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
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
2. Radiative Stability and Deviations from
w = -1 From a particle physics perspective, the
potential V(O) is extraordinarily flat . 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 to
min9 - H0.
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 form
V(O, X) AG(X) c1t + h.c.
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 potential
V(0, p') p cos +
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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.