Physics Letters B 663 (2008) Contents lists available at ScienceDirect. Physics Letters B.

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1 Physics Letters B ) Cotets lists available at ScieceDirect Physics Letters B Holographic cosmological costat ad dark eergy Chao-Ju Feg a,b, a Istitute of Theoretical Physics, CAS, Beijig , PR Chia b Iterdiscipliary Ceter of Theoretical Studies, USTC, Hefei, Ahui 3006, PR Chia article ifo abstract Article history: Received 10 April 008 Accepted April 008 Available olie 7 April 008 Editor: A. Rigwald A geeral holographic relatio betwee UV ad IR cutoff of a effective field theory is proposed. Takig the IR cutoff relevat to the dark eergy as the Hubble scale, we fid that the cosmological costat is highly suppressed by a umerical factor ad the fie tuig problem seems alleviative. We also use differet IR cutoffs to study the case i which the uiverse is composed of matter ad dark eergy. 008 Elsevier B.V. Ope access uder CC BY licese. 1. Itroductio Why cosmological costat observed today is so much smaller tha the Plack scale? This is oe of the most importat problems i moder physics. I history, Eistei first itroduced the cosmological costat i his famous field equatio to achieve a static uiverse i After the discovery of the Hubble s law, the cosmological costat was o loger eeded because the uiverse is expadig. Nowadays, the acceleratig cosmic expasio first iferred from the observatios of distat type Ia superovae [1, has strogly cofirmed by some other idepedet observatios, such as the cosmic microwave backgroud radiatio CMBR) [3 ad Sloa Digital Sky Survey SDSS) [, ad the cosmological costat returs back as a simplest cadidate to explai the acceleratio of theuiversei1990 s. I particle physics, the cosmological costat aturally arises as a eergy desity of the vacuum, which is evaluated by the sum of zero-poit eergies of quatum fields with mass m as follows Λ ρ Λ = 1 0 πk dk π) 3 k + m Λ 16π, 1) where Λ m is the UV cutoff. Usually the quatum field theory is cosidered to be valid just below the Plack scale: M p GeV, where we used deduced Plack mass M p = 8π G for coveiece. If we pick up Λ = M p, we fid that the eergy desity of the vacuum i this case is estimated as GeV, which is about orders of magitude larger tha the observatio value 10 7 GeV. Oe may try to cacel it by itroducig couter terms, however, this requires a fie tuig to adjust the eergy desity of the vacuum to the preset eergy desity of the uiverse for a classic * Correspodig author at: Istitute of Theoretical Physics, CAS, Beijig , PR Chia. address: fegcj@itp.ac.c. review see [5, for a recet ice review see [6, ad for a recet discussio see [7,8). It seems that the umber of the idepedet degrees of freedom of the quatum fields should ot be very large [9,10. Holographic priciple [11 regards black holes as the maximally etropic objects of a give regio ad postulates that the maximum etropy iside this regio behaves o-extesively, growig oly as its surface area. Hece the umber of idepedet degrees of freedom is bouded by the surface area i Plack uits, so a effective field theory with UV cutoff Λ i a box with size L will make sese if it satisfies the Bekestei etropy boud [1,13 LΛ) 3 S BH = π L M pl, ) where M pl G is the Plack mass ad S BH is the etropy of a black hole of radius L which acts as a IR cutoff. Cohe ad collaborators [1 suggested that the total eergy i a regio of size L should ot exceed the mass of a black hole of the same size L 3 Λ LM p, 3) which ca be simply rewritte as LΛ) L M p. ) This boud is much more striget tha the boud ): whe Eq. ) is ear saturatio, the etropy of the quatum field is S max S 3/ BH. 5) Sice we have limited kowledge about the holographic priciple ad we have ot eve kow whether the holographic priciple is right or ot because we have oly a few examples to realize it. The oly successful example to my kowledge is the AdS/CFT correspodece. Mostly, oe believes the holographic priciple is right because it does ot coflict with ay observatios so far. As a result we caot claim whether the bouds metioed above as a cosequece of the holographic priciple is correct or ot, ad it may be too striget or too loose due to some ukow reasos Elsevier B.V. Ope access uder CC BY licese. doi: /j.physletb

2 368 C.-J. Feg / Physics Letters B ) or some uderlyig theory. I Sectio we postulate a geeral boud which provides a mechaism to derive a very small vacuum eergy from the priciple of holography. Whe the matter presets i the uiverse, the evolutio of the dark eergy i this ote we shall use terms the cosmological costat ad the dark eergy exchageably) is sesitive to the chose of the IR cutoff. Whe we take the evet horizo as the IR cutoff, the result is very similar to the case studied i Refs. [15 19 up to some correctios. As log as the vacuum domiates the eergy desity i the later time, it should be small as we discussed i Sectio. This Letter is orgaized as follows. I Sectio we postulate a geeral relatio betwee the UV a IR cutoff ad the smalless of the cosmological costat shall be explaied. I the ext sectio we use three differet IR cutoffs to study the evolutio ad the equatio of state of the dark eergy. I the fial sectio we will give some discussios.. Geeral boud ad the cosmological costat I this ote we postulate a geeral relatio of UV ad IR cutoff as follows LΛ) L M p, 6) where is a dimesioless parameter that comes from some uderlyig theory. Whe = 3, Eq.6) is reduced to ) ad ), respectively, ad we shall see that the fial cosistet is slightly deviatio from but without ay fie tuig. Of course we caot say aythig about this ukow theory yet, sice we do ot eve kow whether there really exists such a theory or ot, but we shall see that if the relatio 6) is correct it will provide a mechaism to derive a small cosmological costat without ay fie tuig. There are some works tryig to solve the cosmological costat problem from the holographic priciple, for istace see [15,0, but they do ot cosider the geeral case of the relatio 6) betwee UV ad IR cutoff. The largest Λ allowed here is the oe saturatig the iequality 6): Λ = L 1 M p. 7) The the eergy desity of the vacuum ρ Λ Λ is ρ Λ = 3c L 8 8 M p, 8) where a umerical costat 3c is itroduced i the above equatio for coveiece. From 1) oe ca see the value of c is aturally either very large or very small. Of course there is also a costat i Eq. 7), but aturally such a costat could ot be very large or very small ad it will ot affect the fial coclusio sice oe ca absorb this costat ito c. The dyamics of the uiverse is described by the Eistei field equatios. The observatios idicate that the uiverse is homogeeous ad isotropic o large scales ad the geeric metric respectig these symmetries is the Friedma Robertso Walker FRW) metric give by [ dr ds = dt + a t) 1 Kr + r dθ + si θdφ ), 9) where at) is scale factor with cosmic time t. The coordiates r, θ ad φ are kow as comovig coordiates. A freely movig particle comes to rest i these coordiates. The costat K i the metric 9) describes the geometry of the spatial sectio of spacetime, ad K =+1, 0, 1 correspods to closed, flat ad ope uiverse, respectively. Cosider a ideal perfect fluid with eergy desity ρ ad pressure p as the source of the eergy mometum tesor ad solve the Eistei equatio with the metric 9), we fid the famous Friedma equatios H = ρ K 3M p a 10) ad Ḣ = ρ + p M p + K a, 11) where H ȧ/a is the Hubble costat. Sice observatios idicate the uiverse is flat, i.e., the critical eergy desity is almost equal to 1, we will oly cosider the flat case K = 0 i the followig. I fact 11) ca be derived with the help of the cotiuity equatios respectig the coservatio of the eergy mometum as the cosequece of the Biachi idetities: ρ + 3Hρ + p) = 0. 1) To idicate the applicatio of the relatio 6), we cosider such a situatio i which we assumed that the IR cutoff is the Hubble scale H 1 ad the vacuum domiates the uiverse: 3M p H = 3c H 8 8 M p, 13) which ca be easily solved ) 8 H 1 = M p c. 1) For a give ) H is a costat, so the uiverse is de Sitter space. While =, the c = 1 ad H could be ay value sice = is a ustable poit, thus we ca get very small value of cosmological costat whe is slightly deviatio from, ad there is ot ay fie tuig problem i such a differece, we shall see this i the followig. We would like to emphasize that ad c here are determied by some uderlyig reasos as a ormal umber, by ormal umber we mea the umber is ot very large like or small like 10 10, so there is o eed for us to adjust them to produced a small vacuum eergy. If our postulatio is right, it should make it. Usually c is ot equal to 1, so the eergy desity of the vacuum is ρ Λ = 3c H 8 8 M p = 3 c ) M p, 15) where we have used 1). Ifc < 1 ad <, the eergy desity ca be highly suppressed ad much smaller tha the Plack scale eergy desity M p i the limit of. If c > 1 ad >, oe ca also get a very small eergy desity. I the other case the value of eergy desity is ruled out by the observatios because it is too large. To illustrate that has the slight but ot fie-tuig differece from, we give a cocrete example i the followig. Assumig c = 0.1, ad if = 0.1 = 3.966, 16) l c l 10 the eergy desity is roughly M p.let = + ɛ, oecasee ɛ = 0.03 i this example. I fact, such a differece ɛ would ot be a extreme small umber, amely, a umber like 10 10,aslog as c is ot very closed to 1, so there is o fie tuig problem here, oe ca see this property i Fig. 1. At first glace it seems that there is a fie tuig problem here: for a give c, oe should adjust the value of to be very closed to 0.1 for example. But it is ot the case, because we do ot eed to adjust c ad i fact, the resultig eergy desity is a cosequece or a predictio of the theory rather tha a iput. The figure above oly idicates that the uderlyig theory should cotai a costat umber whose value is ear without ay fie tuig.

3 C.-J. Feg / Physics Letters B ) Derivative the logarithm of ) with respect to l a as follows l Ω Λ = ) l L + l H [ l LH) + ɛ l L. ) From 1), 3) ad ) we ca derive a equatio of H ad L l H ɛ ) Ω Λ l L + 3 ) 1 Ω Λ = 0 5) Fig. 1. The differece ɛ vs. c.thevalueofc ad ɛ o the curve will deduce a small eergy desity, amely, M p. The black poit o the curve is the poit c = 0.1, ɛ Let us have a look at Eq. 6) LΛ) +ɛ L M p. 17) It seems that we are livig i the fractal dimesio spacetime rather tha, if oe regards the power of LΛ as the dimesio of the world we livig due to the simple fact that the max etropy i dimesio spacetime is roughly L 3 Λ 3 ad i 3 dimesio spacetime is L Λ. It is amazig if this explaatio is correct because it is so couterituitive. But we are forced to recosider the stability of the orbits of plaets like the earth because there are o stable solutios to keep the earth roudig the su, i other words the Iverse Square Law is ot hold i high dimesios. Thigs is also bad i low dimesioal worlds because o oe ca live i dimesio space. Problems may be disappeared here because of the fractal dimesio. Aother possibility is that i the followig. Take the correctio term to the R.H.S. of 17) as L 3 Λ LΛ) ɛ LM p. 18) The factor LΛ) ɛ may come from the some ukow theory. Sice this factor has somethig to do with the cosmological costat which ca be cosidered as a cosequece of quatum gravity [5, oe could guess this factor may come from the correctio of quatum gravity theory. Maybe these two possibilities are the same thig, but there is o evidece here ow. 3. Dark eergy with the presece of matter With matter preset, the Friedma equatio reads 3M p H = ρ m + ρ Λ, 19) where ρ m, the eergy desity of matter, satisfies the cotiuity equatio: l ρ m + 3 = 0. 0) where prime deotes the derivative with respect to l a. The dimesioless eergy desity of matter is defied as Ω m = ρ m /3M p H ), so it satisfies the followig equatio l Ω m = 3 l H, 1) where we have used the cotiuity equatio 0). Defie a dimesioless eergy desity Ω Λ ρ Λ 3M p H = c LM p ) 8 H M p. ) The the Friedma equatio 19) is Ω m + Ω Λ = 1. 3) ad oce we kow the aother relatio of L ad H, wecasolve this equatio. Measurig w as i ρ Λ a 31+w),wehavetheidexw give by w = 1 1 d l ρλ 3 d l a + 1 d ) l ρ Λ dl a) l a, 6) up to the secod order ad the derivatives are take at the preset time a 0 = 1. From 8) we fid d l ρ Λ d l a = ρ ) Λ 8 L = Hρ Λ LH 1 + ɛ ) l L, 7) thus we get the ratio of pressure to eergy desity w as w ɛ ) l L. 8) 3 For the acceleratio of the uiverse w < 1/3, the R.H.S. of 8) should satisfy 1 + ɛ ) l L < 1 9) ad for a icreasig of Ω Λ the R.H.S. of ) should be positive l LH) + ɛ l L < 0. 30) From 9) oe ca see that, whe the IR cutoff L is smaller tha 1, the uiverse is acceleratio defiitely ad this vacuum eergy will evetually domiate the uiverse. This happes whe we regards the evet horizo as a atural cutoff as the IR cutoff, ad Miao s work i [15 has already idicated such a character. I the followig we will study the property of the vacuum eergy with three differet IR cutoffs: Hubble scale H 1, particle horizo R p ad evet horizo R h, sice these IR cutoffs aturally arise whe oe studies the uiverse. The defiitio of R p ad R h is give by t R p t) = at) R h t) = at) 3.1. Case 1: L = H 1 dt at ), 31) 0 dt at ). 3) t I this case, the vacuum eergy behaves almost like the matter, which meas it is equatio of state is very similar to that of the matter up to some correctios, ad we fid this correctio will lead to a evolutio of w, but it will take a log time for w to be 1. I other words, the vacuum eergy will ot accelerate the uiverse util it almost completely domiates the uiverse. Sice the calculatio discussed above is straightforward, we simply give the fial result as follows.

4 370 C.-J. Feg / Physics Letters B ) From ) ad 5) the equatio of the eergy desity is foud to be Ω Λ = 3ɛ [ 1 Ω Λ Ω Λ ɛ )Ω, 33) Λ where L = H 1 was used. Whe ɛ < 0 the dimesioless eergy desity of the vacuum is icreasig with time. This equatio ca be solved easily as l Ω Λ + ɛ l1 Ω Λ) = 3ɛ l a + x 0. 3) If we set a 0 = 1 at the preset time, x 0 is equal to the L.H.S. of 3) with Ω Λ replaced by ΩΛ 0, amely, x 0 = l ΩΛ 0 + ɛ l1 Ω0 Λ ).As time draws by, Ω Λ icreases to 1, the secod term o the L.H.S. of 3) is the importat term, we fid, for large a Ω Λ = 1 e x 0/ɛ a 3. 35) Sice the uiverse is domiated by the dark eergy for large a, we have ρ Λ ρ c = ρ m = ρ0 m a 3. 36) 1 Ω Λ 1 Ω Λ Thus, usig 35) i the above relatio ρ Λ = e x0/ɛ ρm 0, 37) which is too large compared with the observatio value of ΩΛ 0,if we require a acceleratio uiverse, amely ɛ < 0. For small a, matter domiates, the importat term o the L.H.S. of 3) is the first term, we fid Ω Λ = a 3ɛ/ e x 0, 38) thus ρ Λ = Ω Λ ρ c = Ω Λ ρ m = e x 0 ρm 0 a 31+ɛ/), 39) here ɛ is much smaller tha 1, so the evolutio of the vacuum eergy is roughly a 3 the same as the matter whe a is small. I other words w is almost zero whe matter presets. Up to the secod order, the equatio of sate is described by ɛ w = + 3ɛ 1 + ɛ/)ωλ 0 1 Ω0 Λ ) z, + ɛ)ωλ ɛ/)ωλ 0 0) )3 where we used l a = l1 + z) z. Specifyig to the case ɛ = 0.03 ad pluggig the optioal value ΩΛ 0 = 0.73 ito 0), w = z. 1) It seems that the Hubble scale is ot a suitable IR cutoff. 3.. Case : L = R p If we take the particle horizo as the IR cutoff, the situatio is ot much chaged from the Hubble scale case. Sice l L = l R p = R p H, ) the equatio of state from 8) is w = ɛ ɛ ) 3R p H, 3) which is larger tha 1/3 + ɛ/6). It seems that the uiverse is hardly to accelerate i this case. From ) ad 5) we fid l Ω Λ = 1 Ω Λ ) 1 + ɛ ) R p H ) 3, ) so the vacuum eergy will domiates at later time if R p H > 1 + 3ɛ ). 5) It seems that the particle horizo is ot a suitable IR cutoff either Case 3: L = R h I this case the situatio is chaged, amely we ca get a acceleratig uiverse as follows ad firstly oe ca simply see that l L = l R h = 1 1 R h H, 6) where the mius sig i 6) is the mai differece from ), so the equatio of state 7) is chaged to be w = ɛ ɛ ) 3R h H, 7) which is smaller tha 1/3 + ɛ/6). It seems that the uiverse is able to accelerate i this case. Here the term ɛ/6 will slightly chage the value of w, ad this is a correctio to that i [15. From ) ad 5) we fid l Ω Λ = 1 Ω Λ ) 1 + ɛ ) 1 1 ) 3, 8) R h H so the vacuum eergy will domiates if 1 R p H > 1 1 3ɛ ), 9) which is always hold. Use the defiitio of Ω Λ i ), we fid 8) becomes l Ω Λ = 1 Ω Λ ) 1 + ɛ ) ) ΩΛ 1 R h M p ) ɛ/ 3, 50) c which caot be solved aalytically. The approximate solutio whe ɛ is small will reduce to the result i [15. The correspodig equatio of state from 7) will be w = ɛ ɛ ) ΩΛ R h M p ) ɛ/. 51) 3c Wherewehaveused). Ifɛ = 0, the above equatio is the same as that i [15, so terms cotaiig ɛ are correctios with slight effects. Whe the vacuum completely domiates the uiverse at last, the uiverse is a de Sitter space ad the evet horizo is roughly the iverse of the Hubble costat at that time t 0, amely R h H 1 0. If we take today s value of the Hubble costat H M p, the factor R h M p ) ɛ/ i 51) is roughly 0.60 where we have used ɛ Takig the preset value of ΩΛ 0 = 0.73 ad c = 0.5 thew Fig. shows the relatio betwee w 0 ad c, but otice that here c as a costat is from some ukow theory rather tha adjusted.. Discussios The applicatio of holographic priciple discussed i the preset Letter alleviates the cosmological costat problem. Whe the vacuum domiates the uiverse, the eergy desity could be very small due to the umber of ad c come from some uderlyig reasos. This provide a mechaism to explai why the cosmological costat is so small. I other words, oe ca get a very small eergy desity cosistet with the observatio value by this mechaism. We give a example ad argue that there is o fie tuig problem i this mechaism. It should be emphasized that ad c are give umbers rather tha adjusted.

5 C.-J. Feg / Physics Letters B ) Ackowledgemets We are grateful to Xia Gao, Wei Sog, Yushu Sog, Tower Wag, Yi Wag, Wei Xue, ad Xi Zhag for useful discussios. The author ackowledges Miao Li for a careful readig of the mauscript ad valuable suggestios. Refereces Fig.. Thedashliedeotestheliew 0 = 1. The solid lie idicates w 0 vs. c whe ɛ = 0.03 ad the factor R h M p ) ɛ/ 0.6. Whe matter presets as a compoet i the uiverse, this vacuum eergy play a role as the dark eergy. The evolutio of the dark matter here is sesitive to the IR cutoff. We have used three differet cutoff ad fid the result is cosistet with [15, amely the evet horizo is a suitable IR cutoff for the eergy desity to accelerate the uiverse. At first glace it seems that if l L is o-positive the uiverse is defiitely acceleratig from 9). However,ifl L < 0, it meas there is a shrikig IR cutoff, so the cutoff will be smaller ad smaller as time draws by ad we ca see less ad less stars ad galaxies. This is absurd. If l L = 0, it meas there is a uiversal IR cutoff of the uiverse, by uiversal we mea the cutoff is idepedet of cosmic time, the the eergy desity of the dark matter is a costat, amely w = 1, but there is o evidece that we have such a uiversal IR cutoff. I a word, l L should be positive. [1 A.G. Riess, et al., Superova Search Team Collaboratio, Astro. J ) 1009, astro-ph/ [ S. Perlmutter, et al., Superova Cosmology Project Collaboratio, Astrophys. J ) 565, astro-ph/ [3D.N.Spergel,etal.,WMAPCollaboratio,Astrophys.J.Suppl )377, astro-ph/ [ J.K. Adelma-McCarthy, et al., SDSS Collaboratio, arxiv: [astro-ph. [5 S. Weiberg, Rev. Mod. Phys ) 1. [6 S.M. Carroll, Livig Rev. Rel. 001) 1, astro-ph/ [7 J. Polchiski, hep-th/ [8 R. Bousso, Ge. Relativ. Gravit ) 607, arxiv: [hep-th. [9 G. t Hooft, gr-qc/ [10 L. Susskid, J. Math. Phys ) 6377, hep-th/ [11 R. Bousso, Rev. Mod. Phys. 7 00) 85, hep-th/ [1 J.D. Bekestei, Phys. Rev. D ) 333. [13 J.D. Bekestei, Phys. Rev. D ) 87. [1 A.G. Cohe, D.B. Kapla, A.E. Nelso, Phys. Rev. Lett ) 971, hep-th/ [15 M. Li, Phys. Lett. B ) 1, hep-th/ [16 Q.G. Huag, M. Li, JCAP ) 001, hep-th/ [17 Q.G. Huag, M. Li, JCAP ) 013, astro-ph/009. [18 B. Che, M. Li, Y. Wag, Nucl. Phys. B ) 56, astro-ph/ [19 J.f. Zhag, X. Zhag, H.y. Liu, Eur. Phys. J. C 5 007) 693, arxiv: [hep-th. [0 P. Horava, D. Miic, Phys. Rev. Lett ) 1610, hep-th/ [1 Y.S. Myug, Phys. Lett. B ) 7, hep-th/ [ R. Horvat, Phys. Rev. D 70 00) , astro-ph/000. [3 M.M. Sheikh-Jabbari, Phys. Lett. B 6 006) 119, hep-th/ [D.Podolsky,K.Eqvist,arXiv:070.01[hep-th. [5 E. Witte, hep-ph/00097.