# Sustainable Data Management: Indefinite Static Data at Rest with Machine-
Readable Printed Optical Data Sheets (MRPODS)

Artem R. Doll
Independent Researcher
<EMAIL_ADDRESS>

###### Abstract

In an era where both commercial and private sectors place a premium on the
longevity of digital data storage, the imperative to bolster resilience of
digital information while simultaneously curbing costs and reducing failure
rates becomes paramount. This study delves into the unique attributes of
optical encoding methodologies, which are poised to offer enduring stability
for digital data. Despite their promising potential, there remains a notable
dearth of comprehensive analyses comparing various optical encoding techniques
in terms of their durability. This research is thus dedicated to exploring the
financial and environmental implications of employing technology to transcribe
digital data into a machine-readable optical format, assessing both the
advantages and limitations inherent in this approach. Our empirical findings
reveal a marked increase in the efficiency of machine-readable optical
encoding over conventional digital storage methods, particularly as the volume
of data diminishes and the expected lifespan of storage extends indefinitely.
This paper aims to illuminate key aspects of long-term digital data storage
within business contexts, focusing on aspects such as cost, dependability,
legibility, and confidentiality of optically encoded digital information.

_Keywords_ Data Management $\cdot$ Data Archival $\cdot$ Machine-Readable
Formats

##  1 Introduction

In contemporary business environments, documents containing sensitive
information are essential for legal and competitive confidentiality. This
category encompasses, but is not restricted to, confidential agreements
(NDAs), trade secrets, legal contracts, financial records, client and customer
data, intellectual property, employee records, strategic plans, research and
development documents, corporate communications, IT infrastructure details,
access codes, password databases, and asset valuations. These documents, while
occupying minimal storage space, are perpetually at risk of cyber threats,
including cyberattacks, digital corruption, ransomware (leading to data loss),
and unauthorized alterations. Pre-emptively encrypted physical reproductions
of such digital data present a less vulnerable alternative. In scenarios where
digital infrastructures are compromised, the physical storage of critical
information could be pivotal in maintaining operational continuity with
minimal disruption.

Optical disks, traditionally considered robust, exhibit less durability than
previously assumed, suffering from issues such as the degradation of their
reflective layers, sensitivity to light exposure, and a lifespan falling short
of two decades. Additionally, their storage capacity is comparatively limited,
and they necessitate complex playback equipment reliant on motors and lasers,
components with a propensity for failure and potentially limited manufacturing
lifespan.

Conversely, printers and scanners, including basic consumer models, are
expected to remain prevalent in the foreseeable technological landscape.
Notably, while a CD/DVD reader inherently includes burning capabilities for
data retrieval, paper scanners operate independently of printers. This
independence, coupled with the ongoing evolution of digital storage
technologies, necessitates the continued production of scanners for the
foreseeable digitization of paper-based and printed texts. Such a scenario
suggests that even if environmental considerations lead to the eventual
obsolescence of printers within the next half-century to a century, previously
printed data would remain accessible through scanning technology.

##  2 Proof of Concept

Oleh Yuschuk pioneered the printing of digital data in bitmap format onto
paper, integrating Reed-Solomon error correction coding as a practical
demonstration of concept (Yuschuk, 2007). Initially conceived with a semblance
of programming humor, Yuschuk’s innovation may not have fully captured the
extensive potential of Machine-Readable Printed Optical Data Sheets (MRPODS).
A notable advantage identified by Yuschuk is the visual verifiability of data
integrity; the state of the data can be assessed without necessitating the
activation of any electronic storage medium. This facet of physical encoding
allows for the storage and assessment of data devoid of electricity or
advanced technological interfaces, contingent upon the longevity and
compatibility of the encoding and decoding software.

In an endeavor to ensure the future utility of his implementation, Yuschuk
designed the software specifically for the 32-bit Windows platform, and
elected to distribute it under the GPLv3 license. This licensing decision
ensures the freedom to modify the source code, a critical consideration for
future adaptability. Remarkably, the PaperBack software, despite its inception
in 2007, remains functional on contemporary operating systems, including
Windows 11 Pro 23H2 as of 2024, thereby encompassing nearly two decades of
backward compatibility. This enduring functionality resonates particularly in
business settings, where the integration and sustained operation of legacy
software systems are prevalent.

However, should there be a paradigm shift in hardware away from x86
architecture without a corresponding compatibility framework, it is
conceivable that physical encoding and decoding software might operate within
a virtual machine environment, circumventing the need for direct source code
alterations. Nonetheless, such a significant transition in hardware
architecture would likely be accompanied by a prolonged adaptation phase,
offering ample opportunity for entities to modify or port existing source code
to align with future technological standards.

##  3 Security Features Unique to MRPODS

Data Sheets can be output at a sufficiently high dots per inch (DPI)
resolution, effectively safeguarding them from unauthorized replication using
standard equipment. In an era where smartphone cameras are ubiquitous, the
risk of scanning and duplicating human-readable text is a pertinent concern.
However, the fine granularity of physically encoded data in documents renders
them impervious to replication by conventional camera technology. This
attribute enables secure interdepartmental transfer of documents, mitigating
the risk of data alteration or unauthorized copying during transit.

Figure 1: Zoomed out view (left), and closer view (right)

The robustness of documents against duplication can be further enhanced by
meticulously adjusting various parameters such as the data DPI, dot size, and
redundancy levels. Such optimization ensures that any attempt at scanning and
reprinting these documents introduces an excessive amount of artifacts,
thereby thwarting duplication efforts. Intrinsically, data sheets possess a
layer of obfuscation; the bitmap data encoded in these documents is not
decipherable by human cognition, providing a form of security even in the
absence of additional encryption measures.

The structure of data sheets can be further tailored to include headers that
detail the content or the file format, as well as the totality of pieces
necessary for complete data retrieval. These headers are customizable and may
deliberately omit information regarding the aggregate number of documents
required for full data reconstruction until the commencement of the scanning
process. This aspect serves as a strategic ambiguity, ensuring that
individuals in possession of data sheets might remain oblivious to whether the
document is encoded in its entirety, thereby augmenting security measures and
deterring potential information theft.

##  4 Data Density

## Data Compression Using bzip2

The bzip2 compression algorithm, as implemented in the Printer.cpp file,
significantly increases data density through a combination of the Burrows-
Wheeler Transform (BWT), the Move-to-Front (MTF) transform, and Huffman coding
(Lauther and Lukovszki, 2009).

Let $D$ be the original data stream. The BWT rearranges $D$ into $D^{\prime}$,
where similar characters are grouped together, thus forming runs of
characters. This can be represented as:

$D^{\prime}=BWT(D)$ (1)

Subsequently, the MTF transform converts these runs into a more compressible
form, denoted as $D^{\prime\prime}$:

$D^{\prime\prime}=MTF(D^{\prime})$ (2)

Finally, Huffman coding is applied to $D^{\prime\prime}$, resulting in the
compressed data stream $C$:

$C=Huffman(D^{\prime\prime})$ (3)

The overall compression process can be represented as:

$C=Huffman(MTF(BWT(D)))$ (4)

In the context of data compression as implemented in Printer.cpp111The source
code is available from: https://github.com/artem-r-d/mrpods of MRPODS using
the bzip2 algorithm, the compression ratio (CR) is a crucial metric. This
ratio is defined as:

$CR=\frac{ODS}{CDS}$ (5)

where $ODS$ is the Original Data Size and $CDS$ is the Compressed Data Size.
The effectiveness of bzip2 compression is measured by the increase in data
density (DD), which is inversely proportional to the compressed data size.
Thus, the data density can be expressed as:

$DD=\frac{1}{CDS}$ (6)

This mathematical representation underscores the efficiency of bzip2 in MRPODS
applications. In a practical demonstration using a widely available consumer-
grade HP OfficeJet 3830 printer, the efficacy of bzip2 was evident when
printing a WinRar compressed archive of the Project Gutenberg eBook version of
’War and Peace’ (excluding the JPPEG cover) on a mere five sheets of paper.
The specific settings employed in this process were as follows: a dot density
set at 200 dots per inch (dpi), dot size adjusted to 70%, and a redundancy
ratio of 1:5. The printing was executed in high DPI grayscale mode with the
borderless printing option activated.

222The presented chart furnishes a comprehensive depiction of the correlation
between the quantity of printed pages in the machine-readable format and the
ensuing ramifications on both data size and printing costs. The dual-axis
chart methodically juxtaposes two essential variables: the data size (measured
in megabytes) and the associated printing costs (denominated in dollars). The
first axis, rendered in green, delineates the escalation of data size in
direct proportion to the number of printed pages. This representation is
pivotal in elucidating the capacity of data sheets to accommodate varying data
volumes. It underscores the capability of this storage method to scale
linearly with the amount of information, thereby offering a versatile solution
adaptable to diverse data storage needs. The second axis, depicted in blue,
articulates the corresponding financial implications of printing these data
volumes. The cost per page, a critical determinant in this calculation, is
standardized, allowing for a direct and unambiguous interpretation of the cost
implications. As the number of pages increases, so does the total cost,
depicting a linear relationship that highlights the predictability and
uniformity of expenses. This chart serves as an analytical tool, offering
insights into the relationship between data volume and the economic aspects of
MRPODS and the trade-offs involved.

Figure 2: How data density affects page output

For comparison, when the same eBook was formatted for printing via Microsoft
Word, utilizing a standard Times New Roman font at a 12-point size and narrow
margins, the document spanned 748 pages. This stark contrast in page count
between the two methods highlights the significant efficiency and space-saving
potential of MRPODS for the storage of substantial textual data, provided the
data does not exceed 4MB.

##  5 Sustainability

333The graphical representation delineated in the provided chart cogently
illustrates the evolving financial implications of utilizing hard drives for
data storage vis-à-vis the fixed costs associated with machine-readable
printed data over a century. This comparison is anchored on a foundational
premise: the dynamic nature of the cumulative costs incurred in maintaining
hard drive-based storage systems against the static expenditure of printing
data in a machine-readable format. In the analysis, two primary cost
components of hard drive usage are identified. The first is the recurrent
expense of hardware replacement, necessitated by the finite operational
lifespan of hard drives, approximated here at five years. Over the span of a
century, this translates into a series of replacements, each incurring a cost
that cumulatively amplifies the total financial outlay. The second component
pertains to the electricity costs associated with the continuous operation of
the hard drive. This aspect encapsulates the power consumption metrics of the
storage device and the prevailing electricity tariffs, culminating in an
annual cost that, when aggregated over a hundred years, represents a
substantial financial commitment. Conversely, the cost of printing machine-
readable data is presented as a one-time expenditure. This is premised on the
assumption that the act of printing a predefined quantity of data (in this
case, 20 pages) entails an immutable cost, irrespective of the temporal
dimension. This cost, once incurred, remains unchanged throughout the century,
establishing a stark contrast with the escalating costs associated with hard
drive usage. The chart effectively juxtaposes these divergent cost
trajectories, demonstrating a pronounced increase in the cumulative expense of
utilizing hard drives over time. This upward trend is attributable to the
summation of periodic hardware replacement costs and the sustained electricity
charges. In stark contrast, the printing cost is represented as a constant
value across the century, underscoring the cost-effectiveness of this method
for long-term data preservation. This analysis, therefore, elucidates a
crucial insight: while hard drives may offer a seemingly economical solution
in the short term, their long-term financial viability is significantly
undermined by ongoing operational and replacement costs. In contrast, the
machine-readable printing approach, despite an initial investment, emerges as
a more economically sustainable option over an extended period. This
revelation holds profound implications for businesses and organizations
strategizing on data storage methodologies, especially those contemplating the
long-term fiscal and environmental sustainability of their data management
practices.

Figure 3: Traditional storage vs. Data sheets cost

The current paradigm of data storage and dissemination necessitates a
sophisticated comprehension of the longitudinal cost implications inherent in
diverse storage methodologies. Recent explorations have contrasted the
aggregate financial burden of digital storage mediums, such as hard drives,
against the upfront expenditures associated with machine-readable printing
formats. A constructed model, incorporating the recurrent costs of hard drive
replacements (estimated at a 5-year interval) and the unceasing electricity
expenses over a century, unveils a telling dynamic. This model delineates that
the initial investment for printing, as exemplified by the expense incurred in
printing 20 pages in a machine-readable format, remains constant. Conversely,
the total cost associated with hard drive utilization follows an upward linear
progression over time, a consequence of the amalgamated effects of habitual
hardware renewals and persistent energy usage.

This analytical framework brings to light a pivotal inflection point: the
moment where the escalating expenditure of digital storage eclipses the
unchanging outlay of physical printing. This revelation holds substantial
weight in shaping long-term data management strategies, especially in
scenarios where archival durability and cost-efficiency are of utmost
significance. These results accentuate the importance of adopting a
comprehensive perspective when evaluating the financial viability of data
storage options. Such an approach should rigorously account for both the
immediate and protracted cost factors, thereby ensuring a balanced and
thorough assessment of fiscal sustainability in data storage practices.

##  6 Conclusion

In summary, the research presented in this paper elucidates the multifaceted
advantages of Machine-Readable Printed Optical Data Sheets (MRPODS) in the
context of secure and sustainable business data management. The data sheet
paradigm emerges as a highly viable alternative to traditional digital storage
methods, particularly in scenarios demanding long-term preservation and
security of sensitive data. This approach effectively addresses key concerns
such as data longevity, resistance to cyber threats, and the overarching issue
of environmental sustainability.

The comparative analysis of MRPODS against conventional storage mediums
underscores its superiority in various critical aspects. Firstly, the
robustness of data sheets in safeguarding data against unauthorized access and
duplication is unparalleled, primarily due to its high-resolution optical
encoding which resists standard replication techniques. The physical nature of
data sheets inherently reduces the susceptibility of data to cyber threats,
thus offering a more secure storage medium for sensitive and confidential
business documents.

In terms of sustainability, the findings of this study reveal that MRPODS
offers a more economically viable solution over an extended timeframe. The
static cost associated with data sheets printing contrasts sharply with the
escalating expenses of digital storage, which include recurrent hardware
replacements and ongoing electricity costs. This attribute of MRPODS makes it
particularly attractive for long-term data archiving, where cost-effectiveness
is as crucial as data integrity.

Moreover, the application of sophisticated compression algorithms like bzip2
within the MRPODS framework significantly enhances data density, allowing for
the efficient storage of large volumes of information on minimal physical
space. This aspect not only saves physical resources but also aligns with the
broader environmental goals of reducing electronic waste and energy
consumption.

In conclusion, the research presented herein firmly establishes MRPODS as a
formidable contender in the realm of data storage, particularly for businesses
seeking secure, sustainable, and cost-effective methods for managing their
critical data. The future of data storage, as suggested by this study, lies in
the innovative use of machine-readable optical formats that balance security,
durability, and environmental responsibility. As businesses continue to
navigate the complexities of data management in an increasingly digital world,
MRPODS stands out as a beacon of potential, guiding the way towards a more
secure and sustainable future.

## References

  * Yuschuk [2007] O. Yuschuk.  Paperback.  https://ollydbg.de/Paperbak/, 2007.
  * Lauther and Lukovszki [2009] U. Lauther and T. Lukovszki.  Space efficient algorithms for the burrows-wheeler backtransformation.  _Algorithmica_ , 58(2):339–351, 2009.  doi:10.1007/s00453-008-9269-9.


USTC-ICTS/PCFT-21-17

# Revisiting a parity violating gravity model without ghost instability: Local
Lorentz covariance

Mingzhe Li  Haomin Rao  Yeheng Tong  Interdisciplinary Center for Theoretical
Study, University of Science and Technology of China, Hefei, Anhui 230026,
China  Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China

###### Abstract

Recently, based on the theory of teleparallel gravity, a simple and ghost free
parity violating gravity model was proposed in [M. Li, H. Rao, and D. Zhao, J.
Cosmol. Astropart. Phys. 11 (2020) 023], where the Weitzenböck connection was
adopted for simplifying the calculations but breaks the local Lorentz symmetry
explicitly. In this paper, we restore the local Lorentz symmetry of this model
by giving up the Weitzenböck condition on the spin connection. With full local
Lorentz covariance, this model is not a pure tetrad theory any more. We also
apply the new version of this model to the universe with general Friedmann-
Robertson-Walker background. This further generalizes the studies of [M. Li,
H. Rao, and D. Zhao, J. Cosmol. Astropart. Phys. 11 (2020) 023] where only the
spatially flat background was considered. Through the investigations of this
paper, we confirm the results obtained in [M. Li, H. Rao, and D. Zhao, J.
Cosmol. Astropart. Phys. 11 (2020) 023] and in addition get some new results.

##  I introduction

Parity violating (PV) gravities attracted a lot of interests in recent years.
A famous and frequently studied PV gravity is the so-called Chern-Simons (CS)
modified gravity Jackiw:2003pm ; Alexander:2009tp , in which general
relativity (GR) is modified by a gravitational CS term; $S_{CS}\sim\int
d^{4}x\sqrt{-g}\theta(x)\varepsilon^{\mu\nu\rho\sigma}R_{\mu\nu}^{~{}~{}~{}\alpha\beta}R_{\rho\sigma\alpha\beta}$,
where $g$ is the determinant of the metric, $\theta(x)$ is a coupling scalar
field, $\varepsilon^{\mu\nu\rho\sigma}$ is the four-dimensional Levi-Civita
tensor and $R_{\rho\sigma\alpha\beta}$ is the Riemann tensor constructed from
the metric. The CS modified gravity makes a difference between the amplitudes
of the left- and right-handed polarized components of gravitational waves
(GWs), but there is no difference between their velocities. This is the so-
called amplitude birefringence phenomenon. However, the CS gravity suffers
from the problem of ghost instability Dyda:2012rj  which essentially
originated from the higher derivatives in the CS term. To circumvent this
problem, further extensions to the CS gravity were explored in Ref.
Crisostomi:2017ugk  where several terms including higher derivatives of the
coupling scalar field $\theta(x)$ were taken into account. Though each term
contains higher derivatives, combining them together in a special way can
eliminate the ghost modes. The price one must to pay is that all such kind of
models have much more complex forms Gao:2019liu ; Zhao:2019xmm .

Recently we proposed a new PV gravity model which is healthy and simple in
form PVtele1 . This model is based on the theory of teleparallel gravity (TG)
Tele  which is equivalent to GR but formulated in flat spacetime with
vanishing curvature and vanishing nonmetricity. The gravity in TG theory is
identical to the spacetime torsion. Our model PVtele1  modified the GR
equivalent TG theory by an extra PV term: $S_{NY}\sim\int
d^{4}x\sqrt{-g}~{}\theta(x)\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}_{A\mu\nu}\mathcal{T}^{A}_{~{}~{}\rho\sigma}$,
here $\mathcal{T}^{A}_{~{}~{}\mu\nu}$ is the torsion two form and its exact
expression and other details will be presented in the next section. Except for
the coupling with a scalar field, this PV term is in fact the reduced Nieh-Yan
term Nieh:1981ww  within the framework of teleparallelism. Different from the
CS modified gravity, the PV term in this model hides no higher derivatives and
successfully avoids the ghost mode. The Nieh-Yan modification to GR equivalent
TG has been also considered in Ref. Chatzistavrakidis:2020wum  where its
consequences in gravitoelectromagnetism was studied with a nondynamical
$\theta$ field. Still within the framework of TG theory, generalizations
containing more PV couplings beyond Nieh-Yan were made in Ref. Hohmann:2020dgy
where the cosmological dynamics in the Friedmann-Robertson-Walker (FRW)
universe was derived.

In Ref. PVtele1 , our PV gravity model has also been applied to cosmology,
where two additional conditions were set by hand. The first one is the so-
called Weitzenböck condition, i.e., the spin connection vanishes,
$\omega^{A}_{~{}B\nu}=0$. This had been frequently adopted for simplifying the
calculations in many studies of TG theory and its extensions, such as
$f(\mathbb{T})$ models fT . The unsatisfactory aspect of taking the
Weitzenböck connection is the explicit violation of the local Lorentz
symmetry. This is not a difficulty for TG theory because fixing any possible
connection there (including the Weitzenböck one) which satisfies the
teleparallel constraints will only contribute a surface term in the action;
this will not affect the equation of motion. It is not so, however, for
modified TG theories in which the equation of motion indeed depends on the
spin connection. The second additional condition adopted in Ref. PVtele1  is
that the application to cosmology was limited to the case where the universe
has a spatially flat FRW background. With these, we have studied the
background evolution and computed the linear cosmological perturbations. We
found that the PV modification has no effect on the background evolution but
indeed affects the perturbations. For tensor perturbations the left- and
right-handed polarized components of GWs have different propagating
velocities, but their amplitudes are the same. This phenomenon is called
velocity birefringence. For scalar perturbations, the coupled scalar field
$\theta(x)$ surprisingly does not contribute an independent dynamical
component at the linear level—it behaves like an imperfect fluid and causes
viscosities in the stress-energy tensor.

In this paper, we will revisit our PV gravity model PVtele1  and its
applications in cosmology by relaxing the two additional conditions mentioned
above. Without the Weitzenböck condition, we will leave the spin connection
arbitrary as long as the teleparallel constraints (no curvature and no
nonmetricity) are satisfied. This restores the local Lorentz symmetry of our
model and in this new version both the tetrad and the spin connection are
treated as fundamental variables in the action principle. Secondly, we will
apply the new version of our model to the universe which has a general FRW
background, not just the spatially flat one. We will focus on the linear
cosmological perturbations. With these improvements and generalizations, we
will confirm the results obtained in Ref. PVtele1  and get some new results:
for instance, the linear perturbation of the scalar field $\theta(x)$ is
indeed counted as a dynamical degree of freedom in the case of a curved FRW
background.

This paper is organized as follow. In Sec. II, after a brief review of the PV
gravity model proposed in PVtele1 , we consider its new version in which the
local Lorentz symmetry is restored by giving up any extra constraint on the
connection. Then we apply this new version of our model with full local
Lorentz covariance to cosmology in the next sections. In Sec. III, we study
the evolution of the background which is a general FRW spacetime. We
investigate case by case the effects of the PV term on the evolution equations
for the flat, closed and open universes. In section IV, we discuss the linear
perturbations around the FRW background and the gauge transformations. This
ensures that some gauge conditions are available to simplify subsequent
calculations. In Sec. V, we calculate the quadratic actions for scalar and
tensor perturbations. We will confirm the results obtained in PVtele1  and
show some new results—among them the most important one is that the dynamical
scalar field $\theta$ acquires its independent dynamics at the linear-
perturbation level around the spatially curved FRW background.

##  II the healthy party violating gravity model based on TG theory

In this paper, we adopt the unit $8\pi G=1/M_{p}^{2}=1$ and the convention of
most negative signatures for metrics. The local space tensor indices are
denoted by $A,B,C,...=0,1,2,3$ and by $a,b,c,...=1,2,3$ when limiting to
spatial components. They are are lowered and raised by the Minkowski metric
$\eta_{AB}$ and its inverse $\eta^{AB}$. The spacetime tensor indices are
denoted by Greek $\mu,\nu,\rho,...=0,1,2,3$ and by Latin $i,j,k,...=1,2,3$
when limiting to spatial components. These are lowered and raised by the
spacetime metric $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. The tetrad is
denoted as $e^{A}_{~{}\mu}$ and considered as the square root of the metric,
i.e., $g_{\mu\nu}=\eta_{AB}e^{A}_{~{}\mu}e^{B}_{~{}\nu}$. The equation
$\hat{\Gamma}^{\mu}_{~{}\rho\sigma}=e_{A}^{~{}\mu}(\partial_{\rho}e^{A}_{~{}\sigma}+\omega^{A}_{~{}B\rho}e^{B}_{~{}\sigma})$
relates the affine connection $\hat{\Gamma}^{\mu}_{~{}\rho\sigma}$ and the
spin connection $\omega^{A}_{~{}B\mu}$. Furthermore, we have the Levi-Civita
connection from the metric,
$\Gamma^{\mu}_{~{}\rho\sigma}=(1/2)g^{\mu\nu}(\partial_{\rho}g_{\sigma\nu}+\partial_{\sigma}g_{\rho\nu}-\partial_{\nu}g_{\rho\sigma})$
and its associated covariant derivative operator $\nabla=\partial+\Gamma$. The
Levi-Civita tensor
$\varepsilon^{\mu\nu\rho\sigma}=(1/\sqrt{-g})\epsilon^{\mu\nu\rho\sigma}$ is
related with a totally antisymmetric symbol $\epsilon^{\mu\nu\rho\sigma}$
which satisfies $\epsilon^{0123}=1$. This will be used to define the dual
tensors.

GR equivalent TG theory is formulated in a flat spacetime with metricity, so
both the curvature two form and the nonmetricity tensor vanish, i.e.,
$\hat{R}^{A}_{~{}B\mu\nu}=\partial_{\mu}\omega^{A}_{~{}B\nu}-\partial_{\nu}\omega^{A}_{~{}B\mu}+\omega^{A}_{~{}C\mu}\omega^{C}_{~{}B\nu}-\omega^{A}_{~{}C\nu}\omega^{C}_{~{}B\mu}=0$
and $Q_{\rho}^{~{}\mu\nu}=\hat{\nabla}_{\rho}g^{\mu\nu}=0$. The metricity
requires that $\omega_{AB\mu}=-\omega_{BA\mu}$; it together with the vanishing
curvature dictate that the spin connection can be generally expressed as
follows:

$\omega_{~{}B\mu}^{A}=\left(\Lambda^{-1}\right)_{~{}C}^{A}\partial_{\mu}\Lambda_{~{}B}^{C}~{},$
(1)

where $\Lambda^{A}_{~{}B}$ represents the element of an arbitrary Lorentz
transformation matrix which is position dependent and satisfies the relation
$\eta_{AB}\Lambda^{A}_{~{}C}\Lambda^{B}_{~{}D}=\eta_{CD}$ at any spacetime
point. In TG theory, the gravity is attributed to the spacetime torsion
instead of the curvature. The torsion two form generally depends on both the
tetrad and the spin connection,

$\mathcal{T}^{A}_{~{}~{}\mu\nu}=2(\partial_{[\mu}e^{A}_{~{}\nu]}+\omega^{A}_{~{}B[\mu}e^{B}_{~{}\nu]})~{},$
(2)

where the subscript brackets represent the antisymmetrization, so that the
torsion tensor in spacetime is
$\mathcal{T}^{\rho}_{~{}~{}\mu\nu}=e_{A}^{~{}\rho}\mathcal{T}^{A}_{~{}~{}\mu\nu}=2\hat{\Gamma}^{\rho}_{[\mu\nu]}$.
With these building blocks, the action of TG theory is written as

$\displaystyle S_{g}=\frac{1}{2}\int d^{4}x~{}{\|e\|}\mathbb{T}\equiv\int
d^{4}x~{}{\|e\|}\left(-\frac{1}{2}\mathcal{T}_{\mu}\mathcal{T}^{\mu}+\frac{1}{8}\mathcal{T}_{\alpha\beta\mu}\mathcal{T}^{\alpha\beta\mu}+\frac{1}{4}\mathcal{T}_{\alpha\beta\mu}\mathcal{T}^{\beta\alpha\mu}\right)~{},$
(3)

where ${\|e\|}=\sqrt{-g}$ is the determinant of the tetrad, $\mathbb{T}$ is
the torsion scalar, and
$\mathcal{T}_{\mu}=\mathcal{T}^{\alpha}_{~{}~{}\mu\alpha}$ is the torsion
vector. This action is invariant under diffeomorphism and local Lorentz
transformation. Furthermore, it is identical to the Einstein-Hilbert action up
to a surface term,

$S_{g}=\int
d^{4}x\sqrt{-g}[-\frac{1}{2}R(e)-\nabla_{\mu}\mathcal{T}^{\mu}]~{},$ (4)

where the curvature scalar $R(e)$ is defined by the Levi-Civita connection and
considered as being fully constructed from the metric, and in turn from the
tetrad. The surface term does not affect the equation of motion and so TG
theory can considered as a pure tetrad theory. The spin connection only
contributes to the surface term, so it represents pure gauge in the TG action
(3), and in practice we may fix a spin connection [as long as it satisfies the
constraint of Eq. (1)] and this imposes no effect on the equation of motion.
The simplest choice is fixing to the Weitzenböck connection,
$\omega^{A}_{~{}B\nu}=0$, which was frequently adopted in the literature. With
the Weitzenböck connection, the torsion two form is simply expressed as

$\mathcal{T}^{A}_{~{}~{}\mu\nu}=\partial_{\mu}e^{A}_{~{}\nu}-\partial_{\nu}e^{A}_{~{}\mu}~{}.$
(5)

It deserves pointing out that in TG theory fixing a spin connection does not
break the local Lorentz symmetry. One can prove that, after taking the
Weitzenböck connection, the action (3) is unchanged up to a surface term under
the local Lorentz transformation,
$e^{A}_{~{}\mu}\rightarrow\Lambda^{A}_{~{}B}(x)e^{B}_{~{}\mu}$.

The Weitzenböck connection has been continually used in many generalized or
modified TG theories, such as the $f(\mathbb{T})$ models fT  where the torsion
scalar in the action (3) is replaced by its arbitrary function. In Ref.
PVtele1  we have also adopted the Weitzenböck connection in our PV gravity
model which modifies TG theory by adding a Nieh-Yan term in the action (3),

$S_{NY}=\frac{c}{4}\int
d^{4}x\sqrt{-g}~{}\theta\,\mathcal{T}_{A\mu\nu}\widetilde{\mathcal{T}}^{A\mu\nu}~{},$
(6)

where $c$ is the coupling constant and
$\widetilde{\mathcal{T}}^{A\mu\nu}=(1/2)\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}^{A}_{~{}~{}\rho\sigma}$
is the dual of the torsion two form. We also took into account the kinetic and
potential terms of the scalar field and the action of other matter $S_{m}$
which coupled minimally through the metric (or the tetrad). The full action in
its equivalent form is

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[-\frac{R(e)}{2}+\frac{c}{4}\,\theta\,\mathcal{T}_{A\mu\nu}\widetilde{\mathcal{T}}^{A\mu\nu}+\frac{1}{2}\nabla_{\mu}\theta\nabla^{\mu}\theta-V(\theta)\right]+S_{m}~{}.$
(7)

Taking the Weitzenböck connection, as we have done in Ref. PVtele1 , the
torsion two form $\mathcal{T}^{A}_{~{}~{}\mu\nu}$ is given by Eq. (5) and this
gravity model is still a pure tetrad one. The corresponding equations of
motion of this model and its applications to the spatially flat FRW background
universe and the cosmological perturbations around it are presented in detail
in Ref. PVtele1 . Here we only outline its main results: (i) the background
evolution is the same as in GR, (ii) GWs present the velocity birefringence
phenomenon, this is the direct effect of parity violation, (iii) the scalar
field $\theta(x)$ is surprisingly not counted as an independent dynamical
degree of freedom at the linear-perturbation level.

We should point out here that the PV term (6) under the Weitzenböck condition
breaks the local Lorentz symmetry explicitly. Similar to many $f(\mathbb{T})$
models, when using the Weitzenböck connection, the PV gravity model (7) is a
pure tetrad theory, which generally does not have local Lorentz symmetry. This
is because the change in the tetrad induced by a local Lorentz transformation
cannot be compensated for by the change of the spin connection which is absent
in the pure tetrad theory. The only exception is the TG theory, in which, as
mentioned above, the change of the tetrad just brings a surface term into the
action. Even though the explicit local Lorentz symmetry violation does not
contradict with current experiments, it will make the conclusion less
convincible because in practice the physical consequences obtained seem to
rely on some special choices of local frames. For example, when we apply the
model to cosmology, we should first assume some forms for the tetrad
components; however, these forms are not uniquely determined by the metric as
seen from the relation $g_{\mu\nu}=\eta_{AB}e^{A}_{~{}\mu}e^{B}_{~{}\nu}$. In
other words, various (in fact infinite) choices of the tetrad forms will lead
to the same metric. Before the calculation, we must make one specific choice
on the tetrad—usually we choose the tetrads which have simple forms to
simplify the subsequent calculations, this amounts to picking a special local
frame out. Due to the lack of local Lorentz symmetry, different local frames
which are related by local Lorentz transformations are not equivalent any
more. One may doubt that the results from the calculations will depend on the
specific choice of the local frame at the beginning, and cannot have general
meanings. So it is questionable that one can do a more general analysis based
on a particular choice of the tetrad form.

To avoid this problem, in this paper we give up the Weitzenböck condition in
our model. In fact we will not impose any other constraint than Eq. (1) on the
spin connection, this means we will restore the local Lorentz symmetry for our
model (7) by supplementing the eligible spin connections to the torsion so
that it has the full expression (2). With the local Lorentz symmetry we can
always choose the tetrad we prefer in calculations, because if at the
beginning the tetrad at hand is not in the favored form we can obtain the
tetrad what we want through a local Lorentz transformation. The price one must
pay is that the spin connection is not fixed. The tetrads combined with spin
connections are equivalent if they are connected by local Lorentz
transformations. Further in next sections, we will apply this new version of
the model to cosmology in which the background universe has the general FRW
spacetime, not just the spatially flat one. We want to see whether the
conclusions drawn in Ref. PVtele1  are robust and want to know what will
happen for the cases which haven’t been studied in Ref. PVtele1 .

Now the torsion two form depends on the tetrad as well as the spin connection,
so does the action (7), which is a scalar in the spacetime as well as in the
local internal space. Since the spin connection is constrained to be in the
form of Eq. (1), we will use the Lorentz matrix element $\Lambda^{A}_{~{}B}$
to replace the spin connection $\omega^{A}_{~{}B\mu}$ as the fundamental
variable in the action principle. So during the variation of the action,
$\Lambda^{A}_{~{}B}$ and the tetrad $e^{A}_{~{}\mu}$ would be varied
independently. Furthermore, one can see that only the first-order derivatives
of the fundamental variables appeared in the action, it is expected that this
model is free from the ghost instability caused by higher-order derivatives.

The model has two kinds of gauge symmetries—the diffeomorphism invariance and
the local Lorentz invariance—the latter transformation makes the following
change

$e^{A}_{~{}\mu}\rightarrow(L^{-1})^{A}_{~{}B}e^{B}_{~{}\mu}~{},~{}\Lambda^{A}_{~{}B}\rightarrow\Lambda^{A}_{~{}C}L^{C}_{~{}B}~{},$
(8)

where $L^{A}_{~{}B}(x)$ is also the element of Lorentz matrix. We would like
to use different notations to distinguish two kinds of Lorentz matrices,
$\Lambda^{A}_{~{}B}(x)$ is used to express the spin connection as in Eq. (1),
but $L^{A}_{~{}B}(x)$ represents the local transformation which makes a shift
from one local frame to another. It’s easy to prove that the metric
$g_{\mu\nu}$ and torsion tensor $\mathcal{T}_{~{}\mu\nu}^{\rho}$ are invariant
under the local Lorentz transformation (8), so is the action (7).

The equations of motion follow from the variation of the action (7) with
respect to $e^{A}_{~{}\mu}$ and $\Lambda^{A}_{~{}B}$ separately

$\displaystyle G^{\mu\nu}+N^{\mu\nu}$ $\displaystyle=$ $\displaystyle
T^{\mu\nu}+T^{\mu\nu}_{\theta}~{},$ (9) $\displaystyle N^{[\mu\nu]}$
$\displaystyle=$ $\displaystyle 0~{},$ (10)

where $G^{\mu\nu}$ is the Einstein tensor, $T^{\mu\nu}=-(2/\sqrt{-g})(\delta
S_{m}/\delta g_{\mu\nu})$ and
$T^{\mu\nu}_{\theta}=[V(\theta)-\nabla_{\alpha}\theta\nabla^{\alpha}\theta/2]g^{\mu\nu}+\nabla^{\mu}\theta\nabla^{\nu}\theta$
are the energy-momentum tensors for the matter and the scalar field $\theta$
respectively, and
$N^{\mu\nu}=c\,e_{A}^{~{}\,\nu}\partial_{\rho}\theta\,\widetilde{\mathcal{T}}^{A\mu\rho}$.
These equations with same forms had also been obtained in Ref. PVtele1 , but
there are two differences needed to be pointed out. First, in Ref. PVtele1
the torsion $\mathcal{T}^{A}_{~{}~{}\mu\nu}$ contained in the tensor
$N^{\mu\nu}$ was only determined by the tetrad, as expressed in Eq. (5), but
here it depends on both of them and has the full expression (2); Second, in
Ref. PVtele1  the second equation (10) was not obtained from the action
principle but deduced from the first equation (9) that requires all the
tensors appeared in it should be symmetric under the permutation of indices
$\mu$ and $\nu$, but here it is obtained from the variation of the action with
respect to $\Lambda^{A}_{~{}B}$, and it is certainly consistent with Eq. (9).

From another viewpoint, the consistency showed that the equation of motion
(10) from the variation of $\Lambda^{A}_{~{}B}$ is not independent of Eq. (9),
it is just the antisymmetric part of the latter. This subtlety can be
explained as follows. First, the local Lorentz transformation (8) can make
$\Lambda^{A}_{~{}B}$ go through all possible values, because all Lorentz
matrices form a group. In consequence, no matter what change
$\Lambda^{A}_{~{}B}$ undergoes, it can change back through a local Lorentz
transformation. That is to say, for a local Lorentz invariant quantity
$F=F(e,\Lambda)$, any change in $F$ caused only by a change in
$\Lambda^{A}_{~{}B}$ can be equivalent to a change in $F$ caused only by a
change in $e^{A}_{~{}\mu}$. This is pictured as

$F(e,\Lambda+\delta\Lambda)\overset{(\ref{LT})}{=}F(e+\delta
e,\Lambda)~{}~{}\text{for}~{}\forall\delta\Lambda~{}.$

But its inverse statement does not hold because the local Lorentz
transformation cannot make $e^{A}_{~{}\mu}$ to go through all possible values.
The action $S$ of our model is invariant under the local Lorentz
transformation (8), therefore any $\delta S$ caused by
$\delta\Lambda^{A}_{~{}B}$ can always be equal to that due to $\delta
e^{A}_{~{}\mu}$. It means that from the viewpoint of variational principle,
requiring $S$ to take the extremum under $\delta e^{A}_{~{}\mu}$ already
includes the case where $S$ takes the extremum under
$\delta\Lambda^{A}_{~{}B}$. Hence, the equations obtained for the variations
of $e^{A}_{~{}\mu}$ already contains the equations obtained for the variation
of $\Lambda^{A}_{~{}B}$.

Compared with GR, the model has 16+6 basic variables of gravity, while GR has
only ten. At the same time, the model has six more constraint equations and 6
local Lorentz gauge symmetries. Since both theories are diffeomorphism
invariant, it can be expected that our model will have as many physical
dynamical degrees of freedom as GR.

In addition, the Bianchi identity $\nabla_{\mu}G^{\mu\nu}=0$ and the covariant
conservation law, $\nabla_{\mu}T^{\mu\nu}=0$, demand that
$\nabla_{\mu}N^{\mu\nu}=\nabla_{\mu}T_{\theta}^{\mu\nu}$. This brings no
further constraint, in stead it is consistent with the Klein-Gordon equation

$\Box\theta+V_{\theta}-\frac{c}{4}\mathcal{T}_{A\mu\nu}\widetilde{\mathcal{T}}^{A\mu\nu}=0~{},$
(11)

which is obtained from the variation of the action (7) with respect to the
scalar field $\theta$. Here and in the following we will use $V_{\theta}$ to
denote the first derivative of the potential to the scalar field.

Finally, we briefly explain why we introduce the kinetic and potential terms
of the scalar field $\theta$ into the action. Without them, the dynamics and
the energy momentum tensor of $\theta$ will be absent, it can be seen from (9)
that the Bianchi identities and conservation of energy momentum tensor of
matter impose an additional constraint $\nabla_{\mu}N^{\mu\nu}=0$. This extra
constraint together with the constraint (10) put too strong restrictions on
the space of solutions, which makes some common solutions in GR nonexistent.
This is similar to the case of nondynamical CS gravity Jackiw:2003pm ;
Alexander:2009tp . Later in this paper we will see that if the dynamics of
$\theta$ are not turned on, there will be no real closed FRW solution.

##  III Cosmology background solution

Now we apply the new version of our model with full local Lorentz covariance
to cosmology. In this section, we only consider the background evolutions and
leave the discussions on the cosmological perturbations to next sections.

In Ref. PVtele1  we only considered the spatially flat FRW background, and
assumed that the tetrad was diagonal in form. Here we will discuss the more
general FRW background, the metric has the following well-known form in
spherical coordinate system

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=a(\eta)\left(d\eta^{2}-\frac{dr^{2}}{1-Kr^{2}}-r^{2}d\Omega^{2}\right)~{},$
(12)

where $a(\eta)$ is the scale factor of the universe, $\eta$ is the conformal
time and $d\Omega^{2}=d\varphi^{2}+(\sin\varphi)^{2}d\phi^{2}$ is the metric
on the unit sphere. We have used $\varphi$ and $\phi$ to represent the polar
angle and azimuth angle respectively. The subspace of $\eta={\rm constant}$ is
homogeneous and isotropic, so there are six Killing vector fields $\hat{\xi}$
in total. When applying our model to this spacetime and for the convenience of
looking for solutions, we consider the affine connection which is homogeneous
and isotropic,

$\mathcal{L}_{\hat{\xi}}\hat{\Gamma}^{\rho}_{~{}\mu\nu}=0~{},$ (13)

where $\mathcal{L}_{\hat{\xi}}$ is the Lie derivative along the Killing vector
field $\hat{\xi}$. This condition has been used in TG theory telesymmetry1 ;
telesymmetry2 . One may find solutions without the above requirement on the
connection, we leave this for future work. As mentioned before, the affine
connection relates to the spin connection via the equation
$\hat{\Gamma}^{\rho}_{~{}\mu\nu}=e_{A}^{~{}\,\rho}\partial_{\mu}e^{A}_{~{}\nu}+e_{A}^{~{}\,\rho}e^{B}_{~{}\nu}\omega^{A}_{~{}B\mu}$.
Although $\hat{\Gamma}^{\rho}_{~{}\mu\nu}$ is coordinate dependent, the Lie
derivative of $\hat{\Gamma}^{\rho}_{~{}\mu\nu}$ does not depend on the
coordinate. Hence the condition (13) is unambiguous. There is a benefit to
requesting this condition (13), for any tensor $T$ constructed from the
metric, torsion and their covariant derivatives associated with the Levi-
Civita connection or the affine connection, it always has the properties of
homogeneity and isotropy, i.e., $\mathcal{L}_{\hat{\xi}}T=0$.

The general solutions to Eqs. (12) and (13) have been discussed in Ref.
telesymmetry1 ; telesymmetry2  within the framework of TG. Below we briefly
introduce the solution and its application to our model. In this paper, we
only consider the case where both $e^{A}_{~{}\mu}$ and $\Lambda^{A}_{~{}B}$
are real numbers.

###  III.1 Spatially flat FRW universe

In the case of spatially flat FRW universe, $K=0$, one of the solutions can be
parametrized as

${e}^{A}_{~{}\mu}=a(\eta)\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\sin\varphi\cos\phi&r\cos\varphi\cos\phi&-r\sin\varphi\sin\phi\\\
0&\sin\varphi\sin\phi&r\cos\varphi\sin\phi&r\sin\varphi\cos\phi\\\
0&\cos\varphi&-r\sin\varphi&0\end{array}\right)~{},~{}~{}~{}\Lambda=\mathring{\Lambda}~{}.$
(14)

Where $\mathring{\Lambda}$ is a global Lorentz matrix, which does not depend
on spacetime. But no matter which global Lorentz matrix is taken, it will not
affect any result discussed below. So the specific expression of
$\mathring{\Lambda}$ does not matter and it is not necessary to write its
concrete form out explicitly.

All other solutions under the same constraints on the metric and affine
connection differ from the solution (14) only by local Lorentz transformations
(8). The local Lorentz symmetry is a gauge symmetry in our model. So
physically, solution (14) is the only solution when $K=0$. It can be verified
that solution (14) satisfies the constraint equation (10). Putting Eq. (14)
into Eqs. (9) and (11), we can get

$\displaystyle 3\mathcal{H}^{2}=a^{2}\left(\rho_{\theta}+\rho\right)~{},$ (15)
$\displaystyle
2\mathcal{H}^{\prime}+\mathcal{H}^{2}=-a^{2}\left(p_{\theta}+p\right)~{},$
(16)
$\displaystyle\theta^{\prime\prime}+2\mathcal{H}\theta^{\prime}+a^{2}V_{\theta}=0~{},$
(17)

where prime represents the derivative with respect to the conformal time,
$\mathcal{H}=a^{\prime}/a$ is the conformal Hubble rate,
$\rho_{\theta}=\theta^{\prime 2}/\left(2a^{2}\right)+V$ and
$p_{\theta}=\theta^{\prime 2}/\left(2a^{2}\right)-V$ are the energy density
and pressure of the $\theta$ field, and $\rho$ and $p$ denote the energy
density and pressure of other matter. All of them only depend on time. The
background equations are exactly the same as in GR. This means the PV term has
no effect on the flat FRW background. This result is the same as that of Ref.
PVtele1 .

###  III.2 Closed FRW universe

In the case of closed FRW universe, $K>0$, the solution is parametrized as

$e^{A}_{~{}\mu}=a(\eta)\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\sin\varphi\cos\phi/\mathcal{R}&r\mathcal{R}_{1}&r\sin\varphi\mathcal{R}_{2}\\\
0&\sin\varphi\sin\phi/\mathcal{R}&r\mathcal{R}_{3}&r\sin\varphi\mathcal{R}_{4}\\\
0&\cos\varphi/\mathcal{R}&-r\mathcal{R}\sin\varphi&-\mathcal{K}r^{2}(\sin\varphi)^{2}\end{array}\right),~{}\Lambda=\mathring{\Lambda}~{}.$
(18)

Here $\mathcal{R}=\sqrt{1-Kr^{2}}$,
$\mathcal{R}_{1}=\mathcal{R}\cos\varphi\cos\phi+\mathcal{K}r\sin\phi$,
$\mathcal{R}_{2}=\mathcal{K}r\cos\varphi\cos\phi-\mathcal{R}\sin\phi$,
$\mathcal{R}_{3}=\mathcal{R}\cos\varphi\sin\phi-\mathcal{K}r\cos\phi$,
$\mathcal{R}_{4}=\mathcal{R}\cos\phi+\mathcal{K}r\cos\varphi\sin\phi$, and the
real parameter $\mathcal{K}$ satisfies $\mathcal{K}^{2}=K$. One can see that
when $K\rightarrow 0$, the Eq. (18) returns to (14). All other solutions
differ from Eq. (18) only by local Lorentz transformations (8). So physically
Eq. (18) is the only solution when $K>0$. It can be verified that this
solution (18) satisfies the constraint equation (10). Substitute this solution
into the Eqs. (9) and (11), one obtains the background equations,

$\displaystyle
3(\mathcal{H}^{2}+\mathcal{K}^{2})=a^{2}\left(\rho_{\theta}+\rho\right)~{},$
(19) $\displaystyle
2\mathcal{H}^{\prime}+\mathcal{H}^{2}+\mathcal{K}^{2}-2c\mathcal{K}\theta^{\prime}=-a^{2}\left(p_{\theta}+p\right)~{},$
(20)
$\displaystyle\theta^{\prime\prime}+2\mathcal{H}\theta^{\prime}+a^{2}V_{\theta}-6c\mathcal{K}\mathcal{H}=0~{}.$
(21)

These modified corresponding equations in GR and showed the effects of the PV
term on the closed FRW background.

Now, we can see that if the scalar $\theta$ is not a dynamical field, its
kinetic and potential terms are absent in the action at the beginning, an
additional constraint $\nabla_{\mu}N^{\mu\nu}=0$ emerged, but this is violated
by the solution (18). So if $\theta(x)$ is not a dynamical field, we are not
able to have a closed FRW solution under Eq. (13) in our model.

###  III.3 Open FRW universe

In the case of open FRW universe, $K<0$, the solution is parametrized as

${e}^{A}_{~{}\mu}=a(\eta)\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1/\mathcal{R}&0&0\\\ 0&0&r&0\\\
0&0&0&r\sin\varphi\end{array}\right)~{},~{}~{}~{}\Lambda=\mathring{\Lambda}\cdot\left(\begin{array}[]{cccc}\mathcal{R}&\mathcal{K}r&0&0\\\
\mathcal{K}r\sin\varphi\cos\phi&\mathcal{R}\sin\varphi\cos\phi&\cos\varphi\cos\phi&-\sin\phi\\\
\mathcal{K}r\sin\varphi\sin\phi&\mathcal{R}\sin\varphi\sin\phi&\cos\varphi\sin\phi&\cos\phi\\\
\mathcal{K}r\cos\varphi&\mathcal{R}\cos\varphi&-\sin\varphi&0\end{array}\right)$
(22)

Here $\mathcal{R}=\sqrt{1-Kr^{2}}$ and the real parameter $\mathcal{K}$
satisfies $\mathcal{K}^{2}=-K$. It can be proved that when $K\rightarrow 0$,
the solution (22) also returns to (14) through a local Lorentz transformation
(8). All the other solutions differ from solution (22) merely by local Lorentz
transformations (8). So physically Eq. (22) is the only solution when $K<0$.
It can be verified that this solution (22) satisfies the constraint Eq. (10).
Substitute it into the Eqs. (9) and (11), one obtains the background
equations,

$\displaystyle
3(\mathcal{H}^{2}-\mathcal{K}^{2})=a^{2}\left(\rho_{\theta}+\rho\right)~{},$
(23) $\displaystyle
2\mathcal{H}^{\prime}+\mathcal{H}^{2}-\mathcal{K}^{2}=-a^{2}\left(p_{\theta}+p\right)~{},$
(24)
$\displaystyle\theta^{\prime\prime}+2\mathcal{H}\theta^{\prime}+a^{2}V_{\theta}=0~{}.$
(25)

These equations are the same as those in GR, so the PV term has no effect on
the open FRW background.

By the way, we can indeed do a local Lorentz transformation on Eq. (22) to
make $\Lambda=\mathring{\Lambda}$, just like the cases of $K=0$ and $K>0$. But
this will make $e^{0}_{~{}i}\neq 0~{},~{}e^{a}_{~{}0}\neq 0$ and make it
inconvenient to perform decomposition of the tetrad perturbation which will be
discussed in the next section. So we use Eq. (22) instead of the equivalent
solution satisfying $\Lambda=\mathring{\Lambda}$.

The above equations of the background evolutions are consistent with the
results of Ref. Hohmann:2020dgy , where the tetrad field was parametrized
separately as the “vector branch” and “axial branch” {see the Eqs. (29) and
(30) in Hohmann:2020dgy }, and these two branches match in the case of a
spatially-flat universe ($K=0$). If we keep all the components of the tetrad
field and all the terms in the equations real, as we required in this paper,
one can only consider the vector branch for the open universe ($K<0$) and the
axial branch for the closed universe ($K>0$). It was shown in Ref.
Hohmann:2020dgy  (Sec. IV. B) that the PV modifications bring changes only for
the axial branch, so they can only have effects on the background evolutions
of the closed universe.

###  III.4 Additional comments

This subsection is mainly prepared for the following section and for declaring
of some conventions. The FRW metric can always be expressed as

$ds^{2}=a^{2}(d\eta^{2}-\gamma_{ij}dx^{i}dx^{j})~{},$ (26)

where $\gamma_{ij}$ can be regarded as the metric of three-dimensional
hypersurface, and its inverse is denoted as $\gamma^{ij}$. The corresponding
spatial covariant derivative will be denoted by $D_{i}$. From (14), (18) and
(22), we can see that ${e}^{A}_{~{}\mu}$ can always be expressed as

${e}^{0}_{~{}0}=a~{},~{}{e}^{0}_{~{}i}=0~{},~{}{e}^{a}_{~{}0}=0~{},~{}{e}^{a}_{~{}i}=a\gamma^{a}_{~{}i}~{},$
(27)

where $\gamma^{a}_{~{}i}$ can be regarded as the spatial tetrad on the three-
dimensional hypersurface (sometimes it is called dreibein). Its dual tetrad is
denoted as $\gamma_{a}^{~{}i}$. From
$g_{\mu\nu}=\eta_{AB}e_{~{}\mu}^{A}e_{~{}\nu}^{B}$, we have
$\gamma_{ij}=\delta_{ab}\gamma^{a}_{~{}i}\gamma^{b}_{~{}j}$ and therefore
$\gamma^{ij}=\delta^{ab}\gamma_{a}^{~{}i}\gamma_{b}^{~{}j}$. We also denote
$\varepsilon_{ijk}$ as spatial volume element compatible with the spatial
metric $\gamma_{ij}$.

In the rest of this paper, the spatial coordinate indices “i, j, …” of
variables will be raised and lowered by $\gamma^{ij}$ and $\gamma_{ij}$ and
the spatial internal indices “a, b, …” will be raised and lowered by
$\delta^{ab}$ and $\delta_{ab}$. For example,
$\gamma^{ai}=\gamma^{a}_{~{}j}\gamma^{ji}=\delta^{ab}\gamma_{b}^{~{}i}$. And
all the transformations between these two kinds of indices will be performed
by $\gamma^{a}_{~{}i}$ and $\gamma_{a}^{~{}i}$. For example,
$\omega^{i}_{~{}jk}=\gamma_{a}^{~{}i}\gamma^{b}_{~{}j}\omega^{a}_{~{}bk}$.

Here, we would like to stress again the advantage of the model with full local
Lorentz covariance. As mentioned before, given a metric the form for the
tetrad is not uniquely determined. For FRW universe, we have infinite choices
on the tetrad forms, all of them lead to the same metric (12). We just
mentioned that we can always take the simple diagonal form of Eq. (27) for the
tetrad. This is because, if it is not so at the beginning, we can always get
the form of Eq. (27) through a local Lorentz transformation which is now the
gauge symmetry of the model. Due to this advantage, we can also safely take
simple ansatz for the perturbed tetrad components, this is what we will do in
next sections.

##  IV perturbations and gauge transformations

With the FRW background solutions obtained in previous section, we are ready
to study the linear cosmological perturbations around the FRW background. In
this section we will focus on the scalar-vector-tensor (SVT) decomposition and
the gauge transformations of the cosmological perturbations.

###  IV.1 Perturbations and SVT decomposition

The tetrad perturbations in cosmology have been studied extensively in the
literature, such as tetradper1 ; tetradper2 . We will also consider the
cosmological perturbations of the spin connection for a consistent and
complete study. Here we briefly introduce how to do the SVT decomposition of
perturbations to the tetrad and Lorentz matrices, taking the general FRW
universe as the background.

A spatial covariant vector $\mathcal{V}_{i}$ can always be decomposed into

$\mathcal{V}_{i}=D_{i}\mathcal{V}+\mathcal{V}^{V}_{i}~{}.$ (28)

The superscript $V$ represents that it satisfies the transverse condition
$D^{i}\mathcal{V}^{V}_{i}=0$. Because a spatial covariant antisymmetric tensor
field $\mathcal{A}_{ij}$ can always be Hodge-dual to a covariant vector
$\mathcal{V}_{i}=(1/2)\varepsilon_{ijk}\mathcal{A}^{jk}$, a spatial covariant
antisymmetric tensor $\mathcal{A}_{ij}$ can be decomposed into

$\mathcal{A}_{ij}=\varepsilon_{ij}^{~{}~{}k}(D_{k}\mathcal{V}+\mathcal{V}^{V}_{k})~{}.$
(29)

A symmetric spatial tensor $S_{ij}$ can always be decomposed as

$S_{ij}=\hat{S}\gamma_{ij}+D_{i}D_{j}S+D_{(i}S^{V}_{j)}+S^{T}_{ij}~{},$ (30)

where the subscript parentheses denotes the symmetrization, and the
superscript $T$ means traceless and transverse. Finally, we know that any
second-rank tensor $T_{ij}$ can always be expressed as the sum of a symmetric
tensor $S_{ij}$ and an antisymmetric tensor $\mathcal{A}_{ij}$, so it can be
decomposed as

$T_{ij}=S_{ij}+\mathcal{A}_{ij}=\hat{S}\gamma_{ij}+D_{i}D_{j}S+D_{i}S^{V}_{j}+\varepsilon_{ij}^{~{}\,k}(D_{i}\mathcal{V}+\mathcal{V}^{V}_{i})+S^{T}_{ij}~{}.$
(31)

Note that in the last expression, we have absorbed the antisymmetric part of
$D_{i}S^{V}_{j}$ into $\mathcal{V}$ and $\mathcal{V}^{V}_{i}$ in terms of the
method of Hodge duality:
$D_{[i}S^{V}_{j]}=\varepsilon_{ij}^{~{}~{}k}(D_{k}\mathcal{U}+\mathcal{U}^{V}_{k})$.

####  IV.1.1 Tetrad perturbations

Note that $e^{0}_{~{}0}$ can be regarded as spatial scalar field,
$e^{0}_{~{}i}$ and $\gamma_{ij}\gamma_{a}^{~{}j}e^{a}_{~{}0}$ can be regarded
as spatial covariant vector fields, and
$\gamma_{jk}\gamma_{a}^{~{}k}e^{a}_{~{}i}$ can be regarded as spatial
covariant second-rank tensor field. Hence, all the tetrad components can be
written as

$\displaystyle
e^{0}_{~{}0}=a(1+A)~{},~{}e^{0}_{~{}i}=a(D_{i}\beta+\beta^{V}_{~{}i})~{},~{}e^{a}_{~{}0}=a\gamma^{ai}(D_{i}\chi+\chi^{v}_{i})~{},$
$\displaystyle
e^{a}_{~{}i}=a\gamma^{aj}\left[(1-\psi)\gamma_{ij}+D_{i}D_{j}\alpha+D_{i}\alpha^{V}_{j}-\varepsilon_{ij}^{~{}\,k}(D_{k}\lambda+\lambda^{V}_{k})+\frac{1}{2}h^{T}_{ij}\right]~{},$
(32)

so that the perturbed metric components have the following standard forms

$\displaystyle g_{00}=a^{2}(1+2A)~{},~{}g_{0i}=-a^{2}(D_{i}B+B^{V}_{i})~{},$
$\displaystyle
g_{ij}=-a^{2}\left[(1-2\psi)\gamma_{ij}+2D_{i}D_{j}\alpha+D_{i}\alpha_{j}^{V}+D_{j}\alpha_{i}^{V}+h^{T}_{ij}\right]~{},$
(33)

where $B=\chi-\beta$ and $B^{V}_{i}=\chi^{V}_{i}-\beta^{V}_{i}$. Besides the
familiar scalar perturbations ($A,\chi-\beta,\psi,\alpha$), vector
perturbations ($\chi_{i}^{V}-\beta_{i}^{V},\alpha^{V}_{i}$), and tensor
perturbation ($h^{T}_{ij}$) in the metric, the parametrization of tetrad
brings six extra variables which are scalar perturbation $\lambda,\chi+\beta$
and vector perturbation $\lambda_{i}^{V},\chi_{i}^{V}+\beta_{i}^{V}$.

Again, we have assumed the forms in Eq. (IV.1.1) for the tetrad components in
the perturbed universe. This ansatz leads to the metric (IV.1.1). But the
tetrad can have other forms leading to the same metric (IV.1.1). We do not
need to worry about the special status of the tetrad we have chosen in Eq.
(IV.1.1). Because the model has the local Lorentz symmetry, if at the
beginning we have a tetrad different from that in Eq. (IV.1.1), we can always
shift it to the one in Eq. (IV.1.1) by a local Lorentz transformation.

####  IV.1.2 Perturbations to Lorentz matrix elements

As mentioned before, in the covariant version of our model we can always take
simple forms for the tetrad, but there is a price that the spin connection or
the $\Lambda$-matrix element cannot be fixed. In the cosmological perturbation
theory one should also consider the perturbation to the Lorentz matrix
elements. In the following we denote the unperturbed Lorentz matrix element as
$\bar{\Lambda}^{A}_{~{}B}$, and the perturbed one as $\Lambda^{A}_{~{}B}$.
Since the Lorentz matrices form a group,
$(\bar{\Lambda}^{-1})^{A}_{~{}C}\Lambda^{C}_{~{}B}$ is also a Lorentz matrix
element. Thus $(\bar{\Lambda}^{-1})^{A}_{~{}C}\Lambda^{C}_{~{}B}$ can be
exponentialized as $\exp(\epsilon)^{A}_{~{}B}$, where
$\epsilon_{AB}=\eta_{AC}\epsilon^{C}_{~{}B}$ is an antisymmetric matrix and
has six independent components. At the linear perturbation level, the
difference between $\bar{\Lambda}^{A}_{~{}B}$ and $\Lambda^{A}_{~{}B}$ is very
small, so $\epsilon_{AB}$ is a small quantity. Thus the perturbed Lorentz
matrix can be parametrized as

$\Lambda=\bar{\Lambda}\exp(\epsilon)~{}.$ (34)

Note that $\gamma^{a}_{~{}i}\epsilon^{0}_{~{}a}$ can be regarded as a spatial
covariant vector, and $\gamma_{ai}\gamma^{b}_{~{}j}\epsilon^{a}_{~{}b}$ can be
regarded as an antisymmetric spatial covariant tensor, $\epsilon$ can be
decomposed as

$\epsilon^{0}_{~{}a}=\gamma_{a}^{~{}i}(D_{i}\kappa+\kappa^{V}_{i})~{},~{}\epsilon^{a}_{~{}b}=\gamma^{ai}\gamma_{b}^{~{}j}\varepsilon_{ij}^{~{}\,k}(D_{k}\tau+\tau^{V}_{k})~{}.$
(35)

According to this method, the total six independent perturbative degrees of
freedom of the local Lorentz matrices can be decomposed into two scalar
components $\kappa,\tau$ and four vector components
$\kappa^{V}_{i},\tau^{V}_{i}$. There is no tensor component appeared in this
decomposition.

###  IV.2 Gauge transformations

In this subsection, we discuss how the above perturbative variables change
under the infinitesimal gauge transformations. Our model has two kinds of
gauge symmetries: Diffeomorphism invariance and local Lorentz symmetry.

####  IV.2.1 Infinitesimal diffeomorphism

The diffeomorphism induced by the infinitesimal vector field $\xi^{\mu}$ leads
following changes for the tetrad and local Lorentz matrix $\delta
e^{A}_{~{}\mu}=-\mathcal{L}_{\xi}e^{A}_{~{}\mu}$ and
$\delta\Lambda^{A}_{~{}B}=-\mathcal{L}_{\xi}\Lambda^{A}_{~{}B}$. The vector
field $\xi^{\mu}$ can be further decomposed into two scalar degrees of freedom
$\xi^{0},\xi$ and two vector degrees of freedom $\xi^{V}_{i}$, where
$\xi^{\mu}=(\xi^{0},\xi^{i}$),
$\xi_{i}=\gamma_{ij}\xi^{j}=D_{i}\xi+\xi^{V}_{i}$. For convenience, we also
define
$\tilde{\xi}_{i}=(1/2)\varepsilon_{ik}^{~{}~{}l}\gamma_{a}^{~{}k}\gamma^{b}_{~{}l}\omega^{a}_{~{}bj}\xi^{j}$,
and decompose it into $\tilde{\xi}_{i}=D_{i}\tilde{\xi}+\tilde{\xi}^{V}_{i}$.
Through calculations, it is not difficult to obtain the transformation laws of
the perturbations under the infinitesimal diffeomorphism

$\displaystyle A\rightarrow
A-{\xi^{0}}^{\prime}-\mathcal{H}\xi^{0}~{},~{}\psi\rightarrow\psi+\mathcal{H}\xi^{0}~{},~{}\beta\rightarrow\beta-\xi^{0}~{},~{}\chi\rightarrow\chi-\xi^{\prime}~{},$
$\displaystyle\alpha\rightarrow\alpha-\xi~{},~{}\beta^{V}_{i}\rightarrow\beta^{V}_{i}~{},~{}\chi^{V}_{i}\rightarrow\chi^{V}_{i}-{\xi^{V}_{i}}^{\prime}~{},~{}\alpha^{V}_{i}\rightarrow\alpha^{V}_{i}-\xi^{V}_{i}~{},~{}h^{T}_{ij}\rightarrow
h^{T}_{ij}~{},$
$\displaystyle\kappa\rightarrow{\left\\{\begin{array}[]{ll}\kappa&(K\geq 0)\\\
\kappa-\mathcal{K}\xi&(K<0)\end{array}\right.}~{},~{}\kappa^{V}_{i}\rightarrow{\left\\{\begin{array}[]{ll}\kappa^{V}_{i}&(K\geq
0)\\\
\kappa^{V}_{i}-\mathcal{K}\xi^{V}_{i}&(K<0)\end{array}\right.}~{},~{}\tau\rightarrow{\left\\{\begin{array}[]{ll}\tau&(K\geq
0)\\\ \tau-\tilde{\xi}&(K<0)\end{array}\right.}~{},$ (42)
$\displaystyle\tau^{V}_{i}\rightarrow{\left\\{\begin{array}[]{ll}\tau^{V}_{i}&(K\geq
0)\\\
\tau^{V}_{i}-\tilde{\xi}^{V}_{i}&(K<0)\end{array}\right.}~{},~{}\lambda\rightarrow{\left\\{\begin{array}[]{ll}\lambda-\mathcal{K}\xi&(K\geq
0)\\\
\lambda+\tilde{\xi}&(K<0)\end{array}\right.}~{},\lambda^{V}_{i}\rightarrow{\left\\{\begin{array}[]{ll}\lambda^{V}_{i}-\mathcal{K}\xi^{V}_{i}&(K\geq
0)\\\ \lambda^{V}_{i}+\tilde{\xi}^{V}_{i}&(K<0)\end{array}\right.}~{}.$ (49)

By the way, the scalar field $\theta$ can be decomposed into
$\theta(\eta,\vec{x})=\theta(\eta)+\delta\theta(\eta,\vec{x})$, its
fluctuation is not diffeomorphism invariant, i.e.,
$\delta\theta\rightarrow\delta\theta-\theta^{\prime}\xi^{0}$.

####  IV.2.2 Infinitesimal Local Lorentz Transformation

The infinitesimal Lorentz transformation (8) can be expressed as
$L^{A}_{~{}B}=\delta^{A}_{~{}B}+\delta L^{A}_{~{}B}$, and it is easy to see
that $\delta L_{AB}=\eta_{AC}\delta L^{C}_{~{}B}=-\delta L_{BA}$ is
antisymmetric. Similar to the previous analyses, $\delta L_{AB}$ can be
decomposed as

$\delta L^{0}_{~{}a}=\gamma_{a}^{~{}i}(D_{i}\ell+\ell^{V}_{i})~{},~{}\delta
L^{a}_{~{}b}=\gamma^{ai}\gamma_{b}^{~{}j}\varepsilon_{ij}^{~{}\,k}(D_{i}l+l^{V}_{i})~{}.$
(50)

The transformation laws of the perturbations under the infinitesimal local
Lorentz transformation can be obtained in terms of Eq. (8):

$\displaystyle A\rightarrow
A~{},~{}\beta\rightarrow\beta-\ell~{},~{}\chi\rightarrow\chi-\ell~{},~{}\lambda\rightarrow\lambda-l~{},~{}\psi\rightarrow\psi~{},~{}\alpha\rightarrow\alpha~{},~{}\kappa\rightarrow\kappa+\ell~{},~{}\tau\rightarrow\tau+l~{},~{}h^{T}_{ij}\rightarrow
h^{T}_{ij}~{},$
$\displaystyle\beta^{V}_{i}\rightarrow\beta^{V}_{i}-\ell^{V}_{i}~{},~{}\chi^{V}_{i}\rightarrow\chi^{V}_{i}-\ell^{V}_{i}~{},~{}\lambda^{V}_{i}\rightarrow\lambda^{V}_{i}-l^{V}_{i}~{},~{}\alpha^{V}_{i}\rightarrow\alpha^{V}_{i}~{},~{}\kappa^{V}_{i}\rightarrow\kappa^{V}_{i}+\ell^{V}_{i}~{},~{}\tau^{V}_{i}\rightarrow\tau^{V}_{i}+l^{V}_{i}~{}.$
(51)

One can see that all perturbations which enter the metric, such as
$A,\psi,B=\chi-\beta,\alpha,B^{V}_{i}=\chi^{V}_{i}-\beta^{V}_{i},\alpha^{V}_{i},h^{T}_{ij}$,
are local Lorentz invariant.

####  IV.2.3 A special gauge

In this subsection, we consider a specific gauge which will be fixed in the
next section to simplify the calculations. Firstly, from (IV.2.1), we can
always choose $\xi$ and $\xi^{V}_{i}$ to make $\alpha=0,\alpha^{V}_{i}=0$.
Secondly, from (IV.2.2), we choose $\ell$, $l$, $\ell^{V}_{i}$ and $l^{V}_{i}$
to make $\kappa=0,\tau=0,\kappa^{V}_{i}=0,\tau^{V}_{i}=0$. In this process,
the conditions $\alpha=0,\alpha^{V}_{i}=0$ do not change. Finally we adjust
$\xi^{0}$ to make $\delta\theta=0$. In this process, the conditions
$\alpha=0,~{}\alpha^{V}_{i}=0,~{}\kappa=0,~{}\tau=0,~{}\kappa^{V}_{i}=0,~{}\tau^{V}_{i}=0$
do not change. Therefore, we can always go to the gauge:
$\delta\theta=0,~{}\alpha=0,~{}\alpha^{V}_{i}=0,~{}\kappa=0,~{}\tau=0,~{}\kappa^{V}_{i}=0,~{}\tau^{V}_{i}=0$.
In next section, we will do calculations under this gauge.

##  V quadratic actions for scalar and tensor perturbations

When applying our PV gravity model to the early universe, such as the
inflationary epoch, we should attach much importance to the quantum origins of
primordial perturbations. For this purpose, one need to quanitze the
cosmological perturbations, and during which the quadratic actions of
perturbations are indispensable. In this section, we will discuss the
quadratic actions for perturbations in our model. For convenience, we ignore
other mater, so that $S_{m}=0$.

The direct calculations show that in the quadratic action, scalar
perturbations, vector perturbations and tensor perturbations are decoupled.
The vector perturbations are not dynamical degrees of freedom, this is the
same with the case in GR. Therefore we will neglect vector perturbations in
the rest of this paper and only discuss the quadratic actions of scalar and
tensor perturbations.

In the following calculations, we will take the gauge
$\delta\theta=0,~{}\alpha=0,~{}\kappa=0,~{}\tau=0$. We also introduce the
gauge invariant variable
$\zeta=-(\psi+\mathcal{H}\delta\theta/\theta^{\prime})$ which is the curvature
perturbation of the hypersurfaces with constant $\theta$ field. Then, we can
choose $A$, $\zeta$, $B$, $\beta$, $\lambda$ and $h^{T}_{ij}$ as independent
variables. Under this gauge condition, the dynamics of the scalar field at the
linear level are totally shifted to the gauge invariant quantity $\zeta$.

###  V.1 Quadratic Action For Scalar Perturbations

####  V.1.1 Flat FRW

In the flat FRW case, the quadratic action for scalar perturbations is

$S^{(2)}=-\int
d^{4}x~{}a^{2}\gamma~{}\Big{\\{}3{\zeta^{\prime}}^{2}-6\mathcal{H}\zeta^{\prime}A+(2A+\zeta)D^{2}\zeta+a^{2}VA^{2}+2(\mathcal{H}A-\zeta^{\prime})D^{2}B+2c\theta^{\prime}\zeta
D^{2}\lambda\Big{\\}}~{},$ (52)

where $D^{2}=\gamma^{ij}D_{i}D_{j}$ and
$\gamma=\det(\gamma^{a}_{~{}i})=\sqrt{\det(\gamma_{ij})}$. One can see that
none of $\lambda$, $B$ and $A$ is dynamical, they induce the following
constraints:

$\displaystyle\zeta=0~{},$ (53)
$\displaystyle\zeta^{\prime}-\mathcal{H}A=0~{},$ (54)
$\displaystyle-3\mathcal{H}\zeta^{\prime}+D^{2}\zeta+a^{2}VA+\mathcal{H}D^{2}B=0~{}.$
(55)

Compared with GR, there is one more constraint (53) here. Substituting these
constraints back into the action (52), we get $S^{(2)}=0$. Hence there is no
scalar dynamical degree of freedom at the linear perturbation level, this is
the same with the result of Ref. PVtele1 . This result does not depend on the
gauge, because $\zeta$ is gauge invariant.

####  V.1.2 Closed FRW

In the closed FRW background, the quadratic action for scalar perturbations is

$\displaystyle S^{(2)}=$ $\displaystyle-$ $\displaystyle\int
d^{4}x~{}a^{2}\gamma~{}\Big{\\{}3{\zeta^{\prime}}^{2}-6\mathcal{H}\zeta^{\prime}A+(2A+\zeta)D^{2}\zeta+(a^{2}V-3\mathcal{K}^{2})A^{2}+3(\mathcal{K}^{2}-3c\mathcal{K}\theta^{\prime})\zeta^{2}$
(56) $\displaystyle+$ $\displaystyle
6\mathcal{K}^{2}A\zeta+2(\mathcal{H}A-\zeta^{\prime})D^{2}B-\mathcal{K}^{2}BD^{2}B+c\theta^{\prime}(2\zeta+\mathcal{K}\lambda)D^{2}\lambda-c\mathcal{K}\theta^{\prime}\beta
D^{2}\beta\Big{\\}}~{}.$

One can see that $\beta$, $\lambda$, $A$ and $B$ are all non-dynamical fields.
All these non-dynamical fields induce the following constraints:

$\displaystyle\beta=0~{},$ (57) $\displaystyle\zeta+\mathcal{K}\lambda=0~{},$
(58) $\displaystyle\zeta^{\prime}-\mathcal{H}A+\mathcal{K}^{2}B=0~{},$ (59)
$\displaystyle-3\mathcal{H}\zeta^{\prime}+D^{2}\zeta+3\mathcal{K}^{2}\zeta+(a^{2}V-3\mathcal{K}^{2})A+\mathcal{H}D^{2}B=0~{}.~{}$
(60)

Compared with GR, there are two more constraints (57) and (58) here. These two
constraints are used to solve $\beta$ and $\lambda$ respectively. And the
constraints (59) and (60) can be used to solve $A$ and $B$. So it can be
expected that there will be one scalar dynamical degree of freedom $\zeta$ in
the end. But it is worth noting that when $K=0$, that is, the constraint (58)
will become $\zeta=0$. This returns to the case of $K=0$ in the previous
subsection.

In closed FRW universe, the spatial line element can be expressed as
$d\hat{s}^{2}=\gamma_{ij}dx^{i}dx^{j}=(1/K)[d\chi^{2}+\sin^{2}\chi(d\varphi^{2}+\sin^{2}\varphi\,d\phi^{2})]$.
Then, the hyperspherical harmonics $\mathcal{S}_{nlm}(\vec{x})$ (for more
details, see spherescalar1 ; spherescalar2 ) can be expressed as

$\mathcal{S}_{nlm}(\vec{x})=K^{\frac{3}{4}}\sin^{l}\chi\,\frac{d^{l+1}(\cos
n\chi)}{d(\cos\chi)^{l+1}}\,Y_{lm}(\varphi,\phi)~{},$ (61)

where $Y_{lm}(\varphi,\phi)$ is the spherical harmonic, $n$ is a positive
integer, $l$ is an integer satisfying $0\leq l\leq(n-1)$ and $m$ is an integer
satisfying $|m|\leq l$. These hyperspherical harmonics
$\mathcal{S}_{nlm}(\vec{x})$ constitute a complete and orthogonal set for the
expression of a scalar field on three-sphere. These hyperspherical harmonics
satisfy the following properties:

$\displaystyle\quad\quad
D^{2}\mathcal{S}_{nlm}(\vec{x})=-K(n^{2}-1)\mathcal{S}_{nlm}(\vec{x})~{},$
(62) $\displaystyle\int
d^{3}x\,\gamma\,\mathcal{S}_{nlm}(\vec{x})\mathcal{S}^{*}_{n^{\prime}l^{\prime}m^{\prime}}(\vec{x})=\delta_{nn^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}}~{}.$
(63)

In order to solve the constraints and finally obtain the quadratic action
after the constraints are released, we use hyperspherical harmonics
$\mathcal{S}_{nlm}(\vec{x})$ to expand the perturbation variables.

$\zeta(\eta,\vec{x})=\sum^{\infty}_{n\geq
2}\sum^{n-1}_{l=0}\sum^{l}_{m=-l}\zeta_{nlm}(\eta)\mathcal{S}_{nlm}(\vec{x})~{}.$
(64)

Note that the sum starts at $n=2$ since the $n=1$ mode is homogeneous and thus
a part of the background. Then apply the same expansions to the perturbation
variables $A$ and $B$ which represent constraints, and substitute these
constraints back into the action (56), we get the final form of the quadratic
action for scalar perturbations:

$S^{(2)}=\sum^{\infty}_{n\geq 2}\sum^{n-1}_{l=0}\sum^{l}_{m=-l}\int
d\eta~{}z^{2}\left(\frac{1}{2}|\zeta_{nlm}^{\prime}|^{2}-\frac{1}{2}\omega^{2}|\zeta_{nlm}|^{2}\right)~{},$
(65)
$\text{where}~{}\left\\{\begin{array}[]{lll}\displaystyle{z^{2}=\frac{a^{2}\theta^{\prime
2}}{\mathcal{H}^{2}}\left(1-\frac{\theta^{\prime
2}}{\mathcal{F}_{n}}\right)}\\\ \\\
\displaystyle{\omega^{2}=k^{2}+\frac{4\mathcal{K}^{2}}{3\mathcal{F}_{n}}\left[a^{2}V_{n}-(n^{2}-1)\theta^{\prime
2}\right]+\frac{2c\mathcal{K}}{\theta^{\prime}}\left(\mathcal{F}_{n}-12\mathcal{K}^{2}+\frac{4\mathcal{K}^{2}\theta^{\prime
2}}{\mathcal{F}_{n}}\right)}\end{array}\right.$

In above expression, $k^{2}=(n^{2}-1)\mathcal{K}^{2}$,
$V_{n}=(n^{2}-4)(V+3\mathcal{H}V_{\theta}/\theta^{\prime})$ and
$\mathcal{F}_{n}=2(n^{2}-4)\mathcal{H}^{2}+\theta^{\prime 2}$. This quadratic
action shows clearly that there is one dynamical scalar degree of freedom. It
comes from the dynamical scalar field $\theta(x)$. We introduced its dynamics
at the beginning. So the loss of scalar dynamical degree of freedom is only a
special phenomenon in the case of flat FRW background. Finally,
$\mathcal{F}_{n}\geq\theta^{\prime 2}$, so $z^{2}\geq 0$, thereby avoids the
ghost instability.

####  V.1.3 Open FRW

In the case of open FRW background, the quadratic action for scalar
perturbations is

$\displaystyle S^{(2)}=$ $\displaystyle-$ $\displaystyle\int
d^{4}x~{}a^{2}\gamma~{}\Big{\\{}3{\zeta^{\prime}}^{2}-6\mathcal{H}\zeta^{\prime}A+(2A+\zeta)D^{2}\zeta+(a^{2}V+3\mathcal{K}^{2})A^{2}-3\mathcal{K}^{2}\zeta^{2}$
(66) $\displaystyle-$ $\displaystyle
6\mathcal{K}^{2}A\zeta+2(\mathcal{H}A-\zeta^{\prime})D^{2}B+\mathcal{K}^{2}BD^{2}B+2c\theta^{\prime}(\zeta-\mathcal{K}\beta)D^{2}\lambda\Big{\\}}~{}.$

This action (66) has a very different form compared with the one (56) for
closed universe. This is because the corresponding background solutions (18)
and (22) have much different forms. It can be seen that $\beta$, $\lambda$,
$A$ and $B$ are all non-dynamical fields. All these non-dynamical fields
induce the following constraints:

$\displaystyle\lambda=0~{},$ (67) $\displaystyle\zeta-\mathcal{K}\beta=0~{},$
(68) $\displaystyle\zeta^{\prime}-\mathcal{H}A-\mathcal{K}^{2}B=0~{},$ (69)
$\displaystyle-3\mathcal{H}\zeta^{\prime}+D^{2}\zeta-3\mathcal{K}^{2}\zeta+(a^{2}V+3\mathcal{K}^{2})A+\mathcal{H}D^{2}B=0~{}.$
(70)

Similarly, in comparison with GR, there are two more constraints (67) and (68)
here. These two constraints can be used to solve $\beta$ and $\lambda$
respectively. And the constraints (59) and (60) can be used to solve $A$ and
$B$. Again, when $\mathcal{K}=0$, the constraint (58) will become $\zeta=0$
and this returns to the case of flat universe.

In the open FRW universe, the spatial line element can be expressed as
$d\hat{s}^{2}=\gamma_{ij}dx^{i}dx^{j}=(1/K)[d\chi^{2}+\sinh^{2}\chi(d\varphi^{2}+\sin^{2}\varphi\,d\phi^{2})]$.
Then, the $D^{2}$ eigenfunctions $\mathcal{Z}_{nlm}(\vec{x})$ (for more
details, see openscalar ) can be expressed as

$\mathcal{Z}_{nlm}(\vec{x})=K^{\frac{3}{4}}\frac{\Gamma(in+l+1)}{\Gamma(in)}\left(\sinh\chi\right)^{-\frac{1}{2}}P^{-l-\frac{1}{2}}_{in-\frac{1}{2}}(\cosh\chi)\,Y_{lm}(\varphi,\phi)~{},$

where $\Gamma(x)$ is the gamma function, $P^{\alpha}_{\beta}(x)$ is the
associated Legendre function of the first kind and $n$ is a positive real
number. These functions $\mathcal{Z}_{nlm}(\vec{x})$ constitute a complete and
orthogonal set for the expression of a scalar field on three-dimensional
hyperboloid. These functions $\mathcal{Z}_{nlm}(\vec{x})$ satisfy the
following properties:

$\displaystyle\quad\quad
D^{2}\mathcal{Z}_{nlm}(\vec{x})=-|K|(n^{2}+1)\mathcal{Z}_{nlm}~{},$ (71)
$\displaystyle\int
d^{3}x\,\gamma\,\mathcal{Z}_{nlm}(\vec{x})\mathcal{Z}^{*}_{nlm}(\vec{x})=\delta(n-n^{\prime})\delta_{ll^{\prime}}\delta_{mm^{\prime}}~{}.$
(72)

In order to solve the constraints and finally obtain the quadratic action
after the constraints are released, we use the $D^{2}$ eigenfunctions
$\mathcal{Z}_{nlm}(\vec{x})$ to expand the perturbation, i.e.,

$\zeta(\eta,\vec{x})=\sum_{l=0}^{\infty}\sum_{m=l}^{l}\int_{0}^{\infty}dn~{}\zeta_{nlm}(\eta){\mathcal{Z}}_{nlm}(\vec{x})~{},$
(73)

and make similar expansions for the perturbation variables $A$ and $B$ which
represent constraints. Then substitute these constraints back into the action
(66), we get the final form of the quadratic action for scalar perturbations:

$S^{(2)}=\sum^{\infty}_{l=0}\sum^{l}_{m=-l}\int_{0}^{\infty}dn\int
d\eta~{}z^{2}\left(\frac{1}{2}|\zeta_{nlm}^{\prime}|^{2}-\frac{1}{2}\omega^{2}|\zeta_{nlm}|^{2}\right)~{}$
(74)
$\text{where}~{}\left\\{\begin{array}[]{lll}\displaystyle{z^{2}=\frac{a^{2}\theta^{\prime
2}}{\mathcal{H}^{2}}\left(1+\frac{\theta^{\prime
2}}{\mathcal{F}_{n}}\right)}\\\ \\\
\displaystyle{\omega^{2}=k^{2}+\frac{4\mathcal{K}^{2}}{3\mathcal{F}_{n}}\left[(n^{2}+1)\theta^{\prime
2}-a^{2}V_{n}\right]}\end{array}\right.$

and $k^{2}=(n^{2}+1)\mathcal{K}^{2}$,
$V_{n}=(n^{2}+4)(V+3\mathcal{H}V_{\theta}/\theta^{\prime})$ and
$\mathcal{F}_{n}=2[3\mathcal{K}^{2}+(n^{2}+1)\mathcal{H}^{2}+a^{2}V]$. This
quadratic action also shows clearly that there is one dynamical scalar degree
of freedom. Finally, for the case of $V(\theta)\geq 0$ as in most inflationary
models, we always have $\mathcal{F}_{n}>0$, and thus $z^{2}>0$, thereby the
quadratic action avoid the ghost instability.

###  V.2 Quadratic Actions For Tensor Perturbations

####  V.2.1 Flat FRW

In the flat FRW case, we can use the circular polarization bases
$\hat{e}^{L}_{ij}$, $\hat{e}^{R}_{ij}$ and plane wave
$e^{i\vec{k}\cdot\vec{x}}$ to expand tensor perturbation

$h_{ij}^{T}(t,\vec{x})=\sum_{A}\int\frac{d^{3}k}{(2\pi)^{3/2}}h^{A}(t,\vec{k})\hat{e}_{ij}^{A}(\vec{k})e^{ik_{j}x^{j}}~{},$
(75)

The bases satisfy the relation:
$n^{l}\varepsilon_{li}^{~{}\,k}\hat{e}_{jk}^{A}=i\lambda_{A}\hat{e}_{ij}^{A}$,
here $A=L,R$ and $\lambda_{L}=-1$, $\lambda_{R}=1$, $\vec{n}$ is the unit
vector of $\vec{k}$. The final form of the quadratic action for tensor
perturbation is

$S=\sum_{A}\int d\eta
d^{3}k~{}\frac{a^{2}}{4}\left[h^{A*^{\prime}}h^{A^{\prime}}-\left(k^{2}+c\theta^{\prime}\lambda_{A}k\right)h^{A*}h^{A}\right]~{}.$
(76)

The quadratic action (76) is exactly the same as the one obtained in Ref.
PVtele1 . This is because the tensor perturbations only come from the tetrad,
they have no origin from the spin connection or the local Lorentz matrices.
This action shows clearly that there is no ghost instability, contrary to the
CS gravity. And it can be seen that the modified dispersion relation
$\omega_{A}^{2}=k^{2}+\lambda_{A}c\theta^{\prime}k=k^{2}\left(1+\lambda_{A}c\theta^{\prime}/k\right)\equiv
k^{2}\left(1+\mu_{A}\right)$ is helicity dependent. In terms of the notation
of Ref. Zhao:2019xmm , the deviation from the standard dispersion relation is
characterized by the parameter $\mu_{A}$. Consider small coupling and slow
evolution of $\theta$, one can find that GWs with different helicities will
have different phase velocities $v_{p}^{A}=\omega_{A}/k\simeq
1+\lambda_{A}c\theta^{\prime}/(2k)$ and same group velocity
$v_{g}=d\omega_{A}/dk\simeq 1+c^{2}\theta^{\prime 2}/\left(8k^{2}\right)$ up
to the order $\mathcal{O}\left(c^{2}\right)$. This is the so-called velocity
birefringence phenomenon of GWs, which is a direct reflection of the parity
violation in this model. The velocity birefringence here is very similar to
the cosmic birefringence induced by electromagnetic Chern-Simons coupling ECS1
; ECS2 ; ECS3 ; ECS4 . Since the deviation $\mu_{A}$ is inversely proportional
to $k$, this is an infrared effect, contrary to most PV gravity models in the
literature. We can also see that the phase velocity difference or the
deviation of the group velocity from the speed of light in vacuum become
important only at the region of small $k$ (large scales). Besides the velocity
birefringence, the amplitudes for both helicities of GWs are the same, this is
different from the CS modified gravity either. It is expectable that from the
quadratic action (76), the produced primordial GWs in the early universe will
have different power spectra for left- and right-handed polarizations. This
will result in the correlation between the E- and B-modes polarizations of the
cosmic microwave background radiation (CMB), and may have effects on the
planned CMB experiments CMB1 ; CMB2.

####  V.2.2 Curved FRW

In the curved FRW case, we can use the symmetric traceless and transverse
tensor harmonics $\\{Q^{(nlm)}_{ij}(\vec{x}),\bar{Q}^{(nlm)}_{ij}(\vec{x})\\}$
(for more details, see Challinor:1999xz ) to expand the tensor perturbations.
Below we briefly introduce how to construct the symmetric traceless and
transverse tensor harmonics.

In the three-dimensional maximum symmetric space
$d\hat{s}^{2}=\gamma_{ij}dx^{i}dx^{j}$, starting at some arbitrary point $O$,
we construct all geodesics with unit tangent vectors $b^{i}(\chi)$, where
$\chi\geq 0$ (the equality holding at $O$) is the parameter of the geodesic. A
vector field $b^{i}$ and a scalar field $\chi$ can then be constructed. Next,
we consider the rank-$l$ symmetric trace-free tensor field
$\mathcal{Q}_{i_{1}...i_{l}}$ satisfying
$b^{j}D_{j}\mathcal{Q}_{i_{1}...i_{l}}=0$. Since the
$\mathcal{Q}_{i_{1}...i_{l}}$ have only $2l+1$ degrees of freedom, it is
convenient to introduce a set of orthogonal basis tensor
$\\{\mathcal{Q}^{(lm)}_{i_{1}...i_{l}}\\}$, with $m=-l,...,l$. For
convenience, we define
$\mathcal{Q}^{(lm)}\equiv\mathcal{Q}^{(lm)}_{i_{1}...i_{l}}b^{i_{1}}...b^{i_{l}}$,
$\mathcal{Q}^{(lm)}_{i}\equiv\mathcal{Q}^{(lm)}_{i,j_{2}...j_{l}}b^{j_{2}}...b^{j_{l}}$
and
$\mathcal{Q}^{(lm)}_{ij}\equiv\mathcal{Q}^{(lm)}_{i,j,k_{3}...k_{l}}b^{k_{3}}...b^{k_{l}}$.
We also define $\mathcal{H}_{ij}\equiv\gamma_{ij}-b_{i}b_{i}$ and
$[A_{ij}]^{TT}\equiv(\mathcal{H}^{k_{1}}_{i}\mathcal{H}^{k_{2}}_{j}-\mathcal{H}_{ij}H^{k_{1}k_{2}}/2)A_{k_{1}k_{2}}$.
Then the symmetric traceless and transverse tensor harmonics can be expressed
as

$\displaystyle\displaystyle{Q^{(nlm)}_{ij}(\vec{x})=T^{(nl)}_{1}(\sqrt{|K|}\chi)\left(b_{i}b_{j}-\frac{1}{2}\mathcal{H}_{ij}\right)\mathcal{Q}^{(lm)}-T^{(nl)}_{2}(\sqrt{|K|}\chi)b_{(i}\mathcal{H}^{k_{l}}_{j)}\mathcal{Q}^{(lm)}_{k_{l}}+T^{(nl)}_{3}(\sqrt{|K|}\chi)[\mathcal{Q}^{(lm)}_{ij}]^{TT},}$
(77)
$\displaystyle\bar{Q}^{(nlm)}_{ij}(\vec{x})=\bar{T}^{(nl)}_{1}(\sqrt{|K|}\chi)[b_{k_{1}}\varepsilon^{k_{1}k_{2}}_{~{}~{}~{}~{}(i}\mathcal{Q}^{(lm)}_{j)}]^{TT}+\bar{T}^{(nl)}_{2}(\sqrt{|K|}\chi)b_{k_{1}}b_{(i}\varepsilon_{j)}^{\
k_{1}k_{2}}\mathcal{Q}^{(lm)}_{k_{2}},$ (78)

where

$\displaystyle\displaystyle{T^{(nl)}_{1}(x)=\frac{\Phi^{n}_{l}(x)}{S_{K}^{2}(x)}~{},}$
$\displaystyle\displaystyle{T^{(nl)}_{2}(x)=\frac{-2}{(l+1)S_{K}^{2}(x)}\frac{d}{dx}[S_{K}^{3}(x)T^{(nl)}_{1}(x)]~{},}$
$\displaystyle\displaystyle{T^{(nl)}_{3}(x)=\frac{-l}{l+2}T^{(nl)}_{1}(x)-\frac{1}{(l+2)S_{K}^{2}(x)}\frac{d}{dx}[S_{K}^{3}(x)T^{(nl)}_{2}(x)]~{},}$
$\displaystyle\displaystyle{\bar{T}^{(nl)}_{2}(x)=\frac{-2n}{l+1}\frac{\Phi^{n}_{l}(x)}{S_{K}(x)}~{},}$
$\displaystyle\displaystyle{\bar{T}^{(nl)}_{1}(x)=\frac{-1}{(l+2)S_{K}^{2}(x)}\frac{d}{dx}[S_{K}^{3}(x)\bar{T}^{(nl)}_{2}(x)]~{},}$

and $\Phi^{n}_{l}(x)$ is the ultra-spherical Bessel function. When $K>0$,
$S_{K}(x)=\sin(x)$, $n\geq 3$ is an integer, $l$ is an integer satisfying
$2\leq l\leq(n-1)$ and $m$ is an integer satisfying $|m|\leq l$. When $K<0$,
$S_{K}(x)=\sinh(x)$, $n$ is a positive real number, $l\geq 2$ is an integer
and $m$ is an integer satisfying $|m|\leq l$.

For convenience, we recombine these symmetric traceless and transverse tensor
harmonics into the circular polarization bases:

$(Q^{A}_{nlm})_{ij}=\mathcal{N}_{nl}\left[Q^{(nlm)}_{ij}-\lambda_{A}\bar{Q}^{(nlm)}_{ij}\right]~{},$
(79)

where $\mathcal{N}_{nl}$ is a normalization coefficient, $A=L,R$ and
$\lambda_{L}=-1$, $\lambda_{R}=1$. When $K>0$, we define
$k=\sqrt{\mathcal{K}(n^{2}-3)}$; and for $K<0$,
$k=\sqrt{\mathcal{K}(n^{2}+3)}$. It can be proved that these bases satisfy the
following relations:

$D^{2}(Q^{A}_{nlm})_{ij}=-k^{2}(Q^{A}_{nlm})_{ij}~{},~{}~{}\varepsilon_{kl(i}D^{k}(Q^{A}_{nlm})^{l}_{~{}j)}=-\lambda_{A}k\sqrt{1+\frac{3K}{k^{2}}}\,(Q^{A}_{nlm})_{ij}~{}.$
(80)

We can always choose an appropriate normalization coefficient
$\mathcal{N}_{nl}$ so that the bases are normalized to

$\int
d^{3}x~{}\gamma~{}(Q^{A}_{nlm})_{ij}(Q^{B}_{n^{\prime}l^{\prime}m^{\prime}})^{ij}=\left\\{\begin{array}[]{lll}\displaystyle{2\delta^{AB}\delta_{nn^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}}}&~{}(K>0)\\\
\\\
\displaystyle{2\delta^{AB}\delta(n-n^{\prime})\delta_{ll^{\prime}}\delta_{mm^{\prime}}}&~{}(K<0)\end{array}\right.$
(81)

The tensor perturbations can be expanded as

$h^{T}_{ij}(\eta,\vec{x})=\sum_{A=L,R}\sum_{{k}}h^{A}_{nlm}(\eta)(Q^{A}_{nlm})_{ij}(\vec{x})~{},\quad\text{where}~{}\sum_{{k}}\equiv\left\\{\begin{array}[]{lll}\displaystyle{\sum_{n=3}^{\infty}\sum_{l=2}^{n-1}\sum_{m=-l}^{l}}&~{}(K>0)\\\
\\\
\displaystyle{\sum_{l=2}^{\infty}\sum_{m=-l}^{l}}\int^{\infty}_{0}dn&~{}(K<0)\end{array}\right.$
(82)

Hence, we get the final form of the quadratic action for tensor perturbation

$S^{(2)}=\sum_{A=L,R}\sum_{k}\int
d\eta~{}\frac{a^{2}}{4}\left[(h^{A^{\prime}}_{nlm})^{2}-\omega_{A}^{2}(h^{A}_{nlm})^{2}\right]~{},$
(83)

where

$\omega_{A}^{2}=\left\\{\begin{array}[]{lll}\displaystyle{k^{2}+2\mathcal{K}^{2}+c\theta^{\prime}\mathcal{K}+\lambda_{A}c\theta^{\prime}k\sqrt{1+\frac{3\mathcal{K}^{2}}{k^{2}}}}&~{}(K>0)\\\
\\\
\displaystyle{k^{2}-2\mathcal{K}^{2}+\lambda_{A}c\theta^{\prime}k\sqrt{1-\frac{3\mathcal{K}^{2}}{k^{2}}}}&~{}(K<0)\end{array}\right.~{}$

The quadratic action (83) shows no ghost instability. Again, it produces the
velocity birefringence phenomenon rather than the amplitude birefringence
phenomenon. If the three curvature is very small, $\mathcal{K}\approx 0$, the
space of the background is approximately flat, the modified dispersion
relation is approximately $\omega_{A}^{2}\approx
k^{2}+\lambda_{A}c\theta^{\prime}k$, returning to the dispersion relation
obtained in the universe with spatially flat FRW background.

##  VI conclusion

In this paper we revisited the recently proposed parity violating gravity
model PVtele1 , which is based on the teleparallel gravity theory and has a
simple form. More importantly it is ghost free. In this revisit, we gave up
the Weitzenböck connection and restored the local Lorentz symmetry of this
model. With the full local Lorentz covariance, this model is not a pure tetrad
theory any more. We applied this new version of the model to the universe
which has a general FRW background. We studied the background evolution and
the linear cosmological perturbations. We found that in the case of flat FRW
background the background evolution is the same as GR, but the scalar field
$\theta(x)$ which couples to gravity through the PV term is not counted as a
dynamical degree of freedom at the linear perturbation level, and the tensor
perturbations present velocity birefringence in stead of amplitude
birefringence phenomenon. These confirmed the results obtained in PVtele1 . We
also considered the cases of curved FRW backgrounds. We found that, the PV
modification only affects the background evolution of the closed universe, in
both the open and closed universes the fluctuation of the scalar field
$\theta$ remained as a dynamical degree of freedom, the tensor perturbations
are similar to those in flat universe except that in the curved universe their
dispersion relations have further dependences on the spatial curvature. By the
way, the phenomenon of degrees of freedom being hidden under special
backgrounds also appears in $f(\mathbb{T})$ models Ong:2013qja  and the models
of massive gravity DeFelice:2012mx .

Acknowledgement: This work is supported by NSFC under Grant No. 12075231,
11653002, 12047502 and 11947301.

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