Constraints on running vacuum model with $H(z)$ and $f \sigma_8$

TL;DR

This paper tests a running vacuum cosmological model against observational data, finding a small but non-zero parameter value that slightly improves the fit over the standard Lambda-CDM model.

Contribution

It provides observational constraints on the running vacuum model using multiple cosmological data sets, including $H(z)$ and $f \sigma_8(z)$ measurements.

Findings

The parameter $ u$ is constrained to $(1.37^{+0.72}_{-0.95})\times 10^{-4}$.

The model achieves a slightly better fit than $\\Lambda$CDM.

The data supports a small running of the vacuum energy.

Abstract

We examine the running vacuum model with $\Lambda (H) = 3 \nu H^2 + \Lambda_0$, where $\nu$ is the model parameter and $\Lambda_0$ is the cosmological constant. From the data of the cosmic microwave background radiation, weak lensing and baryon acoustic oscillation along with the time dependent Hubble parameter $H(z)$ and weighted linear growth $f (z)\sigma_8(z)$ measurements, we find that $\nu=(1.37^{+0.72}_{-0.95})\times 10^{-4}$ with the best fitted $\chi^2$ value slightly smaller than that in the $\Lambda$CDM model.

Tables

Table 1. Table 1: H ​ ( z ) 𝐻 𝑧 H(z) data points

	$z$	$H\left(z\right)$	Ref.		$z$	$H\left(z\right)$	Ref.		$z$	$H\left(z\right)$	Ref.
		($km/s/Mpc$)				($km/s/Mpc)$				($km/s/Mpc$)
1	0.07	69.0$\pm$19.6	Zhang:2012mp	13	0.4	95.0$\pm$17.0	Simon:2004tf	25	0.9	117.0$\pm$23.0	Simon:2004tf
2	0.09	69.0$\pm$12.0	Jimenez:2003iv	14	0.4004	77.0$\pm$10.2	Moresco:2016mzx	26	1.037	154.0$\pm$20.0	Moresco:2012jh
3	0.12	68.6$\pm$26.2	Zhang:2012mp	15	0.4247	87.1$\pm$11.2	Moresco:2016mzx	27	1.3	168.0$\pm$17.0	Simon:2004tf
4	0.17	83.0$\pm$8.0	Simon:2004tf	16	0.4497	92.8$\pm$12.9	Moresco:2016mzx	28	1.363	160.0$\pm$33.6	Moresco:2015cya
5	0.179	75.0$\pm$4.0	Moresco:2012jh	17	0.4783	80.9$\pm$9.0	Moresco:2016mzx	29	1.43	177.0$\pm$18.0	Simon:2004tf
6	0.199	75.0$\pm$5.0	Moresco:2012jh	18	0.48	97.0$\pm$62.0	Stern:2009ep	30	1.53	140.0$\pm$14.0	Simon:2004tf
7	0.2	72.9$\pm$29.6	Zhang:2012mp	19	0.57	92.4$\pm$4.5	Reid:2012sw	31	1.75	202.0$\pm$40.0	Simon:2004tf
8	0.27	77.0$\pm$14.0	Simon:2004tf	20	0.5929	104.0$\pm$13.0	Moresco:2012jh	32	1.965	186.5$\pm$50.4	Moresco:2015cya
9	0.24	79.69$\pm$2.65	Gaztanaga:2008xz	21	0.6797	92.0$\pm$8.0	Moresco:2012jh	33	2.3	224$\pm$8	Busca:2012bu
10	0.28	88.8$\pm$36.6	Zhang:2012mp	22	0.7812	105.0$\pm$12.0	Moresco:2012jh	34	2.34	222$\pm$7	Hu:2014vua
11	0.352	83.0$\pm$14.0	Moresco:2012jh	23	0.8754	125.0$\pm$17.0	Moresco:2012jh	35	2.36	226$\pm$8	Font-Ribera:2013wce
12	0.3802	83.0$\pm$13.5	Moresco:2016mzx	24	0.88	90.0$\pm$40.0	Stern:2009ep

Table 2. Table 2: f ​ σ 8 𝑓 subscript 𝜎 8 f\sigma_{8} data points

	$z$	$f{\sigma }_8$	Ref.		$z$	$f{\sigma }_8$	Ref.		$z$	$f{\sigma }_8$	Ref.
$1$	$1.36$	$0.482\pm 0.116$	Okada:2015vfa	$10$	$0.59$	$0.488\pm 0.06$	Chuang:2013wga	$19$	$0.35$	$0.440\pm 0.05$	Song:2008qt ; Tegmark:2006az
$2$	$0.8$	$0.470\pm 0.08$	delaTorre:2013rpa	$11$	$0.57$	$0.444\pm 0.038$	Gil-Marin:2015sqa	$20$	$0.32$	$0.394\pm 0.062$	Gil-Marin:2015sqa
$3$	$0.78$	$0.38\pm 0.04$	Blake:2011rj	$12$	$0.51$	$0.452\pm 0.057$	Satpathy:2016tct	$21$	$0.3$	$0.407\pm 0.055$	Tojeiro:2012rp
$4$	$0.77$	$0.490\pm 0.18$	Song:2008qt ; Guzzo:2008ac	$13$	$0.5$	$0.427\pm 0.043$	Tojeiro:2012rp	$22$	$0.25$	$0.351\pm 0.058$	Samushia:2011cs
$5$	$0.73$	$0.437\pm 0.072$	Blake:2012pj	$14$	$0.44$	$0.413\pm 0.080$	Blake:2012pj	$23$	$0.22$	$0.42\pm 0.07$	Blake:2011rj
$6$	$0.61$	$0.457\pm 0.052$	Satpathy:2016tct	$15$	$0.41$	$0.45\pm 0.04$	Blake:2011rj	$24$	$0.17$	$0.51\pm 0.06$	Song:2008qt ; Percival:2004fs
$7$	$0.60$	$0.390\pm 0.063$	Blake:2012pj	$16$	$0.4$	$0.419\pm 0.041$	Tojeiro:2012rp	$25$	$0.15$	$0.49\pm 0.15$	Howlett:2014opa
$8$	$0.6$	$0.433\pm 0.067$	Tojeiro:2012rp	$17$	$0.38$	$0.430\pm 0.054$	Satpathy:2016tct	$26$	$0.067$	$0.423\pm 0.055$	Beutler:2012px
$9$	$0.60$	$0.43\pm 0.04$	Blake:2011rj	$18$	$0.37$	$0.460\pm 0.038$	Samushia:2011cs	$27$	$0.02$	$0.36\pm 0.04$	Hudson:2012gt

Table 3. Table 3: Priors for cosmological parameters with Λ = 3 ​ ν ​ H 2 + Λ 0 Λ 3 𝜈 superscript 𝐻 2 subscript Λ 0 \Lambda=3\nu H^{2}+\Lambda_{0} .

Parameter	Prior
Model parameter $\nu$	$0\le \nu \le 3\times {10}^{-4}$
Baryon density	$0.5\le 100{\Omega }_b{h}^2\le 10$
CDM density	${10}^{-3}\le {\Omega }_c{h}^2\le 0.99$
Optical depth	$0.01\le \tau \le 0.8$
Neutrino mass sum	$0\le \Sigma m_{\nu }\le 2$ eV
$\frac{\mathrm{Sound}\mathrm{horizon}}{\mathrm{Angular}\mathrm{diameter}\mathrm{distance}}$	$0.5\le 100{\theta }_{MC}\le 10$
Scalar power spectrum amplitude	$2\le \ln \left({10}^{10}{A}_s\right)\le 4$
Spectral index	$0.8\le n_s\le 1.2$

Table 4. Table 4: The 1 ​ D 1 𝐷 1D marginalized fitting results for the RVM with Λ Λ \Lambda = 3 ν ​ H 2 𝜈 superscript 𝐻 2 \nu H^{2} + Λ 0 subscript Λ 0 \Lambda_{0} , where the limits are given at 95 % percent \% C.L. with ν 𝜈 \nu calculated within 68 % percent \% C.L., and the numbers in the bracket represent the central values in the Λ Λ \Lambda CDM model.

Parameter	(A) Planck + WL + BAO	(B) Planck + WL + BAO + $f{\sigma }_8$	(C) Planck + WL + BAO + $H\left(z\right)$	(D) Planck + WL+ BAO + $f{\sigma }_8$ + $H\left(z\right)$
Model parameter ${10}^4\nu$
Baryon density $100{\Omega }_b{h}^2$	$2.23\pm 0.03$ (2.23)	${2.23}_{-0.03}^{+0.04}$ (2.24)	${2.23}_{-0.03}^{+0.02}$ (2.23)	$2.22\pm 0.03$ (2.24)
CDM density $100{\Omega }_c{h}^2$	$11.8\pm 0.2$ (11.8)	$11.7\pm 0.2$ (11.7)	$11.7\pm 0.2$ (11.7)	${11.7}_{-0.3}^{+0.2}$ (11.7)
Optical depth $100\tau$	${6.67}_{-2.70}^{+2.83}$ (6.96)	${6.48}_{-3.03}^{+3.23}$ (6.99)	${6.84}_{-2.61}^{+2.76}$ (7.13)	${6.49}_{-2.91}^{+3.08}$ (6.96)
${\sigma }_8$	${0.806}_{-0.026}^{+0.025}$ (0.810)	${0.787}_{-0.028}^{+0.027}$ (0.788)	${0.809}_{-0.024}^{+0.023}$ (0.812)	${0.792}_{-0.026}^{+0.025}$ (0.793)
Neutrino mass $\Sigma m_{\nu }/\mathrm{eV}$	()	()	()	()
${\chi }_{best-fit}^2$	$13487.7$ (13488.9)	$13509.9$ (13512.2)	$13511.3$ (13512.8)	$13531.2$ (13534.7)

Equations

$$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G{T}_{\mu \nu }^M,$$

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

$$H^2=\frac{8\pi G}{3}({\rho }_m+{\rho }_r+{\rho }_{\Lambda }),$$

$$\dot{H}=-4\pi G({\rho }_m+{\rho }_r+{\rho }_{\Lambda }+P_m+P_r+P_{\Lambda }),$$

$$w_{m,r,\Lambda }=\frac{P_{m,r,\Lambda }}{{\rho }_{m,r,\Lambda }}=0,\frac{1}{3},-1.$$

$${\dot{\rho }}_M+3H(1+w_M){\rho }_M=Q,$$

$${\dot{\rho }}_{\Lambda }+3H(1+w_{\Lambda }){\rho }_{\Lambda }=-Q,$$

$$\Lambda =3\nu H^2+{\Lambda }_0,$$

$$Q_{\alpha }=-\frac{{\dot{\rho }}_{\Lambda }({\rho }_{\alpha }+P_{\alpha })}{{\rho }_M+P_M}=3\nu H(1+w_{\alpha }){\rho }_{\alpha },$$

$${\rho }_{\alpha }={\rho }_{\alpha }^{(0)}{a}^{-3(1+w_{\alpha })\xi },$$

$$d{s}^2=a^2(\tau )[-d{\tau }^2+({\delta }_{ij}+h_{ij})d{x}^{i}d{x}^j],$$

$$h_{ij}=\int d^{3}k{e}^{i\vec{k}\vec{x}}[{\hat{k}}_i{\hat{k}}_{j}h(\vec{k},\tau )+6({\hat{k}}_i{\hat{k}}_j-\frac{1}{3}{\delta }_{ij})\eta (\vec{k},\tau )],$$

$$\dot{{\delta }_{\alpha }}=-(1+w_{\alpha })({\theta }_{\alpha }+\frac{\dot{h}}{2})-3H(\frac{\delta P_{\alpha }}{\delta {\rho }_{\alpha }}-w_{\alpha }){\delta }_{\alpha }-\frac{Q_{\alpha }}{{\rho }_{\alpha }}{\delta }_{\alpha },$$

$${\dot{\theta }}_{\alpha }=-H(1-3w_{\alpha }){\theta }_{\alpha }-\frac{{\dot{w}}_{\alpha }}{1+w_{\alpha }}{\theta }_{\alpha }+\frac{\delta P_{\alpha }/\delta {\rho }_{\alpha }}{1+w_{\alpha }}\frac{k^2}{a^2}{\delta }_{\alpha }-\frac{Q_{\alpha }}{{\rho }_{\alpha }}{\theta }_{\alpha },$$

$$\frac{H^2}{H_0^2}=\frac{{\Omega }_m{a}^{-3\xi }+{\Omega }_r{a}^{-4\xi }+{\Omega }_{\Lambda }^{\star }}{1-\nu },$$

$$f(z)$$

$${\sigma }_8^2(z)={\delta }_m^2(z)\int \frac{d^{3}k}{(2\pi {)}^3}P(k,\vec{p})W^2(k{R}_8),$$

$${\chi }_c^2=\sum _{i=1}^n\frac{(T_c(z_i)-O_c(z_i){)}^2}{E_c^i},$$

Full text

I Introduction

To understand the accelerated expansion of the universe discovered in 1998 Riess:1998cb ; Perlmutter:1998np , dark energy is introduced  Copeland:2006wr . In a variety scenarios of dark energy, the Lambda cold dark matter ($\Lambda$CDM) model is the simplest one, which can explain the current cosmological observations very well. Unfortunately, this model accompanies several theoretical difficulties, such as “fine-tuning” Weinberg:1988cp ; WBook and “coincidence” Ostriker:1995rn ; ArkaniHamed:2000tc problems.

The running vacuum model (RVM) is one of the popular attempts to solve the “coincidence” problem Ozer:1985ws ; Carvalho:1991ut ; Lima:1994gi ; Lima:1995ea ; Overduin:1998zv ; Dymnikova:2001ga ; Carneiro:2004iz ; Bauer:2005rpa ; Alcaniz:2005dg ; Barrow:2006hia ; Shapiro:2009dh ; Geng:2016epb ; Geng:2016dqe . In this model, the cosmological constant $\Lambda$ is described by a function of the Hubble parameter and decays to matter and radiation in the expansion of the universe Ozer:1985ws . It has been shown that the RVM is suitable in describing the cosmological evolution on both background and linear perturbation levels Shapiro:2004ch ; Arcuri:1993pb ; EspanaBonet:2003vk ; Basilakos:2009wi ; Sola:2013gha ; Costa:2012xw ; Gomez-Valent:2014rxa ; Sola:2014tta ; Tamayo:2015qla ; Fritzsch:2016ewd .

In this work, we focus on the specific model with the running cosmological constant $\Lambda=3\nu H^{2}+\Lambda_{0}$, where $\nu$ is the model parameter and $\Lambda_{0}$ is the cosmological constant in the $\Lambda$CDM model. Clearly, in this RVM, the $\Lambda$CDM limit corresponds to $\nu=0$. Naively, it is expected that the value of $\nu$ should be arbitrarily close to zero in order to fit the current cosmological observations. However, it has been recently shown that $\nu\sim\mathcal{O}(10^{-2})-\mathcal{O}(10^{-3})$ in this RVM with the exclusion of the $\Lambda$CDM model within $2\sigma$ confidence level in the literature Sola:2016vis ; Sola:2016jky ; Gomez-Valent:2014fda ; Sola:2015wwa ; Gomez-Valent:2015pia ; Sola:2016ecz , which indicates that this model is a better theory to describe the evolution history of our universe. In this study, we plan to reexamine this RVM by using the latest observational data. In particular, we include the measurements of the time dependent Hubble parameter $H(z)$ and weighted linear growth $f(z)\sigma_{8}(z)$ in our analysis. We will use the CAMB Lewis:1999bs and CosmoMC Lewis:2002ah packages with the Markov chain Monte Carlo (MCMC) method.

This paper is organized as follows. In Sec. II, we introduce the RVM and derive the evolution equations for matter and radiation in the linear perturbation theory. In Sec. III, we perform the numerical calculations to obtain the observational constraints on the model parameter $\nu$ and cosmological observables in several datasets. Finally, our conclusions are given in Sec. IV.

II THE RUNNING VACUUM MODEL

II.1 Formalism

We start from the Einstein equation,

[TABLE]

where $R=g^{\mu\nu}R_{\mu\nu}$, $\Lambda$ and $T^{M}_{\mu\nu}$ are the Ricci scalar, cosmological constant and energy-momentum tensor of matter and radiation, respectively. Considering the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric

[TABLE]

we obtain the Friedmann equations,

[TABLE]

where $H=da/(adt)$ is the Hubble parameter, $\rho_{\alpha}$ ($P_{\alpha}$) with $\alpha=r,m$ and $\Lambda$ represent the energy densities (pressures) of matter, radiation and dark energy, respectively. Furthermore, it is convenient to define the corresponding equations of state, given by

[TABLE]

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,

[TABLE]

where $\rho_{M}=\rho_{m}+\rho_{r}$, $w_{M}=(P_{m}+P_{r})/\rho_{M}$ and $Q=Q_{m}+Q_{r}$ with $Q_{m(r)}$ the decay rate of dark energy to matter (radiation). In this work, we consider $\Lambda$ to be a function of the Hubble parameter, which might originate from the cosmological renormalization group Sola:2013gha , given by

[TABLE]

where $\nu$ and $\Lambda_{0}$ are two free parameters with the $\Lambda$CDM model recovered by taking $\nu=0$. If dark energy only couples to matter (radiation), there will be too many non-relativistic (relativistic) particles created in the early (late) time of the universe in terms of the current observations. By combining Eqs. (6) and (8), the coupling $Q_{\alpha}$ ($\alpha$ = $m$ or $r)$ is given by

[TABLE]

with $P_{M}=P_{m}+P_{r}$, where $\alpha$ represents matter or radiation. As a result, the energy density of $\alpha$ can be derived from Eq. (6) as,

[TABLE]

where $\xi=1-\nu$ and $\rho^{(0)}_{\alpha}$ is the energy density at $z=0$. Note that $\nu\geq 0$ is chosen to avoid the negative dark energy density in the early universe.

II.2 Linear perturbation theory

We follow the standard linear perturbation theory Ma:1995ey and derive the growth equation of the density perturbation in the RVM. The metric with the synchronous gauge is given by,

[TABLE]

with $i,j=1,2,3$, $\tau$ the conformal time and

[TABLE]

where $h(\vec{k},\tau)$ and $\eta(\vec{k},\tau)$ are two scalar perturbations, and $\hat{k}=\vec{k}/k$ is the $k$-space unit vector. By using the conservation equation $\nabla^{\nu}(T^{M}_{\mu\nu}+T^{\Lambda}_{\mu\nu})=0$ with $\delta T^{0}_{0}=\delta\rho_{M}$, $\delta T^{0}_{i}=-T^{i}_{0}=(\rho_{M}+P_{M})v^{i}_{M}$ and $\delta T^{i}_{j}=\delta P_{M}\delta^{i}_{j}$, one gets the growth equations of the matter and radiation as follows,

[TABLE]

where $\delta_{\alpha}\equiv\delta\rho_{\alpha}/\rho_{\alpha}$ and $\theta_{\alpha}=ik_{i}v^{i}_{\alpha}$ are the density fluctuation and the divergence of fluid velocity, respectively.

In principle, the dark energy density fluctuation should be taken into account when the dynamical model is considered. To explore the dynamics of dark energy, the time dependent cosmological constant should be rewritten as a Lorentz scalar at the field equation level in Eq. (1). For example, in Refs. Fabris:2006gt ; Borges:2008ii ; VelasquezToribio:2009qp ; Geng:2016fql ; Gomez-Valent:2014rxa ; Sola:2015rra ; Sola:2015wwa ; Gomez-Valent:2015pia ; Basilakos:2015vra ; Sola:2016jky ; Sola:2016ecz ; Sola:2016zeg ; Sola:2017jbl , the vacuum energy is given as $\Lambda=\Lambda(H)$ with $H=\nabla_{\mu}U^{\mu}/3$ and $U^{\mu}=dx^{\mu}/ds$. However, the dark energy perturbation plays an important role to the evolution of $\delta_{m}$ at the subhorizon scale, leading to a strong interaction between dark energy and matter fields, which must be ruled out by the astrophysical observations. In addition, the expression of $H$ is not unique, and the cosmological behavior significantly depends on the explicit form. Due to these two reasons and without losing the generality, we concentrate on the homogeneous and isotropic dark energy model, i.e., $\delta\rho_{\Lambda}=0$. Consequently, we have $\delta_{\Lambda}=\theta_{\Lambda}=0$, so that the particles, created from the dark energy decays, homogeneously distribute to the universe, smoothing the density fluctuation by the factor $Q_{\alpha}/\rho_{\alpha}$.

In the RVM, due to the background evolution of the Hubble parameter, one has

[TABLE]

where $\Omega_{m(r)}=\rho_{m(r)}^{(0)}/3H_{0}^{2}$, $\Omega_{\Lambda}^{\star}=\Omega_{\Lambda}-\nu=\rho_{\Lambda}^{(z=0)}/3H_{0}^{2}-\nu$ and $\Omega_{m}+\Omega_{r}+\Omega_{\Lambda}=1$. As discussed in Ref. Fritzsch:2016ewd , the larger $\nu$ is, the smaller $H(z)$ behaves in the high redshift regime.

It is known that the spectrum of the cosmic matter fluctuations can give important constraints on theoretical models about the structure formation  Press:1973iz ; Bond:1990iw ; Bower:1991kf ; Kauffmann:1993jn ; Lacey:1993iv . These fluctuations can be described by the weighted linear growth $f(z)\sigma_{8}(z)$, where

[TABLE]

is the growth factor and $\sigma_{8}(z)$ is the root-mean-square matter fluctuation amplitude on the scale of $R_{8}=8h^{-1}$ Mpc at the redshift $z$, given by

[TABLE]

with $P(k,\vec{p})$ the ordinary linear matter power spectrum and $W(kR_{8})$ the top-hat smoothing function (see e.g.Gomez-Valent:2014rxa for details). Several methods have been used to estimate $\sigma_{8}$, such as the measurements of the abundance of galaxy clusters Eke:1996ds ; Borgani:2001ir ; Reiprich:2001zv ; Pierpaoli:2000ip , cosmic shear analyses Bacon:2002va ; Brown:2002wt , combined analysis of galaxy redshift survey Bond:2002tp and CMB data Lahav:2001sg .

We note that the RVM modifies not only the background evolution but also the perturbation one. The creations of matter and radiation from the decays of dark energy suppress the growths of the density fluctuations, as demonstrated in Eqs. (13) and (14). If $\nu$ is large, the suppression effect on $\delta_{m}$ should be significant, leading to a “lowering effect” on $f(z)\sigma_{8}(z)$ Gomez-Valent:2014rxa ; Sola:2015rra ; Sola:2015wwa ; Gomez-Valent:2015pia ; Basilakos:2015vra ; Sola:2016jky ; Sola:2016ecz ; Sola:2016zeg ; Sola:2017jbl . Clearly, it is interesting to examine the RVM by using the data from the large scale structure observations, such as the baryon acoustic oscillation (BAO) and $f\sigma_{8}$.

III Numerical calculations

In Tables 1 and 2, we list 35 and 27 points for $H(z)$ and $f(z)\sigma_{8}(z)$ from the time dependent Hubble parameter and large scale structure formation measurements, respectively. By performing the CosmoMC program Lewis:2002ah , we fit the RVM from the observational data with the MCMC method. The dataset includes those from $H(z)$ and $f(z)\sigma_{8}(z)$ along with the CMB temperature fluctuation from Planck 2015 with TT, TE, EE, low-$l$ polarization from SMICA Adam:2015wua ; Aghanim:2015xee ; Ade:2015zua , the weak lensing (WL) data from CFHTLenS  Heymans:2013fya and the BAO data from 6dF Galaxy Survey Beutler:2011hx and BOSS Anderson:2013zyy . In addition, the $\chi^{2}$ function for the data from $H(z)$ or $f(z)\sigma_{8}(z)$ is taken to be

[TABLE]

where the subscript $c$, representing $H(z)$ or $f(z)\sigma_{8}(z)$, denotes the category of the data at the redshift $z_{i}$ from different data, $n$ is the number of the data in each dataset, $T_{c}$ is the theoretical prediction, calculated from CAMB, and $O_{c}$ ($E_{c}$) is the observational value (covariance). The priors of the various parameters are listed in Table 3.

In Fig. 1 and Table 4, we present our global fit result from various datasets, and the values in the brackets correspond to the best-fit results in the $\Lambda$CDM model, where the values of $\sigma_{8}$ are given at $z=0$. From the combined datasets of (A), (B), (C) and (D), we find that $\nu=1.35$, 1.91,1.55 and $1.23\times 10^{-4}$ with the best fitted $\chi^{2}$ values being 13487.7, 13509.9, 13511.3 and 13531.2 in the RVM, which are all slightly smaller than 13488.9, 13512.2, 13512.8 and 13534.7 in the $\Lambda$CDM model, respectively. This combined dataset leads to the lowest $\chi^{2}$ value in comparison with that in the $\Lambda$CDM model as shown in Table 4. We note that a part of the $H(z)$ data in Table 1 is derived by using the BC03 stellar models Bruzual:2003tq , which shows tensions from those by the MaStro models Maraston:2011sq at high redshift regime, as pointed out in Ref. Moresco:2012jh . Specifically, the $H(z)$ data point at $z=1.037$ in the MaStro models is $1.6\sigma$ lower than that in the BC03 one. From Eq. (15), we can estimate that with the same initial conditions $H(z)$ with $\nu_{1}$ is smaller than that with $\nu_{2}$ if $\nu_{1}>\nu_{2}$, so that the constraint on $\nu$ should be slightly relaxed when the BC03 data are replaced by the MaStro ones.

Although the cosmological observables in the RVM do not significantly deviate from those in $\Lambda$CDM, the best $\chi^{2}$ fits in the RVM are better than those in the $\Lambda$CDM model in all the datasets. It clearly indicates that the RVM is good in describing the evolution history of our universe. It should be noted that our result of $\nu\sim\mathcal{O}(10^{-4})$ is about one to two orders stronger that those of $\nu\sim\mathcal{O}(10^{-3})-\mathcal{O}(10^{-2})$ in the literature Sola:2016vis ; Sola:2016jky ; Gomez-Valent:2014fda ; Sola:2015wwa ; Gomez-Valent:2015pia ; Sola:2016ecz . Because the constraints on $\nu$ only slightly change when the $f(z)\sigma_{8}(z)$ and $H(z)$ data are taken into account, the RVM might be strongly constrained by the CMB temperature fluctuation. Moreover, we are unable to exclude the $\Lambda$CDM model more than $1.5\sigma$ confidence level, which is different from the $2\sigma$ exclusion statement in Refs. Sola:2016vis ; Sola:2016jky ; Gomez-Valent:2014fda ; Sola:2015wwa ; Gomez-Valent:2015pia ; Sola:2016ecz . Since the creation of particles from the decaying dark energy restrains the growths of $\delta_{m}$ is also suppressed in the RVM. In addition, it is known that the free streaming massive neutrinos suppress the matter density fluctuation, which also smoothen the density fluctuation. As a result, as shown in Table. 4, the allowed window for $\Sigma m_{\nu}$ is further restricted.

IV Conclusions

We have explored the allowed window for the model parameter in the RVM with $\Lambda(H)=3\nu H^{2}+\Lambda_{0}$ and shown that the constraint on the RVM becomes much very stronger with $\nu\sim\mathcal{O}(10^{-4})$ after considering the CMB temperature fluctuation along with the $H(z)$ and $f(z)\sigma_{8}(z)$ measurements, which is different from the results in the previous studies in the literature, which might be due to the accurate measurement on the photon power spectrum. Clearly, this statement should be further studied in the future. Explicitly, the value of $\nu=(1.37^{+0.72}_{-0.95})\times 10^{-4}$ is found at 68% C.L. by fitting the combined data of the CMB, WL, BAO, $H(z)$ and $f(z)\sigma_{8}(z)$. We have also found that $\chi^{2}_{RVM}\lesssim\chi^{2}_{\Lambda\mathrm{CDM}}$ in all the datasets of our discussions, denoting that the RVM is a good theory to describe the evolution of the universe at both background and linear perturbation levels. However, concerning the existence of an extra degree of freedom in the RVM to $\Lambda$CDM, it is hard to claim that the RVM is preferred by the observational data at the current stage yet. In addition, since dark energy decays to matter and radiation in the evolution of the universe, the matter density fluctuation $\delta_{m}$ is suppressed, leading to the best fitted value of $\Sigma m_{\nu}$ is relatively smaller than the corresponding one in the $\Lambda$CDM model.

Acknowledgments

We thank Prof. J. Sola for useful comments and discussions. This work was partially supported by National Center for Theoretical Sciences, MoST (MoST-104-2112-M-007-003-MY3 and MoST-106-2917-I-564-055) and the Newton International Fellowship (NF160058) from the Royal Society (UK).