Dark Energy and Normalization of the Cosmological Wave Function

Dark energy is investigated from the perspective of quantum cosmology. It is found that, together with an appropriate normal ordering factor $q$, only when there is dark energy then can the cosmological wave function be normalized. This interesting observation may require further attention.

The accelerating expansion of our universe has presented a very challenging problem in theoretical physics as well as in cosmology [1,2]. We believe that the character of dark energy is deeply associated with the nature of quantum gravity. In fact, one of the authors showed that dark energy can be well described with the help of the holographic principle, a characteristic feature of any viable theory of quantum gravity. This holographic dark energy model [3] becomes one of the most competitive and popular dark energy candidates, which also implies that a deep relation between dark energy and quantum gravity is well appreciated in the community.
On the other hand, as an application of quantum physics to the dynamical systems describing closed universes, quantum cosmology is also tightly related to quantum gravity.
In spite of some misgivings, interesting hints on fundamental mathematical and physical questions can be obtained from quantum cosmology. The studies on quantum cosmology as a generally covariant and highly interacting quantum theory may also provide answers to questions concerning the interplay of symmetries, discrete structures and so forth.
Having noticed the common connection between dark energy, quantum cosmology and quantum gravity, a natural and interesting question to ask is whether there is relation between dark energy and quantum cosmology. An answer to this problem may deepen our understanding of the dark energy and shed light on problems related to it.
For a theory which still retains its highly speculative and controversial nature, it is necessary to clarify the notation we will use in the following at first. For a general Wheeler-DeWitt equation for quantum cosmology where we assume a positive spatial curvature as refelcted in the term proportional to a 2 .
We define the inner product of the wave function as and interpret dP as the probability that the universe stays within scale factor between a and a + da when the wave function is normalizable. This quantity will be of our central interest. Of course, the probability interpretation is not simple, for instance to return to usual quantum mechanics one need to introduce time, as proposed in [1], one can imagine introducing a scalar field to play the role of time.
To focus on the key problem without lose of generality, it is convenient to suppose that the universe is closed and filled only with a special dark energy which is just the cosmological constant. In this setup, the Wheeler-DeWitt equation has a simpler form as 1 2 corresponding to Planck scale and length scale introduced by the cosmological constant, respectively.
In the region a ≪ l Λ , the energy induced by spatial curvature dominates the universe, has the form as It is well known that, for 0 <| z |≪ √ α + 1, the asymptotic behavior for modified Bessel functions is These results imply three following asymptotic behaviors for modified Bessel functions which are of our concern: After some tediously but straightforward calculations, one can find that the second term of the wave function (4) will always make the inner product P =´ǫ 0 a q |Ψ| 2 da (ǫ is an arbitrary upper limit of the integral satisfying ǫ ≪ l Λ ) divergent unless the normal ordering ambiguity factor q is in the domain of (−1, 3).
Another way to investigate the Wheeler-DeWitt equation in the region a ≪ l Λ is to use WKB method. Since the Wheeler-DeWitt equation is a second order differential function, boundary conditions are used in this method to select a particular solution. Two most famous wave functions are Hartle-Hawking no-boundary wave function [4] and Vilenkin's tunnelling wave function [5,6]. It will be interesting to find out to which solutions their proposals correspond in the general solution space.
To proceed, one can write Ψ(a) as Ψ(a) = e i S l 2 Inserting this into Eq.(3) and arranging terms according to order in l p , one obtains the zeroth order and second order equations, The above two equations can be solved to obtain the WKB wave function For the asymptotic solution (4), one needs to use the asymptotic form of modified Bessel functions for large arguments So Eq.(4) approximates to Now it can be seen that Hartle-Hawking's choice corresponds to the modified Bessel function of second kind I while Vilenkin's choice corresponds to the modified Bessel function of first kind K. It could be seen that Hartle-Hawking's wave function leads to a convergent probability whatever q is, while the inner product of Vilenkin's wave function is convergent only when −1 < q < 3.
In the region a ≫ l Λ , the cosmological constant term prevails over spatial curvature, the term a 2 in Eq. (3) can be neglected. The general solution of Eq.(3) then is Using the asymptotic expansion of Bessel functions at large argument as one can get Then it is easy to find that the inner product P =´+ ∞ λ a q |Ψ| 2 da (λ is an arbitrary lower limit of the integral satisfying λ ≫ l Λ ) always converges regardless of the normal ordering factor ambiguity q.
The interesting thing is, together with an appropriate normal ordering factor q, cosmological-constant dark energy causes a convergent and thus normalizable cosmological wave function. It is natural to wonder whether the relation between cosmological-constant dark energy and normalization of cosmological wave function is only a coincidence or a profound observation which deserves more attention. To investigate this, we consider following with g( a lp ) an function of a lp which will prevail over a 2 when a is large enough. Denoting Ψ(a) = e i( S 0 l 2 p +S 1 ) , and supposing that the condition |S ′ | 2 ≫ l 2 p |S ′′ + q x S ′ |, which takes the following form for our case is satisfied by g(x), then through a general procedure of the WKB method, one get the wave function and the inner product corresponding to it as following with x = a lp and the prime denoting differentiation with respect to x. Since g( a lp ) prevails over a 2 when a is large enough, Eq.(28) can be simplified further into Now, considering a special energy component whose appearance in Wheeler-DeWitt equation is g(x) = x 2 ln x, the special character of this kind of energy component is that its index of equation of state is w = − 1 3 − 1 3 ln x which will always be less than − 1 3 (for a is large) except at the point where the scale factor a is infinite. One can check that the term g(x) = x 2 ln x indeed prevails over the term a 2 in Eq.(25) and it also satisfies condition (26), thus, one can safely insert it into (30) to find that the inner product is divergent. Now, considering another energy component for which the condition g(x) > x 2+ǫ for a large scale factor (ǫ is an arbitrary given positive constant) is satisfied. This function corresponds to an energy component with index of equation of state w = − 1+ǫ 3 which is less than − 1 3 when a is large enough. Then, after a similar process, one can find which is convergent.
The difference between the first and the second energy component considered above is that, while the second energy component always satisfies the condition w < − 1 3 for large scale factor, the first energy component violates it at point a = ∞. Then, despite the first energy component behaves precisely like ordinary dark energy most of the time in the sense that it can accelerate the universe, it still leads to a divergent inner product of the wave function because it deviates from dark energy at the point with a = ∞. Thus, as having been stressed, the claim that dark energy causes a normalizable cosmological wave function only holds in the circumstances that w < − 1 3 is strictly satisfied when a is large enough.
To conclude, we study dark energy from the perspective of quantum cosmology in this work. It is found that, together with an appropriate normal ordering factor q (take value on domain (−1, 3), dark energy leads to a normalizable cosmological wave function while other energy components can not. We think that the interesting relation between dark energy and normalization of cosmological wave function may imply some deep characters of dark energy which deserves more attention. For further investigation, it is an interesting question whether or not quantum cosmology gives us some hints on choosing the dark energy models. There might not be a good answer based on the cases considered in this paper as the functions representing dark energy are rather arbitrary. However, we believe that there exists an interesting answer to this question when the dynamics of the dark energy is known (in our case, when the dynamics of h is known). Furthermore, considering the fact that the exact meaning of the wave function of the universe and the problem of measurement is still under careful investigation in quantum cosmology, it may be interesting to consider this work from the light of a more precise version of quantum theory, like many worlds or de Broglie-Bohm theory.