Neutrino Models of Dark Energy

LEOFEST

Ringberg Castle

April 25, 2005

R. D. PecceiUCLA

Neutrino Models of Dark energy

• Observational Surprises

• Theoretical Considerations

• The FNW Scenario

• Two Illustrative Examples

• Discussion and Future Directions

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 qo=1/2, found qo -1/2. Universe’s expansion is accelerating, not decelerating!

2R

RRqo

• 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

• 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

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.

33

82

2

2

R

kG

R

RH N

)3(3

4

3p

G

R

R N

• In a flat Universe [k=0], as predicted by inflation and confirmed observationally by WMAP, the Universe accelerates if > 4GN 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:

H2 = 8GN /3 + 8GN 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

• The Hubble parameter now Ho=(1.5 ± 0.1) 10-33 eV is a tiny scale. We know that, at the present time, Ho

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 =Eo

4, then Ho GN Eo2 Eo

2 / 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: Eo

QCD ~QCD 1 GeV]

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 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

• 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 = mn+ 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

• One can compute the equation of state for the dark sector by looking at energy conservation equation

dark / t=-3H(dark + pdark)= -3Hdark ( +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

• 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

1

1F(

22

0

22

ye

ξydyy

πξ

• 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 = mn, 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

• The equation of state becomes

p +pdark energy + dark energy = [1-h()] /3 but

[1-h()] /3=[T4/32] = 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!

)1(22

4

yo eξy

ydy

• 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

• 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 K=0 i. e. running cosmological constant

Two Illustrative Examples

• FNW scenario is characterized by 2 equations:

T3 F() / + V(m) / m =0 [1]

+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 ]

• 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 / mo ) –1/9

Ve(m) = 0.63 c exp{1/9 [(mo /m )-1]}

• For both potentials can show that

(T) o as T To Nonrelativistic limit

and

(T) 1/3 for T >> To Relativistic limit

• 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 To

m(T) 1.12 10-5eV / T(eV)0.95 Relativistic regime

Exponential potential

= m(T*) /T*=1 at T*=4.57 10-2 eV 300 To

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 Tfix*=3.09 eV

•Behavior of m / mo with T for the two different

potentials is shown below. Here z=T/To-1

Power-law Potential Exponential Potential

•Different behaviour of m(T ) implies different evolution of (T) from o to 1/3

Power-law Potential Exponential Potential

• 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

= (72/120)T4 =1.48 1010 c [T(eV)]4

while

Vp= 2.52 c[T(eV)]0.105

and

Ve 4.33 10 4 c[T(eV)]2

•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

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 Eo

4f(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)

• 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 FF* 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: <T[So(T)]>

• Effectively, at each temperature scale the cosmon would find a new minimum So(T), and the cosmological constant would obtain a different value: <T

[So(T)]>

• Even in this scheme, however, it is difficult to understand why the cosmological constant is so small now.

• In QCD for instance,

<T>QCD = <

>QCD + mq<qq> QCD

Naively, even if < >QCD 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!