# Running of the spectral index in deformed matter bounce scenarios with Hubble-rate-dependent dark energy

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## Abstract

As a deformed matter bounce scenario with a dark energy component, we propose a deformed one with running vacuum model (RVM) in which the dark energy density \(\rho _{\Lambda }\) is written as a power series of \(H^2\) and \(\dot{H}\) with a constant equation of state parameter, same as the cosmological constant, \(w=-1\). Our results in analytical and numerical point of views show that in some cases same as \(\Lambda \)CDM bounce scenario, although the spectral index may achieve a good consistency with observations, a positive value of running of spectral index (\(\alpha _s\)) is obtained which is not compatible with inflationary paradigm where it predicts a small negative value for \(\alpha _s\). However, by extending the power series up to \(H^4\), \(\rho _{\Lambda }=n_0+n_2 H^2+n_4 H^4\), and estimating a set of consistent parameters, we obtain the spectral index \(n_s\), a small negative value of running \(\alpha _s\) and tensor to scalar ratio *r*, which these reveal a degeneracy between deformed matter bounce scenario with RVM-DE and inflationary cosmology.

## 1 Introduction

The idea of bouncing cosmology, mainly was suggested for replacing the big bang singularity to a non-singular cosmology. More recent observations of cosmic microwave background (CMB) give us some evidence in which the scalar perturbations is nearly scale-invariant at the early universe [1, 2]. Although the inflationary scenario is the most currently paradigm of the early universe and can solve several problems in standard big bang cosmology, it faced with two basically problems. One key challenge is the singularity problem before the beginning of inflation, which is arisen from an extent of the Hawking-Penrose singularity theorems which show that an inflationary universe is geodesically past incomplete and it cannot reveal the history of the very early universe [3, 4].

The second one is the *trans-Planckian* problem which reveals that the wavelength of all scales of cosmological interest today originate in sub-Planckian values where the general relativity and quantum field theory is broken down. Therefore, it leads to important modifications of the predicted spectrum of cosmological perturbations [5] (more details are referred to a good informative review [6]). These problems however have been avoided in the bouncing cosmology [6]. At a bounce time (\(t=0\)), the space gets a non-vanishing volume and also the wavelength of cosmological perturbations is minimum in which their values correspond to the end of inflation in cosmology. Due to this fact, the bouncing scenario is usually considered as an alternative to inflationary cosmology [7].

In light of cosmological perturbations, three familiar classes of bouncing model which are differences in contracting phase have been introduced. One of the most interested is a matter bounce scenario [8]. The others are Pre-Big-Bang [9] or Ekpyrotic [10] type, matter Ekpyrotic-bounce [11], matter bounce inflation scenario [11], and string gas cosmology [12, 13]. In matter bounce scenario which have been widely discussed in literatures [14, 15, 16, 17], some authors have considered one or two scalar fields [18], others work with a semi matter (a matter with a dark energy component) [19], and many efforts have been done in modified gravity and scalar tensor gravity [20, 21, 22, 23]. In all of them the dynamical behavior is described by loop quantum cosmology (LQC) [19, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] around bouncing point which is arisen from quantum gravity in high energy physics. Despite the success of LQC in non-singular bounce cosmology, it is important to note that the dynamical mechanisms that trigger non-singular bounce at high energy scales are not always provided by the LQC. For instance in [34] and [35], one can find an effective field theory model including the Horndesky operators to give rise to a non-singular bounce without much pathologies; additionally, the curvature corrections appeared at high energy scales can also yield a non-singular bounce, such as in [36, 37] and very recently revisited in [38].

The LQC is also applied around turning point in a cyclic universe scenario, where the universe goes to a contraction after an expansion phase [39]. Although in many models of matter bounce the power spectral index of cosmological perturbation may be consistent with observations, they often obtain a positive running of scalar spectral index (\(\alpha _s\)) which may be irreconcilable with some observational bounds.^{1}

However, future observations may allow one to discriminate between models (inflationary, Ekpyrotic and matter bounce scenarios), in this time, we are interested to introduce some deformed models of matter bounce to obtain a negative running \(\alpha _s\), like the inflationary scenario [42].

After introducing \(\Lambda \)CDM matter bounce scenario by Cai et al. [43], new insights into the deformed matter bounce scenario is provided. In this paper authors considered a cosmological constant (vacuum energy) as a dark energy term with a constant equation of state (EoS) parameter (\(w_{\Lambda }=-1\), accompanied with a pressureless cold dark matter (CDM)). The effective EoS parameter does not remain constant (slightly increasingly negative) in this setting and it eventually provides a slight red tilt in spectral index (an small value less than unity of spectral index \(n_s\)), according to observations. Finally, they obtained a positive value for the running of scalar spectral index. Another model of deformed matter bounce with dark energy introduced by Odintsov et al. in order to describe the late time acceleration of the universe [44]. In this model authors considered a deformation that affects the cosmological evolution, only at the late time not at beginning of contraction phase. They showed that the big rip singularity can also be avoided in their model. These models give us a great motivation to consider some model of matter bounce with various forms of dark energy to solve another remained problem.

*H*and its first time derivative [47, 52], but also it gets a constant equation of state parameter same as the cosmological constant. Then the energy density of RVM-DE reads:

This paper is organized as follows: In Sect. 2, we give a brief review on bouncing cosmology with a dynamical vacuum energy. Then, as a simple example we study on the standard \(\Lambda \)CDM cosmology in bouncing scenario in Sect. 3. We extend this model with RVM-DE in Sect. 4. In Sect. 5, the study of cosmological perturbation theory takes placed analytically for a simple case. The spectral index and its running are calculated numerically for some other cases of (RVM) model in Sect. 6 and at last we finished our paper by some concluding and remarks.

## 2 Cosmological bounce with dynamical vacuum energy

First we give a brief review on the dynamics of bouncing cosmology in a flat FLRW universe with time varying \(\Lambda (t)\) model. The matter contents are composed of radiation and cold dark matter (CDM).

*w*is the effective equation of state parameter \(P_{tot}=w\rho _{tot}\). The continuity equation (5) can be decomposed by the following equations for all components of energy as

## 3 Bouncing with the standard \(\Lambda \)CDM

*i*’ refers to initial condition. Taking critical energy density at bounce point and initial conditions in reduced Planck mass unit, same as [43], as follow

## 4 Bouncing with running vacuum model

*H*can not come from any covariant QFT and it does not even have a well-defined \(\Lambda \)CDM limit and the worst, it is also excluded from the data on structure formation [66].

In following we are also interested to add another term \(n_1 H\) into Eq. (15) and studying on the role of each terms on evolution of the scale factor, density parameters, Hubble parameter and equation of state parameter in some cases and compare them with the \(\Lambda \)CDM bouncing model.

## 5 Cosmological perturbation theory

### 5.1 Analytical solutions with case \(\rho _{\Lambda }=n_2 H^2+\beta \dot{H}\)

Following [43], we consider three continues era for studying on scalar perturbation and power spectrum in analytical method. Modes of interest are those that reach the long wave length limit during the first era where it is very far from the bounce in which the evolution of the contracting universe treats as matter-dark energy domination. Then after equality of radiation with previous pair components, the universe enters to a radiation domination epoch and at last, goes through the bounce where the evolution of the universe governed by LQC.

*z*quantity in (16) reduced to

In this section we study the Fourier modes evolution which they come from initial quantum vacuum state far away the bounce. Any mode that exit the sound Hubble radius in matter-dark domination period becomes scale invariant and they return to sound Hubble radius after the bounce, in expanding branch.

## 6 Spectral index and its running

*w*around \(\eta _0\) up to first order

*w*at \(\eta _0\). Since the changes of

*w*is very small, \(|\Delta w| \approx |w-w_0||\kappa (\eta -\eta _0)|<<1\), therefore during the semi-matter dominated epoch in a contraction phase, at low curvature and energies, in Eq. (17) we have \(\frac{z''}{z}\cong \frac{a''}{a}\) (details are referred to [74]).

*W*and

*M*are Whittaker functions and \(J_i\)’s are constants of integration. By considering the asymptotic behavior of

*W*at large \(|k \eta |\),

*w*is neglected and \(\alpha _s\) becomes

For a constant effective EoS parameter, same as previous case (Eq. (22)), running of spectral index is vanishing which is in contrast with Planck bound [40, 41]. It is worthwhile to mention that also in the standard cosmology this type of RVM-DE (model of Sect. 5) has been already excluded on account of its inability to correct description of the data on structure formation [66, 76, 77]. Thus this fact give us an alternative reason to exclude this type of RVM-DE.

In order to have a negative value of running of spectral index, (\(\alpha _s<0\)), which is compatible with the inflationary paradigm, it is required that \(H \frac{d}{d H} (\rho _{\Lambda }/H^2)>0\) (see Eq. (64)).

At following we will give two other cases of RVM and will calculate the spectral index and running.

### 6.1 Case \(\rho _{\Lambda }=n_0+n_2H^2\)

### 6.2 \(\rho _{\Lambda }=n_0 + n_2 H^2+n_4 H^4\)

It is important to note that in cosmological perturbation theory, the power spectrum, spectral index and its running essentially depend on the effective equation of state and its derivative at time of horizon-crossing. In other words, in the contracting phase, the space-time curvature is not felt by the Fourier modes inside the horizon, so they oscillate until exiting inside the horizon. The background spacetime evolution, equation of state and the time derivative of *w* at the horizon-crossing in matter dominate epoch have an important role to appropriate predict of nearly scale-invariant power spectrum and the running of the spectral index. Now we ask, is it possible to have a negative running of the spectral index in matter bounce scenario in this case?

To answer this question, firstly, we interested to solve numerically the background differential equations (8, 9, 10).

#### 6.2.1 Background numerical calculation

*w*is very close to a constant small negative value \(w \sim -0.003\). This is very good condition for getting a red tilt in the spectral index as indicated in (59). In continuing along the conformal time about \(\eta _d\approx -3.4 \times 10^5\),

*w*decreases and after this point the time derivative of

*w*becomes negative. It is reasonable to expect that after this point, dark energy has been dominated again gradually. This decreasing behavior is continuing until

*w*reaches to a minimum value (\(w=-0.65\)) at \( \eta \approx -6 \times 10^4\). After this point, the radiation component will be dominated. It should be noted that in contracting phase (\(\mathcal {H}<0\)), Eqs. (63) and (64) yields

*w*at the crossing time gets a negative value. In this time, \(w=w_{cr}\approx -0.0029\), \(n_s=12 w_{cr}+1 \approx 0.96\) and after some numerical calculation we obtain \(\alpha _{sc} \approx -0.003 \), which has a very good consistency with constrained results (\(\alpha _{sc}=-0.003 \pm 0.007 \) by \(68 \% \) CL, Planck+TT+LowP+Lensing [41]). In Fig. 4, using Eq. (60), the behavior of the \(\alpha _s \) has been clearly shown around the crossing time (solid red line). As it is shown in this figure, at the horizon-crossing time \(\eta _{c} \), the running \( \alpha _s\) gets a negative small value (\(\alpha _s \approx -0.003\), see the horizontal green dash-dot line in Fig. 4).

#### 6.2.2 Perturbation in numerical calculation

Summary of all cases in RVM-bounce scenario . Note that in all cases, *w* has a small negative value (see Eq. (61)). Also the running \(\alpha _{sc}\) is the value of \(\alpha _s\) at crossing, \(H=H_{cr}=- 8.8\times 10^{-8}\) (in contracting phase) and \(w=w_{cr}=-0.003\) provided that \(\rho _{\Lambda }>0\) even at the bounce point (\(\mathcal {H}=0\))

\(\rho _{\Lambda }\) | \( w =(n_s-1)/12\) | \(\dot{w}\) | \(\alpha _s\) | \(\alpha _{sc}\) | |
---|---|---|---|---|---|

1 | \(n_0\) | \(\dfrac{-n_0}{3H^2}\) | \(3 H w(1+w)\) | \(-72 w\) | 0.22 |

2 | \(n_0+n_2 H^2\) | \( -\dfrac{1}{3}(\dfrac{n_0}{H^2}+n_2)\) | \(\dfrac{3 n_0H}{n_0+n_2 H^2}w(1+w)\) | \(\dfrac{n_0}{n0+n_2H^2}(-72 w) \) | 0.22 |

3 | \(n_0+n_1 H\) | \(-\dfrac{1}{3}(\dfrac{n_0}{ H^2}+\dfrac{n_1}{ H})\) | \(\dfrac{ 3 H (n_1 H +2 n_0)}{2 n_0 +n_1 H}w(1+w)\) | \(\dfrac{n_1 H+2 n_0}{n_1 H+n_0}(-72 w)\) | 0.44 |

4 | \(n_0+\beta \dot{H}\) | \( \dfrac{2 n_0/H^2-3 \beta }{-6+3\beta }\) | \(\dfrac{3 H n_0 w (1+w)}{n_0-3\beta H^2/2} \) | \(\dfrac{n_0}{n_0-3 \beta H^2 /2} (-72 w)\) | 0.001 |

5 | \(n_2 H^2 +\beta \dot{H}\) | \( \dfrac{-2 n_2+3 \beta }{ 6-3 \beta }\) | 0 | 0 | 0 |

6 | \(n_0+n_2 H^2 +n_4 H^4\) | \( -\dfrac{1}{3}(\dfrac{n_0}{H^2}+n_2+n_4 H^2)\) | \( \dfrac{3 H( -n_4 H^4 +n_0)w(1+w)}{n_0+n_2 H^2 +n_4 H^4} \) | \( \dfrac{(-n_4 H^4+n_0)(-72 w)}{n_0+n_2 H^2 +n_4 H^4} \) | − 0.003 |

The evolution of the scalar cosmological perturbation (blue dash line) has been depicted in Fig. 5. As it can be seen, curvature perturbation oscillates from the sub-Hubble scale to super-Hubble region at \(\eta _{c}\approx -3.1 \times 10^5\), in which the oscillation finished. In fact *k*-mode curvature perturbation exits from the sound Hubble horizon at the crossing point \(\eta _{c}\). As it shows, after equality time (vertical green line) when \(\rho _r= \rho _m+\rho _{\Lambda }\), we can consider the radiation begins to dominate or in analytically point of view, \( {z''}/{z} \rightarrow 0\), and consequently the amplitude of the scalar perturbation approximately becomes constant.

The red solid line shows the oscillation of the tensor cosmological perturbation or gravitational wave. The sound speed of the tensor perturbation is \(c_s^T=1\). So clearly, tensor perturbation continue to oscillates even after vertical black dot line and eventually will damp to a constant value after equality time.

Besides the nearly scale invariant power spectrum with a negative running, a small tensor to scalar ratio is predicted by a cosmological bounce scenario. The tensor-to-scalar ratio *r* is constrained by the observational bound (\( r < 0.12\)) [83]. There are some known mechanisms for predicting a small tensor to scalar ratio [18, 30, 84, 85].

*r*is shown in Fig. 6. As one can see, before the equality time (dot green line), the amount of tensor to scalar ratio

*r*slowly decreases from high to low.

In order to compare all studied cases of RVM-bounce scenario, we summarize all of them in Table 1. As we found, for the cases 1-4, some physical requirements such as \(\rho _{\Lambda }>0\) (or \(n_0>0\)) at any time and \(w<0\) at the crossing time, required that \(\alpha _s>0\). In the final column of the Table 1, the value of \(\alpha _{sc}\) is calculated for all cases at crossing time, where \(w=w_{cr}=-0.003\), and \(H=H_{cr}=-8.8 \times 10^{-8}\) for \(n_0>0\). At last, except the case contained the term \(H^4\), other cases cannot satisfy the weak energy condition and observational evidence simultaneously. Therefore we excluded them in further numerical calculation analysis (such as tensor to scalar ratio *r*).

## 7 Conclusion

In this work we introduced a deformed matter bounce scenario with the running vacuum model (RVM). This model could be considered as a viable alternative to the inflationary paradigm both in observational and theoretical aspects. Based on RVM-DE, the standard cosmological constant not more constant, but may consider as series of powers of \(H^2\) and \(\dot{H}\). By introducing some cases of RVM, we calculated the spectral index \(n_s\) and its running \(\alpha _s\) in order to compare with observational data. In fact in the contracting phase, before a bouncing, when the EoS parameter is slightly negative, Fourier modes of perturbations exit from the sound horizon. Thus power spectrum of cosmological perturbation for long wavelength modes is not exactly scale invariant and consequently it gets a slightly red tilt.

The process of creating of red tilt is obviously indicated in the analytical treatment for the case \(\Lambda =n_2 H^2 + \alpha \dot{H}\). In this case the running of spectral index become vanishing, \(\alpha _s=0\), which is inconsistent with inflationary paradigm. Some models with expansion up to \(H^2\) got positive running and for a model \(\Lambda (H)=n_0+n_2 H^2+n_4 H^4\), by estimating a set of parameters, we obtained the spectral index \(n_s\approx 0.96\), running of spectral index \(\alpha _s<0\) and tensor to scalar ratio \(r<0.12\). We found that this model had the best consistency with the cosmological observations and reveals a degeneracy between deformed matter bounce scenario with RVM-DE and inflation. As a work in the future, the observational constraint of this model and comparison with cosmological observations are suggested.

## Footnotes

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