1. Neutrino Models of Dark Energy LEOFEST Ringberg Castle April 25, 2005 R. D. Peccei UCLA

2. Neutrino Models of Dark energy Observational Surprises Theoretical Considerations The FNW Scenario Two Illustrative Examples Discussion and Future Directions

3. 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 Universe’s deceleration parameter Expected q o =1/2, found q o  -1/2. Universe’s expansion is accelerating, not decelerating!

5. 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

6. 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  < -0.78

7. Theoretical considerations A significant challenge is to try to understand from the point of view of particle physics the Dark Energy in the Universe Einstein’s equations determine H and the Universe’s acceleration once , p, k, and  are specified.

8. In a flat Universe [k=0], as predicted by inflation and confirmed observationally by WMAP, the Universe accelerates if  > 4  G N  matter, or, if  =0, a dominant component of the Universe has negative pressure and  +3p < 0. The observed acceleration is evidence for this Dark Energy It is convenient to set  =0 and write the first Einstein equation simply as: H 2 = 8  G N  /3 + 8  G N  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

9. The Hubble parameter now H o =(1.5 ± 0.1) 10 -33 eV is a tiny scale. We know that, at the present time, H o 2 gets about 30% contribution from the first term and 70% from the second term, while -1<  =p/  <-0.8. What is the physics associated with this dark energy? If indeed one has a cosmological constant, so that  dark energy =  vacuum =E o 4, then H o  G N E o 2  E o 2 / M P gives E o  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: E o QCD ~  QCD  1 GeV]

10. 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  o dark energy   o matter 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

11. If indeed  dark energy tracks  then can perhaps also understand scale issue: E o ~ 2 10 -3 eV  m ~ v F 2 /M N ~ 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

12. One can compute the equation of state for the dark sector by looking at energy conservation equation   dark /  t=-3H(  dark + p dark )= -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,  dark ~R - 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

13. 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  = T 4 F(  ) with  = m /T and Stationarity w.r.t. m variations implies T 3  F(  ) /   +   dark energy /  m =0

14. 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 >>1) 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 = p dark +  dark, one finds a surprise

15. The equation of state becomes p +p dark energy +  dark energy =  [1-h(  )] /3 but  [1-h(  )] /3=[T 4 /3  2 ] = p Hence, it follows that p dark energy +  dark energy = 0. which implies that  dark energy = -p dark energy  V(m ) and one sees that the dark energy is just a “running” cosmological constant!

16. 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 p dark energy +  dark energy = (T/3)  K(T) /  T

17. However, p dark energy = K(T) -V(m ) and thus one finds that K(T) = (T/6)  K(T) /  T Thus one deduces that K(T) = K o (T/T o ) 6 which is the behavior you expect from a free massless scalar field Although one can choose K o 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 K=0 i. e. running cosmological constant

18. Two Illustrative Examples FNW scenario is characterized by 2 equations: T 3  F(  ) /   +  V(m ) /  m =0 [1]  +1= [4-h(  )] / 3 [1 + V(m ) /T 4 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: V p (m ) ~ m -  ; V e (m ) ~ exp[  /m ]

19. General assumptions and features:  o matter = 0.3  c ; V o = 0.63  c ;  = 0.07  c ;  o = - 0.9  c =2.46 10 -11 eV 4 ; T o = 1.9 o K; m o = 3.09 eV Then V p (m ) = 0.63  c (m / m o ) –1/9 V e (m ) = 0.63  c exp{1/9 [(m o /m )-1]} For both potentials can show that  (T)   o as T  T o Nonrelativistic limit and  (T)  1/3 for T >> T o Relativistic limit

20. However, models differ on where NR/Rel. transition occurs and in dependence of m on T: Power-law potential  = m (T*) /T*=1 at T*=3.06 10 -3 eV  20 T o m (T)  1.12 10 -5 eV / T(eV) 0.95 Relativistic regime Exponential potential  = m (T*) /T*=1 at T*=4.57 10 -2 eV  300 T o m (T)  0.028 eV /[ 1+0.16 lnT(eV) ] Rel. regime Note that NR/Rel. transition occurs much later than for fixed mass neutrinos, where T fix *=3.09 eV

21. Behavior of m / m o with T for the two different potentials is shown below. Here z=T/T o -1 Power-law Potential Exponential Potential

22. Different behaviour of m (T ) implies different evolution of  (T) from  o to 1/3 Power-law PotentialExponential Potential

23. 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)T 4 =1.48 10 10  c [T(eV)] 4 while V p = 2.52  c [T(eV)] 0.105 and V e  4.33 10 4  c [T(eV)] 2

24. Below we show the behaviour of various components of the Universe’s energy density in units of  /  c. Here solid=matter; dashed=neutrinos; dotted=dark energy Power-law Potential Exponential Potential

25. 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 E o ~ 2 10 -3 eV, which is “put in by hand” (thru m ~ 3 eV) as boundary condition in present epoch: V  E o 4 f(T/T o ) Also difficult to imagine that a running cosmological constant would depend only on the neutrino mass scale. More likely: V(m i ), with all masses being environment dependent m i = m i (T)

26. 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 | S=So = 0 is analogue of FNW equation and should set | S=So = 0, fixing the VEV of the full trace T  , which is the cosmological constant:

27. Effectively, at each temperature scale the cosmon would find a new minimum S o (T), and the cosmological constant would obtain a different value: Even in this scheme, however, it is difficult to understand why the cosmological constant is so small now. In QCD for instance, QCD = QCD + m q QCD Naively, even if QCD were to vanish, what remains is still of O(0.1 GeV) 4. However, m q 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!