Interacting Cosmological Fluids and the Coincidence Problem
aa r X i v : . [ g r- q c ] N ov Interacting Cosmological Fluids and theCoincidence Problem
Sean Z.W. Lip ∗ DAMTP, Centre for Mathematical Sciences,Cambridge University, Wilberforce Road,Cambridge CB3 0WA, UK
Abstract
We examine the evolution of a universe comprising two interactingfluids, which interact via a term proportional to the product of theirdensities. In the case of two matter fluids it is shown that the ratio ofthe densities tends to a constant after an initial cooling-off period. Wethen obtain a complete solution for the cosmological constant ( w = − p = wρ , where w is a constant, and show that periodicsolutions can occur if w < −
1. We further demonstrate that the ratio ofthe dark matter to dark energy densities is confined to a bounded interval,and that this ratio can be O (1) at infinitely many times in the history ofthe universe, thus solving the coincidence problem.PACS: 98.80.-k, 95.36.+x I. INTRODUCTION
Observations based on Type Ia supernovae and the cosmic microwave back-ground suggest that the universe consists mainly of a non-gravitating type ofmatter called ‘dark energy’, as well as a substantial amount of gravitating non-baryonic ‘dark matter’ [1–5]. Surprisingly, although the density of dark matteris expected to decrease at a faster rate than the density of dark energy through-out the history of the universe, their magnitudes are comparable today. This isknown as the ‘coincidence problem’, and various attempts at its solution includethe use of tracker fields [6] and oscillating dark energy models [7].We will discuss a third possibility that has gained some attention recently,which is that dark energy and dark matter interact via an additional couplingterm in the fluid equations [8–69]. The interaction is usually assumed to takethe form
AHρ m + BHρ Λ , where H is the Hubble parameter, ρ m and ρ Λ are thedark matter and dark energy densities respectively, and A, B are dimensionlessconstants. However, in this paper we will instead consider an interaction of the ∗ [email protected] γρ m ρ Λ , where γ is a (non-dimensionless) constant [8]. Such an interactionis natural and physically viable, since we would expect the interaction rate tovanish if one of the densities is zero, and to increase with each of the densi-ties. This form of interaction has also been used to model systems ranging fromstandard two-body chemical reactions to predator-prey systems in biology. Sta-tistical fits to observed data suggest that this form of coupling helps to alleviatethe coincidence problem [9], and it has been shown that among holographic darkenergy models with an interaction term γρ αm ρ β Λ (where α, β > α = β = 1 gives the best fit to observations [10].We propose to investigate this model in more detail. Our analysis will con-sider the interaction of a dust fluid and a second fluid with an equation of state p = wρ , where w is a constant. If the second fluid is also dust ( w = 0), we showthat the ratio of the densities tends to a constant after an initial cooling-off pe-riod. Thus, matter fluids coupled in this way can be considered, at late times,to evolve as a single non-self-interacting matter fluid. We then go on to obtaina complete solution of the system for conventional ( p = − ρ ) dark energy, butshow that such a model cannot address the coincidence problem. Finally, weexhibit the various scenarios that can arise for other forms of dark energy, anddemonstrate that if w < − II. SETTING THE STAGE
We work in a spatially flat Friedmann-Robertson-Walker universe, and as-sume that it contains the following perfect fluids: dark matter, with density ρ m , pressure p m and equation of state p m = 0; ‘dark energy’, with density ρ Λ ,pressure p Λ and an equation of state of the form p Λ = wρ Λ c , where w is a con-stant (in the case w = −
1, we have a standard cosmological constant); baryonicmatter, with density ρ b , pressure p b and equation of state p b = 0; and radiation,with density ρ r , pressure p r and equation of state p r = ρ r c / ρ m = − Hρ m + γρ m ρ Λ (1)˙ ρ Λ = − w ) Hρ Λ − γρ m ρ Λ (2)˙ ρ b = − Hρ b (3)˙ ρ r = − Hρ r (4)3 H = 8 πG ( ρ m + ρ Λ + ρ b + ρ r ) (5)where H = ˙ a/a is the Hubble parameter, and a ( t ) is the scale-factor. Here, anoverdot indicates a derivative with respect to (cosmic) time t . Note that theconstant γ is not dimensionless; it has dimensions of volume per unit mass perunit time. If γ >
0, energy is transferred from dark energy to dark matter; theopposite occurs if γ <
0. In the case γ = 0, the two fluids do not interact. In [11],it is pointed out that γ < γ > ρ b and ρ r decrease as a − and a − respectively. In the next three sections of this paper, we will restrict ourselvesto a late-time analysis, where ρ b and ρ r are small and can be neglected. Thisleads to the reduced set of equations:˙ ρ m = − Hρ m + γρ m ρ Λ (6)˙ ρ Λ = − w ) Hρ Λ − γρ m ρ Λ (7)3 H = 8 πG ( ρ m + ρ Λ ) (8)For simplicity, we will also adopt units in which 8 πG = 1 and c = 1 in thesesections, which contain a qualitative, mathematical analysis of the system. Wewill return to the full set of equations in Section VI when attempting to constrainthe parameters of the system using observations.Differentiating (8) with respect to t and substituting for ˙ ρ m and ˙ ρ Λ givesthe auxiliary equation 2 ˙ H = − ρ m − (1 + w ) ρ Λ (9)which will be useful later. Note that this equation is independent of the couplingparameter γ ; this is a consequence of energy conservation.It will be of interest to consider the acceleration of the universe, which isgiven by the relation ¨ a = a ( ˙ H + H ). From (8) and (9), we have¨ a = − a ρ m + (1 + 3 w ) ρ Λ ) . (10)Hence, since we assume that a ( t ), ρ m and ρ Λ are always non-negative, it isnecessary (but not sufficient) that w < − in order for the universe to acceleratetoday. III. A SIMPLE CASE: TWO DUST FLUIDS
We start by considering a model in which our fluids are both dust. Theequations of the system are: ˙ ρ = − Hρ + γρ ρ (11)˙ ρ = − γρ ρ − Hρ (12)3 H = ρ + ρ (13)Define u ≡ ρ + ρ . Then ˙ u = −√ u / , and integrating gives √ u = √ t + C ),where C is an arbitrary constant. Therefore, u = t + C ) , and by shifting thetime coordinate ( t → t − C ) we can set C = 0. Hence H = u √ = t (as wouldbe expected for a matter-dominated universe).3ubstituting this result into (11) and (13) yields the equation˙ ρ = − t ρ + γρ (cid:18) t − ρ (cid:19) (14)which integrates to ρ = t (4 Ae γ/ t +3) , ρ = t (4 Be − γ/ t +3) . The condition ρ + ρ = t leads to the requirement that 16 AB = 9, which leads to thegeneral solution ρ = 4 t (4 Ae γ/ t + 3) , ρ = 4 t (cid:0) A e − γ/ t + 3 (cid:1) . The ratio of these densities is therefore ρ ρ = 34 A e − γ t and, as t → ∞ , this tends exponentially quickly to the constant 3 / A , with theindividual densities evolving as ρ i ∼ t − .This suggests that even if there were several types of ‘dust’ in the universemutually interacting in this manner, we could approximate the evolution of theirdensities by treating them as non-interacting after a short initial period. (Inother words, we might write ρ i ≈ C i t − for t > t ∗ , where the relative magnitudesof the C i are determined from a relatively short initial evolution up to time t ∗ .) IV. ‘CONVENTIONAL’ DARK ENERGY ( w = − ) We now consider a universe containing a dark matter fluid and a dark energyfluid, in which the dark energy behaves like a cosmological constant (that is, itsatisfies the equation of state p Λ = − ρ Λ ). The dynamical equations simplify to:˙ ρ m = − Hρ m + γρ m ρ Λ (15)˙ ρ Λ = − γρ m ρ Λ (16) H = 13 ( ρ m + ρ Λ ) (17)2 ˙ H = − ρ m (18)In the non-interacting case γ = 0, it is easy to see that the dark energy den-sity ρ Λ stays constant, and ˙ ρ m < ρ m ∝ a − ∝ t − ), so the darkmatter becomes more diffuse. The ratio ρ m ρ Λ decreases as t − , and so the coinci-dence problem remains. Since ρ m →
0, we can see from (10) that the universeaccelerates at an increasing rate as time progresses.Now suppose γ = 0. First, observe that H = 0 implies ρ m = ρ Λ = 0, so werestrict attention to positive H . This implies an eternally expanding universe.From (16) and (18) it follows that ρ Λ = Ae γH , where A is a positive constant.Given γ , we can find A based on the values of ρ Λ and H observed today. Using(17), we can then parametrize the whole system in terms of H : ρ m = 3 H − Ae γH (19) ρ Λ = Ae γH (20)4rom (20) we can see that ρ Λ never vanishes, so dark energy is a perpetual com-ponent of the universe. Now, using (18), we can obtain a first-order separabledifferential equation for H , and its solution is t = Z dHAe γH − H . Unfortunately, the integral on the right hand side cannot be integrated usinganalytic methods.We now examine the acceleration of the universe in the two cases dependingon the sign of γ . We can see that ˙ H + H = − H + A e γH , so the universeaccelerates iff Ae γH > H . In the case γ <
0, we have ˙ ρ m < ρ Λ > ρ Λ − ρ m , the acceleration occursas a faster rate as time proceeds, just as in the non-interacting case.In the case γ >
0, ˙ ρ Λ < ρ m is more com-plicated. A representative example with γ = 1 is given in Figure 1; for othervalues of γ , the qualitative behaviour is similar (in the sense that ρ m generallyincreases to a maximum and then decreases again). The straight line in thediagram corresponds to 2 ρ Λ = ρ m ; in the region of phase space above this linethe universe is accelerating, and in the region below this line it is decelerating.It appears that in this model the universe can only go through at most a singledecelerating phase. 5 Ρ m Ρ L Figure 1: Phase-plane diagram showing the evolution of the dark matterdensity ρ m and the dark energy density ρ Λ , for an interaction term ρ m ρ Λ ( γ = 1). In this case, the dark energy is assumed to behave like acosmological constant, and obeys the equation of state p = − ρ . Thestraight line represents universes with zero acceleration. The region ofphase space above the line corresponds to an accelerating universe, andthe region below corresponds to a decelerating universe. Note that thisfigure (and the next four) are intended only to display the qualitativebehaviour of the model, so the units on the axes are arbitrary.From the expressions above we can see that r ≡ ρ m ρ Λ = H Ae γH −
1, and thisquantity lies in the interval [ − , Ae γ − y = x Ae γx − Aγ < e − .In order for this model to address the coincidence problem, we require that r ∼ O (1), and thus H and e γH should be of comparable magnitude. We argue,however, that this cannot happen.In the case γ <
0, it is easy to see that the ratio r always decreases withtime, because ˙ ρ m ρ Λ > γ > ρ m > c > H will decrease at a rate of at least c per unit time (by (18)), and will reach 0 eventually – but this corresponds to ρ m = 0, which is a contradiction. Therefore ρ m must approach 0 asymptoticallyat late times, so that the quantity 3 H − Ae γH becomes arbitrarily small, andcomparable to ρ Λ = Ae γH (which must therefore also be arbitrarily small).6herefore 3 H , and therefore H , becomes arbitrarily small. This is impossiblebecause 3 H − Ae γH → − A as H → V. GENERAL DARK ENERGY
Now suppose that the dark energy has equation of state p = wρ , where w is a constant, and define K ≡ w + 1. The previous case corresponds to K = 0,and the case in Section 3 (two dust fluids) can be regarded as the case K = 1.The general interacting case is then described by the following equations:˙ ρ m = − Hρ m + γρ m ρ Λ (21)˙ ρ Λ = − γρ m ρ Λ − KHρ Λ (22)3 H = ρ m + ρ Λ (23)We assume that γ = 0. Adding (21) and (22) yields˙ ρ m + ˙ ρ Λ = − p ρ m + ρ Λ )( ρ m + Kρ Λ ) (24)and subtracting appropriate multiples of (21) and (22) yields˙ ρ m Kρ Λ − γKρ m ρ = ˙ ρ Λ ρ m + γρ m ρ Λ . (25)Combining (24) and (25), and integrating, gives the following relation betweenthe dark energy and dark matter densities: ρ Λ = ρ Km e γ √ ρ m + ρ Λ / √ . (26)Due to this rather awkward relation between the two densities, it is difficultto make further progress using purely analytical methods. However, we canstill examine the cosmic evolution by treating (21) – (23) as a two-dimensionaldynamical system: ˙ ρ m = −√ √ ρ m + ρ Λ ρ m + γρ m ρ Λ ˙ ρ Λ = − γρ m ρ Λ − √ √ ρ m + ρ Λ Kρ Λ . We consider the fixed points of this dynamical system. The values of ρ m and ρ Λ at these fixed points must satisfy ρ m ( γρ Λ − p ρ m + ρ Λ )) = 0 and ρ Λ ( γρ m + K p ρ m + ρ Λ )) = 0. It is clear that the origin ( ρ m , ρ Λ ) = (0 ,
0) isalways a fixed point. For fixed points other than the origin, we need the othertwo factors to both vanish, and this leads to the consideration of the followingfour cases based on the signs of γ and K : A. Case 1:
K > , γ > , ρ Λ ρ Λ is sufficiently great to begin with, the7atter density increases and then decreases. The condition ˙ ρ m = 0 correspondsto the parabolic nullcline ρ m = γ ρ − ρ Λ .An illustration of the evolution is shown in Figure 2. In this, as well as theother cases, the actual values of γ and K do not seem to affect the qualitativebehaviour of the orbits, although their signs do.This scenario is unlikely to represent our present universe. Although formost trajectories there is an initial dark energy dominated phase which givesway to a dark-matter dominated phase, the model does not exhibit the late darkenergy dominated phase which we are currently experiencing. Ρ m Ρ L Figure 2: As in Figure 1, with the same interaction term ρ m ρ Λ ( γ = 1),but now assuming a dark energy component whose equation of state is p = − ρ/ B. Case 2:
K > , γ < , ρ m ρ m is sufficiently great to begin with,the dark energy density increases and then decreases. The condition ˙ ρ Λ = 0corresponds to the parabolic nullcline ρ Λ = γ ρ m K − ρ m .8n this scenario, an initial matter-dominated phase gives way to a dark en-ergy dominated phase, and then the density of dark energy falls steeply. How-ever, this model does not address the coincidence problem, because the ratio r ≡ ρ m ρ Λ keeps decreasing (if K Ρ m Ρ L Figure 3: As in Figure 1, but with an interaction term − ρ m ρ Λ ( γ = − p = − ρ/ C. Case 3:
K < , γ < , ρ m ρ Λ >
0, matter is continually being converted into dark energy, and ρ m tends monotonically to 0 (as ρ Λ tends monotonically to ∞ ). As the ratio of thedensities ρ m ρ Λ is always decreasing, this aggravates the coincidence problem andsubstantiates the results from [11] and [12] alluded to earlier in Section 2.9 Ρ m Ρ L Figure 4: As in Figure 1, but with an interaction term − ρ m ρ Λ ( γ = −
1) and a phantom dark energy component with equation of state p = − ρ/ D. Case 4:
K < , γ > ,
0) and (¯ ρ m , ¯ ρ Λ ) ≡ (cid:18) K ( K − γ , − K ) γ (cid:19) . The eigenvalues of the Jacobian at the second fixed point are ± √ K (1 − K );they are both purely imaginary, and so (¯ ρ m , ¯ ρ Λ ) is a center . Phase-plane plotsindicate the existence of stable periodic orbits around this center, with no con-vergence to nor divergence from (¯ ρ m , ¯ ρ Λ ). The two nullclines are the parabola ρ m = γ ρ − ρ m (on which ˙ ρ m = 0) and the parabola ρ Λ = γ ρ m K − ρ m (on which˙ ρ Λ = 0).A linearization about the centre (¯ ρ m , ¯ ρ Λ ) gives the equations˙ x = 3 K γ x + 3 K (2 K − γ y ˙ y = 3( K − γ x − K γ y x = ρ m − ¯ ρ m , y = ρ Λ − ¯ ρ Λ and made the assumption that x, y are both small. Fitting the equation of a conic section Ax + By + xy =const and differentiating gives a solution A = K − , B = K − . In order forthe trajectories to be ellipses, we require that 4 AB − >
0, but this is true (asexpected) since 4 AB − (cid:16) √− K + √− K (cid:17) .Thus, for all K <
0, the orbits of the system are bounded. For a giventrajectory, the ratio ρ m ρ Λ will oscillate between two extremes given by the gradi-ents of the tangents from the origin to this trajectory. Note that the intervaldefined by these two extremes contains the value ¯ ρ Λ ¯ ρ m = − K , which is O (1) if K is not too close to 0. In such a model the dark energy and dark matter will becomparable at infinitely many times, and this would provide a solution to thecoincidence problem. An example of the evolution is shown in Figure 5 for theparameter values γ = 1 and K = − . Ρ m Ρ L Figure 5: As in Figure 1, with an interaction term ρ m ρ Λ ( γ = 1) buta phantom dark energy component with equation of state p = − ρ/ VI. CONSTRAINING THE MODEL PARAMETERS
In the previous two sections, we have seen that, in the class of interactingmodels with an interaction of the form γρ m ρ Λ , the only model that could pos-11ibly help to alleviate the late-time coincidence problem is the one in SectionV.D. Thus, we will now consider this model in more detail.The analysis in the previous sections holds only at late times, when thebaryon and radiation densities are small and can be neglected. In order toconstrain the parameters γ and K of the model using observations of the pastuniverse, we need to examine its past evolution. It is therefore necessary toinclude baryons and radiation in the analysis, since their densities are significantat early times.Before we proceed, we should emphasize that the following discussion is notan attempt to obtain the best-fit parameters, but merely to show that the modelunder consideration is a viable fit to observations for non-zero γ and K . Wehave seen that the strength of this model is that it addresses the coincidenceproblem (which the standard concordance model cannot do), and we will findthat it appears to fit observations at least as well as the concordance model.In order to obtain best-fit parameters and make a more detailed comparison toobservations, more careful work is needed, including a full stability analysis ofperturbations. This is, however, outside the scope of the present paper, and isleft as a topic for future investigation.The complete system of equations to be solved is:(1 + z ) ρ ′ m = 3 ρ m − γρ m ρ Λ H (1 + z ) ρ ′ Λ = 3 Kρ Λ + γρ m ρ Λ H H = 8 πG ( ρ m + ρ Λ + ρ b (1 + z ) + ρ r (1 + z ) )where we have transformed the independent coordinate from cosmic time t toredshift z , and a prime indicates a derivative with respect to z . The quantities ρ b and ρ r are the present densities of baryons and photons, respectively.In order to find suitable choices of parameters for γ and K , we can integratethese equations numerically for various values of these parameters, and comparethe results to observations. In particular, we shall attempt to examine the effectof three observational constraints which are claimed to be model-independent[72]: (i) the shift parameter R , related to the angular scale of the first acousticpeak in the CMB power spectrum, (ii) the distance parameter A , related to themeasurement of the BAO peak from a sample of SDSS luminous red galaxies,and (iii) Type IA Supernovae data.The shift parameter R is defined as follows [72, 73]: R ≡ (Ω m + Ω b ) / H Z z recomb dz ′ H ( z ′ )where z recomb = 1091 . m = 0 .
227 for the present frac-tional density of dark matter, Ω b = 0 . H = 70 . km s − Mpc − for the present value of the Hubbleparameter. Note that this is only an approximation, because the derivation of12hese values from the WMAP7 data assumes a standard, non-interacting ΛCDMmodel. The value of R obtained from the WMAP7 data is 1 . ± . A is defined as follows [75, 76]: A ≡ (Ω m + Ω b ) / H H ( z b ) / (cid:20) z b Z z b dz ′ H ( z ′ ) (cid:21) / where z b = 0 .
35. The value of A has been determined to be 0 . n s / . − . ± .
017 (1 σ constraints) [76], and we will take the scalar spectral index n s to be0 .
963 [74] (WMAP7). We will also assume the WMAP7 parameters for Ω m and Ω b (discussed above).Using these parameters, we have calculated R and A for various values of γ > K <
0. The observational constraints from R put a crude upperbound on γ of about 1 . × m kg − s − , and requires K > − .
48. Theconstraint from A suggests that K > − .
27 and requires γ to be less thanabout 1 . × m kg − s − . The closer K is to 0, the higher the values of γ permitted by either test: a more negative value of K tends to increase thevalues of both A and R .In the remainder of this paper, we will take the parameters γ = 100 and K = − .
15. Note that, although these are not necessarily the best-fit parameters,they are in accord with the observed values of R and A .We have fitted this particular model to SNIa data [77]. At the outset, wemight expect that there will not be a large deviation from the behaviour of theconcordance model in this regime, since the densities of either dark matter ordark energy would be very low at small redshifts, and the interaction term wouldtherefore be negligible. The recent evolution of the system would therefore besimilar to that of a standard ΛCDM universe. The results confirm this: for asample of 608 Union2 supernovae, the model with γ = 100 , K = − .
15 gives a χ value of 756 .
4, which is slightly better than the one obtained for the γ = 0 , K = 0model (780 . VII. DISCUSSION
We now consider the concrete instance of our model with γ = 100 and K = − .
15. The evolution of the fractional densities with redshift is shown inFigure 6: 13 e−02 1e+00 1e+02 . . . . . . Redshift, z F r a c t i ona l den s i t i e s , W i Dark matterDark energyBaryonsRadiation
Figure 6: The past evolution of the fractional densities Ω i with redshift, for auniverse in which γ = 100 and K = − .
15. The universe undergoes the usualsequence of radiation, dark matter and dark energy dominated eras.From Figure 6, we can see that this model provides a suitable sequenceof cosmological eras corresponding to what we know about our universe. Theuniverse is radiation-dominated at very early times. This is followed by a longmatter-dominated era, with Ω Λ ≪ Ω m , up to around redshift z ≈ .
5. At latetimes, we encounter a period of dark-energy-dominated acceleration.The past history of the universe is somewhat dependent on the parameter γ .In particular, increasing γ to larger than about 10 induces a peak in the baryonfractional density at about redshift 2500 and leads to a brief baryon-dominatedera around that time, whilst moving the transition from the radiation-dominatedera to the matter-dominated era later in time (i.e., to smaller redshift).Also, note the peak in the fractional density of dark energy at around z ≈ ρ m and ρ Λ are swamped by the radiationdensity. We can, however, observe this effect on a logarithmic plot (Figure 7):14 e−03 1e+04 1e+11 1e+18 1e+25 − − − − − Redshift, z F r a c t i ona l den s i t i e s , W i Dark matter
Dark energy
BaryonsRadiation
Figure 7: A log-log plot of the fractional densities Ω i against redshift, for auniverse with γ = 100 and K = − .
15. This plot shows that many cycles mayoccur before the current one.Having established that the model adequately describes the past universe,let us consider its implications for the late universe. At late times, the densitiesof the non-interacting baryons and photons will decrease and become negligible.The evolution of the universe thus converges to a stable limit cycle, in whichthe following four phases repeatedly occur: • Dark energy is converted quickly into dark matter, while H remains large. • When the density of dark energy is sufficiently low, the interaction termin the evolution equation for ˙ ρ m is overwhelmed by the − Hρ m term,and the density of dark matter starts to decrease, while the dark energydensity stays low. The value of H decreases, and the universe decelerates. • The dark matter and dark energy densities are small and comparable. Thedark energy density slowly starts to increase while the dark matter densitycontinues to decrease. This marks the transition from a dark-matter-dominated universe to a dark-energy-dominated universe, and the universebegins to accelerate. The universe that we are currently experiencing isin the later stages of this transition. • When the density of dark energy then becomes sufficiently large, it con-tinues to increase until the interaction term becomes non-negligible.15his process goes on forever, with eternally alternating dark-matter- and dark-energy-dominated eras. Also, the universe undergoes infinitely many periods ofacceleration and deceleration, as can be seen from equation (10).In this limit cycle, the dark matter and dark energy densities are comparablefor a significant proportion of the time. The following graph shows a plotdemonstrating this: . . . . . . Cosmic time, t F r a c t i ona l den s i t i e s , W i Dark matterDark energyBaryonsRadiation
Figure 8: The future evolution of the fractional densities Ω i with cosmic time t , for γ = 100 and K = − .
15. In this plot, t is measured in seconds, and thetime t = 0 corresponds to the present.The duration of an entire cycle is dependent on the choice of parameters:choosing a more negative value of K leads to more and shorter cycles. FromFigure 8, the densities are comparable for about a quarter of each cycle, andthat dark energy is dominant at most other times. This suggests that, in thismodel, it is not unnatural for the universe to be in a state at which the darkmatter and dark energies are comparable.This model, however, has a limitation: it does not explain the apparentcoincidence of the baryon and dark matter densities being comparable at thepresent time. In particular, the baryon density continues to decrease steadilyas the cycles progress, whereas the minimum value of the dark matter densitystays at roughly the same level over all future cycles, because of its periodicinteraction with dark energy. VIII. CONCLUSIONS
16e have considered a model with an interaction term proportional to theproduct of the densities of the interacting fluids, and applied it to the interactionof a dust fluid and another fluid with equation of state p = wρ , for various valuesof the constant w . In the case of two dust fluids, we have obtained a generalsolution, and have shown that their density evolution is similar to that in thenon-interacting case, after an initial short cooling-off period.We then proceeded to show that such an interaction between dark matterand dark energy in the form of a cosmological constant would not solve thecoincidence problem. In the case γ <
0, energy is transferred from dark matterto dark energy, and the ratio of the energy densities r = ρ m /ρ Λ always de-creases with time. We would therefore expect the dark matter density todayto be negligible compared to the dark energy density. However, this does notcorrespond to what we observe in the universe today. On the other hand, inthe case γ > γ < w is allowed tovary arbitrarily, or if w > − γ is allowed to vary arbitrarily.However, it is interesting to note that if the dark energy follows a ‘phantom’equation of state ( w < −
1) and energy is transferred from dark energy todark matter, it is possible to obtain periodic orbit solutions. These correspondto cyclic situations in which the ratio of the dark densities is comparable atinfinitely many times. We have also shown that suitable parameters γ and K can be chosen so that the model is consistent with observations.Such a scenario could alleviate the coincidence problem, because it can beshown that if w is not too close to −
1, the periodic orbits would enclose a fixedpoint corresponding to a density ratio r that is O (1), and thus the value of r on these trajectories would be O (1) at infinitely many times in the evolution ofthe universe. ACKNOWLEDGMENTS
I would like to thank my supervisor, John Barrow, for helpful comments;Simeon Bird, David Essex, Hiro Funakoshi, Baojiu Li, Yin-Zhe Ma and ananonymous referee for useful discussions; and the Gates Cambridge Trust forits support.
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