Stellar Streams in Chameleon Gravity

A. P. Naik∗ and N. W. Evans† Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK

E. Puchwein Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany

H. Zhao‡ Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK and Scottish Universities Physics Alliance, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK

A.-C. Davis Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Cambridge CB2 0WA, UK (Dated: November 24, 2020) Theories of gravity that incorporate new scalar degrees of freedom typically require ‘screening mechanisms’ to ensure consistency with Solar System tests. One widely-studied mechanism—the chameleon mechanism—can lead to violations of the equivalence principle (EP), as screened and unscreened objects fall differently. If the stars are screened but the surrounding dark matter is not, EP-violation can lead to asymmetry between leading and trailing streams from tidally disrupted dwarf galaxies in the Milky Way halo. We provide analytic estimates of the magnitude of this effect for realistic Galactic mass distributions, demonstrating that it is an even more sensitive probe than suggested previously. Using a restricted N-body code, we simulate 4 satellites with a range of masses and orbits, together with a variety of strengths of the fifth force and screening levels of the Milky Way and satellite. The ratio of the cumulative number function of stars in the leading and trailing stream as a function of longitude from the satellite is computable from simulations, measurable from the stellar data and can provide a direct test of chameleon gravity. We forecast constraints for streams at large Galactocentric distances, which probe deeper into chameleon parameter space, using the specific example case of Hu-Sawicki f(R) gravity. Streams in the outer reaches of the Milky Way halo (with apocentres between 100 and 200 kpc) provide easily attainable constraints 7 7.5 8 at the level of fR0 = 10− . Still more stringent constraints at the level of 10− or even 10− are | | plausible provided the environmental screening of the satellite is accounted for, and screening of the Milky Way’s outer halo by the Local Group is not yet triggered in this range. These would be among the tightest astrophysical constraints to date. We note three further signatures of chameleon gravity: (i) the trailing stellar stream may become detached from the dark matter progenitor if all the stars are lost, (ii) in the extreme fifth force regime, striations in the stellar trailing tail may develop from material liberated at successive pericentric passages, (iii) if the satellite is fully screened, its orbital frequency is lower than that of the associated dark matter, which is preferentially liberated into the leading tidal tail.

I. INTRODUCTION have ceased to be recognisable in star counts against the stellar background of the Galaxy [2]. The debris persists Stellar streams and substructures are the wreckage of for a large fraction of a Hubble time, sometimes longer, dwarf galaxies and globular clusters that have fallen into so substructures in phase space remain to the present and are being torn apart by the Milky Way’s tidal field. day. Searches in phase space for streams are much more In the past, such substructures have usually been iden- powerful than searches in configuration space. arXiv:2002.05738v2 [astro-ph.GA] 23 Nov 2020 tified as over-densities of resolved stars, as in the ‘Field The Gaia satellite is a scanning satellite of the Euro- of Streams’ image from the Sloan Digital Sky Survey [1]. pean Space Agency that is monitoring all objects brighter There, using multi-band photometry, the stellar halo of than magnitude G ≈ around 70 times over a period of the Milky Way was revealed as being composed of criss- 20 5 years (though the mission lifetime has recently been ex- crossing stellar streams, the detritus of satellite galaxies. tended) [3, 4]. Its telescopes are providing magnitudes, However, streams and substructure remain kinematically parallaxes, proper motions and broad band colours for cold and so identifiable in phase space long after they over a billion stars in the Galaxy (≈ 1 per cent of the Milky Way stellar population) within the Gaia-sphere – or within roughly 20 kpc of the Sun for main sequence

∗ [email protected] stars, 100 kpc for giants. We now possess detailed phase † [email protected] space information, often with spectroscopic and chemical ‡ [email protected] data from cross-matches with other surveys. This has led 2 to the discovery of abundant streams and substructures A widely-studied class of modified gravity theories [5–8]. Streams discovered by Gaia are already being fol- is f(R) gravity [23]. Here, the Ricci scalar R in the lowed up spectroscopically to give six-dimensional (6D) Einstein-Hilbert action is generalised to R + f(R). The phase space data [9]. Bright tracers such as blue hori- Hu-Sawicki form of f(R) [24] exhibits the chameleon zontal branch stars or RR Lyraes can be seen out to dis- mechanism and has been shown to be formally equiva- tances of 250 kpc, assuming Gaia’s limiting magnitude of lent to a subclass of scalar-tensor theories of gravity [25]. G 20.5. Stars near the tip of the red giant branch can The key parameter is the present-day cosmic background be seen even further out to at least 600 kpc. In future, value of the scalar field, fR0. In the present work, we do this should enable Gaia to provide astrometry for very not assume Hu-Sawicki f(R) gravity, but sometimes use distant streams, perhaps beyond the edge of the Milky the parameter fR0 as a concrete example to illustrate Way’s dark halo. the possible constraints achievable from stellar streams, If a stream were a simple orbit, then the positions and noting that constraints are also obtainable in the wider velocities of stars would permit the acceleration and force chameleon space. field to be derived directly from the 6D data. Streams are more complex than orbits [10, 11], but the principle remains the same – their evolution constrains the matter A complete compendium of current constraints on distribution and theory of gravity. Although this idea f(R) gravity and chameleon gravity more generally can has been in the literature for some years, exploitation has be found in the review article by Burrage and Sakstein been sparse primarily because of the limited number of [26]. It is worth noting that some of the strongest streams with 6D data before Gaia. This field is therefore constraints to date have come from weak-field astro- ripe for further exploitation in the Gaia Era. physical probes. Moreover, Baker et al. [27] identify a Because of their different ages and different positions ‘desert’ in modified gravity parameter space accessible in phase space, different streams may tell us different only to galaxy-scale probes, and have launched the ‘Novel things about the theory of gravity. For example, Thomas Probes’ project, aimed at connecting theorists with ob- et al. [12] show that streams from globular clusters are servers in order to devise tests to probe this region. Ac- lopsided in Modified Newtonian Dynamics or MOND be- cordingly, several recent works [28–33] have studied im- cause the ‘external field effect’ violates the strong equiv- prints of screened modified gravity on galaxy scales. alence principle. Meanwhile, Kesden and Kamionkowski [13, 14] demonstrated that if a so-called ‘fifth force’ couples to dark matter but not to baryons, this violation In chameleon theories, main sequence stars will have of the equivalence principle (EP) leads to large, observ- sufficiently deep potential wells to self-screen against the able asymmetries in stellar streams from dark matter fifth force. A diffuse dark matter or gaseous component dominated dwarf galaxies. Specifically, the preponder- of sufficiently low mass, however, will be unscreened. As ance of stars are disrupted via the outer Lagrange point a result, the EP is effectively violated, leading to a num- rather than the inner one, and the trailing stream is con- ber of distinct signatures, as listed by Hui et al. [34]. sequently significantly more populated than the leading Indeed, several of the galaxy-scale studies mentioned in one. Building on that work, Keselman et al. [15] explored the previous paragraph searched for signatures in this the regime of fifth forces much stronger than those inves- list, as well as other signatures of EP-violation. tigated by Kesden and Kamionkowski and found a number of interesting results, including plausible formation scenarios for the Sagittarius stream, the Draco satellite, The present work explores the idea that effective EP- and progenitor-less ‘orphan’ streams around the Milky violation of chameleon gravity should give rise to the Way. stellar stream asymmetries predicted by Kesden and In the intervening years since the work of Kesden and Kamionkowski [13, 14]. We will show that tidal streams Kamionkowski, screened modified gravity theories have in the Milky Way, observable with Gaia, can provide con- become the subject of increasing attention [16–19]. In straints that are comparable to, or stronger than, other these theories, a scalar field coupled to gravity is intro- astrophysical probes. Section II gives a brief introduction duced, giving rise to gravitational-strength ‘fifth forces’. to the fifth force in chameleon theories before providing For the field to retain cosmological relevance while also a new calculation of the magnitude of the effect, extend- avoiding violations of stringent Solar System tests of ing the original work of Kesden and Kamionkowski [13]. gravity, ‘screening mechanisms’ are introduced [20, 21]. Next, Section III describes the Milky Way and satellite There are several varieties of screening mechanism, but models that we use in our simulation code, the methodol- in the one studied here—the chameleon mechanism—the ogy and validation of which are in turn described respec- mass of the scalar field is environment-dependent, such tively in Sections IV and V. Section VI describes results that the fifth force is suppressed within deep potential for a range of tidal streams, inspired by examples discov- wells [22]. In other words, in dense environments like our ered recently in large photometric surveys or the Gaia Solar System, the chameleon becomes invisible to fifth datasets. Finally, Section VII gives some concluding re- force searches, hence its name. marks. 3

Dark Matter Stars

15 15

10 10

5 5

0 0 [kpc] [kpc] y y

5 5 − −

10 10 − −

15 15 − −

15 10 5 0 5 10 15 15 10 5 0 5 10 15 − − − − − − x [kpc] x [kpc]

FIG. 1. Left: Contour map of the effective potential for the dark matter Φeff,DM as given by Eq. (8). Right: Contour map of the effective potential for the stars Φeff, as given by Eq. (9). In both panels, the satellite is at the origin and the Galactic ∗ centre is at (0, 50, 0) kpc. The inner and outer Lagrange points are marked by crosses. To guide the eye, black dashed lines − marking the positions of the DM Lagrange points span the figure. The asymmetry of the Lagrange points for the stellar effective potential illustrates the cause of the stream asymmetries under chameleon gravity. (Parameters: M = 1012M , m = 1010M , and β = 0.5.)

II. THEORY is sourced only by the mass lying between the screening radius and the test particle. We have also assumed A. Chameleon Fifth Forces here that the Compton wavelength of the theory is much larger than the characteristic length scales of the system. In scalar-tensor gravity theories, new scalar degrees of Eq. (1) gives the fifth force experienced by an un- freedom in the gravitational sector couple to matter, giv- screened test particle outside the screening radius of an ing rise to gravitational strength ‘fifth forces’. Chameleon overdense object, but the situation is complicated fur- theories are a class of scalar-tensor theories in which these ther in the case where instead of a test particle, we have fifth forces are suppressed in regions of high-density or another extended object – for example, a star or dwarf deep gravitational potential [22]. In this section, we cover galaxy situated outside the screening radius of its host only the most salient aspects of chameleon theories, and galaxy. In this case, the acceleration of object i (mass refer the reader elsewhere [26] for a more complete de- Mi, radius ri, screening radius rscr,i) due to the fifth scription of the formalism and summary of existing con- force is given by straints. 2 G (M(r) − M(rscr)) Consider a spherical overdensity embedded within a a5(r) = 2β Qi , (2) r2 region of average cosmic density. If the gravitational well of the object is sufficiently deep, a central region of radius where Qi is the ‘scalar charge’ of object i, given in turn rscr will be ‘screened’, such that no fifth forces act within by the region; rscr is the ‘screening radius’ of the object. Outside the screening radius, an unscreened test particle Mi(rscr,i) Qi = 1 − . (3) will experience an acceleration due to the fifth force given Mi by Eq. (3.6) of Ref [26] Thus, the test object experiences the full fifth force only G (M(r) − M(r )) if it is fully unscreened (i.e., rscr = 0) and experiences a (r) = 2β2 scr , (1) 5 r2 no fifth force if it is fully screened (rscr = ri). In the intermediate case where the object is partially screened, where M(r) is the mass enclosed within radius r, and β it experiences a reduced fifth force. In this work, we is the coupling strength. In other words, the fifth force assume stars to be fully self-screened (i.e. Q = 0), and 4 dark matter to be fully unscreened (i.e. Q = 1). For the In practice, the mass of a typical satellite m is at least two satellite galaxies (i.e. the stream progenitors), we explore orders of magnitude less than the mass of the Milky Way, a number of regimes, spanning fully screened, partially and so its contribution to the frequency can be neglected. screened, and fully unscreened. The stationary points of the effective potential Φeff are A commonly used parametrisation of chameleon theo- the Lagrange points or equilibria at which the net force ries is in terms of the coupling strength β and the ‘self- on a star at rest vanishes. In the circular restricted three- screening parameter’ χc. The latter parameter deter- body problem, there are five Lagrange points. Matter mines which astrophysical objects are fully or partially is pulled out of the satellite at the ‘L1’ and ‘L2’ sad- screened, and can be used to calculate their screening dle points, henceforth the ‘inner’ and ‘outer’ Lagrange radii. Note that in the case of Hu-Sawicki f(R) gravity, points. These are situated either side of the satellite, p β is fixed to 1/6, while χc = −fR0. co-linear with the satellite and Milky Way. Leading In order to derive constraints in the β/χc plane or (trailing) streams originate at the inner (outer) Lagrange fR0 space from stellar streams around the Milky Way, points, which lie at (see Section 8.3.1 of Binney and we would need to adopt some prescription to convert χc Tremaine [38]) to Milky Way and satellite screening radii. Analytical formulae exist in the case of an isolated spherical body m 1/3 rt ≈ rh, (6) [35, 36], but such a treatment would neglect the envi- 3M ronmental contribution of the Local Group to the Milky with respect to the satellite centre. Way’s screening, the environmental contribution of the Now consider how the system behaves if a fifth force Milky Way to the satellite’s screening, and the impact acts on the dark matter. Neglecting any screening, and of the non-sphericity of the Milky Way. The calculation assuming the satellite is dark matter dominated, the orbit therefore requires numerical methods in more realistic will circle more quickly with frequency given by scenarios [28]. In this work, we instead use β, rscr,MW, and r as input parameters for reasons of computa- s scr,sat 0 r 0 tional ease. However, in Section VI D, we investigate the 0 G (M + m) G M Ω = 3 ≈ 3 (7) connection between fR0 and rscr,MW in order to forecast rh rh constraints from future data. where G0 ≡ (1 + 2β2)G. The effective potential experienced by a dark matter particle in this system is B. Stream Asymmetries 0 0 G m G M 1 02 2 Φeff,DM(rs) = − − − Ω |rs − rcm| . (8) 1. A Physical Picture rs |rh − rs| 2 This is tantamount to a linear rescaling of Eq. (4), We begin with a physical picture of the cause of stream and the locations of the critical points are therefore un- asymmetries. Consider a satellite represented by a point changed relative to the standard gravity case. However, mass m. For the moment, let us neglect any fifth forces the effective potential is different for a star which does and assume that the Milky Way and can also be rep- not feel the fifth force, namely resented as a point mass M, so both satellite and the Milky Way are moving on circular orbits with frequency Gm GM 1 02 2 Ω around their common center of mass. Φeff,∗(rs) = − − − Ω |rs − rcm| . (9) rs |rh − rs| 2 We use a coordinate system whose origin is at the centre of the satellite. Then, a star at position rs moves in This is not a linear multiple of Eq. (4), and the loca- an ‘effective’ gravitational potential given by [e.g., 37, 38] tions of the Lagrange points are consequently altered. The two panels of Figure 1 shows contour maps of the Gm GM 1 2 2 Φeff (rs) = − − − Ω |rs − rcm| . (4) effective potentials for dark matter and stars, for M = 12 10 rs |rh − rs| 2 10 M , m = 10 M , rh = 50 kpc, and β = 0.5. Also indicated on the diagram are the locations of the inner where rh is the position of the point mass representing the and outer Lagrange points of the potentials. Milky Way and rcm is the position of the centre of mass. In the dark matter case, the points are approximately We use the convention rs = |rs| to denote the modulus equidistant from the satellite centre. However, a signifi- of any vector. The first two terms are the gravitational cant asymmetry is visible in the stellar effective potential, potentials of the satellite and Milky Way respectively, with the outer Lagrange point being much closer to the while the final term provides the centrifugal force due to satellite and at a lower effective potential. Thus, stars the frame of reference, which is rotating about the centre are much more likely to be stripped from the satellite at of mass with frequency Ω the outer Lagrange point, and the trailing stream will s consequently be more populated than the leading one. G(M + m) Physically, we can understand this effect in terms of Ω = 3 . (5) rh force balance. The stars are being dragged along by the 5 satellite, which is orbiting at an enhanced rotation speed come due to the fifth force. This enhanced speed means that the outward centrifugal force on the stars is greater than GM Gm 2 2 Mrh − 2 + 2 + Ω (1 + 2β ) − rt = 0, the inward gravitational attraction by the Milky Way. (rh − rt) rt M + m The consequence of this net outward force is that stars (17) can be stripped from the satellite more easily if they are GM Gm 2 2 Mrh at larger Galactocentric radii than the satellite, and less − 2 − 2 + Ω (1 + 2β ) + rt = 0. (rh + rt) rt M + m easily if they are at smaller radii. This is reflected in the (18) positions of the Lagrange points. Stars unbound from the satellite will be on a slower Proceeding as before orbit around the Milky Way than their progenitor. If β is sufficiently large, then stars that are initially in the m 1/3 1 u 2β2 M u ≈ 1 − + u2 , leading stream can fall behind and end in the trailing 3M (1 + 2β2)1/3 3 3 m stream. (19) m 1/3 1 u0 2β2 M u0 ≈ 1 + − u02 . 2. Circular Restricted Three-Body Problem 3M (1 + 2β2)1/3 3 3 m (20) We now solve for the stream asymmetries in the circu- The last term on the right-hand side produces an asym- lar restricted three-body problem, following and correct- metry with opposite sign to the natural asymmetry. Note ing Ref. [13]. This is a useful preliminary before passing that as u ∝ (m/M)1/3, the M/m factor makes this term to the general case. In Newtonian gravity, the forces bal- actually the largest. The condition for the asymmetry ance at the inner and outer Lagrange points, and so due to the fifth force to overwhelm the Newtonian one is then just GM Gm G(M + m) Mrh − 2 + 2 + 3 − rt = 0, (rh − rt) rt rh M + m m 2/3 2β2 31/3 , (21) (10) & M GM Gm G(M + m) Mrh − − + + r = 0. where only leading terms are kept. This result can be r r 2 r2 r 3 M m t ( h + t) t h + compared with Eq. (29) of Ref. [13]. Although the scaling (11) is the same, the numerical factor is different (remember on comparing results that β2 in our paper corresponds We recall that the inertial frame is rotating about the 2 to β2f f in theirs). In fact, the changes are very much centre of mass, and so the centrifugal terms in Eqs. (10) R sat to the advantage of the fifth force, as smaller values of β and (11) depend on the distance of the Lagrange point to now give detectable asymmetries. the centre of mass, not the Galactic centre (cf. Eqs. (14) The two most massive of the MW dwarf spheroidals and (15) of Ref. [13]). 8 0 0 are Sagittarius with dark matter mass 2.8 × 10 M and We now define u = rt/rh and u = rt/rh for the inner 8 Fornax at 1.3 × 10 M [39]. These will allow values of and outer Lagrange points respectively, and obtain 2 −3 β & 2 × 10 to be probed. For the smallest dwarf 5 m (1 − u3)(1 − u)2 spheroidals such as Segue 1 with a mass of 6 × 10 M , u3 = , (12) then values of β2 2×10−4 are in principle accessible. It M − u u2 & 3 3 + should be noted that the Segue 1 is an ambiguous object, m (1 − u03)(1 + u0)2 u03 = . (13) and it is not entirely clear if it is a dark matter dominated M 3 + 3u0 + u02 dwarf or a globular cluster [40]. Solving, we find that

m 1/3 u 3. General Case u ≈ 1 − , (14) 3M 3 m 1/3 u0 The circular restricted three-body problem is some- u0 ≈ 1 + , (15) what unrealistic, as the Galaxy’s matter distribution is M 3 3 extended. In particular, there is a significant difference so the natural asymmetry is in the enclosed host mass within the inner and outer La- grange points and this plays a role in the strength of 2/3 the asymmetry. We now proceed to give a mathematical 0 2 m ∆rnat = (u − u)rh ≈ rh. (16) 3 3M analysis of the general case. The satellite is now moving on a orbit with instanta- Now introducing a fifth force, the force balance equa- neous angular frequency Ω. We work in a (non-inertial) tions for stars not directly coupling to the fifth force be- reference frame rotating at Ω with origin at the centre of 6

0 the satellite. A star at location rs now feels the follow- the satellite centre u − u. This is the natural stream ing forces: (i) a gravitational attraction by the satellite, asymmetry (ii) a gravitational attraction by the host galaxy; (iii) an inertial force because the satellite is falling into the host m(r ) 2/3 (2 − n)(3 − n)r r ≈ u0 − u r s h ∆ nat ( ) h = 5/3 and so the reference frame is not inertial and (iv) the M(rh) 3(1 − n + 2Ωs/Ω) Euler, Coriolis and centrifugal forces because the refer- (27) ence frame is rotating. Note that (iii) was not necessary In the restricted three-body problem, n = 0 and Ω = Ωs, in our earlier treatment of the circular restricted three- so we recover our previous result in Eq. (16). body problem because there we chose an inertial frame We wish to compare this asymmetry to the asymmetry tied to the centre of mass. produced by adding the modified gravity acceleration of The equation of motion for a star or dark matter par- the satellite to the equation of motion. Now specialising ticle is to the case Ω = Ωs to reduce complexity, we find the asymmetry due to the fifth force is rs (rs − rh) ¨rs = −Gm(rs) 3 − GM(|rs − rh|) 3 rs |rs − rh| 4 2 ∆r5 ≈ − β rh. (28) rh ˙ 3(3 − n) − GM(rh) 3 − Ω × rs − 2Ω × ˙rs (22) rh − Ω × (Ω × r ) So, the requirement that the dark matter asymmetry s overwhelms the natural asymmetry is Save for the assumption that the matter distributions in 2/3 2 (2 − n)(3 − n) m the satellite m(rs) and the host M(rh) are spherically 2β & , (29) symmetric, this expression is general. 2(3 − n)2/3 M We now assume that the star or dark matter particle is which again reduces to Eq. (21) in the restricted three following a circular orbit around the satellite with orbital body case, as it should. For galactic dynamics, a reason- frequency Ω and that r /r 1. By careful Taylor s s h able choice is n , which corresponds to a galaxy with expansion, we obtain = 1 a flat rotation curve, i.e. an isothermal sphere. Assum-

rs (3 − n)(rs · rh)rh ing the stars in the satellite satisfy Ωs = Ω, then tidal ¨rs = −Gm(rs) 3 + GM(rh) 5 streams in galaxies with flat rotation curves are much rs |rh| more sensitive probes of the dark matter asymmetry. As rs − GM(rh) − Ω˙ × rs − 2Ω × (Ωs × rs) (23) we move from n = 0 (the point mass case) to n = 1 (the r 3 h isothermal sphere), we gain an additional factor of ≈ 2.3 + Ω × (Ω × rs) in sensitivity. The changes are again in our favour. The asymmetries in tidal streams are therefore an even more where n(r ) is the logarithmic gradient of M(r ). h h delicate probe of the fifth force than suggested by the To calculate the tidal radius, we now specialise to the analysis in Ref. [13]. case of a particle whose orbit lies in the same plane as the satellite’s orbit. The satellite’s circular frequency is 2 GM r /r 3. The tidal radius is defined as the Ω = ( h) h III. MILKY WAY AND SATELLITE MODELS distance from the centre of the satellite at which there is no net acceleration, i.e., the forces on the particle towards the host and the satellite balance. This gives the tidal In our simulations, we follow the evolution of a large radius as number of tracer particles, stripped from a satellite galaxy and forming tidal tails. The test particles are 1 m(r ) 1/3 accelerated by the gravity field of both the Milky Way r s r . (24) t = 1/3 h and the satellite, together with any fifth force contribu- (1 − n + 2Ωs/Ω) M(rh) tions. We begin by describing our models for the Milky When satellite and host are point masses, then n = 0 Way and satellite. and Ω = Ωs, and we recover the result previously found in Eq. (6). 0 0 We now define u = rt/rh and u = rt/rh for the inner A. Milky Way Model and outer Lagrange points respectively, and obtain

2−n For the Milky Way, we adopt the axisymmetric mass 3 m(rs) (1 − u) u model of Ref. [41], which is designed to ﬁt a number of u = 2−n 2−n ; (25) M(rh) 1 − (1 − u) + α(1 − u) u recent kinematic constraints on the Milky Way matter m(r ) (1 + u0)2−nu0 distribution. The model comprises six distinct compo- u03 s , (26) = 0 2−n 0 2−n 0 nents: a central bulge, a dark matter halo, and four discs M(rh) (1 + u ) − 1 + α(1 + u ) u (thin and thick stellar discs, and atomic and molecular where α ≡ 2Ωs/Ω − 1. We now solve for the diﬀerence gas discs). The functional form of the density distribu- in the positions of the Lagrange points with respect to tion of each of these components is given as follows. The 7

while the two gas discs are given by an exponential disc TABLE I. Milky Way model parameters from Ref. [41]. The model with a central hole and a ‘sech-squared’ vertical first three columns respectively give the symbol representing profile, a given parameter, the number of the equation in which it appears, and a physical description. The final column lists the Σ0 z parameter values; in most cases these are best-fitting values ρ(R, z) = e−(Rh/R+R/R0) sech2 . (33) inferred by Ref. [41], but some were instead fixed a priori, 4z0 2z0 such as the various disc scale heights. Further details can be found in that article. With this axisymmetric mass model in hand, the gravitational potential is then calculated with a Poisson solver Symbol Eq. Parameter Value 3 [45] utilising a spherical harmonic technique similar to ρ0,h 30 Halo scale density 0.00853702 M /pc that described by Dehnen and Binney [46]. The solver r0,h 30 Halo scale radius 19.5725 kpc 3 calculates the potential and its gradients on a spherical ρ0,b 31 Bulge scale density 98.351 M /pc r0,b 31 Bulge scale radius 0.075 kpc grid; for our simulations we use a grid with 2000 log- −4 4 rcut 31 Bulge cutoff radius 2.1 kpc spaced radial cells between 10 kpc 10 kpc, and 2500 thin 2 Σ0 32 Thin disc normalisation 895.679 M /pc (polar) angular cells, and truncate the spherical harmonic thin z0 32 Thin disc scale height 300 pc expansion at multipole l = 80. These settings were found thin R0 32 Thin disc scale radius 2.49955 kpc to yield converged solutions for the Milky Way potential thick 2 Σ0 32 Thick disc normalisation 183.444 M /pc corresponding to the mass model described above. thick z0 32 Thick disc scale height 900 pc The gravitational acceleration on a test particle (ne- thick R0 32 Thick disc scale radius 3.02134 kpc glecting any fifth forces for the moment) due to the Milky HI 2 Σ0 33 HI disc normalisation 53.1319 M /pc Way is then calculated by interpolating the potential gra- HI z0 33 HI disc scale height 85 pc dient at the position of the particle, employing a cubic HI R0 33 HI disc scale radius 7 kpc spline. HI Rh 33 HI disc hole radius 4 kpc H2 2 Σ 33 H2 disc normalisation 2179.95 M /pc 0 H2 z0 33 H2 disc scale height 45 pc B. Satellite Model H2 R0 33 H2 disc scale radius 1.5 kpc H2 R 2 h 33 H disc hole radius 12 kpc We model the satellite with a truncated Hernquist sphere with the density cut off at a radius rt. The reason for this sharp truncation will become clear in the discus- various undefined symbols are parameters of the model, sion of the fifth force in §III C. Defining a reduced radius for all of which we adopt the values of Ref. [41]. We x ≡ r/aS (thus xt ≡ rt/aS) where aS is the scale radius reproduce these values in Table I. of the profile, the density-potential pair is given by For the DM halo, the model employs a Navarro-Frenk-  h 2 i Gm (1+xt) 1 1 White (NFW) profile [42], − 1 + − , x ≤ xt.  rt xt 1+x 1+xt x ρ0,h Φ( ) = ρ(r) = . (30)  2 −Gm, x > x . r 1 + r r t r0,h r0,h  A , x ≤ x .  3 t We have experimented with non-spherical oblate and pro- x(1 + x) ρ(x) = late halo profiles, as well as steeper inner slopes (cf. [43]),  and found no significant impact on our results. However, 0, x > xt. it would be interesting in future to investigate the impact (34) of a truly triaxial dark matter halo. The density normalisation A is related to the total satel- Meanwhile, the bulge is represented by an axisym- lite mass m by metrised version of the model of Bissantz and Gerhard 2 [44], (1 + xt) m A = 2 3 , (35) xt 2πa 0 2 ρ0,b − r ρ(R, z) = e rcut , (31) 1.8 The mass enclosed within a reduced radius x is then 1 + r0 r0,b  2 2 x (1 + xt) √ m 2 2 , if x ≤ xt. 0 2 2  x (1 + x) where r ≡ R + 4z . m(x) = t (36) The two stellar discs are represented by a simple ex-  ponential disc with an exponential vertical profile, m, otherwise.

Σ0 For all satellites, we adopt truncation radii of rt = 10aS, ρ(R, z) = e−R/R0 e−|z|/z0 , (32) 2z0 or equivalently xt = 10. 8

The acceleration on any given test particle due to the the dark matter tracer particles, which we assume to be satellite can then be calculated from the above relations. a diffuse, unscreened component. For self-consistency, the initial phase-space distribution Similarly, the modified gravity acceleration due to the of the tracer particles is that of a truncated Hernquist Milky Way on particle i (which can now also represent profile (see Section IV A for further details). Of course, the satellite) at x is given by this self-consistency is lost as the simulation advances in time, as many of the tracer particles are tidally removed i 2 a (x) = 2β QiQMW(r)aN,MW(x), (39) by the Milky Way, but our assumed satellite potential 5,MW remains unchanged in mass and shape. However, we will where the symbols have analogous meanings to those show in §V that this assumption of an unchanging satel- above. The scalar charge of the Milky Way is given by lite potential is largely harmless.  M(rscr,MW) 1 − , if r ≥ rscr,MW. C. Fifth Forces  M(r) QMW(r) = (40)  In addition to gravity, the satellite and the tracer par- 0, otherwise. ticles also experience accelerations due to the fifth force. The satellite feels a fifth force sourced by the Milky Way, If particle i represents the satellite, then we take the lim- while the tracer particles also feel a fifth force sourced iting value of the satellite scalar charge Qi = Qsat(r = by the satellite. We assume spherical fifth force profiles rt). This is valid as long as the the Milky Way centre in both cases. For the satellite, this is consistent with does not fall within the truncation radius of the satellite its gravitational potential, although the sphericity of the centre, which does not happen in any of our simulations. satellite may be distorted by its tidal disruption. For The formalism given in this subsection demonstrates the Milky Way, the spherical symmetry is inconsistent the utility of truncating the mass profile of the satel- with the presence of the disc. The scalar field profiles lite. By so doing, we have made it straightforward to of disc galaxies have correspondingly discoid shapes [28]. model the satellite as being fully screened (rscr,sat = rt), However, the scalar field profile is roughly spherical when fully unscreened (rscr,sat = 0), or partially screened rscr,MW is much larger than the disc scale radius of 6.5 (0 < rscr,sat < rt). kpc. In particular, using the f(R) scalar field solver de- It is worth remarking that we have used the superpo- scribed in §VI D, we find that fifth force profiles in the sition principle to compute the joint fifth force of Milky Milky Way (assuming a spherical dark matter halo) only Way and satellite on the tracer particles. Strictly speak- become appreciably aspherical for log10 |fR0| & −6.2, ing, the superposition is not valid in highly non-linear and so the spherical approximation is robust in the pa- theories of gravity like chameleon gravity. In particular, rameter regimes we mostly focus on in this article. environmental screening can affect the screening radii of Eq. (2) can be rewritten to give the expression for the objects. Linearity is, however, restored once the screen- modified gravity acceleration due to the satellite on tracer ing radii are fixed (as we do by hand), so that from particle i, situated at position x, that point on we can apply the superposition principle i 2 for computing the joint fifth force. a5,sat(x) = 2β QiQsat(r)aN,sat(x), (37) where β is the coupling strength of the fifth force (an TABLE II. Parameters for each of the 4 progenitors. Here, input parameter of our simulations), aN,sat is the Newto- x0 and v0 give the present position and velocity respectively nian acceleration due to the satellite, and Q and Q r i sat( ) (note that we run the simulations backwards then forwards are the scalar charges of particle i and the satellite re- again, so that the satellites end at x0 and v0), a and m are spectively. The latter is given by the Hernquist scale radius and total mass of the satellite, and tmax is the total time over which each simulation is run; the  m(r ) − scr,sat , if r ≥ r . farther orbits require more time to undergo an appreciable 1 m t 17  number of orbital periods. Note that 10 seconds is 3 Gyr.  ∼  The parameters for Satellite A resemble the Pal-5 stream, m(rscr,sat) Qsat(r) = 1 − , if r > r ≥ r . (38) B the Sagittarius stream, C the Orphan stream, and D a m(r) t scr,sat  hypothetical stream at large distance.   x v  ID 0 0 a m tmax 0, otherwise. (kpc) (km/s) (kpc) (108M )(1017 s)

A (7.7, 0.2, 16.4) (-44, -117, -16) 0.01 0.0003 1 Here, m(r) is the satellite mass enclosed by radius r, B (19.0, 2.7, -6.9) (230, -35, 195) 0.5 5 1 and rscr,sat is its screening radius. Qi, meanwhile, diﬀers C (90, 0, 0) (0, 0, 80) 0.5 2.5 1.5 between the particle types. As we assume the stars are D (150, 0, 0) (0, 0 , 100) 1 5 2.5 fully screened against the ﬁfth force, Qi = 0 for the star tracer particles. On the other hand, we take Qi = 1 for 9

A C A. Tracer Particles B D To generate the initial phase space distribution of N tracer particles, we use a Markov Chain Monte Carlo 140 technique to generate 2N samples from possible equi- 7.2 librium distribution functions (DFs) for the Hernquist 120 − model. The choice of equilibrium includes the isotropic DF [52] 100 |

0 p R 1 E˜ f 80 | f(E˜) = √ 3 0 3/2 2

6.8 10 − 2 (2π) (GM aS) 1 − E˜ log 60   (41) −1 p ˜  2 3 sin E  40 ×  1 − 2E˜ 8E˜ − 8E˜ − 3 +  , 6.4  r  Galactocentric Distance [kpc] − E˜ 1 − E˜ 20 6 and the radially anisotropic DF [53] 0 − 3.0 2.5 2.0 1.5 1.0 0.5 0.0 − − − − − − 3 E˜ Time [Gyr] ˜ f(E) = 3 . (42) 4π aS GL FIG. 2. For the 4 satellites described in Table II, we show Here, E is the specific (binding) energy of a particle, the orbital evolution over 1017 seconds ( 3 Gyr) under stan- 0 2 2 ∼ aS is the scale radius, M = (1 + xt) m/xt is the un- dard gravity. The horizontal dashed lines indicate the Milky truncated mass of the satellite, while E˜ = Ea /GM 0 is f(R) S Way screening radii under Hu-Sawicki gravity for vari- the dimensionless binding energy. The DFs differ in the ous different values of the theory parameter fR0 (the values anisotropy of the velocity distributions. In fact, our sim- of log fR0 are shown at the right hand side of the panel 10 | | (see Section IV C for details about the calculation of these ulations show similar results for stream generation, irre- screening radii). This figure illustrates the range of distances spective of the anisotropy, so the choice of equilibrium is probed by tidal streams, and gives an idea of the possible not so important. constraints achievable for chameleon gravity theories. Given these 2N samples, we integrate the orbits of the particles in the satellite potential (i.e. neglecting fifth forces and the Milky Way) for 1017 seconds (≈ 3 Gyr). At the end of this relaxation phase, we randomly down- sample N of these particles, excluding any particles for IV. METHODS which the orbit ever strayed beyond the truncation radius. This gives a suitable equilibrium distribution of positions and velocities for the test particles in our sim- Approximate methods for quickly generating realistic ulations. streams by stripping stars at the tidal radius of a pro- All of our simulations incorporate 10000 DM particles genitor are now well established [47–49]. The methods and 10000 star particles. This equality does not encode work as restricted N-body simulations, in which we fol- any assumptions about the underlying stellar/DM mass low the orbital evolution of a large number of massless fraction of the satellite; the particles are merely massless tracer particles. The stream particles are integrated in samples of the distribution, and the satellite is assumed a fixed Galactic potential, together with the potential of to be dark matter dominated. the moving satellite. This method robustly reproduces This procedure has omitted the fifth force altogether. the morphology of streams, in particular the locations This is appropriate for the star particles which, by as- of the apocentres of the leading and trailing branches, sumption, do not experience the fifth force. For the dark yet provides two to three orders of magnitude speed-up matter however, it is less self-consistent. We have ex- compared to conventional N-body experiments [49]. The perimented with including a fifth force, both in the dis- main extension of our code here is that it incorporates tribution function and in the relaxation phase described an optional fifth force due to the chameleon field. in the previous paragraph. This leads overall to ∼ 10% increases in the number of dark matter particles being All of our code is made publicly available as the python stripped from the progenitor during the main simula- 3 package smoggy (Streams under MOdified GravitY) tion, but no appreciable morphological change to the [50]. Animations of the simulations depicted in Figures dark matter streams. 4, 6, 10, and 14 are given as Supplemental Material ac- Note also that this procedure makes the unrealistic as- companying this article [51]. sumption that the stars and dark matter have the same 10 spatial distribution. However, we have experimented Way model described in III A. We demonstrate later that with drawing the stars from more compact initial distri- significant stream asymmetries develop when the orbit is butions than the dark matter, and found no appreciable mostly outside the Milky Way screening radius, so these difference in our results. lines give a preview of the modified gravity constraints achievable. For each satellite, we explore a variety of modified B. Orbit Integration gravity scenarios by varying 3 input parameters: the coupling strength β, the satellite screening radius rscr,sat, To calculate the trajectories of the various particles, and the Milky Way screening radius rscr,MW. First, we we use a second-order leapfrog integrator. Under such a consider 4 coupling strengths: β = {0.1, 0.2, 0.3, 0.4}. scheme, the velocities v and positions x of the particles The strength of the fifth force relative to gravity is given are updated at each timestep i via by 2β2, so this corresponds to the range from 2% − 32%. The most extreme case can therefore be used as an ap- vi+1/2 = vi−1/2 + a(xi)∆t, proximate analogue for f(R) gravity, where the strength (43) xi+1 = xi + vi+1/2∆t, of the fifth force is 1/3 that of gravity. For the satellite screening radius, we explore a range where ∆t represents the timestep size, and a(x) repre- of regimes, from fully screened to fully unscreened, and sents the accelerations calculated using the expressions encompassing a variety of partially screened regimes in given in Sections III A, III B, and III C. At the start of between. Using the upper case of Eq. (38), we recast the the simulation (i.e. timestep i = 0), the ‘desynchronised’ screening radius rscr,sat as the scalar charge Qsat, and velocities v−1/2 are obtained using consider a range of values of Qsat from 0 to 1 in steps of 0.1. We recall that Qsat = 0 corresponds to the fully 1 screened case, so here r = 10a, where a is the Hern- v−1/2 = v0 − a(x0)∆t. (44) scr,sat 2 quist scale radius of the satellite in question. Qsat = 1 is From here, Eq. (43) can be used repeatedly to advance the fully unscreened case, so rscr,sat = 0. the system in time. Finally, we consider a range of values for the Milky Our method for choosing the timestep size ∆t is as Way screening radius rscr,MW. As the orbital distances follows. We calculate the total energies of all particles of each satellite are different, it is useful to select a dif- at the start and end of the relaxation phase described in ferent range of values for rscr,MW for each satellite. For Section IV A, in which the orbits are integrated in the each satellite, we define a maximum screening radius satellite potential for 1017 seconds. We repeat the relax- rscr,max, approximately equal to the apocentric distance ation phase, iteratively reducing the timestep size, until of the orbit under standard gravity. These values are the energies of all particles are conserved to within 2%. rscr,max = 20, 50, 90, 150 kpc for satellites A, B, C, and D Through experimentation, we found that energy conser- respectively. Then, we choose a range of 11 values such vation is a good proxy for numerical convergence and this that rscr,MW/rscr,max runs from 0 to 1 in steps of 0.1. 2% criterion gives accurate, converged results. With this Altogether, we run 485 simulations for each satellite: criterion, we find that angular momentum is conserved 4 × 11 × 11 = 484 modified gravity simulations plus one to an even greater precision, with a maximum fractional standard gravity (β = 0) simulation. deviation of ∼10−4. The final timestep size chosen by this process is then used again for the main simulation. In practice, we find ∆t ∼ O(1011) seconds typically. D. Assumptions

The previous subsections have given details about the C. Simulations various parts of our code, but for clarity we provide a list of all of our simplifying assumptions: We simulate the generation of streams from 4 progenitors. Satellite A is inspired by the Palomar 5 stream 1. We neglect self-gravity between the tracer particles, [54], B the Sagittarius stream [55], C the Orphan stream both before and after they are stripped from the [8], and D is a hypothetical stream at large Galactocen- satellite, as is typical in Lagrange stripping codes tric distance, of the kind that is likely to be found in [11, 49]. the later Gaia data releases. The parameters for these 4 progenitors are given in Table II. 2. We assume the gravitational attraction on the Figure 2 shows the evolution of the orbits over ∼ 3 tracer particles due to the satellite can be ap- Gyr for each of the 4 satellites, under standard gravity. proximated as that due to a (truncated) Hernquist Also shown are lines indicating the disc-plane Milky Way sphere, whose orbit is only governed by the Milky screening radii for a range of values of fR0. These calcula- Way potential. This assumption has been veriﬁed tions were performed using the scalar ﬁeld solver within against full N-body simulations of stream formation the f(R) N-body code mg-gadget [56] for the Milky by others [47–49]. 11

Time [Gyr] 8 7 6 5 4 3 2 1 0 − − − − − − − − 60

30 Distance [kpc] 20

10 Leading Trailing

0 [kpc] z 25 −

50 −

75 −

75 50 25 0 25 50 75 50 25 0 25 50 − − − − − − x [kpc] x [kpc]

FIG. 3. Our reproduction of a simulation from Law and Majewski [55] Top: Distance of the simulated Sagittarius dwarf from the Galactic centre over 8 Gyr (to be compared to the results in Figure 7 from Ref. [55]). Bottom left and right: First wrap of the leading and trailing streams respectively (to be compared to the results in the two left-hand panels of Figure 8 of Ref. [55]). The curve represents the orbital path of the satellite, culminating in the current position of the Sagittarius dwarf, represented by the ﬁlled circle. The green points are the positions of the simulation particles. The satellite orbit has been integrated over 3 Gyr up to the present day, so the morphology of the streams should resemble only the orange and magenta particles from the original ﬁgure. This successful reproduction of literature results serves as a test of our code, and checks several of our simplifying assumptions.

3. We assume the depth and radial extent of the lites (cf. [8]) satellite potential well does not change over time. While this assumption could be relaxed in the stan- 5. While we typically sample equal numbers of stel- dard gravity case, it is a greatly helpful one in the lar and dark matter particles, we assume the mass chameleon case. Thus, to allow a fair comparison proﬁles of our satellites to be dark matter domi- between results in the two cases, we make the as- nated. So, the satellites feel the full ﬁfth force in sumption universally. the absence of screening.

4. We assume a static, axisymmetric model for the 6. The initial density profile and kinematics of the Milky Way potential, composed of a disc, bulge, stellar and dark matter particles in the satellites and halo. Dynamical friction is therefore not mod- are assumed to be the same. This simplifies the elled, though the effect is negligible at these low fifth force calculation, and allows us to ensure any mass ratios [57]. We neglect any effects due to the difference in the stellar and dark matter streams is Large Magellanic Cloud or other Milky Way satel- due to the fifth force rather than initial conditions. 12

As described in Section IV A, we have experimented we integrate the orbit of the satellite backwards for with sampling the star particles from radially more 1017 seconds (∼ 3 Gyr), and then forwards again with compact distributions than the dark matter, and 16000 tracer particles. The resulting leading and trailing found no significant difference in results. streams from this simulation are shown in the lower pair of panels in Figure 3. The detailed morphologies of these Assumptions (1)-(6) apply equally in the standard streams closely resemble those of the streams depicted in gravity and modified gravity simulations. The following Figure 8 of Law and Majewski [55], considering only the three assumptions, however, apply only in the simula- orange and magenta particles in that figure (i.e., particles tions including a fifth force. liberated within the last 3 Gyr). Despite this reassuring agreement between the results 7. We adopt spherical fifth force profiles around both from our simplified code and those from full N-body sim- the Milky Way and the satellite, despite the Milky ulations, it is worth noting that several of the assump- Way potential being non-spherical. As discussed tions stated in §IV D are not addressed by this test. In in §III C, this is valid when the the MW screening particular, this test does not validate the assumptions radius is larger than the Galactic disc, log |f | 10 R0 . made in the treatment of the fifth force. However, the −6.2. aim of the present work is to provide a qualitative un- 8. Furthermore, we assume this spherical screening derstanding of the effects of chameleon gravity on stellar surface of the satellite remains fixed throughout the streams. Future work aiming to derive quantitative con- satellite’s orbit. In reality, the radius would vary as straints from observational data will likely require either the Galactocentric distance of the satellite changes, a relaxation or a more careful justification of some of due to environmental screening, and the shape of those assumptions. the screening surface (and surrounding fifth force profile) would likely become aspherical as the satellite approached the Milky Way’s screening radius VI. RESULTS and non-linear effects warp the screening surface. A. Standard Gravity 9. The Compton wavelength of the scalar field is assumed to be much larger than relevant length scales. In the context of Hu-Sawicki f(R) grav- Figure 4 shows the images from the standard gravity simulations for all 4 satellites listed in Table II. Each of ity, the Compton wavelength is given by λC ≈ p −4 the four quarters of the figure represents one of the satel- 32 |fR0|/10 Mpc [58], so this assumption starts to break down at around f ∼ 10−8. lites, as labelled in the top corner. The large subpanel in R0 each quarter shows an image of the stream particles at the end of the simulation. As the stellar and dark matter particles are sampled from the same probability distribu- V. CODE VALIDATION tion initially (see assumption 6 in §IV) and there is no EP-violation by a fifth force in these standard gravity As validation, we compare the results of our code for simulations, the stars and dark matter particles are in- disruption of the Sagittarius dwarf galaxy under standard distinguishable and are thus not plotted separately in this Newtonian gravity with the results of Law and Majewski figure. The three smaller subpanels in each quarter show [55]. They simulate the formation of the stream using the average velocity along the stream, velocity dispersion a full N-body disintegration of the satellite in a static along the stream, and velocity dispersion perpendicular Milky Way potential, so assumptions (1)-(3) in the list to the stream, all as a function of stream longitude and in §IV are not made in their work. In other words, the all calculated in bins of particles along the stream. The gravitational attractions of the satellite and stream are bins are created adaptively, such that each bin contains there treated in fully self-consistent manner. 25 particles, including only the particles which have been To set up this test, we adopt the Milky Way potential stripped from the progenitor. Within each bin, the unit of Ref. [55], i.e. a Hernquist bulge, a Miyamoto-Nagai vector giving the direction ‘along the stream’ is taken disc, and a triaxial logarithmic dark matter halo. The as the (normed) average velocity vector of all particles parameters and initial conditions for the satellite are the in the bin. This figure illustrates the diversity of our same as those for Satellite B, given in Table II. simulated streams, with a variety of morphologies and As a first test, we integrate the orbit of the satellite in Galactocentric distances represented. this potential backwards for 2.5×1017 seconds (∼ 8 Gyr). The distance of the satellite from the Galactic centre as a function of time is shown in the upper panel of Figure B. Unscreened Fifth Force 3. This shows excellent agreement with Figure 7 from Law and Majewski [55]. Turning to fifth forces, we first discuss results from an It is also desirable to check the morphology of the unscreened, EP-violating fifth force coupling only to dark streams generated with our method. As a second test, matter (rscr,sat = rscr,MW = 0). This is the case studied 13

x [kpc] Longitude [◦] Longitude [◦] x [kpc] 10 15 20 25 0 25 350 0 350 50 25 0 25 50 − − − − [km/s] 7.5 A 3 B 25 2 σ 30 5.0 ⊥ 20 1 10 2.5 0 4.5 40 0.0

[kpc] 2.5 σ [kpc] k 25 25

y − y 2.5 0.5 10 − 300 50 5.0 − − 175 v¯ k 200 7.5 75 − 125 100 − 250 C 15 D 25 σ 10 200 100 15 ⊥ 5 5 150

50 40 20 100 25 15

[kpc] σ [kpc] k 10 50 y 10 y 0 5 0 350 300 250 50 50 v¯ 200 − − k 150 100 100 − 50 0 50 100 250 0 250 [km/s] 250 0 250 100 0 100 200 − − − − x [kpc] Longitude [◦] Longitude [◦] x [kpc]

FIG. 4. The simulated streams under standard gravity. The four quarters represent our 4 satellites: A (upper left), B (upper right), C (lower left), and D (lower right). In each quarter, the largest subpanel shows an image of all stream particles in the orbital plane, at the end of the simulation. No distinction is made between star and dark matter particles. The colours differentiate leading and trailing streams, with the darker shade being the trailing stream. For Satellite B, additional shades are used to distinguish multiple wraps. The black cross shows the position of the centre of the Milky Way, while the filled circle shows the final position of the Satellite, with an arrow indicating its instantaneous direction of travel. The side-panels show three quantities calculated in bins of particles: average velocity along the stream, velocity dispersion along the stream, and velocity dispersion perpendicular to the stream. Here again, the colours differentiate leading and trailing streams. In every case, the orbital plane is defined such that the Satellite is on the x-axis, moving in the positive y-direction. Animations of the 4 simulations depicted in this figure are included in the Supplemental Material accompanying this article. by Kesden and Kamionkowski [13, 14]. This case also 5. applies in screened modified gravity with a (formally) universal coupling if stars self-screen, but screening is not Figure 6 shows the positions of the dark matter and triggered otherwise. In our work, the strength of the fifth star particles in the simulation with rscr,sat = rscr,MW = 0 and β . for Satellite C, at 11 equally spaced snap- force relative to gravity is given by 2β2, in keeping with = 0 2 the recent modified gravity literature, whereas Kesden shots over time (recall that animations of selected simu- and Kamionkowski used β2. Thus, the simulation de- lations are available online). The most striking feature is the asymmetry of the stellar stream. The preponderance picted in Figure 6 for example (β = 0.2,F /F = 0.08), 5 N of star particles populate the trailing stream, rather than is most comparable to the ‘β = 0.3’(F5/FN = 0.09) simulation in Refs. [13, 14]. the leading stream. The enhanced rotation speed of the satellite due to the fifth force means that the outward Figure 5 shows the shape of Satellite B’s orbit for a centrifugal acceleration of the stars outweighs the in- variety of values of β. In the absence of screening, the ward gravitational acceleration by the Milky Way. Con- introduction of a fifth force as in Eq. (1) is tantamount sequently, stars are more likely to leave the satellite via to an overall linear rescaling of the Milky Way mass or the outer Lagrange point. Also, even some of the stars gravitational constant by a factor of 1 + 2β2. As a conse- which are disrupted from the inner Lagrange point can quence, the orbital period of the satellite is shorter and eventually end up in the trailing stream, once sufficient the apocentric distance smaller, as is apparent in Figure time has passed for them to be overtaken by the satel- 14

30 potential), may begin to break down when the disruption of the satellite due to the Milky Way is so severe. 20 However, all of our satellites are, by assumption, dark matter dominated. Even in the simulations where the 10 β satellites lose all of their stars, they still retain a large fraction of their dark matter particles, and thus most of 0 0 their assumed mass. 0.1 10 This result echoes a key ﬁnding of Keselman et al. [15], [kpc] − 0.2

y who argued that this prediction of stellar streams without 20 0.3 − associated progenitors could be related to the observed 0.4 ‘orphan’ streams of the Milky Way. 30 − 40 − C. Chameleon Screening 50 − 40 30 20 10 0 10 20 30 We now show results from the chameleon simulations, − − − − x [kpc] i.e. the simulations with screening. Unlike the dark matter force investigated in the previous subsection, the fifth force here is universally coupled. However, as discussed FIG. 5. Satellite B’s orbit in its orbital plane, shown for a in the Sections I and II, an effective EP-violation arises range of β with rscr,sat = rscr,MW = 0. The cross indicates the Galactic centre and the filled circle shows the final posi- because main sequence stars are self-screened against the tion of the satellite, i.e. the current observed position of the fifth force in parameter regimes of interest. Sagittarius dwarf galaxy. This figure illustrates the effect of Figure 8 is the analogue of Figure 5, now showing the an unscreened fifth force on orbital shapes for a fixed final effect on the satellite’s orbit of a varying Milky Way position. screening radius. In the case of the outermost screening radius of 45 kpc, nearly the entire orbit is situated within rscr,MW and is therefore almost equivalent to the lite. Meanwhile, the dark matter particles experience the standard gravity case. Following along this orbit from same fifth force as the satellite, and so there is (almost) plotted position of the progenitor, the other orbits peel no preferential disruption via either Lagrange point. The away one by one, in order of increasing screening radius. dark matter stream that forms, is consequently almost In other words, once the orbit passes outside the screen- symmetric around the progenitor. ing radius, the fifth force becomes active and the orbit These effects are also apparent in Figure 7, which starts to diverge from the standard gravity case. Recall- ing from Eq. (1) that the fifth force is proportional to the shows the longitude difference ∆Λ = Λ − Λsat as a function of time for random subsamples of particles in mass between the test particle and the screening radius, the simulations without screening, with β increasing in the divergences do not become noticeable as soon as the strength from 0.0 to 0.4 in steps of 0.1 for all 4 satellites. orbit passes out of a given screening radius, but some Here, Λ is the longitude in the instantaneous orbital plane time after, once this enclosed mass is large enough for an of the satellite and increases in the direction of the satel- appreciable fifth force. lite’s motion, so particles in the leading stream have pos- Looking instead at the impact of the Milky Way screen- itive ∆Λ. The dark matter particles are stripped almost ing radius on stream asymmetries, one observable quan- equally into the leading and trailing streams, leading to tity is the ratio of the number of stars in the leading to streams that are nearly symmetric about the progenitor the trailing stream, for all values of β. For the star however, as β increases, N the particles are increasingly disrupted into negative lon- α = lead . (45) N gitudes, i.e. the trailing streams. trail Sometimes, the satellite can be stripped completely of Figure 9 shows this quantity as a function of Milky Way all of its stars. Then, the spatial separation between screening radius for all satellites, assuming Qsat = 1, satellite and stream can be very large indeed, as no i.e. fully unscreened satellites. To ensure a fair com- new stars become unbound from the satellite in order parison between simulations, α is computed in each case to bridge the gap. This occurs in Satellite A for both at the moment of the satellite’s third pericentric pas- β = 0.3 and 0.4, as it loses all of its stars at its first peri- sage. As the MW screening radius increases, the asym- centric passage. Satellite A, which is significantly less metry is progressively reduced. This appears to partic- massive than our other satellites, does not have a suffi- ularly be the case when rscr,MW lies between the peri- ciently deep potential well for its stars to remain bound centre and apocentre of the orbit. This makes sense, as under the enhanced centrifugal force from the Milky Way. most tidal disruption occurs at and around pericentric Some caution is needed because assumption 3 for exam- passage. Therefore, screening the pericentre has the con- ple (the assumption of an unchanging satellite mass and sequence of reducing the asymmetry of this disruption 15

x [kpc] 100 0 100 100 0 100 100 0 100 100 0 100 100 0 100 − − − − − 0.00 Gyr 0.48 Gyr 0.95 Gyr 1.43 Gyr 1.90 Gyr 100

0 [kpc] z 100 − 2.38 Gyr 2.85 Gyr 3.33 Gyr 3.81 Gyr 4.28 Gyr 100

0 [kpc] z 100 −

100 0 100 100 0 100 100 0 100 100 0 100 100 40.76100 Gyr − Dark Matter− − − − Stars 150

100

0 [kpc] z

50 −

100 −

150 −

150 100 50 0 50 100 150 − − − x [kpc]

FIG. 6. The simulation depicted here is Satellite C with no screening and a fifth force coupling only to dark matter with β = 0.2. The large panel shows an image of the stellar (purple) and dark matter (green) streams at the end of the simulation, while the smaller panels above show the evolution over time. The interval between images is 1.5 1016 seconds ( 0.48 Gyr, × ∼ as labelled). The cross and large filled circle respectively indicate the positions of the Milky Way and satellite centres. In the large panel, 50 unbound particles have been randomly chosen from each species, and arrows of the corresponding colour are shown indicating their velocities. An animation of this simulation is included in the Supplemental Material accompanying this article. This figure shows the formation of an asymmetric stellar stream over time. 16

β = 0 0.1 0.2 0.3 0.4 720 Dark Matter ]

◦ 360 0 360 ∆Λ [ −720 − A 720 Stars ]

◦ 360 0 360 ∆Λ [ −720 − 12 6 0 12 6 0 12 6 0 12 6 0 12 6 0 − − − − − − − − − − 720 ]

◦ 360 0 360 ∆Λ [ −720 − B 720 ]

◦ 360 0 360 ∆Λ [ −720 − 6 4 2 0 6 4 2 0 6 4 2 0 6 4 2 0 6 4 2 0 − − − − − − − − − − − − − − − 720 ]

◦ 360 0 360 ∆Λ [ −720 − C 720 ]

◦ 360 0 360 ∆Λ [ −720 − 4 3 2 1 0 4 3 2 1 0 4 3 2 1 0 4 3 2 1 0 4 3 2 1 0 − − − − − − − − − − − − − − − − − − − − 720 ]

◦ 360 0 360 ∆Λ [ −720 − D 720 ]

◦ 360 0 360 ∆Λ [ −720 − 3 2 1 0 3 2 1 0 3 2 1 0 3 2 1 0 3 2 1 0 − − − − − − − − − − − − − − − Orbital Periods

FIG. 7. The longitude difference ∆Λ = Λ Λsat as a function of time for all 4 satellites without screening. Each column − shows a different fifth force coupling from β = 0 to 0.4 in steps of 0.1. Here, Λ is longitude in the orbital plane of the satellite, increasing in the direction of the satellite’s orbit. Lines are drawn for 500 star and 500 DM particles in each simulation, i.e. 1 in 200 particles are randomly sampled. Complementing Figure 6, this figure shows the development over time of the asymmetry of the stellar streams, and the increased magnitude of this effect with β. 17

6 6.4 6.8 7.2 − − − − 40 1.00 0.75

20 0.50 A rscr,MW [kpc] 0.25 0 0 15 0.00 [kpc] 1.00 z 30 45 0.75 20 − 0.50 B β 0.25 0

40 trail − 0.00 0.1 /N 1.00 0.2 40 20 0 20 40 − − lead x [kpc] N 0.75 0.3

≡ 0.4

α 0.50 C FIG. 8. Satellite B’s orbit in its orbital plane, shown for rscr,MW = 0, 8, 24, 50, 85 kpc, and rscr,sat = 0, β = 0.4. The 0.25 dotted circles indicate the position of the screening radius 0.00 in each case. The cross indicates the Galactic centre and 1.00 the filled circle shows the final position of the satellite, i.e. the current observed position of the Sagittarius dwarf galaxy. 0.75 This figure illustrates the effect of a Milky Way screening radius on the satellite orbital shapes. 0.50 D

0.25

0.00 0 25 50 75 100 125 process. For all of our satellites, the streams are indis- rscr,MW [kpc] tinguishable from those in the standard gravity case once rscr,MW exceeds the apocentric distance. FIG. 9. The asymmetry parameter α Nlead/Ntrail, for all ≡ simulations with Qsat = 1. The 4 panels correspond to the 4 We have observed in our simulations interesting signa- satellites and the different textures of line correspond to dif- tures of chameleon gravity other than the stellar asymme- ferent values of β. In each panel, the shaded region indicates try. Examples of these are depicted in Figure 10. First, the radial range of the satellite’s orbit. As with the horizon- in the extreme (high β) fifth force regime, the orbital tal lines in Figure 2, the vertical dashed lines here show the locations of Milky Way screening radii for various values of paths of released stars around the Milky Way differ ap- log fR0 . This figure shows the Milky Way screening radius 10 | | preciably from their progenitor. However, because stars can affect the stream asymmetry. Streams at larger Galac- are released from the progenitor at different times, this tocentric distances are sensitive to larger screening radii, and also means that the liberated stars can be on different therefore weaker modified gravity regimes. Milky Way orbits from each other. If most releases oc- cur at pericentric passages, this can lead to a ‘striping’ effect, with neighbouring undulations of stars on the sky, D. Future Constraints corresponding to streams of stars released at successive pericentric passages. This effect is visible in the upper panel of Figure 10. The later Gaia data releases will likely enable the discovery of stellar streams at large distances from the Secondly, if the satellite itself is fully screened or al- Galactic centre. As shown in Figure 9, such streams are able to probe larger Milky Way screening radii, and most so (i.e. low Qsat), then it orbits the Milky Way more slowly than the dark matter that has been released therefore ‘weaker’, or more screened, regions of modified and inhabits unscreened space. Then, we observe the op- gravity parameter space. posite asymmetry to that of the stars: the dark matter Figure 11 shows α evaluated for all of our simulations is preferentially disrupted into the leading stream rather of satellite D, as a function of rscr,MW, Qsat, and β. As than the trailing stream. This effect is shown in the lower with Figure 9, α is computed in each simulation at the panel of Figure 10. While interesting, this effect is of moment of the satellite’s third pericentric passage. This course not readily accessible to observations. figure illustrates many of our earlier points; increasing β 18

Figure 12 shows how the Milky Way screening radius depends on the parameter fR0. These calculations were 0 performed using the scalar field solver within the f(R) N- body code mg-gadget [56]. mg-gadget uses a Newton- Gauss-Seidel relaxation method to solve the f(R) equa- tions of motion, calculating the scalar fields and fifth 5 − forces everywhere across a given mass distribution or within a given simulation volume. Such methods were

[kpc] first explored in the work of Oyaizu [59], and the subse- y 10 − quent years have seen a proliferation of codes simulating a myriad of modified gravity cosmologies [60–67]. Given a mass model for the Galaxy, this can then 15 − be sampled with a large number of particles, which are in turn fed to mg-gadget to calculate the scalar field profile—and thus the Galaxy screening radius—for a 20 given value of fR0. In each case, we truncate the mass − 5 0 5 10 15 − model at r100, the radius encompassing a region with density 100 times the cosmic critical density. In the case of the Milky Way, this radius has been shown to lie close 10 to the ‘splashback radius’, a reasonable definition for the edge of the Galaxy’s halo [68]. The solid purple line in Figure 12 shows the screen- 5 ing radii for the ‘fiducial’ Milky Way model described in § III A, i.e. the model used throughout our simulations. 0 This illustrates the sensitivity of stream asymmetries as [kpc]

y a probe of chameleon gravity. However, the overall mass of the Milky Way is highly 5 − uncertain, and the primary source of uncertainty is the dark matter halo, particularly in its outer regions [69]. Thus, it is interesting to explore how this predicted sen- 10 Dark Matter − sitivity depends on the overall mass of the Galaxy. The Stars radius r100 for our fiducial model is ∼300 kpc, and the total mass M (including baryons) enclosed within this 5 0 5 10 15 100 − radius is approximately 1.5 × 1012M . Figure 12 addi- x [kpc] tionally shows screening radii calculated for less massive (dotted) and more massive (dashed) Milky Way models. FIG. 10. Top: An image from a simulation of Satellite A, In these models the scale density ρ0 of the dark matter β = 0.4, Qsat = 1, and rscr,MW = 10 kpc. Bottom: An image halo has been rescaled by factors of 0.75 and 1.25 re- from another simulation of Satellite A, β = 0.1, Qsat = 0, and spectively. These rescalings still lead to sensible values rscr,MW = 4 kpc. Animations of the 2 simulations depicted for the halo concentration, and correspond to masses of in this figure are included in the Supplemental Material ac- 12 companying this article. This figure shows some interesting M100 = 1.1 and 2.0 × 10 M (note the overall masses signatures of screened modified gravity other than the stellar are not rescaled by exactly 0.75 and 1.25, because r100 asymmetry we have discussed in previous figures. changes along with the density normalisation). As expected, at fixed fR0, increasing (decreasing) the mass leads to an expansion (reduction) of the screening ra- increases the magnitude of the asymmetry, but the asym- dius. These masses bracket a large range of reasonable metry is reduced by increasing rscr,sat (reducing Qsat) or estimates for the Milky Way mass, and so the region of rscr,MW. In the β = 0.4 case, approximately compara- the figure enclosed by these lines should in principle in- ble to f(R) gravity, the asymmetries grow large when clude the ‘true’ screening radius of the Milky Way in an rscr,MW . 100 kpc, assuming the satellite is fully un- f(R) Universe. screened (Qsat = 1). Notably, this lies between the apoc- One additional caveat, however, is that this treatment entre and pericentre of the satellite’s orbit. Most tidal has ignored the environmental contributions to the scalar disruption occurs at pericentric passage, but here there field by the Local Group. As a first approximation of is still enough disruption outside the screening radius, this effect, the green lines in Figure 12 show the Milky and sufficient numbers of leading stars lagging behind Way screening radii when a mass distribution for M31 is the satellite, that a large asymmetry develops anyway. added to the mg-gadget input. The model used for M31 We can again use Hu-Sawicki f(R) gravity to give an is identical to our fiducial Milky Way model, centred at indication of the kinds of constraints attainable here. (−380, 620, −280) kpc in Galactocentric coordinates [70]. 19

β = 0.1 0.2 0.3 0.4 1.0 0

0.2 0.8 trail

0.4 0.6 /N sat lead Q 0.6 0.4 N ≡ 0.8 0.2 α 1 0.0 0 30 60 90 120 1500 30 60 90 120 1500 30 60 90 120 1500 30 60 90 120 150

rscr,MW [kpc] rscr,MW [kpc] rscr,MW [kpc] rscr,MW [kpc]

FIG. 11. The asymmetry parameter α Nlead/Ntrail for the unbound stellar particles in all simulations of satellite D with ≡ screening, shown here as a function of Qsat and rscr,MW, with different panels corresponding to different values of β. This figure shows the effects of varying all of our parameters on the stream asymmetries.

300 seems likely that environmental screening will remain a subdominant eﬀect. 250 We see from Figure 12 that Satellite D is able to probe the region log10 |fR0| & −7.2. However, if the satellite itself is partially screened, the sensitivity is greatly re- 200 duced. It is natural therefore to wonder about the degree to which a satellite would be screened at these values of [kpc] 150 fR0 and this region of the Milky Way’s halo. MW

, Figure 13 shows the scalar ﬁeld proﬁle around the −7 scr Milky Way for f = −10 , again inferred using mg-

r R0 100 Exc. M31 gadget. A Hernquist sphere identical to Satellite D has Inc. M31 been inserted at Galactocentric (X = 100,Y = 0,Z = 50 Fiducial MW Model 100) kpc. There is a clear screened region in the centre Lighter MW of the Milky Way halo, with rscr,MW ≈ 100 kpc. The Heavier MW satellite is also partially screened, with a screened re- 0 gion of rscr,sat ≈ 0.6 kpc at its centre. This corresponds 9.0 8.5 8.0 7.5 7.0 6.5 6.0 to Q . . Comparing to Figure 11, the suggestion − − − − − − − sat = 0 8 log10 fR0 is that in an f(R) Universe, this satellite would provide | | very asymmetric streams. This is demonstrated in Figure 14, which shows a simulation with a similar setup: Satel- FIG. 12. The disc-plane Milky Way screening radius as a lite D with β = 0.4, rscr,MW = 105 kpc, and Qsat = 0.8. function of log fR0 . The solid lines show screening radii 10 | | for the ‘fiducial’ Milky Way model, i.e. the model described The left-hand panel shows the stream, while the right- in Section III A. Meanwhile, the dotted and dashed lines rep- panel shows a more sophisticated observable signature resent Galaxy models in which the scale density ρ0 of the dark than the asymmetry parameter: the cumulative number matter halo has been rescaled by factors of 0.75 and 1.25 re- function of stars in each stream as a function of longi- spectively. The colours of the lines indicate the environment tude in the orbital plane of the satellite. The difference around the Galaxy, where violet lines are for an isolated Milky in the two curves is rather striking, and should be clearly Way model, while the green lines additionally incorporate the discernible in the data. contribution of M31. In every case, the screening radius is cal- The examples shown in Figures 13 and 14 serve as culated with mg-gadget. proof of concept, demonstrating that stellar streams in the outer reaches of our Galaxy’s halo are a sensitive probe of modified gravity. The observation of highly sym- There is a systematic upward shift in the Milky Way metric streams at large Galactocentric distances would screening radii at all fR0 values except at log10 |fR0| . rule out sizeable fifth forces that couple differently to −8.5, where rMW −M31 > λC . However, the magnitude dark matter and stars in the outskirts of the Milky of this shift is typically rather small, on the order of a few Way. This in turn would provide sensitive constraints kpc. This may increase with a more sophisticated model on screened modified gravity theories. For instance, of the Local Group incorporating M33 and various other looking at Figure 12, symmetric streams at distances of −7.5 smaller galaxies, as well as the smooth intervening mass ∼ 150 − 300 kpc would require |fR0| ∼ 10 or even distribution. However, as the Milky Way and M31 are 10−8 to avoid sizeable fifth forces in that radial range. by far the most massive members of the Local Group, it This would be among the tightest constraints on f(R) 20

2 Σ[M /pc ] fR/f¯R0 37 39 5 4 3 2 1 0 10 10 10− 10− 10− 10− 10− 10

10 100 100

5 50 50

0 0 0 [kpc] [kpc] z z

50 50 − − 5 − 100 100 − − 10 100 50 0 50 100 100 50 0 50 100 10 5 0 5 10− − − − − − − x [kpc] x [kpc] x [kpc]

FIG. 13. Left: Edge-on particle density of Galaxy+satellite system fed to mg-gadget to calculate the scalar field profile. The 7 location of the satellite is indicated by the inset box. Middle: Scalar field profile for fR0 = 10− across the same system. − The Milky Way’s screened region is clearly discernible, while the satellite also has a small central screened region, shown in the right-hand panel, which shows a magnified image of the scalar field profile in a 20 kpc region centred around the satellite.

Dark Matter 2000 150 Stars

Trailing stream 1750 100 1500

50 1250 [kpc]

z 1000 0

Leading stream 750 Number of stars 50 − 500

100 250 − 0 150 100 50 0 50 100 0 50 100 150 200 250 − − − x [kpc] ∆Λ [◦] | |

FIG. 14. Left: An image of a simulation of Satellite D, with rscr,MW = 105 kpc, Qsat = 0.8, β = 0.4. The dotted circle shows the location of the Milky Way screening radius, while the cross and filled circle show the locations of the Milky Way and satellite centres respectively. The arrow shows the current direction of motion of the satellite. Right: Cumulative number of stars in either stream, as a function of longitude in the instantaneous orbital plane of the satellite. An Animation of this simulation is included in the Supplemental Material accompanying this article. This figure, taken together with Figure 13, shows that f(R) 7 gravity with fR0 10− should give a clear observational signature in stellar streams between 100 and 200 kpc. ∼ − gravity achievable to date. However, we caution that en- So, given the observation of a large number of symmetric vironmental screening of the satellite may play a more streams, and if there is little environmental screening by significant role at these levels, but Figure 11 suggests the Local Group, then constraints down to these levels that only if the satellite is fully screened does the signal are feasible. disappear entirely. Even when Qsat = 0.1, i.e. 90% of On the other hand, observations of highly asymmetric the mass is screened, there is still an appreciable signal. streams would strengthen the case for screened modified 21 gravity theories. It should be noted, however, that mild of β give large asymmetries, but these are reduced with asymmetries can arise due to dynamical effects. Indeed, increasing rscr,MW and rscr,sat. an asymmetry between the leading and trailing streams Our simulations – the most comprehensive to date for is expected from Eq. (27). This may be compounded the formation of tidal streams under chameleon gravity – by dynamical interactions with dark subhaloes or other have revealed further interesting effects. First, the trail- satellites [71], asymmetries in the stellar populations in ing stellar stream may become detached from the dark the progenitor satellite [72, 73], effects of the Galactic matter progenitor if all the stars are exhausted by ear- bar [54, 74, 75] and regions of chaos in the Galactic poten- lier pericentric stripping (cf. [15]). As an example, this tial [76]. Such effects would have to be carefully weighed effect is visible in Figure 7 and occurs for low mass satel- before a modified gravity interpretation could be seri- lites in the extreme fifth force regime. Second, prominent ously considered for such observations. striations in the stellar trailing tail may exist if stars are stripped at repeated pericentric passages by a strong fifth force. Thirdly, if the satellite is fully screened, its orbital VII. CONCLUSIONS frequency is lower than that of its associated dark matter. This leads to strong asymmetries in the dark mat- We have investigated possible imprints of chameleon ter distribution, which is preferentially liberated into the gravity on stellar streams from dwarf galaxies around the leading tidal tail. −7 Milky Way. While canonical chameleon theories are uni- Taking Hu-Sawicki f(R) gravity with fR0 = −10 as versally coupled, an effective violation of the equivalence an example, we derive a Milky Way screening radius of principle (EP) arises because of the self-screening of main around 100 kpc. A massive dwarf spheroidal galaxy at sequence stars, as noted by Hui et al. [34]. Consequently, a distance of ≈ 150 kpc – such as Fornax – would be stars are preferentially stripped from the progenitor into partially screened but would nonetheless produce highly the trailing stream rather than the leading stream. asymmetric streams under tidal disruption. We have created a restricted N-body code (smoggy; The ratio of the cumulative number function of stars made publicly available [77]), and used it to simulate the in the leading and trailing stream as a function of longi- formation of tidal streams from progenitors with a variety tude from the satellite is computable from simulations, of masses and Galactocentric distances. We considered measurable from the observational data and can provide a range of modified gravity scenarios (coupling strength, a direct test of theories with screening mechanisms like Milky Way screening level, satellite screening level) in chameleon gravity. The later Gaia data releases may lead each case. to discoveries of stellar streams at distances & 100 kpc As found by Kesden and Kamionkowski [13, 14], an from the Galactic centre. These streams will be a sen- EP-violating fifth force that couples to dark matter but sitive probe of modified gravity; such highly asymmetric not baryons causes asymmetries to develop in stellar streams at these distances would be tell-tale signatures streams with dark matter-dominated progenitors. The of modified gravity. stars are preferentially disrupted via the outer Lagrange On the other hand, if the data uncover a number of points into the trailing streams. We have corrected very symmetric streams, then constraints down to the and augmented the analytic calculations of Ref. [13] for −8 level of fR0 ∼ −10 —the tightest constraints to date— point masses so that they are also applicable to extended could be attainable if the screening of the satellite and Galactic mass distributions like isothermal spheres. The other nuisance parameters are carefully taken into ac- effect of these changes is to make the test more sensitive count. Also, our assumption that the Compton wave- to EP-violating fifth forces. For the most massive dwarf length is much larger than relevant length scales begins spheroidals, like the Sagittarius or Fornax, the criterion to break down at such values of fR0, and Yukawa sup- given in Eq. (29) suggests values of β2 −3 can be −8 & 10 pression will become appreciable below fR0 ∼ −10 . probed. For the smallest dwarf spheroidals such as Segue Of course, the investigation need not be limited to Hu- 5 2 −4 1 with a mass of 6×10 M , then values of β & 10 are Sawicki f(R) gravity. Sensitive constraints will also be in principle accessible. As a rule of thumb for a satellite attainable in the general chameleon parameter space, and with mass m at a location enclosing a Milky Way mass we merely use f(R) gravity as a fiducial theory. M, the form of the criterion suitable for a flat rotation This desirability of streams at large distances sug- curve galaxy is gests another interesting avenue for exploration: stel- 2/3 lar streams around other galaxies. Streams have already 2 −5/3 m β & 2 . (46) been observed around other galaxies (e.g. [78, 79]), and it M seems likely that future wide-field surveys such as LSST This asymmetry also occurs in the chameleon context, [80] will uncover more streams at large distances from when screening radii are introduced to the Milky Way their host galaxies. This, combined with a calculation of and satellite, and with stars self-screening. The magni- the host galaxy screening properties (e.g. via the screen- tude of the asymmetry depends on the coupling strength ing maps of Ref. [81]) could also be a sensitive probe of β, the Milky Way screening radius, as well as the de- screened modified gravity. gree of screening of the stream progenitor; large values Finally, we note that other screened modified gravity 22 theories can also be probed with stellar streams. For in- referee for reviewing and suggesting various improve- stance, the symmetron screening mechanism [82, 83] has ments to the manuscript. a simple density threshold as a screening criterion. Con- APN thanks the Science and Technology Facilities sequently, there will necessarily be a region of parameter Council (STFC) for their PhD studentship. HZ ac- space in which the stars are screened, but the surround- knowledges support by the Kavli Foundation. ACD ing diffuse dark matter component is not. In this regime, acknowledges partial support from STFC under grants stream asymmetries will also be present and are worthy ST/L000385 and ST/L000636. This work used the of future investigation. DiRAC Data Analytic system at the University of Cambridge, operated by the University of Cambridge High Performance Computing Service on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This ACKNOWLEDGMENTS equipment was funded by BIS National E-infrastructure capital grant (ST/K001590/1), STFC capital grants The authors would like to acknowledge Matt Auger, ST/H008861/1 and ST/H00887X/1, and STFC DiRAC Vasily Belokurov, Sergey Koposov, Jason Sanders, and Operations grant ST/K00333X/1. DiRAC is part of the Denis Erkal for helpful discussions, and the anonymous National E-Infrastructure.

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