Observational Constraints in Nonlocal Gravity: the Deser-Woodard Case
OObservational Constraints in NonlocalGravity: the Deser-Woodard Case Luca Amendola a Yves Dirian V b,c Henrik Nersisyan a Sohyun Park d a Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Hei-delberg, Germany b Department of Theoretical Physics and Center for Astroparticle Physics, University of Geneva, Quai Anser-met 24, CH–1211 Genève 4, Switzerland c Center for Theoretical Astrophysics and Cosmology, Institute for Computational Science, University ofZürich, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland d CEICO, Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, 182 21 Praha 8, CzechiaE-mail: [email protected], [email protected],[email protected], [email protected]
Abstract.
We study the cosmology of a specific class of nonlocal model of modified gravity, the so–calledDeser–Woodard (DW) model, modifying the Einstein–Hilbert action by a term ∼ Rf ( (cid:3) − R ), where f is afree function. Choosing f so as to reproduce the ΛCDM cosmological background expansion history withinthe nonlocal model, we implement the model in a cosmological linear Einstein–Boltzmann solver and studythe deviations to GR the model induces in the scalar and tensor perturbations. We observe that the DWnonlocal model describes a modified propagation for the gravitational waves, as well as a lower linear growthrate and a stronger lensing power as compared to ΛCDM, up to several percents. Such prominent growth andlensing features lead to the inference of a significantly smaller value of σ with respect to the one in ΛCDM,given Planck
CMB+lensing data. The prediction for the linear growth rate f σ within the DW model istherefore significantly smaller than the one in ΛCDM and the addition of growth rate data f σ from Redshift-space distortion measurements to Planck
CMB+lensing, opens a (dominant) tension between Redshift-spacedistortion data and the reconstructed
Planck
CMB lensing potential. However, model selection issues onlyresult in “weak” evidences for ΛCDM against the DW model given the data. Such a fact shows that the joineddatasets we consider are not constraining enough for distinguishing between the models on firm grounds.As we discuss, the addition of galaxy WL data or the consideration of cosmological constraints from futuregalaxy clustering, weak lensing surveys, but also third generation gravitational wave interferometers, proveto be useful for discriminating modified gravity models such as the DW one from ΛCDM, within the closefuture. a r X i v : . [ a s t r o - ph . C O ] A p r ontents The observations of a variety of complementary cosmological probes such as distant Type Ia supernovae(SNIa), the Cosmic Microwave Background (CMB) or the clustering properties [e.g. Baryonic AcousticOscillations (BAO), Redshift-Space Distortions (RSD)] and the weak lensing (WL) of galaxies have provideddata of unpreceding quality. Given these data, the inferred constraints on the parameter space of thecurrent standard model of cosmology ΛCDM, have reached exquisite accuracy – up to percent-level in thecase of its six-dimensional “minimal” or “base” cosmological parametrisation (see e.g. Refs. [1–3]). As aconsequence, the base ΛCDM model has been shown to be able to explain these data with high significanceand consistently within each probe (internal) and also when the latter are joined together (external, alsoknown as concordance).Despite such astonishing capabilities, the model still suffers from theoretical flaws as well as observa-tional weaknesses that obscure its credibility on fundamental theoretical and empirical grounds. On thetheoretical side, Λ is a simple, dimensionful number that lacks justifications about its nature and late-timedomination (see for example Ref. [4–7] for reviews). On the observational one, A. Riess et al. recently re-ported in Ref. [8] that the direct measurement of H from nearby SNIa is significantly larger (at 3 . σ ) thanthe value inferred from the Planck H such as the ones of Refs. [9–12] also generically prefer values higher than the onepinpointed by Planck , but see Ref. [13] for a different conclusion. In addition, cosmic shear measurementsin the σ –Ω M plane from weakly lensed galaxy maps collected by the CFHTLenS [14] or
KiDS-450 [15]survey were also shown to display a tension with the constraints inferred by
Planck [16]. The same remarkalso potentially applies to σ –Ω M constraints given non- Planck (e.g. X–ray) and
Planck – SZ selected clustercounts [3, 17]. However, all these empirical results are still possibly driven by uncontrolled systematics andrefined analyses of the data are first needed before meaningful conclusions can be established. Indeed, thisfact is well illustrated by the recent controvercy triggered by the low-valued distance scale estimate of H from Ref. [18], which proposed an alternative pipeline including the use of GAIA DR2 “quasar”-correctedparallaxes as compared with the work of Ref. [8] (see also Ref. [19, 20] for further discussions) . Another ex-ample comes from Ref. [23], which improved the modelisation of the noise covariance within the data analysis The significance of the high– z (CMB) and low– z tension on H –derived inferences however mildens when building an inversedistance ladder [21]. This fixes the comoving sound horizon r s at the drag epoch using joined low–z BAO and SNIa distancescale data which can then be compared with independent inferences from CMB observations [22]. – 1 –ipeline of the KiDS-450 survey and increased the agreement between galaxy WL and
Planck
CMB. More-over, comparing cluster counts constraints between
Planck and external experiments requires the knowledgeof the relation between the X–ray mass and the total cluster mass, which is currently parametrized by anapproximate hydrostatic mass linear bias (see for example Refs. [3, 17, 24]).These facts support the development of phenomenological models of dark energy/modified gravity (seefor instance Refs. [25–29] for reviews and Ref. [30] for a standard textbook) that aim to explain better currentdata and, at the same time, can provide hints for solutions to the problems of Λ. Modified gravity modelsall have a common thread in that they evade Lovelock’s theorem [31, 32] and, given that General Relativity(GR) is well-tested on solar system scales, they must modify its dynamics in its infrared regime (IR). Thiscan be realized in various fashions and examples of IR modifications are provided by scalar-tensor theoriesincluding one extra degree-of-freedom such as quintessence (see e.g. Ref. [33] for a review), the so-calledbeyond Horndeski “degenerate higher order scalar-tensor theories” [34] or the more general effective fieldtheory of dark energy [35, 36]. In order to reproduce the predictions of GR on solar system scales (highdensity regions), scalar-tensor theories must either be effectively decoupled to baryons (see e.g. Ref. [37]),or exhibit a screening mechanism of, for example, the chameleon [38, 39], symmetron [40] or Vainstein[41] type, suppressing their fifth force on solar system scales. When reasonably close from ΛCDM for thesame cosmological parameter values, alternative cosmological models can then legitimately and efficiently beconstrained by using the aforementioned cosmological probes. This is well illustrated by the recent works ofRef. [42] on quintessence theories and of Ref. [43] on Horndeski theories. Furthermore, these models can alsobe used for forecasting cosmological constraints and model selection issues from future cosmological surveyssuch as
Euclid [44, 45],
DESI [46, 47],
LSST [48],
SKA [49, 50] or
Stage-4 CMB experiments [51] (see e.g.Refs. [52, 53]). Moreover, prospects of future observations of the Gravitational Waves (GWs) produced bybinary mergers from third (or 2 .
5) generation interferometers such as
LISA [54], the
Einstein Telescope [55]or
Cosmic Explorer [56], will also provide exceptional complementary information to constrain the expansionhistory as well as deviations to GR, such as in the propagation properties of GWs – in particular in the amountof their damping under the Hubble flow (see e.g. Refs. [57–61] for constraints on dark energy/modified gravityusing standard sirens).Another class of IR modifications to GR consists in scalar nonlocal modifications . In the beginningof this century, it has been reported that such corrections can be induced from higher dimensions such aswithin the Dvali-Gabadadze-Porrati (DGP) braneworld model [66, 67], but the model has been shown notto be phenomenologically viable because of the presence of ghosts in the self-accelerating branch (see e.g.Refs. [68, 69] and references therein). Nevertheless, it is still believed that such corrections can also resultfrom quantum effective non-perturbative corrections. In this context, prototypical mechanisms are providedby renormalisation group flow corrections to the bare gravitational couplings, characterized by a fixed pointin the ultraviolet (UV) [70, 71] (see Ref. [72] for a recent cosmological study), or by dynamical mass-scalegeneration in the IR, such as a mass for the conformal mode of the graviton as suggested in Ref. [73] (seealso Ref. [74] for further details along these lines). Quite intriguingly, the background independent latticequantum gravity computations of Ref. [75] have recently shown a “first-hand evidence for the presence ofnonlocal terms [in the gravitational quantum effective action] which could affect the gravitational dynamicsat cosmic scales”.Phenomenologically, the nonlocal models inspired by quantum averaging processes, such as the onesstudied in Refs. [72, 74, 76], exhibit self-accelerating solutions that are generically driven by an effective darkenergy component whose equation of state w lies on the phantom side, i.e. w < − H and will there- Scalar nonlocal modifications refer to the use of composite operators made of diffeomorphism–scalar quantities includingoperators that are non-polynomial in their derivatives to modify GR. As will be discussed below, on phenomenological grounds,most of the scalar nonlocal gravity theories considered to date involve combinaisons of the Ricci scalar R and a Green’sfunction of the d’Alembert operator (cid:3) − , such as ∼ (cid:3) − R, ∼ R (cid:3) − R, ∼ (cid:3) − R, etc. This is opposed to tensor nonlocalmodifications where higher rank tensors/operators are also included. Several models of tensor nonlocalities were shown togenerically exhibit growing modes at the cosmological background or linear perturbation level, preventing them from modellinga phenomenologically viable cosmological dynamics (see e.g. Refs. [62–64]). Nonetheless, exceptions might still exist (see e.g.Ref. [65]). – 2 –ore be in better agreement with direct measurements as compared to ΛCDM. However, if the dark energyis too phantom, a tension between CMB and distant SNIa measurements can arise. Moreover, a smallerHubble expansion rate at late-time also reduces the Hubble friction to matter fluctuations and thereforegives raise to a higher growth of linear as well as nonlinear structures as compared to ΛCDM, degradingthe agreement with growth data. One can however exploit the degeneracy between dark energy/modifiedgravity and the (absolute) neutrino mass to reestablish the compatibility of the model with the data [78](see also Refs. [79–81]). These facts are well illustrated in Refs. [78, 82, 83], where the phenomenology of theso-called RR nonlocal gravity model has been analysed. This model has recently been put under extensiveobservational constraints [78, 82, 84–86] and was shown to explain CMB+BAO+SNIa+RSD data as well asΛCDM, when the absolute neutrino mass is left as a free parameter [78]. Furthermore, the RR model hasalso been used for developing future experiments’ data analysis pipelines in forecasting cosmological con-straints from galaxy clustering and WL surveys in Ref. [87] and from third generation GWs interferometersin Ref. [60].A nonlocal model that has become popular in the past decade has been proposed by S. Deser andR. Woodard in Ref. [88]. In the Deser–Woodard (DW) model, GR is modified by an extra term of the form Rf ( (cid:3) − R ) to the Einstein–Hilbert action, where f is a dimensionless free function. As for f ( R ) theories, thismodel has no predictive power as long as the function f is left unspecified. However, it has been shown inRef. [89] (whose results are reproduced in Sec. 2 below), that once a given ΛCDM Hubble expansion historyis specified as H ΛCDM ( z ), one can solve for X ≡ (cid:3) − R and f ( X ) in terms of H ΛCDM ( z ), so as to reconstructthe same ΛCDM expansion history within the DW model, i.e. for obtaining H DW ( z ) ≡ H ΛCDM ( z ), at anyredshift. The same reconstruction technique can also be carried out for non-standard ΛCDM cosmologiessuch as w CDM models [90] and other simple choices for f were explored as well in Ref. [91]. Once f ( X )is fixed in such a way, no extra freedom is left and the distinction between ΛCDM and DW cosmologiesexclusively lies in the linear and nonlinear observables they describe. The impact on the linear growthof structures has first been studied in Ref. [92, 93]. As recognized in Ref. [94], the authors of Ref. [93]erroneously concluded that RSD data favour ΛCDM over the DW model at a significance level of ∼ σ ,because of an excess of growth described in DW. However, the work of Ref. [95] conducted an equivalentanalysis but at the so-called “localized level” in the equations of motion and concluded that, for the samecosmological parameter values, the DW model actually predicts a linear growth of structures that is weakerthan in ΛCDM, only up to several percent in the quasi-static approximation. This fact was confirmedin Ref. [94], which moreover established the equivalence between the nonlocal and localised versions on aparticular set of initial conditions and the validity of the quasi-static approximation studied in Ref. [95].In the present work, we complement the past analyses of Refs. [94, 95] in solving the full equations ofmotion at background and linear perturbation levels in the scalar and tensor sectors, within the cosmologicalcontext. For doing so, we implement the DW nonlocal gravity model in a modified version of the linearEinstein–Boltzmann code CLASS [96] and study its cosmological phenomenology from a modified gravityperspective by analysing relevant indicators of deviations from GR. We then perform cosmological parameterinference and model selection with the Monte Carlo Markov Chain (MCMC) sampler MONTEPYTHON[97] and confront ΛCDM against DW given high precision CMB+SNIa+RSD data. The paper is organizedas follows. In Sec. 2, we introduce the DW model and present the full set of modified Einstein equationsneeded to evolve its linear cosmological perturbations. In Sec. 3, we display the deviations of the DWmodel to GR given the same parameter values through the use of relevant indicators. Thereafter, we presentthe CMB+SNIa+RSD datasets we use in Sec. 4.1 and perform observational constraints on the ΛCDMand DW models to which we display the inferred cosmological parameter distributions in Sec. 4. We thencompare both models against each other in a (approximate) Bayesian perspective. Our conclusions andfuture perspectives are discussed in Sec. 5. We present the most relevant properties of the DW model of Ref. [88] and present its FLRW cosmologicalbackground and linear perturbation evolution equations. We also reproduce the resulting prescription ofRef. [89], for fixing the distortion function f so as to reconstruct the ΛCDM FLRW background expansionhistory within the DW model. – 3 – .1 Model and Cosmological Background The DW model is given by the action [88], S DW = 116 πG ˆ d x √− g R (cid:20) f (cid:18) (cid:3) R (cid:19)(cid:21) , (2.1)where (cid:3) − is the Green’s function of the d’Alembert operator (cid:3) ≡ g µν ∇ µ ∇ ν and f is an arbitrary dimen-sionless function. The modified Einstein equations are obtained by varying the action with respect to the(inverse) metric g µν , G µν + ∆ G µν = 8 πG T µν , (2.2)where ∆ G µν is the correction to Einstein’s equations induced by the nonlocal distortion function f . Theenergy-momentum tensor of matter is, T µν ( x ) ≡ − √− g δS M δg µν ( x ) . (2.3)For a perfect fluid, when expressed in the frame of an observer u µ comoving with it, the stress tensor takesthe form, T µν = ( ρ + p ) u µ u ν + p g µν + π µν , u µ ≡ d x µ d s , (2.4)where the infinitesimal d s ≡ √− d s is the observer proper time, ρ and p are the energy and pressure densityof the fluid probed by the observer, respectively, and π µν is the traceless–transverse anisotropic stress tensorwhose FLRW background value vanishes.On a flat Friedmann-Lemaître-Robertson-Walker (FLRW) background in conformal time τ , one canwrite the metric as, d s = a ( τ ) (cid:0) − d τ + d ~x (cid:1) , H ≡ ∂ τ aa , (2.5)In that setting, the components ∆ G µν take the form,∆ G = 1 a h a H + 3 aH∂ τ i(cid:26) f (cid:0) X (cid:1) + 1 (cid:3) h Rf , (cid:0) X (cid:1)i(cid:27) + 12 a ∂ τ X ∂ τ (cid:18) (cid:3) h Rf , (cid:0) X (cid:1)i(cid:19) , (2.6)∆ G ij = δ ij " ∂ τ X ∂ τ (cid:18) (cid:3) h Rf , (cid:0) X (cid:1)i(cid:19) − (cid:16) aH + 3 a H + ∂ τ + aH∂ τ (cid:17)(cid:26) f (cid:0) X (cid:1) + 1 (cid:3) h Rf , (cid:0) X (cid:1)i(cid:27) , (2.7)where primes denote derivatives with respect to conformal time, overbars ¯ denote background quantities, f, is the derivative of f with respect to, ¯ X ≡ (cid:3) − ¯ R . (2.8)In order to numerically evolve the system, it is convenient to write the equations of motion in the local form.This can be done by inverting the (cid:3) − operator in Eq. (2.8) and introducing another auxiliary field U suchas at the fully covariant level, (cid:3) X ≡ R , (2.9) (cid:3) U ≡ R f , . (2.10)Given some initial spacelike hypersurface at τ , one can for instance solve for X as, X = (cid:3) − R ≡ ˆ ττ d y G ret ( x, y ) R ( y ) + X hom ( τ, ~x ) , (2.11)– 4 –here the Green’s function is of the retarded kind G ( x, y ) ≡ G ret ( x, y ), and X hom is the homogeneoussolution (cid:3) X hom = 0 (see e.g. Ref. [98, 99] for more details). Once τ and the type of the Green’s functionare fixed, the initial conditions of X are determined by the choice of the homogeneous solution X hom andits first derivative. As the Ricci scalar is negligible compared to the typical energy scale in the radiationdominated era (RD), i.e. | ¯ R/H | (cid:12)(cid:12) RD (cid:28)
1, we consider the case where these initial conditions are vanishing .Equations (2.6),(2.7) therefore take the form,∆ G = 1 a (cid:20) (3 a H + 3 aH∂ τ )( f + U ) + 12 X U (cid:21) , (2.12)∆ G ij = δ ij " X U − (2 aH + 3 a H + ∂ τ + aH∂ τ )( f + U ) . (2.13)The cosmological dynamics of the background auxiliary fields is provided by Eqs. (2.9),(2.10) on a flat FLRWbackground, X + 2 aHX = − (cid:0) aH + 2 a H (cid:1) , (2.14) U + 2 aHU = − f , (cid:0) aH + 2 a H (cid:1) , (2.15)and the modified Friedmann equations read, H (cid:0) f + U (cid:1) + Ha ( f + U ) + 16 a X U = 8 πG ρ , (2.16) (cid:0) f + U (cid:1) H = − (cid:26) πG ¯ p + 32 H (cid:0) f + U (cid:1) + H a (cid:16) f + U (cid:17) + 12 a (cid:16) f + U (cid:17) − a X U (cid:27) . (2.17)Once the function f ( X ) is provided, these equations can be numerically integrated. As discussed above, wefocus here on a particular class of DW models where f ( X ) is chosen so as the ΛCDM expansion historyis reproduced within the DW model, for non-trivial f [89]. Before presenting the reconstruction methodof Ref. [89], we display the set of modified Einstein equations of the DW model within linear cosmologicalperturbation theory. We study the linear cosmological perturbations of the DW model in the scalar and tensor sectors. Theperturbed FLRW metric is taken in the conformal Newtonian (longitudinal) gauge,d s = a (cid:2) − (1 + 2Ψ) d τ + (cid:2) (1 − δ ij + h ij (cid:3) d x i d x j (cid:3) . (2.18)where h ij is traceless and transverse with respect to ∂ i and the adopted convention agrees with the ones ofMa & Bertschinger in Ref. [100] and CLASS [96]. In the scalar sector, the longitudinal trace of the perturbed ij component of Einstein equations Eq. (2.2),yields, Ψ = Φ −
11 + f + U (cid:20) πG a k (¯ ρ + ¯ p ) σ + δf + δU (cid:21) , (2.19)where we have defined, (¯ ρ + ¯ p ) σ ≡ a δ im δ jn (cid:18) ∂ i ∂ j ∂ − δ ij (cid:19) π mn , (2.20) Ref. [99] studied the localised version on various initial conditions and found that instabilities were emerging for non–trivialinitial conditions. – 5 –ith the flat Laplace operator ∂ ≡ δ ij ∂ i ∂ j , and we have written, δf ≡ f , δX . According to the integrationscheme adopted in CLASS, we then solve the longitudinal part of the perturbed i component of Eq. (2.2),to obtain Φ ,Φ = − H Ψ + 11 + f + U " πG a (¯ p + ¯ ρ ) θk − (cid:16) f + U (cid:17) Ψ+ 12 ( δf + δU ) − H δf + δU ) − (cid:16) δXU + δU X (cid:17) , (2.21)where H ≡ a /a and θ ≡ ¯ ∇ i v i . The dynamics of the linear perturbations of the auxiliary fields δX and δU is provided by, δX + 2 H δX + k δX = 6Φ + (6 H + X )(Ψ + 3Φ ) − k (Ψ − , (2.22) δU + 2 H δU + k δU = f , h + (6 H + U )(Ψ + 3Φ ) − k (Ψ − i − f ,, (cid:0) H + H (cid:1) δX , (2.23)where we have used the background equations of motion to replace X and U . To complete the above setof equations, we still need to find expressions for Φ and Ψ . The former can be obtained from the trace ofthe perturbed ij component of Eq. (2.2),Φ = 1(1 + f + U − f , ) ( πG a δp + 16 (cid:0) H + 6 H + 2 k (cid:1) ( δf + δU ) + 12 H ( δf + δU ) + 12 (cid:0) δf − f , δX (cid:1) + f , (cid:20) − H ( δX + δU /f , ) − k ( δX + δU/f , ) − k (Ψ − H + X + U /f , )(Ψ + 3Φ ) − f ,, (cid:0) H + H (cid:1) δX/f , (cid:21) − (cid:16) X δU + U δX (cid:17) − (cid:20) H + 2 H )(1 + f + U ) + 12 H ( f + U ) + 2 (cid:0) f − H U − f , ( H + H ) (cid:1) − X U (cid:21) Ψ+ k f + U )(Ψ − Φ) − h H (1 + f + U ) + f + U i (Ψ + 2Φ ) ) , (2.24)where we have replaced the quantities by using their equations of motion and we have written, δf ≡ f ,, X δX + f , δX , (2.25) δf − f , δX ≡ δ f , δτ δX + 2 δf , δτ δX , (2.26) f ,, ≡ (cid:0) X (cid:1) − (cid:2) f (cid:0) X (cid:1) − − f (cid:0) X (cid:1) − X (cid:3) , (2.27)= f (cid:0) X (cid:1) − + f (cid:2) a H (cid:0) X (cid:1) − + 6 (cid:0) X (cid:1) − (cid:0) a H + 2 a H (cid:1)(cid:3) . (2.28)To find an expression for Ψ , we can take the derivative of Eq. (2.19) to get,Ψ = Φ + f + U (cid:0) f + U (cid:1) (cid:20) πG a k (¯ ρ + ¯ p ) σ + δf + δU (cid:21) −
11 + f + U (cid:20) πG a k H (¯ ρ + ¯ p ) σ + 12 πG a k [(¯ ρ + ¯ p ) σ ] + δf + δU (cid:21) . (2.29)The full evolution system of linear cosmological perturbation equations is closed with the addition of theenergy-momentum conservation equations of each individual matter species considered. We start the evolu-tion in deep radiation era and provide vanishing initial conditions for the linear perturbations of the auxiliaryfields δU, δX , in agreement with our “minimal” choice for the boundary conditions.– 6 – .2.2 Tensor Sector The evolution equations for the linear cosmological perturbations of the traceless-transverse part of thespatial 3–metric are given by (see also Ref. [101]), h ij + 2 H (cid:18) − H ∂ τ log (cid:0) G eff , gw ( τ ) /G (cid:1)(cid:19) h ij + k h ij = 16 πG eff , gw ( τ ) a π ij . (2.30)where we have defined, G eff , gw ( τ ) /G ≡ (cid:18) f (cid:0) X ( τ ) (cid:1) + U ( τ ) (cid:19) − . (2.31)Several comments are in order at that point. First, we observe that within the DW model, the propagationequations for GWs are modified in their Hubble friction term (i.e., the coefficient of h A ) as well as in theircoupling to matter with respect to the one described in GR. More precisely, the extra quantities modifyingsuch a behaviour identify themselves in terms of the “Newton constant” for GWs G eff , gw ( τ ) /G , which is infact the | k | → + ∞ asymptotic behaviour of the effective Newton’s constant G eff ( z, k ) within the DW model.The latter is related to the modified growth and lensing features as compared to the same observables ofthe ΛCDM model, as we will discuss in detail in Secs. 3 and 4. Such a structure including G eff , gw ( τ ) /G , isa quite typical fact in modified gravity theories, as is witnessed by the appearance of the same structure inthe RR nonlocal gravity model of Ref. [76] (see Refs. [60, 102]) and generically in Horndeski models, see e.g.Refs. [103, 104].Second, and as a consequence, this structure in particular implies that the GWs propagate at the speedof light, as we will see below. Moreover, the fact that the Hubble friction is altered modifies the amplitudeof the GWs as they propagate though spacetime, e.g. from inspiralling binaries of compact objects toobservers . Noticing that, on a cosmological background in GR, the GW amplitude is inversely proportionalto the luminosity distance for electromagnetic sources 1 /D L ( z ), the accurate knowledge of the amplitudeof their (polarised) strain and of their redshift, obtained for instance from an electromagnetic counterpart(and modulo systematic proportionality factors), therefore allows one to build up a Hubble diagram forcompact binaries emitting GWs, making them “standard sirens” [106, 107]. Modifying the friction termin Eq. (2.30), makes the relations between GWs amplitude and luminosity distance different, forming aluminosity distance for GWs D gw L ( z ), whose Hubble diagram therefore differs from the electromagnetic one.When the propagation equations for GWs are modified such as in Eq. (2.30), both notions are related as[59, 102, 104], D gw L ( z ) ≡ D em L ( z ) × s G eff , gw ( z ) G eff , gw ( z = 0) . (2.32)As we will discuss in more details in Sec. 3.2, the ratio between the Hubble diagrams for electromagneticand gravitational signals provides a powerful way for testing deviations to GR with future third generationGW interferometers such as the Einstein Telescope [108] or the
Cosmic Explorer [109].We now display the result of the reconstruction procedure conduced in Ref. [89].
Following Ref. [89], for reproducing the ΛCDM expansion history within the DW model, for non–trivialdistortion function f (cid:0) X (cid:1) , the latter is expressed in terms of the reduced ΛCDM Hubble expansion rate, h ( ζ ) ≡ H ( ζ ) /H = (cid:2) Ω Λ + Ω( ζ ) (cid:3) . (2.33)where ζ ≡ /a = 1 + z and z is the redshift, H is the Hubble constant today and the total energy densityfraction of the different matter species reads Ω( ζ ) = P i Ω i ( ζ ) ≡ P i πG ¯ ρ i ( ζ ) / (3 H ), and for the dark Actually, the amplitude of the GWs is rescaled as (1 + f (0)) − even in asymptotically flat space, in which X = 0 = U , asnoted in Ref. [105]. – 7 –nergy Ω Λ ≡ Λ / (3 H ). The reconstruction procedure leads to, f ( ζ ) = − ˆ ∞ ζ d ζ ζ φ ( ζ ) − Λ ˆ ∞ ζ d ζ ζ h Λ ( ζ ) I ( ζ ) ˆ ∞ ζ d ζ I ( ζ ) h Λ ( ζ ) ζ + 2 ˆ ∞ ζ d ζ ζ h Λ ( ζ ) I ( ζ ) ˆ ∞ ζ d ζ r ( ζ ) φ ( ζ ) ζ , (2.34)with r ( ζ ) ≡ ¯ R/H = 6 (cid:18) h Λ aH + 2 h (cid:19) , (2.35)and φ ( ζ ) = − Λ ˆ ∞ ζ d ζ h Λ ( ζ ) ˆ ∞ ζ d ζ h Λ ( ζ ) ζ , I ( ζ ) = ˆ ∞ ζ d ζ r ( ζ ) ζ h Λ ( ζ ) . (2.36)In order to solve the background system of equations, one can numerically integrate Eq. (2.34) and, at anytimestep, obtain f as a function of ζ . One can then compose it with the inverse function of, X ( ζ ) = − ˆ ∞ ζ d ζ ζ h Λ ( ζ ) I ( ζ ) , (2.37)to obtain f ( X ). In the left panel of Fig. 1, we show two reconstructions of the function f ( X ) for differentsets of cosmological parameters. We also display the points ( f, X )( z i ) as a function of several redshift values z i , marked by black crosses. The ΛCDM expansion history is reproduced within the DW model via theeffective dark energy it describes, which features a constant equation of state w de = − ∼ Rf ( (cid:3) − R ) can behave exactly thesame way as a cosmological constant, provided the above results of the reconstruction procedure worked outin Ref. [89] are used. As we will see in the next section, the fact that the background expansion history is thesame as ΛCDM, does not imply that the linear cosmological perturbations described by the DW model arenecessarily the same than in ΛCDM. Later, we will also see that this fact makes the DW models attractivein the light of current high precision cosmological data.
16 15 14 13 12 11 X f ( X ) z=10z=5z=4z=3z=2z=1z=0.5z=0.25z=0 f(X) DWDW
CDMfid z w d e ( z ) w de ( z ) DWDW
CDMfid
Figure 1 . Left panel: the distortion function f as a function of X . The black crosses indicate the functionscorresponding values for different values of z . Right panel: effective dark energy equation of state. In both plots, theorange solid lines shows the prediction from DW on its bestfit to the CMB+SNIa+RSD data we describe in Sec. 4.1,while the green dashed line shows the prediction from DW on the ΛCDM best fitting parameter values to the samedata, a cosmological model that we quote DW Λ CDM fid in the following. – 8 –
Cosmological Phenomenology
In this section, we integrate the background and linear perturbation systems using a modified version ofthe linear Boltzmann-Einstein solver CLASS [96]. To do so, we fix both models’ cosmological setting to theso–called
Planck baseline described in detail in Refs. [16, 110] which, in particular, is parametrized by sixcosmological parameters (see Sec. 4 for more details). We first focus on deviations to GR in the scalar sectorand then on deviations in the tensor sector.
From the evolution equations presented in Sec. 2.2, we integrate the scalar linear cosmological perturbationsin the DW model. We use adiabatic initial conditions from a Gaussian random field with a slightly red tiltedflat power spectrum. To evaluate the extent to which the DW model deviates from GR within the scalarsector, it is convenient to introduce the following indicative functions [111–113] , η ( z, k ) ≡ ΨΦ , (3.1) G eff /G ( z, k ) ≡ k Φ4 πG ¯ ρ a δ , (3.2)Ψ ≡ (cid:0) µ ( z, k ) (cid:1) Ψ ΛCDM , (3.3) (cid:0) Ψ + Φ (cid:1) ≡ (cid:0) z, k ) (cid:1)(cid:0) Ψ ΛCDM + Φ
ΛCDM (cid:1) . (3.4)where δ is the linear gauge invariant matter density contrast of total matter, η is the gravitational slip z G e ff / G ( k , z ) DWk =10 Mpc , DWk =1 Mpc , DWk =1 Mpc , RR z ( k , z ) DWk =10 Mpc k =1 Mpc Figure 2 . The effective Newton constant G eff (left panel) and the inverse of the gravitational slip 1 /η for k =10 − Mpc − (green dashed) and for k = 1 Mpc − (solid orange). and G eff is the effective Newton constant. The quantity µ is the deviation of the gravitational potential Ψin the DW model with respect to the one in ΛCDM, whereas Σ measures deviations in the lensing (Weyl)potential. The quantity µ therefore relates to the modification of the motion of non-relativistic matter andis therefore probed through clustering properties of structures (growth) while Σ relates to the motion ofrelativistic particles (e.g. light) and is probed through WL. Another set of alternative indicators to { η, µ, Σ } is given by, η p ( z, k ) ≡ η − ( z, k ) , (3.5) µ p ( z, k ) ≡ − k Ψ( z, k )4 πG a ¯ ρδ = − ηG eff /G ( z, k ) , (3.6)Σ p ( z, k ) ≡ − k (Ψ + Φ)8 πG a ¯ ρδ , (3.7) For these four functions, we work on standard ΛCDM best fitting cosmological parameter values inferred from observationalconstraints given the CMB+SNIa+RSD joined data presented in Sec. 4 (see also Table 2). – 9 – .00.10.20.30.40.50.6 ( k , z ) k =10 Mpc CDMDW
CDMfid z ( k , z ) ( k , z ) k =1 Mpc CDMDW
CDMfid z ( k , z ) Figure 3 . The gravitational potential Ψ in ΛCDM (black dashed) and in DW Λ CDM fid (green solid) for k = 10 − Mpc − (left upper panel) and for k = 1 Mpc − (right upper panel). The lower panels reproduces the indicator µ as a functionof redshift for both chosen length scales, respectively. where we denote with a subscript P the quantities used in Ref. [114]. In that case, all the terms on the righthand sides of the three above equations are evaluated within a given model and no comparisons betweenthe true and alternative hypothesis are made. The latter quantities probe the response of the gravitationalpotential Ψ, and of the lensing potential Ψ + Φ, to the total distribution of matter fluctuations within agiven gravity model. We display the results of both parametrisations for completeness. P ( k )[( M p c / h ) ] P ( k , z = 0) CDMDWDW
CDMfid k [ h / Mpc ]1.00.50.0 P ( k ) Figure 4 . Upper panel:
Gauge invariant linear matter power spectrum at z = 0 computed for the ΛCDM (blackdashed), the DW (orange solid) on their respective best fit to Planck
CMB data and for the DW Λ CDM fid model (browndashed).
Lower panel: corresponding relative differences.
The quantities in Eqs. (3.1)–(3.4) are evaluated on the same cosmological parameter values, the bestfitting of standard ΛCDM to CMB+SNIa+RSD data described in Sec. 4.1, so that the predicted backgroundcosmologies of both models are similar and the relevant differences lie in their linear cosmological perturba-tions. In the scalar sector of the theory, these are conveniently parametrised by pairs drawn from the set { η, G eff , µ, Σ } , and very similarly in the tensor sector as will be discussed below. Such a setting is convenientto express “how far” the alternative model deviates from GR for given features probed by cosmologicalsurveys, such as in the CMB, SNIa or distribution of galaxies. The second one, formed by pairs drawn fromthe set { η P , G eff , µ P , Σ P } , which is more model independent, is most conveniently used for forecasting [53]or constraining [114] deviations from GR given future or current data respectively.– 10 – .951.001.051.101.151.201.25 ( + )( k , z ) k =10 Mpc CDMDW
CDMfid z ( k , z ) ( + )( k , z ) k =1 Mpc CDMDW
CDMfid z ( k , z ) Figure 5 . The lensing potential Ψ − Φ in ΛCDM (black dashed) and in DW Λ CDM fid (green solid) for k = 10 − Mpc − (left upper panel) and for k = 1 Mpc − (right upper panel). The lower panels show the ratio between both minusone, which reproduces Σ as a function of redshift for both chosen wave numbers. In what follows, we work a posteriori and anticipate the observational constraints results providedCMB+SNIa+RSD data presented in Sec. 4. When showing the quantities in Eqs. (3.1)–(3.4), we fix bothmodels’ cosmological parameters to the ΛCDM best fitting values given CMB+SNIa+RSD data of Sec. 4.1and we quote such DW cosmological model as DW Λ CDM fid . The indicators in Eqs. (3.5)–(3.7) are displayedon the CMB+SNIa+RSD best fitting parameter values of each respective model. The left panel of Fig. 2shows the effective Newton constant G eff /G , as a function of redshift at large and small scales. The functionis the same for both scales and modifies the response of the gravitational potential Φ to the fluctuations ofmatter. In the case of the DW model, such a response is enhanced and can leave significant imprints in thepredictions of galaxy clustering features. More precisely, given that the growth is mostly controlled by Ψ(as is seen from the growth equation, see e.g. Eq. 4.45 of Ref. [82]) which relates to Φ through Eq. (2.19),which in turn relates to η , the non–trivial behaviour of η implies that deviations in clustering and lensingpredictions are of opposite trends. Indeed, a trivial behaviour for η , i.e. η ( z, k ) ≈
1, implies that an enhanced G eff ( z ) directly translates into an enhanced growth of structures probed by µ , as well as a stronger lensingpower probed by Σ (examples of such models are provided in Ref. [82]). However, in the case for the DWmodel, η has a non-trivial behaviour as seen from the right panel of Fig. 2. The trend (this is not an exactlimit) η ( z → , k ) →
0, reflects the fact that the anisotropic stress of the effective dark energy describedby the DW model drives the linear perturbation of the potential Ψ to small values at small redshifts. Thisshows that, although G eff /G ( z ) is greater than unity in the DW model, the clustering of linear structuresis lowered compared to the one described by ΛCDM. This is induced by a non-trivial behaviour of theanisotropic stress associated with the effective dark energy described by the DW model.This fact is illustrated in Fig. 3, which shows the deviations of DW Λ CDM fid to ΛCDM in Ψ( z ). Wesee that the latter is lower in the DW model at large and small scales, and the deviation increases at verylate time. This has a significant impact on the growth of structures as is seen from Fig. 4, which shows thetotal matter power spectrum within the DW model as compared to the one described by ΛCDM (on theirrespective best fit values given CMB+SNIa+RSD data), or DW Λ CDM fid . The power spectrum predicted bythe DW Λ CDM fid model (i.e. DW on the same cosmological parameter values as ΛCDM) is lower than theone given by ΛCDM by a constant factor of 10% down to scales of about k ≈ − Mpc − , below which thedeviation of the DW model to GR drops further down. Turning to the lensing potential shown in Fig. 5,we see that a non-negligible anisotropic stress affects the lensing considerably. The latter is pushed in theopposite direction as compared to the deviation in the growth. While a lower growth of structures wouldintuitively involve a deficit in the lensing response as well, or vice versa (such as in the RR nonlocal gravitymodel studied in Ref. [82]), we can see that in the DW model the lensing potential is in fact enhanced by afew tens of percent at late time as compared to the one in ΛCDM. The behaviour of the indicators µ and Σcan be compared with the ones defined in Eqs. (3.5)–(3.7) and which are shown in Fig. 6. We can see thatthe scale dependence is more pronounced for the P –quantities, but the results remain qualitatively the same,up to a few percents. The similarity of both parametrisations partly ties to the fact that the cosmological– 11 – .0 0.5 1.0 1.5 2.0 2.5 3.0 z p ( k , z ) DW CDMfid k =10 Mpc k =1 Mpc z p ( k , z ) DW CDMfid k =10 Mpc k =1 Mpc Figure 6 . The quantities µ P ( z, k ) (left panel) and Σ P ( z, k ) (right panel), in the DW Λ CDM fid model. These are definedaccording to the convention adopted in Ref. [114]. The cosmological parameters are chosen on the respective models’best fitting values to CMB+SNIa+RSD data. Both plots show these quantities for k = 10 − Mpc − (green dashed)and for k = 1 Mpc − (orange solid). background in the DW model is the same as the one of ΛCDM. In Refs. [3, 114], the authors inferred con-straints on the present time values µ ≡ µ ( z = 0) and Σ ≡ Σ( z = 0) (see their Fig. 15) and their possiblescale dependence, through the use of phenomenological functions together with complementary cosmologicaldata. In the case of the DW Λ CDM fid model, the present time values correspond to ( µ , Σ ) = ( − . , . k = 10 − Mpc − , and therefore lie into the “sweet quadrant” for galaxy WL and galaxy clustering datasuch as RSD. – 12 – olar System Constraints. One of the possible drawbacks that arose in nonlocally modified gravity theoriesis the potential remaining of a FLRW background–time dependence in the small scale limit of the Newtonconstant, G eff ( z, k (cid:29) ’ G eff ( z ) = G , (3.8)that is, a lack of screening mechanism that spoils the predictions of the theory at solar system scales.In effects, as originally noticed in Ref. [83], such a residual time-dependence can expose these models todangerous conflicts with Lunar Laser Ranging experiments (LLR), that put bounds on the time variation ofthe Newton constant such as, ˙ G eff /G = (4 ± × − yr − [115]. An example of such a model is providedby the RR model introduced in Ref. [73], where the original Newton constant G remains multiplied by a termdepending on FLRW background quantities. In the case of the DW model, we see that, from the variationof the action Eq. (2.1) evaluated in cosmological perturbation theory, the small scale asymptotics of theeffective Newton constant is a background dependent function too (see also the discussion of Ref. [83]), G eff /G ( z, k (cid:29)
1) = (cid:18) f ( X ) + 1 (cid:3) ( ¯ Rf , ) (cid:19) − , (3.9)and its asymptotic behaviour on small scales is basically the one seen in the left panel of Fig. 2, on a FLRWcosmological background. In Ref. [116] (see also Ref. [117], for similar views), the authors argue that insidebound objects the auxiliary field X = (cid:3) − R is positive, whereas it is negative in cosmology (see left panelof Fig. 1). Then, their point is that, as one is free to choose the distortion function, one can set it so thatit vanishes for positive values of X , i.e. f ( X ) ∼ θ ( − X ), where θ is the Heaviside step function. Hence inthat case, a “perfect” screening mechanism makes the DW model reproduce the well established predictionsof GR on solar system scales.However, as explicitly outlined in Ref. [118], the value of X is actually also negative at solar systemscale, therefore this procedure cannot be applied. The DW model therefore presents the same pathology asthe RR nonlocal one, i.e. the remaining of a time dependence in the small scale limit of G eff ( k, z ), Eq. (3.8).In that case, the question reduces to asking if it is realistic to consider the limiting value G eff ( k, z ),i.e. comprising FLRW background quantities, as valid in the solar system. Indeed, once the k −→ ∞ limitis taken on FLRW, one probes regions where the matter fluctuations becomes nonlinear and virialised, solinear perturbation theory on FLRW can in principle break down. Nevertheless, Ref. [118] argues that thelinear perturbation theory on FLRW background, based on a metric expansion of the form of Eq. (2.18), isstill valid at solar system scales. Hence, any of the models exposing the same pathology as the DW and theRR one are ruled out by LLR.From our point of view, we believe that the conclusions of Ref. [118] are too strong in the view of theapproximations made within their study. Indeed, here the problem consists in being able to understand howthe FLRW background (i.e. averaged) quantities behave when evaluated from cosmological scales down tosolar system ones, where the system “decouples” from the Hubble flow (such as within virialised objects).To do so, in contrast with respect to the method proposed in Ref. [118], we believe that a full non-lineartime– and scale–dependent solution around a non–linear structure would need to be studied. The screeningproperties in the DW and RR models should then be addressed in this framework. Lacking such a solutionat the moment, we will not consider this issue any longer in this paper.Nevertheless, if validated on conceptual ground, the conclusions of LLR constraints are severe for theDW model. Indeed, in Ref. [83], the authors found that this quantity within the RR nonlocal gravity modelis about ˙ G eff /G = 92 × − yr − , putting the model under serious pressure. For the case of the DW modelon its best fit to the CMB+SNIa+RSD data we find ,˙ G eff /G = 3780 × − yr − , (3.10)which strongly rules out the model in the view of the measurements of Ref. [115] at ˙ G eff /G = (4 ± × − yr − , but again, provided the cosmological background solution remains valid on (very) small scales.In any case, when seen from a cosmological perspective, the results of this section motivate the useof current, high precision complementary cosmological data from CMB observations, growth measurementsfrom RSD and WL for constraining the DW nonlocal model. After presenting the relevant GR–deviationsof the DW model in the tensor sector, such a study is addressed in the rest of this article. This result quantifies the “three orders of magnitude deviation” estimated at the end of Sec. 4 of Ref. [118]. – 13 – .2 Indicators from GR–deviations: Tensor Sector
The window of GW astronomy has recently been opened from the observations of several GW signals emittedfrom different coalescing binary systems in the sky. Most of the sources are thought to consist of black holes(see Refs. [119–123]), but GWs from a binary neutron stars (BNS) merger has also been observed [124]. Theadvantage of detecting BNS mergers is the potential of observing an associated electromagnetic counterpartin the IR, optical or UV wavelength. In that case, one can precisely determine the redshift of the source thatfixes the redshift–mass degeneracy in the chirp mass, obtained from the time variation of the frequency of thesignal. On the other hand, the detection of the strain amplitude of the GWs together with an estimation ofthe inclination angle of the binary system (that can for instance be obtained from polarisation–differentiatedmeasurements) allows one to derive the luminosity distance, and therefore a Hubble diagram for GWs, makingthem “standard sirens” (see e.g. Refs. [106, 107] and Ref. [125] for a standard textbook). One of such multi-messenger astronomical signals has been detected last year from the observations of the GWs from the binaryneutron star merger GW170817 by the
LIGO/Virgo collaboration, as reported in Ref. [124], together withthe (quasi–)simultaneous detection of the associated γ -ray burst GRB170817A, presented in Refs. [126–128],and follow up studies of the electromagnetic counterpart in Ref. [129]. Accessing this measurement allowedto put strong constraints on the lower bound of the speed of GWs, which has been found to be equal tothe speed of light within an error of O (10 − ). This constraint implied dramatic restrictions for modifiedtheories, such as those belonging to the (beyond) Horndeski class, for which several operators into the actionare now forbidden (see e.g. Refs. [130–135], Ref. [136] for a review and [137] for a possible way out), butalso in nonlocal modified gravity, as shown in Ref. [74].Another source of deviations to GR in the tensor sector is the way gravitational waves are dampedunder the cosmic Hubble flow. Indeed, the usual propagation equation for tensor modes in GR is given by, h ij + 2 H h ij − ∆ h ij = 16 πG a π ij . (3.11)where π ij is the traceless-tranverse (helicity–2) anisotropic stress matter tensor, non-vanishing when rela-tivistic particles are present in the course of the evolution [138, 139]. In going to Fourier space, writing h ij and π ij on a basis of helicity–2 polarisation tensors, e.g. h ij ≡ h A Q Aij , where A = + , × (see e.g. Ref. [125]),redefining the fields as h A ( τ, k ) ≡ χ A ( τ, k ) /a ( τ ) and discarding the source, one gets, χ A + (cid:18) k − a a (cid:19) χ A = 0 . (3.12)At small scales, k (cid:29) a /a and the first term in the parenthesis dominates. As such, these equations showthat the GWs propagate with dispersion relation k = ω in GR, i.e. at the speed of light. For binaryinspirals on a cosmological FLRW background, the amplitude of the GWs is related to the inverse of theluminosity distance to the source, h A ( τ, k ) ∼ D L (cid:0) z ( τ ) (cid:1) , D L ( z ) ≡ (1 + z ) ˆ z d z H ( z ) . (3.13)Up to systematic factors, the precise knowledge of the amplitude of the GWs and redshift of the source there-fore allows one to reconstruct a Hubble diagram from GW sources, making them “standard sirens” [106, 107].As a GW interferometry is independent from any experiments observing electromagnetic signals, such as theCMB, closeby or distant SNIa or galaxy cluster observations, their constraints are complementary. Regardingthe expansion history, cosmological constraints from current observations of GWs still remain quite loose, asonly one event with the electromagnetic counterpart has been recorded, viz . GW170817/GRB170817A [124].However, GW observations prove to be very useful in the close future, especially within the light of third(or “2.5”) generation GW interometers such as the LISA [140] and
DECIGO [141] space telescopes or theground–based
Einstein Telescope [108] and
Cosmic Explorer [109]. Examples are provided by the study inRef. [60], where the constraints on H from GW170817/GRB170817A and 3G–detector forecast have beenstudied (see also the forthcoming, more realistic, Ref. [142]).If GR is modified, the coefficient of the friction term h A in Eq. (3.11), can possibly be altered in thegeneric fashion (here we follow the discussion of Ref. [60], but see also e.g. Refs. [59, 136, 143–145] for relateddiscussions), h ij + 2 H (cid:0) − Ξ( τ ) (cid:1) h ij + k h ij = 0 , (3.14)– 14 –hich implies that the transformation to get rid of the Hubble friction term for obtaining Eq. (3.12), nowreads h A ( τ, k ) ≡ χ A ( τ, k ) / ˜ a ( τ ), with, ˜ a ˜ a ≡ H (cid:0) − Ξ( τ ) (cid:1) . (3.15)This shows that, in such a class of models, GWs propagate at the speed of light and the above condition inturn implies that the strain of the GWs is now proportional to, h A ∼ ˜ a ( z ) a ( z ) 1 D L ( z ) , (3.16)from which one defines the luminosity distance to GWs sources, D gw L ( z ) ≡ a ( z )˜ a ( z ) D em L ( z ) . (3.17)From Eq. (3.15), one can solve for ˜ a ( z ) in terms of Ξ( z ) to obtain [60, 74], D gw L ( z ) ≡ D em L ( z ) × exp (cid:18) − ˆ z d z z Ξ( z ) (cid:19) . (3.18)In modified gravity theories, the Hubble diagram built up from standard candles (i.e. electromagneticsources) can therefore be different from the one constructed from standard sirens. Measuring the deviationsbetween the Hubble diagram from electromagnetic and GW sources can therefore provide a clear signaturefor modification to GR. Such a fact has been studied in Ref. [60] which, rewriting Eq. (3.18) by using afunction parametrised by the pair (cid:0) Ξ , n (cid:1) such as, D gw L ( z ) D em L ( z ) = Ξ + a ( z ) n (cid:0) − Ξ (cid:1) , (3.19)found that the parameter Ξ can be constrained four times more accurately than the equation of state today w from the well–known Chevallier-Polarski-Linder parametrisation [146, 147]), given current CMB+SNIa+BAOconstraints joined with forecast constraints from third generation GW experiments. The perspectives toprobe modifications to gravity from future generation of GW intereferometers are therefore more optimisticthan originally expected. Similar cosmological constraints including refined data analyses for the assumedcoincident electromagnetic+GWs signals detection rate will be presented in Ref. [142].In Sec. 2.2, we have seen that the DW model features a similar modification in the Hubble friction termthan in Eq. (3.14). Such a modification involves the quasi–static limit of the effective Newton’s constant(see Eq. (2.30)) and this structure is found in several theories of modified gravity [102–104]. Likewise, in theDW model, the modified luminosity distance for GWs, as compared to the one provided by electromagneticsources, is given by, D gw L ( z ) ≡ D em L ( z ) × s G eff , gw ( z ) G eff , gw ( z = 0) , (3.20)where G eff , gw ( z ) /G ≡ G eff ( z, k ) /G | | k |(cid:29) and is seen from the orange solid curve in the left panel of Fig. 2.The luminosity distance for GWs is shown in the upper panel of Fig. 7, while the lower panel shows its ratio tothe electromagnetic luminosity distance, for the ΛCDM (black solid) and DW (orange solid) model on theirrespective bestfit to the CMB+SNIa+BAO data that we describe in Sec. 4.1. Following the parametrisationof Ref. [102] in Eq. 3.19, and fixing the cosmological parameters to the best fit to CMB+SNIa+RSD data,we find that the DW model provides the parameter values Ξ = 0 .
843 and n = 11 / .
75. These valuescan be compared to the preferred ones in the RR nonlocal model being Ξ
RR0 = 0 . n RR = 5 / . In Ref. [60], The cosmological parameter values used for the RR model are the best fitting values inferred from the constraints givenCMB+SNIa+BAO data performed in Ref. [85]. – 15 –he authors worked out an estimation of the number of standard sirens needed to discriminate betweenstandard ΛCDM and the RR nonlocal model, given current high precision CMB+SNIa+BAO data joinedto the forecast constraints of third generation GW interferometers. Within the (although greatly simplified,see e.g. Ref. [142]) framework of their study, the authors find that ∼ ∼
200 and ∼
400 standard sirensare needed to distinguish between ΛCDM and RR in the “optimistic”, “realistic” or “pessimistic” scenario,respectively. As we can see from the values of (cid:0) Ξ , n (cid:1) for DW as compared to the ones of RR, the deviationsfrom GR within the former are much more prominent than those in the RR model. This means that lessstandard sirens will be needed to distinguish between the ΛCDM model and DW. As GW interferometry(cosmological) pipelines are still under development (see e.g. Ref. [142]), a quantitative estimation of thatnumber within our context is behind the scope of the present paper. Similarly, constraints on the cosmologicalparameter space in the DW, given the forecast sensitivity of future GWs experiments is left for future work.In what follows, we carry out observational constraints and model selection given current complementarycosmological data. Figure 7 . Upper panel: gravitational waves luminosity distance as a function of redshift in ΛCDM (black solid),DW (orange solid) on their best fit to CMB+SNIa+RSD and for the RR model (brown dashed).
Lower panel: theratio between luminosity distance from standard sirens to the one from standard candles.
We place observational constraints on the cosmological models made out of the DW nonlocal gravity andGR plus a cosmological constant Λ, given high precision complementary data from CMB, SNIa and RSDobservations. In the previous section, we have displayed a set of relevant indicators to conveniently quantifythe extent to which the DW model deviates from GR. On these grounds, we motivate the cosmological probeswe use in the constraints and introduce the associated datasets. Then, given different joined combinationsof these datasets, we analyze the impact induced by the GR–deviations of the DW model on the preferredcosmological parameter subspace and compare it with the one preferred by ΛCDM. Finally, we also apply(approximate Bayesian) model selection for comparing the ΛCDM and the DW models at each step. Wethen draw our statistical conclusion and discuss future perspectives.
Here, we discuss the type of cosmological probes we choose to efficiently constrain the DW model and pointout the associated observational datasets. In Sec. 3, we have shown several relevant indicators of deviations toGR in the scalar and tensor sectors of the theory. In the scalar sector, we have seen that the set { G eff , µ, Σ , η } in Eqs. (3.1)–(3.4) predicted by both models, offered a convenient way to appreciate the deviations of theDW model to GR. For the same fiducial cosmology, the DW model has a background cosmology similar to– 16 –he one in ΛCDM and their deviations therefore only lie in their linear (and nonlinear) perturbations. Wehave seen that the DW model features a deficit of growth of linear structures – about −
10% in the totalmatter power spectrum, at the scales of interest – but a higher lensing power as compared to ΛCDM. Botheffects begin at late time, when the dark energy density starts to dominate the energy density of the Universe(at z (cid:46) z fσ ± σ fσ Survey z fσ ± σ fσ Survey0.001 0.505 0.085 2MTF [148] 0.52 0.488 0.065 BOSS DR12 [149]0.02 0.428 0.0465 6dFGS+SNIa [150] 0.56 0.482 0.067 BOSS DR12 [149]0.02 0.314 0.048 2MRS [151, 152] 0.57 0.417 0.045 SDSS DR10 and DR11 [153]0.02 0.398 0.065 SNIa+IRAS [152, 154] 0.57 0.426 0.029 BOSS CMASS [155]0.067 0.423 0.055 6dFGS [156] 0.59 0.481 0.066 BOSS DR12 [149]0.1 0.370 0.130 SDSS-veloc [157] 0.59 0.488 0.060 SDSS-CMASS [158]0.1 0.48 0.16 SDSS DR13 [159] 0.60 0.390 0.063 WiggleZ [160]0.1 0.376 0.038 SDSS DR7 [161] 0.60 0.433 0.067 SDSS-BOSS [162]0.15 0.490 0.145 SDSS-MGS [163] 0.60 0.550 0.120 VIPERS PDR-2 [164]0.17 0.510 0.060 2dFGRS [165] 0.64 0.486 0.070 BOSS DR12 [149]0.18 0.360 0.090 GAMA [166] 0.727 0.296 0.0765 VIPERS [167]0.25 0.3512 0.0583 SDSS-LRG-200 [168] 0.73 0.437 0.072 WiggleZ [160]0.3 0.407 0.055 SDSS-BOSS [162] 0.76 0.440 0.040 VIPERS v7 [169]0.31 0.469 0.098 BOSS DR12 [149] 0.77 0.490 0.18 VVDS [165]0.32 0.427 0.056 BOSS-LOWZ [153] 0.80 0.470 0.08 VIPERS [170]0.32 0.48 0.10 SDSS DR10 and DR11 [153] 0.85 0.45 0.11 VIPERS PDR-2 [171]0.35 0.429 0.089 SDSS-DR7-LRG [172] 0.86 0.48 0.10 VIPERS [173]0.36 0.474 0.097 BOSS DR12 [149] 0.86 0.400 0.110 VIPERS PDR-2 [164]0.37 0.4602 0.0378 SDSS-LRG-200 [168] 0.978 0.379 0.176 SDSS-IV [174]0.38 0.440 0.060 GAMA [175] 1.05 0.280 0.080 VIPERS v7 [169]0.40 0.419 0.041 SDSS-BOSS [162] 1.23 0.385 0.099 SDSS-IV [174]0.40 0.473 0.086 BOSS DR12 [149] 1.40 0.482 0.116 FastSound [176]0.44 0.413 0.080 WiggleZ [160] 1.52 0.420 0.076 SDSS-IV [177]0.44 0.481 0.076 BOSS DR12 [149] 1.52 0.396 0.079 SDSS-IV [178]0.48 0.482 0.067 BOSS DR12 [149] 1.526 0.342 0.070 SDSS-IV [174]0.5 0.427 0.043 SDSS-BOSS [162] 1.944 0.364 0.106 SDSS-IV [174]
Table 1 . Redshifts, means, standard deviations and names with corresponding references of the various RSDmeasurements we consider in that work.
We therefore need cosmological surveys that efficiently probe the growth history of the Universe. To date,robust data are provided by growth features such as the linear growth rate f σ , accurately measured by RSDobservations, as well as the WL (of e.g CMB or galaxies), that usually measure the combination ∼ σ (Ω M ) α ,where α is a number depending on the survey considered. To that end, we constrain the linear growth ofstructures with several measurements of f σ , from various surveys that resolve RSD (see Table 1 for details),while the lensing power is constrained with the (lensed) CMB temperature and polarisation (cross) spectra,as well as the tri–spectrum extracted CMB WL measurements of Planck σ within a given model, while RSD measurements form anadditional and complementary probe. Additionally, at the background level, the CMB constrains the acousticdistance–scale ratio θ ∗ ≡ r s ( z ∗ ) /D A ( z ∗ ) with great accuracy and in turn fixes the value of the sound horizonto recombination, given additional data are provided to break the geometrical degeneracy between Ω M and Ω Λ in D A ( z ∗ ) (see e.g. Ref. [179]). For instance, one could make use of the galaxy clustering BAOfor building up a distance ladder for constraining further the expansion history at late time. However,geometrical distortions between longitudinal and transversal BAOs (Alcock-Palschinsky effect), induced bye.g. modifications of gravity, are degenerated with RSD. Therefore, for all the RSD measurements that weconsider in that work (see Table 1), we would need to access and implement the full covariance matrix thatincludes BAO shape measurements for controlling that degeneracy. Since such a requirement is hard for usto realise in practice, we use RSD measurements where the BAO shape information has been marginalisedout. Although this is a ΛCDM–dependent procedure, it is not worrisome in our case since the DW modeldescribes the same FLRW background as ΛCDM. Moreover, we see from Fig. 4 that, around the scale ofinterest for the RSD ( ∼
100 Mpc), no scale dependence is introduced in the growth by the DW modificationto GR, as the matter power spectrum solely undergoes a constant shift. This is also true for the redshiftrange of interest for RSD measurements, as we have explicitly checked. We are therefore safe to use currentRSD data to constrain the DW model.Lacking the BAO shape information associated with these RSD data, we adopt the simplest (but notless worth) strategy to fix the angular distance to the last scattering surface by using distance scale datafrom the distant SNIa of the
JLA compilation described in Ref. [180]. This partially breaks the geometricaldegeneracy from CMB independent data and makes the inferred constraints especially robust.– 17 –irect H measurements which prefer higher values as compared to Planck
CMB (as the ones mentionedin Sec. 1) will exert the same tension within the DW model as compared to within ΛCDM, as both have thesame background expansion history.Finally, we notice that the combination of the amplitude of fluctuation σ and density fraction Ω M ofmatter, can also be efficiently probed by observations of the cosmic shear from galaxy WL or from X–ray orSZ selected cluster counts. However, as mentioned in Sec. 1, these data are still subject to current debatesabout systematic issues that makes the interpretation of the resulting constraints delicate. We thereforemake the conservative choice not to include such probes into our analysis, but nevertheless discuss theirpotential impact at the end of our study.We now present the datasets we use in the observational constraint study that follows. CMB.
For the CMB, we use the likelihoods of
Planck ‘ ≤
29) and the high- ‘ Plik TT,TE,EE (cross-half-mission) ones for the high multipoles ( ‘ >
29) of the power spectra [114, 182].Moreover, in order to further constrain the excess of lensing in the DW model, we also include the powerspectrum of the lensing potential extracted from the CMB trispectrum (where only the conservative multipolerange ‘ = 40 −
400 is used). Such a consideration allows to break degeneracies in the primary CMBanisotropies and to place further constraints on the growth history at late times (see e.g. [183, 184]). Inwhat follow, this set of CMB data will be quoted as
Planck and the same dataset excluding the reconstructedlensing maps as
Planck wo lensing . Distant SNIa.
For the distant SNIa, we consider the data of the
SDSS-II/SNLS3
Joint Light-curve Analysis(JLA) of Ref. [180], and make use of the complete (non-compressed) corresponding likelihoods.
RSD.
Finally, the RSD measurements that we consider are listed in Table 1. To use this data, we implementthe computation of the growth rate, f σ ( z ) = d log δ M d log a σ ( z ) , (4.1)where the perturbation quantities are evaluated at the wave number of interest for RSD surveys, i.e. k ’ − Mpc − . We then construct the RSD likelihood in considering the data points provided in Table 2. Inour constraints, we compress these data in forming weighted averages of points close in redshift, i.e. in binsof ∆ z ’ . Priors.
Regarding the cosmological setup we use, the primordial fluctuations are considered to be adiabaticand Gaussian, with a slightly red tilted power spectrum, together with which, the reionisation historyand matter content in the Universe are taken according to the
Planck baseline (see e.g. Ref [110]). Theprescription for f in the DW model we consider, i.e. reproducing the ΛCDM expansion history (see Sec. 2.3for more details), precisely requires the input of the ΛCDM Hubble expansion rate H ΛCDM ( z ), so the DWmodel has the same number of parameters as ΛCDM. Within the baseline, the models can be parametrizedby the six–dimensional vector, θ base = (cid:0) H , ω b , ω c , ln(10 A s ) , n s , τ re (cid:1) , (4.2)where H is the Hubble parameter today, ω b ≡ Ω b h and ω c ≡ Ω c h are the physical baryon and cold darkmatter density fractions today, respectively, A s is the amplitude and n s the spectral tilt of the power ofprimordial scalar perturbations, and τ re is the reionization optical depth. We choose improper flat priors onall these parameters, except for τ re which is bounded from below at 0 .
01, in accordance with Gunn-Petersontrough observations (see e.g. Ref. [185]). The baseline also assumes the presence of two massless neutrinostogether with a third one which is massive. The massive neutrino mass therefore plays the role of the absoluteneutrino mass M ν and is fixed to the lowest value allowed by terrestrial experiments within the baseline,that is, M ν = 0 .
06 eV [110]. However, first, considering a varying M ν is perfectly legitimate given the presentbounds from neutrino oscillation experiments (see e.g. Sec. F of Ref. [78] and Refs. therein) and second, theabsolute neutrino mass is known to be degenerated with effects of modified gravity at background, linearand nonlinear perturbation level (see for example Refs. [78, 79, 81]), and fixing the prior M ν = 0 .
06 eV, couldtherefore result in biased inferences. This is for instance the case in particular classes of modified gravity– 18 –heories, as for the RR nonlocal model considered in Refs. [78, 85], where the authors show that a dark energyof the phantom nature, i.e. with equation of state w de ( z ) < −
1, can typically give raise to a tension between
Planck
CMB,
JLA distant SNIa and RSD data, in the framework of the baseline Eq. (4.2). Nevertheless,the authors then also show that this discrepancy can be healed by allowing the absolute neutrino mass M ν to vary. For the DW model, the background cosmology is the same as in ΛCDM and the growth of linearstructures is lower as compared to the one in ΛCDM, thus fixing M ν to its lowest allowed value is, a priori,sufficient in the present framework for consistently confronting the DW model against ΛCDM. Planck Planck +RSD
Planck +RSD+
JLA
Param ΛCDM DW ΛCDM DW ΛCDM DW100 ω b . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . ω c . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . H . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . ln(10 A s ) 3 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . n s . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . τ re . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . σ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . z re . +1 . − . . +1 . − . . +1 . − . . +1 . − . . +1 . − . . +1 . − . χ − − . . . . χ − − . . χ JLA − − − − . . χ JLA − − − − . χ . . . . . . χ . . . B . . α × . × . × . Table 2 . Means and standard deviations of the inferred cosmological parameters given the associated dataset and(effective) χ goodness-of-fit. The ∆ χ values are taken with respect to the lowest value within each dataset and χ ≡ − L , where L is the likelihood function. The quantity ∆ B and α are defined in Eq. (4.4) and Eq. (4.7),respectively. In this section, we perform observational constraints on the DW and ΛCDM models given three joinedcombinations of the high precision complementary datasets presented above.To begin with, we start by evaluating the posterior distribution within the six–dimensional parameterspace Eq. (4.2), provided the priors and the
Planck
CMB dataset described in Sec. 4.1. The second andthird columns of Table 2, show the inferred cosmological parameter means and standard deviations for theΛCDM and the DW model, respectively. As we can see, the background–related parameters (cid:8) H , ω b , ω c (cid:9) ,do not significantly change from ΛCDM to DW ( (cid:46) . σ shift). Of course, this results from the fact thatthe DW model under consideration is designed so as to reproduce the ΛCDM cosmological backgroundhistory. The mild shifts in the background–parameters are mostly due to correlations with other parametersor statistical fluctuations. Nevertheless, among the quantities mostly related to the linear perturbations (cid:8) ln(10 A s ) , n s , τ re (cid:9) , A s and τ re are significantly different from the preferred ones in ΛCDM.We show the angular power spectrum of the CMB temperature anisotropies predicted from the ΛCDM,DW and DW Λ CDM fid models and compare them with Planck ‘ inthe TT power for DW Λ CDM fid . This region being dominated by the integrated Sachs–Wolfe (ISW) effect, thisresults from a lower gravitational potential Ψ at late-time and can be seen in Fig. 3. Moreover, on higher ‘ ’s in all the spectra, one can see a mismatch in the prediction of the CMB peaks and troughs amplitude inthe DW Λ CDM fid model. We see that peaks are lower and troughs less deep, i.e. the lensing smoothing of theCMB temperature and polarisation (cross) power spectra is increased within the DW model. Indeed, thisagrees with the results of Sec. 3 [see e.g the indicator Σ( z, k ) in Fig. 5], where we saw that the DW model Recall this is the DW model on ΛCDM–best fitting cosmological parameters to the data under consideration. – 19 – ( + ) C TT / ( )[ K ] CDM PlanckDW PlanckDW
CDMfid
Planck
Planck10 low - C TT
500 1000 1500 2000 2500 high - Figure 8 . Upper panel: temperature power spectrum for the ΛCDM (black dashed), the DW (orange solid) ontheir best fit to
Planck
CMB and for the DW Λ CDM fid model (brown dashed).
Lower panel: residuals for ΛCDM andthe difference between the prediction in DW (orange solid) and DW Λ CDM fid (brown dashed) as compared to the oneof ΛCDM. Green data points are from the
Planck ± σ uncertainty.
250 500 750 1000 1250 1500 1750 200015010050050100150 ( + ) C T E / ( )[ K ] CDM PlanckDW PlanckDW
CDMfid
Planck
Planck250 500 750 1000 1250 1500 1750 200010010 C T E ( + ) C EE / ( )[ K ] CDM PlanckDW PlanckDW
CDMfid
Planck
Planck10 C EE Figure 9 . Upper left panel:
TE cross–correlation power spectrum for the ΛCDM (black dashed) and the DW (orangesolid) on their best fit to
Planck
CMB, and for the DW Λ CDM fid model (brown dashed).
Lower left panel: residuals forΛCDM and corresponding difference in the TE–power spectra.
Upper right panel:
E–mode CMB polarization powerspectrum for the ΛCDM (black dashed) and the DW (orange solid) on their best fit to
Planck
CMB, and for theDW Λ CDM fid model (brown dashed).
Lower right panel: residuals for ΛCDM and the corresponding difference in theEE–power spectra. Data points are from
Planck ± σ uncertainty. describes a higher lensing power as compared to ΛCDM, for fixed cosmological parameters. This means thatfor accessing a given amount of lensing smoothing for the CMB TT, TE or EE power spectra, or of the– 20 –econstructed lensing potential, the amplitude of the fluctuations ( ∼ A s ∼ σ ) therefore needs to be smallerin DW as compared to ΛCDM. This explains why the amplitude of primordial fluctuations A s significantlyshifts, or equivalently σ that undergoes a ∼ σ lower shift, and in turn also why the TT, TE and EE powerspectra are lowered once the DW model is fit to Planck
CMB (see Figs. 8 and 9).The excess of lensing power within the DW model is better illustrated in Fig. 10 which displays the CMBlensing potential power spectrum ‘ ( ‘ +1) C φφ‘ and Fig. 11 that shows the difference between the unlensed (ul.)and the lensed (l.) CMB TT angular power spectrum, as predicted from the ΛCDM, DW and DW Λ CDM fid models. From the latter, we can indeed see that the lensing smoothing to the CMB TT power spectrum inthe DW model is ∼ +10%, when the DW model parameters are fixed to the ΛCDM Planck best fit.On Figs. 10 and 11, we also show the prediction of the DW model given
Planck
CMB data whenexcluding the reconstructed C φφ‘ . We can see that the lensing smoothing of the TT power spectrum isalso ∼ +10% stronger in that case. This is caused by the fact that the CMB power spectra genericallyprefer stronger lensing smoothing (i.e. overestimes σ ) than the weak lensing potential extracted from thetemperature anisotropies four–point function (see Fig. 10, and e.g. Refs. [3, 16] for more details). Such afact implies that the addition of CMB WL C φφ‘ data to the fit to Planck
CMB TT+TE+EE power spectradecreases the amplitude of matter fluctuations. Moreover, Fig. 12 clearly displays the distinction betweenthe predictions of the DW and ΛCDM models in the Ω M – S [ ≡ σ (Ω M / . . ] plane. Another way tounderstand why the amplitude of matter fluctuation is smaller in DW is to refer to the left panel of Fig. 2,where we can see that the effective Newton constant G eff ( z ) is significantly increased at small redshifts inthe DW model as compared to ΛCDM, as well as to the RR nonlocal model. ( + ) C / ( )[ K ] CDM PlanckDW PlanckDW
CDMfid
PlanckDW CMB + SNIa + RSDDW Planck w / o C Planck50 100 150 200 250 300 350 400 4501012 C Figure 10 . Upper panel: lensing potential power spectrum in ΛCDM (black dashed), DW (orange solid)and DW Λ CDM fid (brown dashed) given
Planck
Planck w/o lensing (orange dotted).
Lower panel: residual datapoints for ΛCDM and the difference between the prediction in DW (orange solid) and in DW Λ CDM fid (brown dashed)with respect to the one in ΛCDM. Data points are from
Planck ± σ uncertainty. In turn, as the damping tail of the CMB accurately measures the combination ∼ A s e − τ re , togetherwith a decrease in the amplitude A s , the optical depth to reionisation τ re also decreases (i.e. the smaller thefluctuations, the later the reionisation). The spectral tilt n s is however almost unaffected.From the two last lines of the second and third columns of Table 2, we can read the χ goodness-of-fit obtained from both distributions . We can see that, despite having very different behaviour in theirlinear scalar perturbations, and therefore different preferred cosmological parameter values, both models The sampling method for obtaining such values and associated parameters bestfits is detailed in Ref. [85]. – 21 – ( + )( C TT , n . l . C TT , l . ) / ( )[ K ] CDM PlanckDW PlanckDW
CDMfid
PlanckDW Planck w / o C ( C TT , n . l . C TT , l . ) Figure 11 . Upper panel: differences between the unlensed (ul.) and lensed (l.) temperature angular power spectrafor the same models as in Fig. 10.
Lower panel: differences of C TT, ul.‘ − C TT, l.‘ between the prediction of the DWmodel (orange solid) and of DW Λ CDM fid (brown dashed) with respect to the one in ΛCDM. are statistically indistinguishable given the
Planck
CMB data, as they have almost equal χ goodness-of-fitvalues. Figure 12 . Two dimensional marginalised posterior distribution in the Ω M – S plane for DW (orange contour) andΛCDM (grey contour) provided Planck
CMB data and individual constraints from the
KIDS-450 survey of Ref. [186](green contour). The shaded regions correspond to 1 σ and 2 σ confidence level. The corresponding black encircledlighter regions correspond to the 1 σ contour given the CMB+SNIa+RSD data we describe in Sec. 4.1. As discussed in Sec. 4.1, we use the RSD measurements reported in Table 1 and displayed in Fig. 13,to further constrain the CMB–calibrated growth history described by the DW model, and compare itsperformance to the one of standard ΛCDM given
Planck
CMB+RSD data. The results are shown in thefourth and fifth columns of Table 2. Concerning parameter shifts, we can see that the addition of RSDdata to
Planck
CMB has the opposite effects on the models. Indeed, while σ is pushed towards slightlylower values in ΛCDM ( ∼ . σ ), it is preferred higher in the DW model. This fact can also be been in– 22 – .30.40.50.60.7 f ( z ) CDM PlanckDW PlanckCDM CMB + SNIa + RSDDW CMB + SNIa + RSD
RSD0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 z f Figure 13 . Upper panel: growth rate computed for the ΛCDM (black dashed) and the DW (orange solid) on their
Planck
CMB bestfit, and for the DW Λ CDM fid model (brown dashed), together with the data points detailed in Table. 1.
Lower panel: corresponding residuals and relative differences. the Ω M – S plane, as illustrated by the two extra lighter 1 σ contours in Fig. 12, that are produced usingCMB+SNIa+RSD data . This fact can also be understood in referring to Fig. 13, where we can see that the Planck –CMB calibrated growth rate described by the DW model underestimates most of the f σ data valuesfrom RSD. The data therefore favour a higher amplitude σ , so as to compensate for the too large deficitin the growth. On the contrary, ΛCDM slightly overestimates the data and the RSD constraints thereforefavour lower amplitudes to matter fluctuations than Planck
CMB data alone. As a consequence, because theamplitude of fluctuations is anti-correlated with the total matter density fraction Ω M , the latter tends to(although quite slightly) increase in the DW model, while it tends to decrease within ΛCDM. Furthermore,as the CMB shape information accurately constrains the combination ω M ≡ Ω M h , the Hubble constant H therefore increases accordingly in ΛCDM, and decreases within the DW model, a (not significant) tendencythat is however not preferred by the direct measurements of H evoked in Sec. 1.From the two last lines of the fourth and fifth columns of Table 2, we see that the addition of RSDmeasurements to Planck
CMB data creates a tension within the DW model as compared to ΛCDM (with∆ χ = 8 . χ = 5 . Planck
CMB within DW, the trend for lower σ induced by boosted CMB lensingfeatures of the DW model, in addition to a milder lensing power favoured by the reconstructed Planck
CMB WL data, then competes with the preference for higher growth of the RSD data, i.e. for a larger σ .Reciprocally, such a fact unavoidably comes together with an increased CMB lensing power. In the DWmodel, this tends to push the predicted CMB lensing potential C φφ‘ away from Planck
CMB 1 σ errorbarsat low– ‘ , as can be seen from the left panel of Fig. 10. This is precisely where the CMB–RSD tension is atplay. The standard ΛCDM model turns out to be favoured over DW with a Bayes Information Criterion(BIC) (see e.g. Ref. [187] for a comprehensive discussion) of ∆ χ = 5 .
3, given
Planck
CMB+RSD data.This value becomes ∆ χ = 5 .
4, when also including the SNIa for refining further the constraints. Accordingto the Jeffrey scale reported in Ref. [187], such BICs can be interpreted as “weak” evidences for ΛCDM As will be discussed below, the addition of distant SNIa data from Ref. [180] does not affect our argument on growth andlensing properties of the DW model. – 23 –gainst the DW model given the data. The conclusion is that the joined dataset described in Sec. 4.1, doesnot possess enough constraining power for significantly distinguishing between standard ΛCDM and DW.A better approximation to the Bayesian factor B , which encapsulates a simplified version of theOccam’s razor, has been proposed in Ref. [188] and reads,2 ln B ≈ ∆ B = ∆ χ − ln P F P F (4.3)where P , , F , are the determinants of the prior and of the likelihood parameter inverse covariance matrices(i.e. of the Fisher matrices), respectively, of the two models 1 , P = P . Then, if model 1 is ΛCDM and model2 is DW, ∆ B = ∆ χ − ln F F . (4.4)Since B is to be interpreted as the odds of model 1 with respect to model 2, the expression P = B B , (4.5)is the probability that model 1 is the correct one, in a space of models represented only by 1 or 2. Then, P = 11 + e − ∆ B/ , (4.6)and Jeffrey’s scale can be replaced by the usual 1 , , σ probability levels of P . That is, model 1 is betterthan model 2 at a σ level of, α = −√ − [ 11 + e − ∆ B/ + 1] . (4.7)In the framework of our study, the corresponding values of ∆ B are found in the second last column of Table 2and can be compared with the BIC ∆ χ . We can see that the contribution of the supplemental term inEq. (4.4), are non–trivial but not significant as compared to the contributions of the BIC ∆ χ , found in ourcase.The associated values of α are reported in the last line of Table 2. Interpreted in such terms the standardΛCDM model is favoured against the DW model as ∼ . σ given the Planck
CMB+RSD data describedin Sec. 4.1, while the discrepancy reduces to ∼ . σ , when the JLA
SNIa lightcurve compilation is addedto the latter. As already discussed, this discrepancy is not stringent enough for ruling out the DW modelagainst ΛCDM on firm statistical grounds, and additional data are needed for potentially being able to tellthe difference between both cosmologies.
We study the cosmological phenomenology of a particular class of nonlocal modification to gravity providedby the DW nonlocal gravity model given in Eq. (2.1). Within this class, we consider a particular type ofmodels where the free distortion function f is fixed so as to reproduce a given ΛCDM expansion historywithin the DW model. Such a fact implies that the DW model only deviates from GR at the cosmologicalperturbation level that we have within the linear scalar and tensor sectors.In the former case, by using a set of relevant indicators, we have seen that the DW model genericallydescribes a lower linear growth rate but a stronger lensing power, as compared to ΛCDM on the samecosmological parameter values.Within the tensor sector, we saw (see Sec. 3.2) that the DW model modifies the way GWs are damped whenpropagating on a cosmological background, and this fact makes the Hubble diagram that can be constructedfrom standard sirens deviate from the one obtained from standard candles (i.e. electromagnetic sources).Such a deviation is known to be efficiently constrained by future third generation GW experiments andrelated forecast constraints are left for future work.In Sec. 4, we perform observational constraints and use model selection techniques to confront the DWmodel against standard ΛCDM, given high precision cosmological data. Provided the studied GR–deviations– 24 –nduced by the DW model, we choose to constrain both cosmological models by using the (lensed) Planck
CMB temperature and polarisation (auto and cross) power spectra together with the tri–spectrum extractedCMB WL potential power spectrum. In addition, we join complementary RSD data to the CMB ones forexerting further constraints on the linear growth, as well as SNIa data for further increasing the constrainingpower.As a result, the DW model is statistically equivalent to ΛCDM in the light of the
Planck
CMB dataset,but prefers significantly lower values for the amplitude of matter fluctuations σ , as well as for the opticaldepth to reionisation τ re . We have argued that such a fact results from the higher lensing power in DW ascompared to ΛCDM, that pushes σ to lower values. This trend is however not favoured by the linear growthrate data from RSD measurements we consider (see Table 1), as most of the f σ data points lie above theCMB–calibrated prediction of the DW model (see Fig. 13). The RSD data therefore prefer higher valuesfor σ and this explains the mechanism driving the tension appearing within the DW model when joining Planck
CMB together with RSD data.According to the Jeffrey scale, such a tension (with ∆ χ ’ .
35 or ∼ . σ , see Sec. 4.2) is qualified as“weak” evidence against the DW model as compared with ΛCDM. Thus, the cosmological data considered isnot accurate enough to discriminate between the ΛCDM and DW models, and complementary informationis needed for eventually being able to tell the difference.On the contrary, lower values of σ are favoured by the galaxy WL data from the KiDS-450 survey.Indeed, in referring to the two–dimensional marginalised constraints in Ω M – S plane Fig. 12, we see thatthe lower value of S ∼ σ preferred by the DW model makes it more consistent with the galaxy WL datafrom KiDS-450 , as compared with ΛCDM. This implies that adding
KiDS-450
WL data to our global fitwould tend to favour the DW model over ΛCDM, and therefore to pull the BIC towards negative values (i.e.where DW is favoured by the data) . However, as mentioned in Sec. 1, it is not yet clear that these data arefree from uncontrolled systematics, so we decide to only briefly and qualitatively comment on their potentialimpact on our results, rather than including them within the full analysis.Moreover, in Sec. 3.2, we have seen that DW modifies the way GWs propagate as compared to ΛCDMand such a feature is known to be efficiently probed by future third generation GW interferometers (seee.g. Ref. [60]). Furthermore, the characteristic linear growth and lensing features of the DW model exposethe need to be further constrained by current galaxy clustering and WL data, as for instance the ones from
DES [2], or future galaxy and WL surveys such as
Euclid , DESI , LSST , SKA [49, 50] or
Stage-4 CMB experiments. Future experiments such as galaxy and WL surveys, or GW interferometers therefore appearof prime interest for efficiently constraining the modifications to GR induced by the DW model, and forultimately being able to distinguish it from standard ΛCDM.Finally, we should not forget that this analysis is legitimate only in the case where one shows that thesmall-scale limit of the effective Newton constant G eff ( z ) within the DW model, tends toward G in boundobjects (virialised systems). Indeed, when evaluating the effective Newton constant in DW in the subhorizonlimit G eff ( z, k ) | | k |(cid:29) , a FLRW background time–dependence persists within the latter and G is not reached.However, FLRW background quantities only make sense on cosmological scales, and the nature of theirbehaviour when transposed to solar system scales is not clear at all.If one can show that G eff ( z, k ) | | k |(cid:29) = G , the model possesses a screening mechanism that makes itagree with solar system constraints. However, if this is not the case, the model is severely ruled out by LunarLaser Ranging constraints on the time variation of the Newton constant, as the prediction of the DW modelgiven in Eq. (3.10) lies significantly off from the current errorbars. Acknowledgments.
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