Some aspects of tachyon field cosmology

We study inflation, perturbations, non-Gaussianity, and late-time cosmological dynamics of a tachyon field both minimally and nonminimally coupled to gravity. By analyzing the parameter space of the model, the viability of the model in confrontation with recent observational data is considered. In a dynamical system technique, we study the phase space dynamics of both a minimally and nonminimally coupled tachyon field. We find the fixed points (lines in our setup) and explore their stability. Also, we perform a statefinder diagnostic to both cases and show that the trajectories of the statefinder pairs reach a stable state which corresponds to a Lambda cold dark matter scenario.

Figure 1

Evolution of the scalar spectral index (left panel) and tensor-to-scalar ratio (right panel) versus the number of e-folds for the quadratic potential. For a minimally coupled tachyon field, the spectral index is red tilted and nearly scale invariant.

Figure 2

Behavior of the tensor-to-scalar ratio with respect to the scalar spectral index for various $N$ and for quadratic potential in the background of $\mathrm{WMAP}9+\mathrm{eCMB}+\mathrm{BAO}+{\mathrm{H}}_0$ data. The two contours are corresponding to the 68% and 95% levels of confidence. $N$ increases in the direction of the arrow.

Figure 3

Evolution of the scalar spectral index (left panel) and tensor-to-scalar ratio (right panel) versus the number of e-folds, for an exponential-type potential.

Figure 4

Behavior of the tensor-to-scalar ratio with respect to the scalar spectral index for various $N$ and for the exponential-type potential in the background of $\mathrm{WMAP}9+\mathrm{eCMB}+\mathrm{BAO}+{\mathrm{H}}_0$ data. The two contours are corresponding to the 68% and 95% levels of confidence.

Figure 5

Evolution of the scalar spectral index (left panel) and tensor-to-scalar ratio (right panel) versus the number of e-folds and the intermediate parameter $l$, for the intermediate potential.

Figure 6

Behavior of the tensor-to-scalar ratio with respect to the scalar spectral index for various $l$ for the intermediate potential in the background of $\mathrm{WMAP}9+\mathrm{eCMB}+\mathrm{BAO}+{\mathrm{H}}_0$ data. The two contours are corresponding to the 68% and 95% levels of confidence. $l$ increases in the direction of the arrow.

Figure 7

Evolution of the non-Gaussianity with respect to $N$ and $\dot{\varphi }$ for a quadratic potential (left panel), an exponential potential (middle panel), and an intermediate potential (right panel) in a minimal coupling setup.

Figure 8

The phase space trajectories of the minimally coupled tachyon field model. The critical point $O$ is an effectively ordinary components (matter or radiation) dominated solution and is an unstable saddle point. Points $P$ and $Q$ are unstable too. The points $N$ and $M$ are effectively tachyon field dominated solutions and are stable critical points. There is no accelerated phase of expansion.

Figure 9

Behavior of the effective equation of state parameter versus the cosmic time. The left panel is corresponding to $\omega =0$ (dust) and the right panel is corresponding to $\omega =\frac{1}{3}$ (radiation). ${\omega }_{\mathrm{eff}}$ evolves and tends to the value $-0.286$. These figures have been plotted with $x(0)=0.99$ and $y(0)=0.01$ (the magenta curve) and $x(0)=0.75$ and $y(0)=0.05$ (the blue dashed curve).

Figure 10

Trajectories in the $\{r,q\}$ phase plan with $\omega =0$ (left panel, corresponding to the dust) and $\omega =\frac{1}{3}$ (right panel, corresponding to the radiation). The initial conditions are $x(0)=0.99$ and $y(0)=0.01$ for the magenta (solid line) trajectory and $x(0)=0.75$ and $y(0)=0.05$ for the blue (dashed) trajectory. The magenta and blue highlighted dots (specified by $T$) are current values of $\{r,q\}$ in the model. The black dot (specified by $F$) is the stable state of $\{r,q\}$ in the future. Also the red dot (remarked by $\Lambda \mathrm{CDM}$) is the value of statefinder $\{r,q\}$ in a $\Lambda \mathrm{CDM}$ scenario.

Figure 11

Trajectories in the $\{s,q\}$ phase plan with $\omega =0$ (left panel) and $\omega =\frac{1}{3}$ (right panel). The initial conditions are $x(0)=0.99$ and $y(0)=0.01$ for the magenta (solid line) trajectory and $x(0)=0.75$ and $y(0)=0.05$ for the blue (dashed) trajectory. The magenta and blue highlighted dots (specified by $T$) are current values of $\{s,q\}$ in the model. The black dot (specified by $F$) is the stable state of $\{s,q\}$ in the future. Also the red dot (remarked by $\Lambda \mathrm{CDM}$) is the value of statefinder $\{s,q\}$ in a $\Lambda \mathrm{CDM}$ scenario.

Figure 12

The trajectories in $\{r,s\}$ phase plan with $\omega =0$ (the left panel, corresponding to dust) and $\omega =\frac{1}{3}$ (the right panel, corresponding to radiation). The initial conditions are $x(0)=0.99$ and $y(0)=0.01$ for magenta (solid line) trajectory and $x(0)=0.75$ and $y(0)=0.05$ for blue (dashed) trajectory. The magenta and blue bold dots (specified by $T$) are current values of $\{r,s\}$ in the model. The black dot (specified by $F$) is the stable state of $\{r,s\}$ in the future. Also the red dot (remarked by $\Lambda \mathrm{CDM}$) is the value of statefinder $\{r,s\}$ in a $\Lambda \mathrm{CDM}$ scenario.

Figure 13

Evolution of the scalar spectral index (left panel) and the tensor-to-scalar ratio (right panel) versus the nonminimal coupling and number of e-folds parameters.

Figure 14

Behavior of the tensor-to-scalar ratio with respect to the scalar spectral index for various $\xi$ and for quadratic potential in the background of $\mathrm{WMAP}9+\mathrm{eCMB}+\mathrm{BAO}+{\mathrm{H}}_0$ data. The two contours are corresponding to the 68% and 95% levels of confidence.

Figure 15

Evolution of the scalar spectral index (left panel) and the tensor-to-scalar ratio (right panel) with respect to the nonminimal coupling and number of e-folds parameters.

Figure 16

Behavior of the tensor-to-scalar ratio with respect to the scalar spectral index for various $\xi$ and for exponential potential in the background of $\mathrm{WMAP}9+\mathrm{eCMB}+\mathrm{BAO}+{\mathrm{H}}_0$ data. The two contours are corresponding to the 68% and 95% levels of confidence.

Figure 17

Evolution of the scalar spectral index with $l=0.4$ (left panel) and $l=0.87$ (right panel) with respect to the nonminimal coupling and number of e-folds parameters.

Figure 18

Evolution of the tensor-to-scalar ratio with $l=0.4$ (left panel) and $l=0.87$ (right panel) with respect to the nonminimal coupling and the number of e-folds parameters.

Figure 19

Behavior of the tensor-to-scalar ratio with respect to the scalar spectral index for various $\xi$ and for exponential potential in the background of $\mathrm{WMAP}9+\mathrm{eCMB}+\mathrm{BAO}+{\mathrm{H}}_0$ data. The two contours are corresponding to the 68% and 95% levels of confidence. The left panel is corresponding to $l=0.4$ and right panel with $l=0.87$.

Figure 20

Non-Gaussianity versus $\xi$ and $\dot{\varphi }$ for $N=50$ for quadratic (left panel), exponential (middle panel), and intermediate potentials (right panel) in the nonminimal coupling setup.

Figure 21

The phase space trajectories of a model with ordinary component(s) and a nonminimally coupled tachyon field. We assumed that the ordinary component is dust with $\omega =0$. There is one stable critical line ${\mathcal{L}}_1$ and 3 unstable critical lines (${\mathcal{L}}_2$, ${\mathcal{L}}_3$, and ${\mathcal{L}}_4$). The trajectories are plotted for just three values of $z$: $z(0)=0.02$, $z(0)=0.6$, and $z(0)=0.98$. We have not plotted the critical line ${\mathcal{L}}_5$ to avoid complexity in the figure.

Figure 22

The behavior of the effective equation of state parameter in the nonminimal coupling case. The left panel is corresponding to $\omega =0$ (dust) and the right one is corresponding to $\omega =\frac{1}{3}$ (radiation). As the figure shows, ${\omega }_{\mathrm{eff}}$ evolves and tends to the stable state with ${\omega }_{\mathrm{eff}}=-1$. The figure is plotted with $x(0)=0.99$, $y(0)=0.01$, $z(0)=0.2$, and $u(0)=0.1$ for the magenta (solid line) curve and $x(0)=0.75$, $y(0)=0.05$, $z(0)=0.2$ and $u(0)=0.1$ for the blue dashed curve.

Figure 23

The trajectories in the $\{r,q\}$ phase plan with $\omega =0$ (left panel) and $\omega =\frac{1}{3}$ (right panel). The initial conditions are $x(0)=0.99$, $y(0)=0.01$, $z(0)=0.2$ and $u(0)=0.1$ for the magenta (solid line) trajectory and $x(0)=0.75$, $y(0)=0.05$, $z(0)=0.2$ and $u(0)=0.1$ for the blue dashed trajectory. The magenta and blue highlighted dots (specified by $T$) are current values of $\{r,q\}$ in the model with a nonminimally coupled tachyon field. The black dot (specified by $F$) is the stable state of $\{r,q\}$ in the future. Also the red dot (marked by $\Lambda \mathrm{CDM}$) is the value of the statefinder $\{r,q\}$ in a $\Lambda \mathrm{CDM}$ scenario.

Figure 24

The trajectories in the $\{s,q\}$ phase plan with $\omega =0$ (left panel) and $\omega =\frac{1}{3}$ (right panel). The initial conditions are $x(0)=0.99$, $y(0)=0.01$, $z(0)=0.2$, and $u(0)=0.1$ for the magenta (solid line) trajectory and $x(0)=0.75$, $y(0)=0.05$, $z(0)=0.2$, and $u(0)=0.1$ for the blue dashed trajectory. The magenta and blue dots (specified by $T$) are current values of $\{s,q\}$ in the model with a nonminimally coupled tachyon field. The black dot ($F$) is the stable state of $\{s,q\}$ in the future. Also the red dot (remarked by $\Lambda \mathrm{CDM}$) is the value of statefinder $\{s,q\}$ in a $\Lambda \mathrm{CDM}$ scenario.

Figure 25

The trajectories in the $\{r,s\}$ phase plan with $\omega =0$ (left panel) and $\omega =\frac{1}{3}$ (right panel). The initial conditions are $x(0)=0.99$, $y(0)=0.01$, $z(0)=0.2$, and $u(0)=0.1$ for the magenta (solid line) trajectory and $x(0)=0.75$, $y(0)=0.05$, $z(0)=0.2$ and $u(0)=0.1$ for the blue dashed trajectory. The magenta and blue highlighted dots (specified by $T$) are current values of $\{r,s\}$ in the model with a nonminimally coupled tachyon field. The black dot ($F$) is the stable state of $\{r,s\}$ in the future. Also the red dot (marked by $\Lambda \mathrm{CDM}$) is the value of statefinder $\{r,s\}$ in a $\Lambda \mathrm{CDM}$ scenario.