# Bare Cosmological Constant and Fine-Tuning Problem

Whenever anyone talks about the cosmological constant, they usually cite the difference of $\approx 120$ orders of magnitude between the observed and calculated values for the vacuum energy ($\rho_\text{obs} \approx 10^{-8}\text{erg}/\text{cm}^3$ and $\rho_\text{calc} \approx 10^{112}\text{erg}/\text{cm}^3$) and then claim that this leads to a serious fine-tuning problem. I've been trying to look into this, and, as best I can tell, the root of it appears to be that people claim that there is a bare cosmological constant: $$\Lambda_{\text{eff}} = \Lambda_{\text{bare}} + \Lambda_{\text{vac}}.$$ Where it appears that $\Lambda_\text{eff}$ is the "actual" cosmological constant that gets introduced into the Einstein Field Equations: $$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda_\text{eff}g_{\mu\nu} = 8\pi G T_{\mu\nu},$$ $\Lambda_\text{vac}$ corresponds to the roughly calculated value of the vacuum energy from all of the zero-point energies, and $\Lambda_\text{bare}$ is the the bare cosmological constant. The claim is that in order to get the extremely small observed value, the bare constant must cancel the vacuum constant out to $120$ orders of magnitude, which is a pretty serious fine-tuning problem.

My question, then, is this: What is the nature of the bare cosmological constant? My understanding of the "actual" cosmological constant that enters the Einstein Field Equations is that it arises from the vacuum energy (I guess, in my mind, I've always thought $\Lambda_\text{eff} = \Lambda_\text{vac}$), which leaves me confused about where the bare constant comes from. I've seen several papers make mention of the bare constant, but no one ever actually explains why it's present or how it is justified, so a reference would be greatly appreciated. Thanks!

• The vacuum energy in quantum field theory is simply intellectual nonsense. All it tells us is that we a) don't have the right theory or b) that we are too dumb to calculate its real ground state. I would actually bet on a combination of a) and b). One can make the same statement about dark energy/a cosmological constant, by the way. There is no deep idea behind any of this, it pops out of the math, but so do other things (like torsion) which we neglect, so it's all just random choices to fit the data right now. – CuriousOne Aug 2 '16 at 21:53
• So the bare constant is introduced ad hoc precisely to give a cancellation with the vacuum value in order to provide an "explanation" of the observed value, even though all that does is introduce a new problem (the fine-tuning problem)? – TimeFall Aug 2 '16 at 22:38
• I recommend arxiv.org/abs/0708.4231 – innisfree Aug 3 '16 at 9:33
• That was a great paper! As a follow-up question: Everyone talks about how the vacuum energy as related to the cosmological constant is due to empty space, but then they proceed, when discussing the cosmological constant problem, to factor in contributions from the zero-point energies of all of the fields in the Universe. Why would these fields be related to the energy of empty space? Isn't that comparing apples and oranges, in a sense? I guess what I'm asking is this: Isn't the vacuum energy due to particle-fields and such different from the vacuum of spacetime? – TimeFall Aug 5 '16 at 20:33

There are several sources that can contribute to $\Lambda_{eff}$:

1) A cosmological constant in general relativity. This doesn't need a source it can just be a modification to the Lagrangian. If you think of the General Relativity Lagrangian as series expansion of the Ricci tensor, its natural to have a constant term.

2) Vacuum fluctuations.

3) The potential of a scalar field, such as the Higgs field. The Higgs' potential energy could be quite large, it's just another contribution to $\Lambda_{eff}$.

The cosmological constant problem is why all these terms almost cancel to such high precision when they appear unrelated. See http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.61.1 for more details.