1. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 Dark energy from a quadratic equation of state Marco Bruni ICG, Portsmouth & Dipartimento di Fisica, Tor Vergata (Rome) & Kishore Ananda ICG, Portsmouth

2. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 Outline  Motivations  Non-linear EoS and energy conservation  RW dynamics with a quadratic EoS  Conclusions

3. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 Motivations  Acceleration (see Bean and other talks): –modified gravity; –cosmological constant  ; –modified matter.  Why quadratic, P=P o +  +   /  c ? –simplest non-linear EoS, introduces energy scale(s); –Mostly in general, energy scale -> effective cosmological constant   ; –qualitative dynamics is representative of more general non-linear EoS’s; –truncated Taylor expansion of any P(  ) (3 parameters); –explore singularities (brane inspired). 2 “…my biggest blunder.” A. Einstein

4. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 Energy cons. & effective    RW dynamics:  Friedman constraint:  Remarks: 1.If for a given EoS function P=P(  ) there exists a   such that P(   ) = -  , then   has the dynamical role of an effective cosmological constant. 2.A given non-linear EoS P(  ) may admit more than one point  . If these points exist, they are fixed points of energy conservation equation.

5. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 Energy cons. & effective    Further remarks: 3.From Raychaudhury eq., since, an accelerated phase is achieved whenever P(  ) < -  /3. 4.Remark 3 is only valid in GR. Remarks 1 and 2, however, are only based on conservation of energy. This is also valid (locally) in inhomogeneous models along flow lines. Thus Remarks 1 and 2 are valid in any gravity theory, as well as (locally) in inhomogeneous models. 5.Any point   is a de Sitter attractor (repeller) of the evolution during expansion if  +P(  ) 0) for  0 (  .

6. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 Energy cons. & effective   1.For a given P(  ), assume a   exists. 2.Taylor expand around   : 3.Keep O(1) in  =  -   and integrate energy conservation to get:

7. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 Energy cons. & effective   4.Note that:, thus. 5.Assume and Taylor expand: 6.Then: a)At O(1) in   and O(0) in , in any theory of gravity, any P(  ) that admits an effective   behaves as  -CDM; b)For  > -1  ->  , i.e.   is a de Sitter attractor. ¯ ¯ From energy cons. -> Cosmic No-Hair for non-linear EoS.

8. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 P=  (  +   /  c )  P o =0,  = ± 1  dimensionless variables:  Energy cons. and Raychaudhuri:  Friedman:

9. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 P=  (  +  /  c )  parabola: K=0; above K=+1, below K=-1  dots: various fixed points; thick lines: separatrices  a:  > -1/3, no acc., qualitatively similar to linear EoS (different singularity)  b: -1<  <-1/3, acceleration and loitering below a threshold   c:  < -1,  , de Sitter attractor, phantom for  <   abc

10. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 P=  (  -  /  c )  a:  < -1, all phantom, M in the past, singular in the future  b: -1    c:  >-1/3, similar to b, but with oscillating closed models  b and c: for  <   first acc., then deceleration bac

11. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 P=P o +   dimensionless variables:  Energy cons. and Raychaudhuri:  Friedman:

12. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 P=P o +   a: P o >0,   , recollapsing flat and oscillating closed models  b: P o >0, -1<  <-1/3: similar to lower part of a  c: P o <0, -1/3<  : phantom for  <  , de Sitter attractor, closed loitering models. abc

13. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 Full quadratic EoS  Left:  =1,  <-1, two  , phantom in between  Right:  =-1,  >-1/3, two  , phantom outside

14. Marco Bruni, ICG, University of Portsmouth & Dipartimento di Fisica, Roma ``Tor Vergata” Paris 8/12/05 Conclusions  Non-linear EoS: –worth exploring as dark energy or UDM (but has other motivations); –dynamical, effective cosmological constant(s) mostly natural; –Cosmic No-Hair from energy conservation: evolution a-la  -CDM at O(0) in dP/d  (   ) and O(1) in  =  -  , in any theory gravity.  Quadratic EoS: –simplest choice beyond linear; –represents truncated Taylor expansion of any P(  ) (3 parameters); –very reach dynamics:  allows for acceleration with and without   ;  Standard and phantom evolution, phantom -> de Sitter (no “Big Rip”);  Closed models with loitering, or oscillating with no singularity; –singularities are isotropic (as in brane models, in progress ).  Constraints: high z, nucleosynthesis (  >0), perturbations.