 Spacetime singularities have been discovered which are physically much weaker than those predicted by the classical singularity theorems. Geodesics evolve through them and they only display infinities in the derivatives of their curvature invariants. So far, these singularities have appeared to require rather exotic and unphysical matter for their occurrence. Here, we show that a large class of singularities of this form can be found in a simple Friedmann cosmology containing only a scalar-field with a power-law self-interaction potential. Their existence challenges several preconceived ideas about the nature of spacetime singularities and has an impact upon the end of inflation in the early universe. This is the author accepted manuscript. The final version is available from World Scientific via http://dx.doi.org/10.1142/S021827181544012

Barrow, JD

Graham, AAH

Apollo (Cambridge)

arXiv:1505.04003v1  [gr-qc]  15 May 2015New Singularities in Unexpected PlacesJohn D. Barrow∗ and Alexander A. H. Graham†Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeWilberforce Road, Cambridge, CB3 0WA, UKMay 18, 2015AbstractSpacetime singularities have been discovered which are physically much weaker than those predicted by theclassical singularity theorems. Geodesics evolve through them and they only display infinities in the derivatives oftheir curvature invariants. So far, these singularities have appeared to require rather exotic and unphysical matter fortheir occurrence. Here we show that a large class of singularities of this form can be found in a simple Friedmanncosmology containing only a scalar-field with a power-law self-interaction potential. Their existence challengesseveral preconceived ideas about the nature of spacetime singularities and impacts upon the end of inflation in theearly universe.A striking feature of relativistic cosmology is the prediction that past and future singularities can occur. Originally,singularities were defined by the existence of incomplete geodesics, and a variety of sufficient conditions for geodesicincompleteness were established by a series of important theorems from 1965-1972 [1]. More recently, by using theEinstein equations, new types of physical singularities have been identified which can occur at finite time and areunaccompanied by geodesic incompleteness [2, 3, 4]. Many quantities, such as the density and the expansion rate,which diverge at traditional ’big bang’ singularities, remain finite whilst other physical quantities, like the pressure,diverge in finite proper time. The simplest example of what is termed a ’sudden’ singularity occurs in the zero-curvature Friedmann universe with scale factor a(t) and Hubble rate H = a˙/a, containing matter with density ρ andpressure p. The field equations are (8piG = 1 = c)3H2 = ρ, (1)ρ˙ = −3H(ρ+ p), (2)a¨ = −(ρ+ 3p)a6, (3)These equations permit there to be a finite time, ts, at which a,H, and ρ all remain finite, in accord with Eq. (1),but where p, ρ˙ and a¨ all become infinite, in accord with Eqs. (2)-(3). The key to their existence is in not assumingany functional link between p and ρ, nor any boundedness condition on p, and this freedom allows an accelerationsingularity a¨→∞ to arise at finite time as t→ ts because of a divergence in the matter pressure, p→∞. Here is anexplicit example. On the time interval 0 ≤ t ≤ ts, we can choose a solution for the scale factor a(t) of the forma(t) =(tts)q(as − 1) + 1−(1−tts)n, (4)where as ≡ a(ts), q and n are positive constants. If t → ts from below then a → as, H → Hs and ρ → ρs > 0,where as, Hs, and ρs are all finite, but p → ∞ and a¨ → −∞ whenever 1 < n < 2 and 0 < q ≤ 1. As t → 0 we∗Email: J.D.Barrow@damtp.cam.ac.uk†Email: A.A.H.Graham@damtp.cam.ac.uk1have a big bang singularity with ρ→∞ and a(t) ∝ tq but, as t→ ts is approached from below, a sudden singularityoccurs with a¨→ −∞ but a and a˙ finite.Nothing singular happens to geodesics as t → ts [5] and we always have ρ + 3p > 0 because a¨/a < 0. Thesesingularities are notable because they obey all the classical energy conditions bar the dominant energy condition, incontrast to most other exotic singularities discovered so far [6]. We can also create a divergence in any higher (N+1)storder time-derivative of the scale factor, with all lower-order derivatives staying finite, by choosing n ∈ (N,N + 1)for integer N > 2 in Eq. (4). Adding the curvature term to the Friedmann equation makes no difference to theseconclusions, and Eq. (4) is actually a leading-order approximation to part of the general solution of the Einsteinequations [7].Notice that no equation of state linking ρ and p has been assumed, and in fact the relation between them inthese solutions tends to be pathological, as P diverges at finite ρ. It is natural to ask whether this type of finite-time singularity can occur when there is a physically motivated choice of p and ρ which doesn’t allow them to beindependent variables? We shall show that the answer to this question is ’yes’.There are in fact very simple examples that can arise in the study of inflation in the early stages of the universe, orthe universe’s more recent phase of dark-energy driven acceleration. They arise for the simple case of a cosmologicalscalar field, φ, with a positive power-law self-interaction potential, V (φ), contributing a density and pressureρ =12φ˙2+ V (φ) and p = 12φ˙2− V (φ),withV (φ) = Aφn, A > 0. (5)The field equations (1)-(3) are therefore3H2 =12φ˙2+ V (φ), (6)φ¨ = −3Hφ˙−Anφn−1, (7)2H˙ = −φ˙2. (8)When n is a positive even integer they describe the classic model of large-field inflation in a potential with a singleminimum [8]. When n is a positive odd integer the universe appears to recollapse under the influence of the scalarfield (for the n = 1 case see Ref. [9]). We will be interested in the case where n > 0, with n not an integer.We first examine the case where 0 < n < 1. We choose initial conditions so that the universe is expanding initiallyand φ0 > 0, but the value of φ˙0 is unconstrained. It is not difficult to see how the system evolves in time. Since φ˙0 > 0both terms on the right-hand side of Eq. (7) are negative, so in finite time φ˙ becomes negative. Hence, in finite timethe scalar field starts to decrease, and since φ˙ continues to decrease, as the second term on the right-hand side of Eq.(7) increases as φ decreases, it will reach φ = 0 in finite time. When this happens φ˙ can also be shown to be finite andstrictly negative using Eq. (6), but from Eq. (7) we have that φ¨ → −∞ as φ → 0. From Eqs. (6)-(8), we see that Hand H˙ are both finite at this point but H¨ diverges becauseH¨ = −φ˙φ¨→ −∞ as φ→ 0. (9)This divergence is not a scalar polynomial curvature singularity [10], as both H and H˙ are finite at this point. For ourspatially-flat Friedmann universe, the Ricci scalar, R, may be written asR = 6(2H2 + H˙), (10)which is clearly finite as φ → 0. However, higher scalar derivatives of the curvature (like ∂aR∂aR or R) are notregular sinceR˙ = 6(4HH˙ + H¨)→ −∞ as φ→ 0. (11)It is easy to check also that these singularities satisfy all the classical energy conditions in the vicinity of the singularity.This is the first example of a finite-time singularity for a simple and realistic matter model.2Similar singularities can also be shown to exist when n > 1, although in this case only higher-order derivatives ofφ will diverge at the singularity. This may be seen as differentiating Eq. (7) once gives...φ − 9H2φ˙−32φ˙3− 3HV ′(φ) + V ′′(φ)φ˙ = 0. (12)For 1 < n < 2, every term except the first and last on the left-hand side is finite as φ→ 0, so...φ →∞ as φ→ 0. Thismeans that the first divergence in the scale factor occurs at fourth order in its derivatives, since...H = −φ¨2− φ˙...φ →∞ as φ→ 0. (13)Hence, R and higher derivatives of the curvature are divergent on approach to this singularity.It is not difficult to generalise these conclusions to scalar-field potentials of the form of Eq. (4) with arbitrarilylarge non-integer values of n. If k < n < k + 1, where k is a positive integer, then as φ → 0 we have φ(k+2) →(−1)k+1 ×∞, with all lower derivatives of φ finite. This implies that the first divergence of the Hubble rate occursfor the (k + 2)th derivative: H(k+2) → (−1)k+1 ×∞ as φ→ 0. But if n is an integer these singularities never occurbecause V (φ) is smooth at φ = 0. By similar arguments any potential which is not smooth at φ = 0 should createsingularities of a similar type.These singularities have a number of remarkable properties. They are remarkably weak in that they exhibit nodivergence of the curvature on approach to the singularity and all polynomial curvature invariants are finite: the onlydivergence occurs in derivatives of the curvature. Due to the weakness of these singularities the spacetime remainsgeodesically complete. To our knowledge they are the first examples of such weak singularities in a Friedmannspacetime with a realistic matter model.Evolving these spacetimes beyond φ = 0 is not always simple, as in some cases (for instance n = 12 ) the mattermodel breaks down beyond the singular point φ = 0, since the naive evolution would push φ to become strictlynegative, which makes the expansion rate become complex. However, this is only the case for some choices of n, andthere are many choices (e.g. n = 13 ) for which V (φ) is always real-valued. Numerical evidence suggests that if V (φ)is negative-definite for φ < 0 the spacetime collapses to a ’big crunch’ singularity, while if V (φ) is positive-definitefor φ < 0 no such collapse occurs. Instead, the universe passes through φ = 0 an infinite number of times. Thesespacetimes illustrate that the distinction between singular and non-singular spacetimes is by no means clear-cut, andthere are many spacetimes, sourced by realistic matter, which are geodesically complete yet possess observables whichcan evolve to finite-time singularities.The formation of these singularities is completely generic. Indeed, once can rigorously show that they formfrom any homogeneous and isotropic initial data when 0 < n < 1. They can also be shown to be stable to smallperturbations using standard perturbation theory [11], and one can adapt the arguments above to show they form inexpanding, homogeneous universes with large anisotropies. They are even stable to quantum corrections since one canconstruct examples where the divergence only occurs at an arbitrarily high order of the scale factor derivative [12, 13].This is in contrast to most other examples of weak singularities discovered so far.Finally, the model studied in this essay offers an intriguing alternative to conventional models of inflation. Ifwe choose initial conditions so that the system starts high enough up the potential, and its early evolution is potential-dominated, then inflation occurs as usual for as long as the strong-energy condition is violated. Inflation ends as φ→ 0,whereupon the universe enters the reheating phase [14]. However, the evolution will differs from standard models ofinflation when the system reaches φ = 0 deep in the reheating phase. When that happens, the spacetime developsthe weak singularity described in this essay. Since predictions for the power spectrum of the cosmic microwavebackground (CMB) are insensitive to the behaviour at reheating these models will give the same predictions for CMBobservables as conventional large-field inflation models. Indeed, our monomial potentials with n < 2 give a betterfit to current CMB data than those with large integer values of n [15]. However, these models ultimately have verydifferent dynamics from conventional reheating models. They bring inflation to a singular but timely end.References[1] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, (CUP, Cambridge, 1973)3[2] J. D. Barrow, G. J. Galloway and F. J. Tipler, Mon. Not. Roy. Astron. Soc. 223, 835-844 (1986)[3] J. D. Barrow, Class. Quantum Grav. 21, L79-L82 (2004)[4] J. D. Barrow, Class. Quantum Grav. 21, 5619-5622 (2004)[5] L. Ferna´ndez-Jambrina and R. Lazkoz, Phys. Rev. D 70, 121503(R) (2004)[6] R. R. Caldwell, Phys. Lett. B 545, 23 (2002)[7] J. D. Barrow, S. Cotsakis and A. 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