Relic gravitons at intermediate frequencies and the expansion history of the Universe

The early expansion history of the Universe is constrained by combining the most recent limits on the cosmic gravitons in the audio band and the claimed evidences of the nHz domain. The simplest scenario stipulates that between the end of inflation and the formation of light nuclei the evolution consists of a single phase expanding at a rate that is either faster or slower than the one of radiation. If there are instead multiple postinflationary stages evolving at different rates, then the spectral energy density always undershoots the signals potentially attributed to relic gravitons by the pulsar timing arrays at intermediate frequencies but ultimately develops a local maximum. After examining further complementary possibilities (like the presence of a secondary stage of inflation at low scales), we analyze the early modifications of the effective expansion rate and argue that if the refractive index of the relic gravitons increases during a conventional inflationary epoch, then the spectral energy density is blue above the fHz and then flattens out in the $\mu \mathrm{Hz}$ region. In this instance the signal is compatible with the unconfirmed nHz observations, with the most recent limits of the wide-band interferometers and with the further constraints customarily imposed on the backgrounds of relic gravitons produced during inflation.

Figure 1

The effective expansion rate is qualitatively illustrated in terms of the scale factor; common logarithms are employed on both axes. The shaded region extends from the end of the inflationary epoch down to the nucleosynthesis curvature scale, which is here considered as a strict lower bound on the duration of a postinflationary phase potentially different from radiation.

Figure 2

In the plot at the left the effective horizon is illustrated when the inflationary evolution is modified. In the plot at the right we instead illustrate the most general situation where the evolution of the effective horizon is modified during and after inflation; the plot at the right is, in practice, the combination of Fig. 1 with the plot at the left. Common logarithms are employed on both axes and in both cartoons.

Figure 3

The profiles of $aH/M_P$ are illustrated in terms of the scale factor; common logarithms are employed on both axes. After an inflationary stage, where $aH/M_P$ evolves linearly with the scale factor, the background decelerates and, in the left panel, the expansion rate is faster than radiation while in the plot at the right, it is slower than radiation. The two rectangles approximately define the region where the expansion rate may differ from radiation. A swift comparison suggests that the same gap in $aH$ is covered in different redshifts depending on the expansion rate. The two profiles also define the pivotal frequencies of the spectrum (i.e., ${\nu }_{\max }$ and ${\nu }_r$) that are associated, respectively, with $a_1{H}_1$ and with $a_r{H}_r$ [see, in this respect, Eqs. (3.7)–(3.8) and (3.9)].

Figure 4

The common logarithm of $h_0^2{\mathrm{\Omega }}_{gw}(\nu ,{\tau }_0)$ is illustrated as a function of the common logarithm of the frequency in the cases $\delta >1$ (panel at the left) and $\delta <1$ (panel at the right). The selected parameters correspond to the last Planck release supplemented by the more constraining bounds on $r_T$ obtained later on [5, 6, 7].

Figure 5

In the plot at the left the BBN bound of Eqs. (2.34)–(2.35) is illustrated in the $(\xi ,\delta )$ plane and for $0<\delta <1$. The dashed line corresponds to the analytic approximation of Eq. (3.23). The shaded area corresponds to the allowed region of the $(\delta ,\xi )$ plane where the BBN bound is enforced. In the plot at the right we illustrate the LVK bound of Eqs. (2.36) and (2.37)–(2.38). By comparing the two allowed regions, the BBN bound turns out to be the most constraining.

Figure 6

When ${\delta }_1>1$ and ${\delta }_2<1$ the two successive phases expand, respectively, faster and slower than radiation (see the left panel). In the plot at the right the hierarchy is inverted and ${\delta }_1<1$ while ${\delta }_2>1$. In both panels ${\delta }_1>0$ and ${\delta }_2>0$ have exactly the same meaning of $\delta$ appearing in Fig. 3 with the difference that there are now two postinflationary stages.

Figure 7

The two plots are obtained from Fig. 6 by setting ${\delta }_1\to 1$. In the left panel ${\delta }_2>1$ while in the right panel ${\delta }_2<1$. The case ${\delta }_1\to 1$ and ${\delta }_2\ne 1$ is dynamically different from the one already examined in Sec. 3. On the contrary the choice ${\delta }_1\ne 1$ and ${\delta }_2\to 1$ gives again the time line of Fig. 3 with the only difference that radiation dominates earlier.

Figure 8

The evolution of $aH/M_P$ is approximately illustrated when a second stage of inflation takes place after the radiation-dominated phase. In the intermediate (decelerated) epoch of expansion we may have that ${\delta }_1\to 1$ (as it happens in the case of radiation) even if, in these two panels, we illustrate the more general situation.

Figure 9

The spectral energy density is reported when ${\delta }_2<1$ and ${\delta }_1\ge 1$. The spectral energy develops a maximum for $\nu \simeq {\nu }_2$. For the parameters chosen here, the typical frequencies ${\nu }_r$, ${\nu }_2$, and ${\nu }_{\max }$ (see Eqs. (4.13)–(4.14) and discussion thereafter) have been explicitly illustrated. For consistency the late-time parameters correspond to the ones already discussed in Fig. 4.

Figure 10

In the plot at the left we illustrate the claimed signal of the PTA collaborations (see Eqs. (2.30)–(2.33) and discussions therein) in the case $\beta =-2/3$ which implies, according to Eq. (2.33), that the spectral energy density scales as ${\nu }^{2/3}$. In the plot at the right the spectral energy density is computed by estimating ${\xi }_1$ and ${\xi }_2$ from Eqs. (4.16)–(4.17). Since the dashed region in the right panel accounts for the potential signal of the PTA collaborations, the spectral energy density computed for different frequencies and as a function of ${\delta }_2$ always undershoots the PTA data.

Figure 11

The KAGRA-LIGO-Virgo bound is applied to the case where the postinflationary stage contains two successive phases where ${\delta }_1\ge 1$ and $0<{\delta }_2\le 1$. The spectral energy density associated with this dynamical profile is illustrated in Fig. 9. The labels on the different curves in both plots correspond to the value of the common logarithm of $h_0^2{\mathrm{\Omega }}_{gw}(\nu ,{\tau }_0)$ which is constant along each of the contours.

Figure 12

The spectral energy density is illustrated in the situation where ${\delta }_2>1$ and ${\delta }_1\le 1$. In this case the first postinflationary phase is slower than radiation while the second stage expands at a rate which is faster than radiation. The spectral energy density is standard for $\nu <{\nu }_r$, it decreases at intermediate frequencies (i.e., for ${\nu }_r<\nu <{\nu }_2$), and it increases again in the high-frequency branch (i.e., for ${\nu }_2<\nu <{\nu }_{\max }$). However, since the increase always occurs after the absolute minimum, it is clear that the maximal $h_0^2{\mathrm{\Omega }}_{gw}(\nu ,{\tau }_0)$ always undershoots the profile obtained in the case of a single phase with $\delta <1$ (see Fig. 4) for the same choice of the other late-time parameters.

Figure 13

In the plot at the left we schematically illustrate $aH/M_P$ and the different frequency ranges. In the right panel the corresponding spectral energy density is reported. While ${\delta }_1$ changes, the value of ${\delta }_2$ is now fixed, in both plots, by the occurrence of a second inflationary phase implying ${\delta }_2\to -1$. For the illustrative choice of the parameters of the plot at the right we have ${\nu }_r=\mathrm{kHz}$ while ${\nu }_2=\mathrm{mHz}$.

Figure 14

In the plot at the left we report a profile for the effective expansion rate where the refractive phase takes place during inflation and then the radiation background dominates after inflation. In the right plot the dominance of radiation is delayed. We note that on the vertical axis we report $F=\dot{b}/b$ where the overdot now denotes a derivation with respect to the $\eta$ time that differs from the conformal time during the refractive stage but coincides with $\tau$ time after inflation.

Figure 15

The shaded areas in both plots illustrate the allowed regions of the parameter space where the spectral energy density is compatible with the PTA measurements (see Eq. (2.30) and also the left plot in Fig. 10). The different contours appearing in the plots correspond to the values of $h_0^2{\mathrm{\Omega }}_{gw}({\nu }_{\mathrm{LVK}},{\tau }_0)$ where ${\nu }_{\mathrm{LVK}}$ approximately denotes the most sensitive frequency domain of the wide-band interferometers [see Eqs. (2.36) and (2.37)–(2.38)]. Within the allowed regions the BBN constraints of Eqs. (2.34)–(2.35) are always satisfied.

Figure 16

We illustrate the spectral energy density for a set of parameters selected within the shaded region appearing in the exclusion plot of Fig. 15. In the plot at the left we consider the situation where the postinflationary evolution is dominated by radiation, as illustrated in the left cartoon of Fig. 14. In the right plot the postinflationary evolution includes, instead, a first stage expanding at a rate slower than radiation (see also the right profile of Fig. 14).