Constraints on warm power-law inflation in light of Planck and results
We want to constrain the exponents of the potential and the dissipative coefficients for warm power-law inflation considering Planck 2015 results. To do this end, we intro- duce a power-law potential with a general exponent beside a general dissipative coefficient. Then the parameters of the model are constrained to estimate best fits for them. In addition related inflationary observables such as tensor-to-scalar ratio, power-spectra of density perturbations and gravitational waves, scalar and tensor spectral indices, running spectral index and the number of e-fold in both weak and strong regimes are obtained. Moreover, our results for free parameters are compatible with data originating from cosmic microwave background and Planck 2015.
It is well-known, inflationary scenario is the most acceptable paradigm to explain the evolution of early Universe in the modern cosmology Sta80 (); Gut81 (); Lin82 (); Alb (); Lin83 (); Lin86a (); Lin86b (). This theory can solve some drawbacks of the standard big bang theory such as the horizon, the flatness and the relic problem Lid00 (); Bas (); Lem (); Kin (); Bau09 (); Bau14 (). Additionally, inflation theory provides a mechanism to explain structure formation and the source of the observed anisotropy in the Cosmic Microwave Background (CMB) radiation Larson (); Bennettc (); Jarosik (). Beside, inflation scenario evaluates correct tensor to scalar perturbation ratio
Bas (); Liddle0 (); Langl (); Lyth1 (); Guth00 (); Lidsey97 (); Mukhanov-etal (); Haidar0 (); Haidar (); Haidar2 ().
Many various inflationary models have been investigated in (Robert, ; Linde, ), that could be divided into two parts cold and warm inflationary models. In the scenario of warm inflation
Berera (); Taylor (); Oliveira (); Oliveira2 (); Hall (); Bastero (); Cid (), during the initial rapid expansion of the Universe the continuous radiation production was happened and also there exist interaction between radiation and scalar fields Oliveira (); Oliveira2 (). So that energy density remains roughly constantTaylor (). Thanks to this interaction the inflation ends and smoothly entered into radiation dominate era, without need to reheating phase. As advantage, the warm inflation gets rid of the graceful exit problem Oliveira2 (); Hall (); Antonella (). Additionally, this interaction is worked out by dissipative effect that appears in the evolution equation of scalar field and initially was introduced by (Fang, ), beside this another important term, i.e. the friction term, that appears in the continuity equations of scalar field and radiation was studied in Ref (Moss, ). As supplementary discussion, a principal condition in order to warm inflation happen is that the radiation temperature should satisfy , where and are temperature and well-known Hubble parameter respectively Herrera:2015aja (). In Ref RefBerera (); Hall (), it was noticed that both quantum and thermal perturbations are related to and respectively. According to condition , the thermal fluctuations play a crucial role in producing the primary density perturbations, as the seeds for large-scale structure formation. In this case, the thermal fluctuations are dominant compared to the quantum portion Hall (); Berera2 ().
On the other hand, in 1983, A. Linde has introduced a new proposal for inflation chaotic scalar field namely ”chaotic inflation” Linde1 (). In this theory the power law potential, as same as quantum field theory, plays an basic role. Hence the power law potentials attract a lot of interests; these models are interesting for their simplicity and compatibility with observations. But this simple question that which exponent can run warm inflation in a better way remains unsolved Lyth ().
This paper is organized as follows: In Sec. II, we will express the main dynamic equations for warm inflation and evaluate the inflationary parameters. Sec. II is related to power-law potential in weak dissipative regime IV. Also sub-Sec IV is dedicated to power-law potential in strong regime. We will compare our results with planck data2015. At last, Sec. V is devoted to conclusion.
Ii General framwork
We consider a spatially flat Friedmann-Robertson-Walker (FRW) metric. Our model consists of two sources as a self-interacting scalar field and a perfect fluid. The energy density and pressure of scalar field are expressed respectively as
The energy density for perfect fluid, i.e. almost radiation, is presented as . Also we suppose that the Friedmann equation appeared as follows
where is the reduced Planck mass. The evolution equation of scalar field in warm inflation can be expressed as (sayar, )
where the parameter is introduced as an anomalous dissipation function inwhich is the dissipation coefficient. The continuity equation of and is described by the equations (Berera2, )
where the dissipative coefficient , and from above equations it is understood that the flow of energy is from scalar field to the radiation. We consider an ansatz for dissipation coefficient as
here the dissipation coefficient is a function of both temperature and scalar field (Zhang:2009ge, ; BasteroGil:2011xd, ; warm). During warm inflation, the energy density of scalar field is dominant compared to the radiation energy density i.e. (62526, ; 6252602, ; 6252603, ; 6252604, ). In other words, the expansion rate is smaller than the radiation energy density, i.e. or , which is the requirement of happening the warm inflation. Beside, if we consider the slow roll conditions, it is assumed that during inflation the radiation production becomes quasi-stable i.e. (62526, ; 6252602, ; 6252603, ; 6252604, ) and according to (2) . Then, the equations (2) and (4) could be approximated as
The necessity condition to occur inflation and its persist is that the slow-roll parameters in terms of potential should appear as
As a result, whenever one of these parameters becomes equal to the inflation process terminates. Another important parameters to investigate inflation evolutions is the number of e-fold. This parameter plays role in solving the horizon problem and is defined as
Additionally to constrain the free parameters of the model we need to calculate the scalar and tensor spectra, the tensor-to-scalar spectrum ratio, the scalar spectral index and running parameter which are expressed respectively as
From the above equations and slow roll conditions it ends up that at sound horizon exit , i.e. , one obtains (Unn13, )
To receive above relation we consider this fact that for the slow roll inflation and sound speed are constant.
Iii Consistency Of Power-Law Potential In Weak dissipative Regime
In this section, we want to separate the calculation into two separated sections namely the weak and strong dissipation regimes, i.e. and respectively. Then we introduce a general form for the power-law potential and investigate the behaviour of such a potential in the warm inflation scenario. We shall see in the weak dissipation regime the Hubble parameter governs the process of inflation; and in the strong regime, the evolution of inflation is managed by the dissipation coefficient . The different types of dissipation coefficients were investigated extensively in the literature and for example we refer the reader to (BasteroGil:2012cm, ; Herrera:2015aja, ). In this regime we interested to find best estimation for the exponents of potential considering related dissipation coefficient. Given that in this case our model grows based on the weak dissipative regime in which . Hence, by introducing the potential as
where . In addition, the first slow-roll parameter (8) for this regime reduces to
Generally, the inflation period terminates when , where using Eq.(21) one achieves . Beside, from Eq.(10) the value of scalar field at the time of exit of the horizon is given by the following equation
In what following, we want to calculate the perturbation parameters based on our model and then compare them with observational data. To do so we consider Eq.(11) in which for weak regime we have . Whereas the fluctuations in the inflation are created by the thermal fluctuations instead of the quantum fluctuations, (Herrera:2015aja, ), accordingly the amplitude of scalar perturbation becomes
From Eq.(12), the tensor power spectrum in terms of the number of e-fold obtained as
In addition, from Eq. (13), the tensor to scalar ratio is expressed by
Now to make the constraints on free parameters of the model, we need to compare them with observational data originated from Planck and data (Planck2015, ). For this propose, by virtue of Eqs.(28) and (24) we plot the diagram as one of the more important instruments to test the accuracy of a theoretical model. So the figure 1 shows the diagram in which Confidence Levels (CLs) 68% and 95% CL allowed by Planck 2015, TT, TE, EE+lowP data (Planck2015, ) are illustrated. The black line shows prediction of our model in which free parameters are obtained as , , and , and . As it is seen figure 1 indicates that our results are in a good agreement compared to the observations.
The latest observational data suggest that the amplitude of scalar perturbation at the horizon crossing is very close to amount , and the tensor-to-scalar ratio has an upper limit as at 68% (Planck2015, ). Therefore, to satisfy the above conditions we will specify what values for the parameters and are in good agreement. Also based on these values for the parameters and the amount of the parameters and are calculated, one can see table (1).
In what following, by means of Eqs.(25) and (24) we want to plot the diagram and then compare the results with the observational data originated by Planck. After some algebra the free parameters of the model can be obtained as , , , , , and . Figure 2 which indicates the prediction of this model can lie insides the joint 68% CL region of Planck 2015 TT, TE, EE+lowP data (Planck2015, ), and so is in good compatibility with observations.
Iv Consistency Of Power-Law Potential In strong dissipative Regime
Now we want to repeat our calculations but for strong dissipation regime, . So, by considering Eqs. (1), (7) and (17) and slow-roll approximation, the Friedman and evolution scalar field equations take the following form
where . Therefore based on above equations the first slow-roll parameter and dissipation term are appeared respectively as
The inflation in the strong regime ends up when , where using Eq.(33) the scalar field related to that era are obtained as
Additionally, from Eq.(10) the value of the scalar field at the time which perturbations cross the horizon in the strong regime is given by
Now to evaluate consistency of our model in strong regimes with observations originated from Planck 2013 and 2015 data (Planck2015, ), similar to the weak case, we do recourse to the plot and the diagram. The figure 3 shows the diagram in which CLs 68% and 95% CL allowed by Planck 2015 data, TT, TE, EE+lowP data (Planck2015, ) are illustrated and the black line indicates our results. We obtain the free parameters as , , , , and . As it is seen from the figure 3, our results are in a good agreement comparing with with observations.
As aforementioned said, the latest observational data imply that the amplitude of the scalar perturbation at the horizon crossing is very close to , and the tensor-to-scalar ratio has an upper bound as at 68% (Planck2015, ). Therefore, in table (2) we will specify that what exponents of the and parameters satisfy the constraints on parameters , and . Then based on the different values of the parameters and the amount of the parameters and are obtained.
By considering Eqs.(39) and (38) we depict the diagram and compare its contrast with the observations originated from Planck 2015. Here we find out that the free parameters should be considered as , , , and . The Fig.4 indicates the prediction of our model which can lie insides the joint 68% CL region of Planck 2015, TT, TE, EE+lowP data (Planck2015, ) in which we see a good agreement.
By introducing a general form for the power-law potential we have studied warm power-law inflation. In this work we tried to put constraints on the power of the potential beside the dissipation coefficient, i.e. , by virtue of data originated from the Planck and .
To do so, at first the different inflationary observables such as tensor-to-scalar ratio, power spectrum indices of density perturbations and gravitational waves, scalar spectral index and running spectral index in both the weak and strong regimes were calculated. Then, whereas one of the important results of the Planck data is diagram, we compared our results with Planck data and found out they are in good agreement. In more detailes the Figs. (1) and (3) for both the weak and strong regimes respectively have can be considered as a justification for our claims. In addition, the Figs.(1) and (3) have shown that in the weak regime the suitable adaption is happened for the parameters and . This results bring in our mind the well-known theory in quantum field theory. But in the strong regime the parameters appeared as and in which are similar to the theory. The other suitable values for the parameters and that could derive inflation have given for both the weak and strong regimes in the Tabs. (I) and (II) respectively. Beside these, in Figs.(2) and (4) the predictions for the running spectral index are also in satisfactory agreement with observational data. In which in the weak regime the adaption is happened in the exponent parameter of inflation potential and dissipation coefficient and respectively. But in strong case it was happened by taking the parameters as and . They are lie inside the joint 68% CL region of Planck 2015 TT, TE, EE+lowP data (Planck2015, ). The coefficient of scalar potential, in both weak and strong regimes are roughly the same order but the coefficient of dissipative coefficient in the weak regime is very lager than it in strong case, one can see Tabs. (I) and (II). At last, it was considerable that only some determinate values of the parameters and are allowed to derive inflation for the weak and strong warm inflation.
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