Neutrino Models of Dark Energy

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Neutrino Models of Dark Energy

• LEOFEST
• Ringberg Castle
• April 25, 2005

R. D. Peccei UCLA
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Neutrino Models of Dark energy

• Observational Surprises
• Theoretical Considerations
• The FNW Scenario
• Two Illustrative Examples
• Discussion and Future Directions

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Observational Surprises

• In the late 1990s two groups Supernova
  Cosmology Project and High-z Supernova Team
  using supernovas as standard candles set out to
  measure the Universes deceleration parameter
• Expected qo1/2, found qo ? -1/2. Universes
  expansion is accelerating, not decelerating!

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• Early data interpreted acceleration as being due
  to a cosmological constant ? and found, in an
  assumed flat Universe ?1, that ?? ? 0.7 and ?M
  ? 0.3

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• The WMAP experiment, measuring the angular
  dependence of the temperature fluctuations in the
  cosmic microwave background in the last year
  confirmed this result with much more accuracy,
  finding
• ? 1.02 0.02 Flat
  Universe
• ?? 0.73 0.04 Dark
  energy
• ?M 0.27 0.04 Matter
• ?B 0.044 0.004 Baryons
• Most matter is not baryonic, but some form of
• non-luminous matter- Dark Matter
• Equation of state of dark energy gives ? lt -0.78

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Theoretical considerations

• A significant challenge is to try to understand
  from the point of view of particle physics the
  Dark Energy in the Universe
• Einsteins equations
• determine H and the Universes acceleration
  once ?, p, k, and ? are specified.

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• In a flat Universe k0, as predicted by
  inflation and confirmed observationally by WMAP,
  the Universe accelerates if ? gt 4?GN ?matter ,
  or, if ?0, a dominant component of the Universe
  has negative pressure and ? 3p lt 0. The
  observed acceleration is evidence for this Dark
  Energy
• It is convenient to set ?0 and write the first
  Einstein equation simply as
• H2 8?GN ? /3 8?GN ?dark energy /3.
  
• Then using an equation of state ?p/? , the
  pure cosmological constant case, where the
  density is a pure vacuum energy density,
  corresponds to ? -1
• ?dark energy -p dark energy
  ?vacuum ? constant

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• The Hubble parameter now Ho(1.5 0.1) 10-33 eV
  is a tiny scale. We know that, at the present
  time, Ho2 gets about 30 contribution from the
  first term and 70 from the second term, while
  -1lt ?p/? lt-0.8. What is the
  physics associated with this dark energy?
• If indeed one has a cosmological constant, so
  that ?dark energy ?vacuum Eo4, then Ho ??GN
  Eo2 ? Eo2 / MP
• gives Eo ? 2 10-3 eV. What physics is
  associated with this very small scale? All
  particle physics vacuum energies are enormously
  bigger e.g. for QCD EoQCD ?QCD ? 1 GeV

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The FNW Scenario

• Can one understand ?dark energy as arising
  dynamically from a particle physics scale?
• A very interesting suggestion along these lines
  has been put forward recently by Fardon, Nelson
  and Weiner.
• Coincidence of having in present epoch
• ?odark energy ? ?omatter
• is resolved dynamically if the dark energy
  tracks some component of matter
• Easy to convince oneself that the best component
  of matter for ?dark energy to track are the
  neutrinos

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• If indeed ?dark energy tracks ?? then can perhaps
  also understand scale issue
• Eo 2 10-3 eV ? m? vF2 /MN 10-1
  -10 -3 eV
• In FNW picture neutrinos and dark energy are
  coupled. In NR regime examined by FNW
• ?dark m?n? ?dark energy (m?)
• with the neutrino masses being fixed by
  minimizing the above
• n? ?'dark energy (m?) 0
• Thus neutrino masses are variable depending on
  the neutrino density m? m?(n?). This is the
  principal assumption of FNW

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• One can compute the equation of state for the
  dark sector by looking at energy conservation
  equation
• ? ?dark /? t-3H(?dark pdark) -3H?dark (?
  1) ()
• and in NR limit one finds
• ? 1 m? n?/ ?dark m? n?/ m? n? ?dark
  energy
• We see that if ? ? -1 the neutrino contribution
  to ?dark is a small fraction of ?dark energy.
  Further, we expect from () that, if ? does not
  change much with R, ?darkR - 3(1 ?). But n?
  R-3, so from the equation of state the neutrino
  mass must be nearly inversely proportional to the
  neutrino density
• m? R-3 ? n? ? ? n? -1

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• I have examined this scenario for neutrinos of
  arbitrary velocity obtaining an important result
• FNW scenario ? Running cosmological const.
• In general,
• ?dark ?? ?dark energy (m?)
• where ?? T4F(?) with ? m? /T and
• Stationarity w.r.t. m? variations implies
• T3 ?F(?) / ? ? ? ?dark energy /? m?
  0

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• Using the conservation of energy equation one can
  show that, in the general case, the equation of
  state is given by
• ? 1 ??4-h(?) /3 ?dark ()
• where h(?) ? ? F(?) / ? ? / F(?)
• In the non-relativistic limit (? m? /T gtgt1)
  where ?? m?n?, one can check that h(?) ? 1 so
  that () indeed reduces to the FNW equation
• ? 1 m? n?/ ?dark
• However, working out the () expression, using
  that (? 1) ?dark pdark ?dark, one finds a
  surprise

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• The equation of state becomes
• p? pdark energy ?dark energy ??
  1-h(?) /3 but
• ?? 1-h(?) /3T4/3?2
  p?
• Hence, it follows that
• pdark energy ?dark energy 0.
• which implies that
• ?dark energy -pdark energy ? V(m?)
  
• and one sees that the dark energy is just a
  running cosmological constant!

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• This result perhaps is not so surprising, since
  we assumed ?dark energy (m?), so that all
  T-dependence comes through m?(T)
• If, however,
• ?dark energy K(T) V(m?)
• one finds a modified equation of state
• 1 ??4-h(?) T ? K(T) / ? T /3 ?dark
• One deduces from the above that
• pdark energy ?dark energy (T/3) ? K(T)
  / ? T

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• However,
• pdark energy K(T) -V(m?)
• and thus one finds that
• K(T) (T/6) ? K(T) / ? T
• Thus one deduces that
• K(T) Ko (T/To )6
• which is the behavior you expect from a free
  massless scalar field
• Although one can choose Ko small enough so that K
  is negligible compared to V in the present epoch,
  in earlier times K(T) totally dominates and
  distorts the evolution of the Universe
• Therefore, FNW scenario consistent only if K0
• i. e. running cosmological constant

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Two Illustrative Examples

• FNW scenario is characterized by 2 equations
• T3 ?F(?) / ? ? ? V(m?) /? m? 0
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• ? 1 4-h(?) / 3 1 V(m?) /T4 F(?) 2
• 1 determines m?(T), while 2 determines the
  evolution of equation of state ?(T) for any given
  potential V(m?)
• Studied two examples
• Vp(m?) m?-? Ve(m?) exp? /m?

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• General assumptions and features
• ?omatter 0.3 ?c Vo 0.63 ?c ?? 0.07
  ?c ?o - 0.9 ?c 2.46 10-11 eV4 To 1.9 oK
  m?o 3.09 eV Then
• Vp(m?) 0.63 ?c (m? / m?o ) 1/9
• Ve(m?) 0.63 ?c exp1/9 (m?o /m? )-1
• For both potentials can show that
• ?(T) ? ?o as T ? To Nonrelativistic
  limit
• and
• ?(T) ? 1/3 for T gtgt To Relativistic
  limit

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• However, models differ on where NR/Rel.
  transition occurs and in dependence of m? on T
• Power-law potential
• ? m?(T) /T1 at T3.06 10-3 eV ? 20 To
• m?(T) ?1.12 10-5eV / T(eV)0.95
  Relativistic regime
• Exponential potential
• ? m?(T) /T1 at T4.57 10-2 eV ? 300
  To
• m?(T) ?0.028 eV / 10.16 lnT(eV)
  Rel. regime
• Note that NR/Rel. transition occurs much later
  than for fixed mass neutrinos, where Tfix3.09
  eV

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Behavior of m? / mo? with T for the two different
potentials is shown below. Here zT/To-1
Exponential Potential
Power-law Potential
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Different behaviour of m?(T ) implies different
evolution of ?(T) from ?o to 1/3
Power-law Potential
Exponential Potential
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• Other significant difference is in behavior of
  potentials with temperature. In both cases, V is
  only important in the NR regime
• In relativistic regime dark sector is always
  dominated by neutrino contribution, rather than
  by the running cosmological constant. One finds
• ?? (7?2/120)T4 1.48 1010 ?c T(eV)4
• while
• Vp 2.52 ?cT(eV)0.105
• and
• Ve ? 4.33 10 4 ?cT(eV)2

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Below we show the behaviour of various components
of the Universes energy density in units of
?/?c. Here solidmatter dashedneutrinos
dotteddark energy
Exponential Potential
Power-law Potential
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Discussion and Future Directions

• Speculative idea of tying the dark energy sector
  with the neutrino sector gives rise to appealing
  idea of a running cosmological constant V(m?),
  but requires bold new dynamics
• However, scenario does not explain the dark
  energy scale Eo 2 10-3 eV, which is put in by
  hand (thru m? 3 eV) as boundary condition in
  present epoch V? Eo4f(T/To)
• Also difficult to imagine that a running
  cosmological constant would depend only on the
  neutrino mass scale. More likely V(mi) , with
  all masses being environment dependent mi mi(T)

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• Old idea of RDP, Sola and Wetterich may be worth
  reviving cosmological constant changes as
  function of a dynamical dilaton field- the cosmon
  S
• S ? S ?M Dilatations
• Cosmon couples to anomalous energy momentum trace
  ??? and adjusts its VEV to zero in same way
  axion which couples to F??F?? adjusts ? to zero
• Equation
• M ?? /? S SSo 0
• is analogue of FNW equation and should set
  lt???gtSSo 0, fixing the VEV of the full trace
  T?? , which is the cosmological constant
  ltT??So(T)gt

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• Effectively, at each temperature scale the cosmon
  would find a new minimum So(T), and the
  cosmological constant would obtain a different
  value ltT??So(T)gt
• Even in this scheme, however, it is difficult to
  understand why the cosmological constant is so
  small now.
• In QCD for instance,
• ltT??gtQCD lt??? gtQCD mqlt?qqgt QCD
• Naively, even if lt??? gtQCD were to vanish,
  what remains is still of O(0.1 GeV)4. However, mq
  is itself the result of another VEV, coming from
  the electroweak theory, so perhaps it cannot be
  treated as a hard mass.
• Correct conclusion to draw is that there is still
  much to understand in this difficult problem!