The \(\Lambda \)-term in Einstein’s equations is a fundamental building block of the ‘concordance’ \(\Lambda \)CDM model of cosmology. Even though the model is not free of fundamental problems, they have not been circumvented by any alternative dark energy proposal either. Here we stick to the \(\Lambda \)-term, but we contend that it can be a ‘running quantity’ in quantum field theory (QFT) in curved space time. A plethora of phenomenological works have shown that this option can be highly competitive with the \(\Lambda \)CDM with a rigid cosmological term. The, so-called, ‘running vacuum models’ (RVM’s) are characterized by the vacuum energy density, \(\rho _{vac}\), being a series of (even) powers of the Hubble parameter and its time derivatives. Such theoretical form has been motivated by general renormalization group arguments, which look plausible. Here we dwell further upon the origin of the RVM structure within QFT in FLRW spacetime. We compute the renormalized energy-momentum tensor with the help of the adiabatic regularization procedure and find that it leads essentially to the RVM form. This means that \(\rho _{vac}(H)\) evolves as a constant term plus dynamical components \({{\mathcal {O}}}(H^2)\) and \(\mathcal{O}(H^4)\), the latter being relevant for the early universe only. However, the renormalized \(\rho _{vac}(H)\) does not carry dangerous terms proportional to the quartic power of the masses (\(\sim m^4\)) of the fields, these terms being a well-known source of exceedingly large contributions. At present, \(\rho _{vac}(H)\) is dominated by the additive constant term accompanied by a mild dynamical component \(\sim \nu H^2\) (\(|\nu |\ll 1\)), which mimics quintessence.

的\（\ LAMBDA \） -term在爱因斯坦方程是“一致”的基本构建块\（\ LAMBDA \） CDM宇宙学模型。即使该模型没有基本问题，也没有被任何其他暗能量提议所规避。在这里，我们坚持\（\ Lambda \）-项，但我们认为它可能是弯曲时空中量子场论（QFT）中的“运行量”。大量的现象学研究表明，该选项与具有严格宇宙学术语的\（\ Lambda \） CDM可能具有很高的竞争力。所谓的“运行真空模型”（RVM）的特征在于真空能量密度\（\ rho _ {vac} \），是哈勃参数及其时间导数的一系列（偶数）次幂。这种理论形式是受到一般的重新规范化组论证的推动的，这些论证似乎是合理的。在这里，我们进一步讨论FLRW时空中QFT中RVM结构的起源。我们借助绝热正则化程序计算重新归一化的能量动量张量，发现它本质上导致了RVM形式。这意味着\（\ rho _ {vac}（H）\）作为常数项加上动态分量\（{{\ mathcal {O}}}（H ^ 2）\）和\（\ mathcal {O} （H ^ 4）\），后者仅与早期宇宙有关。但是，重新归一化的\（\ rho _ {vac}（H）\）并不包含与场的质量（\（\ sim m ^ 4 \）的四次幂成比例的危险术语，这些术语是众所周知的贡献极大的来源。目前，\（\ rho _ {vac}（H）\）以加法常数项为主导，并伴随着轻微的动力学分量\（\ sim \ nu H ^ 2 \）（\（| \ nu | \ ll 1 \）），它模仿精髓。