HYM-flation: Yang-Mills cosmology with Horndeski coupling

1. 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

2. 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

3. 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

4. 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…)

5. 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…)

6. 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 non-minimal 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 low-energy limits. Thus one is led to Horndeski • Vector Horndeski coupling is much simpler and unique than the scalar Horndeski (Fab Four etc)

7. 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

8. Equivalent forms • Using the dual strength the Horndeski action can 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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.

15. 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

16. 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)

17. 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

18. 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

20. 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

21. Thanks for attention!