Dark Energy and Tracker Solution- A Review

Transcription

1 Dark Energy and Tracker Soluton- A Revew Rakh R., Indulekha K. School of Pure and Appled Physcs Mahatma Gandh Unversty, Pryadarshn Hlls P.O. Kottayam, Kerala, Inda PACS Numbers: k, Cq, Qc Key Words: Cosmology, Dark energy, Dark matter, Tracker Model Abstract In ths paper, bascs and some theoretcal models of dark energy are revewed. Theoretcal models nclude cosmologcal constant, vacuum fluctuatons of quantum felds, scalar feld models, chaplygn gas, vector feld models and brane world models. Besdes ths, some alternate models of dark energy are also ncluded. Fnally, some elementary deas on tracker models are also dscussed.

2 Contents: 1. Introducton: Dark Energy &Dark Matter 3. What s Dark Energy? 4 3. Evdence for Dark Energy 6-7. Supernovae 6. Cosmc Mcrowave Background (CMB) 6. Large Scale Structure 7 v. Late-tme Integrated Sachs-Wolfe Effect 7 4. Canddates of Dark Energy Cosmologcal Constant 7. A non-zero vacuum energy provded by vacuum fluctuatons of quantum felds => quantum feld theory 8. Scalar Feld Models 9-14 v. Chaplygn gas 14 v. Vector Feld Models 16 v. Brane World Models 17 v. Alternate Ideas Tracker Soluton for Dark Energy [Elementary deas] 0-7. Why Tracker Soluton? 1. Tracker Feld. Famly of Tracker Solutons v. Trackng equaton and Trackng Potentals 3

3 3 1. Introducton: Dark Energy and Dark matter In the early 1990's, one thng was farly certan about the expanson of the Unverse. It mght have enough energy densty to stop ts expanson and recollapse, t mght have so lttle energy densty that t would never stop expandng, but gravty was certan to slow the expanson as tme went on. Granted, the slowng had not been observed, but, theoretcally, the Unverse had to slow. The Unverse s full of matter and the attractve force of gravty pulls all matter together. Then came 1998 and the Hubble Space Telescope (HST) observatons of very dstant supernovae that showed that, a long tme ago, the Unverse was actually expandng more slowly than t s today. So the expanson of the Unverse has not been slowng due to gravty, as everyone thought, t has been acceleratng. No one expected ths; no one knew how to explan t. But somethng was causng t. Eventually theorsts came up wth three sorts of explanatons. Maybe t was a result of a long-dscarded verson of Ensten's theory of gravty, one that contaned what was called a "cosmologcal constant." Maybe there was some strange knd of energy-flud that flled space. Maybe there s somethng wrong wth Ensten's theory of gravty and a new theory could nclude some knd of feld that creates ths cosmc acceleraton. Theorsts stll don't know what the correct explanaton s, but they have gven the soluton a name. It s called dark energy. Dark Matter:- a cold, non-relatvstc materal most lkely n the form of exotc partcles that nteract very weakly wth atoms and lght. In astronomy and cosmology, dark matter s hypothetcal matter that does not nteract wth the electromagnetc force, but whose presence can be nferred from gravtatonal effects on vsble matter. Accordng to present observatons of structures larger than galaxes, as well as Bg Bang cosmology, dark matter and dark energy account for the vast majorty of the mass n the observable unverse. The observed phenomena whch mply the presence of dark matter nclude the rotatonal speeds of galaxes, orbtal veloctes of galaxes n clusters, gravtatonal lensng of background objects by galaxy clusters such as the Bullet cluster, and the temperature dstrbuton of hot gas n galaxes and clusters of galaxes. Dark matter also plays a central role n structure formaton and galaxy evoluton, and has measurable effects on the ansotropy of the cosmc mcrowave background. All these lnes of evdence suggest that galaxes, clusters of galaxes, and the unverse as a whole contan far more matter than that whch nteracts wth electromagnetc radaton: the remander s called the "dark matter component." The dark matter component has much more mass than the "vsble" component of the unverse. At present, the densty of ordnary baryons and radaton n the unverse s estmated to be equvalent to about one hydrogen atom per cubc meter of space. Only about 4% of the total energy densty

4 4 n the unverse (as nferred from gravtatonal effects) can be seen drectly. About % s thought to be composed of dark matter. The remanng 74% s thought to consst of dark energy, an even stranger component, dstrbuted dffusely n space. Some hard-to-detect baryonc matter s beleved to make a contrbuton to dark matter but would consttute only a small porton. Determnng the nature of ths mssng mass s one of the most mportant problems n modern cosmology and partcle physcs.. What s Dark Energy? Dark energy s a repulsve force that opposes the self-attracton of matter and causes the expanson of the unverse to accelerate. In physcal cosmology, dark energy s a hypothetcal form of energy that permeates all of space and tends to ncrease the rate of expanson of the unverse. Strangely, dark energy causes expanson because t has strong negatve pressure. A substance has postve pressure when t pushes outward on ts surroundngs. Ths s the usual stuaton for fluds. Negatve pressure, or tenson, exsts when the substance nstead pulls on ts surroundngs. A common example of negatve pressure occurs when a sold s stretched to support a hangng weght. Accordng to the FLRW metrc, whch s an applcaton of General Relatvty to cosmology, the pressure wthn a substance contrbutes to ts gravtatonal attracton for other thngs just as ts mass densty does. Negatve pressure causes a gravtatonal repulson. The gravtatonal repulsve effect of dark energy's negatve pressure s greater than the gravtatonal attracton caused by the energy tself. At the cosmologcal scale, t also overwhelms all other forms of gravtatonal attracton, resultng n the acceleratng expanson of the unverse. Fg.1 Unverse Dark Energy-1 Expandng Unverse. Ths dagram shows changes n the rate of expanson snce the Unverse's brth 14 bllon years ago. The more shallow the curve, the faster the rate of expanson. The curve changes notceably about 7.5 bllon years ago, when objects n the Unverse began flyng apart at a faster rate. [Credt: NASA/STSc/Ann Feld] One explanaton for dark energy s that t s a property of space. Albert Ensten was the frst person to realze that empty space s not anythng. Space has amazng propertes, many of whch are just begnnng to be understood. The frst property that Ensten dscovered s that t s possble for more space to come nto exstence. Then one verson of Ensten's gravty theory, the

5 5 verson that contans a cosmologcal constant, makes a second predcton: "empty space" can possess ts own energy. Because ths energy s a property of space tself, t would not be dluted as space expands. As more space comes nto exstence, more of ths energy-of space would appear. As a result, ths form of energy would cause the Unverse to expand faster and faster. Unfortunately, no one understands why the cosmologcal constant should even be there, much less why t would have exactly the rght value to cause the observed acceleraton of the Unverse. Another explanaton for how space acqures energy comes from the quantum theory of matter. In ths theory, "empty space" s actually full of temporary ("vrtual") partcles that contnually form and then dsappear. But when physcsts tred to calculate how much energy ths would gve empty space, the answer came out wrong - wrong by a lot. The number came out tmes too bg. It's hard to get an answer that bad. Another explanaton for dark energy s that t s a new knd of dynamcal energy flud or feld, somethng that flls all of space but somethng whose effect on the expanson of the Unverse s the opposte of that of matter and normal energy. A last possblty s that Ensten's theory of gravty s not correct. That would not only affect the expanson of the Unverse, but t would also affect the way that normal matter n galaxes and clusters of galaxes behaved. Ths fact would provde a way to decde f the soluton to the dark energy problem s a new gravty theory or not: we could observe how galaxes come together n clusters. But f t does turn out that a new theory of gravty s needed, what knd of theory would t be? How could t correctly descrbe the moton of the bodes n the Solar System, as Ensten's theory s known to do, and stll gve us the dfferent predcton for the Unverse that we need? There are canddate theores, but none are compellng. So the mystery contnues. 3. Evdence for Dark energy. Supernovae In 1998, publshed observatons of Type Ia supernovae by the Hgh-z Supernova Search Team followed n 1999 by the Supernova Cosmology Project suggested that the expanson of the unverse s acceleratng. Snce then, these observatons have been corroborated by several ndependent sources. Measurements of the cosmc mcrowave background, gravtatonal lensng, and the large scale structure of the cosmos as well as mproved measurements of supernovae have been consstent wth the Lambda-CDM model. Supernovae are useful for cosmology because they are excellent standard candles across cosmologcal dstances. They allow the expanson hstory of the Unverse to be measured by lookng at the relatonshp between the dstance to an object and ts redshft, whch gves how fast t s recedng from us. The relatonshp s roughly lnear, accordng to Hubble's law. It s relatvely easy to measure redshft, but fndng the dstance to an object s more dffcult. Usually,

6 6 astronomers use standard candles: objects for whch the ntrnsc brghtness, the absolute magntude, s known. Ths allows the object's dstance to be measured from ts actually observed brghtness, or apparent magntude. Type Ia supernovae are the best-known standard candles across cosmologcal dstances because of ther extreme, and extremely consstent, brghtness.. Cosmc Mcrowave background (CMB) The exstence of dark energy, n whatever form, s needed to reconcle the measured geometry of space wth the total amount of matter n the unverse. Measurements of cosmc mcrowave background (CMB) ansotropes, most recently by the WMAP satellte, ndcate that the unverse s very close to flat. For the shape of the unverse to be flat, the mass/energy densty of the unverse must be equal to a certan crtcal densty. The total amount of matter n the unverse (ncludng baryons and dark matter), as measured by the CMB, accounts for only about 30% of the crtcal densty. Ths mples the exstence of an addtonal form of energy to account for the remanng 70%. The most recent WMAP observatons are consstent wth a unverse made up of 74% dark energy, % dark matter, and 4% ordnary matter.. Large scale structure The theory of large scale structure, whch governs the formaton of structure n the unverse (stars, quasars, galaxes and galaxy clusters), also suggests that the densty of baryonc matter n the unverse s only 30% of the crtcal densty. v. Late tme Integrated Sachs-Wolfe Effect (ISW) Accelerated cosmc expanson causes gravtatonal potental wells and hlls to flatten as photons pass through them, producng cold spots and hot spots on the CMB algned wth vast super vods and super clusters. Ths so-called late-tme Integrated Sachs-Wolfe effect (ISW) s a drect sgnal of dark energy n a flat unverse. 4. Canddates of Dark energy. Cosmologcal Constant The smplest canddate for dark energy s provded by cosmologcal constant. The cosmologcal constant corresponds to a flud wth a constant equaton of state w = 1.But there are certan theoretcal ssues assocate wth t: a) the smallest numercal value of lambda leads to fne tunng problem [Fne tunng refers to crcumstances when the parameters of the model must be adjusted very precsely n order to agree wth observatons. Theores requrng fne tunng are regarded as problematc n the absence of a known mechansm to explan why the parameters of the model happen to have precsely the needed values]. b) t leads to concdence problem[throughout the hstory of the Unverse, the scalar feld densty and matter feld denstes decrease at dfferent

7 7 rates, so t appears that the condtons n the early unverse must be set very carefully n order for the energy denstes to be comparable today. Ths ssue of ntal condtons s known as concdence problem ]. Table 1: Cosmologcal Constant Vs. Ordnary Matter Perfect flud of densty, Pressure p, and 4-velocty u. Cosmologcal constant Energy-momentum tensor T = c + p u u pg ( 0 ) µν µ ν µν Equaton of state lnks densty and pressure, e.g., p = w, w = 0, 1/3 for non-rel. matter, radaton Energy-momentum tensor 4 c T = Λ g 8π G Energy Densty µν µν Λ = c 8π G Λ Classcal Λ acts as a gas wth equaton of state w = -1; but may also comprse contrbutons wth dfferent equaton of state. Adabatc expanson: non-relatvstc matter: a radaton: a 4 3 Λ can be a constant durng expanson; but more complex contrbutons evolve dfferently All contrbutons to Λ are called dark energy. A non-zero vacuum energy provded by vacuum fluctuatons of quantum felds => quantum feld theory Energy assocated wth space tself (spontaneous creaton and destructon of vrtual partcles; evdence: Casmr effect). Comparson wth classcal pdv work yelds an equaton of state wth w = - 1. Quantum felds can be vewed as a set of harmonc oscllators n momentum space. In ther ground state (n=0), these oscllators have a non-zero energy E ( )( 1 n = ħ ω k n + ), the total vacuum energy s then gven by the sum over all oscllators. The resultng can be transformed nto a densty of 4 V k max ħ, k max s the maxmum wave vector of the feld, taken to be the energy scale at whch QFT fals.

8 8 The Λ problem: take the nverse Planck scale (~10 19 GeV) for k max => V ~ 10 9 gcm -3 whch s larger than the cosmologcally acceptable value by a mere 10 orders of magntude.. Scalar Feld Models The cosmologcal constant corresponds to a flud wth a constant equaton of state w = -1. Now, the observatons whch constran the value of w today to be close to that of the cosmologcal constant, these observatons actually say relatvely lttle about the tme evoluton of w, and so we can broaden our horzons and consder a stuaton n whch the equaton of state of dark energy changes wth tme, such as n nflatonary cosmology. Scalar felds naturally arse n partcle physcs ncludng strng theory and these can act as canddates for dark energy. So far a wde varety of scalar-feld dark energy models have been proposed. These nclude quntessence, phantoms, K-essence, tachyon, ghost condensates and dlatonc dark energy amongst many. [1][] Quntessence: - In physcs, quntessence s a hypothetcal form of dark energy postulated as an explanaton of observatons of an acceleratng unverse. Quntessence s a scalar feld whch has an equaton of state (relatng ts pressure p and densty ) of p = w, where w s equal to the equaton of state of the energy component domnatng the unverse (.e. equal to 1/3 durng radaton domnaton and 0 durng matter domnaton) untl w undergoes a transton to less than -1/3 whch ntates the accelerated expanson of the unverse. Quntessence s dynamc, and generally has a densty and equaton of state ( w > 1) that vares through tme and space. By contrast, a cosmologcal constant s statc, wth a fxed energy densty and w = 1. In quntessence models of dark energy, the observed acceleraton of the scale factor s caused by the potental energy of a dynamcal feld, referred to as quntessence feld. Although the cosmc concdence ssue remans unresolved, the fne tunng problem facng dark energy/quntessence models wth a constant equaton of state can be sgnfcantly allevated f we assume that the equaton of state s tme dependent. Quntessence dffers from the cosmologcal constant n that t can vary n space and tme. In order for t not to clump and form structure lke matter, the feld must be very lght so that t has a large Compton wavelength. The Quntessence feld must couple to ordnary matter, whch even f suppressed by the Planck scale wll lead to long range forces and tme dependence of the constants of nature. There are tght constrants on such forces and varatons and any successful model must satsfy them.

9 9 Acton Equaton of moton Energy-Momentum tensor Energy Densty Pressure Equaton-of-State Table : Quntessence (Summary) 4 1 ( ) ( ) S = d x g V dv + 3H + = 0 d 1 αβ Tµν = µ ν gµν g α β + V ( ) 0 1 = T0 = + V ( ) 1 p = T = V ( ) Parameter p V ( ) w = = + V Hubble s Constant (H) 8π G 1 H = + V ( ) 3 Acceleraton a 8π G = V ( ) a 3 Condton for Acceleraton < V, whch means that one requres a flat potental to ( ) ( ) gve rse to an accelerated expanson K-essence: - Quntessence reles on the potental energy of scalar felds to lead to the late tme acceleraton of the unverse. It s possble to have a stuaton where the accelerated expanson arses out of modfcatons to the knetc energy of the scalar felds. Orgnally knetc energy drven nflaton, called K-nflaton, was proposed (by Armendarz- Pcon et al) to explan early unverse nflaton at hgh energes. The analyss was extended to a more general Lagrangan (by Armendarz-Pcon et al) and ths scenaro was called K-essence.In general, K-essence can be defned as any scalar feld wth a non-canoncal knetc energy.

10 10 Table 3: K-essence (Summary) Acton Pressure Densty (accordng to the new defnton of feld) Energy Densty Feld redefnton Equaton-of- State Parameter Condton for Acceleraton 4 1 (, ) S = d x g R + p X 1 and Lag., where X ( )( ) p, X = f p X = K X + L X Densty ( ) ( ) ( ) ( ) ( ) corresponds to a pressure densty new old = d L K p(, X ) = f ( )( X + X ) where, X X L = X K K ( ) ( old ) f = L ( ) old p = X p = f X + 3X X p 1 X w = = 1 3X ( )( ) w 1 <, whch translates nto the condton X < 3 3 new new old and Equaton-of state parameter shows that the knetc term X plays a crucal role n determnng the equaton of state of. As long as X belongs n the range 1/ < X < /3, the feld behaves as ( ) 1+ w dark energy for 0 α whereα = + w 1 m background flud, durng matter/radaton domnant era., wm beng the equaton-of-state of the Tachyon feld: - Recently t has been suggested that rollng tachyon condensates, n a class of strng theores, may have nterestng cosmologcal consequences. A rollng tachyon has an nterestng equaton of state whose parameter smoothly nterpolates between 1 and 0. Ths has led to a flurry of attempts beng made to construct vable cosmologcal models usng the tachyon as a sutable canddate for the nflaton at hgh energy. However tachyon nflaton n open strng models s typcally plagued by several dffcultes assocated wth densty perturbatons and reheatng. Meanwhle the tachyon can also act as a source of dark energy dependng upon the form of the tachyon potental.

11 11 Table 4: Tachyon Feld (Summary) Acton S = d 4 xv ( ) det ( gab + a b ) Tachyon Potental V ( ) V0 V ( ) = where 0 = [from open strng theory] cosh ( ) 0 super strng and 0 = for the bosonc strng for non-bps D-brane n Tachyon potental gvng the power-law expanson, p a t V ( ) p = 1 4π G 3p 1 are not steep compared to ( ) expanson] [Tachyon potentals whch V lead to an accelerated Equaton of moton 1 dv + 3H + = 0 V d 1 Energy-Momentum tensor V ( ) Energy Densty V ( ) Pressure µ ν αβ Tµν = gµνv ( ) 1+ g αβ α β 1+ g = T = 0 0 α 1 β ( ) 1 p = T = V Equaton-of-State Parameter w p = = 1 Hubble s Constant (H) H 8π GV ( ) = Acceleraton a a ( ) 8π GV 3 = Condton for Acceleraton < 3

12 1 Irrespectve of the steepness of the tachyon potental, the equaton of state vares between 0 and m 1, n whch case the tachyon energy densty behaves as a wth 0 < m < 3. Phatom (Ghost) feld: - Recent observatonal data ndcates that the equaton of state parameter w les n a narrow strp around w = 1 and s qute consstent wth beng below ths value. The scalar feld models dscussed n the prevous subsectons corresponds to an equaton of state w 1. The regon where the equaton of state s less than 1 s typcally referred to as some form of phantom (ghost) dark energy. Meanwhle the smplest explanaton for the phantom dark energy s provded by a scalar feld wth a negatve knetc energy. Table 5: Phantom feld (Summary) Acton Equaton of moton Energy-Momentum tensor Energy Densty 4 1 ( ) ( ) S = d x g V dv + 3H = 0 d 1 αβ Tµν = µ ν + gµν g α β V ( ) 0 1 = T0 = + V ( ) Pressure Equaton-of-State 1 p = T = V ( ) Parameter p + V ( ) w = = V ( ) Hubble s Constant (H) 8π G 1 H = + V 3 Acceleraton a 8π G = + V ( ) a 3 ( ) The curvature of the unverse grows toward nfnty wthn a fnte tme n the unverse domnated by a phantom flud. Thus, a unverse domnated by Phantom energy culmnates n a future curvature sngularty ( Bg Rp ) at whch the noton of a classcal space-tme breaks down. In the case of a phantom scalar feld ths Bg Rp sngularty may be avoded f the potental has a maxmum, e.g,

13 13 V ( ) α = V0 cosh m pl 1, where α s constant. Snce the energy densty of a phantom feld s unbounded from below, the vacuum becomes unstable aganst the producton of ghosts and normal (postve energy) felds. Even when ghosts are decoupled from matter felds, they couple to gravtons whch medate vacuum decay processes of the type: vacuum ghosts + γ. It was shown by Clne et al. that we requre an unnatural Lorenz nvarance breakng term wth cut off of order MeV to prevent an overproducton of cosmc gamma rays. Also phantom felds are generally plagued by severe Ultra-Volet (UV) quantum nstabltes. Hence the fundamental orgn of the phantom feld stll poses an nterestng challenge for theoretcans. Dlatonc feld: - It s mentoned n the prevous secton that the phantom feld wth a negatve knetc term has a problem wth quantum nstabltes. Dlatonc model solves ths problem. (Dlaton s a hypothetcal partcle that appears n Kaluza-Klen theory and strng theory..e, a dlaton s a partcle of a scalar feld ; a scalar feld (followng the Klen-Gordon equaton) that always comes wth gravty). Ths model s also an nterestng attempt to explan the orgn of dark energy usng strng theory. In 008, A Carbo et.al showed that the form of potental for the Dlaton suggests that after solvng for the cosmologcal evoluton of the model, the thermal energy of the unverse could be gradually transformed n energy of the Dlaton, whch then could play the role of a quntessence feld descrbng dark energy. v. Chaplygn gas So far we have dscussed a number of scalar-feld models of dark energy. There exsts another nterestng class of dark energy models nvolvng a flud known as a Chaplygn gas. Ths flud also leads to the acceleraton of the unverse at late tmes. Remarkably, the Chaplygn gas appears lke pressure-less dust at early tmes and lke a cosmologcal constant durng very late tmes, thus leadng to an accelerated expanson.the Chaplygn gas can be regarded as a specal case of a tachyon wth a constant potental. However t was shown that the Chaplygn gas models are under strong observatonal pressure from CMB ansotropes. Ths comes from the fact that the Jeans nstablty of perturbatons n Chaplygn gas models behaves smlarly to cold dark matter fluctuatons n the dust-domnant stage but dsappears n the acceleraton stage. The combned effect of the suppresson of perturbatons and the presence of a non-zero Jeans length gves rse

14 14 to a strong ntegrated Sachs-Wolfe (ISW) effect, thereby leadng to the loss of power n CMB ansotropes. Table 6: Chaplygn gas (Summary) Lagrangan densty, µ L = V 1, where, x µ 0, µ µ Four-velocty, µ µ u =, α, α Energy-Momentum tensor ( ) T = + p u u pg µν µ ν µν Energy Densty V0 =, 1, µ µ Pressure, p = V 1 0, µ µ Equaton-of-State Hubble s Parameter [H(z)] p c A =, where ( ) 6 c c = A + B 1+ z,z s redshft 1 m and A B Ω κ 1 = or A = V 0 Ω m Ω A H z H0 1 z 1 z κ B κ = 0m B ( ) = Ω ( + ) + + ( + ) 3 6 1, where A generalzed Chaplygn gas has also been proposed for whch p 1 state n ths case s ( ) w a w 0 = 1 w0 w a ( +α ) p α. The equaton of, whch nterpolates between w=0 at early tmes ( a 1) and w = 1 at very late tmes( a 1). w0 s the current equaton of state when a = 1. (The constant α regulates the transton tme n the equaton of state). WMAP, supernovae and large scale structure data have all been used to test Chaplygn gas models.

15 15 v. Vector Feld Models Nowadays scentsts consder several new classes of vable vector feld alternatves to the nflaton and quntessence scalar felds. In 008, Tom Kovsto and Davd F. Mota presented a paper enttles Vector feld models of nflaton and Dark Energy [3]. In ther work, spatal vector felds are shown to be compatble wth the cosmologcal ansotropy bounds f only slghtly dsplaced from the potental mnmum whle domnant, or f drvng an ansotropc expanson wth nearly vanshng quadropole today. The Banch I model wth a spatal feld and an sotropc flud s studed as a dynamcal system, and several types of scalng solutons are found. On the other hand, tme-lke felds are automatcally compatble wth large-scale sotropy. They show that they can be dynamcally mportant f non-mnmal gravty couplngs are taken nto account. As an example, they reconstruct a vector-gauss-bonnet model whch generates the concordance model acceleraton at late tmes and supports an nflatonary epoch at hgh curvatures. The evoluton of the vortcal perturbatons n these models s computed. Jose Beltra n Jme'nez and Antono L. Maroto, n 008, explored the possblty that the present stage of accelerated expanson of the unverse s due to the presence of a cosmc vector feld [4]. They had showed that vector theores allow for the generaton of an accelerated phase wthout the ntroducton of potental terms or unnatural scales n the Lagrangan. They proposed a partcular model wth the same number of parameters as Λ CDM and excellent fts to SNIa data. The model s scalng durng radaton era, wth natural ntal condtons, thus avodng the cosmc concdence problem. They concluded that vector theores offer an accurate phenomenologcal descrpton of dark energy n whch fne-tunng problems could be easly avoded. v. Brane World Models Here, the dea rests on the noton that space-tme s hgher-dmensonal, and that our observable unverse s a (3+1)-dmensonal brane whch s embedded n a (4+1)-dmensonal braneworld models allow the expanson dynamcs to be radcally dfferent from that predcted by conventonal Ensten s gravty n 3+1 dmensons. Some cosmologcal surprses whch sprng from Braneworld models nclude [1]: Both early and late tme acceleraton can be successfully unfed wthn a sngle scheme (Quntessental Inflaton) n whch the very same scalar feld whch drves Inflaton at early tmes becomes Quntessence at late tmes The (effectve) equaton of state of dark energy n the braneworld scenaro can be phantom-lke (w<-1) or Quntessence-lke (w>-1).these two possbltes are

16 16 essentally related to the two dstnct ways n whch the brane can be embedded n the bulk. The acceleraton of the unverse can be a transent phenomenon: braneworld models accelerate durng the present epoch but return to matter-domnated expanson at late tmes. A class of braneworld models encounter a Quescent future sngularty, at whch a const, but a. The surprsng feature of ths sngularty s that whle the Hubble parameter, densty and pressure reman fnte, the deceleraton parameter and all curvature nvarants dverge as the sngularty s approached. A spatally flat Braneworld can mmck a closed unverse and loter at large redshfts. A braneworld embedded n a fve dmensonal space n whch the extra (bulk) dmenson s tme-lke can bounce at early tmes, thereby genercally avodng the bg bang sngularty. Cyclc models of the unverse wth successve expanson-contracton cycles can be constructed based on such a bouncng braneworld. v. Alternatve deas Some theorsts thnk that dark energy and cosmc acceleraton are a falure of general relatvty on very large scales, larger than super-clusters. It s a tremendous extrapolaton to thnk that our law of gravty, whch works so well n the solar system, should work wthout correcton on the scale of the unverse. Most attempts at modfyng general relatvty, however, have turned out to be ether equvalent to theores of quntessence, or nconsstent wth observatons. It s of nterest to note that f the equaton for gravty were to approach r nstead of r at large, ntergalactc dstances, then the acceleraton of the expanson of the unverse becomes a mathematcal artfact, negatng the need for the exstence of Dark Energy. Alternatve deas for dark energy have come from strng theory, DGP model, the holographc prncple, Gravty correctons etc, but have not yet proved as compellng as quntessence and the cosmologcal constant. Strng curvature correctons: - It s nterestng to nvestgate the strng curvature correctons to Ensten gravty amongst whch the Gauss-Bonnet correcton enjoys specal status. These models, however, suffer from several problems. Most of these models do not nclude tracker lke soluton and those whch do are heavly constraned by the thermal hstory of unverse. For nstance, the Gauss-Bonnet gravty wth dynamcal dlaton mght cause transton from matter scalng regme to late tme acceleraton allowng to allevate the fne tunng and concdence problems.

17 17 DGP Model: - In DGP model, gravty behaves as four dmensonal at small dstances but manfests ts hgher dmensonal effects at large dstances. The modfed Fredmann equatons on the brane lead to late tme acceleraton. The model has serous theoretcal problems related to ghost modes superlumnal fluctuatons. The combned observatons on background dynamcs and large angle ansotropes reveal that the model performs worse than Λ CDM. Non-Local Cosmology: - An nterestng proposal on non-locally corrected gravty nvolvng a 1 functon of the nverse d Almbertan of the Rcc scalar, f ( R) functon ( 1 f R) exp( α 1 R). For a generc, the model can lead to de-stter soluton at late tmes. The range of stablty of the soluton s gven by 1 < α < 3 1 correspondng to the effectve EoS parameter weff rangng as w eff 3 < <. For 1 < α < 3 1 and 1 < α <, the underlyng 3 system s shown to exhbt phantom and non-phantom behavor respectvely; the de Stter soluton corresponds toα = 1. For a wde range of ntal condtons, the system mmcs dust lke behavor before reachng the stable fxed pont at late tmes. The late tme phantom phase s acheved wthout nvolvng negatve knetc energy felds. Unfortunately, the soluton becomes unstable n presence of the background radaton/matter. f ( ) R Theores of gravty: - On purely phenomenologcal grounds, one could seek a modfcaton of Ensten gravty by replacng the Rcc scalar by f(r). The f(r) gravty theores gvng rse to cosmologcal constant n low curvature regme are plagued wth nstabltes and on observatonal grounds they are not dstngushed from cosmologcal constant. The acton of f ( R) gravty s gven by ( R) f 16πG 4 S = gd x + Lm.The functonal form of f ( R ) should satsfy certan requrements for the consstency of the modfed theory of gravty. The stablty of f ( R) theory would be ensured provded that, f R > 0 gravton s not ghost, ' ( ) f '' ( R ) > 0 scalaron s not tachyon. The f ( R) models whch satsfy the stablty requrements can broadly be classfed nto categores: () Models n whch f ( R ) dverge for R R0 where R 0 fnte or f ( R ) s non analytcal functon of the Rcc scalar. These models ether can not be dstngushable from ΛCDM or are not vable cosmologcally. () Models wth f ( R) 0 for R 0 and reduce to

18 18 cosmologcal constant n hgh curvature regme. These models reduce to ΛCDM n hgh redshft regme and gve rse to cosmologcal constant n regons of hgh densty and dffer from the latter otherwse; n prncpal these models can be dstngushed from cosmologcal constant. Unfortunately, the f ( R ) models wth chameleon mechansm are plagued wth curvature sngularty problem whch may have mportant mplcatons for relatvstc stars. The model could be remeded wth the ncluson of hgher curvature correctons. At the onset, t seems that one needs to nvoke fne tunngs to address the problem. The presence of curvature sngularty certanly throws a new challenge to f ( R) gravty models. 5. Tracker Soluton for Dark Energy A substantal fracton of the energy densty of the unverse may consst of quntessence n the form of a slowly rollng scalar feld. Snce the energy densty of the scalar feld generally decreases more slowly than the matter energy densty, t appears that the rato of the two denstes must be set to a specal, nfntesmal value n the early unverse n order to have the two denstes nearly concde today. Recently, Stenhardt et.al ntroduced the noton of tracker felds to avod ths ntal condtons problem. The term tracker s meant to refer to solutons jonng a common evolutonary track, as opposed to followng closely the background energy densty and equaton-of-state. The tracker models are smlar to nflaton n that they funnel a dverse range of ntal condtons nto a common fnal state. Although trackng s a useful tool to promote quntessence as a lkely source of the mssng energy n the unverse, the concept of trackng as gven by Stenhardt et al does not ensure the physcal vablty of quntessence n the observable unverse. It smply provdes for synchronzed scalng of the scalar feld wth the matter/radaton feld n the expandng unverse n such a way that at some stage (undefned and unrelated to observatons), the scalar feld energy starts domnatng over matter and may nduce acceleraton n the Hubble expanson. Snce there s no control over the slow roll-down and the growth of the scalar feld energy durng trackng, the transton to the scalar feld domnated phase may take place much later than observed. Moreover, any addtonal contrbuton to the energy densty of the unverse, such as quntessence, s bound to affect the dynamcs of expanson and structure formaton n the unverse. As such, any physcally vable scalar feld must comply wth the cosmologcal observatons related to helum abundance, cosmc mcrowave background and galaxy formaton, whch are the pllars of the success of the standard cosmologcal model. A realstc theory of trackng of scalar felds must, therefore, take nto account the astrophyscal constrants arsng from the cosmologcal observatons.

19 19. Why Tracker Soluton? To overcome the fne tunng or the ntal value problem, the noton of tracker felds was ntroduced. A key problem wth the quntessence proposal s explanng why and the matter energy densty should be comparable today. There are two aspects to ths problem. Frst of all, throughout the hstory of the unverse, the two denstes decrease at dfferent rates; so t appears that the condtons n the early unverse have to be set very carefully n order for the energy denstes to be comparable today. We refer to ths ssue of ntal condtons as the concdence problem. The very same ssue arses wth a cosmologcal constant as well. A second aspect, whch we call the fne-tunng problem, s that the value of the quntessence energy densty (or vacuum energy or curvature) s very tny compared to typcal partcle physcs scales. The fne-tunng condton s forced by drect measurements; however, the ntal condtons or concdence problem depends on the theoretcal canddate for the mssng energy. Recently, Stenhardt et.al ntroduced a form of quntessence called tracker felds whch avods the concdence problem. It permts the quntessence felds wth a wde range of ntal values of to roll down along a common evolutonary track wth m and end up n the observable unverse wth comparable to m at the present epoch. Thus, the tracker felds can get around both the concdence problem and the fne tunng problem wthout the need for defnng a new energy scale for Λ. eff An mportant consequence of the tracker solutons s the predcton of a relaton between w and Ω today. Because tracker solutons are nsenstve to ntal condtons, both w and depend on V ( ). Hence, for any gven V ( ), once Ω only Ω s measured, w s determned. In general, the closer Ω s to unty, the closer wq s to 1. However, snce Ω 0. today, there m s a suffcent gap between Ω and unty that w cannot be so close to -1. We fnd that w 0.8 for practcal models. Ths w - Ω relaton makes the tracker feld proposal dstngushable from the cosmologcal constant.. Tracker Feld:- a feld whose evoluton accordng to ts equaton-of-moton converges to the same soluton the tracker soluton for a wde range of ntal condtons for the feld and ts tme dervatve. Tracker felds have an equaton-of-moton wth attractor-lke solutons n the sense that a very wde range of ntal condtons rapdly converge to a common, cosmc evolutonary track of

20 0 ( t) and w ( t) soluton because nether. Techncally, the tracker soluton dffers from a classcal dynamcs attractor Ω nor any other parameters are fxed n tme. The ntal value of can vary by nearly 100 orders of magntude wthout alterng the cosmc hstory. The acceptable ntal condtons nclude the natural possblty of equ-partton after nflaton nearly equal energy densty n as n the other degrees of freedom (e.g., 3 Ω 10 ). Furthermore, the resultng cosmology has desrable propertes. The equaton-of-state w vares accordng to the background equaton-of-state w B. When the unverse s radatondomnated ( w 1 B = ), then w 3 s less than or equal to 1/3 and decreases less rapdly than the radaton densty. When the unverse s matter-domnated ( w B = 0 ), then w s less than zero and decreases less rapdly than the matter densty. Eventually, surpasses the matter densty and becomes the domnant component. At ths pont, slows to a crawl and w 1as Ω 1 and the unverse s drven nto an acceleratng phase. These propertes seem to match current observatons well.. Famly of Tracker Solutons 4 For a potental V ( ) M v( M ) = (where ṽ s a dmensonless functon of M ), there s a famly of tracker solutons parameterzed by M. The value of M s determned by the measured value of Ω m today (assumng a flat unverse). v. Tracker Equaton (wth quntessence as an eg.) Many models of quntessence have a tracker behavor, whch partly solves the cosmologcal constant problem. In these models, the quntessence feld has a densty whch closely tracks (but s less than) the radaton densty untl matter-radaton equalty, whch trggers quntessence to start havng characterstcs smlar to dark energy, eventually domnatng the unverse. Ths naturally sets the low scale of the dark energy. Although the cosmc concdence ssue remans unresolved, the fne tunng problem facng dark energy/quntessence models wth a constant equaton of state can be sgnfcantly allevated f we assume that the equaton of state s tme dependent. An mportant class of models havng ths property are scalar felds whch couple mnmally to gravty and whose energy momentum tensor s

21 1 0 1 T0 = + V 1 ( ), p T = V ( ) (1) The equaton-of-moton for the feld s ' H + V = and the equaton-of-state s w p V = = + V ( ) ( ) equaton-of-moton as,.it s extremely useful to combne these relatons nto new form for the ' V 1 d ln x 3 1 w 1 V 6 d ln a κ ± = + + Ω () 1+ w 1 x = = 1 w ( ), where ( ) V s the rato of the knetc to potental energy densty for and a prme means a dervatve wth respect to. The ± sgn depends on whether ' V > 0 or ' V > 0 respectvely. The trackng soluton (to whch general solutons converge) has the property that w s nearly constant and les between w and -1. For 1+ w = Ο( 1 ), Ω H B and the ' V 1 H equaton-of-moton [Eqn. ()] dctates that, for a trackng soluton; ths s V Ω referred to as the tracker condton. [5] A scalar feld rollng down ts potental slowly generates a tme-dependent -term snce P - - V( ) f << V( ). Potentals whch satsfy V"V / (V') 1 have the nterestng property that scalar felds approach a common evolutonary path from a wde range of ntal condtons. In these so-called `tracker' models the scalar feld densty (and ts equaton of state) remans close to that of the domnant background matter durng most of cosmologcal evoluton. Trackng behavor wth constant d ( Γ Hdt) w < w occurs for any potental n whch V"V / (V') >1 and s nearly B 1/ Γ 1 over the range of plausble ntal. Ths case s relevant to tracker models of quntessence snce we want w < 0 today. The range of ntal condtons extends from V ( ) equal to the ntal background energy densty B down to V ( ) equal to the background densty at matter-radaton equalty, a span of over 100 orders of magntude. The testng for the exstence of trackng solutons reduces to a smple condton on V ( ) wthout

22 havng to solve the equaton-of-moton drectly. The condton Γ > 1 s equvalent to the constrant that ' V be decreasng as V decreases. These condtons encompass an extremely V broad range of potentals. A good example s provded by the exponental potental ( ) = ( ) 1 V V0 exp 8π λ M pl for whch ( + w ) 3 1 B = + λ B =constant<0. (3) B s the background energy densty whle w B s the assocated equaton of state. The lower lmt / total < 0. arses because of nucleosynthess constrants whch prevent the energy densty n quntessence from beng large ntally (at t ~ few sec.). Snce the rato remans fxed, exponental potentals on ther own cannot supply us wth a means of generatng dark energy/quntessence at the present epoch. However a sutable modfcaton of the exponental acheves ths. For nstance the class of potentals has the property that w ( ) [ λ ] V = V cosh 1 p 0 (4) wb at early tmes whereas w ( p 1 ) ( p 1 = + ) at late tmes. Consequently Eqn. (4) descrbes quntessence for p (CDM) for p = 1. A second example of a tracker-potental s provded by ( ) 0 total 1/ and pressure-less `cold' dark matter V α = V. Durng trackng the rato of the energy densty of the scalar feld (quntessence) to that of radaton/matter gradually ncreases t 4 + α B whle ts equaton of state remans margnally smaller than the w = αw B α +. These propertes allow the scalar feld to eventually background value ( ) ( ) domnate the densty of the unverse, gvng rse to a late-tme epoch of accelerated expanson. (Current observatons place the strong constransα.) Several of the quntessental potentals lsted n Table 7 have been nspred by feld theoretc deas ncludng super-symmetrc gauge theores and super-gravty, pseudo-goldstone boson models, etc [1]. However accelerated expanson can also arse n models wth: () topologcal defects such as a frustrated network of cosmc strngs (w - 1/3) and doman walls (w - /3); () scalar feld Lagrangans wth non-lnear knetc terms and no potental term (k-essence); () vacuum polarzaton assocated wth an ultra-lght scalar feld; (v) non-mnmally coupled scalar

23 3 felds; (v) felds that couple to matter; (v) scalar-tensor theores of gravty; (v) brane-world models etc. Quntessence Potental Table 7 Reference V0 exp( λ ) Ratra & Peebles (1988), Wetterch (1988), Ferrera & Joyce (1998) 4 m, λ Freman et al (1995) α V 0, α > 0 Ratra & Peebles (1988) ( ) V exp α, 0 0 λ α > Brax & Martn (1999,000) V0 ( cosh λ 1) p 0 V 0 snh α ( λ ) ( ) ακ βκ V e e Sahn & Wang (000) Sahn & Starobnsky (000), Ureña-López & Matos (000) + Barrero, Copeland & Nunes (000) ( p ) V0 exp M 1 Zlatev, Wang & Stenhardt (1999) ( ) α V 0 B + A e λ Albrecht & Skords (000) FIG.. A plot comparng two tracker solutons 6 for the case of a V 1 potental (sold lne) and a V exp( 1 ) potental (dot dashed lne). The dashed lne s the background densty. The two tracker solutons were chosen to have the same energy densty ntally. [Credt: - Stenhardt et.al (1999)]

24 4 From the Fg., t s seen that the tracker soluton for the generc example [ V exp( 1 ) ] reaches the background densty much later than for the pure nverse-power law potental. Hence, Ω s more lkely to domnate late n the hstory of the unverse n the generc case. Scalar feld based quntessence models can be broadly dvded nto two classes: () those for whch M pl 1 as t t 0, () those for whch M pl 1 as t t 0 (t 0 s the present tme). An mportant ssue concernng the second class of models s whether quantum correctons become mportant when 1 and ther possble effect on the quntessence potental. One can also M pl ask whether a gven choce of parameter values s `natural'. Consder for nstance the potental 4+ V M α α =, current observatons ndcate V GeV 4 and α, whch together suggest M 0.1 GeV (smaller values of M arse for smallerα ) t s not clear whether such small parameter values can be motvated by current models of hgh energy physcs. References: [1] Sahn, V, Dark Energy, COSMOLOGY AND GRAVITATION: XI th Brazlan School of Cosmology and Gravtaton. AIP Conference Proceedngs,Vol.78, pp (005). [] Copeland, Edmund J; Sam, M; Tsujkawa, Shnj, Dynamcs of Dark Energy, Internatonal Journal of Modern Physcs D, Vol. 15, Issue 11, pp (006). [3] Kovsto, Tom; Mota, Davd F., Journal of Cosmology and Astropartcle Physcs, Issue 08, pp. 01 (008). [4 Beltran Jmenez, Jose; Maroto, Antono L., arxv:astro-ph/ [5] Stenhardt, P J; Wang, L; Zlatev, I, Cosmologcal Trackng Solutons, Phys.Rev D, Vol. 59, (1999)