Origin of structure in the Universe
Abstract
It is assumed that the Universe is in the quantum state defined by a path integral over compact four-metrics. This can be regarded as a boundary condition for the wave function of the Universe on superspace, the space of all three-metrics and matter field configurations on a three-surface. We extend previous work on finite-dimensional approximations to superspace to the full infinite-dimensional space. We treat the two homogeneous and isotropic degrees of freedom exactly and the others to second order. We justify this approximation by showing that the inhomogeneous or anisotropic modes start off in their ground state. We derive time-dependent Schr\"odinger equations for each mode. The modes remain in their ground state until their wavelength exceeds the horizon size in the period of exponential expansion. The ground-state fluctuations are then amplified by the subsequent expansion and the modes reenter the horizon in the matter- or radiation-dominated era in a highly excited state. We obtain a scale-free spectrum of density perturbations which could account for the origin of galaxies and all other structure in the Universe. The fluctuations would be compatible with observations of the microwave background if the mass of the scalar field that drives the inflation is ${10}^{14}$ GeV or less.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Unlike standard WKB derivations, our assumptions do not require the clock to be large or massive, so non-clock degrees can be more than a small perturbation and the clock need not have nearly zero energy.
“In particular, in contrast to standard derivations [8][9][10][11][12][13][14] of evolution through the use of the WKB approximation, our assumptions do not imply that the clock is large or massive so that the non-clock degrees of freedom may, in this approach, represent more than a small perturbation and the clock need not be in a state with nearly zero energy †.”
From Almost ideal clocks in quantum cosmology: a brief derivation of time (1995) · cites this paper as [9]
Cites
Inflationary universe: A possible solution to the horizon and flatness problems
Guth, A. H. (1981) · Physical Review D · doi:10.1103/physrevd.23.347 · Cited by 9,983 · see citing works
Linde, A. D. (1982) · Physics Letters B · doi:10.1016/0370-2693(82)91219-9 · Cited by 6,405 · see citing works
Hartle, J. B., & Hawking, S. W. (1983) · Physical Review D · doi:10.1103/physrevd.28.2960 · Cited by 3,027 · see citing works
Self-citationFluctuations in the New Inflationary Universe
Guth, A. H., & Pi, S.-Y. (1982) · Physical Review Letters · doi:10.1103/physrevlett.49.1110 · Cited by 2,392 · see citing works
Spontaneous creation of almost scale-free density perturbations in an inflationary universe
Bardeen, J. M., Steinhardt, P. J., & Turner, M. S. (1983) · Physical Review D · doi:10.1103/physrevd.28.679 · Cited by 2,073 · see citing works
The development of irregularities in a single bubble inflationary universe
Hawking, S. W. (1982) · Physics Letters B · doi:10.1016/0370-2693(82)90373-2 · Cited by 1,991 · see citing works
Self-citationInvestigations in relativistic cosmology
LIFSHITZ, E. M., & Khalatnikov, I. M. (1963) · Advances In Physics · doi:10.1080/00018736300101283 · Cited by 913 · see citing works
Supercooled phase transitions in the very early universe
Hawking, S. W., & Moss, I. L. (1982) · Physics Letters B · doi:10.1016/0370-2693(82)90946-7 · Cited by 816 · see citing works
Self-citationThe quantum state of the universe
Hawking, S. W. (1984) · Nuclear Physics B · doi:10.1016/0550-3213(84)90093-2 · Cited by 689 · see citing works
Self-citationGraviton creation in the inflationary universe and the grand unification scale
Rubakov, V. A., Sazhin, M. V., & Veryaskin, A. V. (1982) · Physics Letters B · doi:10.1016/0370-2693(82)90641-4 · Cited by 675 · see citing works
Graviton creation in the early universe
Grishchuk, L. P. (1977) · Annals of the New York Academy of Sciences · doi:10.1111/j.1749-6632.1977.tb37064.x · Cited by 210 · see citing works
Higher derivatives in quantum cosmology
Hawking, S. W., & Luttrell, J. C. (1984) · Nuclear Physics B · doi:10.1016/0550-3213(84)90380-8 · Cited by 159 · see citing works
Self-citationLimits on inflationary models of the universe
Hawking, S. W. (1985) · Physics Letters B · doi:10.1016/0370-2693(85)90989-x · Cited by 77 · see citing works
Self-citationHomogeneous collapsing star: Tensor and vector harmonics for matter and field asymmetries
Gerlach, U. H., & Sengupta, U. K. (1978) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.18.1773 · Cited by 67 · see citing works
Numerical calculations of minisuperspace cosmological models
Hawking, S. W., & Wu, Z. (1985) · Physics Letters B · doi:10.1016/0370-2693(85)90815-9 · Cited by 57 · see citing works
Self-citation
Flagged first, then by citations
Cited by
Sorted by citations
Bojowald, M. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-11 · Cited by 723 · see citing works
Two dimensional nearly de Sitter gravity
Maldacena, J., Turiaci, G. J., & Yang, Z. (2021) · Journal of High Energy Physics · doi:10.1007/jhep01(2021)139 · Cited by 190 · see citing works
Almost ideal clocks in quantum cosmology: a brief derivation of time
Marolf, D. (1995) · Classical and Quantum Gravity · doi:10.1088/0264-9381/12/10/007 · Cited by 77 · see citing works
What is the geometry of superspace?
Giulini, D. (1995) · Physical Review D · doi:10.1103/physrevd.51.5630 · Cited by 74 · see citing works
Weyl Curvature Hypothesis in Light of Quantum Backreaction at Cosmological Singularities or Bounces
Hu, B.-L. (2021) · Universe · doi:10.3390/universe7110424 · Cited by 8 · see citing works
- Authors
- Halliwell, J. J., & Hawking, S. W.
- Affiliations
- Max Planck Society (DE)
- ORCID
- Halliwell, J. J.
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 1985-04-15
- Volume
- 31
- Issue
- 8
- Pages
- 1777-1791
- ISSN
- 0556-2821 (ISSN-L)
- OpenAlex
- W2017483000
- Cited by
- 665
- References
- 15
- Citations by year
- 2026: 14 · 2025: 43 · 2024: 9 · 2023: 12 · 2022: 15 · 2021: 23 · 2020: 35 · 2019: 21 · 2018: 20 · 2017: 22 · 2016: 16 · 2015: 9 · 2014: 19 · 2013: 20 · 2012: 19
- FWCI
- 30.00
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- wave function of the universe, superspace, time-dependent Schrödinger equation, inflation (cosmology), galaxy formation, cosmic microwave background, scalar field mass, horizon crossing
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
