Weighing neutrinos in the scenario of vacuum energy interacting with
  cold dark matter: application of the parameterized post-Friedmann approach
  We constrain the neutrino mass in the scenario of vacuum energy interacting
with cold dark matter by using current cosmological observations. To avoid the
large-scale instability problem in interacting dark energy models, we employ
the parameterized post-Friedmann (PPF) approach to do the calculation of
perturbation evolution, for the $Q=\beta H\rho_{\rm c}$ and $Q=\beta
H\rho_{\Lambda}$ models. The current observational data sets used in this work
include Planck (cosmic microwave background), BSH (baryon acoustic
oscillations, type Ia supernovae, and Hubble constant), and LSS (redshift space
distortions and weak lensing). According to the constraint results, we find
that $\beta>0$ at more than $1\sigma$ level for the $Q=\beta H\rho_{\rm c}$
model, which indicates that cold dark matter decays into vacuum energy; while
$\beta=0$ is consistent with the current data at $1\sigma$ level for the
$Q=\beta H\rho_{\Lambda}$ model. Taking the $\Lambda$CDM model as a baseline
model, we find that a smaller upper limit, $\sum m_{\nu}<0.11$ eV ($2\sigma$),
is induced by the latest BAO BOSS DR12 data and the Hubble constant measurement
$H_{0} = 73.00 \pm 1.75$ km~s$^{-1}$~Mpc$^{-1}$. For the $Q=\beta H\rho_{\rm
c}$ model, we obtain $\sum m_{\nu}<0.20$ eV ($2\sigma$) from Planck+BSH. For
the $Q=\beta H\rho_{\Lambda}$ model, $\sum m_{\nu}<0.10$ eV ($2\sigma$) and
$\sum m_{\nu}<0.14$ eV ($2\sigma$) are derived from Planck+BSH and
Planck+BSH+LSS, respectively. We show that these smaller upper limits on $\sum
m_{\nu}$ are affected more or less by the tension between $H_{0}$ and other
observational data.
