A Possible Solution of the Cosmological Constant Problem based on Minimal Length Uncertainty and GW170817 and PLANCK Observations
aa r X i v : . [ phy s i c s . g e n - ph ] M a y ECTP-2020-05WLCAPP-2020-05
A Possible Solution of the Cosmological Constant Problem based onMinimal Length Uncertainty and GW170817 and PLANCKObservations
Abdel Magied Diab ∗ Modern University for Technology and Information (MTI),Faculty of Engineering, 11571 Cairo, Egypt
Abdel Nasser Tawfik † Nile University, Egyptian Center for Theoretical Physics (ECTP),Juhayna Square of 26th-July-Corridor, 12588 Giza, Egypt (Dated: May 11, 2020)
Abstract
We propose the generalized uncertainty principle (GUP) with an additional term of quadratic mo-mentum motivated by string theory and black hole physics as a quantum mechanical framework for theminimal length uncertainty at the Planck scale. We demonstrate that the GUP parameter, β , couldbe best constrained by the the gravitational waves observations; GW170817 event. Also, we suggestanother proposal based on the modified dispersion relations (MDRs) in order to calculate the differ-ence between the group velocity of gravitons and that of photons. We conclude that the upper boundreads β ≃ . Utilizing features of the UV/IR correspondence and the obvious similarities betweenGUP (including non-gravitating and gravitating impacts on Heisenberg uncertainty principle) and thediscrepancy between the theoretical and the observed cosmological constant Λ (apparently manifest-ing gravitational influences on the vacuum energy density), known as catastrophe of non-gravitatingvacuum , we suggest a possible solution for this long-standing physical problem, Λ ≃ − GeV / ¯ h c . PACS numbers: 04.30.-w, 04.60.-m, 02.40.Gh, 98.80.EsKeywords: Gravitational waves, Quantum gravity, Noncommutative geometry, Observational cosmology ∗ Electronic address: [email protected] † Electronic address: tawfi[email protected] . INTRODUCTION The cosmological constant, Λ, an essential ingredient of the theory of general relativity (GR)[1], was guided by the idea that the evolution of the Universe should be static [2, 3]. This modelwas subsequently refuted and accordingly the Λ-term was abandoned from the Einstein fieldequation (EFE), especially after the confirmation of the celebrated Hubble obervations in 1929[4], which also have verified the consequences of Friedmann solutions for EFE with vanishing Λ[5]. Nearly immediate after publishing GR, a matter-free solution for EFE with finite Λ-termwas obtained by de Sitter [6]. Later on when it has been realised that the Einstein static
Universe was found unstable for small perturbations [7–9], it was argued that the inclusion ofthe Λ-term remarkably contributes to the stability and simultaniously supports the expansionof the Universe, especially that the initial singularity of Friedmann-Lemˆaitre-Robertson-Walker(FLRW) models could be improved, as well [10, 11]. Furthermore, the observations of type-Iahigh redshift supernovae in late ninteeth of the last century [12, 13] indicated that the expandingUniverse is also accelerating, especially at a small Λ-value, which obviously contributes to thecosmic negative pressure [14, 15]. With this regard, we recall that the cosmological constantcan be related to the vacuum energy density, ρ , as Λ = 8 πGρ/c , where c is the speed oflight in vacuum and G is the gravitational constant. In 2018, the PLANCK observations haveprovided us with a precise estimation of Λ, namely ΛPlanck ≃ − GeV / ¯ h c [16]. Whencomparing this tiny value with the theoretical estimation based on quantum field theory inweakly- or non-gravitating vacuum, ΛQFT ≃ GeV / ¯ h c , there is, at least, a 121-orders-of-magnitude-difference to be fixed [17–19].The disagreement between both values is one of the greatest mysteries in physics and knownas the cosmological constant problem or catastrophe of non-gravitating vacuum . Here, wepresent an attempt to solve this problem. To this end, we utilize the generalized uncertaintyprinciple (GUP), which is an extended version of Heisenberg uncertainty principle (HUP), wherea correction term encompassing the gravitational impacts is added, and thus an alternativequantum gravity approach emerges [20, 21]. To summarize, the present attempt is motivatedby the similarity of GUP (including non-gravitating and gravitating impacts on HUP) and thedisagreement between theoretical and observed estimations for Λ (manifesting gravitationalinfluences on the vacuum energy density) and by the remarkable impacts of Λ on early andlate evolution of the Universe [2, 3, 22]. So far, there are various quantum gravity approaches2resenting quantum descriptions for different physical phenomena in presence of gravitationalfields to be achnowledged, here [20, 21].The GUP offers a quantum mechanical framework for a potential minimal length uncertaintyin terms of the Planck scale [23–26]. The minimal length uncertainty, as proposed by GUP,exhibits some features of the UV/IR correspondence [27–29], which has been performed inviewpoint of local quantum field theory. Thus, it is argued that the UV/IR correspondence isrelevant to revealing several aspects of short-distance physics, such as, the cosmological constantproblem [18, 30–32]. Therefore, a precise estimation of the minimal length uncertainty stronglydepends on the proposed upper bound of the GUP parameter, β [25, 33].Various ratings for the upper bound of β have been proposed, for example, by comparingquantum gravity corrections to various quantum phenomena with electroweak [34, 35] andastronomical [36, 37] observations. Accordingly, β ranges between 10 to 10 [36–38]. As apreamble of the present study, we present a novel estimation for β from the binary neutronstars merger, the gravitational wave event GW170817 reported by the Laser InterferometerGravitational-Wave Observatory (LIGO) and the Advanced Virgo collaborations [39]. Withthis regard, there are different efforts based on the features of the UV/IR correspondence inorder to interpret the Λ problem [40–44] with Liouville theorem in the classical limit [40, 45, 46].Having a novel estimation of β , a solution of the Λ problem, catastrophe of non-gravitatingvacuum , could be best proposed.The present paper is organized as follows. Section II reviews the basic concepts of the GUPapproach with quadratic momentum. The associated modifications of the energy-momentumdispersion relations related to GR and rainbow gravity are also outlined in this section. Insection III, we show that the dimensionless GUP parameter, β o , could be, for instance, con-strained to the gravitational wave event GW170817. Section IV is devoted to calculating thevacuum energy density of states and shows how this contributes to understanding the cosmo-logical constant problem with an quantum gravity approach, the GUP. The final conclusionsare outlined in section V. 3 I. GENERALIZED UNCERTAINTY PRINCIPLE AND MODIFIED DISPERSIONRELATIONS
Several approaches to the quantum gravity, such as GUP, predict a minimal length un-certainties that could be related to the Planck scale [20, 21]. There were various laboratoryexperiments conducted to examine the GUP effects [47–50]. In this section, we focus the dis-cussion on GUP with a quadratic momentum uncertainty [20, 21]. This version of GUP wasobtained from black hole physics [51] and supported by gedanken experiments [52], which havebeen proposed Kempf, Mangano, and Mann (KMM), [53]∆ x ∆ p ≥ ¯ h (cid:2) β (∆ p ) (cid:3) , (1)where ∆ x and ∆ p are the uncertainties in position and momentum, respectively. The GUPparameter can be exressed as β = β ( ℓ p / ¯ h ) = β / ( M p c ) , where β is a dimensionless param-eter, ℓ p = 1 . × − GeV − is the Planck length, and M p = 1 . × GeV /c is thePlanck mass. Equation (1) implies the existence of a minimum length uncertainty, which isrelated to the Planck scale, ∆ x min ≈ ¯ h √ β = ℓ p √ β . It should be noticed that the minimumlength uncertainty exhibits features of the UV/IR correspondence [27–29]. ∆ x is obviouslyproportional to ∆ p , where large ∆ p (UV) becomes proportional to large ∆ x (IR). Equation (1)is a noncommutative relation; [ˆ x i , ˆ p j ] = δ ij i ¯ h [1 + βp ], where both position and momentumoperators can be defined as ˆ x i = ˆ x i , ˆ p j = ˆ p j (1 + βp ) , (2)where ˆ x i and ˆ p j are corresponding operators obtained from the canonical commutation rela-tions [ˆ x i , ˆ p j ] = δ ij i ¯ h, and p = g ij p i p j .We can now construct the modified dispersion relation (MDR) due to quadratic GUP. Westart with the background metric in GR gravitational spacetime ds = g µν dx µ dx ν = g c dt + g ij dx i dx j , (3)with g µν is the Minkowski spacetime metric tensor ( − , + , + , +). Accordingly, the modifiedfour-momentum squared is given by p µ p µ = g µµ p µ p µ = g ( p ) + g ij p i p j (1 + βp )= − ( p ) + p + 2 β p · p . (4)4omparing this with the conventional dispersion relation, p µ p µ = − m c , the time componentof the momentum can then be written as( p ) = m c + p (1 + βp ) . (5)The energy of the particle ω can be defined as ω/c = − ζ µ p µ = − g µν ζ µ p ν , where the killingvector is given as ζ µ = (1 , , , ω = − g c ( p ) = c ( p ) and the modified dispersion relation in GR gravity reads ω = m c + p c (1 + 2 βp ) . GR Gravity (6)For β →
0, the standard dispersion can be obtained.The rainbow gravity generalizes the MDR in doubly special relativity to curved spacetime[54], where the geometry spacetime is explored by a test particle with energy ω [55, 56], ω f (cid:18) ωω p (cid:19) − ( pc ) f (cid:18) ωω p (cid:19) = (cid:0) mc (cid:1) , (7)where ω p is the Planck energy and f ( ω/ω p ) and f ( ω/ω p ) are known as the rainbow functionswhich are model-depending. The rainbow functions can be defined as [57, 58], f ( ω/ω p ) = 1 , f ( ω/ω p ) = q − η ( ω/ω p ) n , (8)where η and n are free positive parameters. It was argued that for the logarithmic correctionsof black hole entropy [59], the integer n is limited as n = 1 , n = 2. Thus, the MDR for rainbow gravity with GUP can be writtenas, ω = ( mc ) + p c (1 + 2 βp )1 + η h pcω p i (1 + 2 βp ) . Rainbow Gravity (9)Again, as β →
0, Eq. (9) goes back to the standard dispersion relation.We have constructed two different MDRs for quadratic GUP, namely Eqs (6) and (9) inGR and rainbow gravity, respectively. Bounds on GUP parameter from GW170817 shall beoutlined in the section that follows.
III. BOUNDS ON GUP PARAMETER FROM GW170817
Instead of violating Lorentz invariance [61], we intend to investigate the speed of the gravitonfrom the GW170817 event. To this end, we use MDRs obtained from the quadratic GUP5pproaches, section II. Thus, defining an upper bound on the dimensionless GUP parameter β for given bounds on mass and energy of the graviton, where m g < ∼ . × − eV /c and ω = 8 . × − eV, respectively, plays an essential role. Assuming that the gravitational wavespropagate as free waves, we could, therefore, determine the speed of the mediator, that of thegraviton, from the group velocity of the accompanying wavefront, i.e. v g = ∂ω/∂p , where ω and p are the energy and momentum of the graviton, respectively [62]. The idea is that thegroup velocity of the graviton can be simply deduced from the MDRs, Eqs. (6) for the GRgravity and (9) and the rainbow gravity, in presence and then in absence of the GUP impacts,which have been discussed in section II. Accordingly, Eq. (6) implies that the group velocityreads v g = ∂ω∂p = pc ω (cid:0) βp (cid:1) . (10)The unmodified momentum p in terms of the modified parameters up to O ( β ), can beexpressed as p = a + bβ , where a and b are arbitrary parameters. By substituting this expressioninto Eq. (6), we find that p = ( ω g /c ) − m c . Thus, Eq. (10) can be rewritten as v g = c nh − (cid:16) mc ω g (cid:17) i / + 4 β ω g c h − (cid:16) mc ω g (cid:17) i / o , (11)where ω g is the energy of the graviton. It is obvious that for β →
0, i.e. in absence of GUPimpacts, the group velocity reads v g = c h − (cid:16) mc ω g (cid:17) i . (12)Then, the difference between the speed of photon (light) and that of graviton without GUPimpacts is given as (cid:12)(cid:12)(cid:12) δv (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) c − v g (cid:12)(cid:12)(cid:12) = c (cid:12)(cid:12)(cid:12) (cid:16) mc ω g (cid:17) (cid:12)(cid:12)(cid:12) < ∼ . × − c. (13)Although the small difference obtained, we are - in the gravitational waves epoch - technicallyable to measure even a such tiny difference! In light of this, we could use the results associatedwith the GW170817 event, such as the graviton velocity, in order to set an upper bound on theGUP parameter, β .For a massless graviton, the difference between the speed of photons (light) and that of thegravitons in presence of the GUP impacts reads (cid:12)(cid:12)(cid:12) δv GUP (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) β ω c (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) β (cid:16) ω M p c (cid:17) (cid:12)(cid:12)(cid:12) < ∼ . × − β c. (14)6hus, the upper bound on the dimensionless parameter, β , of the quadratic GUP can be simplydeduced from Eqs. (13) and (14), β < ∼ . × . (15)The group velocity of the graviton due to MDR and rainbow gravity when applying thequadratic GUP approach, Eq. (9), can be expressed as v g = ∂ω∂p = (cid:16) pc ω g (cid:17) (cid:16) − ηω p ( mc ) (cid:17)(cid:16) βp (cid:17)(cid:20) η (cid:16) cpω p (cid:17) (1 + 2 βp ) (cid:21) . (16)Similarly, one can for a massless graviton express the conventional momentum in terms of theGUP parameter. In order of O ( β ), we get cp = ω g h(cid:16) − η (cid:16) ω g ω p (cid:17) (cid:17) − / − β ω g c (cid:16) − η (cid:16) ω g ω p (cid:17) (cid:17) − / i . (17)The unmodified momentum can be expressed in GUP-terms up to O ( β ); p = a + a β , where a and a are arbitrary parameters. Nevertheless, the investigation of the speed of the gravitonfrom the GW150914 observations [63] specifies the rainbow gravity parameter, η ( ω g /ω p ) ≤ . × − [64]. Accordingly, Eq. (17) can be reduced to cp = ω g (1 − βω g /c ) and the groupvelocity of the massless graviton becomes v g = c h − βω c + O ( β ) i . (18)Then, the difference between the speed of photons and that of the gravitons reads (cid:12)(cid:12)(cid:12) δv GUP (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) β ω c (cid:12)(cid:12)(cid:12) < ∼ . × − β c. (19)By comparing Eqs. (19) and (13), the upper bound of the GUP parameter β can be estimatedas β < ∼ . × . (20)It is obvious that both results, Eqs. (15) and (20), are very close to each other; β < ∼ .The improved upper bound of β is very similar to the ones reported in refs. [36, 37], which - aswell - are depending on astronomical observations. The present results are based on mergers ofspinning neutron stars. Thus, it is believed that more accurate observations, the more preciseshall be β .Having set a upper bound on the GUP parameter and counting on the spoken similaritiesbetween GUP and the catastrophe of non-gravitating vacuum, we can now propose a possiblesolution of the cosmological constant problem.7 V. A POSSIBLE SOLUTION OF THE COSMOLOGICAL CONSTANT PROBLEM
The cosmological constant can be given as Λ = 3 H Ω Λ , where H and Ω Λ are the Hubbleparameter and the dark energy density, respectively [65]. On the other hand, the origin of thecatastrophe of non-gravitating vacuum would be understood from the disproportion of the valueof Λ in the theoretical calculations, while this is apparently impacting the GW observations[66]. From the most updated PLANCK observations, the values of Ω Λ = 0 . ± . H = 67 . ± .
42 Km · s − · Mpc − [16]. Then, the vacuum energy density c πG Λ = (cid:18) H c πG (cid:19) Ω Λ = 3¯ hc πℓ p ℓ Ω Λ , (21)where the scale of the visible light, ℓ = c/H ≃ . × Km [16]. Therefore, one can useEq. (21) to esiamte the vacuum energy density in order of 10 − GeV / (¯ h c ). In quantumfield theory, the cosmological constant is to be calculated from sum over the vacuum fluctuationenergies corresponding to all particle momentum states [65]. For a massless particle, we obtain1(2 π ¯ h ) Z d ~p (¯ hω p / ≃ . × GeV / (¯ h c ) . (22)This is clearly infinite integral. But, it is usually cut off, at the Planck scale, µ p = ¯ h/ℓ p . Weassume ω p is the vacuum energy of quantum harmonic state ¯ hω p = [ p c + m g c ] / .To propose a possible solution of the cosmological constant problem, it is initially needed todetermine the number of states in the phase space volume taking into account GUP, Eq. (1).An analogy can be found in Liouville theorem in the classical limit. We need to make sure thatthe size of each quantum mechanical state in phase space volume is depending on the modifiedmomentum p , especially when taking GUP into consideration, Eq. (1). In other words, thenumber of quantum states in the phase space volume is assumed not depending on time.In the classical limit, the relation of the quantum commutation relations and the Poissonbrackets is given as [ ˆ A, ˆ B ] = i ¯ h { A, B } . Details on the Poisson bracket in D-Dimensions are out-lined in appendix A. Consequently, the modified density of states implies different implicationson quantum field theory, such as, the cosmological constant problem.In D-dimensional spherical coordinate systems, the density of states in momentum space isgiven as [40, 45, 46] V d D ~p (1 + βp ) D , (23)8here V is the volume of space. It should be noticed that in quantum mechanics, the numberof quantum stated per unit volume is given as V / (2 π ¯ h ) D . Therefore, for Liouville theorem, theweight factor in 3-D dimension reads [40, 45, 46] (review appendix A)1(2 π ¯ h ) d ~p (1 + βp ) . (24)In quantum field theory, the modification in the quantum number of state of the phase spacevolume should have consequences on different quantum phenomena, such as, the cosmologicalconstant problem and the black body radiation. At finite weight factor of GUP, the sum overall momentum states per unit volume of the phase space modifies the vacuum energy density.The cosmological constant, on the other hand, is determined by summing over the vacuumfluctuations, the energies, corresponding to a particular momentum stateΛGUP( m ) = 1(2 π ¯ h ) Z d ~pρ ( p )(¯ hω p /
2) = 12(2 π ¯ h ) Z d ~p (1 + βp ) q p c + m g c (25)For a massless particle, the vacuum energy density, which is directly related to Λ, readsΛGUP( m = 0) = c π ¯ h Z p (1 + βp ) dp = c ( M p c ) π ¯ h β = 1 . × − GeV / (¯ h c ) . (26)The agreement between the observed value of the cosmological constant, Λ ≃ − GeV / ¯ h c ,and our calculations based on quantum gravity approach, Eq. (26), is very convincing. Weconclude that the connection between the estimated upper bound on β , Eqs. (19) and (13),from GW170817 event [39] and the most updated observations of the PLANCK collaboration[16] for the cosmological constant Λ, Eq. (22), and our estimated value of Λ( m = 0), Eq. (26),gives an interpretation for the cosmological constant problem in presence of the minimal lengthuncertainty. V. CONCLUSIONS
In the present study, we have proposed the generalized uncertainty principle (GUP) withan addition term of quadratic momentum, from which we have driven the modified dispersionrelations for GR and rainbow gravity, Eq. (6) and Eq. (9), respectively. Counting on thesimilarities between GUP (manifesting gravitational impacts on HUP) and the likely origin ofthe great discrepancy between the theoretical and observed values of the cosmological constantthat in the gravitational impacts on the vacuum energy density, the present study suggests9 possible solution for the long-standing cosmological constant problem ( catastrophe of non-gravitating vacuum ) that Λ ≃ − GeV / ¯ h c .We have assumed that the gravitational waves propagate as a free wave. Therefore, wecould drive the group velocity in terms of the GUP parameter β for GR and rainbow gravity,Eq. (15) and Eq. (20), respectively. Moreover, we have used recent results on gravitationalwaves, the binary neutron stars merger, GW170817 event, in order to determine the speed ofthe gravitons. Then, we have calculated the difference between the speed of gravitons andthat of (photons) light, at finite and visnishing GUP parameter. We have shown that theupper bound on the dimensionless GUP parameter, β ∼ , is merely constrained by such aspeed difference. We have concluded that the speed of graviton is directly related to the GUPapproach utilized in.The cosmological constant problem, which is stemming from the large discrepancy betweenthe QFT-based calculations and the cosmological observations, is tagged as Λ QF T / Λ exp ∼ .This quite large ratio can be interpreted by features of the UV/IR correspondence and theimpacts of gravity. For the earlier, the large ∆ x (IR) corresponds to a large ∆ p (UV) inscale of Planck momentum. For the later, the GUP approach, for instance, Eq. (1), plays anessential role. We have assumed that in calculating the density of states where GUP approachis taken into account, a possible solution of the cosmological constant problem, Eq. (24), canbe proposed. At Planck scale, the resulting density of the states seems to impact the vacuumenergy density of each quantum state, Eq. (26). A refined value of the cosmological constantwe have obtained for a novel upper bound on β , which - in turn - was determined from theGW170817 observations. Finally, the possible matching between the estimation of the upperbound on the GUP parameter deduced from the gravitational waves, GW170817 event, and theone estimated from the PLANCK 2018 observations seems to support the conclusion about thegreat importance of constructing a theory for quantum gravity. This likely helps in explainingvarious still-mysterious phenomena in physics.10 ppendix A: Algebra of quantum mechanical commutators and Poisson brackets For a binary set of anticommutative functions on position and momentum, for instance, inD-dimensions, the Poisson bracket expresses their binary operation { F ( x , · · · x D ; p , · · · p D ) , G ( x , · · · x D ; p , · · · p D ) } = (cid:18) ∂F∂x i ∂G∂p j − ∂F∂p i ∂G∂x j (cid:19) { x i , p j } + ∂F∂x i ∂G∂x j { x i , x j } . (A1)During a time duration, δt , the Hamilton’s equations of motion for position and momentumcan be given as x ′ i = x i + δx i , p ′ i = p i + δp i , (A2)where, δx i , = { x i , H } δt = { x i , p j } ∂H∂p j + { x i , x j } Hxj , (A3) δp i , = { p i , H } δt = −{ x i , p j } ∂H∂x j , (A4)where H ≡ H ( x, p ; t ) is the Hamiltonian, itself.The estimation of the change in the phase space volume during the time evolutionrequires to determine the Jacobain of the transformation from ( x , · · · x D ; p , · · · p D ) to( x ′ , · · · x ′ D ; p ′ , · · · p ′ D ), i.e. d D x ′ d D p ′ = d D x d D p J , (A5)where J is the Jacobain of the transformation, which can be expressed as J = (cid:13)(cid:13)(cid:13) ∂ ( x ′ , · · · x ′ D ; p ′ , · · · p ′ D ) ∂ ( x , · · · x D ; p , · · · p D ) (cid:13)(cid:13)(cid:13) = 1 + (cid:18) ∂∂x i ∂ ( δx i ) ∂t + ∂∂p i ∂ ( δp i ) ∂t (cid:19) × δt. 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