# Performance of the new RD51 VMM3a/SRS beam telescope—studying MPGDs
simultaneously in energy, space and time at high rates

L. Scharenberg,11footnotetext: Corresponding author. J. Bortfeldt  F.
Brunbauer  K. Desch  K. Flöthner  F. Garcia  D. Janssens  M. Lisowska  H.
Muller  E. Oliveri  G. Orlandini  D. Pfeiffer  L. Ropelewski  J. Samarati  D.
Sorvisto  M. van Stenis  and R. Veenhof

###### Abstract

The RD51 collaboration maintains a common infrastructure at CERN for its R & D
activities, including two beam telescopes for test beam campaigns. Recently,
one of the beam telescopes has been equipped and commissioned with new multi-
channel and charge-sensitive front-end electronics based on the ATLAS/BNL
VMM3a front-end ASIC and the RD51 Scalable Readout System (SRS). This allows
to read out the detectors at high rates (up to the MHz regime) with
electronics time resolutions of the order of $1\text{\,}\mathrm{n}\mathrm{s}$
and the ability to handle different detector types and sizes, due to a larger
dynamic range compared to the previous front-end electronics based on the
APV25 ASIC. Having studied and improved the beam telescope’s performance over
the course of three test beam campaigns, the results are presented in this
paper.

##  1 Introduction and system overview

As part of its activities, the RD51 collaboration [1] organises up to three
test beam campaigns per year at the H4 extraction beam line of the CERN Super
Proton Synchrotron (SPS). For this, two beam telescopes are provided, one
being based on COMPASS-like triple-GEM detectors [2] and the other on
resistive-strip MicroMegas detectors [3].

Both telescopes are equipped with the common RD51 readout electronics—the
Scalable Readout System (SRS) [4]—allowing to read out multiple detectors
(several thousand readout channels) simultaneously. Following the integration
of the ATLAS/BNL VMM3a front-end ASIC [5] into the SRS [6, 7], it replaced the
APV25 [8] electronics of the GEM-based telescope. This enables the acquisition
of data at much higher rates (MHz regime instead of kHz), with the electronics
providing charge and time of the induced signal simultaneously, with the
latter being in the nanosecond scale. In addition, different peaking times
($25$ to $200\text{\,}\mathrm{n}\mathrm{s}$) and electronics gains ($0.5$ to
$16\text{\,}\mathrm{m}\mathrm{V}\mathrm{/}\mathrm{f}\mathrm{C}$) can be
selected, being useful in the R & D context, as it allows tuning the
electronics’ settings to the specific detector types. The VMM3a is operated in
a self-triggered continuous readout mode, requiring precise clock
synchronisation between the front-end cards. This is achieved by using the
SRS’ Clock and Trigger Fanout (CTF) card.

The telescope contains three triple-GEM detectors
($10\text{\times}10\text{\,}\mathrm{c}\mathrm{m}$ active area, $x$-$y$-strip
anode, 256+256 strips at $400\text{\,}\mu\mathrm{m}$ pitch) over a distance of
about $1\text{\,}\mathrm{m}$ to provide information on the beam particles’
trajectories. In addition, it contains three scintillators (two in front of
the tracking detectors and one behind them) which are read out with Photo-
Multiplier Tubes (PMTs). The output pulses from the PMTs are processed by a
NIM-logic including a coincidence unit, which is also read out with VMM3a/SRS.
This provides the arrival time of the interacting particles.

##  2 Performance evaluation

###  2.1 Electronics’ rate-capability

VMM3a/SRS can acquire particle interaction rates in the MHz regime, following
studies with X-rays [7]. The same behaviour was observed with Minimum Ionising
Particles (MIPs), where the number of particle interactions that can be
recorded saturates at around $1\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$
(figure 1(a)).

(a) Rate-capability

(b) Beam profile

Figure 1: Measured number of particles within a $5\text{\,}\mathrm{s}$ beam
extraction versus the expected one (a). Transverse profile of an
$80\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ pion beam with
$10^{6}$ particles (b).

This MHz rate-capability—in combination with the triple-GEM detectors—helps
for example in understanding the beam structure. While the expected number of
particles is provided by the scintillators of the beamline instrumentation
even at higher rates, the beam profile measurement of the beamline’s delay
wire chambers fails at rates
$\gtrsim$100\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ (figure 1(b)) due to the
rate limits.

###  2.2 Time resolution

The VMM3a provides the signal arrival time with a resolution between $0.5$ and
$2\text{\,}\mathrm{n}\mathrm{s}$ [9]. This allows measuring the intrinsic time
resolution of MPGDs for tracking without the need of external high-resolution
electronics. Using the pulses from the NIM coincidence unit as reference
timestamps $t_{\mathrm{ref}}$, it is determined if a particle interaction was
recorded by the detector of interest within a previously defined time window
(e.g. $\pm$500\text{\,}\mathrm{n}\mathrm{s}$$) around $t_{\mathrm{ref}}$. The
time of the particle interaction is defined as the arrival time of the signal
with the largest amplitude $t_{\mathrm{LA}}$ [10], as the charge signal is
typically spread over several adjacent readout channels. For recorded
interactions within the time window, $\Delta
t=t_{\mathrm{ref}}-t_{\mathrm{LA}}$ is calculated, resulting in a Gaussian
distribution. From the width $\sigma_{\Delta t}$ of this distribution, the
detector resolution can be extracted by quadratically subtracting the
contributions from the front-end electronics, the NIM-logic and the
scintillator/PMTs (figure 2).

(a) Tracking detectors

(b) Detector under test

Figure 2: Time resolution of the three tracking detectors, depending on their
gain (a). Time resolution of a COMPASS-like triple-GEM detector, depending on
the drift field (b).

###  2.3 Energy/charge information

The matching in time between the coincidence unit’s signals and the tracking
detectors allows also for studying the charge/gain/energy behaviour of the
detectors. In the given example (figure 2(b)), the time resolution and charge
collection behaviour of a triple-GEM detector are shown.

###  2.4 Particle trajectories: spatial resolution and efficiency

In the following, the spatial information is added, making use of the three
tracking detectors and the reconstruction of the particles’ trajectories with
a Kalman filter. Due to the self-triggered readout, the events for the track
reconstruction are built during the offline analysis, using the recorded
interactions in each detector and a time window, similar to the time
resolution studies (section 2.2).

To determine the spatial resolution of the tracking detectors, the expected
interaction point in the detector of interest $x_{\mathrm{ref}}$ is compared
with the reconstructed one $x^{\prime}$, leading to the distribution of
position residuals $\Delta x=x_{\mathrm{ref}}-x^{\prime}$ (figure 3(a)).

(a) Residual distribution

(b) Spatial resolution (Res.) and efficiency (Eff.)

Figure 3: Example of the $\Delta x$ distribution (a). Spatial resolution and
detector efficiency of the three tracking detectors (b).

The distribution is fitted with a double-Gaussian function to account for the
tails of the residual distribution. The combined width of the residual
distribution is determined by $\sigma_{\Delta
x}^{2}=w\sigma_{\mathrm{centre}}^{2}+(1-w)\sigma_{\mathrm{tail}}^{2}$, with
the weighting factor $w$ being the relative scale between the centre and the
tail distribution. The detector resolution (figure 3(b)) is determined by
subtracting quadratically the uncertainty of the track reconstruction.

The efficiency of the telescope’s detectors is determined via
$\epsilon_{\mathrm{det}}=N_{\mathrm{comb}}/N_{\mathrm{ref}}$, where
$N_{\mathrm{ref}}$ is the number of expected particles (provided by the
reference detectors) and $N_{\mathrm{comb}}$ is the number of recorded
particle interactions where both, the reference detectors and the detector of
interest, participate in the track fit. Only at effective gains $>10^{4}$,
efficiencies of $>$90\text{\,}\%$$ can be reached. The reason for this is the
electronics threshold, which in this case was set to
$1.5\text{\,}\mathrm{f}\mathrm{C}$. Due to the system being self-triggered at
the threshold level (THL), each signal surpassing the THL is going to be
processed, taking part of the available bandwidth. As such, lowering the THL
will lead at a certain point to a situation with electronic noise dominating
the number of recorded signals. A potential method to overcome the issue with
noise acquisition at low threshold levels—still requiring further
investigation—is the sub-hysteresis discrimination of the VMM (for details see
[11]).

##  3 Improving the spatial resolution

The telescope provides a precise reference position, making it the ultimate
tool for studying improvements in spatial resolution; with a discrete readout
structure and threshold-based electronics, charge information is lost, which
decreases the accuracy of the position reconstruction when using the centre-
of-gravity (COG) method (for details it is referred to [12]). Two
approaches—tested previously with X-rays [12]—are the VMM3a’s neighbouring-
logic222When the neighbouring-logic gets enabled and the signal on one channel
surpasses the THL, the two adjacent channels are read out as well, even if the
signal does not surpass the threshold level. (hardware approach) and changing
the $Q^{2}$ weighting method333In the centre-of-gravity formula, the position
is calculated via $x=\sum_{i}Q_{i}^{n}x_{i}/\sum_{i}Q_{i}^{n}$ with $n=1$. For
the $Q^{2}$ weighting, $n=2$ is used in order to reduce the weight of the
tails of the charge distribution (for more details it is referred to [12]).
(software approach).

A comparison between the two approaches and the COG results is shown in figure
4(a). The neighbouring-logic improves the spatial resolution only in the low
signal-to-threshold regime, i.e. at low detector efficiencies. In these
situations, the relative amount of charge recovered from below the THL is
large, while at higher gains it is significantly lower, also with an increased
probability of potentially acquiring noise which would decrease the spatial
resolution. The $Q^{2}$ weighting on the other hand improves the resolution
over the full gain range, also for different threshold levels (figure 4(b)),
allowing to reach resolutions around $50\text{\,}\mu\mathrm{m}$.

(a) Spatial resolution

(b) COG vs $Q^{2}$ weighting

Figure 4: Spatial resolution at an electronics THL of
$1.5\text{\,}\mathrm{f}\mathrm{C}$ (a). Comparison between COG and $Q^{2}$
reconstruction at different THLs (b).

##  4 Conclusion

With the integration of the VMM3a front-end ASIC into the RD51 Scalable
Readout System, new possibilities opened for detector characterisation
studies. The commissioning of a beam telescope with VMM3a/SRS electronics
allows studying detectors’ time resolution (better than
$10\text{\,}\mathrm{n}\mathrm{s}$) and position resolution (around
$50\text{\,}\mu\mathrm{m}$) simultaneously with a single readout system and
all data being contained intrinsically in a single data stream. At the same
time, due to the increased rate-capability up to the MHz regime, the detector
characterisation studies can be performed much faster, allowing to scan a
larger range of parameters within a single test beam. It should be noted that
while the electronics’ commissioning was performed with COMPASS-like triple-
GEM detectors, other detector types ($\mu$RWELL [13], resistive-plane
MicroMegas, triple-GEM detector with three-layer readout [14] or straw tubes
[15]) have been also successfully studied with the new beam telescope and its
electronics.

## Acknowledgments

This work has been sponsored by the Wolfgang Gentner Programme of the German
Federal Ministry of Education and Research (grant no. 13E18CHA). This project
has received funding from the European Union’s Horizon 2020 Research and
Innovation programme under Grant Agreement No 101004761. The work has been
supported by the CERN Strategic Programme on Technologies for Future
Experiments (https://ep-rnd.web.cern.ch/). The authors would like to thank
Alexandru Rusu (CERN, SRS Technology) for the development and production of
the readout electronics.

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Intermediate and Power-Law Inflation in the Tachyon Model with Constant Sound
Speed
Narges Rashidia,b<EMAIL_ADDRESS>
(a)Department of Theoretical Physics, Faculty of Science, University of
Mazandaran,
P. O. Box 47416-95447, Babolsar, IRAN
(b) ICRANet-Mazandaran, University of Mazandaran,
P. O. Box 47416-95447, Babolsar, IRAN

###### Abstract

By adopting the intermediate and power-law scale factors, we study the tachyon
inflation with constant sound speed. We perform some numerical analysis on the
perturbation and non-gaussianity parameters in this model and compare the
results with observational data. By using the constraints on the scalar
spectral index and tensor-to-scalar-ratio, obtained from Planck2018 TT, TE,
EE+lowE+lensing+BAO+BK14 data, the constraint on the running of the scalar
spectral index obtained from Planck2018 TT, TE, EE+lowEB+lensing data, and
constraint on tensor spectral index obtained from Planck2018 TT, TE, EE
+lowE+lensing+BK14+BAO+LIGO and Virgo2016 data, we find the observationally
viable ranges of the model’s parameters at both $68\%$ CL and $95\%$ CL. We
also analyze the non-gaussian features of the model in the equilateral and
orthogonal configurations. Based on Planck2018 TTT, EEE, TTE and EET data, we
find the constraints on the sound speed as $0.276\leq c_{s}\leq 1$ at $68\%$
CL, $0.213\leq c_{s}\leq 1$ at $95\%$ CL, and $0.186\leq c_{s}\leq 1$ at
$97\%$ CL.

##  1 Introduction

Considering that the early time inflation in the history of the universe seems
to have occurred around Planck’s scale, the M/String theory inspired fields
can be a possible candidate for the field responsible for inflation. Tachyon
inflation is one of these fields leading to interesting cosmological results.
Tachyon field, which has attained a lot of attention, is associated with the
D-branes in string theory. One interesting implication of this field is that,
its slow-rolling down to the potential leads to the smooth evolution of the
universe from the accelerating phase of the expansion to the era dominated by
the non-relativistic fluid [44, 45, 46, 19].

On the other hand, we know that in the simple inflation model with canonical
scalar fields, we find the primordial perturbation to be scale-invariant and
gaussian [20, 27, 5, 28, 25, 26, 42, 29, 30]. When we consider an inflation
model with a canonical scalar field, we find a gaussian perturbation that the
values of its tensor-to-scalar ratio for $\phi$, $\phi^{2}$, $\phi^{4/3}$ are
not consistent with observational data [3]. Therefore we have to seek some
other specific potential like hilltop potential to get an observationally
viable single canonical field inflation model [3]. However, with every
potential, the amplitude of the primordial perturbation in the single
canonical field inflation is almost gaussian. Or, we should consider a
nonminimal inflation model to give a better explanation of the early time
cosmological inflation. However, with a non-canonical scalar field like
tachyon, it is possible to get the scale-dependent and non-gaussian
distribution for the amplitude of the primordial perturbations. Another
interesting point about the tachyon field is corresponding to the equation of
state of the tachyon field. This parameter in the tachyon field can be $-1$,
which describes both very early time and very late time accelerating expansion
of the universe. Also, its value can be $0$, describing the matter/dark matter
dominated era. Therefore, with the tachyon field it is possible to explain the
thermal history of the universe in a reasonable way [19]. These are
interesting issues that motivate cosmologists to consider and study the
inflation models driven by a tachyon field as a non-canonical scalar field. In
this regard, a lot of interesting works based on tachyon inflation have been
done. For instance, the authors of Ref. [23] have considered a tachyon field
in the context of quantum loop gravity. In this way, they have obtained the
inflation and perturbation parameters and tested the observational viability
of some inflation models. In ref. [12], by considering the holographic
cosmology, the tachyon inflation has been studied and the results have been
compared to the observational data. The authors of Ref. [31] have studied an
inflation model where a tachyon field interacts with photon gas. They have
shown that under some assumptions and conditions this model shows some
agreement with observational data. In [38], the tachyon model with a
superpotential, a potential based on supersymmetry, has been studied. It has
been shown that tachyon inflation with a superpotential, at least in some
ranges of the model’s parameter space, is consistent with observational data.
There are other works on tachyon inflation leading to interesting cosmological
results [32, 36, 13, 41, 39, 40] and all of these works show that the tachyon
field can be considered as a field running the inflation.

One of the important parameters in the inflation models is the sound speed of
the primordial perturbation, shown by $c_{s}$. The sound speed is
corresponding to the Lorentz factor $\gamma$ as $c_{s}=\frac{1}{\gamma}$. For
the canonical scalar field, we get the sound speed equal to unity. However,
with the non-canonical scalar fields, we have $c_{s}^{2}\neq 1$. The value of
the parameter $\gamma$ and correspondingly sound speed determines the
deformation of the field’s kinetic energy from the canonical one. The constant
sound speed is an interesting idea motivated by the authors in Refs. [47, 48].
Considering that in the non-canonical scalar field the sound speed is related
to the time variation of the field, the constant sound speed is corresponding
to the constant field’s variation in times. This can lead to interesting
results. Now, the question is what constant values of $c_{s}$ we should adopt,
to study the model numerically. Considering that the sound speed is related to
the amplitude of the non-gaussianity, by using the observational constraints
on the amplitude of the non-gaussianity, we can find suitable ranges of the
sound speed. Also, there are some other parameters, such as the scalar
spectral index, tensor spectral index, and tensor-to-scalar ratio that help us
to find some constraints on the sound speed values.

To perform a numerical analysis, we use the Planck2018 constraints on the
inflation parameters. Based on the $\Lambda CDM+r+\frac{dn_{s}}{d\ln k}$
model, the Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data gives the value of
the scalar spectral index as $n_{s}=0.9658\pm 0.0038$ and implies a constraint
on the tensor-to-scalar ratio as $r<0.072$ [2, 3]. The constraint on the
tensor spectral index, released by Planck2018 TT, TE, EE
+lowE+lensing+BK14+BAO+LIGO and Virgo2016 data is $-0.62<n_{T}<0.53$ [2, 3].
Also, from Planck2018 TT, TE, EE+lowEB+lensing data, we have the value of the
running of the scalar spectral index as $\alpha_{s}=-0.0085\pm 0.0073$ [2, 3].
Other useful constraints are corresponding to the amplitude of the non-
gaussianity. The planck2018 combined temperature and polarization analysis
gives the constraints on the amplitude of the non-gaussianity in the
equilateral configuration as $f_{NL}^{equil}=-26\pm 47$ and in the orthogonal
configuration as $f_{NL}^{ortho}=-38\pm 24$ [4]. These several constraints
determine the observational viability of every inflation model.

With these preliminaries, this paper is organized as follows. In section 2, we
review the inflation in a tachyon model with constant sound speed. In this
section, we present the main equations governing the dynamics of the model in
terms of the sound speed. In section 3, we consider an intermediate scale
factor and study the perturbation and non-gaussianity parameters in this model
numerically. We compare the results with several data sets to find some
constraints on the model’s parameter space. In section 4 we study the power-
law tachyon inflation with constant sound speed and compare the results with
observational data. In section 5, we perform some discussion on unifying the
initial inflation with late time dark energy in the tachyon model with
constant sound speed. In section 6, we present a summary of this work.

##  2 Review on the Tachyon Inflation with Constant Sound Speed

For the tachyon field, we have the following Dirac-Born-Infeld type effective
4-dimensional action

$\displaystyle S=\int
d^{4}x\,\sqrt{-g}\Bigg{[}\frac{1}{2\kappa^{4}}R-V(\phi)\,\sqrt{1-2\,X}\Bigg{]}\,,$
(1)

with $\kappa$ being the gravitational constant, $R$ as the Ricci scalar, and
$X=-\frac{1}{2}\,\partial_{\mu}\phi\,\partial^{\mu}\phi$. Also, the tachyon
field $\phi$ has the potential $V(\phi)$. Cosmologists believe that the
physics of the tachyon condensation can be described by such an action.
Einstein’s field equations, corresponding to action (1), are given by

$\displaystyle
G_{\mu\nu}=\kappa^{2}\Bigg{[}-g_{\mu\nu}V(\phi)\sqrt{1-2\,X}+\frac{V(\phi)\partial_{\mu}\phi\,\partial_{\nu}\phi}{\sqrt{1-2\,X}}\Bigg{]}\,,$
(2)

which have been obtained by varying action (1) with respect to the metric.
Considering the FRW metric as

$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}\,,$ (3)

we find the following Friedmann equation

$\displaystyle 3H^{2}=\frac{\kappa^{2}\,V(\phi)}{\sqrt{1-\dot{\phi}^{2}}}\,.$
(4)

corresponding to the $(0,0)$ component of Einstein’s field equations. Note
that, a dot on the parameter shows the derivative of the parameter with
respect to the time. Also, the $(i,i)$ component of Einstein’s field equations
give the following second Friedmann equation

$\displaystyle
2\dot{H}+3H^{2}=\kappa^{2}\bigg{[}V(\phi)\,\sqrt{1-\dot{\phi}^{2}}\,\bigg{]}\,.$
(5)

We find the equation of motion of the tachyon field by varying the action (1)
with respect to the field as

$\frac{\ddot{\phi}}{1-\dot{\phi}^{2}}+3\,H\dot{\phi}+\frac{V^{\prime}(\phi)}{V(\phi)}=0\,,$
(6)

where a prime shows the derivative with respect to the tachyon field. Since
our purpose in this paper is to study the tachyon model with constant sound
speed, we should re-write the above main equations in terms of the sound
speed. To this end, we need to define the sound speed of the tachyon field. In
general, the square of the sound speed is given by
$c_{s}^{2}\equiv\frac{P_{,X}}{\rho_{,X}}$, that in the tachyon model takes the
following form

$\displaystyle c_{s}=\sqrt{1-\dot{\phi}^{2}}\,.$ (7)

In this regard, the equations (4)-(6) take the following forms

$\displaystyle 3H^{2}=\frac{\kappa^{2}\,V(\phi)}{c_{s}}\,,$ (8) $\displaystyle
2\dot{H}+3H^{2}=\kappa^{2}\,V(\phi)\,c_{s}\,,$ (9)

and

$3\,H\,\sqrt{1-c_{s}^{2}}+\frac{V^{\prime}(\phi)}{V(\phi)}=0\,,$ (10)

where we have considered the sound speed as a constant parameter. To study
cosmological inflation, we need the following slow-roll parameters

$\displaystyle\epsilon=-\frac{\dot{H}}{H^{2}}\,,$ (11)
$\displaystyle\eta=-\frac{1}{H}\frac{\ddot{H}}{\dot{H}}\,.$ (12)

Note that, in the case of the constant sound speed, the third slow-roll
parameter $s=\frac{1}{H}\frac{\dot{c_{s}}}{c_{s}}$ is zero. Another needed
parameter to study inflation is the number of e-folds parameter defined as

$\displaystyle N=\int H\,dt\,.$ (13)

These parameters are useful to study the primordial perturbations in the model
and compare the results with observational data. The observational data that
we use in our work is the data released by the Planck2018 team [2, 3, 4].
Based on the fact that, under the assumption of statistical isotropy, the two-
point correlations of the CMB anisotropies are described by the angular power
spectra $C_{l}^{TT}$, $C_{l}^{TE}$, $C_{l}^{EE}$, and $C_{l}^{BB}$ (where the
subscript $l$ shows the multipole moment number) [24, 49, 43, 22, 21], the
Planck team has obtained some constraints on the important perturbation
parameters. To include the contributions from the scalar and tensor
perturbations in the CMB angular power spectra, the Planck team has used the
following expressions [1]

$\displaystyle
C_{l}^{ab,s}=\int_{0}^{\infty}\frac{dk}{k}\Delta_{l,a}^{s}(k)\,\Delta_{l,b}^{s}(k)\,{\cal{A}}_{s}(k)\,,$
(14)

$\displaystyle
C_{l}^{ab,T}=\int_{0}^{\infty}\frac{dk}{k}\Delta_{l,a}^{T}(k)\,\Delta_{l,b}^{T}(k)\,{\cal{A}}_{T}(k)\,.$
(15)

In equations (14) and (15), we have shown the transfer functions by
$\Delta_{l,a}^{s}(k)$ and $\Delta_{l,a}^{T}(k)$. The parameter $a$ and $b$ are
given as $a,b=T,E,B$. Also, the primordial power spectrum, the parameter
identified by the physics of the primordial universe [1], is shown by
${\cal{A}}_{i}(k)$ (where, $i=s,T$). Model-independent forms of the scalar and
tensor power spectra are given by

$\displaystyle{\cal{A}}_{s}(k)=A_{s}\left(\frac{k}{k_{*}}\right)^{n_{s}-1+\frac{1}{2}\frac{dn_{s}}{d\ln
k}\ln\big{(}\frac{k}{k_{*}}\big{)}+\frac{1}{6}\frac{d^{2}n_{s}}{d\ln
k^{2}}\ln\big{(}\frac{k}{k_{*}}\big{)}^{2}+...}\,,$ (16)
$\displaystyle{\cal{A}}_{T}(k)=A_{T}\left(\frac{k}{k_{*}}\right)^{n_{T}+\frac{1}{2}\frac{dn_{T}}{d\ln
k}\ln\big{(}\frac{k}{k_{*}}\big{)}+...}\,.$ (17)

These forms of the scalar and tensor power spectra have been used by the
Planck team to compare the perturbation parameters with data. Note that, in
equations (16) and (17), $A_{j}$ shows the scalar (corresponding to $j=s$) and
tensor (corresponding to $j=T$) perturbations. Also, the running of the scalar
or tensor spectral index and the running of the running of the scalar spectral
index are given by $\frac{dn_{i}}{d\ln k}$ and $\frac{d^{2}n_{s}}{d\ln
k^{2}}$, respectively. One can also find the tensor-to-scalar ratio by using
the power spectra as

$\displaystyle r=\frac{{\cal{A}}_{T}(k_{*})}{{\cal{A}}_{s}(k_{*})}\,,$ (18)

which is a very important parameter in studying the inflation models. The
parameters $n_{s}$ and $n_{T}$ are the scalar and tensor spectral indices
showing the scale dependence of the primordial perturbations. The scalar
spectral index, at the time of sound horizon exit of the physical scales, is
defined as

$\displaystyle n_{s}-1=\frac{d\ln{\cal{A}}_{s}}{d\ln
k}\Bigg{|}_{c_{s}k=aH}\,,$ (19)

where the parameter ${\cal{A}}_{s}$, the amplitude of the scalar spectral
index, is given by the following definition

${\cal{A}}_{s}=\frac{H^{2}}{8\pi^{2}{\cal{W}}_{s}c_{s}^{3}}\,,$ (20)

with

${\cal{W}}_{s}=\frac{V\,(1-c_{s}^{2})}{2{H}^{2}c_{s}^{3}}\,.$ (21)

The scalar spectral index, in terms of the slow-roll parameters, is written as

$\displaystyle n_{s}=1-6\epsilon+2\eta\,.$ (22)

This is one of the parameters that can be compared with observational data.
Another important parameter in studying the inflation models is the running of
the scalar spectral index, defined as

$\displaystyle\alpha_{s}=\frac{dn_{s}}{d\ln
k}=8\epsilon\,\eta-12\epsilon^{2}-2\zeta+2\eta^{2}\,,$ (23)

where, the parameter $\zeta$ is given by

$\displaystyle\zeta=\frac{\dddot{H}}{H^{2}\dot{H}}\,.$ (24)

The tensor spectral index, obtained from equation (17), is defined as follows

$\displaystyle n_{T}=\frac{d\ln{\cal{A}}_{T}}{d\ln k}\Bigg{|}_{k=aH}\,,$ (25)

with

$\displaystyle{\cal{A}}_{T}=\frac{2\kappa^{2}H^{2}}{\pi^{2}}\,,$ (26)

being the amplitude of the tensor perturbations. In terms of the slow-roll
parameters, the tensor spectral index is expressed as

$\displaystyle n_{T}=-2\epsilon\,.$ (27)

Finally, we have the following expression for the tensor-to-scalar ratio

$\displaystyle r=16\,c_{s}\,\epsilon\,.$ (28)

As we see in equation (28), the tensor-to-scalar ratio depends explicitly on
the sound speed of the perturbations. Therefore, from the observationally
viable value of $r$, we can constrain the values of the sound speed.

Another important parameter that depends on the sound speed parameter is the
nonlinearity parameter. The nonlinearity parameter demonstrates the non-
gaussian property of the amplitude of the primordial perturbation. The
Gaussian distributed primordial perturbations are characterized by a two-point
correlation. To seek the additional statistical information, corresponding to
the non-Gaussian distributed perturbation, it is necessary to consider the
higher-order correlations. The three-point correlation function for the
spatial curvature perturbation in the interaction picture is given by [30]

$\displaystyle\langle{\Phi}(\textbf{k}_{1})\,{\Phi}(\textbf{k}_{2})\,{\Phi}(\textbf{k}_{3})\rangle=(2\pi)^{3}\delta^{3}(\textbf{k}_{1}+\textbf{k}_{2}+\textbf{k}_{3}){\cal{B}}_{\Phi}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})\,,$
(29)

with the following definition for the parameter ${\cal{B}}_{\Phi}$

${\cal{B}}_{\Phi}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})=\frac{(2\pi)^{4}{\cal{A}}_{s}^{2}}{\prod_{i=1}^{3}k_{i}^{3}}\,{\cal{D}}_{\Phi}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})\,.$
(30)

The parameter ${\cal{D}}_{\Phi}$ in equation (40) is expressed as follows

$\displaystyle{\cal{D}}_{\Phi}=\Bigg{(}1-\frac{1}{c_{s}^{2}}\Bigg{)}\Bigg{[}\frac{3}{4}{\cal{J}}_{1}-\frac{3}{2}{\cal{J}}_{2}-\frac{1}{4}{\cal{J}}_{3}\Bigg{]}\,,$
(31)

where

$\displaystyle{\cal{J}}_{1}=\frac{2\sum_{i>j}k_{i}^{2}\,k_{j}^{2}}{k_{1}+k_{2}+k_{3}}-\frac{\sum_{i\neq
j}k_{i}^{2}\,k_{j}^{3}}{(k_{1}+k_{2}+k_{3})^{2}}\,,$ (32)
$\displaystyle{\cal{J}}_{2}=\frac{\left(k_{1}\,k_{2}\,k_{3}\right)^{2}}{(k_{1}+k_{2}+k_{3})^{3}}\,,$
(33)

and

$\displaystyle{\cal{J}}_{3}=\frac{2\sum_{i>j}k_{i}^{2}\,k_{j}^{2}}{k_{1}+k_{2}+k_{3}}-\frac{\sum_{i\neq
j}k_{i}^{2}\,k_{j}^{3}}{(k_{1}+k_{2}+k_{3})^{2}}+\frac{1}{2}\sum_{i}k_{i}^{3}\,.$
(34)

By using the parameter ${\cal{D}}_{\Phi}$, one can find the non-linear
parameter as

$f_{NL}=\frac{10}{3}\frac{{\cal{D}}_{\Phi}}{\sum_{i=1}^{3}k_{i}^{3}}\,.$ (35)

Note that, in equation (35), there is a parameter $k_{i}$ which is the
momentum. Depending on different values of the momenta ($k_{1}$, $k_{2}$, and
$k_{3}$), we get different shapes of the non-gaussianity. In every shape,
there is a maximal peak in the amplitude of the perturbations that defines the
kind of the corresponding shape. We are interested in the case with a peak at
$k_{1}=k_{2}=k_{3}$ limit, leading to an equilateral shape, and the case
orthogonal to it [15, 6, 17, 11]. It is possible to write the bispectrum (31)
in terms of the equilateral and orthogonal shapes basis as follows [17]

${\cal{D}}_{\Phi}={\cal{M}}_{1}\,\breve{{\cal{J}}}^{equil}+{\cal{M}}_{2}\,\breve{{\cal{J}}}^{ortho}\,,$
(36)

where,

$\breve{{\cal{J}}}^{equil}=-\frac{12}{13}\Big{(}3{\cal{J}}_{1}-{\cal{J}}_{2}\Big{)}\,.$
(37)
$\breve{{\cal{J}}}^{ortho}=\frac{12}{14-13{\cal{C}}}\Big{(}{\cal{C}}\big{(}3{\cal{J}}_{1}-{\cal{J}}_{2}\big{)}+3{\cal{J}}_{1}-{\cal{J}}_{2}\Big{)}\,,$
(38)
${\cal{M}}_{1}=\frac{13}{12}\Bigg{[}\frac{1}{24}\bigg{(}1-\frac{1}{c_{s}^{2}}\bigg{)}\bigg{(}2+3{\cal{C}}\bigg{)}\Bigg{]}\,,$
(39)

and

${\cal{M}}_{2}=\frac{14-13{\cal{C}}}{12}\Bigg{[}\frac{1}{8}\bigg{(}1-\frac{1}{c_{s}^{2}}\bigg{)}\Bigg{]}\,,$
(40)

with ${\cal{C}}\simeq 1.1967996$. Too see more details to obtain the non-
linear parameter, see Refs. [17, 18, 14]. Now, we find the amplitudes of the
non-Gaussianity in the equilateral and orthogonal configurations as

$f_{{}_{NL}}^{equil}=\frac{130}{36\sum_{i=1}^{3}k_{i}^{3}}\Bigg{[}\frac{1}{24}\bigg{(}1-\frac{1}{c_{s}^{2}}\bigg{)}\bigg{(}2+3{\cal{C}}\bigg{)}\Bigg{]}\breve{\zeta}^{equil}\,,$
(41)

and

$f_{{}_{NL}}^{ortho}=\frac{140-130{\cal{C}}}{36\,\sum_{i=1}^{3}k_{i}^{3}}\Bigg{[}\frac{1}{8}\bigg{(}1-\frac{1}{c_{s}^{2}}\bigg{)}\Bigg{]}\breve{\zeta}^{ortho}\,.$
(42)

The equations (41) and (42) in the $k_{1}=k_{2}=k_{3}$ limit become

$f_{{}_{NL}}^{equil}=\frac{325}{18}\Bigg{[}\frac{1}{24}\bigg{(}\frac{1}{c_{s}^{2}}-1\bigg{)}\bigg{(}2+3{\cal{C}}\bigg{)}\Bigg{]}\,,$
(43)

and

$f_{{}_{NL}}^{ortho}=\frac{10}{9}\Big{(}\frac{65}{4}{\cal{C}}+\frac{7}{6}\Big{)}\Bigg{[}\frac{1}{8}\bigg{(}1-\frac{1}{c_{s}^{2}}\bigg{)}\Bigg{]}\,.$
(44)

Up to this point, we have presented the main equations describing the
cosmological dynamics of the tachyon model. In the following, we adopt two
types of scale factor (power-law and intermediate) for the tachyon model with
constant sound speed and study its observational viability.

##  3 Intermediate Tachyon Inflation with Constant-Sound Speed

In this section, we study the intermediate inflation in the tachyon model with
constant sound speed. In the intermediate inflation, we have the following
scale factor [8, 9, 10]

$\displaystyle a=a_{0}\,\exp\left(b\,t^{\beta}\right)\,,$ (45)

where, $0<\beta<1$ and $b$ is a constant. This scale factor demonstrates that
in this case, the expansion of the universe is slower than an exponential
expansion and faster than a power-law expansion. From the intermediate scale
factor (41) we find the Hubble parameter as

$\displaystyle H(N)=N\left({\frac{N}{b}}\right)^{-{\frac{1}{\beta}}}\beta\,.$
(46)

To use this Hubble parameter and study the observational viability of the
model, we need to reconstruct the potential and the inflation parameters in
terms of the Hubble parameter and its derivatives [7, 37]. From equation (8)
we obtain the potential as

$\displaystyle V=3\,{\frac{{H}^{2}(N)\,{cs}}{{\kappa}^{2}}}\,.$ (47)

Also, the slow-roll parameters take the following form

$\displaystyle\epsilon=-{\frac{{\frac{d}{{d}N}}H\left(N\right)}{H\left(N\right)}}\,,$
(48)

and

$\displaystyle\eta=\frac{\left(\left({\frac{d}{{d}N}}H\left(N\right)\right)^{2}\sqrt{1-c_{s}^{2}}+\left({\it
cs}-1\right)\left({c_{s}}+1\right)\left(H\left(N\right){\frac{{d}^{2}}{{d}{N}^{2}}}H\left(N\right)+2\,\left({\frac{d}{{d}N}}H\left(N\right)\right)^{2}\right)\right)}{\sqrt{1-{c_{s}^{2}}\,H\left(N\right){\frac{\rm
d}{{\rm d}N}}H\left(N\right)}}\,.$ (49)

By using the above equations, we can rewrite equations (22), (23), (27) and
(28), in terms of the Hubble parameter and its derivatives. After that, by
using equation (46), we can obtain the scalar spectral index, its running,
tensor spectral index, and tensor-to-scalar ratio in terms of the intermediate
parameters and also the sound speed. Then we perform some numerical analysis
and, to obtain some constraints on the model’s parameters, we compare the
results with observational data. The results have been shown in several
figures. Figure 1 shows the tensor-to-scalar ratio of the intermediate tachyon
model with constant sound speed versus the scalar spectral index in this
model, in the background of the Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14
data at $68\%$ CL and $95\%$ CL. As we can see from figure 1, this model in
some ranges of the model’s parameter space is consistent with observational
data. Our numerical analysis shows that the intermediate tachyon model with
constant sound speed is consistent with Planck2018 TT, TE,
EE+lowE+lensing+BAO+BK14 data at $95\%$ CL, if $0<c_{s}\leq 1$ and
$0.787\leq\beta\leq 1$. Also, this model is consistent with Planck2018 TT, TE,
EE+lowE+lensing+BAO+BK14 data at $68\%$ CL, if $0<c_{s}\leq 0.997$ and
$0.833\leq\beta\leq 1$. Figure 2 shows the running of the scalar spectral
index of the intermediate tachyon model with constant sound speed versus the
scalar spectral index in this model, in the background of the Planck2018 TT,
TE, EE+lowE+lensing data at $68\%$ CL and $95\%$ CL. By studying these
parameters, we find that $\alpha_{s}-n_{s}$ of the intermediate tachyon model
with constant sound speed is consistent with Planck2018 TT, TE,
EE+lowE+lensing data at $68\%$ CL, if $0<c_{s}\leq 1$ and $0.756\leq\beta\leq
1$ and at $95\%$ CL, if $0<c_{s}\leq 1$ and $0.812\leq\beta\leq 1$. The
evolution of the tensor-to-scalar ratio of the intermediate tachyon model with
constant sound speed versus the tensor spectral index in this model, in the
background of the Planck2018 TT, TE, EE +lowE+lensing+BK14 +BAO+LIGO and
Virgo2016 data at $68\%$ CL and $95\%$ CL, has been shown in figure 3. The
obtained constraints in this case are $0<c_{s}\leq 1$ and $0.081\leq\beta\leq
0.984$ at $68\%$ CL and $0<c_{s}\leq 1$ and $0.027\leq\beta\leq 1$ at $95\%$
CL.

Figure 1: Tensor-to-scalar ratio versus the scalar spectral index in the
intermediate tachyon model with constant sound speed.  Figure 2: Running of
the scalar spectral index versus the scalar spectral index in the intermediate
tachyon model with constant sound speed. Figure 3: Tensor-to-scalar ratio
versus the tensor spectral index in the intermediate tachyon model with
constant sound speed.

Figure 4: Left panel: ranges of the parameters $c_{s}$ and $\beta$ in the
intermediate tachyon inflation with constant sound speed, leading to
observationally viable values of $r$ versus $n_{s}$, obtained from Planck2018
TT, TE, EE+lowE+lensing+BAO+BK14 data at $68\%$ CL (dark region) and $95\%$ CL
(light region). Right panel: ranges of the parameters $c_{s}$ and $\beta$ in
the intermediate tachyon inflation with constant sound speed, leading to
observationally viable values of $r$ versus $n_{T}$, obtained from Planck2018
TT, TE, EE +lowE+lensing+BK14 +BAO+LIGO and Virgo2016 data at $68\%$ CL (dark
region) and $95\%$ CL (light region).  Figure 5: Ranges of the parameters
$c_{s}$ and $\beta$ in the intermediate tachyon inflation with constant sound
speed, leading to observationally viable values of the amplitude of the scalar
perturbation.

Figure 6: Left panel: orthogonal amplitude of the non-gaussianity versus the
equilateral amplitude of the non-gaussianity in the intermediate tachyon model
with constant sound speed. Right panel: observationally viable range of the
tensor-to-scalar ratio and sound speed squared, based on the observationally
viable values of the orthogonal and equilateral amplitudes of the non-
gaussianity.

We have also performed some numerical analysis to find the domain of $\beta$
and $c_{s}$, leading to observationally viable values of scalar spectral
index, tensor spectral index, and tensor-to-scalar ratio. The results are
shown in figure 4, where we have used both Planck2018 TT, TE, EE+lowE+lensing
+BAO+BK14 and Planck2018 TT, TE, EE +lowE+lensing+BK14 +BAO+LIGO and Virgo2016
data sets at $68\%$ CL and $95\%$ CL. Both panels confirm that, depending on
the value of sound speed, the observationally viable values of $\beta$ start
from almost $0.75$. The interesting point is that for these ranges of
parameter space, the amplitude of the scalar perturbation is consistent with
observational data too. The constraint on the amplitude of the scalar
perturbation, obtained from Planck2018 TT, TE, EE+lowE+lensing data, is
$\ln(10^{10}{\cal{}_{A}}_{s})=3.044\pm 0.014$. By this observational
constraint, we have obtained the observational viable domain of $c_{s}$ and
$\beta$, shown in figure 5. Another way to check the observational viability
of the model is to study the amplitudes of the non-gaussianity numerically and
compare the results with observational data. To this end, we consider the
amplitudes of the non-gaussianity in equilateral and orthogonal
configurations, given by equations (43) and (44). In this way, we plot the
behavior of the orthogonal amplitude of the non-gaussianity versus the
equilateral amplitude of the non-gaussianity, in the background of Planck2018
TTT, EEE, TTE and EET data. The result is shown in the left panel of figure 6.
Our numerical analysis shows that the orthogonal and equilateral amplitudes of
the non-gaussianity in the intermediate tachyon model with constant sound
speed are consistent with Planck2018 TTT, EEE, TTE and EET data at $68\%$ CL
if $0.276\leq c_{s}$, at $95\%$ CL if $0.213\leq c_{s}$, and at $97\%$ CL if
$0.186\leq c_{s}$. These ranges of the sound speed lead to the observationally
viable values of the tensor-to-scalar ratio too. In the right panel of figure
6, we have plotted the phase space of the tensor-to-scalar ratio and the sound
speed squared, based on the domain of the sound speed leading to the
observationally viable values of the amplitudes of the non-gaussianity. As the
figure shows, the values of the tensor-to-scalar ratio are in the domain
consistent with Planck2018 TT, TE, EE+lowE+lensing +BAO+BK14 data. We have
summarized the prediction of the model for some sample values of the sound
speed in table 1. According to our analysis, it seems that the intermediate
tachyon model with constant sound speed is an observationally viable inflation
model.

Table 1:  Ranges of the parameter $\beta$ in which the tensor-to-scalar ratio, the scalar spectral index, and the tensor spectral index of the intermediate tachyon model with constant sound speed are consistent with different data sets. | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE
---|---|---|---|---
| +lensing+BK14+BAO | +lensing+BK14+BAO | lensing+BK14+BAO | lensing+BK14+BAO
|  |  |  +LIGO$\&$Virgo2016 |  LIGO$\&$Virgo2016
$c_{s}$ |  $68\%$ CL |  $95\%$ CL |  $68\%$ CL |  $95\%$ CL
$0.1$ | $0.981<\beta<1$ | $0.965<\beta<1$ | $0.312<\beta<0.859$ | $0.250<\beta<1$
$0.4$ | $0.968<\beta<1$ | $0.953<\beta<1$ | $0.630<\beta<0.960$ | $0.554<\beta<1$
$0.7$ | $0.935<\beta<0.970$ | $0.920<\beta<0.995$ | $0.748<\beta<0.976$ | $0.684<\beta<1$
$0.9$ | $0.885<\beta<0.931$ | $0.871<\beta<0.931$ | $0.792<\beta<0.982$ | $0.731<\beta<1$

##  4 Power-Law Tachyon Inflation with Constant-Sound Speed

Now, we study the power-law inflation in the tachyon model with constant sound
speed. In the power-law inflation, the scale factor is given by

$\displaystyle a=a_{0}\,t^{n}\,,$ (50)

leading to the following Hubble parameter

$\displaystyle H(N)=N\,e^{-\frac{N}{n}}\,.$ (51)

We use this Hubble parameter and substitute it in equations (47)-(49) to find
the inflation parameter in the power-law case. In this way, we can obtain
equations (22), (23), (27) and (28) in terms of the model’s parameters and
perform some numerical analysis on the power-law tachyon inflation with
constant sound speed. To obtain some constraints on the model’s parameters, we
compare the numerical results with observational data. According to our
numerical analysis, the power-law tachyon model with constant sound speed is
consistent with Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data at $95\%$ CL
if $0<c_{s}\leq 1$ and $222\leq n\leq 565$. Also, this model is consistent
with Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data at $68\%$ CL if
$0<c_{s}<0.990$ and $293\leq n\leq 485$. The behavior of the tensor-to-scalar
ratio has been shown in figure 7. The running of the scalar spectral index of
the power-law tachyon model with constant sound speed versus the scalar
spectral index in this model is shown in figure 8. By studying these
parameters, we find that in this case, the model is consistent with Planck2018
TT, TE, EE+lowE+lensing data at $68\%$ CL if $0<c_{s}\leq 1$ and $299\leq
n\leq 473$ and at $95\%$ CL if $0<c_{s}\leq 1$ and $231\leq n\leq 551$. Figure
9 shows the evolution of the tensor-to-scalar ratio of the power-law tachyon
model with constant sound speed versus the tensor spectral index in this
model, in the background of the Planck2018 TT, TE, EE +lowE+lensing+BK14
+BAO+LIGO and Virgo2016 data. In this case, the model has observational
viability if $0<c_{s}\leq 1$ and $27.3\leq n\leq 3200$ at $68\%$ CL, and
$0<c_{s}\leq 1$ and $1.67\leq n$ at $95\%$ CL.

Figure 7: Tensor-to-scalar ratio versus the scalar spectral index in the
power-law tachyon model with constant sound speed. Figure 8: Running of the
scalar spectral index versus the scalar spectral index in the power-law
tachyon model with constant sound speed. Figure 9: Tensor-to-scalar ratio
versus the tensor spectral index in the power-law tachyon model with constant
sound speed.

Figure 10: Left panel: ranges of the parameters $c_{s}$ and $n$ in the power-law tachyon inflation with constant sound speed, leading to observationally viable values of $r$ versus $n_{s}$, obtained from Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data at $68\%$ CL (dark region) and $95\%$ CL (light region). Right panel: ranges of the parameter $c_{s}$ and $n$ in the power-law tachyon inflation with constant sound speed, leading to observationally viable values of $r$ versus $n_{T}$, obtained from Planck2018 TT, TE, EE +lowE+lensing+BK14 +BAO+LIGO, and Virgo2016 data at $68\%$ CL (dark region) and $95\%$ CL (light region). Figure 11: Ranges of parameter $c_{s}$ and $n$ in the power-law tachyon inflation with constant sound speed, leading to observationally viable values of the amplitude of the scalar perturbation. Table 2:  Ranges of the parameter $n$ in which the tensor-to-scalar ratio, the scalar spectral index, and the tensor spectral index of the power-law tachyon model with constant sound speed are consistent with different data sets. | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE
---|---|---|---|---
| +lensing+BK14+BAO | +lensing+BK14+BAO | lensing+BK14+BAO | lensing+BK14+BAO
|  |  |  +LIGO$\&$Virgo2016 |  LIGO$\&$Virgo2016
$c_{s}$ |  $68\%$ CL |  $95\%$ CL |  $68\%$ CL |  $95\%$ CL
$0.1$ | $368<n<492$ | $336<n<569$ | $27.3<n<300$ | $20.1<n$
$0.4$ | $354<n<488$ | $324<n<559$ | $102<n<1600$ | $76.0<n$
$0.7$ | $330<n<447$ | $301<n<511$ | $177<n<2930$ | $131<n$
$0.9$ | $305<n<373$ | $270<n<483$ | $228<n<3130$ | $167<n$

The domain of $n$ and $c_{s}$, leading to observationally viable values of
scalar spectral index, tensor spectral index, and the tensor-to-scalar ratio
is shown in figure 10, where we have used both Planck2018 TT, TE,
EE+lowE+lensing +BAO+BK14 and Planck2018 TT, TE, EE +lowE+lensing+BK14
+BAO+LIGO and Virgo2016 data sets at $68\%$ CL and $95\%$ CL. It is clear
that, although from the observationally viable values of the tensor spectral
index we can’t find an upper bound on the parameter $n$, the observational
values of the scalar spectral index imply an upper limit on it. These
observationally viable ranges of $c_{s}$ and $n$ lead to the value of the
amplitude of the scalar perturbations released by planck2018 data
($\ln(10^{10}{\cal{}_{A}}_{s})=3.044\pm 0.014$). Figure 11 shows this issue
clearly. Considering that the sound speed is constant, the behavior of the
orthogonal amplitude of the non-gaussianity versus the equilateral amplitude
of the non-gaussianity in the power-law tachyon case is the same as the one in
the intermediate tachyon case. This means that in this case also we have
consistency with Planck2018 TTT, EEE, TTE and EET data at $68\%$ CL if
$0.276\leq c_{s}$, at $95\%$ CL if $0.213\leq c_{s}$, and at $97\%$ CL if
$0.186\leq c_{s}$. These ranges are compatible with the ranges obtained from
$n_{s}$, $\alpha_{s}$, $n_{T}$ and $r$. Therefore, the power-law tachyon
inflation with constant sound speed seems to be a fine model that in some
ranges of its parameter space is consistent with observational data. In table
2, we have summarized some constraints obtained in studying the power-law
tachyon inflation with constant sound speed.

##  5 A Discussion on Unifying the Inflation with Dark Energy

Currently, cosmologists have been interested in the models covering a larger
domain in the thermal history of the universe. In this way, the models that
can describe both very early time and very late time accelerating expansion of
the universe have attracted a lot of attention. In this respect, in some
viable models of $f(R)$ gravity it is possible to describe both early time
inflation and late time acceleration of the universe [33, 34, 16, 35]. For
instance, in Ref. [34] the authors have considered a model of $f(R)$ gravity
where the universe effectively starts with a large cosmological constant at
the early universe (leading to inflation) and after passing the
radiation/matter dominated era, reaches the small values of the cosmological
constant (leading to late time accelerating expansion). In Ref. [35] another
model of $f(R)$ gravity that can explain the initial inflation, and at the
late time can reproduce the behavior of the $\Lambda$CDM model. As we have
mentioned in the “Introduction” section, the equation of state of the tachyon
field can be $-1$, corresponding to late time dark energy and early time
inflationary phases of the universe. Therefore, in our model also, it seems
possible to unify the inflation with dark energy. As we know, the tachyon
inflation with constant sound speed is an observationally viable inflation
model, at least in some domains of its parameter space. After inflation ends,
and the tachyon field reaches the non-zero minimum of the potential, the
universe becomes radiation dominated. During the radiation and matter
dominated universe, the minimum value of the potential doesn’t disturb the
thermal history of the universe. With more expansion of the universe, the
energy densities of the radiation and matter become small and eventually at
the late time the potential become the dominant component in the energy
density of the universe. In fact, this dominant component of the energy
density can be considered as an effective cosmological constant, leading to
late time accelerating expansion of the universe. It is also possible to
consider some extension of the tachyon field so we can get both inflation and
late time acceleration in the model. For instance, by considering the non-
minimal coupling or non-minimal derivative coupling between the tachyon field
and the gravity, depending on the values of the non-minimal parameter, it may
possible to have a model covering both initial inflation and late time
acceleration.

##  6 Conclusion

In this paper, we have considered tachyon inflation where the sound speed is
constant. We have obtained the Friedmann equations and equation of motion in
terms of sound speed. After that, we have presented the main perturbation
parameters, like the scalar spectral index, its running, tensor spectral
index, and tensor-to-scalar ratio. We have also, by considering the three-
point correlations, presented the amplitude of the non-gaussianity in both
equilateral and orthogonal configurations. Then, we have adopted two types of
scale factor: Intermediate and power-law. We have studied both cases
separately. To study these cases, we have obtained the potential and inflation
parameters in terms of the Hubble parameter, its derivative, and also the
sound speed. In this way, we were able to study the model based on two types
of scale factor. By considering the intermediate scale factor, we have studied
$r-n_{s}$ behavior in the background of Planck2018 TT, TE, EE+lowE+lensing
+BAO+BK14 data in both $68\%$ CL and $95\%$ CL and found some constraints on
the model’s parameters. The behavior of $\alpha_{s}-n_{s}$ in the background
of Planck2018 TT, TE, EE+lowE+lensing data has been studied too. Another
important parameter is the tensor scalar spectral index corresponding to the
primordial perturbations. In this way, we have studied $r-n_{T}$ behavior in
the background of Planck2018 TT, TE, EE +lowE+lensing+BK14 +BAO+LIGO and
Virgo2016 data to find some constraints on the model’s parameter space. We
have also analyzed $c_{s}$ and $\beta$ phase space in both $68\%$ CL and
$95\%$ CL. By studying these parameters numerically, we have found that
intermediate tachyon inflation with constant sound speed is observationally
viable if $0<c_{s}\leq 0.997$ and $0.833\leq\beta\leq 0.984$ at $68\%$ CL, and
$0<c_{s}\leq 1$ and $0.787\leq\beta\leq 1$ at $95\%$ CL. To check the
viability of the model more precisely, we have studied the non-gaussian
feature of the primordial perturbations. In this regard, we have considered
the Planck2018 TTT, EEE, TTE and EET constraints on the equilateral and
orthogonal configurations of the non-gaussianity. In this way, we have
obtained the constraints on the sound speed as $0.276\leq c_{s}\leq 1$ at
$68\%$ CL, $0.213\leq c_{s}\leq 1$ at $95\%$ CL, and $0.186\leq c_{s}\leq 1$
at $97\%$ CL. We have also studied $c_{s}^{2}-r$ phase space and found that
the ranges of the sound speed, obtained from the observational constraints on
the non-gaussianity, lead to the observationally viable values of the tensor-
to-scalar ratio.

The power-law tachyon inflation with constant sound speed is another model
that has been studied in this paper. Similar to the intermediate tachyon
inflation case, we have studied the perturbation parameters in this model, but
here, based on the power-law scale factor. In this case, by using different
data sets, we have studied $r-n_{s}$, $\alpha-n_{s}$, and $r-n_{T}$ behavior.
The phase space of the parameters $c_{s}$ and $n$, based on the
observationally viable ranges of the scalar and tensor spectral indices has
been studied too. Our numerical analysis has shown that the power-law tachyon
inflation with constant sound speed is observationally viable for $0<c_{s}\leq
0.990$ and $299\leq n<\leq 473$ at $68\%$ CL, and $0<c_{s}\leq 1$ and $231\leq
n<\leq 551$ at $95\%$ CL. For these obtained ranges, the amplitude of the
scalar spectral index is consistent with observational data. We have also
discussed the issue of unifying the inflation with dark energy in the tachyon
model with constant sound speed. We have argued that the non-zero minimum
value of the tachyon potential becomes important at the late time, after the
energy density of the radiation/matter decreases. Based on our analysis, it
seems that both intermediate and power-law tachyon models in some ranges of
their parameter space are consistent with observational data.

Acknowledgement
I thank the referee for the very insightful comments that have improved the
quality of the paper considerably.

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