# Evolving Dark Energy with w =/ -1 Page: 3 of 5

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cosmology allows a solution to an otherwise intractable

problem: the dynamical mass scale causing the evolution

of 0 must be given parametrically by

mo 3 v4/M (2)

In quintessence theories, we can expect to observe de-

viations from w= -1 if the mass scales in V(O) are

appropriately related to the electroweak scale v. If

the mass parameters of V(O) are not related to those

of known particle physics, it does not appear possible

to answer this problem, except perhaps with anthropic

arguments [7].

In this letter we study quintessence in the seesaw

cosmology framework. We exhibit a large class of theories

that are radiatively stable and automatically solve the

"Quintessence Why Now?" problem. It is much more

constraining to also solve the usual "Dark Energy Why

Now?" problem, and we are led to a particular class of

axion-like models.

2. Radiative Stability and Deviations from

w = -1 From a particle physics perspective, the

potential V(O) is extraordinarily flat [8]. Supersymmetry

is commonly used to protect scalar masses at the mass

scale v, and can even protect certain scalars to v2/M as

needed for acceleressence theories, but this is far from

the desired scale of (2). Factors of 1/167r2 from quantum

loops are hardly likely to help. We are thus led to

introduce a small parameter p4 which explicitly breaks

the shift symmetry p -> p+ c:

V(0) =4F(0 + h.c. (3)

The dimensionless function F is arbitrary, and for sim-

plicity we have assumed that it depends on only a single

dimensionful parameter f. Throughout, we assume

that the approximate global symmetries of interest are

sufficiently protected from any corrections involving non-

perturbative quantum gravity. In general F depends on

many dimensionless parameters that are taken to be of

order unity. We assume that the initial value of 0 is

of order f, and that, since today p is at most slowly

evolving, 00 is also of order f. The observed size of

PDE then implies that p must be taken of order the meV

scale. To solve the "Dark Energy Why Now?" problem

we will later seek theories that lead to p , v2/M. In

the limit that p4 -> 0, shift symmetry requires the

potential to vanish. Hence all radiative corrections to

V are proportional to p4 the potential is radiatively

stable. A pseudo-Goldstone boson provides a well-known

example of quintessence with radiative stability, in which

case F is a cosine [4, 9].

The dynamical mass scale for p evolution is mo i9

p2/f. Once the dark energy dominates, the Friedmann

equation gives Ho O p p2/M, leading toM

min9 - H0.(4)

The slow role condition becomes f > M. In the

framework of seesaw cosmology, there are only two

fundamental mass scales M and v, and so we must choose

f , M. This gives m i9 Ho so that the "Quintessence

Why Now?" problem is solved; the slow roll condition is

lost during the present era and deviations from w= -1

are generically expected. With f , M, one immediately

finds m n9 p2/M, and with p , v2/M the double seesaw

mo , (v2/M)2/M leads to the desired relation (2). To

explain why p v2/M, and to be more precise about the

prediction for w(z), we must address the "Dark Energy

Why Now?" problem.

3. A Dynamical p4 As long as p4 appears as

an independent free parameter of the theory, the "Dark

Energy Why Now?" problem will remain unsolved. To

make progress, p4 must itself be understood to arise

dynamically p4 -> AG(X), with G a product of fields

X which may include scalars and fermions. A simple

example is G X4, with X a scalar. The introduction

of propagating fields X changes the radiative structure

of the theory the parameter which explicitly breaks

the shift symmetry on p is now A, which we take to be

dimensionless and order unity. For example, integrating

over internal X fields induces a radiative correction to

the potential at order A12: AV() 2M4F(f)2

giving a p mass of order AM2/f. Indeed, treating A as the

spurion for shift symmetry breaking, such a term cannot

be forbidden. By making p4 dynamical, m9 is generically

changed from order Ho to order AM! Even if the loop

integrals are cutoff by supersymmetry, m9 can only be

protected to v2/M, sufficient for acceleressence, but very

far from the requirements of dynamical quintessence.

This disastrous radiative correction, however, is easily

removed by taking F = 1f. In this case the potential is

periodic, and p is understood to be the pseudo-Goldstone

boson of some symmetry U(1)9 that is spontaneously

broken at scale f near the Planck scale. Our potential V

then takes the formV(O, X) AG(X) c1t + h.c.

(5)

There are other potentially problematic radiative cor-

rections to the potential for p from diagrams involving X

loops. For example, if X is a scalar and G X14, then

there are radiative corrections at order A in which the

four X fields are contracted into a two loop diagram. To

avoid such contributions G must carry some charge under

some symmetry U(1)X. For example, with X a complex

scalar and G X4 it is not possible to contract the x

fields into loops as long as there are no other interactions

which violate U(1)X. In such theories the interaction (5)

explicitly breaks one combination of U(1)9 and U(1)X.

The parameter p4 is generated by having X develop

an expectation value f', so that AG -> A (G)e24'/f'

p42 '/f', giving a potentialV(0, p') p cos +

(f f,(6)

2

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Hall, Lawrence J.; Nomura, Yasunori & Oliver, Steven J. Evolving Dark Energy with w =/ -1, article, March 31, 2005; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc928064/m1/3/: accessed April 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.