Is it possible to obtain cosmic accelerated expansion through energy transfer between different energy densities?
aa r X i v : . [ g r- q c ] D ec Is it possible to obtain cosmic accelerated expansion throughenergy transfer between different energy densities?
Recai Erdem ∗ Department of Physics˙Izmir Institute of TechnologyG¨ulbah¸ce, Urla 35430, ˙Izmir, Turkey (Dated: November 15, 2018)The equation of state of an energy density may be significantly modified by cou-pling it to another energy density. In the light of this observation we check thepossibility of producing cosmic accelerated expansion in this way. In particular weconsider the case where matter is converted to radiation (or vice versa by particlephysics processes). We find that cosmic accelerated expansion can be obtained inthis way only if an intermediate state with negative equation of state forms duringthe conversion.
I. INTRODUCTION
Observations show that the universe is undergoing accelerated expansion at present, andmany theoretical arguments and observational evidence suggest that the universe must haveundergone an accelerated expansion period at the early times as well [1, 2]. Although thestandard explanations for these accelerated expansions are cosmological constant at presentera and inflationary models at early times there are many alternative ways; for example,quintessence, f ( R ) models [2], and gravitational particle production [3, 4]. However all thesemodels have some problems. There is a problem associated with cosmological constant calledthe cosmological constant problem, and it seems that the best way may be the use of somesymmetry to make it cancel and seek another method for late time cosmic acceleration [5].Inflationary models usually employ at least one new postulated scalar, and need specialinitial conditions, a similar situation (although less severe) is true for quintessence models. f ( R ) type modified gravity models use an extension of general relativity, in gravitational ∗ [email protected] particle production the energy density of the universe is an open thermodynamical systemthat is assumed to acquire energy from gravitational field while the question of if the universeis a closed system in this case is not clear enough. Therefore it is useful to seek the possibilityof additional alternative ways for accelerated expansion. In particular it would be desirableto have a model where the accelerated expansion is achieved with a minimal extension ofthe standard models of particle physics and cosmology. In this paper, in the light of the factthat coupling an energy density to another one modifies its equation of state [6] we seek ifan energy density transfer due to elementary particle processes may have the potential ofproviding a source for cosmic accelerated expansion. Although the analysis in this paper,in principle, is applicable to all types of particle physics processes, we specify it to thecase of conversion of heavy particles to light particles i.e. to the conversion of matter toradiation. In fact there must be an era of the creation of matter and radiation not onlybecause the ordinary matter and radiation must be produced anyway but also to have awell defined model that may serve at all eras of the universe [7]. Moreover in the standardlore of cosmology the ordinary matter and radiation are assumed to be produced by thedecays or the collisions of some other particles such as Higgs particle, curvaton etc. at earlytimes [8]. Particle physics processes ranging from high energies to atomic physics have animportant role at present as well. Therefore the possibility of using just matter and radiation(as in this paper) interacting through the particle physics processes for cosmic accelerationwith minimal need for exotic matter is interesting. The results of the following analysisshows that obtaining cosmic acceleration through conversion of matter to radiation (or viceversa) seems impossible except through formation of an intermediate state with negativeequation of state (e.g. a QCD-like condensate formed by intermediate particles produced inthe particle physics processes).In this study we consider the Robertson-Walker metric ds = − dt + a ( t )[ dr − kr + r ( dθ + sin θdφ )] (1)and for simplicity we take k = 0 which is in agreement with observations [9]. For theillustration of the method we consider a simple case; a universe that consists of matterand radiation. We assume that, at some time t , the energy density of either of matter orradiation starts to be transferred to the other through some particle physics processes suchas those given in Figure 1. There may be two cases: i) The conversion of each matter particle to radiation may be instantaneous e.g. as in thethree particle (decay) process in Figure 1 or the four particle process in the figure where theinternal line is deleted (i.e. a four particle contact interaction) or the four particle processin the figure where the process takes place in t or u channels. This option is considered inthe next section. ii)
There may be an on-shell intermediate particle in the process e.g. the four point processin the figure where the process takes place in s channel i.e. a resonance forms in theintermediate state in the conversion of matter to radiation. This option is considered inthe section after the next section. We consider the two cases mentioned above separately.In both cases above we take the matter to be extremely non-relativistic and approximateits cosmological evolution by that of dust while we assume that the matter particles haveyet non-vanishing small velocities enabling them to participate in particle processes such asscattering.
II. THE CASE OF INSTANTANEOUS INTERACTIONS
In this case the interaction between individual particles is instantaneous as in the threeparticle process or the four particle process in t or u channels in Fig. 1. In this case thetotal energy density of the universe may be written as ρ ( t ) = ρ m ( t ) + ρ r ( t ) (2) ρ m ( t ) = C m ( t )[ a ( t )] , ρ r ( t ) = C r ( t )[ a ( t )] (3)where a ( t ) is the scale factor (of Robertson-Walker metric) at time t , and (see Appendix A) C m ( t ) = ρ m − Z tt d ˜ ρ ( t ′ ) dt ′ ! a ( t ′ ) dt ′ (4) C r ( t ) = ρ r + Z tt d ˜ ρ ( t ′ ) dt ′ ! a ( t ′ ) dt ′ (5)where ˜ ρ ( t ) dt is the rate of the energy density transfer from matter to radiation (in the com-moving frame). In this case the effect of interaction is absorbed into the change in evolutionof the energy densities of matter and radiation through time dependence of C m ( t ) and C r ( t ).We may easily see from (3), (4), (5) that ρ m ( t ), ρ r ( t ) and ρ ( t ) = ρ m ( t ) + ρ r ( t ) satisfythe following equations (see Appendix A)˙ ρ m ( t ) + 3 H ρ m ( t ) = ˙ C m ( t )[ a ( t )] = − d ˜ ρ ( t ) dt (6)˙ ρ r ( t ) + 4 H ρ r ( t ) = ˙ C r ( t )[ a ( t )] = d ˜ ρ ( t ) dt (7)˙ C m ( t ) a + ˙ C r ( t ) a = 0 (8)˙ ρ ( t ) + 3 H " ρ r ( t ) ρ ( t ) ρ ( t ) = 0 (9)In other words the interaction that converts matter to radiation does not induce an additionalenergy density or pressure, so it does not affect the overall evolution rate of the universewhile it affects the evolution of each component, matter and radiation separately. In factthis conclusion holds in a more general context. Consider two energy densities, ρ and ρ with equations of states ω and ω , respectively in the absence of any interaction between ρ and ρ ; then allow an energy density transfer Q between ρ and ρ . The situation is similarto Eqs. (6), (7), and (9) where d ˜ ρdt is replaced by Q , namely,˙ ρ ( t ) + 3 H ( t ) (1 + ω ( t )) ρ ( t ) = Q ( t ) (10)˙ ρ ( t ) + 3 H ( t ) (1 + ω ( t )) ρ ( t ) = − Q ( t ) (11)˙ ρ ( t ) + 3 H (1 + ω ( t )) ρ ( t ) = 0 (12)where ρ ( t ) = ρ ( t ) + ρ ( t ) , ω ( t ) = ω ρ ( t ) + ω ρ ( t ) ρ ( t ) + ρ ( t ) (13)In fact the form of the equations (10) and (11) are familiar from the models of dark matter- dark energy interactions [10] where dark matter and dark energy are replaced by (dark)matter and radiation. It is evident from the form of (13) that ω is positive for all times ifboth ω and ω are positive at some initial time and ω is negative for all times if both ω and ω are negative at some initial time (provided that ρ is positive). In other words ω and ω should have different signs if one wants to construct a model where the dark energyis subdominant initially and at later time becomes dominant or vice versa (and introductionof Q does not have any effect on the time of transition between ω > ω < − d ˜ ρ ( t ) dt =3 Hρ m . Then (6), (7) become˙ ρ m ( t ) + 3 H ρ m ( t ) = 3 Hρ m ( t ) ⇒ ρ m ( t ) = constant = ρ (14)˙ ρ r ( t ) + 4 H ρ r ( t ) = − Hρ m ( t ) = − Hρ (15)Hence 43 ρ r + ρ = A e − R H dt (16)where A is some integration constant. In other words ρ m in this example behaves as cosmo-logical constant (although in the absence of an energy density transfer between matter andradiation ρ m scales as a ), and ρ r behaves as an exotic fluid with the energy density givenin (16). Therefore introduction of an energy transfer between matter and radiation mayconsiderably change their cosmic evolution and an otherwise standard energy density mayseem exotic due to the coupling of that energy density to another energy density. Howeverif one adds the differential equations in (14) and (15) one finds that the equation is thesame as (9) i.e. it is the same as the evolution of an energy density composed of matterand radiation that do not interact with each other. In other words, although the evolutionof each energy density component in total energy density may drastically change due tointroduction of an energy transfer between different components of the total energy density,the overall evolution of the total energy density remains the same after the introduction ofthe energy transfer. III. THE CASE OF A RESONANCE AS THE INTERMEDIATE STATE IN THECONVERSION OF MATTER TO RADIATIONA. General discussion
This is the case where the internal line in the four particle process in Fig. 1 is an on-shells-channel intermediate particle (i.e. resonance). In this case the total energy density of theuniverse may be written as ρ ( t ) = ρ m ( t ) + ρ r ( t ) + ρ R ( t ) (17)where ρ m ( t ) and ρ r ( t ) are the energy densities of matter and radiation given by (2) where C m ( t ) and C r ( t ) are replaced by C m ( t ) = ρ m − Z tt d ˜ ρ m ( t ′ ) dt ′ ! a ( t ′ ) dt ′ (18) C r ( t ) = ρ r + Z tt d ˜ ρ r ( t ′ ) dt ′ ! a ( t ′ ) dt ′ (19)Because the form of the energy density of the intermediate state i.e. the resonance ρ R ( t ) isnot specified it has a general form given by ρ R ( t ) = e − Γ( t − t ) C R ( t ) exp " − Z a ( t ) a ( t ) (1 + ω R ( a ′ )) da ′ a ′ (20)Note that e − Γ( t − t ) term would not be present in (20) if we had taken the resonance as stable,where Γ is the width of the resonance. C R in (20) may be obtained in the same way as donein Appendix A as C R ( t ) = Z tt dt ′ [( d ˜ ρ m ( t ′ ) dt ′ ) e Γ( t ′ − t ) exp " Z a ( t ′ ) a ( t i ) (1 + ω R (˜ a )) d ˜ a ˜ a (21)where we have used d ˜ ρ R ( t ′ ) dt ′ = d ˜ ρ m ( t ′ ) dt ′ since the amount of energy transferred from the matteris equal to the one acquired by the resonance. ρ R satisfies˙ ρ R + 3 H (1 + ω R ) ρ R = e − Γ( t − t ) ˙ C R ( t ) exp " − Z a ( t ) a ( t i ) (1 + ω ( a ′ )) a ′ − Γ e − Γ( t − t ) C R ( t ) exp " − Z a ( t ) a ( t i ) (1 + ω ( a ′ )) a ′ = d ˜ ρ R ( t ) dt − Γ ρ R (22)which is equivalent to ˙ ρ R + 3 H (1 + ω R + ∆ ω R ) ρ R = o (23)where ∆ ω R = 13 H − d ˜ ρ R ( t ) dt ρ R + Γ (24)Eq.(24) tells us that the effect induced by the formation of the resonance will act like darkenergy if the ratio of the rate of conversion of the energy (of the matter to the resonance)to the energy density of the resonance is higher than the decay rate of the resonance. Notethat this effect is independent of the background effect produced by the resonance i.e. ∆ ω R is independent of ω R in general.Note that we had taken d ˜ ρ m dt = d ˜ ρ r dt in the previous case because the conversion of eachmatter particle to radiation was instantaneous while here d ˜ ρ m dt = d ˜ ρ r dt since there is a reso-nance as the intermediate state in this case hence the rate of the conversion of matter tothe resonance, d ˜ ρ m dt may not be equal to the rate of the conversion of the energy density ofthe resonance to radiation d ˜ ρ r dt at a specific time. Instead we should have − d ˜ ρ m dt + d ˜ ρ r dt + d ˜ ρ m dt − Γ ρ R = ˙ C m ( t ) a + ˙ C r ( t ) a + d (cid:16) e − Γ( t − t ) C R ( t ) (cid:17) dt exp " − Z a ( t ) a ( t ) (1 + ω ( a ′ )) a ′ = 0 (25)i.e. d ˜ ρ r dt = Γ ρ R (26)In other words the net change in the total energy due to the energy transfers between matter,radiation, and the resonance state should be zero at each time t since we assume that systemis a closed system with no energy transfer from outside, and the source of radiation is thedecay of the resonance. Eq.(25) tells us that there is no direct effect of the interaction (of theconversion of energy densities) on the total energy density, instead the effect of conversion isindirect and through the evolutions of ˙ C m ( t ), ˙ C r ( t ), ˙ C R ( t ) as in the preceding subsection. Inother words by taking different sets of ˙ C m ( t ), ˙ C r ( t ), ˙ C R ( t ) one may mimic different effectiveenergy densities in this case as well while the evolution of the total energy density does notchange after introduction of conversion between energy densities of matter and radiation.The only way to induce a dark energy effect in this way is to assume that the resonancestate has ω < ρ with ω < ω that becomes ω < ω may change sign during the evolution of the energy densities while itis impossible in the case of a universe that only consist of matter and radiation with ω > ω > ρ > ρ >
0. In this subsection we want to focus on the effect of the resonanceand possibility of taking it as an energy density inducing an accelerated cosmic expansion.One may describe the general evolution of this system as follows: The evolutions of C m and C R are given by (18) and (21), respectively. The evolution of C r is given by (19) where(26) is employed. For simplicity we take ρ r = 0 i.e. C r ( t ) = Γ Z tt ρ R ( t ′ ) a ( t ′ ) dt ′ (27)Then at t = t C m (so, ρ m ) is maximum, and C R (so, ρ R ) and C r (so, ρ r ) are zero. Bytime C m decreases till t = t mf when C m becomes small or C m ≃ C R is maximum and C r has a small finite value. At t > t mf C m gets even closer to zero (if it is not zero yet), e − Γ ( t − t ) C R ( t ) gets smaller by time and becomes zero as t → ∞ while C r gets its maximumat t → ∞ . In other words the evolution of the system may be summarized as: Initiallythe universe is matter dominated then it becomes resonance dominated and eventually theuniverse becomes radiation dominated. In the next subsection we will describe the resonancestate by resonance particles produced by the particle physics process given in the seconddiagram in Fig.1 for relatively simple cases. B. A particle physics description
We may adopt the derivation for the formula of energy density of matter in Appendix A(i.e. Eq.(18)) to find an expression for number densities since both have the same dependenceon scale factor i.e. both vary as ∝ a − . To find the corresponding expression for numberdensities we replace d ˜ ρ ( t ) dt by d ˜ n ( t ) dt and ρ m ( t ) by n ( t ). Then we may employ the methodwe had used for energy densities in Appendix A to write a similar expression for numberdensities of stable particles n ( t ) = C ( n ) ( t )[ a ( t )] , C ( n ) = n − Z t − t d ˜ n ( x ) dx ! x =( t − u ) a ( t − u ) θ ( u ) du (28)where we have put the superscript ( n ) to the coefficient C to distinguish it from the corre-sponding symbol for energy densities, n stands for the initial value of n ( t ) at the initial time t = t of the energy transfer, and d ˜ n ( t ) dt stands for the number density transferred per timein per comoving volume i.e. the rate of the number density transfer in the comoving frame,and θ ( u ) is the Heaviside function (i.e. unit step function) given by θ = 0 if u < θ = 1 if u >
0. Eq.(28) (after using a derivation similar to the one for matter in AppendixA) implies that d ˜ n ( t ) dt = − ˙ C ( n ) ( t )[ a ( t )] (29)The rate of the conversion of the matter to resonance particles by collisions of two beamsof matter particles, each with number densities n m may be expressed as − d ˜ n m ( t ) dt = ˙ C ( n ) m ( t )[ a ( t )] = − β ′ n m ( t ) σ ( t ) v ( t ) (30)where > β ′ > n m to a target of n m particles in unit comoving volume, σ is the cross section of the four-point Feynman diagram in Fig.1, v is the average velocity of the (non-relativistic) particlesrelative to the comoving frame. For later reference we need dependence of cross section onscale factor. The differential cross section dσ of two initial particles of 4-momenta p , p and masses m , m going to a final state of two particles of 4-momenta p ′ and p ′ in localMinkowski space is given by [11] dσ = (2 π ) δ ( p + p − p ′ − p ′ )4 E E v d p ′ (2 π ) E ′ d p ′ (2 π ) E ′ | M fi | (31)where bold face quantities denote 3-dimensional quantities, E denotes energies, the subindices1 and 2 denote the 1st and the 2nd particles, unprimed letters correspond to initial stateparticles and the primed ones correspond to final state particles. E j denotes energy of thej’th particle, and M fi is the transition amplitude from initial state to the final state. Thedependence of the cross section on scale factor is through its dependence on momenta in(31). After integrating over final momenta the apparent dependence of the cross section on p ′ disappears. Therefore the dependence of the cross section on scale factor is through E , E ′ , and | M ( p , p ; k i ) | . In this case E are extremely non-relativistic, so theymay essentially be taken constant. E ′ are extremely relativistic, so may be taken to varyproportional to a . M if corresponding to the four-point process in figure in the perturbativelimit for an on-shell intermediate state (in s-channel) is given by [11, 12] M fi = ( − ig )( − ig ) s − m R + i √ s Γ R ! (32)where g , g are coupling constants of the interactions in the vertices, s = ( p + p ) , and m R , Γ R are the mass, the decay width of the resonance, respectively. Because the initialstate consist of extremely non-relativistic particles one may write( p + p ) ≃ ( E + E ) ≃ (2 m + mv ) , m R ≃ m (33)Therefore for the values of s near the pole of the resonance s − m R + i √ s Γ R ( s ) ≃ m R v + im R Γ R ( s ) (34)Γ R is also a function of s in general, as emphasized in (34), so the form of dependence of Γ R on s must be also known to determine the dependence of cross section on scale factor. Inthe case of perturbative field theory Γ R is identified as the imaginary part of self interactionand can be calculated in principle. In the case of a perturbative quantum field theoryconsisting of only scalars with 3-point interactions the leading order contribution is foundto be proportional to √ s [13]. When the theory gets non-perturbative higher order termsthat depend on higher powers a are expected to become dominant. In fact in this study we0consider extremely narrow resonances (i.e. Γ << m R ) since the lifetime of the resonancemust be long enough to be cosmologically relevant, so the last term in (34) may be neglected.Therefore M fi ∝ a p ( t ) (35)where p = 2 in the case of an extremely narrow resonance. Hence we find that dσ ∝ a r ( t ) where in the case of an extremely narrow resonance r = 7 (36)The same result may be obtained from the general expression M fi for the values of the totalenergy close to the resonance pole [9, 14] M fi = − Γ Γ s − m R + i √ s Γ R ! (37)where Γ are the partial widths of the resonance associated with the first and second vertexin Figure 1. While (37) is more general in the sense that it is not restricted to perturbativeinteractions, it may not be evident that Γ do not vary with scale factor.The energy density of dust particles is proportional to its number density. We assume thematter particles are extremely non-relativistic so their energy density may be approximatedby that of dust. Therefore (30) may be rewritten as d ˜ ρ m ( t ) dt = E d ˜ n m ( t ) dt = − ˙ C m ( t )[ a ( t )] = β ρ m ( t ) a r − ( t ) , β = β ′ σ ( t ) v ( t ) E a ( t ) (38)where E ≃ m is the average energy of the matter particles. In (38) we have used Eq.(35) i.e. σ ( t )= σ ( t ) (cid:16) a ( t ) a ( t ) (cid:17) r (where we have used the expression for cross section in local Minkowskispace), and the fact that v ( t )= v ( t ) a ( t ) a ( t ) , and E is taken to be constant since particles aretaken to be extremely non-relativistic and it becomes even more non-relativistic by time dueto redshift. After using ρ m ( t ) in (3), (38) may be rearranged as˙ C m ( t )[ a ( t )] = − θ ( t − t ) β C m ( t )[ a ( t )] ! a r − ( t ) (39) ⇒ dC m C m = − θ ( t − t ) β a r − ( t ) dt = − θ [ a ( t ) − a ( t )] β daa − r H (40)Eq.(40) may be used to determine C m ( t ). This, in turn, may be used to find d ˜ ρ m ( t ) dt from(38), and ρ R from (20), and so ∆ ω R from (24). This will be done below for a relativelysimple, yet phenomenologically relevant case.1Next we find C m ( t ) for some choices of H by using (40). For the sake of simplicity considerthe cases where H = ξ a s ( t ) (41)Here ξ and s are some constants. In fact (41) contains most of the phenomenologicallyrelevant simple cases i.e. radiation, matter, cosmological constant dominated eras. Afterusing (41), (40) becomes − C m ( t ) = − θ ( t − t ) βξ Z daa − r + s − C m ( t )= β θ ( t − t ) ξ (4 − r + s ) " a − r + s ( t ) − a − r + s ( t ) − ρ m (42)where we have used C m ( t ) = ρ m . (42) results in C m = ξ ρ m (4 − r + s ) a − r + s ( t ) a − r + s ( t ) − βρ m θ ( t − t ) [ a − r + s ( t ) − a − r + s ( t )] + ξ (4 − r + s ) a − r + s ( t ) a − r + s ( t )= ξ ρ m (4 − r + s ) ξ (4 − r + s ) + βρ m θ ( t − t ) [ a r − s − ( t ) − a r − s − ( t )] (43)Eq.(43) implies that s → r − β = 0 and is constant, andbecomes not-well defined if β = 0. This suggests that at a universe where s = r − C m → a ( t ) → ∞ for r − s − > C m goes a non-zero constant value as a ( t ) → ∞ for r − s − <
0. This implies thatthis process is not feasible in the case r − s − < − d ˜ ρ m ( t ) dt = ˙ C m ( t ) a ( t ) = − θ ( t − t ) ρ m βξ (4 − r + s ) a r − ( t ) { ξ (4 − r + s ) + β ρ m θ ( t − t ) [ a r − s − ( t ) − a r − s − ( t )] } (44) ρ m ( t ) = C m a = ξ ρ m (4 − r + s ) a − ( t ) ξ (4 − r + s ) + βρ m θ ( t − t ) [ a r − s − ( t ) − a r − s − ( t )] (45)
1. Evolution of the energy densities through an example
Now we consider an example to check if one may realize the correct evolution of the totalenergy density in a consistent way. To check the possibility of the resonance obtained duringconversion for matter to radiation that serve as dark energy we take the simplest choice i.e.the case where the resonance particles induce a condensate that behaves as cosmologicalconstant. We consider, first, a matter dominated universe for t < t < t , where s = − s = 0 for t < t < t , finally, the radiation dominated era at the end where s = − t < t . For simplicity we assume abrupt changes in the equations of states as one passesfrom one era to the other although in a more realistic case the changes would be smooth.However this choice is enough to give the essentials points. We take r = 7 i.e. we take theintermediate state resonance be an extremely narrow resonance (to have a sufficiently longlifetime). In other words we consider the case ω R = − , r = 7 andInitially s = − , in between s = 0 , at late times s = − β is constant in all eras. We give C m , ρ R , C r in these eras belowby using the formulas derived above. (Please refer to Appendix B for the details of thederivations). ρ m and ρ r are evident from C m and C r . C m ( t ) at different eras are found asFor t < t < t C m = ρ m βρ m ξ (cid:18) a − a (cid:19) (47)where a = a ( t ), a = a ( t ), and (47) is directly found from (43) for r = 7, s = s = − , ξ = ξ . For t < t < t C m ( t ) = ρ m { ρ m β [ 29 ξ − ( a − a ) + 13 ξ − ( a − a )] } − (48)where a = a ( t ), and to find C m in this case we have divided the integral in (42) into twoparts; one for the era when s = − between t and t , and then the era when s = 0, ξ = ξ .between t and t . In a similar way as in (48) we findFor t < t C m ( t ) = ρ m A ′ + B ′ a (49)Here A ′ , B ′ are some constant whose explicit forms are given in (B21). We notice that C m starts from ρ m and goes to zero as a → ∞ as expected. ρ R ’s in these eras are found asFor t < t < t ρ R ( t ) = e − Γ( t − t ) C R a ≃ ρ m βξ a ( t ) (cid:20) a − a (cid:21) (50)3For t < t < t ρ R ( t ) = (cid:18) a a (cid:19) " ρ m βξ { a − a } + ρ m βBξ { A − A + B a } (51)where have taken γ = Γ H = 3 in (B12) as a generic case where the integration is simpler,and A , B are some constants whose explicit forms are given in (B13) and (B14). For t > t ρ R ( t ) = e − Γ( t − t ) C R ( t ) ≃ ξ ρ m β e Γ ξ (cid:16) a − a (cid:17) − (cid:18) a − a (cid:19) + a B −
11 +
B a ! (52)where we have essentially employed (6), (21), (20), and (47), (48), (49). One notices that ρ R starts from zero at t = 0 and keeps rising at intermediate times and goes to zero dueto its decay as a → ∞ as expected. In fact one may also refer to the behavior of C R inAppendix B to see that C R is zero first and goes to a constant value as a → ∞ . ρ R wouldbehave like a cosmological constant at large scales factors if it had not decayed.The corresponding C r ’s areFor t < t < t C r ( t ) = 2Γ ρ m βξ Z aa ( a ′ − a a ′ ) da ′ = Γ ρ m β ξ a − a a + 35 a (53)For t < t < t C r ( t ) ≃ Γ ρ m ξ ρ m β ξ a − a a + 35 a + ρ m β a ( a − a )3 ξ ( a − a ) (54)For t < tC r ( t ) ≃ Γ ρ m ξ ρ m β ξ a − a a + 35 a + ρ m β a ( a − a )3 ξ + a ( a − a ) + C ′′ " a − a a + 3 a a − a ) − a − a a + 3 a a − a ) (55)where C ′′ is a constant whose explicit form is given in (B33), and C r ’s are found by using(27) and (50), (51), (52). We notice that C r starts from zero and goes to a constant valueas a increases as expected.4
2. A concrete particle physics model
In this sub-subsection, through a concrete particle physics model, we illustrate the mainlines of the results of our analysis on the use of on-shell intermediate state particle conden-sates for cosmic accelerated expansion. In fact many scalar fields in cosmology includinginflaton or quintessence fields may be considered as Bose-Einstein condensates in the lightof their homogeneity and coherence [15]. This approach is elaborated and detailed in manymodels of dark matter [16, 17], and a few models of dark energy [18–20] and inflation [21, 22].We study a specific model of dark energy similar to a model in literature [19] to see how theformulation developed in the previous parts of this subsection provides additional insightand improvement. The model we consider is simple yet sophisticated enough to be able tostudy main lines of this scheme in a proper way.We consider the following Lagrangian L = √− g {− g µν (cid:18) ∂ µ φ ∗ ∂ ν φ + ∂ µ χ ∗ m ∂ ν χ m + 12 ∂ µ χ m ∂ ν χ m + ∂ µ χ ∗ r ∂ ν χ r + 12 ∂ µ χ r ∂ ν χ r (cid:19) − m φ φ ∗ φ − λ φ ∗ φ ) − m m χ ∗ m χ m − m m χ m − m r χ ∗ r χ r − m r χ r − µ m φ ∗ χ m χ m − µ ∗ m χ ∗ m φχ m − µ r φ ∗ χ r χ r − µ ∗ r χ ∗ r φχ r } (56)Here φ , χ m , χ r are complex scalars while χ m , χ r are real scalars. These are the particlesin the two body scattering process in Figure 1, namely, χ m , χ m denote the incoming matterparticles with momenta p , p while χ r , χ r denote the outgoing radiation particles withmomenta p , p , and φ denotes the on-shell intermediate state particle in the figure.The Klein-Gordon equation for φ is obtained from (56) as ∂ φ∂ t + 3 H ∂φ∂ t − c a ∆ φ + m φ c ¯ h φ + c λφ ( φ ∗ φ ) + c µ m χ m χ m + c µ r χ r χ r = 0(57)where ∆ is the 3-dimensional Laplace operator, we have assumed the metric to be thespatially flat case of (1) i.e. we have assumed the perturbations of φ due to its motion benegligible with respect to the background, and we have introduced c and ¯ h into the equationthat has been taken as c = ¯ h = 1 in (56) to see the reduction of (57) to Schr¨odinger equationin the non-relativistic limit more clearly.To obtain the non-relativistic limit of (57) (to compare this case with the standard liter-ature at atomic scales where cosmic expansion is not considered) we assume that the second5term is negligible with respect to the others. One may neglect the second term providedthat m φ c ¯ h ≫ H (that may be seen more easily after transforming φ to ψ as given below).Consider the transformation φ = e − i mφ c h t ψ . Note that this transformation essentially sub-tracts the rest mass energy from the total energy of φ . If φ is non-relativistic for a long timethen its time evolution should be slow. These together, in turn, imply | ¨ ψ ˙ ψ | ≪ M φ c ¯ h . Then(57) reduces to − i ¯ h ˙ ψ − ¯ h a m φ ∆ ψ + ˜ λψ ( ψ ∗ ψ ) + ˜ µ m ˜ χ m χ m + ˜ µ r ˜ χ r χ r = 0 (58)Here ψ = e i mφ c h t φ , ˜ χ m = e i mφ c h t χ m , ˜ χ r = e i mφ c h t χ r , ˜ λ = ¯ h m φ λ , ˜ µ m = ¯ h m φ µ m , ˜ µ r = ¯ h m φ µ r . Eq.(58) is the Gross-Pitaevskii equation [23, 24] for the metric in (1) where thetrapping potential is zero and there is an external source term [25], namely, µ m ˜ χ m χ m + µ r ˜ χ r χ r . Note that Gross-Pitaevskii equation is the equation for a particle in a condensate,so it is the equation for the condensate state since all particles are in the same state in aBose-Einstein condensate. The external source term in (58) accounts for the conversion of χ m , χ m to ψ and the decay of ψ to χ r and χ r .We let the coupling constants µ m and µ r satisfy µ m ≫ µ r since we want a quick formationand a long lifetime for the condensate to simplify the situation. A short condensate formationtime together with the condition that χ m , χ m are non-relativistic and the mass of φ beingcomparable to those of χ m , χ m guarantees smallness of any entropy production throughthe condensation process [15, 26]. To be specific; the short duration of the condensationand small velocities guarantee smallness of any work done during the process, and the non-relativistic nature of χ m , χ m , φ guarantees the smallness of any heat production, so anyprobable entropy production due to mismatch of these terms will be small. In fact in thecase of Bose-Einstein condensation in atomic gases the formation time of the condensateis very small compared to its lifetime [27–29]. Note that the condensates described byGross-Pitaevskii equation for λ > T = 0 where thereare no inelastic two and three body collisions in the condensate that limits its life-time [29],hence a condensate described by Gross-Pitaevskii equation form more easily and is stable i.e.has an infinite lifetime. Therefore the assumption of a short time for the formation of thecondensate in this case is a reasonable assumption. Moreover we require a long lifetime toobtain a simple model compatible with the present energy densities of dark energy and darkmatter as we will see at the end of this sub-subsection. This requirement is easily satisfied6provided that we take µ r sufficiently small. The equations (56) and (58) are similar to thosein [19] except the additional scalars χ mi , χ ri , i=1,2; hence the presence of an external sourceterm in this case. We let m m χ ∗ m χ m ≫ µ m χ ∗ m φχ m , m m χ ∗ m χ m ≫ µ m χ ∗ m φχ m so thatone may consider χ m , χ m as (almost) free particles, and in their non-relativistic limitstheir equations of state become (almost) zero, so considering them as matter particles isinsured. In a similar way we let m r χ ∗ r χ r ≫ µ m χ ∗ r φχ r , m r χ ∗ r χ r ≫ µ m χ ∗ r φχ r sothat one may consider χ r , χ r as (almost) free particles, and hence we may consider themas radiation provided that and m r ≃ m r ≃
0. Another important difference between[19] and this one is that [19] takes λ < λ > φ ∗ χ r χ r term) while in [19] the condensateexperiences few collapse and re-formation cycles. In fact the condensate in this case (evenin the presence of the φ ∗ χ r χ r term) is rather stable since we take µ r very small. Anotherpotential point that makes the condensate more stable is the use of complex scalars i.e.charged scalars that makes the condensate more stable than a naive expectation [15]. Weassume that the particle number density is sufficiently high enough so that the de Brogliewave length λ dB = q π ¯ h mkT is larger than the mean separation n − between the particles.This condition may be converted to an upper bound on the temperature of φ ’s T < T c = 2 π ¯ h n k (cid:16) ζ ( ) (cid:17) m φ (59)where k is the Boltzmann constant, n is the number density. In the case where the φ ’s werein thermal equilibrium with baryonic matter and radiation at some initial time, Eq.(59) putsan upper bound on m φ while there is no bound on m φ in the case where φ ’s never had athermal equilibrium with baryonic matter and radiation. In the case of non-relativistic φ ’sthe temperature of φ ’s remain below T c at later times once it is below T c at some initial timein the past [19]. In this study we assume that φ ’s are non-relativistic and they had neverthermal equilibrium with baryonic matter and radiation in the past.After the basic elements of condensate formation for (56), now we are ready to derive itsresults at macroscopic level by using the general analysis we have given. In another wordswe will relate the microscopic view (i.e. the particle physics view) of the condensate givenabove to its macroscopic view at the level of energy densities in the light of the analysis wehave given before this sub-subsection. The evolution of the energy densities may be derived7from (6), (23), and (38), (26), as H = 8 π G ρ m + ρ φ + ρ r ) (60)˙ ρ m + 3 Hρ m = − d ˜ ρ m dt = − β ( a ( t )) r − ρ m (61)˙ ρ φ + 3 H (1 + ω φ ) ρ φ = d ˜ ρ m dt − Γ ρ φ = β ( a ( t )) r − ρ m − Γ ρ φ (62)˙ ρ r + 4 Hρ m = d ˜ ρ r dt = Γ ρ φ (63)where ρ m consists of χ m and χ m , ρ r consists of χ r and χ r , and ρ φ consists of the φ particles. Here we do not make any distinction between the resonance particles produced asthe intermediate state of the 4-point function in Fig. 1 and the particles in the condensatebecause we assume that the condensate is formed almost simultaneously after the formationof the resonance particles as we have mentioned before. The equations (61), (62) somewhatdiffer from those of [19] since the d ˜ ρ m dt term in the above equations is proportional to ρ m in[19] while it is proportional to ρ m here. Taking d ˜ ρ m dt proportional to ρ m (due to (38) seemsmore correct in the view that Bose-Einstein condensation involves interaction of particles ina Bose gas rather than their decay (where decay rate would be proportional to the energydensity of the decaying particles). [19] also considers a possible dependence of the decaywidths on the scale factor in an ad hoc way. While the dependence of d ˜ ρ m dt on scale factorin this study and in [19] are similar their origins are different. The origin of the dependenceof d ˜ ρ m dt in [19] is through Γ’s while in our case it results from the propagator part of thecross section, not through Γ’s. We have found that dependence of the self energies on themomentum in perturbative quantum field theory suggests that the leading order contributionto Γ is independent of scale factor. Moreover the explicit form of ρ m in terms of the crosssection is derived in this study.One may see the general lines of the formation of the condensate by following a proceduresimilar to that of [19]. Eq.(62) be written as˙ ρ φ + 6 H (cid:20) ˙ φ ∗ ˙ φ + 1 a ( ~ ∇ φ ∗ ) . ( ~ ∇ φ ) (cid:21) = β [ a ( t )] r − ρ m − Γ ρ φ (64)Here we have used ρ φ + p φ = 2 h ˙ φ ∗ ˙ φ + a ( ~ ∇ φ ∗ ) . ( ~ ∇ φ ) i . Note that in a case where φ ishomogenous we would have ~ ∇ φ = 0 while here we take it non-zero at initial times of theformation of the condensate (although the background is taken as homogeneous). Otherwise8there will be no condensate formation starting from ρ φ = 0 at initial time t since ρ φ = ˙ φ ∗ ˙ φ + 1 a ( ~ ∇ φ ∗ ) . ( ~ ∇ φ ) + V (65)implies φ ( t ) = 0 , ˙ φ ( t ) = 0 , ~ ∇ φ ( t ) = 0 (66)because V = m φ φ ∗ φ + λ φ ∗ φ ) + µ m φ ∗ χ m χ m + µ r φ ∗ χ r χ r + h.c. (67)(where h.c. implies taking the Hermitean conjugate of the proceeding terms) is positive forall values of φ since we take all coefficients positive. (In fact even in the case where µ m , µ r are negative V is still positive since we take the corresponding terms very small comparedto the rest of the terms). These together with Eq.(57) imply that we should have∆ φ ( t ) > ∂ φ ( t ) ∂ t > ρ φ >
0. Therefore Eq. (64) impliesthat during the formation of the condensate one has one must have β ( a ( t )) r − ρ m − Γ ρ φ > H ( ρ φ − V ) = 6 H (cid:18) | ˙ φ | + 1 a | ~ ∇ φ | (cid:19) > φ , ~ ∇ φ start with the values given in (66) and increase because of (68) till the bound in (69)is saturated. Eq.(57) implies that first ∂ φ∂ t becomes zero i.e. ˙ φ reaches its maximum valuewhile ~ ∇ φ keeps rising i.e. still ∆ φ > ~ ∇ φ keeps rising till the bound in (69) is saturatedonce again. (Note that, as the right hand side of the inequality in (69) increases its lefthand side decreases by time since the energy density of ρ m gets smaller, that of ρ φ getslarger.) Then ∆ φ changes sign and | ~ ∇ φ | decreases too till it eventually becomes zero, sothe right hand-side (69) becomes zero. Finally the left-hand side of (69) becomes zero tooby evolution of the energy densities, and the final values of ρ m and ρ φ are reached (providedthat the decay of φ is neglected i.e. the term with coefficient µ r is neglected since it verysmall). Therefore one may get a rough picture of the evolution of the energy densities.Now let us see if this model is compatible with the present day energy densities, at least,at an order of magnitude level. We will identify the condensate by dark energy and theleftover part of ρ m after formation of the condensate by dark matter. We will neglect thecontributions due to baryonic matter and (the observable) radiation for simplicity. We take9the time that passed since the start of the condensation t till the present time t be muchsmaller than the lifetime of the condensate i.e. we let ( t − t ) ≪ Γ − to ensure that theenergy density of χ r and χ r is below the observational bounds on dark radiation [30].This condition is expressed in a more concrete form in the next equation. To derive thatequation we make use of the following approximations and equations: We take ρ φ almostconstant from t to t , and make use of (3) and (27) (where ρ R = ρ φ ). We approximatethe integral in (27) as the sum of two parts; the first part being from t to t q where weassume the universe was wholly matter dominated and from t q to t where we assume thatthe universe was wholly ρ φ dominated which is a rough approximation to the evolution ofthe universe. We neglect the radiation dominated era because we neglect the contributionof (the observable) radiation in this study. (In fact even we had included the contributionof the usual observable radiation its effect would turn out to be negligible in the integralgiven below.) Then (27) and (3) imply Ω γ Ω DE ≃ − > Ω r Ω DE = ρ r ( t ) ρ φ ( t ) ≃ Γ Z t t a ( t ′ ) θ ( t − t ′ ) dt ′ ≃ Γ H Z a ( t q ) a ( t ) ( a ′ ) √ Ω DM da ′ + Z a ( t ) a ( t q ) ( a ′ ) √ Ω DE da ′ ≃ . H ≃ . t Γ (70)where we used the fact that the present energy density of a radiation component other thanphotons and neutrinos, ρ r is, at most, at the level of a tenth of the energy density of photons ρ γ [30]; ρ DE is the present energy density for dark energy; Ω’s denote the correspondingdensity parameters, and we have used the numerical values of the density parameters in [9]where Ω DE = Ω Λ and that the time for the beginning of the dark energy dominated era isat about the redshift z = 0 .
65 i.e. at a ( t q )=0.6. One may obtain a similar result by usinga more heuristic approach as well: Provided that Γ ( t − t ) ≪
1, the ratio of the energydensity of the decayed part of the condensate to its energy density at present ρ r ρ DE roughlysatisfies Ω γ Ω DE ≃ − > ρ r ρ DE = 1 − e − Γ ( t − t ) e − Γ ( t − t ) ≃ Γ ( t − t ) (71)In the notation of the preceding sub-subsection we have t < t ≪ t . If φ never hada thermal equilibrium with baryonic matter and photons then the condensation must havestarted before matter dominated era if we want to mimic ΛCDM (because of its observationalsuccess). On the other hand, if φ had thermal equilibrium with baryonic matter and photonsin the past then φ must have decoupled from baryonic matter and radiation before the time0of nucleosynthesis to keep the successful nucleosynthesis scenario of the standard modelintact. In either case t − t must be in the order of the age of the universe i.e. in the orderof 10 seconds. Hence (70) or (71) implies that > seconds. In the light of theseobservations we derive a rough lower bound on the value of the production cross section of φ ’s by using (69). Since > seconds, and after the end of the formation process ofthe condensate (as discussed after (69)) we have β ( a ( t )) r − ρ m − Γ ρ φ = 0, we obtain110 sec. × β < Γ β = ρ m ( t ) ρ φ ( t ) ρ m ( t ) ≃ (cid:18) . . (cid:19) × . ρ crit ≃ . ρ crit (72)Here ρ crit is the critical energy density at present. After using (72) and the expression for β in (38) and the numerical value of ρ crit we may write β = β ′ σ ( t ) v ( t ) E a ( t ) > ( 169 ) × − ( eV ) − ( sec ) − (73)i.e. σ ( t ) cm > (cid:18) EeV (cid:19) cm/secv ( t ) ! a ( t ) β ′ (cid:18) × − (cid:19) (74)It is evident from (74) that one can produce the values of the present dark matter and darkenergy density parameters Ω DM ≃ .
26 and Ω DE ≃ .
69 for a wide range of cross sections.For example if we take v ( t ) = × − c , a ( t ) = 10 − , β ′ = , (cid:16) EeV (cid:17) = 1 then wefind σ ( t ) > × − cm , and σ ( t ) > − cm if (cid:16) EeV (cid:17) = 10 , a ( t ) = 10 − and allother parameters kept the same. We may also check if the formula we have derived for ρ R in this section is consistent with a constant value for ρ φ at present provided that we adoptthe values of parameters given above. In this case one can not directly use the results of thepreceding sub-subsection because in this case γ = Γ H ≪ γ = 3 to simplify the integrals for ρ r . Instead we should go back to the originalequation in (B11) and make use of the fact that ρ m βξ i ≪ i = 1 , ξ i arein the order of H . Therefore (cid:16) d ˜ ρ m ( t ) dt (cid:17) ≃ θ ( t − t ) ρ m β . Then we insert this value in (B12)while we keep the exponential factor, and take the integral, and finally we use γ = Γ H ≪ C R is essentially constant at present, so the same is true for ρ R = ρ φ . C. Some additional remarks for an intermediate state resonance
In the preceding subsection we have given a particle physics description of an intermediatestate with negative equation of state in the conversion of matter to radiation in terms1of resonance particles that form a Bose-Einstein condensate. We have also considered aconcrete model in this context to see the situation more explicitly. However such a specificmodel is not enough to see all aspects of these types of models. We have neglected theera for formation of the condensate since it is expected to be very small compared to itslifetime and to be able focus on the more relevant points. A similar situation is true forthe era of the decay of the condensate since this era corresponds to far future in our model,so phenomenologically is less interesting. Although the study of these eras in detail is notessential in the context of the model we have considered for late time cosmic accelerationit would be relevant in models of inflation, especially in those where the condensate decaysinto usual radiation rather than to dark radiation. Therefore in the following paragraphswe give the overall picture in terms of some simple generic hypothetical energy distributionto see the overall picture more clearly.After adding (6), (7), (22), and using (25) one obtains˙ ρ ( t ) + 3 H " ρ r ( t ) ρ ( t ) + ω R ρ R ( t ) ρ ( t ) ρ ( t ) = 0 (75) ρ ( t ) = ρ m ( t ) + ρ r ( t ) + ρ R ( t )Now we consider some specific cases in the light of the equation (75). Note that ρ R ρ maybe close to 1, especially if we identify ρ R by a resonance or a condensate formed as anintermediate state in the transition of matter to radiation as in the second graph in Figure1. Therefore the effect of ρ R may be sizeable for some time interval in the evolution ofthe universe. Moreover, since ρ R corresponds to some intermediate state in the conversionof matter to radiation it must be localized in time. In the light of these observations weconsider some possible forms for ρ R . This will give us an idea about the possible effects ofconversion of matter to radiation (or vice versa) on the expansion rate of the universe. Infact, to have a concrete, realistic picture one needs to calculate the cross sections and therates of different such process in an expanding universe for different possible scenarios. Weleave this ambitious program to detailed studies in future. In this subsection we give anidea on the range of possibilities by considering some hypothetical choices of ρ R ’s that arelocalized in time, and this will be sufficient for our purpose. First consider the example, ρ R = σ [1 − tanh β ( a − a i )] where σ , β are some constants and a i denotes the scale factor2where ρ R is maximum. Then ˙ ρ R ρ R = − β ˙ a tanh β ( a − a i )]. After using (24) one gets ω R + ∆ ω R = − ˙ ρ R Hρ R − β a tanh β ( a − a i ) − ω R + ∆ ω R → − a → a → a i ,ω R + ∆ ω R ≤ − < a < a i and ω R + ∆ ω R ≥ − a > a i ω R → ∞ as a → ∞ , and ω R + ∆ ω R ≃ .
28 when ( a − a i ) β ≃ ρ R = σ β ( a − a i ) . Then ˙ ρ R ρ R = − β ˙ a ( a − a i )[1+ β ( a − a i ) ] . After using (24)one gets ω R + ∆ ω R = − ˙ ρ R Hρ R − β a ( a − a i )3[1 + β ( a − a i ) ] − a = a i ω R + ∆ ω R = ( ω R + ∆ ω R ) min = − β ( a i ) β ( a i ) ] − ω R + ∆ ω R > ( ω R + ∆ ω R ) min ω R + ∆ ω R → − a → a → a i and ω R + ∆ ω R → −
13 as a → ∞ where the subindex min denotes the minimum value of ω R + ∆ ω R .In both examples we have ω R + ∆ ω R = − a = a i and ω R + ∆ ω R < − a < a i , ω R + ∆ ω R > − a > a i . This is not surprising, it is just a result of the requiring ρ R be localized. Therefore most of the general lines of the above examples are valid in all casesprovided that ρ R is localized in time. Note that both in this set-up and in the standardmodel of cosmology, the universe is radiation dominated at the time just after the time ofcreation of radiation. Moreover ρ R ρ ≃ a ≃ a i . Thus in a model of inflation in thiscontext one may take the ω R + ∆ ω R values obtained for the times close to the time a = a i as a sufficiently realistic of the universe for the time just before creation of radiation. On theother hand the same is not true for other times e.g. for much later times since, at much latertimes, the universe almost exhaustively is composed of components other than radiation e.g.matter (and possibly dark energy as well). Therefore a more realistic modification of thecontent of the universe is needed to extend this scheme to account for early time cosmicaccelerated expansion. However regarding dark energy even these simple examples may givesome insight to the problem: A residual effect of the ρ R that survives may have some effect3for later times. Although tiny, ρ R may still have a tail for large scale factor, a ( t ). The effectof this tail is determined by how well ρ R is localized, and how big ρ R and ω R + ∆ ω R are.In fact such a model is considered in literature [21]. IV. CONCLUSION
In this paper we have studied, as a sub-case of two interacting energy densities, if theaccelerated cosmic expansion may be induced by conversion of extremely non-relativisticparticles to radiation. We have seen that cosmic accelerated expansion can not be obtainedin conversion between matter and radiation through instantaneous interactions. In factthermo dynamical studies give a similar conclusion [31]. It seems that the only way to obtaincosmic accelerated expansion by particle physics interactions is through some intermediatestate with a negative equation of state that forms during the conversion. It seems difficult toobtain the present cosmic accelerated expansion wholly through the usual particle physicsinteractions in this way since the localization scales of corresponding ρ R ’s for the usualparticle physics processes are at the order of atomic scales i.e. at scales much smaller thanthe cosmological scales. Even when they have such an effect, these interactions will firstaccelerate the universe and then decelerate it in the time scale of the interaction time (whichis smaller than ∼ − sec ), hence the net effect would be zero. This type of interactions maybe relevant cosmologically only at early times (if they involve the usual particles) providedthat a significant redshift takes place during their interaction time e.g. during the lifetimeof the resonance particle. A very early time acceleration may be induced by fast out ofequilibrium processes as those given in Figure 1 provided an intermediate state with ω < ρ R and ˜ ρ ( t ) dt are specified and their theoretical origins discussed and whose theresults are confronted with observational data may be considered in future.4 Appendix A: Details of the derivations of (4), (5), (6), (7)
Let at time t < t the energy density of the universe, ρ consists of matter (i.e. dust) andradiation, that is, ρ = ρ m + ρ r (A1) ρ m = ρ m [ a ( t )] , ρ r = ρ r [ a ( t )] (A2)Assume that at t < t < t + ∆ t (where ∆ t is very small) some energy density (∆ ˜ ρ ) is transferred from either of the dust or the radiation to the other, say, from the dust tothe radiation e.g. through some decay or other particle physics process such as in Figure 1.Then the new energy density becomes ρ ( t ) = { ρ m [ a ( t )] − (∆ ˜ ρ ) θ [ t − ( t + ∆ t )] " a ( t + ∆ t ) a ( t ) } + { ρ r [ a ( t )] + (∆ ˜ ρ ) θ [ t − ( t + ∆ t )] " a ( t + ∆ t ) a ( t ) } = ρ m − (∆ ˜ ρ ) θ [ t − ( t + ∆ t )][ a ( t + ∆ t )] [ a ( t )] + ρ r + (∆ ˜ ρ ) θ [ t − ( t + ∆ t )][ a ( t + ∆ t )] [ a ( t )] (A3)where θ ( t ) is the Heaviside function (i.e. unit step function) with θ ( t ) = 0 if t <
0, and θ ( t ) = 1 if t ≥
0. Next assume that at t + ∆ t < t < t + 2∆ t some other energy density(∆ ˜ ρ ) is transferred from the dust to the radiation, and so on. Hence at t = t + n ∆ t theenergy density is ρ ( t ) = ρ M ∆ [ a ( t )] + ρ r ∆ [ a ( t )] (A4)where ρ M ∆ = ρ m − (∆ ˜ ρ ) θ [ t − ( t + ∆ t )] a ( t + ∆ t ) − (∆ ˜ ρ ) θ [ t − ( t + 2∆ t )] a ( t + 2∆ t ) . . .. . . − (∆ ˜ ρ ) n θ [ t − ( t + n ∆ t )] a [ t + n ∆ t )] (A5) ρ r ∆ = ρ r + (∆ ˜ ρ ) θ [ t − ( t + ∆ t )] a ( t + ∆ t ) + (∆ ˜ ρ ) θ [ t − ( t + 2∆ t )] a ( t + 2∆ t ) . . .. . . + (∆ ˜ ρ ) n θ [ t − ( t + n ∆ t )] a [ t + n ∆ t )] (A6)As ∆ t →
0, (∆ ˜ ρ ) i → i = 1 , , ..... ) the energy density in (A4) at t = t > t becomes ρ ( t ) = ρ m ( t ) + ρ r ( t ) (A7) ρ m ( t ) = C m ( t )[ a ( t )] , ρ r ( t ) = C r ( t )[ a ( t )] (A8)5where C m ( t ) = ρ m − lim ∆ t → , n ∆ t → t − t { (∆ ˜ ρ ) ∆ t ! θ [ t − ( t + ∆ t )] a ( t + ∆ t )+ (∆ ˜ ρ ) ∆ t ! θ [ t − ( t + 2∆ t )] a ( t + 2∆ t ) . . . + (∆ ˜ ρ ) n ∆ t ! θ [ t − ( t + n ∆ t )] a [ t + n ∆ t )] } = ρ m − Z t t d ˜ ρ ( t ′ ) dt ′ ! a ( t ′ ) θ ( t − t ′ ) dt ′ (A9)= ρ m − Z t − t d ˜ ρ ( x ) dx ! x =( t − u ) a ( t − u ) θ ( u ) du (A10) C r ( t ) = ρ r + lim ∆ t → , n ∆ t → t − t { (∆ ˜ ρ ) ∆ t ! θ [ t − ( t + ∆ t )] a ( t + ∆ t )+ (∆ ˜ ρ ) ∆ t ! θ [ t − ( t + 2∆ t )] a ( t + 2∆ t ) . . . + (∆ ˜ ρ ) n ∆ t ! θ [ t − ( t + n ∆ t )] a [ t + n ∆ t )] } = ρ r + Z t t d ˜ ρ ( t ′ ) dt ′ ! a ( t ′ ) θ ( t − t ′ ) dt ′ (A11)= ρ r + Z t − t d ˜ ρ ( x ) dx ! x =( t − u ) a ( t − u ) θ ( u ) du (A12)From (A8) we see that (for t > t )˙ ρ m ( t ) = − H ρ m ( t ) + ˙ C m ( t )[ a ( t )] (A13)˙ ρ r ( t ) = − H ρ r ( t ) + ˙ C r ( t )[ a ( t )] (A14)where over dot denotes derivative with respect to t. Eqs. (A13) and (A14) imply that ρ m and ρ r satisfy the following equations˙ ρ m ( t ) + 3 H ρ m ( t ) = ˙ C m ( t )[ a ( t )] (A15)˙ ρ r ( t ) + 4 H ρ r ( t ) = ˙ C r ( t )[ a ( t )] (A16)We see that the energy-momentum tensors for the dust and the radiation in this case areconserved. They have source terms on the right-hand sides of (A15) and (A16). This is notsurprising since there is an energy transfer from the dust to the radiation. If one adds (A15)to (A16) one obtains˙ ρ m ( t ) + ˙ ρ r ( t ) + 3 H (cid:20) ρ m ( t ) + 43 ρ r ( t ) (cid:21) = ˙ C m ( t )[ a ( t )] + ˙ C r ( t )[ a ( t )] (A17)First calculate ˙ C m ( t ), that may be found from˙ C m ( t ) = lim ∆ t → C m ( t + ∆ t ) − C m ( t )∆ t (A18)6In the above formula we employ Eqs. (A10) and (A12) (rather than Eqs. (A9) and (A11))not to deal with variation the θ function that may lead to ambiguity since we do not cover t ′ > t in the integration. Hence C m ( t + ∆ t ) = ρ m − Z t +∆ t − t d ˜ ρ ( x ) dx ! x =( t +∆ t − u ) a ( t + ∆ t − u ) θ ( u ) du (A19) C r ( t + ∆ t ) = ρ r + Z t +∆ t − t d ˜ ρ ( x ) dx ! x =( t +∆ t − u ) a ( t + ∆ t − u ) θ ( u ) du (A20)Let us calculate these explicitly. Z t +∆ t − t d ˜ ρ ( x ) dx ! x =( t +∆ t − u ) a ( t + ∆ t − u ) θ ( u ) du = Z t − t d ˜ ρdx ! x =( t +∆ t − u ) a ( t + ∆ t − u ) θ ( u ) du + Z t − t +∆ tt − t d ˜ ρ ( x ) dx ! x =( t +∆ t − u ) a ( t + ∆ t − u ) θ ( u ) du (A21)For small ∆ t a ( t + ∆ t − u ) = " a ( t − u ) + da ( x ) dx ! x = t − u ∆ t + . . . ≃ a ( t − u ) + 3 a ( t − u ) da ( x ) dx ! x = t − u ∆ t (A22) d ˜ ρ ( x ) dx ! x = t +∆ t − u ≃ d ˜ ρ ( x ) dx ! x = t − u + d ˜ ρ ( x ) dx ! x = t − u ∆ t (A23) Z t − t +∆ tt − t d ˜ ρdu ! ( t + ∆ t − u ) a ( t + ∆ t − u ) θ ( u ) du ≃ d ˜ ρdx ! x = t +∆ t a ( t + ∆ t ) θ ( t − t ) ∆ t ≃ d ˜ ρdx ! x = t a ( t ) θ ( t − t ) ∆ t (A24)Hence we find C m ( t + ∆ t ) = ρ m − Z t − t { " d ˜ ρ ( x ) dx ! x = t − u + d ˜ ρ ( x ) dx ! x = t − u ∆ t × " a ( t − u ) + 3 a ( t − u ) da ( x ) dx ! x = t − u ∆ t θ ( u ) du }− d ˜ ρ ( x ) dx ! x = t a ( t ) θ ( t − t ) ∆ t ≃ C m ( t )7+ (∆ t ) Z t − t { d ˜ ρ ( x ) dx ! x = t − u a ( t − u ) da ( x ) dx ! x = t − u + d ˜ ρ ( x ) dx ! x = t − u a ( t − u ) } θ ( u ) du + (∆ t ) d ˜ ρ ( x ) dx ! x = t a ( t ) θ ( t − t ) (A25)Note that Z t − t { d ˜ ρ ( x ) dx ! x = t − u a ( t − u ) da ( x ) dx ! x = t − u + d ˜ ρ ( x ) dx ! x = t − u a ( t − u ) } θ ( u ) du = Z t − t { ddx " d ˜ ρ ( x ) dx ! a ( x ) x = t − u } θ ( u ) du = Z tt { ddx " d ˜ ρ ( x ) dx ! a ( x ) θ ( t − x ) dx = − d ˜ ρ ( x ) dx ! a ( x ) | x = tx = t θ ( t − t ) (A26)After using (A25) and (A26) in (A18) we find˙ C m ( t ) = − d ˜ ρ ( t ) dt ! a ( t ) θ ( t − t ) (A27)In a similar way one may find ˙ C r ( t ) as˙ C r ( t ) = Z t − t { d ˜ ρ ( x ) dx ! x = t − u a ( t − u ) da ( x ) dx ! x = t − u + d ˜ ρ ( x ) dx ! x = t − u a ( t − u ) } θ ( u ) du + d ˜ ρ ( x ) dx ! x = t a ( t ) θ ( t − t )= d ˜ ρ ( t ) dt ! a ( t ) θ ( t − t ) (A28) Appendix B: An example for the evolution of the energy densities of matter, aresonance with ω = − , and radiation as a function of scale factor In this appendix we derive the evolutions of C m , ρ R , and C r and the related quantitiesin the case ω R = − , r = 7 andInitially s = − , in between s = 0 , at late times s = − t < t < t , s = − i.e. H = ξ a − where ξ is some constant. Then a cosmologicalconstant-like resonance dominated universe where s = 0 for t < t < t , Finally, theradiation dominated era at the end where s = − t < t . For simplicity We will assume8that β is constant in all eras. We give C m , ρ R , C r in these eras below by using the formulasderived in the text. ρ m and ρ r are evident from C m and C r . For simplicity we assume abruptchanges in the equations of states as one passes from one era to the other although in a morerealistic case the changes would be smooth. C m for t < t may be directly found from (43) for r = 7, s = s = − , ξ = ξ . Theresult is C m = ρ m βρ m ξ θ ( t − t ) (cid:16) a r − s − − a r − s − (cid:17) (B2)where a = a ( t ), a = a ( t ). d ˜ ρ m ( t ) dt in this case directly follows from Eq.(44) d ˜ ρ m ( t ) dt ! = 81 θ ( t − t ) ρ m βξ {− ξ + β ρ m θ ( t − t ) h a ( t ) − a ( t ) i } (B3)After using Eq.(21), and Eq.(B3) we get C R ( t ) = 814 θ ( t − t ) ρ m βξ − Z aa e Γ( t ′ − t ) a ′ da ′ {− ξ + β ρ m (cid:20) a − a ′ (cid:21) } (B4)where a = a ( t ), a = a ( t ), and we have used dt = daa H . One may let | Γ( t − t ) | ≃ a − a ≃ ρ R have just started. Hence(B4) may be approximated by C R ( t ) = 23 θ ( t − t ) ρ m βξ (cid:20) a − a (cid:21) (B5) ρ R ( t ), by (20), becomes ρ R ( t ) = e − Γ( t − t ) C R a ≃ θ ( t − t ) ρ m βξ a ( t ) (cid:20) a − a (cid:21) (B6)where t < t < t (B7)The corresponding C r is found by using (27) and (B6) as C r ( t ) = 2Γ ρ m βξ Z aa ( a ′ − a a ′ ) da ′ = Γ ρ m β ξ a − a a + 35 a (B8)At later times we assume the condensate of the resonance dominates, so s = 0 i.e. H = ξ .To determine ρ R we should first find C m in this case. To find C m in this case we should9divide the integral in (42) into two parts; one for the era when s = − between t and t ,and then the era when s = 0 between t and t . Then the corresponding C m ( t ) is found as − C m ( t ) = − θ ( t − t ) β " ξ Z a a da ′ a ′ − r + s + 1 ξ Z aa da ′ a ′ − r + s − C m ( t )= β θ ( t − t ) { ξ (4 − r + s ) " a − r + s ( t ) − a − r + s ( t ) + 1 ξ (4 − r + s ) " a − r + s ( t ) − a − r + s ( t ) − ρ m = { ρ m β [ θ ( t − t ) ξ (4 − r + s ) (cid:16) a − r + s − a − r + s (cid:17) a − r + s a − r + s + θ ( t − t ) ξ (4 − r + s ) (cid:16) a − r + s − a − r + s (cid:17) a − r + s a − r + s ] − ξ ξ (4 − r + s )(4 − r + s ) a − r + s a − r + s a − r + s a − r + s }× h ρ m ξ ξ (4 − r + s )(4 − r + s ) a − r + s a − r + s a − r + s a − r + s i − (B9)Hence we find C m in this case as C m ( t ) = ρ m { − ρ m β [ θ ( t − t ) ξ − (4 − r + s ) − ( a r − − s − a r − − s )+ θ ( t − t ) ξ − (4 − r + s ) − ( a r − − s − a r − − s )] } − (B10)Thus, the corresponding d ˜ ρ m ( t ) dt for r = r = 7, s = − , s = 0, and t > t is d ˜ ρ m ( t ) dt ! = − ˙ C m ( t ) a ( t )= θ ( t − t ) ρ m β { β ρ m (cid:20) θ ( t − t ) ξ − ( a − a ) + θ ( t − t ) ξ − ( a − a ) (cid:21) } (B11)where we have skipped the term proportional to the delta function δ ( t − t ) since it doesnot contribute to C R below. C R ( t ) = Z t t dt ′ d ˜ ρ m ( t ′ ) dt ′ ! θ ( t − t ′ ) e Γ( t ′ − t ) + Z tt dt ′ d ˜ ρ m ( t ′ ) dt ′ ! θ ( t − t ′ ) e Γ( t ′ − t ) ≃ ρ m βξ (cid:20) a − a (cid:21) + θ ( t − t ) ρ m βξ − Z aa (cid:16) a ′ a (cid:17) γ da ′ a ′ { A + Bθ ( t − t ) a ′ } (B12)where γ = Γ H , a ( t )= e Ht , and A = 1 + 13 β ρ m (cid:20) ξ − ( a − a ) − ξ − a (cid:21) (B13) B = 13 ξ − β ρ m (B14)0In general the result of (B12) is complicated, so analyzing the result is not easy to see.However one may have an idea on the form of ρ R by analyzing some specific cases. Onemust have Γ > H to insure that the system stays coupled, so the decay may take place.One must also take Γ in the same order of magnitude as the Hubble parameter H to insurethat the decay process is relevant at cosmological scales. In the light of these observationswe may, for example, let γ = 3. Hence (B12) becomes C R ( t ) = 23 ρ m βξ (cid:20) a − a (cid:21) + θ ( t − t ) ρ m β Bξ (cid:20) A − A + B a (cid:21) (B15)It is evident from (B15) that C R approaches the constant value ρ m a − and its rate ofincrease decreases as the scale factor a increases, as expected. ρ R ( t ), by (20), becomes ρ R ( t ) = (cid:18) a a (cid:19) " ρ m βξ { a − a } + θ ( t − t ) ρ m βBξ { A − A + B a } (B16)where t < t < t here the (cid:16) a a (cid:17) term in the front of (B16) is due to the e − Γ( t − t ) term in (20). The corre-sponding C r is found as C r ( t ) = Γ (cid:20)Z t t ρ R ( t ′ ) a ( t ′ ) θ ( t − t ′ ) dt ′ + Z tt ρ R ( t ′ ) a ( t ′ ) θ ( t − t ′ ) dt ′ (cid:21) = Γ ρ m β ξ a − a a + 35 a + Γ ρ m β a ξ Z aa " ξ { a − a } + 1 Bξ { A − A + B a ′ } da ′ = Γ ρ m β ξ a − a a + 35 a + 2Γ ρ m β a ξ ξ ( a − a )( a − a )+ Γ ρ m β a Bξ [ a − a A + (6 A B ) − { √ tan − − B A a √ − √ tan − − B A a √ − A + B aA + B a + ln A − A B a + B a A − A B a + B a } ] (B17)The rate of change C r in (B17), and if it increases for reasonable values of parameters isnot evident from(B17) since the form of the expression is rather complicated. Thereforeone may check the rate of change of C r in (B17) for a few values of phenomenologicallyviable sets of parameters. The results of the corresponding C r versus a plots suggest that C r increases as ∝ a with the scale factor in this era, t < t < t . For example, for ρ m ≃ ρ c = 9 × − g cm − , β ′ = , σ ( t ) = 10 − cm , v ( t ) = 10 cm/s , n = 4 × − cm − (where β ′ , σ ( t ), v ( t ) are defined in (30) and n is the number density of matter particles attime t ) gives ξ = q π G ρ m ≃ . × − s − , βρ m =10 − s − . Moreover one may relate ξ to ξ by ξ = a ξ where we have used the fact that at time t the total energy density in aunit comoving volume a lost by the matter i.e. ρ m = ξ π G is equal to the one obtained bythe resonance i.e. to ξ a π G . ξ = a ξ implies that ξ > . × − s − . Therefore A ≃ B ≃
0, and from the definition of B we have ξ B = βρ m . In this approximation the lasttwo lines in (B17) cancel after expanding tan − and ln by Taylor expansion and keeping theleading order terms and then letting B = 0, A = 1. Thus (B17) may be approximated by C r ( t ) ≃ Γ ρ m ξ ρ m β ξ a − a a + 35 a + ρ m β a ( a − a )3 ξ ( a − a ) (B18)We see that C r continues to increase in the era, t < t < t while its rate of increase getssmaller as expected.Finally we consider the last era of the conversion of matter to radiation through formationof a resonance i.e. the radiation dominated era after conversion of most of matter to radiationi.e. when s = −
2. We find C − m in this case in a similar way as done in (B9), that is,1 C m ( t ) = βξ Z a a da ′ a ′ − r + s + βξ Z a a da ′ a ′ − r + s + 1 C m ( t )+ θ ( t − t ) βξ Z aa da ′ a ′ − r + s (B19)Then C m for t > t is found as C m ( t ) = ρ m A ′ + θ ( t − t ) B ′ a (B20)where A ′ , B ′ are some constants that, for r = r = r = 7, s = − , s = 0, s = −
2, givenby A ′ = 1 + 13 βρ m (cid:20) ξ − (cid:18) a − a (cid:19) + ξ − (cid:16) a − a (cid:17)(cid:21) − a θ ( t − t )5 ξ , B ′ = 15 ξ − β ρ m (B21)The corresponding d ˜ ρ m ( t ) dt is d ˜ ρ m ( t ) dt ! = − ˙ C m ( t ) a ( t ) = θ ( t − t ) ρ m B H a ( A + B a ) (B22)2where we have skipped the term proportional to the delta function δ ( t − t ) since it doesnot contribute to C R below. Thus C R is found as C R ( t ) = 23 ρ m βξ (cid:20) a − a (cid:21) + θ ( t − t ) 13 Bξ ρ m β " A − A + B a + Bρ m θ ( t − t ) Z aa e Γ2 ξ ( a ′ − a ) a ′ da ′ { A ′ + B ′ a ′ } (B23)where we have used the fact that H = ξ a − implies t − t = ξ ( a − a ). The integral in(B23) can not be evaluated exactly even with use of Mathematica. However one may givean approximate result as follows. At the final stages of the resonance we have Γ( t − t ) ∼ x = 1 − Γ( t − t ) <<
1. Therefore e Γ( t − t ) = e − x = e e − x ≃ e (1 − x ) = e Γ( t − t ) (B24)Hence the integral in (B23) may be approximated as Z aa e Γ2 ξ ( a ′ − a ) a ′ da ′ { A ′ + B ′ a ′ } ≃ Z aa e Γ2 ξ ( a ′ − a ) a ′ da ′ { A ′ + B ′ a ′ } ≃ e Γ2 ξ Z aa a ′ da ′ { A ′ + B ′ a ′ } (B25)The result of (B25) is rather complicated so that it makes it difficult to get concrete resultsfrom the expression. However one may draw its plots for several values of A ′ and B ′ . Theindefinite form of (B25) for all tried values of A ′ , B ′ give flat graphs for almost all valuesof a ( t ). For example, for reasonable values of parameters discussed after Eq. (B17) andusing the conservation of energy in a comoving volume at time t i.e. ξ a = ξ a − i.e. ξ = ξ a = ξ a − a if the resonance lives a sufficiently long time so that a >> a then we have ξ < ξ < ξ . Otherwise ξ and ξ must be comparable. Therefore A ′ ≃ B ′ ≃
0. We have given the result of this integration for two three different ratios of A ′ and B ′ in the figures Fig.2, Fig.3, Fig.4. Therefore the result of (B25) is almost zero for allphenomenologically viable cases. In other words in this final era i.e. for t > t C R does notincrease any more and ρ R decays exponentially ρ R ( t ) = e Γ( t − t ) C R ( t ) ≃ ξ ρ m β e Γ ξ (cid:16) a − a (cid:17) − a − a + θ ( t − t ) a B −
11 +
B a ! (B26)3Then the corresponding C r becomes C r ( t ) ≃ Γ ρ m ξ ρ m β ξ a − a a + 35 a + ρ m β a ( a − a )3 ξ + a ( a − a ) + C ′ Z aa a ′ da ′ a ′ − a (B27)where C ′ = 2 ρ m β e ξ (cid:18) a − a (cid:19) + a B −
11 +
B a ! for t > t (B28)After integration of the integral in (B27) we obtain C r ( t ) ≃ Γ ρ m ξ ρ m β ξ a − a a + 35 a + ρ m β a ( a − a )3 ξ + a ( a − a ) + C ′ " a (cid:16) a − a (cid:17) + a − a a a − a a − a ! (B29)We notice that there is a term proportional to a . Although the coefficient of this muchsmaller than the others it is anomalous since it makes the production of radiation seem getsfaster while it should be go to a constant value at the end of the decay. In fact this anomalousbehavior is due to identifying the time when Γ( t − t ) ≃ s = 0 survive at this time and the increase in C r due todecay of resonance makes it effectively behave as an energy density with equation of statesmaller than . Therefore taking s = − s = − t − t ) ≃ t − t )4 ∼ ⇒ x = 1 − Γ( t − t )4 << ⇒ e Γ( t − t ) = e − x ) = e (cid:16) e − x (cid:17) ≃ e (1 − x ) = e Γ( t − t )4 ! = e Γ8 ξ ! ( a − a ) (B30)After substituting (B30) in the first term in (B25) and using Mathematica to evaluate theintegral and draw the corresponding plots we get almost zero for the result of the integrationas before for reasonable values of parameters mentioned before. Next we use (B30) to find ρ R . Then (B26) is replaced by ρ R ( t ) = e Γ( t − t ) C R ( t ) ≃ ξ ρ m β e Γ ξ (cid:16) a − a (cid:17) − × (cid:18) a − a (cid:19) + θ ( t − t ) a B −
11 +
B a ! for t > t (B31)Then the corresponding C r becomes C r ( t ) ≃ Γ ρ m ξ ρ m β ξ a − a a + 35 a + ρ m β a ( a − a )3 ξ + a ( a − a ) + C ′′ Z aa a ′ da ′ ( a ′ − a ) (B32)where C ′′ = 4096 ξ ρ m β e Γ ξ (cid:18) a − a (cid:19) + a B −
11 +
B a ! for t > t (B33)The result of the integral in (B32) is Z aa a ′ da ′ ( a ′ − a ) = a − a a + 3 a a − a ) − a − a a + 3 a a − a ) (B34)which goes to a constant value (i.e. C r goes to a constant value) as a increases as expected.One may check that later times (that may be taken as the time that satisfy Γ( t − t ) ≃ k where k > C r goes to the constant value even faster. ACKNOWLEDGMENTS
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Decaying particles do not ”heat up” the Universe , Phys. Rev.D , 681 (1985). p (cid:0)(cid:0)(cid:0)✒(cid:0)(cid:0)(cid:0) p ❅❅❅❘❅❅❅ p ❅❅❅❘❅❅❅ p (cid:0)(cid:0)(cid:0)✒(cid:0)(cid:0)(cid:0) p (cid:0)(cid:0)(cid:0)✒(cid:0)(cid:0)(cid:0) p ❅❅❅❘❅❅❅ p FIG. 1. The diagram on the left-hand side shows the decay of a particle with momentum p intotwo particles with momenta p and p e.g. the decay of a non-relativistic particle to two relativisticparticles while the diagram on the right-hand side shows the (inelastic) collision of two particleswith momenta p and p into two other particles with momenta p and p e.g. the collision of twonon-relativistic particles into two relativistic particles through formation of an intermediate state FIG. 2. The value of the integral in Equation (B25) as a function of scale factor for A ′ = 1, B ′ = 10 − FIG. 3. The value of the integral in Equation (B25) as a function of scale factor for A ′ = 1, B ′ = 1 FIG. 4. The value of the integral in Equation (B25) as a function of scale factor for A ′ = 1, B ′ = 0 ..