# How Generic Is Eternal Inflation?

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Three Roads to Eternal Inflation

#### 1.1.1. Stochastic Inflation

#### 1.1.2. Long-Lived Metastable de Sitter Vacua

#### 1.1.3. Inflating Topological Defects

#### 1.2. The Case for Generic

#### 1.3. The Dissent

#### 1.4. Measures on Cosmologies

## 2. Monte Carlo Methods

#### 2.1. Desperate Measures

#### 2.1.1. Sampling Potential Functions

#### 2.1.2. Sampling Initial Conditions

- A
- Sample field values maximizing $V\left(\phi \right)$, weighted by the distance in field space between the two adjacent minima. (This is equivalent to sampling uniformly and then going uphill to the peak.) Discard instances in which $|{\eta}_{V}|>1$ at ${\phi}_{0}$.
- B
- Sample field values uniformly. Discard instances in which ${\u03f5}_{V}>1$ or $|{\eta}_{V}|>1$ at ${\phi}_{0}$.

- C
- Sample field values a distance in field space equal to ${H}_{\mathrm{max}}/2\pi =\sqrt{2V\left({\varphi}_{\mathrm{max}}\right)/3\pi}$ from local maxima of $V\left(\phi \right)$, weighted by the distance in field space between the two adjacent local minima. Discard instances in which ${\u03f5}_{V}>1$ or $|{\eta}_{V}|>1$ at ${\phi}_{0}$.

#### Inflation Below the Peak

**A**${}^{*}$- Sample field values maximizing $V\left(\phi \right)$, weighted by the distance in field space between the two adjacent minima. If ${\eta}_{V}>1$ at the peak, then assume inflation starts where ${\u03f5}_{V},\left|{\eta}_{V}\right|<1$ first becomes valid $\left({\phi}_{\mathrm{sr}}\right)$, if ${\dot{\phi}}_{\mathrm{sr}}<\sqrt{2V({\phi}_{\mathrm{sr}}})$ along a trajectory approaching the peak as $t\to -\infty $. (For details of the calculation, see Appendix A.)

#### 2.2. Simulation Design

## 3. Results and Discussion

#### 3.1. Measure A: Summits

- Let A denote the set of all models in the sample from Measure A.
- Let $S\subset A$ denote the set of models that have successful inflation, meaning greater than 70 e-folds accrued in an interval in which the potential slow roll conditions are satisfied.
- Let $D\subset S$ denote the set of models that are successful AND in which the only sustained bout of inflation occurs in a field space interval that is not contiguous with the peak.
- Let ${D}^{\prime}\subset D$ denote the set of models in D for which the stochastic inflation criteria are never satisfied. (All models in S but not in ${D}^{\prime}$ are stochastically eternal.)

#### 3.1.1. Stochastic Eternality

#### Conditioning on Spectral Shape

#### Conditioning on Spectral Amplitudes

#### 3.1.2. Topological Eternality

#### 3.2. Measure B: Uniform

#### 3.2.1. Initialized in a True-Vacuum Basin

#### Stochastic Eternality

#### Topological Eternality

#### 3.2.2. Initialized in a False-Vacuum Basin

#### Coleman–de Luccia

#### Hawking–Moss

#### 3.3. Measure C: Hilltops

## 4. Concluding Remarks

#### Further Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Monte Carlo Methods Continued

#### Appendix A.1. Simulation Design

**Initialization**- (a)
- (b)
- Initialize the inflaton at ${\phi}_{\mathrm{start}}$ according to one of the Measures A, B, or C outlined above.
- (c)
- Determine the potential energy ${\rho}_{\Lambda}$ of the minimum of the starting basin and in one neighboring basin in both directions. (For Measure A, the starting basin adjacent to the initial peak is chosen randomly weighted by width.) If $|{\rho}_{\Lambda}|$ in any basin in this search space is below a threshold, shift the potential so that ${\rho}_{\Lambda}=0$ in that basin. If ${\rho}_{\Lambda}$ is negative and less than this threshold in the starting basin, abort.

**Instanton Pre-selection**Computing instanton profiles is time consuming, so we take the following steps to determine if a tunneling event is likely to be followed by sufficient inflation to produce a possibly observable universe in another basin of the potential.- (a)
- If initialized in the true vacuum with ${\rho}_{\Lambda}=0$, continue to Equation (4).
- (b)
- If the thin-wall or Hawking–Moss approximations hold, continue to Equation (3).
- (c)
- Taking the cutoff CDL instanton terminus on the true-vacuum side ${\phi}_{\mathrm{edge}}$ to coincide with $V\left({\phi}_{\mathrm{edge}}\right)=0.05\phantom{\rule{0.166667em}{0ex}}{V}_{T}+0.95\phantom{\rule{0.166667em}{0ex}}{V}_{\mathrm{bar}}$, compute the maximum number of e-folds of inflation accrued over any field space interval in which the potential slow roll conditions are met between ${\phi}_{\mathrm{edge}}$ and the true minimum. If the maximum e-fold count is less than 70, abort.

**Check for Quantum Tunneling**- (a)
- If the thin-wall approximation is strongly valid or ${m}_{h}\gg {m}_{\mathrm{P}}$ (Hawking–Moss eminent), compute the transition rate, otherwise
- (b)
- Compute the Coleman–de Luccia tunneling profile; determine the instanton terminus on the true-vacuum side; compute the number of e-folds of inflation, assuming inflation takes place anywhere below the terminus where the potential slow roll conditions are weakly met (${\u03f5}_{V},{\eta}_{V}<1)$.

**Characterize Slow Roll**- (a)
- Look downhill from ${\phi}_{\mathrm{start}}$ for breakdown of the slow roll approximation, ${\phi}_{\mathrm{end}}$.
- (b)
- Compute the number of e-folds ${\mathcal{N}}_{e}$ in the current basin. If ${\mathcal{N}}_{e}<70$, skip to Equation (6).
- (c)
- Find ${\phi}_{\mathrm{exit}}$, the field value at the horizon exit scale for CMB fluctuations, taken to be 55 e-folds before the end of inflation. Our e-fold cutoffs (70 for successful inflation, 55 for imprinting of CMB fluctuations) of course depend on the fiducial reheating model. One could include those models in the input space, but we opt not to include that freedom in this analysis as doing so would likely obscure our conclusions.

**Check for Eternal Inflation**- (a)
- Evaluate the stochastic inflation criterion Equation (1) between ${\phi}_{\mathrm{start}}$ and ${\phi}_{\mathrm{end}}$ in each basin.
- (b)
- Check the second potential slow roll condition at all local maxima along the trajectory; compare to the upper bound for topological inflation.
- (c)
- If a transition into the basin with ${\rho}_{\Lambda}=0$ is followed by enough e-folds, compute the transition rate $\lambda $ and compare to the upper bound in Equation (3).

**Data Collection**Record observables if inflation ends with ${\mathcal{N}}_{e}>70$ in a vacuum with ${\rho}_{\Lambda}\ll 1$, along with indicators for eternal inflation:$$\begin{array}{cc}\hfill {\mathbf{p}}_{\mathrm{obs}}& =({Q}_{s},r,{n}_{s},\alpha ,{n}_{t},\delta \rho /\rho ,log|\Omega -1\left|\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathbf{p}}_{\mathrm{eternal}}& =({N}_{s},\langle {\mathcal{N}}_{e,\mathrm{stoch}}\rangle ,{b}_{t},{\lambda}_{\mathrm{fv}},{H}_{\mathrm{F}},{b}_{\mathrm{HM}})\hfill \end{array}$$

#### Appendix A.1.1. Criterion for Inflation Discontiguous with an Initial Peak

**Figure A1.**Examples of models drawn from Measure A that fall into subset D (as defined in Section 3.1). The potential on the left represents the spirit of successful inflation taking place in field space interval discontiguous with an initial non-inflating peak. The potential on the right has ${\eta}_{V}<-4/3$ at the peak, but the curvature quickly shrinks to within the slow roll attractor; initialized with a small field velocity, inflation really continues uninterrupted between the peak and the interval in which the potential slow roll conditions are met, with the kinetic energy never rivaling $V\left(\varphi \right)$. Both models are treated the same in our simulations, raising the question of how many such models are actually free of stochastic eternal inflation.

#### Appendix A.2. Instanton Computation

#### Appendix A.2.1. Instanton Pre-Selection

**Figure A2.**Fraction of simulated models in which a Coleman–de Luccia instanton solution exists. At larger horizontal mass scales (flatter potential peaks), the Hawking–Moss instanton is dominant.

#### Appendix A.2.2. Obtaining the Profile

- Guess a starting field value on the true-vacuum side of the barrier.
- Integrate equations of motion for the scalar field $\phi \left(\xi \right)$ and Euclidean radius $\rho \left(\xi \right)$ of the bubble as a function of the radial coordinate $\xi $.
- Stop integrating when one of the follow events occurs:
- (a)
- $\phi \left(\xi \right)$ approaches ${\phi}_{\mathrm{F}}$ with $\dot{\phi}\left(\xi \right)\approx 0$ (Converge)
- (b)
- $\dot{\phi}\left(\xi \right)$ approaches 0 with $\phi \approx {\phi}_{\mathrm{F}}$ or $\dot{\rho}\left(\xi \right)\approx 0$ (Converge)
- (c)
- $\dot{\phi}\left(\xi \right)$ changes sign with $\phi \ne {\phi}_{\mathrm{F}}$ (Undershoot)
- (d)
- $\phi \left(\xi \right)$ passes ${\phi}_{\mathrm{F}}$ (Overshoot)
- (e)
- $\dot{\rho}\left(\xi \right)$ approaches $-1$ with $\dot{\phi}\left(\xi \right)\approx 0$ (Converge) or $\dot{\phi}\left(\xi \right)\not\approx 0$ (Overshoot)
- (f)
- $\rho \left(\xi \right)$ changes sign (Converge)

- If converged, we’re done; return the profile.
- If within a tolerance value of the top the barrier, report a single data point that fully characterizes the Hawking–Moss profile.$$\left(\right)open="\{"\; close="\}">\phi ,\dot{\phi},\rho ,\dot{\rho},\ddot{\rho}=\left(\right)open="\{"\; close="\}">{\phi}_{\mathrm{top}},\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}{w}_{\mathrm{top}}^{-1},\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}-{w}_{\mathrm{top}}\phantom{\dot{\phi}}$$
- If the integration overshoots, move the guess closer to the maximum; if it undershoots, move the guess closer to the true minimum.
- Go to Step 2.

#### Appendix A.2.3. Transition Rates with Gravity

#### Appendix A.3. Statistical Methods

#### Appendix A.3.1. Mass Scale Weighting Schemes

- The epektacratic weighting scheme (rule by expansion) samples an equal number of potentials for each pairing of mass scales ${m}_{v}$ and ${m}_{h}$, and lets them succeed or fail at producing sufficient e-folds of inflation. The total population is aggregated from successful inflation models at all mass scales, and that population is used to determine rates. Naturally this scheme will tend to give more representation to large field models.
- The democratic scheme gives every mass pairing within the specified range equal weight in informing ${f}_{m}({\mathbf{p}}_{\mathrm{eternal}}\mid {\mathbf{p}}_{\mathrm{obs}})$ in Equation (7), regardless of how common or rare it is for models comprising each to produce enough inflation. From each pairing, we sample as many potentials as it takes to get an equal number of successful models, or we give lower-expansion mass pairings extra weight to the same effect.

#### Appendix A.3.2. Rate Estimation

**Figure A3.**Distributions of log likelihood ratios, comparing maximizing over one rate parameter versus independent rate parameters for each ${m}_{v}$, from Monte Carlo simulation assuming the former null hypothesis. Ratios for the simulated data are indicated in black. Only for ${m}_{h}=0.25$, with p-value 0.016, should we consider rejecting the hypothesis of a well defined rate of incidence of delayed non-stochastic inflation independent of ${m}_{v}$.

#### Appendix A.3.3. Testing Scale Invariance of Stochastic Eternal Inflation in Measure A

## Appendix B. Matching Observables

#### Appendix B.1. Measure A

**Figure A4.**Marginal distributions of spectral parameters in Measures A (

**left**) and B (

**right**). Foreground (magma): 95% confidence upper bound on the rate of incidence of ${Q}_{s}$ falling within Planck 2018 68% confidence interval. The vertical striation pattern emerging on the left-hand side reflects the shortage of samples with successful inflation at low ${m}_{\mathrm{h}}$ – due to slower accrual of e-folds and the second slow roll criterion not being met at the peak – resulting in a weaker bound. Background: rates of incidence of ${n}_{s}$ (blue, upper) and r (green, lower) falling within Planck’s 68% confidence intervals, with higher color saturation (darker gray) indicating a higher rate.

#### Appendix B.2. Measure B

#### Appendix B.3. Measure C

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**Figure 1.**Conceptual flowchart illustrating our high-level methodology. To generate Monte Carlo samples with which to address the question “How generic is eternal inflation?”, we select a measure on initial conditions (either A, B, or C), scan values for the scales of the potential spanning the region of tractability, and sample random variables (the potential—via the Fourier components of a Gaussian random field—and the initial field value—via the potential and the choice of measure). We then perform various checks for sufficient inflation and presence of three modes of eternal inflation, logging the computed outputs for later analysis. Refer to Section 2 and Appendix A for elaboration.

**Figure 2.**Incidence rate of models with slow roll starting below the peak and no stochastic inflation, among models in Measure A with 70+ e-folds. Each data point represents one batch of simulations with particular ${m}_{v},{m}_{\mathrm{h}}$ (only showing batches with at least one positive event per sample). All models within a vertical stratum have the same value of ${m}_{h}$ (the center of the stratum on the left axis); vertical position within the stratum reflects ${log}_{10}{m}_{v}$ (range shown on the right axis). Data points reflect 95% confidence upper bounds. In the left plot, green data points are derived from samples conditioned on ${n}_{s}$ and $\alpha $; purple data points are conditioned only on minimal e-folds; the shaded bars indicate 90% confidence intervals taking models from all values of ${m}_{v}$ as belonging to one sample. In the right plot, the blue lines show 90% confidence intervals derived from samples of successful models conditioned further on ${Q}_{s}$. Darker lines reflect samples that have at least one non-stochastic model in the sample, whereas lighter points are determined only by sample size.

**Figure 3.**95% confidence upper bound on the rate of incidence of stochastic inflation in Measure B binned with respect to scalar tilt (${n}_{s}$) and tensor-to-scalar ratio (r), with at least 70 e-folds (

**left**) and 200 e-folds (

**right**) of slow roll inflation, subject to epektacratic field scale weighting.

**Figure 4.**(

**Top**) Total number of successful models with ${Q}_{s}<{10}^{-3}$ in each bin. (

**Bottom**) 95% confidence upper bound on the rate of incidence of stochastic inflation in Measure B, binned with respect to scalar tilt (${n}_{s}$) and tensor-to-scalar ratio (r), with at least 55 e-folds (left) and 200 e-folds (right) of slow roll inflation, conditioned on scalar amplitude ${Q}_{s}<{10}^{-3}$ and subject to epektacratic field scale weighting.

**Figure 5.**95% confidence lower bound on incidence rate of models with high probability of topological inflation, among models in Measure A with 70+ e-folds. In these models, quantum fluctuations are comparable in size to $|{\phi}_{0}-{\phi}_{\mathrm{max}}|$, allowing $\gtrsim 1$ Hubble volume to descend toward the opposite local minimum after a Hubble time with high probability, and produce a persisting topological defect. Each line represents one batch of simulations with particular ${m}_{v},{m}_{\mathrm{h}}$ (only showing batches with at least one positive event per sample or a sample size of 100). All models within a vertical stratum have the same value of ${m}_{h}$ (the center of the stratum on the left axis); vertical position within the stratum reflects ${log}_{10}{m}_{v}$ (range shown on the right axis). In the left plot, green data points are derived from samples conditioned on ${n}_{s}$ and $\alpha $; the purple 90% confidence intervals are conditioned only on minimal e-folds; and the shaded bars indicate 90% confidence intervals taking models from all values of ${m}_{v}$ as belonging to one sample. In the right plot, the blue lines show 90% confidence intervals derived from samples of successful models conditioned further on ${Q}_{s}$. Darker lines reflect samples that have at least one non-stochastic model in the sample, whereas lighter points are determined only by sample size.

**Figure 6.**The bimodal distribution of ${S}_{\mathrm{E},\mathrm{bkg}}-{S}_{\mathrm{E}}\left[\varphi \left(r\right)\right]$ for Coleman–de Luccia transitions in models initialized in the false vacuum, for ${m}_{v}=0.0025$ (red) and $0.0042$ (blue). The normalized counts for values of the tunneling suppression are plotted with respect to its absolute value; the solid lines correspond to slow tunneling, for which inflation is eternal, while the dashed lines are fast tunneling.

**Figure 7.**(

**Top**) Distributions of the number of slow roll e-folds in the true vacuum basin after CDL tunneling, for two values of the shape parameter $\gamma $ characterizing the potential in Equation (10). (

**Bottom**) Moments of the distribution of number of e-folds after CDL tunneling when inflation ends in the tunneled-to basin, as a function of ${m}_{\mathrm{h}}$. The blue plot (solid line with circular markers, left axis) shows the mean of ${log}_{10}{N}_{e}$ and the shaded 2-$\sigma $ confidence interval. The red (dashed line with square markers, right axis) is the number of standard deviations between the mean and ${log}_{10}55$.

**Figure 8.**(

**Left**) Bin counts. (

**Right**) 95% confidence lower bound on the rate of incidence of models with fluctuations smaller than the width of the stochastic inflation interval around the maximum, under epektacratic weighting in Measure C, binned with respect to scalar tilt and (top) vertical or (bottom) horizontal mass scale, and conditioned on $r<0.064$, and $-4.5<{log}_{10}Q<-4.1$. The grid points indicate the centers of bins with a non-zero number of non-eternal models. Where grid points are absent, the reported bound is determined only by sample size; a small, uninformative lower bound in regions with few samples.

**Figure 9.**95% confidence lower bound on the rate of incidence of stochastic inflation in Measure C, binned with respect to scalar tilt and tensor-to-scalar ratio, and conditioned on scalar amplitude $-4.5<{log}_{10}{Q}_{s}<-4.1$. The left plot includes all mass scales with epektacratic weighting. The right includes only ${m}_{\mathrm{h}}\le 1$.

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Greenwood, R.; Aguirre, A.
How Generic Is Eternal Inflation? *Sci* **2022**, *4*, 23.
https://doi.org/10.3390/sci4020023

**AMA Style**

Greenwood R, Aguirre A.
How Generic Is Eternal Inflation? *Sci*. 2022; 4(2):23.
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**Chicago/Turabian Style**

Greenwood, Ross, and Anthony Aguirre.
2022. "How Generic Is Eternal Inflation?" *Sci* 4, no. 2: 23.
https://doi.org/10.3390/sci4020023