A Solution to the Cosmological Constant Problem in Two Time Dimensions

For the last hundred years, the existence and the value of the cosmological constant Λ has been a great enigma. So far, any theoretical model has failed to predict the value of Λ by several orders of magnitude. We here offer a solution to the cosmological constant problem by extending the Einstein-Friedmann equations by one additional time dimension. Solving these equations, we find that the Universe is flat on a global scale and that the cosmological constant lies between 10 m and 10 m which is in range observed by experiments. It also proposes a mean to explain the Planck length and to mitigate the singularity at the Big Bang.

or on the gravity's rainbow effect [13] of black holes on the propagation of photons with different wave lengths where the cosmological constant is treated equivalent to the thermodynamic pressure [14] [15]. This latter property of the cosmological constant is, for example, also used to derive a relation amongst geometry, thermodynamics and information theory by using complexity [16].
There have been a plethora of attempts to derive the value of the cosmological constant theoretically. However, such calculations yield predictions which are many orders of magnitude larger than ≈10 −52 m −2 which is also known as the "cosmological constant problem". Zel'dovich related all elementary particles and the quantum fluctuations in the Universe to the background energy which manifests as the Dark Energy in General Relativity. By this approach, he, however, found a value of the cosmological constant which is approximately 20 orders of magnitude larger than the measured value [17] [18]; other authors quantize these discrepancy to even 120 orders of magnitude [19] [20]. Another approach to explain Dark Energy are Extended Theories of Gravity where Dark Energy does not result from the cosmological constant and thus makes it obsolete, but becomes manifest from higher order curvature terms which are added to the standard Einstein-Hilbert action, see [21] and references therein. More recently, Marcolli and Perpaoli coupled gravity to matter with methods of noncommutative geometry and found a variable cosmological constant. They give an analytic solution for the effective cosmological constant with values of 10 60 m −2 , again larger than the measured value [22]. Garattini, Kruglov and Faizal showed that the cosmological constant might originate from a deformation of the Wheeler-DeWitt equation [23] [24]. Furthermore, this approach shows that the existence of the cosmological constant might exclude the Big Bang singularity.
In 1996 Bars showed that a certain class of string theories contains more than one time dimension [25]. This idea was further developed by Bars and Kounnas [26] [27] constructing actions for interacting p-branes within two dimensions. This yields a phase transition where the additional time dimension becomes part of the compactified universe. Additionally, they present a new Kaluza-Klein like dimensional reduction mechanism together with an action for a string in two time dimensions. Based on such a second time dimension, Araya and Bars showed that the Lagrangians and Hamiltonians of one-time systems which do not appear to be connected to each other, can be dualised by introducing a second time dimensions [28]. As an example, they show that the Lagrangians of massive relativistic and massless relativistic or massless relativistic and massive non-relativistic systems are dual to each other through such hidden relations.
Chen [29] later interpreted two extra time dimensions as hidden quantum variables where the non-local properties of quantum physics or the idea of matter waves using the de Broglie wave length are natural consequences from the action of a free particle in several time dimensions. Combining two-time physics with noncommutative ( ) 2, Sp R gauge theories leads to theories containing all known results of two-time physics including the reduction of physical spacetime with the associated "holography" and "duality" properties [30]. Based on such an then allows to construct a field theoretic formulation of two-time physics including interactions. Such a scenario determines the interaction at the level of the action uniquely by the interacting BRST gauge symmetry and opens a way to study two-time physics on the quantum level through the path integral approach [31]. Based on Romero's and Zamora's work on the relation between two-time physics and the Snyder noncommutative Space [32], Carrisi and Mignemi extended this model to seven dimensions deriving the symplectic structure of the Snyder model on a de-Sitter background [33].
Although theoretically possible, causality puts certain constraints on the properties of time dimensions summarized by Tegmark [34]. In the case of three spatial dimensions, more than one macroscopic time dimension would lead to an unpredictable Universe. Therefore, any additional time dimension must be small or at least be irrelevant for the macroscopic observables of the Universe. However, even in this case, a second time dimension puts certain constraints on macroscopic Newtonian and quantum physics: Whilst a second time dimension puts strong constraints on the connections coming from gauge interactions in Newtonian physics, it leads to a generalized uncertainty relation involving level spacings and Planck's constant on the quantum level [35].
In previous work [36], we coupled the Lagrangian of a free, relativistic particle to a second time dimension, small enough to not violate the causality of the observable Universe. Starting from this Lagrangian, we showed that a small length scale, the Planck length, emerged from this theory and that the speed of light was not constant in the Early Universe, as predicted by Albrecht and Magueijo [37].
We now proceed one step further and extend the Robertson-Walker-Lemaître-Friedmann metric by one additional time dimension.

The Friedmann Equations in Five-Dimensional Space-Time
Our starting point is the extension of the four-vector where a is the scaling parameter of the Universe, K is the inverse square of the curvature and ƒ the characteristic speed for the second time dimension such as c for the first time dimension [36]. Here, K is chosen such that the Universe is hyperbolic for In order to formulate Einstein's field equations, we also extend the energy-momentum tensor with the density ρ and the pressure p for an isotropic, perfect fluid.
Finally, inserting the metric (3), the Ricci tensor and Ricci scalar as well as the energy-momentum tensor (5) into the Einstein equation where the dot (  ) denotes the time derivative with respect to t and (') with respect to τ .
In comparison to the 4D Friedmann equations, we now have two equations more allowing us to estimate the cosmological constant Λ and the curvature of Universe. This is a similar approach to the original idea by Kaluza and Klein [40] [41] where they used a five-dimensional space time with four space dimensions to decouple 4D electromagnetism from 5D relativity.

Discussion and Limits of the Five-Dimensional Friedmann Equations
In this section, we are going to reduce Equations (6)-(9) and derive analytic expressions for the cosmological constant Λ and the curvature 1 K . Combining Equations (6)-(8) leads to the conservation of energy which relates the scale factor a, the pressure p and the density ρ to each other.
We divide the pressure into two terms coming from the matter and radiation in the Universe. We assume that the motion of particles in the Universe on a global scale is isotropic and collisionless, therefore the matter pressure is negligible, hence 0 mat p = Solving (10) leads to − + + = with a lower time limit l t , and an integration constant mat C . Since ƒ depends on τ only, ƒ needs to be constant, hence The radiation pressure is connected to the density through 2 3 rad p c ρ = [42]. Using that ƒ is constant, the solution of the resultant differential Equation (10) gives The total density is thus given through where the lower limit l t is a time close to the origin of the Universe, hence we . We will show later (20) that we can assume a τ to be constant and therefore for all t. Note that the conservation of energy (10) is equivalent to the acceleration Equation (6), therefore we are left with the three Equations (7)-(9) relating the two velocities ƒ, c, the curvature 1 K , the cosmological constant Λ and the scaling parameter a to each other. Generally the scaling parameter depends on both time dimensions t and τ .
However, we know that we only observe one time dimension together with three spatial dimensions. Because of this and as a standard method to solve differential equations with two independent variables, we here choose a separation ansatz for ( ) , a t τ which is usually either a product ansatz or a sum ansatz. As we will discuss below the system of equations considered here can be solved using a sum ansatz. Since the second time dimension cannot be observed at the present stage of the Universe's evolution, ( ) [37].
Inserting (12) and (13) C We now use this system of equations to derive analytic expressions for the cosmological constant and the Universe's curvature. But before doing so, we discuss this system for very large and very small t.
In the following, we estimate the values of a τ , of K and of Λ . Since we do not know the exact values for the density ρ as well as for the constants 0 t , c  and β , we here limit ourselves to give ranges for K and Λ . Furthermore, we We finally need to give estimates for c  , 0 t and β before we can continue our discussion. Since we know that the contribution of a τ to the whole scaling factor ( ) , a t τ (12) needs to vanish not only for t → ∞ , but also for the observable Universe, we require  [37], it needs to be constant for the Universe today.
We therefore choose Ω . It illustrates that the growth of a τ is negligible relative to its size; therefore we assume a τ to be constant for all τ , hence 0 a τ ′ ≡ . Thus, retrospectively, we see indeed that the integral in (11) disappears. Since we have no direct measurements of the second time dimension τ , a τ needs to be small; from previous work [36], we know which is the approximate scale factor for the Planck length ≈10 −35 m.

The Universe's Curvature and the Value of the Cosmological Constant
Assuming a τ to be constant, hence 0 a a τ τ ′ ′′ = ≈ , we use the remaining Equations (18) and (19) to determine Λ and K for which we still need to approximate ( ) These two equations give two sets of solutions ( )  Ω . In contrast, the dependence on β and c  is much more significant with bridging several orders of magnitude. Hence, K varies from approx. 10 174 m 2 to 10 216 m 2 for all considered cases which implies that the curvature 1 K is approximately 0. Additionally, Λ varies between 10 −90 m −2 and 10 −51 m −2 which is well in agreement with measurements determining Λ to approx. 10 −52 m −2 [11]. In addition, we have performed parameter studies how Λ and K depend on considered cases, we observe that there is not a significant deviation from the values of Λ and K as shown in Figure 2, therefore not altering our conclusions.

Conclusions and Outlook
Extending the Friedmann equations by one time dimension allows us to solve several cosmological mysteries simultaneously: 1) The second time dimension leads to a small expansion of the Universe which we interpret as Planck length.
2) This small extension exists for all times t, including the limit 0 t → . Therefore, the size of the Universe does not shrink to zero for small t avoiding a singularity of infinite mass and energy density at the Big Bang.  3) The Universe's curvature is almost zero; hence the Universe is flat.
The overall scenario is thus that the second time dimension expands the Universe on a very small length scale becoming manifest as the Planck length. Subsequently, from this small length scale, the Universe's curvature and the size of the cosmological constant follow. Note that the value of the cosmological constant is sometimes related to the zero-point-energy or vacuum-energy in quantum physics. However, in contrast to a quantum theory approach, we have presented a mechanism, which allows to estimate the value of the cosmological