Horizon-flow off-track for inflation

Inflation can be parametrized by means of truncated flow equations. In this “horizon-flow” setup, generic results have been obtained, such as typical values for $r/(1-n_{\mathrm{S}})$. They are sometimes referred to as intrinsic features of inflation itself. In this paper we first show that the phenomenological class of inflationary potentials sampled by horizon flow is directly responsible for such predictions. They are therefore anything but generic. Furthermore, the horizon-flow setup is shown to rely on trajectories in phase space that differ from the slow roll. For a given potential, we demonstrate that this renders horizon flow blind to entire relevant inflationary regimes, for which the horizon-flow trajectory is shown to be unstable. This makes horizon flow a biased parametrization of inflation.

Figure 1

Relations between $H$ and $V$ functions. Starting from a given $H_{\mathrm{HF}}$ function, the associated potential $V$ can be obtained using Eq. (37) from the right-hand side to the left-hand side [hence the direction of the arrow above (37)]. Starting now from this potential $V$, several $H$ functions can be obtained using Eq. (37) from the left-hand side to the right-hand side (i.e. solving a first order differential equation, that involves one integration constant), depending on the initial condition $H({\varphi }_{\text{in}})$ one chooses. The one corresponding to slow roll, obtained by setting $H({\varphi }_{\text{in}})=H_{\mathrm{SR}}({\varphi }_{\text{in}})$, has a priori no reason to mach the initial $H_{\mathrm{HF}}$ function corresponding to $H({\varphi }_{\text{in}})=H_{\mathrm{HF}}({\varphi }_{\text{in}})$.

Figure 2

Left panel: Potential $V/M^4=1+(\varphi /M_{\mathrm{Pl}}{)}^2$ as a function of $\varphi$. The green disk and arrow stand for the initial value and sign of velocity of the inflaton field for the trajectory displayed in the middle and right panels. Middle panel: Numerical integration of equation (5) from ${\varphi }_{\text{in}}/M_{\mathrm{Pl}}=0.8$ and ${\dot{\varphi }}_{\text{in}}/M_{\mathrm{Pl}}^2=2$ (colored line), displayed in the phase plane $(\varphi ,\dot{\varphi })$. Each color corresponds to a different piece of it, with alternate signs of $\dot{\varphi }$. The black dashed line stands for the slow-roll leading order trajectory $\dot{\varphi }=-V^{\prime }/(3H^{\mathrm{SR},\mathrm{LO}})$. Right panel: Hubble parameter $H(\varphi ,\dot{\varphi })$ evaluated along this trajectory (same color code) with Eq. (4). Note that the logarithmic scale is used for $H$ for display convenience. Again, the black dashed line stands for the slow-roll leading order solution $H^{\mathrm{SR},\mathrm{LO}}(\varphi )=V(\varphi )/(3M_{\mathrm{Pl}}^2)$. In the middle and right panels, the arrows indicate in which direction inflation proceeds.

Figure 3

Inflationary regimes for a few potential examples (sketch). On each panel, the functional form of $V$ is the same but its coefficients differ (blue and red curves). Regions supporting slow-roll inflation are thickened in green. The horizon-flow trajectory is displayed with the arrows and starts from the location of the disk.

Figure 4

Hubble function $H(\varphi )$ and first slow-roll parameter ${\epsilon }_1$ in the case $b<0$ ($a=1$ and $b=-1$).

Figure 5

Hubble function $H(\varphi )$ and first slow-roll parameter ${\epsilon }_1$ in the case $b\ge 0$ and $a^2>4b$ ($a=2.5$ and $b=1$).

Figure 6

Hubble function $H(\varphi )$ and first slow-roll parameter ${\epsilon }_1$ in the case $a^2/4=b$ ($a=2$ and $b=1$).

Figure 7

Hubble function $H(\varphi )$ and first slow-roll parameter ${\epsilon }_1$ in the case $b>0$ and $b-2b^2<a^2/4<b$ ($a=1$ and $b=1$).

Figure 8

Hubble function $H(\varphi )$ and first slow-roll parameter ${\epsilon }_1$ in the case $b>0$ and $a^2/4<b-2b^2$ ($a=0.3$ and $b=0.4$).

Figure 9

Ratio $r_{16}/(1-n_{\mathrm{S}})$ for the model $H/H_0=1+ax+b{x}^2$ as a function of $a$ and $b$, with $\mathrm{\Delta }{N}_{*}=50$. The light blue surface stands for the regimes of inflation where it ends naturally, i.e. by slow-roll violation. The “hole” in this surface corresponds to case (v) where there is not such a regime. In this case the predictions are calculated at the late-time attractor $x_0$ that yields $r_{16}=0$, which is displayed by the black ellipse behind the light blue surface (and which is the projection of the hole onto the plane $r_{16}=0$). In the same manner, the dark blue surface corresponds to the case ($\mathrm{iv}\text{−}\beta$) where $r_{16}=0$ for the same reason. The blue straight line corresponds to $b=0$ and $r_{16}/(1-n_{\mathrm{S}})=1/4$, which matches the calculation of Sec. 2f, and the blue curved line corresponds to $a^2=4b$, i.e. case (iii), for which $r_{16}/(1-n_{\mathrm{S}})=1/3$ exactly.

Figure 10

Ratio $r_{16}/|1-n_{\mathrm{S}}|$ corresponding to the model $H/H_0=1+\alpha x^p$ in the regime of ending inflation, as a function of $\alpha$, for different values of $p$ and with $\mathrm{\Delta }{N}_{*}=50$. Note that the absolute value of the ratio is displayed (continuous lines when it is positive, and dashed lines when it is negative). When $p$ is even, such a regime exists only when $\alpha$ is larger than ${\alpha }_{\mathrm{c}}$. The black dotted line stands for the $p$-parametrized points $({\alpha }_{\mathrm{c}},r_{16}/(1-n_{\mathrm{S}}){|}_{\alpha \gg {\alpha }_{\mathrm{c}}})$, where ${\alpha }_c$ is given by Eq. (73) and $r_{16}/(1-n_{\mathrm{S}}){|}_{\alpha \gg {\alpha }_{\mathrm{c}}}$ is given by Eq. (75).

Figure 11

Large field inflation [see Eq. (79)]. The colored lines stand for solutions of Eq. (37) for a few different initial conditions, while the black dashed line represents the slow-roll leading order solution $H_{\mathrm{SR},\mathrm{LO}}^2\simeq V/3M_{\mathrm{Pl}}^2$ (left panel) and ${\dot{\varphi }}_{\mathrm{SR},\mathrm{LO}}\simeq -V^{\prime }/(3H_{\mathrm{SR},\mathrm{LO}})$ (right panel). In the left panel are displayed the $H(\varphi )$ functions, and the corresponding trajectories in phase space $(\varphi ,\dot{\varphi })$ are shown in the right panel. There, grey lines represent numerical solutions of the Klein-Gordon equation (5) for a few different initial conditions. In both panels, inflation proceeds in the blue surface, defined by ${\epsilon }_1<1$.

Figure 12

Small field inflation [see Eq. (77)] for $\mu =M_{\mathrm{Pl}}$. The colored lines stand for solutions of Eq. (37) for a few different initial conditions, while the black dashed line represents the slow-roll leading order solution $H_{\mathrm{SR},\mathrm{LO}}^2\simeq V/3M_{\mathrm{Pl}}^2$ (left panel) and ${\dot{\varphi }}_{\mathrm{SR},\mathrm{LO}}\simeq -V^{\prime }/(3H_{\mathrm{SR},\mathrm{LO}})$ (right panel). In the left panel are displayed the $H(\varphi )$ functions, and the corresponding trajectories in phase space $(\varphi ,\dot{\varphi })$ are shown in the right panel. There, grey lines represent numerical solutions of the Klein-Gordon equation (5) for a few different initial conditions. In both panels, inflation proceeds in the blue surface, defined by ${\epsilon }_1<1$.

Figure 13

Higgs inflation [see Eq. (78)]. The colored lines stand for solutions of Eq. (37) for a few different initial conditions, while the black dashed line represents the slow-roll leading order solution $H_{\mathrm{SR},\mathrm{LO}}^2\simeq V/3M_{\mathrm{Pl}}^2$ (left panel) and ${\dot{\varphi }}_{\mathrm{SR},\mathrm{LO}}\simeq -V^{\prime }/(3H_{\mathrm{SR},\mathrm{LO}})$ (right panel). In the left panel are displayed the $H(\varphi )$ functions, and the corresponding trajectories in phase space $(\varphi ,\dot{\varphi })$ are shown in the right panel. There, grey lines represent numerical solutions of the Klein-Gordon equation (5) for a few different initial conditions. In both panels, inflation proceeds in the blue surface, defined by ${\epsilon }_1<1$.

Figure 14

Potential $V(\varphi )$, first slow-roll parameter ${\epsilon }_1^{\mathrm{LO}}$ and $H(\varphi )$ function in the case $b<0$ ($a=1$ and $b=-1$).

Figure 15

Potential $V(\varphi )$, first slow-roll parameter ${\epsilon }_1^{\mathrm{LO}}$ and $H(\varphi )$ function in the case $b\ge 0$ and $a^2>4b$ ($a=2.5$ and $b=1$).

Figure 16

Potential $V(\varphi )$, first slow-roll parameter ${\epsilon }_1^{\mathrm{LO}}$ and $H(\varphi )$ function in the case $b>0$ and $b-2b^2/3\le a^2/4<b$ ($a=1.5$ and $b=1$).

Figure 17

Potential $V(\varphi )$, first slow-roll parameter ${\epsilon }_1^{\mathrm{LO}}$ and $H(\varphi )$ function in the case $b>0$ and $b-10b^2/9<a^2/4<b-2b^2/3$ ($a=1.12$ and $b=0.5$).

Figure 18

Potential $V(\varphi )$, first slow-roll parameter ${\epsilon }_1^{\mathrm{LO}}$ and $H(\varphi )$ function in the case $4(b-4b^2/3)<a^2<4(b-10b^2/9)$ ($a=0.9$ and $b=0.4$).

Figure 19

Potential $V(\varphi )$, first slow-roll parameter ${\epsilon }_1^{\mathrm{LO}}$ and $H(\varphi )$ function in the case $b>0$ and $b-2b^2<a^2/4<b-4b^2/3$ ($a=0.7$ and $b=0.4$).

Figure 20

Potential $V(\varphi )$, first slow-roll parameter ${\epsilon }_1^{\mathrm{LO}}$ and $H(\varphi )$ function in the case $b>0$ and $a^2<4(b-2b^2)$ ($a=0.55$ and $b=0.4$).

Figure 21

Reheating consistent predictions for the model (58) in the case $b>0$ (left panel), and $b<0$ (right panel), making use of the horizon-flow setup (yellow area) and the slow-roll one (blue area in the hilltop regime, green area in the chaotic one), and computed at second order in slow roll. The parameters $a$ and $b$ are continuously varied in the ranges $|a|,|b|\in [{10}^{-3},{10}^3]$, and $\mathrm{\Delta }{N}_{*}$ samples values such that ${\rho }_{\text{nuc}}<{\overline{\rho }}_{\text{reh}}<{\rho }_{\text{end}}$ and ${\overline{w}}_{\text{reh}}=0$. The two red solid contours are the one- and two-sigma Planck confidence intervals (marginalized over second order in slow roll). The black dashed line stands for $r_{16}/(1-n_{\mathrm{S}})=1/3$ ($b\gg 1$ limit at first order in slow roll), while the dash-dotted one is for $r_{16}/(1-n_{\mathrm{S}})=1/4$ ($b\to 0$ limit at first order in slow roll). One can check that in agreement with Fig. 9, this ratio $r_{16}/(1-n_{\mathrm{S}})$ continuously varies in the range $[0,1/3]$.