Brief history of curvature

The trace of the stress-energy tensor of the cosmological fluid, proportional to the Ricci scalar curvature in general relativity, is determined on cosmic scales for times ranging from the inflationary epoch to the present day in the expanding Universe. The post-inflationary epoch and the thermal history of the relativistic fluid, in particular the QCD transition from asymptotic freedom to confinement and the electroweak phase transition, leave significant imprints on the scalar curvature. These imprints can be of either sign and are orders of magnitude larger than the values that would be obtained by naively extrapolating the pressureless matter of the present epoch back into the radiation-dominated epoch.

Figure 1

The Ricci scalar curvature in units of ${\mathrm{Mpc}}^{-2}$, or equivalently the trace of the cosmological fluid stress-energy tensor in units of ${\mathrm{GeV}}^4$, is shown by the thick (blue) curve for ${10}^{-4}<a/a_0<1$ or a run of redshift $0<z<{10}^4$. At late times, the trace approaches a constant value due to the onset of dark energy domination. For reference, the energy density is shown by the thin, long dashed (black) curve. The pressure is shown by the thin, short dashed (red) curve. Note that the pressure is negative after $a/a_0\gtrsim 0.08$, so the absolute value is plotted. In the lower panel, the trace in units of the total energy density is shown, $1-3p/\rho$.

Figure 2

The contribution to the ratio $\Theta /T^4$ by a species ($g=1$) cooling below its rest mass is shown. The solid (dashed) line is for fermions (bosons). According to a free field treatment, the freeze-out of each species of the Standard Model results in a bump to the trace.

Figure 3

Comparison of the thermodynamics of lattice QCD and of a theory of free quarks and gluons. In all cases, three quark species ($u$, $d$, and $s$) have been included. The curve labeled “hotQCD” is based on Ref. 4, while the curve labeled “Wuppertal” is from Ref. 5; see the main text for details. The free field theory curve was obtained by setting the masses of all the quarks and gluons artificially to 500 MeV. This counter-factual assumption is made in order to obtain the narrowest possible peak in $\Theta /T^4$ in the vicinity of the QCD transition, illustrating that interactions have a significant effect beyond introducing dynamically generated masses.

Figure 4

Comparison of the ring-improved one-loop treatment of electroweak thermodynamics [15] with free field expectations, shown by the solid black and dashed red lines, respectively. Both curves are based on the minimal Standard Model with the Higgs of mass $m_H=125\mathrm{GeV}$ [6, 7]. The sharp feature at 160 GeV, corresponding to the temperature at which the symmetry of the Higgs potential is broken, is a shortcoming of the ring-improved one-loop treatment.

Figure 5

The degrees of freedom of the Standard Model and a supersymmetric extension are illustrated as functions of equilibrium temperature. In the top panel, the degrees of freedom of the energy density, $g_{*}$ (heavy curves), and pressure $g_{*p}$ (light curves), are shown for the Standard Model (solid curves) and a representative supersymmetric extension (dashed curves). In the lower panel, the difference $g_{*}-g_{*p}$ is shown. Note that adding a single dark matter species to the Standard Model curves does not produce a discernible change in these figures.

Figure 6

The Ricci scalar curvature in units of ${\mathrm{m}}^{-2}$, or equivalently the trace of the cosmological fluid stress-energy tensor in units of ${\mathrm{GeV}}^4$, as in Fig. 1, for ${10}^{-18}<a/a_0<{10}^{-4}$. The energy density and the pressure are also shown as long dashed (black) and short dashed (red) lines. Except as noted below, $\Theta =\rho -3p$ is positive. For reference, a continuation of the energy density in pressureless matter is shown by the dotted black line; naive treatments that equate the trace with the matter density at early times are easily off by $\sim 5-10$ orders of magnitude. The solid (blue) curve shows $\Theta$ for the Standard Model plus a dark matter particle of mass $m_{\mathrm{DM}}=100\mathrm{TeV}$. The thin dashed (blue) curve shows the case of the SM through the electroweak transition as described in the text, with $\Theta$ becoming negative at $a/a_0<{10}^{-16}$. The long dashed (blue) curve near the left edge of the plot shows the case of the cMSSM. In the lower panel, the trace in units of the critical density is shown; the spikes at $a/a_0={10}^{-16}$, ${10}^{-15}$, $4\times {10}^{-13}$, ${10}^{-9}$ correspond to particles of the cMSSM becoming nonrelativistic, the electroweak transition, the QCD transition, and the departure of electrons from equilibrium as they become nonrelativistic, respectively.

Figure 7

The Ricci scalar curvature in units of ${\mathrm{fm}}^{-2}$, or equivalently the trace of the cosmological fluid stress-energy tensor in units of ${\mathrm{GeV}}^4$, as in Fig. 1, for ${10}^{-32}<a/a_0<{10}^{-18}$. Except as noted below, $\Theta =\rho -3p$ is positive. The energy density and the absolute value of the pressure are also shown as thin long dashed (black) and thin short dashed (red) lines. Inflation continues at earlier times than shown in the plot, with all curves continuing as horizontal lines. The solid (blue) line shows the extrapolation of the SM with a DM species with mass 100 TeV back to the post-inflationary epoch. The thick long dashed (blue) line shows the extrapolation of the cMSSM where the LSP is at 363 GeV. In the cMSSM scenario, a putative decay product of the inflaton having equation-of-state $+1$ is considered, for which the contribution to the trace is negative. In such a case, the curvature and trace may become negative, as shown in the range $3\times {10}^{-27}<a/a_0<3\times {10}^{-23}$ by the thin dotted (blue) line. $\Theta$ also becomes negative during oscillations at the end of inflation. In the lower panel, the trace in units of the critical density is shown. After the inflaton makes several oscillations across the bottom of its potential, and until the Universe is fully radiation dominated, we plot the time average of the trace for clarity of display.

Figure 8

A cosmic history of the Ricci scalar curvature in units of ${\mathrm{m}}^{-2}$, or equivalently the trace of the cosmological fluid stress-energy tensor in units of ${\mathrm{GeV}}^4$. The run of time scales is from the end stages of inflation to the present day, ${10}^{-32}<a/a_0<1$. The energy density and the absolute value of the pressure are shown by the thin long dashed (black) and thin short dashed (red) curves. Also shown is an extrapolation of the energy density in pressureless matter into the radiation era, for reference, which terminates near the possible freeze-out of a dark matter particle of mass $m_{\mathrm{DM}}=100\mathrm{TeV}$. This plot is a composite of Figs. 1, 6, 7, and the explanatory comments of previous captions apply to this figure as well. In the lower panel, the trace in units of the critical density is shown. The inset box is magnified by a factor of three to show spikes in curvature, due to the detailed thermal history of the SM and a supersymmetric extension, the electroweak transition, the QCD transition, and the departure from equilibrium of electrons. Simple treatments that attribute the curvature due to the contribution to the trace by the energy density in pressureless matter at early times underestimate the amplitude of the trace by many orders of magnitude, and miss out on episodes in which the curvature and trace are negative.