Cosmology with Ricci dark energy
We assume the cosmological dark sector to consist of pressureless matter and holographic dark energy with a cutoff length proportional to the Ricci scale. The requirement of separate energy-momentum conservation of the components is shown to establish a relation between the matter fraction and the (necessarily time-dependent) equation-of-state parameter of the dark energy. Focusing on intrinsically adiabatic pressure perturbations of the dark-energy component, the matter perturbations are found as linear combinations of the total energy-density perturbations of the cosmic medium and the relative (nonadiabatic) perturbations of the components. The resulting background dynamics is consistent with observations from supernovae of type Ia, baryonic acoustic oscillations and the differential age of old objects. The perturbation dynamics, on the other hand, is plagued by instabilities which excludes any phantom-type equation of state. The only stable configuration is singled out by a fixed relation between the present matter fraction and the present value of the equation-of-state parameter of the dark energy. However, this instability-avoiding configuration is only marginally consistent with the observationally preferred background values of the mentioned parameters.
Despite the many efforts to consistently explain the results of the observations of supernovae of type Ia (SNIa) in SN (), the physical nature of the dark sector of the Universe, assumed to consist of dark matter (DM) and dark energy (DE), remains largely mysterious. The favored cosmological model is the -cold-dark-matter (CDM) model which also serves as a reference for alternative approaches to the DE problem. Grosso modo, the CDM model does well in fitting most observational data (see, e.g., the recent WMAP 9 results wmap9 ()). Nevertheless, there is an ongoing interest in alternative models within General Relativity (GR) itself and beyond it. Although no serious contender seems to be around at the present time, these efforts continue to make sense not only because of the notorious cosmological constant and coincidence problems (see, e.g. problems ()) but also to test as many potential deviations from the “standard” description as possible in order to constrain additional parameter sets which are usually introduced in alternative approaches in order to quantify these deviations. Typically, these approaches “dynamize” the cosmological constant in terms of scalar fields or fluids with a generally time-dependent equation of state (EoS). Constraining a potential time dependence by the data, e.g., is then crucial for a comparison with the CDM model. Holographic models represent a specific class of dynamic approaches to the DE problem. These models are characterized by a relation between an ultraviolet cutoff and an infrared cutoff cohen (); li (); Hsu (). Such relation guarantees that the energy in a given volume does not exceed the energy of a black hole of the same size. The infrared cutoff has to be a cosmological length scale. For the most obvious choice, the Hubble radius, only models in which dark matter and dark energy are interacting with each other also nongravitationally, give rise to a suitable dynamics DW (); HDE (). Following li (), there has been a considerable number of investigations based on the future event horizon as cutoff scale futureEH (). However, all models with a cutoff at the future event horizon suffer from the serious drawback that they cannot describe a transition from decelerated to accelerated expansion. A future event horizon does not exist during the period of decelerated expansion. A further option that has received attention more recently and which will be the subject of the present paper is a model based on a cutoff length proportional to the Ricci scale. The role of a distance proportional to the Ricci scale as a causal connection scale for perturbations was noticed in brustein (). As a cutoff length in DE models it was first used in gao (). Afterwards, investigations along this line have been carried out in cai (). In xuliluchang (); xuwang (); xinzhang (); rong () observational constraints were obtained on the basis of which Ricci DE was compared with the CDM model. The dynamics of perturbations was considered in fengli (); karwan (); yuting (). A number of studies performed in yi (); zhang3 (); ivdi (); luis11 (); tian-fu (); jamil1 (); zhenhui (); luis (); jamil2 (); luis13 () include interactions between DM and Ricci DE. A relation to quantum field theory has been claimed in broda (). Various generalizations rely on the cutoff introduced in granda (). Other previous work in the field includes statef (); fengzhang (); xululi (); granda2 (); suwa (); feiyu (); yuzhang ().
In the present paper we reconsider the dynamics of a two-component system of pressureless DM and Ricci-type DE both in the homogeneous and isotropic background and on the perturbative level. While most dynamic DE scenarios start with an assumption for the EoS parameter for the DE, the starting point of holographic models is an expression for the DE energy density from which the EoS is then derived. Moreover, as was pointed out in SRJW (), the mere definition of the holographic DE density generally implies an interaction with the DM component. Requiring this interaction to vanish imposes an additional condition on the dynamics. Noninteracting Ricci-type DE, in particular, is characterized by a simple relation between the matter fraction and the necessarily time-dependent EoS parameter. Therefore it is not compatible with a cosmological constant. We shall confront the resulting background dynamics with recent SNIa data, results from baryonic acoustic oscillations (BAO) and from the history of the Hubble parameter. A crucial issue for Ricci-type DE is the perturbation dynamics karwan (). Based on a gauge-invariant analysis, the matter perturbations are found as a combination of the total and the relative energy-density perturbations. In general, the perturbation dynamics suffers from instabilities. For a phantom-type EoS these should have occurred already before the present time. Consequently, phantom DE is not consistent with our approach. For present EoS parameters one obtains growth-rate oscillations and instabilities as well, this time at finite future values of the scale factor . We shall show that there exists just one situation without instabilities at finite values of . It is characterized by a DE saturation parameter already obtained in karwan (). We show that for this configuration to be realized, a certain relationship between the current matter content of the Universe and the EoS parameter is required. Remarkably, under the corresponding condition the pressure perturbations vanish and the mentioned (unobserved) oscillation disappear. Moreover, the cosmic coincidence problem is substantially alleviated since holographic Ricci DE itself behaves as nonrelativistic matter at high redshift. There remain, however, tensions between the observationally favored values of and and the values that are necessary to avoid instabilities of the perturbation dynamics.
The structure of the paper is as follows. In Sec. II we recall basic relations for holographic models of DE. The resulting homogeneous and isotropic background dynamics is confronted with observational data in Sec. III. Sec. IV provides us with the general two-component dynamics of the cosmic medium. The first-order perturbation theory of the model is presented in Secs. V, VI and VII. On this basis, the final set of coupled equations for the nonadiabatic perturbation dynamics is found in Sec. VIII. In Sec. IX we consider issues of stability and single out a model which is stable for any finite value of the scale factor. A summary of the paper is given in the final Sec. X.
Ii Background dynamics for Ricci dark energy
We start by recalling basic features of holographic DE models in a homogeneous and isotropic background SRJW (). The cosmic medium is assumed to be describable by pressureless DM with energy density and a holographic DE component with energy density . In the spatially flat case Friedmann’s equation is
In general, both components are not necessarily conserved separately but obey the balance equations
such that the total energy is conserved. Here, is the equation-of-state (EoS) parameter of the DE and is the pressure associated with the holographic component. The acceleration equation can be written
where is the ratio of the energy densities. The total effective EoS of the cosmic medium is
According to the balance equations (2), the ratio changes as
The quantity is the infrared (IR) cutoff scale and is the reduced Planck mass. The numerical constant determines the degree of saturation of the condition
which is crucial for any holographic DE model. It states that the energy in a box of size should not exceed the energy of a black hole of the same size cohen ().
In general, there is no reason for to vanish. Assuming provides us with a specific relationship between and the ratio of the rates and . Any nonvanishing will modify this relationship.
The case without interaction is characterized by [cf. Eq. (5)]
with a generally time-dependent . Different choices of the cutoff scale give rise to different expressions for the total effective EoS parameter in (4) and to different relations between and . Our interest in the present paper will be the Ricci-scale cutoff. The role of a distance proportional to the Ricci scale as a causal connection scale for perturbations was noticed in brustein (). In gao () it was used for the first time as a DE cutoff scale. The Ricci scalar is . For the corresponding cutoff scale one has , i.e.,
where . Upon using (3) we obtain
for the holographic DE density. Notice that the (not yet known) EoS parameter explicitly enters . Use of Friedmann’s equation provides us with
which coincides with the result in karwan (). Obviously, a constant value of necessarily implies a constant and vice versa. The time derivatives of and are related by . The second relation in (13) can be used to express in terms of the present values (subindex ) of and :
The parameter is related both to and . In a next step we differentiate in (11) which yields
Relation (16) with (17), which implies that in the general case one has , i.e., both dark components do interact with each other also nongravitationally, is a direct consequence of the ansatz (11). The DE balance in Eq. (2) may then also be written as with an effective EoS parameter
The present ratio is related to the present matter fraction of the Universe by .
According to relation (17), a constant EoS parameter is compatible with only for , i.e., if behaves as dust. If we admit , however, there exists a non trivial case :
where we have normalized the present value of the cosmic scale factor to . The expression for is included here for later reference. There is no freedom left to choose the equation of state. It is fixed by the choice of together with the requirement . Notice that this is different from the more familiar procedure to deal with (nonholographic) DE, where one starts with an assumption for the EoS parameter and afterwards finds an expression for the DE density by integrating the corresponding balance equation. Here, the starting point is the energy density and the EoS parameter has to be derived.
i.e., is fixed as well. At high redshifts we have
The property that noninteracting Ricci-DE behaves as dust at high redshift was already pointed out in gao (). The values in the far-future limit are
The limits in (22) imply that this model naturally reproduces an early matter-dominated era. For and , the ratio approaches for . This value is only roughly ten times larger than the present value . For the CDM model the corresponding difference is about nine orders of magnitude. In this sense, the coincidence problem is considerably alleviated for the the present model. On the other hand, in the opposite limit the ratio approaches zero as for the CDM model. Apparently, the far-future EoS can be of the phantom type for . However, as we shall demonstrate below, such configuration is unstable and does not represent a realistic scenario.
The Hubble rate of our model turns out to be
For we recover the Einstein-de Sitter behavior . The total effective EoS is
For the adiabatic sound speed of the DE component we find
and the corresponding quantity of the total cosmic medium is
With the solutions (20) and (21) all these quantities are explicitly known, i.e., the background dynamics is completely solved analytically. In the following section we perform an actualized confrontation of the background dynamics with recent observational data.
Iii Observational analysis
Our observational analysis of the background dynamics uses the following three tests: the differential age of old objects based on the dependence as well as the data from SNIa and from BAO. A fourth test could potentially be added: the position of the first peak of the anisotropy spectrum of the cosmic microwave background radiation (CMB). However, the CMB test implies integration of the background equations until which requires the introduction of the radiative component. But the inclusion of such radiative component considerably changes the structure of the equations and no analytic expression for is available. Hence, we shall limit ourselves to the mentioned three tests for which a reliable estimation is possible.
Based on the evaluation of the age of old galaxies that have evolved passively jimenez (), there are 13 observational data available for the differential age verde (); stern (); verdebis (); ma (); mabis (). Recently, a new set of 21 data has been considered moresco (); ratra (). The basic relation is
The value of the Hubble parameter today can be added to these data, leading to 14 or 22 observational points, depending on the sample used.
The SNIa test is based on the distance modulus which is related to the luminosity distance by
In this expression we have restored the velocity of light . The quantities and denote the apparent and the absolute magnitudes, respectively.
Two decisions have to be taken for this test. The first one concerns the choice of the sample. There are many different SNIa data sets, obtained with different techniques. In some cases, these different samples may give very different results. The second point is the existence of two different calibration methods. One of them uses cosmological relations and takes into account SNIa with high (Salt 2), the other one, using astrophysical methods, is suitable for small (MLCS2k2) ioav (). In some cases, the application of different calibrations can lead to different results also. All this makes the SNIa analysis very delicate. Here, we use the Union 2 sample union (), calibrated by the Salt 2 method.
Baryonic acoustic oscillations have their origin in oscillations in the photon-baryon plasma at the moment of the decoupling at about . They can be characterized by the distance scale eise (),
We shall use the WiggleZ-data blake () and for the redshifts and , respectively.
Generally, the key quantity of a statistical analysis is the parameter
where is the theoretical evaluation of a given observable, depending on free parameters, is the corresponding observational value with an error bar and is the total number of observational data for the given test. In terms of the parameter one defines the probability distribution function (PDF) by
where is a normalization constant. The estimations for one or for two given parameters are obtained by integrating over the remaining ones. For a combination of all tests we use the total -value ,
Assuming a spatially flat universe, the three free parameters of the model are the density-ratio parameter , the EoS and the reduced Hubble parameter , defined by km/s/Mpc.
In Fig. 1 we display the one-dimensional PDFs for each of the tests and for their combination. The results for the density parameter are different for each test. The combination of all tests leads to a value of , corresponding to , roughly in agreement with the CDM model. For the equation of state parameter we obtain , consistent with the CDM model as well. According to the first relation of (14), the parameter turns out to be . This value is coincides with the result in gao (). The two-dimensional PDFs at ( of confidence level), ( of confidence level) and ( of confidence level) are shown in Fig. 2. The estimation for , based on a combination of the three tests at , is , while for we find . The straight line represents the combination which is singled out by the stability analysis of the perturbation dynamics in Sec. IX below. The tension to the results for the background dynamics is obvious, an agreement is possible only at the level.
Iv General two-component dynamics
To study the dynamics of inhomogeneities on the homogeneous and isotropic background of the previous sections we first consider the general description of the two-component system. It is characterized by a total energy-momentum tensor
where and . The quantity denotes the total four-velocity of the cosmic substratum. Latin indices run from to . The total splits into a matter component (subindex m) and a (holographic) DE component (subindex H),
For separately conserved fluids we have
Then, the separate energy conservation equations are
In general, each component has its own four-velocity, with . The quantities are defined as . For the homogeneous and isotropic background we assume . Likewise, the momentum conservations are written as
Denoting first-order perturbations about the homogeneous and isotropic background by a hat symbol, the perturbed time components of the four-velocities are
According to the perfect-fluid structure of both the total energy-momentum tensor (34) and the energy-momentum tensors of the components in (36), and with in the background, we have first-order energy-density perturbations , pressure perturbations and
In linear order the spatial components of the accelerations are
For the first-order pressure gradient terms we find (recall )
From the matter-momentum conservation (40) it follows that
According to (43) the differences between and the corresponding quantities of the components are
Restricting ourselves to scalar perturbations, the perturbed line element can be written
Furthermore, we define the (three-) scalar quantities , and by
From the definitions of , and it follows that
where is the three-dimensional Laplacian,
V Conservation equations in linear order
respectively. The total first-order energy conservation takes the form
For the total energy-density perturbations equation (56) yields
where . In terms of the fractional perturbation , the matter energy conservation (54) can be written as
With the energy conservation (55) for the DE component is
In the following it will be convenient to introduce the quantity
In terms of eq. (63) then takes the form
Our final goal in this paper is to calculate the matter-density perturbations. To this purpose we shall solve the coupled system of total energy perturbations and relative energy-density perturbations . In the following section we start by establishing the equation for the total energy-density perturbations.
Vi Total energy-density perturbations
We consider Eq. (59) and introduce therein the gauge-invariant quantities
Then, Eq. (59) is rewritten as
The perturbation has to be determined from the perturbed Raychaudhuri equation for . Neglecting shear and vorticity, the Raychaudhuri equation is
In terms of the gauge-invariant variables one finds, at linear order,
the equation for becomes
Changing to as independent variable () and transforming into the space, we arrive at
Equation (74) governs the behavior of the total energy-density perturbations. As we shall see, via the term the perturbations are coupled to the relative perturbations .
Vii Combining the separate conservation equations
Because of the structure of the first-order expressions in (53) one has
Equation (75) then becomes
Differentiation of (78) yields
The difference describes the deviation of the DE pressure perturbations from being adiabatic. It is zero for purely adiabatic DE perturbations. We discuss this issue in more detail in the following section.
Viii Nonadiabaticity and final set of equations
Generally, the deviation from adiabaticity in a two-component system with components (pressureless) and is
Let us consider the combination . With one has
which is a gauge-invariant expression. In general, now an assumption for the perturbed EoS parameter is necessary to proceed. We shall consider here the adiabatic case
This assumption of an adiabatic DE component allows us to relate the otherwise undetermined perturbation of the EoS parameter to the DE energy perturbation . Under these circumstances Eq. (80) reduces to
We emphasize that the total perturbation dynamics remains nonadiabatic due to the two-component nature of the medium. With an adiabatic DE component, the general relation (81) simplifies to