A natural cosmological constant from chameleons

We present a simple model where the effective cosmological constant appears from chameleon scalar fields. For a Kachru-Kallosh-Linde-Trivedi (KKLT)-inspired form of the potential and a particular chameleon coupling to the local density, patches of approximately constant scalar field potential cluster around regions of matter with density above a certain value, generating the effect of a cosmological constant on large scales. This construction addresses both the cosmological constant problem (why $\Lambda$ is so small, yet nonzero) and the coincidence problem (why $\Lambda$ is comparable to the matter density now).


Introduction
The cosmological constant problem is one of the most challenging problems in theoretical physics today. Indeed, the problem is twofold. Firstly the observed cosmological constant today is about 123 orders of magnitude lower than the natural value implied by quantum loop corrections, the uppermost cut-off for effective field theory, namely the Planck scale. In the days before the observation of the cosmological constant, when it was thought to be zero, the problem was easier, since some kind of yet undiscovered symmetry could perhaps force it to be zero. Supersymmetry for instance aleviates the problem a bit, since in exact global supersymmetry Λ = 0, so broken supersymmetry requires that the cosmological constant be of the order of the supersymmetry breaking scale, instead of the Planck scale. Moreover unbroken supergravity (local supersymmetry) requires that any cosmological constant be negative.
The observation of a non-zero, positive cosmological constant dashed these hopes, and introduced yet a second more philosophical puzzle, the coincidence problem: why is the cosmological constant (which should be constant for all times) of the order of the matter density today, when cosmologically there is nothing particularly special about the moment in time we happen to live in. Experimentally, ρ Λ is about twice the density of dark matter today. In Copernican terms, we appear to live at a special time in the history of the universe when ρ Λ , ρ DM and ρ matter are all comparable.
It is then perhaps natural to think of an evolving scalar field which happens now to be in a region of the potential which is almost constant very small (though explaining that very small value is not easy), an idea known generically as quintessence. Yet such a scalar must be very light, and there are very strong constraints on light scalars from gravity, as these would generate an, as of yet unobserved, fifth force. Chameleon scalars [1,2] were introduced as a way to avoid those constraints: they are scalars whose mass depends on the local matter density, so on Earth the scalars are very massive, avoiding laboratory gravity experimental constraints, as well as those from lunar laser ranging etc, (see [2] for further discussions). The coupling of chameleons with the local density takes the form and we will see that rather generally the coupling function A(φ) can be written as where M Pl is the reduced Planck mass and g is the coupling between the scalar φ and the matter type in the energy density ρ. On the other hand, on planetary and Solar System scales, the scalar force is suppressed due to the fact that the scalar profile only varies within a thin shell inside large bodies, hence only the mass within this thin shell effectively interacts via the scalar fifth force. This leaves the possibility that the chameleon scalars have some interesting behaviour on lower density scales such as vacuum [3,4], space [2] and on cosmological scales.
Generically, quintessence is strongly constrained both by the allowed variation of the masses of fundamental particles and the low mass required of a quintessence field to drive dark energy.Theoretically as well, one would still need to explain why is the value of the potential so small, which is non-trivial.
In this paper we present a new approach to the cosmological constant problem, based on the chameleon scalar idea. The potential for the chameleon scalar that we will choose is a phenomenological one, based on the KKLT-inspired models used in [5][6][7]. However, unlike [5], when the coupling g between the chameleon and the local matter density was a fixed number of order one given by string theory, we will allow an arbitrary value for g, and find that we need a very large value for it in our model, which is allowed in chameleon models, see [8]. The potential has a minimum which we set exactly at V = 0.
Within this model, we will find that the value for the scalar field potential is approximately constant near concentrations of matter (with density greater than a minimum density), and the fact that the value of this potential is so small and comparable with the density of matter arises simply from the condition of minimization of the effective potential (1.1). 1 In this way we translate both the issue of the smallness of the cosmological constant and the coincidence problem (ρ Λ ∼ ρ DM ) into just choosing the shape of the potential, which however we argue that is quite natural. Of course, we still have the old cosmological constant problem: why don't quantum corrections affect the V = 0 value of the minimum?

The model
The universal coupling to the matter density in (1.1) appears because the metric that couples universally to matter is not the Einstein frame metric g µν , but the metric Such a situation appears naturally in KK compactifications, when the relation betwen the higher dimensional metric ds 2 D and the lower dimensional metric ds 2 d is of the type so R is a modulus for the volume of compactification. If matter couples naturally to the D-dimensional metric ds 2 D , we obtain where R * is a particular value for R, close to the average value of R in the Universe, to be defined shortly. The exact form of the kinetic term for R depends on the details of the compactification, but in general it is such that it leads to (1.2), therefore the canonical scalar is The mechanism described here depends only on having (1.2) and (2.3), not on the fact that R is obtained from KK compactification as the volume modulus, but we can use the KK compactification ansatz to motivate the form of the potential V (R). Indeed the volume of the extra dimensions must be stabilized at some value, therefore we can approximate the potential by a quadratic where α, β > 0 are both dimensionless, around the minimum at The constant in the potential (2.5) was chosen such that the minimum is at V (R min ) = 0, and the value R * was introduced such that (2.5) is valid only for R > R * . For R < R * , we assume that the potential is well approximated by a very steep exponential, with γ, k > 0 and v > 0 all dimensionless. This form is for instance the leading exponential (at small R) arising from the KKLT-like potential generated by a superpotential W = W 0 + Ae −ia̺ , with a < 0 instead of KKLT's a > 0, as explained in [5], so it is a rather natural possibility. The parameter v is included here to allow us to fix the value of dV dR(R * ) for R < R * independently of R * .
Minimizing the effective potential (1.1), one finds so for R < R * , in the steep exponential side of the potential, using the fact that R/R * ≃ 1, we find where we have used the implicit dependence ρ(R) to denote by V (ρ) the potential at the minimum value of the effective potential, and we have denoted by ρ * = ρ(R * ), i.e. the minimum density so that we are in the region R ≤ R * . We then have Pl α 2 /4β. At R = R * , we can equate both the value of the potential, and of the derivative in (2.5) and in (2.7). Using the potential for R > R * , we have Using the potential for R < R * on the other hand, we obtain We can now input experimental constraints on the parameters. In [5], a constraint on γkR −k * was found from Earth laboratory experiments. The constraint was given for g ∼ O(1), but we now write it for general g, as R k * γk log 10 γk R k * − 24 + log 10 g 2 < ∼ g10 −29 . (2.14) On the other hand, from the condition that the Milky Way Galaxy be screened (it has a thin shell), where Φ G ∼ 10 −6 is the Newtonian potential of the galaxy, φ cosmo and φ solar are the values of the scalar field on cosmological and solar system scales respectively. We find that ln R min R * < ∼ 2g 2 10 −6 (2.16) Consider now the case g ∼ c × 10 3 , with c =a few, then (2.14) becomes γkR −k * > ∼ g −1 10 30 = c −1 10 27 (2.17) and (2.16) becomes α βR * < ∼ 4g 2 10 −6 ∼ 4c 2 (2.18) We finally get leading to a potential in the R < R * -i.e., ρ > ρ * -region (in the case we are close to saturating the bounds) with b ∼ a few, and d < ∼ c 2 ∼ a few, to be constrained better from experiments shortly. Note that the value of b is irrelevant, all that matters is that changing ρ has almost no effect on the value of V for ρ > ρ * , which stays close to dρ * . Finally, note that in order to have a consistent picture, there must be regions in the Universe which are on the quadratic piece of the potential, and that requires ρ * > ρ Λ . (2.11) (and (2.13), (2.20) in the last inequality) then implies that We now look to understand the consequences of the potential (2.20). Consider the case where ρ galaxy cluster ∼ a few ρ * , (where galaxy clusters were chosen as the largest matter structure, and their density means the density of the dark and normal matter inside them), which means that the chameleon inside galaxy clusters is in the R < R * region, but a bit away from R * , and moreover that More precisely, consider that where the right hand side is the experimental value (and matter includes dark and normal matter). Then outside the galaxy clusters, where the density falls to almost zero, the chameleon drops down to R = R min , where V (R min ) = 0. In this way, the chameleon creates a potential V that is almost constant in regions with matter (independent of the distribution of matter inside the patch), and almost zero outside. If the ratio to the matter density is taken as in (2.23), this creates an extra energy contribution that accounts for the energy of the observed cosmological constant.
Then, as the Universe expands, ρ galaxy cluster drops due to the Hubble expansion (the volume of the galaxy cluster expands), but as long as we are still in the R < R * region, we still have therefore the value of this extra energy contribution is constant in time, i.e. it is effectively a cosmological constant.
So we have simulated the effect of the cosmological constant with this chameleon field. Considering our original motivation, we can ask: is this construction natural, that is, was it natural to obtain a very small value for the observed cosmological constant, and was it natural to have a value for ρ Λ so close to ρ matter today? The smallness of the cosmological constant is related to the smallness of ρ * = R * αM 4 Pl . In [5][6][7] it was shown that we can obtain a potential like the desired phenomenological potential from the KKLT construction with large extra dimensions, and the required values for R * and α, β, v are all obtained due to the e ia̺ exponentials in the superpotential and the large extra dimensions. So in that particular case, the naturalness of the small cosmological constant would be reduced to the naturalness of large extra dimensions.
As for the coincidence problem, an explanation for the fact that ρ Λ is close to the matter density ρ matter today is more nuanced. The potential is correlated to the matter energy density, V = V (ρ), but is approximately constant, at the value d ρ * , and the fact that this is close to ρ matter today is still somewhat coincidental.
However, we also have another feature that is different from other constructions of an effective cosmological constant. Though it is was well approximated by a cosmological constant until now, in the near future, this potential contribution will drop to zero. Indeed, once ρ galaxy cluster drops below ρ * , (2.20) will not be valid anymore. In fact, asymptotically, as ρ galaxy cluster becomes very small, eventually R will settle at R min , with V (R min ) = 0, i.e. with no dark energy at all. So from this point of view, the coincidence is less drastic: as soon as ρ Λ becomes comparable to ρ matter we can observe it, but it also means that we are close to the point where it will start disappearing. After that, we will still have some dark energy, though it will take the form of a decreasing potential energy contribution.
To close the discussion of the model, we will review some experimental constraints on the model which were derived in [5] for the case g ∼ O(1), to apply in our case of g ∼ 10 3 . We already discussed (2.14) and (2.16 independent of g. One thing which does change with respect to [5] is the range of the chameleon in various environments for ρ > ρ * , i.e., a factor of g smaller range.

Conclusions
In this paper we have given an alternative to a simple cosmological constant based on chameleon scalars. Because of the chameleon coupling, a value for the chameleon potential energy as a function of the matter density was approximately constant above a certain value ρ * , namely (2.20), but zero for sufficiently small ρ (close to vacuum). That means that for certain choices of parameters, we can have patches of approximately constant energy density around the largest matter structures, and zero outside them. The patches have constant energy in time, thus effectively simulating a cosmological constant, as long as the matter density of the largest matter structures stays above a certain value. Eventually, as matter will get diluted, the potential energy will start to drop, leaving no cosmological constant in the far future.
This mechanism generates a small cosmological constant despite the vacuum being at V = 0, thus aleviating the cosmological constant problem. The small value of the effective cosmological constant can be in principle obtained rather naturally, like in a large extra dimensions scenario. It would be interesting to see whether this scenario can be embedded in a consistent fundamental theory. We took the coupling g to be ∼ a few ×10 3 , though in string theory g is usually a fixed number of order 1.