Late-time acceleration and inflation in a Poincar\'e gauge cosmological model

TL;DR

This paper presents a Poincaré gauge cosmological model that unifies early inflation and late-time acceleration without dark energy or inflaton, fitting observational data with specific parameter estimates.

Contribution

It introduces a self-consistent Poincaré gauge cosmology model that unifies Starobinsky inflation and the $ m{ extLambda}$CDM model without additional fields.

Findings

Two analytical solutions under ghost- and tachyon-free conditions.

Early-time solution aligns with Starobinsky cosmology.

Late-time solution naturally includes a dark energy component.

Abstract

A self-consistent model which can unify Starobinsky inflation and the $\Lambda$CDM model in the framework of Poincar\'e gauge cosmology (PGC) is studied in this work, without extra inflaton and ``dark energy''. We start from the general nine-parameter PGC Lagrangian and get two Friedmann-like analytical solutions with the certain ghost- and tachyon-free conditions on the Lagrangian parameters. The scalar torsion $h$-determined solution is consistent with Starobinsky cosmology in the early-time and the pseudo-scalar torsion $f$-determined solution contains naturally a constant ``dark energy'' density, $3(\alpha-1)^2/8B_0$, which covers the $\Lambda$CDM model in the late-time. According to the latest observations, we estimate the magnitudes of parameters: $\alpha-1\simeq 8.07\times10^{-56}$, $B_0\simeq5.76\times10^{-28}GeV^{-2}$ in the natural units.

Tables

Table 1. Table 1 : The ghost- and tachyon-free conditions on parameters for six spin modes, respectively.

spin modes	conditions on parameters
$2^{-}$
$1^{-}$
$0^{-}$	$-2b_1+b_5>0$
	$\alpha -4a_1+4a_2>0$
$2^{+}$	$4b_1+4b_2+b_3+b_4+2b_5>0$
$1^{+}$
	$\left(2a_1-a_2\right)\left(-\alpha +4a_1-4a_2\right)\left(\alpha +2a_1+a_2\right)>0$
$0^{+}$	$b_1+b_2+b_3+b_4+b_5/2>0$
	$\alpha \left(2a_1+a_2+3a_3\right)\left(-2\alpha +2a_1+a_2+3a_3\right)>0$

Equations

$${\mathcal{L}}_G\sim \Lambda +curvature+torsio{n}^2+\frac{1}{\varrho }curvatur{e}^2,$$

$${\theta }^a(e_b)={\delta }_b^a.$$

$$e_a=e_a{}^{\mu }{\partial }_{\mu }.$$

$${\eta }_{ab}=e_a{}^{\mu }{e}_b{}^{\nu }{g}_{\mu \nu }.$$

$${\nabla }_{e_a}{e}_b$$

$${\nabla }_{\mu }{\partial }_{\nu }$$

$${\omega }^b{}_{\lambda a}d{x}^{\lambda }=(e^b{}_{\nu }{\partial }_{\lambda }{e}_a{}^{\nu }+e^b{}_{\nu }{\Gamma }^{\nu }{}_{\lambda \mu }{e}_a{}^{\mu })d{x}^{\lambda },$$

$$T^c$$

$$R^d{}_c$$

$$T^{\rho }{}_{\mu \nu }:={\Gamma }^{\rho }{}_{\mu \nu }-{\Gamma }^{\rho }{}_{\nu \mu }.$$

$${\Gamma }^{\rho }{}_{\mu \nu }={\{}^{\rho }{}_{\mu \nu }\}+\frac{1}{2}\left(T^{\rho }{}_{\mu \nu }+T_{\mu }{}^{\rho }{}_{\nu }+T_{\nu }{}^{\rho }{}_{\mu }\right),$$

$$R_{\mu \nu }{}^{\rho }{}_{\sigma }=2{\partial }_{[\mu }{\Gamma }^{\rho }{}_{\nu ]\sigma }+2{\Gamma }^{\rho }{}_{[\mu |\lambda |}{\Gamma }^{\lambda }{}_{\nu ]\sigma }.$$

$$d{s}^2=-d{t}^2+a^2(t)\left[d{\mathbf{x}}^2+K\frac{{\left(\mathbf{x}\cdot d\mathbf{x}\right)}^2}{1-K{\mathbf{x}}^2}\right],$$

$$T_{ij0}=a^{2}h(t){\delta }_{ij},T_{ijk}=a^{3}f(t){ϵ}_{ijk},i,j,k=1,2,3.$$

$$S_{\rho }{}^{\mu \nu }:=\frac{1}{\sqrt{|g|}}\frac{\delta \left(\sqrt{|g|}{\mathcal{L}}_m\right)}{\delta T^{\rho }{}_{\mu \nu }},$$

$$T_m^{\mu \nu }:=\frac{1}{\sqrt{|g|}}\frac{\delta \left(\sqrt{|g|}{\mathcal{L}}_m\right)}{\delta g_{\mu \nu }},$$

$${\nabla }_{\mu }{T}_m^{\mu }{}_{\nu }+T_{\mu }{T}_m^{\mu }{}_{\nu }+T_{\mu }{}^{\rho }{}_{\nu }{T}_m^{\mu }{}_{\rho }=0,$$

$$\dot{\rho }+3(1+w)H\rho =0,$$

$${\rho }_r={\rho }_{r0}{a}^{-4},{\rho }_b={\rho }_{b0}{a}^{-3},$$

$${\rho }_{\Lambda }=const.$$

$$I=$$

$${\mathcal{L}}_G=$$

$${\mathcal{L}}_T\equiv$$

$${\mathcal{L}}_R\equiv$$

$$+b_4{R}_{\mu \nu }{R}^{\nu \mu }+b_5{R}_{\mu \nu \rho \sigma }{R}^{\mu \rho \nu \sigma },$$

$$\int d^{4}x\sqrt{|g|}(R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }-4R_{\mu \nu }{R}^{\mu \nu }+R^2)=0,$$

$$\frac{1}{3}\kappa \rho$$

$$-\kappa w\rho$$

$$0$$

$$A_0$$

$$B_0$$

$$B_1$$

$$B_0>0,B_1<0,\alpha -4A_0>0,\alpha A_1(A_1-\alpha )>0.$$

$$A_1=B_1=B_2=0,4A_0=1,$$

$$H^2$$

Full text

1 Introduction

Inflation and late-time acceleration are two significant accelerated expansion periods of the Universe in the standard model (SM) framework of cosmology based on Einstein’s general relativity (GR) and observations. Latest observations [1] indicate good consistency with the standard spatially-flat $6$-parameter $\Lambda$CDM cosmology having a power-law spectrum of adiabatic scalar perturbations, from polarization, temperature, and lensing, separately and in combination. The joint constraint with baryon acoustic oscillation (BAO) measurements on spatial curvature is consistent with a flat universe, $\Omega_{K}=0.001\pm 0.002$. Also combining with Type Ia supernovae (SNe), the equation of state (EOS) parameter of “dark energy” is measured to be $w_{0}=-1.03\pm 0.03$, which is consistent with a cosmological constant. However, although the $\Lambda$CDM cosmology can accurately describe the evolution of the universe from a phenomenological perspective, the value of vacuum energy density estimated from quantum field theory is $10^{121}$ times larger than the observed value [2], and there is still no evidence of the existence of “dark energy”. On the other hand, the constraints on inflation [3] support the key prediction of the standard single-field (inflaton) inflation models. However, any single-field inflation model faces the origin of the scalar field. As a single-field model, Starobinsky inflation given by a Lagrangian $\tilde{R}+\tilde{R}^{2}/6M^{2}$ plus some small non-local terms (which are crucial for reheating after inflation) is an internally self-consistent cosmological model, which possess a (quasi-)de Sitter stage in the early Universe with slow-roll decay, and a graceful exit to the subsequent radiation-dominated Friedmann-Lemaître-Robertson-Walker (FLRW) stage [4, 5, 6]. This is one of the most appealing from both theoretical and observational perspectives among different models of inflation [7]. The motivation is natural to unify both Starobinsky inflation and the $\Lambda$CDM in one model, as people tried in [8, 9, 10, 11, 12, 13, 14, 15], from a perspective of $f(R)$ gravity.

Besides adding directly higher-order curvature invariants to the Hilbert-Einstein (HE) action, another more fundamental way to generalize GR from the geometric and gauge perspectives was introduced systematically since 1970’s [16, 17], which is called the Poincaré gauge gravity (PGG). As the maximum group of Minkowski spacetime isometrics, the Poincaré group is the semidirect product of the translation group and the rotation group, which has $10$ degrees of freedom in total. If constructing a gauge field theory based on the local invariance of the Poincaré group, gravity will be represented by two sets of independent gauge fields: the canonical $1$-forms $\theta$s (the dual co-vectors of tetrads $e$s) and the spin-connections $\omega$s, corresponding to the translations and the rotations, respectively. Analogous to the Yang-Mills theory, one can verify that torsion $T$ and curvature $R$ are just their gauge field strengths. According to Noether’s theorems, the symmetries of translations and rotations lead to two sets of conserved objects: the energy-momentums and the spin-angular momentums. Furthermore, the energy-momentum can be connected through Einstein’s equation with curvature, and the spin-angular momentum with torsion through Cartan’s equation, which mean that the sources of spacetime curvature and torsion are energy-momentum and spin of matter, respectively. The above is the basic idea of PGG, which follows the schemes of the Yang-Mills theory. From the geometrical perspective, the spacetime extends from Riemann’s to Riemann-Cartan’s, where curvature measures the difference of a vector after parallel transporting along an infinitesimal loop, and torsion measures the failure of closure of the parallelogram made of the infinitesimal displacement. To show the extension of PGG to GR, we plot the following diagram:

General speaking, the crucial difference between PGG and GR-based theories, such as $f(R)$ gravity, is that the former removed the restriction of torsion-free on the connections. However, the direct generalization from HE action will be back to Einstein’s theory, when the spin tensor (2.14) of matter vanishes because of the algebraic Cartan equation, i.e. torsion can not propagate in this case. This makes it necessary for us to generalize the action to obtain the propagating torsion in the vacuum. The standard PGG Lagrangian with quadratic-strength reads: [18]:

[TABLE]

where $\Lambda$ is the cosmological constant, and $\varrho$ the parameter with certain dimension. The additional quadratic terms come from the traces of field strengths in the internal space, which are at most second derivative if one regards the canonical $1$-forms (or the tetrads) and the spin-connections as the fundamental variables. It seems that these terms will introduce the ghosts degrees of freedom when one considers the particle substance of gravity. That would be something troublesome even for a simple modified gravity theory. The existence of the ghosts is closely related to the fact that the modified equation of motion has orders of time-derivative higher than two, for example, scale factor $a$ will be fourth-order over time in the general quadratic curvature case in FLRW cosmology. Due to Ostrogradsky’s theorem [19], a system is not (kinematically) stable if it is described by a non-degenerate higher time-derivative Lagrangian. To avoid the ghosts, a bunch of scalar-tensor theories of gravity was introduced, such as the Horndeski theory and beyond [20, 21]. Another way to evade Ostrogradsky’s theorem is to break Lorentz invariance in the ultraviolet and include only high-order spatial derivative terms in the Lagrangian, while still keeping the time derivative terms to the second order. This is exactly what Hořava did recently [22, 23]. Besides, another recipe to treat the ghosts is not removing them from the action, while focusing on the higher-order instability in the equations of motion [24]. For the general second-order Lagrangian with propagating torsion, a systematical way to remove the ghosts and the tachyons was introduced in [25, 26] using spin projection operators. The gauge fields $(\theta,\omega)$ can be decomposed irreducibly by $su(2)$ group into different spin modes by means of the weak-field approximation. In addition to the graviton, three classes spin-$0^{\pm},1^{\pm},2^{\pm}$ modes of torsion were introduced. [26] studied the general quadratic Lagrangian with nine parameters and obtained the conditions on the parameters for not having ghosts and tachyons at massive and massless sectors, respectively. In this work, to develop a good cosmology based on PGG, we will adopt their nine-parameter Lagrangian with the ghost- and tachyons-free conditions on parameters. The Hamiltonian analysis of PGG for different modes can be found in [27, 28], which tell us that the only safe modes of torsion are spin-$0^{\pm}$, corresponding to the scalar and pseudo-scalar components of torsion, respectively.

It’s natural to apply the corresponding Poincaré gauge cosmology (PGC) on understanding the evolution of the Universe. The last decade, a series of work [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] (from both analytical and numerical approach) proved that it is possible to reproduce the late-time acceleration in PGC without “dark energy”. In Ref. [40], the authors discussed the early-time behaviors of the expanding solution of PGC with a scalar field, while in Ref. [41], power-law inflation was studied in an $R+R^{2}$ model of PGC without inflaton. The motivation of our work is naturally raised, that is, to seek a model in PGC which can unify both the early- and the late-time evolutions without any priori hypothesis of inflaton and “dark energy”.

This paper is organized as follows. In Sec. 2 we briefly summarize the fundamental notions of PGG, and introduce the corresponding cosmology by means of some assumptions, then we get the conservation law in this framework. In Sec. 3 we derive the general cosmological equations of the nine-parameter Lagrangian with ghost- and tachyon-free conditions, then by choosing certain combinations of parameters, we obtain two analytical solutions which fit the early- and the late-time evolutions of the Universe, respectively; we also estimate the magnitudes of the model parameters according to the latest observations. We conclude and discuss our work in Sec. 4.

2 Poincaré gauge cosmology

In a Riemann-Cartan spacetime, the local invariance with respect to the internal symmetries of Poincaré group induces two sets of gauge potentials: the canonical $1$-forms $\theta$s corresponding to the translations and the spin-connections $\omega$s corresponding to the rotations. The canonical $1$-form is the dual co-vector of the tetrad, i.e.

[TABLE]

The tetrad can be decomposed with respect to an ordinary coordinate basis $\{\partial_{\mu}\}$, with a set of coefficients $e_{a}{}^{\mu}$,

[TABLE]

Here, we use $a,b,c,d,e$ to represent the internal indices, and $\mu,\nu,\rho,\lambda,\sigma$ to represent the external (spacetime) indices. In PGG, one chooses the orthonormal tetrad basis, which induces the following relation between the spacetime metric and the Minkowski metric:

[TABLE]

Therefore, two formalisms, the spacetime metric and the tetrad, are consistent. On the other hand, according to the definitions of the spin-connection $\omega$ and the affine-connection $\Gamma$:

[TABLE]

one has the relation between the spin-connection and the affine-connection:

[TABLE]

where $dx^{\lambda}$ is the dual co-vector of $\partial_{\lambda}$. Thus the spin-connection $1$-form $\omega^{b}{}_{a}\equiv\omega^{b}{}_{\lambda a}dx^{\lambda}$ and the affine-connection $1$-form $\Gamma^{\nu}{}_{\mu}\equiv\Gamma^{\nu}{}_{\lambda\mu}dx^{\lambda}$ are consistent.

The gauge field strengths, torsion and curvature, which are $2$-forms, can be constructed by means of the gauge potentials in the following way:

[TABLE]

which are also called the Cartan $1$st- and $2$nd-structure, respectively. On the other hand, the same geometric objects can be characterized by the metric $g$ and the affine-connection $\Gamma$ using the external indices, where the torsion tensor is defined as the anti-symmetric part of the affine-connection $\Gamma$:

[TABLE]

If we omit the non-metricity part of the affine-connection, i.e. $Q\equiv-\nabla g=0$, metric, torsion and affine-connection fulfill the following relation:

[TABLE]

where $\{^{\rho}{}_{\mu\nu}\}$ is the Levi-Civita connection constructed by means of the metric directly. Eq. (2.10) shows that instead of the pair of canonical $1$-form and spin-connection, the combination of metric and torsion can also represent the independent structures in Riemann-Cartan spacetime. The definition of curvature by means of the affine-connection reads:

[TABLE]

which implies torsion in it. In the subsequent context, $\nabla$ represents the covariant derivative with respect to the affine-connection. The torsion-free curvatures are labeled by a tilde, such as $\tilde{R}$.

In PGC, we assume that the cosmological principle is still valid, namely, our Universe is homogeneous and isotropic when viewed on a large enough scale. This assumption alone determines the spacetime metric up to FLRW form:

[TABLE]

where $a(t)$ is the scale factor with $t$ being the cosmic time. In this work, we consider the spatially flat case, $K=0$, which is consistent with the observations mentioned earlier in this article. Meanwhile, according to [42], it turns out that the only torsion tensors compatible with FLRW Universe are the time-like vector torsion and the time-like axial torsion [43], which can be expressed in terms of a scalar $h(t)$ and a pseudo-scalar $f(t)$,

[TABLE]

In addition, we assume that the spin effect of matter doesn’t appear on the cosmological scales. Since the spin orientation for ordinary matter is random, the macroscopic space average of the spin vanishes. The Weyssenhoff fluid was introduced into Einstein-Cartan theory [44] to describe the spin source of torsion. However, [45, 46] indicate that the Weyssenhoff fluid is incompatible with the cosmological principle. So we do not consider the Weyssenhoff fluid in our work. Therefore, the spin tensor, defined as

[TABLE]

vanishes [47]. There is still no direct evidence to support the existence of “dark energy”. When we say “dark energy”, it means that a priori hypothesis in our phenomenological models, such as the $\Lambda$CDM (cosmological constant with cold dark matter model) and the $w$CDM (“dark energy” with parameterized EOS $w$ and cold dark matter model). Here, we do not assume a priori the existence of “dark energy”, while we just start from our current knowledge of the Universe: radiation, (baryonic and dark) matter, and the late-time acceleration. From the perspectives of geometry, the power caused the acceleration is called the geometric “dark energy” in some references [48, 2, 49, 50]. Nonetheless, in our work, the energy-momentum tensor defined as

[TABLE]

contains radiation, baryonic and dark matter.

Noether’s theorem in PGG implies the conservation laws for energy-momentum and angular momentum currents, respectively [16, 17]. The vanishing of spin tensor $S_{\rho}{}^{\mu\nu}$ (2.14) leads to the conservation law of energy-momentum tensor as

[TABLE]

where $T_{\mu}\equiv T^{\nu}{}_{\mu\nu}$ is the trace of the torsion tensor. It is obvious that (2.16) is independent of the choice of the gravitational Lagrangian. Its time-component equation on FLRW background reads

[TABLE]

which is same as the one in SM. The EOS $w=\tfrac{1}{3},0$ correspond to relativistic particles (photons and massless neutrinos, or say radiation, labeled by “r”), and baryons (“b”), respectively. Eq. (2.17) can be solved immediately:

[TABLE]

where the subscript “[math]” represents values at present, $a_{0}=1$. In the $\Lambda$CDM model, the cosmological constant $\Lambda$ is regarded as the constant “dark energy” with $w=-1$. To understand the phenomenon of the late-time acceleration, we solve Eq. (2.17) for $\Lambda$ also:

[TABLE]

3 The solutions of PGC on FLRW background

3.1 The general cosmological equations

According to [26, 51], we consider a nine-parameter gravitational Lagrangian $\mathcal{L}_{G}$, at most quadratic in torsion and curvature, which reads:

[TABLE]

where $\kappa\equiv 8\pi G$, and $\alpha$, $a_{1}\sim a_{3}$, $b_{1}\sim b_{5}$ are free Lagrangian parameters with appropriate units. The $R^{2}$ term need not be included due to the use of the Chern-Gauss-Bonnet theorem [52]:

[TABLE]

for spacetime topologically equivalent to flat space. For the pair of gauge fields ($\theta,\omega$) as the dynamical variables, the field equations are up to $2$nd-order. However, the gauge fields ($\theta,\omega$) can be decomposed irreducibly by $su(2)$ group into different spin modes by means of the weak-field approximation. In addition to the graviton, three classes spin-$0^{\pm},1^{\pm},2^{\pm}$ modes of torsion are introduced. It is obvious that in such a general quadratic, the ghosts and tachyons are inevitable for certain modes. Fortunately, the authors studied this Lagrangian in [26] using the spin projection operators and obtained the conditions on parameters for not having ghosts and tachyons at massive and massless sectors, respectively.

According to [26], we summarize the ghost- and tachyon-free conditions on parameters for action (3.1) in TABLE I:

For the massless sector, the ghost-free condition is just: $\alpha>0$.

The ghost- and tachyon-free conditions in Table 1 are derived by means of the fundamental variables ($\theta,\omega$), but it’s general in any case. According to the previous context, it is convenient for us to treat $(g,T)$ as the fundamental variables to derive the field equations. Varying the action (3.1) with respect to $g_{\mu\nu}$ and $T^{\rho}{}_{\mu\nu}$, respectively, as well as considering (2.14) and (2.15), one can get the modified Einstein and the modified Cartan field equations. We do the calculations with the help of xAct: Efficient tensor computer algebra for the Wolfram Language111Authors: José M. Martín-García et. al. Homepage: http://www.xact.es/. We integrate our calculations in a Wolfram package PGC: Symbolic computing package for Poincare Gauge Cosmology222PGC version 1.2.1: https://github.com/zhanghc0537/Poincare-Gauge-Cosmology, which is available on Github. Since the field equations are too paper-consuming, we don’t intend to copy them here, but feel free to download and install our package if you want to check the field equations and their components on FLRW background. The README file will indicate you how to use it. Here, we just sort out the cosmological equations on FLRW background which read:

[TABLE]

with the combinations of parameters

[TABLE]

(3.3) is the time-component of the modified Einstein field equation, and (3.4) is the trace of its space-component. While (3.5,3.6) are the non-vanishing components of the modified Cartan field equation corresponding to the evolution of $h$ and $f$, respectively. The degeneracy among these Lagrangian parameters on the background makes the inequalities in Table 1 can not be solved completely. However, it is obvious that ghost- and tachyon-free spin-$0^{\pm}$ “particles” require:

[TABLE]

The above constraints for not having ghosts and tachyons follow from the requirement of having real mass and positive-definite residue matrix at the pole [26]. If one changes the inequality to an equal sign, the corresponding term may be lost the mass or momentum, therefore, the “particle” becomes massless or non-dynamic. We will consider this case by forcing the spin-$0^{-}$ mode non-dynamic and the spin-$0^{+}$ mode massless, such that the system can be solved analytically.

(3.3) and (3.4) correspond to the generalized Friedmann equations. For more discussions on the cosmological aspects of these field equations, please see [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].

3.2 The analytical solutions

Two independent Friedmann-like solutions can be obtained from (3.3) $\sim$ (3.6), if we just set:

[TABLE]

which corresponding to the spin-$0^{-}$ mode non-dynamic and the spin-$0^{+}$ mode massless case. Then the cosmological solutions read:

[TABLE]

or

[TABLE]

It’s obvious that if $\alpha\rightarrow 1$ and $B_{0}\rightarrow 0$, the first solution ($h$-determined) reduces to the SM. However, this solution doesn’t contain a natural “dark energy” ($\rho_{\Lambda}$ arises only when we set $w=-1$ artificially, which means the “dark energy” is introduced in the same way as SM), while the second solution ($f$-determined) implies a constant $\frac{3(\alpha-1)^{2}}{8B_{0}}$ if $\alpha\neq 1$ and $B_{0}\neq 0$. Therefore, $\alpha$ must be greater (according to (3.8) and (3.9)) but very close to $1$, and $B_{0}$ can be estimated by the value of “dark energy” density parameterized in the $\Lambda$CDM. $\alpha$ and $B_{0}$ represent the weights of Ricci scalar and the bulk effect of the quadratic curvatures, respectively. And this constant $\frac{3(\alpha-1)^{2}}{8B_{0}}$ can be regarded as the coupling strength between the curvature field and its quadratic terms. On the other hand, the first solution is more consistent with the evolution in the early-time. Since during the reheating (at any given temperature $T$), the energy density of non-relativistic particles in thermal equilibrium is exponentially suppressed with respect to that of relativistic particles, which means that during the reheating:

[TABLE]

The first solution shows $h=0,f=0$ naturally when $w=\tfrac{1}{3}$. Substituting $h=0,f=0$ into the general field equations and considering (3.9), (3.3) reduces to

[TABLE]

which is consistent with the Starobinsky cosmology. And the Starobinsky inflation arises if $\rho$ vanishes. $\tfrac{3\alpha}{B_{0}}$ is equivalent to $6M^{2}$ in the standard Starobinsky inflation:

[TABLE]

where $M$ is the scalaron rest-mass, determined from the normalization of the primordial scalar spectrum [53].

3.3 The numerical magnitudes of parameters

The combined WMAP3-SDSS measurements [54] constrain $M=2.8\times 10^{-6}(N/50)^{-1}M_{Pl}$ where $M_{Pl}\equiv 1/\sqrt{G}$ is the Planck mass 333The natural units are used in this subsection, where $\hbar=c=1$, $N$ the number of e-folds between the first Hubble radius crossing of the present inverse comoving scale $0.05Mpc^{-1}$ and the end of inflation. The model predictions for the slope of the primordial spectrum of scalar perturbations $n_{s}$ is $n_{s}-1=-2N^{-1}$. Planck 2018 [3] temperature, polarization, and lensing data determine the spectral index of scalar perturbations to be $n_{s}=0.9649\pm 0.0042$ at $68\%$ CL. Planck 2018 also constrain the inferred late-Universe parameters of the $\Lambda$CDM are: Hubble constant $H_{0}=(67.4\pm 0.5)km~{}sec^{-1}Mpc^{-1}$; matter density parameter $\Omega_{m}=0.315\pm 0.007$. Ignoring the spatial curvature $\Omega_{K}$ and radiation $\Omega_{r}$, the “dark energy” $\kappa\rho_{\Lambda}\simeq(1-\Omega_{m})\kappa\rho_{crit}=(1-\Omega_{m})(3H^{2}_{0})\simeq 4.25\times 10^{-84}GeV^{2}$. Combining the above data, we get the constraints (omitting the error bars):

[TABLE]

which derive

[TABLE]

According to these estimations, $B_{0}\kappa\rho_{b}\ll\alpha^{2}$, thus $H^{2}\simeq\frac{\kappa\rho_{b}}{3\alpha}$, $h^{2}\simeq\frac{B_{0}^{2}(\kappa\rho_{b})^{3}}{3\alpha^{4}}$ when $w=0$ in (3.10,3.11). $\alpha$ is $-54$ order of magnitudes close to $1$, which hit our previous guess. (3.11,3.14) show that the evolution of the Universe on the background is influenced by the spacetime torsion besides the effects from curvature. Conversely, such an evolution of the Universe proves indirectly the existence of torsion. In Ref. [55, 56, 57], authors discussed the practicability of the direct measurement of the spacetime torsion, which will help us further study the evolutionary mechanism of the Universe in PGC.

4 Conclusion

In this work, we start from the construction of Poincaré gauge cosmology based on certain assumptions and derive the conservation law (2.17) in this framework, which is still the usual one in SM. Then by varying the general nine-parameter PGC Lagrangian (3.1), we get the general cosmological equations (3.3$\sim$3.6). To remove the ghosts and tachyons in this Lagrangian, we summarized the ghost- and tachyon-free conditions on parameters in TABLE I according to [26]. Two Friedmann-like analytical solutions are obtained when the constraints for Lagrangian parameters (3.9) are imposed. The $h$-determined solution is consistent with the Starobinsky cosmology in the early-time, and the $f$-determined one contains naturally a constant $3(\alpha-1)^{2}/8B_{0}$, which can be regarded as the “dark energy” density. These two solutions are consistent because they are obtained from the same gravitational Lagrangian. According to the latest observations, and the behaviors in the early- and late-time respectively, we estimate the magnitudes of parameters: $\alpha-1\simeq 8.07\times 10^{-56}$, $B_{0}\simeq 5.76\times 10^{-28}GeV^{-2}$ in the natural units. (3.6) shows that $B_{1}=0$ leads to the second-order term of $f$ vanished, i.e. (3.6) is just a constraint, which is the key reason to get these two solutions. Loosing the constraint on $B_{1}$ to an infinitesimal level, such that the $h$-determined solution is an asymptotic solution in the early-time and the $f$-determined solution is an asymptotic solution in the late-time, which will be the methodology to pursue a continuous model of PGC, where the inflation and late-time acceleration can be unified smoothly. For more general case (without the constraint (3.9) on parameters) and the inflation in PGC, please see our recent preprint [58]. We expect that further analysis PGC can be consistent with the recent evidence of dynamic “dark energy” [59] from the perspective of geometry and gauge field theory.

Acknowledgments

We thank Prof. Abhay Ashtekar for helpful comments. Lixin Xu is supported in part by National Natural Science Foundation of China under Grant No. 11675032. Hongchao Zhang is supported by the program of China Scholarships Council No. 201706060084. Our calculations are accomplished with the help of a series of Wolfram packages under the name xAct, written by José M. Martín-García et. al.