Vacuum models with a linear and a quadratic term in H: structure formation and number counts analysis

Adrià Gómez-Valent, and Joan Solà

High Energy Physics Group, Dept. ECM Univ. de Barcelona,

Av. Diagonal 647, E-08028 Barcelona, Catalonia, Spain

and

Institut de Ciències del Cosmos

Univ. de Barcelona, Av. Diagonal 647, E-08028 Barcelona

E-mails: adriagova@ecm.ub.edu, sola@ecm.ub.edu

Abstract. We focus on the class of cosmological models with a time-evolving vacuum energy density of the form , where is the Hubble rate. Higher powers of could be important for the early inflationary epoch, but are irrelevant afterwards. We study these models at the background level and at the perturbations level, both at the linear and at the nonlinear regime. We find that those with are seriously hampered, as they are unable to fit simultaneously the current observational data on Hubble expansion and the linear growth rate of clustering. This is in contrast to the models, including the concordance CDM model. We also compute the redshift distribution of clusters predicted by all these models, in which the analysis of the nonlinear perturbations becomes crucial. The outcome is that the models with predict a number of counts with respect to the concordance model which is much larger, or much smaller, than the CDM and the dynamical models with . The particular case (the pure lineal model), which in the past was repeatedly motivated by several authors from QCD arguments applied to cosmology, is also addressed and we assess in detail its phenomenological status. We conclude that the most favored models are those with , and we show how to discriminate them from the CDM.

## 1 Introduction

The accurate measurement of the luminosity-redshift curve of distant type Ia supernovae carried out at late 1990s by The Supernova Cosmology Project (Perlmutter et al. 1998) and The High-z Supernova Search Team (Riess et al. 1998) showed that our universe is speeding up. This positive acceleration could be produced by the presence of a tiny cosmological constant (CC) in Einstein’s field equations, . This framework, the so-called concordance or CDM model, seems to describe quite well the available cosmological data (Ade et al. 2013). Despite this, the CC, which is usually associated to the energy density carried by the vacuum, through the parameter (in which is the Newtonian constant), has also been the origin of two of the most important current open problems in physics, namely the old CC problem (Weinberg 1989) and the Cosmic Coincidence problem (see e.g. the reviews by Padmanabhan 2003, Peebles & Ratra 2003, Copeland, Sami & Tsujikawa 2006). The severity of these problems are the main motivation to search for alternative frameworks capable to offer a more satisfactory explanation for them while still keeping a good fit to the observational data.

Different scenarios have been proposed in order to alleviate this
situation, to wit: scalar fields, e.g. quintessence, modified
gravity theories, decaying vacuum models, etc (cf. the previous
review articles and references therein). The present work takes the
point of view that the vacuum energy density is a dynamical variable
in QFT in curved spacetime as in such framework it should be
possible to better tackle the basic CC problems ^{1}

In this article we focus on the dynamical vacuum models that include
a linear and a quadratic term in , i.e. . We discuss the various possibilities, in particular we
examine the phenomenological status of the models where no additive
term is present. Of especial significance is to check out the
purely linear model , which is a particular case of
the models. The linear model was amply discussed several
times in the literature by different authors from different points
of view. It was theoretically motivated as a possible fundamental
description of the cosmological vacuum energy in terms of QCD – see
e.g. (Schutzhold 2002; Klinkhamer & Volovik 2009; Thomas, Urban &
Zhitnitsky 2009; Ohta 2011). Phenomenological analysis claiming its
possible interest for the description of the current Universe were
carried out e.g. in (Borges et al. 2008; Alcaniz et al. 2012;
Chandrachani et al. 2014). We shall revisit the linear
model here, but only as a particular case of the larger class
of dynamical vacuum models that we analyze. We put to the test all
these models in the light of the recent observational data and
assess which are the most favored ones ^{2}

The layout of this paper is as follows. We address the background solution of the different models in Section 2. The matter perturbations (linear and nonlinear) are considered in Section 3. The confrontation with the linear structure data is performed in Sect. 4, whereas in section 5 we probe the models with the number counts method, which requires the nonlinear analysis of structure formation. In Section 6, we present our conclusions. Finally, in the Appendix A we briefly extend the discussion of the linear model.

## 2 Different types of vacuum models with linear term in H

Let us consider a (spatially) flat FLRW universe. From Einstein’s field equations, i.e. , one can derive Friedmann’s equation and the pressure equation by taking the and the component, respectively:

(1) |

(2) |

where the overdot denotes a derivative with respect to the cosmic time. If we are interested in describing the structure formation process, we can limit ourselves to consider only the contributions of cold matter, , and a true dynamical vacuum term, . Effects of radiation will be included in a subsequent stage, when they will be necessary. Combining Eqs. (1) and (2) it is easy to obtain the equation of local covariant conservation of the energy for a pressureless matter fluid:

(3) |

From these equations we can also obtain the evolution law for the Hubble function:

(4) |

The universe’s dynamics depends on the specific dynamical
nature of . In the present paper we study the
following dynamical vacuum models ^{3}

(5) | |||||

Notice that the parameter has dimension (i.e. mass squared) in natural units. We have introduced the dimensionful constant (the value of the Hubble function at present) as a part of the linear term, and in this way the parameter in front of it can be dimensionless. Similarly is dimensionless since it is the coefficient of . Obviously models I, II and III are particular cases of IV, i.e. they can be obtained from IV just by setting , and , respectively. However, it is convenient to study the different implementations separately because they are not phenomenological alike, and some of them are actually unfavored.

On inspecting the structure of these models, it is even questionable that theoretically all these possibilities are admissible. For example, the presence of a linear term in the structure of deserves some considerations. This term does not respect the general covariance of the effective action of QFT in curved spacetime (Shapiro & Solà, 2009). The reason is that it involves only one time derivative with respect to the scale factor. In contrast, the quadratic terms involve two derivatives and hence it can be consistent with covariance. From this point of view one expects that the term is a primary structure in a dynamical model, whereas is not. Still, we cannot exclude a priori the presence of the linear terms since they can be of phenomenological interest. For example, they could mimic bulk viscosity effects (cf. Barrow 1983; Zimdahl 1996; Ren & Meng 2006; Komatsu & Kimura 2013).

The first task to do in order to analyze the above models is to solve the background cosmological equations. In the following we provide the solution of model IV, which is the more general one. However, in this model the Hubble function and the energy densities cannot be solved explicitly in terms of the scale factor, only in terms of the cosmic time. This feature also holds for model III. Models I and II, however, can be solved analytically also as a function of the scale factor and we will do it because it is more convenient.

For all these models we have the following relation between the basic parameters , and , which follows from imposing that the dynamical vacuum energy density must coincide with the current value at present (the point where ):

(6) |

If we apply Eq. (4) to the general type-IV models, we find

(7) |

where we have defined . Upon direct integration we obtain the Hubble function as a hyperbolic function of the cosmic time:

(8) |

with

(9) |

Using (8) in the expression of for type-IV models we can infer the vacuum energy density in terms of

(10) |

Inserting equations (8) and (10) into Friedmann’s equation (1) we can also derive the time evolution of the pressureless matter density:

(11) |

The scale factor can be obtained by integration of Eq. (8):

(12) |

with the normalization constant ():

(13) |

From (12) one sees that, in general, it is not possible to eliminate the cosmic time in terms of the scale factor. It is only possible if and/or .

Let us remark that for models III and IV the values of and should necessarily be small since they parametrize a mild dynamical departure from the CDM which we know it fits reasonably well the data. The dynamical vacuum function in these models stays around the constant value for near and hence and must be small in absolute value. This situation is of course possible because for these models . Put another way: models III and IV have a smooth CDM limit for , in contrast to models I and II. As we shall see in Sect. 4, the confrontation of models III and IV against observations does indeed confirm that and are in the ballpark of (see Table 1). Quite in contrast, for models I and II and cannot be arbitrarily small since for these models is not protected by the nonvanishing additive term . For them, the constraint (6) implies that the following relations must hold:

(14) |

It is thus clear that in model I there is no free parameter (apart from or ), and for model II we find that if is small, cannot be small, and vice versa.

Obviously models I and II satisfy one of the aforementioned conditions for which the solution in terms of the scale factor is possible, so let us provide such analytical solution in this case. The constraints (14) entail that the quantity defined in (9) boils down to , and this allows to combine the exponential factor and the hyperbolic function in (12). The result for model II reads:

(15) |

From here we can invert and derive , and then substitute in (8) to obtain the normalized Hubble rate to its current value, i.e. , for type-II models:

(16) |

We can also furnish analytical expressions for the matter and vacuum energy densities as a function of the scale factor:

(17) |

where

(18) |

and

(19) |

Here is the current critical density, i.e. . The corresponding expressions for type-I model can be directly extracted from (16), (17) and (19 by setting ):

(20) | |||

(21) | |||

(22) |

The first term on the r.h.s. of Eq. (17) is the dominant one at high redshifts (, equivalently ). Therefore, in this regime the matter density evolves like . A similar scaling law is found for type-I models, with .

The following observation is in order. In the concordance model we have the standard behavior of the matter density , but when we compare it with (17) and (21) we observe that for type-I and II models there is in an extra factor of . This factor stands out maximally in the remote past where for the same value of the type-I and type-II models should predict a larger matter density at present (cf. Appendix A). The reason is obvious: if a term of order should mimic the standard value , the value itself of must be of order and hence significantly larger () than the standard one. This situation does not occur so acutely for the low and intermediate redshift range, as can be seen e.g. from Eq. (21) for model I, where for the two terms in the square brackets add up approximately to and we recover the CDM behavior . As we will comment in Sect. 4, we find more appropriate to test these “anomalous” models near the region where they can mimic the CDM to some reasonable extent, i.e. at relatively low redshifts.

Up to now, we have not included the effect of relativistic matter since we were interested in studying the background solutions near our time and the physics of cosmological perturbations. In spite of this, when we will put our models to the test the radiation correction must be taken into account in our overall fit to the main cosmological data, especially in regard to the data on Baryonic Acoustic Oscillations (BAOs) and Cosmic Microwave Background (CMB). In fact, when the CMB was released () the amount of radiation was not negligible, so we had better include the relativistic matter component in our analysis. The generalized energy conservation law involving also radiation reads as follows:

(23) |

We may compute in this expression from the explicit form of the general vacuum energy in Eq. (2), i.e. using model IV. Since relativistic and non-relativistic matter are in interaction one can split the obtained expression with the aid of an interaction source :

(24) |

Notice that when one of the matter components dominates over the other we are allowed to turn the source off and solve the decoupled system. The corresponding results for non-relativistic and relativistic matter are as follows:

(25) |

(26) |

The presence of the time dependence in the exponential, which is triggered by the -parameter of the linear term in the vacuum function, is reminiscent of the fact that for models III and IV the energy densities cannot be expressed fully in terms of the scale factor.

In good approximation we can assume that the evolution of the scale factor as of the time when the CMB was released corresponds to the cold matter epoch, i.e. we suppose it is evolving as indicated in Eq. (12). In this way we can determine the energy densities (25) and (26) in terms of the cosmic time only. This last step can be performed analytically only if we suppose that the effects of radiation are sufficiently small, as it is indeed the case under consideration. We find:

(27) |

(28) |

where the constant is the same as that in (13).

The following normalization condition must be fulfilled so that the energy densities take the present value at :

(29) |

Note that we can actually determine in terms of the remaining parameters by matching equations (27) and (11):

(30) |

The Hubble function of the matter-dominated epoch including the radiation contribution can be calculated from the generalized Friedmann’s equation for model IV:

(31) |

It can be checked that thanks to the condition (29) the implicit formula (31) for the Hubble function leads to the extended cosmic sum rule , as expected. After some rearrangement and making use of (27) and (28), we can bring Eq. (31) into the form

(32) |

where

(33) |

Thus, the sought-for Hubble function in the presence of a relatively small amount of relativistic matter can be computed by solving Eq. (32):

Model | ||||
---|---|---|---|---|

CDM | - | - | ||

I | - | |||

II | ||||

III | - | |||

(34) |

where is defined as

(35) |

In Eq. (34) we have made use of the extended cosmic sum rule mentioned above to establish the relation

(36) |

The numerical integration of (34) provides the improved form of the scale factor for a general type-IV model, namely . Thus, we can obtain the points of the curves , , and computationally using the results presented before. With this strategy we can better confront the model with observations since the data inputs are given in terms of the cosmological redshift variable .

On comparing equations (8) with
(34) we immediately recognize that
represents the correction term introduced by the effect of the
radiation upon the original expression (8). Obviously
at the present time. However, this is not so at
the decoupling time. Indeed, taking ^{4}

As we have seen, for type-II model (and in particular for type-I) one can derive the energy densities (25) and (26) in terms of the scale factor. The answer for nonrelativistic matter is given in Eq. (17). For radiation we can proceed in a similar way, and the result is:

(37) |

where was defined in (18). The modified Hubble rate for type-II models can be expressed in terms of the scale factor by solving (31) after setting and using (17) and (37):

(38) |

Substituting this expression in (2)/ type II we find the corresponding vacuum energy density including the effect of radiation, which is a cumbersome expression.

We can also estimate the equality time, , between the radiation and the non-relativistic matter energy densities for models of type I and II. Equating (17) and (37) and taking into account that we obtain:

(39) |

For the typical values that and take in Table 1, deviates significantly from the CDM prediction value . In contrast, for models of type-III and IV (which have ) one can use the concordance value as a very good approximation. In these cases one can show that , where and the deviations from the CDM model value are only at the few percent level. Needless to say, additional important differences of the models are expected to appear in connection to the photon decoupling and baryon drag epochs, and in the value of the comoving Hubble scale, , at the redshift of matter-radiation equality (cf. sections 4 and 5 for additional considerations on these matters).

As to the behavior of the energy densities deep in the radiation epoch for type-I and II models, let us note that it can be relevant for the primordial big bang nucleosynthesis (BBN). The ratio between the vacuum and radiation energy densities for can be estimated from the foregoing analysis, with the result:

(40) |

where use has been made of (39). Notice that the term enclosed in the square brackets provides the correction to the result that can be inferred from (17) and (19) in the matter dominated epoch at high redshift, but without radiation. Taking into account the fitted values of the parameters presented in Table 1, we find . Therefore, at the ratio (40) yields , and at the BBN epoch (where ) we have . We learn from these estimates that type-II models predict a negative value of in the past and, moreover, it is a non-negligible faction of at the BBN time. This fraction could of course be made smaller by decreasing (i.e. approaching ) but this would worsen the quality of the fit since model II provides a better fit to low energy data than model I (cf. Table I). In compensation for its poorer description of the current data, model I satisfies when , similar to the CDM, and therefore its vacuum energy is, in principle, harmless for the BBN.

## 3 Linear and nonlinear structure formation

For type-IV models the nonlinear equation for the growth
factor can be derived after
some lengthy calculations leading to the following final
result ^{5}

(41) |

where the variable is related to the cosmic time through , and . The primes indicate derivatives with respect to . The expressions for , and for type-IV models are, respectively:

(42) |

(43) |

and

(44) |

The numerical solution of the above nonlinear equation is used to compute the collapse density threshold , an important model-dependent quantity that is used in the number counts analysis of Sect.5. Once more we refer the reader to (Grande, Solà, Basilakos & Plionis 2011) for details (see also Pace, Waizmann & Bartelmann 2010).

If we are, however, interested only in the linear growth factor we can throw away the nonlinear terms from (41), i.e. the terms. Let us dispense with these terms at this point, as we wish to focus on the large scale linear perturbations. In practice, to solve the resulting linear differential equation we have to fix the initial conditions for and . We take them at very high redshift . The scale factor (12) can be expressed in terms of :

(45) |

For the general model IV (with ) we normalize the growth factor with the value , i.e. , and we take at very high redshifts. The initial conditions at , corresponding to for type-IV models, are the following. For the growth factor we have , and for its derivative with respect to the -variable, we obtain

(46) |

Unfortunately, the differential equation for cannot be solved analytically neither for type-III nor for type-IV models. We are forced to use numerical techniques, for instance the method of finite differences, which is anyway necessary for tackling the original nonlinear equation (41). For type-II models the perturbation equations can be readily obtained by setting . In this limit, the -variable reads . Introducing now the differential equation for the linear perturbations becomes

(47) |

This result is consistent with that of (Basilakos & Solà 2014). A power-like solution of Eq. (47) immediately ensues: . While an explicit relation of the variable with the scale factor is impossible for models III and IV, for type-II models the variable defined above permits such relation:

(48) |

Thanks to this feature the previously found solution can be rewritten as . The latter is the decaying mode solution (since ) and, therefore, must be rejected. From it we can generate the growing mode solution for the type-II model:

(49) |

with a constant. The behavior of Eq. (49) in the early epoch, namely when , is . In the case of model I, for which , we have . This is the same limiting behavior as that of the CDM model, with the proviso that that for both models with there is an extra factor of in the matter density. Such anomaly is not innocuous; it has dramatic consequences that will be analyzed in the next sections.

Before closing this section we should like to point out that the
pure quadratic model (corresponding to
is excluded since such model does not have an
inflection point from deceleration to acceleration (cf. Basilakos,
Polarski & Solà 2012). In addition, it has no growing modes for
structure formation. This last part can be immediately inferred from
Eq. (49) using the fact that for that
model. As a result, one can easily check that the growing mode
exists only for (equivalently, for ) and in
this case the Universe is always decelerating. Thus we shall not
consider this model any longer in our analysis. While the pure
model is excluded, we should emphasize that when it is complemented
with the term, i.e. when we consider in
type-IV models, the resulting expression takes on the general form
. This structure for the vacuum energy density
is perfectly viable from the phenomenological point of view, and in
fact it is one of the simplest and more attractive formulations of
the dynamical vacuum compatible with the general form of the
effective action in QFT since now both terms (the constant terms and
the term) are allowed by general covariance ^{6}

## 4 Vacuum models and linear growth

In Table 1 we show the best-fit values for the models
we are considering. For type-III and type-IV models we have used a
joint statistical analysis involving the latest data, i.e.
SNIa-Union2.1 (Suzuki et al. 2011), BAO measurements in terms of the
parameter (Blake et al. 2011) and the
CMB shift parameter (Ade et al. 2013; Shaefer and Huterer,
2013) ^{7}

For this reason, for the non CDM-like models I and II we have implemented the fitting procedure by just concentrating on the low and intermediate redshifts, that is to say, we have used the type Ia supernovae data but avoided using CMB data. At the same time for these models we have used Eisenstein’s BAO parameter (Eisenstein, 2005), tabulated as in (Blake et al. 2011). It is given as follows:

(50) |

For models I and II we have avoided to use the BAO -parameter, which requires the computation of the comoving distance that light can travel up to the baryon drag epoch (at redshift ), i.e. the quantity

(51) |

where

(52) |

is the sound speed in the baryon-photon plasma. Note that this quantity is model-dependent because is so. However, for models III and IV (the ones which are CDM-like) we can safely use the BAO -parameter, also tabulated in (Blake et al. 2011), and in fact we have adopted it in such cases. The necessary corrections for these models amount to the following expression, which is obtained after using equations (27) and (28):

(53) |

One can easily check that for we retrieve the corresponding CDM result:

(54) |

As already warned, the situation for models I and II is different as we cannot use the standard formulae for estimating owing to the anomalous behavior of at very high redshift. For this reason we have used only the BAO data for them (based on the aforementioned acoustic parameter whose computation does not involve any integration in the very high redshift range), and of course the SNIa data. For models III and IV, in contrast, we have used SNIa and CMB data collected from the aforementioned references, and BAO data based on the -parameter, tabulated also in (Blake et al. 2011).

Proceeding in this way we can see from Table 1 that the fitting values of associated to models I and II are not very different from those of models III and IV, and all of them are reasonably close to the CDM model (which is also included in that table and fitted from the same data). From this point of view (and attending also to the values per d.o.f.) we can say that these models perform an acceptable fit to the cosmological data. For models I and II, however, we can attest this fact only for the low and intermediate redshift data. If we include the CMB shift parameter and the BAO data, models I and II then peak at around (and with a bad fit quality, see Appendix A). Such poor performance is caused by the aforementioned anomalous behavior of these models at large redshift.

Even if we restrain to the low and intermediate redshift data for models I and II, which as we have seen lead to an acceptable value of (cf. Table 1), they nevertheless clash violently with a serious difficulty, namely they are bluntly unable to account for the linear structure formation data, as it is plain at a glance on Fig. 1 (plot on the right). The observational data in that plot have been taken from Table of (Jesus et al. 2011) and references therein.

To better understand the meaning of Fig. 1, let us recall that from the standard definition of the density contrast one can define the linear growth rate of clustering (Peebles 1993), as follows:

(55) |

Both and have been plotted in Fig. 1 for the models under study together with the CDM.

The obvious departure of models I and II from the linear growth data is an important drawback for these models. It implies that the initial success in fitting the Hubble expansion data cannot be generalized to all low redshift data. Such situation is in contrast to type III and IV models, which are able to successfully fit the linear growth data at a similar quality level as the CDM, as can also be appreciated in Fig. 1. In fact, the three curves corresponding to models III, IV and the CDM (for the best fit values of the parameters in Table 1) lie almost on top of each other in that figure, whereas the curves for models I and II depart very openly from the group of CDM-like models. For the former there is an evident defect of structure formation with respect to the CDM, whilst for the latter there is a notable excess.

The large differences can be explained as follows. As we have seen before the ratio for type-II models is far from 0, and negative, in the far past. Now, from the basic equations in Sect. 2 we find that during the matter-dominated epoch the acceleration of the expansion is given by . Thus, a negative value of the vacuum energy density, , helps to slow down the expansion (it actually cooperates with gravitation and enhances the aggregation of matter into clusters). Actually, the vacuum energy of model II did not become positive until , what corresponds to a redshift . This is why we obtain larger values of the density contrast in comparison with the models that take (cf. Fig. 1). Later on the universe started to speed up, and the transition value from deceleration to acceleration is given by

(56) |

From the values of the fitted parameters in Table 1, we find . Numerically, it is significantly larger than in the CDM (, for the central fit value of quoted in Table 1). From this point onwards the type-II vacuum has been accelerating the universe and restraining the gravitational collapse, but it has left behind a busy history of structure formation triggered by the large growth rate (cf. the fit value from Table 1). Such history is difficult to reconcile with the (much more moderate) one indicated by observations.

In the other extreme we have type-I model, showing a serious lack of structure formation as compared to the CDM (cf. Fig. 1), despite for both models . We can also understand the reason as follows. Let us assume a common value of the density parameter (which is a good approximation under the fitting strategy we have followed in Table 1). In that case Eq. (22) tells us that the ratio of their vacuum energy densities is: . Thus, during the past cosmic history the vacuum energy density for the type-I model is positive and always larger than in the concordance model, so we should expect a reduced growth rate as compared to the CDM. This is confirmed in Fig. 1.

In the next section, we analyze the nonlinear perturbation effects at small scales and consider the different capability of the vacuum models under study to produce cluster-size halo structures in the universe. This study will give strength to the results obtained at the linear level.

## 5 Number counts analysis

In the previous section we have shown that the CDM-like vacuum models III and IV deviate mildly from the concordance model when we consider the linear structure formation. While in the future it may be possible to resolve better these differences there is another useful strategy that can be adopted to magnify the differences to a larger degree. It is based on the clustering properties of the nonlinear regime at smaller scales and on counting the number of formed structures in each vacuum framework. Present X-ray and Sunyaev-Zeldovich surveys, such as eROSITA (Merloni et al, 2012) and SPT (Bleem et al, 2014), can be very helpful to test these models. The method ultimately relies on the Press and Schechter (PSc) formalism (Press & Schechter 1974) and generalizations thereof. We will apply it to the various models under study.

From that formalism one can predict the abundance of bound structures that have been formed by gravitational collapse. The comoving number density of collapsed objects at redshift within the mass interval and takes on the form

(57) |

where is the comoving background density and is the PSc-function. An important parameter in it is the collapse density threshold , which we have computed numerically for our models in Fig. 2. In the original PSc-form, . However, in the present work we adopt the improved one proposed by (Reed et al. 2007), which depends on several additional parameters. Finally, is the mass variance of the smoothed linear density field. In Fourier space it is given by:

(58) |

In this expression, is the linear growth factor of perturbations, i.e. , which we have computed before for our models, is the CDM power-spectrum of the linear density field and finally we have the smoothing function , which is the Fourier transform of the following geometric top-hat function with spherical symmetry: , where is the Heaviside function. It contains on average a mass within a comoving radius .

The CDM power spectrum is used, where is a normalization constant, and is the spectral index given by as measured by Planck+WP (Ade et al. 2013). Finally, is the BBKS transfer function (Bardeen, Bond, Kaiser & Szalay 1986; Liddle & Lyth 2000). Introducing the dimensionless variable , in which is the value of the wave number at the equality scale of matter and radiation, we can write the transfer function as follows:

It is important to emphasize that is a model dependent quantity. For type-III and type-IV models one can use the same formula that is obtained in the CDM, due to the fact that the deviations are negligible in these cases, as we have checked. On the contrary, with type-I and type-II models we are not allowed to do that. We must derive the corresponding expression for by applying (38) and (39). The final results for each model read as follows:

(59) |

and

(60) |

We normalize the power spectrum using , the rms mass fluctuation amplitude on scales of