˜ Finally, in the absence of the Hubble expansion, the Poisson (2λ) δi ∂t + ∂t c0(∂i ∂j + ∂i ξ∂j ) æj Si , (C16) equation becomes λ where 2 bary ˆ i 1 2 α 2 − − 2∂ Ψ = 8πGρ + ∇ (2λB )+ ϕ˙ + V − (θ + η) , 2 (1 λ)c2 i i 2 c = , (C17) 2N 6 0 (2 + 3c )λ (C8) 2 − where −Ψ = Φ, and the right-hand side terms in square brackets ≡ c2 ˜ = Φ˙ ϕ bary ∼ 2 ˙ ξ ln 2+3c , and Si ∂i O( ) O(N λ∂i ln λ)+ ϕÆ ˆ i 2 are from 8πGρ (cf. Equation (B20)), where ∇ =−∂ , bary j i O(8πGρ vi ). Note that δ equals unity if i = j or zero 2 ˙ i Bi = (−∂i Φ + æ˙ i ), and the quadratic term (θ + η) ∼ [O(Φ)+ − otherwise, and the vector field æi appears to track the evolution O(N 2ϕ˙)]2. For the study of the (small) galactic scale dynamics, of the baryons and the scalar field λ. ˙ we can safely drop all source terms involving ϕ˙2/N 2, Φ2, V, To see the tracking to baryons, we take can take the limit that ∼ Λ ˙ which are all relativistic corrections of the order of V 0. N →∞, hence treat the scalar field as constant, so λ = 0 = ∂i ξ, Therefore, the Poisson equation in time-dependent systems can the EOM of the vector perturbation can be approximated as be approximated as (2λ +3c )8πG 2 − 2∇2 ∼ 2 bary →∞ 2Φ = bary − − Φ ˙ ∂t c0 æi ρ vi , if N . (C18) ∂i 4πGρ ∂i [λ( ∂i + æi )] . (C9) 2λ(2 + 3c2) 140 ZHAO & LI Vol. 712
2 = 2 = 2 ∇2 = 2∇2 → 2 − 2 −1/2 2 2 We used c0∂i ∂j æj c0∂i ∂j ∂j Y c0∂i Y c0 æi since (ϕ ) if ϕ < . (D3) æi = ∂i Y for the spin-0 perturbations. Hence, we recover the 2− 2∇2 = a propagation equation [∂t c0 ]æi 0 of Jacobson’s Einstein– So, our dynamical norm vector field Z can be approximated 2 as Jacobson’s fixed norm field with a small positive norm 2 in Aether theory in vacuum. Indeed, c0 plays the role of sound speed squared for the spin-0 wave in vacuum. The vector field the regime where gravity is high, such as the solar system; the bary = − 2 æi tracks baryon current vector ρ vi with Equation (C18) like effective Newtonian gravitational constant GN G/(1 ), the magnetic vector potential tracks the moving electric charges. which is virtually the same as the bare G here for a small 2. −1 To estimate η, we apply −(2λ) ∂i to both sides of the EOM of Note that Jacobson’s fixed norm theory is also the limiting case the vector (Equation (C16) or Equation (C18)), and sum over i, for a wide class of time-like fields Æa with a simple algebraic and we get an equation for η: potential function
(2λ +3c )8πG − 2 − 2∇2 ∼ 2 bary n − 2n n ∂t c0 η ∂t ρ , (C19) 2 1/k 3 ϕ ϕ 2λ(2 + 3c ) F (ϕ ) = [|f | − 1] B0,f≡ , (D4) 2 1 − 2n bary =− bary ∼ ∇2Φ where we used ∂i (4πGρ vi ) ∂t 4πGρ O( ). ˙ 3 η ∼ O Φ where the minimum is at φ = , and B = 1 2k for Thus, one estimates that ( c2 ) if purely sourced by baryon 0 3 n 0 an appropriate normalization at φ = 1, and 0