Cosmological histories from the Friedmann equation: The universe as a particle
aa r X i v : . [ phy s i c s . g e n - ph ] J un Cosmological histories from the Friedmannequation: The universe as a particle
Edvard M¨ortsell , Oskar Klein Centre, Stockholm University, AlbaNova University Center, 106 91Stockholm, Sweden Department of Physics, Stockholm University, AlbaNova University Center, 106 91Stockholm, SwedenE-mail: [email protected]
Abstract.
In this note, we discuss how possible expansion histories of the universecan be inferred in a simple way, for arbitrary energy contents. No new physical resultsare obtained, but the goal is rather to discuss an alternative way of writing theFriedmann equation in order to facilitate an intuitive understanding of the possiblesolutions; for students and researchers alike. As has been noted in passing by others,this specific form of the Friedmann equation allows us to view the universal expansionas a particle rolling along a frictionless track. Specific examples depicted include thecurrent concordance cosmological model as well as a stable static universal model.
Submitted to:
Eur. J. Phys.
PACS numbers: 01.40.-d,04.20.-q
1. Introduction
In 1917, Albert Einstein [1] and Willem de Sitter [2] both suggested that our Universecould be described in terms of the relativistic field equations proposed by Einsteintwo years earlier [3]. Being guided by the principle that inertia could be definedonly in relation to other matter sources, Einstein’s model had finite spatial extentand introduced the cosmological constant in order to achieve a static (background)distribution of matter. de Sitter on the other hand avoided the assumption of theuniverse being static since “we only have a snapshot of the world, and we cannot andmust not conclude (. . . ) that everything will always remain as at that instant whenthe picture was taken” [4]. Instead, de Sitter’s model was devoid of matter and anytest particle initially at rest with respect to an observer would not remain so but ratherrequire a positive radial velocity. At this point, the choice between Einstein’s model(with matter but no motion) and de Sitter’s model (with motion but no matter) waspurely a matter of taste. he universe as a particle ‡ but that the evolution of this velocity,i.e. the acceleration, is set by the energy content of the universe. An empty universe willexpand with constant velocity, whereas pressureless matter decelerates the expansionand a cosmological constant gives accelerated expansion. Einstein’s static solution to thefield equations is accomplished by having exact specific amounts of these counteractingenergy components. The effect on the expansion velocity from an energy componentis set by the relation between its density ρ and pressure p , or the equation of state ω ,defined for a perfect fluid as p ≡ ωc ρ. (1)The limiting case for an energy component to accelerate or decelerate the universalexpansion is p = − ρc /
3, or ω = − / ‡ Unless we have complete knowledge of both the energy densities and the curvature of the universe. he universe as a particle § .Although since long obsolete in terms of the actual confidence contours derivedfrom observed distances to Type Ia supernovae, it also differentiates between regionslabeled as expanding forever, recollapsing eventually, having no Big Bang, acceleratingand decelerating. For a student recently introduced to the Friedmann and accelerationequations, understanding how these regions come about can be challenging.In the right panel of figure 1, a small ball is pictured being released from rest atheight h = 5 at zero horizontal position, x = 0. Assuming no friction is acting on theball, it is easy to qualitatively get a full picture of the dynamics of the rolling ball. Thetotal energy of the ball will be the constant sum of the kinetic and potential energy ofthe ball where the potential energy is given by U = mgh ( x ). Here, m is the mass of theball, g the gravitational acceleration at Earth and h the (arbitrarily normalized) heightof the ball. When the potential energy decreases, the kinetic energy increases and viceversa.In the following, we will show how the different regions in the left panel can betrivially understood using the mechanical picture of a rolling ball depicted in the rightpanel.
2. Method
Einstein’s field equations with a homogeneous and isotropic metric ansatz give twodifferential equations for the time evolution of the scale factor a ( t ) of the universe. The § A discovery subsequently awarded the Nobel prize in Physics in 2011. he universe as a particle Figure 1.
Left panel:
Figure from [11], differentiating between regions expandingforever, recollapsing eventually, having no Big Bang, accelerating and decelerating, asa function of the densities in matter and a cosmological constant. Note that currentconstraints on Ω M and Ω Λ as derived from Type Ia supernovae are significantly better[14]. Right panel: A particle rolling frictionless along a track, maximally reachingthe height indicated by the dotted line. The potential energy of the ball U = mgh follows the track of the particle. When the potential energy decreases (i.e., the particlerolls down the track), the kinetic energy of the particle increases. When the potentialenergy increases (i.e., the particle rolls up the track), the kinetic energy decreases. Inthis paper, we show how the different regions of the left panel can be understood usinga similar picture. first Friedmann equation is given by H (cid:18) ˙ aa (cid:19) ≡ H = 8 πGρ − kc R a , (2)and the second, often denoted the acceleration equation, by H ¨ aa = − πG (cid:18) ρ + 3 pc (cid:19) = − πGρ ω ) . (3)Here, Λ is a cosmological constant, k = [ − , , R is the radius of curvature and ρ , p and ω denote the total energy density, pressure and equation of state, respectively.The current value of the scale factor a is normalized to unity. Dots denote derivativeswith respect to the dimensionless time coordinate τ ≡ H t . H can generally be anyconstant of dimension t − , but for models for which the Hubble parameter today is notzero, corresponds to this value, i.e. H = H ( a = 1). From the Friedmann equations, we he universe as a particle ρ + 3 HH (cid:18) ρ + pc (cid:19) = 0 , (4)or ρ = ρ a − ω ) , (5)where a subscript zero denotes the current value. Equations (4) and (5) holdindependently for all energy species as long as they do not convert into each other.So far, equations have been written in the standard textbook form.In order to make contact to the familiar picture of a particle rolling along africtionless track, we define Ω ≡ πGρ/ (3 H ), Ω Λ ≡ Λ / (3 H ) and Ω k ≡ − kc / ( H R ),and rewrite equation (2) as˙ a − Ω a = Ω k , (6)where, in a universe containing radiation, pressureless matter and a cosmologicalconstant Ω = Ω R a + Ω M a + Ω Λ . (7)Here, Ω R and Ω M are the current, dimensionless energy densities in radiation and matterrespectively. Equation 6 is the energy equation K + U = E for a particle k movingone dimensionally along coordinate a with kinetic energy K ≡ ˙ a , potential energy U ≡ − Ω a and total energy E ≡ Ω k . If ˙ a = 0 today, that is corresponding to ˙ a = 1,the total energy is given by E = Ω k = 1 − Ω . For a static universe however, thetotal energy and curvature does not need to obey this relation. Taking the derivative ofequation 6 with respect to τ , we obtain the acceleration equation in the form familiarfrom conservative systems in classical mechanics ¶ ¨ a = − dU ( a ) da . (8)In the standard model of the universe, containing radiation, pressureless matterand a cosmological constant, the potential and total energy and is given by U = − (cid:18) Ω R a + Ω M a + Ω Λ a (cid:19) , (9) E = Ω k = 1 − Ω R − Ω M − Ω Λ . (10)All we have to do to understand the allowed expansion histories of a given model is toplot the potential energy function. The expansion history will be given by the motion ofa rolling particle that can maximally reach the height Ω k . Since in general, energydensities are positive, the shape of the potential energy function has some generalproperties. For any energy density component with w > − /
3, the contribution tothe potential energy will go to minus infinity as the scale factor goes to zero and tozero as the scale factor become infinitely large. For any energy density component with k The corresponding mass for the particle is 2. Note that
K, U and E are dimensionless. ¶ The unfamiliar factor of 1 / a / he universe as a particle w < − /
3, the contribution to the potential energy will go to zero as the scale factorgoes to zero and to minus infinity as the scale factor become infinitely large.If w = − /
3, the contribution to the potential energy will be constant. Since it willaffect the total energy in the same way as the curvature term, any such energy componentwill not affect the expansion history of the universe, given the expansion velocity today + .The only exception to energy densities being positive is the cosmological constant( ω = −
1) that could have any sign. The contribution to the potential from a positivecosmological constant will go to minus infinity as the scale factor become infinitelylarge whereas the potential contribution from a negative cosmological constant will goto infinity in the same limit.
3. Dynamical models
We will first study the case of the so called concordance model, see e.g. [15]. It hasΩ M = 0 .
3, Ω Λ = 0 . a ∼ . Λ = 2, we getthe case depicted in the right panel of figure 2. The total energy, indicated by thedotted line, is E = 1 − Ω M − Ω Λ = − .
3. At a = 1, we are either in a state ofdecelerated contraction if the particle is rolling up the slope from to the left, or in astate of accelerated expansion if the particle is rolling down the slope to the right afterturning around at a ∼ /
3. In either case, the particle will never reach a = 0 and thecorresponding universe does not have a Big Bang. The region to the left with 0 < ∼ a < ∼ / a ∼ / M = 0 . Λ = 0 . M = 2 and negative total energy E = 1 − Ω M = −
1. A particle rollingin from the left will reach a = 2 at which the velocity is zero and the particle startsrolling back again. This corresponds to the case of a universe first expanding with ever + On the other hand, as noted in the section 1, it will affect cosmological observations through thegeometrical curvature that enters into distance measures. he universe as a particle Figure 2.
Left panel:
The concordance cosmological model with Ω M = 0 .
3, Ω Λ = 0 . a = 1, we are currently living in a period of accelerated expansion. Right panel:
The potential energy function (solid line) of a model with Ω M = 0 . Λ = 2 and total energy (dotted line) of E = 1 − Ω M − Ω Λ = − .
3. In this case wehave two different solutions: First, the expansion history is constrained to a > ∼ / a = 0. Second, a universe originating froma Big Bang will expand up to a ∼ / Figure 3.
Left panel:
The potential energy (solid line) for a matter dominatedoverclosed universe with Ω M = 2 and total energy E t = 1 − Ω M = − a = 2, thekinetic energy becomes zero and the particle starts rolling back towards a Big Crunch. Right panel:
The potential energy (solid line) for a matter density of Ω M = 0 .
05, anegative cosmological constant Ω Λ = −
1, and Ω X = 3 with ω X = − /
3. The universewill oscillate back and forth between a ∼ . a ∼ . decreasing velocity and then contracting with ever increasing velocity down to a BigCrunch.Including a negative cosmological constant will necessarily make the universe enter he universe as a particle Figure 4.
Left panel:
The potential (solid line) and total (dotted line) energyfor Einsteins (unstable) static universe with Ω Λ = − Ω k / M / Right panel:
The potential (solid line) and total (dotted line) energy for a stable static universedominated by domain walls with energy density Ω X = 2 and equation of state ω X = − / Λ = Ω k = − Ω X / a contracting phase at some point ∗ . An interesting possibility is shown in the rightpanel of figure 3, where in addition to a matter density of Ω M = 0 .
05 and a negativecosmological constant Ω Λ = −
1, we have added a component Ω X = 3 with ω X = − / a ∼ . a ∼ .
4. Static solutions
Einstein first introduced the cosmological constant in order to find static solutions forthe scale factor a with ˙ a = ¨ a = 0. In a universe with matter, Ω M , and a cosmologicalconstant, Ω Λ , this corresponds toΩ Λ = − Ω k M , (11)as shown in the left panel of figure 4. It is obvious that this is not a stable situation;the slightest perturbation and the particle will start rolling down the potential, eitherto left or the right depending on the nature of the perturbation.The only way to obtain a stable static solution is to make the static point a minimumin the potential. This can be accomplished, e.g. as in figure 4, by having zero matterdensity, a component Ω X = 2 with ω X = − / ∗ The exception is if there is a positive energy component with ω < − ω = − he universe as a particle Λ = Ω k = − Ω X . (12)If perturbed, the solution will oscillate around the minimum of the potential.
5. Summary and conclusion
In this paper, it is argued that a simple mechanical picture of the background expansionallows us to obtain a full, qualitative understanding of possible expansion histories giventhe energy density components of the universe. The cases depicted in sections 3 and 4nicely illustrates how this can be done trivially also for models with quite complicatedenergy contents, yielding such diverse behaviour as oscillations between expansion andcontraction as well as stable static solutions.How then can the regions in the left panel of figure 1 from [11] be triviallyunderstood, as advertised in the introduction?The division between open and closed models is determined by the total energy, E ,being positive (open) or negative (closed). Except for the concordance model in the leftpanel of figure 2, all models discussed in this paper have closed spatial geometries.Whether the universe is accelerating or decelerating is determined by if the particleis rolling upwards a slope (decelerating) or downwards (accelerating). Note that thedashed line in figure 1 denotes the division line determined by the state of the universetoday. In the mechanical picture, we can trivially determine the dynamical state atany given redshift. Also note that any model displaying acceleration at some redshiftalso would have deceleration at the same point would the direction of the particle bereversed, that is if the direction of the expansion is reversed and vice versa.Whether the universe will expand forever (e.g. the concordance model) or recollapseeventually (e.g. a matter dominated universe as in the left panel of figure 3) will bedetermined by if the potential energy is larger than the total energy at some redshift.Finally, the universe will not have a Big Bang if the rolling particle is confined toa region not including a = 0, e.g. as in the solution with a > ∼ / Acknowledgments
I acknowledge support for this study from the Swedish Research Council. Also, I amgrateful to Jonas Enander and the referees for careful readings of the manuscript anduseful suggestions. he universe as a particle Figure A1.
The kinetic energy of the universal expansion as a function of the scalefactor. We assume that the Universe was dominated by radiation at a < − andthat inflation took place at 10 − < a < − . We live at log a = 0. Note that thekinetic energy today can be comparable to that at the onset of inflation. Appendix A. Inflation
In the inflationary scenario [16–19], at some point in the early Universe, the energydensity was dominated by a component, Ω i behaving similarly to the cosmologicalconstant, say between scale factors a b and a e . During this epoch, the potential energyfunction was given by U ∼ − Ω i a , (A.1) E ∼ , (A.2)after which some mechanism converted the energy density in Ω i to radiation and matter.We need approximately 60 e-foldings of inflation in order to successfully obtain the initialconditions for the subsequent universal evolution, for which the theory was devised. Thescale factor will then increase by a factor of ∼ during inflation, the potential energywill decrease by a factor of ∼ and the kinetic energy will increase by the same factor.Assuming that inflation took place at 10 − < a < − and that the Universe beforethat was dominated by a radiation like component ♯ , the kinetic energy of the expansionas a function of the scale factor is depicted in figure A1. Since the observable Universe(given by the particle horizon), today has a radius of ∼ · m, at the end of inflation,the corresponding radius was ∼
40 m, and at the start of inflation ∼ · − m, or ∼ · l P . In this picture, at a ∼ − , we reach the Planck density and we thereforedo not extend the plot to smaller values of the scale factor. ♯ This assumption of course being completely hypothetical.
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