A Possible Solution to the Cosmological Constant Problem By Discrete Space-time Hypothesis

Transcription

1 A Possible Solution to te Cosological Constant Proble By Discrete Space-tie Hypotesis H.M.Mok Radiation Healt Unit, 3/F., Saiwano Healt Centre, Hong Kong SAR Got, 28 Tai Hong St., Saiwano, Hong Kong, Cina. Abstract Te cosological constant proble is explained by a teory based on te discrete space-tie ypotesis. Te calculated λalue is of te order of [ ] or equialent to about Ω λ =0.. It is in excellent agreeent wit te Type Ia SN obserational data and recent results of BOOMERANG and MAXIMA. Our teory also iplies tat te quantization of te space-tie etric g µν is not necessary since it is not a fundaental field. Te diergence proble of quantu graity is ten of no interest. Cosic inflation is gien out as a consequence of te teory and te unierse is found to be alternatiely doinated by te cosological constant and te ass density at different cosic tie period. Our calculation also sows tat ρ is of siilar order of agnitude as ρ in te present unierse but it is just a coincidence. Tis result supports te antropic principle. PACS nubers: k, Cq, Te international collaboration on te Hig-Z SN Ia obseration, wic was aied at easuring te cosic deceleration and global curature, found tat te unierse is accelerating instead of decelerating. Te obserational results fro bot teas, Perlutter [1] and Scidt [2], indicated tat tere is a non-anising cosological constant ( λ) in 2 2 our unierse. Te alue of Ω ( λc / 3H ) is a few tents of te critical ass density and, Λ 0

2 wen copared wit te Planck scale or electroweak scale, is any order of agnitudes saller tan tat expected in quantu field teory. Recent obserations by te BOOMERANG and MAXIMA also support suc finding [3]. Tis cosological constant proble is one of te ysteries of bot cosology and particle field teory. Viable approac, suc as quintessence, antropic principle and iger diensional brane world solution are under inestigation (Recent concise coents on suc approaces can be found on [4]). Howeer, up to now, tere is still no satisfactory prediction on te λalue using suc attepts. On te oter and, te teoretical estiate of te λ alue by te quantu field teory is based on te spontaneous syetry breaking process proided by te Higgs ecanis []. Altoug suc teory is successful in ost of te particle experients, it is an ad oc ecanis and lacks detail understanding. For exaple, te reason of existence of a scalar field in te acuu is still not known. Te present situation on te λ proble urges us to consider bot probles togeter in a new direction. Altoug it is strange tat tere exists so large discrepancy between te teoretical and experiental λ alue, we ay not be too unfailiar wit suc situation. If we copare te λ proble wit te property of ass density, siilar caracteristic can be found. Due to te atoic structure of atter, ass density is relatiely sall in acroscopic scale (around kg / for coon aterials) but extreely large inside te atoic nucleus (around kg / ). Te case for te cosological constant, also, is ery sall in acroscopic scale but extreely large wen quantu field consideration (i.e. icroscopic scale) as been put in. If te λ proble is analogous to te case of atter density, it indicates tat soe kind of discrete structure ay exist in te acuu. Since te acuu and space-tie are indistinguisable, suc discrete acuu properties ay furter iply tat te space-tie itself is also discrete in nature. Besides, tere is no eidence tat space-tie is soot and continuous in extree icroscopic scale. In quantu graity, space-tie is expected to ae a ery different geoetrical structure in Planck scale, suc as te existence of woroles and space-tie foas. Also, te concept of discrete space-tie is not new. T.D.Lee [6], G.t Hooft [] and oters [8] ad considered suc possibility in resoling te UV/IR diergence proble in quantu graity but tey ad not related it to te proble of te cosological constant at tat tie.

3 If space-tie is really discrete in nature, te cosological constant described by te quantu field teory can be just te situation inside te basic constituents of space-tie and is any order of agnitudes larger tan te acroscopic obserational data. Te quantu field teory description and te cosological obserations of te acuu energy density ay ten bot be correct on its corresponding lengt scales. Furterore, te spontaneous syetry breaking of te scalar field ay be corresponding to te pase transition of suc space-tie condensate like structure. Let us sow te aboe idea in a paradig. If we iagine our 3-D space as a 2-D elastic ebrane, wic is coonly use for te illustration of cosic expansion, te discrete space-tie is corresponding to a ebrane wic is not soot but as its atoic structure (in fact, a pysical ebrane is ade of atos). Te graitational field for te 2-D creatures liing on te ebrane is te properties of deforing te ebrane by ass and general relatiity is ten a kind of continuous teory of elasticity. Tey ay find tat te space is soot in acroscopic scale but as its icroscopic structure. Also, te internal energy of te ebrane (It ay be iewed as te acuu energy by te 2- D creatures.) is extreely large in icroscopic scale (i.e. te nuclear energy or te atoic bonding energy) but sall in acroscopic scale and suc internal energy of is space does not cure te space-tie structure. Tis siple odel gies te properties of our cosological constant. Based on te aboe arguents, we postulate tat : (1) Space-tie is discrete in nature and its fundaental unit is of te order of Planck scale; (2) Te space-tie fors a kind of pase (or say condensate ) wit its constituents; (3) Te scalar field plays bot te role as te order paraeter of suc space-tie pase and te waefunction of its constituents as Cooper pairs in superconductiity [9,10] (We ae to reark tat suc condensate ay not be exactly te sae as in usual understanding but as siilar properties tat is useful to draw analogy between te). Soe iportant consequences can be directly followed fro tese postulates. Firstly, te Higgs particle will be just a kind of excited state of te indiidual space-tie constituent. Secondly, te diergence proble of te quantu graity is of no interest since te space-tie etric g µν is a

4 collectie effect of te space-tie constituents (like te strain tensor of elasticity of a ebrane) and is not a fundaental field itself. General Relatiity is also not a fundaental field teory but is just a collectie description. Using te language of te paradig described aboe, tere is no need to insist in quantizing te wae propagation of te strain tensor on a 2-D ebrane since te atoic lattice ibration is te one tat needed to be quantized (ponons). Back to our case, it is te field of te space-tie constituents (or say te scalar field) and its lattice like ibration energy tat need to be quantized. Terefore, te proble of quantu graity is reduced to te quantization of te ibration energy of te space-tie constituents and in soe sense we can say tat te quantization of Higgs field is already part of a quantu graity teory! It sees tat te concept in condensed atter pysics is useful in building te quantized odel of te space-tie structure. If we take te for of te scalar field potential as V = V g (1) µ φ φ + ( φ φ) were µ 2 > 0, g > 0 and assuing tat V 0 (Tey as its usual eaning as in te 0 = electroweak teory. SM Higgs is assued ere for siplicity and also to aoid unnecessary coplexity). Te energy density at broken syetry ρ (wic will be sown to be different fro te acroscopic acuu energy density in cosological obseration) of te scalar field is gien by 4 ρ = [], were is te electroweak ass scale. Since te sape of te potential around te iniu φ 0 deterines te ass of te quantu particle and, in our case, it is equal to te Higgs ass, te energy density ρ expression can ten be interpreted as te contribution of te scalar field of ass wit nuber density icroscopic acuu energy density wit alue 3. Te negatie sign of ρ akes it beaes as te 4 (We use te sign conention tat +e acuu energy density corresponding to e noral energy density.). Also, because te scalar field is te waefunction of te space-tie constituents, ρ acts as a kind of

5 binding energy (or internal energy) of te constituents and, as entioned aboe, suc binding energy will not cure te space-tie since it beaes as te internal energy of te space-tie structure. If pase transition appens on te space-tie condensate, tis binding energy can be released as positie energy and is iportant in early unierse eolution tat will be discussed later. Due to te energy density constituents and wit te alue equals to ρ, pressure will be created on eac space-tie P = ρ = (2) 4 Te sign on te RHS of equation (2) sows tat te pressure is negatie (i.e. beae as a stretcing force on indiidual space-tie constituent but as a binding force between te space-tie constituents.). Tis alue is wat we expect fro quantu field teory []. Howeer, as discussed aboe, te space-tie is postulated to be discrete and its fundaental unit is of te order of Planck scale wit estiated density of about 3. Terefore, te acroscopic acuu energy density sould be weiged by a factor of ( M 3 / ) to aerage it out on an ideally soot acroscopic space scale (just as aerage out te nuclear ass density to get te acroscopic ass density of atter by considering te atoic spacing), were M is te Planck ass. Te acroscopic acuu energy density ρ ν will becoe 3 4 ρ = = (3) 3 M M Te cosological constant is ten equal to λ = κρ ~ (4) M

6 (Tis expression was first appeared in te autor s 1999 e-print paper [11] in te attepts on te explanation of te cosological constant proble by te electroweak and Planck scale.). If we put to about 100 GeV, te alue of λ is λ ~ 10 [ GeV] = 10 [ ] () (we use GeV ~ 10 ). If we use a diensionless Hubble constant = 0. 6 (recent obserational results gie te Hubble constant in te range [12] ), our calculated cosological constant will be equialent to about 0.ρc ( ρc = 3H 2 0 / 8πG is te critical ass density wen λ = 0 ). It is in excellent agreeent wit te Type Ia SN obseration data. Besides te explanation of te present λ alue, tis teory as iportant iplications on te early deelopent of te unierse. We can first put equation (4) into a ore general for as λ = κρ ~ T (6) M were T is te VEV ass scale of te scalar field in different transition stage in te early unierse. Te λ alue for te 3 transition stages, te Planck stage, GUT stage and te electroweak stage were ten equal to 2 M, GUT / M and / M respectiely. If we beliee tat te ass energy density (including radiation and atter) of te unierse is cae fro te cange of te acroscopic acuu energy density, we expect tat at te oent just after te GUT and te electroweak transition te atter density were as and 3 GUT / M respectiely. One of te point reains uncertain and cannot be gien by tis teory is te ass energy density at Planck tie. If it was coparable to te acuu energy at tat tie, te λ effect would doinated te ass energy effect at about M s (estiated by Friedann odel) and te unierse would be fast cooled to GUT transition

7 teperature by te accelerated expansion due to te cosological constant. Tis ade te GUT transition stages cae earlier tan in te ot Friedann unierse (at 3 10 s ). λ = GUT After GUT transition, te ass energy density would be equal to / M 4 M and. Te unierse was ten doinated by ass and continuous cooling by te usual Friedann expansion. Assuing tat te unierse was radiation doinated and te relation between te cosic teperature and tie was as T 1/ 2 t (we neglect te cosological constant effect before it doinated te expansion for siplicity), we expect tat te unierse would be doinated by te cosological constant at te tie around s (Tis is estiated by te calculation tat te ass energy density is greater tan 42 acuu energy density by 28 order of agnitude at around 10 s and tis corresponding to te cange of teperature of about order of agnitude. Terefore, te λ would be doinated at s = 10 s.). Te unierse would ten be fast cooled by te accelerated expansion in a sort tie. At te tie just before te electroweak transition, te λ alue was ten about 40 order of agnitude larger tan te ass energy density and 91 order of agnitude larger tan te present λ alue. Suc uge cosological constant effect igt cause an extree large expansion of te unierse in tat period. Tis is wat we coonly called te inflation! Altoug it is also drien by a uge λ alue, te acceleration process is not due to te false acuu as oter inflation odels [9] but is an intrinsic property of te space-tie condensate. As in te aboe estiated tie of transition, we ae not consider te additional acceleration effect by te λ before it becoe doinated. It is ten reasonable to expect tat te inflation also occurred at around te tie order s. One can find tat 1/ 2 t ~ 1 λ at tat tie. Te inflation would be ended at te copletion of te electroweak transition. Tis eans tat it also cae earlier tan te expected tie of about s in te ot Friedann unierse. As in te usual inflation odel, te inflation also occurred in te period between te GUT and electroweak transition but te cosic tie was different.

8 After te electroweak transition, te ass energy becae 3 GUT / M and λ = / M. Te cosological constant was dropped to te present alue, te inflation ten stopped and te unierse becae atter doinated again. Wen te atter density of te unierse continuously decreased by te expansion of te unierse, te λ alue becoes doinate again as te present cosological obseration [1,2]. If te radiation doinated unierse ended at around s and enter te atter doinate stage, te dilution factor for te ass energy density due to te cosic expansion fro s contribute about GUT was about / M Te additional factor in te atter stage up to now Tat eans te estiated present ass energy density is = 10 / M = 10λ. It is pretty close to te obseration tat te ass energy density about te sae order as te acuu energy density in present unierse. Te estiation is a bit large and ay due to uncertainty in estiating te ending tie of te electroweak transition. It is because only alf an order of agnitude cange can cause suc deiation. If we beliee tat tere is no furter pase transition of te space-tie condensate in future, our unierse will ten doinate by λ alue foreer. Up to now, we know tat te unierse eolution stages starting fro Big Bang igt be alternatiely doinated by cosological constant and atter density in different transition stages and te cosological constant will win te process finally. Also, fro te ass energy density estiation aboe, it sees tat ρ ~ ρ in te present unierse ay be just a coincidence, not due to underlying pysical teory. Tis result supports te antropic principle []. Fro te aboe teory, we find tat te cosological constant proble can be resoled by postulating tat space-tie is discrete in nature wit its pase transition properties described by te scalar field. Te cosological constant calculated by our teory is in excellent agreeent wit te Type Ia SN obseration data. Since te λ alue is dependent on te VEV of te scalar field and terefore it is closely related to te pase transition of te space-tie condensate. Te eolution of te unierse including te Big Bang igt be a series of pase transition of tis condensate. Our teory autoatically gies out te inflation process in te early unierse but te stage of acceleration is

9 different fro oter odels of inflation. It explains te cosological constant proble and te inflation ecanis togeter in a single siple teory. Te eolution of te unierse is found to be alternatiely doinated by te cosological constant and te ass density at different transition stages. Our calculation sows tat ρ ~ ρ in te present unierse. It is ten just a coincidence, not due to underlying pysical teory. Proided tat no furter pase transition will occur, our teory predicts tat te unierse will be doinated by te λ alue foreer. One ay find tat all te aboe results is not acieed by fine tuning paraeters but follows autoatically fro our postulates. In our teory, te scalar field also becoe ore pysical tan just an unknown acuu potential but is te pase paraeter and waefunction of te space-tie condensate. Te diergence proble of quantu graity can be autoatically soled because te space-tie itself is not a fundaental field but a collectie effect of a ore fundaental Higgs process. Te aboe arguents sows tat te space-tie structure can be a ore coplicated structure ten just a continuous ateatical space so tat an eolution on te space-tie concept is terefore necessary. Reference [1] S.Perlutter et al., Astropys. J. 1, 6 (1999) [2] B.P.Scidt et al., Astropys. J 0, 46 (1998) [3] J.R.Bond et al., astro-p/ [4] S.Weinberg, astro-p/ [] S.Weinberg, Re. Mod. Pys (1989) [6] T.D.Lee, Pys. Lett. 122B, 21 (1983)

10 [] G.t Hooft, Recent Deelopent in Graitation, edited by M.Ley & S.Denser, (Plenu, New York, 199); Under te Spell of te Gauge Principle (World Scientific, 1994) [8] L.Bobelli et al, Pys. Re. Lett. 9, 21 (198) [9] A.Linde, Particle Pysics and Inflationary Cosology (Harwood Acadeic, GbH, 1990) [10] I.Dynikoa et al, Graitation and Cosology 4, 0 (1998) [11] H.M.Mok, ttp://publis.aps.org/eprint/grateway/eplist/aps1999aug1_001 [12] W.L.Freedan, astro-p/001236; B.R.Parodi et al., astro-p/