Observational constraints on generalized Proca theories
aa r X i v : . [ a s t r o - ph . C O ] J un YITP-17-58
Observational constraints on generalized Proca theories
Antonio De Felice , Lavinia Heisenberg , and Shinji Tsujikawa Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,Kyoto University, 606-8502, Kyoto, Japan Institute for Theoretical Studies,ETH Zurich, Clausiusstrasse 47, 8092 Zurich, Switzerland Department of Physics, Faculty of Science, Tokyo University of Science,1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan (Dated: June 27, 2017)In a model of the late-time cosmic acceleration within the framework of generalized Proca theo-ries, there exists a de Sitter attractor preceded by the dark energy equation of state w DE = − − s ,where s is a positive constant. We run the Markov-Chain-Monte-Carlo code to confront the modelwith the observational data of Cosmic Microwave Background (CMB), baryon acoustic oscillations,supernovae type Ia, and local measurements of the Hubble expansion rate for the background cos-mological solutions and obtain the bound s = 0 . +0 . − . at 95 % confidence level (CL). Existenceof the additional parameter s to those in the Λ-Cold-Dark-Matter (ΛCDM) model allows to reducetensions of the Hubble constant H between the CMB and the low-redshift measurements. Includingthe cosmic growth data of redshift-space distortions in the galaxy power spectrum and taking intoaccount no-ghost and stability conditions of cosmological perturbations, we find that the bound on s is shifted to s = 0 . +0 . − . (95 % CL) and hence the model with s > s, H and the today’s matter density parameter Ω m , theconstraints on other model parameters associated with perturbations are less stringent, reflectingthe fact that there are different sets of parameters that give rise to a similar cosmic expansion andgrowth history. I. INTRODUCTION
Two fundamental pillars are used in the standard model of Big Bang cosmology for describing the physics oncosmological scales: the cosmological principle and General Relativity (GR). The first is the notion of homogeneityand isotropy. Even if the fundamental theory behind GR is very elegant and simple, the problems of cosmologicalconstant and dark energy imply that we may need some modifications of GR in both infrared and ultraviolet scales.The cosmological constant problem represents the enormous discrepancy between observations and the expectationsfrom a field theory point of view [1], whereas the dark energy problem stands for the observed late-time accelerationof the Universe. Another tenacious challenge is the successful construction of a consistent theory of quantum gravity.To address such problems, there have been numerous attempts for modifying gravity in the infrared and ultravioletregimes [2–9].In the context of infrared modifications of gravity, theories based on scalar fields are the most extensively exploredones. One essential reason for these considerations is the natural provision of isotropic accelerated expansion. Besidesthis practical reasoning, we know that scalar fields do exist in nature. The Higgs field is a fundamental ingredient ofthe Standard Model of particle physics. Accepting an additional scalar degree of freedom in the gravity sector, thescalar field has to be very light to drive the late-time cosmic acceleration. This new scalar degree of freedom generallygives rise to long-range forces with baryonic matter, but such fifth forces have never been detected in solar-systemtests of gravity [10]. Therefore, one has to rely on some successful implementations of screening mechanisms, whichhide the scalar field on small scales whereas being unleashed on large scales to produce desired cosmological effects.In particular, the Vainshtein mechanism [11] in the presence of non-linear field self-interactions can efficiently screenthe propagation of fifth forces within a radius much larger than the solar-system scale [12–17].An interesting class of the self-interacting scalar field with a Galileon symmetry was proposed in Ref. [18]. TheseGalileon interactions involve explicit dependence on second derivatives of the scalar field in their Lagrangians, butthey maintain the second-order nature of field equations such that the Ostrogradski instability is avoided. A naivecovariantization of these Galileon interactions would result in higher-order equations of motion. This can be preventedby including corresponding non-minimal derivative couplings with the Ricci scalar and the Einstein tensor [19, 20].The generalization of covariant Galileons led to the construction of theories respecting the Galilean symmetry on thede Sitter background [21] and the rediscovery of the Horndeski action [22].Horndeski theories constitute the most general scalar-tensor theories leading to second-order equations of motionwith one scalar propagating degree of freedom besides two tensor polarizations [22–25]. Similar scalar-tensor theorieswith second-order equations of motion also arise from the decoupling limit of massive gravity [26–29]. Even outsidethe second-order domain, it is possible to construct more general scalar-tensor theories with one scalar propagatingdegree of freedom [30–32]. All these new realms of possibilities have been giving rise to a plethora of attempts fordescribing dark energy. These attempts also shed light to how classical field theories can be constructed in a consistentway as to keep the theory sensible and viable, i.e., without introducing ghost instabilities and removing unwanteddegrees of freedom.The Standard Model of particle physics contains both abelian and non-abelian vector fields as the fundamentalcarriers of gauge interactions. Consequently, it is comprehensible to wonder whether bosonic vector fields may alsoplay an important role in the cosmological evolution besides scalar fields. Similarly to the scalar counterpart, vectorfields being part of the gravitational interactions could naturally result in an accelerated Universe on large scaleswhile being screened on small scales. Learned from the lessons for constructing consistent theories for scalar-tensorinteractions, one can apply the same approach to vector-tensor theories. In Minkowski space-time, allowing for a massof the vector field leads to the propagation of a longitudinal scalar mode besides two transverse vector polarizationsdue to the breaking of U (1) gauge invariance. One can generalize this massive Proca theory to that in curved space-time in such a way that the propagating degrees of freedom remain three besides two tensor polarizations. A specifictype of massive Proca theories naturally arise in the framework of Weyl geometries [33, 34].The generalized Proca theories with second-order equations of motion constitute Galileon type vector self-interactions with non-minimal derivative couplings to gravity. The systematic construction of the action of generalizedProca theories was carried out in Ref. [35], where it was shown that despite the derivative self-interactions only threephysical degrees of freedom propagate. The specific case where the longitudinal mode of the vector field has the scalarGalilean self-interactions was considered in Ref. [36]. These generalized Proca interactions were further investigatedin Refs. [37, 38]. Even outside the domain of second-order theories, one can construct more general vector-tensorinteractions without increasing the number of propagating degrees of freedom relative to that in generalized Procatheories [39, 40].Recently, the background cosmological solutions and the stabilities of perturbations were studied for concrete darkenergy models that belong to generalized Proca theories [41, 42]. These models can be also compatible with solar-system constraints for a wide range of parameter space [43]. At the background level, there exists a de Sitter attractorresponsible for the late-time cosmic acceleration. Moreover, it was illustrated how dark energy models in the frameworkof generalized Proca theories can be observationally distinguished from the standard cosmological model accordingto both expansion history and cosmic growth [41, 42]. Within this framework of generalized Proca interactions,the de Sitter solution arises from the temporal component of the vector field compatible with the symmetries ofhomogeneity and isotropy of the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) background. Even if the temporalcomponent does not correspond to a propagating degree of freedom by construction, it does have a non-trivial effecton the cosmological dynamics by behaving as an auxiliary field. Another way of constructing homogeneous andisotropic cosmological solutions within this class of theories consists of considering triads configuration [44, 45]. Athird possibility is a combination of the temporal configuration with the triads as proposed in Ref. [44]. It wouldbe worthwhile to investigate the cosmological implications of this type of multi-Proca interactions and extend theexisting studies [44, 46–50].In this work, we go along the lines of Refs. [41, 42] and place constraints on these models by using several differentcosmological data sets: the Cosmic Microwave Background (CMB) shift parameters (from the Planck data), BaryonAcoustic Oscillations (BAO), Supernova Type Ia (SN Ia) standard candles (from the Union 2.1 data), local measure-ments of the Hubble expansion rate, and Redshift-Space-Distortions (RSD). The Λ-Cold-Dark-Matter (ΛCDM) modelrequires typical values of today’s density parameter of non-relativistic matter Ω m to be around 0.31, whereas, ingeneralized Proca theories, smaller values of Ω m are reached due to the existence of an extra parameter, s . Moreover,the Hubble constant H tends to be higher in generalized Proca theories with a lower total χ relative to that in theΛCDM model. Therefore, generalized Proca theories can help to reduce the tension between early-time and late-timedata sets, compared to the ΛCDM model.This paper is organized as follows. In Sec. II we review the background cosmological dynamics for a class of darkenergy models in the framework of generalized Proca theories. In Sec. III we discuss stability conditions and theevolution of matter density perturbations relevant to the growth of large-scale structures. In Sec. IV we explainthe data sets used for the likelihood analysis in later sections. In Sec. V we place observational bounds on modelparameters associated with the background expansion history by using the data of CMB, BAO, SN Ia, and Hubbleexpansion rate. In Sec. VI we put constraints on model parameters related to linear cosmological perturbations byadding the RSD data in the analysis. Sec. VII is devoted to conclusions. II. GENERALIZED PROCA THEORIES AND THE BACKGROUND DYNAMICS
In generalized Proca theories, the vector field A µ possesses two transverse polarizations and one longitudinal scalarmode non-minimally coupled to gravity. To keep the equations of motion up to second order, the field self-interactionsneed to be of specific forms. The action of generalized Proca theories is given by [35, 38] S = Z d x √− g ( L + L M ) , L = X i =2 L i , (2.1)where g is the determinant of the metric tensor g µν , L M is the matter Lagrangian density, and L , , , , are thevector-tensor interactions given by L = G ( X, F, Y ) , (2.2) L = G ( X ) ∇ µ A µ , (2.3) L = G ( X ) R + G ,X ( X ) (cid:2) ( ∇ µ A µ ) − ∇ ρ A σ ∇ σ A ρ (cid:3) , (2.4) L = G ( X ) G µν ∇ µ A ν − G ,X ( X )[( ∇ µ A µ ) − ∇ µ A µ ∇ ρ A σ ∇ σ A ρ + 2 ∇ ρ A σ ∇ γ A ρ ∇ σ A γ ] − g ( X ) ˜ F αµ ˜ F βµ ∇ α A β , (2.5) L = G ( X ) L µναβ ∇ µ A ν ∇ α A β + 12 G ,X ( X ) ˜ F αβ ˜ F µν ∇ α A µ ∇ β A ν . (2.6)The field strength is F µν = ∇ µ A ν − ∇ ν A µ and its dual is ˜ F µν = ǫ µναβ F αβ /
2, where ∇ µ stands for the covariantderivative operator. The function G can depend in general on the quantities X = − A µ A µ , F = − F µν F µν , Y = A µ A ν F µα F να , (2.7)while the remaining functions G , , , and g depend only on X . The partial derivatives of the functions are denoted by G i,X ≡ ∂G i /∂X . In the same way as in scalar Horndeski theories, the vector field is coupled only to the divergencelesstensors and their corresponding versions at the level of the equations of motion. Hence, the vector field is directlycoupled to the Ricci scalar R , the Einstein tensor G µν = R µν − Rg µν /
2, and the double dual Riemann tensor L µναβ = 14 ǫ µνρσ ǫ αβγδ R ρσγδ , (2.8)where ǫ µνρσ is the Levi-Civita tensor and R ρσγδ is the Riemann tensor. The specific case with G = m X and G , , , = 0 corresponds to the standard Proca theory, in which case two transverse vector modes and the longitudinalscalar propagate. These three propagating degrees of freedom are not altered by the derivative interactions (2.2)-(2.6),apart from the appearance of two tensor polarizations from the gravity sector [35, 40]. The non-minimal derivativecouplings (2.4)-(2.6) are needed to keep the equations of motion up to second order. The gauge-invariant vector-tensorinteraction introduced by Horndeski in 1976 corresponds to L = F + L + L with constant functions G and G [51]. A. Background equations of motion
For the purpose of cosmological applications, we take the flat FLRW metric with the line element ds = − dt + a ( t ) δ ij dx i dx j , where a ( t ) stands for the time-dependent scale factor with the cosmic time t . We consider the vectorfield A µ with a time-dependent temporal component φ ( t ) alone, i.e., A µ = ( φ ( t ) , , , , (2.9)which is compatible with the background symmetry. Assuming that the matter field in the Lagrangian density L M (with energy density ρ M and pressure P M ) is minimally coupled to gravity, they obey the continuity equation˙ ρ M + 3 H ( ρ M + P M ) = 0 , (2.10)where H ≡ ˙ a/a is the Hubble expansion rate, and a dot represents a derivative with respect to t . Varying the action(2.1) with respect to g µν , we obtain the modified Einstein field equations G − G ,X φ − G ,X Hφ + 6 G H − G ,X + G ,XX φ ) H φ + G ,XX H φ + 5 G ,X H φ = ρ M , (2.11) G − ˙ φφ G ,X + 2 G (3 H + 2 ˙ H ) − G ,X φ (3 H φ + 2 H ˙ φ + 2 ˙ Hφ ) − G ,XX H ˙ φφ + G ,XX H ˙ φφ + G ,X Hφ (2 ˙ Hφ + 2 H φ + 3 H ˙ φ ) = − P M . (2.12)Variation of the action (2.1) with respect to φ leads to φ (cid:0) G ,X + 3 G ,X Hφ + 6 G ,X H + 6 G ,XX H φ − G ,X H φ − G ,XX H φ (cid:1) = 0 . (2.13)The functions g , G and the additional dependence of F and Y in the function G , which correspond to intrinsicvector modes, do not contribute to the background equations of motion as expected. From Eq. (2.13) one immediatelyobserves that, for the branch φ = 0, there exist interesting de Sitter solutions with constant values of φ and H [41]. B. Concrete models
In a previous work, we have shown that viable dark energy models exist in the framework of generalized Procatheories [41]. As we mentioned before, the temporal vector component is not dynamical and can be expressed in termsof the Hubble parameter H . In order for the energy density of the temporal component φ to start dominating overthe background matter densities at the late cosmological epoch, the amplitude of the field φ should increase with thedecrease of H . Thus, the relation should be of the form φ p ∝ H − , (2.14)with a positive constant p . In the following, we assume that φ is positive. To guarantee the scaling (2.14) between φ and H , the functions G , , , in Eq. (2.13) should be chosen with the following specific scaling of X [41]: G ( X ) = b X p + F , G ( X ) = b X p , G ( X ) = M b X p , G ( X ) = b X p , (2.15)with the powers p , , of the form p = 12 ( p + 2 p − , p = p + p , p = 12 (3 p + 2 p − , (2.16)where M pl is the reduced Planck mass and b , , , are constants. Note that the specific case with p = 1 and p = 1 corresponds to vector Galileons [35, 36], where φ ∝ H − . The models given by the functions (2.15) are thegeneralization of vector Galileons.For the matter fields, we will assume the existence of non-relativistic matter (energy density ρ m and pressure P m = 0) and radiation (energy density ρ r and pressure P r = ρ r /
3) together with their respective continuity equations˙ ρ m + 3 Hρ m = 0 and ˙ ρ r + 4 Hρ r = 0. Then we have ρ M = ρ m + ρ r and P M = ρ r / i = 3 , , y ≡ b φ p M H p , β i ≡ p i b i p i − p p b ( φ p H ) i − , (2.17)and the density parametersΩ r ≡ ρ r M H , Ω m ≡ ρ m M H , Ω DE ≡ − Ω r − Ω m . (2.18)The variables β i ’s are constants due to the relation (2.14). For the branch φ = 0, Eq. (2.13) is expressed in a simpleform 1 + 3 β + 6(2 p + 2 p − β − (3 p + 2 p ) β = 0 , (2.19)which can be exploited to express β in terms of β and β . On using Eq. (2.11), the dark energy density parameteris related to the quantity y , as Ω DE = βyp ( p + p ) , (2.20)where the constant β is defined by β ≡ − p ( p + p )(1 + 4 p β ) + 6 p (2 p + 2 p − β . (2.21)For the constant b appearing in G ( X ), we choose it to be negative, i.e., b = − m M − p )pl , where m is a mass term.This is for avoiding the appearance of tensor ghosts in the limit that G → λ ≡ (cid:18) φM pl (cid:19) p Hm , (2.22)which is constant from Eq. (2.14). On using Eqs. (2.17) and (2.21), the temporal vector component can be expressedas φ = M pl [ − p · λ p ( p + p )Ω DE /β ] / [2( p + p )] . We will focus on the case β <
0, under which φ > λ > p ( p + p ) > φ = 0 branch of Eq. (2.13) with respect to t and together with Eq. (2.12) we can solve them for ˙ φ and ˙ H . Taking the derivatives of Ω DE and Ω r with respect to N ≡ ln a (denoted by a prime), we obtain the followingautonomous equations Ω ′ DE = (1 + s )Ω DE (3 + Ω r − DE )1 + s Ω DE , (2.23)Ω ′ r = − Ω r [1 − Ω r + (3 + 4 s )Ω DE ]1 + s Ω DE , (2.24)where we introduced the variable s ≡ p p . (2.25)After integrating Eqs. (2.23) and (2.24) for given initial conditions of Ω DE and Ω r , the three density parametersΩ DE , Ω r and Ω m = 1 − Ω DE − Ω r are known accordingly. Furthermore, we impose the condition s > − DE from diverging in the interval 0 ≤ Ω DE ≤
1. We also define the effective equation of state of the system by w eff ≡ − − H/ (3 H ), which can be expressed as w eff = Ω r − s )Ω DE s Ω DE ) . (2.26)We write Eqs. (2.11) and (2.12) in the forms 3 M H = ρ DE + ρ M and M (3 H + 2 ˙ H ) = − P DE − P M , respectively,where ρ DE = − G + G ,X φ + 3 G ,X φ H − g H + 6 φ H (2 G ,X + G ,XX φ ) − H G ,XX φ − H G ,X φ , (2.27) P DE = G − ˙ φφ G ,X + 2 g (3 H + 2 ˙ H ) − φG ,X (3 φH + 2 ˙ φH + 2 φ ˙ H ) − H ˙ φφ G ,XX + ˙ φφ H G ,XX + G ,X φ H (2 φ ˙ H + 2 φH + 3 ˙ φH ) , (2.28)with g ( X ) = b X p . Defining the dark energy equation of state as w DE = P DE /ρ DE , it follows that w DE = − s ) + s Ω r s Ω DE ) . (2.29)Using Eqs. (2.23) and (2.24), we obtain a single differential equation of the formΩ ′ DE Ω DE = (1 + s ) (cid:18) Ω ′ r Ω r + 4 (cid:19) . (2.30)This equation is easily integrated to give Ω DE Ω sr = Ω DE0 Ω sr (cid:18) aa (cid:19) s ) , (2.31)where the lower subscript “0” denotes today’s values. III. COSMOLOGICAL PERTURBATIONS
By considering linear cosmological perturbations on the flat FLRW background, the conditions for avoiding ghostsand Laplacian instabilities in the small-scale limit were already derived in Refs. [41, 42]. Here, we briefly review sixno-ghost and stability conditions arising from tensor, vector, and scalar perturbations. We also discuss the equationsof motion for matter perturbations to confront generalized Proca theories with the observations of RSD in Sec. VI.
A. Stability conditions
For the metric we take the perturbed line element in the flat gauge ds = − (1 + 2 α ) dt + 2 ( ∂ i χ + V i ) dt dx i + a ( t ) ( δ ij + h ij ) dx i dx j , (3.1)where α, χ are scalar metric perturbations, V i is the vector perturbation obeying the transverse condition ∂ i V i = 0,and h ij is the tensor perturbation satisfying the transverse and traceless conditions ∂ i h ij = 0 and h ii = 0. Thetemporal and spatial components of the vector field can be decomposed into the background and perturbed parts, as A = φ ( t ) + δφ , A i = 1 a ( t ) δ ij ( ∂ j χ V + E j ) , (3.2)where δφ and χ V are scalar perturbations, and E j is the vector perturbation satisfying ∂ j E j = 0. The Schutz-Sorkin action [52] allows us to describe both vector and scalar perturbations of the matter perfect fluid. For scalarperturbations, the key observables are the matter density perturbation δρ M and the velocity potential v .First of all, the tensor perturbation h ij has two polarization modes h + and h × , which can be expressed as h ij = h + e + ij + h × e × ij in terms of the unit tensors obeying the normalizations e + ij ( k ) e + ij ( − k ) ∗ = 1, e × ij ( k ) e × ij ( − k ) ∗ = 1, e + ij ( k ) e × ij ( − k ) ∗ = 0. Expanding the action (2.1) in h ij up to quadratic order, the second-order action yields S T = X λ =+ , × Z dt d x a q T (cid:20) ˙ h λ − c T a ( ∂h λ ) (cid:21) . (3.3)The quantities q T and c T determine no-ghost and stability conditions, respectively, whose explicit forms are given by q T = 2 G − φ G ,X + Hφ G ,X > , (3.4) c T = 2 G + φ ˙ φ G ,X q T > . (3.5)For vector perturbations, the dynamical field is given by the combination Z i = E i + φ ( t ) V i . (3.6)Due to the transverse condition ∂ i Z i = 0, there are two propagating degrees of freedom for Z i , e.g., Z i =( Z ( z ) , Z ( z ) ,
0) for the vector field whose wavenumber k is along the z direction. Expanding the action (2.1) up toquadratic order and taking the small-scale limit, the resulting second-order action for Z and Z can be written asthe form analogous to Eq. (3.3) with the no-ghost and stability conditions q V = G ,F + 2 G ,Y φ − g Hφ + 2 G H + 2 G ,X H φ > , (3.7) c V = 1 + φ (2 G ,X − G ,X Hφ ) q T q V + 2[ G ˙ H − G ,Y φ − ( Hφ − ˙ φ )( Hφ G ,X − g )] q V > . (3.8)For scalar perturbations, the dynamical field arising from the vector field is given by ψ = χ V + φ ( t ) χ . (3.9)The matter perturbation δρ M , which arises from the Schutz-Sorkin action, also works as a dynamical scalar field.The second-order action of scalar perturbations is given in Eq. (4.6) of Ref. [42]. Varying this action with respect to α, χ, δφ, ∂ψ, v, and δρ M , the equations of motion in Fourier space are given, respectively, by δρ M − w α + (3 Hw − w ) δφφ + k a ( Y + w χ − w ψ ) = 0 , (3.10)( ρ M + P M ) v + w α + w φ δφ = 0 , (3.11)(3 Hw − w ) α − w δφφ + k a (cid:20) Y + w χ − (cid:18) w φ + w (cid:19) ψ (cid:21) = 0 , (3.12)˙ Y + H − ˙ φφ ! Y + 2 φ ( w α + w ψ ) + (cid:18) w φ + w (cid:19) δφ = 0 , (3.13)˙ δρ M + 3 H (cid:0) c M (cid:1) δρ M + k a ( ρ M + P M ) ( χ + v ) = 0 , (3.14)˙ v − Hc M v − c M δρ M ρ M + P M − α = 0 , (3.15)where c M is the matter propagation speed squared, and w = H φ ( G ,X + φ G , XX ) − H ( G + φ G , XX ) − φ G ,X , (3.16) w = w + 2 Hq T , (3.17) w = − φ q V , (3.18) w = 12 H φ (9 G ,X − φ G , XXX ) − H (2 G + 2 φ G ,X + φ G , XX − φ G , XXX ) − Hφ ( G ,X − φ G , XX ) + 12 φ G , XX , (3.19) w = w − H ( w + w ) , (3.20) w = − φ (cid:2) H φ ( G ,X − φ G , XX ) − H ( G ,X − φ G , XX ) + φG ,X (cid:3) , (3.21) w = 2( HφG ,X − G ,X ) ˙ H + (cid:2) H ( G ,X + φ G , XX ) − Hφ G , XX − G ,X (cid:3) ˙ φ , (3.22) Y = w φ (cid:16) ˙ ψ + δφ + 2 αφ (cid:17) . (3.23)On using Eqs. (3.10)-(3.12) and (3.14), we can express α, χ, δφ, v in terms of ψ, δρ M and their derivatives. Then, thesecond-order action of scalar perturbations is written in terms of the two dynamical fields ψ and δρ M . This allowsone to derive no-ghost and stability conditions in the small-scale limit. The no-ghost and stability conditions of thematter field δρ M are trivially satisfied for ρ M + P M > c M >
0. For the perturbation ψ , we require the followingconditions [41, 42] Q S = a H q T q S φ ( w − w ) > , (3.24) c S = µ S H φ q T q V q S > , (3.25)where q S = 3 w + 4 q T w , (3.26) µ S = [ w φ ( w − w ) + w w ] − w (cid:0) w ˙ w − w ˙ w (cid:1) + φ ( w − w ) w ˙ w + w ( w − w ) h(cid:16) H − φ/φ (cid:17) w w + ( w − w ) n w (cid:16) Hφ − ˙ φ (cid:17) + 2 w φ oi +2 w w ( ρ M + P M ) . (3.27)Under the no-ghost conditions of tensor and vector perturbations ( q T > , q V > q S > µ S >
0. There are specific cases where the quantity w − w in Eq. (3.24) crosses 0, at which Q S exhibits the divergence [41]. We will exclude such cases for constraining the viable parameter space. B. Effective gravitational coupling with matter perturbations
To confront generalized Proca theories with the observations of large-scale structures and weak lensing, we considernon-relativistic matter (labeled by m ) with the equation of state w m = P m /ρ m = 0 + and the sound speed squared c m = 0 + . We introduce the matter density contrast δ and the gauge-invariant gravitational potentials δ = δρ m ρ m + 3 Hv ,
Ψ = α + ˙ χ , Φ =
Hχ . (3.28)Taking the time derivative of Eq. (3.14) and using Eq. (3.15), we obtain¨ δ + 2 H ˙ δ + k a Ψ = 3 ¨ B + 6 H ˙ B , (3.29)where B ≡ Hv . The effective gravitational coupling G eff is defined by k a Ψ = − πG eff ρ m δ . (3.30)For the perturbations deep inside the sound horizon ( c S k /a ≫ H ), we can resort to the quasi-static approxi-mation for deriving the relations between Ψ , Φ and δ [53–55]. Provided that c S is not very much smaller than 1, thedominant contributions to the perturbation equations of motion are the terms containing k /a and δρ m . The termson the r.h.s. of Eq. (3.29) are negligible relative to those on the l.h.s., so that¨ δ + 2 H ˙ δ − πG eff ρ m δ ≃ . (3.31)On using the quasi-static approximation for Eqs. (3.10)-(3.13), the effective gravitational coupling is analyticallyknown as [42] G eff = ξ + ξ ξ , (3.32)where ξ = 4 πφ ( w + 2 Hq T ) , (3.33) ξ = [ H ( w + 2 Hq T ) − ˙ w + 2 ˙ w + ρ m ] φ − w q V , (3.34) ξ = 18 H φ q S q T c S (cid:20) φ { q S [ w ˙ w − ( w − Hq T ) ˙ w ] + ρ m w [3 w ( w + 2 Hq T ) − q S ] } + q S q V w { w ( w − Hq T ) − w φ ( w + 2 Hq T ) } (cid:21) . (3.35)The solutions to δ derived by solving Eq. (3.31) with Eq. (3.32) can reproduce full numerical results at high accuracy[42].Besides G eff , the gravitational slip parameter η = − Φ / Ψ is also an important quantity for describing the deviationof light rays in weak lensing observations [56]. Under the quasi-static approximation it follows that [42] η = ξ ξ + ξ , (3.36)where ξ = w + 2 Hq T Hq S q V q T c S (cid:20) H φ q S q V q T c S + 2 φ q S q V w ˙ w ( w − Hq T ) + w { φq S w ( w + 2 Hq T ) − w q S ( w − Hq T ) − φ q S q V ˙ w + 2 φ q V [ q S − w ( w + 2 Hq T )] ρ m } (cid:21) . (3.37)Although we do not use the information of the gravitational slip parameter η in our likelihood analysis, this can beimportant for confronting the model with future high-precision observations of weak lensing. IV. OBSERVATIONAL DATA
We would like to test for the model (2.15) with different observational data from CMB, BAO, SN Ia, the Hubbleexpansion rate, and RSD measurements. In this section, we will explain the data sets used in the likelihood analysisperformed in Secs. V and VI.
A. CMB
The CMB power spectrum is affected by the presence of dark energy in at least two ways [2]. First, the positionsof CMB acoustic peaks are shifted by the change of the angular diameter distance from the last scattering surface totoday. Second, the variation of gravitational potentials induced by the presence of dark energy leads to the late-timeintegrated Sachs-Wolfe effect. The latter mostly affects the temperature anisotropies on large scales, at which theobservational data are prone to uncertainties induced by the cosmic variance. Since the first effect is usually moreimportant to constrain the property of dark energy, we will focus on the CMB distance measurements in the following.The comoving distance to the CMB decoupling surface r ( z ∗ ) (the redshift z ∗ ≃ r s ( z ∗ ) can be constrained from CMB measurements. In particular, the CMB shiftparameters R = p Ω m H r ( z ∗ ) , (4.1) l a = πr ( z ∗ ) r s ( z ∗ ) (4.2)are the two key quantities for placing constraints on dark energy [57–59]. The shift parameter R is associated withthe overall amplitude of CMB acoustic peaks, whereas l a determines the average acoustic structure. We need toemploy both R and l a for extracting the necessary information to constrain dark energy models from the CMB powerspectrum.The comoving distance on the flat FLRW background is given by r ( z ∗ ) = R z ∗ dz/H ( z ), where z = a /a − R = p Ω m Z z ∗ dzE ( z ) , (4.3)where E ( z ) represents the Hubble ratio given by E ( z ) ≡ H ( z ) H = r Ω r Ω r (1 + z ) . (4.4)We can promote R to a function of z satisfying the condition R (0) = 0. Defining the ratios ¯Ω r = Ω r / Ω r and¯ z = z/z ∗ , it follows that d R (¯ z ) d ¯ z = z ∗ p Ω m ¯Ω r (1 + z ∗ ¯ z ) , (4.5)which should be integrated from ¯ z = 0 to ¯ z = 1 with R (0) = 0. In doing so, we need to know ¯Ω r as a function of ¯ z .For the model (2.15) with Eq. (2.16), the dark energy density parameter is known from Eq. (2.31), asΩ DE = (1 − Ω m − Ω r ) ¯Ω sr (1 + z ∗ ¯ z ) − s ) . (4.6)On using Eq. (2.24), the radiation density parameter obeys d ¯Ω r d ¯ z = z ∗ ¯Ω r [1 − Ω r ¯Ω r + (3 + 4 s ) (1 − Ω m − Ω r ) ¯Ω sr (1 + z ∗ ¯ z ) − s ) ](1 + z ∗ ¯ z )[1 + s (1 − Ω m − Ω r ) ¯Ω sr (1 + z ∗ ¯ z ) − s ) ] , (4.7)with ¯Ω r (0) = 1. We integrate Eqs. (4.5) and (4.7) to compute the value of R at ¯ z = 1.The dimensionless comoving distance ¯ r ( z ) = H r ( z ), where r ( z ) = R z dz ′ /H ( z ′ ), obeys the differential equation d ¯ r (¯ z ) d ¯ z = z ∗ p ¯Ω r (1 + z ∗ ¯ z ) , (4.8)where ¯ r (0) = 0. The comoving sound horizon at the redshift z is defined by r s ( z ) = Z t c s dta = 1 H Z ∞ z dz ′ c s ( z ′ ) E ( z ′ ) , (4.9)where we used the normalization a = 1 in the second equality. The sound speed squared c s of the coupled system ofbaryons (density ρ b ) and photons (density ρ γ ) is given by [60] c s = 1 p R b / (1 + z )] , (4.10) The quantity c s is different from the sound speed c S of the scalar degree of freedom ψ arising from the vector field. Since the density ofthe vector field is much smaller than those of the background fluids before the CMB decoupling epoch, the existence of the vector fielddoes not affect c s . R b = 3 ρ b ρ γ = 31500Ω b h (cid:18) . . (cid:19) − . (4.11)Introducing the dimensionless quantity ¯ r s = H r s and taking the z derivative of Eq. (4.9), we have d ¯ r s /dz = − c s ( z ) /E ( z ). Upon the change of variable, ¯ a = a/a ∗ , it follows that 1 + z = 1 /a = 1 / ( a ∗ ¯ a ) = (1 + z ∗ ) / ¯ a . Onusing Eqs. (4.4) and (4.10), the dimensionless distance ¯ r s obeys the differential equation d ¯ r s d ¯ a = 11 + z ∗ p Ω r (¯ a ) p r [1 + R b ¯ a/ (1 + z ∗ )] , (4.12)with r s (¯ a = 0) = 0.The radiation density parameter is known from Eq. (2.24), such that d Ω r d ¯ a = − Ω r [1 − Ω r + (3 + 4 s )Ω DE ]¯ a (1 + s Ω DE ) . (4.13)At first glance the r.h.s. of Eq. (4.13) looks divergent in the limit ¯ a →
0, but this is not the case because the numeratoralso approaches 0. For numerical purposes, we will rewrite the r.h.s. of Eq. (4.13) in a more convenient form. Indoing so, we introduce the following quantityΓ ≡ ρ r ρ r + ρ m = Ω r Ω r + Ω m = Ω r − Ω DE = Ω r (1 + z ∗ )Ω r (1 + z ∗ ) + Ω m ¯ a . (4.14)Since Ω r = Γ(1 − Ω DE ), the quantity 1 − Ω r in Eq. (4.13) can be expressed as1 − Ω r = 1 − Γ + ΓΩ DE = Ω m ¯ a Ω r (1 + z ∗ ) + Ω m ¯ a + Ω r (1 + z ∗ )Ω r (1 + z ∗ ) + Ω m ¯ a Ω DE , (4.15)where Ω DE = (1 − Ω m − Ω r )(1 + z ∗ ) − s ) (Ω r / Ω r ) s ¯ a s ) from Eq. (2.31). Then, we can express Eq. (4.13) inthe form d Ω r d ¯ a = − Ω r [1 + s (1 − Ω m − Ω r )(Ω r / Ω r ) s (1 + z ∗ ) − s ) ¯ a s ) ] × (cid:26) Ω m Ω r (1 + z ∗ ) + Ω m ¯ a + (cid:20) Ω r (1 + z ∗ )Ω r (1 + z ∗ ) + Ω m ¯ a + 3 + 4 s (cid:21) (1 − Ω m − Ω r )Ω sr ¯ a s (1 + z ∗ ) s ) Ω sr (cid:27) , (4.16)with Ω r (¯ a →
0) = 1. Provided that 3 + 4 s > d Ω r /d ¯ a → − Ω m / [Ω r (1 + z ∗ )] as ¯ a →
0, so the r.h.s.of Eq. (4.16) remains finite. Integrating Eqs. (4.12) and (4.16) with Eq. (4.8), the second CMB shift parameter l a = π ¯ r ( z ∗ ) / ¯ r s ( z ∗ ) can be computed accordingly.The CMB shift parameters extracted from the Planck 2015 data have the mean values h l a i = 301 .
77 and hRi =1 . σ ( l a ) = 0 .
090 and σ ( R ) = 0 . b h = 0 . h is the normalized Hubble constant ( H = 100 h km s − Mpc − ). The components ofthe normalized covariance matrix C are given by C = C = 1 and C = C = 0 . χ statisticsassociated with the CMB shift parameters is given by χ = ( l a − . × .
916 + (
R − . × .
31 + 2(
R − . l a − . × ( − . . (4.17)The dependence on the parameters Ω m , h and s are encoded in R and l a . For the test parameters Ω m = 0 . h = 0 . s = 0 .
1, we find that χ = 1519 . The Planck 2015 team [62] provided the values l a = 301 . ± .
14 and R = 1 . ± . C = C = 1 and C = C = 0 .
54. We confirm that the corresponding χ analysis gives very similar likelihood results as thosederived from Eq. (4.17). B. BAO
We also use the BAO data to constrain our model further. The BAO represent periodic fluctuations of the densityof baryonic matter as a result of the counteracting forces of pressure and gravity. The photons release a pressureafter the decoupling, which on the other hand creates a shell of baryonic matter at the sound horizon. From the BAOmeasurements we can deduce the distance-redshift relation at the observed redshifts. One of the important quantitiesis the sound horizon r s ( z d ), where z d is the redshift at which baryons are released from photons. There is a fittingformula for the drag redshift z d , as [63] z d = 1291(Ω m h ) . . m h ) . (cid:2) b (Ω b h ) b (cid:3) , (4.18)where the parameters b and b are b = 0 . m h ) − . (cid:2) . m h ) . (cid:3) , (4.19) b = 0 . m h ) . . (4.20)The sound horizon r s ( z ) is given by Eq. (4.9) with the sound speed (4.10). We perform a change of the variable˜ a = a/a d with 1 + z = a / ( a d ˜ a ) = (1 + z d ) / ˜ a . This helps us to make use of some of the results in the CMB distancemeasurements. Since ˜ a/ ¯ a = (1 + z d ) / (1 + z ∗ ), it follows that ¯ a ( a = a d ) = (1 + z ∗ ) / (1 + z d ). With this change ofvariables, we can now integrate Eq. (4.16) from ¯ a = 0 to ¯ a = (1 + z ∗ ) / (1 + z d ). For the test parameters Ω m = 0 . h = 0 . s = 0 .
1, we obtain the value ¯ r s ( z d ) = 0 . D A ( z ) = 1 H (1 + z ) Z z dz ′ E ( z ′ ) . (4.21)The dimensionless quantity ¯ D A = H D A is known by integrating the following differential equations: d ¯ D A dz = − ¯ D A z + p ¯Ω r (1 + z ) , (4.22) d ¯Ω r dz = ¯Ω r [1 − Ω r ¯Ω r + (3 + 4 s )(1 − Ω m − Ω r ) ¯Ω sr (1 + z ) − s ) ](1 + z )[1 + s (1 − Ω m − Ω r ) ¯Ω sr (1 + z ) − s ) ] , (4.23)with ¯ D A ( z = 0) = 0. With the diameter distance, we are now at a place to compute the dilation scale [64] D V ( z ) = (cid:2) (1 + z ) D A ( z ) z H − ( z ) (cid:3) / = (cid:20) D A ( z ) z H − q ¯Ω r ( z ) (cid:21) / . (4.24)The important observable is the ratio between the sound horizon at the drag redshift and the dilation scale r s ( z d ) D V ( z ) = ¯ r s ( z d )( ¯ D A ( z ) z p ¯Ω r ) / , (4.25)which is dimensionless and does not depend on H .We use the BAO data from the 6dFGS [65], SDSS-MGS [66], BOSS [67], BOSS CMASS [68], and Wiggle Z [69]surveys. Then, the χ statistics in BAO measurements is given by χ = 10 . (cid:20) r s ( z d ) D V ( z = 0 . − . (cid:21) + 1 (cid:0) . (cid:1) (cid:20) D V ( z = 0 . r s ( z d ) − . (cid:21) + 1 (cid:0) . (cid:1) (cid:20) D V ( z = 0 . r s ( z d ) − . (cid:21) + 1 (cid:0) . (cid:1) (cid:20) D V ( z = 0 . r s ( z d ) − . (cid:21) + 10 . (cid:20) r s ( z d ) D V ( z = 0 . − . (cid:21) + 1 (cid:0) . (cid:1) (cid:20) D V ( z = 0 . r s ( z d ) − . (cid:21) + 1 (cid:0) . (cid:1) (cid:20) D V ( z = 0 . r s ( z d ) − . (cid:21) + 10 . (cid:20) r s ( z d ) D V ( z = 0 . − . (cid:21) + 1 (cid:0) . (cid:1) (cid:20) D V ( z = 0 . r s ( z d ) − . (cid:21) + 10 . (cid:20) r s ( z d ) D V ( z = 0 . − . (cid:21) . (4.26)For the test parameters mentioned above Eq. (4.21), we find that χ = 10 . C. SN Ia
The SN Ia can be used as standard candles with known brightness to refer to physical distances. This is based onthe fact that the logarithm of an astronomical object’s luminosity seen from a distance of 10 parsecs gives its absolutemagnitude M , which on the other hand enables us to refer to its brightness. For SN Ia the absolute magnitude at thepeak of brightness is nearly constant ( M ≃ − z is observed with the apparentmagnitude m , then the difference between m and M is related to a luminosity distance d L ( z ), as µ ( z ) ≡ m ( z ) − M = 5 log ¯ d L ( z ) − h + µ , (4.27)where µ = 42 .
38, and ¯ d L ( z ) ≡ H d L ( z ) = (1 + z ) Z z dz ′ E ( z ′ ) . (4.28)The Hubble expansion rate H ( z ) is known by measuring m ( z ) for many different redshifts ( z . χ estimator χ = N X i =1 [ µ obs ( z i ) − µ th ( z i )] σ i , (4.29)where N is the number of the SN Ia data set, µ obs ( z i ) and µ th ( z i ) are the observed and theoretical values of thedistance modulus µ ( z i ), respectively, and σ i are the errors on the data. Since the SN Ia data are in the low-redshiftregime, we can neglect the contribution of radiation and set Ω r = 0. Furthermore, due to the degeneracy betweenthe absolute magnitude and h , we will marginalize over h . For the likelihood analysis, we use the Union 2.1 data sets[72]. D. Local measurements of the Hubble expansion rate
The recent observations of Cepheids in galaxies of SN Ia placed the bound on the local value of normalized Hubbleconstant H , as [73] h = 0 . ± . . (4.30)With the knowledge of the comoving sound horizon it is possible to extract the information of the Hubble expansionrate from BAO measurements, as H ( z ) r s ( z d ) = E ( z ) ¯ r s ( z d ) = (1 + z ) p ¯Ω r ¯ r s ( z d ) . (4.31)For this purpose we use the recent BOSS data [67] in addition to the bound (4.30). Thus, we define the χ statisticsassociated with the local measurements of H as follows χ H = ( h − . . + [ H ( z = 0 . r s ( z d ) − . × . / . (2 . × . / . + [ H ( z = 0 . r s ( z d ) − . × . / . (2 . × . / . + [ H ( z = 0 . r s ( z d ) − . × . / . (2 . × . / . . (4.32)For the test parameters Ω m = 0 . h = 0 . s = 0 .
1, we obtain χ H = 30 . E. RSD
For the perturbations relevant to the RSD measurements (sub-horizon modes with k ≫ aH ) it was shown inRef. [42] that the quasi-static approximation is sufficiently accurate to describe the evolution of the matter densitycontrast δ , so we resort to Eq. (3.31) without taking into account the radiation (Ω r = 0). In terms of the derivativewith respect to N = ln a , we can rewrite Eq. (3.31) in the form δ ′′ + 1 − w eff δ ′ − πG eff ρ m H δ = 0 , (4.33)where w eff = − − H/ (3 H ) is the effective equation of state whose explicit form in our model is given by Eq. (2.26).On using the matter density parameter Ω m = 8 πGρ m / (3 H ), where G = 1 / (8 πM ) is the bare gravitational constant,Eq. (4.33) reduces to δ ′′ + 1 + (3 + 4 s )Ω DE s Ω DE ) δ ′ − G eff G (1 − Ω DE ) δ = 0 . (4.34)For the integration of this differential equation, we use the effective gravitational coupling given by Eq. (3.32).The ΛCDM model corresponds to s = 0 and G eff = G in Eq. (4.34), in which case there is the growing-modesolution δ ∝ a during the matter dominance (Ω DE ≃ DE ≃ G eff ≃ G in the early matterera, so the evolution of δ at high redshifts ( z ≫
1) is very similar to that in the ΛCDM model. Hence we choosethe initial conditions satisfying δ = δ ′ in the deep matter era. The difference from the ΛCDM model arises at lowredshifts where Ω DE and G eff deviate from 0 and G , respectively.We define σ as the amplitude of over-density at the comoving 8 h − Mpc scale. To compute the χ estimator ofRSD measurements, we introduce the following quantity y ( z ) ≡ f ( z ) σ ( z ) , f ( z ) ≡ δ ′ δ . (4.35)Since the behavior of perturbations in our model is very close to that in the ΛCDM model at high redshifts, the twomodels should have similar initial conditions of δ . This implies that σ Proca8 ( z i ) ≈ σ ΛCDM8 ( z i ) at an initial redshift z = z i ≫
1. Since we assume that the radiation is negligible, we choose such an initial redshift to be in the deepmatter era, that is, at N i = − z i ≃ f σ data extracted from RSD measurements, we use those listed in Table I of Ref. [74]. This includes therecent data of FastSound at the redshift z = 1 .
36 [75]. We define the χ estimator, as χ = N X i =1 [ y obs ( z i ) − y th ( z i )] σ i , (4.36)where N is the number of the RSD data, y obs ( z i ) and y th ( z i ) are the observed and theoretical values of y ( z i ) atredshift z i respectively, and σ i ’s are the errors on the data.Among other measurements discussed in this section, the RSD data are only those strictly connected with thegrowth of perturbations. Since the effective gravitational coupling (3.32) depends on many model parameters like q V and c S , there should be some level of degeneracy of model parameters compared to the analysis based on thebackground expansion history. Reflecting this situation, we will focus on the case β = 0 for the likelihood analysisincluding the RSD data. V. BACKGROUND CONSTRAINTS
As a first analysis of the theory, we study how the background cosmic expansion history can fit the observationaldata sets of CMB, BAO, SN Ia, and the Hubble expansion rate. In this section we do not take into account the RSDdata, as they directly deal with the growth of matter perturbations. Then, we focus on the likelihood analysis for thefollowing parameters: Ω m , h, s . (5.1)The analysis of the background alone is simpler than the one including the perturbations, as the space of parametersreduces to a three-dimensional one. It should be noticed that, compared to the ΛCDM model, our model has only4one additional background parameter, s . Furthermore, the ΛCDM model corresponds to s = 0 at the backgroundlevel, so the direct comparison between the two models is straightforward.For the Monte-Carlo-Markov-Chain (MCMC) sampling, we need to put some priors on the allowed parameter spaceof the three parameters. We will choose sensible priors for the parameters (5.1) as follows: • For the density parameter of non-relativistic matter, 0 . ≤ Ω m ≤ . • For the normalized Hubble constant, 0 . ≤ h ≤ . • For the deviation parameter s from the ΛCDM model, − . ≤ s ≤ . q V > c S > χ = χ + χ + χ + χ H . (5.2)The best fit corresponds to the case in which χ is minimized.In Fig. 1 we plot one-dimensional probability distributions of the parameters (5.1) and two-dimensional observationalcontours for the combination of these three parameters. The one-dimensional probability distributions show that theminimum value of χ does exist for the following (approximated) values:Ω m = 0 . , h = 0 . , s = 0 . , (5.3)for which χ , min ≈ . . (5.4)The existence of a minimum around s = 0 .
25 shows that the model with s > σ constraints on the three parameters are given byΩ m = 0 . +0 . − . , (5.5) h = 0 . +0 . − . , (5.6) s = 0 . +0 . − . . (5.7)This means that the ΛCDM model is disfavored over the model with s > σ level. We note that theextended scalar Galileon model [76] has a tracker solution whose background evolution is the same as that in themodel under consideration. In Ref. [77] two of the present authors performed the likelihood analysis by using thedata of the CMB (WMAP7), BAO, SN Ia, and derived the bound s = 0 . +0 . − . (95 % CL). With the new data ofthe CMB (Planck), BAO, SN Ia, and the Hubble expansion rate, the constraint on s is shifted toward larger values.In the ΛCDM model, the best-fit values of Ω m and h constrained by the Planck CMB data are around Ω m ≈ . h ≈ .
68, respectively [78]. These best-fit values are in tension with their low-redshift measurements, whichgenerally favor lower Ω m and higher h , see, e.g., Eq. (4.30). The model with s > m and larger h . We can confirm this property in the probability distributions of Ω m and h in Fig. 1. VI. OBSERVATIONAL CONSTRAINTS INCLUDING THE RSD DATA
If we take into account the evolution of matter perturbations, the likelihood results can be subject to change intwo different ways: 1) the stability conditions of perturbations, which need to hold at all times, generally reduce theallowed parameter space; and 2) the RSD contribution can shift observational bounds of model parameters. We setthe following additional priors to those used in Sec. V: • The no-ghost conditions for scalar, vector, and tensor perturbations to apply at all times, i.e., q T > , q V > , Q S > • The stability conditions associated with the propagation speeds of scalar, vector, and tensor perturbations,i.e., c T > , c V > , c S >
0. We also put the priors that c T , c V , c S are initially smaller than 10 to avoid thedivergences of these quantities in the asymptotic past.5 FIG. 1: Observational bounds on the three parameters Ω m , s, h constrained by the data of the CMB, BAO, SN Ia, and theHubble expansion rate. The RSD data are not taken into account in the analysis. Since the analysis is based on the backgroundcosmic expansion history, we do not consider the conditions associated with ghosts and stabilities of perturbations. From top tobottom, the right panels in each column are one-dimensional probability distributions of the parameters Ω m , s, h , respectively.The vertical dashed lines correspond to the best fit (central) and the 2 σ confidence limits (outside). The other panels are thetwo-dimensional likelihood contours in the ( s, Ω m ), ( h, Ω m ), and ( h, s ) planes with 1 σ (inside) and 2 σ (outside) boundaries.The ΛCDM model, which corresponds to s = 0, is disfavored over the model with s > • The condition c T > today . This is for evading the Cherenkov radiation bound 1 − c T < × − today [79, 80]. • < p ≤
25 for keeping the parameter p positive and of order unity. • − ≤ β ≤ − − , 10 − ≤ λ ≤
15. The reason for the choices of negative β and positive λ were explained inSec. II B. We choose flat distributions for the natural logarithms of these variables.6 • − ≤ ln( q V ) ≤
15. We choose the values of q V to be not very close to 0 to avoid the strong coupling problem.For simplicity we set the parameter β to zero, as keeping it non-zero in the analysis does not change the final resultssignificantly. The likelihood results seem to be flat in this direction, such that the observational data do not notablyconstrain this parameter.We perform a MCMC analysis and compute χ = χ + χ + χ + χ H + χ , (6.1)in order to quantify how the RSD data affect the available parameter space constrained at the background level. Weverify that the role of no-ghost and stability conditions is important, because some of the parameter space preferred bythe background analysis (performed in Sec. V) does not possess a stable cosmological evolution. For example, the bestfit obtained by the background analysis does not in general satisfy no-ghost conditions of perturbations. Furthermorewe find that the RSD data affect constraints on some parameters like p , but they only give mild bounds on otherparameters such as β, λ, q V . Nonetheless, the RSD data also contribute to shifting/reducing the parameter space forthe background parameters Ω m , h, s , because the matter perturbation equation depends on those parameters.The MCMC likelihood analysis shows that there is a quite large degeneracy in terms of the minimum χ , i.e.,several different parameters associated with perturbations can lead to similar values of χ . In other words, the modeldoes not seem to have one unique global minimum of χ , but there are several of those minima for different sets ofparameters. We can pick up one example of such a low value of χ . For instance, we obtain the minimum value χ = 625 . , (6.2)for the following (approximated) values:Ω m = 0 . , h = 0 . , s = 0 . ,p = 2 . , ln( − β ) = 0 . , ln q V = − . , ln λ = − . . (6.3)The corresponding 2 σ bounds on these parameters are given, respectively, byΩ m = 0 . +0 . − . , (6.4) h = 0 . +0 . − . , (6.5) s = 0 . +0 . − . , (6.6) p = 2 . +19 . − . , (6.7) β = − . +0 . − . , (6.8)¯ q V ≤ q V < , (6.9)¯ λ ≤ λ < . , (6.10)where ¯ q V and ¯ λ are the lower limits from the assumed prior/numerical precision.The probability distributions and the likelihood contours for the background parameters Ω m , s, h and the fullparameters Ω m , s, h, p, β, q V , λ are plotted in Figs. 2 and 3, respectively. The RSD data together with the stabilityconditions influence the parameter space in several ways. We summarize the main results in the following. • Inclusion of the RSD data tends to reduce the best-fit value of s (compared to s ≃ .
26 constrained from thebackground), but still a positive value of s around 0.16 is favored, see Fig. 2. This implies that the RSD datagenerally prefer the model with lower s . Indeed, the RSD data alone can be consistent with the case s = 0. Thisis mostly attributed to the fact that the models with a larger s tend to give rise to a larger effective gravitationalcoupling with G eff > G . There is a tendency that the current RSD measurements favor the cosmic growth ratesmaller than that predicted by GR [62]. In generalized Proca theories it is possible to realize G eff < G , but thisoccurs at the expense of choosing the value of q V close to 0 [42]. To avoid the strong coupling problem of vectorperturbations we set the prior q V . − , in which case G eff cannot be significantly smaller than G . Since theeffect of weak gravity arising from small q V is limited, the modification of G eff induced by the change of s tendsto be more important. In Fig. 3 we observe that the parameter q V is loosely constrained. • The data other than RSD favor a non-zero positive value of s . Therefore, combining all these different contri-butions, we obtain the bound (6.6), which is smaller than s = 0 .
25. On performing a MCMC sampling for theΛCDM model with the two parameters Ω m and h , we find that the best-fit case corresponds to χ = 642 . m = 0 .
298 and h = 0 . χ is larger than (6.2), the model with s ≈ .
16 can fit thejoint observational constraints of the CMB, BAO, SN Ia, the Hubble expansion rate, and RSD better than theΛCDM model.7
FIG. 2: Observational bounds on the three parameters Ω m , s, h derived by adding the RSD data to the data of the CMB,BAO, SN Ia, and the Hubble expansion rate. The no-ghost and stability conditions are also taken into account. The meaningsof one-dimensional probability distributions and two-dimensional likelihood contours are the same as those explained in thecaption of Fig. 1. The ΛCDM model is still disfavored over the model with s > • Comparing the bounds (6.4) and (6.5) with Eqs. (5.5) and (5.6), the constraints on Ω m and h derived byadding the RSD data to the data associated with the background expansion history are not subject to significantmodifications relative to those obtained without the RSD data. • Since only the RSD data are affected by the four parameters p, β, q V , λ associated with perturbations, we findthat the constraints on them are mild and that some degeneracy of χ exists among different model parameters.We expect that this degeneracy may be reduced by including other independent observational data relevant toperturbations.8 FIG. 3: Observational constraints on the seven model parameters Ω m , s, h, p, β, q V , λ derived by the joint data analysis of theCMB, BAO, SN Ia, the Hubble expansion rate, and RSD data, with no-ghost and stability conditions taken into account. Themeanings of one-dimensional probability distributions and two-dimensional likelihood contours are the same as those explainedin the caption of Fig. 1. The three parameters Ω m , s, h associated with the background are tightly constrained, but the boundson the four parameters p, β, q V , λ are quite weak. VII. CONCLUSIONS
The recently proposed generalized Proca interactions constitute promising alternative theories of gravity on largescales. These derivative vector-tensor interactions are constructed in such a way that the resulting theories containonly five propagating degrees of freedom, three of them originating from the massive vector field [35, 37, 38]. Theyestablish a consistent framework for the late-time cosmic acceleration. On the FLRW background, the temporalcomponent of the vector field gives rise to interesting de Sitter solutions relevant to dark energy. Even if the temporalcomponent is not dynamical, its auxiliary role results in promising de Sitter attractors as it was shown in Ref. [41].In this work, we have placed observational constraints on a class of dark energy models in the framework of9generalized Proca theories. We have first summarized the key findings of the background evolution and stabilityanalysis of perturbations performed in Refs. [41, 42]. The background dynamics is rather simple and dictated by thethree parameters Ω m , h, s , where s represents the deviation from the ΛCDM model. For the evolution of matterperturbations, we have used the equation derived under the quasi-static approximation on sub-horizon scales, whosevalidity was explicitly checked in Ref. [42]. We have also taken into account conditions for avoiding ghosts andLaplacian instabilities of tensor, vector, and scalar perturbations. The perturbations carry four additional parameters p, β, q V , λ than those associated with the background.At the background level, we have exploited the data sets of CMB distance priors, BAO, SN Ia (Union 2.1), andlocal measurements of the Hubble expansion rate. We have found that the MCMC analysis constrains the parameter s to be s = 0 . +0 . − . (95 % CL) from the background cosmic expansion history, so the model with s > s can reduce the tensions of the parameters Ω m and h between early-time and late-time data sets.Including the RSD data as well as no-ghost and stability conditions of perturbations, the bound on the parameter s is shifted to s = 0 . +0 . − . (95 % CL). This shift is mostly related to the fact that the RSD data tend to favor lowervalues of s for realizing G eff close to G . Existence of the intrinsic vector mode can lead to a G eff smaller than G for q V close to 0, but this effect is limited by the fact that the vector perturbation has a strong coupling problem for suchsmall values of q V . Since the data other than RSD prefer positive values of s away from 0, the joint data analysisincluding both the background and the RSD data still favor the model with s > m , h, s , the observational bounds on other parameters are notstringent.As we have seen in this work, the derivative interactions in generalized Proca theories facilitate the backgroundevolutions quite generically, which give rise to the dynamics for alleviating the tension between early-time and late-time data sets. It is still possible that the tension present in the ΛCDM model may be related to some systematicerrors in one (or more) data sets. Nonetheless, we find it interesting at least to have shown that the cosmologicalbackground fitting the data well can naturally follow from generalized Proca theories.We note that beyond-generalized Proca theories recently proposed in Ref. [39] share the same background evolutionas the model (2.15). Nevertheless, the presence of additional terms yields distinctive features at the level of pertur-bations. For instance, it is possible to weaken the gravitational coupling with non-relativistic matter further, e.g., G eff ≈ . G today [81]. The smaller effective gravitational coupling, which may fit the RSD data better than themodel studied in this paper, arise from beyond-generalized Proca interactions rather than from intrinsic vector modeswith q V close to 0. It will be of interest to place observational constraints on such models as well by adding otherdata associated with perturbations, e.g., the integrated Sachs-Wolfe effect of CMB and weak lensing. We hope thatfuture high-precision observations will allow us to distinguish the models in the framework of (beyond-)generalizedProca theories from the ΛCDM model further. Acknowledgments
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