Tracker Solutions Research Articles

3D View Factor Power Output Modelling of Bifacial Fixed, Single, and Dual-Axis Agrivoltaic Systems

This study investigates the performance of agrivoltaic systems employing bifacial photovoltaic modules. A comparison between yield in Sweden and Italy was carried out. Three agrivoltaic system designs were evaluated: vertical fixed, single-axis tracker, and dual-axis tracker. The results showed that the specific production varied between 1090 to 1440 kWh/kWp/yr in Sweden and 1584 to 2112 kWh/kWp/yr in Italy, where the lowest production was obtained with the vertical fixed agrivoltaic system while the highest production was obtained with the dual-axis tracking agrivoltaic system. The vertical fixed design had a higher electricity production during low solar elevation angles, while the single-axis and dual-axis tracking designs had significantly higher power production during mid-day. The electricity production gain using a dual-axis tracker design was mostly during mid-day, but the increase compared to the single-axis tracker was only 1-2%. The study concludes that low-height, fixed agrivoltaic systems without tracking are well-suited for high-latitude countries like Sweden, while elevated systems with tracker solutions are more suitable for locations like Italy. The findings suggest that the performance of agrivoltaic systems with bifacial photovoltaic modules is highly dependent on geographical location and the specific characteristics of the crops grown beneath them.

Catch-me-if-you-can: the overshoot problem and the weak/inflation hierarchy

We study the overshoot problem in the context of post-inflationary string cosmology (in particular LVS). LVS cosmology features a long kination epoch as the volume modulus rolls down the exponential slope towards the final minimum, with an energy density that scales as {m}_s^4 . It is a known fact that such a roll admits attractor tracker solutions, and if these are located the overshoot problem is solved. We show that, provided a sufficiently large hierarchy exists between the inflationary scale and the weak scale, this will always occur in LVS as initial seed radiation grows into the tracker solution. The consistency requirement of ending in a stable vacuum containing the weak hierarchy therefore gives a preference for high inflationary scales — an anthropic argument, if one likes, for a large inflation/weak hierarchy. We discuss various origins, both universal and model-dependent, of the initial seed radiation (or matter). One particularly interesting case is that of a fundamental string network arising from brane inflation — this may lead to an early epoch in which the universe energy density principally consists of gravitational waves, while an LVS fundamental string network survives into the present universe.

Qualitative dynamics of interacting vacuum cosmologies

We present a phase-space analysis of the qualitative dynamics cosmologies where dark matter exchanges energy with the vacuum component. We find fixed points corresponding to power-law solutions where the different components remain a constant fraction of the total energy density and given an existence condition for any fixed points with nonvanishing energy transfer. For some interaction models we find novel fixed points in the presence of a third noninteracting fluid with constant equation of state, such as radiation, where the interacting vacuum+matter tracks the evolution of the third fluid, analogous to tracker solutions previously found for self-interacting scalar fields. We illustrate the phase-plane behavior, determining the equation of state and stability of the fixed points in the case of a simple linear interaction model, for interacting vacuum and dark matter, including the presence of noninteracting radiation. We give approximate solutions for the equation of state in matter- or vacuum-dominated solutions in the case of small interaction parameters.

Phenomenology of the generalized cubic covariant Galileon model and cosmological bounds

We investigate the generalized cubic covariant Galileon model, a kinetically driven dark energy model within the Horndeski class of theories. The model extends the cubic covariant Galileon by including power laws of the field derivatives in the K-essence and cubic terms which still allow for tracker solutions. We study the shape of the viable parameter space by enforcing stability conditions which include the absence of ghost, gradient and tachyon instabilities and the avoidance of strong coupling at early time. We study here the relevant effects of the modifications induced by the model on some cosmological observables such as the cosmic microwave background (CMB), the lensing potential auto-correlation and the matter power spectrum. For this goal, we perform parameter estimation using data of CMB temperature and polarization, baryonic acoustic oscillations (BAO), redshift-space distortions (RSD), supernovae type Ia (SNIa) and Cepheids. Data analysis with CMB alone finds that the today's Hubble parameter $H_0$ is consistent with its determination from Cepheids at $1\sigma$, resolving the famous tension of the cosmological standard models. The joint analysis of CMB, BAO, RSD and SNIa sets a lower bound for the sum of neutrino masses which is $\Sigma m_\nu >0.11$ eV at 1$\sigma$, in addition to the usual upper limit. The model selection analysis based on the effective $\chi_\text{eff}^2$ and Deviance Information Criterion is not able to clearly identify the statistically favored model between $\Lambda$CDM and the generalized cubic covariant Galileon, from which we conclude that the latter model deserves further studies.

Growth of matter perturbations in the bi-Galileons field model

We study a dark energy cubic bi-Galileons field model based on truncation of the recently proposed generalized covariant multi-Galileons model. We investigate the cosmological dynamic of the model by the theory of dynamical systems through the analysis of the properties of the fixed points in each cosmological epoch. We show the existence of two tracker solutions, one of which is that of the cubic single Galileon model and the other solution is the signature of the second Galileon field. Exploiting the competition between the two Galileon fields, we find a dark energy solution that avoids the approach to the tracker solution with dark energy equation of state $w_{DE}=-2$ during the matter epoch which is disfavored by the observational data. We study also the growth rate of matter perturbations. Using recent $f\sigma_{8}$ redshift space distortion (RSD) and model-independent observational Hubble (OHD) data sets, we place observational constraints on the coupling constant and cosmological parameters of the bi-Galileons model through Monte Carlo numerical method based on the Metropolis-Hastings algorithm. We find that the amplitude of growth matter fluctuations is consistent with the Planck15 data and ease the tension between early and later clustering, and fits better the data from the DES survey over the data from KiDS-450 survey. We also find that the best fit value for the Hubble constant is compatible with new measurements of Cepheid-supernovae distance scale. Finally, we perform a model selection through the Bayes factor and found that the bi-Galileons model is disfavored in comparison to the $\Lambda$CDM model, but slightly preferred to $w$CDM model.

Observational status of the Galileon model general solution from cosmological data and gravitational waves

The Galileon model is a tensor-scalar theory of gravity which explains the late acceleration of the Universe expansion with no instabilities and recovers General Relativity in the strong field limit. Most constraints obtained so far on Galileon model parameters from cosmological data were derived for the limited subset of tracker solutions and reported tensions between the model and data. This paper presents the first exploration of the general solution of the Galileon model, which is confronted against recent cosmological data for both background observables and linear perturbations, using Monte-Carlo Markov chains. As representative scenarios of the Galileon models, we study the Full galileon model with disformal coupling to matter and the uncoupled Cubic galileon model. We find that the general solution of the Full galileon model provides a good fit to CMB spectra, while the Cubic galileon model does not. When extending the comparison to BAO and SNIa data, even the general solution of the Full galileon model fails at providing a good fit to all datasets simultaneously. Tensions remain if the models are extended with an additional free parameter, such as the sum of active neutrino masses or the normalization of the CMB lensing spectrum. Finally, the multi-messenger observation of GW170817 is also discussed in the framework of the scenarios considered. The time delay between the gravitational wave signal and its electromagnetic counterpart was computed a posteriori in every scenario of the Full galileon model cosmological fit chains and found to be ruled out by this observation.

Tracker and scaling solutions in DHOST theories

In quadratic-order degenerate higher-order scalar–tensor (DHOST) theories compatible with gravitational-wave constraints, we derive the most general Lagrangian allowing for tracker solutions characterized by ϕ˙/Hp=constant, where ϕ˙ is the time derivative of a scalar field ϕ, H is the Hubble expansion rate, and p is a constant. While the tracker is present up to the cubic-order Horndeski Lagrangian L=c2X−c3X(p−1)/(2p)□ϕ, where c2,c3 are constants and X is the kinetic energy of ϕ, the DHOST interaction breaks this structure for p≠1. Even in the latter case, however, there exists an approximate tracker solution in the early cosmological epoch with the nearly constant field equation of state wϕ=−1−2pH˙/(3H2). The scaling solution, which corresponds to p=1, is the unique case in which all the terms in the field density ρϕ and the pressure Pϕ obey the scaling relation ρϕ∝Pϕ∝H2. Extending the analysis to the coupled DHOST theories with the field-dependent coupling Q(ϕ) between the scalar field and matter, we show that the scaling solution exists for Q(ϕ)=1/(μ1ϕ+μ2), where μ1 and μ2 are constants. For the constant Q, i.e., μ1=0, we derive fixed points of the dynamical system by using the general Lagrangian with scaling solutions. This result can be applied to the model construction of late-time cosmic acceleration preceded by the scaling ϕ-matter-dominated epoch.

Thawing versus tracker solutions: a dynamical systems approach

A comparative study of thawing and tracking models of dark energy is carried out with the help of a dynamical systems analysis. It is found that both of them have stable solutions which are consistent with the requirement of a dark energy. So none of them is actually favored from the consideration of stability. The trackers have the interesting possibility that the present acceleration is a transient phenomenon.

Tracker coupled quintessence

In this paper two tracker solutions for conformally coupled quintessence model is obtained. The first solution is determined by dividing the evolution of the universe to matter and dark energy dominated epochs and then the allowed bound on the equation of state parameter is obtained, such that a tracker behaviour is resulted. For the other solution, the quintessence potential is assumed to be a linear combination of two exponential functions. Then we have found an exact solution which can be tracker by adjusting some parameters.

Cosmological constraints on extended Galileon models

The extended Galileon models possess tracker solutions with de Sitterattractors along which the dark energy equation of state is constantduring the matter-dominated epoch, i.e. wDE = −1−s, wheres is a positive constant. Even with this phantom equation of statethere are viable parameter spaces in which the ghosts and Laplacianinstabilities are absent. Using the observational data of the supernovaetype Ia, the cosmic microwave background (CMB), and baryon acousticoscillations, we place constraints on the tracker solutions at thebackground level and find that the parameter s is constrained tobe s = 0.034−0.034+0.327 (95 % CL) in the flat Universe. Inorder to break the degeneracy between the models we also study theevolution of cosmological density perturbations relevant to the large-scalestructure (LSS) and the Integrated-Sachs-Wolfe (ISW) effect in CMB.We show that, depending on the model parameters, the LSS and the ISWeffect is either positively or negatively correlated. It is then possibleto constrain viable parameter spaces further from the observationaldata of the ISW-LSS cross-correlation as well as from the matter powerspectrum.

Quintom evolution in power-law potentials

We investigate quintom evolution in power-law potentials. We extract the early-time, tracker solutions, under the assumption of matter domination. Additionally, we derive analytical solutions at intermediate times, that is at low redshifts, which is the period during the transition from matter to dark-energy domination. A comparison with exact evolution reveals that the tracker solutions are valid within 98% for z ≳ 1.5 , while the intermediate ones are accurate within 98% up to z ≈ 0.5 . Using these expressions we extract two new w-parametrizations, one in terms of the redshift and one in terms of the dark-energy density-parameter, and we present various quintom evolution sub-classes, including quintessence-like or phantom-like cases, realization of the −1-crossing, and non-monotonic w-evolution.

Trackingf(R)cosmology

Metric $f(R)$ gravity theories are conformally equivalent to models of quintessence in which matter is coupled to dark energy. We derive a condition for stable tracker solution for metric $f(R)$ gravity in the Einstein frame. We find that tracker solutions with $-0.361<\omega_{\varphi}<1$ exist if $0<\Gamma<0.217$ and $\frac{d}{dt} \ln f'(\tilde{R})>0$, where $\Gamma=\frac{V_{\varphi\varphi}V}{V_{\varphi}^{2}}$ is dimensionless function, $\omega_{\varphi}$ is the equation of state parameter of the scalar field and $\tilde{R}$ refers to Jordan frame's curvature scalar. Also, we show that there exists $f(\tilde{R})$ gravity models which have tracking behavior in the Einstein frame and so the curvature of space time is decreasing with time while they lead to the solutions in the Jordan frame that the curvature of space time can be increasing with time.

Tracking quintessence andk-essence in a general cosmological background

We derive conditions for stable tracker solutions for both quintessence and $k$-essence in a general cosmological background, ${H}^{2}\ensuremath{\propto}f(\ensuremath{\rho})$. We find that tracker solutions are possible only when $\ensuremath{\eta}\ensuremath{\equiv}d\mathrm{ln}f/d\mathrm{ln}\ensuremath{\rho}\ensuremath{\approx}\mathrm{\text{constant}}$, aside from a few special cases, which are enumerated. Expressions for the quintessence or $k$-essence equation of state are derived as a function of $\ensuremath{\eta}$ and the equation of state of the dominant background component.

K-essence and the coincidence problem

$k$-essence has been proposed as a possible means of explaining the coincidence problem of the Universe beginning to accelerate only at the present epoch. We carry out a comprehensive dynamical systems analysis of the k-essence models given so far in the literature. We numerically study the basin of attraction of the tracker solutions and we highlight the behavior of the field close to sound speed divergences. We find that, when written in terms of parameters with a simple dynamical interpretation, the basins of attraction represent only a small region of the phase space.

Trackingk-essence

We derive a condition for converging a common evolutionary track for k-essence (a scalar field dark energy with noncanonical kinetic terms). For the Lagrangian density $V(\ensuremath{\varphi})W(X)$ with $X={\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}}^{2}/2,$ we find that tracker solutions with ${w}_{\ensuremath{\varphi}}l{w}_{B}$ exist if $\ensuremath{\Gamma}\ensuremath{\equiv}{V}^{\ensuremath{''}}{V/(V}^{\ensuremath{'}}{)}^{2}g3/2.$ Here ${w}_{\ensuremath{\varphi}}{(w}_{B})$ is the equation of state of the scalar field (background radiation or matter). Our condition may be useful for examining the existence of attractorlike behavior in cosmology with k-essence (for example, a rolling tachyon).

Tracking extended quintessence

We investigate the cosmological role of a tracking field $\ensuremath{\varphi}$ in extended quintessence scenarios, where the dynamical vacuum energy driving the acceleration of the universe today possesses an explicit coupling with the Ricci scalar R of the form $F(\ensuremath{\varphi})R/2,$ where $F(\ensuremath{\varphi})$ mimics general relativity today, $F({\ensuremath{\varphi}}_{0})=1/8\ensuremath{\pi}G.$ We analyze explicit nonminimally coupled (NMC) models where $F(\ensuremath{\varphi})=1/8\ensuremath{\pi}G+\ensuremath{\xi}({\ensuremath{\varphi}}^{2}\ensuremath{-}{\ensuremath{\varphi}}_{0}^{2}),$ with $\ensuremath{\xi}$ is the coupling constant and ${\ensuremath{\varphi}}_{0}$ is the Q value today. Tracker solutions for these NMC models, with inverse power-law potentials, possess an initial enhancement of the scalar field dynamics, named the R-boost, caused by the effective potential generated by the Ricci scalar in the Klein-Gordon equation. During this phase the field performs a ``gravitational'' slow rolling until the true potential becomes important. We give accurate analytic formulas describing the R-boost, showing that the quintessence energy in this phase scales with the redshift z as ${(1+z)}^{2}.$ When the R-boost ends, the field trajectory matches the tracker solution in minimally coupled theories. We compute perturbations in these tracking extended quintessence models, by integrating the full set of equations for the evolution of linear fluctuations in scalar-tensor theories of gravity, and assuming Gaussian scale-invariant initial perturbations. The integrated Sachs-Wolfe (ISW) effect on the cosmic microwave background (CMB) angular spectrum causes a change $\ensuremath{\delta}{C}_{l}{/C}_{l}\ensuremath{\simeq}6[1\ensuremath{-}8\ensuremath{\pi}GF({\ensuremath{\varphi}}_{\mathrm{dec}})]$ at $l\ensuremath{\lesssim}10,$ where ``dec'' stands for decoupling. Similarly, the CMB acoustic peak multipoles shift compared to ordinary tracking quintessence models by roughly an amount $\ensuremath{\delta}l/l\ensuremath{\simeq}[8\ensuremath{\pi}GF({\ensuremath{\varphi}}_{\mathrm{dec}})\ensuremath{-}1]/8.$ The turnover wave number ${k}_{\mathrm{turn}}$ in the matter power spectrum shifts by an amount $\ensuremath{\delta}{k}_{\mathrm{turn}}{/k}_{\mathrm{turn}}\ensuremath{\simeq}[1\ensuremath{-}8\ensuremath{\pi}GF({\ensuremath{\varphi}}_{\mathrm{eq}})]/2,$ where ``eq'' stands for matter-radiation equivalence. All these corrections may assume positive as well as negative values, depending on the sign of the NMC parameter $\ensuremath{\xi}.$ We show that the above effects can be as large as 10--30 % with respect to equivalent cosmological constant and ordinary tracking quintessence models, respecting all the existing experimental constraints on scalar-tensor theories of gravity. These results demonstrate that the playground where the data of the next decade will have their impact includes the nature of the dark energy in the Universe, as well as the structure of the theory of gravity.

Cosmological tracking solutions

A substantial fraction of the energy density of the universe may consist of quintessence in the form of a slowly rolling scalar field. Since the energy density of the scalar field generally decreases more slowly than the matter energy density, it appears that the ratio of the two densities must be set to a special, infinitesimal value in the early universe in order to have the two densities nearly coincide today. Recently, we introduced the notion of tracker fields to avoid this initial conditions problem. In the paper, we address the following questions: What is the general condition to have tracker fields? What is the relation between the matter energy density and the equation-of-state of the universe imposed by tracker solutions? And can tracker solutions help to explain why quintessence is becoming important today rather than during the early universe?