Imprints of the QCD phase transition on the spectrum of gravitational waves

We have investigated effects of the QCD phase transition on the relic GW spectrum applying several equations of state for the strongly interacting matter: Besides the bag model, which describes a first-order transition, we use recent data from lattice calculations featuring a crossover. Finally, we include a short period of inflation during the transition which allows for a first-order phase transition at finite baryon density. Our results show that the QCD transition imprints a step into the spectrum of GWs. Within the first two scenarios, entropy conservation leads to a step-size determined by the relativistic degrees of freedom before and after the transition. The inflation of the third scenario much stronger attenuates the high frequency modes: An inflationary model being consistent with observation entails suppression of the spectral energy density by a factor of $\lesssim {10}^{-12}$.

Figure 1

Sketch of a possible phase diagram for strongly interacting matter. The early Universe is generally assumed to evolve at very small baryon chemical potential, where lattice calculations predict a crossover. At higher ${\mu }_b$ a first-order phase transition is possible. The evolution of the Universe is indicated by arrows, see also 24.

Figure 2

Possible track of the background fluid through the phase diagram for strongly interacting matter. Within this scenario the system intersects the first-order line starting from high chemical potential. It is supercooled during inflation after which the actual phase transition takes place and reheating sets in. Afterwards, the evolution follows the commonly accepted path.

Figure 3

Behavior of a GW entering the horizon. For this calculation the background was assumed to be purely radiative. A phase transition is not included. The frequency of the mode was chosen to be the critical frequency: $|\mathbf{k}|=2\pi {\nu }^{*}$. The dashed line indicates a function proportional to $1/a$ fit to the amplitude of the GW after horizon entry.

Figure 4

The trace anomaly of strongly interacting matter, normalized to the temperature. Within the bag model, the only contribution comes from the bag constant $B$. The dashed curves depict parameterizations of results of lattice calculations. More explanation is provided in the text.

Figure 5

The pressure of strongly interacting matter, normalized to the temperature. The dashed lines correspond to lattice data. For bag model 1 we included pions in the low temperature phase and we excluded strange quarks from the high temperature phase: $g_{\mathrm{quarks}}=37$, $g_{\mathrm{hadrons}}=3$. For bag model 2 it is the opposite way around: $g_{\mathrm{quarks}}=47.5$, $g_{\mathrm{hadrons}}=0$. The lattice results tend to the Stefan-Boltzmann limit of bag model 2.

Figure 6

The energy density of strongly interacting matter within the bag model and according to lattice data. More information is provided in the caption of Fig. 5.

Figure 7

The energy density of GWs per logarithmic frequency interval after different types of QCD phase transition. Modes with high frequency are damped by a factor of $\sim 0.7$ compared to low frequency modes. The solid line represents the result of a bag model calculation with $T_c=180\mathrm{MeV}$. The dashed curves are calculated using lattice data. In three cases the hadron resonance gas ansatz has also been taken into account.

Figure 8

Energy density and pressure during a QCD phase transition with a short period of inflation. The total energy density decreases proportional to $a^{-4}$ until the bag constant $B\approx (250\mathrm{MeV}{)}^4$ is no longer negligible. When the vacuum contribution is dominant, the total energy density does not depend on the scale factor $a$. The same is true for the pressure which is negative during inflation: $p=-B$. The radiative energy density becomes negligible soon after inflation begins.

Figure 9

The energy spectrum of GWs after a QCD phase transition with a short period of inflation. The inflationary phase leads to a much stronger dilution than an ordinary bag model transition.

Figure 10

Temporal evolution of two modes during a QCD phase transition with a period of inflation. The mode with frequency ${\nu }_1/{\nu }_2$ is at maximal/small displacement when leaving the horizon. Therefore, the former appears as a maximum in the spectrum and the latter as a minimum.

Figure 11

The energy spectrum of GWs after a QCD phase transition including a little inflation followed by a period which is dominated by the kinetic energy density of a scalar field. The spectrum of modes which reenter the horizon during this stage falls like ${\nu }^{-3}$, whereas the energy density in modes which reenter during the subsequent radiation dominated epoch is proportional to ${\nu }^{-4}$.

Figure 12

Energy spectra of relic GWs after different types of QCD transition. The parameter $s$ indicates the duration of the oscillation dominated period within the corresponding scenario. In the case $s=0$ there is no oscillation domination included.

Figure 13

GWs after the QCD transition and their detectability. The (predicted) sensitivities of three GW detectors are displayed. PPTA (Parkes Pulsar Timing Array) already constrains the GW production during the QCD phase transition. The corresponding spectrum is displayed on the basis of 3 (Kamionkowski et al.) and 4 (Caprini et al.). See also 21, 25. For the amplitude of relic GWs, we use the largest amplitude consistent with the COBE constraint given in 26. The solid line in the upper panel sketches the result of a calculation including a little inflation and a subsequent oscillation dominated period with $s=15$.