# Exiting Inflation with a Smooth Scale Factor

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Inflation Solution

#### 1.2. Reheating

## 2. Period Scale Factors

#### 2.1. Quantifying the Discontinuity

## 3. The Transition

- R—the radius of the circular arc;
- $\Delta $—the time between ${t}_{f}$ and the tangent point at which the circular arc ${a}_{\mathrm{cir}}\left(t\right)$ meets the displaced radiation era ${a}_{2}^{\prime}\left(t\right)$;
- $a({t}_{f}-\delta )$—the a-axis value at ${t}_{f}-\delta $, aligned with the center of the arc;
- $\delta $—the measure of the t-axis displacement corresponding to the difference between $a\left({t}_{f}\right)$ and $a({t}_{f}-\delta )$;
- $a({t}_{f}+\Delta )$—the scale factor at ${t}_{f}+\Delta $, the t-axis point of tangency for ${a}_{\mathrm{cir}}\left(t\right)$ and ${a}_{2}^{\prime}\left(t\right)$.

**Figure 3.**Five unknown parameters characterize the two points of tangency of ${a}_{\mathrm{cir}}\left(t\right)$ with the period scale factors (not to scale).

## 4. Power-Law Transitions

#### 4.1. Interpolating Power Laws with $n<1$

#### 4.1.1. Smoothness at ${t}_{f}$

#### 4.1.2. Smoothness at ${t}_{f}+\Delta $

#### 4.2. Power Laws with $n>1$

## 5. Additional Constraints

#### 5.1. The Equation of State

#### 5.2. Speed of Sound Constraints

#### 5.3. Continuity of the Equation of State

## 6. Summary of Numerical Results

## 7. The Smooth Scale Factor in the Preheating Model

#### 7.1. Occupation Numbers

#### 7.2. Number Density

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## Appendix D

**Figure A1.**Stability–instability chart. The areas highlighted in gray are the regions of instability in the q-${A}_{k}$ parameter space. The plot also depicts the Mathieu equation parameters associated with the equation of motion solutions. Note that q and ${A}_{k}$ decrease as time progresses.

**Figure A2.**The final three instability regions superimposed on the $log{n}_{k}$ resonance growth. As the number of oscillations increases, we see exponential growth in the occupation number as the q and ${A}_{k}$ of Figure A1 decline toward zero and the equation of motion crosses the last three instability regions.

## Appendix E

**Figure A3.**Plots of scalar field X and its time derivative $\dot{X}$ with the cusped scale factor for 10 oscillations following the end of broad parametric resonance.

**Figure A4.**Plots of scalar field X and its time derivative $\dot{X}$ with the smooth scale factor for 10 oscillations following the end of broad parametric resonance.

**Table A1.**Data in support of the differences in appearance between Figure 12b and Figure 13b. The last column represents the increase in occupation number $log{n}_{k}\left(36\right)$ compared to the average value over 4 oscillations from oscillation 34 to 38, which are $43.9$ and $36.8$ for the KLS and smooth scale factor models, respectively.

$\mathit{a}\left(\mathit{t}\right)\phantom{\rule{0.166667em}{0ex}}\mathbf{Model}$ | ${\mathit{\omega}}_{\mathit{k}}$ | ${\mathit{\omega}}_{\mathit{k}}{|\mathit{X}|}^{2}$ | $\frac{|\dot{\mathit{X}}{|}^{2}}{{\mathit{\omega}}_{\mathit{k}}}$ | $log{\mathit{n}}_{\mathit{k}}\left(36\right)$ | Increase |
---|---|---|---|---|---|

cusped | $0.91$ | $4.62\times {10}^{17}$ | $9.56\times {10}^{19}$ | $45.3$ | $0.03$ |

smooth | $0.091$ | $8.71\times {10}^{13}$ | $5.79\times {10}^{17}$ | $40.2$ | $0.09$ |

## Appendix F

**Figure A5.**The outer and inner pairs of dotted lines represent the ranges of $\varphi \left(t\right)$ that participate in parametric resonance. The wider outer band corresponds to the values of $\varphi \left(t\right)$ that participate in the resonance for the minimal Fourier component—that is, as $k\to 0$. The inner band, $-\frac{1}{2}\sqrt{m\Phi /g}\le {\varphi}_{*}\le \frac{1}{2}\sqrt{m\Phi /g}$, corresponds to the participating $\varphi \left(t\right)$ associated for the modes with ${k}_{*}=\sqrt{gm\Phi}$. This is the preheating band of broad parametric resonance. Explosive growth in the number of particles occurs as $\varphi \left(t\right)\to 0$.

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**Figure 1.**The contrast between the potentials of slow-roll inflation, shown in (

**a**), and chaotic inflation, shown in (

**b**). Slow-roll inflation requires a plateau to generate enough e-folds of inflationary expansion to solve the horizon, flatness, and monopole problems. In chaotic inflation, the effects of the friction term in the equation of motion replace that of the plateau in keeping the inflaton from moving to the true vacuum too quickly. After inflation, both models involve the inflaton oscillating around a minimum potential during a period of reheating, which we review in Section 1.2.

**Figure 2.**(

**a**) The graph depicts the discontinuity between inflationary and radiation-era scale factors at the end of inflation. (

**b**) The circular arc defined by the transitional scale factor ${a}_{\mathrm{cir}}\left(t\right)$ intersects tangentially with the inflationary scale factor ${a}_{1}\left(t\right)$ and the now-displaced radiation-era scale factor noted with a prime, ${a}_{2}^{\prime}\left(t\right)$. The parameter $\Delta $ is the time period from the end of inflation until the time of continuity between ${a}_{\mathrm{cir}}\left(t\right)$ and ${a}_{2}^{\prime}\left(t\right)$. The a functions are not to scale.

**Figure 4.**The displacement $\delta $ necessarily places the vertex of the representative inverted parabola with power-law index $n=2$ later than the end of inflation at ${t}_{f}$ and the tangency point at ${t}_{f}+\Delta $ (not to scale). Increasing the duration over which the the interpolating scale factor applies also shifts the vertex similarly.

**Figure 5.**The three ${a}_{v}\left(t\right)$ scale factors with power laws ${t}^{3/2}$, ${t}^{2}$, and ${t}^{5/2}$ essentially overlay each other, and the ${a}_{p}\left(t\right)$ scale factor proportional to ${t}^{1/4}$ approaches those of the power laws with $n>1$. The graph also shows the unsmoothed scale factor defined by Equation (13), which exceeds that of ${a}_{p}\left(t\right)$ at ${t}^{3/4}$. The t-axis timeline begins at $t={10}^{-43}$ s after inflation terminates, while the vertical dotted line at $t={10}^{-35}$ s marks the nominal start of the radiation era.

**Figure 6.**The Hubble parameters corresponding to the scale factors shown in Figure 5. Once again, the graphs based on the scale factors ${a}_{v}\left(t\right)$ for ${t}^{3/2}$, ${t}^{2}$, and ${t}^{5/2}$ essentially overlay each other. The discontinuity of the radiation-era Hubble parameter with ${H}_{\mathrm{in}}$ exceeds two orders of magnitude. The graph based on scale factor ${a}_{p}\left(t\right)$ for ${t}^{3/4}$ fails to display asymptotic behavior with ${H}_{\mathrm{in}}$ at small times, because the timeline has the same t-axis translation and does not start at ${t}_{f}$. Again, the vertical dotted line marks the start of the radiation era at $t={10}^{-35}$ s.

**Figure 7.**The slopes of the graphs equal the parameter ${\u03f5}_{H}=\frac{d\phantom{\rule{0.166667em}{0ex}}log\left({H}^{-1}\right)}{dlog\phantom{\rule{0.166667em}{0ex}}a}$.

**Figure 8.**The parameter ${\u03f5}_{H}=-\frac{\dot{H}}{{H}^{2}}$ with power law ${a}_{p}\left(t\right)\propto {t}^{n}$ for $n<1$ at the start of the radiation era, with the transition period $\Delta ={10}^{-35}$ s.

**Figure 9.**Values of the parameter ${\u03f5}_{H}=1+\frac{(1-n)}{H\delta}$ at the end of inflation, ${t}_{f}\approx 4\times {10}^{-39}$ s, for ${a}_{p}\left(t\right)\propto {t}^{n}$ with $n\approx \frac{1}{2}$. The gray band depicts the permissible values of ${\u03f5}_{H}$, $\frac{3}{2}\le {\u03f5}_{H}\le 3$, subject to the assumption that the inflaton condensate at the end of inflation is a single-component perfect fluid with the equation of state $p=\omega \rho $.

**Figure 10.**The growth in scale factors for single-component universes with smoothness enforced at ${t}_{f}+\Delta $ with $\Delta ={10}^{-35}$ s and ${10}^{-22}$ s in (

**a**,

**b**), respectively. We note that the approximate order-of-magnitude increases in the power-law scale factors occur at around ${10}^{-37}$ s in all cases.

**Figure 11.**The scale factor ratios $\frac{{a}_{p}\left(t\right)}{{a}_{2}^{\prime}\left(t\right)}$. At approximately $t={10}^{-37}$ s, the ratios reach greater than $98\%$ of the asymptotic values of $9.8$ and $11.2$ for ${a}_{p}\left(t\right)\propto {t}^{1/2}$ and ${a}_{p}\left(t\right)\propto {t}^{2/3}$, respectively.

**Figure 12.**(

**a**) The scalar field and (

**b**) occupation number for the first 60 oscillations, in the model with a cusped scale factor, and for the inflaton mass $m={10}^{-6}$${M}_{P}$. The t-axis is in units of the number of oscillations, $2\pi /m$. We selected the specific mode of the KLS model with the wave number $k=4m$ to maximize the growth of the occupation number. To reproduce the broad-resonance exponential growth, we used parameters $g=6.25\times {10}^{-4}$, $\dot{X}\left({t}_{f}\right)=0.045$, and $\ddot{X}\left({t}_{f}\right)\approx 0$; these were identified empirically, and varying the parameter values away from these will decrease the observable resonance effect. The scalar field derivative $\dot{X}\left({t}_{f}\right)$ approximates what KLS advise—namely, that the positive-frequency solution ${X}_{k}\left(t\right)\approx exp\left(-i{\omega}_{k}t/\sqrt{2{\omega}_{k}}\right)$ be applied as an initial condition.

**Figure 13.**(

**a**) The scalar field and (

**b**) occupation number for the first 60 oscillations in a model with the smooth scale factor. The graphs show reduced values of ${X}_{k}$ and $log{n}_{k}$ compared to Figure 12 because of the effect of the order-of-magnitude increase in $a\left(t\right)$.

**Figure 14.**These plots compare 10 oscillations of $log{n}_{k}$ after the end of broad resonance at ∼34 oscillations for the two functional forms of the scale factor.

**Figure 15.**The ratio ${R}_{\chi}$ of ${n}_{\chi}$ for the smooth scale factor, to that with $a\left(t\right)$ with a discontinuous derivative at the end of inflation, as used by KLS. The increase in volume in the smooth model, resulting from the extra time given for space to expand as inflation tails off, dilutes ${n}_{\chi}$ by ∼10${}^{-3}$, which the broad parametric resonance term ${e}^{2\mu mt}/\sqrt{\mu mt}$ partially tends to offset. Maximizing the offset with the total phase $sin\left({\theta}_{\mathrm{tot}}^{j}\right)=-1$ in Equation (88) minimizes the dilution, and that is what is shown in this figure. Thus, with the smooth scale factor, ${n}_{\chi}$ should be diluted by at least the ratios shown here.

**Table 1.**The vertex displacements $\delta $ for power laws $a\left(t\right)\propto {t}^{1/2}$ and ${t}^{2}$ for a sample set of transition durations $\Delta $ between the end of inflation and the beginning of the radiation era. Increasing the duration $\Delta $ for the power law $n=\frac{1}{2}$ tends to set the displacement of the vertex. However, because of the difficulty of establishing continuity with the inflationary slope at ${t}_{f}$, the displacement $\delta $ for the $n=2$ interpolator is many orders of magnitude greater. With its vertex located later on the timeline than ${t}_{f}$, the table shows that increasing the transition has the effect of shifting the vertex of ${a}_{v}\left(t\right)$ farther away from the end of inflation.

Transition | n | $\mathbf{\Delta}\phantom{\rule{0.166667em}{0ex}}\left(\mathbf{s}\right)$ | $\mathit{\delta}\phantom{\rule{0.166667em}{0ex}}\left(\mathbf{s}\right)$ |
---|---|---|---|

${a}_{p}\left(t\right)$ | $\frac{1}{2}$ | ${10}^{-35}$ | $2.65\times {10}^{-41}$ |

${10}^{-33}$ | $3.22\times {10}^{-41}$ | ||

${10}^{-30}$ | $3.29\times {10}^{-41}$ | ||

${10}^{-22}$ | $3.29\times {10}^{-41}$ | ||

${a}_{v}\left(t\right)$ | 2 | ${10}^{-35}$ | $1.50\times {10}^{-35}$ |

${10}^{-22}$ | $1.50\times {10}^{-22}$ |

**Table 2.**The displacement $\delta $, parameter ${\u03f5}_{H}$, and equation-of-state parameter $\omega =\frac{2}{3}{\u03f5}_{H}-1$ for power laws ${a}_{p}\left(t\right)\propto {t}^{n}$, with values of n ranging between 0 and 1. ${}^{\left(\mathrm{a}\right)}$ Computation sets this value more precisely at $\approx 2.00039$. After interpolation using the linear relation associated with Equation (46), the expected ${\u03f5}_{H}=2$ for a radiation-dominated scale factor occurs at $n\approx 0.5002$. ${}^{\left(\mathrm{b}\right)}$ Values of ${\u03f5}_{H}\to 2.{6}^{-}$ and $\omega \approx 0.73$ signify unphysical, exotic tachyon-like particles with velocities greater than the speed of light, which Section 5.2 discusses in detail. ${}^{\left(\mathrm{c}\right)}$ For ${\u03f5}_{H}\approx 1.50$ and $\omega \approx 0.00$, we have a transition from inflation to an equation of state that would be consistent with a matter-dominated universe. We take up consideration of the single-component matter-dominated universe in Section 5.3. ${}^{\left(\mathrm{d}\right)}$ As ${\u03f5}_{H}\to 1.{0}^{+}$ and $\omega \to -\frac{1}{3}$, the scale factor remains inflationary, effectively eliminating the transition.

n | $\mathit{\delta}\phantom{\rule{0.166667em}{0ex}}\left(\mathbf{s}\right)$ | ${\mathit{\u03f5}}_{\mathit{H}}$ | $\mathit{\omega}$ |
---|---|---|---|

0.002 | $2.53\times {10}^{-36}$ | $2.59$ | $0.{73}^{\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{b}\right)}$ |

0.25 | $9.55\times {10}^{-37}$ | $2.37$ | $0.58$ |

0.50 | $2.65\times {10}^{-41}$ | $2.{00}^{\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{a}\right)}$ | $0.33$ |

0.75 | $9.46\times {10}^{-56}$ | $1.50$ | $0.{00}^{\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{c}\right)}$ |

0.98 | $2.31\times {10}^{-294}$ | $1.04$ | $-0.{31}^{\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{d}\right)}$ |

**Table 3.**These parameters correspond to power laws ${a}_{p}\left(t\right)$ with indices $n=\frac{1}{2}$ and $n=\frac{2}{3}$ transitioning to radiation-dominated and matter-dominated eras with scale factors similarly proportional to ${t}^{1/2}$ and ${t}^{2/3}$. The column ${\u03f5}_{H\phantom{\rule{0.166667em}{0ex}}\mathrm{derived}}$ reconstructs the parameter as the sum of 1 and the contribution from the displacement and the time components. We conclude that the displacement causes the difference from ${\u03f5}_{H\phantom{\rule{0.166667em}{0ex}}\mathrm{expected}}$.

n | $\mathit{\delta}$ (s) | ${\mathit{\u03f5}}_{\mathit{H}\phantom{\rule{0.166667em}{0ex}}\mathbf{expected}}$ | ${\mathit{\u03f5}}_{\mathit{H}\phantom{\rule{0.166667em}{0ex}}\mathbf{derived}}$ | ${\mathit{\u03f5}}_{\mathit{H}\phantom{\rule{0.166667em}{0ex}}\mathbf{displacement}}$ | ${\mathit{\u03f5}}_{\mathit{H}\phantom{\rule{0.166667em}{0ex}}\mathbf{time}}$ |
---|---|---|---|---|---|

$\frac{1}{2}$ | $2.65\times {10}^{-41}$ | 2 | $2.00039$ | $0.00039$ | $1.00000$ |

$\frac{2}{3}$ | $7.71\times {10}^{-42}$ | $1.5$ | $1.50020$ | $0.00020$ | $0.50000$ |

**Table 4.**The ratios of ${a}_{p}\left(t\right)$ to ${a}_{2}^{\prime}\left(t\right)$ at different times during the interpolation period. The percentages represent the degree to which the ratios have approached the asymptotic values reached at ${10}^{-34}$ s.

$\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\mathit{a}}_{\mathit{p}}\left(\mathit{t}\right)\propto {\mathit{t}}^{1/2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{a}}_{\mathit{p}}\left(\mathit{t}\right)\propto {\mathit{t}}^{2/3}$ | |||
---|---|---|---|---|

Time (s) | Ratio | % | Ratio | % |

${10}^{-34}$ | 9.837 | 100 | 11.227 | 100 |

${10}^{-36}$ | 9.828 | 99.9 | 11.217 | 99.9 |

${10}^{-37}$ | 9.69 | 98.5 | 11.03 | 98.3 |

${10}^{-38}$ | 8.4 | 85.8 | 9.4 | 83.3 |

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Oslislo, H.; Altschul, B.
Exiting Inflation with a Smooth Scale Factor. *Symmetry* **2023**, *15*, 2042.
https://doi.org/10.3390/sym15112042

**AMA Style**

Oslislo H, Altschul B.
Exiting Inflation with a Smooth Scale Factor. *Symmetry*. 2023; 15(11):2042.
https://doi.org/10.3390/sym15112042

**Chicago/Turabian Style**

Oslislo, Harry, and Brett Altschul.
2023. "Exiting Inflation with a Smooth Scale Factor" *Symmetry* 15, no. 11: 2042.
https://doi.org/10.3390/sym15112042