Coupled vector dark energy

TL;DR

This paper develops a framework for analyzing vector field interactions with dark matter, showing how such couplings influence cosmological perturbations and can produce dark energy behavior close to a cosmological constant.

Contribution

It introduces a general coupled vector dark energy model, analyzing its effects on perturbations and demonstrating how coupling can make dark energy resemble a cosmological constant.

Findings

Coupling affects scalar perturbation stability and sound speed.

Interaction can produce a dark energy equation of state close to -1.

Tensor and vector perturbations remain unaffected by the coupling.

Abstract

We provide a general framework for studying the evolution of background and cosmological perturbations in the presence of a vector field $A_{\mu}$ coupled to cold dark matter (CDM). We consider an interacting Lagrangian of the form $Q f(X) T_c$, where $Q$ is a coupling constant, $f$ is an arbitrary function of $X=-A_{\mu}A^{\mu}/2$, and $T_c$ is a trace of the CDM energy-momentum tensor. The matter coupling affects the no-ghost condition and sound speed of linear scalar perturbations deep inside the sound horizon, while those of tensor and vector perturbations are not subject to modifications. The existence of interactions also modifies the no-ghost condition of CDM density perturbations. We propose a concrete model of coupled vector dark energy with the tensor propagation speed equivalent to that of light. In comparison to the $Q=0$ case, we show that the decay of CDM to the vector field leads to the phantom dark energy equation of state $w_{\rm DE}$ closer to $-1$. This alleviates the problem of observational incompatibility of uncoupled models in which $w_{\rm DE}$ significantly deviates from $-1$. The maximum values of $w_{\rm DE}$ reached during the matter era are bounded from the CDM no-ghost condition of future de Sitter solutions.

Equations

$$\mathcal{S}={\mathcal{S}}_{\mathrm{GP}}+{\mathcal{S}}_M+{\mathcal{S}}_{\mathrm{int}},$$

$${\mathcal{S}}_{\mathrm{GP}}=\int {\mathrm{d}}^{4}x\sqrt{-g}\left[\frac{M_{\mathrm{pl}}^2}{2}R+G_2(X,F)+G_3(X){\nabla }_{\mu }{A}^{\mu }\right],$$

$${\cal S}_{M}=-\sum_{I=c,b,r}\int{\rm d}^{4}x\Bigl{[}\sqrt{-g}\,\rho_{I}(n_{I})+J_{I}^{\mu}\bigl{(}\partial_{\mu}\ell_{I}+\mathcal{A}_{I1}\partial_{\mu}\mathcal{B}_{I1}+\mathcal{A}_{I2}\partial_{\mu}\mathcal{B}_{I2}\bigr{)}\Bigr{]}\,,$$

$$n_I=\sqrt{\frac{J_I^{\mu }{J}_I^{\nu }{g}_{\mu \nu }}{g}}.$$

$$(T_I{)}_{\nu }^{\mu }=\left({\rho }_I+P_I\right)u_I^{\mu }{u}_{I\nu }+P_I{\delta }_{\nu }^{\mu },$$

$$u_{I\mu }\equiv \frac{J_{I\mu }}{n_I\sqrt{-g}}.$$

$$u_I^{\mu }{u}_{I\mu }=-1.$$

$${\mathcal{S}}_{\mathrm{int}}=\int {\mathrm{d}}^{4}x\sqrt{-g}Qf(X)T_c,$$

$$P_c=0.$$

$${\mathcal{S}}_{\mathrm{int}}=-\int {\mathrm{d}}^{4}x\sqrt{-g}Qf(X){\rho }_c(n_c).$$

$${\partial }_{\mu }{\ell }_I+{\mathcal{A}}_{I1}{\partial }_{\mu }{\mathcal{B}}_{I1}+{\mathcal{A}}_{I2}{\partial }_{\mu }{\mathcal{B}}_{I2}=u_{I\mu }{\rho }_{I,n_I},\mathrm{for}I=b,r,$$

$${\partial }_{\mu }{\ell }_c+{\mathcal{A}}_{c1}{\partial }_{\mu }{\mathcal{B}}_{c1}+{\mathcal{A}}_{c2}{\partial }_{\mu }{\mathcal{B}}_{c2}=u_{c\mu }{\rho }_{c,n_c}\left[1+Qf(X)\right].$$

$$\mathrm{d}{s}^2=-N^2(t)\mathrm{d}{t}^2+a^2(t){\delta }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j,$$

$$A^{\mu }=\left(\frac{\varphi (t)}{N(t)},0,0,0\right),$$

$${\mathcal{S}}_{\mathrm{GP}}=\int {\mathrm{d}}^{4}x{a}^3\left(N{G}_2+G_3\dot{\varphi }+3G_{3}H\varphi -\frac{3M_{\mathrm{pl}}^2{H}^2}{N}\right),$$

$$J_I^0=n_I{a}^3.$$

$${\mathcal{N}}_I\equiv J_I^0=n_I{a}^3=\mathrm{constant},$$

$${\dot{n}}_I+3H{n}_I=0.$$

$${\mathcal{S}}_M+{\mathcal{S}}_{\mathrm{int}}=-\int {\mathrm{d}}^{4}x\left[N{a}^3\left\{[1+Qf(X)]{\rho }_c+{\rho }_b+{\rho }_r\right\}+a^3\left(n_c{\dot{\ell }}_c+n_b{\dot{\ell }}_b+n_r{\dot{\ell }}_r\right)\right].$$

$${\dot{\ell }}_I$$

$${\dot{\ell }}_c$$

$$P_I\equiv n_I{\rho }_{I,n_I}-{\rho }_I.$$

$${\dot{\rho }}_I+3H\left({\rho }_I+P_I\right)=0.$$

$${\dot{\rho }}_b+3H{\rho }_b=0,$$

$${\dot{\rho }}_r+4H{\rho }_r=0,$$

$${\dot{\rho }}_c+3H{\rho }_c=0.$$

$$3M_{\mathrm{pl}}^2{H}^2=-G_2+\left(1+Qf\right){\rho }_c+{\rho }_b+{\rho }_r,$$

$$M_{\mathrm{pl}}^2\left(2\dot{H}+3H^2\right)=-G_2+G_{3,X}{\varphi }^2\dot{\varphi }-\frac{1}{3}{\rho }_r,$$

$$\varphi \left(G_{2,X}+3G_{3,X}H\varphi -Q{f}_{,X}{\rho }_c\right)=0,$$

$$G_{2,X}+3G_{3,X}H\varphi -Q{f}_{,X}{\rho }_c=0.$$

$${\tilde{\rho }}_c\equiv \left(1+Qf\right){\rho }_c.$$

$${\dot{\tilde{\rho }}}_c+3H{\tilde{\rho }}_c=\frac{Q{f}_{,X}\varphi \dot{\varphi }}{1+Qf}{\tilde{\rho }}_c.$$

$${\rho }_{\mathrm{DE}}$$

Full text

I Introduction

The energy density of today’s universe is dominated by two unknown components– dark energy and dark matter dreview1 ; dreview2 ; dreview3 ; dreview4 ; dreview5 ; dmreview1 ; dmreview2 . Dark energy gives rise to the late-time cosmic acceleration through an effective negative pressure, while dark matter leads to the growth of structures through gravitational clusterings. Although the two dark components have different characteristics, they can be potentially coupled to each other. The existence of such interactions generally modifies the cosmic expansion and growth histories, so the coupled models can be distinguished from the $\Lambda$-cold-dark-matter ($\Lambda$CDM) model by exploiting numerous observational data including supernovae type Ia (SN Ia) SN1 ; SN2 , cosmic microwave background (CMB) temperature anisotropies WMAP ; Planck13 , baryon acoustic oscillations (BAOs) BAO , and redshift-space distortions (RSDs) RSD1 ; RSD2 .

For a canonical scalar field $\varphi$, Wetterich Wetterich first proposed a coupled quintessence scenario in which $\varphi$ interacts with CDM. In this model, the continuity equation of dark energy is sourced by the term $\beta\rho_{c}\dot{\varphi}$, where $\beta$ is a coupling constant, $\rho_{c}$ is the dark matter density, and $\dot{\varphi}$ is the time derivative of $\varphi$. Such a coupling arises in Brans-Dicke theories Brans after a conformal transformation to the Einstein frame FM ; fRreview . The quintessence field $\varphi$ drives the cosmic acceleration in the presence of a shallow potential $V(\varphi)$, e.g., the exponential potential $V(\varphi)=V_{0}e^{-\lambda\varphi/M_{\rm pl}}$ with $|\lambda|<{\cal O}(1)$ (where $M_{\rm pl}$ is the reduced Planck mass) quin1 ; quin2 ; quin3 ; quin4 ; quin5 . Amendola Amendola99 showed that there exists a scaling $\varphi$-matter-dominated epoch ($\varphi$MDE) during which the coupling gives rise to a constant dark energy density parameter $\Omega_{\varphi}=2\beta^{2}/3$.

The coupling $\beta$ in interacting quintessence is constrained to be $\beta<0.062$ at 95 % CL from the Planck CMB data alone, but the joint datasets of CMB, SN Ia, BAOs, and today’s Hubble constant $H_{0}$ lead to the marginalized posterior distribution with a peak around $\beta=0.036$ Ade15 . Thus, the possibility for sizable interactions between the two dark components remains in current observations.

There are also dark energy models in which a noncanonical scalar field is coupled to dark matter, including k-essence Piazza ; Tsuji04 ; Gumjudpai , Horndeski Gomes1 ; Gomes2 ; ABDG ; Frusciante:2018aew , and DHOST theories Koyama . They are mostly based on the interacting term $\beta\rho_{c}\dot{\varphi}$ in the dark energy continuity equation. In such noncanonical theories, it is also possible to realize the $\varphi$MDE followed by late-time cosmic acceleration Amendola06 ; Frusciante:2018aew . There are also models with more phenomenological choices of couplings between two dark sectors Dalal:2001dt ; Zimdahl:2001ar ; Chimento:2003iea ; Wei:2006ut ; Amendola:2006dg ; Guo:2007zk ; Wang1 ; Wang2 , e.g., $\beta H\rho_{c}$, where $H$ is the Hubble expansion rate. In the latter approach, it is generally difficult to identify corresponding Lagrangians and associated stability conditions (e.g., no ghosts) of perturbations.

The scalar field is not only the possibility for realizing late-time cosmic acceleration, but a massive vector field can also be the source for dark energy. In generalized Proca (GP) theories with a vector field $A_{\mu}$ breaking the $U(1)$ gauge symmetry Heisenberg ; Tasinato1 ; Tasinato2 ; Fleury ; Hull ; Allys ; Jimenez16 ; Allys2 , the time-dependent temporal component of $A_{\mu}$ can give rise to self-accelerating de Sitter attractors preceded by a constant phantom dark energy equation of state $w_{\rm DE}$ during the matter era DeFelice:2016yws ; DeFelice:2016uil . The dark energy models given by the Lagrangian $L=M_{\rm pl}^{2}R/2-(1/4)F_{\mu\nu}F^{\mu\nu}+b_{2}X^{p_{2}}+b_{3}X^{p_{3}}\nabla_{\mu}A^{\mu}$, where $R$ is the Ricci scalar, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength, and $b_{2},b_{3},p_{2},p_{3}$ are constants with $X=-A_{\mu}A^{\mu}/2$, exhibit a better compatibility with the datasets of SN Ia, CMB, BAOs, RSDs, and $H_{0}$ in comparison to the $\Lambda$CDM model deFelice:2017paw . This property persists even with the integrated-Sachs-Wolfe (ISW) effect and galaxy cross-correlation data, by reflecting the fact that the existence of intrinsic vector modes can generate positive cross-correlations Nakamura:2018oyy .

The natural question arises as to what happens in the presence of couplings between the massive vector field and CDM. For this purpose, we introduce the interacting Lagrangian of the form $L_{\rm int}=-Qf(X)\rho_{c}$ in this paper, where the vector field is coupled to the CDM density $\rho_{c}$ with an arbitrary coupling $f(X)$. We consider cubic-order GP theories with baryons and radiations taken into account and obtain the background equations of motion on the flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime to study the dynamics of coupled dark energy from the radiation era to today (see Refs. Wei:2006tn ; Wei:2006gv for other works about coupled vector dark energy). We compute the second-order actions of tensor, vector, and scalar perturbations and derive conditions for the absence of their ghost and Laplacian instabilities.

For the power-law coupling $f(X)=(X/M_{\rm pl}^{2})^{q}$ with the Lagrangian $L=-(1/4)F_{\mu\nu}F^{\mu\nu}+b_{2}X+b_{3}X^{p_{3}}\nabla_{\mu}A^{\mu}$ in the vector sector, we show that, for $Q<0$, the dark energy equation of state can be larger than that in the uncoupled case during the matter era. When $Q=0$ the models with $p_{3}\leq 2$ are in tension with observational datasets due to the large deviation of $w_{\rm DE}$ from $-1$ deFelice:2017paw ; Nakamura:2018oyy . For example, the vector Galileon ($p_{3}=1$) gives the value $w_{\rm DE}=-2$ in the matter era, which corresponds to the tracker solution for the scalar Galileon DeFelice:2010pv ; DeFelice:2010nf . Existence of the negative coupling $Q$ allows an interesting possibility for reducing such tensions. As we will show in this paper, the no-ghost condition of the matter sector on future de Sitter solutions places the upper limit on $|Qf(X)|$ as well as the maximum value of $w_{\rm DE}$ during the matter dominance. In particular, the model with $p_{3}=2$ can be compatible with observational data of the background expansion history.

We note that the negative coupling $Q$ corresponds to the decay of dark matter to dark energy. There are other phenomenological coupled dark energy models in which the decay of dark matter can reduce the tensions of today’s Hubble constant $H_{0}$ and the amplitude of matter perturbations $\sigma_{8}$ between CMB and low-redshift measurements DiValentino:2017iww ; DiValentino:2019ffd . This gives us the further motivation to compute the effective gravitational coupling for CDM in coupled vector dark energy and to confront the model with observations. We will address these issues in a future separate publication.

This paper is organized as follows. In Sec. II, we derive the background equations of motion and discuss how the vector field and CDM are coupled to each other. In Sec. III, we identify conditions for the absence of ghosts and Laplacian instabilities of tensor, vector, and scalar perturbations for linear cosmological perturbations. In Sec. IV, we propose a concrete coupled vector model of late-time cosmic acceleration and study the dynamics of dark energy together with the stability conditions. Sec. V is devoted to conclusions.

Throughout the paper, the Greek and Latin indices are used to represent four-and three-dimensional quantities, respectively. For the partial and covariant derivatives with respect to $x^{\mu}$, we adopt the notations $\partial_{\mu}$ and $\nabla_{\mu}$, respectively. We also use the natural unit in which the speed of light $c$ and the reduced Planck constant $\hbar$ are equivalent to 1. The capital label “$I$” represents different matter species (CDM, baryons, radiations).

II Coupled vector dark energy model

and background equations

We study the cosmology of cubic-order GP theories Heisenberg ; Tasinato1 ; Tasinato2 in which a vector field $A_{\mu}$ is coupled to CDM. We also take baryons and radiations into account and assume that they are not directly coupled to $A_{\mu}$. The total action is then given by

[TABLE]

where

[TABLE]

with $g$ being the determinant of metric tensor $g_{\mu\nu}$. The function $G_{2}$ depends on $X=-A_{\mu}A^{\mu}/2$ and $F=-F_{\mu\nu}F^{\mu\nu}/4$, while $G_{3}$ is a function of $X$ alone. The massive vector field with the standard Maxwell Lagrangian corresponds to $G_{2}(X,F)=m^{2}X+F$, where the mass squared $m^{2}$ can be either positive or negative. Existence of the Lagrangian $G_{3}(X)\nabla_{\mu}A^{\mu}$ allows the possibility for realizing a de Sitter solution with constant $X$ DeFelice:2016yws ; DeFelice:2016uil . Besides two tensor polarizations arising from the Ricci scalar $R$, there are two transverse vector modes and one longitudinal scalar arising from the breaking of $U(1)$ gauge symmetry.

For the matter action ${\cal S}_{M}$, we consider perfect fluids described by the Schutz-Sorkin action Sorkin ; Brown ; DGS

[TABLE]

where the subscripts $I=c,b,r$ represent CDM, baryons, and radiations, respectively. The energy density $\rho_{I}$ depends on the fluid number density $n_{I}$. We note that the perturbation $\delta\rho_{I}$ of energy density plays the role of a dynamical scalar degree of freedom in the matter sector. The vector field $J_{I}^{\mu}$ is related to $n_{I}$ according to

[TABLE]

The vector field $J_{I}^{\mu}$ is related to the four-velocity of each matter, whereas the scalar field $\ell_{I}$ is a Lagrange multiplier corresponding to a constraint of the particle conservation. The quantities ${\cal A}_{1,2}$ and ${\cal B}_{1,2}$ are the Lagrange multipliers and Lagrange coordinates of fluids, respectively, both of which are associated with nondynamical intrinsic vector modes. In Sec. III, we vary the second-order actions of vector and scalar perturbations with respect to these nondynamical variables and eliminate them from the corresponding actions. This is for the purpose of deriving stability conditions of dynamical vector and scalar degrees of freedom.

The energy-momentum tensor of each perfect fluid is given by

[TABLE]

where $\rho_{I}$ and $P_{I}$ correspond to the energy density and pressure, respectively, and $u_{I{\mu}}$ is the four-velocity related to $J_{I{\mu}}$, as

[TABLE]

From Eq. (4), it follows that

[TABLE]

We consider the case in which both baryons and radiations are uncoupled to the vector field. Even in this case, the cubic interaction $G_{3}(X)\nabla_{\mu}A^{\mu}$ leads to the gravitational coupling for baryons different from the Newton gravitational constant for linear cosmological perturbations Nakamura:2018oyy . In over-density regions of the Universe, however, the fifth force is suppressed by the same cubic interaction through the operation of the Vainshtein mechanism DeFelice:2016cri .

On the other hand, we assume that $A_{\mu}$ is coupled to CDM with the interacting action

[TABLE]

where $Q$ is a dimensionless coupling constant, $f$ is a function of $X$, and $T_{c}$ is the trace $(T_{c})^{\mu}_{\mu}$ of Eq. (5). We focus on the case in which the CDM pressure vanishes, i.e.,

[TABLE]

On using Eq. (5) with Eq. (7), the action (8) reduces to

[TABLE]

Taking the variation of the action (1) with respect to $J_{I}^{\mu}$ and employing the relation $\partial n_{I}/\partial J_{I}^{\mu}=J_{I{\mu}}/(n_{I}g)$, we obtain

[TABLE]

where $\rho_{I,n_{I}}\equiv\partial\rho_{I}/\partial n_{I}$. For CDM, we have

[TABLE]

Let us consider the flat FLRW background described by the line element

[TABLE]

where $N(t)$ is the lapse, $a(t)$ is the scale factor, and $t$ is the cosmic time. The vector field profile compatible with the line element (13) is given by

[TABLE]

where $\phi$ depends on $t$. Then, we have that $X=\phi^{2}(t)/2$ and $F=0$. The temporal vector component $\phi(t)$ is an auxiliary field playing the role of dark energy at the background level DeFelice:2016yws ; DeFelice:2016uil . Replacing $A_{\mu}$ for $\partial_{\mu}\varphi$, the GP theories discussed below have the analogue to shift-symmetric scalar-tensor theories with the scalar field $\varphi$. At the background level, integrating the relation $A_{0}=\partial_{0}\varphi$ gives the additional integration constant for $\varphi$. In other words, the scalar-tensor counterpart of GP theories has an additional degree of freedom for the choice of initial conditions DeFelice:2010pv ; DeFelice:2010nf . This means that the background dynamics in GP theories is not generally the same as the corresponding analogue of scalar-tensor theories.

Up to boundary terms, the action (2) of the gravity and vector-field sectors yields

[TABLE]

where a dot represents the derivative with respect to $t$, and $H\equiv\dot{a}/a$ is the Hubble expansion rate.

Since the fluid four-velocity in its rest frame is given by $u_{I}^{\mu}=(N^{-1},0,0,0)$, Eq. (6) leads to

[TABLE]

Varying the action (3) with respect to $\ell_{I}$, we obtain

[TABLE]

which means that the particle number ${\cal N}_{I}$ is conserved. In other words, the number density of each matter species (including CDM) obeys

[TABLE]

On using Eq. (16), the action in the matter sector reduces to

[TABLE]

On the FLRW background (13) the vector modes are absent in Eqs. (11)-(12), with the four velocity $u_{I{\mu}}=(-N,0,0,0)$, Then, we obtain

[TABLE]

The pressure $P_{I}$ associated with the energy density $\rho_{I}$ is given by DeFelice:2016yws ; DeFelice:2016uil

[TABLE]

Substituting Eqs. (20) and (21) into Eq. (19) and taking the limit $Q\to 0$, the action (19) reduces to the sum of pressures, i.e., $\int{\rm d}^{4}x\sqrt{-g}\sum_{I=c,b,r}P_{I}$. For $Q\neq 0$, the additional term $Qf(X)$ is present for CDM.

On using Eqs. (18), (22) and the property $\dot{\rho}_{I}(n_{I})=\rho_{I,n_{I}}\dot{n}_{I}$, the energy density $\rho_{I}(n_{I})$ of each matter component obeys

[TABLE]

We consider the case in which the pressures of baryons and radiations satisfy $P_{b}=0$ and $P_{r}=\rho_{r}/3$, respectively, with the vanishing CDM pressure (9). Then, the energy densities of three matter components obey

[TABLE]

Varying the sum of actions (15) and (19) with respect to $N$, $a$, $\phi$ and setting $N=1$ at the end, we obtain the background equations:

[TABLE]

where $G_{i,X}\equiv\partial G_{i}/\partial X$. We focus on the branch $\phi\neq 0$ in Eq. (29), i.e.,

[TABLE]

We define the energy density of CDM containing the effect of interactions with the vector field, such that

[TABLE]

On using Eq. (26), we find that $\tilde{\rho}_{c}$ obeys the differential equation:

[TABLE]

Unlike the conserved CDM density $\rho_{c}$, the effective CDM density $\tilde{\rho}_{c}$ is a physical quantity whose continuity equation contains the effect of couplings on the right hand side of Eq. (32). In spite of the conservation of CDM particle number (18), i.e., $n_{c}a^{3}={\rm constant}$, the CDM density $\tilde{\rho}_{c}$ acquires the effective mass term $m_{c}(Q_{f})$ by the coupling $Q_{f}=Qf$, such that $\tilde{\rho}_{c}=m_{c}(Q_{f})n_{c}$. This means that, unlike $\rho_{c}$, the effective density $\tilde{\rho}_{c}$ does not obey the standard continuity equation. We also observe that the matter action (19) contains the term proportional to $\tilde{\rho}_{c}$.

We also define the energy density $\rho_{\rm DE}$ and pressure $P_{\rm DE}$ of dark energy (arising from the vector field), as

[TABLE]

with the equation of state

[TABLE]

Taking the time derivative of Eq. (33) and using Eqs. (30) and (34), it follows that

[TABLE]

Comparing Eq. (32) with Eq. (36), it is clear that the vector field and CDM interact with each other through the couplings with opposite signs.

III Conditions for avoiding ghosts and gradient instabilities

We derive conditions for the absence of ghosts and Laplacian instabilities for tensor, vector, and scalar perturbations. Throughout the paper, we focus on the evolution of linear cosmological perturbations without considering the nonlinear regime, e.g., the small-scale region in which dark matter is concentrated in halos of galaxies. Even for nonlinear perturbations today, they were in the linear regime in the past cosmic growth history. The stability conditions derived in this section should be consistently satisfied to ensure the stability of perturbations from the past to today for linear perturbations deep inside the sound horizons.

We consider the perturbed line element in the flat gauge on the flat FLRW background:

[TABLE]

where $\alpha$ and $\chi$ are scalar perturbations, $V_{i}$ is the vector perturbation satisfying the transverse condition $\partial^{i}V_{i}=0$, and $h_{ij}$ is the tensor perturbation obeying the transverse and traceless conditions $\partial^{i}h_{ij}=0$ and $h^{i}_{i}=0$. All the perturbed quantities depend on both $t$ and $x^{i}$.

We decompose the temporal and spatial components of $A^{\mu}=(A^{0},A^{i})$ into the background and perturbed parts:

[TABLE]

where $\delta\phi$ and $\chi_{V}$ are scalar perturbations, and $E_{j}$ is the vector perturbation obeying the transverse condition $\partial^{j}E_{j}=0$. Similarly, the temporal and spatial components of the vector $J_{I}^{\mu}=(J_{I}^{0},J_{I}^{i})$ (with $I=b,r,c$) in the Schutz-Sorkin action (3) are decomposed as

[TABLE]

where $\delta J_{I}$ and $\delta j_{I}$ are scalar perturbations, and $W_{Ij}$ is the vector perturbation obeying the transverse condition $\partial^{j}W_{Ij}=0$. We express the quantities $\ell_{b,r,c}$ in the form

[TABLE]

where $\rho_{I,{n_{I}}}(t)$ and $Qf(t)$ are evaluated on the background, $v_{I}$ and $v_{c}$ are velocity potentials having the dependence of both $t$ and $x^{i}$. We observe that the background Eqs. (20) and (21) with $N=1$ are consistent with Eqs. (40) and (41).

As for the vector perturbations $V_{i}$, $E_{i}$ and $W_{Ii}$, we choose

[TABLE]

whose $x$ and $y$ components depend on $t$ and $z$. They are consistent with the transverse conditions mentioned above. For the quantities ${\cal A}_{Ii}$ and ${\cal B}_{Ii}$ in Eq. (3), we choose

[TABLE]

where $\delta{\cal A}_{Ii}$ and $\delta{\cal B}_{Ii}$ are linearly perturbed quantities. We recall that the four-velocities of baryons and radiations have the relation (11), while the CDM four-velocity satisfies the relation (12). On defining the intrinsic velocity vectors $v_{Ii}$ as

[TABLE]

the spatial components of $u_{I{\mu}}$ yield

[TABLE]

where $v_{b,r,c}$ are the scalar velocity potentials appearing in Eqs. (40) and (41).

III.1 Tensor perturbations

The tensor perturbation is expressed in terms of the sum of two polarization modes, as $h_{ij}=h_{+}e_{ij}^{+}+h_{\times}e_{ij}^{\times}$. In Fourier space with the comoving wavenumber $\bm{k}$, the unit vectors $e_{ij}^{+}$ and $e_{ij}^{\times}$ satisfy the normalizations $e_{ij}^{+}(\bm{k})e_{ij}^{+}(-\bm{k})^{*}=1$, $e_{ij}^{\times}(\bm{k})e_{ij}^{\times}(-\bm{k})^{*}=1$, and $e_{ij}^{+}(\bm{k})e_{ij}^{\times}(-\bm{k})^{*}=0$.

Expanding the action (1) up to second order in $h_{ij}$ and using the background Eq. (28), it follows that the terms containing $h_{+}^{2}$ and $h_{\times}^{2}$ identically vanish. The resulting second-order action of tensor perturbations is given by

[TABLE]

where

[TABLE]

The action (48) is the same as that in general relativity. Hence the propagation of tensor perturbations is not modified by the nonvanishing coupling $Q$. Since the speed $c_{T}$ of gravitational waves is equivalent to 1, the coupled dark energy theories given by (1) are consistent with the bound arising from the GW170817 event GW170817 .

III.2 Vector perturbations

As for vector perturbations, we first expand the actions (3) and (8) up to second order. The resulting quadratic-order actions are given, respectively, by

[TABLE]

Varying the action $({\cal S}_{M}^{(2)})_{V}+({\cal S}_{\rm int}^{(2)})_{V}$ with respect to $W_{Ii}$ and using Eqs. (45) and (46), it follows that

[TABLE]

which hold for $I=b,r,c$. Substituting this relation into Eqs. (50)-(51) and varying $({\cal S}_{M}^{(2)})_{V}+({\cal S}_{\rm int}^{(2)})_{V}$ with respect to $v_{Ii}$ and $\delta{\cal B}_{Ii}$, we obtain

[TABLE]

where $C_{Ii}$ are constants in time.

After integrating out the perturbations $W_{Ii}$ and $\delta{\cal A}_{Ii}$, the resulting second-order action in the matter sector yields

[TABLE]

We now expand the action (1) up to second order in vector perturbations. In doing so, it is convenient to introduce the combination

[TABLE]

which correspond to $A_{i}$. Then, the total quadratic-order action for vector perturbations yields

[TABLE]

where

[TABLE]

Varying the action (57) with respect to $V_{i}$ in Fourier space with the comoving wavenumber $k=|{\bm{k}}|$, we obtain

[TABLE]

which can be used to eliminate the fourth term in Eq. (57). For linear perturbations deep inside the Hubble radius, the action (57) reduces to

[TABLE]

where

[TABLE]

The two dynamical fields $Z_{1}$ and $Z_{2}$ propagate with the speed $c_{V}$ equivalent to 1, so there are no Laplacian instabilities of vector perturbations. The no-ghost condition corresponds to $q_{V}>0$, i.e.,

[TABLE]

From the above discussion, it is clear that the coupling $Q$ does not affect the stability conditions of linear vector perturbations.

III.3 Scalar perturbations

To study the propagation of scalar perturbations, we first define the density perturbation $\delta\rho_{I}$ of each matter fluid ($I=c,b,r$), as

[TABLE]

where $\rho_{I,n_{I}}$ solely depends on the number density $n_{I}$. By defining $\delta\rho_{I}$ in this way, the perturbation of number density $n_{I}$, expanded up to second order, can be expressed as

[TABLE]

whose first term on the right hand side is consistent with the left hand side.

Expanding ${\cal S}_{M}+{\cal S}_{\rm int}$ up to second order in scalar perturbations and varying the resulting quadratic-order action with respect to $\delta j_{I}$, it follows that

[TABLE]

which can be used to eliminate the nondynamical fields $\delta j_{I}$ from the matter action. We note that the relations (65) also follow from the spatial components of Eqs. (11) and (12).

The propagation speed squares of matter perfect fluids are defined as

[TABLE]

with $I=c,b,r$. We focus on the case in which $c_{I}^{2}$ for CDM, baryons, and radiations are given, respectively, by

[TABLE]

To expand the action ${\cal S}_{\rm GP}$ up to quadratic order in scalar perturbations, we introduce the combination

[TABLE]

so that $A_{i}=\partial_{i}\psi$ in the scalar sector. Using the background Eqs. (27)-(29), the second-order action arising from (1) reads

[TABLE]

where

[TABLE]

and

[TABLE]

with

[TABLE]

The action ${\cal S}_{Q=0}^{(2)}$ coincides with that derived in Refs. DeFelice:2016yws ; DeFelice:2016uil in the single-fluid limit. The coupling $Q$ gives rise to the additional action ${\cal S}_{Q}^{(2)}$ to ${\cal S}_{Q=0}^{(2)}$. We note that the intrinsic vector mode affects the second-order scalar action through the quantity $w_{3}=-2\phi^{2}q_{V}$ in Eq. (74). Hence the scalar perturbation evolves differently compared to the corresponding analogue of scalar-tensor theories. Indeed, this difference manifests itself in the observations of ISW-galaxy cross-correlations Nakamura:2018oyy .

Varying (69) with respect to nondynamical fields $\alpha$, $\chi$, $\delta\phi$, $v_{b}$, $v_{r}$, and $v_{c}$ in Fourier space, respectively, it follows that

[TABLE]

where

[TABLE]

The dynamical perturbations correspond to the four fields $\psi$ and $\delta\rho_{I}$ ($I=c,b,r$). Under the gauge transformation $\tilde{t}=t+\xi^{0}$ and $\tilde{x}^{i}=x^{i}+\delta^{ij}\partial_{j}\xi$, these fields transform as $\tilde{\psi}=\psi+\phi\,\xi^{0}$ and $\widetilde{\delta\rho_{I}}=\delta\rho_{I}-\dot{\rho}_{I}\xi^{0}$, respectively. If we consider two scalar metric perturbations $\zeta$ and $E$ in the spatial part of the line element (37), as in the form $a^{2}(t)[(1+2\zeta)\delta_{ij}+2\partial_{i}\partial_{j}E]{\rm d}x^{i}{\rm d}x^{j}$, they transform as $\tilde{\zeta}=\zeta-H\xi^{0}$ and $\tilde{E}=E-\xi$, respectively Heisenberg:2018wye . The spatial gauge-transformation scalar $\xi$ is fixed by choosing $E=0$. From the temporal gauge-transformation, we can construct the gauge-invariant variables $\psi_{\zeta}=\psi+\phi\zeta/H$ and $\delta\rho_{I\zeta}=\delta\rho_{I}+3(\rho_{I}+P_{I})\zeta$, where we used the continuity Eq. (23). Since $\zeta=0$ in the flat gauge, the perturbations $\psi_{\zeta}$ and $\delta\rho_{I\zeta}$ simply reduce to $\psi$ and $\delta\rho_{I}$, respectively.

Solving Eqs. (79)-(82) for $\alpha$, $\chi$, $\delta\phi$, $v_{b}$, $v_{r}$, $v_{c}$ and substituting them into Eq. (69), the second-order action in Fourier space is expressed in the form

[TABLE]

where $\bm{K}$, $\bm{G}$, $\bm{M}$ and $\bm{B}$ are $4\times 4$ matrices, and the vector field $\vec{\mathcal{X}}^{t}$ is given by

[TABLE]

Neither $\bm{M}$ nor ${\bm{B}}$ contains the $k^{2}/a^{2}$ term. If there are the terms including the $k^{2}/a^{2}$ dependence in ${\bm{B}}$, it can be absorbed into $\bm{G}$ after the integration by parts. For linear perturbations deep inside the sound horizon, the nonvanishing matrix components are

[TABLE]

and

[TABLE]

where

[TABLE]

The scalar ghosts are absent for $K_{11}>0$, $K_{22}>0$, $K_{33}>0$, and $K_{44}>0$. Since $n_{b}\rho_{b,n_{b}}=\rho_{b}>0$ and $n_{r}\rho_{r,n_{r}}=4\rho_{r}/3>0$, the last two conditions trivially hold. The first condition is satisfied for

[TABLE]

Since $n_{c}\rho_{c,n_{c}}=\rho_{c}>0$, the second condition translates to

[TABLE]

For negative value of $Qf$, this gives the upper bound on $|Qf|$.

For linear perturbations deep inside the sound horizon, the second-order action (84) gives rise to the dispersion relation

[TABLE]

with the frequency $\omega$. The scalar propagation speed $c_{s}$ is defined as $c_{s}^{2}=\omega^{2}a^{2}/k^{2}$. The propagation speed squared associated with the perturbation $\psi$ is given by $c_{S}^{2}=G_{11}/K_{11}$. To avoid the Laplacian instability, we require that

[TABLE]

The matter propagation speed squares, which correspond to the ratios $G_{22}/K_{22}$, $G_{33}/K_{33}$, $G_{44}/K_{44}$ for CDM, baryons, and radiations respectively, reduce to $c_{c}^{2}=0$, $c_{b}^{2}=0$, and $c_{r}^{2}=1/3$. Hence there are no Laplacian instabilities for the three matter perfect fluids.

The CDM perturbation $\delta\rho_{c}$ is associated with the perturbation of number-density dependent quantity $\rho_{c}(n_{c})$, which is not identical to the perturbation $\tilde{\delta\rho_{c}}$ absorbing the contribution of coupling $Qf$ (related to the background CDM density $\tilde{\rho_{c}}=(1+Qf)\rho_{c}$). In Eqs. (79) and (81), however, we observe that $\delta\rho_{c}$ is coupled to the scalar perturbation $\psi$ arising from the vector field. Indeed, the effect of coupling appears in the expressions of $q_{S}$ and $c_{S}^{2}$ derived above. Unlike the background CDM density $\rho_{c}$ obeying Eq. (26), the interaction between $\delta\rho_{c}$ and $\psi$ manifests itself at the level of linear perturbations.

IV A concrete dark energy model

In this section, we propose a concrete coupled vector dark energy model and study the background cosmological dynamics by paying particular attention to the evolution of $w_{\rm DE}$. Let us consider the model given by the functions

[TABLE]

where $b_{2}$, $b_{3}$, $p_{3}$, and $q$ are constants. Since $G_{2,F}=1$ in this model, the no-ghost condition (62) of vector perturbations is automatically satisfied.

For the functions (95), Eq. (30) reduces to

[TABLE]

For $Q=0$, there is the solution where $\phi$ solely depends on $H$, such that $\phi\propto H^{-1/(2p_{3}-1)}$. As long as the energy density of $A_{\mu}$ is subdominant to that of background fluids during the radiation and matter eras, the Hubble parameter evolves as $H\propto 1/t$ and hence the temporal vector component grows as $\phi\propto t^{1/(2p_{3}-1)}$ for $p_{3}>1/2$. Finally, the solutions approach stable de Sitter attractors characterized by constant $\phi$ DeFelice:2016yws .

For $Q\neq 0$, we would like to focus on the case where the third term in Eq. (96) is subdominant to the constant $b_{2}$ in the radiation era and temporally approaches a constant after the onset of matter dominance. In doing so, we deal with the second term in Eq. (96) as a constant during the matter era (as in the case $Q=0$), in which case $\phi\propto t^{1/(2p_{3}-1)}$. On using the property $\rho_{c}\propto a^{-3}\propto t^{-2}$ in this epoch, the third term in Eq. (96) is proportional to $t^{(2q-4p_{3})/(2p_{3}-1)}$. For $q$ satisfying the relation

[TABLE]

all the terms in Eq. (96) are constants during the matter era.

For the choice (97), the third term in Eq. (96) is subdominant to other two terms during the radiation dominance. Indeed, exploiting the solutions $\phi\propto t^{1/(2p_{3}-1)}$ and $\rho_{c}\propto a^{-3}\propto t^{-3/2}$ in this epoch, the third term in Eq. (96) grows in proportion to $t^{1/2}$ toward a constant value in the matter era. After the onset of late-time cosmic acceleration, the coupling term in Eq. (96) starts to decrease toward 0 by reflecting the fact that $\rho_{c}$ decreases faster than $t^{-2}$. Finally, the solutions approach the de Sitter fixed point satisfying

[TABLE]

where the subscript “dS” represents the values on the de Sitter point.

IV.1 Autonomous system

In the following, we focus on the background cosmological dynamics for the power $q$ satisfying the relation (97). In doing so, it is convenient to define

[TABLE]

where $I=r,b,c$. From Eq. (27), the CDM density parameter $\tilde{\Omega}_{c}$, which accommodates the interaction with the vector field, can be expressed as

[TABLE]

Taking the derivatives of $\Omega_{\rm DE}$, $\Omega_{b}$, $\Omega_{r}$, and $\Omega_{c}$ with respect to $\mathcal{N}\equiv\ln a$ and using the continuity Eqs. (24)-(26), it follows that

[TABLE]

where

[TABLE]

After differentiating Eq. (30) with respect to $t$ and using Eq. (28), we can solve them for $\dot{\phi}$ and $\dot{H}$. In doing so, we exploit Eqs. (100) and (30) to eliminate $\rho_{c}$ and $b_{3}$. Defining the dimensionless variable

[TABLE]

we obtain

[TABLE]

where

[TABLE]

In what follows, we focus on the theories with $p_{3}>1/2$, i.e., $s>0$. We also consider the case in which $\Omega_{\rm DE}$ is positive, i.e., $b_{2}<0$.

When $Q=0$, the parameter $s$ characterizes the deviation of $w_{\rm DE}$ from $-1$ DeFelice:2016yws ; DeFelice:2016uil . At the background level, our coupled dark energy model has two additional parameters $Q$ and $s$ relative to those in the $\Lambda$CDM model. The variable $r_{Q}$, which corresponds to the ratio between the third and first terms on the left hand side of Eq. (30), obeys the differential equation

[TABLE]

For given constants $Q$ and $s$, the background cosmological dynamics is known by integrating Eqs. (101)-(103) and Eq. (110) with Eqs. (107) and (108). In doing so, we need to specify the initial conditions of $\Omega_{\rm DE}$, $\Omega_{b}$, $\Omega_{r}$, and $u$ at some redshift $z=1/a-1$. The initial value of $\Omega_{c}$ is determined by using Eq. (100) and the correspondence

[TABLE]

From Eq. (106), the initial condition of $r_{Q}$ is known accordingly. Instead of solving Eq. (110), we can also integrate Eq. (107) for the dimensionless temporal vector component $u=\phi/M_{\rm pl}$. Nevertheless, using the variable $r_{Q}$ is convenient to study the effect of coupling $Q$ on the background cosmological dynamics. Indeed, the dark energy equation of state (35) is simply expressed as

[TABLE]

which shows that $w_{\rm DE}$ is determined by the two quantities $r_{Q}$ and $\epsilon_{\phi}$.

IV.2 Analytic estimation for each cosmological epoch

Before solving the above autonomous system numerically, we analytically estimate the evolution of background quantities during the radiation, matter, and accelerated epochs.

Let us begin with the early Universe in which $\Omega_{\rm DE}$ is much smaller than 1. Expanding $\epsilon_{\phi}$ and $\epsilon_{h}$ around $\Omega_{\rm DE}=0$, it follows that

[TABLE]

In this regime, the dark energy equation of state can be estimated as

[TABLE]

In addition, the quantity $r_{Q}$ approximately obeys

[TABLE]

We would like to consider the case in which the effect of coupling $Q$ on Eq. (30) is unimportant in the early radiation era, so that $r_{Q}\ll 1$. From the above estimation, the fixed point corresponding to the radiation dominance is given by

[TABLE]

with $\tilde{\Omega}_{c}=0$ and $u=0$. In this epoch, the dark energy equation of state (114) reduces to

[TABLE]

This value of $w_{\rm DE}$ is identical to that derived for $Q=0$ in Ref. DeFelice:2016yws ; DeFelice:2016uil , so the effect of coupling $Q$ on $w_{\rm DE}$ does not appear in the deep radiation epoch. However, we have $r_{Q}^{\prime}\simeq r_{Q}$ around the point ${\rm P}_{r}$ and hence the nonvanishing coupling $Q$ leads to the increase of $r_{Q}$ proportional to $a$. In the late radiation era, this growth of $r_{Q}$, together with the decrease of $\Omega_{r}$, gives rise to the departure of $w_{\rm DE}$ from the constant value (117). From Eq. (107) the temporal vector component obeys $u^{\prime}\simeq 2su$ around the point ${\rm P}_{r}$, so it grows as $u\propto a^{2s}$.

After $\Omega_{r}$ becomes much smaller than 1 during the matter era, we have $\epsilon_{\phi}\simeq 3s/2$ and $\epsilon_{h}\simeq-3/2$ from Eq. (113). On using Eqs. (102), (104), and (110), it follows that $\Omega_{b}=\Omega_{b}^{(m)}={\rm constant}$, $\Omega_{c}=\Omega_{c}^{(m)}={\rm constant}$, and $r_{Q}=r_{Q}^{(m)}={\rm constant}$ in this epoch. Then, the fixed point corresponding to the matter dominance is

[TABLE]

with $\tilde{\Omega}_{c}=1-\Omega_{b}^{(m)}$ and $u=0$. From Eq. (107), the quantity $u$ approximately obeys $u^{\prime}\simeq 3su/2$ around the point ${\rm P}_{m}$, so that the temporal vector component grows as $u\propto a^{3s/2}$. Provided that $|Q(u^{2}/2)^{2p_{3}}|\ll 1$, $\Omega_{c}$ is approximately equivalent to $\tilde{\Omega}_{c}$. Around the fixed point ${\rm P}_{m}$, the dark energy equation of state is simply given by

[TABLE]

where $w_{\rm DE}^{(m)}$ is a constant. In the limit that $Q\to 0$, this result coincides with the value $w_{\rm DE}^{(m)}=-1-s$ derived in Refs. DeFelice:2016yws ; DeFelice:2016uil . For $Q>0$, $r_{Q}^{(m)}$ is a negative constant and hence $w_{\rm DE}^{(m)}<-1-s$. On the other hand, the negative coupling $Q$ gives rise to the value $0<r_{Q}^{(m)}<1$, so $w_{\rm DE}^{(m)}$ gets closer to $-1$ in comparison to the $Q=0$ case.

The magnitude of constant $r_{Q}^{(m)}$ depends on the initial condition of $r_{Q}$ in the radiation era. In terms of the quantity $Qf$ equivalent to the second term in the square bracket of Eq. (111), we can express $r_{Q}$ as $r_{Q}=-2p_{3}(\Omega_{c}/\Omega_{\rm DE})Qf$. Even if $|Qf|$ is very much smaller than 1, the quantity $r_{Q}$ can be as large as the order of 0.1 due to the large term $\Omega_{c}/\Omega_{\rm DE}\gg 1$ in the matter era. In Sec. IV.3, we will show that the no ghost condition (92) puts upper limits on the value $r_{Q}^{(m)}$.

The matter fixed point ${\rm P}_{m}$ is different from the $\varphi$MDE Amendola99 characterized by the nonvanishing constant $\Omega_{\rm DE}$ proportional to the coupling-constant squared $\beta^{2}$, in that $\Omega_{\rm DE}=0$ on the point ${\rm P}_{m}$. From Eq. (101), the dark energy density parameter around ${\rm P}_{m}$ approximately obeys $\Omega_{\rm DE}^{\prime}=3(1+s)\Omega_{\rm DE}$, so $\Omega_{\rm DE}$ grows as

[TABLE]

Eventually, the contribution of $\Omega_{\rm DE}$ to Eq. (110) becomes nonnegligible at the late cosmological epoch. Assuming that $r_{Q}\ll 1$ in this epoch and expanding Eq. (110) around $r_{Q}=0$, it follows that

[TABLE]

This means that, after the dominance of dark energy, $r_{Q}$ starts to decrease from the value $r_{Q}^{(m)}$. Finally, the solutions approach the dS fixed point characterized by

[TABLE]

with $\tilde{\Omega}_{c}=0$ and $u=u_{\rm dS}={\rm constant}$. Since $\epsilon_{\phi}=0=\epsilon_{h}$ on this point, both $u$ and $H$ are constants, with the dark energy equation of state

[TABLE]

During the cosmological sequence of radiation, matter, and dS epochs, $w_{\rm DE}$ changes as (117) $\to$ (119) $\to$ (123).

IV.3 Numerical analysis

To study the background cosmological dynamics in more detail, we numerically integrate Eqs. (101)-(103) and Eq. (110) for several different values of $s$ and $Q$. We also discuss whether the conditions for the absence of ghosts and Laplacian instabilities for linear cosmological perturbations are consistently satisfied. The background initial conditions in the deep radiation era (around the redshift $z=10^{7}$) are chosen to realize today’s values $\Omega_{\rm DE}^{(0)}=0.68$, $\Omega_{b}^{(0)}\simeq 0.05$, and $\Omega_{r}^{(0)}\simeq 10^{-4}$. We consider the case in which today’s temporal vector component $u^{(0)}$ is of order 1.

In Fig. 1, we plot the evolution of $w_{\rm DE}$ and $r_{Q}$ versus $z+1$ for $s=1$ and four different values of $Q$. When $Q=0$, the dark energy equation of state evolves as $w_{\rm DE}=-7/3\to-2\to-1$ during the radiation, matter, and dS epochs, respectively. The joint observational constraint based on SN Ia, CMB, BAOs, RSDs, $H_{0}$, and ISW-galaxy cross-correlation data give the bound $s=0.185^{+0.100}_{-0.089}$ (95 % CL) Nakamura:2018oyy , so the model with $s=1$ and $Q=0$ is excluded due to the large deviation of $w_{\rm DE}$ from $-1$ before the onset of cosmic acceleration.

If $Q>0$, we observe in Fig. 1 that $w_{\rm DE}$ is smaller than $-2$ during the matter era. This is consistent with the analytic estimation (119), which gives $w_{\rm DE}^{(m)}=-2+r_{Q}^{(m)}<-2$ for $s=1$ and $Q>0$. In case (A4) of Fig. 1, the numerical value of $r_{Q}^{(m)}$ is about $-0.2$ and hence $w_{\rm DE}^{(m)}\simeq-2.2$. This behavior of $w_{\rm DE}$, which occurs through the decay of dark energy to CDM, is in more tension with the observational data in comparison to the $Q=0$ case.

When $Q<0$, the quantity $r_{Q}^{(m)}$ is positive. In cases (A1) and (A2) of Fig. 1, we can confirm that $r_{Q}$ temporally approaches the positive constant $r_{Q}^{(m)}$ after its increase during the radiation dominance ($r_{Q}\propto a$). This leads to $w_{\rm DE}$ larger than $-2$ in the matter era, whose behavior is attributed to the decay of CDM to dark energy. After the matter-dominated epoch ends, $w_{\rm DE}$ starts to approach the asymptotic value $-1$ with the decrease of $r_{Q}$. The case (A1) in Fig. 1 corresponds to the marginal one in which the quantity $q_{c}=1+Qf$ is close to $+0$ at the dS point. For $Q<0$, the no-ghost condition $q_{c}>0$ constrains the field value $u_{\rm dS}=\phi_{\rm dS}/M_{\rm pl}$ in the range

[TABLE]

This also puts the upper limit on the magnitude of $r_{Q}^{(m)}$. When $s=1$, we numerically find the bound $r_{Q}^{(m)}<0.48$, which translates to $w_{\rm DE}^{(m)}<-1.52$. Thus, the negative coupling $Q$ allows the possibility for realizing $w_{\rm DE}^{(m)}$ closer to $-1$ relative to the $Q=0$ case.

Although the model with $s=1$ should be still difficult to be compatible with the observational data due to the upper bound $w_{\rm DE}^{(m)}<-1.52$, the situation is different for the models with $s<1$ (i.e., $p_{3}>1$). In Fig. 2, we depict the evolution of $w_{\rm DE}$ and $r_{Q}$ for $s=1/3$ (i.e., $p_{3}=2$) and four different values of $Q$. When $Q=0$, we have $w_{\rm DE}^{(m)}=-1.33$, in which case the model is outside the 95 % CL observational boundary Nakamura:2018oyy . As we observe in Fig. 2, the negative coupling $Q$ leads to larger values of $w_{\rm DE}^{(m)}$ compatible with the data. In case (B2), the numerical values of $w_{\rm DE}$ and $r_{Q}$ in the matter era are $-1.22$ and $0.35$, respectively, which are consistent with the analytic estimation (119). The case (B1) is the marginal situation in which the quantity $q_{c}$ at the dS point is close to $+0$. This corresponds to the maximum dark energy equation of state $w_{\rm DE}^{(m)}=-1.14$ with $r_{Q}^{(m)}=0.59$. Thus, for $s=1/3$, the negative coupling $Q$ gives rise to $w_{\rm DE}^{(m)}$ in the range $-1.33<w_{\rm DE}^{(m)}<-1.14$. This alleviates the problem of observational incompatibility of the model with $s=1/3$ and $Q=0$. The positive coupling does not improve the situation, see case (B4) of Fig. 2.

In the above discussion, the no-ghost condition $q_{c}>0$ on the dS solution is crucial to put an upper bound on the value of $w_{\rm DE}^{(m)}$. As we showed in Sec. IV.2, the temporal vector component $u=\phi/M_{\rm pl}$ increases during the radiation and matter epochs. Around the dS fixed point there is the approximate relation $u^{\prime}/u\simeq 3(1-\Omega_{\rm DE})s/[2(1+s\Omega_{\rm DE})]$, so $u$ also grows toward the constant value $u_{\rm dS}$. This means that, under the condition (124), the no-ghost condition $q_{c}>0$ of CDM is satisfied during the whole cosmic expansion history. The other no-ghost condition (91) of the vector field translates to

[TABLE]

Provided that $r_{Q}>-1$, the condition (125) is always satisfied. This is the case for $Q<0$, under which $r_{Q}>0$.

On using the background equations of motion, the vector propagation speed squared (94) can be expressed as

[TABLE]

where $\tilde{s}=(1-r_{Q})s$, and $q_{V}=1$ for the model under consideration. On the fixed points Pr, Pm, and PdS, Eq. (126) reduces, respectively, to

[TABLE]

Since we are considering the case $s>0$, both $(c_{S}^{2})_{r}$ and $(c_{S}^{2})_{\rm dS}$ are positive under the absence of vector ghosts ($q_{V}>0$). During the matter era, the condition $(c_{S}^{2})_{m}\geq 0$ gives

[TABLE]

which is satisfied for all the cases shown in Figs. 1 and 2. The quantity $|r_{Q}|$ reaches the maximum value $|r_{Q}^{(m)}|$ during the matter era, so the no-ghost condition (125) automatically holds under the bound (128).

In order to confirm the above analytic estimations, we compute the quantities $q_{c}$, $q_{S}$ and $c_{S}^{2}$ by numerically integrating the autonomous equations. Figure 3 is such an example, which corresponds to the case (B1) in Fig. 2. We recall that this is close to the marginal case in which the condition $q_{c}>0$ is satisfied on the dS solution. In Fig. 3, we observe that $q_{c}$ starts to deviate from 1 at low redshifts and it finally approaches the asymptotic value $0.02$. As estimated from Eq. (125), $q_{S}$ is always positive during the whole cosmological evolution. Since $s=1/3$, $q_{V}=1$, $r_{Q}^{(m)}=0.59$, and $u_{\rm dS}=1.41$ for the case (B1) in Fig. 2, the analytic estimations (127) give the values $(c_{S}^{2})_{r}=0.407$, $(c_{S}^{2})_{m}=0.086$, and $(c_{S}^{2})_{\rm dS}=0.084$, respectively. They are in good agreement with their numerical values computed in Fig. 3. We note that $c_{S}^{2}$ also remains positive during the transition from the matter era to the dS epoch.

As long as the no-ghost condition $q_{c}>0$ of CDM is satisfied on the dS solution, the numerical simulations of Figs. 1 and 2 show that $|r_{Q}|$ does not exceed 1. In this case, there are neither ghosts ($q_{S}>0$) nor Laplacian instabilities ($c_{S}^{2}>0$) for the longitudinal scalar mode of $A_{\mu}$. In other words, under the condition (124), all the stability conditions associated with scalar perturbations are consistently satisfied. We found that the maximum allowed values of $w_{\rm DE}$ consistent with the stability conditions are $w_{\rm DE}^{(m)}=-1.52$ for $s=1$ and $w_{\rm DE}^{(m)}=-1.14$ for $s=1/3$. For smaller $s$, $w_{\rm DE}^{(m)}$ increases further, e.g., $w_{\rm DE}^{(m)}=-1.07$ for $s=1/5$ (i.e., $p_{3}=3$). This upper limit of $w_{\rm DE}^{(m)}$ is mostly determined by the product $Qf=Q(u^{2}/2)^{2p_{3}}$, which means that both coupling constant $Q$ and temporal vector component $u$ affect the evolution of $w_{\rm DE}$ after the onset of matter dominance.

V Conclusions

We studied the cosmology of GP theories in which a massive vector field $A_{\mu}$ is coupled to CDM with the interacting action (10). We deal with the matter sector as perfect fluids described by the Schutz-Sorkin action (3). In this approach, the conserved part of the CDM energy density $\rho_{c}$ is associated with the Schutz-Sorkin action, while the additional interacting energy density $Qf\rho_{c}$ arises from the interaction (10). As a result, the total effective CDM energy density $\tilde{\rho}_{c}=(1+Qf)\rho_{c}$ contains the effect of coupling $Q$ on the right hand side of Eq. (32). Defining the dark energy density and pressure arising from the vector field according to Eqs. (33) and (34), they obey the modified continuity Eq. (36), whose sign of the interacting term is opposite to that of $\tilde{\rho}_{c}$. This clearly shows the consistency of our approach of dealing with the coupling between two dark sectors.

In Sec. III, we provided a general framework for studying the dynamics of linear cosmological perturbations in coupled vector dark energy theories given by the action (1). We derived the second-order actions of tensor, vector, and scalar perturbations and studied conditions for the absence of ghosts and Laplacian instabilities. The quadratic tensor action is of the same form as that in general relativity, so the theories automatically pass the bound on the propagation speed of gravitational waves. Deep inside the Hubble radius, the second-order action of vector perturbations reduces to the form (60) with $q_{V}=G_{2,F}$ and $c_{V}^{2}=1$, so the vector ghost is absent under the condition $G_{2,F}>0$. For scalar perturbations, there are two no-ghost conditions (91) and (92) associated with the vector field and CDM, respectively. The scalar propagation speed squared $c_{S}^{2}$ of the vector field is affected by the coupling $Q$ as Eq. (94), which must be positive to avoid the Laplacian instability. We also showed that the Laplacian instability is absent in the matter sector.

In Sec. IV, we proposed a viable model of coupled vector dark energy given by the functions (95). For the power $q=2p_{3}$, the coupling term in Eq. (96) grows in proportion to $t^{1/2}$ in the radiation era and it reaches a constant value during the matter dominance. This interacting term starts to decrease after the onset of cosmic acceleration, which is followed by the approach to de Sitter solutions with $\rho_{c}=0$. In other words, the effect of interactions between the vector field and CDM on cosmological observables mostly manifests itself from the onset of matter era to today. During the matter dominance, the dark energy equation of state $w_{\rm DE}$ is a constant smaller than $-1$, so the corresponding density parameter $\Omega_{\rm DE}$ grows in time. This property is different from the $\varphi$MDE of coupled scalar dark energy models in which $\Omega_{\rm DE}$ is a nonvanishing constant affected by the coupling $\beta$.

We found that the negative coupling $Q$ leads to $w_{\rm DE}$ closer to $-1$ relative to the uncoupled case. This is attributed to the fact that, for $Q<0$, CDM decays to the vector field. The maximum value of $w_{\rm DE}$ reached during the matter era is determined by the CDM no-ghost condition (92). In this case, the other stability conditions (91) and (94) of scalar perturbations are satisfied from the radiation era to the de Sitter attractor. For $p_{3}=1,2,3$ the maximum values are given by $w_{\rm DE}^{(m)}=-1.52,-1.14,-1.07$, respectively, which are larger than their corresponding values $w_{\rm DE}=-2,-1.33,-1.2$ for $Q=0$. Thus, our coupled dark energy model alleviates the problem of observational incompatibility of uncoupled models with $p_{3}\leq 2$.

We thus showed that the coupled vector dark energy allows the phantom dark energy equation of state being compatible with the observational data, while satisfying all the stability conditions of linear cosmological perturbations. This property is very different from the standard coupled quintessence in which $w_{\rm DE}=1$ during the $\varphi$MDE. It will be of interest to place observational constraints on the coupling and the power $p_{3}$ along the line of Ref. deFelice:2017paw . On scale relevant to the linear growth of cosmological perturbations, the effective gravitational coupling for baryons is different from the Newtown gravitational constant due to the existence of cubic interactions $G_{3}(X)\nabla_{\mu}A^{\mu}$ Nakamura:2018oyy . The gravitational interaction for CDM should be further modified by the direct coupling to the vector field. The derivation of effective gravitational couplings felt by CDM and light is the next important step for probing the signatures of coupled vector dark energy in the observations of RSDs and ISW-galaxy cross-correlations. We leave these interesting issues for future works.

Acknowledgements

RK is supported by the Grant-in-Aid for Young Scientists B of the JSPS No. 17K14297. ST is supported by the Grant-in-Aid for Scientific Research Fund of the JSPS No. 19K03854 and MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas “Cosmic Acceleration” (No. 15H05890).