Exotic singularities in the Universe and supernovae observations
Mariusz P. Da¸browski
Institute of Physics University of Szczecin Poland
Szczecin Cosmology Group http://cosmo.fiz.univ.szczecin.pl Grassmannian Conference in Fundamental Cosmology, 14-19.09.2009
Exotic singularities in the Universe and supernovae observations – p. 1/53 References
MPD, T. Stachowiak and M. Szydłowski, PRD 68, 103519 (2003) (hep-th/0307128)
MPD, and T. Stachowiak, Annals of Physics (N. Y.) 321 (2006) (hep-th/0411199)
MPD, PLB 625, 184 (2005) (gr-qc/0505069)
MPD, Ann. Phys. (Leipzig) 15, 362 (2006) (astro-ph/0606574)
A. Balcerzak and MPD, PRD 73, 101301(R) (2006) (hep-th/0604034)
MPD, C. Kiefer and B. Sandhoefer, PRD 74, 044022 (2006) (hep-th/0605229)
MPD, T. Denkiewicz, M.A. Hendry, PRD 75, 123524 (2007) (astro-ph/0704.1383)
MPD, T. Denkiewicz, PRD 79, 063521 (2009); arXiv:0902.3107
MPD, T. Denkiewicz, M.A. Hendry, H. Ghoshi, in progress
Exotic singularities in the Universe and supernovae observations – p. 2/53 Content:
1. Supernovae, dark energy and statefinders. Standard cosmological singularities. 2. Phantom dark energy. 3. Big-Rip (type I) as an exotic singularity. Phantom cosmologies and phantom duality. 4. Sudden Future Singularity (type II) as an exotic singularity. 5. Type III (Finite Scale Factor) exotic singularity. 6. Generalized Sudden Future, type IV and w singularities. − 7. Exotic cosmological singularities and statefinders blow-up. 8. Exotic singularities versus supernovae data.
9. Conclusions Exotic singularities in the Universe and supernovae observations – p. 3/53 1. Supernovae, dark energy and statefinders. Standard cosmo- logical singularities.
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 (cz) 5 log H log e − 10 − 10 0 2 10 1 q 9 7 (1 q )z + 0 q + 5 j Ω z2 − 0 3 2 2 0 − 0 − 4 − K0 1 2j (8q + 5) 2k q 7q2 + 11q + 23 + 25 24 0 0 − 0 − 0 0 0 3 4 +4Ω K0 (2q0 + 1)] z + O(z ) .
Here we also admitted statefinders j, k.
Exotic singularities in the Universe and supernovae observations – p. 4/53 Statefinders (jerk, snap etc.)
Statefinders are the higher-order characteristics of the universe expansion which go beyond the Hubble parameter and the deceleration parameter
a˙ 1 a¨ aa¨ H = , q = = . (1) a −H2 a − a˙ 2 They can generally be expressed as (i 2) ≥ 1 a(i) a(i)ai−1 x(i) = ( 1)i+1 =( 1)i+1 , (2) − Hi a − a˙ i and the lowest order of them are known as: jerk, snap ("kerk"), crack ("lerk") ...... 1 a aa2 1 a a a3 1 a(5) a(5)a4 j = = , k = = , l = = (3), H3 a a˙ 3 −H4 a − a˙ 4 H5 a a˙ 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) Exotic singularities in the Universe and supernovae observations – p. 5/53 It changed our view of cosmology based on Einstein-Friedmann equations
which are two equations for three unknown functions of time a(t),p(t),%(t)
a˙ 2 K % = 3 + , (4) a2 a2 a¨ a˙ 2 K p = − 2 + + . (5) a a2 a2 plus an equation of state, e.g., of a barotropic type (w = const. ≥−1):
p(t)= w%(t) (6)
gives three standard solutions - each of them starts with Big-Bang singularity in which a → 0, %,p → ∞ – one of them (of K = +1) terminates at the second singularity (Big-Crunch) where a → 0, %,p → ∞ – the other two (K = 0, −1) continue to an asymptotic emptiness %,p → 0 for a → ∞.
Exotic singularities in the Universe and supernovae observations – p. 6/53 “pre-supernovae” cosmology
based on a paradigm of the obedience of the strong energy condition (SEC) of Hawking and Penrose
µ ν µ Rµν V V ≥ 0, V − a timelike vector , (7)
(Rµν - Ricci tensor) which in terms of the energy density and pressure it is equivalent to
% + 3p ≥ 0, % + p ≥ 0 . (8)
From (1) and (2) one has
4πG a¨ 2 aa¨ a˙ − (% + 3p) = = −qH , (q = − ; H = ) , (9) 3 a a˙ 2 a
which together with (5) means that
a¨ ≤ 0, or q ≥ 0 , (10)
so that the universe should decelerate its expansion in this “pre-supernovae” case.
Exotic singularities in the Universe and supernovae observations – p. 7/53 SCP + High-z Team (’98-’99) made the plot to determine q0 ( j0 = k0 = 0)
(ΩΜ,ΩΛ) = 26 ( 0, 1 ) (0.5,0.5) (0, 0) ( 1, 0 ) (1, 0) 24 (1.5,–0.5) (2, 0) Supernova Cosmology Flat Project Λ = 0 22 B m
20 Calan/Tololo (Hamuy et al, A.J. 1996) effective 18
16
14 0.02 0.05 0.1 0.2 0.5 1.0 redshift z
and showed that the best-fit model is for q = Ω /2 Ω < 0, so that a>¨ 0 0 M − Λ (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. Exotic singularities in the Universe and supernovae observations – p. 8/53 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)
Any combination of dark energy with w = const. 1 leads to standard ≥− cosmological singularities: Big-Bang or Big-Crunch (or to emptiness - de Sitter).
Exotic singularities in the Universe and supernovae observations – p. 9/53 2. Phantom dark energy.
WMAP + SDSS + Supernovae combined bound on the dark energy barotropic index w (Tegmark et al. (2004)):
There was no sharp cut-off of the data at p = %!!! − Dark energy with p< % (phantom) can be admitted! − Exotic singularities in the Universe and supernovae observations – p. 10/53 More supernovae data:
Knop et al. 2003 (from SNe + CMB + 2dFGRS combined) – w = 1.05+0.15 (statistical) 0.09 (systematic) − −0.20 ± Riess et al. 2004 (w< 1) − Seljak et al. astro-ph/0604335 – w = 1.04 0.06 − ± though more recently Kowalski et al. (arXiv:0804.4142) analyzed 307 supernovae (Sne + BAO + CMB) – w = 1.001+0.059 (statistical) +0.063 − −0.063 −0.066 (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) Of course the equation of state may evolve and so w = w(t) during the evolution
Exotic singularities in the Universe and supernovae observations – p. 11/53 Phantom dark energy.
Phantom is dark energy of a very large negative pressure (Caldwell astro-ph/9908168 - published in PLB 2002; after 2002 tens of refs which are not listed
p< %, or w< 1 , (11) − − which violates all the remained energy conditions, i.e., the null (NEC)
T kµkν 0, kµ a null vector , i.e., % + p 0 , (12) µν ≥ − ≥ the weak (WEC)
T V µV ν 0, V µ a timelike vector , i.e., % + p 0, ρ 0 , (13) µν ≥ − ≥ ≥ and the dominant energy (DEC)
µ ν µ Tµν V V 0,Tµν V not spacelike , i.e., p %, % 0 . (14) ≥ − Exotic singularities in| the Universe|≤ and supernovae≥ observations – p. 12/53 Simplest example of phantom:
a scalar field with negative kinetic energy (l = 1 – phantom; l = +1 – a standard scalar field) − l L = ∂ φ∂µφ V (φ) (15) −2 µ − so that
φ˙2 ρ l + V (φ) , (16) ≡ 2 φ˙2 p l V (φ) , (17) ≡ 2 − and so % + p = lφ˙2 . (18)
Null energy condition is violated for l = 1! −
Exotic singularities in the Universe and supernovae observations – p. 13/53 Phantom trouble:
classical and quantum instabilities (e.g. Buniy and Hsu hep-th/0502203) - 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/0407421; Pereira and Lima 0806.0682) cosmic censorship conjecture fails (all three above rely on the energy conditions)
But see recent papers: Cline et al. PRD 70 (2004), 043543; Buniy, Hsu, Murray hep-th/0606091; Rubakov hep-th/0604153; (no negative kinetic terms, NEC violated) Creminelli et al. hep-th/0606090 (no instabilities, NEC violated)
Exotic singularities in the Universe and supernovae observations – p. 14/53 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. − Exotic singularities in the Universe and supernovae observations – p. 15/53 3. Big-Rip (type I) as an exotic singularity. Phantom cosmologies and phantom duality.
For convenience take w + 1 = (w + 1) > 0 , (19) | | − so that the conservation law for phantom gives
% a3|w+1| . (20) ∝ Conclusion: the bigger the universe grows, the denser it is, and it becomes dominated by phantom (overcomes Λ-term) – an exotic future singularity appears – Big-Rip %, p for a → ∞ → ∞ 2 µν µνρσ Curvature invariants R , Rµν R , RµνρσR diverge at Big-Rip Only for 5/3 Exotic singularities in the Universe and supernovae observations – p. 16/53 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 10−19 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 systemp at radius R, mass M. Exotic singularities in the Universe and supernovae observations – p. 17/53 Phantom duality BR to BR model as dual to standard BB to BC model Scale factor, Scale factor, a(t) a(t) a max time, t a min a(t) Big Rip Big Rip Big Bang time, t Big Crunch time, t Duality: Standard matter (p> %) Phantom (p< %) − ↔ − w (w +2) or better γ γ γ = w + 1 (21) ↔− ↔− i.e. 1 a(t) (22) ↔ a(t) Exotic singularities in the Universe and supernovae observations – p. 18/53 Phantom duality is similar to scale factor duality in superstring (pre-big- bang) cosmology (Meissner, Veneziano ’92) 3 4 a(t) 1 2 curva ture and strong t < 0 t > 0 coupling singularit y at t = 0 t Dual branches: pre-big-bang superinflationary (1) and post-big-bang radiation-dominated (4) 1 6 a1(t) , φ(t) φ(t) ln a (t) (23) ↔ a4(t) ↔ − Exotic singularities in the Universe and supernovae observations – p. 19/53 Explicit examples of phantom duality, form-invariance – exact dual classical solution standard/phantom matter + negative Λ-term (MPD et al ’03) 2 3(w+1) √3 1 1 aw+1 = A sin w + 1 ( Λ) 2 t = , (24) " 2 | | − !# a−(w+1) 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, % + p (% + p) (25) ↔− ↔− Exotic singularities in the Universe and supernovae observations – p. 20/53 Phantom triality - for a scalar field (Wick rotated φ¯ iφ) ↔ φ¯˙2 φ˙2, (26) ↔ − V¯ (φ¯) φ˙2 + V (φ). (27) ↔ – phantom triality (Lidsey ’04) - phantom duality extended to a correspondence between standard, brane (cyclic) and phantom models - generalizes conservation of the spectral index ns w.r.t. a change from rapid accelerated expansion ( H/H˙ 2 1) to slow decelerated contraction ( 1) ≡− (Boyle et al. hep-th/0403026) (λ2 < 6) 2 2 2 a = t2/λ a˜ = tλ /2 aˆ =( t)−λ /2, (28) ↔ ↔ − φ = (2/λ)ln t φ˜ = λ ln t φˆ = λ ln( t), (29) ↔ − ↔ − − V = 2λ−4(6 λ2)e−λφ V˜ = 4−1λ2(3λ2 2)e2φ/λ − ↔ − Vˆ = 4−1λ2(3λ2 + 2)e2φ/λ. (30) ↔ Exotic singularities in the Universe and supernovae observations – p. 21/53 Best phantom cosmology option 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 tf Big Beginning Big time, t Bang of phantom Rip domination z=0,46 −+ 0,13 Advantage: preserves standard Hot-Big-Bang results (turning point at z = 0.46 (since j0 > 0) Riess et al. 2004) Exotic singularities in the Universe and supernovae observations – p. 22/53 Quintom. Phantom accretion. Quintom is a combination of quintessence and phantom (Feng et al. astro-ph/0404224; Wei et al. gr-qc/0705.4002): 1 1 L = ∂ φ ∂µφ + ∂ φ ∂µφ V (φ ,φ ) (31) −2 µ q q 2 µ p p − q p which gives 2 2 p φ˙ φ˙ 2V (φp,φq) w = = q − p − (32) ρ φ˙2 φ˙2 + 2V (φ ,φ ) q − p p q ˙2 ˙ 2 and leads to both phantom domination (φp > φq ) and quintessence domination (...<...) during the evolution of the universe together with “phantom divide crossing”. Phantom matter accretion onto a black hole may cause its mass diminishing (Babichev, Dokuchaev, Eroshenko ’2004). Exotic singularities in the Universe and supernovae observations – p. 23/53 Phantom in brane cosmology Phantom brane (RSII) cosmologies (MPD et al. IJMPD ’04, Gergely et al. ’06) 8π ρ2 k Λ 2m H2 = ρ + + 4 + pl(4)U , (33) 3m2 2λ − a2 3 8πλ pl(4) where λ is brane tension and is dark radiation. Phantom in brane cosmology U helps to fit supernovae, but not CMBR and nucleosynthesis. (For DGP brane model fit see e.g. Lazkoz et al. astro-ph/0605701) Note: there is a possibility for a bounce H = 0 in these models provided brane tension is negative at the energy density ρ = 2 λ and λ = λ> 0. (34) b | | | | − This means that non-singular oscillating cosmologies are possible in brane cosmology (e.g. MPD et al. ’04, Freese et al. astro-ph/0405353). Exotic singularities in the Universe and supernovae observations – p. 24/53 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/0601085, Ashtekar et al. ’06, Copeland et al. gr-qc/0510022 etc.): 1 ρ2 k H2 = ρ , (35) 3m2 − ρ − a2 pl c where the critical density is √3 ρ , (36) c ≡ 16πγ3G2~2 and γ is the Barbero-Immirzi parameter (γ 0.2375, see Meissner ≈ gr-qc/0407052). Exotic singularities in the Universe and supernovae observations – p. 25/53 4. Sudden Future Singularity (type II) as an exotic singularity. Big -Rip appears at some finite future time t = t 20 Gyr in analogy to BR ≈ Big-Crunch at t = tBC . Some people call it “sudden”, but we can have sth really “sudden” (as we will learn later). Sudden Future Singularity (or type II - Nojiri et al. 2005) – manifests as a singularity of pressure (or a¨) 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(%), no analytic form of this relation is given (37) 6 Choose a special form of the scale factor (may be motivated in fundamental cosmologies) as: m n t a(t)= as [1+(1 δ) y δ (1 y) ] , y (38) − − − ≡ ts Exotic singularities in the Universe and supernovae observations – p. 26/53 where a a(t )= const. and δ, A, m, n = const. s ≡ s But what about t = ts? One has to look onto the derivatives of a! m n a˙ = a (1 δ) ym−1 + δ (1 y)n−1 , (39) s t − t − s s a s m−2 n−2 (40) a¨ = 2 m (m 1) (1 δ) y δn (n 1) (1 y) . ts − − − − − h i If we assume that 1 then using Einstein equations (38)-(40) we get a = const., a˙ = const. % = const. a¨ p for t t (42) → ∓∞ → ±∞ → s Exotic singularities in the Universe and supernovae observations – p. 27/53 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 Sudden time, t Bang Future Singularity (or upwards in a similar way as in phantom scenario with an inflection point) Exotic singularities in the Universe and supernovae observations – p. 28/53 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: – 0 < m 1 when w 1/3 (standard matter); ≤ ≥− – m> 1 when 1 Exotic singularities in the Universe and supernovae observations – p. 29/53 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 as = a(ts)= const. (Fernandez-Jambrina, Lazkoz PRD 74, 064030 (2006) ((gr-qc/0607073)) dt 2 P 2 + KL2 = A + , (44) dτ a2(t) dr P cosφ + P sin φ = 1 2 1 Kr2 , (45) dτ a2(t) − dφ L p = . (46) dτ a2(t)r2 Geodesic deviation equation D2nα + Rα uβnγ uδ = 0 , (47) dλ2 βγδ feels SFS since at t = t we have the Riemann tensor Rα . s Exotic singularitiesβγδ in the→ Universe ∞ and supernovae observations – p. 30/53 No geodesic incompletness. Weak singularities. 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 curvature singularities i.e. in-falling observers or detectors are not destroyed by tidal forces: Tipler’s (Phys. Lett. A64, 8 (1977)) definition (of a weak singularity): 0 τ 0 τ 00 a b 0 dτ 0 dτ Rabu u does not diverge on the approach to a singularity at R R τ = τs Królak’s (CQG 3, 267 (1988)) definition (of a weak singularity): τ 0 a b 0 dτ Rabu u does not diverge on the approach to a singularity at R τ = τs A singularity may be weak acc. to Tipler, but strong according to Królak Exotic singularities in the Universe and supernovae observations – p. 31/53 Extended objects through exotic singularities. 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. One may extend the evolution behind an SFS (e.g. strings may survive these future singularities - Balcerzak and MPD ’06) but not behind a Big-Rip Conclusion: a Big-Rip and an SFS are of different nature. Big-Rip (similar as Big-Crunch) - is really the final state of the universe, though for a the 4-velocity uµ 0 (particles slow) → ∞ → Interesting point: – Schwarzschild spacetime at r = rg - metric singular, curvature invariants regular – Sudden singularity models - metric regular, curvature invariants diverge SFS with a = a = const., a˙ = 0 and a¨ , so p were also b → −∞ → ∞ termed Big-Brake (Gorini, KamenschchikExotic et al. singularities PRD in 69the Universe (2004) and supernovae, 123512) observations – p. 32/53 Strings at future singularities Polyakov action in conformal gauge: T S = dτdσηabg ∂ Xµ∂ Xν (48) − 2 µν a b Z Equations of motion ¨ µ µ ˙ ν ˙ ρ 00µ µ 0ν 0ρ X +ΓνρX X = λ X +ΓνρX X (49) ˙ µ ˙ ν 0ν 0ρ gµν X X = λgµν X X (50) − µ 0ν gµν X˙ X = 0 (51) Invariant string size (λ = 1 - tensile strings, λ = 0 - null strings 2π 0ν 0ρ S(τ)= gµν X X dσ (52) Z0 p Exotic singularities in the Universe and supernovae observations – p. 33/53 Strings at future singularities Circular string ansatz t = t(τ),X = R(τ)cos σ, Y = R(τ)sin σ, Z = const. (53) gives simple equations of motion in conformal time a η¨ + 2 ,η R˙ 2 = 0 (54) a a R¨ + 2 ,η η˙R˙ + λR = 0 (55) a η˙2 R˙ 2 λR2 = 0 (56) − − and the string size S(τ) = 2πa(η(τ))R(τ) (57) For phantom a(η)= η−2/(3|γ|+2) (58) Exotic singularities in the Universe and supernovae observations – p. 34/53 Strings at future singularities EOMs solve by 3 γ +2 η(τ) η(τ) = exp | | η ,R(τ)= 3 γ +2 (59) ±s2 3 γ ! 2 | | − | | p so 3 γ S(τ)= π 3 γ +2exp | | τ (60) | | ± 4 9γ2 ! p − Big-Rip for η 0 a(η) (or τ for ’+’p and τ for ’-’) → → → ∞ → ∞ → −∞ Conclusion (Balcerzak, MPD ’06): Strings are infinitely stretched S at → ∞ Big-Rip For SFS the scale factor is finite at η-time at SFS so that the invariant string size is also finite. The same is true for type III, IV and generalized SFS. Exotic singularities in the Universe and supernovae observations – p. 35/53 5. Type III (Finite Scale Factor) exotic singularity. Type III singularities (Nojiri et al. PRD 71, 063004 (2005)) which we will call Finite Scale Factor singularities are characterized by the following conditions: a = a = const., %, a˙ , p , a¨ s s → ∞ | | s → ∞ The simplest way to get them is to apply the scale factor as given previously for SFS, i.e., m n t a(t)= as [1+(1 δ) y δ (1 y) ] , y (61) − − − ≡ ts where a a(t )= const. and δ, A, m, n = const., but with the range of f ≡ f parameter n changed from 1 Exotic singularities in the Universe and supernovae observations – p. 36/53 6. Generalized Sudden Future, type IV and w singularities. − Sudden future singularities may be generalized to GSFS. Take a general scale factor time derivative of an order r: m(m 1)...(m r + 1) a(r) = a − − (1 δ) ym−r s tr − s n(n 1)...(n r + 1) + ( 1)r−1δ − − (1 y)n−r , (62) − tr − s Choosing (Barrow ’04, Lake ’04) r 1 Exotic singularities in the Universe and supernovae observations – p. 37/53 Type IV Type IV singularity according to Nojiri et al. ’05 is when: ... a = a = const., % 0, p 0, a, H¨ etc. s → → → ∞ and so it is similar to Generalized Sudden Future singularity with only one exception: it also gives the divergence of the barotropic index in the barotropic equation of state p(t)= w(t)%(t) i.e., w(t) → ∞ Exotic singularities in the Universe and supernovae observations – p. 38/53 Barotropic index w singularity − Another exotic is a w singularity only (without the divergence of the − higher-derivatives of the scale factor). (Apparently, it appears in physical theories such as f(R) gravity (Starobinsky ’80), scalar field gravity (Setare, Saridakis ’09, and brane gravity Sahni, Shtanov ’05)). We choose 2 n t 3γ t a(t)= A + B + C D , (63) t − t s s where A, B, C, D, γ, n, and ts are constants and impose the conditions: a(0) = 0, a(t )= const. a , a˙(t ) = 0, a¨(t ) = 0 , (64) s ≡ s s s which finally leads to the following form of the scale factor: Exotic singularities in the Universe and supernovae observations – p. 39/53 w singularity − as a(t) = n−1 3γ n−1 1 2 n− 2 − 3γ 2 2 1 na t 3γ + − 3γ s 2 2 n−1 n n− ts − 3γ 1 2 3γ − 3γ n−1 2 n a 1 t s − 3γ + n−1 1 2 , (65) 3γ n−1 − n 3γ ts ! 2 n− 2 1 − 3γ − with the admissible values of the parameters: γ > 0 and n = 1. 6 Exotic singularities in the Universe and supernovae observations – p. 40/53 w duality − We have a blow-up of the effective barotropic index, i.e., c2 w(t )= [2q(t ) 1] , (66) s 3 s − → ∞ accompanied by p(t ) 0; %(t ) 0 . (67) s → s → There is an amazing duality between the Big-Bang and the w-singularity in the form 1 1 1 pBB , %BB , wBB . (68) ↔ pw ↔ %w ↔ ww In other words: p ; % ; w 0; a 0 BB → ∞ BB → ∞ BB → BB → p 0; % 0; w ; a a = const. w → w → w → ∞ w → s Exotic singularities in the Universe and supernovae observations – p. 41/53 w duality − Exotic singularities in the Universe and supernovae observations – p. 42/53 7. Exotic cosmological singularities and statefinders blow-up. 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, type IV, w singularities do not lead to any violation of − the energy conditions. Conclusion: the application 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. Exotic singularities in the Universe and supernovae observations – p. 43/53 Statefinders and generalized energy conditions As an example assume the first-order dominant energy condition in the form K %˙ +p ˙ = 2H3 j + 3q +2+2 0 , (69) − a2H2 ≥ K %˙ p˙ = 2H3 j + 3q +4+4 0 , (70) − − − a2H2 ≥ 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 BC, BR, q SFS, j GenSFS, etc. | 0|→∞⇒ | 0 |→∞⇒ | 0 |→∞⇒ Exotic singularities in the Universe and supernovae observations – p. 44/53 8. Exotic singularities versus supernovae data. 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. PRD 75, 123524 (2007) (astro-ph/0704.1383))? In order to do so one should first compare SFS models with observational data from supernovae. The redshift formula in these models reads as a(t ) δ + (1 δ) ym δ (1 y )n 1+ z = 0 = − 0 − − 0 , (71) a(t ) δ + (1 δ) ym δ (1 y )n 1 − 1 − − 1 where y0 = y(t0) and y1 = y(t1). Exotic singularities in the Universe and supernovae observations – p. 45/53 A redshift-magnitude relation for SFS cosmology. We have m(z)= M 5 log H + 25 + 5 log [r a(t )(1 + z)], (72) − 10 0 10 1 0 where r1 comes from null geodesic equation r1 dr t0 cdt = . (73) 2 0 √1 kr t a(t) Z − Z 1 For a rough estimation of the dark energy models (acceleration) we study the product a¨ q H = = (74) 0 0 − a˙ 0 m−2 n−2 t0 m(m 1)(1 δ)y0 δn(n 1) (1 y0) − − m−1 − − n−−1 , − y0 m(1 δ)y + nδ (1 y ) − 0 − 0 which should be negative. Exotic singularities in the Universe and supernovae observations – p. 46/53 Parameter space for pressure-driven dark energy 20 ∆ 10 0 -10 -20 0.5 0 H0*q0 -0.5 -1 0.2 0.4 0.6 0.8 1 y0 Parameter space (H0q0, δ, y0) for fixed values of m = 2/3, n = 1.9993, t0 = 13.3547 Gyr of the sudden future singularity models. There are large regions of the parameter space which admit cosmic acceleration q0H0 < 0 . (75) Exotic singularities in the Universe and supernovae observations – p. 47/53 SFS dark energy versus Λ-term dark energy (concordance cosmology - CC) Distance modulus µ = m M for the CC model (H = 72kms−1Mpc−1, L − 0 Ωm0 = 0.26, ΩΛ0 = 0.74) (dashed curve) and SFS model (m = 2/3, n = 1.9999,δ = 0.471, y = 0.99936) (solid curve). Open circles − 0 are for the ‘Gold’ data and filled circles are for SNLS data. Exotic singularities in the Universe and supernovae observations – p. 48/53 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. t0 = 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. PRL ’08 ;arXiv: 0808.0190. Exotic singularities in the Universe and supernovae observations – p. 49/53 Big-Brake is much further in future! SFS with a = a = const., a˙ = 0 (% 0), and a¨ (p ) were also b → → −∞ → ∞ termed Big-Brake (Gorini, Kamenschchik et al. PRD 69 (2004), 123512). They fulfill an anti-Chaplygin gas equation of state of the form A p = A = const. . (76) % They were studied in the context of the tachyon cosmology by Keresztes et al. (0901. 2292). However, due to the imposition of different values of parameters which are given by tachyon constraints (plus anti-Chaplygin gas constraints) the closest singularity in their model appears – 1 Gyr in future – and the furthest even 44 Gyr in future. Exotic singularities in the Universe and supernovae observations – p. 50/53 FSF (type III) v. supernovae (work in progress) 46 44 42 µ 40 38 δ=0.53938, n=0.87558, χ2/dof=1.01319 t0 in Gyrs=15.29571 36 Riess gold set SNIa fit 34 0.01 0.1 1 10 z Type III singularity. a>¨ 0 for δ > 0: n = 0.87558 δ = 0.53938 t0 = 15.29571 Gyrs χ2/dof = 1.01319 Time to a type III singularity t t 0.3Gyrs (about 30 times larger than the s − 0 ≈ time to an SFS) Exotic singularities in the Universe and supernovae observations – p. 51/53 8. Conclusions Apart from quintessence 0 >p> %, astronomical data gives a direct − motivation to a non-standard type of matter phantom with p< % as a − candidate for dark energy. Phantom produces an exotic singularity – Big-Rip in which (a and → ∞ % ) which is different from Big-Crunch. → ∞ Investigations of phantom inspired other searches for non-standard singularities in future (sudden future, generalized sudden future (=Big-Brake), type III (Finite Scale Factor), type IV, w singularities etc.) − which, in fact, are not necessarily the “true” singularities (according to Hawking and Penrose definition) as there are 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. Exotic singularities in the Universe and supernovae observations – p. 52/53 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. Exotic singularities in the Universe and supernovae observations – p. 53/53