Inflation-Dark Matter unified through Quantum Generation
We unify inflation and dark matter via a single scalar field . One of the main difficulties for this unification is that between inflation and dark matter one needs a successful reheating process and a long lasting period of radiation. Therefore the amount of energy density in the inflaton-dark matter field must be fine tune after reheating to give dark matter. Here we show an alternative scheme in which the inflaton decays completely, disappearing entirely from the spectrum. However, at low energies, before matter-radiation equality, the same interaction term that leads to the inflaton decay, regenerates . An essential feature is that the transition between the intermediate radiation dominated to the dark matter phase is related to a quantum generation of the scalar field instead to purely classical dynamics. Thanks to this quantum transition the inflation-dark matter unification can take place naturally without fine tuning. The unification scheme presented here has three parameters, the mass of the dark matter particle , the inflation parameter and the coupling for the inflaton interaction. Phenomenology fixes the value for and gives a constraint between and , leaving only the mass of the dark matter particle as a free parameter. These same three parameters are present in models with inflation and a dark matter wimp particle but without unification. Therefore our unification scheme does not increase the number of parameters and it accomplishes the desired unification between inflaton and dark matter for free.
Inflation has now become part of the standard model of the cosmology inflationarycosmology (). With new data coming soon, and in particular with the Planck satellite mission, the different inflationary models will need to pass a strong test. Inflation set up the initial perturbations from which gravity forms the large scale structures LSS in the universe. At the same time, a huge amount of evidence is gathering in structure formation via the large galaxy maps that are currently under way sdss (). These surveys have detected not long ago the Baryon Acoustic Oscillation BAO sdss (), which have the same origin as the CMB wmap () but are measured via galaxy distribution. The formation structure requires the existence of dark matter and therefore the nature of inflation and of dark matter are then essential building blocks to understand the observations.
Inflation is associated with a scalar field, the ”inflaton”, and the energy scale at which inflation occurs is typically of the order of inflationarycosmology () but it is possible to have consistent inflationary models with as low as lowinflation (). On the other hand dark matter is described by an energy density which redshifts as and is described by particles where its mass with its temperature. These particles can be either fermions or scalar fields. In the case of scalar fields the scalar potential must be and independently on the value of the its mass the classical equation of motion ensures that .
Here, since we want to unify inflation with dark matter we will assume that DM is made out of the same field, a scalar field . inf-dm (). A scalar field can easily give inflation and DM if the potential is flat at high energies and at low energies the potential approaches the limit . However, most of the time our universe was dominated by radiation. Therefore, any realistic model must not only explain the two stages of inflation and dark matter but must also allow for a long period of radiation domination. Typically the inflaton decays while it oscillates around the minimum of its quadratic potential inflationarycosmology (). If the inflaton decay is not complete then the remaining energy density of the inflaton after reheating must be fine tuned to give the correct amount dark matter.
In order to avoid a fine tuning problem we follow the quantum generation work presented in fabios (). In fabios () the quantum re-generation process was used to unify inflation with dark energy but here we use the same idea to unify inflation with dark matter. To reheat the universe we couple to a relativistic field via an interaction term . This field may be a standard model ”SM” particle, as for example neutrinos, but it could also be an extra relativistic particle not contained in the SM. An extra relativistic degree of freedom is fine with the cosmological data rel.d (). After inflation the decays into this field at and to reheat the universe with ”SM” particles we couple them to at high energies, e.g. the same energy as the decay of . Thermal equilibrium ”TE” between and SM particles will be maintained as long as and the SM particles remain relativistic. At low energies we show that we can re-generate the dark matter field at using the same interaction term as for the high energy inflaton decay. The appearance of the dark matter field at late times is then via a quantum transition and not due to a classical evolution. A requirement on the mass of the inflaton-dark matter field is that so the simplest potential will not work.
The unification scheme presented here has three parameters, the mass of the dark matter particle , the parameter in the inflation potential and the coupling for the inflaton decay. Density perturbations normalized to COBE fixes the value for and the correct amount of dark matter determines the cross section of dark matter at decoupling (wimp particle) gives a constraint between and , leaving only the mass of the dark matter particle as a free parameter. We will show that the same coupling strength that gives the inflaton decay gives the dark matter re-generation at low scales and sets in combination with the wimp decoupling cross section. These same three parameters are present in models with inflation and a dark matter wimp particle wimp () but without unification. This implies that our unification scheme does not increase the number of parameters and it accomplishes the desired unification between inflaton and dark matter for free.
Ii General Framework
Our starting point is a flat FRW universe with the inflaton-dark matter field coupled to a relativistic scalar . This field can be either a fermion field or scalar field and we only require that it is relativistic at least for energies larger than the mass of the dark matter particle . For presentation purposes we take as a scalar field, but generalizing this work to a fermion field is straightforward and the fermion field could well be part of the standard model SM as for example a neutrino. We take the lagrangian , where is the standard model SM lagrangian and
are the scalar potentials for and is the interaction potential. The classical evolution of and are given by the equations of motion,
with , a prime denotes derivative w.r.t. , and we take natural units . The mass of is given by
and it is in general a field and time dependent quantity.
ii.1 Inflaton-Dark Matter Potential
ii.1.1 Inflaton Potential
There are many potentials that lead to an inflation epoch. Inflation with a single scalar field can be classified in small or large field models. Small fields are potentials that inflate for values of the inflaton as in new inflation models, e.g. , while large field models are when inflation occurs for as in chaotic models, e.g. . Here we will assume the simplest inflaton potential because we are more interesting in showing how the inflaton-dark matter unification scheme takes place in the context of quantum re-generation process. However, it is simple to work with other inflaton potentials.
The potential must satisfy at the inflation scale the slows roll conditions and the constraint on the energy density perturbation normalized to COBE wmap ()
For example for chaotic potentials or inflation occurs for and the value of the mass and the dimensionless parameter and the scale of inflation are given by
ii.1.2 Dark Matter Potential
To obtain Dark Matter at low energies the scalar potential must have the limit
with a constant mass term. A scalar field with potential redshifts as with (for n-even) and only for do we have a energy density redshifting as matter, while for mio.Q (). The constraint on is that
where we used that the pressure vanishes and is the present time dark matter density (from here on the subscript gives present time quantities).
ii.2 Interaction Term
The interaction term between and should allow the following processes:
i) at high energies the inflaton must decay into efficiently,
ii) at much lower energies it should re-generate (with ),
iii) it must inhibit the decay of dark matter particle .
The point (iii) is needed because dark matter must last for a long period of time, however, depending on the ratio of its decay it may be relax and we could have for example a dark matter dark energy interaction. However, we will not follow this interesting path here and will be presented elsewhere dm-de.fabio (). There are many possible interaction terms between and and generalization of our work presented here is straight forward. If we take only renormalizable terms with dimensionless constant we have with . We show in the Appendix that the terms and , i.e. and do not satisfy the conditions (i-iii). The term does not allow for to decay into if and our working hypothesis has a massive inflaton field and a relativistic field , i.e. . On the other hand the term does not work because it gives a decaying dark matter particle. Therefore, we are left with the interaction term
To produce all SM particle the field must be either part of the SM or it must couple to at least one field of the SM. It is standard to assume the interaction term as inflationarycosmology ()
where are SM particles. We will choose such that the transition takes place as soon as the are produced. If the fields acquire a large mass then they will decouple from at since below this temperature are exponentially suppressed and will be smaller than one. However, as long as remains relativistic the temperature redshifts as and since it was in thermal equilibrium with the SM we can determine with and the number of relativistic degrees of freedom of the SM at decoupling.
ii.2.1 Interaction rates
The differential transition rate is given by G ()
where is the initial (final) momentum, is the volume (normalized to one particle per volume) and is the transition amplitude. The conservation of energy-momentum requires that initial and final energies are equal, and , and this ensures that the amount of homogeneity of the universe is preserved by the interaction. In a process of initial particles with the same energy and a final state consisting of particles with the same energy and one has and the differential transition rate is given by
ii.2.2 Interaction rates for
The decay rate for using eq.(15) with is given by
with and where is the cross section times the relative velocity of the initial particles with
and the in last equality in eq.(18) we used eq.(12). If one of the initial particles becomes non-relativistic then in eq.(17) is exponentially suppressed by with . However, if the two initial particles are relativistic than the number density is given by with and and eq.(17) becomes
with , and for a real scalar field.
ii.2.3 Decay Rate for
The decay process takes place as long as and the decay rate for is given by
with and it is field and time dependent trough the terms . The field in eq.(20) may evolve with time or it may be constant if the scalar potential gives a nonvanishing v.e.v. , as in new inflation models. In order to have a decay we require that
is larger than one, . If and is constant then the decay in eq.(57) will be efficient since is constant and . If however, the scalar potential is , as for dark matter, than the decay is not efficient since and . In general, the evolution of , and will depend on the choice of scalar potential .
ii.3 Decay Efficiency
The interaction or decay process depends on the transition rate and and it takes place if the ratio . The functional form of depends on the interaction term and it may be field and time dependent. The classical evolution of the fields depends on the scalar potentials and it is then the combination of and which determines the transition process.
The inflaton decay can be efficient or not efficient. The process is not efficient when for , where is the decoupling time when . In this case we will have a remanent energy density . The energy density will evolve classically depending on the form of the potential . If the potential evolves as matter then the amount of energy density at decoupling is easily determined and it is given by
with and the present time relativistic energy density. Clearly one requires a huge amount of fine tuning in the value of at a high decoupling energy since our universe had a large radiation domination epoch. We conclude that if we only take into account a classical evolution of after reheating the inflaton-dark matter unification requires a large fine tuning of initial conditions.
On the other hand the transition process is efficient when the inflaton decays completely and this requires that for , where is the time when . In this case the particles decay completely and disappear from the spectrum and . A simple example is when is constant since and . However if becomes smaller than one for then we will also say that the decay is efficient if the residual energy density is subdominant. For example if redshifts as matter then that the decay is efficient if for with the dark matter energy density. An efficient decay would clearly not allow to account for dark matter.
Iii Inflaton-Dark Matter Unification with Quantum Generation
iii.1 Generic Quantum Transitions
Another possibility to achieve inflaton-dark matter unification is if the particles are re-generated via a quantum transition fabios () at some late time but before , the matter-radiation equivalence scale factor. In this case, the decay at high energies , i.e. below inflation, must be efficient and disappears from the spectrum. At much lower scales , with , the can be re-generated by .
After inflation the energy of the particles is since the momentum redshifted with the expansion of the universe, i.e. with the number of e-folds of inflation. If decays into at we will produce relativistic particles with energy . On the other hand at low energies when we have the inverse process of production from we must have that the energy of the particles must satisfy and therefore we must have
which implies that the mass of after inflation must be much larger than at generation scale . Since clearly the simplest inflationary potential , with constant, will not work. However, potentials such as or of the new inflation type, e.g. , where the field rolls down a flat region for small and then oscillates around the minimum of the potential at may work in this scenario.
Iv Inflaton-Dark Matter Unification Model
We will work out the inflation-dark matter unification through a simple example. Of course, the whole scheme is much more general and other inflaton-dark matter potentials or interaction terms may be used, however in all cases for dark matter. We take the following potential
with a mass
and we coupled with via the interaction term . For
we have . The energy at is
So, if the potential dominates at high energies, during inflation, while at low energies, when dark matter prevails. If either potential or dominates then its classical evolution is with and with . From eq.(9) we know that inflation requires and inflation ends at an energy and a mass and . At low energies the mass of the dark matter particle is given by and for CDM the mass of must be while warm DM requires a smaller mass with warm ().
We will use in this model the interaction term
This term will allow the inflaton to decay efficiently after inflation at high energies into the relativistic particles and will disappear from the spectrum (at most is negligible). At a much later time the same interaction term in eq.(28) will produce relativistic particles at energies below with . Eventually the particles become non-relativistic and they will decouple from as a WIMP particle. The constraint to give the correct amount of dark matter today fixes the cross section . Finally, since becomes massive we also show that at low energies does not decay into and allows for dark matter to dominate a long period as in a standard cosmological scenario. We show in fig.1 the evolution of the inflaton-dark matter and relativistic energy densities. We see that for a long period of time the field disappears from the spectrum from the inflaton decay to the re-generation scale. Since the is re-generated while relativistic redshifts as radiation first and then when it becomes non-relativistic it redshifts as matter and decouples from as any WIMP particle.
iv.1 Quantum Transitions
The inflaton-dark matter field should decay efficiently at high energies, after inflation, to reheat the universe but at low energies when dark matter dominates we do no longer want the to decay since the period of dark matter domination must last from to when dark energy starts to dominate.
iv.1.1 Decay at Inflation:
with . Using we have
where is an energy scale which depends on the field and
is a constant quantity with energy dimensions and set the scale of the decay. We have a decay for energies . The evolution of and grows with time giving an efficient decay. Once the decay of takes place and does not stop (as long as dominates ).
iv.1.2 Quantum Re-generation:
If the field decays completely after inflation than there will no left to account for dark matter. In order to re-generate the field we follow fabios () and we use the same interaction term as for the inflaton decay but now the universe contains relativistic particles and no particles. As long as the energy of the relativistic particles is larger than the mass of , i.e. , we can produce particles. If then both fields are relativistic and the transition rate for the process is given by eq.(19) with
with , and with . We have taken in eq.(32) that is proportional to since is relativistic. The process takes place for energies of the relativistic particles below the constant scale with or
Since the particles are relativistic and they are in thermal equilibrium with SM particles we have . As long as we will , but once we reach the region with the two fields will decouple since will be exponentially suppressed.
iv.1.3 Non-relativistic Decoupling:
If two relativistic particles are in thermal equilibrium and one (in our case ) becomes non-relativistic then the density number is exponentially suppressed and and decouple. This is just the standard WIMP particle decoupling. The transition rate for a process is given by eq.(17)
In order to have the correct amount of dark matter a WIMP must decouple at wimp ()
For the transition this implies a cross section
with with at . For energies the fields and are no longer coupled and evolves classically as matter with . If , where is the radiation-matter equality, the decoupling takes place while the universe is radiation dominated and the constraint on warm dark matter sets a lower scale warm ().
iv.1.4 Dark Matter Decay?:
We have seen that at the field ceases to maintain thermal equilibrium with through the process. However, the field may decay into since . Of course, we do not want to decay since it must account for dark matter. In this case the decay rate is the same is in eq.(16)
but the mass is now constant and evolves as since now the potential that dominates is and . For radiation dominated epoch we have and
with . Since , then we can write
We will not have a decay for , i.e. energies below the constant energy ,
so that after decoupling at there is no decay.
iv.2 Universe Reheating
The reheating of the universe takes place via a process (or ) with a cross section for relativistic particles (we take the same strength for the and ) and an interaction rate
with , and . For we have and . Clearly eq.(45) maintains a TE for . A good choice of is such that reheating takes as soon as the particles are produced at with at but if can take any values as in the range which is the lower limit for reheating lowinflation ()
iv.2.1 Summary of Energies
We present the different energy scales relevant in the process of our inflation-dark matter unification scheme. From high energy to low energy we have the following energy scales: Inflation occurs at , then decays efficiently into at via the interaction term and disappears from the spectrum. Using the same interaction term the field is re-generated at a much lower scale . The field becomes none relativistic while in thermal equilibrium with and decouples at . We also show that below the field does not decay again into with , which ensures that after thermal decoupling the field does not further decay. We have then the following order of energies
SM particles are produced at via the coupling with and the constraint is lowinflation ().
Concerning the inflaton-dark matter unification scheme we have 3 different parameters in the potential and a coupling between and . The seven energies in eqs.(49)-(IV.2.1) are given in terms of these three parameters. Inflation fixes one parameter, , and the amount of dark matter today gives a constraint between and . We are left with one single free parameter which we take it to be the mass . We show in fig.(2) the dependence of the on and in table 1 we give the values for . We see that and that the values of and all other energies are phenomenologically viable. This implies that it is feasible to implement the inflation-dark matter unification. We would like to point out when , or even larger than this only means that as soon as ends inflation at , decays immediately since the condition for its decay is satisfied. We resume the definitions and values of these energies