Download HYM-flation: Yang-Mills cosmology with Horndeski coupling

HYM-flation: Yang-Mills
cosmology with Horndeski
coupling
E.A. Davydov (JINR, Dubna)
and
D.V. Gal’tsov (Moscow State University)
II FLAG meeting “The Quantum and Gravity”
Trento 6-8 June 2016
Motivation and overview
•
GUT-scale inflation, favored by PLANCK-15, raises the problem of
initial conditions. Possible solution is pre-inflation caused by an
additional inflaton
•
Our proposal for this is the Yang-Mills component non-minimally
coupled to gravity via Horndeski prescription
•
Adding YM component is natural in the context of gauge and
sugra/superstring theories
•
Horndeski YM coupling is a unique gauge invariant and ghost-free
curvature dependent coupling with natural de Sitter attractor
•
In addition, this theory contains natural mechanism of exit, ensuring
finite duration of de Sitter stage
Classical YM minimally coupled
to gravity
•
Non-abelian baldness of EYM black holes DG and Ershov ‘88
•
Bartnik-MacKinnon particle-like solutions: gravitating sphalerons DG
and Volkov ’91, Sudarski and Wald ‘92
•
Hairy black holes Volkov and DG ’89, Bizon ’90, Maison et al ‘91
•
Cosmological ansatz: homogeneous and isotropic mode
•
Cosmological sphaleron DGibbons and Steif ‘92
•
Cosmological instanton and Euclidean wormholes: de Sitter – FRW
quantum transitions Donets and DG ’92
Cold “matter” for hot Universe
• Standard TM lagrangian is conformally invariant, so the
Universe driven by classical HI mode of YM will have hot EOS
(Cervero and Jakob’ 79, Hosotani ’80, DG and Volkov ‘91
• YM admixture to photon gas breaks parity, and can be
distinguished measuring polarization of primodrial GW
(Bielefeld and Caldwell ’15)
• Coupled to Higgs inflaton, as prescribed by gauge and/or
SUGRA/superstring theories, introduces novel intriguing
features to inflation scenario (DG and Davydov ’11, Rinaldi ’13…)
Cosmic acceleration as CSB
In YM-driven cosmology inflation is manifestation of breaking
of the conformal symmetry of the YM field. Various
mechanisms were explored:
•
Born-Infeld modification of the lagrangian (strings). EOS
interpolates bewtween string gas and hot (DG, Dyadichev, Zorin,
Zotov ’02, Fuzfa and Alimi ’06…)
•
Non-linear dependence of the lagrangian on the
pseudoscalar invariant (quantum corrections) denerates
effective cosmological constant ( DG and Davydov ’10, Maleknejad
and Sheikh-Jabbari ’12, Soda…)
•
•
Coupling to dilaton
Coupling to axion: “chromo-natural” inflation (Adshead and
Wyman ’12…))
•
Coupling to Higgs ( DG and Davydov ’11, Rinaldi ’13…)
Gravitational CSB (non-minimal)
• Like in the scalar case, for many reasons non-minimal
gravitational interaction may be favored as compared with
modificaltion of ‘matter’. Similarly to ‘Higgs’ inflation via nonminimal coupling one is led to explore possible non-minimal
couplings of YM (Balakin et al ’08, Davydov and DG ’13…)
•
But generically this leads to Ostrogradski ghosts, once EOM-s
become higher order than the second. It is presumed, however,
that ‘good’ theories (like superstrings) avoid ghosts in the lowenergy limits. Thus one is led to Horndeski
• Vector Horndeski coupling is much simpler and unique than the
scalar Horndeski (Fab Four etc)
Vector Horndeski
General gauge-invariant curvature-dependent action, quadratic in the
vector field strength and linear in the curvature
Horndeski choice: q1=q2=q3, leading to
Crucial property is zero divergence
Equivalent forms
•
Using the dual strength
be rewritten as
•
This structure is reminiscent of the Gauss-Bonnet lagrangian
•
from which it can be obtained replacing the Riemann tensor by the
product of two field tensors. In fact Horndeski action can be derived
form the higher-dimensional Gauss-Bonnet by Kaluza-Klein zero mode
reduction
the Horndeski action can
Induction tensor
Adding non-minimal Horndeski term with dimensionful coupling to
standard YM lagrangian, on can rewrite the total action in the effective
‘media’ YM(electro)dynamics terms
introducing the induction field tensor
satisfying the ‘media’ Yang-Mills equations
where the YM (and gravity) –covariant derivative is introduced whose
action on the ordinary field tensor is
Energy-momentum tensor
Variation of non-minimal coupling action over the metric is non-trivial
and demands using the YM equations to put the result in a simple
form
in which the absence of higher-derivative terms is manifest. The
Einstein equations then can be put into the familiar form
moving all the remaining terms as the energy momentum tensor. This
can be presented as follows using the induction tensor
De Sitter boundary
The induction tensor vanishes in the de Sitter space with
In this case
so
Similar boundary is present for generic q1, q2, q3 theory, then
the induction tenor vanishes if
This is not the stationary point of the dynamical system, but the
boundary of the physical subspace, which marks the region where
solutions are singular. Solutions starting in the non-singular region
never cross this bounday, though are attracted to it for finite time
FRW ansatz
•
Consider the FRW space-time with flat 3-metric
with the ansatz for YM potential
parametrized by a single function. This lead to the effective electric
(kinetic term) and magnetic
(potential)
fields (in the gauge N=1). The standard YM lagrangian then reads
while the full lagrangian of the system reads
indicating again on de Sitter boundary. The YM equation is
Friedmann equations
The Hubble parameter satisfies the Friedmann equations
with the following energy density and pressure
In the limit of vanishing coupling
this reproduces the EOS
Physical domain
An important characteristic of the system of differential equations is
the determinant of the matrix of coefficients before the highest
derivatives, in our case
. This determinant reads
When it vanishes, the solution of the system develops a singularity.
One can show that the de Sitter boundary
just separates
the domain of non-singular solutions from that of singular ones.
Suppose at some t=t1,
. Then YM equations give
. This implies vanishing of the determinant
indicating on the singularity. Thus non-singular curves do not
intersect de Sitter boundary
. Since solutions must reach
the flat asymptotic
, where D=1, initial conditions
must lie in the domain
The de Sitter bound
separate the region of phantom states.
HYM- flation
Near the de Sitter boundary the systen can be solved analytically.
Indeed, the YM equation is satisfied if
. Then the first
Friemdann equation can be solved with respect to psi-derivative as
This is valid if the expression under the square root is positive implying
with some critical value depending on HYM coupling
parameter. This is possible only in non-Abelian case due to non-linear
terms. For large psi two above branches simplify and can be integrated
They correspond to electric and magnetic dominance respectively
Considering small deviations from these solutions
one can find the eigenvalues of the corresponding linearized
systems (local Lyapunov exponents):
Thus the first solution is unstable, while the second is stable for
some finite time, since the solution itself is exponentially dying.
This second (marked by minus) solution acts as ‘inflationary
attractor’, since it is valid until the Hubble parameter is
approximately constant. Strictly speaking this is not an attractor
of the dynamical system, but it is a regime which is met for a wide
variety of integral curves starting within the physical domain. As
expansion is going on, the YM field exponentially decays and
eventially drops below the critical value when the expression
under the square root becomes negative. This marks transition to
the regime of minimal YM cosmology (hot EOS)
Numerical solutions
Starting with initial data lying in the physical domain one see that the
trajectories have qualitatively similar behavior. Even with zero initial
value of YM, and the derivatives being in the domain one observes
that the YM field rapidly grows, while metric approaches de Sitter
HYM-flation vs chaotic inflation
Comparing with scalar inflation in power-law potential one observes
an important difference. In the scalar slow-roll regime H grows with
increasing field value. In the HYM case the potential term is quatric but
with increasing field
resembling conformal attractors.
The regime of rolling down id ‘constant-roll’ with
In our case n=-1/4 , while slow-roll corresponds to
(for ‘ultra
slow-roll n=-3). Pecuiar feature of this regime is that quantum
fluctuations remain relatively small. Indeed,
, while
the YM field decreases on
. Since for most of
this stage
quantum fluctuations are small
Contrary to the chaotic inflation, H=const while the potential terms grows
as
so the field can not climb up due to quantum fluctuations.
Only at the end of inflation fluctuations become more significant, this
might give LSS, but not eternal inflation
HYM-flation as pre-inflation
Trying to obtain sufficient number of e-folds with Planck-scale
parameters, one encounter the problem of perturbations, however.
The power spectrum turns out to be blue-tilted, contradicting Planck
data. But the model can serve as pre-inflation preparing the Universe
to GUT-scale observed inflation. In this respect it looks natural, since
uses the field naturally present in gauge or sugra/superstring models.
In this case the grows of scale-factor by 100-1000 can be achieved
without contradiction with observed perturbation spectrum. Such a
model would combine advantages of the Higgs conformal inflation and
the preliminary chaotic-like inflation in a natural and economic way.
Replacing non-minimal Higgs coupling by Horndeski non-minimal
coupling of Yang-Mills component provides an alternative to Higgs
inflation with intrinsic preinflationary mechanism
Thanks for attention!