Marco Bruni, ICG, University of Portsmouth

1
Dark energy from a quadratic equation of state

• Marco Bruni
• ICG, Portsmouth Dipartimento di Fisica, Tor
  Vergata (Rome)
• Kishore Ananda
• ICG, Portsmouth

2
Outline

• Motivations
• Non-linear EoS and energy conservation
• RW dynamics with a quadratic EoS
• Conclusions

3
Motivations

• Acceleration (see Bean and other talks)
• modified gravity
• cosmological constant ?
• modified matter.
• Why quadratic, PPo ?? ? ? /?c ?
• simplest non-linear EoS, introduces energy
  scale(s)
• Mostly in general, energy scale -gt effective
  cosmological constant ??
• qualitative dynamics is representative of more
  general non-linear EoSs
• truncated Taylor expansion of any P(?) (3
  parameters)
• explore singularities (brane inspired).

my biggest blunder. A. Einstein
2
4
Energy cons. effective ??

• RW dynamics
• Friedman constraint
• Remarks
• If for a given EoS function PP(?) there exists a
  ?? such that P(??) - ??, then ?? has the
  dynamical role of an effective cosmological
  constant.
• A given non-linear EoS P(?) may admit more than
  one point ??. If these points exist, they are
  fixed points of energy conservation equation.

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Energy cons. effective ??

• Further remarks
• From Raychaudhury eq., since
  , an accelerated phase is achieved
  whenever P(?) lt -?/3.
• 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.
• Any point ?? is a de Sitter attractor (repeller)
  of the evolution during expansion if ?P(?)lt0
  (gt0) for ?lt ?? and ?P(?)gt0 (lt0) for ?gt ??.

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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

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Energy cons. effective ??

• Note that , thus
  .
• Assume and Taylor expand
• Then
• At O(1) in ?? and O(0) in ?, in any theory of
  gravity, any P(?) that admits an effective ??
  behaves as ?-CDM
• For ? gt -1 ? -gt ??, i.e. ?? is a de Sitter
  attractor.

From energy cons. -gt Cosmic No-Hair for
non-linear EoS.
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P?(? ? ?/?c)

• Po 0, ? 1
• dimensionless variables
• Energy cons. and Raychaudhuri
• Friedman

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P?(? ?/?c)
a
b
c

• parabola K0 above K1, below K-1
• dots various fixed points thick lines
  separatrices
• a ? gt -1/3, no acc., qualitatively similar to
  linear EoS (different singularity)
• b -1lt ? lt-1/3, acceleration and loitering below
  a threshold ?
• c ? lt -1, ?? , de Sitter attractor, phantom for
  ? lt ??

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P?(? - ?/?c)
b
a
c

• a ? lt -1, all phantom, M in the past, singular
  in the future
• b -1lt ? lt-1/3, ?? , de Sitter saddles, phantom
  for ? gt ??
• c ? gt-1/3, similar to b, but with oscillating
  closed models
• b and c for ? lt ?? first acc., then deceleration

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PPo??

• dimensionless variables
• Energy cons. and Raychaudhuri
• Friedman

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PPo??
a
b
c

• a Pogt0, ?lt-1 phantom for ? gt ??, recollapsing
  flat and oscillating closed models
• b Pogt0, -1lt?lt-1/3 similar to lower part of a
• c Polt0, -1/3lt? phantom for ? lt ??, de Sitter
  attractor, closed loitering models.

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Full quadratic EoS

• Left ?1, ?lt-1, two ?? , phantom in between
• Right ?-1, ?gt-1/3, two ?? , phantom outside

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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 -gt 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 (?gt0),
  perturbations.