H₀ Tension and the Phantom Regime: A Case Study in Terms of an Infrared f(T) Gravity

Varying ${{ \mathcal S }}_{m}$ with respect to the tetrad fields (which has been shown to be equivalent to varying with respect to the metric; de Andrade & Pereira 1998) enables one to define the stress–energy tensor of a perfect fluid as

$\begin{eqnarray}&&{{\mathfrak{T}}}_{\mu \nu }={e}_{a\mu }\left(-\displaystyle \frac{1}{e}\displaystyle \frac{\delta {{ \mathcal S }}_{m}}{\delta {{e}_{a}}^{\nu }}\right)=\rho {u}_{\mu }{u}_{\nu }+p({u}_{\mu }{u}_{\nu }+{g}_{\mu \nu }),\end{eqnarray} \tag{ 9 }$

where u^μ is the 4-velocity unit vector of the fluid.

In the Einstein–Hilbert action, the TEGR has been generalized by replacing T by an arbitrary f(T) function (Bengochea & Ferraro 2009; Linder 2010; Bamba et al. 2010, 2011) similar to the f(R) generalization. The f(T) Lagrangian is

$\begin{eqnarray}&&{{ \mathcal L }}_{g}=\displaystyle \frac{1}{2{\kappa }^{2}}\,f(T).\end{eqnarray} \tag{ 10 }$

By varying the action ${{ \mathcal S }}_{g}$ with respect to the tetrad fields, we obtain the tensor

$\begin{eqnarray}&&{{\mathfrak{H}}}_{\mu \nu }={e}_{a\mu }\left(\displaystyle \frac{1}{e}\displaystyle \frac{\delta {{ \mathcal S }}_{g}}{\delta {{e}_{a}}^{\nu }}\right).\end{eqnarray} \tag{ 11 }$

This gives rise to (Li et al. 2011)

$\begin{eqnarray}&&{{\mathfrak{H}}}_{\mu \nu }=\displaystyle \frac{1}{{\kappa }^{2}}\left({f}_{T}{{\mathfrak{G}}}_{\mu \nu }+\displaystyle \frac{1}{2}{g}_{\mu \nu }\left(f-{{Tf}}_{T}\right)+{f}_{{TT}}{S}_{\nu \mu \rho }{{\rm{\nabla }}}^{\rho }T\right),\end{eqnarray} \tag{ 12 }$

where f_T and f_TT stand for ${f}_{T}=\tfrac{{df}(T)}{{dT}}$ and ${f}_{{TT}}=\tfrac{{d}^{2}f(T)}{{{dT}}^{2}}$, respectively.

Using Equations (9) and (12), the variation of the total action (8) with respect to the tetrad fields gives the f(T) gravity field equations

$\begin{eqnarray}&&{{{\mathfrak{H}}}_{\mu }}^{\nu }={{{\mathfrak{T}}}_{\mu }}^{\nu }.\end{eqnarray} \tag{ 13 }$

Equivalently, by substituting from Equation (12), it can be written as

$\begin{eqnarray}&&\displaystyle \frac{1}{{\kappa }_{\mathrm{eff}}^{2}}{{\mathfrak{G}}}_{\mu \nu }={{\mathfrak{T}}}_{\mu \nu }+{{\mathfrak{T}}}_{\mu \nu }^{{DE}},\end{eqnarray} \tag{ 14 }$

where

$\begin{eqnarray}\begin{array}{rcl}{\kappa }_{\mathrm{eff}}^{2} & = & \displaystyle \frac{{\kappa }^{2}}{{f}_{T}},\\ {{\mathfrak{T}}}_{\mu \nu }^{{DE}} & = & \displaystyle \frac{1}{{\kappa }^{2}}\left(\displaystyle \frac{1}{2}{g}_{\mu \nu }\left({{Tf}}_{T}-f\right)-{f}_{{TT}}{S}_{\nu \mu \rho }{{\rm{\nabla }}}^{\rho }T\right).\end{array}\end{eqnarray} \tag{ 15 }$

It is clear that the general relativistic limit is recovered by setting f(T) = T, where κ_eff → κ and ${{\mathfrak{T}}}_{\mu \nu }^{{DE}}$ vanishes. This allows one to deal with the torsional dark energy on equal footing with the physical one. Although the teleparallel torsion scalar is not local Lorentz invariant, the field equations in the TEGR limit are invariant under local Lorentz transformation (LLT). On the contrary, the field equations of the nonlinear f(T) are not in general invariant under LLT (Li et al. 2011; Sotiriou et al. 2011). This crucial property makes the f(T) teleparallel gravity different from f(R) gravity. However, it has been shown that a covariant formulation of f(T) gravity can be obtained by including the nontrivial spin connection (see Krššák & Saridakis 2016); in addition, the determination of the spin connection associated with a certain vierbein has been investigated (see Golovnev et al. 2017; Krššák 2017).

We assume that the background geometry of the universe is a flat Friedmann–Lemaître–Robertson–Walker (FLRW). Hence, we take the Cartesian coordinate system (t; x, y, z) and the diagonal vierbein

$\begin{eqnarray}&&{{e}_{\mu }}^{a}=\mathrm{diag}\left(1,a(t),a(t),a(t)\right),\end{eqnarray} \tag{ 16 }$

where a(t) is the scale factor of the universe. Using Equations (5) and (16), this gives rise to the flat FLRW metric

$\begin{eqnarray}&&{{ds}}^{2}={{dt}}^{2}-a{\left(t\right)}^{2}{\delta }_{{ij}}{{dx}}^{i}{{dx}}^{j},\end{eqnarray} \tag{ 17 }$

where the Minkowskian signature is ${\eta }_{{ab}}=(+;-,-,-)$. We note that this choice of the vierbein (16) leads to consistent field equations without involving any unphysical degrees of freedom for any f(T) theory (Ferraro & Fiorini 2011; Krššák & Saridakis 2016). The diagonal vierbein (16) directly relates the teleparallel torsion scalar (6) to Hubble rate as follows:

$\begin{eqnarray}&&T=-6H{\left(t\right)}^{2},\end{eqnarray} \tag{ 18 }$

where $H(t)\equiv \dot{a}/a$ is the Hubble parameter, and the "dot" denotes differentiation with respect to the cosmic time t. Inserting the vierbein (16) into the field equations (14) for the matter fluid (9), the modified Friedmann equations of the f(T) gravity are

$\begin{eqnarray}&&\displaystyle \frac{3}{{\kappa }^{2}}{H}^{2}=\rho +{\rho }_{T}\equiv {\rho }_{\mathrm{eff}},\end{eqnarray} \tag{ 19 }$

$\begin{eqnarray}&&-\displaystyle \frac{1}{{\kappa }^{2}}\left(3{H}^{2}+2\dot{H}\right)=p+{p}_{T}\equiv {p}_{\mathrm{eff}},\end{eqnarray} \tag{ 20 }$

where ρ and p are the energy density and pressure of the matter sector, respectively, considered to correspond to a perfect fluid. Independently of the above equations, one should choose an EOS to relate ρ and p. Here we choose the simple linear barotropic case p = wρ, where w is the EOS parameter. We are interested in evolution during (pressureless) matter domination; we therefore in practice set w = 0. Additionally, the torsional density and pressure in the above equations are

$\begin{eqnarray}&&{\rho }_{T}=\displaystyle \frac{1}{2{\kappa }^{2}}\left[2{{Tf}}_{T}-T-f(T)\right],\end{eqnarray} \tag{ 21 }$

$\begin{eqnarray}&&{p}_{T}=\displaystyle \frac{1}{2{\kappa }^{2}}\left[\displaystyle \frac{f(T)-{{Tf}}_{T}+2{T}^{2}{f}_{{TT}}}{{f}_{T}+2{{Tf}}_{{TT}}}\right],\end{eqnarray} \tag{ 22 }$

By acquiring the standard matter conservation, we write the continuity equation

$\begin{eqnarray}&&\dot{\rho }+3H(\rho +p)=0.\end{eqnarray} \tag{ 23 }$

This in turn implies the continuity equation of the torsional fluid

$\begin{eqnarray}&&{\dot{\rho }}_{T}+3H({\rho }_{T}+{p}_{T})=0,\end{eqnarray} \tag{ 24 }$

in order to have a conservative universe. We additionally take an EOS parameter w_T ≡ p_T/ρ_T of the torsional fluid, which incorporates the dark energy sector. Hence, we write

$\begin{eqnarray}\begin{array}{rcl}{w}_{{DE}} & \equiv & {w}_{T}\\ & = & \displaystyle \frac{{p}_{T}}{{\rho }_{T}}\\ & = & -1+\displaystyle \frac{[f(T)-2{{Tf}}_{T}]({f}_{T}+2{{Tf}}_{{TT}}-1)}{[f(T)+T-2{{Tf}}_{T}]({f}_{T}+2{{Tf}}_{{TT}})}.\end{array}\end{eqnarray} \tag{ 25 }$

It is useful to define the effective EOS parameter

$\begin{eqnarray}&&{w}_{\mathrm{eff}}\equiv \displaystyle \frac{{p}_{\mathrm{eff}}}{{\rho }_{\mathrm{eff}}}=-1-\displaystyle \frac{2}{3}\displaystyle \frac{\dot{H}}{{H}^{2}},\end{eqnarray} \tag{ 26 }$

which can be related to the deceleration parameter q by the following expression:

$\begin{eqnarray}&&q\equiv -1-\displaystyle \frac{\dot{H}}{{H}^{2}}=\displaystyle \frac{1}{2}\left(1+3{w}_{\mathrm{eff}}\right).\end{eqnarray} \tag{ 27 }$

Thanks to the nice feature of the f(T) theory being that its field equations are of second order, currently there are viable f(T) theories of gravity that give good results with a wide range of cosmological observations (Nunes et al. 2016; Xu et al. 2018; Nunes 2018). In the following sections, we focus on a specific model with IR torsional gravity, which is in principle an alternative to phantom dark energy.

In a recent study (Awad et al. 2018b; see also Hohmann et al. 2017), the dynamical system approach was applied to the ordinary differential equations arising in the context of f(T) cosmologies. This showed that the modified Friedmann equations can be reduced to a 1D autonomous system, where $\dot{H}={ \mathcal F }(H)$. This allows us to utilize some geometrical procedures to analyze the dynamical behavior of the set of all solutions and its stability just by visualizing it as trajectories in an (H, $\dot{H}$) phase space. As seen in Figure 1, the (H, $\dot{H}$) phase space has a Minkowskian origin at (0, 0), while by identifying the zero acceleration boundary curve $q\equiv -1-\dot{H}/{H}^{2}=0$ (given as a dotted curve) the phase space is divided into four dynamical regions according to the values of H and q in each region: The unshaded region (I) represents an accelerated contraction, since H < 0 and q < 0. The shaded region (II) represents a decelerated contraction, since H < 0 and q > 0. The shaded region (III) represents a decelerated expansion, since H > 0 and q > 0. The unshaded region (IV) represents an accelerated expansion, since H > 0 and q < 0. Notably, one can thus examine and evaluate complicated cosmological models by following their phase trajectories and studying their qualitative behavior, such as their ability to cross between different regions of the phase space.

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It has been proven that the f(T) phase portraits can be analyzed easily and information can be extracted in a clear way (for more applications of this approach to f(T) gravity cosmology see Bamba et al. 2016; El Hanafy & Nashed 2017a, 2017b; Awad et al. 2018a). In particular, the governing equation is given by

$\begin{eqnarray}&&\dot{H}=3(1+w)\displaystyle \frac{f-{{Hf}}_{H}}{{f}_{{HH}}}={ \mathcal F }(H),\end{eqnarray} \tag{ 34 }$

where $f=f(H)$, ${f}_{H}=\tfrac{{df}}{{dH}}$, and ${f}_{{HH}}=\tfrac{{d}^{2}f}{{{dH}}^{2}}$. Inserting the torsional IR correction (28) into the governing Equation (34), we can determine the phase portrait equation of the model:

$\begin{eqnarray}&&\dot{H}=-\displaystyle \frac{3}{2}(1+w){H}^{2}\left[\displaystyle \frac{{\left(H/{H}_{0}\right)}^{2(n+1)}-(1-{{\rm{\Omega }}}_{m,0})}{{\left(H/{H}_{0}\right)}^{2(n+1)}+n(1-{{\rm{\Omega }}}_{m,0})}\right].\end{eqnarray} \tag{ 35 }$

At large Hubble regime, the above equation reduces to

$\begin{eqnarray*}&&\dot{H}\approx -\displaystyle \frac{3}{2}(1+w){H}^{2},\end{eqnarray*}$

which characterizes the phase portrait in GR. The torsional IR correction model thus matches standard predictions of matter domination (w = 0), prior to cosmic acceleration, as well as the earlier radiation domination era, w = 1/3. Indeed, from Equations (29) and (30), it is not difficult to show that ρ_T → 0 and p_T → 0 as $z\to \infty$ ($H\to \infty$). We thus expect that our torsion correction to the teleparallel equivalent to GR will not affect the thermal history and structure formation up to the transition to cosmic acceleration.

In Figure 1, we visualize the phase portrait (34) for different values of n versus the ΛCDM (n = 0) using Planck parameters. As is clear, the portrait is unbounded from below, where $\dot{H}\to -\infty$ as $H\to \infty$, which indicates an initial singularity (big bang); asymptotically the portrait matches the sCDM one in the shaded region III (decelerated expansion). However, it cuts the zero acceleration curve, q = 0 (i.e., $\dot{H}=-{H}^{2}$), which determines the value of the Hubble parameter at transition H_tr. Using Equation (34), we find

$\begin{eqnarray}&&{H}_{\mathrm{tr}}={\left[(2n+3)(1-{{\rm{\Omega }}}_{m,0})\right]}^{\tfrac{1}{2(n+1)}}{H}_{0}.\end{eqnarray} \tag{ 36 }$

Using the values H₀ = 73.5 km s⁻¹ Mpc⁻¹ and Ω_m,0 = 0.262, we obtain H_tr = 107 (102) km s⁻¹ Mpc⁻¹ for n = 1/3 (n = 1). For n = 0 with Planck parameters (ΛCDM), we find H_tr = 98 km s⁻¹ Mpc⁻¹. Plugging these results into Equation (33), we determine the transitional redshift z_tr ∼ 0.797 (∼0.798) for n = 1/3 (n = 1). However, for ΛCDM with Planck parameters, it is z_tr ∼ 0.649.

For comparison, we can also pin the predicted transition to cosmic acceleration directly from the evolution of the deceleration parameter. Plugging Equation (35) into Equation (27), we write the deceleration parameter

$\begin{eqnarray}&&q(z)=-1+\displaystyle \frac{3}{2}\displaystyle \frac{E{\left(z\right)}^{2n+2}-(1-{{\rm{\Omega }}}_{m,0})}{E{\left(z\right)}^{2n+2}+n(1-{{\rm{\Omega }}}_{m,0})}.\end{eqnarray} \tag{ 37 }$

In Figure 2, we plot the evolution of the deceleration parameter for different values of n versus the ΛCDM using Planck parameters. The plots show that the deceleration parameter q → 0.5 (w_eff → 0) at high redshift, which is in agreement with the sCDM domination. In addition, the transition from deceleration to acceleration occurred at redshift z_tr ≈ 0.8, which is in agreement with the measured value (Farooq et al. 2017). Also, the current value of the deceleration parameter q₀ ≈ −0.68 (w_eff ≈ −0.79).

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The portrait crosses the zero acceleration curve to the unshaded region IV (accelerated expansion) and evolves toward a fixed point H_f at $\dot{H}=0$. This determines the Hubble value at the fixed point

$\begin{eqnarray}&&{H}_{f}={\left(1-{{\rm{\Omega }}}_{m,0}\right)}^{\tfrac{1}{2(n+1)}}{H}_{0}.\end{eqnarray} \tag{ 38 }$

Notably, this fixed point cannot be reached in finite time, i.e., H → H_f = constant as $t\to \infty$; this indicates a pseudo-rip fate (Frampton et al. 2012). In the following we show that this is associated with a phantom regime.

In order to investigate the physics of the torsional IR correction, we define its EOS, Equation (25). Substituting from Equations (29) and (30), we obtain

$\begin{eqnarray}&&{w}_{T}(z)\equiv \displaystyle \frac{{p}_{T}}{{\rho }_{T}}=-1-n\left[\displaystyle \frac{E{\left(z\right)}^{2n+2}-(1-{{\rm{\Omega }}}_{m,0})}{E{\left(z\right)}^{2n+2}+n(1-{{\rm{\Omega }}}_{m,0})}\right].\end{eqnarray} \tag{ 39 }$

The inverse relation of Equation (33) allows us to express the torsional EOS in terms of redshift, w_T(z), explicitly. In Figure 3, we plot the torsional EOS for different choices of the parameter n.

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We thus determine the current value of the torsional EOS, w_T(z = 0) = −1.07 (−1.15) where n = 1/3 (1), which is in agreement with observations (Sahni et al. 2014; Di Valentino et al. 2016a, 2017). We find that the torsional fluid in the past fixed to ${w}_{T}\to -1\tfrac{1}{3}$ (−2) where n = 1/3 (1) as the redshift $z\to \infty$, while it is evolving toward the cosmological constant with w_T → −1 in the far future as z → −1. This confirms that the torsional IR correction incorporates phantom-like dark energy.

As mentioned in the Introduction, dynamical phantom-like dark energy is in fact favored by recent observations. Modified gravity can provide for a framework for such scenarios without introducing ghost instabilities associated with scalar field models of phantom dark energy.

Finally, it is worth noting that the invoked phantom regime does not violate age constraints (e.g., those derived from old globular clusters) even while using the locally measured value of H₀. For the proposed model (28), the age of the universe is

$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{age}} & = & -{\int }_{{H}_{0}}^{\infty }{\dot{H}}^{-1}\,{dH}\\ & = & \displaystyle \frac{2}{3{H}_{0}}{\int }_{1}^{\infty }{E}^{-2}\displaystyle \frac{{E}^{2(n+1)}+n(1-{{\rm{\Omega }}}_{m,0})}{{E}^{2(n+1)}-(1-{{\rm{\Omega }}}_{m,0})}\,{dE}.\end{array}\end{eqnarray} \tag{ 40 }$

Even for a large Hubble constant, e.g., H₀ = 73.5 km s⁻¹ Mpc⁻¹ as measured by local observations (Riess et al. 2018a, hereafter referred to as R18) and Ω_m,0 ∼ 0.262 (so as to keep Ω_m,0h² = 0.1417 constant as we discuss below), the model predicts an age t_age ∼ 13.6 (13.9) billion years for n = 1/3 (1). In conclusion, the model predicts an age of the universe compatible with current observations.

As is now well known, there exists significant tension between the locally measured value of the Hubble constant and that inferred from the CMB. For example, Riess et al. (2018a) recently measured H₀ = 73.52 ± 1.62 km s⁻¹ Mpc⁻¹, while Planck Collaboration XIII (2016) estimate H₀ = 67.8 ± 0.9 km s⁻¹ Mpc⁻¹. While the debate continues as to whether the discrepancy is due to new physics or simply observational systematics, it is straightforward to show that the values can in principle be reconciled by invoking a phantom acceleration regime, as we now outline.

Given a primordial fluctuation spectrum and an FLRW cosmology, the relative height of the CMB peaks is essentially determined by the dimensionless physical dark matter and baryon densities Ω_ch² and Ω_bh², respectively. Fixing, in addition, the number of effective relativistic degrees of freedom in turn fixes the era of matter radiation equality and recombination, and with these the intrinsic physical scale of the CMB peaks (e.g., Hu & Sugiyama 1995; Percival et al. 2002), as well as light-element production in the context of big bang nucleosynthesis (BBN). We will assume that all these parameters are fixed to standard values (namely, as quoted in Planck Collaboration XIII 2016) and that the cosmological evolution is practically indistinguishable from the standard scenario up to late times, when the dark-energy-like component becomes significant. For the specific case of the modified gravity models used here, the latter assumption is justified by the fact that the IR correction theory tends to the teleparallel equivalent to GR at such redshifts; we thus expect the evolution, including the growth of perturbations, to be similar.

In this context, a measurement of the angular diameter (transverse) distance to the CMB,

$\begin{eqnarray}&&{D}_{A}(z)=\displaystyle \frac{1}{1+z}{\int }_{0}^{z}\displaystyle \frac{{dz}^{\prime} }{H(z^{\prime} )},\end{eqnarray} \tag{ 41 }$

with z = z_lss referring to the redshift of the last scattering surface, determines H₀, given a cosmological model (i.e., $H(z)$). In the standard ΛCDM model, such a measurement should yield a value that is smaller than locally measured values (similar to the one obtained by Planck Collaboration XIII 2016 by fitting the full CMB spectrum). Nevertheless, as H(z) can be written as H(z) = H₀E(z), it is easy to see that one can increase H₀, while keeping the angular distance constant, by choosing a model where E(z) is smaller than that associated with ΛCDM in the redshift range 0 ≤ z ≤ z_lss. This state of affairs would also be reflected in the invariance of the "shift parameter" of Efstathiou & Bond (1999),

$\begin{eqnarray}&&{{ \mathcal R }}_{\mathrm{lss}}=\sqrt{{{\rm{\Omega }}}_{m,0}{H}_{0}^{2}}{\int }_{0}^{{z}_{\mathrm{lss}}}\displaystyle \frac{{dz}}{H(z)}=\sqrt{{{\rm{\Omega }}}_{m,0}}{\int }_{0}^{{z}_{\mathrm{lss}}}\displaystyle \frac{{dz}}{E(z)}.\end{eqnarray} \tag{ 42 }$

Clearly, if one keeps Ω_m,0h² constant while increasing H₀, then ${{\rm{\Omega }}}_{m,0}$ becomes smaller. It is then sufficient for E(z) to be always below its ΛCDM value E_Λ(z) for 0 ≤ z ≤ z_lss for it to be possible to keep ${{ \mathcal R }}_{\mathrm{lss}}$ constant. We now show that it is not only possible but necessary, in the phantom regime, that E_p(z) ≤ E_Λ(z).

Friedmann evolution in a flat universe with matter and DE (or DE-like, as in torsion gravity) components implies

$\begin{eqnarray}&&\displaystyle \frac{{E}_{p}^{2}(z)}{{E}_{{\rm{\Lambda }}}^{2}(z)}=\displaystyle \frac{{{\rm{\Omega }}}_{{mp}}{a}^{-3}+{{\rm{\Omega }}}_{p}(z)}{{{\rm{\Omega }}}_{m{\rm{\Lambda }}}{a}^{-3},+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}\end{eqnarray} \tag{ 43 }$

where Ω_mp and Ω_mΛ refer to the contributions of the matter densities to the critical density at z = 0 in the phantom and ΛCDM cases, respectively. If one requires a larger value for H₀ in the phantom case, while keeping Ω_m,0h² the same in the two cases, then ${{\rm{\Omega }}}_{{mp}}\lt {{\rm{\Omega }}}_{m{\rm{\Lambda }}}$. The contribution Ω_Λ to the current critical density is constant, while ${{\rm{\Omega }}}_{p}={{\rm{\Omega }}}_{p}(z)$, being a phantom DE contribution, necessarily increases in time (with decreasing z).

By definition E_p(z = 0)/E_Λ(z = 0) = 1. But since Ω_p (z = 0) > Ω_p (z > 0) and Ω_Λ is constant, it follows that $\tfrac{{E}_{p}(z)}{{E}_{{\rm{\Lambda }}}(z)}\lt 1$ for z > 0, even if Ω_mp = Ω_mΛ. If we require that ${{\rm{\Omega }}}_{{mp}}\lt {{\rm{\Omega }}}_{m{\rm{\Lambda }}}$, so as to keep Ω_m,0h² the same while increasing H₀, then the ratio $\tfrac{{E}_{p}(z)}{{E}_{{\rm{\Lambda }}}(z)}$ becomes smaller still. It is thus apparent that in the presence of phantom-like dark energy, it is not only possible but necessary to decrease E(z) relative to the standard case, which in turn necessitates an increase in H₀ if CMB angular distance, shift parameter, and physical matter densities are to be kept constant.

Figure 4 illustrates this in the context of our torsion gravity models. Here we vary H₀, keeping Ω_m,0h² = 0.1417 (as measured in Planck Collaboration XIII 2016), and evaluate the shift parameter ${{ \mathcal R }}_{\mathrm{lss}}$, substituting H(z), namely, the inverse of Equation (33), into Equation (42), where z_lss = 1089.9 (Planck Collaboration XIII 2016). We then subtract this from the measured value of ${{ \mathcal R }}_{\mathrm{lss}}=1.7488$, retrieved from PlanckTT+lowP (Planck Collaboration XIV 2016), and divide by the error estimate quoted therein (±0.0074). As can be seen, as one deviates from the cosmological constant scenario (n = 0) and further into the phantom regime, the lines intersect the zero error horizontal at larger values of H₀; as expected, these larger values are thus necessary in order to fit the CMB data embodied in the shift parameter. In particular, a value of about n = 1/3 fits the shift parameter with H₀ = 73.5 km s⁻¹ Mpc⁻¹, as locally measured by Riess et al. (2018a).

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The electron scattering optical depth ${\tau }_{e}$ of the CMB provides a direct probe of the reionization epoch and its redshift z_re; it places constraints on the cosmological model, as it depends on H(z) at redshifts intermediate between z_lss and local measurements. The optical depth can be evaluated from

$\begin{eqnarray}&&{\tau }_{e}({z}_{{re}})=c{\int }_{0}^{{z}_{{re}}}\displaystyle \frac{{n}_{e}(z){\sigma }_{T}\,{dz}}{(1+z)H(z)},\end{eqnarray} \tag{ 44 }$

where n_e(z) is the electron density and σ_T is the Thomson cross section describing scattering between electrons and CMB photons. Here we take the densities of hydrogen, helium, and electrons, respectively, as ${n}_{H}=\left[(1-{Y}_{p}){{\rm{\Omega }}}_{b}{\rho }_{{cr},0}/{m}_{H}\right]{\left(1+z\right)}^{3}$, n_He = y n_H, and n_e = (1 + y)n_H, where $y=\tfrac{{Y}_{p}}{4(1-{Y}_{p})}$ and m_H is the hydrogen mass (Shull & Venkatesan 2008). We use the Planck constraint ${{\rm{\Omega }}}_{b,0}{h}^{2}=0.02230$ (Planck Collaboration XIII 2016), which gives the baryon density parameter Ω_b,0 = 0.0413 for H₀ = 73.5 km s⁻¹ Mpc⁻¹, the helium mass fraction Y_p = 0.247 (Peimbert et al. 2007), and the current critical density ρ_c,0 = 1.88 × 10⁻²⁹ h² g cm⁻³. Then, using the inverse function of Equation (33) and by evaluating the integral (44), we get τ_e(z_re) ≈ 0.058 at z_re = 8.1, which is in agreement with Planck Collaboration XLVII (2016) (lollipop + PlanckTT + lensing) observations,⁴ τ_e(z_re) = 0.058 ± 0.012, where 7.8 < z_re < 8.8.

As the resolution of the H₀ tension in terms of phantom dark energy described above involves changing the evolution of H(z)—through changing E(z)—it is natural to inquire whether this change can be actually distinguished directly from local H(z) measurements. Figure 5 collects such measurements. These include the 43 Hubble measurements given in Cao et al. (2018), which lists a number of CC and BAO measurements (including two Lyα observations). Also, we include four BAO measurements H(z = 0.978) = 113.72 ± 14.63, H(z =1.23) = 131.44 ± 12.42, H(z = 1.526) = 148.11 ± 12.75, and H(z = 1.944) = 172.63 ± 14.79 km s⁻¹ Mpc⁻¹ (Zhao et al. 2018) and one BAO Lyα observation H(z = 2.33) = 224 ± 8 km s⁻¹ Mpc⁻¹ (du Mas des Bourboux et al. 2017), in addition to these observations we include the R18 observation of H₀ = 73.52 ± 1.62 km s⁻¹ Mpc⁻¹ as measured by Riess et al. (2018a). Figure 5 shows clearly the capability of the torsional IR gravity (with n = 1/3) to fit with R18 and Lyα better than the ΛCDM model.

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The χ² statistics of these 49 Hubble measurements are

$\begin{eqnarray}&&{\chi }^{2}(n,\alpha )=\displaystyle \sum _{i}\displaystyle \frac{{\left({H}_{i}^{t}-{H}_{i}^{o}\right)}^{2}}{{\sigma }_{{H}_{i}^{o}}^{2}},\end{eqnarray} \tag{ 45 }$

where the subscript i = 1, 2, ..., 49, the superscripts t and o denote the theoretical and observed values of H(z_i), respectively, and ${\sigma }_{{H}_{i}^{o}}$ denote the one standard deviations in the measured values. As can be inferred from Table 1, both ΛCDM and the model with n = 1/3 and H₀ = 73.5 km s⁻¹ Mpc⁻¹ display ${\chi }^{2}/\mathrm{dof}\lesssim 1$, where "dof" is the number of degrees of freedom given that there are two model parameters. But the model associated with a phantom-like effective dark energy component performs better than that invoking the cosmological constant. This remains the case as long as the Lyα and R18 data are included. The BAO data on their own, on the other hand, favor the standard model. As we see below, this conclusion is definitely consolidated by BAO distance measurements, combined with the CMB.

Table 1.  The ${\chi }^{2}$ Calculations of H(z)

Data Set	n	χ2/dof		${\chi }_{\mathrm{dof}}^{2}$
CC	0	14.77/29	≈	0.51
	1/3	16.78/29	≈	0.58
BAO	0	9.59/12	≈	0.80
	1/3	11.39/12	≈	0.95
CC+BAO+R18	0	35.97/17	≈	0.82
	1/3	28.17/17	≈	0.64
CC+BAO+R18+Lyα	0	44.98/47	≈	0.96
	1/3	33.67/47	≈	0.72

Note. For R18, we take H₀ = 73.52 ± 1.62 km s⁻¹ Mpc⁻¹ as measured by Milky Way 50 Gaia + HST, Long P Parallaxes at redshift (Riess et al. 2018a).

For n = 0 (ΛCDM), we take H₀ = 68 km s⁻¹ Mpc⁻¹. For n = 1/3 (torsional IR gravity), we take H₀ = 73.5 km s⁻¹ Mpc⁻¹.

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We note that the Hubble function is related to the luminosity–distance D_L(z) by

$\begin{eqnarray}&&H(z)={\left[\displaystyle \frac{d}{{dz}}\left(\displaystyle \frac{{D}_{L}(z)}{1+z}\right)\right]}^{-1}.\end{eqnarray} \tag{ 46 }$

Since the Hubble function is related to the first derivative of D_L, one expects the measured values of H(z) to be much noisier than D_L(z) measurements. In other words, distances are in principle integrable quantities, which makes them relatively more precise. In the following we confront the torsional IR gravity with the BAO angular distance measurements.

BAO can be used as standard rulers; from isotropic measurements one can infer ${D}_{V}={\left[{cz}{(1+z)}^{2}{D}_{A}^{2}(z)/H(z)\right]}^{1/3}$ and from anisotropic measurements D_A itself (given the sound horizon at the baryon drag epoch r_d). These measurements rely on the same principle as that used to infer the angular diameter distance to the CMB (as once the physical densities and eras of recombination and matter radiation equality are determined, the physical scale of the peaks is fixed). It turns out that such measurements are highly constraining and essentially rule out solutions of the H₀ tension invoking phantom-like dark energy.

We show this by using the observations from various independent data sets: Beutler et al. (2011), which use 6dFGS data, Kazin et al. (2014) for reconstructed WiggleZ, Ross et al. (2015) for the SDSS MGS data, Alam et al. (2017a) for BOSS, Ata et al. (2018) for eBOSS quasar data, and Abbott et al. (2019) for the DES survey. We use r_d = 147.5 Mpc when the results are given as ratios involving r_d. We calculate the relevant distances to the observed redshift of the BAO peaks of each observation for our torsion models, deriving D_A and D_V for values of 0 ≤ n ≤ 1. For each such value, we vary Ω_m to search for the minimum of

$\begin{eqnarray}&&{\chi }^{2}(n,{H}_{0})=\displaystyle \sum _{i}\displaystyle \frac{{\left({D}_{i}^{t}-{D}_{i}^{o}\right)}^{2}}{{\sigma }_{i}^{2}}+\displaystyle \frac{{\left({{ \mathcal R }}_{\mathrm{lss}}^{t}-{{ \mathcal R }}_{\mathrm{lss}}^{o}\right)}^{2}}{{\sigma }_{\mathrm{lss}}^{2}},\end{eqnarray} \tag{ 47 }$

where D_i refers to the different BAO measurements (either D_V or D_A, depending on the particular set of observations), σ denotes the one standard deviations in measurements, and the superscripts t and o refer to the theoretical and observed values of the different quantities. For each n there is then a unique Ω_m that minimizes the χ². Assuming, as we do, that Ω_m,0h² is held fixed (at 0.1417), one can also associate a unique H₀ with each Ω_m, and hence for each Ω_m that minimizes χ² at each n.

The results are shown in Figure 6. As can be seen, the deviation between the observed and inferred distances, as measured using Equation (47), is smallest for n = 0. Values of n ≳ 1/5 are ruled out at the 99% confidence level. Moreover, for n = 1/3, the corresponding H₀ that minimizes χ² is significantly smaller than that inferred when fitting the CMB alone in Section 5.1. The reason for this failure is discussed in the next section.

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In Section 5.1, we argued that the angular diameter distance to the CMB and the shift parameter can be kept constant if one increases the value of H₀ while invoking cosmic evolution in the phantom regime. This was because E(z) = H(z)/H₀ is smaller up to z = 0 for such scenarios than in the case when the DE contribution comes from a cosmological constant. Requiring a larger value of H₀ evidently implies that the H(z) associated with phantom dark energy becomes larger than that of ΛCDM at some redshift z_c ≥ 0. From the Friedmann equations

$\begin{eqnarray}&&\displaystyle \frac{{H}_{p}^{2}(z)}{{H}_{{\rm{\Lambda }}}^{2}(z)}=\displaystyle \frac{{\rho }_{{mp}}(z)+{\rho }_{p}(z)}{{\rho }_{m{\rm{\Lambda }}}(z)+{\rho }_{{\rm{\Lambda }}}},\end{eqnarray} \tag{ 48 }$

one can find this redshift. The physical matter densities remain such that ρ_mp = ρ_mΛ if we keep Ω_m,0h² fixed, so that the ratio in Equation (48) is smaller than 1 when the phantom dark energy density is less than that associated with a cosmological constant: ${\rho }_{p}(z)\lt {\rho }_{{\rm{\Lambda }}}$. The ratio then increases to finally reach ${H}_{p}^{2}(z=0)/{H}_{{\rm{\Lambda }}}^{2}(z=0)$ at z = 0, which is greater than unity if we assume a larger value of the Hubble constant to be associated with the phantom case. The critical value z_c corresponds to a ratio of 1. If this occurs during matter domination, then the epoch where $\tfrac{{H}_{p}^{2}(z)}{{H}_{{\rm{\Lambda }}}^{2}(z)}\lt 1$ has negligible effect on the evolution and ${H}_{p}^{2}(z)\gt {H}_{{\rm{\Lambda }}}^{2}(z)$ for all practical purposes (that is, during DE domination). In this case, the angular diameter distance to the CMB will increase. If this distance (and shift parameter) is to be kept in line with observed values, then ${H}_{p}^{2}(z)\gt {H}_{{\rm{\Lambda }}}^{2}(z)$ should become unity at z_c ∼ 1.

In the case of torsion gravity models studied here this is illustrated in Figure 7, where we plot 1/H(z)—the integrand in the formula for the angular diameter distance—for the standard model with n = 0 and for n = 1. If H₀ is assumed to be the same in the two cases, H(n = 0) is always smaller than or equal to H(z = 1), which simply reflects the fact that E_p ≤ E_Λ up to z = 0 as expected from the discussion following Equation (43). When H₀ associated with the n = 1 case is larger, the lines cross at z = z_c. What this implies is that the angular diameter distance will be smaller than in the standard case for objects at z < z_c. And if the distance to the CMB D_A(z = z_lss) is to be kept fixed, while increasing H₀ and invoking the phantom regime, then D_A(z) to any object at 0 ≤ z ≤ z_lss will be larger than or equal to that predicted by ΛCDM. If the distance to the CMB is overestimated, then the distances to objects can be either overestimated or underestimated depending on its redshift.

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This implies that, in order to fit CMB and BAO distances simultaneously using a larger H₀ and $n\gt 0$, the standard model should systematically overestimate distances to BAO measurements, with the discrepancy being maximal for redshifts around z_c. This is not observed, as can be seen from Figure 7 (right panel). To further illustrate the point, we plot the errors associated with the different observations, which were used to estimate the χ² in Figure 8. As can be seen, at n = 0 some distances are overestimated and some underestimated, with no clear trend in terms of redshift dependence. As n is varied, the critical redshift z_c changes, and the χ² minimization procedure causes the distance to the CMB to also shift. As a result, there is another critical redshift below which BAO measurements are underestimated relative to the standard case and beyond which they are overestimated. Since there is no systematic deviation with respect to ΛCDM predictions in the BAO data used, this process means that some distances that were initially underestimated at n = 0 become even more so for n > 0, and conversely some overestimates are increased.

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Current CMB and BAO measurements seem to therefore rule out a significant phantom-like regime in the redshift range of the BAO data included here. This is the case even if one keeps H₀ at a small value, for this would shift the distance to the CMB and also the BOA points owing to the smaller E(z) and hence larger associated 1/H(z) (as discussed in Section 5.1 and reflected in Figure 7). We note, nevertheless, that there seems to be a systematic underestimate of the BAO distances inferred from Lyα measurements in the context of ΛCDM. We have not included these points here, as they lead to worse ΛCDM fits and do not lead to much improvement for the cases with $n\ne 0$, given that the models studied here are close to ΛCDM for the relevant redshifts (z ≳ 2) and the relatively large observational errors. Possible modest phantom evolution confined to redshifts z ≳ 2 is therefore not ruled out and can be tested by upcoming data.

For the n = 1 case, the torsion gravity model (28) reads

$\begin{eqnarray}&&f(T)=T+\alpha \displaystyle \frac{\,{T}_{0}^{2}}{T}.\end{eqnarray} \tag{ 49 }$

The modified Friedmann's equations, Equations (19) and (20), become

$\begin{eqnarray}&&\rho =\displaystyle \frac{3}{{\kappa }^{2}}\left[{H}^{2}-3\alpha {H}_{0}^{2}{\left(\displaystyle \frac{{H}_{0}}{H}\right)}^{2}\right],\end{eqnarray} \tag{ 50 }$

$\begin{eqnarray}&&p=-\displaystyle \frac{2}{{\kappa }^{2}}\dot{H}\left[1+3\alpha {\left(\displaystyle \frac{{H}_{0}}{H}\right)}^{4}\right]-\rho .\end{eqnarray} \tag{ 51 }$

By constraining the above to the linear EOS choice p = wρ, the solution is given as

$\begin{eqnarray}\begin{array}{rcl}t & = & {t}_{0}+\displaystyle \frac{2}{3(1+w)H}\\ & & +\,\displaystyle \frac{{3}^{\tfrac{3}{4}}}{9}\displaystyle \frac{\left[\mathrm{ln}\left(\tfrac{H+{\left(3\alpha \right)}^{\tfrac{1}{4}}{H}_{0}}{H-{\left(3\alpha \right)}^{\tfrac{1}{4}}{H}_{0}}\right)-2\arctan \left(\tfrac{H}{{\left(3\alpha \right)}^{\tfrac{1}{4}}{H}_{0}}\right)\right]}{(1+w){\alpha }^{\tfrac{1}{4}}{H}_{0}},\end{array}\end{eqnarray} \tag{ 52 }$

where t₀ is an integration constant. Although the above solution is exact, it is hard to extract information about the system from Equation (52). For example, its not clear how the system could behave at $t\to \infty$, or how sensitive it is to the choice of initial conditions. On the contrary, as we have shown, the graphical analysis of its phase portrait represents an adequate description of the qualitative features of the global dynamics. For the n = 1 model, the phase portrait (35) reads

$\begin{eqnarray}&&\dot{H}=-\displaystyle \frac{3}{2}(1+w){H}^{2}\left[\displaystyle \frac{{\left(H/{H}_{0}\right)}^{4}-3\alpha }{{\left(H/{H}_{0}\right)}^{4}+3\alpha }\right],\end{eqnarray} \tag{ 53 }$

which has been drawn in Figure 1. As clear from Equations (50) and (51), the torsional counterpart has density and pressure,

$\begin{eqnarray}&&{\rho }_{T}=\displaystyle \frac{9\alpha {H}_{0}^{4}}{{\kappa }^{2}{H}^{2}},\end{eqnarray} \tag{ 54 }$

$\begin{eqnarray}&&{p}_{T}=-\displaystyle \frac{18\alpha {H}_{0}^{4}{H}^{2}}{{\kappa }^{2}({H}^{4}+3\alpha {H}_{0}^{4})}.\end{eqnarray} \tag{ 55 }$

It is useful to represent the Friedmann Equation (50) in dimensionless form:

$\begin{eqnarray}&&{{\rm{\Omega }}}_{m}+{{\rm{\Omega }}}_{T}=1,\end{eqnarray} \tag{ 56 }$

where Ω_m = ρ/ρ_eff and Ω_T = ρ_T/ρ_eff are the matter and the torsion density parameters, respectively. Also, we note that the model parameter α, namely, Equation (31), is related to the current matter density parameter,

$\begin{eqnarray}&&\alpha =\displaystyle \frac{1}{3}(1-{{\rm{\Omega }}}_{m,0})=\displaystyle \frac{1}{3}{{\rm{\Omega }}}_{T,0}.\end{eqnarray} \tag{ 57 }$

Using the above equation and the useful relation

$\begin{eqnarray}&&\dot{H}=-(1+z)H(z)\displaystyle \frac{{dH}}{{dz}},\end{eqnarray} \tag{ 58 }$

one can solve Equation (35) for Hubble,

$\begin{eqnarray}&&H(z)=\displaystyle \frac{{H}_{0}}{\sqrt{2}}\sqrt{{{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}+\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}+4{{\rm{\Omega }}}_{T,0}}}.\end{eqnarray} \tag{ 59 }$

One of the important results that can be directly extracted from Equation (59) is the age–redshift relation,

$\begin{eqnarray}&&\begin{array}{l}t(z)\\ =\displaystyle \frac{\sqrt{2}}{{H}_{0}}{\int }_{z}^{\infty }\displaystyle \frac{{dz}^{\prime} /(1+z^{\prime} )}{\sqrt{{{\rm{\Omega }}}_{m,0}{\left(1+z^{\prime} \right)}^{3}+\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z^{\prime} \right)}^{6}+4{{\rm{\Omega }}}_{T,0}}}}.\end{array}\end{eqnarray} \tag{ 60 }$

Next, we evaluate the matter density parameter by substituting from Equation (59) into Equation (50), which yields

$\begin{eqnarray}&&{{\rm{\Omega }}}_{m}(z)=\displaystyle \frac{2\,{{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}}{{{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}+\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}+4{{\rm{\Omega }}}_{T,0}}}.\end{eqnarray} \tag{ 61 }$

Thus, the torsional density parameter is

$\begin{eqnarray}&&{{\rm{\Omega }}}_{T}(z)=1-\displaystyle \frac{2\,{{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}}{{{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}+\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}+4{{\rm{\Omega }}}_{T,0}}}.\end{eqnarray} \tag{ 62 }$

We plot the evolution of Ω_m(z) and Ω_T(z) in Figure 9 (left panel). It shows that Ω_m → 1 at large z while Ω_T → 0, which indicates the CDM domination. On the contrary, Ω_m drops to zero and Ω_T → 1 at z → −1 ($t\to \infty$), where the evolution is dominated by the dark torsion with a pseudo-rip cosmology as a final fate. The pattern shown in Figure 9 (left panel) is in agreement with basic requirements of the viable scenario.

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Using Equations (58) and (59), the deceleration parameter of the torsional IR model is given by

$\begin{eqnarray}&&q(z)=-1+\displaystyle \frac{3{{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}}{2\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}+4{{\rm{\Omega }}}_{T,0}}}.\end{eqnarray} \tag{ 63 }$

Alternatively, using Equation (27), we write the effective (total) EOS

$\begin{eqnarray}&&{w}_{\mathrm{eff}}(z)=-1+\displaystyle \frac{{{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}}{\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}+4{{\rm{\Omega }}}_{T,0}}},\end{eqnarray} \tag{ 64 }$

which is plotted as in Figure 9 (middle panel), showing that −1 ≥ w_eff ≤ −1/3 at −1 ≥ z ≲ 0.7 in agreement with observations. However, to express the torsional counterpart EOS in terms of redshift, w_T(z), we substitute Equation (59) into Equations (29) and (30), to write its density and pressure

$\begin{eqnarray}\begin{array}{rcl}{\rho }_{T}(z) & = & \displaystyle \frac{6{{\rm{\Omega }}}_{T,0}{H}_{0}^{2}}{{\kappa }^{2}\left[{{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}+\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}+4{{\rm{\Omega }}}_{T,0}}\right]},\\ {p}_{T}(z) & = & -\displaystyle \frac{6{{\rm{\Omega }}}_{T,0}{H}_{0}^{2}}{{\kappa }^{2}\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}\,+\,4{{\rm{\Omega }}}_{T,0}}}.\end{array}\end{eqnarray} \tag{ 65 }$

Hence, we obtain the torsional EOS

$\begin{eqnarray}&&{w}_{T}(z)=-1-\displaystyle \frac{{{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}}{\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}+4{{\rm{\Omega }}}_{T,0}}}.\end{eqnarray} \tag{ 66 }$

At present, z = 0, the above equation reduces to

$\begin{eqnarray*}&&{w}_{T,0}=-1-\displaystyle \frac{{{\rm{\Omega }}}_{m,0}}{2-{{\rm{\Omega }}}_{m,0}}.\end{eqnarray*}$

For any value ${{\rm{\Omega }}}_{m,0}\gt 0$, the torsional EOS goes below −1. This clarifies the phantom-like nature of the torsional IR corrections. Also, we note that the angular distance, namely, Equation (41), allows us to perform an important qualitative test, that is, the evolution of the comoving volume element within solid angle dΩ and redshift dz,

$\begin{eqnarray}&&{dV}=\displaystyle \frac{{\left(1+z\right)}^{2}{D}_{A}^{2}}{H(z)}d{\rm{\Omega }}\,{dz}.\end{eqnarray} \tag{ 67 }$

This quantity provides a useful test for computing the source counts (Newman & Davis 2000). Using Equations (59) and (41), the evolution of the volume element (up to a factor of Hubble volume ${H}_{0}^{3}$) is plotted in Figure 9 (right panel). The plot shows that the comoving volume element reaches a maximum value at z ≳ 2 very similar to the ΛCDM pattern.

In addition, we perform a basic test on the perturbation level of the theory that should be carried out for any modified gravity theory, that is, the propagation of the sound speed of the scalar fluctuations. As a matter of fact, a considerable array of modified gravity theories can describe the late transition of the cosmic acceleration fulfilling the basic requirements on the background level. However, any such theory remains at risk until its description on the perturbation level also fulfills some physical conditions. A necessary condition is for the sound speed of scalar fluctuations to be constrained between $0\leqslant {c}_{s}^{2}\leqslant 1$. This is required in order to have a stable and causal theory.

To calculate the sound speed, we take the longitudinal gauge with two-scalar metric fluctuation, that is,

$\begin{eqnarray}&&{{ds}}^{2}=(1+2{\rm{\Phi }}){{dt}}^{2}-{a}^{2}(1-2{\rm{\Psi }}){{dx}}^{2}.\end{eqnarray} \tag{ 68 }$

This leads to a fluctuation in the teleparallel torsion scalar (Cai et al. 2011)

$\begin{eqnarray*}&&\delta T=12H(\dot{{\rm{\Phi }}}+H{\rm{\Psi }}).\end{eqnarray*}$

Just as in GR theory, the weak-field limit about Minkowski space clarifies that the scalar metric fluctuation Φ plays the role of the gravitational potential. We follow the perturbation equations (Cai et al. 2011) up to the linear order, assuming that the matter sector is a canonical scalar field ϕ with a Lagrangian

$\begin{eqnarray}&&{{ \mathcal L }}_{m}\to {{ \mathcal L }}_{\phi }=\displaystyle \frac{1}{2}{\partial }_{\mu }\phi \,{\partial }^{\mu }\phi -V(\phi ).\end{eqnarray} \tag{ 69 }$

For the choice of the vierbein (16), it has been shown that (see Cai et al. 2011; Chen et al. 2011), in the f(T) gravity, we have only a single degree of freedom minimally coupled to a canonical scalar field ϕ, since the scalar field fluctuation δϕ can fully determine the gravitational potential Φ in the absence of anisotropic stress, i.e., Φ = Ψ. Using relation (18), we find that the square of the sound speed⁵ for the general form of the IR f(T) theory (28) is

$\begin{eqnarray}&&{c}_{s}^{2}=\displaystyle \frac{{f}_{H}}{{{Hf}}_{{HH}}}=1-\displaystyle \frac{2n(1+n)(1-{{\rm{\Omega }}}_{m,0})}{(1+2n)[{\tilde{H}}^{2(n+1)}+n(1-{{\rm{\Omega }}}_{m,0})]}.\end{eqnarray} \tag{ 70 }$

As is clear, for n = 0, the model reduces to ΛCDM where the speed of sound is fixed to the value c_s = 1. For the n = 1 case, substituting from Equation (59) into Equation (70), we write the square of the sound speed in terms of the redshift,

$\begin{eqnarray}&&{c}_{s}^{2}(z)=1-\displaystyle \frac{8{{\rm{\Omega }}}_{T,0}/\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}+4{{\rm{\Omega }}}_{T,0}}}{3\left({{\rm{\Omega }}}_{m,0}{\left(1+z\right)}^{3}+\sqrt{{{\rm{\Omega }}}_{m,0}^{2}{\left(1+z\right)}^{6}+4{{\rm{\Omega }}}_{T,0}}\right)}.\end{eqnarray} \tag{ 71 }$

We thus can verify that the square of the sound speed of the primordial scalar fluctuation ${c}_{s}^{2}\to 1$ in the past as $z\to \infty$, while its current value ${c}_{s}^{2}(z=0)\sim 0.43$. However, in the far future ${c}_{s}^{2}\to \tfrac{1}{3}$ as  z → − 1. The detailed evolution is given in Figure 10, which shows that the square of the sound speed is $\tfrac{1}{3}\leqslant {c}_{s}^{2}\leqslant 1$. Also, we include the evolution of ${c}_{s}^{2}(z)$ in the n = 1/3 case for completeness. This result confirms that the torsional IR correction theory is free from ghost/gradiant instabilities and acausality problems at all times.

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