Cherenkov Radiation from Stars Constrains Hybrid MOND-Dark-Matter Models

Home , Cold dark matter, Sabine Hossenfelder

Cherenkov radiation from stars constrains hybrid MOND-dark-matter models

Essay written for the Gravity Research Foundation 2021 Awards for Essays on Gravitation.

Tobias Mistele*

Frankfurt Institute for Advanced Studies Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany

Submission date: March 31, 2021

Abstract We propose a new method to constrain alternative models for dark matter with observations. Specifically, we consider hybrid models in which cold dark matter (CDM) phenomena on cosmological scales and Modified Newtonian Dynamics (MOND) phenomena on galactic scales share a common origin. Various such models were recently proposed. They typically contain a mode that is directly coupled to matter (for MOND) and has a non-relativistic sound speed (for CDM). This allows even non-relativistic objects like stars to lose energy through Cherenkov radiation. This is unusual. Most modified gravity models have a relativistic sound speed, so that only high-energy cosmic rays emit Cherenkov radiation. We discuss the consequences of this Cherenkov radiation from stars. arXiv:2103.16954v1 [gr-qc] 31 Mar 2021

*mistele@ﬁas.uni-frankfurt.de

1 Cherenkov radiation in hybrid MOND-dark-matter models

1 Introduction

A collisionless fluid is the simplest explanation for the missing mass problem on cosmological scales. On galactic scales, the simplest explanation is in terms of Modified Newtonian Dynamics (MOND) [1–4], i.e. a modified force law. A natural idea is to find a common origin for both in a single model. Various such hybrid models have been proposed. For example, the superfluid dark matter (SFDM) model [5,6] and the model by Skordis and Złosnik´ (SZ) [7]. These contain a component that plays a role in both CDM and MOND phenomenology. For example, in SFDM, the collisionless fluid on cosmological scales is a non-relativistic superfluid, which forms a cored halo around galaxies and whose phonons carry a MOND-like force. These phonons constitute a massless mode which is directly coupled to matter and has a non-relativistic sound speed cs. Thus, if they move faster than cs, even non-relativistic objects like stars may emit Cherenkov radiation. This is different from most modified gravity models. Usually, only relativistic objects emit Cherenkov radiation since cs is relativistic [8–12]. We show how this Cherenkov radiation from stars constrains hybrid MOND-dark-matter models.

2 Toy model

Consider a star with mass M and velocity V at a distance Rp from the center of a galaxy with baryonic mass Mgal, and a real scalar field ϕ that carries a MOND force, ϕ ∝ p GMgala0 ln(R) with the MOND acceleration scale a0. In hybrid MOND-dark-matter models with a common origin for MOND and CDM phenomenology, perturbations δ around a background field ϕ0 typically have the Lagrangian 1 1 1 g 2 ~ 2 ~ 2 √ m L = 2 (∂tδ) − (∇δ) + (â∇δ) − δ δb , (1) 2 c¯ 2 2MPl after a possible rescaling. Here, gm and c¯ are constants that depend on ϕ0, and δb is a perturbation of the baryonic density ρb. The unit vector aˆ points into the direction of the background MOND force. The dispersion relation is ω = cs|~k| with 2 2 2 cs =c ¯ (1 + γ ) , (2) where γ is the cosine of the angle between the perturbation’s wavevector and aˆ. There is a direct coupling to matter (for MOND) and cs is typically non-relativistic (for CDM). Thus, even non-relativistic objects like stars may emit Cherenkov radiation, if V is larger than a critical velocity Vcrit = O(cs). For V > Vcrit, we can calculate the associated energy loss E˙ Ch following Refs. [13, 14]. The timescale on which stars lose a significant amount of their energy E is roughly E/|E˙ Ch|. We define [15] 1 2 3 2 Ekin 2 MV 8πV MPl 1 τE ≡ ≡ = 2 2 2 2 . (3) |E˙ Ch| |E˙ Ch| fac¯ gmMkmax 1 − (kmin/kmax)

2 Cherenkov radiation in hybrid MOND-dark-matter models

The cutoffs kmax and kmin ensure that the Lagrangian Eq. (1) is valid, and fa depends on the direction of aˆ relative to V~ . For circular orbits, V~ ⊥ aˆ,

⊥ 1 ⊥ fa = ,Vcrit =c ¯ . (4) p1 + (¯c/V )2

Stars have both kinetic and potential energy. For simplicity, τE includes only the kinetic energy. Still, τE is a useful quantity. Indeed, for a star with V > Vcrit,

∂t (Ekin + Egrav) = E˙ Ch , (5) with the gravitational energy Egrav. This gives

R˙ 1 p = − , (6) Rp 2τE for approximately circular orbits in the MOND-regime with a flat rotation curve, i.e. 2 p V = GMgala0 [15]. Thus, if τE depends only weakly on Rp, stars transition to smaller galactic radii as exp(−t/2τE) due to Cherenkov radiation. A numerical analysis confirms this, see Fig.1. There, we numerically solve the equation of motion of a star 9 with a friction force corresponding to an energy loss E˙ Ch with, initially, τE = 5 · 10 yr. The initial values are such that the orbit is circular without Cherenkov radiation. We can treat this Cherenkov radiation as a friction force because the energy loss happens through a large number of emissions, each carrying only a small fraction of the star’s energy. This is due to the strict cutoffs we impose below. In Fig.1, the friction force acts in the direction of V~ . We have numerically verified that other directions give similar results. We now choose the cutoffs kmin and kmax. Since the background galaxy’s field varies on kpc scales, we choose

−26 kmin ∼ 1/kpc ∼ 10 eV . (7)

For kmax, we choose [15] s s r gal gal a0 ab −22 ab kmax ∼ fp · ∼ 10 eV · fp · , (8) GM a0 a0

gal where ab is the Newtonian baryonic acceleration of the galaxy at the star’s position, and fp parametrizes additional model-dependent cutoffs. For standard SFDM and V~ ⊥ aˆ, we can take fp = 1 [15]. This choice of kmax avoids complications close to the star. First, the star’s ﬁeld is no longer a perturbation of the galaxy’s ﬁeld. Second, the acceleration due to the star becomes larger than a0, so that the model may behave very different from its MOND limit.

3 Cherenkov radiation in hybrid MOND-dark-matter models

1.0 1.04 Numerical Analytical 1.02 0.8 0 0 R V /

/ 1.00 p V R 0.6 0.98

0.96 0.4

0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0 t in 109yr t in 109yr

Figure 1: Left: Velocity of a star with initial values R0 = 30 kpc and V0 = 200 km/s in the MOND regime of a galaxy. The star loses energy through Cherenkov radiation 3 9 3 corresponding to τE/V = 5 · 10 yr/(200 km/s) . The galaxy mass is chosen such that the orbit is circular without Cherenkov radiation. Right: The star’s position Rp in the galaxy for the same orbit.

With our particular choice of kmax, using kmin kmax,

8 2 ! −10 2 2 · 10 yr V/c¯ a0 V 1.2 · 10 m/s τE = 2 2 · · gal · · . (9) faf g 2 200 km/s a0 p m ab

For gm of order 1 and V > Vcrit, stars lose a signiﬁcant fraction of their energy on roughly galactic timescales. This is a conservative estimate. The actual energy loss may be higher, e.g. k > kmax modes may contribute, but are not considered here.

3 Application to standard SFDM

The SFDM model from Ref. [5] introduces a new type of particle which behaves simply as cold dark matter on cosmological scales. On galactic scales, it condenses to a superfluid whose phonons mediate a MOND-like force. The effective Lagrangian for the field θ, that carries both the superfluid and the MOND force, is

2Λ 3/2p α¯Λ L = (2m) |X − βY |X − ρb θ , (10) 3 MPl with

˙ 2 ˙ X = θ +µ ˆ − (∇~ θ) /(2m) ,Y = θ +µ ˆ , µˆ = µnr − mφN . (11)

Here, m is the mass of the particles, µnr is the non-relativistic chemical potential, φN is the Newtonian gravitational potential, β parametrizes ﬁnite-temperature effects, and Λ

4 Cherenkov radiation in hybrid MOND-dark-matter models

Vcrit

600 Vrot

Vrot, asympt 500

400 km/s 300

200

100

10 20 30 40 50 R in kpc

10 Figure 2: Critical velocity and rotation curve of a galaxy with mass Mgal = 5 · 10 M concentrated at its center in standard SFDM with the fiducial parameters from Ref. [6]. gal The shaded region is not in the MOND regime since ab > a0. Here, Vrot,asympt is the asymptotic rotation velocity. and α¯ are constants. To avoid an instability and for a positive superfluid energy density, we need β ∈ (3/2, 3) [5]. Consider a galaxy in the MOND limit, (∇~ θ)2 2mµˆ [5]. Up to a term that mixes spatial and time derivatives, the Lagrangian for perturbations on top of such a galaxy has the form of Eq. (1) with |~a | α¯2Λ r a ∇~ θ 1 c¯ = 3f¯ θ0 , g = 0 , aˆ = 0 , f¯ = , (12) β m ~ β p a0 m |aθ0 | |∇θ0| 3(β − 1)(β + 3) 3 2 where a0 =α ¯ Λ /MPl and ~aθ = −λ∇~ θ [15]. The sound speed is typically non- relativistic. For the fiducial parameters from Ref. [6], c¯ = 375 km/s · (aθ0 /a0). The term that mixes spatial and time derivatives requires a more careful calculation [15]. The result has the same form as before, but with adjusted fa and Vcrit. For V~ ⊥ aˆ, s ⊥ 2 ⊥ 1 1 Vcrit =c ¯ 2 , fa = √ 2 , (13) 2 + fβ 2 1 + fβ ¯ where fβ = (3 − β)fβ. Because of a more conservative approximation to keep the ⊥ calculation with fβ 6= 0 simple [15], fa does not reproduce the previous fβ = 0 result. ⊥ In the MOND limit, c¯ ∝ 1/Rp. Thus, also Vcrit ∝ 1/Rp. Since rotation curves are flat at large radii, there is a critical radius where Vcrit drops below the rotation curve Vrot. Beyond this radius, stars with velocity Vrot lose energy on timescales τE, see Fig.2.

5 Cherenkov radiation in hybrid MOND-dark-matter models

Rp V (ql, qh) for β = 3/2 (ql, qh) for β = 2 (ql, qh) for β = 3 kpc km/s +1 15.2 220−1 (0.25, 1.56) (0.34, 2.19) (0.51, 3.34) +3 20.3 203−3 (0.35, 1.92) (0.46, 2.70) (0.69, 4.11) +6 24.8 202−6 (0.47, 2.34) (0.62, 3.29) (0.93, 5.01)

√ −1 Table 1: Excluded α/m¯ intervals (ql, qh) · eV from the Milky Way rotation curve at different radii.

The Milky Way stellar rotation curve rules out standard SFDM unless either c¯is large ⊥ enough to kinematically forbid Cherenkov radiation, i.e. Vrot < Vcrit. Or τE is larger 2 than galactic timescales, i.e. τE > τmin for some τmin. Since τE ∝ 1/c¯ , the latter can 10 be achieved by making c¯ small. Here, we assume Mgal = 6 · 10 M [16, 17] and the 10 rotation curve from Refs. [18, 19]. We choose τmin = 10 yr, i.e. stars should not lose 10 much energy in√10 yr. Then, for a given β, the rotation curve at each point Rp excludes an interval of α/m¯ [15]. For β ∈ {3√/2, 2, 3}, we list these intervals from different radii in Table1. Together, they rule out α/m¯ in an interval [15] s  s 10 10 10 1/4 6 · 10 M 10 yr 6 · 10 M −1  ql, qh · eV , (14) Mgal τmin Mgal for some ql and qh. Concretely,

ql = 0.25 , qh = 2.34 , for β = 3/2 ,

ql = 0.34 , qh = 3.29 , for β = 2 , (15)

ql = 0.51 , qh = 5.01 , for β = 3 . √ This also excludes the ﬁducial value α/m¯ ≈ 2.4 eV−1 for β = 2 from Ref. [6].

4 Application to two-ﬁeld SFDM

To avoid tensions within standard SFDM, Ref. [16] proposed a model with phenomenology close to the original SFDM model, but in which the√ two roles of the field θ are split −iθ between two different fields. The field φ− = ρ−e − / 2 carries the superfluid’s energy density, but is not directly coupled to normal matter. In contrast, the field θ+ is coupled directly to normal matter and carries a MOND-like force in equilibrium. The Lagrangian reads

2 α¯Λ L = L− + f(K+ + K− − m ) − θ+ ρb , (16) MPl α where K± = ∇αθ±∇ θ± and ∗ α 2 2 4 L− = (∇αφ) (∇ φ) − m |φ| − λ4|φ| . (17)

6 Cherenkov radiation in hybrid MOND-dark-matter models

p The function f(K) ≡ |K|K is similar to standard SFDM but contains both θ+ and θ−. There are two gapless modes, roughly corresponding to θ+ and θ−. Only the θ− mode is relevant for us because it has a non-relativistic sound speed, r µˆ c = 1 . (18) s m 2 This mode couples to normal matter only because θ− and θ+ mix in f(K+ + K− − m ). This suppresses Cherenkov radiation, [15] √ λ g = O 4 1 , (19) m α¯ so that 2 2 2 r −2 α¯ V V 16 V V 10 a0 τE ∼ ∼ 10 yr . (20) λ4 cs a0 200 km/s cs a¯

Here, a¯ a0 is an acceleration below which the equilibrium becomes unstable [16]. Thus, τE is much larger than the age of the universe and does not constrain the model. The reason is that the non-relativistic gapless mode couples to normal matter only indi- rectly through a mixing.

5 Application to the SZ model

Skordis and Złosnik´ have proposed a hybrid MOND-dark-matter model based on a scalar field φ and a unit vector field Aµ [7]. On cosmological scales, the scalar field φ(t) is responsible for a CDM-like fluid. In galaxies, φ = Q0 ·t+ϕ where Q0 is constant and ϕ carries the MOND-like force. Thus, φ plays a double role so that, potentially, stars can emit Cherenkov radiation. For simplicity, like Ref. [7], we consider perturbations on top of the late-time Min- kowski limit φ = Q0t and not on top of a galaxy. We assume that our results are qualita- tively valid also in galaxies. The Lagrangian contains terms [7] ˙ 2 ˙ L = KB A~ + ∇~ φN + 2(2 − KB) A~ + ∇~ φN · ∇~ ϕ + ... (21) with constant KB. In the static limit, ω = 0, the ϕ equation of motion contains 2 0 = ··· + ∇~ φN , (22) ~ 2 2 where ∇ φN ∝ ρb/MPl + ... . This gives the coupling of ϕ to normal matter, which allows it to mediate a MOND-like force. For ω 6= 0, one scalar mode involving φ has a potentially non-relativistic sound speed [7] c0 cs = √ , (23) K2KB

7 Cherenkov radiation in hybrid MOND-dark-matter models

0 with constant K2 and c = O(1). Surprisingly, this mode’s coupling to normal matter 2 √ is not always ∼ ρb/MPl, but is suppressed for ω Q0/ KB [15]. Indeed, in the ϕ equation, ˙ 0 = ··· + ∇~ A~ + ∇~ φN . (24)

˙ The combination A~ + ∇~ φN also occurs in the A~ equation of motion, ˙ 0 = ··· + ∂t A~ + ∇~ φN , (25)

¨ where, for scalar perturbations, all k2A terms cancel [7]. Then, the A~ term may dominate ˙ in this equation despite ω = csk k. Thus, A~ ≈ −∇~ φN + ... , which cancels the ~ 2 ∇ φN term, and therefore the matter coupling, in the ϕ√equation of motion. A more careful calculation shows that this happens for√ω Q0/ KB [15]. This condition is satisﬁed here because K2Q0 . 1/Mpc [7] and we have k > kmin ∼ 1/kpc, √ Q Q K √ 0 ∼ 0 2 1 . (26) KBω |~k| Thus, the matter coupling is suppressed, [15] √ K Q 3 g ∝ 2 0 , (27) m k which gives 2 2 6 8 V kmax kmin V τE ∼ 10 yr · · · √ · . (28) cs kmin K2Q0 200 km/s √ Since kmax kmin and kmin K2Q0, this is much larger than the age of the universe and does not constrain the model. The reason is the suppressed matter coupling in dynamical situations.

6 Conclusion

One usually avoids superluminal sound speeds for theoretical reasons. Yet, for empiri- cal reasons, subluminal sound speeds are also dangerous, since these allow Cherenkov radiation. Hybrid MOND-dark-matter models with a common origin for cosmological and galactic phenomena allow Cherenkov radiation even for non-relativistic objects like stars. Above, we showed how this rules out some of the parameter space of standard SFDM despite restrictive cuts to avoid technical complications. We also discussed how one may evade these constraints, namely by mixing (two-ﬁeld SFDM) or a suppressed coupling in dynamical situations (SZ model).

8 Cherenkov radiation in hybrid MOND-dark-matter models

Acknowledgements

I thank Sabine Hossenfelder for valuable discussions. I am grateful for ﬁnancial support from FIAS.

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