We present results for the equation of state in ($2+1$)-flavor QCD using the highly improved staggered quark action and lattices with temporal extent $N_{\tau }=6$, 8, 10, and 12. We show that these data can be reliably extrapolated to the continuum limit and obtain a number of thermodynamic quantities and the speed of sound in the temperature range 130–400 MeV. We compare our results with previous calculations and provide an analytic parameterization of the pressure, from which other thermodynamic quantities can be calculated, for use in phenomenology. We show that the energy density in the crossover region, $145\mathrm{MeV}\le T\le 163\mathrm{MeV}$, defined by the chiral transition, is ${\epsilon }_c=(0.18–0.5)\mathrm{GeV}/{\mathrm{fm}}^3$, i.e., $(1.2–3.1){\epsilon }_{\text{nuclear}}$. At high temperatures, we compare our results with resummed and dimensionally reduced perturbation theory calculations. As a byproduct of our analyses, we obtain the values of the scale parameters $r_0$ from the static quark potential and $w_0$ from the gradient flow.

We present results on the chiral and deconfinement properties of the QCD transition at finite temperature. Calculations are performed with $2+1$ flavors of quarks using the p4, asqtad, and HISQ/tree actions. Lattices with temporal extent $N_{\tau }=6$, 8, and 12 are used to understand and control discretization errors and to reliably extrapolate estimates obtained at finite lattice spacings to the continuum limit. The chiral transition temperature is defined in terms of the phase transition in a theory with two massless flavors and analyzed using $O(N)$ scaling fits to the chiral condensate and susceptibility. We find consistent estimates from the HISQ/tree and asqtad actions and our main result is $T_c=154\pm 9\mathrm{MeV}$.

We calculate the equation of state in $2+1$ flavor QCD at finite temperature with physical strange quark mass and almost physical light quark masses using lattices with temporal extent $N_{\tau }=8$. Calculations have been performed with two different improved staggered fermion actions, the asqtad and p4 actions. Overall, we find good agreement between results obtained with these two $O(a^2)$ improved staggered fermion discretization schemes. A comparison with earlier calculations on coarser lattices is performed to quantify systematic errors in current studies of the equation of state. We also present results for observables that are sensitive to deconfining and chiral aspects of the QCD transition on $N_{\tau }=6$ and 8 lattices. We find that deconfinement and chiral symmetry restoration happen in the same narrow temperature interval. In an appendix we present a simple parametrization of the equation of state that can easily be used in hydrodynamic model calculations. In this parametrization we include an estimate of current uncertainties in the lattice calculations which arise from cutoff and quark mass effects.

We calculated the QCD equation of state using Taylor expansions that include contributions from up to sixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have been performed with the Highly Improved Staggered Quark action in the temperature range $T\in [135\mathrm{MeV},330\mathrm{MeV}]$ using up to four different sets of lattice cutoffs corresponding to lattices of size $N_{\sigma }^3\times N_{\tau }$ with aspect ratio $N_{\sigma }/N_{\tau }=4$ and $N_{\tau }=6-16$. The strange quark mass is tuned to its physical value, and we use two strange to light quark mass ratios $m_s/m_l=20$ and 27, which in the continuum limit correspond to a pion mass of about 160 and 140 MeV, respectively. Sixth-order results for Taylor expansion coefficients are used to estimate truncation errors of the fourth-order expansion. We show that truncation errors are small for baryon chemical potentials less then twice the temperature (${\mu }_B\le 2T$). The fourth-order equation of state thus is suitable for the modeling of dense matter created in heavy ion collisions with center-of-mass energies down to $\sqrt{s_{NN}}\sim 12\mathrm{GeV}$. We provide a parametrization of basic thermodynamic quantities that can be readily used in hydrodynamic simulation codes. The results on up to sixth-order expansion coefficients of bulk thermodynamics are used for the calculation of lines of constant pressure, energy and entropy densities in the $T\text{−}{\mu }_B$ plane and are compared with the crossover line for the QCD chiral transition as well as with experimental results on freeze-out parameters in heavy ion collisions. These coefficients also provide estimates for the location of a possible critical point. We argue that results on sixth-order expansion coefficients disfavor the existence of a critical point in the QCD phase diagram for ${\mu }_B/T\le 2$ and $T/T_c({\mu }_B=0)>0.9$.

We present results on the equation of state in QCD with two light quark flavors and a heavier strange quark. Calculations with improved staggered fermions have been performed on lattices with temporal extent $N_{\tau }=4$ and 6 on a line of constant physics with almost physical quark mass values; the pion mass is about 220 MeV, and the strange quark mass is adjusted to its physical value. High statistics results on large lattices are obtained for bulk thermodynamic observables, i.e. pressure, energy and entropy density, at vanishing quark chemical potential for a wide range of temperatures, $140\mathrm{MeV}\le T\le 800\mathrm{MeV}$. We present a detailed discussion of finite cutoff effects which become particularly significant for temperatures larger than about twice the transition temperature. At these high temperatures we also performed calculations of the trace anomaly on lattices with temporal extent $N_{\tau }=8$. Furthermore, we have performed an extensive analysis of zero temperature observables including the light and strange quark condensates and the static quark potential at zero temperature. These are used to set the temperature scale for thermodynamic observables and to calculate renormalized observables that are sensitive to deconfinement and chiral symmetry restoration and become order parameters in the infinite and zero quark mass limits, respectively.