Quintessential Enhancement of Dark Matter Abundance

The presence of a dynamical scalar field in the early universe could significantly affect the `freeze-out' time of particle species. In particular, it was recently shown that an enhancement of the relic abundance of neutralinos can be produced in this way. We examine under which conditions this primordial scalar field could be identified with the Quintessence scalar and find, through concrete examples, that modifications to the standard paradigm are necessary. We discuss two possible cases: the presence of more scalars and the switching on of an interaction.


Introduction
As it is well known, according to the standard paradigm [1] a particle species goes through two main regimes during the cosmological evolution. At early times each constituent of the universe is in thermal equilibrium, a condition which is maintained until the particle interaction rate Γ remains larger than the expansion rate H. At some time, however, H will overcome Γ because the particles will be so diluted by the expansion that they will not interact anymore. The epoch at which Γ = H is called 'freeze-out', and after that time the number of particles per comoving volume for that given species will stay constant for the remaining cosmological history. This is how cold dark matter particle relics (neutralinos, for example) are generated. This framework, although simple in principle, can be very complicated in practice. On one hand, we need a particle theory in order to compute Γ, on the other we have to choose a cosmological model to specify H. Even a small change in Γ and/or H would result in a delay or anticipation of the 'freeze-out' time of a given particle species and, as a consequence, in a measurable change in the relic abundance observed today.
The standard scenario [1] assumes that before Big Bang Nucleosynthesis (BBN), the dominant cosmological component was radiation and so the Hubble parameter was evolving according to H 2 ∼ ρ r ∼ a −4 , where ρ r is the energy density of radiation and a is the scale factor of the universe. This is a reasonable assumption, but the available data do not exclude modifications to this scenario. For example, it is conceivable that in the pre-BBN era, a scalar field had dominated the expansion for some time 1 , leaving room to radiation only afterwards. To be more concrete, if we imagine to add a significant fraction of scalar energy density to the background radiation at some time in the past, this would produce a variation in H 2 , depending on the scalar equation of state w φ . 2 If w φ > w r = 1/3, the scalar energy density would decay more rapidly than radiation, but temporarily increase the global expansion rate. This possibility was explicitly considered in Ref. [2], where it was calculated that a huge enhancement of the relic abundance of neutralinos could be produced in this way.
The effect of an early scalar field dominance on electroweak baryogenesis is discussed in Ref. [4]. An alternative possibility for non-standard 'freeze-out' is proposed in Ref. [5].
In this paper we will recall how a period of scalar 'kination' (see below) could affect the relic density of neutralinos and discuss if this primordial scalar field could be identified with the Quintessence scalar, i.e. the field thought to be responsible for the present acceleration of the universe [3]. We will find that modifications to the standard Quintessence paradigm are necessary and discuss some concrete examples.

Scalar field 'kination'
The early evolution of a cosmological scalar field φ with a runaway potential V (φ) is typically characterised by a period of so-called 'kination' [6,7,8], during which the scalar energy density is dominated by the kinetic contribution E k =φ 2 /2 ≫ V (φ). After this initial phase, the field comes to a stop and remains nearly constant for some time ('freezing' phase), until it eventually reaches an attractor solution 3 . A simple and interesting example is that of inverse power law potentials: with M a mass scale and n a positive number. These potentials exhibit the attractive feature of a stable attractor solution characterised by a constant scalar equation of state [6,9] w n = nw − 2 n + 2 which depends only on the exponent n in the potential and on the background equation of state w. Since n is positive, the condition w n < w always holds and the scalar field, which can be sub-dominant at the beginning, will eventually overtake the background energy density. This is a welcome feature if we are modelling the present acceleration of the universe through the scalar field dynamics (Quintessence), since during matter domination (w = w m = 0) the attractor has negative equation of state for any n.
In general, a scalar field in a cosmological setting obeys the evolution equation and, for any given time during the cosmological evolution, the relative importance of the scalar energy density w.r.t. to matter and radiation in the total energy density ρ 2 Remember that the energy density of each cosmological component scales as ρx/ρ o x = (a/ao) −3(wx +1) , if wx is the corresponding equation of state. 3 For a detailed discussion of the existence and stability of attractor solutions for general potentials see [7,8]. it can be easily seen that after an initial stage of 'kination' (w φ = 1), the field is 'freezing' (w φ = −1) and subsequently joins the attractor until it overtakes the background energy density. On the attractor the scalar equation of state is depends on the initial conditions, and is constrained by the available cosmological data on the expansion rate and large scale structure. As it is well known, the cosmological expansion rate is governed by the Friedmann equation where ρ includes all the contributions of eq.(5), and we have assumed a spatially flat universe. Then, if we modify the standard picture according to which only radiation plays a role in the post-inflationary era and suppose that at some timet the scalar contribution was small but non negligible w.r.t. radiation, then at that time the expansion rate H(t) should be correspondingly modified 4 . Since during the kination phase the scalar to radiation energy density ratio evolves like ρ φ /ρ r ∼ a −3(w φ −wr) = a −2 , the scalar contribution would rapidly fall off and leave room to radiation domination. Is this of any interest to us? The answer is yes, because there is a clear cosmological signature of this early phase: the relic density of neutralinos [2]. The reasoning goes as follows: since the fall off of ρ φ is so rapid during kination, we can respect the BBN bounds and at the same time keep a significant scalar contribution to the total energy density just few red-shifts before. For example, a scalar to radiation ratio ρ φ /ρ r = 0.01 at BBN (z ≃ 10 9 ) would imply ρ φ /ρ r = 0.1 at z ≃ 3.16 × 10 9 and ρ φ /ρ r = 1 at z ≃ 10 10 , if the scalar field is undergoing a kination phase. As extensively discussed in [2], calculations of the relic densities of dark matter (DM) particles are usually done under the assumption that the universe is dominated by radiation while they decouple from the primordial plasma and reach their final relic abundance. However, as we have seen, it is conceivable that the scalar energy density respects the BBN bounds and at the same time contributes significantly to the total energy density at the time DM particles decouple. Indeed, an increase in the expansion rate H due to the additional scalar contribution would anticipate the decoupling of particle species and result in a net increase of the corresponding relic densities. As shown in [2], a scalar to radiation energy density ratio ρ φ /ρ r ≃ 0.01 at BBN would give an enhancement of the neutralino codensity of roughly three orders of magnitude.

Quintessence?
As discussed in the previous section, the enhancement of the relic density of neutralinos requires that at some early time the scalar energy density was dominating the Universe. This fact raises a problem if we want to identify the scalar contribution responsible for this phenomenon with the Quintessence field [3] which (we suppose) accelerates the Universe today. Indeed, the scalar initial conditions are crucial to establish the scalar energy density contribution at any time.
In particular, the range of initial conditions which give rise to a non-negligible Quintessence contribution at present is huge but nonetheless does not include the case of a dominating scalar field at the beginning. In other words, the initial conditions must be such that the scalar energy density is sub-dominant (or, at most, of the same order of magnitude of ρ r ) at the beginning, if we want the Quintessence field to reach the cosmological attractor in time to be responsible for the presently observed acceleration of the expansion [6]. For initial conditions ρ φ > ∼ ρ r we obtain the so-called 'overshooting' behaviour: the scalar field rapidly rolls down the potential and after the kination stage remains frozen at an energy density which is much smaller than the critical one. The larger is the ratio ρ φ /ρ r at the beginning, the smaller will be the ratio ρ φ /ρ o c today.
There is also another situation in which the attractor is not reached in time. If the initial conditions are such that ρ φ < ∼ ρ o c (the initial scalar density is smaller than the present critical energy density), then the scalar field would remain frozen throughout the whole cosmological history and join the attractor only beyond the present time. In this case the ratio ρ φ /ρ o c remains unchanged and smaller than one until today (this the so-called 'undershooting' behaviour).
We should notice, however, that these rules strictly apply only to the standard case of a single uncoupled field with an inverse power law potential V ∼ φ −n . As shown in [10] more complicated dynamics are possible if we relax this hypothesis and consider more general situations. The presence of several scalars and/or of a small coupling with the dark matter fields could modify the dynamics in such a way that the attractor is reached in time even if we started, for example, in the overshooting region.
More fields. Consider a potential of the form with M a constant of dimension mass. In this case, as discussed first in [10], the two fields' dynamics enlarges the range of possible initial conditions for obtaining a quintessential behaviour today. This is due to the fact that the presence of more fields allows to play with the initial conditions in the fields' values, while maintaining the total initial scalar energy density fixed. Doing so, it is possible to obtain a situation in which for a fixed ρ in φ in the overshooting region, if we keep initially φ 1 = φ 2 we actually produce an overshooting behaviour, while if we choose to start with φ 1 = φ 2 (and the same ρ in φ ) it is possible to reach the attractor in time. This different behaviour emerges from the fact that, if at the beginning φ 2 ≪ φ 1 then, in the example at hand, ∂V /∂φ 2 ≫ ∂V /∂φ 1 and so φ 2 (the smaller field) will run away more rapidly and tend to overshoot the attractor, while φ 1 (the larger field) will move more slowly, join the attractor trajectory well before the present epoch and drive the total scalar energy density towards the required value 5 . In figure 2 the comparison between different cosmological evolutions depending on the fields' initial conditions, keeping ρ in φ fixed, is illustrated.
Interaction. Suppose now that the Quintessence scalar is not completely decoupled from the rest of the Universe. Among the possible interactions, as will be discussed below, two interesting cases are the following: If we add V b or V c to the potential V = M n+4 φ −n , the cosmological evolution will be accordingly modified. The main effect is that now the potential acquires a (time-dependent) minimum and so the scalar field is prevented from running freely to infinity. As a result, the long freezing phase that characterises the evolution of a scalar field with initial conditions in the overshooting Coupling constants two orders of magnitude smaller than the ones considered here are sufficient to ensure the desired effect. region can be avoided. As can be seen in figure 3, the interactions in eq. (8) drive the scalar field trajectory towards the attractor (in the case of V b ) or towards ρ m (in the case of V c ) well before the non-interacting case.
Effective interaction terms like V b in eq. (8) were first introduced in Ref. [11] and are more recently discussed in Ref. [12]. The point is that supersymmetry breaking effects in the early universe can induce mass corrections to the scalar Lagrangian of order H 2 . Indeed, if a term like δK ∼ χ * χφ * φ is present in the Kahler potential, where χ is a field whose energy density is dominating the universe, this will result in a correction proportional to ρ χ φ * φ in the Lagrangian. Then, if the universe is critical ρ χ ∼ H 2 and we obtain a mass correction for φ which goes like δV ∼ H 2 φ 2 .
The second type of interaction (V c in eq. (8)) emerges in the context of scalar-tensor theories of gravity, in which a metric coupling exists between matter fields and massless scalars 6 . These theories, expressed in the so-called 'Einstein frame' are defined by the action (see, for example, [14]): where S m = S m [Ψ m , A 2 (φ)g µν ] is the matter action which includes the scalar interaction via the multiplicative factor A 2 (φ) before the metric tensor g µν , and the gravitational action reads The scalar field equation in this context is modified w.r.t. eq.(4) by the presence of an additional 6 For a detailed discussion of these theories in the context of Quintessence cosmology see, for example, Refs. [13].
source termφ + 3Hφ + 1 2 where α(φ) ≡ d log A(φ)/dφ and T is the trace of the matter energy-momentum tensor T µν . The case α(φ) = 0 (i.e. A(φ) = const.) corresponds to the standard scenario with the scalar field decoupled from matter fields; while it can be easily seen that eq.(11) is equivalent to switching on an interaction term like V c of eq. (8), if we choose the function A(φ) to be an exponential.
As it is well known, introducing an interaction between the matter fields and a light scalar should always done with great care in order to avoid unwanted effects like time variation of constants and modification of gravitational laws (for discussions of these issues in the Quintessence context see Refs. [10,15,16]). Limits on the possible values of the couplings b and c in eq. (8) depend on the details of the theory that originates them and on the cosmological epoch we are considering. Just as a rough estimate, we recall that at the present time solar system measurements impose on metric theories of gravity (see [14]) an upper bound for c of order 10 −1 .

Conclusions
In this letter we have shown that modifications to the standard Quintessence paradigm are possible in order to make the Quintessence scalar responsible of an enhancement of the relic density of neutralinos. We have illustrated through specific examples that this can be obtained in two different ways: by considering more scalar fields in the Quintessence fluid and introducing an interaction term in the scalar potential.