Constraints on a special running vacuum model

We study a special running vacuum model (RVM) with $\Lambda = 3 \alpha H^2+3\beta H_0^4 H^{-2}+\Lambda_0$, where $\alpha$, $\beta$ and $\Lambda_0$ are the model parameters and $H$ is the Hubble one. This RVM has non-analytic background solutions for the energy densities of matter and radiation, which can only be evaluated numerically. From the analysis of the CMB power spectrum and baryon acoustic oscillation along with the prior of $\alpha>0$ to avoid having a negative dark energy density, we find that $\alpha<2.83\times 10^{-4}$ and $\beta=(-0.2^{+3.9}_{-4.5})\times 10^{-4}$ (95$\%$ C.L.). We show that the RVM fits the cosmological data comparably to the $\Lambda$CDM. In addition, we relate the fluctuation amplitude $\sigma_8$ to the neutrino mass sum $\Sigma m_\nu$.


I. INTRODUCTION
Since the discovery of the accelerated expanding universe in 1998 [1,2], dark energy has been the most popular scenario to explain this phenomenon [3]. Among the various theories, the Lambda Cold Dark Matter (ΛCDM) model is the simplest one to reveal the nature of our universe, which also fits well with all cosmological observational data. Unfortunately, the ΛCDM model has some theoretical unsatisfactories, such as "fine-tuning" [4,5] and "coincidence" [6,7] problems.
In this study, we plan to fit this RVM by using the most recent observational data. In particular, we use the CAMB [34] and CosmoMC [35] packages with the Markov chain Monte Carlo (MCMC) method. Since this model has no analytical solution for the energy density of matter or radiation, we modify the CAMB program to get the background evolution.
This paper is organized as follows. In Sec. II, we introduce our special RVM. We also derive the evolution equations for matter and radiation in the linear perturbation theory.
In Sec. III, we present our numerical calculations. In particular, we show the CMB and matter power spectra and the constraints on the model-parameters from several cosmological observation datasets. Finally, our conclusions are given in Sec. IV.

II. RUNNING VACUUM MODEL
We start with the Einstein equation, written as where R = g µν R µν is the Ricci scalar, Λ is the cosmological constant, G is the gravitational constant and T M µν is the energy-momentum tensor of matter and radiation. For the homogeneous and isotropic universe, we use the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, given by Consequently, the Friedmann equations are found to be where H = da/(adt) is the Hubble parameter and ρ m,r,Λ (P m,r,Λ ) represent the energy densities (pressures) of matter, radiation and dark energy, respectively. In this work, we consider Λ to be the specific function of the Hubble parameter, given by Here, α and β are dimensionless model-parameters. It is clear that the ΛCDM model is recovered by taking α = 0 and β = 0. This special model is inspired by the studies of [22,23]. It is convenient to define the equations of state for matter, radiation and dark energy by respectively.
In the RVM, dark energy decays to radiation and matter in the evolution of the universe, so that the continuity equations can be written as, where ρ Λ = Λ/(8πG), ρ M = ρ m + ρ r , w M = (P m + P r )/ρ M and Q = Q m + Q r with Q m(r) the decay rate of dark energy to matter (radiation). By combining Eqs. (5) and (7), the coupling Q µ with µ = m or r is given by with P M = P m + P r .
The energy densities of matter and radiation can be evaluated from derived from Eq. (7), where "′" stands for the derivative with respective to ln a and ρ ′ µ = ρ µ /H. However, there are no analytical solutions for ρ m,r in Eq. (10). From the modified CAMB program, we can solve Eq. (10) numerically. Note that α ≥ 0 is chosen to avoid the negative dark energy density in the early universe.
In our calculation, we use the conformal time τ in order to perform the perturbation theory in the synchronous gauge. From the standard linear perturbation theory [36], we can derive the growth equation of the density perturbation in the RVM. In the synchronous gauge, the metric is given by where i, j = 1, 2, 3 and with the k-space unit vector ofk = k/k and two scalar perturbations of h( k, τ ) and η( k, τ ).

The conservation equation is given by
As shown in Refs. [37,38], there are two basic perturbation equations, given by i=Λ,Mθ where δρ i represent the density fluctuations, and θ i are the corresponding velocities. As there is no peculiar velocity for dark energy, we take θ Λ = 0. In addition, we assume that δρ M ≫ δρ Λ and δρ M ≫ δρ Λ in our model. As a result, we can ignore the discussion for the dark energy perturbation. For the matter perturbation, the growth equations are given bẏ where δ µ ≡ δρ µ /ρ µ and µ = m, r.

III. NUMERICAL CALCULATIONS
As mentioned in the previous section, we modify the CAMB program to solve Eq. (10).
In our calculation, ρ m and ρ r are evaluated in terms of log a from the current universe to the past. By performing the CosmoMC program [35], we fit the RVM from the observational data with the MCMC method. The dataset includes those of the CMB temperature fluctuation from Planck 2015 with TT, TE, EE and low-l polarization [39][40][41], the baryon acoustic oscillation (BAO) data from 6dF Galaxy Survey [42,43], the WiggleZ Dark Energy Survey [44] and BOSS [45][46][47] and the redshift space distortion (RSD) data from SDSS-III BOSS [48]. The BAO data points are shown in Table I.
In addition, the χ 2 fit is given by For the BAO, the observation measures the distance ratio of d z ≡ r s (z d )/D V (z), where D V is the volume-averaged distance and r s (z d ) is the comoving sound horizon with z d the redshift at the drag epoch [49]. Here, where D A (z) is the proper angular diameter distance, given by while r s (z) is described by where Ω 0 b and Ω 0 γ are the present values of baryon and photon density parameters, respectively. The χ 2 value for the BAO data is given by where n is the number of the BAO data points and σ i correspond to the errors of the data, given by Table I. Here, the subscripts of "th" and "obs" represent the theoretical and observational values of the volume-averaged distance, respectively.
The CMB is sensitive to the distance to the decoupling epoch z * . It constrains the model in the high redshift region of z ∼ 1000. The χ 2 value of the CMB data can be calculated by where and respectively.
For the RSD measurements, we use where σ 8 is the amplitude of the over-density at the comoving 8h −1 Mpc scale and f (z)=δ ′ /δ with δ the evolution of the matter density contrast.
The data points in RSD are given by respectively. The priors of the various cosmological parameters are listed in Table II. Here, we have set α to be a positive number to avoid having a negative dark energy density.
In Fig. 1, we show the CMB power spectra in the ΛCDM and RVM with several different sets of α and β. In the figure, we see that both model parameters α and β in Eq. (5) are expected to be smaller than 0.0001 as illustrated by the blue line, which almost coincides with the ΛCDM one (black). In the green and red lines with α=0.01, the first acoustic peaks are reduced, which could result from too much contribution from dark energy to the total energy density to suppress the baryon part [51]. In this case, the overall shift of the Doppler peaks towards lower multipoles as a consequence of the increased sound speed of the plasma [51]. Compared with α, the effect of β in the CMB is not obvious, because H(z) is very large due to the H 2 term in the early universe. In Fig. 2, we give the ratio of ∆C ℓ /C ℓ , where ∆C ℓ is the change between the RVM and ΛCDM for the TT mode of the CMB power spectra, while C ℓ corresponds to the one in ΛCDM. This figure illustrates the effects from the model-parameter of β from -0.01 to 0.01. It is clear that the changes in the CMB power spectra due to β are small, so that the results in the RVM will be only slightly different from those in ΛCDM in the early universe. We present the matter power spectra of the RVM in Fig. 3, which behavior similar to those in Figs. 1 and 2. In addition, we demonstrate that the matter power spectra for the RVM and ΛCDM do not have similar evolution paths in the early universe until k≈5.
In Fig. 4 and Table III, we present our results of the global fits from several datasets, where the values in the brackets correspond to the best-fit values in the ΛCDM model. In particular, we find that (α, β) = (< 2.83, −0.2 +3.9 −4.5 ) × 10 −4 and (< 1.57, −0.2 ± 2.6) × 10 −4 with 95% and 68% C.L., respectively. Here, Ω Λ = (ρ Λ /ρ C )| z=0 = α + β + Λ 0 /(3H 2 0 ) is the fractional dark energy density with ρ Λ ≡ Λ/(8πG) and ρ C = 3H 2 /(8πG). It is interesting to note that the value of σ 8 =0.835 ± 0.038 (95% C.L.) in the RVM is smaller than that of 0.838 +0.038 −0.040 (95% C.L.) in ΛCDM. As shown in Table III, the best fitted χ 2 value in the RVM is 2543.259, which is smaller than 2546.662 in the ΛCDM model. Although the cosmological observables for the best χ 2 fit in the RVM do not significantly deviate from those in ΛCDM, they look better in all datasets. It implies that the RVM is favored by the cosmological observations. However, we remark that our results can only be viewed as in the literature [15][16][17][18][19]. However, this difference may be due to the fact that the models are actually different as there is no term proportional to negative powers of H in the cases of the other authors.
Another interesting result is about the correlation between the fluctuation amplitude  σ 8 and the neutrino mass sum Σm ν shown in Fig. 5. Many local observations [52][53][54] have claimed that the value of σ 8 should be smaller than the one given by the Planck measurement [55,56]. We remark that our fitted value of σ 8 is much higher than those in Refs. [55,56] due to the different data set. It is known that the cosmic shear data are important as well and several surveys usually provide values of σ 8 much smaller than those from the Planck data [57]. Nevertheless, we would examine the tendency of σ 8 in our model.
In order to reduce σ 8 , we should have a smaller matter amplitude, which is consistent with our fitting result in Fig. 4, in which the RVM has lower values of Ω b and σ 8 than those in the ΛCDM model. In Fig. 5 which leads to H 2 = H 2 0 β/(1 − α), pointing out that the existence of the de-Sitter space appears only if β/(1−α) > 0. As we have discussed [20,21], α ∼ 1 gives us a large abundance of the dark energy density in the early universe, so that the observations require the value of α to be small. On the other hand, the negative value of α induces ρ Λ < 0 at a high z, which should be avoided. Thus, the allowed window for α is tiny with 1 ≫ α ≥ 0. As a result, the de-Sitter space can exist if we have a suitable positive value of β ∼ O(1). This special case not only keeps the late-time accelerating universe but also further reduces the model parameters by one, i.e., Λ 0 = 0. We show the evolution of f σ 8 in Fig. 6 with α = 0 and different values of Λ 0 and β. Here, to illustrate the behavior of f σ 8 , we have fixed Ω Λ to be the best fitted value of 0.68 shown in Table III. As we can see, the smaller value for Λ 0 is, the more significant deviation from that of the ΛCDM prediction behaves, indicating that the vanishment of the Λ 0 term in the specific RVM does not work well at the linear perturbation level. This result is clearly due to the strong interaction between matter and dark energy in the late time of the universe. A bunch of relativistic and non-relativistic matter decay into dark energy, which further enhance the matter density perturbation δ M and change f σ 8 in our universe. Therefore, this specific case with Λ 0 = 0 is available in the background evolution history but, of course, unacceptable at linear perturbation observations.

IV. CONCLUSIONS
We have studied the RVM with Λ = 3αH 2 + 3βH 4 0 H −2 + Λ 0 . By modifying the program in CAMB, we have solved the equations for the energy densities of matter and radiation and obtain the numerical solutions. In the CMB and matter power spectra, we have used several different sets of α and β to show the cosmological evolutions of the model in the early universe. With the data of BAO, RSD and CMB, we have found that α and β are (< 2.83, −0.2 +3.9 −4.5 )×10 −4 (95% C.L.) and (< 1.57, −0.2±2.6)×10 −4 (68% C.L.), respectively. The best fitted χ 2 value is 2543.259 in the RVM, which is in the same order but a little smaller than 2546.662 in the ΛCDM model. In addition, the fitting result of σ 8 has been found to be also smaller than that in ΛCDM. The results in the RVM are comparable to those in ΛCDM to explain the observational data, especially consistent with the local data in the σ 8 problem.