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Cosmology With Negative Potentials

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Gary Felder, Andrei Frolov, Lev Kofman, Andrei Linde

High Energy Physics - Theory · Submitted 4 Feb 2002 · 48 references

Pith's verdict: The no-AdS result and the Section 6 turning-point calculation are solid and worth citing; the cyclic-scenario critique is the speculative half and leans on order-of-magnitude estimates the authors themselves flag.

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arXiv IDhep-th/0202017 (version 2)
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CitationAndrei Frolov, Andrei Linde, Gary Felder, Lev Kofman. "Cosmology With Negative Potentials." arXiv:hep-th/0202017 (2002). https://pith.science/paper/O3J5TMUA
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The paper

The paper claims

A universe whose potential dips below zero cannot settle into anti-de Sitter space; it turns around and collapses, and that fact constrains the cyclic universe scenario.

The reading

The paper asks what happens to a Friedmann universe whose scalar-field potential has a minimum below zero, as assumed in the cyclic and ekpyrotic scenarios. The naive expectation is that such a universe relaxes into anti-de Sitter space dominated by negative vacuum energy. The authors show this cannot happen for a flat universe that has inflated: the constraint $6H^2-2V_0 = 6H^2 - 2V_0$ written in the form $6H^2-2V_0 = 6H^2 - 2V_0$ becomes, at $V_0<0$, a one-sheeted hyperboloid in the space of $(H,dot varphi, varphi)$, joining the expanding and contracting branches, while $H^2=rho/3$ admits no solution with $rho<0$. Instead the universe reaches a turning point of zero total energy density, reverses, and falls into a singularity whose properties are nearly independent of $V(varphi)$ because kinetic energy always outruns potential energy there. The paper then uses these phase portraits to audit the cyclic universe, finding that a successful bounce must supply a specific, small energy increment and that further energy losses from particle production make the required jump harder still; the authors' recommended repair is to put ordinary inflation after the singularity and let it generate the density perturbations.

What carries the argument

The load-bearing device is the three-dimensional phase portrait in $(varphi, dotvarphi, H)$, drawn with a Poincare projection so trajectories at infinity are visible, combined with the constraint equation that fixes a two-dimensional surface for a flat universe. That surface's topology, two sheets versus one, is what encodes whether expansion can turn into contraction. The same portraits supply the audit of the cyclic model: separatrices running from saddle points to attractors divide the trajectories into disconnected classes, and only trajectories lying sufficiently far to one side of the repulsive (deflationary) separatrix can re-enter the inflationary regime after a bounce. The authors also use a patching argument, matching the quadratic-potential motion between $pmvarphi_0$ onto the kinetic-dominated solution $a propto t^{1/3}$, $varphi - varphi_p = sqrt{2/3},ln(sqrt{3},t)$ outside it, to translate the bounce energy budget into a condition on the radiation density at the Planck time.

Extended reading notes

Core claim

The central claim is topological. For $V(varphi)=tfrac{1}{2}m^2varphi^2+V_0$ with $V_0>0$, the flat-universe constraint surface is a hyperboloid of two sheets, so an expanding flat universe can never begin contracting. With $V_0<0$ the same equation describes a hyperboloid of one sheet: expansion and contraction are connected through a throat, and every trajectory that starts at large $H>0$ ends at large $|H|$ with $H<0$. Because $H^2=rho/3$ has no solution with negative total density, the universe cannot sit in an AdS phase; it reaches a turning point where $rho=0$, then collapses. The authors compute the turning point analytically for a quadratic potential, finding $t_c approx m^{-1}(12Delta V/(m^2varphi_0^4))^{1/6}$ and $varphi_c approx varphi_0 - (3Delta V/(2m^2varphi_0))^{1/3}$, and they show that during a half-oscillation the field arriving at $-varphi_0$ acquires kinetic energy $dotvarphi^2/2 approx (3pi^2/16),m^2varphi_0^4$ whether the universe was expanding or collapsing, the two cases differing only by the tiny term $deltarho = pisqrt{3},Delta V,mvarphi_0^2$. Section 7 extends the conclusion to inverted potentials and to symmetry-breaking potentials with a negative vacuum energy: minima below zero are as dangerous as potentials unbounded from below, and once contraction begins the null energy condition forbids a return to expansion.

Load-bearing premise

The critique of the cyclic scenario depends on order-of-magnitude estimates that particle production at the bounce and the rapid change of the field's mass near the potential minimum inject energy of order $m^4$, and that gravitational production gives density of order one at the Planck time; if the real production rate is much smaller, the required energy jump becomes easier to achieve.

Editorial extensions

If this is right

  • A flat universe that inflated and whose potential has a minimum at $V<0$ has a finite lifetime, turning around at $t sim 2sqrt{3}/sqrt{|V_0|}$, so an observed age of order $10^{10}$ years forces $|V_0| lesssim 10^{-120}$ in Planck units unless the potential is unusually flat.
  • Potentials with a stable minimum below zero are, from the standpoint of cosmic destiny, no safer than potentials unbounded from below: both end in a crunch, and the equation of state near the singularity is $p=rho$ for any power-law potential.
  • For the cyclic scenario to close, the kinetic energy of the field at $varphi=-varphi_0$ after the bounce must exceed $Deltarho_+$ by an amount of order $m^4$ generated by the nonadiabatic change of the field mass, and the radiation density at the Planck time must satisfy $rho_{rp} lesssim (Deltarho_+)^{1/3}$.
  • Realizing the cyclic regime as published requires potentials of super-Planckian depth $|V|<0$ with $|V(varphi)|>1$ and a field mass above the Planck mass, together with percent-level fine-tuning of the potential's shape parameters.
  • Adding an ordinary inflationary stage after the singularity removes the need to track perturbations through the bounce, at the cost of making the pre-bounce history unobservable except through whatever relics survive inflation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The topology argument is more general than the quadratic example: any model in which the total energy density can pass through zero in a flat universe will show the same one-sheet structure, so the conclusion should transfer to multi-field and brane-motivated effective potentials without redoing the analysis.
  • If the recommended repair is adopted, the resulting cosmology becomes observationally close to standard chaotic inflation; the only surviving handle on the preceding cycle would be a stochastic background of gravitational waves or a nonstandard pre-inflationary equation of state, both of which are in principle searchable.
  • The turning-point formula suggests a separate test: a universe currently accelerating with a potential that will eventually go negative should show a specific relation between the present dark-energy density and the time remaining before turnaround, which is a statement about $V_0$ alone and does not depend on the shape of the potential.
  • The paper's own energy-budget arithmetic implies that the bounce, if it happens at all, is a strongly dissipative event; that suggests framing the cyclic scenario as a claim about entropy production at the bounce rather than about a reversible rewind of the scalar field.
Original abstract

We investigate cosmological evolution in models where the effective potential V(\phi) may become negative for some values of the field \phi. Phase portraits of such theories in space of variables (\phi,\dot\phi,H) have several qualitatively new features as compared with phase portraits in the theories with V(\phi) > 0. Cosmological evolution in models with potentials with a "stable" minimum at V(\phi)<0 is similar in some respects to the evolution in models with potentials unbounded from below. Instead of reaching an AdS regime dominated by the negative vacuum energy, the universe reaches a turning point where its energy density vanishes, and then it contracts to a singularity with properties that are practically independent of V(\phi). We apply our methods to investigation of the recently proposed cyclic universe scenario. We show that in addition to the singularity problem there are other problems that need to be resolved in order to realize a cyclic regime in this scenario. We propose several modifications of this scenario and conclude that the best way to improve it is to add a usual stage of inflation after the singularity and use that inflationary stage to generate perturbations in the standard way.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.33 · human participants 0.01 · controlled comparison 0.19 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.76 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We investigate cosmological evolution in models where the effective potential V(\phi) may become negative for some values of the field \phi. Phase portraits of such theories in space of variables (\phi,\dot\phi,H) have several qualitatively new features as compared with phase portraits in the theories with V(\phi) > 0. Cosmological evolution in models with potentials with a "stable" minimum at V(\phi)<0 is similar in some respects to the evolution in models with potentials unbounded from below. Instead of reaching an AdS regime dominated by the negative vacuum energy, the universe reaches a turning point where its energy density vanishes, and then it contracts to a singularity with properties that are practically independent of V(\phi). We apply our methods to investigation of the recently proposed cyclic universe scenario. We show that in addition to the singularity problem there are other problems that need to be resolved in order to realize a cyclic regime in this scenario. We propose several modifications of this scenario and conclude that the best way to improve it is to add a usual stage of inflation after the singularity and use that inflationary stage to generate perturbations in the standard way.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate cosmological evolution in models where the effective potential V(\phi) may become negative for some values of the field \phi. contribution 0.95 0.86 0.03 hedged possibility 0.79
s1 Phase portraits of such theories in space of variables (\phi,\dot\phi,H) have several qualitatively new features as compared with phase portraits in the theories with V(\phi) > 0. result 0.84 0.66 0.05 asserted 1.00
s2 Cosmological evolution in models with potentials with a "stable" minimum at V(\phi)<0 is similar in some respects to the evolution in models with potentials unbounded from below. result 0.86 0.72 0.09 asserted 0.99
s3 Instead of reaching an AdS regime dominated by the negative vacuum energy, the universe reaches a turning point where its energy density vanishes, and then it contracts to a singularity with properties that are practically independent of V(\phi). result 1.00 0.86 0.54 asserted 1.00
s4 We apply our methods to investigation of the recently proposed cyclic universe scenario. contribution 0.92 0.83 0.03 asserted 0.93
s5 We show that in addition to the singularity problem there are other problems that need to be resolved in order to realize a cyclic regime in this scenario. result 0.83 0.88 0.08 asserted 0.99
s6 We propose several modifications of this scenario and conclude that the best way to improve it is to add a usual stage of inflation after the singularity and use that inflationary stage to generate perturbations in the standard way. contribution 0.51 0.92 0.70 asserted 0.86

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.90.

What was examined, and what was not

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Evolution of the scalar field and the scale factor in the model V (φ) = m2 2 φ 2 + V0 with V0 > 0. In the beginning we have a stage of inflation with the field φ linearly decreasing at φ > 1. At this stage the equation of state is p ≈ −ρ. Then the field enters a stage of oscillations with a gradually decreasing amplitude of the field; p ≪ ρ. When the energy of the oscillations becomes smaller than V0, the universe enters a second stage of inflation. It is amazing that this simple model with V0 > 0 can describe not only inflation in the early universe, but also the present stage of…
Figure 2. Evolution of the scalar field and the scale factor in the model V (φ) = m2 2 φ 2 + V0 with V0 < 0. In the beginning we have a stage of inflation with the field φ linearly decreasing at φ > 1. At this stage the equation of state is p ≈ −ρ. Then the field enters a stage of oscillations with a gradually decreasing amplitude of the field; p ≪ ρ. When the energy of the oscillations becomes equal to |V0|, the universe stops expanding and begins to contract. At this stage the amplitude of oscillations grows. When it becomes greater than O(1), the field stops oscillating, the energy density
Figure 3. Phase portrait for the theory V (φ) = 1 2m2φ 2 + V0 with V0 > 0 in rescaled coordinates (φ, φ, H˙ ). The branches describing stages of expansion and contraction (upper and lower parts of the hyperboloid) are disconnected. matter, p ≪ ρ. During the oscillations the scalar field decreases as φ(t) ≈ 1 4N sin mt , (4.6) where N is the number of oscillations, see Eq. (2.21). Although particle production is not included in these phase portraits, this evolution will typically end with the scalar field decaying into other forms of matter, thus finishing the evolution in the fourth regime,
Figure 4. Projection of the upper branch of the full phase portrait for the theory V (φ) = 1 2m2φ 2 + V0 with V0 > 0 in rescaled coordinates (φ, φ˙). of the hyperboloid, respectively [23]. For an open universe nearly all trajectories would asymptotically approach the separatrices on the flat universe hypersurface. This tendency reflects the fact that for most initial conditions inflation will occur and drive the universe towards flatness. Once this has occurred the trajectories spiral in towards the focus at the bottom of the hyperboloid. For a closed universe there are also many trajectories…
Figure 5. Phase portrait for the theory V (φ) = 1 2m2φ 2 without Poincar´e mapping. The blue (thick) lines show trajectories describing the universe without radiation. The scalar field has half Planck density at the beginning of the simulations. The red (thin) lines show trajectories where an equal amount of energy in radiation was added to the system. As we see, in the presence of radiation the velocity of the scalar field rapidly decreases, which usually leads to the onset of inflation. slightly diminished by the presence of radiation, but the probability of inflation is nearly unity. 5.…
Figure 6. Phase portrait for the theory V (φ) = 1 2m2φ 2 + V0 for V0 < 0. The branches describing stages of expansion and contraction (upper and lower parts of the hyperboloid) are connected by a throat. To show a 2d projection of the flat universe hypersurface for this model, we have to plot both the expanding and contracting branches, as depicted on
Figure 7. Left: (φ, φ˙) projection of the H > 0 branch. Right: (φ, φ˙) projection of the H < 0 branch. Trajectories from the left panel continue on the right panel. Aside from this “wormhole” connecting the expanding and contracting branches this phase portrait looks a lot like the one for V0 > 0 shown in
Figure 8. A different version of the phase portrait for the theory V (φ) = 1 2m2φ 2 + V0 for V0 < 0. We begin with the trajectories evenly distributed with respect to the initial values of φ in the early universe (upper part of the hyperboloid) and see what happens to them in the lower part. These trajectories are concentrated near the red separatrices and repulsed from the green ones. the initial value of the field φ at the Planck time and follows their evolution from the region with H > 0 to the region with H < 0. If we do so, the phase portrait shown in
Figure 9. As in the previous figure, we begin with the trajectories evenly distributed with respect to the initial values of φ in the early universe. However, now we show a 2d projection of these trajectories, without Poincar´e mapping. Left: (φ, φ˙) projection of the H > 0 branch. Right: (φ, φ˙) projection of the H < 0 branch. Trajectories from the left panel continue on the right panel. An important (and obvious) feature of the 3d phase portraits
Figure 10. Evolution of the scalar field and scale factor in the model V (φ) = V0 − m2φ 2 2 . Another example is the standard potential used for the description of spontaneous symmetry breaking, with the addition of a negative cosmological constant V0 < 0: V (φ) = λ 4 (φ 2 − v 2 ) 2 + V0 = − 1 2 m2φ 2 + m2 4v 2 φ 4 + 1 4 m2 v 2 + V0 . (7.2) Here m2 ≡ λv2 and the point φ = v corresponds to the minimum of V (φ) with symmetry breaking. The potential V (φ) becomes equal to V0 < 0 in the minimum of V (φ) at φ = v. As we see in
Figure 11. Evolution of the scalar field and the scale factor in the model V (φ) = λ 4 (φ 2 −v 2 ) 2 +V0, with V0 < 0. When the field accelerates enough it enters the regime φ˙2 ≫ V (φ) and continues growing with a speed practically independent of V (φ): φ ∼ ln t, φ˙ ∼ t −1 , see Eq. (2.18). So for all potentials V (φ) growing at large φ no faster than some power of φ one has φ˙2/2 growing much faster than V (φ) (a power law singularity versus a logarithmic singularity). This means that one can indeed neglect V (φ) in the investigation of the singularity, virtually independently of the ch
Figure 12. Scalar field potential in the cyclic scenario. The minimum of the potential may occur at any value of φ; in this section for simplicity we will assume that it occurs at φ = 0; we will consider a more general situation later. The effective potential of the scalar field in the cyclic scenario has the shape shown in
Figure 13. The 3d phase portrait for the cyclic scenario. All trajectories (blue lines) begin at H > 0 and end in a singularity at H < 0. right panel). This might seem surprising because this line is a solution of the equations of motion, so there must be other solutions nearby. The reason is that the deflationary universe regime described by this line is a strong repulsor, just opposite to the fact that the inflationary red line at H > 0 (the right separatrix at the left panel) is a strong attractor. As a result, the density of trajectories near the green line at H < 0 is very small; that is…
Figure 14. The 2d phase portrait for the cyclic scenario. All trajectories begin at the bounding box of the left panel (H > 0) and end at the bounding box of the right panel (H < 0). should jump sufficiently far away to the right of the green separatrix. As we already men￾tioned, Ref. [10] assumes that this jump may occur due to an increase in the energy of the scalar field bouncing back from the singularity. This increase in energy is supposed to happen due to particle production. Only if this jump is sufficiently large can these trajec￾tories reach the red inflationary separatrix going from…
Figure 15. An example of cyclic scenario potential used in Ref. [10]. Let us try to understand the origin of the parameters c = 10, γ ≈ 1/8 used in [10]. According to [14], the amplitude of density perturbations in this scenario in the limit c ≫ 1 can be estimated as δρ ρ ∼ 10−5p −Vj ξ 4 , (9.18) – 39 –
Figure 16. Effective mass squared m2 = V ′′ of the scalar field in the vicinity of the minimum of V (φ) in the cyclic scenario
Figure 17. Symmetric scalar field potential in the new cyclic scenario. At large values of |φ| one has V (φ) ≈ V0 ∼ 10−120 and there is a minimum at φ = 0. In the beginning, the scalar field is large and positive and it slowly moves towards the minimum. When it falls to the minimum the universe begins to contract and the field is rapidly accelerated towards the singularity at φ = −∞. As we already mentioned, the structure of the singularity is not sensitive to the existence of the potential, especially if it is as small as V0 ∼ 10−120. Suppose in the vicinity of its minimum the potential is…
Figure 18. Scalar field potential in the cyclic scenario incorporating a stage of chaotic inflation. Inflationary perturbations are generated and the large-scale structure of the universe is produced at φ < 0. Gradually the density of ordinary matter decreases, and the energy density of the universe becomes determined by V (φ) ≈ V0. The universe enters a stage of low energy inflation (quintessence), which may result in a regime of self-reproduction if V (φ) is flat enough. In those exponentially large domains of the universe where the field eventually falls down to the minimum of V (φ), it…
Figure 19. Scalar field potential in the cyclic scenario incorporating a stage of chaotic inflation. Inflationary perturbations are generated and the large-scale structure of the universe is produced both at φ < 0 and at φ > 0. use the mechanism of instant preheating [47, 48]. In order to combine the stage of chaotic inflation and the stage of low-scale inflation (quintessence) the potential must be rather complicated. To avoid this complication one may need to consider two-field models of the type of hybrid inflation. φ V
Figure 20. The scalar field potential that appears after the step-by-step simplification of the cyclic scenario. The main problem of this model is that one still must assume that somehow the universe can go through the singularity. But now this assumption is no longer required for the success of the scenario since the large scale structure of the universe in this scenario does not depend on processes near the singularity. This allows us to remove the remaining epicycles of this model. Indeed, the main source of all the problems in this model is the existence of the minimum of the effective…

Figures reproduced from arXiv:hep-th/0202017 by the authors.

References
How the literature uses it

42 arXiv papers cite this in 70 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 40 sentences without a confident label

Contests it 2 papers
  • When this scenario was analyzed using the particular potential given in [ref] and taking into account the effect of particle production in the early universe, a very different cosmological regime was found [ref] .

    The New Ekpyrotic Ghost · Introduction: Inflation versus Ekpyrosis · contest 0.73

  • When we analyzed this scenario using the particular potential given in [ref] , and took into account the effect of particle production in the early universe, we found a very different cosmological regime [ref] .

    Inflationary Cosmology · Ekpyrotic/cyclic scenario · contest 0.80

Builds on it 1 paper
  • We will also see that the DBI kinetic term of brane inflation models leads to an extension of the standard slow roll inflationary attractor solution [ref] to the DBI inflationary regime where the DBI speed limit is saturated.

    Brane Inflation is Attractive · Introduction · 0.97

Uses its method or tool 5 papers
Compares with it 2 papers
Cites it as support 2 papers
Cites it as background 12 papers
  • The problem of inflationary initial conditions has been studied before [ref] , with the result that large-field inflationary models, such as chaotic inflation [ref] , are typically stable to changes in initial conditions, while small-field models, such as new inflation [ref] , are sensitive to initial conditions [ref] (for more information on the distinction between large- and small-field models, see [ref] ).

    Brane Inflation is Attractive · Introduction · 0.91

  • Geodesically incomplete singularities include the Big-Bang [ref] , the Big-Rip [ref] where the scale factor diverges at a finite time due to infinite repulsive forces of phantom dark energy, the Little-Rip [ref] and the Pseudo-Rip [ref] singularities where the scale factor diverges at a infinite time and the Big-Crunch [ref] where the scale factor vanishes due to the strong attractive gravity of future envolved dark energy, as in quintessence models with negative potential.

    Sudden Future Singularities in Quintessence and Scalar-Tensor Quintessence Models · Introduction · 0.96

  • Such diagrams have found prolific use in the study of the late universe dominated by dark energy [ref] , and have also been used in the study of inflationary scalar fields from early on [ref] , but only occasionally.

    Global Portraits of Nonminimal Inflation: Metric and Palatini · Introduction · 0.80

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