Absence of a Singularity in Loop Quantum Cosmology
Open accessAbstract
It is shown that the cosmological singularity in isotropic minisuperspaces is naturally removed by quantum geometry. Already at the kinematical level, this is indicated by the fact that the inverse scale factor is represented by a bounded operator even though the classical quantity diverges at the initial singularity. The full demonstration comes from an analysis of quantum dynamics. Because of quantum geometry, the quantum evolution occurs in discrete time steps and does not break down when the volume becomes zero. Instead, space-time can be extended to a branch preceding the classical singularity independently of the matter coupled to the model. For large volume the correct semiclassical behavior is obtained.
AI snapshot
Written from the paper’s full text.
TL;DR
Loop quantum cosmology removes the classical isotropic cosmological singularity: the inverse scale factor is quantized to a bounded operator, and the Hamiltonian constraint becomes a discrete difference equation that can evolve through zero volume. The result is a pure quantum-geometry effect, with correct semiclassical behavior at large volume.
Key findings
- The inverse scale factor is represented by a bounded operator despite classical divergence; its eigenvalues approach the classical \(V^{-1/3}\) for large \(j\), with corrections only of fourth order, and only the lowest three eigenvalues show large deviations (even \(j=1\) is very close to classical). The volume operator has a three-fold degenerate zero eigenvalue, but the inverse scale factor remains finite.
- Quantum dynamics occurs in discrete time steps; the constraint is a difference equation. For \(n<-4\) coefficients are determined by initial data; \(n=0\) gives a consistency condition and \(s_0\) drops out; all \(s_n\) for \(n>0\) are determined, e.g. \(s_4\) is determined solely by \(s_{-4}\). Evolving solutions have \(s_0=0\).
- There is a contracting branch for \(n<0\), zero volume at \(n=\pm1\) (corresponding to \(j=0\)), and an expanding branch for \(n>0\); the conclusion holds for any matter and cosmological constant, without violating energy conditions.
- For large volume and small curvature, the difference operator approximates the classical constraint and WKB gives correct classical behavior; in the simplest Euclidean flat no-matter case there is a unique (up to a constant factor) solution with a classical regime.
Methods
- First quantizes full quantum geometry and then reduces to isotropic minisuperspace using generalized symmetric states; uses connection variables \(\{c,p\}=\kappa\gamma/3\), kinematical Hilbert \(L^2(SU(2),d\mu_H)\), and volume eigenstates.
- Quantizes the inverse scale factor via Thiemann’s strategy of absorbing inverse powers of volume into Poisson brackets/commutators; quantizes the Hamiltonian constraint for spatially flat isotropic models (key Euclidean term; qualitative results stated for the full constraint and positive curvature).
- Introduces internal time \(p\), expands states in eigenstates \(|n\rangle\), derives a discrete evolution difference equation with matter \(\phi\); analyzes the singularity via recurrence, semiclassical interpolation/WKB, and an explicit Euclidean solution.
Limitations
- Stated: the result crucially depends on the chosen factor ordering of the constraint (all triad components ordered to the right).
- Stated: temporal observables have not been included in the full theory; the explicit solution is only for the simplest Euclidean flat no-matter case.
- Evident from the text: the analysis is restricted to isotropic minisuperspace models and does not establish the same behavior in full quantum gravity.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
0 of 12 cited sources checked · 0 supported · 0 flagged
- Findingnone given§ Absence of Singularity in Loop Quantum Cosmology · p. 1
For large volume the correct semiclassical behavior is obtained.
“For large volume the correct semiclassical behavior is obtained.”
Quote in the paperSubstantial claim- The authors state this as an assumption: In the second step of the expansion, j (and hence V_j) is large.
- Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 1
Quantum evolution occurs in discrete time steps.
“Because of quantum geometry, the quantum evolution occurs in discrete time steps and does not break down when the volume becomes zero.”
Quote in the paperSubstantial claim - Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 1
Quantum evolution does not break down when the volume becomes zero.
“Because of quantum geometry, the quantum evolution occurs in discrete time steps and does not break down when the volume becomes zero.”
Quote in the paperSubstantial claim - Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3
All s_n for n > 0 are determined by (7) from the initial data.
“But this is not the case; in fact all s n for n > 0 are determined by (7) from the initial data.”
Quote in the paperSubstantial claim - Findingnone given§ Absence of Singularity in Loop Quantum Cosmology · p. 4
In the simplest case, the Euclidean constraint for a spatially flat model without matter, it is possible to find an explicit solution to the constraint.
“In the simplest case, the Euclidean constraint for a spatially flat model without matter, it is possible to find an explicit solution to the constraint.”
Quote in the paperNot a substantial claim
- Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3
Given initial data s_n(φ) for some negative n, we can use (7) to determine later values for higher n.
“Given initial data s n ( φ ) for some negative n , we can use (7) in order to determine later values for higher n .”
Quote in the paperSubstantial claim - Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3
All coefficients for n < -4 are determined by the initial data.
“So all coefficients for n < -4 are determined by the initial data.”
Quote in the paperSubstantial claim - Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4
Under the condition of a classical regime, there is a unique (up to a constant factor) solution.
“Under this condition one can see that there is a unique (up to a constant factor) solution”
Quote in the paperNot a substantial claim - Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4
We have a unique solution which incorporates the characterization of Euclidean space to have vanishing extrinsic curvature of its flat spatial slices.
“Therefore, we have a unique solution which incorporates the characterization of Euclidean space to have vanishing extrinsic curvature of its flat spatial slices.”
Quote in the paperSubstantial claim - Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4
It also does not avoid the zero volume state present in the classical singularity because in general the wave function is not orthogonal to states with zero volume eigenvalue.
“It also does not avoid the zero volume state present in the classical singularity because in general the wave function is not orthogonal to states with zero volume eigenvalue.”
Quote in the paperNot a substantial claim - Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3
s_0 completely drops out of the iterative evolution.
“thus s 0 completely drops out of the iterative evolution.”
Quote in the paperSubstantial claim - Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3
The approximate constraint operator obtained by interpolation matches the classical constraint in standard quantum cosmology for large scale factors.
“This is exactly what one obtains from the classical constraint H (E) = -6 c 2 √ | p | in standard quantum cosmology [17] by quantizing 3ˆ c = -iγl 2 P d / d p . In our framework, however, this is only an approximate equation valid for large scale factors.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [17] H. Kodama, Phys. Rev. D 42 , 2548 (1990). source
Not checked in the free run.
- Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 1
The cosmological singularity in isotropic minisuperspaces is naturally removed by quantum geometry.
“It is shown that the cosmological singularity in isotropic minisuperspaces is naturally removed by quantum geometry.”
Quote in the paperSubstantial claim - Premisecitation§ Absence of Singularity in Loop Quantum Cosmology · p. 2
The Hamiltonian constraints for cosmological models can be quantized with some adaptations to the symmetry.
“Following the basic steps of the quantization in the full theory [11], the Hamiltonian constraints for cosmological models can be quantized with some adaptations to the symmetry [12].”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationcitation§ Absence of Singularity in Loop Quantum Cosmology · p. 1
The evolution equation is not a differential equation in time, but a difference equation manifesting the discreteness of time.
“This implies another difference, namely that the evolution equation is not a differential equation in time, but a difference equation manifesting the discreteness of time [13].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13] source
Not checked in the free run.
- Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 1
Space-time can be extended to a branch preceding the classical singularity independently of the matter coupled to the model.
“Instead, space-time can be extended to a branch preceding the classical singularity independently of the matter coupled to the model.”
Quote in the paperSubstantial claim - Premisecitation§ Absence of Singularity in Loop Quantum Cosmology · p. 1
Quantum geometry predicts a discrete geometry because spectra of geometric operators such as area and volume are discrete.
“The results reported in this letter are obtained in the framework of quantum geometry [4] which does predict a discrete geometry because, e.g., the spectra of geometric operators as area and volume are discrete [5-7].”
Quote in the paperSubstantial claim0 of 3 sources checked - Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3
We do not need to introduce matter violating energy conditions and thereby evade the singularity theorems.
“In particular, we do not need to introduce matter violating energy conditions and thereby evade the singularity theorems.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4
This leads to a discrete structure of space and time which cannot be seen in standard quantum cosmology.
“This leads to a discrete structure of space and time which cannot be seen in standard quantum cosmology.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3
At the singularity, all corrections must add up to yield the discrete time behavior.
“The closer we come to the classical singularity, the more corrections we have to include; and at the singularity we need to know all corrections which, as we know from our nonperturbative solution, have to add up to yield the discrete time behavior.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4
It is the same mechanism which regularizes ultraviolet divergences in matter field theories and which removes the classical cosmological singularity.
“So it is the same mechanism which regularizes ultraviolet divergences in matter field theories and which removes the classical cosmological singularity.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3
Higher order terms arise from the non-locality in discrete time of the fundamental theory.
“So in these models higher order terms arise from the non-locality in discrete time of the fundamental theory.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4
In our framework, the standard quantum cosmological description arises only as a limit for large volume where the discreteness is unimportant.
“In our framework, the standard quantum osmological description arises only as a limit for large volume where the discreteness is unimportant.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4
Canonical quantum gravity is well-suited to analyze the behavior close to the classical singularity.
“We have shown in this paper that canonical quantum gravity is well-suited to analyze the behavior close to the classical singularity.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4
For small volume, quantum geometry leads to new effects which are responsible for the removal of the classical singularity.
“For small volume, quantum geometry leads to new effects which are responsible for the removal of the classical singularity.”
Quote in the paperSubstantial claim - Premisecitation§ Absence of Singularity in Loop Quantum Cosmology · p. 4
In standard quantum cosmology the constraint equation has a solution which is not unique.
“In standard quantum cosmology the constraint equation is ˆ c 2 √ | ˆ p | ξ ( c ) = 0 with a solution √ | ˆ p | ξ ( c ) = δ ( c ) which is not unique.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [17] H. Kodama, Phys. Rev. D 42 , 2548 (1990). source
Not checked in the free run.
- Study design and conductnone given§ Absence of Singularity in Loop Quantum Cosmology · p. 3
Our result crucially depends on the factor ordering of the constraint which was chosen as one of the standard possibilities ordering all triad components to the right.
“However, our result crucially depends on the factor ordering of the constraint which was chosen as one of the standard possibilities ordering all triad components to the right.”
Quote in the paperSubstantial claim
- May combine two claims. 20
- Classified as probably not a substantial claim. 2
May combine two claims.
Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 1The inverse scale factor is represented by a bounded operator even though the classical quantity diverges at the initial singularity.
“Already at the kinematical level, this is indicated by the fact that the inverse scale factor is represented by a bounded operator even though the classical quantity diverges at the initial singularity.”
- Also stated in: § Absence of Singularity in Loop Quantum Cosmology · p. 2 · plain
“we arrive at the bounded operator”
Quote in the paperSubstantial claim- Also stated in: § Absence of Singularity in Loop Quantum Cosmology · p. 2 · plain
May combine two claims.
Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 2For large j, the leading term of the inverse scale factor eigenvalues is the classical value V^{-1/3} δ_IJ, and corrections are of fourth order.
“Thus, for large j , the leading term is the classical value V -1 3 δ IJ , and the corrections (which are not necessarily isotropic) are of only the fourth order.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: In the second step of the expansion, j (and hence V_j) is large.
May combine two claims.
Findingown data§ Absence of Singularity in Loop Quantum Cosmology · p. 2Even for j=1, the eigenvalues of the inverse scale factor are very close to the classical expectation, and only the lowest three eigenvalues show large deviations.
“As Fig. 1 demonstrates, even for j = 1 are the eigenvalues very close to the classical expectation, and only the lowest three eigenvalues show large deviations.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 2The volume operator has eigenvalue zero (three-fold degenerate), but even in the corresponding eigenstates the quantization of the inverse scale factor is perfectly finite.
“Note that the volume operator has the eigenvalue zero (three-fold degenerate), but even in the corresponding eigenstates is the quantization of the inverse scale factor perfectly finite.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3We can evolve through the singularity and determine all s_n for n=0.
“So we can evolve through the singularity and determine all s n for n = 0.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3Equation (7) does not determine s_0 and instead leads to a consistency condition for the initial data.
“However, (7) does not determine s 0 and instead leads to a consistency condition for the initial data.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3All evolving solutions are orthogonal to the zero-volume eigenstate and have s_0 = 0.
“All evolving solutions are orthogonal to this state and have s 0 = 0 which already fixes the coefficient s 0 left undetermined by using the evolution equation.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4The constraint equation is of order eight with one consistency condition, so one expects seven independent solutions.
“The constraint equation is of order eight with one consistency condition as described above, so one expects seven independent solutions.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based). May combine two claims.
Interpretationnone given§ Absence of Singularity in Loop Quantum Cosmology · p. 2All qualitative results remain true for the full constraint and also for isotropic models with positive curvature.
“However, all our qualitative results remain true for the full constraint and also for isotropic models with positive curvature.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 2This may be taken as a first indication for a removal of the classical singularity, although only at the kinematical level.
“This may be taken as a first indication for a removal of the classical singularity, although only at the kinematical level.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationcitation§ Absence of Singularity in Loop Quantum Cosmology · p. 1In contrast to standard quantum cosmological models, spatial Riemannian geometry is discrete leading to a discrete volume spectrum.
“An immediate and striking consequence is that, in contrast to standard quantum cosmological models, spatial Riemannian geometry is discrete leading to a discrete volume spectrum [10].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10] source
Not checked in the free run.
Classified as probably not a substantial claim. May combine two claims.
Premisecitation§ Absence of Singularity in Loop Quantum Cosmology · p. 1Symmetric states can be defined as generalized states of quantum geometry and used for reduction to minisuperspace models.
“But symmetric states can be defined as generalized states of quantum geometry [8] which can be used for a reduction to minisuperspace models [9].”
Quote in the paperSubstantial claim0 of 2 sources checkedMay combine two claims.
Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4In contrast to earlier attempts this is not achieved by introducing matter which violates energy conditions; it is a pure quantum gravity effect.
“In contrast to earlier attempts this is not achieved by introducing matter which violates energy conditions; it is a pure quantum gravity effect.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3There is a contracting branch for negative n leading to a state of zero volume in which the universe bounces off leading to the expanding branch for positive n.
“The intuitive picture is as follows: Since for n < 0 the volume eigenvalues V ( | n |-1) / 2 decrease with increasing n , there is a contracting branch for negative n leading to a state of zero volume (in general, s ± 1 = 0 and the volume vanishes for n = ± 1 which corresponds to j = 0) in which the universe bounces off leading to the expanding branch for positive n which only can be seen in”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationcitation§ Absence of Singularity in Loop Quantum Cosmology · p. 3Discrete geometry implies that eigenvalues of p are discrete, whence time evolution is now discrete.
“Furthermore, discrete geometry imples that eigenvalues of p are discrerte, whence time evolution is now discrete (see [13] for details).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13] source
Not checked in the free run.
May combine two claims.
Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4Non-perturbative effects are solely responsible for this behavior and a purely perturbative analysis could not lead to these conclusions.
“We have also seen that non-perturbative effects are solely responsible for this behavior and a purely perturbative analysis could not lead to these conclusions.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3This conclusion holds true for any kind of matter and cosmological constant, and is a purely quantum gravitational effect.
“This conclusion holds true for any kind of matter and cosmological constant, and is a purely quantum gravitational effect.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3The complete state is determined by initial data for negative n, and so there is no singularity in isotropic loop quantum cosmology.
“We see that the complete state is determined by initial data for negative n , and so there is no singularity in isotropic loop quantum cosmology.”
Quote in the paperSubstantial claimMay combine two claims.
Premisereasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 3Classical behavior can only be present for large volume and small extrinsic curvature, i.e. if |n| is large, c is small and the wave function does not vary strongly between successive times n.
“Classical behavior can only be present for large volume and small extrinsic curvature, i.e. if | n | is large, c is small and the wave function does not vary strongly between successive times n (otherwise the state would have access to the Planck scale).”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ Absence of Singularity in Loop Quantum Cosmology · p. 4Nevertheless there is no sign of a singularity because in quantum geometry it is possible to have vanishing volume but non-diverging inverse scale factor, which in isotropic models dictates all curvature blow-ups.
“Nevertheless there is no sign of a singularity because in quantum geometry it is possible to have vanishing volume but non-diverging inverse scale factor, which in isotropic models dictates all curvature blow-ups.”
Quote in the paperSubstantial claim
The authors state this as an assumption.
In the second step of the expansion, j (and hence V_j) is large.
“where in the second step we have assumed that j -and hence V j - is large.”
Cites
Discreteness of area and volume in quantum gravity
Rovelli, C., & Smolin, L. (1995) · Nuclear Physics B · doi:10.1016/0550-3213(95)00150-q · Cited by 1,365 · see citing works
Lectures on Non-Perturbative Canonical Gravity
Ashtekar, A. (1991) · Advanced Series in Astrophysics and Cosmology · doi:10.1142/1321 · Cited by 872 · see citing works
Quantum theory of geometry: I. Area operators
Ashtekar, A., & Lewandowski, J. (1997) · Classical and Quantum Gravity · doi:10.1088/0264-9381/14/1a/006 · Cited by 852 · see citing works
Flagged first, then by citations
Research works printed but not linked: 15
- S. W. Hawking, 1973, The Large Scale Structure of Space-Time, 10.1017/CBO9780511524646 · DOI · Paper page
- B. S. DeWitt, 1967, Phys. Rev., 160, 1113, 10.1103/PhysRev.160.1113 · DOI · Paper page
- C. W. Misner, 1969, Phys. Rev., 186, 1319, 10.1103/PhysRev.186.1319 · DOI · Paper page
- C. Rovelli, 1998, Living Rev. Relativity, 1, 10.12942/lrr-1998-1 · DOI
- C. Rovelli, 1995, Nucl. Phys., B456, 753
- A. Ashtekar, 1997, Adv. Theor. Math. Phys., 1, 388, 10.4310/ATMP.1997.v1.n2.a8 · DOI · Paper page
- M. Bojowald, 2000, Classical Quantum Gravity, 17, 3009, 10.1088/0264-9381/17/15/311 · DOI · Paper page
- M. Bojowald, 2000, Classical Quantum Gravity, 17, 1489, 10.1088/0264-9381/17/6/312 · DOI · Paper page
- M. Bojowald, 2000, Classical Quantum Gravity, 17, 1509, 10.1088/0264-9381/17/6/313 · DOI · Paper page
- T. Thiemann, 1998, Classical Quantum Gravity, 15, 839, 10.1088/0264-9381/15/4/011 · DOI · Paper page
- M. Bojowald, 2001, Classical Quantum Gravity, 18, 1055, 10.1088/0264-9381/18/6/307 · DOI · Paper page
- M. Bojowald, 2001, Classical Quantum Gravity, 18, 1071, 10.1088/0264-9381/18/6/308 · DOI
- T. Thiemann, 1998, Classical Quantum Gravity, 15, 1281, 10.1088/0264-9381/15/5/012 · DOI · Paper page
- K. V. Kuchař, 1992, Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics,
- H. Kodama, 1990, Phys. Rev. D, 42, 2548, 10.1103/PhysRevD.42.2548 · DOI
Cited by
Sorted by citations
Background independent quantum gravity: a status report
Ashtekar, A., & Lewandowski, J. (2004) · Classical and Quantum Gravity · doi:10.1088/0264-9381/21/15/r01 · Cited by 1,951 · see citing works
Quantum nature of the big bang: Improved dynamics
Ashtekar, A., Pawlowski, T., & Singh, P. (2006) · Physical Review D · doi:10.1103/physrevd.74.084003 · Cited by 1,147 · see citing works
“These results are attractive and add to the growing evidence [3, 4, 5] suggesting that the quantum geometry effects of loop quantum gravity (LQG) hold a key to many long standing questions.”
ResultQuantum Nature of the Big Bang
Ashtekar, A., Pawlowski, T., & Singh, P. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.141301 · Cited by 825 · see citing works
Mathematical structure of loop quantum cosmology
Ashtekar, A., Bojowald, M., & Lewandowski, J. (2003) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.2003.v7.n2.a2 · Cited by 762 · see citing works
Self-citationQuantum nature of the big bang: An analytical and numerical investigation
Ashtekar, A., Pawlowski, T., & Singh, P. (2006) · Physical Review D · doi:10.1103/physrevd.73.124038 · Cited by 609 · see citing works
Bojowald, M. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-11 · Cited by 723 · see citing works
Self-citationQiu, T., Evslin, J., Cai, Y.-F., Li, M., & Zhang, X. (2011) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2011/10/036 · Cited by 206 · see citing works
Loop quantum cosmology of Bianchi type IX models
Wilson-Ewing, E. (2010) · Physical Review D · doi:10.1103/physrevd.82.043508 · Cited by 179 · see citing works
What happened before the big bang?
Bojowald, M. (2007) · Nature Physics · doi:10.1038/nphys654 · Cited by 104 · see citing works
Dynamical coherent states and physical solutions of quantum cosmological bounces
Bojowald, M. (2007) · Physical Review D · doi:10.1103/physrevd.75.123512 · Cited by 84 · see citing works
Self-citationde Cesare, M., Pithis, A. G. A., & Sakellariadou, M. (2016) · Physical Review D · doi:10.1103/physrevd.94.064051 · Cited by 72 · see citing works
Bojowald, M. (2008) · Physical Review Letters · doi:10.1103/physrevlett.100.221301 · Cited by 54 · see citing works
Self-citationThe loop quantum cosmology bounce as a Kasner transition
Wilson-Ewing, E. (2018) · Classical and Quantum Gravity · doi:10.1088/1361-6382/aaab8b · Cited by 48 · see citing works
String tension between de Sitter vacua and curvature corrections
Basile, I., & Platania, A. (2021) · Physical Review D · doi:10.1103/physrevd.104.l121901 · Cited by 31 · see citing works
Emerging singularities in the bouncing loop cosmology
Mielczarek, J., & Szydłowski, M. (2008) · Physical Review D · doi:10.1103/physrevd.77.124008 · Cited by 26 · see citing works
- Authors
- Bojowald, M.
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2001-06-04
- Volume
- 86
- Issue
- 23
- Pages
- 5227-5230
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2061134346
- Cited by
- 769
- References
- 3
- Citations by year
- 2026: 45 · 2025: 61 · 2024: 41 · 2023: 20 · 2022: 24 · 2021: 26 · 2020: 34 · 2019: 24 · 2018: 23 · 2017: 26 · 2016: 36 · 2015: 25 · 2014: 22 · 2013: 27 · 2012: 31
- FWCI
- 14.44
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Technische Universität Dortmund Eldorado (Technische Universität Dortmund) (submittedVersion)
- arXiv (Cornell University)
- Technische Universität Dortmund Eldorado (Technische Universität Dortmund)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- loop quantum cosmology, cosmological singularity, quantum geometry, discrete time evolution, semiclassical limit, quantum dynamics
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.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.
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
