Gravitational wave generation in loop quantum cosmology

We calculate the full spectrum, as observed today, of the cosmological gravitational waves generated within a model based on loop quantum cosmology, in which the corrections to the dynamical equations describing the evolution of the universe during the semiclassical regime are of the inverse-volume type. It is assumed that the universe, after the transition to the classical regime, undergoes a period of inflation driven by a scalar field with a chaotic-type potential. Our analysis shows that, for certain conditions, loop quantum effects leave a clear signature on the spectrum, namely, an overproduction of low-frequency gravitational waves. One of the aims of our work is to show that loop quantum cosmology models can be tested and that, more generally, preinflationary physical processes leave their imprint in gravitational-wave spectra and can also be tested.

Figure 1

Time evolution of the scalar field $\varphi$ for the case $j=100$ and $\ell =3/4$. Because of loop quantum effects, the scalar field increases naturally from its initial value of about $3.6\times {10}^{-4}{m}_{\mathrm{P}}$ to a maximum value of about $3m_{\mathrm{P}}$, which guarantees a long enough inflationary period.

Figure 2

The shaded region corresponds to values of the ambiguity parameters $\ell$ and $j$ for which the universe expands at least 60 e-folds during the inflationary period. Such an expansion is achieved if the scalar field grows from ${\varphi }_i\ll m_{\mathrm{P}}$ to ${\varphi }_{\max }\gtrsim 3m_{\mathrm{P}}$ during the preinflationary epoch. The requirement that ${\varphi }_i\lesssim {10}^{-3}{m}_{\mathrm{P}}$, together with the choice $m_{\varphi }={10}^{-6}{m}_{\mathrm{P}}$ and ${\dot{\varphi }}_i=2\times {10}^{-6}{m}_{\mathrm{P}}^2$, imposes the constraint $j\gtrsim 87.3$.

Figure 3

Potential $U$ as a function of (cosmic) time $t$ for the case $\ell =3/4$ and $j=100$. The inflationary barrier is located on the right (for $t\gtrsim {10}^6{t}_{\mathrm{P}}$), while the much smaller barrier arising due to loop quantum effects is located on the left (for $t\lesssim t_{\mathrm{P}}$).

Figure 4

Gravitational-wave spectra for $\ell =3/4$ and $\ell =5/6$ (dashed line). In both cases $j=100$.

Figure 5

Today’s frequency of a gravitational wave as a function of the time at which this wave first crossed the Hubble horizon. For $\ell =3/4$, gravitational waves generated in the early universe have frequencies, today, of the order of $({10}^{-11}–{10}^{-17})\mathrm{rad}/\mathrm{s}$ (shaded region), corresponding to wavelengths of the order or smaller than the Hubble horizon today. For $\ell =5/6$, gravitational waves generated in the early universe have, today, frequencies of the order of $({10}^{-38}–{10}^{-45})\mathrm{rad}/\mathrm{s}$, corresponding to wavelengths much bigger than the Hubble horizon today. For both cases today’s maximum frequency of the gravitational waves is of the order of ${10}^9\mathrm{rad}/\mathrm{s}$, corresponding to waves that crossed the Hubble horizon at the end of the inflationary period ($t\sim 3\times {10}^7{t}_{\mathrm{P}}$).

Figure 6

Gravitational-wave spectra for $\ell =1/2$ and $\ell =3/4$ (dashed line). In both cases $j=2000$.

Figure 7

Gravitational-wave spectra for $\ell =0.70$, 0.73, 0.75, and 0.77 ($j=100$ in all cases). Loop quantum effects leave a signature on the spectra, namely, an overproduction of gravitational waves at low frequencies. As $\ell$ increases this signature is located at increasingly lower frequencies. For $\ell =0.77$ the signature moved to frequencies so low, that it becomes almost unnoticeable in the spectrum.

Figure 8

Gravitational-wave spectra for $j=100$, 560 and 750 ($\ell =0.6$ in all cases). As $j$ increases the loop quantum signature shows up at increasingly lower frequencies.