Tachyon warm inflation with the effects of Loop Quantum Cosmology in the light of Planck 2015

TL;DR

This paper explores how tachyon warm inflation models within loop quantum cosmology align with Planck 2015 data, analyzing cosmological functions, slow-roll parameters, and fluctuation spectra.

Contribution

It introduces a novel combination of tachyon warm inflation with loop quantum cosmology and tests its observational viability against Planck data.

Findings

Certain models fit the Planck data well

The framework provides consistent cosmological functions

Predictions align with observed scalar and tensor fluctuations

Abstract

We investigate the observational signatures of quantum cosmology in the Cosmic Microwave Background data provided by Planck collaboration. We apply the warm inflationary paradigm with a tachyon scalar field to the loop quantum cosmology. In this context, we first provide the basic cosmological functions in terms of the tachyon field. We then obtain the slow-roll parameters and the power spectrum of scalar and tensor fluctuations respectively. Finally, we study the performance of various warm inflationary scenarios against the latest Planck data and we find a family of models which are in agreement with the observations.

Equations

$$\{c,p\}=\frac{8\pi \gamma }{3M_p^2}$$

$$c=\gamma \dot{a}p=a^2,$$

$${\mathcal{H}}_{cl}=-\frac{3\sqrt{p}}{{\gamma }^2}+c^2+{\mathcal{H}}_m$$

$${\mathcal{H}}_{\mathrm{eff}}=-\frac{3\sqrt{p}}{\gamma {\overline{\mu }}^2}{\sin }^2(\overline{\mu }c)+{\mathcal{H}}_m$$

$$\dot{p}=\{p,{\mathcal{H}}_{\mathrm{eff}}\}=-\frac{\gamma }{3}\frac{\partial {\mathcal{H}}_{\mathrm{eff}}}{\partial c}$$

$$\dot{a}=\frac{1}{\gamma \overline{\mu }}\sin (\overline{\mu }c)\cos (\overline{\mu }c)$$

$${\sin }^2(\overline{\mu }c)=\frac{8\pi }{3M_p^{2}a}{\mathcal{H}}_m.$$

$$H^2=\frac{\kappa }{3}\rho \left(1-\frac{\rho }{{\rho }_c}\right),$$

$$S=\int d^{4}x\sqrt{-g}\left(\frac{\mathcal{R}}{2\kappa }+L_{\varphi }+L_{\gamma }\right)$$

$$L_{\varphi }=-V(\varphi )\sqrt{1-g^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi }$$

$${\dot{\rho }}_{\varphi }+3H({\rho }_{\varphi }+p_{\varphi })+{\dot{\rho }}_{\gamma }+4H{\rho }_{\gamma }=0$$

$$\dot{\rho }+H(3p_{\varphi }+3{\rho }_{\varphi }+4{\rho }_{\gamma })=0,$$

$$\dot{{\rho }_{\varphi }}+3H({\rho }_{\varphi }+p_{\varphi })=-\Gamma {\dot{\varphi }}^2,$$

$${\dot{\rho }}_{\gamma }+4H{\rho }_{\gamma }=\Gamma {\dot{\varphi }}^2,$$

$${\dot{\rho }}_{\varphi }=-3H{\dot{\varphi }}^2{\rho }_{\varphi }\left(1+R\right).$$

$$T_{\nu }^{\mu }(\varphi )=\frac{\partial L}{\partial ({\partial }_{\mu }\varphi )}{\partial }_{\nu }\varphi -g_{\nu }^{\mu }{L}_{\varphi }=\mathrm{diag}(-{\rho }_{\varphi },p_{\varphi },p_{\varphi },p_{\varphi })$$

$${\rho }_{\varphi }=\frac{V(\varphi )}{\sqrt{1-{\dot{\varphi }}^2}},p_{\varphi }=-V(\varphi )\sqrt{1-{\dot{\varphi }}^2}.$$

$$\dot{H}=-\frac{\kappa }{6}\left(3p_{\varphi }+3{\rho }_{\varphi }+4{\rho }_{\gamma }\right)\left(1-\frac{2\rho }{{\rho }_c}\right).$$

$$\dot{H}=-\frac{\kappa }{6}\left(3{\rho }_{\varphi }{\dot{\varphi }}^2+4{\rho }_{\gamma }\right)\left(1-\frac{2\rho }{{\rho }_c}\right).$$

$$\frac{\ddot{\varphi }}{1-{\dot{\varphi }}^2}+3H\dot{\varphi }+\frac{1}{V}\frac{dV}{d\varphi }=-\frac{\Gamma }{V}\dot{\varphi }\sqrt{1-{\dot{\varphi }}^2}.$$

$$\ddot{\varphi }+(3H+\frac{\Gamma }{V})\dot{\varphi }+\frac{1}{V}\frac{dV}{d\varphi }=0.$$

$$H^2=\frac{\kappa }{3}{\rho }_{\varphi }\left(1-\frac{{\rho }_{\varphi }}{{\rho }_c}\right).$$

$${\dot{\varphi }}^2$$

$${\rho }_{\gamma }=\frac{\Gamma {\dot{\varphi }}^2}{4H}=\frac{-\Gamma \dot{H}}{2\kappa H(1+R){\rho }_{\varphi }}{\left(1-\frac{12H^2}{\kappa {\rho }_c}\right)}^{-\frac{1}{2}}.$$

$${\rho }_{\gamma }=C_{\gamma }{T}^4$$

$$T={\left[\frac{-\Gamma \dot{H}}{2\kappa C_{\gamma }H(1+R){\rho }_{\varphi }}\right]}^{\frac{1}{4}}{\left(1-\frac{12H^2}{\kappa {\rho }_c}\right)}^{-\frac{1}{8}},$$

$${\Gamma }^{1-\frac{m}{4}}$$

$$V(t)$$

$$\times {\left[1+\frac{2\dot{H}}{\kappa {\rho }_{\varphi }(1+R)}{\left(1-\frac{12H^2}{\kappa {\rho }_c}\right)}^{-\frac{1}{2}}\right]}^{\frac{1}{2}}.$$

$$V(t)\simeq \frac{{\rho }_c}{2}\left(1-\sqrt{1-\frac{12H^2}{\kappa {\rho }_c}}\right).$$

$$\Gamma (\varphi ,T)=C_{\varphi }\frac{T^m}{{\varphi }^{m-1}},$$

Full text

I Introduction

Recent studies of the Cosmic Microwave Background (hereafter CMB) have placed tight constraints on single scalar-field models of slow-roll inflation. Specifically, based on Planck data Ade et al. (2016) it has been found that the inflationary models which are in agreement with the data are those with very low tensor-to-scalar fluctuation ratio $r=P_{T}/P_{s}\ll 1$, with a scalar spectral index $n_{s}\simeq 0.96$. Actually, the upper bound found by Planck team Ade et al. (2016), on this ratio, as a result of the non-observation of B-modes, is $r<0.11$.

In the standard picture of slow-roll inflation Guth (1981); Albrecht and Steinhardt (1982) the kinetic energy (${\dot{\phi}}^{2}/2$) and the potential energy $V(\phi)$ of the inflaton field satisfies ${\dot{\phi}}^{2}/2\ll V(\phi)$. The latter condition imposes the de Sitter expansion in the early universe, since the corresponding equation of state parameter $w\equiv\frac{(\dot{\phi}^{2}/2)-V(\phi)}{(\dot{\phi}^{2}/2)+V(\phi)}$ tends to -1. Moreover, in the standard inflation possible interactions among other fields with the inflaton must be neglected. Subsequently, after the slow-roll era the potential energy becomes comparable to the kinetic energy. This simply means that we are dealing with the so called reheating period in which the inflaton starts to oscillate around the minimum of the potential and progressively the universe is filled by radiation Shtanov et al. (1995); Kofman et al. (1997).

In the literature one may find other theoretical approaches in order to treat the nature of the early universe. Specifically, over the last two decades a lot of attention has been paid on the so-called warm inflationary pattern. Unlike to standard inflation, in this scenario the inflaton field is allowed to interact with other light fields, implying that radiation production occurs during the slow-roll period and hence reheating is avoided Berera (1995, 1997). Obviously, the main idea of the warm inflationary model is quite different with that of the standard cold inflation. Indeed, warm inflation satisfies the condition $T>H$, where $T$ is the temperature and $H$ is the Hubble parameter, which implies that the fluctuations of the inflaton field are thermal instead of fluctuations of the ground state. These thermal fluctuations play an eminent role in large scale structure studies because they are the initial seeds of density perturbations Hall et al. (2004); Moss (1985); Berera (2000).

From the scalar-field viewpoint, warm inflation is characterized by a tachyon scalar field with positive potential $V(\phi)$, while the corresponding kinetic energy does not obey the standard form (k-inflation Armendariz-Picon et al. (1999)). Also the potential energy has two special properties, namely a maximum at $\phi\rightarrow 0$ and a minimum when $\phi\rightarrow\infty$. For more details regarding the warm tachyon inflationary model we refer the reader to Refs.Herrera et al. (2006); Setare and Kamali (2012); Deshamukhya and Panda (2009). In this framework, it is well known that under specific conditions tachyon fields which are related with unstable D-branes Sen (2002a); Chingangbam et al. (2005) provide cosmic acceleration Sen (2002b); Sami et al. (2002); Armendariz-Picon et al. (1999); Mazumdar et al. (2001); Deshamukhya and Panda (2009); Panda et al. (2006); Chingangbam et al. (2005); Choudhury et al. (2002) during early times.

In the current article we present the dynamical behavior of the effective Loop Quantum Gravity (LQG) theory via the Hubble expansion, and investigate the compatibility of this scenario with warm tachyon-like inflationary scenarios. The basic techniques of LQG Thiemann (2003); Bojowald (2005); Ashtekar and Lewandowski (2004); Singh and Vandersloot (2005), can be applied in homogeneous and isotropic space-times in order to build a Loop Quantum Cosmology (LQC). The layout of the paper is as follows: At the beginning of Sec. II we present the main points of the LQC and then we discuss the scalar field description of the warm tachyon inflationary paradigm in the context of LQC. In Sec. III we estimate the slow roll parameters and in Sec, IV we calculate the temperature at the end of warm inflation. In Sec V we test the performance of the slow-roll predictions against the latest Planck 2015 data and finally, we summarize our conclusions in Sec. VI.

II Loop Quantum Cosmology versus Warm Inflation:

In this section we briefly present the basic cosmological features of Loop Quantum Cosmology (LQC). In the literature canonical quantization of gravity is given in terms of the so called Ashtekar-Barbero connection variables (see Ashtekar et al. (2003)). Without wanting to enter into the full details the phase space of classical general relativity can be spanned by conjugate variables $A_{q}^{i}$ (connection) and $E_{i}^{q}$ (triad) on a $3-manifold$ $\mathcal{M}$ which encodes curvature and spatial geometry respectively (labels $q$ and $i$ denote internal indices of $SU(2)$ and space index respectively). At the cosmological level (LQC) due to the isotropic and homogeneous symmetries the phase space is characterized by a single connection $c$ and a single triad $p$. Notice, that the Poisson bracket of LQC variables is given by

[TABLE]

where $\gamma\simeq 0.2375$ is the dimensionless Barbero-Immirzi parameter, derived from the black hole thermodynamics Ashtekar et al. (2000); Corichi (2009); Agullo et al. (2010); Meissner (2004). Considering a spatially flat Friedmann-Robertson- Walker (FRW) metric, the LQC variables become

[TABLE]

in the classical regime, where $a(t)$ is the scale factor of the universe and the overdot denotes derivative with respect to cosmic time $t$. In the variable system $\{c,p\}$ Ashtekar et al. Ashtekar et al. (2003) proposed that the classical Hamiltonian constraint is given by

[TABLE]

where $\mathcal{H}_{m}$ is the matter Hamiltonian. However, in the semi-classical regime various authors Singh and Vandersloot (2005); Ashtekar et al. (2006a, b) introduced an effective theory of loop quantum gravity which is appropriate for cosmology. In this case the effective Hamiltonian constraint is written as

[TABLE]

where the constant $\overline{\mu}$ is related to the minimal area of LQG (for more details see Singh and Vandersloot (2005); Ashtekar et al. (2006a, b)). Now, using the effective Hamilton equation

[TABLE]

and the Hamiltonian constraint ($\mathcal{H}_{\rm eff}\approx 0$) Singh and Vandersloot (2005) we can obtain the following equations of motion:

[TABLE]

Lastly, combining the above set of equations we provide the first Friedmann equation, namely the Hubble parameter

[TABLE]

where $\kappa=8\pi G=M^{-2}_{p}$, $\rho$ is the total energy density and $\rho_{c}=\sqrt{3}\,/(16\pi^{2}G^{2}\gamma^{3})\simeq 0.41\rho_{pl},~{}~{}\rho_{pl}=M_{p}^{4}$ ($M_{p}\simeq 2.43\times 10^{18}$Gev is the reduced Planck mass),appears as the critical loop quantum density Ashtekar and Wilson-Ewing (2008); Ashtekar et al. (2008); Zhang and Zhu (2013).

Before we continue our analysis we would like to mention that in the context of LQC the properties of inflation with a standard scalar field (the so called ”LQC-inflation”)111For a standard scalar field the corresponding density is given by $\rho_{\phi}=\frac{\dot{\phi}}{2}+V(\phi)$, while in the case of a tachyon field we have $\rho_{\phi}=\frac{V(\phi)}{\sqrt{1-{\dot{\phi}}^{2}}}$. The quantity $V(\phi)$ is the potential. has been discussed extensively in the literature (see Zhang and Zhu (2013); Copeland et al. (2008); Ribassin et al. (2011); Xiao et al. (2013); Xiong and Zhu (2007); Li and Zhu (2009); Gibbons (2002); Cai et al. (2011); del Campo et al. (2009); Setare and Kamali (2014a)) Here following a similar to the previous papers methodology we attempt to investigate the basic features of LQC warm inflation with a tachyon scalar field.

Below, following the work of Xiong and Zhu (2007); Li and Zhu (2009) we introduce the analysis of the tachyon field in the framework of LQC. Specifically, up to this point we did not specify the nature of the fluids involved. Let us now consider that we have a mixture of two fluids, radiation and tachyon field. Therefore, the total density takes the form $\rho=\rho_{\phi}+\rho_{\gamma}$, where $\rho_{\phi}$ and $\rho_{\gamma}$ are the corresponding tachyon field and radiation densities. In this framework the overall action Gibbons (2002) is written as

[TABLE]

where ${\cal R}$ is the Ricci scalar and $g$ is determinant of the metric.

At this point it is worth mentioning that the nature of tachyon warm inflation has yet to be found. However, in order to produce warm inflation one can use a tachyon scalar field for which the kinetic term does not follow the canonical form (k-inflation Armendariz-Picon et al. (1999)). Therefore, a possible path towards understanding the underlying mechanism of the current inflationary paradigm is to associate the tachyon fields with unstable D-branes Sen (2002a), for which it is well known that they lead to cosmic acceleration in early times Sen (2002b); Sami et al. (2002); Armendariz-Picon et al. (1999). Alternatively, one may study tachyon field inflation in the context of the Randall-Sundrum II brane (see Ref.Setare and Kamali (2013a) and references therein).

Notice that the Lagrangian of the tachyon field Gibbons (2002) which can be non-minimally coupled to gravity is given by

[TABLE]

The radiation Lagrangian $L_{\gamma}$ is associated with a fluid of photons with pressure $P_{\gamma}=\frac{\rho_{\gamma}}{3}$.

Therefore, using the conservation law for the total energy density we find

[TABLE]

or

[TABLE]

This law is the outcome of imposing the covariant conservation of the total energy density of the combined system of tachyon field and radiation, and thus is a direct reflection of the Bianchi identity satisfied by the geometric side of the Einstein’s equation. Due to the fact that in warm inflationary scenario Cai et al. (2011); Herrera et al. (2006); del Campo et al. (2009); Setare and Kamali (2012, 2013a); Zhang and Zhu (2014), the tachyon-field/photon interaction leads to radiation production one can split Eq.(4) as follows

[TABLE]

[TABLE]

where the positive quantity $\Gamma$ is the dissipation factor in unit of $M^{5}_{pl}$. Usually, in this kind of studies the dissipation term $\Gamma\dot{\phi}^{2}$ is given on a phenomenological basis in order to describe the nearly-thermal radiation bath of the warm inflationary paradigm. Notice, that in several papers Herrera et al. (2006); Setare and Kamali (2012, 2013b, 2014a, 2014b, 2013a) one may find another parameter that characterizes warm inflation, namely the dimensionless dissipation parameter $R\equiv\frac{\Gamma}{3H\rho_{\phi}}$. Within this framework, equation (5) boils down

[TABLE]

Notice, that if $R\gg 1$ then we are in the high dissipation regime, while in the weak dissipation regime the dimensionless dissipation parameter tends to zero ($\Gamma/3H\rho_{\phi}\ll 1$).

Utilizing the energy momentum tensor of the tachyon field

[TABLE]

equation and the corresponding Lagrangian one can obtain

[TABLE]

Now differentiating Eq.(1) with respect to cosmic time $t$ and using simultaneously Eq.(4) we find after some simple calculations

[TABLE]

Introducing the expressions (9) into Eq.(10) we obtain

[TABLE]

Obviously, in order to understand the dynamical behaviour of the current problem we need to combine Eq.(1) with Eq.(11). In particular, from Eq.(1) the quantum bounce point ($H=0$) occurs for $\rho=\rho_{c}$ Ashtekar et al. (2006c, b); Singh and Toporensky (2004); Vereshchagin (2004); Bamba et al. (2013); Ashtekar and Sloan (2011); Corichi and Karami (2011); Singh and Vandersloot (2005); Bojowald (2002); Bojowald and Vandersloot (2003); Copeland et al. (2008); Ranken and Singh (2012); Amorós et al. (2013); Xiao et al. (2013). On the other hand, prior to the bounce point the quantity ${\dot{H}}$ satisfies the restriction ${\dot{H}}>0$ and it remains in the positive regime until $\rho=\frac{\rho_{c}}{2}$. At this point ${\dot{H}}$ vanishes and eventually it becomes negative. Following the notations Ref.Xiao et al. (2013), we verify that LQC accommodates a ”LQC-tachyon inflation” from $\rho=\rho_{c}$ to $\rho=\frac{\rho_{c}}{2}$ ($\frac{\rho_{c}}{2}<\rho<\rho_{c}$). In this regime the Hubble parameter lies in the interval $0<H<H_{I}$, where the characteristic scale $H_{I}=\left(\frac{\kappa\rho_{c}}{12}\right)^{1/2}$ corresponds to the epoch where $\rho=\frac{\rho_{c}}{2}$ and for which ”LQC-tachyon inflation” ends. It is interesting to mention that the same to the above condition (see $\frac{\rho_{c}}{2}<\rho<\rho_{c}$) holds in the case of ”LQC-inflation”Zhang and Zhu (2013); Copeland et al. (2008); Ribassin et al. (2011); Xiao et al. (2013); Xiong and Zhu (2007); Li and Zhu (2009); Gibbons (2002); Cai et al. (2011); del Campo et al. (2009); Setare and Kamali (2014a) in which the scalar field has the standard form ($\rho_{\phi}=\frac{\dot{\phi}}{2}+V$). To this end, the arofementioned analysis points that the $(1-\rho/\rho_{c})$ term, imposes a specific era of inflation $0<H<H_{I}$, where for $\rho\simeq\rho_{c}$ the universe enters in the inflationary phase, while the characteristic scale $H_{I}$ presumably connected to a scale which is around to that of the Grand Unified Theory (GUT). For example, substituting $\kappa=M^{-2}_{p}$, $M_{p}\simeq 2.43\times 10^{18}$Gev and $\rho_{c}\simeq 0.41M_{p}^{4}$ into $H_{I}=\left(\frac{\kappa\rho_{c}}{12}\right)^{1/2}$ we find $H_{I}\sim 10^{17}$ Gev. Well after the primeval inflationary epoch, specifically for $\rho\ll\rho_{c}$, the Universe enters in the nominal radiation era. After this period the radiation component starts to become sub-dominant and the matter dominated era appears.

Overall, the condition $H=0$, ${\dot{H}}>0$ implies that the Universe enters in the LQC inflationary era at the bounce point Xiao et al. (2013) (see also Xiong and Zhu (2007)). Unlike nominal (cold) inflation, in which $H\simeq$const., here the key point is to understand that the Hubble parameter varies with time. Therefore, it has been proposed (see Ref.Xiao et al. (2013) references therein) that the necessary condition towards solving the horizon problem in the context of LQC-inflation is to ensure that the quantity $aH$ grows substantially during the early period of the Universe evolution. Specifically, it has been shown Copeland et al. (2008); Ribassin et al. (2011); Xiao et al. (2013) that the horizon problem can be solved if one introduces a single scalar field (for other alternatives see Ranken and Singh (2012)) in LQC-inflation. Also, Ref.Amorós et al. (2013) found that LQC-inflation can provide the appropriate number of e-folds ($\sim 60$), while the Universe includes a small cosmological constant and matter.

Now, inserting the equations (9) into Eq.(5) we immediately derive the modified Klein-Gordon equation which provides the time-evolution of $\phi$

[TABLE]

In the slow-roll approximation $\dot{\phi}^{2}\ll 1$ for tachyon field, the above equation simplified as:

[TABLE]

As it is obvious from the above equation the energy exchange between the tachyon field and radiation introduces an additional friction term ($\frac{\Gamma}{V}\dot{\phi}$) that modifies the standard Klein-Gordon equation.

At the epoch of warm inflation it is safe to assume that the energy density of the tachyon field is the dominant component of the cosmic fluid ($\rho_{\phi}\gg\rho_{\gamma}$ see Bastero-Gil et al. (2012)). Therefore, the effective Friedmann Eq. (1) reduces to

[TABLE]

On the other hand, utilizing Eqs. (7, 14), one can easily show that

[TABLE]

Moreover, if we consider that the quantity $\Gamma\dot{\phi}^{2}$ varies adiabatically then the radiation component evolves slowly which means that $\dot{\rho}_{\gamma}\ll 4H\rho_{\gamma}$ and $\dot{\rho}_{\gamma}\ll\Gamma\dot{\phi}^{2}$. Under the latter conditions the combination of Eq. (6) and Eq. (15) yields

[TABLE]

On the other hand, under adiabatic condition the above formula can be identified with the expression relating $\rho_{\gamma}$ with the radiation temperature $T$. By adding all the degrees of freedom of the created massless modes, the relationship between the radiation energy density and the temperature is given by Kolb and Turner (1990),

[TABLE]

and combining with Eq.(16) we find

[TABLE]

where $C_{\gamma}=\pi^{2}g_{*}/30$ in which we included the degrees of freedom of the created massless modes via the $g_{*}$ factor. Substituting Eq. (18) in (22) we obtain

[TABLE]

Now, by solving Eq.(14) for the tachyon density and using simultaneously Eq.(15) we find:

[TABLE]

Lastly, including the slow-roll approximation $\dot{\phi}^{2}\ll 1$ in Eq.(9) we have $\rho_{\phi}\simeq V(\phi)$ and thus $R=\frac{\Gamma}{3HV}$. Therefore, Eq. (II) reduces to

[TABLE]

From the above analysis it becomes clear that the aforementioned cosmological functions strongly dependent on the dissipation factor $\Gamma$ and the background expansion history, through the Hubble parameter $H(t)$. As we have already mentioned the tachyon field is exchanging energy with radiation implying that the dissipation factor is a characteristic quantity of warm inflation. Although, the precise form of the dissipation factor has yet to be found, various candidates for $\Gamma$ have been proposed in the literature. In the present study, based on the notations Bastero-Gil et al. (2011); Zhang (2009) we treat the dissipation factor as follows

[TABLE]

where $T$ is the temperature, $C_{\phi}$ is constant. From now on, we will call $\Gamma_{m}$ the parametrization of the dissipation factor where the index $m$ determines the form of $\Gamma$. Under the latter parametrization we immediately recognize the following situations: (a) when $m=-1$ (hereafter $\Gamma_{-1}$) we have $\Gamma=C_{\phi}\frac{\phi^{2}}{T}$ which corresponds to the dissipation rate of the non-SUSY model Berera et al. (1998); Yokoyama and Linde (1999); Herrera et al. (2013), (b) for $m=0$ ($\Gamma_{0}$ parametrization) we get $\Gamma=C_{\phi}\phi$ (SUSY case see Berera et al. (1998)) (c) for $m=1$ ($\Gamma_{1}$ parametrization) we have $\Gamma=C_{\phi}T$ (see Refs.Panotopoulos and Videla (2015); Berera et al. (2009); Moss and Xiong (2008); Bastero-Gil et al. (2016)) and (d) for $m=3$ ($\Gamma_{3}$ parametrization) we obtain $\Gamma=C_{\phi}\frac{T^{3}}{\phi^{2}}$ that corresponds to the low temperature SUSY model Moss and Xiong (2006); Bueno Sanchez et al. (2008); Ramos and da Silva (2013); Cerezo and Rosa (2013).

In this kind of studies it is well known that the precise functional form of the potential $V(t)$ and the Hubble parameter $H(t)$ can not be simultaneously found from first principles. The latter implies that the only way to use the Loop Quantum approach in Cosmology is to phenomenologically select the functional form of either the potential or the Hubble parameter. As an example in the context of LQC with a standard scalar field ($\rho_{\phi}=\frac{\dot{\phi}}{2}+V$) we refer the reader the work of Ref.Zhang and Zhu (2013) These authors have been phenomenologically introduced the well known power law potential $V\propto\phi^{2}$ in LQC towards treating the cosmic expansion. Alternatively, one may impose the functional form of the Hubble parameter in order to derive the potential Barrow and Saich (1990); Barrow and Liddle (1993); Berera (1997); Setare and Kamali (2012, 2013b); Herrera et al. (2013); Sharif and Saleem (2015).

In our work we have decided to phenomenologically select the functional form of $H(t)$ using the well known solution of warm intermediate inflation provided by BarrowBarrow (1996) as a reference model.

Inspired by the work of Herrera et al.Herrera et al. (2014) in the context of warm intermediate inflation, we consider the following form of the scale factor

[TABLE]

where $f$ lies in the interval $0<f<1$. Obviously, the evolution of Hubble parameter and its first time-derivative are given by

[TABLE]

In this scenario the cosmic expansion evolves slower than the standard de Sitter model, $a(t)\propto{\rm exp}(H_{I}t)$ [$H(t)=H_{I}=$const.] and faster than the power-law inflation ($a\propto t^{p}$, $p>1$). The exponential expression of the scale factor Eq.(23) and the corresponding Hubble parameter Eq.(24) are used in order to approximate the cosmic expansion prior to the inflationary era and not near the bounce. Therefore, Eq.(24) is obtained using the classical Einstein’s equations. Similar to our notations have been used in the paper of Herrera et al.Herrera et al. (2014) but in the case of LQC warm inflation with a canonical scalar field.

III The Strong Dissipative Regime and Slow-Roll parameters:

In the rest of the paper we focus our analysis on the high dissipation regime ($R\gg 1$). Hence, Eqs. (15, 19) reduces to

[TABLE]

[TABLE]

If we insert Eq. (26) in Eq. (25) then the evolution of tachyon scalar field is given by

[TABLE]

Performing the integration of Eq. (27) with the aid of Eq. (24) we have

[TABLE]

where $\phi_{\star}$ is the value of $\phi(t_{\star})$ at a characteristic time $t_{\star}$ which obeys the following inequality $t_{\star}>t_{\rm bounce}$ by definition. In the above relations $K_{m}$ and $G_{m}(t)$ are defined as follows

[TABLE]

[TABLE]

Notice, that ${}_{1}F_{2}$ is the hypergeometric function and $S\equiv\frac{12A^{2}f^{2}}{\kappa\rho_{c}}$. Without any loss of generality we set $\phi_{\star}=0$.

Under the above conditions the Hubble parameter can be expressed as a function of the tachyon scalar field as follows

[TABLE]

in which $G_{m}^{-1}$ corresponds to the inverse function of $G_{m}(t)$. Moreover, the variable $\varphi$ is given by

[TABLE]

Clearly, the evolution of the Hubble parameter is affected by the critical loop quantum density since the cosmic time $t$ can be written in terms of $\rho_{c}$ [see Eqs. (III) and (31)]. Therefore, using Eqs.(21, 24, 26, III),the potential and the dissipation factor are calculated as:

[TABLE]

and

[TABLE]

Now, using standard lines we are ready to provide the slow-roll parameters

[TABLE]

Indeed, with the aid of Eqs. (24,26, III) the slow-roll parameters become

[TABLE]

We would like to stress here that for the intermediate inflation the slow-roll parameters are always less than unity which means that inflation never ends. In our model the condition $\epsilon=1$ makes sure that inflation starts at the earliest possible stage Barrow et al. (2006); Barrow and Liddle (1993). As far as the number of e-folds is concerned we have

[TABLE]

or

[TABLE]

where for the last equality we used Eq.(24). Also, $\varphi_{k}=\varphi(t_{k})$ and $\varphi_{in}=\varphi(t_{in})$ denote the values at the horizon crossing and at the beginning of inflation respectively. Equating Eqs. (36) to unity, $\epsilon\left(\varphi_{in}\right)=1$ and using Eqs.(28) and (32) the value of the tachyon scalar field at the beginning of inflation is given by

[TABLE]

where $y=\left(\frac{1-f}{Af}\right)^{1/f}$. To this end, it becomes clear that from (41) one may express Eqs. (31 - LABEL:wr6) as a function of $N$. Indeed, by doing that we find

[TABLE]

where $I(N)=(\frac{1+f(N-1)}{fA})^{\frac{1}{f}}$. One of the most important features of any inflationary scenario is related with the formation of large scale structures. For example, in the case of warm inflationary paradigm thermal fluctuations play a key role because they provide the initial seeds for the formation of cosmic structures. The situation regarding cosmological perturbations within the effective Hamiltonian formalism in LQC has been studied in Bojowald et al. (2006); Wilson-Ewing (2012); Bojowald et al. (2008); Cailleteau et al. (2012), which however is beyond the scope of the present study. Cosmological perturbations in LQC have been explored in several studies Zhang (2009); Herrera (2010); Haro (2013); Setare and Kamali (2014a); Zhang and Ling (2007); Herrera (2010). Following these works the corresponding curvature perturbation was found to be $\delta{\cal R}=\left(\frac{H}{\dot{\phi}}\right)\delta\phi$ and thus the amplitude of scalar fluctuations for LQC is given by

[TABLE]

As we have already defined in the introduction the nature of scalar perturbations in warm inflation is thermal and not quantum as we consider in the standard inflationary model. In particular, it has been found Berera (1995, 1997); Hall et al. (2004) that in the case of $R\gg 1$ warm scalar perturbations obey the following expression

[TABLE]

where the wave number $k_{F}=\sqrt{\frac{\Gamma H}{V}}=H\sqrt{\frac{\Gamma}{3HV}}\geq H$ provides the freeze-out scale at which the dissipation damps out to thermally excited fluctuations of inflaton ($\frac{V^{\prime\prime}}{V^{\prime}}<\frac{\Gamma H}{V}$) Taylor and Berera (2000). Inserting the wave-number $k_{F}$ and Eq.(44) into Eq.(43) we arrive at

[TABLE]

We continue our calculations by substituting Eqs. (31 - LABEL:wr6) and Eqs. (41) in Eq. (45) we obtain the power spectrum in terms of the number of e-folds

[TABLE]

where

[TABLE]

At this point we introduce the spectral spectral index in our analysis which is defined as

[TABLE]

Since $d{\rm ln}k=-dN$ and armed with the analytic expression of ${\cal P}_{\cal R}$ [see Eq.(43)] we derive after some simple algebra the scalar spectral index takes

[TABLE]

with

[TABLE]

[TABLE]

On the other hand, the amplitude of tensor fluctuations is given by

[TABLE]

from which we can define the tensor-to-scalar ratio

[TABLE]

Introducing the appropriate formulas of $H$, $V$, $\Gamma$ and $T$ in the above equation we find

[TABLE]

where

[TABLE]

In order to simplify the computation of the power spectrum we utilize the initial states from the standard inflationary scenario and we plug the LQC effective equations for the background.

Finally, we would like to compare our analytical results with those of previous studies. In particular, Herrera Herrera (2010) studied the canonical warm inflationary model in LQC (for similar studies see Herrera et al. (2013); Zhang and Zhu (2013); Xiao and Zhu (2011); Herrera et al. (2014); Xiao et al. (2014)) under specific conditions, namely (i) the potential has the form $V(\phi)\propto\phi^{2}$ (chaotic potential) and (ii) the dissipation rate $\Gamma$ is constant. Also, Herrera et al. Herrera et al. (2013) extended the analytical solutions of Ref.Herrera (2010) by taking into account a general form of $\Gamma$. However, the aforementioned studies are in the context of standard scalar field theory which means that the density and the corresponding pressure are given by $\rho_{\phi}={\dot{\phi}}^{2}/2+V(\phi)$ and $p_{\phi}={\dot{\phi}}^{2}/2-V(\phi)$. In the current article we investigate, for a first time, the warm LQC-tachyon inflationary scenario [see Eq.(9] for a large family of $\Gamma$ parametrizations ($\Gamma\propto T^{m}/\phi^{m-1}$) and we provide the corresponding class of potentials [see Eq.(33)].

IV Temperature at the end of inflation:

In this section, following the methodology of Mielczarek (2011) we attempt to derive the temperature at the end of warm inflationary scenario. The entire cosmological history contains the following eras: I) from $t_{k}$ (Hubble crossing time) till the end of slow-roll warm inflation which is denoted by $t_{end}$, II) from $t_{end}$ (recombination era) till the recombination epoch which is indicated by $t_{rec}$ and III) from $t_{rec}$ up to present time $t_{0}$, for which we have the matter and dark energy dominated eras. Therefore, it is easy to write

[TABLE]

and thus $N_{tot}=N_{0}+N_{rec}+N$. Notice that in the above expression $a_{0}\equiv 1$ is the scale factor at the present time, $k=a_{k}H_{k}$ is the Fourier mode, $N_{tot}$ is the total number of e-folds, $N_{rec}\equiv\frac{a_{rec}}{a_{end}}$, $N_{0}\equiv\frac{a_{0}}{a_{rec}}$ and $N$ is given by Eq.(39).

At the end of warm inflation, the cosmic expansion enters in the radiation dominated era in which the universe is full of relativistic particles. Under adiabatic circumstances the radiation entropy per comoving volume reads

[TABLE]

from which we find

[TABLE]

or

[TABLE]

In the last step we utilized the entropy conservation law of the adiabatic radiation phase, which means that $g_{end}T^{3}_{end}a^{3}_{end}=g_{rec}T^{3}_{rec}a^{3}_{rec}$ and we have set $g_{rec}=g_{\gamma}=2$. On the other hand, the temperature at the recombination epoch satisfies the well known formula

[TABLE]

Inserting the above in Eq.(61) we find

[TABLE]

where

[TABLE]

Lastly, using Eqs. (31, 40, 41), the Hubble parameter at the Hubble crossing time is written as a function of $N$, namely

[TABLE]

Substituting Eq. (63) in Eq. (58) and utilizing Eq. (65) we obtain the temperature at the end of warm inflation

[TABLE]

Obviously, in order to compute $T_{end}$ we need to know the parameters $A$, $f$ and $N$ or equivalently the pair $(n_{s},r)$. Of course, we expect that the temperature at the end of warm inflation to satisfy the following inequality $T_{BBN}<T_{end}<T_{I}$, where $T_{BBN}\sim 10^{-2}$Gev is the temperature at the Big Bang nucleosynthesis (BBN) and $T_{I}$ is the estimated temperature scale of inflation, an upper bound of which is provided by Planck team Ade et al. (2016), $\sim 10^{16}$Gev.

V Observational constraints:

In order to check the consistency of the above slow-roll predictions with observation, we compare our results against those of Planck Ade et al. (2016). Notice, that the pair $(n_{s},r)$ is given by Eqs.(49,III). Also, regarding the number of e-folds we use $N=50$ and $N=60$ respectively.

In particular we find:

• •

For the $\Gamma_{-1}$ ($m=-1$) parametrization: In fig.(1) we present the $(n_{s},r)$ contours together with our prediction for various values of $f$, $A$ and $C_{\phi}$. Specifically, in this graph the solid, dot-dashed and dashed lines correspond to $(A,f,C_{\phi})=(3.1,0.7,10000)$, $(A,f,C_{\phi})=(2.8,0.63,5000)$ and $(A,f,C_{\phi})=(4,0.63,15000)$ respectively. We observe that in the case of large $C_{\phi}$ the scalar-to-tensor ratio $r$ (55) becomes small with respect to the reference Planck result, while the spectral index lies in the interval $0.955<n_{s}<0.975$ Therefore, we argue that it is always possible to find the appropriate value of $C_{\phi}$ in order to get predictions which are in agreement with those of Planck. For example, using $A=3.3,f=0.7,C_{\phi}=15000$ we obtain $(n_{s},r)=(0.965691,0.0141694)$.

• •

For the $\Gamma_{0}$ ($m=0$) parametrization: here our results can be found in figure (2) for the following special cases: $(A,f,C_{\phi})=(4.6,0.59,10000)$ [see solid curve], $(A,f,C_{\phi})=(3.1,0.59,5000)$ [see dashed curve], and $(A,f,C_{\phi})=(4.3,0.59,15000)$ [see dot-dashed curve]. As in the previous case, also here we need to fine-tune $C_{\phi}$ in order to be consistent with observations. As an example, using $A=4.6,f=0.59,C_{\phi}=15000$ we obtain $(n_{s},r)=(0.960205,0.0191892)$.

• •

For the $\Gamma_{1}$ ($m=1$) parametrization: We argue bellow that this model alleviates the fine-tuning issue of the above parametrizations. We remind the reader warm inflation satisfies the following condition $T>H$ (or $\frac{T}{H}>1$). For $m=1$ the ratio between the $T$ and $H$ is given by (18),(24),(39):

[TABLE]

In Fig.(3) we show $\frac{T}{H}$ in $A-f$ plane. The black solid line corresponds to the boundary limit $\frac{T}{H}=1$, while the top left part of the plane is consistent with the restriction $\frac{T}{H}>1$. Moreover, in Fig.(4) we plot the $A-f$ allowed area in which our $(n_{s},r)$ results are in agreement with those of Planck within $1\sigma$ errors. Notice, that the transparent background (foreground opaque) corresponds to $N=60$ ($N=50$). From this figure we observe that for different values of the dissipation coefficient $C_{\phi}$ we always find a narrow strip in the $A-f$ parameter space which is in agreement with Planck’s priors of $n_{s}$ and $r$. Indeed, in Fig.(5) we plot the Planck confidence contours in the plane of $(n_{s},r)$. On top of that we show the big solid point for the individual pair of $(n_{s},r)$ in the case of $N=60$. Also, we provide the corresponding small solid point for $N=50$.

It is worth noting that concerning the observational signatures of LQC in the CMB data, an intense debate is taking place in the literature about the implementation of LQC to CMB data. Recently, Ashtekar and Gupt (see Ref.Ashtekar and Gupt (2017)) using various correlation functions for scalar perturbations found that LQC is favored by Planck, while standard (cold) inflation can not accommodate the data at large angular scales ($l\leq 30$). The heated discussion is going and the aim of our study is to contribute to this debate. Within this framework in figure 5 we plot the predicted $(n_{s},r)$ in the case non-LQC warm inflation (see stars in figure 5). We then compare the latter $(n_{s},r)$ predictions with those of the warm LQC-tachyon inflation (solid points). In principle, this can help us to understand better the theoretical expectations of the warm LQC-tachyon inflationary model, as well as to identify the differences from the non-LQC warm tachyon inflation. From figure 5 we observe that both inflationary tachyon models provide the same spectral index $n_{s}$. Concerning the tensor-to-scalar fluctuation ratio $r$ the situation is different. Although the predictions are in agreement with Planck observations (within 1$\sigma$), we find that the non-LQC warm tachyon inflation provides a tensor-to-scalar fluctuation ratio which is smaller than that of warm LQC-tachyon inflation. Therefore, in the light of the next generation of $B-$mode data one may use this difference in order to test the performance of LQC in the CMB data.

VI Conclusions:

In this work, we studied the observational signatures of quantum cosmology in the Cosmic Microwave Background data given by Planck2015. We utilized the paradigm of warm inflation with a tachyon scalar field to the loop quantum cosmology. Within this framework, we first derived the main cosmological quantities as a function of the tachyon field. Second, we provided the slow-roll parameters and the power spectrum of scalar and tensor fluctuations respectively. Finally, we checked the performance of various warm inflationary models against the data provided by Planck2015 data and we find a class of patterns which are consistent with the observations.

Acknowledgment: SB acknowledges support by the Research Center for Astronomy of the Academy of Athens in the context of the program ”Testing general relativity on cosmological scales” (ref. number 200/872).