Universal Predictions of Screened Modified Gravity on Cluster Scales

A&A 583, A123 (2015) Astronomy DOI: 10.1051/0004-6361/201526611 & c ESO 2015 Astrophysics

Universal predictions of screened modiﬁed gravity on cluster scales M. Gronke1, D. F. Mota1, and H. A. Winther2

1 Institute of Theoretical Astrophysics, University of Oslo, Postboks 1029, 0315 Oslo, Norway e-mail: [email protected] 2 Astrophysics, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UK Received 27 May 2015 / Accepted 31 August 2015

ABSTRACT

Modified gravity models require a screening mechanism to be able to evade the stringent constraints from local gravity experiments and, at the same time, give rise to observable astrophysical and cosmological signatures. Such screened modified gravity models necessarily have dynamics determined by complex nonlinear equations that usually need to be solved on a model-by-model basis to produce predictions. This makes testing them a cumbersome process. In this paper, we investigate whether there is a common signature for all the different models that is suitable to testing them on cluster scales. To do this we propose an observable related to the fifth force, which can be observationally related to the ratio of dynamical-to-lensing mass of a halo, and then show that the predictions for this observable can be rescaled to a near universal form for a large class of modified gravity models. We demonstrate this using the Hu-Sawicki f (R), the Symmetron, the nDGP, and the Dilaton models, as well as unifying parametrizations. The universal form is determined by only three quantities: a strength, a mass, and a width parameter. We also show how these parameters can be derived from a specific theory. This self-similarity in the predictions can hopefully be used to search for signatures of modified gravity on cluster scales in a model-independent way. Key words. large-scale structure of Universe – dark energy – galaxies: clusters: general – galaxies: kinematics and dynamics – gravitation

1. Introduction – Is it possible to remap observables within one model and/or between different models; i.e., are SMGs self-similar in A possible solution for the observed acceleration of our Universe some way? is that general relativity (GR) is not the correct theory to describe gravity on large scales. Over the past decade, a large number of Previous work has been carried out in this direction. Brax et al. alternative gravity theories have been proposed (Clifton et al. (2012a,b) have developed a theoretical framework in which it 2012). Many of these models cannot alleviate the existing, or is possible to describe certain SMGs with two free functions. they introduce new problems; nevertheless, the idea that cosmic Using this framework Winther & Ferreira(2015) propose an ap- acceleration has a gravitational origin persists. proximate scheme to perform fast N-body simulations of SMGs High precision tests of gravity indicate that general relativ- by combining linear theory with a screening factor obtained ity is the correct theory for describing gravitational physics on from studying spherical symmetric systems. Also, Mead et al. Earth and in the solar system (Bertotti et al. 2003; Will 2006; (2015) have recently published a technique for remapping Λ cold Williams et al. 2004). This places strong constraints on any the- dark matter (CDM) power spectra to the Chameleon f (R) model ory that seeks to modify general relativity. Thus, if gravity is with ∼3% accuracy. Complementary to that, we want to base modified on large scales – giving rise to cosmic acceleration – this work on an observationally focused approach. Specifically, then a way of suppressing the modifications of gravity in the in this paper we focus on the behavior of screened modified well-tested regimes is required (Khoury 2013). A key feature gravity theories inside and in the vicinity of clusters of galax- of screening mechanisms currently being considered is that they ies. In general we expect the strongest signal of modified grav- are fundamentally nonlinear, thereby making the study of their ity to be found in cosmological voids, but clusters are easier to cosmological effects challenging (Vainshtein 1972; Khoury & work with theoretically and observationally, and they remain an Weltman 2004a; Hinterbichler & Khoury 2010). important observational probe of gravity on intermediate scales As previously argued in Gronke et al.(2014, 2015), devia- (Baker et al. 2015). tions from GR occurring in screened-modified gravity theories To answer the questions above, we have structured this pa- (SMGs) can generally be grouped into three categories: no devi- per as follows. In Sect.2, we give a general introduction to the ations (fully screened regime), maximum deviations (unscreened topography of SMGs. We also picked three models as examples, regime), and the intermediate deviations (partially screened which we describe in more detail. This includes the respective regime). As a natural continuation of this idea, the main ques- solving method for halo density profiles. In Sect.3, we present tions we would like to address in this paper follow: the numerical results of these three models. In addition, we study the effects of varying the model parameters on the observables – How do the extents and occurrences of the three regimes and try to empirically find a rescaling that counterbalances this vary with the screened modified gravity model and its variation. Also in this section, we develop a general analytic parameters? rescaling method and compare it to our numerical results. In

Article published by EDP Sciences A123, page 1 of 12 A&A 583, A123 (2015)

Sect.4, we discuss our results and possible observational im- Table 1. Overview of screened modified gravity models. plications. We conclude in Sect.5. −2 ≡ A Throughout this work we use (Ωm0, H0) = (0.3, 0.7), MPl Model Screening type N-body code 2 2 8πG, ρc = 3H MPl, ρm = Ωmρc and denote values today with a f (R) HSa (1) Chameleon (E), (I), (MGG), (O) subscript zero. Dilaton (1) Chameleon-like (E) Symmetron (1) Chameleon-like (E), (I) DGP (2) Vainshtein (E), (S), (K) 2. Methods Galileon (2) Vainshtein (E), (I) k-mouflage (2) Vainshtein-like s– In this section we give a brief introduction to screened modified gravity theories. We present a few example models in more detail Notes. (a) Hu & Sawicki(2007) f (R). (A) (E) ECOSMOG (Li et al. 2012); and then give a general framework for considering a large class (I) ISIS (Llinares et al. 2014); (K) Khoury & Wyman(2009); (MGG) of models. In the end we discuss the numerical implementation MG-GADGET (Puchwein et al. 2013); (O) Oyaizu(2008); (S) Schmidt for solving for the field profiles of the models for the case of a et al.(2009), Schmidt(2009). Navarro-Frenk-White (NFW) density profile.

2. Screening due to derivatives of the scalar field ∂ϕ and/or ∂2ϕ. 2.1. Modified gravity theories In this case, the screening is caused by the relation between 1 2 When we speak about modification of gravity or modified grav- the kinetic term of Lϕ, i.e., − 2 (∂ϕ) , and the rest of the scalar ity theories, we generally mean the addition of terms to the field Lagrangian. A popular mechanisms incorporating this Einstein-Hilbert action. Specifically, in scalar-tensor theories idea is the Vainshtein mechanism (Vainshtein 1972), where 2 one (or multiple) additional scalar field is added with their re- screening depends on the value of ∂ ϕ, and the k-mouflage mechanism (Babichev et al. 2009; Brax & Valageas 2014a), spective Lagrangian Lϕ, and the general action reads as where the magnitude of (∂ϕ)2 decides whether a region is Z √  M2  screened or not. Particular SMGs employing the Vainshtein 4  Pl 2  (i) S = d x −g  R + Lϕ(ϕ, ∂ϕ, ∂ ϕ) + Sm(ψ , g˜µν), (1) Mechanism are massive gravity (de Rham 2014), Galileons 2 (Nicolis et al. 2009), and the DGP model (Dvali et al. 2000). (i) 2 where the matter fields ψ couple to the metricg ˜µν ≡ A gµν, and A selective overview over some models and their employed R is the Ricci scalar. We limit ourselves to one scalar field and a screening mechanism is given in Table1. 1 conformal coupling to matter . Since the scalar field is coupled Over the past decade there have been several studies of mod- to matter, i.e., A = A(ϕ), an extra fifth force arises that is in the ified gravity on nonlinear scales, often using N-body simula- non-relativistic limit, and per unit mass is given by tions. For theories that screen via the first method described above (chameleon-like screening), there have been studies of −∇ Fϕ = log A. (2) f (R) gravity (Li et al. 2012; Llinares et al. 2014; Puchwein et al. The predictions of general relativity have been confirmed to 2013; Zhao et al. 2011a; Oyaizu 2008; He et al. 2014; Lombriser high accuracy in the laboratory and in the solar system, which et al. 2012), other chameleon models (Brax et al. 2013), the severely constrains any new such force in our (Earth’s) local en- Symmetron (Davis et al. 2012; Llinares et al. 2014; Brax et al. vironment (Bertotti et al. 2003; Will 2006; Williams et al. 2004). 2012c), and the environment-dependent dilation model (Brax If the fifth force is weak close to earth, then one would naively et al. 2011, 2012c). For models that employ the second screen- expect it to also be weak on cosmological scales. However, in ing method (Vainshtein-like screening), there have been studies the past decade several theories have been proposed where one of the DGP model (Schmidt et al. 2009; Schmidt 2009; Chan is able to dynamically suppress the effect of such a fifth force in & Scoccimarro 2009; Khoury & Wyman 2009; Li et al. 2013b; high density environments (relative to the cosmic mean) and still Falck et al. 2015, 2014), Galileons (Barreira et al. 2013; Li et al. have interesting cosmological signatures. If this is the case, then 2013a), and k-moflage (Brax & Valageas 2014b). This is by no a theory is said to possess a screening mechanism (for reviews means a complete list. see, e.g., Clifton et al. 2012; Khoury 2010; Joyce et al. 2015). In spite of the differences between the different screening The action Eq. (1) allows a loose mathematical classification mechanisms and the theories employing them, some common of possible screening mechanisms: quantities can be defined. In particular, we focus on the enhancement of the gravitational force 1. Screening due to the scalar field value ϕ. If Lϕ possesses a potential V(ϕ), it is possible to suppress the fifth force thanks γ ≡ Geff/G − 1 (3) to the value of the scalar field moving inside the potential. with its theoretical maximum value γ , given a specific modi- Physically, this can occur if the coupling strength depends max fied gravity model. As described in Sect. 4.2, γ can be related to on the local environment as is the case in the Symmetron the ratio of the dynamical to the lensing mass. mechanism (Hinterbichler & Khoury 2010; Hinterbichler et al. 2011). Another possibility is that the range of the fifth force depends on the environment, which is often called the 2.2. Example models Chameleon mechanism (Khoury & Weltman 2004b; Khoury & Weltman 2004a; Mota & Shaw 2007). A frequently con- The screened modified gravity models that we considered pos- sidered SMG possessing the Chameleon screening is the Hu sess three intrinsically different screening mechanisms: the & Sawicki(2007) f (R) formulation. Vainshtein (Vainshtein 1972) screening where the derivatives of the scalar field play a major role; the Symmetron (Hinterbichler 1 Disformal coupling allows for additional screening mechanisms et al. 2011a,b) screening, where the coupling strength is envi- (Koivisto et al. 2012). ronment dependent and goes to zero in high density regions; and

A123, page 2 of 12 M. Gronke et al.: Universal predictions of screened modified gravity on cluster scales the Chameleon (Khoury & Weltman 2004a,b) screening, where 2.2.2. Hu-Sawicki f(R) model the range of the fifth force approaches zero for regions with high matter density. As an example of the latter, we focus on the Hu & The f (R) gravity models are a group of modified gravity the- Sawicki(2007) f (R) model, which incorporates the Chameleon ories where the Ricci scalar R in the Einstein-Hilbert action is screening mechanism. replaced by a generic function R + f (R). By applying a confor- In the following, we do not present the example models in mal transformation, the f (R) action can be brought in the form of Eq. (1) (see, e.g., Clifton et al. 2012; Brax et al. 2008). This great detail. Instead, we refer the reader to our previous work 2 (Gronke et al. 2014, 2015), where we stated the full equations transformation translates f (R)-gravity into a scalar tensor the- for the Symmetron and the Hu & Sawicki(2007) f (R) model ory. The theory screens via the chameleon mechanism. or – even better – to excellent reviews, such as those by Clifton The fifth force is given by et al.(2012); Khoury(2013); Joyce et al.(2015); and, Koyama 1 (2015). F = − ∇ f (12) ϕ 2 R

2.2.1. Symmetron model where fR ≡ d f (R)/ dR is the scalar field. In a FLRW background spacetime and in the quasi-static limit (Noller et al. 2014), the In the Symmetron model (Hinterbichler & Khoury 2010; field-equation becomes Hinterbichler et al. 2011), both the coupling function and the ! potential are symmetric around ϕ = 0 are given by 2 2 2 ρm 2 2 ∇ fR = − ΩmH0 a − 1 + a H0 Ωm ρm ϕ2 A(ϕ) = 1 + (4)  ! !n+1  2  ΩΛ fR0 −3 ΩΛ  M ×  1 + 4 − a − 4  , (13) 1 1  Ωm fR Ωm  V(ϕ) = − µ2ϕ2 + λϕ4. (5) 2 4 where n and fR0 are dimensionless model parameters. The max- Here, the prefactors M, µ and λ can be rewritten as imum enhancement of the gravitational force, i.e. the enhancement when there is no screening, is simply 2 2 2Ωm0ρc0L M = , (6) 1 a3 γ = 2β2 = , (14) ssb max 3 1 µ = √ , (7) 2L so gravity can be enhanced by up to 33%. How the screening 3 mechanism works is explained in more detail in Sect. 2.3. a MPl λ1/2 = √ ssb , (8) 8L3Ω ρ β m0 c0 2.2.3. DGP which leaves the Symmetron parameters to be the scale factor at Dvali et al.(2000, DGP) model is an example of a braneworld symmetry breaking assb, the range of the fifth force in vacuum L, model where we are confined to live in a four-dimensional brane and the fiducial coupling β. that itself is embedded in a five-dimensional spacetime. The In a Friedmann-Lemaître-Robertson-Walter (FLRW) back- DGP model can give rise to self-acceleration of the Universe ground and in the quasi-static limit (Noller et al. 2014), the field without an explicit cosmological constant, but this branch has equation becomes problems with instabilities. We take the normal branch (n)DGP model as our main example. To get accelerated expansion in this 2  3  a  a ρm  branch, we need to add dark energy, and for simplicity, we as- ∇2ϕ˜ = ϕ˜ ssb − ϕ˜ + ϕ˜3 (9) 2  3  2λ a ρm sume that the background expansion is the same as in ΛCDM. 0 This choice is not expected to change any of our conclusions. √ Since it is a higher dimensional theory, the DGP model only whereϕ ˜ = λϕ/µ. The fifth force is given by fits into the formalism of Eq. (1) as an effective theory. However, ϕ∇ϕ the modifications of gravity in this model can also be described F = · (10) as a fifth force given by ϕ M2

2 2 1 In regions of high ambient matter density (ρm µ M = Fϕ = ∇ϕ, (15) 3 2 ρc0/assb), the field will reside close to φ = 0, and the fifth force will be screened. In low-density regions, on the other hand, the where the evolution of the scalar field ϕ, the co-called brane- fifth force can be in full operation. In the Symmetron model, bending mode, is determined by the field equation3 the maximum enhancement of the gravitational force, γmax, is given by 2 2 r ΩmH ∇2ϕ c ∇ϕ 2 − ∇ ∇ ϕ 2 0 δ , + 2 ( ) ( i j ) = m (16) " 3# 3βDGP(a)a aβDGP(a) 2 assb γmax = 2β 1 − . (11) √ a 2 2 βϕ/M g˜µν = A (ϕ)gµν where A(ϕ) = e Pl with β = 1/ 6. 3 For the self-accelerating branch, the same expression holds but with Since β is a free parameter, the model can, in principle, give rise q 1 −3 2 2 −1 rc → −rc. In this case, H(a)/H0 = + Ωma + (4r H ) with to an arbitrarily large (or small) deviation. How the screening 2rc H0 c 0 1 mechanism works is explained in more detail in Sect. 2.3. Ωm = 1 − . rcH0 A123, page 3 of 12 A&A 583, A123 (2015) where The two functions are, in the case of the Symmetron model, ! given by H(a) H˙ β a r H · DGP( ) = 1 + 2( c 0) 1 + 2 (17) r H0 3H m(a) β(a) a 3 = = 1 − ssb ; (27) The maximum value of the gravitational force in this model is m0 β0 a

1 i.e., they follow the same evolution. The parameters β0 and m0 γmax = · (18) are related to the model parameters described in the previous 3βDGP 1 section via β0 = β and m0 = L . 1 f R For high values of rc, we have βDGP(a = 1) ∝ rc, so γmax ∝ . For the Hu-Sawicki ( ) model, on the other hand, we have rc The larger rc, the weaker the effect of the fifth force. 1 As also stated in Schmidt(2010), in spherical symmetry we β(a) = √ (28) can integrate Eq. (16) directly to obtain 6 R r and 0 2 0 !2 2 δ r2 dr ϕ 2rc ϕ ΩmH0 0 m √ + = · (19) −3 !n/2+2 r 3β (a)a2 r aβ (a) r3 H0 Ωm + 4ΩΛ Ωma + 4ΩΛ DGP DGP m(a) = p , (29) | f |(n + 1) Ωm + 4ΩΛ This translates to a solution for the fifth force profile given by R0 −3(n/2+1) 1 0 " √ # which means that β is constant, and m(a) ∝ a for small a. ϕ 1 2(−1 + 1 + (r)) 2 Note that the nDGP model does not map on to this parametriza- γ = 0 = (20) Φ 3βDGP (r) tion except for the case of linear perturbations where it would p where correspond to m(a) = 0 and β(a) = 1/ 6βDGP. When the scalar field, in a region of density ρ is close to the R r 2 δ r2 r 8(rcH0) Ωm 0 m d minimum of the effective potential (this corresponds to the fully (r) = · (21) ≈ 1/3 3β2 (a)a3 r3 screened regime), we have ϕ ϕ(aρ) where aρ = aenv(ρm/ρ) . DGP The screening condition for an object with Newtonian poten- 4 We have two regimes. For 1 (large r), we get γ = 1 tial ΦN in a region of density ρ can therefore be written as 3βDGP " # and gravity is maximally modified. On the other hand, for 1 |ϕ(aρ) − ϕ(aenv)| (small r), we have (aρ, aenv) = min , 1 . (30) 2β(aenv)MPlΦN 1 2 1 γ = √ , (22) An object is screened whenever (aρ, aenv) 1. Here, aenv = 3βDGP 3βDGP 1/3 a(ρm/ρenv) where ρenv is the density of the environment the and the fifth-force is screened. object is located in. The min condition ensures that we get the correct value, = 1, in the nonscreened regime. The screening condition is related to γ via 2.3. The unified {m(a), β(a)} description of chameleon-like 2 models γ = 2β (aenv)(aρ, aenv). (31) Brax et al.(2012b,a) show that any scalar-tensor theory that We should note that this is a very rough analytical approxima- screens using the Chameleon and/or Symmetron mechanism can tion, but it is usually good enough for order-of-magnitude es- be described uniquely by the evolution of the mass and matter timates. To get accurate predictions, we need to solve the field coupling along the cosmological attractor. Thus parametrizing equation numerically, as discussed in the next section. these two functions, which have a clear intuitive meaning – the range and the strength of the fifth force, respectively – to be ef- fectively parametrized. The mapping between the potential V(ϕ) 2.4. Numerical solving methods and the coupling function A(ϕ) to {m(a), β(a)} has been derived To obtain accurate predictions for the field and force profiles in Brax et al.(2012b) and reads as (in parametric form) in the models we study we need to numerically solve the field Z a 2 equations Eqs. (9) and (13). We solve these equations by dis- 3 β (a) 2 V(a) = V − ρ (a) da (23) cretizing them on a grid, using the NFW density profile for ρm 0 2 am2(a) m MPl ai (see AppendixA for the relevant NFW-equations), and then use 3 Z a β(a) Newton-Gauss Seidel relaxation with multigrid acceleration to ϕ(a) = ϕ + ρ (a) da (24) i M am2(a) m obtain the solution. Pl ai In spherical symmetry the field equation becomes one- Z a 2 3 β (a) 2 d2φ 2 dφ log A(a) = log A(a ) + ρ (a)da, (25) dimensional, φ = φ(r) and ∇ φ = 2 + . For the Symmetron i 2 am2(a) m dr r dr MPl ai model, we implemented a simple fixed-grid multigrid solver for this one-dimensional problem. To do this we write Eq. (9) as where ρ (a) = 3H2 M2 Ω /a3. m 0 Pl m Li = 0 where One of the advantages of using this form is the direct rela- 2  3  tionship between the evolution of linear matter perturbations and a  a ρm(ri)  L = (∇2ϕ) − ϕ ssb − ϕ + ϕ3 , (32) the β, m functions i i 2  i 3 i i  2λ a ρm ! 0 3 2β2(a)k2 δ¨ Hδ˙ a−3δ · 4 This condition is derived under the assumption of a spherical top-hat m + 2 m = Ωm m 1 + 2 2 2 (26) 2 k + a m (a) overdensity.

A123, page 4 of 12 M. Gronke et al.: Universal predictions of screened modiﬁed gravity on cluster scales Symmetron f(R ) nDGP 100 14 ) 13

1 M 10−

12 h / γ ( M 2 11 10− 10

10 log

10 3 9 − 3 2 1 0 1 2 1 0 1 2 1 0 1 10− 10− 10− 10 10 10− 10− 10 10 10− 10− 10 10 R/R200 R/R200 R/R200

9 −1 14.5 −1 Fig. 1. γ vs. R/R200 for NFW halos with masses ranging from 10 M h to 10 M h . The left panel shows the Symmetron solution using −1 −6 (assb, L, β) = (0.7, 1 Mpc h , 1), the central panel the Hu & Sawicki f (R) solution with (| fR0|, n) = (10 , 1), and the right panel the nDGP solution with rc = 1. where subscript i denotes the value at the ith gridpoint and 3. Results

ϕ + ϕ − 2ϕ 2 ϕ − ϕ ∇2ϕ i+1 i−1 i i+1 i−1 In this section we present our results from numerical solutions ( )i = 2 + (33) (∆r) ri ∆r of the ﬁeld equations (Sects. 3.1, 3.2) and the obtained rescaling method for the three example models. The rescaling is ﬁrst is a second-order discretization of the Laplacian. We start by found empirically in Sect. 3.3 and then semi-analytically in the making a guess for the solution, and then we loop through the {m(a), β(a)} formulations in Sect. 3.4. grid updating the solution using

Li 3.1. Force profiles ϕi ← ϕi − · (34) ∂Li/∂ϕi Figure1 shows the radial profiles of γ obtained solving The multigrid technique is used to speed up convergence. For Eqs. ((9), (13), (20)) for the Symmetron, f (R), and, nDGP more details about the methods we have used see Llinares et al. model, respectively (from left to right panels). We display a set (2014). of halo masses for each model using fixed model parameters. The f (R) equation Eq. (13) is stiffer than the Symmetron The differences in shape between the models are striking. equation for low values of | fR0|, and a simple fixed-grid one- dimensional solver does not converge for the whole range of The Symmetron model shows a whole range of solutions: from fully screened in the center for the heaviest clusters to not considered parameters. The main reason for this is that | fR| is required to be strictly larger than zero (but can get very close). screened at all for the lightest ones. The maximum of the fifth force seems to be located ∼1−10R and drops again for larger This means that one has to ensure that | fR| does not become neg- 200 ative in the solving process. To achieve this, we used the already radii. This drop happens due to the fifth force having a finite ∝ 1 −mr well-tested f (R)-module in the ISIS code (Llinares et al. 2014). range, Fφ r2 e . The only modification to the module is the change to a purely The f (R) model presented in the central panel of Fig.1 in- geometrical refinement criterion based on the value of ρm. In to- herits a much more abrupt change from fully screened to not tal, we used 10−15 levels of refinements starting from a base 3 screened in the central halo region. With this particular choice of grid of N = 64 grid nodes in order to get accurate field profiles −6 12 parameters (| f | = 10 , n = 1), halos with a mass of &10 M down to very small radii. The Symmetron is also implemented in R0 seem to be fully screened in the central (R . 0.1R200) region, the ISIS code, and we tested that our Symmetron field profiles whereas in lighter halos, the fifth force stays at its theoretical agree very well with those obtained with our code. maximum 1/3FN . For comparison, in Fig.2 we show the force After having calculated the field profiles, the fifth force can profiles derived using the semi-analytical method described in be calculated by using Eqs. (10) and (12) for the Symmetron Sect. 3.4 for the f (R) (left) and Symmetron (right panel) mod- and f (R) model, respectively. For nDGP the analytical solutions els. The agreement is fairly good around the most significant to the field equation and force profile in spherical symmetry is region R ∼ 1−5R200 where γ peaks. The semi-analytic results given by Eqs. (19) and (20), respectively. This makes it needless are presented in more detail in Sect. 3.4. to solve any differential equations numerically5. The derivation of the analytical solution is shown in AppendixB. In the righthand panel of Fig.1, we show the radial scalar field profile ϕ(r) for a NFW halo in the nDGP model with rc = 1. 5 Generally, this is true for all models that have a derivative shift sym- As is also possible to see from Eq. (B.2), the solution includes metry (like Galileons) and where the field equation is second order. If a curious feature: owing to the concentration-mass relation used this is satisfied, then the equation of motion can be integrated to yield (Eq. (A.3)), heavier halos are less screened in the central region dϕ ∝ an algebraic equation in dr Fϕ. of the halo. Generally, the fifth force is less strong in the nDGP A123, page 5 of 12 A&A 583, A123 (2015) Symmetron f R

100 100

10 1 10 1 Γ Γ

10 2 10 2

10 3 10 3 10 3 10 2 10 1 100 101 10 3 10 2 10 1 100 101

R R200 R R200

Fig. 2. Semi-analytical solutions for γ for the Symmetron (left) and f (R)(right) using the same parameter values (and color coding) as in Fig.1.

Symmetron f(R ) nDGP

100 ¯ γ 1 10−

2 10− 1011 1012 1013 1014 1011 1012 1013 1014 1011 1012 1013 1014 1 1 1 M200(M h− ) M200(M h− ) M200(M h− ) −1 Fig. 3. γ¯ vs halo mass M. The left panel shows the Symmetron curves with (assb, L, β) being (from solid to more interceptions) 0.7, 1 Mpc h , 1 (red), 0.7, 2 Mpc h−1, 1 (blue), 0.3, 1 Mpc h−1, 1 (green), 0.7, 1 Mpc h−1, 2 (purple). The central panel shows the Hu & Sawicki f (R) solution with log10 | fR0| = (−4, −5, −6, −7) (in red, blue, green, and purple, respectively) and the right panel the nDGP solution with rc = (0.5, 1 , 3) (from top to bottom). model and follows a similar functional form that is almost inde- considered in Clampitt et al.(2012) for the symmetron model. 6 pendent of mass . The outer cutoff was fixed to xcut = 10. This value was chosen so that w(xcut) is roughly one-tenth of the maximum weight, meaning that not too much information is lost. As discussed in 3.2. Profile variation with halo mass Sect. 4.2 (see Eq. (55)),γ ¯ can be related to the ratio of the dy- To capture the change of the fifth force profile with halo mass, namical mass to the lensing mass. we chose to show the mass-weighted mean of the fifth force Figure3 shows ¯γ versus the halo mass. However, this mea- sure is not unique. Instead it is also possible, for example, to use Z xcut γ¯ = dxγ(x)w(x) (35) a difference between the force in the center and in the outskirts 0 of the halo. We discuss the advantages and caveats of usingγ ¯ as where w(x) is the normalized mass fraction w(x) dx ≡ an “observable” in Sect. 4.2. The lefthand panel of Fig.3 shows three Symmetron solu- dM/M(

A123, page 6 of 12 M. Gronke et al.: Universal predictions of screened modiﬁed gravity on cluster scales

Fig.3) possess an upper limit at the already mentioned value of 1.0 one-third. For lighter halosγ ¯ falls off rapidly. In the righter-most panel of Fig.3, we show the variation in ¯γ with mass for the 0.8 nDGP model for three different values of rc. Overall, the impact of the fifth force is much weaker than in the other models. As discussed in Sect. 3.1, the variation in the fifth force with halo 0.6 mass is only due to the mass-concentration relation of the NFW max ¯ halo. This feature manifests itself in Fig.3 in the slight increase γ/γ 0.4 unscreened screened inγ ¯ with increasing halo mass. 0.2 3.3. Empirical rescaling 0.0 4 2 0 2 4 To find a rescaling that maps the various characteristics pre- 10− 10− 10 10 10 sented in the last section onto one single curve, we need to M200/µ200 rescale the x and the y axes of Fig.3. The latter is naturally normalized to the theoretical possible maximum of the gravita- Fig. 4. γ¯ versus M/µ200 for the Symmetron model. The solid colored lines indicate the same model parameters as in Fig.3. The gray lines tional enhancement γmax. The rescaling of the halo mass is less are an additional 124 randomly chosen Symmetron parameters. µ for trivial. Therefore, we simply define a characteristic mass µ200 in 200 the following way: the Symmetron is given by Eq. (37). The black dashed lines indicate the point defining µ200 in Eq. (36), and the gray arrows point to the screened 1 and unscreened regimes. γ¯(µ200) ≡ γmax (36) 2 1.0 whereγ ¯ is still the mass-weighted average of the enhancement of the fifth force as defined in Eq. (35) In general µ200 is a function 0.8 of the various model parameters. Halos with mass much higher than µ200 are in the fully screened regime, whereas halos with 0.6 mass much lower than µ are in the unscreened regime.

200 max In this section we find the functional form of µ200 for ¯ γ/γ 0.4 each model merely empirically from our numerical solutions. unscreened screened However, in the next section, we derive semi-analytic predictions for a large class of scalar tensor theories. 0.2 This means that, by looking at Fig.3, it is possible to read off several values of µ . For example, for the f (R) model with 200 0.0 −6 13 −1 4 2 0 2 4 (n, | fR0|) = (1, 10 ) (in green), γ(∼10 M h ) = 1/6 = 10− 10− 10 10 10 13 −1 1/2γmax, i.e., µ200, is in that particular case ∼10 M h . M200/µ200 For the Symmetron model, we found µ200 to be well fit by Fig. 5. γ¯ versus M/µ200 for the Hu-Sawicki f (R) model. The solid col- !3 L ored lines indicate the same model parameters as in Fig.3. The gray µSymmetron × 10 M −1 × a−4.5, 200 = 2 10 h −1 ssb (37) lines are an additional 5 values of fR0. µ200 for this model is given by Mpc h Eq. (38). and γmax is given by Eq. (11). Figure4 shows the resulting rescaled curves, i.e.,γ/γ ¯ max versus M/µ200. In addition to the four parameter sets already presented in the lefthand panel presented in Fig.3 color-coded accordingly. Also here, the scat- of Fig.3 (keeping the color coding), we plot an additional ter around the mean form is quite small. The second striking 125 curves with randomly chosen parameters. Specifically, we feature the similarity in the overall shape of the curves in Figs.4 −1 and5 – even though the solution stems from two independent drew L ∼ [0.2, 2] Mpc h , assb ∼ [0.2, 0.8], and β ∼ [0.2, 2.4] (all uniformly). modified gravity theories. The main differences between the two It is notable that in spite of this wide parameter range, the models are the slightly different transitions widths and, the fact resulting curves in Fig.4 are very similar. All follow the char- that for M µ200, γ ≈ γmax in the f (R) case, whereas the Symmetron curves are limited to a lower value. acteristic shape – from γ ∼ γmax at M µ200 through γ(M ≈ γ µ200) = γmax/2 to γ ∼ 0 for M µ200 – with very little scatter. Rescaling the ¯-mass relation for the nDGP model (pre- We note, however, that for low masses,γ ¯ does not fully approach sented in the right panel of Fig.3) is not as straightforward. γmax but falls ∼10% short. This is due to the drop in γ for larger Since the curves are nearly constant (and do not possess a char- radii as is visible in the lefthand panel of Fig.1. acteristic drop), no empirical formulation for µ200 is apparent. In Fig.5 we present the results of the same rescaling tech- However, since calculatingγ ¯ in the nDGP model is not computa- nique applied to the f (R) results. In this case the rescaling mass tionally expensive, we can numerically solve Eq. (36) for several is found to be values of rc. This let us fit the empirically found relationship

!1.5 b −rc/a f (R) 13 −1 | fR0| log10 µ200 = A − (rc/a) e , (39) µ = 10 M h × (38) 200 10−6 which yields (A, a, b) = (11.14, 6.72, −0.57). The rescaled and γmax = 1/3. The curves in Fig.5 correspond to the numer- γ¯(M200) solutions obtained with rc = (0.5, 1,..., 20) showed so ical solutions for each combination of n = (1, 2) and | fR0| = little variation that we chose not to explicitly show them. Instead, −8 −8 −7 −7 −4 (10 , 3 × 10 , 10 , 3 × 10 ,..., 10 ), with the ones already we display in Fig.6 the average ¯γ/γmax −M200/µ200 curves (solid

A123, page 7 of 12 A&A 583, A123 (2015)

1.0 3.4. Semi-analytical derivation of µ200 Symmetron in the {m(a), β(a)} formulation f(R) 0.8 In the following, we discuss the origin of the empirical rescalings nDGP found. Using the general framework of the {β(a), m(a)} formula- 0.6 tion described in Sect. 2.3, we can derive an analytic approxima-

max tion for µ200. It turns out that we can do this quite generally, so ¯

γ/γ our results below can be applied to any scalar-tensor theory with 0.4 1 2 a Lagrangian L = 2 (∂ϕ) + V(ϕ) and a conformal coupling to matter. We only focus on a very rough derivation, which can be 0.2 seen as the foundation of a more precise treatment in the future. Before we do this, we should check to what extent our semi- 0.0 4 2 0 2 4 analytic approximation, which formally holds only for a spher- 10− 10− 10 10 10 ical top-hat overdensity, is valid for NFW halos. In Fig.2 we M200/µ200 show the semi-analytical force profiles that can be compared to Fig.1, which shows the true numerical results. The qualitative Fig. 6. γ¯ versus M/µ200 for the three example models. The solid agreement is fairly good. We are able to match the shape around [dashed] lines show the mean (minimum and maximum) values for the peak, which is the significant part of the profile that will give the 128 (Symmetron), 16 ( f (R)), and 20 (nDGP) numerical curves obtained. The solid gray lines show the fits of Eq. (41) described in rise to most of the signal inγ ¯, and the amplitude is off by no more Sect. 3.3. than a factor of a few, which is good enough for our purposes. As a first – rather crude – approximation, we assume that a halo is fully screened within a certain radius xscreen. In the un- lines), as well as the maximal deviation found for each (dashed screened regime xscreen → 0, but generally 0 . xscreen . 10. lines). Clearly, the nDGP curves show very little variation. Thus, we approximate γ(x) ' γmaxθ(xscreen − x) where θ is Figure6 once more shows the di fference between the the Heaviside function. This leads to a solution for the mass- Symmetron and f (R) model with their characteristic “inverse- weighted average of γ, which reads as S-shape” on the one side, and, the nDGP – with a log-linear relation – on the other. A simple fitting function that is able to M(

However, it is also clear that the models of the former group do M(

A123, page 8 of 12 M. Gronke et al.: Universal predictions of screened modiﬁed gravity on cluster scales

1.0 1.0

0.8 0.8

0.6 0.6 ma x Γ Γ Γ 0.4 0.4

0.2 0.2

0.0 4 0.01 1 100 4 10 10 0.0 9. 10. 11. 12. 13. 14. 15. M200 Μ200 Log10 M h Mü Fig. 7. Semi-analytical predictions forγ ¯ versus M/µ200 for the Hu- Sawicki f (R) model (red), the Symmetron (blue) and the Dilaton model 9 −1 Fig. 8. γ(M) for NFW halos with masses ranging from 10 M h to (purple). The solid black curve shows the fitting formula Eq. (40). The n 1015 M h−1 using Eq. (52) with G(z) = 2((1+ζz) −1) . The four bands of shaded regions show the results of the full numerical analysis. ζz lines show the results (from top to bottom) ofζ = 0.1, 1.0, 10, 100, and within each band, we have (from top to bottom) n = 0.5, 0.4, 0.3, and 0.2. gives us a factor 1.5 offset in the value we find for µ200. Again the proportionality factor is found numerically and gives us where the last fraction can be identified as gNFW(cx)/gNFW(x) !3/2 | f | (see Eq. (A.2)). For a general model this leads to f (R) ∼ ± × 13 −1 × R0 µ200 = ( 1.1 0.4) 10 M h − , (48) ! 10 6 g (c) γ = γ G NFW (52) max x3g (cx) which is almost spot on the empirical relation found in the pre- NFW vious section. for some function G that√ satisfies G(0) = 1 and G(∞) = 0. For − 2 2 For a general {m(a), β(a)} model, we see from Eq. (23) that 2( 1+ζz 1) 8rc H0 Ωm∆vir β(a) DGP we have G(z) = ζz with ζ = 2 . ∼ 9βDGP we would (again very roughly) expect to have φ(a) a3m2(a) . This in the condition Eq. (45) means that we generally expect For models that only have a shift symmetry, like the 3 −1 k-mouflage screening mechanism, the field equation, µ200 ∝ λφ(a) where λφ(a) = m (a) is the range of the fifth force. " # In Fig.7 we show the semi-analytical profiles for ¯γ in terms 1 d dφ βδρ r2 f (X) = m , (53) of M200/µ200 for f (R) and the symmetron, together with the full r2 dr dr M numerical results. To demonstrate that this rescaling works in Pl general, and not just for the two models presented here, we also can also be integrated up, and from this it follows that the fifth- R r 2 calculated theγ ¯ predictions for the Dilaton model presented in δmr dr force F ∝ ∇φ is a function of 0 for spherical symmetry. Brax et al.(2012c, 2010). This model is characterized by φ r2 This leads to a slightly different form −r ! m(a) = m0a (49) g (c) γ γ F NFW s (a2r−3−1) = max 2 (54) β(a) = β0e 2r−3 (50) x gNFW(cx) for some function F satisfying F(∞) = 0. where r, s, m0, and β0 are free model parameters and where the evolution of β are very different from our two example models. The exact form for the functions F and G are model depen- Figure7 shows that the rescaling also works perfectly for the dent, making it hard to make definite statements here. However, Dilaton model. one general feature we are able to deduce is that, just as we saw for DGP, γ is only sensitive to the concentration of a halo. We We have not been able to get a prediction for the width of the therefore expect a fairly weak mass dependence ofγ ¯ for models partially screened region semi-analytically, so we leave this for in this class. future work. As an example, we can try to parametrize the functional form for F and G and calculateγ ¯. One functional form that satisfies n 3.5. Predictions for other screening mechanisms (1+ζz) −1 the requirements is G(z) = ζz with n ∈ (0, 1). The results For other screening mechanisms that are not encapsulated by we get from this of course depend on the values we take for the {m(a), β(a)} formulation, it is harder to make general pre- the parameters (like ζ); however, taking realistic values for the dictions. For any theory with a derivative shift symmetry (the parameters, which is motivated by the requirement of satisfying Galileon symmetry), second-order equations of motion the field local gravity constraints, the typical result we find is as expected equation in spherical symmetry are integrable, leading to an al- (see Fig.8): a very weak M dependence ofγ ¯. gebraic equation on the form 4. Discussion R r 2 cx ! δmr dr ∆ log(1 + cx) − f (φ0/r) = 0 = vir 1+cx (51) Our numerical force profiles presented in Sects. 3.1 and 3.2 are 3 3 − c r 3x log(1 + c) 1+c in good agreement to what has been previously found in N-body

A123, page 9 of 12 A&A 583, A123 (2015)

Other parameters: In particular, one expects that the majority of halo proper- 1 n = 1, β = 1, L = 1Mpc h− , assb = 1/2 ties are rescaled by ∼γmax. This can make the comparison 3.0 with ΛCDM difficult when using, for example, the dynamical mass as mass estimate. 2.5 – Halos with M200 ∈ [µ200/W, Wµ200] are partially screened. This means that in this mass range, halo properties might a [0.1, 0.9] L ssb ∈ differ from the expected. Since halos in this mass range are 2.0 ∈ 1 [0.1, 10]Mpc h− between screened and unscreened, the environment of these halos has a major effect (see Zhao et al. 2011b; Winther et al. 2012, for the study of this effect using N-body simulations). max 1.5 γ Also, comparing the inner and outer regions of individual ob- 25] . 1 jects can lead to insight (for instance, the ratio M500/M200). ,

1.0 1 . In Fig.9 the partially screened region is shown as a shaded [0 log f area in the matching color. As demonstrated for the nDGP ∈ 10 | R0| 0.5 β [ 8, 4] case, it is possible that – for some SMGs – all halos are al- rc [0 ∈ − − ways partially screened. ∈ .5, 10] 0.0 – For masses &Wµ200, the halo is fully screened, so no devia- 107 108 109 1010 1011 1012 1013 1014 1015 1016 1017 tion from the ΛCDM predictions is expected. 1 µ200 (M h− ) Conclusively, in any observable-mass scaling relation, we pre- dict a peculiar feature at µ , where the observables match the Fig. 9. Rescaling parameter µ200 versus the maximum gravitational en- 200 hancement γmax for several discussed models (solid colored lines) and ΛCDM prediction for higher masses and where they are differ- the respective partially screened regions (colored semi-transparent ar- ent for lower masses. However, the latter statement only holds eas). The red line shows the space spanned by the Hu-Sawicki f (R) if the true mass of the halo is used. This is because mass tracers −8 −4 model when varying | fR0| from 10 (left end) to 10 (right end), while like the dynamical mass can also be affected by the fifth force leaving n = 1 fixed. For the Symmetron, we also varied one parameter (Schmidt 2010). per curve (in blue, green, and violet), while leaving the others fixed at −1 The extent of the partially screened region, which we quan- β = 1, L = 1 Mpc h , assb = 1/2. The orange line indicates the region tified with α (or, W, see Eq. (42), can potentially be used to for nDGP models with rc ∈ [0.5, 10] (left to right) with the associated semi-transparent area spanning wider than the plot range. See Sect. 4.1 distinguish between various screening mechanisms. We found for details. that this extent is constant for each screened-modified gravity considered, but differs significantly between them. In particular, the value of α found in the numerical fits of Sect. 3.3 translates simulations (e.g., Zhao et al. 2011a; Falck et al. 2015). The sim- to log10 W ∼ 16, ∼1, and ∼0.6 for the nDGP, Symmetron, and ple rescaling procedure based on the maximum enhancement the f (R) models, respectively (using d = 0.2, as above). In a of the gravitational force γmax and the intermediately screened γmax-|α| (or equivalently, a γmax-W) diagram, we expect eventual mass µ200 (Sect. 3.3) demonstrates that in spite of their foun- constraints to rule out the area γmax > γconstraint, |α| . αconstraint dational differences, the three screened-modified gravity models (W & W(αconstraint)). investigated show self-similar behavior in cluster environments. 4.2. Caveats 4.1. Observational implications Several assumptions went into deriving the presented formalism. Our rescaling methods allow an abstract characterization of Here, we want to mention and investigate them. screened modified gravity theories, which themselves might possess quite different screening mechanisms. This potentially has – Halo profiles and shape. Throughout this work, we assumed interesting implications for constraining SMGs. spherical symmetry and perfect NFW halos. However, in re- The colored lines in Fig.9 show γmax versus µ200 for some ality clusters of galaxies do not exactly follow an NFW mass selected models. The lines were drawn by altering one model distribution, and halos also possess a nonzero ellipticity (e.g., parameter while leaving the others constant (see figure cap- Oguri et al. 2010). Another phenomenon falling in the same tion for details). Each set of SMG model parameters occu- category is our assignment of a unique mass-concentration pies a specific point on this graph; for example, the combina- mapping, whereas the spread in this relation is fairly wide −1 tion (assb, β, L) = (1/2, 1, 1 Mpc h ) maps to (γmax, µ200) ≈ (Neto et al. 2007). Both are consequences of the fact that 11 −1 (1.75, 4.5 × 10 M h ). The semi-transparent areas in Fig.9 clusters are individual objects with their own histories and denote the extent of the partially screened region using Eq. (42) environments. As a result, individual clusters (as any astro- with d = 0.2. This means that for every selected value of µ200, the physical object) have mostly a limited explanatory power semi-transparent area indicates the range where the fifth force is and must be stacked for analysis. at least 20% but a maximum of 80% active. – Model assumptions. To rescale theγ ¯ − µ200 curves ac- Thus, Fig.9 indicates (i) what halos are partially screened cordingly, an intermediate-mass scale has to exist where and; (ii) what the maximum enhancement of gravity is for this γ¯ ∼ γmax/2. Since this is the scale between the screened model. Physically, µ200/W can be interpreted as the minimum and unscreened regimes and, our work addresses “screened mass where one expects a variation in halo properties; that is to modified gravity theories”, we think that this can be safely say: assumed. The universal rescaling works perfectly for the models we have tested, and we conjecture that it will hold for – For M200 . µ200/W halos in this model are unscreened, all {m(a), β(a)} models. However, this might not be true for thus allowing for comparison with the ΛCDM predictions. cases where m(a) and/or β(a) are very steep functions of a,

A123, page 10 of 12 M. Gronke et al.: Universal predictions of screened modiﬁed gravity on cluster scales

leading to a very rapid evolution of the screening condition astrophysical structures, a fully screened, an unscreened, and a with density. partially screened regime, we investigated whether one can spec- – Choice of parameters. Throughout this work, we fixed a ify, independently of the SMG model, at which halo mass range number of parameters. Probably, the most influential ones these regimes are located and how extended they are. are the halo cutoff radius xcut = 10 and defined µ200 as These two questions can be answered as follows: γ¯(µ200)/γmax = k with k = 1/2. The latter choice is arguably quite natural. In spite of that, we do not expect our results to – It is possible to formulate the mass of the partially screened change dramatically if a (slightly) higher or lower value of k regime as a function of various model parameters. We is taken since the model curves are expected to shift equally. demonstrated this empirically for the Symmetron, the Hu- Sawicki f (R), and the nDGP models (Sect. 3.3). In addition, The choice of xcut affects our results more drastically, as can we found a semi-analytic derivation of this functional form be inferred from Fig.1. A lower value of xcut leads, for instance, to a decrease in the extent of the partially screened in the {m(a), β(a)}-parametrization of Brax et al.(2012b). region (i.e., an increase in α). Nevertheless, since all models This allowed us to expand our method easily to the Dilaton model. are affected by this, the precise choice of xcut (within reason) does not alter our conclusions. – The extent of the partially screened regime is an intrinsic – Practicality. We focused merely on the force profiles di- property of a particular screening mechanism. We found that rectly, leaving the question how practical the study is since the mass range of the partially screened region differs signif- γ(r) andγ ¯ are not directly observable. Although this is not icantly between the three example models but are (approxi- wrong, we caution that the fifth force triggers many observ- mately) constant within a particular model. able phenomena, as has been shown in various N-body sim- In spite of the differences in the nature of their screening mech- ulations. In particular, γ(r) can be related to the ratio of the anisms (see Sect. 2.1), we characterize the Symmetron, the dynamical mass to the lensing mass (Schmidt 2010; Zhao Hu-Sawicki f (R), and the nDGP models with three parame- et al. 2011b; Winther et al. 2012) ters: (i) the center of the partially screened regime µ200; (ii) the MDyn(

Although the density of the NFW proﬁle is diverging for Brax, P., Davis, A.-C., Li, B., Winther, H. A., & Zhao, G.-B. 2012c, J. Cosmol. x → 0, the enclosed mass within a certain radius does not Astropart. Phys., 10, 2 (Lokas & Mamon 2001). This leaves the mass fraction within Brax, P., Davis, A.-C., Li, B., Winther, H. A., & Zhao, G.-B. 2013, J. Cosmol. Astropart. Phys., 4, 29 [x ± dx/2] as Bullock, J. S., Kolatt, T. S., Sigad, Y., et al. 2001, MNRAS, 321, 559 Chan, K. C., & Scoccimarro, R. 2009, Phys. Rev. D, 80, 104005 dM c2 xg (cx ) NFW cut x, Clampitt, J., Jain, B., & Khoury, J. 2012, J. Cosmol. Astropart. Phys., 1, 30 = 2 d (A.4) Clifton, T., Ferreira, P. G., Padilla, A., & Skordis, C. 2012, Phys. Rep., 513, 1 M(

A123, page 12 of 12