Phantom and phantom dark energy

Transcription

1 Phantom and phantom dark energy Mariusz P. Da browski Institute of Physics University of Szczecin Poland Phantom and phantom dark energy p. 1/5

2 References MPD, T. Stachowiak and M. Szydłowski, PRD 68, (2003) (hep-th/ ) MPD, and T. Stachowiak, Annals of Physics (N. Y.) 321 (2006) (hep-th/ ) MPD, PLB 625, 184 (2005) (gr-qc/ ) MPD, Ann. Phys. (Leipzig) 15, 362 (2006) (astro-ph/ ) A. Balcerzak and MPD, PRD 73, (R) (2006) (hep-th/ ) MPD, C. Kiefer and B. Sandhoefer, PRD 74, (2006) (hep-th/ ) MPD, T. Denkiewicz, M.A. Hendry, PRD 75, (2007) (astro-ph/ ) MPD, T. Denkiewicz, arxiv: MPD, T. Denkiewicz, M.A. Hendry, H. Ghoshi, in progress Phantom and phantom dark energy p. 2/5

3 Content: 1. Observational evidence for dark energy. Statefinders. 2. Phantom dark energy. Phantom domination in future - Big-Rip as an exotic singularity. Phantom duality. 3. Classical and quantum phantom cosmologies. 4. Quintom, hessence, phantom in brane cosmology, phantom in loop quantum cosmology. 5. Other exotic cosmological singularities and statefinder blow-up. 6. Timely nearness of pressure-driven singularity in future. 7. Conclusions Phantom and phantom dark energy p. 3/5

4 1. Observational evidence for dark energy and statefinders. A redhift-magnitude relation (generalized Hubble Law) applied to supernovae Ia caused the biggest revolution in cosmology at the turn of 20th century: ( ) 5 m(z) M = 5 log 10 (cz) 5 log 10 H 0 2 log 10 e { (1 q 0 )z [ q0 ( ) 9 2 q j Ω K0 1 [ ( 2j0 (8q 0 + 5) 2k 0 q 0 7q q ) Ω K0 (2q 0 + 1)] z 3 + O(z 4 ) }. Here we also admitted statefinders j, k. ] z 2 Phantom and phantom dark energy p. 4/5

5 Statefinders (jerk, snap etc.) Statefinders are the higher-order characteristics of the universe expansion which go beyond the Hubble parameter and the deceleration parameter They can generally be expressed as (i 2) H = ȧ a, q = 1 H 2 ä a = äa ȧ 2. (1) x (i) = ( 1) i+1 1 H i a (i) a = ( 1)i+1a(i) a i 1 ȧ i, (2) and the lowest order of them are known as: jerk, snap ("kerk"), crack ("lerk") j = a H 3 a = aa 2, k = a ȧ 3 H 4 a = a a 3, l = 1 a (5) ȧ 4 H 5 a = a(5) a 4,(3) ȧ 5 the and pop ("merk"), "nerk", "oerk", "perk" etc. (Harrison 76, Landsberg 76, Chiba 98, Alam et al. 03, Sahni et al. 03, Visser 04, Caldwell, Kamionkowski 04, MPD+Stachowiak 06, Dunajski and Gibbons 08) Phantom and phantom dark energy p. 5/5

6 It changed standard cosmology based on Einstein-Friedmann equations which are two equations for three unknown functions of time a(t), p(t), ϱ(t) ( ȧ2 ϱ = 3 a 2 + K ) a 2, (4) ( p = 2ä a + ȧ2 a 2 + K ) a 2. (5) plus an equation of state, e.g., of a barotropic type (w = const. 1): p(t) = wϱ(t) (6) gives three standard solutions - each of them starts with Big-Bang singularity in which a 0, ϱ, p one of them (of K = +1) terminates at the second singularity (Big-Crunch) where a 0, ϱ, p the other two (K = 0, 1) continue to an asymptotic emptiness ϱ, p 0 for a. Phantom and phantom dark energy p. 6/5

7 standard cosmology by standard we meant the obedience of the strong energy condition (SEC) of Hawking and Penrose R µν V µ V ν 0, V µ a timelike vector, (7) (R µν - Ricci tensor). In terms of the energy density and pressure it is equivalent to ϱ + 3p 0, ϱ + p 0. (8) From (1) and (2) one has 4πG 3 which together with (5) means that (ϱ + 3p) = ä a = qh2, (q = äa ȧ 2 ; H = ȧ a ), (9) ä 0, or q 0, (10) so that the universe should decelerate its expansion in this standard case. Phantom and phantom dark energy p. 7/5

8 SCP + High-z Team ( 98-99) made the plot to determine q 0 ( j 0 = k 0 = 0) Supernova Cosmology Project (Ω Μ, Ω Λ ) = ( 0, 1 ) (0.5,0.5) (0, 0) ( 1, 0 ) (1, 0) (1.5, 0.5) (2, 0) Flat Λ = 0 effective m B Calan/Tololo (Hamuy et al, A.J. 1996) redshift z and showed that the best-fit model is for q 0 = Ω M /2 Ω Λ < 0, so that ä > 0 (Perlmutter et al. (1999)) - favours dark energy with negative pressure such as a Λ term or quintessence ( 1 w 0). It leads to the strong energy condition (5) violation. Phantom and phantom dark energy p. 8/5

9 Λ term dark energy and Λ term domination in future Provided only w = const. 1 matter appears in the universe (just the strong energy condition is violated), the cosmological constant (w = 1) of any small fraction will always dominate Energy density radiation dust domain walls Λ cosmological constant a(t) Despite, as Fillipenko et al. 99 showed, the best fit value of w was 2/3 (domain walls) Phantom and phantom dark energy p. 9/5

10 Recent supernovae data WMAP + SDSS + Supernovae combined bound on the dark energy barotropic index w (Tegmark et al. (2004)): There is no sharp cut-off of the data at p = ϱ!!! Dark energy with p < ϱ is likely! Phantom and phantom dark energy p. 10/5

11 This is confirmed by other data: Knop et al (from SNe + CMB + 2dFGRS combined) w = (statistical) ±0.09 (systematic) Riess et al (w < 1) Seljak et al. astro-ph/ w = 1.04 ± 0.06 Then p < ϱ matter is a serious (though troublesome - see later) candidate for dark energy though more recently Kowalski et al. (arxiv: ) analyzed 307 supernovae (Sne + BAO + CMB) w = (statistical) (systematic) The point is still that even a small fraction of w < 1 dark energy (not the Λ-term) will dominate the evolution (see later) Phantom and phantom dark energy p. 11/5

12 2. Phantom dark energy. Phantom domination in future - Big- Rip as an exotic singularity. Phantom duality. Phantom is dark energy of a very large negative pressure (Caldwell astro-ph/ published in PLB 2002; after 2002 tens of refs which are not listed p < ϱ, or w < 1, (11) which violates all the remained energy conditions, i.e., the null (NEC) the weak (WEC) T µν k µ k ν 0, k µ a null vector, i.e., ϱ + p 0, (12) T µν V µ V ν 0, V µ a timelike vector, i.e., ϱ + p 0, ρ 0, (13) and the dominant energy (DEC) T µν V µ V ν 0, T µν V µ not spacelike, i.e., p ϱ, ϱ 0. (14) Phantom and phantom dark energy p. 12/5

13 Simplest example of phantom: a scalar field with negative kinetic energy (l = 1 phantom; l = +1 a standard scalar field) so that L = l 2 µφ µ φ V (φ) (15) ρ l φ 2 2 p l φ V (φ), (16) V (φ), (17) and so ϱ + p = l φ 2. (18) Null energy condition is violated for l = 1! Phantom and phantom dark energy p. 13/5

14 Interesting points about phantom: easily appears in Brans-Dicke theory (Einstein frame) for ω < 3/2 S = d 4 x [ R ( ω + 3 ) ] µ φ µ φ (19) which for ω = 1 is equivalent to low-energy-effective-string action in string frame also in bulk viscous stress models (due to particle production) in higher-order gravity theories, k-essence; in superstring and brane cosmology etc. transition from a collapse to an expansion impossible without null energy condition violation (i.e. without effective phantom) - cf. in ekpyrotic models (Turok et al. 2002) Phantom and phantom dark energy p. 14/5

15 Phantom trouble: classical and quantum instabilities (e.g. Buniy and Hsu hep-th/ ) - due to negative kinetic energy of phantom particles scattered rather gain than lose energy positive mass theorems fail black hole thermodynamics is under trouble (cf. Gonzalez-Diaz astro-ph/ ; Pereira and Lima ) cosmic censorship conjecture fails (all three above rely on the energy conditions) But see recent papers: Cline et al. PRD 70 (2004), ; Buniy, Hsu, Murray hep-th/ ; Rubakov hep-th/ ; (no negative kinetic terms, NEC violated) Creminelli et al. hep-th/ (no instabilities, NEC violated) Phantom and phantom dark energy p. 15/5

16 Radical approach Proposes alternative explanation of supernovae dimming due to Λ + photon axion conversion. Mechanism affects only SnIa which is most strongly biased towards w < 1. Phantom and phantom dark energy p. 16/5

17 Phantom domination in future - Big-Rip. Phantom duality. Phantom divide. For convenience take w + 1 = (w + 1) > 0, (20) so that the conservation law for phantom gives ϱ a 3 w+1. (21) Conclusion: the bigger the universe grows, the denser it is, and it becomes dominated by phantom (overcomes Λ-term) a future singularity appears Big-Rip ϱ, p for a Curvature invariants R 2, R µν R µν, R µνρσ R µνρσ diverge at Big-Rip Only for 5/3 < w < 1 the null geodesics are geodesically complete; for other values of w, including all timelike geodesics, there is a geodesic incompleteness (Lazkoz et al. gr-qc/ , PRD 07) - the singularity is reached in an infinite proper time. Phantom and phantom dark energy p. 17/5

18 Future evolution towards Big-Rip In a Big-Rip scenario everything is pulled apart on the approach to a Big-Rip in a reverse order (Caldwell et al. PRL 03). Specifically, for w = 3/2 Big-Rip will happen in 20 Gyr from now and the scenario will be as follows: in 1 Gyr before BR - clusters are erased in 60 Myr before BR - Milky Way is destroyed 3 months before BR - Solar System becomes unbound 30 min before BR - Earth explodes s before BR - atoms are dissociated nuclei etc.... All this comes from the formula t P 2 w + 1 /[6π 1 + w ], where P is the period of a circular orbit around the system at radius R, mass M. Phantom and phantom dark energy p. 18/5

19 Phantom duality BR to BR model as dual to standard BB to BC model Scale factor, a(t) a max Scale factor, a(t) time, t a min a(t) Big Bang time, t Big Crunch Big Rip Big Rip time, t Duality: Standard matter (p > ϱ) Phantom (p < ϱ) w (w + 2) or better γ γ γ = w + 1 (22) i.e. a(t) 1 a(t) (23) Phantom and phantom dark energy p. 19/5

20 < 0 t > 0 curvature and strong t singularity at t = 0 coupling Phantom duality is similar to scale factor duality in superstring (pre-bigbang) cosmology (Meissner, Veneziano 92) 3 4 a(t) 1 2 t Dual branches: pre-big-bang superinflationary (1) and post-big-bang radiation-dominated (4) a 1 (t) 1 a 4 (t), φ(t) φ(t) lna6 (t) (24) Phantom and phantom dark energy p. 20/5

21 Phantom divide The value of w = 1 or γ = 0 (the Λ term) plays a role of the border in phantom duality - a border between the standard/quintessential matter and the phantom matter it is often called phantom divide. There are many dynamical system studies of the problem of crossing the phantom divide and possible return of the universe evolution to a non-phantom phase which would allow to avoid a Big-Rip (e.g. Bouhmadi-Lopez, Madrid astro-ph/ , Babichev et al. astro-ph/ etc.). Phantom and phantom dark energy p. 21/5

22 Explicit examples of phantom duality, form-invariance exact dual classical solution standard/phantom matter + negative Λ-term (MPD et al 03) a w+1 = [ A sin ( )] 2 3(w+1) 3 2 w + 1 ( Λ) 1 2 t = 1 a (w+1), (25) Duality: dust (w=0) hyperphantom (w=-2) strings (w=-1/3) superphantom (w =-5/3) walls (w=-2/3) phantom (w=-4/3) Other name: form-invariant symmetry (Lazkoz et al 03) - for a perfect fluid H H, ϱ + p (ϱ + p) (26) Phantom and phantom dark energy p. 22/5

23 Phantom triality - for a scalar field (Wick rotated φ iφ) φ 2 φ 2, (27) V ( φ) φ 2 + V (φ). (28) phantom triality (Lidsey 04) - phantom duality extended to a correspondence between standard, brane (cyclic) and phantom models - generalizes conservation of the spectral index n s w.r.t. a change from rapid accelerated expansion (ɛ H/Ḣ 2 1) to slow decelerated contraction (ɛ 1) (Boyle et al. hep-th/ ) (λ 2 < 6) a = t 2/λ2 ã = t λ2 /2 â = ( t) λ2 /2, (29) φ = (2/λ) ln t φ = λlnt ˆφ = λln( t), (30) V = 2λ 4 (6 λ 2 )e λφ Ṽ = 4 1 λ 2 (3λ 2 2)e 2φ/λ ˆV = 4 1 λ 2 (3λ 2 + 2)e 2φ/λ. (31) Phantom and phantom dark energy p. 23/5

24 3. Classical and quantum phantom cosmologies Admission w < 1 enlarges possible set of cosmological solutions. The most desirable are the solutions which begin at Big-Bang and terminate at Big-Rip. Scale factor, a(t) a f t f Big Bang Beginning of phantom domination z=0,46 + 0,13 Big Rip time, t Advantage: preserves standard Hot-Big-Bang results (turning point at z = 0.46 (since j 0 > 0) Riess et al. 2004) Phantom and phantom dark energy p. 24/5

25 Exact radiation + superphantom (w = 5/3) model, anisotropic models a a 0 = ([ 1 2 Ω sp0(η η 0 ) ] 2 Ωr0 Ω sp0 ) 1 6 dη = ( a a 0 ) 4 H 0 dt (32) in homogeneous (Bianchi type) models shear anisotropy σ cannot dominate over the phantom matter on the approach to a Big-Rip singularity since for a. ϱ ph a 3 w+1 > σ2 a 6 0 (33) Phantom and phantom dark energy p. 25/5

26 Quantum phantom cosmologies In a model of phantom field with negative Λ term and phantom potential V (φ) = V 0 cosh 2 (φ/f) (V 0, F =const.) we have a trajectory in configuration space (no time coordinate) as: (left - phantom, right - an ordinary scalar field) φ(a) = ± 1 D 3 D l ln A +. (34) A 2 a2d a D φ φ + φ φ 0 φ a a Phantom and phantom dark energy p. 26/5

27 Wheeler-DeWitt equation Wheeler-DeWitt equation for phantom (l = 1) becomes elliptic [ 2 2 a a a a l ( φ 2 + a6 V (φ) + Λ ) ] Ka4 ψ(a, φ) = 0. (35) 6 2 In the standard scalar field case (Kiefer 88, Kiefer and Zeh 95) in minisuperspace wavepackets do not follow classical trajectories everywhere in configuration space. They disperse near the region of Big-Bang and Big-Crunch singularity. They also disperse near the point of maximum expansion (for a massive scalar field) showing that there might be large-scale (macroscopic) quantum effects in this region. Near Big-Rip an extremely high energy density (of Planck scale) accompanies an extremely large size of the universe so that it seems that the same is true at that region. Phantom and phantom dark energy p. 27/5

28 Solutions to the Wheeler-DeWitt equation For large φ (where the classical singularities appear), one can approximate the cosh potential by, Ṽ (φ) V 0 4 e ±2φ F, (36) The Wheeler DeWitt equation can be simplified by a transformation of the variables (a = e α ) u l (α, φ) v l (α, φ) V0 3 2 V0 3 2 e 3α± φ F ( 1 l 9F 2 ) 2 e 3α± φ F ( 1 l 9F 2 ) 2 where X l(3φ ± lα/f). In these variables, we get 2 ( 2 Ψ u 2 l ( l cosh(x) ± 1 ) 3F l sinh(x) ( 1 l sinh(x) ± 1 ) 3F cosh(x) l 2 Ψ v 2 l ) + Ψ = 0. (37), Phantom and phantom dark energy p. 28/5,

29 WKB approximation ansatz Making a WKB-approximation ansatz, Ψ = Ce ± i S, one obtains at lowest order the Hamilton Jacobi equation ( ) 2 ( ) 2 S0 S0 l = 1, (39) u l v l which can be solved via a separation ansatz by S 0k = ku l l(k 2 1)v l. The equations of motion obtained for S 0k k k= k = 0 are φ(α) = 1 3 D l α + C k,l. (40) Exact wave packet solution to the Wheeler DeWitt equation reads as Ψ(u, v) = dk A(k) ( C 1 e i (ku l l(k 2 1)v l ) + C 2 e i (ku l l(k 2 1)v l ) ). (41) Phantom and phantom dark energy p. 29/5

30 Spreading of the wavepackets at Big-Rip v u u v The wave packets following classical trajectories (l = 1 phantom) (left), (l = 1 ordinary scalar field) (right) spread near the Big-Rip and Big-Bang, respectively. The former signals large-scale quantum effects near Big-Rip!!! Interesting point: is there any spreading near the inflection point (z = 0.46) in the BB-BR model - this would test if we live near a genuine quantum region!? Phantom and phantom dark energy p. 30/5

31 4. Quintom, hessence, phantom in brane cosmology, phantom in loop quantum cosmology. Quintom is a combination of quintessence and phantom (Feng et al. astro-ph/ ; Wei et al. gr-qc/ ): L = 1 2 µφ q µ φ q µφ p µ φ p V (φ q, φ p ) (42) which gives w = p ρ = φ2 q φ 2 p 2V (φ p, φ q ) φ 2 q φ 2 p + 2V (φ p, φ q ) (43) and leads to both phantom domination ( φ 2 p > φ q 2 ) and quintessence domination (...<...) during the evolution of the universe together with phantom divide crossing. On the other hand, oscillating quintom fulfills the equation of state (Feng et al. astro-ph/ ) w = w 0 + w 1 cos[a lna/a c ]. (44) Phantom and phantom dark energy p. 31/5

32 Hessence is a non-canonical complex scalar field Φ = φ 1 + iφ 2 which by introducing new variables (Weng et al. hep-th/ , astro-ph/ , astro-ph/ ) gives the Lagrangian φ 1 = φ coshθ, φ 2 = φ sinhθ (45) L = 1 2 [ µ φ µ φ φ 2 µ θ µ θ ] V (φ) (46) so that w = p ρ = φ 2 φ 2 θ2 2V (φ) φ 2 φ 2 φ 2 + 2V (φ) (47) and it again plays the role of both phantom for φ 2 < φ 2 θ2 and quintessence (...>...). Phantom and phantom dark energy p. 32/5

33 Phantom in brane cosmology Phantom brane (RSII) cosmologies (MPD et al. IJMPD 04, Gergely et al. 06) H 2 = 8π 3m 2 pl(4) (ρ + ρ2 2λ ) k a 2 + Λ m pl(4)u 8πλ, (48) where λ is brane tension and U is dark radiation. Phantom in brane cosmology helps to fit supernovae, but not CMBR and nucleosynthesis. (For DGP brane model fit see e.g. Lazkoz et al. astro-ph/ ) Note: there is a possibility for a bounce H = 0 in these models provided brane tension is negative at the energy density ρ b = 2 λ and λ = λ > 0. (49) This means that non-singular oscillating cosmologies are possible in brane cosmology (e.g. MPD et al. 04, Freese et al. astro-ph/ ). Phantom and phantom dark energy p. 33/5

34 Phantom accretion onto a black hole Phantom behaviour (pulling apart) may destroy black holes which would start dominating at the contraction. Phantom matter accretion onto a black hole may cause its mass diminishing (Babichev, Dokuchaev, Eroshenko 2004) Effectively same as Hawking radiation. Phantom and phantom dark energy p. 34/5

35 Phantom in loop quantum cosmology However, in standard brane models only extra timelike dimensions give a possibility for negative brane tension (e.g. Shtanov and Sahni PLB, 557 (2003), 1). It is amazing that in loop quantum cosmology the term which simulates negative brane tension appears in a natural way (Bojowald PRL 02, gr-qc/ , Ashtekar et al. 06, Copeland et al. gr-qc/ etc.): H 2 = 1 3m 2 pl ) (ρ ρ2 ρ c k a 2, (50) where the critical density is ρ c 3 16πγ 3 G 2 2, (51) and γ is the Barbero-Immirzi parameter (γ , see Meissner gr-qc/ ). Phantom and phantom dark energy p. 35/5

36 Phantom in loop quantum cosmology Big-Rip singularity can be avoided in LQC (e.g. Sami et al. gr-qc/ ) even if phantom is present - a bounce. Energy exchange phantom-to-matter model (e.g. Gumjudpai gr-qc/ ) in LQC does not change the result though energy exchange may alter the bouncing time. Quintom, hessence in LQC (e.g. Wei et al. gr-qc/ ) admit stable attractors which are non-phantom (w > 1) and there is no Big-Rip singularity. Phantom duality versus PBB scale factor duality - in LQC the dual solutions in PBB are regularized (De Risi et al. hep-th/ ) and so the simple phantom duality may not work in that context (but see also phantom triality, Lidsey 04). Phantom and phantom dark energy p. 36/5

37 5. Other exotic cosmological singularities and statefinder blowup. Big -Rip appears at some finite future time t = t BR 20 Gyr in analogy to Big-Crunch at t = t BC. Some people call it sudden, but we can have sth really sudden (as we will learn later). Sudden Future Singularity manifests as a singularity of pressure (or ä) only leads to the dominant energy condition violation only The hint (Barrow 04): release the assumption about the imposition of an equation of state p p(ϱ), no analytic form of this relation is given (52) Choose a special form of the scale factor (may be motivated in fundamental cosmologies) as: a(t) = a s [1 + (1 δ)y m δ (1 y) n ], where a s a(t s ) = const. and δ, A, m, n = const. Obviously, for t = 0 one has a Big-Bang singularity. y t t s (53) Phantom and phantom dark energy p. 37/5

38 But what about t = t s? One has to look onto the derivatives of a! ȧ = a s [ m t s (1 δ)y m 1 + δ n t s (1 y) n 1 ] ä = If we assume that a s t 2 s, (54) [ m (m 1) (1 δ)y m 2 δn (n 1) (1 y) n 2]. 1 < n < 2, (55) then using Einstein equations (1)-(2) we get a = const., ȧ = const. ϱ = const. ä p ± for t t s (56) Phantom and phantom dark energy p. 38/5

39 Sudden future singularity It is only the pressure singularity with finite scale factor and the energy density. Possible universe evolution scenario: Scale factor, a(t) Big Bang Sudden Future Singularity time, t (or upwards in a similar way as in phantom scenario with an inflection point) Phantom and phantom dark energy p. 39/5

40 Friedmann limit - nonstandardicity parameter δ Friedmann limit is easily obtained in the limit of nonstandardicity parameter δ 0. The parameter m can be taken to be just a form of the w parameter present in the barotropic equation of state: Near to SFS one has 0 < m 1 when w 1/3 (standard matter); m > 1 when 1 < w < 1/3 (quintessence); m < 0 when w < 1 (phantom). and n plays the role in making a pressure blow-up. a SFS = a s [1 δ(1 y) n ], (57) Important point: Unless we take m > 1 or m < 0 as an independent source of energy, pressure-driven dark energy (acceleration) is possible only for δ < 0! We can call it pressure-driven dark energy (whatever it is!). Phantom and phantom dark energy p. 40/5

41 SFS as a weak singularity SFS are determined by a blow-up of the Riemann tensor and its derivatives Geodesics do not feel SFS at all, since geodesic equations are not singular for a s = a(t s ) = const. (Fernandez-Jambrina, Lazkoz 04, also gr-qc/ ) ( dt dτ ) 2 = A + P 2 + KL 2 dr dτ dφ dτ Geodesic deviation equation D 2 n α a 2, (58) (t) = P 1cosφ + P 2 sin φ 1 Kr2 a 2, (59) (t) L = a 2 (t)r. (60) 2 dλ 2 + R α βγδu β n γ u δ = 0, (61) feels SFS since at t = t s we have the Riemann tensor R α βγδ. Phantom and phantom dark energy p. 41/5

42 No geodesic incompletness There is a difference between the behaviour of the point particles and extended objects (for example strings) at SFS. Point particles do not even see SFS while extended objects may suffer instantaneous infinite tidal forces but still may not be crushed. No geodesic incompletness (a = const. and r.h.s. of geodesic eqs. do not diverge) SFS are not the final state of the universe They are weak singularities (Tipler 77, Królak 86) - τ 0 dτ R i 0j0 (τ ) does not diverge on approach to asingularity at τ = τ s Big-Rip and SFS are of different nature Interesting comparison: Schwarzschild spacetime at r = r g - metric singular, curvature invariants regular Sudden singularity models - metric regular, curvature invariants diverge Phantom and phantom dark energy p. 42/5

43 Generalized Sudden Future Singularity, type III and IV, Big-Brake Take a general scale factor time derivative of an order r: [ m(m 1)...(m r + 1) a (r) = a s t r (1 δ)y m r s ] + ( 1) r 1 n(n 1)...(n r + 1) δ t r (1 y) n r s, (62) Choosing (Barrow 04, Lake 04) r 1 < n < r, for any integer r we have a singularity in the scale factor derivative a (r), and consequently in the appropriate pressure derivative p (r 2). Note that it fullfils all the energy conditions for any r 3!!! Type III (Nojiri, Odintsov 04): a = a s = const., ϱ, p Type IV (Nojiri, Odintsov, Tsujikawa 05): a = a s = const., ϱ 0, p 0,... a etc. Big-Brake = SFS (Gorini, Kamenschchik et al. PRD 69 (2004), ): a = a b = const., ȧ = 0 and ä, so p. Phantom and phantom dark energy p. 43/5

44 Quantum cosmology of Big-Brake singularity One finds also an explicit model of pressure singularity with anti-chaplygin equation of state p = A ρ. (63) The study of quantum cosmological Wheeler-DeWitt equation for this model (Kamenschchik et al. gr-qc/ ) shows that, despite Big-Rip, the wave packets vanish in the region of classical Big-Brake (so then SFS) singularity. This means there is no sigularity there at the quantum sense, too. It is consistent with what we have already said about geodesic completness (or mildness) of these singularities in the classical sense. In the context of LQC one obtains similar result for Big-Brake and also for Big-Bang which is expected due to LQC regularization. Phantom and phantom dark energy p. 44/5

45 Energy conditions and exotic dark energy Big-Crunch may emerge despite all the energy conditions are fulfilled Big-Rip may emerge despite the energy conditions are not fulfilled SFS emerges when only the dominant energy condition is violated Generalized SFS do not lead to any violation of the energy conditions. Conclusion: an applicability of the standard energy conditions to cosmological models with more exotic properties is not very useful. Suggestion: formulate some different energy conditions which may be helpful in classification of exotic singularities. Phantom and phantom dark energy p. 45/5

46 Statefinders and generalized energy conditions As an example assume the first-order dominant energy condition in the form ( ϱ + ṗ = 2H 3 j + 3q K ) a 2 H 2 0, (64) ( ϱ ṗ = 2H 3 j + 3q K ) a 2 H 2 0, (65) Result: It is not possible to fulfill this dominant energy condition if a statefinder j is singular. This is because the signs in the appropriate expressions in front of it are the opposite. Conluclusion: The violation of this first-order dominant energy condition can be a good signal for the emergence of the generalized SFS. Interesting point: Blow-up of statefinders (possibly read-off a redshift-magnitude relation) may be linked to an emergence of future singularities: H 0 BC, BR, q 0 SFS, j 0 GenSFS, etc. Phantom and phantom dark energy p. 46/5

47 6. Timely nearness of pressure-driven singularity in future. We have said previously that Big-Rip may appear in 20 Gyr in future (quite safe for us!). Now we ask the question of how far in future may a pressure singularity appear (MPD et al. astro-ph/ )? In order to do so one should first compare SFS models with observational data from supernovae. The redshift formula in these models reads as 1 + z = a(t 0) a(t 1 ) = δ + (1 δ)ym 0 δ (1 y 0 ) n δ + (1 δ)y m 1 δ (1 y 1) n, (66) where y 0 = y(t 0 ) and y 1 = y(t 1 ). Phantom and phantom dark energy p. 47/5

48 A redshift-magnitude relation for SFS cosmology. We have m(z) = M 5 log 10 H log 10 [r 1 a(t 0 )(1 + z)], (67) where r 1 comes from null geodesic equation r1 0 dr 1 kr 2 = t0 t 1 cdt a(t). (68) For a rough estimation of the dark energy models (acceleration) we study the product (ä ) q 0 H 0 = ȧ 0 = (69) t 0 m(m 1)(1 δ)y0 m 2 δn(n 1) (1 y 0 ) n 2 y 0 m(1 δ)y0 m 1 + nδ (1 y 0 ) n 1, which should be negative. Phantom and phantom dark energy p. 48/5

49 Parameter space for pressure-driven dark energy H0 q y Parameter space (H 0 q 0, δ, y 0 ) for fixed values of m = 2/3, n = , t 0 = Gyr of the sudden future singularity models. There are large regions of the parameter space which admit cosmic acceleration q 0 H 0 < 0. (70) Phantom and phantom dark energy p. 49/5

50 SFS dark energy versus Λ-term dark energy (concordance cosmology - CC) Distance modulus µ L = m M for the CC model (H 0 = 72kms 1 Mpc 1, Ω m0 = 0.26, Ω Λ0 = 0.74) (dashed curve) and SFS model (m = 2/3, n = , δ = 0.471, y 0 = ) (solid curve). Open circles are for the Gold data and filled circles are for SNLS data. Phantom and phantom dark energy p. 50/5

51 Pressure singularity in the near future? Surprising remark: If the age of the SFS model is equal to the age of the CC model, i.e. t 0 = 13.6 Gyr, one finds that an SFS is possible in only 8.7 million years!!!. In this context it is no wonder that the singularities were termed sudden. It was checked that GSFS (generalized SFS - no energy conditions violation) are always more distant in future. That means the strongest of SFS type singularities is more likely to become reality. A practical tool to recognize them well in advance is to measure possible large values of statefinders (deceleration parameter, jerk, snap etc.)! Interesting point: SFS plague loop quantum cosmology! - see Wands et al Phantom and phantom dark energy p. 51/5

52 7. Conclusions Apart from quintessence 0 > p > ϱ, astronomical data gives a direct motivation to a non-standard type of matter phantom with p < ϱ as a candidate for dark energy. Phantom produces an exotic singularity Big-Rip in which (a and ϱ ) which is different from Big-Crunch. However, in some scenarios (quintom, hessence, brane cosmology, loop quantum cosmology) these may be avoided. Investigations of phantom inspired other searches for non-standard singularities in future (sudden future, generalized sudden future, type III, type IV, Big-Brake etc.) which, in fact, both classically and quantum mechanically are not true singularities as Big-Bang or Big-Rip. However, despite Big-Rip which may happen in 20 Gyr, these weak singularities (of tidal forces and their derivatives) may appear quite nearly in future. For example an SFS may even appear in 8.7 Myr with no contradiction with supernovae data. Phantom and phantom dark energy p. 52/5

53 conclusions contd. Big-Rip and other exotic singularities appear in fundamental theories of particle physics (scalar-tensor, superstring, brane, loop quantum cosmology etc.). These singularities may manifest themselves in the higher-order characteristics of expansion such as statefinders (jerk, snap, lerk/crack, merk/pop). Statefinders are then useful tools to detect a progress of the exotic singularities in the universe. In other ways they may be the tool to test fundamental theories of unified interactions. Phantom and phantom dark energy p. 53/5