A Dynamical Solution to the Problem of a Small Cosmological Constant and Late-time Cosmic Acceleration

Increasing evidence suggests that most of the energy density of the universe consists of a dark energy component with negative pressure, a “cosmological constant” that causes the cosmic expansion to accelerate. In this paper, we address the puzzle of why this component comes to dominate the universe only recently rather than at some much earlier epoch. We present a class of theories based on an evolving scalar field where the explanation is based entirely on internal dynamical properties of the solutions. In the theories we consider, the dynamics causes the scalar field to lock automatically into a negative pressure state at the onset of matter-domination such that the present epoch is the earliest possible time, consistent with nucleosynthesis restrictions, when it can start to dominate.

Increasing evidence suggests that most of the energy density of the universe consists of a dark energy component with negative pressure, a "cosmological constant" that causes the cosmic expansion to accelerate. In this paper, we address the puzzle of why this component comes to dominate the universe only recently rather than at some much earlier epoch. We present a class of theories based on an evolving scalar field where the explanation is based entirely on internal dynamical properties of the solutions. In the theories we consider, the dynamics causes the scalar field to lock automatically into a negative pressure state at the onset of matter-domination such that the present epoch is the earliest possible time, consistent with nucleosynthesis restrictions, when it can start to dominate.
Introduction. Observations of large scale structure, searches for Type Ia supernovae, and measurements of the cosmic microwave background anisotropy all suggest that the universe is undergoing cosmic acceleration and is dominated by a dark energy component with negative pressure. 1 The dark energy may consist of a cosmological constant (vacuum density) or quintessence, 2 such as a scalar field with negative pressure. In either case, a key challenge is the "cosmic coincidence" problem: Why is it that the vacuum density or scalar field dominates the universe only recently? Until now, either cosmic initial conditions or model parameters (or both) had to be tuned to explain the low density of the dark energy component.
In this paper, we explore a new class of scalar field models with novel dynamical properties that avoid the fine-tuning problem altogether. A feature of these models is that the negative pressure results from the non-linear kinetic energy of the scalar field, which we call, for brevity, k-field or k-essence. (This consideration is inspired by earlier studies of k-inflation, kinetic energy driven inflation. [3][4][5] ). As we will show, for a broad class of theories, there exist attractor solutions which determine the equation-of-state of k-essence during different epochs depending on the equation-ofstate of the background. Effectively, the scalar field changes its speed of evolution in dynamic response to changes in the background equation-of-state. During the radiation-dominated epoch, k-essence is led to be subdominant and to mimic the equation-of-state of radiation. Hence, the ratio of k-essence to radiation density remains fixed. When the universe enters the dust-dominated epoch, though, k-essence is unable to mimic the dust-like equation-of-state for dynamical reasons. Instead, the energy decreases rapidly by several orders of magnitude and freezes at a fixed value. After a period (typically corresponding roughly to the current age of the universe), the field overtakes the matter density and drives the universe into cosmic acceleration. Ultimately, the k-essence equation-of-state slowly relaxes to an asymptotic value between 0 and -1. (The reader may wish to sneak a peek at Fig. 3 which illustrates the behavior in a specific numerical example.) The scenario bears some likeness to the quintessence "tracker models" discussed by Zlatev et al., 6,7 but is in fact very different from them. For a certain class of tracker potentials, the quintessence scalar field converges to an attractor solution in which the energy density in the quintessence field mimics the equation-of-state of the background (matter or radiation) energy density. However, for tracker potentials, it does not make a difference what the equation-of-state of the background is. Only after the scalar field passes a certain critical value does the quintessence develop a negative pressure, and, then, the energy density becomes fixed. The weakness in this model is that the energy density for which the pressure becomes negative is set by an adjustable parameter, which has to be fine tuned to explain why cosmic acceleration is happening at the present epoch in the history of the universe.
The distinctive feature of k-essence models is that tracking of the background energy density can only occur in the radiation epoch. At the matter-radiation equality, a sharp transition of k-essence from positive to negative pressure is automatically triggered by dynamics. The k-essence cannot dominate before matter-radiation equality because it is exactly tracking the radiation background. It also cannot dominate immediately after dustdomination because its energy density necessarily drops several orders of magnitude at the transition to dustdomination. However, since its energy density decreases more slowly than the matter density as the universe expands, k-essence must dominate not too long thereafter, at roughly the current epoch. The resolution of the cosmic coincidence problem boils down to the fact that we live at the "right time" after matter-radiation equality.
As noted above, the remarkable behavior comes at the cost of introducing a non-linear kinetic energy density functional of the scalar field and adjusting it to obtain the desired attractor behavior. This kind of action may describe a fundamental scalar field or be a low-energy effective action. For example, in string and supergravity theories, non-linear kinetic terms appear generically in the effective action describing moduli and massless degrees of freedom (superpartners) due to higher order gravitational corrections to the Einstein action. 8,9 The attractor behavior of our models relies on certain broad conditions on the form of these terms. Our initial examples are admittedly contrived for the purposes of numerical illustration. A systematic study of model-building will appear in a forthcoming paper, 10 although, having seen here the relatively simple basic principles, the reader should be equipped to explore more attractive and better-motivated forms.
Equations. In the theories we consider the Lagrangian density for ϕ is taken to be where R is the Ricci scalar, X ≡ 1 2 (∇ϕ) 2 , L m is the Lagrangian density for dust and radiation and we use units where 8πG/3 = 1. The energy density of the k-field ϕ is ρ k = (2Xp ,X −p)/ϕ 2 ; the pressure is p k =p/ϕ 2 ; and the speed of sound of k-essence is c 2 s = p k,X /ρ k ,X , where the subscript means derivative with respect to X. 3,4 The attractor behavior can be explained most easily by changing variables from X to y = 1/ √ X and rewriting the k-field Lagrangian as: In this case, the energy density and pressure are ρ k = −g ′ /ϕ 2 and p k = g/ϕ 2 y, where prime indicates derivative with respect to y. The equation-of-state is and the sound speed is In order to have a sensible, stable theory, we require ρ k > 0 and c 2 s > 0. These conditions are satisfied if g ′ < 0 and g ′′ > 0 in the region where p ′ k is positive. Therefore, a general, convex, decreasing function g(y), such as shown in Fig. 1, satisfies these necessary conditions. Using the Friedmann equation: H 2 = ρ tot = ρ k + ρ m ,where ρ m is the energy density of matter (radiation and dust), and the energy conservation equations,ρ i = −3ρ i (1+w k ) for the k-essence (i ≡ k) and matter (i ≡ m) components, we obtain the following equations of motioṅ where r(y) ≡ − and dot denotes derivative with respect to N ≡ ln a. These are the master equations describing the dynamics of k-essence models. Once some general properties of g(y) are specified, the attractor behavior described in the introduction follows from these coupled equations. Dynamics. We are seeking a tracker solution y(N ) in which the k-essence equation-of-state is constant and exactly equal to the background equation-of-state, w k (y (N )) = w m , and the ratio ρ k /ρ tot is fixed. Generically, this requires y(N ) be a constant y tr and therefore ρ k /ρ tot = r 2 (y tr ). The last condition can only be satisfied if r(y tr ) is less than unity. Hence, given a convex function such as shown in Fig. 1, we can first identify those ranges of y where r(y) is greater than unity or less than unity. In ranges where r(y) is less than unity, we can seek values of y where w k in Eq. (3)  Although there is no dust attractor, it is quite possible for there to be a radiation attractor. The radiation attractor corresponds to positive pressure, so it can be located only at y < y D . Hence, we must have g(y) such that r(y) is less than unity for some range at y < y D which includes some point y R where w k (y R ) = 1/3. During the radiation-dominated epoch, the ratio of k-essence to the total density remains fixed on this attractor and equal to (ρ k /ρ tot ) = r 2 (y R ).
In Fig. 1, the pressure p k = g/ϕ 2 y is positive above the y-axis, and negative below the y-axis. The dust equationof-state p k = 0 can only be obtained at y = y D where g(y) goes through zero. However, this point can be an attractor, only if, the second condition, r(y D ) < 1, is satisfied. If it so happens that r(y D ) > 1, then there is no dust attractor in the matter-dominated epoch. This is precisely what we want for our scenario, and this is possible for a broad class of functions g.
If g possesses a radiation attractor but no dust attractor, what happens at dust-radiation equality? To answer this question let us study the solutions of the master equations Eqs. (5-7) in two limiting cases, when the energy density of k-essence is either much smaller or much greater than the matter energy density. If ρ k /ρ m ≪ 1, one can neglect the last term in the equation (5) and it is obvious that y (N ) ≃ y S , where y S satisfies the equation r (y S ) = 0, is an approximate solution of the equations of motion. The point S satisfies g(y S ) = g ′ (y S )y S , so the tangent of g at y S passes through the origin, as shown in Fig. 1. Since r ∝ (1 + w k ), the equation of state of k-essence at y S corresponds to w k (y S ) ≈ −1; we call this solution the de Sitter attractor and denote it by S in Fig. 1. From Fig. 1, it is clear that y S nearly always exists for convex decreasing functions g. We stress that y (N ) ≈ y S is an approximate solution to the equationsof-motion only when matter strongly dominates over kessence. So, if ρ k during the radiation dominated epoch is significantly less than the radiation density which it tracks, which is both typical and required to satisfy nucleosynthesis constraints, then k-essence proceeds to the de Sitter attractor immediately after dust-radiation equality. 10 As the transition to dust-domination occurs, ρ k first drops to a small, fixed value, as can be simply understood. Suppose that (ρ k /ρ tot ) R = r 2 (y R ) = α < 10 −2 during the radiation dominated epoch, where the bound is set by nucleosynthesis constraints. From the equation of state, Eq. (3), we have the relation: g(y R ) = −g ′ (y R )y R /3. The condition r(y D ) ≥ 1 is required in order to have no dust attractor solution. Combining these relations, we obtain: On the other hand, it is apparent from Fig. 1 that In particular, the tangent at y D falls below g(y R ), so g ′ D (y R − y D ) ≈ −y D g ′ D ≤ g(y R ) = −y R g ′ R /3. Using this relation, we obtain Since ρ k = −g ′ /ϕ 2 and |g ′ (y S )| ≤ |g ′ (y D )| we conclude that after radiation domination, when the k-field reaches the vicinity of the S-attractor, the ratio of energy densities in k-essence and dust does not exceed (ρ k /ρ tot ) R × g ′ D /g ′ R ; that is, ρ k /ρ dust < α 2 /16 < 7 · 10 −6 . Hence, provided (ρ k /ρ tot ) R ≤ 10 −2 at dust-radiation equality, the k-essence field loses energy density on its way to the S-attractor down to a value below 7 × 10 −6 .
By definition, the S-attractor is one in which w ≈ −1 and the energy density is nearly constant. Hence, once ρ k has reached its small but non-zero value, it freezes. In the further evolution of the universe, the matter density decreases, but the k-essence energy density remains constant, eventually overtaking the matter density of the universe. Note that, as ρ k approaches ρ m , the condition ρ k /ρ m ≪ 1 is necessarily violated and a new attractor solution is found for the case where k -essence itself dominates the background energy density. This attractor is denoted K in Fig. 1.
To prove that that the K-attractor exists, we consider the master equations, Eqs. (5-7), in the limit where ρ k /ρ tot → 1. If y K satisfies the equation r (y K ) = 1, then y (N ) ≃ y K is an approximate solution of the equations of motion. When dust is not a tracker, there always exists a unique attractor y K in the interval y D < y < y S . 10 To prove this, note that, within this interval, the function r(y) has a negative derivative. Recall that r(y S ) = 0 (definition of S-attractor) and r (y D ) > 1 (to avoid a dust attractor). Since r (y) is a monotonically decreasing, continuous function, there exists a unique point y K (y D < y K < y S )) where r(y) becomes equal to unity. At y > y D the pressure of k-essence is negative. Hence, generically the K attractor, located near y K , describes a universe dominated by a negative pressure component which induces power-law cosmic acceleration. As acceleration proceeds, ρ k increasingly dominates and y → y K .
Following along using Fig. 1, the dynamics can be summarized as follows: k-Essence is attracted to y = y R during the radiation dominated epoch; at matterdomination, the energy density drops sharply as kessence skips past y = y D , because there is no dust attractor, and heads towards y ≈ y S . The energy density ρ k freezes and, after a period, overtakes the matter density. As it does so, y relaxes towards y K . In this scenario, our current universe would be making the transition from y S to y K . All this occurs for generic g(y) satisfying broad conditions on its first and second derivatives. If the ratio of ρ k to the radiation density is near the maximum allowed by nucleosynthesis (roughly equipartition initial conditions), the scenario predicts that the ρ k dominates by the present epoch.
Numerical results. We have verified these analytic predictions numerically for a wide class of g(y). As as a strategy, we look for forms which are roughly linear, in the vicinity of radiation attractor R and parabolic in the region y D ≤ y ≤ y S . One can easily check, that the points y R , y D and y S here are, by construction, the places where the corresponding attractors are located and g ′ R , g ′ D are the derivatives of g at the appropriate points. The results are not sensitive to the precise form of g that interpolates between these regimes. The main constraints are that the attractor solution have a small ratio of ρ k /ρ tot during the radiation epoch and that there is no dust attractor. For illustrative purposes, we have used these principles to obtain a sample g(y) and have transformed it to a form forp(X) in the action, Eq. (1): where a = 10 −5 and b = 10 −6 . For small values of X, after a field redefinition, this Lagrangian density reduces to one equivalent to a canonical scalar field with exponential potential (curiously, a tracker model 6 ). The distinctive dynamical attractor in our models relies deviations from linearity at large X.
The results of a numerical integration are presented in Figs. 2 and 3. We see that k-essence tracks the radiation (w k ≈ 1/3) during the radiation-dominated epoch. Then, at the onset of matter-domination, w k starts to change and the energy density of k-essence suddenly drops by several orders of magnitude becoming of the order of 10 −7 times the critical density at red shifts about z ≃ 1000 as the S-attractor is approached and w → −1. At about red shift z ∼ 3 − 5, ρ k becomes non-negligible and w k starts to increase, ultimately reaching w k ≃ −0.77 at z = 0. The ratio of the k-essence energy density to the critical density today is Ω k ≈ 0.74. In the future, w k in this model has to approach the value −0.55, corresponding to the K-attractor solution, and the universe will enter the period of power law k-inflation.
Summary. In this paper, we have presented a scenario in which cosmic acceleration occurs late in the history of the universe due to an inevitable sequence of events caused by attractor dynamics. We view the present work as a demonstration of principle; hence, we have emphasized general conditions and an analytic understanding of the scenario. The specific example illustrated in this paper is admittedly complex, composed to illustrate the concept, but we know of no fine-tuning or other requirement that poses a barrier to finding simpler and bettermotivated forms. By changes of variables and other simple techniques, one can quickly enlarge the class of actions considered here, which may suggest other types of attractive models.
A prediction of k-essence models that distinguishes them from models based on tracker potentials 7 is that w k is in the process of increasing today from -1 towards its asymptotic value at the K attractor, whereas, for trackers, w k is undergoing a transition from w ≈ 0 towards w = −1. A consequence is that the effective value of w k for k-essence -that is, the Ω k weighted average of w k between the present and z = 1 -can be significantly lower than for the tracker potential case, which is bounded below by w ef f ≈ −0.75. 7 In the numerical example above, the effective w ef f = −0.84, for example. The current supernovae data suggest a lower value of w k more consistent with k-essence. 11 Of course, the k-essence range for w k is more difficult to distinguish from a cosmological constant (w = −1).
In future work, 10 we discuss model-building using further examples and generalizations. We also explore interesting variations of the dynamical scenario with different kinds of attractors, including some which can lead to different long-term future outcomes, such as a return to a pressureless, unaccelerated expansion in the long-term future.
This work was initiated at the Isaac Newton Institute for Mathematical Sciences; we thank the organizers and staff of the Institute for their support and kind hospitality. This work was supported in part by the "Sonder-