Vacuum Energy and Renormalized Cosmological Constant

A raw zero-point mode sum is not an independently observable source for cosmic expansion. Its value depends on the regulator and subtraction convention, while gravity responds to the renormalized coefficient multiplying $\int\sqrt{-g}$ together with all vacuum contributions. Semiclassical gravity makes that distinction precise; it does not explain why the measured combination is so small compared with familiar particle-physics thresholds.

Helpful background. Quantum Fields in Curved Spacetime supplies covariant stress renormalization; Einstein Equations supplies the role of $\Lambda$; Black-Hole Geometry and Mechanics supplies gravitational normalization; Semiclassical Gravity supplies effective couplings; and Causality and Global Structure supplies the spacetime domain on which a state is defined.

Vacuum stress and the cosmological term

Lorentz invariance requires the vacuum expectation value in Minkowski space to have the form

$$\langle T_{ab}\rangle_{\rm vac}=-\rho_{\rm vac}g_{ab}.$$

To isolate this issue, hold the already renormalized Newton constant fixed and write the cosmological part before removing a regulator $\epsilon$ as

$$G_{ab}+\Lambda_{\rm B}(\epsilon)g_{ab} =8\pi G_{\rm ren} \left(T^{\rm excit}_{ab}-\rho_{\rm vac}^{\rm reg}g_{ab}\right),$$

where $T^{\rm excit}_{ab}$ contains nonvacuum sources. The finite combination seen by the geometry is

$$\Lambda_{\rm eff} =\Lambda_{\rm B}(\epsilon) +8\pi G_{\rm ren}\rho_{\rm vac}^{\rm reg}(\epsilon) =\Lambda_{\rm ren}(\mu) +8\pi G_{\rm ren}\rho_{\rm vac}^{\rm ren}(\mu).$$

Explicitly, write $\Lambda_{\rm B}=\Lambda_{\rm ren}+\delta\Lambda$ and $\rho_{\rm vac}^{\rm reg}=\rho_{\rm vac}^{\rm ren}+\delta\rho$. The counterterms obey $\delta\Lambda+8\pi G_{\rm ren}\delta\rho=0$. There is no second independent cosmological counterterm to add after the bare coefficient has already been defined this way. A finite subtraction change shifts the two renormalized terms oppositely and leaves $\Lambda_{\rm eff}$ unchanged. One can absorb the finite vacuum term into a single coupling and call that sum $\Lambda$; the notation must say which convention is being used.

What the mode sum does and does not say

For a free scalar in flat spacetime, the formal expression

$$\rho_{\rm vac}^{\rm formal} =\frac12\int\frac{d^3k}{(2\pi)^3}\sqrt{\mathbf k^2+m^2}$$

diverges. A hard three-momentum cutoff displays quartic, quadratic, and logarithmic terms but is not by itself a covariant gravitational regulator. Dimensional or point-splitting regularization packages the divergences into local covariant terms proportional to $g_{ab}$, $G_{ab}$, and curvature-squared tensors. Different schemes change those local pieces; after measured couplings are fixed, predictions agree.

Martin reviews the regularization dependence and the distinction between zero-point energy and the measured cosmological term Martin 2012, §§ 2.2–2.4 and 3.1, PDF. The lesson is not that vacuum energy fails to gravitate. It is that gravity measures the renormalized sum, not either bare summand.

For an FLRW solution,

$$3\left(H^2+\frac{k}{a^2}\right) =\Lambda_{\rm eff}+8\pi G_{\rm ren}\rho_{\rm nonvac}+\cdots ,$$

where the ellipsis denotes controlled local curvature and state-dependent contributions. The observed late-time expansion determines an effective renormalized parameter only after the matter model, geometry, and perturbation analysis are specified.

Radiative sensitivity

Renormalization does not remove the naturalness problem. When a field of mass $M$ is integrated through a threshold, its vacuum contribution generically shifts by a quantity of order $M^4/(16\pi^2)$ times spin, multiplicity, and logarithmic factors. The bare parameter can be readjusted so that the measured sum remains small, but repeated thresholds then require correlated cancellations between parameters with no known protection in ordinary low-energy theory.

Weinberg formulates the old cosmological-constant problem as the absence of a successful adjustment or symmetry mechanism compatible with observations Weinberg 1989, §§ II–VI. The semiclassical equation therefore supports two simultaneous statements:

• the absolute raw mode sum is scheme dependent and cannot be compared directly with observation;
• the small renormalized gravitational response is radiatively sensitive and remains unexplained.

Confusing the first statement with a cancellation mechanism erases the problem; ignoring it makes a regulator artifact look measurable.

Checks and boundaries

A constant shift $\langle T_{ab}\rangle_{\rm ren}\mapsto\langle T_{ab}\rangle_{\rm ren}-C g_{ab}$ is compensated by $\Lambda_{\rm ren}\mapsto\Lambda_{\rm ren}-8\pi G_{\rm ren}C$. The semiclassical equation is unchanged. Differences between states at fixed renormalization, such as a Casimir energy or a particle excitation, are not removed by this freedom and can gravitate.

The flat-space limit also clarifies the domain: one may choose the measured renormalized cosmological term to vanish and obtain Minkowski space, but that renormalization condition does not predict its value in nature. Explaining the observed value, its radiative stability, or a dynamical relaxation mechanism requires physics beyond the bookkeeping established here.

Exercise

Start from $G_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab}$. A new prescription defines $T'_{ab}=T_{ab}-C g_{ab}$. Find $\Lambda'$ such that the same metric solves the primed equation. Does this operation remove a state-dependent radiation density proportional to $a^{-4}$?

Solution

Demanding $G_{ab}+\Lambda' g_{ab}=8\pi G(T_{ab}-C g_{ab})$ and comparing with the original equation gives

$$\Lambda'=\Lambda-8\pi G C.$$

The sign differs from moving a vacuum term from the right to the left because here the stress definition itself was shifted downward by $C$. A radiation term cannot be absorbed: it varies as $a^{-4}$ and has $p=\rho/3$, whereas a cosmological term is constant and has $p=-\rho$.

References

• Martin, J. (2012). “Everything You Always Wanted to Know About the Cosmological Constant Problem (But Were Afraid to Ask).” Comptes Rendus Physique 13, 566–665. DOI. Open PDF.
• Weinberg, S. (1989). “The Cosmological Constant Problem.” Reviews of Modern Physics 61, 1–23. DOI.