Dark energy and normalization of cosmological wave function in modified gravitations
aa r X i v : . [ g r- q c ] M a y Dark energy and normalization of cosmological wavefunction in modified gravitations
Peng Huang ∗ , Yue Huang , † Department of Information, Zhejiang Chinese Medical University,Hangzhou 310053, China Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences,Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences,No.19A Yuquan Road, Beijing 100049, China
Abstract
Based on Wheeler-DeWitt equation derived from general relativity, it had been foundthat only dark energy can lead to a normalizable cosmological wave function. It is shownin the present work that, for dRGT gravity, Eddington-inspired-Born-Infeld gravity andHoˇrava-Lifshitz gravity, the previous conclusion can also stand well in quantum cosmol-ogy induced from these modified gravities. This observation implies that there might bea universal relation between dark energy and normalizability of the cosmological wavefunction. ∗ [email protected] † [email protected] Introduction
The mysterious accelerating expansion of the current universe is of central concern both intheoretical and experimental aspects since its discovery [1,2]. The unknown reason that triggersthe accelerating expansion is dubbed as dark energy. Despite of nearly two decades of intensestudy, it is fair to say that the intrinsic nature of the dark energy is not well understood. Asatisfied solution to dark energy problem may need a complete knowledge of quantum gravitywhich is still under investigation. In this situation, trying new perspective into dark energyproblem is necessary.When expected to be an exotic energy component, dark energy can be well described withthe help of the holographic principle [3], a characteristic feature of any viable theory of quantumgravity. On the other hand, as an application of quantum physics to the classical cosmology,quantum cosmology is also tightly related to (or originates from) quantum gravity. Since bothdark energy and quantum cosmology have connections to quantum gravity, it is is well andnaturally motivated to investigate dark energy from the perspective of quantum cosmology.A trial along this line of thought was done in [4]. It was found that only dark energy (anexotic energy component) can give rise to a normalizable cosmological wave function. Thisresult indicates that, while being responsible for the classical accelerating expansion of theuniverse, dark energy can also lead to very different behaviors at the quantum level comparedwith ordinary energy components.What needs to be emphasized is that the previous conclusion—only dark energy can leadto a normalizable cosmological wave function—is obtained based on the hypothesis that darkenergy is an exotic energy components. However, it is completely possible that dark energycomes from a modification of general relativity on cosmological scale. The key point is, whileenergy component show itself in Wheeler-DeWitt (WD) equation as a potential term (functionof scale factor a multiply by cosmological wave function Ψ( a )), modified gravitation may changethe dynamical part (differential of the cosmological wave function with respect to a ) of WDequation besides introducing nontrivial exotic potential term. Therefore, even though that asuitable exotic energy or an appropriate modification of general relativity can achieve the samepurpose on explaining the accelerating expansion of the current universe, the WD equationsattached to this two different approach may differ from each other distinctly.Then comes the question: Does the conclusion that only dark energy (treated as an exoticenergy component) can lead to a normalizable cosmological wave function also stand wellwhen dark energy is an effect of modified gravitation? At first glance, the answer is morelikely to be negative since WD equations for these two different cases, as explained previously,could be very different to each other. However, it is shown in present work that, for some2epresentative modified gravitations, dark energy also leads to a normalizable cosmologicalwave function, which can be regarded as strong clue for the observation that dark energyleads to a normalizable cosmological wave function is a universal conclusion. This is the maincontent of the present work.The paper is organized as following: Sec.(2) is devoted to a brief review of work that hadbeen done in [4] for later convenience. In Sec.(3), three representative modified theories ofgravity are investigated in detail. One can see that, while WD equation induced from dRGTgravity has a very different dynamical part comparing to the ordinary one (derived from GR),new potential terms will be introduced in WD equations derived from Eddington-inspired-Born-Infeld gravity and Hoˇrava-Lifshitz (HL) gravity. However, in all the three representativemodified theories of gravity, the conclusion that only dark energy can lead to a normalizablecosmological wave function stands well. Dark energy is investigated from the perspective of quantum cosmology derived from generalrelativity in [4]. For a universe filled only with the cosmological constant, the Wheeler-DeWittequation is [ − ∂ ∂a − qa ∂∂a + 1 l p ( ka − a l )]Ψ( a ) = 0 (2.1)with k , l p = ( G ~ π ) , l Λ = ( ) corresponding to the spatial curvature, Planck scale andlength scale introduced by the cosmological constant. In this WD equation, energy componentscoming from spatial curvature and cosmological constant are reflected in term ka l p and a l p l respectively. Then, similar to quantum mechanics, the inner product of the cosmological wavefunction is defined as P = Z + ∞ a q | Ψ | da. (2.2)The integral P can be divided as P = Z ǫ a q | Ψ | da + Z Cǫ a q | Ψ | da + Z + ∞ C a q | Ψ | da, (2.3)the potentially singular behavior of the cosmological wave function comes from the first andthe third part. When P is finite, Ψ is said to be normalizable otherwise non-normalizable.Solving differential equation (2.1) and inserting its solution into (2.3), it can be found thatnormalizability of the cosmological wave function imposes strict constraints on the form ofWD equation: on one hand, the normal ordering ambiguity factor q must take value in thedomain of (-1, 3) to ensure no divergence in R ǫ a q | Ψ | da ; on the other hand, the existence of3osmological constant (the appearance of term a l p l ) is obligatory to ensure the convergence of R + ∞ C a q | Ψ | da .To see whether the present result has a more universal meaning, the WD equation (2.1) isgeneralized to the following form[ − ∂ ∂a − qa ∂∂a + 1 l p ( a − f ( h ) a h l h − p )]Ψ( a ) = 0 , (2.4)in which f ( h ) is a general function of parameter h and f (4) = ( l Λ l p ) is required. By exploringthe classical limit of Eq.(2.4), one can see that term f ( h ) a h l h − p in fact corresponds to a generalenergy component with index of equation of state having the value of w = − h . Apparently, h = 0 , , , a , the divergence in R ǫ a q | Ψ | da can always be avoided by choosing an appropriate normal ordering ambiguityfactor q in region ( − , a , only when h > w < − ) can there be nodivergence in R + ∞ C a q | Ψ | da .Through the above analysis, one has the following conclusion: equipped with an appropri-ate normal ordering ambiguity factor q , only dark energy can lead to a normalizable cosmo-logical wave function. One should notice that such conclusion is obtained on the assumptionthat dark energy is not an effect of modified gravitation but an exotic energy component. dRGT is a healthy massive gravity in which no ghosts exist [5, 6]. Suppose the metric has thefollowing form g µν = η µν + h µν = η ab ∂ µ φ a ( x ) ∂ ν φ b ( x ) + H µν , (3.1)then the action of dRGT is S g = Z d x √− g [ R − m U ( g, H )] (3.2)in which the first term comes from Einstein-Hilbert action and the second term manifests themodification caused by a small but nonzero mass of graviton. U ( g, H ) can be expressed as U ( g, H ) = − L + α L + α L ) (3.3)4ith L , L and L defined as following: L = 12 ( < K > − < K > ) , L = 16 ( < K > − < K >< K > +2 < K > ) , L = 124 ( < K > − < K > < K > +3 < K > +8 < K >< K > − < K > ) , (3.4)where K µν stands for K µν ( g, H ) ≡ δ µν − p η ab ∂ µ φ a ∂ ν φ b (3.5)and symbol < ... > means trace, i.e., < K > = g µν K µν , < K > = g ρσ g µν K ρµ K σν , ... Deriving WD equation from (3.2), a highly nontrivial task, had been carried out in [7]: " a − q ( − i ∂∂a + C ± m p | k | a )[ a q ( − i ∂∂a + C ± m p | k | a )] − | k | a − c ± m a Ψ( a ) = 0 (3.6)with q the normal ordering ambiguity factor, C ± and c ± standing for C ± = X ± (1 − X ± )[3 + 3 α (1 − X ± ) + α (1 − X ± ) ] (3.7) c ± = ( X ± − − X ± ) + α (1 − X ± )(4 − X ± ) + α (1 − X ± ) ] (3.8)with X ± ≡ α + α ± √ α + α − α α + α .To investigate the relation between energy components and normalization of cosmologicalwave function, introducing energy components into Eq.(3.6) is necessary. Since the couplebetween matter and gravity is not impacted by massive graviton in dRGT, one can safelybring energy components into this theory as following: (cid:20) a − q ( − i ∂∂a + C ± m p | k | a )[ a q ( − i ∂∂a + C ± m p | k | a )] − | k | a − c ± m a − ρ ( a ) a (cid:21) Ψ( a ) = 0 . (3.9)This will be the start point of our discussion.Comparing Eq.(3.9) with Eq.(2.4) in Sec.(2), it is apparent that the dynamical part ofWD equation has changed dramatically, which may make solving the WD equation to be adifficult problem. Fortunately, one can circumvent this problem in dRGT: the central concernof the present work is whether the innerproduct (2.2) is convergent, thus one has the freedomto rewrite the cosmological wave function Ψ( a ) as Ψ( a ) = e if ( a ) Φ( a ). Let f ( a ) = − i C ± m √ | k | a ,WD equation (3.9) turns into (cid:20) − a − q ( ∂∂a a q ∂∂a ) − | k | a − c ± m a − ρ ( a ) a (cid:21) Φ( a ) = 0 . (3.10) For present purpose, it suffices to content ourselves with the minimal introduction of dRGT, interestedreaders are referred to the original work in [5, 6]. The introduction of Eddington-inspired-Born-Infeld gravityand Hoˇrava-Lifshitz gravity in later sections will also be in their simplest forms.
5t can be inferred from this WD equation that the square of the mass of graviton offers equiv-alently a nonzero cosmological constant to trigger the accelerating expansion of the universe,a characteristic feature of massive gravity. Similar to the discussions shown in Sec.(2), one canfind that, for large a , the dominant term − c ± m a would lead to a normalizable cosmologicalwave function if there is no matter in the universe ( ρ ( a ) = 0); what’s more, in the limitationof m = 0, ρ ( a ) (an energy component) must be dark energy otherwise the cosmological wavefunction cannot be normalizable. In other words, the conclusion that only dark energy canlead to a normalizable cosmological wave function also stands in dRGT. Eddington-inspired-Born-Infeld gravity (EiBI) was proposed by Ba˜nados and Ferreira as acombination of insights from Einstein and Eddington [8]. It is defined in Palatini formalismas S EiBI [ g, Γ , Φ] = 2 κ (cid:20)q | g µν + κR µν (Γ) | − λ p | g | (cid:21) + S M [ g, Γ , Φ] (3.11)with λ a dimensionless parameter which cannot be zero because of self-consistency. DerivingWD equation from this action had been tactfully carried out in [9]. There the WD equation is (cid:20) b − q ( ∂∂b b q ∂∂b ) + 48 λ κ b + 24 κ ( λ + κρ ( a )) a b − − λ κ a b (cid:21) Ψ( b, a ) = 0 . (3.12)in which ρ = ρ a − ω ) and p are energy density and pressure, furthermore, a and b relate toeach other through ( λ + κρ )( λ − κp ) = λ b a . (3.13)One can find that, new terms are introduced in the EiBI potential λ κ b + κ ( λ + κρ ( a )) a b − − λ κ a b , which is more intricate than its counterpart in general relativity.When ρ = p = 0, WD equation (3.12) has a much more simpler form: (cid:20) b − q ( ∂∂b b q ∂∂b ) + 48 λ ( λ − κ b (cid:21) Ψ( b ) = 0 . (3.14)Comparing this equation with Eq.(2.1), one can see that the intrinsic accommodation of darkenergy in EiBI is reflected in the appearance of term λ ( λ − κ b : λ = 1 corresponds to theexistence of cosmological-constant-like dark energy, otherwise dark energy is eliminated in thistheory. In case of λ = 1, as shown in Sec.(2), the cosmological wave function Ψ( b ) of EiBI isnormalizable if the normal ordering ambiguity factor q takes value in domain of ( − , λ = 1, the WD equation is (cid:20) b − q ( ∂∂b b q ∂∂b ) (cid:21) Ψ( b ) = 0 . (3.15)6eneral solution of this equation isΨ( b ) = ( C b − q + C , q = 1 ,C ln b + C , q = 1 . (3.16)The corresponding integral P = R ∞ C b q | Ψ | db ( C >
0) is Z ∞ C b q | Ψ | db = Z ∞ C ( C b − q + 2 C C b + C b q ) db, q = 1 , Z ∞ C b [ C (ln b ) + 2 C C ln b + C ] db, q = 1 . (3.17)Apparently these integrals cannot be convergent. The conclusion is that, assuming the absenceof matter ( ρ = p = 0), the cosmological wave function cannot be normalized if there is no darkenergy.When ρ = 0, it is easy to obtain the following relations for large a from (3.13): ( b ∼ a − − ω , ω < − b ∼ a, ω ≥ − . (3.18)In case of ω ≥ − ρ ( b ) = ρ b − ω ) can be dark energy ( − > ω ≥ −
1) or ordinary matter( ω ≥ − ), the WD equation corresponds to this situation is (cid:20) b − q ( ∂∂b b q ∂∂b ) + 48 λ ( λ − κ b + 48 λρ ( b ) b + 24 κρ ( b ) b (cid:21) Ψ( b ) = 0 . (3.19)If λ = 1, the power of b in dominant potential term in (3.19) is definitely not smaller than4 which ensures the normalizability of the cosmological wave function. If λ = 1, the build-indark energy of EiBI is eliminated and the WD equation turns into (cid:20) b − q ( ∂∂b b q ∂∂b ) + c · b (1 − ω ) (cid:21) Ψ( b ) = 0 . (3.20)The normalization of wave function requires 1 − ω >
2, i.e., ω < − /
3. In case of ω < − ρ ( a ) must be dark energy. The WD equation is (cid:20) b − q ( ∂∂b b q ∂∂b ) + 48 λ κ b + 24 λ κ b − − ω + c λb − ω + c κb − ω − λ κ b − ω (cid:21) Ψ( b ) = 0 . (3.21)The potential in this equation is complicated and it is difficult to eliminated the intrinsic darkenergy introduced by EiBI. Thus there will always be dark energy in this situation.The power of b has four different values: 4, − − ω , − ω and 2 − ω . They cannotequal to each other when when ω < −
1. When b is large enough, b is the dominant term inthese terms. According to discussions shown in Sec.(2), one can see that the inner product ofthe wave function is convergent. Thus, for modified gravity EiBI, it is also only dark energycan lead to a normalizable cosmological wave function.7 .3 Ho ˇ rava-Lifshitz gravity An important motivation for dRGT, as well as EiBI, is to explain the accelerating expansionof the universe from the perspective of modified gravitation. Nevertheless, the central focus ofHoˇrava-Lifshitz (HL) gravity is to construct a theory of gravity that can be renormalizable [10],no special attention has been paid to dark energy problem. It is interesting to see whether theconclusion—only dark energy can lead to a normalizable cosmological wave function—standswell in HL gravity, a theory whose primary goal is unrelated to dark energy. As a preliminaryattempt, we will not consider the complicated coupling between matter and gravity in HLgravity .Starting with the projectable Hoˇrava-Lifshitz gravity without detailed balance, its actionis [12, 13] S HL = M pl Z d xdtN √ h [ K ij K ij − λK −
2Λ + R − g M − pl R − g M − pl R ij R ij − g M − pl R − g M − pl RR ij R ji − g M − pl R ij R jk R ki − g M − pl R ∇ R − g M − pl ∇ i R jk ∇ i R jk ] , (3.22)with N the shift function, h the determinant of the induced metric on spatial hypersurface, K ij the extrinsic curvature, and λ , g , ... g parameters of the theory. Quantum cosmologybased on this theory of gravity has been given in [14]. The corresponding WD equation is (cid:18) ∂ ∂a + qa ∂∂a − g C a + g Λ a + g r + g s a (cid:19) Ψ( a ) = 0 , (3.23)with g C , g Λ , g r and g s defined as g C = 23 λ − , g Λ = Λ M − pl π (3 λ − ,g r = 24 π (3 g + g ) , g s = 288 π (3 λ − g + 3 g + g ) . (3.24)Recalling discussions shown in Sec.(2), it is easy to figure out that, if the normal orderingambiguity factor q is chosen appropriately in the domain of ( − , g Λ = 0) is required to ensure the normalizability of the cosmological wave function. Investigating dark energy from the perspective of quantum cosmology, it had been found thatonly dark energy can lead to a normalizable cosmological wave function [4]. This conclusionis based on the assumption that dark energy is an exotic energy component. Nevertheless,at the classical level, the accelerating expansion of the universe can also be attributed to an See [11] for a clear overview on this subtle issue.
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