Observable Features of Tachyon-Dominated Cosmology

TL;DR

This paper explores a cosmological model dominated by tachyonic dark matter, showing it can mimic standard models but also has distinct features that could be tested observationally, such as through supernovae data.

Contribution

It demonstrates that tachyon-dominated cosmology can replicate standard expansion features and proposes observational tests to distinguish it from conventional models.

Findings

Tachyonic dark matter can produce expansion histories similar to Lambda-CDM.

Distinct features like the cosmic jerk can differentiate tachyon models from standard cosmology.

Comparison with supernova data shows potential for observational testing.

Abstract

A Friedmann-Robertson-Walker cosmological model dominated by tachyonic -- faster-than-light -- dark matter can exhibit features similar to those of a standard dark energy/dark matter or $Lambda$CDM model. It can undergo expansion which decelerates to a minimum rate, passes through a "cosmic jerk," then accelerates. But some features of a tachyon-dominated model are sufficiently distinct from those of the standard model that the two possibilities might be distinguished observationally. As a demonstration of concept, the distance-redshift relation of such a model is compared here with some observations of Type Ia supernovae. Other measures of the third time derivative of the cosmic scale factor -- the true cosmic jerk -- might be found to test a tachyonic-dark-matter hypothesis.

Tables

Table 1. Table 1: Redshift/Distance Data for Type Ia Supernovae from  Garnavich et al. ( 1998 )

$z$	$m-M$	$D_L$, Gpc	${\sigma }_D$, Gpc
0.014	34.00	0.0629	0.006
0.020	34.80	0.0904	0.008
0.050	37.20	0.2654	0.024
0.100	38.30	0.4205	0.039
0.440	41.83	1.6342	0.151
0.480	42.40	2.0677	0.191
0.500	42.59	2.2268	0.205
0.970	44.15	3.4796	0.641

Equations

$$d{s}^2=-c^{2}d{t}^2+a^2(t)\left[d{\chi }^2+{\sinh }^2\chi \left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right)\right],$$

$${\left(\frac{da}{dt}\right)}^2-\frac{8\pi G}{3c^2}\rho a^2=+c^2,$$

$$\rho (a)=\frac{({\rho }_0+{\mathcal{M}}_0{c}^2)a_0^4}{a^4}-\frac{{\mathcal{M}}_0{c}^2{a}_0^3}{a^3},$$

$${\left(\frac{da}{dt}\right)}^2+\left(\frac{2B}{a}-\frac{A^2}{a^2}\right)=+c^2,$$

$$withA\equiv {\left(\frac{8\pi G({\rho }_0+{\mathcal{M}}_0{c}^2)a_0^4}{3c^2}\right)}^{1/2}$$

$$andB\equiv \frac{4\pi G{\mathcal{M}}_0{a}_0^3}{3}.$$

$${\rho }_{\Lambda \mathrm{CDM}}(a)={\rho }_{\Lambda }+\frac{{\rho }_{m0}{a}_0^3}{a^3},$$

$${\rho }_{\Lambda }=6.06\times 10^{-10}{J/m}^3,$$

$${\rho }_{m0}=2.60\times 10^{-10}{J/m}^3,$$

$$a(\eta )=\frac{A}{c}\sinh \eta -\frac{B}{c^2}(\cosh \eta -1)$$

$$t(\eta )=\frac{A}{c^2}(\cosh \eta -1)-\frac{B}{c^3}(\sinh \eta -\eta ),$$

$$H_0$$

$$=\frac{c^2}{A}\frac{\cosh {\eta }_0-\beta \sinh {\eta }_0}{[\sinh {\eta }_0-\beta (\cosh {\eta }_0-1){]}^2},$$

$$t_0=\frac{A}{c^2}[(\cosh {\eta }_0-1)-\beta (\sinh {\eta }_0-{\eta }_0)],$$

$$1+z_j$$

$$=\beta \sinh {\eta }_0-{\beta }^2(\cosh {\eta }_0-1)$$

$$H_0{t}_0={\left(\frac{\beta }{1+z_j}\right)}^2(\cosh {\eta }_0-\beta \sinh {\eta }_0)[(\cosh {\eta }_0-\beta \sinh {\eta }_0)+(\beta {\eta }_0-1)]$$

$$H_0$$

$$=2.32\times 10^{-18}{s}^{-1}$$

$$t_0$$

$$=4.32\times 10^{17}s,$$

$$z_j=0.500,$$

$$\beta =0.993,$$

$${\eta }_0=5.02,$$

$$A$$

$$=(0.930Gpc)c.$$

$$B$$

$$=(0.924Gpc)c^2$$

$$a_0=1.40Gpc$$

$$\ell =a_0({\eta }_0-{\eta }_{\ast }),$$

$$1+z=\frac{a_0}{a({\eta }_{\ast })}.$$

$$\ell (z)$$

$$=(1.40Gpc)\left\{2.19-{\sinh }^{-1}\left[8.53\left(\frac{1.51}{1+z}-0.993\right)\right]\right\}.$$

$$D_L(z)$$

$$=(1\times 10^{-7}Gpc)\exp \left(\frac{\ln (10)(m-M)}{5}\right)$$

$$=a_0(1+z)\sinh [\ell (z)/a_0]$$

$$=a_0(1+z)\sinh \left[{\eta }_0-\alpha -{\sinh }^{-1}\left(\frac{a_0\cosh \alpha }{(A/c)(1+z)}-\sinh \alpha \right)\right]$$

$$=(1.40Gpc)(1+z)\sinh \left\{2.19-{\sinh }^{-1}\left[8.53\left(\frac{1.51}{1+z}-0.993\right)\right]\right\},$$

$$a_0=(2.97\pm 0.71)Gpc,$$

$${\eta }_0=3.33\pm 0.41,$$

$$\alpha =1.28\pm 0.35,$$

$$\beta =0.856\pm 0.093,$$

$$A$$

$$=[(1.04\pm 0.25)Gpc]c,$$

$$B$$

Full text

cosmology, dark energy, dark matter, tachyons, distance-redshift relation

1 Introduction

Over the past two decades it has become clear that the matter/energy content of the cosmos must be dominated by constituents quite distinct from the ordinary “luminous” matter of stars and planets. These are called “dark matter” if they obey equations of state similar to those of familiar matter and energy, and “dark energy” if they do not—the latter including cosmological-constant or vacuum-energy contributions. Despite considerable and ongoing efforts, however, specific dark-matter and dark-energy components have not yet been identified. Hence, it remains possible to consider even exotic candidates. For example, a gas of tachyons—faster-than-light particles, with spacelike four-momenta—can drive an open (spatially hyperbolic) cosmolgical model which expands from an initial singularity at a rate which decelerates to a minimum value, passes through a “cosmic jerk,” then accelerates (Starke & Redmount, 2022). These are features that, in accord with current observations, characterize what has become the “standard” cosmological-constant/cold-dark-matter or $\Lambda$CDM cosmological model. The behavior of such a tachyon-dominated model differs sufficiently from a $\Lambda$CDM model that the two might be distinguished observationally. As a demonstration of concept, we present here some basic features of such a tachyon-dominated model to be tested against observations. A variety of more rigorous tests (Kramer & Redmount, 2022; Gopal & Redmount, 2022) should be possible.

A suitable tachyon-dominated model universe is described in Sec. 2. Numerical parameters sufficient to specify the model are obtained from a few basic cosmological measurements in Sec. 3. The distance-redshift relation for the model, which might be compared to a more extensive set of observations, is shown in Sec. 4. A tachyon-dominated model fitted to the same observational data, with its own predictions for the basic cosmological measurements, is presented in Sec. 5. Conclusions drawn from these results are described in Sec. 6.

2 Cosmological Model

The model in question is an open—spatially hyperbolic—Friedmann-Robertson-Walker spacetime geometry. It has line element

[TABLE]

with comoving time coordinate $t$, angular coordinates $\chi$, $\theta$, and $\phi$, and scale factor or curvature radius $a(t)$. The Einstein Field Equations applied to this metric imply the Friedmann Equation

[TABLE]

with $\rho$ the energy density of the cosmic fluid and $G$ Newton’s constant. The evolution of the model is determined by the form of the density $\rho$ as a function of $a$. A thermal ensemble of free, noninteracting (i.e., dark-matter) tachyons gives rise to density (Starke & Redmount, 2022)

[TABLE]

with $\rho_{0}$ the energy density and ${\cal M}_{0}$ an invariant-mass111Tachyons, never at rest, are not properly characterized by “rest mass.” density at some fiducial time $t_{0}$—e.g., the present time—at which the scale factor has value $a_{0}$. The Friedmann Equation (2) takes the form

[TABLE]

The “potential energy” term on the left-hand side of Eq. (4a) is illustrated in Figure 1. For parameter values giving $B<Ac$, the model expands from an initial singularity ($a\to 0^{+}$)

at a decreasing rate, passes through a minimum expansion rate at the peak of the potential, then accelerates. Unlike the standard $\Lambda$CDM model, which ultimately continues to accelerate exponentially in time, this tachyon-dominated model asymptotically approaches finite expansion rate $da/dt=c$. The tachyonic model illustrated in the figure has parameter values given in Sec. 3, following. The $\Lambda$CDM model shown has density

[TABLE]

and the current-time scale factor from Eq. (10e), for purposes of comparison.

The Friedmann Equation (4a) can be solved in closed form. The model of interest here, satisfying $B<Ac$, is described via scale factor and time

[TABLE]

in terms of conformal time parameter $\eta\in[0,+\infty)$.

3 Model Parameters

This model is specified by three parameters, e.g., the parameter $A$ setting the overall scale of the model, the dimensionless ratio $\beta\equiv B/(Ac)$, and the current conformal-time value $\eta_{0}$. These are to be determined by fitting predictions of the model to a suitable set of observations. A simple example of such a procedure is shown in Sec. 5. But as a starting point, simply to test the feasibility of the model, the parameters can be set by matching the values of three well-known cosmological quantities, so that other consequences of the model can be compared to other data. For example, the current value of the Hubble parameter

[TABLE]

can be used. Solving Eq. (7c) for $\beta$, then solving the combined equation

[TABLE]

for $\eta_{0}$, then fixing $A$ via Eq. (7b) determines all three model parameters.

Using the following sample values (Hinshaw et al., 2009; Lahav & Liddle, 2010):

[TABLE]

this procedure yields parameters

[TABLE]

for this version of the model.

4 Distance-Redshift Relation

The distance-redshift relation is a feature of the model that might be tested against astronomical observations. Since incoming light rays in a spacetime geometry with line element (1) travel on trajectories satisfying $d\chi=-d\eta$, the metric distance $\ell$ to a source is given by

[TABLE]

with $\eta_{*}$ the conformal-time parameter at emission of the signal, and subscript zero denoting present-time values. The redshift $z$ of the source is related to the scale factor $a(\eta)$ thus:

[TABLE]

These can be combined with the model’s scale factor (6a) to obtain distance-redshift relation

[TABLE]

Here the parameter $\alpha\equiv\tanh^{-1}\beta$ is introduced. Numerical values are taken from results (10a)–(10e); these imply value $\alpha=2.83$.

Metric distance $\ell(z)$ is not measured directly. Observations provide luminosity distances (Misner, Thorne, & Wheeler, 1973; Lightman, Press, Price, & Teukolsky, 1975), e.g.,

[TABLE]

for a source with intensity $S$, luminosity ${\cal L}$, apparent magnitude $m$, and absolute magnitude $M$. The overall factor $1+z$ incorporates redshift and time-dilation effects, and the $\sinh$ function incorporates the curvature correction appropriate to an open model with line element (1).

While this relation may appear opaque, it is actually not so unreasonable. It can be compared to measurements of Type Ia supernovae, e.g., from Garnavich et al. (1998). Data adapted from Figure 3 of that work are shown in Table 1, which displays redshift $z$, distance moduli $m-M$

for apparent magnitudes $m$ and absolute magnitudes $M$, distances $D_{L}$ obtained via the second line of Eq. (14), and uncertainties $\sigma_{D}$ estimated from the error bars shown in Garnavich et al. (1998) and that equation. The comparison of model relation and data is shown in Figure 2.

Here the model relation fits the low-redshift data points reasonably well, but yields greater distances for the larger redshift values. As a demonstration of concept, however, this shows that the tachyon-dominated model could be tested agains such observations. The parameters $A$, $B$, and $\eta_{0}$—that is, the invariant mass, density, and temperature of the tachyon gas, as well as the distance and time scales of the model—could be fixed, as is done to evaluate the features of more familiar models (Hinshaw et al., 2009), by fitting the model to the data.

5 Tachyon-Dominated Model Fitted to Data

Distance-redshift relation (14) can be fitted to the data of Table 1 by adjusting the three model parameters $a_{0}$, $\eta_{0}$, and $\alpha$. The results of a Levenberg-Marquardt $\chi^{2}$-minimization calculation (Press, Flannery, Teukolsky, & Vetterling, 1986a) are shown in Figure 3.

The parameters of the model shown are those obtained from a simple Monte Carlo calculation: Ten iterations of the fitting program are carried out, with normally-distributed variations introduced into the data. (Press et al., 1986b) The means and standard deviations of the resulting parameter values (Press et al., 1986c) yield

[TABLE]

The fit shown in Figure 3 has $\chi^{2}=100.$, or $\chi^{2}$ per degree of freedom $20.0$; the iterations of the Monte Carlo calculation have average $\chi^{2}$ value 144., or $\chi^{2}$ per degree of freedom 28.8. These suggest, as the graph does, at best a modest fit to the data. It may be that the small errors of the low-$z$ distance values force a close fit to these points, leaving larger discrepancies at the higher $z$ values. Clearly a larger, more carefully curated data set is needed for a more thoroughgoing test of the tachyonic model; one such study is in progress. (Kramer & Redmount, 2022)

But results (15g) and (15h), in particular, show that the tachyon-dominated model can yield cosmological features close, but not identical, to those obtained from more standard models—such as Eqs. (9a) and (9b). Hence, these results serve as a “demonstration of concept,” indicating that the tachyonic dark-matter model might ultimately serve as a novel alternative to the now standard $\Lambda$CDM cosmology.

6 Conclusions

Despite their somewhat fanciful nature, tachyons might actually be a viable dark-matter candidate. An open Friedmann-Robertson-Walker spacetime with mass/energy content dominated by a tachyon gas can show the same general features—expansion from an initial singularity decelerating to a minimum rate, then subsequently accelerating—as a standard $\Lambda$CDM model. More specific features of such a tachyon-dominated model can be compared with observations—sufficient perhaps to justify more careful and thorough investigation.

One important distinction between the tachyonic and $\Lambda$CDM models is that the latter continues expanding exponentially in time, while the former asymptotically approaches constant expansion rate $da/dt=c$. Hence, determination of not just the first and second time derivatives of the scale factor $a$, but also its third derivative—the true “cosmic jerk”—might be crucial to distinguishing between the two possibilities. Furthermore, the standard model must asymptotically approach the geometry of de Sitter (de Sitter, 1917a, b) spacetime, while the tachyonic model must approach the Milne (Milne, 1932) universe, which is flat spacetime. This might imply that the quantum-state spaces for elementary-particle fields in the two models are different (Redmount, 1999), hence, that the two possible cosmologies might even be distinguished by close examination of physics at the very smallest scales.