Fluctuations about cosmological instantons

We study the semiclassical fluctuation problem around bounce solutions for a self-interacting scalar field in curved space. As in flat space, the fluctuation problem separates into partial waves labeled by an integer $l$, and we determine the large $l$ behavior of the fluctuation determinants, a quantity needed to define a finite fluctuation prefactor. We also show that while the Coleman-De Luccia bounce solution has a single negative mode in the $l=0$ sector, the oscillating bounce solutions also have negative modes in partial waves higher than the $s$-wave, further evidence that they are not directly related to quantum tunneling.

Figure 1

The scalar field potential $V(\phi )$ with the parameters: $\beta =45$, $b=0.25$, $ϵ=0.23$.

Figure 2

Single-bounce solution to (2.3, 2.4, 2.5, 2.6), for parameter values: $\beta =45$, $b=0.25$, $ϵ=0.23$. The shooting procedure determines the initial scalar field value ${\phi }_1(0)={\phi }_{\mathrm{fv}}-0.0001452232432675769$, and ${\sigma }_{\max }$ is given by: $H_{\mathrm{top}}{\sigma }_{\max }^{1\mathrm{b}}=3.7475$. Here ${sin}_1\equiv \frac{1}{H_1}\sin (H_1\sigma )$, and $H_1\equiv \pi /{\sigma }_{\max }^{1\mathrm{b}}$.

Figure 3

Triple bounce solution to (2.3, 2.4, 2.5, 2.6), for parameter values: $\beta =45$, $b=0.25$, $ϵ=0.23$. The shooting procedure determines the initial scalar field value ${\phi }_3(0)={\phi }_{\mathrm{tv}}-0.008860088903713227$, $H_{\mathrm{top}}{\sigma }_{\max }^{3\mathrm{b}}=3.1243$. Here ${sin}_3\equiv \frac{1}{H_3}\sin (H_3\sigma )$, and ${\sigma }_{\max }$ is given by: $H_3\equiv \pi /{\sigma }_{\max }^{3\mathrm{b}}$.

Figure 4

Quintuple bounce solution to (2.3, 2.4, 2.5, 2.6), for parameter values: $\beta =45$, $b=0.25$, $ϵ=0.23$. The shooting procedure determines the initial scalar field value ${\phi }_5(0)={\phi }_{\mathrm{tv}}-0.3371027481952641$, $H_{\mathrm{top}}{\sigma }_{\max }^{5\mathrm{b}}=3.1264$. Here ${sin}_5\equiv \frac{1}{H_5}\sin (H_5\sigma )$, and ${\sigma }_{\max }$ is given by: $H_5\equiv \pi /{\sigma }_{\max }^{5\mathrm{b}}$.

Figure 5

$\ln [\mathrm{Det}{\Lambda }^{(l)}/\mathrm{Det}{\Lambda }_{\mathrm{free}}^{(l)}]$ for the single bounce, for parameter values: $\beta =45$, $b=0.25$, $ϵ=0.046$, $\phi (0)={\phi }_{\mathrm{tv}}-0.0005517367392784155$, $\phi (\pi )={\phi }_{\mathrm{fv}}+0.0000019291139437721194076985$. Diamond points denote the numerical results using the Gel’fand-Yaglom technique as in (4.10, 4.11); the (blue) dash line denotes the leading large $l$ behavior in (4.12) with $\alpha$ defined in (4.14) and $\gamma =0$; the (red) solid line denotes the leading and subleading large $l$ behavior in (4.14) with $\alpha$ in (4.12) and $\gamma$ defined in (4.15).

Figure 6

$\ln [\mathrm{Det}{\Lambda }^{(l)}/\mathrm{Det}{\Lambda }_{\mathrm{free}}^{(l)}]$ for the triple bounce, for parameter values: $\beta =45$, $b=0.25$, $ϵ=0.046$, $\phi (0)={\phi }_{\mathrm{tv}}-0.007517937804678529$, $\phi (\pi )={\phi }_{\mathrm{fv}}+0.001906935294979618$. Diamond points denote the numerical results using the Gel’fand-Yaglom technique as in (4.10, 4.11); the (blue) dash line denotes the leading large $l$ behavior in (4.12) with $\alpha$ defined in (4.14) and $\gamma =0$; the (red) solid line denotes the leading and subleading large $l$ behavior in (4.12) with $\alpha$ in (4.14) and $\gamma$ defined in (4.15).

Figure 7

$\ln [\mathrm{Det}{\Lambda }^{(l)}/\mathrm{Det}{\Lambda }_{\mathrm{free}}^{(l)}]$ for the quintuple bounce, for parameter values: $\beta =45$, $b=0.25$, $ϵ=0.046$, $\phi (0)={\phi }_{\mathrm{tv}}-0.3037875466774131$, $\phi (\pi )={\phi }_{\mathrm{fv}}+0.1875253487461535$. Diamond points denote the numerical results using the Gel’fand-Yaglom technique as in (4.10, 4.11); the (blue) dash line denotes the leading large $l$ behavior in (4.12) with $\alpha$ defined in (4.14) and $\gamma =0$; the (red) solid line denotes the leading and subleading large $l$ behavior in (4.12) with $\alpha$ in (4.14) and $\gamma$ defined in (4.15).

Figure 8

$\ln [\mathrm{Det}{\Lambda }^{(l)}/\mathrm{Det}{\Lambda }_{\mathrm{free}}^{(l)}]$ for the Hawking-Moss bounce, for parameter values: $\beta =45$, $b=0.25$, $ϵ=0.046$. Diamond points denote the numerical result using the Gel’fand-Yaglom technique as in (4.10, 4.11); the (blue) dash line denotes the leading large $l$ behavior in (4.12) with $\alpha$ defined in (4.14) and $\gamma =0$; the (red) solid line denotes the leading and subleading large $l$ behavior in (4.12) with $\alpha$ in (4.14) and $\gamma$ defined in (4.15).