Cosmological Constant Problem

Prerequisites: General Relativity, Canonical Quantum Field Theory. Full path (30 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

• Why the Tiny Weight of Empty Space Is Such a Huge Mystery by Natalie Wolchover

Dark Energy certainly sounds catchy but is a terrible terminology: It's neither dark nor energy. Instead, it's transparent, and (if you divide it by Newton's constant) it's an energy-density.. Dark energy can be interpreted as the energy-density of empty space, but I don't find this a good interpretation. Better to think of it as the curvature of empty space.

Sabine Hossenfelder

Concrete

We use $c = 1$ and the signature $(-,+,+,+)$. The Einstein equation allows a constant term,

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T_{\mu\nu} .$$

The cosmological constant $\Lambda$ has dimension of inverse length squared. Observations of the expansion of the universe (supernova distances, the cosmic microwave background, galaxy surveys) are fit well by a positive $\Lambda$ of about $1.1\times10^{-52}\,\mathrm{m}^{-2}$, which makes up roughly $70\%$ of the energy density of the universe today. Expressed as an energy density, $\rho_\Lambda = \Lambda/(8\pi G)$, and in natural units ($\hbar = c = 1$) this is $\rho_\Lambda \approx (2\times10^{-3}\,\mathrm{eV})^4$.

Vacuum energy acts like $\Lambda$. Lorentz invariance requires that the stress-energy tensor of the vacuum be proportional to the metric, $T^{\text{vac}}_{\mu\nu} = -\rho_{\text{vac}}\,g_{\mu\nu}$. This is a perfect fluid with energy density $\rho_{\text{vac}}$ and pressure $p = -\rho_{\text{vac}}$. Moving it to the left side of the Einstein equation shows that only the combination

$$\Lambda_{\text{eff}} = \Lambda + 8\pi G\,\rho_{\text{vac}}$$

is observable. Gravity therefore measures the total vacuum energy, which in theories without gravity is an unobservable constant shift of the energy.

The estimate. In quantum field theory every mode of a free field with frequency $\omega_k$ contributes a zero-point energy $\frac12\hbar\omega_k$. For a massless field and a momentum cutoff $k_{\text{max}}$,

$$\rho_{\text{vac}} = \int^{k_{\text{max}}}\frac{d^3k}{(2\pi)^3}\,\frac{k}{2} = \frac{k_{\text{max}}^4}{16\pi^2} \quad (\hbar = c = 1).$$

With $k_{\text{max}}$ at the Planck scale, about $10^{19}\,\mathrm{GeV}$, this exceeds the observed value by roughly 120 orders of magnitude. A sharp cutoff breaks Lorentz invariance, and a more careful calculation in dimensional regularization gives a contribution of order $m^4/(64\pi^2)$ times a logarithm for each field of mass $m$. Even the electron alone then overshoots the observed value by about 30 orders of magnitude, and the shift of the Higgs potential energy at the electroweak phase transition by more than 50.

The problem is that $\Lambda$ must cancel these contributions to the observed precision. The bare parameter can be tuned to do this, but the tuning has to be redone every time we add a contribution, for example each order of perturbation theory or each phase transition in the early universe.

Abstract

In the language of effective field theory, the gravitational action at low energies is

$$S = \int d^4x\,\sqrt{-g}\left(-\frac{\Lambda}{8\pi G} + \frac{R}{16\pi G} + \dots\right) + S_{\text{m}}[g,\phi] ,$$

and the cosmological term is the operator of lowest dimension, the volume form itself. It has a coefficient of mass dimension four. Integrating out a field of mass $M$ shifts this coefficient by an amount of order $M^4$ (up to logarithms and loop factors), and no symmetry of the standard model coupled to gravity forbids such shifts. The cosmological constant is therefore not technically natural in 't Hooft's sense, unlike fermion masses, which are protected by chiral symmetry. Unbroken supersymmetry would set the vacuum energy to zero, but supersymmetry must be broken at a scale of at least about a TeV, which leaves contributions of at least the order $(\mathrm{TeV})^4$.

The problem is usually split into two parts. The old problem asks why the vacuum energy is not of the order of the largest scales in the theory. The new problem asks why the remainder is not exactly zero but of the same order as the matter density today.

Proposed directions include:

• Adjustment mechanisms in which a field dynamically cancels $\Lambda$. Weinberg gave a no-go argument against a broad class of these, based on the requirement that the adjusting field settles in a Poincaré-invariant vacuum.
• Unimodular gravity, where $\sqrt{-g}$ is fixed and $\Lambda$ appears as an integration constant of the field equations. Vacuum energy then does not gravitate directly, but the value of the integration constant is not explained.
• Anthropic selection in a landscape of vacua, where Weinberg's bound states that galaxies can only form if the vacuum energy density is not much larger than the matter density at the time galaxies form.
• Modifications of gravity at large distances, which try to decouple the vacuum energy from the curvature at cosmological scales.

None of these is generally accepted.

Why is it interesting?

From a post-natural perspective, it seems that the problem of the cosmological constant can no longer be ignored. By making a field expansion of the SM effective potential, it appears that the problem with the cosmological constant and the Higgs mass have a common origin. The only difference is that the vacuum energy, unlike the Higgs mass, becomes an observable only through its coupling to gravity. But it seems likely that progress during the post-naturalness era will only come by addressing the two problems simultaneously. The cosmological constant corresponds to a scale of about 2×10−3 eV and affects physics in the deep IR, at astronomical and cosmological distances. Modifications of gravity at large distances are attempts to tackle the problem from an IR perspective [37]. However, the conceptual problem with the cosmological constant comes from quantum effects in the deep UV. Attempts to tackle the UV problem with the traditional methods used for the Higgs seem hopeless because the corresponding UV cutoff is way lower than any energy scale entering those theories. This confusion among scales is at the basis of the problem. It is a big source of confusion because the systematic approach of effective field theories has taught us how to separate energy scales in successive shells and make sense of the theory at each shell separately. The cosmological constant seems to resist this approach. Naturalness is an offspring of effective field theory and so it is not surprising that the difficulty we are encountering with the effective theory description leads to a problem with naturalness. A theory that manifests an active interplay between the IR and the UV would not be simply describable by an effective field theory, as it would violate its inner logic. It is not impossible that quantum gravity will exhibit some kind of IR/UV interplay. An indication could come from the classical behaviour of gravity. Consider the head-on collision of two particles at ever increasing energies. Once you pass the threshold for forming a black hole, the more energy you feed in the system the larger the Schwarzschild radius becomes. In other words, higher energy collisions are less sensitive to short distances, in contrast with our effective-theory intuition for a separation between IR and UV. A radical conclusion that could be derived from these considerations is that we are facing the end of validity of field theory and the solution of the cosmological constant lies beyond our familiar theories. The new framework should incorporate an interplay of physical effects occurring at all scales. Although it is difficult to tell how such a framework would look, one can easily expect that the cosmological constant problem will play a key role in the post-naturalness era.

https://arxiv.org/pdf/1710.07663.pdf

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