No-Boundary Measure of the Universe

Near the bounce the universe approaches a de Sitter state with radius ∼ (μφ0)^-1.

“Near the bounce the universe approaches a de Sitter state with radius ∼ (µφ 0 ) -1 .”

The condition that the universe lasts ∼14 Gyr further restricts the ensemble, requiring φ0 to be larger than a critical value φg0 > φc0.

“The condition that the universe lasts ∼14 Gyr further restricts the ensemble, requiring φ 0 to be larger than a critical value φ g 0 > φ c 0 .”

For five members of the ensemble of classical histories in the μ=3 model for φ0 between 1.3 and 2, when followed back in time to higher H, they all lie within a very narrow band around H=μφ, which corresponds to slow-roll inflationary solutions.

“When we follow the histories back in time to higher values of H, they all lie within a very narrow band around H = µφ.But this is precisely the regime that corresponds to slow roll inflationary solutions”

For quadratic potentials with realistic values of m and Λ, the constraints of classicality and minimum age yield a restricted ensemble of histories whose volume-weighted probabilities slightly favor a large number of e-foldings necessary for explaining the observed spatial flatness.

“For quadratic potentials with realistic values of m and Λ the constraints of classicality and minumum age yield a restricted ensemble of histories whose volume weighted probabilities slightly favor a large number of efoldings [5] that are necessary for explaining the observed spatial flatness.”

• cites [5]Not matched

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

The relative probabilities for histories in the classical ensemble are given by exp[-2I_R(b,χ)/h] in the leading semiclassical approximation.

“The relative probabilities for histories in this classical ensemble are given by exp[-2I R (b, χ)/h] in the leading”

In the quadratic potential minisuperspace model with μ=3 and Λ=.03, there are no classical histories for φ0 below a critical value φc0 at about 1.2.

“There are no classical histories for φ0 below a critical value φ c 0 at about 1.2.”

In the semiclassical limit, the no-boundary proposal for homogeneous isotropic closed universes with a cosmological constant and a quadratic scalar field potential predicts classical behavior at late times if the initial scalar field exceeds a minimum value.

“In the semi-classical limit, it predicts classical behavior at late times if the initial scalar field is more than a certain minimum.”

A critical value φs0 at about 1.5 separates large φ0 histories that bounce at a finite radius when extrapolated back from singular histories for smaller φ0.

“A critical value φ s 0 at about 1.5 separates large φ0 histories that bounce at a finite radius when extrapolated back from singular histories for smaller φ0.”

In the regime μ > 3/2, the NBWF requires the universe to contain a minimum amount of scalar field energy at early times to behave classically at late times.

“in the interesting regime where µ > 3/2 we find the NBWF requires the universe to contain a minimum amount of scalar field energy at early times to behave classically at late times.”

The no-boundary proposal provides a probability measure on classical solutions that selects inflationary histories but is heavily biased towards small amounts of inflation.

“The no-boundary proposal provides a probability measure on the classical solutions which selects inflationary histories but is heavily biased towards small amounts of inflation.”

An important feature of the volume-weighted probability distribution is that there is a wide region where the probability is strongly increasing with N.

“An important feature of the volume weighted probability distribution is that there is a wide region where the probability is strongly increasing with N .”

When classical late-time histories are extended back, they may be singular or bounce at a finite radius.

“If the classical late time histories are extended back, they may be singular or bounce at a finite radius.”

The no-boundary wave function severely restricts the possible classical universes.

“We find the NBWF severely restricts the possible classical universes”

There is a striking contrast between the unconditioned bottom-up probabilities that favor small amounts of inflation and the top-down probabilities conditioned on our data that favor larger amounts.

“Hence, there is a striking contrast between the unconditioned bottom-up probabilities that favor small amounts of inflation and the top-down probabilities conditioned on our data that favor larger amounts.”

Classicality restrictions would act as a strong vacuum selection principle in the string landscape.

“and argue that such classicality restrictions would act as a strong vacuum selection principle in the string landscape.”

• AssumptionRests on an assumption: The string theory landscape contains a vast ensemble of stable and metastable vacua including some with a small positive effective cosmological constant and the low-energy effective field theory of the Standard Model.

The existence of singularities in the extrapolation of some classical approximation in quantum mechanics is not an obstacle to prediction but merely a limitation of the validity of the approximation.

“The existence of singularities in the extrapolation of some classical approximation in quantum mechanics is not an obstacle to prediction but merely a limitation of the validity of the approximation.”

Near a broad maximum, the volume factor more than compensates for the reduction in amplitude due to the higher value of the potential.

“However in contrast with quadratic potentials, near a broad maximum the volume factor more than compensates for the reduction in amplitude due to the higher value of the potential.”

The volume weighting can overcome the gradient action for very long wavelength perturbations that leave the horizon while (6) is satisfied.

“It appears that the volume weighting can overcome the gradient action for very long wavelength perturbations that leave the horizon while (6) is satisfied.”

This means that the thermodynamic arrow of time is likely to point away from the bounce on either side of it.

“This means that the thermodynamic arrow of time is likely to point away from the bounce on either side of it.”

The sum over our location multiplies the NBWF probability for each classical spacetime in the ensemble by the number of Hubble volumes in the total present volume-a factor proportional to exp(3N).

“The sum over our location therefore multiplies the NBWF probability for each classical spacetime in the ensemble by the number of Hubble volumes in the total present volume -a factor proportional to exp(3N ).”

This suggests the NBWF with volume weighting will predict a universe that is very inhomogeneous on very large scales.

“This suggests the NBWF with volume weighting will predict a universe that is very inhomogeneous on very large scales.”

By analogy with quadratic potentials, the classical histories predicted by the NBWF for φ0 near a broad maximum of V are expected to have an early period of inflation, during which the scalar field rolls down to a nearby minimum of V.

“By analogy with quadratic potentials we expect the classical histories predicted by the NBWF for φ 0 near a broad maximum of V to have an early period of inflation, during which the scalar field rolls down to a nearby minimum of V .”

In a landscape potential, the most probable homogeneous history of the universe that is consistent with our data started in an unstable de Sitter-like state near a broad saddle-point of V.

“This leads us to predict that in a landscape potential, the most probable homogeneous history of the universe that is consistent with our data started in an unstable de Sitter like state near a broad saddle-point of V .”

• AssumptionRests on an assumption: A typical landscape potential has several saddle-points besides the quadratic directions discussed.

Volume weighting increases the probability of a large number of e-foldings.

“Volume weighting increases the probability of a large number of efoldings.”

The NBWF predicts the probabilities of entire classical histories.

“The no-boundary wave function provides probabilities for entire classical histories.”

• cites [1]Not matched

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

The universe requires a minimum amount of matter to behave classically at late times.

“The universe therefore requires a minimum amount of matter to behave classically at late times.”

For wave functions of the form (2) to predict an ensemble of Lorentzian histories with high probabilities for classical correlations in time, a necessary condition is the classicality constraint |(∇S)^2| ≫ |(∇I_R)^2|.

“In order for wave functions of the form (2) to predict an ensemble of Lorentzian histories with high probabilities for classical correlations in time further conditions must be satisfied.A necessary one is the classicality constraint”

The probability for a homogeneous universe should be multiplied by exp(3N), where N is the number of e-foldings of slow-roll inflation, to obtain the probability for what we observe in our past light cone.

“However we argue that the probability for a homogeneous universe should be multiplied by exp(3N ) where N is the number of e-foldings of slow roll inflation to obtain the probability for what we observe in our past light cone.”

In a landscape potential, the universe would then have always been in the semi-classical regime.

“The universe would then have always been in the semi-classical regime.”

The bounce for φ0 > φs0 is possible despite the singularity theorems because a scalar field and the cosmological constant violate the strong energy condition.

“This is possible despite the singularity theorems because a scalar field and the cosmological constant violate the strong energy condition.”

For universes that bounce at a minimum radius, it seems likely that the NBWF will predict that fluctuations away from homogeneity and isotropy will be at a minimum at the bounce and grow away from the bounce for at least a while on either side.

“For the universes that bounce at a minimum radius it seems likely that the NBWF will predict that fluctuations away from homogeneity and isotropy will be at a minimum at the bounce and grow away from the bounce for at least a while on either side (cf.[7]).”

• cites [7]Not matched

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The resulting probabilities of past histories consistent with present data significantly favor a large number of e-foldings.

“The resulting probabilities of past histories consistent with present data significantly favor a large number of efoldings.”

• cites [5]Not matched

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

The NBWF plus classicality at late times implies inflation at early times.

“Hence the NBWF plus classicality at late times implies inflation at early times.”

The classicality constraint φ0 > φc0 implies that all histories in the classical ensemble inflate.

“The classicality constraint φ 0 > φ c 0 implies that all histories in the classical ensemble inflate (Figure 2).”

For saddle points with more than one descent direction, there will generally be a lower saddle-point with only one descent direction, and with lower action.

“For saddle points with more than one descent direction, there will generally be a lower saddle-point with only one descent direction, and with lower action.”

• AssumptionRests on an assumption: A typical landscape potential has several saddle-points besides the quadratic directions discussed.

There could be quantum mechanical transitions rather than classical ones across cosmological singularities that connect two classical regimes.

“Indeed, there could be quantum mechanical transitions rather than classical ones across cosmological singularities that connect two classical regimes [10].”

• cites [10]Not matched

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

Only broad saddle-points with a single descent direction will give rise to significant no-boundary amplitudes for universes that behave classically at late times.

“Thus only broad saddlepoints with a single descent direction will give rise to significant no-boundary amplitudes for universes that behave classically at late times.”

The classicality constraint is closely related to the slow-roll condition of scalar field inflation.

“The classicality constraint is closely related to the slow roll condition of scalar field inflation.”

The no-boundary proposal's bias towards small amounts of inflation would not be compatible with observations.

“This would not be compatible with observations.”

Only a few of the saddle-points will satisfy the demanding condition that they be broad, because it requires that the scalar field varies by order the Planck value across them.

“Only a few of the saddlepoints will satisfy the demanding condition that they be broad, because it requires that the scalar field varies by order the Planck value across them.”

The classicality constraint acts as a vacuum selection principle.

“The classicality constraint, therefore, acts as a vacuum selection principle.”

Because the dominant saddle-points are well below the Planck density, the most probable histories are expected to be bouncing solutions of the field equations which lie entirely in the semi-classical regime.

“Because the dominant saddle-points are well below the Planck density we expect the most probable histories are bouncing solutions of the field equations which lie entirely in the semi-classical regime.”

• AssumptionRests on an assumption: A typical landscape potential has several saddle-points besides the quadratic directions discussed.