Running cosmological constant with observational tests

We investigate the running cosmological constant model with dark energy linearly proportional to the Hubble parameter, $\Lambda = \sigma H + \Lambda_0$, in which the $\Lambda$CDM limit is recovered by taking $\sigma=0$. We derive the linear perturbation equations of gravity under the Friedmann-Lema\"itre-Robertson-Walker cosmology, and show the power spectra of the CMB temperature and matter density distribution. By using the Markov chain Monte Carlo method, we fit the model to the current observational data and find that $\sigma H_0/ \Lambda_0 \lesssim 2.63 \times 10^{-2}$ and $6.74 \times 10^{-2}$ for $\Lambda(t)$ coupled to matter and radiation-matter, respectively, along with constraints on other cosmological parameters.


I. INTRODUCTION
The type-Ia supernova observations [1,2] have shown that our universe is undergoing a late-time accelerating expansion, which is caused by Dark Energy [3]. The simplest way to realize such a late-time accelerating mechanism is to introduce a cosmological constant to the gravitational theory, such as that in the ΛCDM model. This model fits current cosmological observations very well, but exists several difficulties, such as the "fine-tuning" [4,5] and "coincidence" [6] problems.
In this work, we will concentrate on the latter problem [7], which has been extensively explored in the literature. One of the popular attempts is the running Λ model, in which the cosmological constant evolves in time and decays to matter in the evolution of the universe [8][9][10][11][12][13][14][15][16][17][18][19][20], so that the present energy densities of dark energy and dark matter are of the same order of magnitude. Its observational applications have been investigated in Ref. [21][22][23]. In our study, we are interested in the specific model with Λ = σH [24][25][26][27][28][29], which would originate from the theory with the QCD vacuum condensation associated with the chiral phase transition [30][31][32][33][34]. In this scenario, the cosmological constant decays to matter (non-relativistic) and radiation (relativistic), leading to a large number of particles created in the cosmological evolution. Without loss of generality, we phenomenologically extend this model to include that Λ additionally couples to radiation with Λ = σH + Λ 0 [35][36][37], in which the ΛCDM limit can be realized if σ = 0. In this scenario, when dark energy dominates the universe, the decay rate of Λ is reduced, and the late-time accelerating phase occurs, describing perfectly the evolution history of the universe. As a result, it is reasonable to go further to analyze the cosmological behavior of this model at the sub-horizon scale.
In this paper, we examine the matter power spectrum P (k) and CMB temperature perturbations in the linear perturbation theory of gravity. By using the Markov chain Monte Carlo (MCMC) method, we perform the global fit from the current observational data and constrain the model. This paper is organized as follows: In Sec. II, we introduce the Λ(t)CDM model and review its background cosmological evolutions. In Sec. III, we calculate the linear perturbation theory and illustrate the power spectra of the matter distribution and CMB temperature by the CAMB program [38]. In Sec. IV, we use the CosmoMC package [39] to fit the model from the observational data and show the constraints on cosmological parameters.
Our conclusions are presented in Sec. V.

II. THE RUNNING COSMOLOGICAL CONSTANT MODEL
We start with the Einstein equation, given by, where κ 2 = 8πG, R = g µν R µν is the Ricci scalar, Λ(t) is the time-dependent cosmological constant, and T M µν is the energy-momentum tensor of matter and radiation. In the Friedmann-Lemaïtre-Robertson-Walker (FLRW) case, we obtain, where τ is the conformal time, H = da/(adτ ) represents the Hubble parameter, ρ M (P M ) corresponds to the energy density (pressure) of matter and radiation, and ρ Λ (P Λ ) is the energy density (pressure) of the cosmological constant. We note that from the relation of ρ Λ = −P Λ = κ −2 Λ(t), derived from Eq. (1), one has the equation of state (EoS) of Λ to be In Eq. (1), we consider Λ(t) to be a linear function of the Hubble parameter, given by [29,[35][36][37]] where σ and Λ 0 are two free parameters. From Eq. (6), we can write ρ Λ with two dimensionless parameters λ 0,1 as, where ρ 0 Λ ≡ ρ Λ | z=0 is the current dark energy density with the condition λ 0 + λ 1 = 1 and λ 1 = σH 0 /(σH 0 + Λ 0 ). Note that λ 0 has been treated as a constant of integration and set to zero in Ref. [29]. Without loss of generality, we will keep λ 0 as a free parameter with the ΛCDM model recovered when λ 0 → 1.
resulting in that dark energy unavoidably couples to matter and radiation, given bẏ where Q m,r are the decay rate from Λ(t) to matter and radiation, taken to be respectively. Note that the analytical solution of Eq. (8)  Under the FLRW background, the metric perturbation is given by [40] where i, j = 1, 2, 3, h and η are two scalar perturbations in the synchronous gauge, andk = k/k is the k-space unit vector. The matter (baryon, cold dark matter and massive neutrino) and radiation (photon and massless neutrino) density perturbations can be derived from the given by [27,41], where also changes in this model, given by In Fig. 3, we present the matter power spectrum P (k) as a function of the wavenumber k  14) with Q m ∝ −Ḣ ≥ 0. As λ 1 increases, the running cosmological model deviates from the ΛCDM limit, while the matter creation gets enhanced due to the dark energy decay as seen in Eq. (9). In addition, since dark energy is considered to be homogeneous and isotropic, δρ Λ = 0, so that the creation of matter smoothly distributes to our universe. Consequently, the density perturbation is diluted and the matter power spectrum is suppressed by the decay of Λ. where legend is the same as Fig. 3.
from that in ΛCDM (black line) when λ 1 ≫ 0. This gives us a hint that λ 1 would be relatively small in order to fit the spectrum. Our results demonstrate that the allowed matter creation from the cosmological constant should be tiny.  data from SDSS DR4 and WiggleZ [48], and (iv) weak lensing data from CFHTLenS [49].

IV. OBSERVATIONAL CONSTRAINTS
With this dataset, we keep the model parameter λ 1 ≥ 0 to avoid the negative dark energy density and explore the constraints on λ 1 and other cosmological parameters. The priors of the various parameters are listed in Table. I, and our results are shown in Fig. 5.

V. CONCLUSIONS
We have investigated the Λ(t)CDM model with the dark energy decaying to both matter and radiation, in which Λ(t) = σH + Λ 0 . Although this scenario is suitable to describe the late-time accelerating universe at the background level, the linear perturbation analyses of the matter power and CMB temperature spectra have set a strong constraint on the model parameter λ 1 in Eq. (7). Explicitly, by performing the global fit from the observational data, we have obtained that λ 1 ≃ σH 0 /Λ 0 6.68 × 10 −2 (2.63 × 10 −2 ) and χ 2 Λ(t)CDM = 13546.5(13545.7) χ 2 ΛCDM = 13545.3 for C r = 1(0), implying that the current data prefers the ΛCDM limit. Constraints on other cosmological parameters in both Λ(t)CDM and ΛCDM models have been also given in Table II.