_{1}

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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
^{-90} m
^{-2} and 10
^{-51} m
^{-2} 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.

The history of the cosmological constant has been inferred with ups and downs. Originally, it was introduced by Einstein [

Since the early 1990s, it has been known that the energy-matter content of the Universe consists not only of visible baryonic matter which contributes ≈ 5% to the Universe’s energy, but also of Dark Matter (≈23%) and a phenomenon called Dark Energy (≈72%) which is related to the expansion of the Universe and probably associated to the cosmological constant [^{−}^{52} m^{−}^{2} [

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 [^{60} m^{−}^{2}, again larger than the measured value [

In 1996 Bars showed that a certain class of string theories contains more than one time dimension [

Chen [

Although theoretically possible, causality puts certain constraints on the properties of time dimensions summarized by Tegmark [

In previous work [

We now proceed one step further and extend the Robertson-Walker-Lemaître-Friedmann metric by one additional time dimension.

Our starting point is the extension of the four-vector

x μ = ( t r θ φ ) → x μ = ( t τ r θ φ ) (1)

to a five-vector with a second time dimension τ expanding four-dimensional space-time with time dimension t to five-dimensional space-time. Similarly, we extend the Robertson-Walker-Lemaître-Friedmann metric [

g μ ν = ( c ( t ) 2 0 0 0 0 − a ( t ) 2 1 − r 2 K 0 0 0 0 − a ( t ) 2 ⋅ r 2 0 0 0 0 − a ( t ) 2 ⋅ r 2 ⋅ s i n ( θ ) 2 ) (2)

to

→ g μ ν = ( c ( t ) 2 0 0 0 0 0 f ( τ ) 2 0 0 0 0 0 − a ( t , τ ) 2 1 − r 2 K 0 0 0 0 0 − a ( t , τ ) 2 ⋅ r 2 0 0 0 0 0 − a ( t , τ ) 2 ⋅ r 2 ⋅ s i n ( θ ) 2 ) (3)

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 [

From metric (3), we derive the Christoffel symbols and subsequently the Ricci tensor R μ ν and the Ricci scalar R (which are summarized in the supplementary Mathematica script).

In order to formulate Einstein’s field equations, we also extend the energy-momentum tensor

T μ ν = ( c ( t ) 2 ρ ( t ) 0 0 0 0 − p ( t ) g 11 0 0 0 0 − p ( t ) g 22 0 0 0 0 − p ( t ) g 33 ) (4)

→ ( c ( t ) 2 ρ ( t , τ ) c ( t ) f ( τ ) ρ ( t , τ ) 0 0 0 c ( t ) f ( τ ) ρ ( t , τ ) f ( τ ) 2 ρ ( t , τ ) 0 0 0 0 0 − p ( t , τ ) g 33 0 0 0 0 0 − p ( t , τ ) g 44 0 0 0 0 0 − p ( t , τ ) g 55 ) (5)

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 R μ ν − 1 / 2 g μ ν R + Λ g μ ν = 8 π G N / c 4 T μ ν leads to a set of four equations

c K f 3 a 2 [ c 3 ( ( 1 + K Λ a 2 ) f 3 − 2 K a f ′ a ′ + K f ( a ′ 2 + 2 a ″ ) ) − 2 K a f 3 c ˙ a ˙ + K c f 3 ( a ˙ 2 + 2 a a ¨ ) ] = − 8 π G N p (6)

− 3 c 2 a ˙ ′ = 8 π G N f 2 ρ a (7)

1 K a 2 f 3 [ c 2 ( ( 3 + K Λ a 2 ) f 3 − 3 K a f ′ a ′ + 3 K f ( a ′ 2 + a a ″ ) ) + 3 K f 3 a ˙ 2 ] = 8 π G N ρ (8)

c K a 2 [ c 3 ( ( 3 + K Λ a 2 ) f 2 + 3 K a ′ 2 ) − 3 K a f 2 c ˙ a ˙ + 3 K c f 2 ( a ˙ 2 + a ¨ ) ] = 8 π G N f 4 ρ (9)

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 [

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

− ( 3 a ˙ p c 2 + 3 a ˙ ρ + a ρ ˙ + 2 c ˙ c a ρ ) = f ′ f a ρ + 3 a ′ ρ + a ρ ′ (10)

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 p m a t = 0 Solving (10) leads to ρ m a t = C m a t 1 a 3 c 2 f ( − t + τ + t l ) f ( τ )

with a lower time limit t l , and an integration constant C m a t . Since ƒ depends on

τ only, ƒ needs to be constant, hence ρ m a t = C m a t 1 a 3 c 2 . The radiation pressure

is connected to the density through p r a d = ρ c 2 / 3 [

ρ r a d = C r a d 1 a 4 c 2 e ∫ t l t d t ¯ a ′ ( t ¯ , − t + τ + t ¯ ) a ( t ¯ , − t + τ + t ¯ ) . The total density is thus given through

ρ = C m a t 1 a 3 c 2 + C r a d 1 a 4 c 2 e ∫ t l t d t ¯ a ′ ( t ¯ , − t + τ + t ¯ ) a ( t ¯ , − t + τ + t ¯ ) (11)

where the lower limit t l is a time close to the origin of the Universe, hence we set t l = t Planck ≈ 10 − 43 s ≈ 0 . We will show later (20) that we can assume a τ to be

constant and therefore e x p ( ∫ t l t d t ¯ a ′ ( t ¯ , − t + τ + t ¯ ) a ( t ¯ , − t + τ + t ¯ ) ) = 1 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, a ( t , τ ) needs to be of the form ~ a t ( t ) + α ( t ) a τ ( τ ) such that l i m t → ∞ α ( t ) = 0 . Therefore, the second time dimension can only operate on a small spatial scale which is not accessible to experiments directly. Therefore, we make the ansatz

a ( t , τ ) = a t ( t ) + e β t a τ ( τ ) , β < 0 (12)

with a contribution a t ( t ) for the first time dimension only and a contribution e β t a τ ( τ ) for the second time dimension such that a ( t , τ ) → a t ( t ) for large t. This implies that the second time dimension expands the Universe by a small value a τ not measurable with current methods. Note that in theory, there is a whole class of functions α ( t ) satisfying l i m t → ∞ α ( t ) = 0 ; however, since Equation (12) needs to be inserted into a system of differential equations, we require a function which is sufficiently smooth and differentiable, which is why we choose the exponential function. On the other hand, we assume that a t ( t ) tends to 0 for t → 0 which is the usual assumption for the Big Bang. Yet, unlike a universe with four space-time dimensions, the overall scaling parameter a ( t , τ ) tends to a τ ( τ ) ≠ 0 for t → 0 ; hence we avoid a singularity for t → 0 [

For the speed of light, we make the ansatz

c ( t ) = c 0 + c ˜ t − t 0 (13)

since the speed of light is time-dependent with t − 1 for small t and becomes c 0 ≈ 3 × 10 8 m ⋅ s − 1 for t ≫ t 0 and reference length and time c ˜ and t 0 [

Inserting (12) and (13) into (7)-(9) leads to the equations

− 3 ( c 0 + c ˜ t − t 0 ) β e β t a ′ τ = 8 π G N f 2 ( a t + e β t a τ ) ( C m a t 1 ( a t + e β t a τ ) 3 ( c 0 + c ˜ t − t 0 ) 2 + C r a d 1 ( a t + e β t a τ ) 4 ( c 0 + c ˜ t − t 0 ) 2 e ∫ t l t d t ¯ a ′ ( t ¯ , − t + τ + t ¯ ) a ( t ¯ , − t + τ + t ¯ ) ) (14)

1 K ( a t + e β t a τ ) 2 f 3 [ ( c 0 + c ˜ t − t 0 ) 2 ( ( 3 + K Λ ( a t + e β t a τ ) 2 ) f 3 + 3 K f ( e 2 β t a ′ τ 2 + ( a t + e β t a τ ) e β t a ″ τ ) ) + 3 K f 3 ( a ˙ t + β e β t a τ ) 2 ] = 8 π G N ( C m a t 1 ( a t + e β t a τ ) 3 ( c 0 + c ˜ t − t 0 ) 2 + C r a d 1 ( a t + e β t a τ ) 4 ( c 0 + c ˜ t − t 0 ) 2 e ∫ t l t d t ¯ a ′ ( t ¯ , − t + τ + t ¯ ) a ( t ¯ , − t + τ + t ¯ ) ) (15)

( c 0 + c ˜ t − t 0 ) K ( a t + e β t a τ ) 2 [ ( c 0 + c ˜ t − t 0 ) 3 ( ( 3 + K Λ ( a t + e β t a τ ) 2 ) f 2 + 3 K e β t a ′ τ 2 ) − 3 K ( a t + e β t a τ ) f 2 ( − c ˜ ( t − t 0 ) 2 ) ( a ˙ t + β e β t a τ ) + 3 K ( c 0 + c ˜ t − t 0 ) f 2 ( ( a ˙ t + β e β t a τ ) 2 + ( a ¨ t + β 2 e β t a τ ) ) ] = 8 π G N f 4 ( C m a t 1 ( a t + e β t a τ ) 3 ( c 0 + c ˜ t − t 0 ) 2 + C r a d 1 ( a t + e β t a τ ) 4 ( c 0 + c ˜ t − t 0 ) 2 e ∫ t l t d t ¯ a ′ ( t ¯ , − t + τ + t ¯ ) a ( t ¯ , − t + τ + t ¯ ) ) . (16)

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.

Since ρ ~ a t − 3 + a t − 4 Equation (14) for large t implies that the scaling factor and hence the Universe are growing. For t → ∞ and identifying f ≈ c 0 2 (15) and (16) simply become the Friedmann equation for four space-time dimensions [

For small t, t → 0 , and using f = c 0 2 , we get

− 3 ( c 0 − c ˜ t 0 ) 2 β a ′ τ = 8 π G N c 0 2 ( C m a t 1 a τ 2 ( c 0 − c ˜ t 0 ) 2 + C r a d 1 a τ 3 ( c 0 − c ˜ t 0 ) 2 ) (17)

3 ( c 0 − c ˜ t 0 ) 2 K a τ 2 + Λ ( c 0 − c ˜ t 0 ) 2 + 3 ( c 0 − c ˜ t 0 ) 2 c 0 2 a τ 2 ( a ′ τ 2 + a τ a ″ τ ) + 3 ( a ˙ t ( t = 0 ) + β a τ ) 2 a τ 2 = 8 π G N ( C m a t 1 a τ 3 ( c 0 − c ˜ t 0 ) 2 + C r a d 1 a τ 4 ( c 0 − c ˜ t 0 ) 2 ) (18)

c 0 − c ˜ t 0 K a τ 2 [ ( c 0 − c ˜ t 0 ) 3 ( ( 3 + K Λ a τ 2 ) c 0 2 + 3 K a ′ τ 2 ) + 3 K c 0 2 c ˜ t 0 2 a τ ( a ˙ t ( t = 0 ) + β a τ ) + 3 K c 0 2 ( c 0 − c ˜ t 0 ) ( ( a ˙ t ( t = 0 ) + β a τ ) 2 + a τ ( a ¨ t ( t = 0 ) + β 2 a τ ) ) ] = 8 π G N c 0 4 ( C m a t 1 a τ 3 c 2 + C r a d 1 a τ 4 c 2 ) . (19)

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 t 0 , c ˜ and β , we here limit ourselves to give ranges for K and Λ . Furthermore, we use G N ≈ 6.674 × 10 − 11 m 3 ⋅ kg − 1 ⋅ s − 2 and c 0 ≈ 3 × 10 8 m ⋅ s − 1 for the gravitational constant and the speed of light. In standard cosmology [

C m a t ≈ 3 H 0 2 8 π G Ω m a t a 3 c 0 2 and C r a d ≈ 3 H 0 2 8 π G Ω r a d a 4 c 0 2

where we use H 0 ≈ 70 km ⋅ s − 1 ⋅ ( Mpc ) − 1 ≈ 2.27 × 10 − 18 s − 1 [

We finally need to give estimates for c ˜ , t 0 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 e x p ( β t o b s ) ≪ 1 for t o b s ≈ 13.8 Ga ≈ 4.35 × 10 17 s , and hence β ≪ l o g ( 1.0 ) / t o b s ≈ 0 s − 1 ; we here choose β ≪ ( − 10 − 5 ) - ( − 10 − 15 ) s − 1 such that e x p ( β t o b s ) ≈ 0 . Although the speed of light might not have been constant in the Early Universe [

For these estimates,

that a second time dimension is responsible for the Planck length. Therefore, for the following, we set

a τ ( τ ) ≈ 4.59 × 10 − 62 = const . (20)

which is the approximate scale factor for the Planck length ≈10^{−}^{35} m.

Assuming a τ to be constant, hence a ′ τ = a ″ τ ≈ 0 , we use the remaining Equations (18) and (19) to determine Λ and K for which we still need to approximate a ˙ t ( t = 0 ) and a ¨ t ( t = 0 ) . We are here stuck with a dilemma since these derivatives couple the two time dimensions to each other, and there is no way to determine a ˙ t ( t = 0 ) and a ¨ t ( t = 0 ) in a self-consistent way. For small t ≈ t Planck ≈ 0 , we are thus limited to use the four-dimensional Friedmann

equation [

Finally using these expressions to calculate a ˙ t ( t = 0 ) and a ¨ t ( t = 0 ) and inserting them into (18) and (19) yield

3 ( c 0 − c ˜ t 0 ) 2 K a τ 2 + Λ ( c 0 − c ˜ t 0 ) 2 + 3 ( 8 π G N 3 ( C m a t 1 a t ( t 0 ) c 0 2 + C r a d 1 a t ( t 0 ) 2 c 0 2 ) − c 0 2 K − Λ c 0 2 3 a t ( t 0 ) 2 + β a τ ) 2 a τ 2 = 8 π G N ( C m a t 1 a τ 3 ( c 0 − c ˜ t 0 ) 2 + C r a d 1 a τ 4 ( c 0 − c ˜ t 0 ) 2 ) (21)

c 0 − c ˜ t 0 K a τ 2 [ ( c 0 − c ˜ t 0 ) 3 ( 3 + K Λ a τ 2 ) f 2 + 3 K f 2 c ˜ t 0 2 a τ ( 8 π G N 3 ( C m a t 1 a t ( t 0 ) c 0 2 + C r a d 1 a t ( t 0 ) 2 c 0 2 ) − c 0 2 K − Λ c 0 2 3 a t ( t 0 ) 2 + β a τ ) + 3 K f 2 ( c 0 − c ˜ t 0 ) ( ( 8 π G N 3 ( C m a t 1 a t ( t 0 ) c 0 2 + C r a d 1 a t ( t 0 ) 2 c 0 2 ) − c 0 2 K − Λ c 0 2 3 a t ( t 0 ) 2 + β a τ ) 2 + a τ ( 4 π G N 3 ( C m a t − 1 a t 2 c 0 2 + C r a d − 2 a t 3 c 0 2 ) − Λ c 0 2 3 a t + β 2 a τ ) ) ] = 8 π G N f 4 ( C m a t 1 a τ 3 ( c 0 − c ˜ t 0 ) 2 + C r a d 1 a τ 4 ( c 0 − c ˜ t 0 ) 2 ) . (22)

These two equations give two sets of solutions ( Λ 1 , K 1 ) and ( Λ 2 , K 2 ) which are very lengthy and which we have therefore added to the supplementary Mathematica script.

^{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} [

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 t → 0 . 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.

4) The cosmological constant varies between 10^{−}^{90} m^{−}^{2} and 10^{−}^{52} m^{−}^{2} which agrees with measurements [

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 constant self-consistently from the Friedmann equations in a (2, 3) space time geometry, and whilst quantum theories cannot predict the value of the cosmological constant correctly [

Furthermore, it is interesting to note that we have not presumed any periodicity or compactness of the second time dimension as normally assumed for additional space dimensions. The effect of the second time dimension, however, is limited to a spatial scale of the size of the Planck length only. Therefore, the second time dimension does not act on the macroscopic Universe and is thus not visible to current experiments so far. However, this does not exclude that the proposed scenario can be tested in the future with advanced technology.

The presented approach gives novel ideas on how to approach cosmological singularities in the future. The connection of vanishing singularities and additional time dimensions gives rise to investigate the Big Bang more thoroughly in the context of extended space-time. Beyond that, we suggest to extend other metrics with additional time dimensions, such as the Schwarzschild metric or the Eddington-Finkelstein metric to study the effect of additional time dimensions on black holes.

I would like to thank Henrik Svensmark, Sara Svendsen and Khaled Alizai for careful reading and useful suggestions. The supplementary material for this publication can be found at 10.5281/zenodo.4007757.

The author declares no conflicts of interest regarding the publication of this paper.

Köhn, C. (2020) A Solution to the Cosmological Constant Problem in Two Time Dimensions. Journal of High Energy Physics, Gravitation and Cosmology, 6, 640-655. https://doi.org/10.4236/jhepgc.2020.64043