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Unscreening Modified Gravity in the Matter Power Spectrum Lucas Lombriser,1 Fergus Simpson,2 and Alexander Mead1 1Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, U.K. 2ICC, University of Barcelona (UB-IEEC), Marti i Franques 1, 08028, Barcelona, Spain (Dated: September22, 2015) Viable modifications of gravity that may produce cosmic acceleration need to be screened in high-density regions such as the Solar System, where general relativity is well tested. Screening mechanisms also prevent strong anomalies in the large-scale structure and limit the constraints 5 that can be inferred on these gravity models from cosmology. We find that by suppressing the 1 contribution of the screened high-density regions in the matter power spectrum, allowing a greater 0 contributionofunscreenedlowdensities,modifiedgravitymodelscanbemorereadilydiscriminated 2 from the concordance cosmology. Moreover, by variation of density thresholds, degeneracies with p other effects may be dealt with more adequately. Specializing to chameleon gravity as a worked e example for screening in modified gravity, employing N-body simulations of f(R) models and the S halo model of chameleon theories, we demonstrate theeffectiveness of this method. Wefind that a 1 percent-levelmeasurementoftheclippedpoweratk<0.3h/Mpccanyieldconstraintsonchameleon 2 modelsthataremorestringentthanwhatisinferredfrom SolarSystemtestsordistanceindicators inunscreeneddwarfgalaxies. Finally,weverifythatourmethodisalsoapplicabletotheVainshtein ] mechanism. O C Introduction.— Determining the nature of the accel- gravitational coupling and exponents of the chameleon . h erated expansion of our Universe is a prime endeavor field potential. We also verify the applicability of clip- p to cosmologists. In the conventional picture, the flat Λ ping to Vainshtein screening [1]. - o cold dark matter (ΛCDM) concordance model based on Chameleon gravity.— We first specialize to the Hu- r general relativity (GR), a cosmological constant Λ con- Sawicki f(R) (n = 1) model, where the nonlinear func- st tributes the bulk of the present energy density in the tion f(R) ≃ −2Λ − fR0R¯02/R of the Ricci scalar R a cosmos and drives the late-time acceleration. While al- is added to the Einstein-Hilbert action. Here, bars [ ternatively, a modification of gravitymay be responsible denote quantities evaluated at the cosmological back- 2 forcosmicacceleration,stringentlimitationsfromexperi- ground, zeros refer to present time, fR ≡ df/dR is the v mentswithinourSolarSystemmustbesatisfied. Anum- additional, scalar degree of freedom of the model, and 1 6 ber of screening mechanisms [1–5] have been identified fR0 ≡ f¯R(z = 0). In the quasistatic approximation and 9 thatcansuppressmodificationsofgravityinhigh-density for |fR0| ≪ 1, the modified Poisson equation becomes 4 regions to recover GR, while still generating significant (see, e.g., Refs. [11, 12]) 0 modifications within lowerdensities onlarger,cosmolog- . 16πG 1 1 ical scales. However, this suppression effect, along with ∇2Ψ= δρm− δR(fR), (1) 0 other nonlinear effects, also prevents strong anomalies 3 6 5 from manifesting in the averagedlarge-scalestructure of where δ denotes perturbations with respect to the cos- 1 : our Universe [6] and limits the constraints that can be mological background, e.g., δR = R − R¯, and Ψ ≡ v inferred on these gravity models from cosmology. i δg00/(2g00). The scalar field equation is given by X Given the density dependence of the screening effect, ar idnentshitisyrLeegtitoenrs, iwnestpartoisptoicsaelothbesedrvoawbnlewsesiguhchtinagstohfehmigaht-- ∇2δfR =−8π3Gδρm+ 31δR(fR). (2) ter power spectrum P(k) to enhance, or unscreen, the The chameleon mechanism works such that in a high- signaturesofmodifiedgravityandimproveobservational density region δR ≃ 8πGδρm and hence Eq. (1) reduces constraints. Such a weighting is conducted in the clip- to the Poisson equation of Newtonian gravity. In con- ping method of Ref. [7], with the original motivation of trast, at low densities δR ≃ ∂R/∂fR|R=R¯ = 3m2δfR, facilitating the modeling of P(k) by reducing contribu- which when applied to Eqs. (1) and (2) in Fourier space tions of high densities, where the assumptions of pertur- yields the unscreened modified Poisson equation bation theory break down. As a worked example, we first focus on Hu-Sawicki [8] f(R) gravity [9], which em- 2 −1 4 1 k ploys the chameleon screening mechanism [2]. We an- k2Ψˆ =−4πG − +1 a2δρ . (3) m alyze effects on the power spectrum from clipping den- 3 3"(cid:18)ma(cid:19) #  sity fields in numerical simulations of the model. Using b the halo model of chameleon theories [10], we then gen- Hence, whereasin high-density regions gravity returns eralize our findings to chameleon models with arbitrary to Newtonian due to the chameleon mechanism, at low 2 densitiesandscalesbelowm−1,gravitationalinteractions We shall also make use of applying a minimum instead remainenhancedbyafactorof4/3. Inparticular,Eq.(3) of a maximum threshold, which is equivalent to clipping applies to the linear perturbation regime. Note that the negative field −δ(x). While this may prove more f(R) models correspond to a Brans-Dicke scalar-tensor challengingto apply to realdata,due to the lowersignal theory with Brans-Dicke parameter ω = 0, Jordan- to noise associated with cosmic voids, it will serve as a frame scalar field ϕ = 1+fR, and scalar field potential useful validation test in our simulations and support for U =(RfR−f)/2. Moregenerally,thechameleonmecha- the concept of weighting to unscreen or screen gravita- nism is realizedfor scalar-tensormodels with scalarfield tional modifications. We quantify the clipping strength potentialU(ϕ)−Λ∝(1−ϕ)αwithα≡n/(n+1)∈(0,1), in terms of the fractional loss of power in the lowest k- where the gravitational coupling is maximally enhanced bin, applying a simple iterative procedure to determine by a factor of (4 + 2ω)/(3+ 2ω) with ω > −3/2 [10]. the threshold δ0 required to establish the desired frac- Importantly,althoughservingas veryusefulexample for tion. Defining clipping strength directly in terms of δ0 screening mechanisms, chameleon models do not yield a is another possibility, but complicates the comparisonof genuine self-acceleration of the cosmic expansion due to results from fields with a different choice of smoothing their gravitationalmodifications [13]. length. In order to obtain accurate results in the nonlinear Given the nature of the chameleon mechanism, the regime of the Hu-Sawicki f(R) model, we use dark mat- clipping transformation promises the extraction of more terN-bodysimulationsruninRef.[14]withtheECOSMOG information on the gravitational physics, as it allows us code of Ref. [15], which uses particles and adaptive tofocusonthelessdense,unscreenedregionsoftheUni- meshes to solve Eqs. (1) and (2). The background ex- verse,wherethereexistsagreaterdifferencefromGR.In pansion is taken to be equivalent to that of ΛCDM, ap- theleftpanelofFig.1,wecomparethedeviationinpower propriate for observationally interesting fR0 values. We between the F6 and ΛCDM simulations as a function of usesimulationsoftheconcordancemodelandf(R)grav- clipping strength. With increased clipping strength, the ity where |fR0| is 10−4 (F4), 10−5 (F5), and 10−6 (F6), difference in P(k) is enhanced. At k = 0.5 h/Mpc this all sharing an initial seed and cosmological parameters. differencemorethandoubles,greatlyfacilitatingthe dis- Each simulation contains 5123 particles in a box with crimination between the two models. In practice, the L = 512h−1Mpc; h = 0.697, Ωm = 0.281, Ωb = 0.046, amount of clipping that can realistically be applied will ΩΛ = 0.719, ns = 0.971, and amplitude of the matter dependonthenoiselevels,limitedbythenumberdensity powerspectrumsuchthatσ8 =0.82inΛCDM.Notethat ofgalaxies. Toseethereverseofthiseffect,wealsoapply σ8 is larger for f(R) gravity due to the enhanced forces the clipping transformation to the negative field −δ(x). and growth of structure. An initial power spectrum for Sincethisisclippingvoids,thisreducesthecontributions the simulations was generated using MPGRAFIC[16]. The fromunscreenedregionstoP(k). Therefore,asexpected, particle mass in eachcase is ≃7.80×1010h−1M⊙. Each strongerclipping leads to convergencebetweenthe mod- simulation has exactly the same initial power spectrum ified gravity and ΛCDM density fields. Note that we do (zi = 49) and differences between models are confined not expect a full unscreening of modified dynamics by to different strengths of enhanced perturbation growth clipping high-density peaks since this only affects self- at late times and different strengths of screening. For screening but not environmental screening [10]. In the the extrapolation of nonlinear physics from f(R) grav- opposite case of clipping low-density troughs, however, ity to more general chameleon models with ω 6= 0, we only self-screened regions with GR dynamics remain. employ the halo model of chameleon theories developed In the middle panel of Fig. 1, we show the ratios of in Ref. [10], which accounts for the density dependence the power spectra of F4-6 to that of ΛCDM simulations of the effective gravitational coupling in the nonlinear and the corresponding ratios after clipping has been ap- regime and provides matter power spectra that are in plied such that the large-scale power is reduced by 50%. goodagreementwithmeasurementsinf(R)N-bodysim- We also show the results of instead applying a logarith- ulations (see Fig. 4 in Ref. [10]). mic density transform [17]. Both local transformations Clipping the density fields.— The spatial distribution suppress contributions from the densest regions of the of matter on cosmological scales may be quantified by field. Note that while the shape recovered from the log- the fractional overdensity field δ(x) ≡ ρm/ρ¯m −1. We arithmic transformis closer to linear theory, for sparsely construct the density fields from the simulations using sampledfields the logarithmictransformbecomes unsta- a cloud-in-cellinterpolationona 2563 Cartesianmeshin ble whereas clipping is insensitive to the number den- eachcell. Clippingisalocaldensitytransformationchar- sity of sources [18]. Regarding the proximity to linear acterizedbyenforcingamaximumfractionaloverdensity theory, Ref. [18] demonstrated that the clipped ΛCDM δ0 such that [7] matterpowerspectrumiswelldescribedbyalinearcom- bination of the linear and one-loop contributions, where δ0, δ(x)>δ0, higher-orderterms are strongly suppressed. Thereby,in- δc(x)= (4) δ(x), δ(x)≤δ0. creasedclippingdownweightstheone-looprelativetothe (cid:26) 3 FIG. 1: Left: Fractional difference between simulated matter power spectra of f(R) gravity (|fR0| = 10−6) and ΛCDM at two different k-bins. By clipping high densities, contributions from self-screened regions are downweighted and unscreened featuresenhanced(dashedcurves). Inthereversecaseofapplyingaminimumthresholdtothedensityfield,contributionsfrom unscreenedlowdensitiesaredownweighted,enhancingscreenedregions,andthepowerspectraconverge(dottedlines). Middle: Fractional departure from the ΛCDM power spectrum in three different f(R) models for unclipped fields (solid lines), linear perturbation theory (dashed lines), clipped fields with power at the largest scales reduced by 50% (dot-dashed curves), and logarithmicdensitytransformationwithmatchingtotheΛCDMpoweratthelowestk-bin(dottedcurves). Right: Clippedf(R) power spectra from simulations (circles) compared to weighting linear and one-loop contributions (dashed curves). Increased clipping strength enhancesthe weight of the unscreened linear (dotted curves) relative to theone-loop power. linear contribution. We verify the recoveryof linear the- variationsin σ8 andfR0. However,the ΛCDMcase with ory for f(R) gravity in the right-hand panel of Fig. 1, a larger value of σ8, will experience a strong reduction using F4 simulations, where chameleon screening affects of the enhancement in power at large k after clipping. P(k) at k >∼ 0.1 h/Mpc (see Fig. 2 in Ref. [6]). With In contrast, the f(R) power spectrum increases the en- increased clipping, P(k) more closely reflects linear, un- hancement at large k after clipping, becoming more lin- screened, theory with a reduced relative weight of the ear and unscreened. Therefore variations in fR0 and σ8 one-loop correction. respond to clipping in a qualitatively different manner, breaking the degeneracy. Modifications in the shape of Differentialclippingtobreakdegeneracies.—Effectson P(k) due to changes in Ωm are qualitatively different thetotalmatterpowerspectrumfromf(R)modifications from f(R) modifications, e.g., changing baryon acous- ofgravity,andhenceotherchameleonmodels,havebeen tic oscillation features. Although a partial suppression showntoyieldsomedegeneracywitheffectsfromvarying in the change of power attributed to Ωm variations is the total neutrino mass [19, 20] or from baryonic feed- seen in nonlinear compared to linear theory, it does not back processes [21]. It is important to note that while reproduce the strong screening effect of chameleon mod- degenerate in P(k), these contributions are not affecting els. Finally, note that since focusing on differences in aparticularrangeofdensitiesinthesamemannerasthe the shape of P(k), effects of linear galaxy bias can be chameleon modification. More precisely, nonvanishing neglected. However, through redshift-space distortions, neutrino masses suppress P(k) predominantly linearly, in chameleon models the ratio between galaxy and dark i.e., with negligible preference on density, and baryonic matterdensitybecomesscaledependent. Sinceaddingto feedback processes emanate from high-density regions. thedeviationsbetweentheshapeofmodifiedandΛCDM Hence,acombinedanalysisofclippedpowerspectraem- galaxypower spectra [23], we conservatively assume this ploying different density thresholds can break degenera- ratio to be constant when estimating potential observa- ciesbetweenthedifferentcontributions. Furthermore,in tional bounds on chameleon models. aparameterestimationanalysisofchameleonmodels,we need to allow for a variationof cosmologicalparameters. Outperforming Solar System constraints.— The re- In particular the growth of structure can be strongly al- quirement that the Milky Way dark matter halo screens tered by variations of σ8 or Ωm. Using HALOFIT [22], theSolarSystemsetsaconstraintonthechameleonfield we estimate effects of varying these parameters on the amplitude of|ϕ¯0−1|<∼5×10−6/(6+4ω)[10]. Similarly clipped power by comparing signatures in the linear and strong constraints can be obtained from the absence of nonlinear P(k). While enhancing the amplitude of P(k) deviations in luminosity distances from different types with increasing σ8, the shape of the nonlinear enhance- of distance indicators in unscreened dwarf galaxies [24]. ment resembles that of a chameleon model. Hence, with Current cosmological constraints are about 2 orders of absent information on the absolute amplitude of P(k) magnitudeweakerthanSolarSystembounds[25–27]. We due to galaxybias,there clearlyis adegeneracybetween refertoRef.[28]forareviewofconstraintsonchameleon 4 gravity. Having analyzed the effect of clipping on the matterpowerspectrum,weestimatetheconstraintsthat can be inferred from applying this method to observa- tions of galaxy clustering. While fractional errors in the measurement can be kept approximately constant, clipping reduces systematic errors from modeling uncertain- ties of the nonlinear structure contributing at large k such that constraints oncosmologicalparameterscan be improved (see Ref. [29]). We assume that a future measurement of the clipped P(k) can discriminate 1% deviations fromthe fiducial ΛCDMshape atk ≤0.3 h/Mpc. Note that this is a conservative estimate with surveys already yielding subpercent-level measurements of the FIG.2: Forecastofupperboundsonthechameleonfieldam- acoustic features [30]. We compute the modified nonlin- plitude 1−ϕ¯0 from the clipped matter power spectrum as a ear power employing the halo model of chameleon the- function of gravitational coupling strength set by the Brans- ories [10]. As clipping chameleon densities mainly re- Dickeparameterω andfordifferentvaluesoftheexponentof moves regions where GR is recovered, we approximate thechameleonfieldpotentialα. Theconstraintsarecompared the clipped chameleon power spectrum by removing the to upper bounds inferred from Solar System [32], astrophysical distance indicator [24], and cosmological (cluster) obser- difference of unclipped to clipped power obtained from vations [25] (cf. Ref. [28]). For a range of chameleon model the ΛCDM simulation. For k ≤ 0.3 h/Mpc, this simple parameters, clipping can providethestrongest constraints. approach reproduces the absolute clipped power for F4 andF6atthefewpercentandpermillelevel,respectively, andrecoversthemeasuredfractionaldifferencetoΛCDM tional coupling at nonlinear scales in models exhibiting within 20%, increasingly underestimating it from F4 to a Vainshtein [1] screening effect [33]. While the modifi- F6. Employingthis method andvarying the background cations of P(k) are comparable, the Vainshtein mecha- fieldvalueϕ¯0,thecouplingstrengthsetbyω,andtheex- nism originates from derivative self-interactions in con- ponent of the scalar field potential α, we set constraints trast to the scalar field potential in chameleon gravity, where the deviation between the clipped chameleon and causing some differences in the screening behavior [34]. ΛCDM P(k) at k =0.3 h/Mpc exceeds 1%. Chameleonscreeningdependsonthegapbetweenanob- We present our results in Fig. 2, comparing them to ject’s internal and exterior scalar field values set by the Solar System, astrophysical distance indicator, and cur- corresponding densities. Hence, an object can be self- rentcosmologicalbounds. Wefindthatforthesimulated screened or environmentally screened by a large local or Hu-Sawicki model (α=0.5), clipping constraints on fR0 environmental density, respectively. In contrast, in the can improve upon existing cosmological constraints, us- Vainshteinmechanism,screeningdependsonmorphology ing clusters [25], the matter powerspectrum [26], or cos- of the source but not on its environment, introducing a micmicrowavebackgroundwithredshift-spacedistortion largescreeningscalearoundanobject,suppressingmodi- data[27],by2–3ordersofmagnitudeandoutperformthe ficationstoitsowngravitationalfieldbutstillresponding Solar System and astrophysical bounds as well as local to external field modifications. We verify that the clip- tests of the equivalence principle [2, 28, 31]. For smaller ping method is also applicable to the Vainshtein mecha- values of ω, corresponding to larger force modifications, nism by employing 50% clipping to the nDGP [35] sim- the improvement of constraints from clipping over Solar ulations of Ref. [34]. We find an enhancement of ∆P/P System bounds is even greater. This is not surprising (cf. Fig. 1) of 70%, 60%, and 20% at screened scales of since larger force modifications also imply a more effi- k ∼1.5h/Mpcformodelswithσ8 valuesequaltotheF4, cientscreeningoftheSolarSystemregionwhereasgravi- F5, and F6 f(R) scenarios, respectively. We also expect tationaldynamicsinunscreenedlowdensitiesismodified our method to be applicable to further screening mecha- evenmorestrongly. Constraintsinthisregionofparame- nisms like the symmetron [4] or k-mouflage [3], however, terspacemayalsoconfirmorruleoutchameleonmodels not to linear shielding [5]. as an explanation of the observed cored density profiles Conclusions.—Modificationsofgravitypotentiallyex- of dwarf spheroidal galaxies in the Milky Way [32]. In plaining cosmic acceleration need to employ a screening the opposite limit of increasing ω, i.e., weakening grav- mechanism that allows modifications in low densities at itational coupling, Solar System bounds strengthen and large scales while suppressing them in high-density re- surpassthedecreasingclippingconstraints. Importantly, gions such as the Solar System. Such screening mech- these power spectrum constraints clearly depend on the anisms also suppress anomalies in the averaged cosmo- exponent of the scalar field potential α. logical large-scale structure, limiting observational con- Vainshtein screening.— A similar density dependence straintsinferredonsuchmodels. Byclipping thedensity to the chameleonmechanismentersthe effective gravita- fields at a maximal threshold, contributions of screened 5 high-density regions to the matter power spectrum can M. Formisano and S. Odintsov, Phys.Rev. D85, 044022 bedownweighted,enhancingmodifiedgravityeffects. 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