Unimodular Approaches to the Cosmological Constant Problem
Abstract
:1. Introduction
2. Classical Formulations of Unimodular Gravity
2.1. The Unimodular Constraint
2.2. Henneaux and Teitelboim UG
2.3. Diffeomorphism Covariant, Weyl Invariant UG
2.4. Degrees of Freedom of UG
3. Quantum Aspects of UG
3.1. Path Integral
3.2. Quantum Fluctuations
4. Vacuum Energy Sequestering
Local Formulation
5. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
CC  cosmological constant 
GR  general relativity 
UG  unimodular gravity 
HT  Henneaux and Teitelboim 
EH  Einstein–Hilbert 
Notes
1  We use the reduced Planck units $8\pi {G}_{N}=\hslash =1$ and signature convention $\left(+,,,\right)$ 
2  A similar equation was written down originally by Einstein himself [34], however, only for a priori traceless energymomentum tensor (of radiation). Only later was it realized that these equations describe UG. 
3  This theory can be very easily rewritten in several other forms that are immediately equivalent. The only difference is that the fields ${V}^{\mu}$ and $\lambda $ can be redefined in such a way that the constraint part of the action becomes
$$\sqrt{g}\lambda ({\nabla}_{\mu}{W}^{\mu}1)\phantom{\rule{4pt}{0ex}},$$
$$\lambda (\frac{1}{4!}{\u03f5}^{\mu \nu \sigma \rho}{F}_{\mu \nu \sigma \rho}\sqrt{g})\phantom{\rule{4pt}{0ex}},$$

4  A possible way to circumvent this limitation is to introduce additional terms dependent on the vector density ${V}^{\mu}$, which modify Equation (42). Such terms provide the necessary couplings to the ‘nonconserved’ matter sector, which allow for spacetime dependence of $\lambda $. However, it is not clear whether such couplings can reproduce the diffusion processes considered in [35,38,39,40]. 
5  Interestingly, the corresponding Hamiltonian is linear in the momentum $\lambda $ and thus it is unbounded from below. Nevertheless, since $\lambda $ becomes a constant onshell, the system is perfectly stable [49]. 
6  In the Abelian case $\mathcal{P}$ represents the Pfaffian of the matrix ${F}_{\mu \nu}$. 
7  This preserves the original solutions while also adding a novel branch, where $\lambda $ becomes dynamical. This addition can be also applied to (40), where the constraint on constancy of $\lambda $ is stricter and no new branch appears. 
8  Note that the Lambert W function ${W}_{1}\left(x{e}^{x}\right)\ne x$ for $x>1$. 
9  Note that a more rigorous approach requires first going to the ADM formalism to work out the canonical structure of the theory and then calculating the associated path integral in the Hamiltonian formalism along with any necessary fixing of gauge symmetries and associated Faddeev–Popov determinants. Such procedure has been carried out in [31], while the ghost sector has been discussed in [28]. Furthermore, the structure of the ghost sector was analyzed in the BRST formalism in [47,59,60,61]. Nevertheless, such considerations do not meaningfully affect the result in comparison to a more naive approach we consider here. 
10 
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Jiroušek, P. Unimodular Approaches to the Cosmological Constant Problem. Universe 2023, 9, 131. https://doi.org/10.3390/universe9030131
Jiroušek P. Unimodular Approaches to the Cosmological Constant Problem. Universe. 2023; 9(3):131. https://doi.org/10.3390/universe9030131
Chicago/Turabian StyleJiroušek, Pavel. 2023. "Unimodular Approaches to the Cosmological Constant Problem" Universe 9, no. 3: 131. https://doi.org/10.3390/universe9030131