Observable effects of anisotropic bubble nucleation

Probably not a substantial claim. The paper’s conclusions may not depend on it.

String theory supports an enormous landscape of metastable vacua.

“Indeed this view of cosmology is supported by the present understanding of string theory, which argues for the existence of an enormous landscape of such metastable vacua [5,6,7,8].”

• cites [5] R. Bousso, 2000, J. High Energy Phys. sourceSupported
• cites [6] L. SusskindNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [7] M.R. Douglas, 2003, J. High Energy Phys. sourcePartly supported

  The General introduction says string/M theory appears to describe a very large number of vacua. Section 3.6, however, says metastability still needs to be checked for a vacuum with the appropriate cosmological constant, although it expects that constraint to be mild. The paper therefore supports an enormous multiplicity of vacua, but does not establish that an enormous number are metastable. (LLM, conf=0.85)

  Experimental — accuracy not yet measured
• cites [8] sourceSupported

It may state two separate claims as one.

These conditions are easy to satisfy; the circumference condition is akin to solving the horizon problem and is accomplished by a sufficiently long period of slow-roll inflation after bubble nucleation along each large spatial dimension.

“In fact these conditions are easy to satisfy-the latter is akin to solving the horizon problem of classical big bang cosmology, and is accomplished by a sufficiently long period of slow-roll inflation (after bubble nucleation) along each large spatial dimension [13].”

• cites [13] sourcePartly supported

  In VI. Conclusion (page unavailable), Guth says that sufficient exponential expansion could resolve the horizon problem. But the paper places that expansion before substantial bubble nucleation, not after it, and V. Problems of the Inflationary Scenario describes bubble nucleation as a source of serious inhomogeneity. It does not support the specified slow-roll mechanism along each large spatial dimension or the assertion that the conditions are easy to satisfy. (LLM, conf=0.88)

  Experimental — accuracy not yet measured

Probably not a substantial claim.

The number of large spatial dimensions may vary from vacuum to vacuum, as expected in string theory.

“Yet the number of large spatial dimensions may vary from vacuum to vacuum [9,10,11,12], as is expected in string theory.”

• cites [9] sourceSupported
• cites [10] J.J. Blanco-Pillado, 2009, J. Cosmol. Astropart. Phys. sourcePartly supported

  In “B. The 4d perspective” (page unavailable), the paper says positive-energy compactified vacua may decay by tunneling, leading to decompactification. This supports the possibility of an extra spatial dimension becoming large, but the paper does not establish that distinct stable vacua have different numbers of large spatial dimensions or that string theory specifically predicts this. (LLM, conf=0.78)

  Experimental — accuracy not yet measured
• cites [11] S.M. Carroll, 2009, J. High Energy Phys. sourceSupported
• cites [12] J.J. Blanco-Pillado, 2010, J. Cosmol. Astropart. Phys. sourceSupported

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The prior distribution from [66] suggests the landscape might strongly prefer fewer e-folds of slow-roll inflation, with dP(N) ∝ N^-4 dN.

“The prior distribution is addressed in [66], where an (admittedly crude) argument is given to suggest that the landscape might strongly prefer fewer e-folds of slow-roll inflation, with the distribution dP (N ) ∝ N -4 dN .”

• cites [66] sourceCould not verify

  The Introduction discusses open-universe inflation and the duration of secondary inflation, but gives no landscape prior. Searches for the e-fold distribution and N^-4 found only unrelated mode-function equations; no passage in our copy addresses the claimed probability distribution. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

It may state two separate claims as one.

Late-time anisotropic expansion induces a quadrupole in the CMB at order Ω0curv, which absent cancellations provides a stronger constraint on Ω0curv.

“however it should be emphasized that late-time anisotropic expansion induces a quadrupole in the CMB, itself appearing at order Ω 0 curv , which absent cancellations provides a stronger constraint on Ω 0 curv [37].”

• cites [37] S. Yokoyama, 2008, J. Cosmol. Astropart. Phys. sourceCould not verify

  The Introduction and Summary discuss primordial statistical anisotropy generated at the end of inflation, not late-time anisotropic expansion. Searches for a CMB quadrupole, Ωcurv, and a resulting constraint found no passage addressing the attributed claim. Page numbers are unavailable in our copy. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Fluctuations in σ may be converted into adiabatic density perturbations by a variety of proposed mechanisms; for instance σ could be a curvaton.

“The fluctuations in σ may be converted into adiabatic density perturbations by a variety of proposed mechanisms; for instance σ could be a curvaton [35].”

• cites [35] sourcePartly supported

  The Introduction (page unavailable) describes a proposed mechanism by which fluctuations of a test field χ could be transferred to primordial density perturbations. It does not identify σ as that field, describe multiple mechanisms, or propose that σ could be a curvaton. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

It may state two separate claims as one.

The metric is of the Kantowski-Sachs form (case 2 of [43]), a Bianchi type III homogeneous but anisotropic geometry.

“We now see that the metric is of the Kantowski-Sachs form (case 2 of [43]), a Bianchi type III homogeneous but anisotropic geometry.”

• cites [43] sourceCould not verify

  The passages labeled Abstract discuss cosmological moduli, CMB perturbations, and an assumed flat universe; they do not classify a metric as Kantowski–Sachs or Bianchi type III. Searches for those terms and for a homogeneous anisotropic metric found no passage addressing the cited classification. No page numbers are available in our copy. (LLM, conf=0.72)

  Experimental — accuracy not yet measured

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Seemingly anomalous correlations have already been detected among the low-multipole CMB anisotropies, which might indicate deviations from statistical isotropy in the inflationary spectrum.

“Indeed, seemingly anomalous correlations have already been detected among the low-multipole CMB anisotropies [20,21,22,23,24], which might indicate deviations from statistical isotropy in the inflationary spectrum.”

• cites [20] C. Pitrou, 2008, J. Cosmol. Astropart. Phys. sourcePartly supported

  The Introduction links possible large-scale CMB anomalies to a preferred direction in primordial power spectra, and the discussion of primordial spectra describes a possible breakdown of statistical isotropy as an indication awaiting confirmation. The paper does not specifically report detected correlations among low-multipole CMB anisotropies; its treatment of the observational evidence is more tentative. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites [21] M.-a. WatanabeNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [22] sourceSupported
• cites [23] sourcePartly supported

  The Abstract reports statistically significant low-multipole correlations and says they are difficult to reconcile with standard inflationary cosmology’s prediction of statistical isotropy. It does not establish that the inflationary spectrum itself is anisotropic; the authors also suggest a local source or an unrecognized systematic as possible explanations. The indexed copy supplies no page number for these passages. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites [24] H.K. Eriksen, 2004, Astrophys. J. sourcePartly supported

  The Conclusions report an anisotropic distribution of large-scale CMB power at multipoles ℓ = 2–40, corroborated by an N-point correlation-function analysis. They allow that unusual large-scale features might require new physics, but do not specifically link the asymmetry to statistical isotropy in the inflationary spectrum. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

It may state two separate claims as one.

Periodicity in one of our large spatial dimensions has not been observed, indicating that the physical circumference of the closed z dimension must be greater than the physical diameter of the surface of last scattering.

“The second important relationship comes from realizing that periodicity in one of our large spatial dimensions has not been observed (see e.g.[20]), indicating that the physical circumference of the closed z dimension must be greater than the physical diameter of the surface of last scattering, or”

• cites [20] C. Pitrou, 2008, J. Cosmol. Astropart. Phys. sourceCould not verify

  The discussion of primordial spectra addresses CMB signatures of anisotropic inflation, not an observed absence of spatial periodicity. Searches for periodicity, a closed spatial dimension, and the diameter of the surface of last scattering found no passage establishing the claimed circumference bound; the cited paper’s pages were not supplied in our copy. (LLM, conf=0.72)

  Experimental — accuracy not yet measured

It may state two separate claims as one.

Anthropic selection gives a mass function of Milky-Way mass galaxies approximately independent of Ω_curv for Ω_curv ≪ O(0.1), falling off exponentially for roughly Ω_curv > 0.7.

“The effect of anthropic selection is most carefully computed in [69], where it is found that the mass function of Milky-Way mass galaxies is approximately independent of Ω curv for Ω curv ≪ O(0.1), and falls off exponentially for roughly Ω curv > 0.7”

• cites [69] sourceCould not verify

  The Introduction discusses open-inflation models, and Section IV analyzes the spectrum of gravity waves. Neither those passages nor searches for anthropic selection, the Milky-Way galaxy mass function, and the stated curvature thresholds found a passage addressing the claimed result. Our copy therefore cannot verify that this source computed it. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

It may state two separate claims as one.

Background anisotropy present at the onset of inflation generates statistical anisotropy in quantum fluctuations as they expand beyond the Hubble radius, and this can modify the spectrum of primordial density perturbations.

“Although the initial anisotropy rapidly redshifts away, background anisotropy present at the onset of inflation will generate statistical anisotropy in quantum fluctuations as they expand beyond the Hubble radius, and this in turn can modify the spectrum of primordial density perturbations [16,17,18,19].”

• cites [16] A.E. GumrukcuogluNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [17] A.E. Gumrukcuoglu, 2007, J. Cosmol. Astropart. Phys. sourceSupported
• cites [18] sourcePartly supported

  The Introduction (page unavailable) says that fluctuations redshift beyond the Hubble radius and that broken rotational invariance changes the primordial power spectrum. Section III (page unavailable) calculates a direction-dependent fluctuation spectrum in an anisotropic inflation model. But the paper models anisotropy persisting through inflation until its end; it does not establish the citing sentence’s claim that initial anisotropy rapidly redshifts away. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites [19] T.S. Pereira, 2007, J. Cosmol. Astropart. Phys. sourceSupported

Probably not a substantial claim.

When the background is homogeneous and isotropic, inflationary perturbations generally decouple into scalar, vector, and tensor modes.

“When the background is homogeneous and isotropic, inflationary perturbations generally decouple into scalar, vector, and tensor modes (for a review of cosmological perturbation theory see e.g.[34]).”

• cites [34] sourcePartly supported

  Section 3 (page unavailable) says that, for a massive vector field with vanishing background value in an isotropic background, transverse vector perturbations decouple from the other vector-field fluctuations. It does not state that inflationary perturbations generally decouple into scalar, vector, and tensor modes; the example is narrower and does not address tensor modes. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

It may state two separate claims as one.

Anisotropic spatial curvature induces anisotropic expansion, which affects the free streaming of photons and deforms the surface of last scattering, along with perception of angular scales on it.

“Indeed, anisotropic spatial curvature induces anisotropic expansion, which affects the free streaming of photons and thus deforms the surface of last scattering, along with our perception of angular scales on it [37].”

• cites [37] S. Yokoyama, 2008, J. Cosmol. Astropart. Phys. sourceCould not verify

  The Introduction and Summary discuss statistical anisotropy in primordial curvature perturbations, not anisotropic spatial curvature causing anisotropic expansion. Searches for photon free streaming, the surface of last scattering, and perceived angular scales found no passage addressing the claimed causal chain; page numbers were unavailable in our copy. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

It may state two separate claims as one.

The combination of prior and anthropic effects places most observers in spacetime regions where Ω_curv is unobservably small, but gives a reasonable probability of order ten percent to observe Ω_curv ~ 10^-5.

“The combination of these effects places most observers in spacetime regions in which Ω 0 curv is unobservably small (i.e.below cosmic variance), but at the same time gives a reasonable probability, roughly of order ten percent chance, to observe Ω 0 curv 10 -5 [68,69].”

• cites [68] sourceCould not verify

  The Introduction discusses an open-universe inflation scenario, and the Conclusion discusses its predicted CMB anisotropies; neither addresses how prior and anthropic effects distribute observers. Searches for observers, anthropic effects, cosmic variance, and a ten-percent probability of observing curvature near 10^-5 found no supporting passage. The roughly 10% figure in Section III refers to possible calculation errors, not an observation probability. (LLM, conf=0.75)

  Experimental — accuracy not yet measured
• cites [69] sourceCould not verify

  The Introduction describes an open-inflation bubble model, and Section III discusses quantizing its perturbations; neither addresses the distribution of observers or the probability of observing curvature. Searches for anthropic and prior effects, cosmic variance, Ω_curv ≈ 10^-5, and a ten-percent probability found no passage supporting the attributed claim. Page numbers were not supplied for these passages. (LLM, conf=0.75)

  Experimental — accuracy not yet measured

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The parent vacuum may transition to any vacuum with winding number n′=n.

“Note that this is not the only channel by which the parent vacuum may decay-in particular it may transition to any vacuum with winding number n ′ = n (see e.g.[10]).”

• cites [10] J.J. Blanco-Pillado, 2009, J. Cosmol. Astropart. Phys. sourcePartly supported

  The Introduction and §V.A (Decompactification tunneling; page unavailable) describe decompactification as an alternative decay channel, supporting the point that the parent vacuum has another way to decay. They do not say it can transition to any vacuum with unchanged winding number n′=n; the flux-tunneling channel described in the Introduction reduces the flux by one unit. (LLM, conf=0.83)

  Experimental — accuracy not yet measured

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Higher-dimensional solutions analogous to this have been discussed by [12].

“Higher dimensional solutions analogous to this have been discussed by [12].”

• cites [12] J.J. Blanco-Pillado, 2010, J. Cosmol. Astropart. Phys. sourceSupported

Probably not a substantial claim. The paper’s conclusions may not depend on it.

This is not unlike the situation in regular (isotropic) 4d open bubble inflation, where the inflationary spectrum is also modified at q O(1).

“This is not unlike the situation in regular (isotropic) 4d open bubble inflation, where the inflationary spectrum is also modified at q O(1) [53,54,55].”

• cites [53] sourceNot checked — no full text of the source
• cites [54] sourceNot checked — no full text of the source
• cites [55] C.B. Collins, 1973, Mon. Not. R. Astron. Soc. sourceCould not verify

  The Introduction (p. 1) and Type IX discussion (p. 8) concern isotropic cosmological models and microwave-background anisotropy, not open bubble inflation. Searches for an inflationary spectrum modified at q of order one and for low-wavenumber effects in open inflation found no passage addressing the claim. (LLM, conf=0.75)

  Experimental — accuracy not yet measured

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The Minkowski coordinates u and v are not periodic with the bubble coordinate z, indicating that the bubble geometry covers only a subset of the hyperboloid, with periodic boundary conditions.

“Indeed, closer inspection reveals that that the Minkowski coordinates u and v are not periodic with the bubble coordinate z (due to the factors of b d H inf ), indicating that the bubble geometry covers only a subset of the hyperboloid, with periodic boundary conditions.”

It may state two separate claims as one.

The power spectrum of inflationary perturbations is given by Eq. (4.25), and in the asymptotic limit of large q and μ it approximates H_inf^2/(2 b_d H_inf) (q^2+μ^2)^(-3/2).

“This gives the power spectrum”

• Also stated in: § 4.1 Power spectrum of a massless scalar · plain

  “≈ H 2 inf 2 b d H inf q 2 + µ 2 -3/2 ,(4.26)”

• AssumptionThe authors state this as an assumption: The inflaton energy density is precisely constant during inflation.
• AssumptionThe authors state this as an assumption: The initial conditions for inflationary perturbations are set by the Bunch-Davies vacuum.

It may state two separate claims as one.

Components of δC_{ll'mm'} with l'-l > 1 and odd are expected to be precisely zero, and components with l'-l > 2 and even appear consistent with zero.

“We have not displayed components of δC ℓℓ ′ mm ′ with ℓ ′ℓ > 1 and odd, which we expect, like ℓ ′ℓ = 1, to give precisely zero.The components with ℓ ′ℓ > 2 and even feature very slow numerical convergence, but also appear consistent with zero.”

• AssumptionThe authors state this as an assumption: The isocurvature fluctuations in the light scalar field σ ultimately directly source the adiabatic CMB perturbations.

It may state two separate claims as one.

As τ→0, a→0 and ȧ→1.

“Meanwhile, referring to (2.14) and (2.15) and being careful to track the various variable redefinitions, we find that as τ → 0, a → 0 , ȧ → 1 .”

It may state two separate claims as one.

These effects are suppressed relative to the statistically isotropic components of C_{ll'mm'} by the present-day curvature parameter Ω_curv, but appear to extend to arbitrarily large l.

“These effects are suppressed relative to the statisticallyisotropic components of C ℓℓ ′ mm ′ by the present-day curvature parameter Ω 0 curv , but appear to extend to arbitrarily large ℓ.”

• AssumptionThe authors state this as an assumption: The isocurvature fluctuations in the light scalar field σ ultimately directly source the adiabatic CMB perturbations.

It may state two separate claims as one. The quote only partly states this claim.

The low-multipole C_{ll'mm'} features off-diagonal elements in l and l' when l-l'=±2, and dependence on the multipole moment m (it is still diagonal in m and m').

“The most salient feature of Table 1 is the existence of off-diagonal terms with respect to ℓ and ℓ ′ .”

• Also stated in: § 4.2 Projection onto spherical harmonics · plain

  “at fixed ℓ ′ = ℓ, the correction δC ℓℓ ′ mm ′ increases with increasing magnitude of m ′ = m.”

• AssumptionThe authors state this as an assumption: The isocurvature fluctuations in the light scalar field σ ultimately directly source the adiabatic CMB perturbations.
• AssumptionThe authors state this as an assumption: Late-time anisotropic expansion h ≡ ȧ/a - ḃ/b is ignored.
• AssumptionThe authors state this as an assumption: The evolution of primordial perturbations after they enter the Hubble radius is ignored.
• AssumptionThe authors state this as an assumption: The physical distance from the origin to the surface defined by (4.29) is fixed, a⋆ ̺⋆ = constant.

It may state two separate claims as one.

The daughter vacuum is created when ψ tunnels through the barrier to ψ=ψd, after which ψ accelerates from rest and rolls down the potential, with ψ→∞ as time x0→∞.

“The daughter vacuum is created when ψ tunnels through the barrier, to some value ψ = ψ d , after which ψ accelerates from rest and rolls down the potential, with ψ → ∞ as time x 0 → ∞.”

It may state two separate claims as one.

The condition |V′′| < 32πG V at the peak of the potential barrier is not hard to satisfy and is satisfied for the choice of parameters given above.

“This condition not hard to satisfy; in particular it is satisfied for the choice of parameters given above.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Periodic boundary conditions on φ and z dictate that r and s must be positive or negative integers.

“where the periodic boundary conditions on φ and z dictate that r and s must be (positive or negative) integers (recall that we scale the z direction so that it is periodic on 0 ≤ z ≤ 2π).”

• AssumptionThe authors state this as an assumption: The compact dimension z has the topology of a circle with periodic boundary conditions, 0 ≤ z ≤ 2π.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The solution resembles a slicing of 4d de Sitter space.

“One might notice that this solution resembles a slicing of 4d de Sitter space.”

The quote only partly states this claim.

The 4d equations of motion for the anisotropic model admit a solution of the form a(σ)=H_p^{-1} sin(H_p σ), b(σ)=b_p, with H_p and b_p constants, provided condition (A.6) holds.

“These equations of motion permit a solution of the form a(σ) = H -1 p sin(H p σ) , b(σ) = b p , (A.5)”

It may state two separate claims as one.

The leading-order comoving distance to the surface of last scattering is ̺⋆ ≃ 3.5/(a0H0) ≃ 6.1√Ω0curv ≤ 0.50.

“The first is the leading-order comoving distance to the surface of last scattering, ̺ ⋆ = τ 0 τ⋆ dτ a(τ ) ≃ 3.5 a 0 H 0 ≃ 6.1 Ω 0 curv ≤ 0.50 , (3.19)”

It may state two separate claims as one.

δC_{ll'mm'} scales as ̺⋆^2 when ̺⋆≪1, and thus the non-zero components scale with the present-day curvature parameter Ω_curv.

“Comparing numerical results for several different values of ̺ ⋆ , we find (roughly) δC ℓℓ ′ mm ′ ∝ ̺ 2 ⋆ when ̺ ⋆ 1. Meanwhile (3.19) gives ̺ 2 ⋆ ≃ 37Ω 0 curv ; thus we conclude the non-zero components of δC ℓℓ ′ mm ′ scale with the present day curvature parameter Ω 0 curv .”

• cites (3.19)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The isocurvature fluctuations in the light scalar field σ ultimately directly source the adiabatic CMB perturbations.

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

In the (4+1)-dimensional Minkowski embedding, the bubble coordinates sit on the hyperboloid -t^2+u^2+v^2+x^2+y^2=Hinf^{-2}, on which (3.9) gives the scale factors of the induced metric.

“the bubble coordinates {τ, χ, φ, z} sit on the hyperboloid -t 2 + u 2 + v 2 + x 2 + y 2 = H -2 inf , on which (3.9) gives the scale factors of the induced metric.”

It may state two separate claims as one.

The anisotropic bubble scenario introduces m-dependence into C, while C remains diagonal in m and m′.

“and introduces m-dependence into C ℓℓ ′ mm ′ (it is still diagonal in m and m ′ ).”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The hypersurface χ=0 is a Cauchy surface.

“Here the coordinate χ provides the spacelike foliation, η c is a spatial coordinate, and the hypersurface χ = 0 is a Cauchy surface.”

Probably not a substantial claim.

All we must do is match the solutions F1 and F2 onto F1 and F2 in the limit η_c → -∞.

“All we must do is match the solutions F 1 and F 2 onto F1 and F2 , in the limit η c → -∞.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The power spectrum can be computed as in Section 4.1, including the extra normalization factor N_q, yielding (B.33).

“The power spectrum can now be computed just as in Section 4.1, except now we must include the extra normalization factor N q .That is, we write”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The solutions F1 and F2 represent two orthogonal spatial modes, and each should be normalized separately, with the two summed over in the final mode expansion.

“The solutions F 1 and F 2 represent two orthogonal spatial modes, and thus each should be normalized separately, with the two summed over in the final mode expansion.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The functions F1 and F2 have been expressed so as to have simple asymptotic behavior, and putting all together gives the pair of mode functions (B.29) and (B.30).

“The functions F 1 and F 2 have been expressed so as to have simple asymptotic behavior, see (4.17).Putting all of this together, we obtain the pair of mode functions”

Probably not a substantial claim.

By matching asymptotic positive and negative frequency modes, Υ_qs and Υ+_qs are given by (B.17) and (B.18).

“By matching the asymptotic positive and negative”

Probably not a substantial claim.

The bubble coordinate system of (4.3) can be smoothly connected to the parent vacuum coordinate system of (B.1) by analytic continuation.

“Note that the bubble coordinate system of (4.3) can be smoothly connected to the parent vacuum coordinate system of (B.1) by using analytic continuation,”

It may state two separate claims as one.

The deviation from the asymptotic curve is itself anisotropic; for example at q=2 the curve with μ=0.1 deviates more significantly than the curve with μ=10.

“In the present situation, it is evident that the deviation from the asymptotic curve is itself anisotropic; for example at q = 2 the curve with µ = 0.1 deviates more significantly than the curve with µ = 10.”

Probably not a substantial claim. The paper may give no reason why it holds.

The normalizations of Φ_r, Z_s, and X_qr are such that, after analytic continuation, each mode maps onto the similarly-denoted mode defined within the bubble.

“The normalizations of Φ r , Z s , and X qr are such that, after analytic continuation, each mode maps onto the similarly-denoted mode defined within the bubble.”

Probably not a substantial claim. It may state two separate claims as one.

The z dimension of the parent vacuum maps directly onto that of the bubble, as does the coordinate φ.

“Instead the z dimension of the parent vacuum maps directly onto that of the bubble (as is the case with the coordinate φ), assuming that b(η c ) → b d and ḃd → 0 as η c → -∞.”

• AssumptionThe authors state this as an assumption: b(η_c) → b_d and ḃ_d → 0 as η_c → -∞.

It may state two separate claims as one.

The Bunch-Davies modes of σ in the parent vacuum are propagated into the bubble by analytic continuation; Φ_r and Z_s are unchanged, and X_qr and Υ± are propagated by analytic continuation of coordinates.

“Now that we have the Bunch-Davies modes of σ in the parent vacuum, the next step is to propagate these modes into the bubble.For the modes Φ r and Z s this is easy: they are unchanged.The modes X qr and Υ ± qs are propagated by analytic continuation of the coordinates in their arguments, as given by (B.2).”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Using orthogonality of mode functions and setting Cauchy hypersurface at χ=0, the normalization condition can be written as (B.20).

“As mentioned before, we set the Cauchy hypersurface Σ µ at χ = 0. Using the orthogonality of the mode functions Υ ± qs , Φ r , and Z s , the above normalization condition can be written”

It may state two separate claims as one.

The only dependence of Υ± on η_c is via tanh(η_c), which is rotated into coth(η_c) when η_c → η_c + iπ/2.

“that the only dependence of the Υ ± qs on η c is via tanh(η c ), which is rotated into coth(η c ) when we take η c → η c + iπ/2.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The normalization of the delta function is determined by noting that all of its support comes from large |η_c|, where the solutions are plane-wave.

“The normalization of the delta function is determined by noting that all of its support comes from large |η c |, where the solutions are plane-wave.”

Probably not a substantial claim.

Evaluating the power spectrum in the limit η_c→0 and simplifying gamma functions gives the power spectrum formula (B.35).

“As described in Section 4.1 we evaluate the power spectrum in the limit η c → 0, in which case the asymptotic behavior of the time-dependent parts is given by (4.18) and (4.19).Putting everything together, and performing some manipulations to eliminate some of the gamma functions, we find the power spectrum to be”

• AssumptionThe authors state this as an assumption: Both effects (bubble collisions and the parent vacuum residing in or containing other bubbles) are negligible.
• AssumptionThe authors state this as an assumption: All of spacetime can be covered by just the two universes (no other bubbles) for the diagram and work below.

It may state two separate claims as one.

The analytic continuation of the scale factor a gives a = H_inf^{-1} sech(η_c), while in the parent vacuum the scale factor b is constant, b = b_d.

“The analytic continuation of the scale factor a gives a = H -1 inf sech(η c ), while in the parent vacuum the scale factor b is constant, b = b d .”

It may state two separate claims as one.

The potential V(η_c) tends to zero as η_c → ±∞, so Υ may be expressed in terms of two linearly independent 'scattering' solutions.

“The potential V (η c ) tends to zero as η c → ±∞; thus Υ υ qs may just as well be expressed in terms of two linearly independent”

It may state two separate claims as one.

Nevertheless they have the correct asymptotic form as η_c → -∞, because b(η_c) ∝ coth(η_c) approaches a constant in that limit.

“Nevertheless they have the correct asymptotic form, as η c → -∞, because the scale factor b(η c ) ∝ coth(η c ) approaches a constant in that limit.”

Probably not a substantial claim. It may state two separate claims as one.

In (B.29) and (B.30), the terms in the first parentheses of (B.29) cancel to zero, as do the terms in the second parentheses of (B.30), and other simplifications yield (B.31) and (B.32).

“Referring to (B.25)-(B.28), we see that the terms in the first parantheses of (B.29) cancel to zero, as do the terms in the second parantheses of (B.30).Referring also to (B.12)-(B.16), we see that other basic simplifications are possible, and in the end we obtain”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

By setting b_d by trial and error, the entire numerical solution is smooth, with a → σ_max σ and b → constant as σ approaches σ_max.

“We then set the value of b d by trial and error, so that the entire solution is smooth, in particular so that as σ approaches some value σ max , a → σ maxσ and b → constant.”

It may state two separate claims as one.

For the daughter vacuum to correspond to our universe, its local evolution should approach an approximately O(3) rotationally-symmetric FRW cosmology, and the circumference of each large spatial dimension should become and/or remain much larger than the Hubble radius.

“In order for the daughter vacuum to correspond to our universe, its local evolution should approach an approximately O(3) rotationally-symmetric Friedmann-Robinson-Walker (FRW) cosmology, and the circumference of each large spatial dimension should become and/or remain much larger than the Hubble radius.”

Probably not a substantial claim. It may state two separate claims as one.

The tunneling process is described by an instanton that interpolates between parent and daughter vacua and appears as bubble nucleation in the inflating spacetime.

“The tunneling process is described by an instanton that interpolates between parent and daughter vacua, and appears as bubble nucleation in the inflating spacetime [3,4].”

• cites [3] sourceNot checked — no full text of the source
• cites [4] sourceNot checked — no full text of the source

Probably not a substantial claim.

In the standard picture, both the parent and daughter vacua have three large expanding spatial dimensions.

“In the standard picture, both the parent and daughter vacua have three large (expanding) spatial dimensions.”

It may state two separate claims as one.

The bubble geometry is expected to be toroidal, with O(2) rotational symmetry in the two large spatial dimensions, uniformly wrapping around the compact space.

“Indeed, we expect the bubble geometry to be toroidal, with O(2) rotational symmetry in the two large spatial dimensions, uniformly wrapping around the compact space.”

• AssumptionThe authors state this as an assumption: One of the large spatial dimensions of the daughter vacuum is compact in the parent vacuum, with a volume modulus that tunnels out of a metastable minimum during bubble nucleation and subsequently grows to large size.

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

During inflation both forms of anisotropy rapidly redshift away.

“During inflation both forms of anisotropy rapidly redshift away.”

Probably not a substantial claim. The paper may give no reason why it holds.

A round of slow-roll inflation within the bubble is sufficient to redshift away the initial anisotropy and curvature, creating an O(3)-symmetric FRW cosmology consistent with the observed universe.

“Yet, as expected, a round of slow-roll inflation within the bubble is sufficient to redshift away the initial anisotropy and curvature, creating an O(3)-symmetric FRW cosmology consistent with the observed universe.”

• AssumptionThe authors state this as an assumption: Slow-roll inflation lasts long enough to conform to present observation, but not so long as to push all effects of initial anisotropy outside our present horizon.

It may state two separate claims as one.

The computation of the inflationary spectrum accounts for spatial curvature anisotropy, but the projection onto spherical harmonics does not.

“While our computation of the inflationary spectrum accounts for this spatial curvature anisotropy, our projection onto spherical harmonics does not.”

Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.

Anisotropic bubble nucleation provides concrete theoretical constraints on the form of initial conditions and serves as a natural extension of the standard inflationary scenario.

“However unlike other approaches, anisotropic bubble nucleation provides concrete theoretical constraints on the form of initial conditions, and serves as a natural extension of the standard inflationary scenario.”

It may state two separate claims as one.

Although the asymptotic limit of (4.26) is congruent with standard flat isotropic inflation, the two inflationary spectra are not the same because the power spectrum is expressed in anisotropic Fourier modes U_qrs while isotropic Cartesian Fourier modes are e^{ik·y}; this difference affects the observed spectrum because the correlations implied by the delta functions differ.

“Although we have just remarked on the congruence between the asymptotic limit of our result, (4.26), and the result from standard, flat, isotropic inflation, the two inflationary spectra are not the same.”

• Also stated in: § 4.1 Power spectrum of a massless scalar · plain

  “This difference affects the observed spectrum because the (lack of) correlations implied by the delta functions δ(qq ′ ) δ rr ′ δ ss ′ in the two-point correlator (4.23) are different than the (lack of) correlations implied by δ (3) (kk ′ ) in the standard isotropic case.”

It may state two separate claims as one. The quote only partly states this claim.

The failure of off-diagonal terms of δC to approach zero at large l is because the anisotropic modes are evaluated on a surface at fixed non-zero ̺⋆; the discrepancy corresponds to statistical anisotropy in the projection onto spherical harmonics.

“The failure of off-diagonal terms of δC ℓℓ ′ mm ′ to approach zero in the limit of large ℓ may at first seem to conflict with the usual interpretation that increasing multipole moments ℓ correspond to probing smaller physical scales.”

• Also stated in: § 4.2 Projection onto spherical harmonics · plain

  “Since the spherical harmonics receive support over the entire two-sphere (regardless of ℓ), this discrepancy corresponds to statistical anisotropy in the projection onto spherical harmonics.”

• AssumptionThe authors state this as an assumption: The isocurvature fluctuations in the light scalar field σ ultimately directly source the adiabatic CMB perturbations.

It may state two separate claims as one.

As bdHinf is decreased, the maximum allowed present-day curvature parameter is decreased, along with the size of all related effects.

“Thus, as the quantity b d H inf is decreased, the maximum allowed present-day curvature parameter is decreased, along with the size of all related effects.”

Probably not a substantial claim. The paper may give no reason why it holds.

The metastable minimum must have positive vacuum energy, so the parent vacuum can decay to our positive-vacuum-energy universe.

“Note that the metastable minimum referred to above must have positive vacuum energy, so the parent vacuum can decay to our positive-vacuum-energy universe.”

Probably not a substantial claim.

During such inflation an initially homogeneous but anisotropic universe rapidly approaches local isotropy.

“Meanwhile, it is well known that during such inflation an initially homogeneous but anisotropic universe rapidly approaches local isotropy [14,15].”

• cites [14] sourceNot checked — no full text of the source
• cites [15] sourceNot checked — no full text of the source

It may state two separate claims as one.

The balance of contributions to the energy density in the bubble affects only the time dependence of the scale factors a and b, which in turn affects only the power spectrum P_qs, not the anisotropic Fourier mode functions U_qrs.

“The balance of contributions to the energy density in the bubble affects only the time dependence of the scale factors a and b, which in turn affects only the power spectrum P qs (not the anisotropic”

• AssumptionThe authors state this as an assumption: The inflaton energy density is precisely constant during inflation.
• AssumptionThe authors state this as an assumption: The inflaton energy density dominates over the modulus effective potential immediately after bubble nucleation.

Probably not a substantial claim. The paper may give no reason why it holds.

Anisotropic bubble nucleation provides an opportunity to confirm aspects of the landscape/multiverse hypothesis.

“Furthermore, it provides an opportunity to confirm aspects of the landscape/multiverse hypothesis.”

Probably not a substantial claim.

A full understanding of the observable signatures of anisotropic bubble nucleation requires combining the effects of spatial curvature and expansion anisotropy on photon free streaming.

“A full understanding of the observable signatures of anisotropic bubble nucleation requires combining both of these effects.”

• AssumptionThe authors state this as an assumption: The effects of spatial curvature and expansion anisotropy on the free streaming of photons from the surface of last scattering to the point of present detection are ignored.

It may state two separate claims as one.

The bubble geometry features two forms of anisotropy: expansion anisotropy from the two distinct scale factors a and b, and spatial curvature anisotropy from the (χ,φ) plane being open while the orthogonal z direction is flat.

“Note that the bubble geometry features two forms of anisotropy: expansion anisotropystemming from the two distinct scale factors a and b-and spatial curvature anisotropystemming from the (χ, φ) plane being open while the orthogonal z direction is flat.”

It may state two separate claims as one.

The state with ψ=ψp is the parent vacuum, which appears as effectively 3d de Sitter space on scales much larger than 2πL e^{αψp/2}.

“The state with ψ = ψ p is the parent vacuum, which appears as effectively 3d de Sitter space on scales much larger than 2πL e αψp/2 .”

It may state two separate claims as one.

In the 3d effective theory, condition (A.6) can be written V'=0, so the compactification solution has been identified from the full 4d theory.

“In the language of the 3d effective theory, this condition can be written V ′ = 0; therefore we have identified the compactification solution but from the vantage of the full 4d theory.”

Probably not a substantial claim.

The approximations (negligible wall perturbation, constant b_d, equal energy density) allow treating the parent vacuum as pure de Sitter space crossed with a circle.

“In particular they allow us to treat the parent vacuum as pure de Sitter space (crossed with a circle), in which case the central, diamond-shaped region of Figure 5 can be covered by coordinates with line element”

• AssumptionThe authors state this as an assumption: The energy density of the bubble wall does not significantly perturb the metric.
• AssumptionThe authors state this as an assumption: The radius b_d of the compact dimension z is essentially constant throughout the parent vacuum.
• AssumptionThe authors state this as an assumption: The energy density immediately inside the bubble is essentially equal to that of the parent vacuum outside (H^2_inf = Λ/3 is essentially the same as H^2_p).

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

The statements below apply equally to more realistic geometries than the simplified diagram with equal vacuum energies inside and outside the bubble.

“For simplicity we have constructed the diagram as if the vacuum energies inside and outside of the bubble are the same; however the statements below apply equally to more realistic geometries.”

It may state two separate claims as one. The quote only partly states this claim.

The same instanton would describe the reverse 'tunneling' process, whose description is less clear in the 3d effective theory because the 3d effective theory breaks down inside the nucleating bubble.

“While the interpretation of this instanton as a tunneling event is not so apparent in the 4d picture, note that the same instanton would describe the reverse”

Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.

Bubble nucleation rates are likely to be exponentially suppressed, so any such disturbances would typically be very far from the physical region of interest.

“However bubble nucleation rates are likely to be exponentially suppressed, so any such disturbances would typically be very far from the physical region of interest.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

The parent vacuum itself likely resides in an open bubble, or contains other bubbles that intersect the Cauchy surface at χ=0.

“Meanwhile, the parent vacuum itself likely resides in an open bubble, or contains other bubbles that intersect the Cauchy surface at χ = 0 (see text below).”

Probably not a substantial claim. The paper may give no reason why it holds.

The phenomenological success of inflationary theory suggests that any effects of such collisions should be small linear perturbations to the standard inflationary background (at least in our past lightcone).

“however the phenomenological success of inflationary theory suggests that any effects of such collisions should be small-linear perturbations to the standard inflationary background (at least in our past lightcone).”