NuPhys2023-Cookman

Measuring Solar Neutrino Oscillations in the SNO+ Detector

Daniel Cookman111 Science and Technologies Facilities Council

King’s College London

> The SNO+ experiment is a large multi-purpose neutrino detector, currently
> filled with liquid scintillator. For the first time in a single experiment,
> SNO+ is able to measure the neutrino oscillation parameters $\theta_{12}$
> and $\Delta m^{2}_{21}$ simultaneously through both reactor anti-neutrinos
> and ^8B solar neutrinos. The latter approach is demonstrated here, with an
> analysis of an initial 80 days of scintillator phase data. A Bayesian
> statistical approach via Markov Chain Monte Carlo is used, allowing for the
> simultaneous fitting of the oscillation parameters, ^8B neutrino flux,
> background components with constraints, and systematic uncertainties. The
> neutrino oscillation parameter $\theta_{12}$ was measured to be
> $38.9^{\circ+8.0^{\circ}}_{\phantom{\circ}-7.9^{\circ}}$, assuming the
> current global fit flux of ^8B solar neutrinos. This is consistent with the
> current global fit result for $\theta_{12}$. A sensitivity study shows that
> this measurement is statistics-limited, and precision could be improved by a
> factor of two with two years of livetime, assuming the same backgrounds and
> selections.

> PRESENTED AT
>
>
>
>
> NuPhys2023, Prospects in Neutrino Physics
> King’s College, London, UK,
> December 18–20, 2023

##  1 Introduction

SNO+ is a large-scale liquid scintillator detector based
$2\text{\,}\mathrm{km}$ underground near Sudbury, Canada. It consists of 780
tonnes of liquid scintillator held within a $6.0\text{\,}\mathrm{m}$ radius
Acrylic Vessel (AV), surrounding which are 9362 inward-facing PMTs held on a
stainless steel PMT Support Structure (PSUP). Between the AV and PSUP is
ultra-pure water shielding; the AV is held in place with a series of hold-down
and hold-up ropes. For more details about the detector, see [1].

It is possible to observe and measure neutrino oscillations through at least
two sources with SNO+: anti-neutrinos from nuclear fission reactors, and
neutrinos from the Sun. In both cases, oscillations lead to a deficit in the
number of interactions observed compared to no oscillations, with the shape of
the measured energy spectrum enabling one to measure the neutrino oscillation
parameters $\theta_{12}$ and $\Delta m^{2}_{21}$. For the case of solar
neutrino oscillations, the survival probability of electron neutrinos
($P_{ee}$) generated within the core of the Sun is impacted by neutrino
oscillations in the vacuum of space, as well as the matter effect as the
neutrinos pass through the Sun and Earth. This leads to a dependence of
$P_{ee}$ on the neutrino’s energy, $\theta_{12}$, and $\Delta m^{2}_{21}$, as
shown in Fig. 1.

Figure 1: ^8B solar neutrino survival probability as a function of true
neutrino energy, $\theta_{12}$, and $\Delta m^{2}_{21}$. For a given
oscillation parameter being scanned over, all other oscillation parameters are
set at the NuFit 5.1 global fit values [4]. Calculations are made using the
algorithm described in [2].

This analysis measures $\theta_{12}$ and $\Delta m^{2}_{21}$ by looking for
interactions of ^8B solar neutrinos in the SNO+ detector via neutrino-electron
elastic scattering, $\nu_{x}+e^{-}\to\nu_{x}+e^{-}$. The difference in cross-
section between $\nu_{e}$ and $\nu_{\mu,\tau}$, and the (mild) correlation
between the energies of the $\nu_{x}$ and scattered electron allow for the
measurement of neutrino oscillations through the observed energy spectrum of
the scattered electrons.

##  2 Method

Data was selected between 17th May–30th November 2022. A series of cuts was
then applied to this data for the purposes of this analysis. After choosing
only periods of running for which the detector was operating stably, a number
of data cleaning cuts were applied to triggered events to remove instrumental
backgrounds. Cosmic ray muon events were also tagged, and a
$20\text{\,}\mathrm{s}$ veto was applied to all subsequent events, to remove
cosmogenic backgrounds. To cut out low-energy radioactive backgrounds,
triggered events were required to have a reconstructed energy in the range
$2.514.0\text{\,}\mathrm{MeV}$. Backgrounds from sources external to the
liquid scintillator, e.g. the AV, external water, and PMTs, were removed
through a fiducial volume cut: the reconstructed radial distance of an event
from the centre of the AV, $R$, must be less than $5\text{\,}\mathrm{m}$.
External backgrounds were further suppresed with the use of two classifiers
that distinguish external backgrounds from the solar signal. To substantially
reduce the remaining ^238U/^232Th-chain and $(\alpha,n)$ backgrounds, a broad
coincidence pair tagging algorithm was applied to all events. A PMT hit time
classifier was used to search for and remove ^212/214BiPo events for which the
two decays occured within one triggered event window.

After all of these cuts, 652 events remained. Considering the application of
the $20\text{\,}\mathrm{s}$ veto, the net livetime of this dataset was 80.615
days. Accompanying this data were full Monte Carlo (MC) simulations of the
signal and background processes expected to contribute to the cut data, with
the same set of cuts applied to the MC events as in data. Events from data and
each MC process after cuts were then binned into 2D histograms in
reconstructed energy and radius. The binned MC distributions were then fit to
the data using Markov-Chain Monte Carlo (MCMC) via the Metropolis algorithm,
with an extended binned maximum likelihood test statistic. This fit allowed
the oscillation parameters $\theta_{12}$ and $\Delta m^{2}_{21}$ to float
within the fit, varying the shape of the ^8B energy distribution. The fit also
allowed all signal and background rates, as well as the reconstructed energy
scale systematic to float within constraints. The flux of ^8B solar neutrinos
is constrained using a global fit constraint [3]:
$\Phi_{\ce{{}^{8}B}{}}=5.16^{+0.13}_{-0.09}\times
10^{6}${\mathrm{cm}}^{-2}\text{\,}{\mathrm{s}}^{-1}$$.

##  3 Results and Future Sensitivity

Fig. 2 shows the reconstructed energy distributions of data for four equal-
volume radial slices, compared to the MC using the fit parameters that
obtained the greatest likelihood. The solar signal can be seen to dominate
above $5\text{\,}\mathrm{MeV}$ at all radii, and also around
$2.5\text{\,}\mathrm{MeV}$ in the most central radial bin.

Figure 2: Distribution of selected of events in data in both reconstructed
energy and radius, compared to the fitted MC, grouped by type of signal or
background. Each subplot corresponds to one equal-volume radial slice.

The 2D marginalised posterior density distribution for the parameters
$\theta_{12}$ and $\Delta m^{2}_{21}$ is shown in Fig. 3. This plot shows that
whilst there is not yet sufficient statistics to constrain $\Delta
m^{2}_{21}$, $\theta_{12}$ can be measured with the limited dataset. After
marginalising onto $\theta_{12}$, a Bayesian $1\sigma$ CI of
$\theta_{12}{}=38.9^{\circ+8.0^{\circ}}_{\phantom{\circ}-7.9^{\circ}}$ was
obtained.

Figure 3: 2D marginalised posterior density distribution of $\theta_{12}$ and
$\Delta m^{2}_{21}$ from this analysis. The associated Bayesian CIs are shown,
with the colour axis corresponding to twice the negative log-likelihood ratio,
$-2\ln{L_{R}}$. Figure 4: Marginalised posterior density distribution of
$\theta_{12}$ for 6 Asimov datasets of increasing livetime. The Bayesian
$1\sigma$ CIs are shown for each livetime scenario.

A sensitivity study was performed to understand the impact of increased
statistics alone on the precision of the oscillation measurement. Asimov fake
datasets were generated from MC, for a variety of livetimes between 150 days
and 5 years. The rates of each process were determined in general by their
fitted rate in the existing dataset. An identical analysis approach was used
on the fake datasets as for the real data. The resulting marginalised
posterior density distributions for $\theta_{12}$ are shown in Fig. 4. These
show that the measured precision of $\theta_{12}$ from this analysis can be
halved with under two years of livetime, assuming no changes to the detector
or analysis.

##  4 Conclusions

An oscillation analysis has been performed with an initial 80 days of data
from the SNO+ scintillator phase, by looking at the change in shape of ^8B
solar neutrino-electron elastic scattering events. Using this data,
$\theta_{12}$ was measured as
$38.9^{\circ+8.0^{\circ}}_{\phantom{\circ}-7.9^{\circ}}$, consistent with the
current global fit value of
$33.41^{\circ+0.75^{\circ}}_{\phantom{\circ}-0.72^{\circ}}$ [4]. A sensitivity
study indicates that substantial improvements in precision can be made just
with an increase in livetime; scintillator-phase data continues to be taken
since the dataset in this analysis. More details of this analysis can be found
in [5]. Further increases in precision for this analysis are being explored,
such as increasing the fiducial volume, as well as supressing any of the
remaining backgrounds.

SNO+ is also measuring $\theta_{12}$ and $\Delta m^{2}_{21}$ through reactor
anti-neutrino oscillations [6]. Because the reactor measurement has greater
precision in $\Delta m^{2}_{21}$, whilst the solar is better in $\theta_{12}$,
a combined fit between the two approaches could lead to even further
improvements in both oscillation parameters compared to each alone.

ACKNOWLEDGEMENTS

This work is supported by ASRIP, CIFAR, CFI, DF, DOE, ERC, FCT, FedNor, NSERC,
NSF, Ontario MRI, Queen’s University, STFC, and UC Berkeley, and have
benefited from services provided by EGI, GridPP, and Compute Canada. Daniel
Cookman is supported by STFC (Science and Technologies Facilities Council). We
thank Vale and SNOLAB for their valuable support.

## References

  * [1] SNO+ Collaboration, Albenese V. et al, “The SNO+ Experiment”, J. Instrum. 16 (2021) P08059
  * [2] Barros, N., “Precision Measurement of Neutrino Oscillation Parameters: Combined Three-Phase Results of the Sudbury Neutrino Observatory”, PhD thesis, Universidade de Lisboa (2011)
  * [3] Bergström, J. et al, “Updated Determination of the Solar Neutrino Fluxes from Solar Neutrino Data”, J. High Energ. Phys. 3 (2016), 132
  * [4] Esteban, I. et al, “The Fate of Hints: Updated Global Analysis of Three-Flavour Neutrino Oscillations”, JHEP (2020); also NuFit 5.2 (2022), www.nu-fit.org
  * [5] Cookman, D., “Measurement of Oscillations in Solar Boron-8 Neutrinos and Studies of Optical Scattering in the SNO+ Detector”, DPhil Thesis, University of Oxford (2024)
  * [6] Morton-Blake, I., “First Measurement of Reactor Antineutrinos in Scintillator at SNO+ and Study of Alternative Designs for Large-Scale Liquid Scintillator Detectors”, DPhil thesis, University of Oxford (2021)


11institutetext: Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge, CB3 0HA, United Kingdom
11email<EMAIL_ADDRESS>22institutetext: Dipartimento di Fisica, Università
degli Studi di Milano, Via Celoria 16, Milano, 20133, Italy 33institutetext:
Department of Physics & Astronomy, University of Victoria, Victoria, BC V8P
5C2, Canada 44institutetext: European Southern Observatory, Karl-
Schwarzschild-Strasse 2, D-85748 Garching bei Munchen, Germany
55institutetext: Department of Physics and Astronomy, The University of
Georgia, Athens, GA 30602, USA 66institutetext: Center for Simulational
Physics, The University of Georgia, Athens, GA 30602, USA 77institutetext:
Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, the
Netherlands

# Angular momentum transport via gravitational instability in the Elias 2-27
disc

C. Longarini 1122 G. Lodato 22 C. J. Clarke 11 J. Speedie 33 T. Paneque-
Carreño 4477 E. Arrigoni 22 P. Curone 22 C. Toci 44 C. Hall 5566

(Received ; accepted )

Gravitational instability is thought to be one of the main drivers of angular
momentum transport in young protoplanetary discs. The disc around Elias 2-27
offers a unique example of gravitational instability at work. It is young and
massive, displaying two prominent spiral arms in dust continuum emission and
global non-axisymmetric kinematic signatures in molecular line data.

In this work, we used archival ALMA observations of 13CO line emission to
measure the efficiency of angular momentum transport in the Elias 2-27 system
through the kinematic signatures generated by gravitational instability, known
as ’GI wiggles’. Assuming the angular momentum is transported by the observed
spiral structure and leveraging previously-derived dynamical disc mass
measurements, the amount of angular momentum transport we found corresponds to
an $\alpha-$viscosity of $\alpha=0.038\pm 0.018$. This value implies an
accretion rate onto the central star of $\log_{10}\dot{M}_{\star}=-6.99\pm
0.17\text{M}_{\odot}/\text{yr, which}$ reproduces the one observed value of
$\log_{10}\dot{M}_{\star,\text{obs}}=-7.2\pm 0.5\text{M}_{\odot}/\text{yr }$
very well. The excellent agreement we have found serves as further proof that
gravitational instability is the main driver of angular momentum transport
acting in this system.

###### Key Words.:

protoplanetary discs – gravitational instability – planet formation

##  1 Introduction

Elias 2-27 is a young ($\sim 0.5$Myr, $M_{\star}=0.46\text{M}_{\odot}$) M0
star (Andrews et al. 2009) located at a distance of 116 pc (Gaia Collaboration
et al. 2018) in the $\rho-$Oph star forming region, hosting a likely
gravitationally unstable disc. The circumstellar disc shows two large-scale
trailing spiral arms in dust continuum emission (Pérez et al. 2016), whose
origins were first attributed to gravitational instability, due to the high
dust mass. An estimate of the total dynamical mass of the disc has been
provided by Veronesi et al. (2021), $M_{d}=0.08\pm 0.04\text{M}_{\odot}$,
implying a disc-to-star mass ratio of 17%. Meru et al. (2017) performed three-
dimensional (3D) numerical SPH simulations to investigate the origin of the
spiral structure: by comparing gravitational instability and internal and
external companion scenarios, they found that GI best reproduces the observed
morphology. Similar results have also been found by Hall et al. (2018). In
addition, due to its high brightness, Elias 2-27 has become part of the the
DSHARP sample (Andrews et al. 2018), allowing for more thorough studies of its
dust morphology. Huang et al. (2018) characterised annular substructures in
the discs within the DSHARP sample, finding that Elias 2-27 has a gap at $\sim
70\text{au}$. Even though the main focus of the DSHARP program was dust
emission, also kinematic data on CO isotopologues were collected. Pinte et al.
(2020) found complex kinematic features in the Elias 2-27 system, showing
perturbations to the velocity field. However, due to the low resolution of the
data, a detailed analysis was not possible. Paneque-Carreño et al. (2021)
presented new data on this system and conducted a detailed analysis of the
morphology and the kinematics. Global perturbations in the velocity field of
13CO and C18O were found and their morphology follows the shape of the spiral,
consistent with kinematic signatures of GI induced density waves, or GI
Wiggles (Hall et al. 2020; Longarini et al. 2021; Terry et al. 2022). From the
same dataset, Veronesi et al. (2021) studied the rotation curve and obtained
the previously mentioned disc mass, by measuring the super-Keplerian
contribution of the disc self-gravity.

In this letter, we study the GI Wiggle in Elias 2-27. Thanks to the disc mass
estimate provided by Veronesi et al. (2021), we have been able to constrain
the amount of angular momentum transported throughout the disc. In Section 2,
we discuss the theoretical framework we use in this Letter. In Section 3, we
present the dataset and the analysis. In Section 4, we discuss the results and
in Section 5 we draw our conclusions.

##  2 Gravitational instability in protoplanetary discs

The onset of gravitational instability is determined by the Toomre parameter

$Q=\frac{c_{s}\kappa}{\pi G\Sigma},$ (1)

where $\kappa$ is the epicyclic frequency, which (for a Keplerian disc) is
just the Keplerian frequency, $\Omega_{k}$; $\Sigma$ is the surface density;
and $c_{s}$ the sound speed. Essentially, $Q$ measures the strength of
stabilising terms (e.g. pressure and rotation, at the numerator) compared to
destabilising ones (e.g. self-gravity, at the denominator). A protoplanetary
disc is marginally unstable when $Q\simeq 1$: in this regime, the disc
develops a spiral structure and, by means of shocks, this leads to energy
dissipation and heating. One of the most important consequences of
gravitational instability is its ability to drive angular momentum transport
throughout the disc, and therefore induce accretion onto the central object
(Lynden-Bell & Kalnajs 1972).

###  2.1 Gravito-turbulence and transport of angular momentum

A disc is in the gravito-turbulent regime when its angular momentum transport
is driven by gravitational instability. In this context, the $R\phi$ component
of the vertically integrated stress tensor can be written as:

$T_{R\phi}=\int\left\langle\frac{g_{R}g_{\phi}}{4\pi
G}\right\rangle\text{d}z,$ (2)

where $g_{R},g_{\phi}$ are the radial and azimuthal component of the perturbed
self-gravitating field, and the brackets indicate azimuthal averaging. To this
stress, it is necessary to also add the induced Reynolds stress, given by:

$T_{R\phi}=\Sigma\langle\delta v_{R}v_{\phi}\rangle,$ (3)

where $\delta v_{R}$ and $\delta v_{\phi}$ are the perturbed fluid velocities
(Balbus & Papaloizou 1999). In the classic viscous scenario, this term is
responsible for angular momentum transport in the accretion disc. The Shakura
and Sunyaev $\alpha-$prescription (Shakura & Sunyaev 1973) relates the stress
tensor to the local disc pressure

$T_{R\phi}=\alpha\Sigma c_{s}^{2}\frac{\text{d}\log\Omega}{\text{d}\log R},$
(4)

where, for a Keplerian disc, ${\text{d}\log\Omega_{k}}/{\text{d}\log R}=-3/2$.
It is possible to show that the transport of energy and angular momentum
through the propagation of the GI spiral density waves can be divided into two
parts (Toomre 1969; Shu 1970): a non-local term and a viscous-like term. Since
the non-local term is important only for very high disc-to-star mass ratios,
$M_{d}/M_{\star}\gtrsim 0.5$ (Lodato & Rice 2005), gravitationally unstable
protostellar discs essentially behave as $\alpha$-discs (Lodato & Rice 2004;
Forgan et al. 2011).

To characterise the transport of angular momentum through spiral density
waves, we need to rely on numerical simulations of gravitationally unstable
discs (Cossins et al. 2009). Usually in numerical simulations of
gravitationally unstable discs, the $\beta$ cooling framework is adopted
(Gammie 2001). We suppose that the disc is cooling with a rate per unit mass
of:

$q^{-}=-\frac{e}{t_{\text{cool}}},$ (5)

where $e=c_{s}^{2}/\gamma(\gamma-1)$ is the internal energy per unit mass,
with the adiabatic index $\gamma=5/3$, and the cooling time is defined in
terms of the dynamical one: $t_{\text{cool}}=\beta\Omega^{-1}$. In the absence
of external heating mechanisms, an initially stable hot disc ($Q>>1$) will
cool down, eventually reaching the marginally stable state ($Q=1$). At this
point, gravitational instability turns on: the disc develops a spiral
structure that, by means of compression and shocks, leads to efficient energy
dissipation and heating. In this sense, the Q-stability condition acts as a
thermostat so that heating turns on only if the system is sufficiently cold,
keeping it in a marginally stable state (Kratter & Lodato 2016). In this
regime, namely, thermal saturation, the cooling is completely balanced by
heating provided by the shocks, and the amplitude of spiral perturbations
saturates at a fixed value, according to Cossins et al. (2009):

$\left(\frac{\delta\Sigma}{\Sigma}\right)=\left[\frac{2}{\epsilon\beta\gamma(\gamma-1)}\frac{1}{\mathcal{M}\tilde{\mathcal{M}}}\right]^{1/2}=\chi\beta^{-1/2},$
(6)

where $\epsilon$ is the heating factor, $\mathcal{M}$ and
$\tilde{\mathcal{M}}$ are the Mach numbers relative to the radial phase speed
and Doppler-shifted radial phase speed of the wave: respectively,
$m\Omega_{p}/k$ and $m(\Omega-\Omega_{p})/k$, with $\Omega_{p}$ as the spiral
pattern frequency. Cossins et al. (2009) showed through numerical simulations
that the relevant scaling of the spiral density perturbation is $\beta^{-1/2}$
and the other terms of the order of unity. For this reason, we introduce
$\chi$, which is of the order of unity. In the thermal saturation regime, the
transport of angular momentum provided by gravitational instability is
described within an $\alpha-$framework(Kratter & Lodato 2016):

$\alpha_{\text{GI}}=\left|\frac{\text{d}\log\Omega}{\text{d}\log
R}\right|^{-2}\frac{1}{\gamma(\gamma-1)\beta}.$ (7)

Combining Eqs. (7) and (6), we obtain the relationship between the amplitude
of the density wave and the amount of angular momentum

$\frac{\delta\Sigma}{\Sigma}=\frac{3\chi}{2}\sqrt{\gamma(\gamma-1)}\alpha_{\text{GI}}^{1/2},$
(8)

where the last equation is valid for a Keplerian disc. In the following, we
assume that the constant of proportionality is $\chi=1$, which is well
justified by numerical simulations (Cossins et al. 2009).

###  2.2 Kinematic signatures of gravitational instability

Kinematics offers a unique opportunity to quantify the transport of angular
momentum in a gravitationally unstable disc. Indeed, when there is a spiral
density wave, also the velocity field is affected. Hall et al. (2020)
predicted that a disc undergoing such instability has clear kinematic
signatures in molecular line observations across the entire disc azimuth and
radius, called ‘GI wiggles’. Longarini et al. (2021) provided an analytical
model to describe such kinematic signatures, implemented in the publicly
available code giggle111https://doi.org/10.5281/zenodo.10205110. Under the
hypothesis of thin disc, in a marginally unstable regime $(Q=1)$ and in
thermal saturation, the amplitude of the velocity perturbation increases with
the disc-to-star mass ratio $(M_{d}/M_{\star})$, as has also been described
with numerical simulations (Terry et al. 2022), and which is proportional to
the cooling factor, $\beta^{-1/2}$. However, from the analysis carried out in
Longarini et al. (2021), the actual quantity that determines the amplitude of
the velocity perturbations $(\delta u_{R},\delta u_{\phi})$ is the amplitude
of the spiral density wave, $\delta\Sigma/\Sigma$. This quantity is
intrinsically linked to the efficiency of angular momentum transported by the
spiral, which can be described within an $\alpha-$viscosity framework through
Eq. (7). Thus, it is possible to constrain the value of $\alpha_{\text{GI}}$
from the amplitude of the wiggle. Rewriting the velocity perturbations (Eq. 22
of Longarini et al. (2021)) as a function of $\alpha_{\text{GI}}$, we obtain:

$\delta
u_{R}=3im\alpha_{\text{GI}}^{1/2}\sqrt{\gamma(\gamma-1)}\left(\frac{M_{d}}{M_{\star}}\right)^{2}u_{k},$
(9) $\delta
u_{\phi}=-\frac{3i\alpha_{\text{GI}}^{1/2}}{4}\sqrt{\gamma(\gamma-1)}\left(\frac{M_{d}}{M_{\star}}\right)u_{k}.$
(10)

The final expressions for the velocity field are:

$u_{\phi}(R,\phi)=R\Omega+\Re\left[\delta
u_{\phi}(R)e^{i(m\phi+\psi)}\right]=R\Omega-|\delta
u_{\phi}(R)|\sin(m\phi+\psi),$ (11) $u_{R}(R,\phi)=\Re\left[\delta
u_{R}(R)e^{i(m\phi+\psi)}\right]=-|\delta u_{R}(R)|\sin(m\phi+\psi).$ (12)

##  3 Gravitational instability in Elias 2-27

###  3.1 Toomre parameter

Veronesi et al. (2021) estimated the dynamical mass of Elias 2-27 from 13CO
and C18O rotation curves, and found $M_{\star}=0.46\text{M}_{\odot}\pm 0.03$
and $M_{d}=0.08\pm 0.04\text{M}_{\odot}$. In their fit, they used a self-
similar surface density given by

$\Sigma=\frac{M_{d}}{2\pi(200\text{au})^{2}}\left(\frac{R}{200\text{au}}\right)^{-1}\exp\left[-\frac{R}{200\text{au}}\right],$
(13)

where the scale radius has been fixed to $R_{c}=200\text{au}$. As for the
thermal structure, they assumed a vertically isothermal disc with
$T(R)=20(R/60\text{au})^{0.5}$K taken from Pérez et al. (2016), which
corresponds to a sound speed of $c_{s}=281\text{m/s}(R/60\text{au})^{-0.25}$.
With this information, it is possible to compute the Toomre parameter profile
for Elias 2-27, as displayed in Fig. 1. Despite not being exactly $Q=1$, the
Toomre profile is close enough to the critical threshold to consider
gravitational instability to be significant. In addition, we are not
considering uncertainties on $T(R)$, that can impact on the $Q-$parameter
estimate. Finally, we know that the disc shows non-axisymmetric features in
gas and dust, making the azimuthally averaged Toomre profile solely an
approximation of the actual value.

Figure 1: Toomre profile of Elias 2-27, where the shaded region indicates the
uncertainties on the disc and star mass from Veronesi et al. (2021).

###  3.2 Dataset

In this work, we use the 13CO $J=3-2$ datacube presented in Paneque-Carreño et
al. (2021). The images have been obtained with a robust parameter of 0.5,
resulting in a beam size of $0.26^{\prime\prime}\times 0.25^{\prime\prime}$
for the 13CO and a spectral resolution of $\Delta v=111$m/s. Further details
of the observations and reduction can be found in Paneque-Carreño et al.
(2021). In this work, we use a Gaussian velocity map obtained with
bettermoments (Teague & Foreman-Mackey 2018). The code also returns a map of
the errors on the velocity field (Teague 2019). In the analysis, we masked the
emission coming from the first two beams ($\sim 50\text{au}$).

###  3.3 Model of the GI wiggle of Elias 2-27

In the analytical model for the GI wiggle of Longarini et al. (2021), the
amplitude of the velocity perturbations is determined by the disc to star mass
ratio and the cooling factor. There is a degeneracy between the two
quantities; however, in the case of Elias 2-27 the value of the disc to star
mass ratio is known (Veronesi et al. 2021). Longarini et al. (2021) showed
that the amplitude of the wiggle in the PP (position-position) space scales as
$\beta^{-1/2}$ (i.e. $\alpha_{\text{GI}}^{1/2}$), and they proposed this
relationship as a way to constrain this unknown parameter. In this paragraph,
we study the wiggle in the PV (position-velocity) space, as done in Speedie et
al. (2024, submitted). We consider the observed velocity field:

$u_{\text{obs}}(R,\phi)=\left[u_{\phi}(R,\phi)\cos\phi+u_{R}(R,\phi)\sin\phi\right]\sin
i,$ (14)

where $u_{\phi}$ and $u_{R}$ are described by the model presented in Eqs. (11)
and (12) and $i$ is the disc inclination. We compute the variation of velocity
along the semi-minor axis of the disc (i.e. $\phi=-\pi/2$), because along this
axis only the radial velocity contributes to the observed velocity field

$u_{\text{obs}}(R,-\pi/2)=-2m\beta^{-1/2}\left(\frac{M_{d}(R)}{M_{\star}}\right)^{2}u_{k}\sin\left(\psi(R)+\psi_{0}\right),$
(15)

where $\psi(R)$ is the phase function of the spiral that is given by

$\frac{\text{d}\psi}{\text{d}R}=\frac{m}{R\tan\alpha_{p}},$ (16)

with $\alpha_{p}$ being the pitch angle of the spiral and $\psi_{0}$ is just a
phase shift. We suppose that the pitch angle is constant over the radial
extent of the disc, that is well justified for GI spirals (Cossins et al.
2009). We note that the last equation can be written as a function of
$\alpha_{\text{GI}}$ as:

$u_{\text{obs}}(R,-\pi/2)=-3m\alpha_{\text{GI}}^{1/2}\sqrt{\gamma(\gamma-1)}\left(\frac{M_{d}(R)}{M_{\star}}\right)^{2}u_{k}\sin\left(\psi(R)+\psi_{0}\right).$
(17)

We then extract the PV wiggle by cutting along the semi minor axis and
considering just the southern part of the disc, on account of cloud
contamination (Paneque-Carreño et al. 2021). The errors on the velocity are
returned by (Teague & Foreman-Mackey 2018) as shown in figure 2. The disc and
spiral parameters we use are taken from literature, namely,
$M_{\star}=0.46\text{M}_{\odot}$, $M_{d}=0.08\text{M}_{\odot}$, $R_{c}=200$au
(Veronesi et al. 2021), $\alpha_{p}=13^{\circ}$, $i=56.2^{\circ}$,
$\text{PA}=118.8^{\circ}$ (Paneque-Carreño et al. 2021), and $m=2$. Hence, the
only free parameters in Eq. (17) are $\alpha_{\text{GI}}$ and $\psi_{0}$. To
fit the curve to the data, we used the method of nonlinear least squares
implemented in Scipy (Virtanen et al. 2020). While the spiral pattern in the
continuum extends to approximately $200\text{au}$, non-axisymmetric kinematic
signatures in the gas emission are observable across the entire radial extent
of the disk. Hence, in the fitting procedure, we analysed the signal up to the
outer edge of the disc. The best-fit values are $\alpha_{\text{GI}}=0.038\pm
0.018$ and $\psi_{0}=43^{\circ}\pm 1^{\circ}$. The value of
$\alpha_{\text{GI}}$ corresponds to a $\beta=10.5$, meaning that the cooling
time of the system is approximately ten times the dynamical one. We estimated
the error on $\alpha_{\text{GI}}$ propagating the uncertainties on star and
disc masses222The fitting procedure implemented in Scipy returns an error for
the best fit parameters. As for the $\alpha_{\text{GI}}$ parameter, the error
provided by the fit is subdominant compared to the one driven by the star and
disc masses uncertainties..

Figure 2 shows the comparison between the extracted PV wiggle from 13CO data
and the model that best describes the data. By comparing the model and the
data, we observe that the overall shape of the perturbation is well
reproduced. However, it appears that there is a radial shift between the two
curves. This effect can be attributed to the fact that in the analytical
model, we assume that a single spiral mode is present, with $m=2$, and we also
suppose that the perturbation wave-number, $k,$ is the most unstable
$(k_{\text{uns}})$. While the $m=2$ mode, with $k=k_{\text{uns}}$ may be the
dominant one, determining the overall morphology of the spiral, this does not
prevent the presence of additional lower amplitude modes that will interfere
with the dominant one and thus create a more complex pattern than the purely
sinusoidal one that we assume here. The scope of this work is to find the
amount of stress generated by gravitational instability; hence, we are
interested in the amplitude of the wiggle, not in the whole shape. For this
reason, the additional lower amplitude modes are not a concern in terms of the
amplitude of the perturbation.

Figure 3 shows the analytical observed velocity field with the aforementioned
parameters. Despite its simplicity, the analytical model matches the shape of
the different channels of the data. In particular, the bending of the
isovelocity contours is described very well by the analytical model,
especially in the blue-shifted region, where the effect of cloud contamination
are negligible. The analytical models are produced using the publicly
available code giggle. It is important to point out that the model presented
in Longarini et al. (2021) is limited to two dimensions and does not consider
the vertical extent of the disc. There is evidence suggesting that the 13CO
emission in Elias 2-27 is optically thick (Paneque-Carreño et al. 2022);
consequently, the received signal does not originate from the midplane, but
from a layer at $z\neq 0$. The analytical model we employ is not able to
reproduce this effect, and, for a proper comparison, hydrodynamical and
radiative transfer simulations would be needed. In any case, assuming that
$z/R<0.3$ for the 13CO (Paneque-Carreño et al. 2022), the difference in
inclination induced by the finite height of the emitting layer is $\delta
i=\arcsin(0.3)\simeq 0.29$. The corresponding geometrical error is roughly
$1-\sin(i+\delta i)/\sin(i)\simeq 15\%$, that is subdominant compared to the
one driven by the mass estimate.

Figure 2: GI wiggle in the PV space: comparison between the the data (red
line) and the model (blue line). The shaded region represents the error on the
model, driven by the the uncertainties on the star and disc masses from
Veronesi et al. (2021).

###  3.4 Angular momentum transport and accretion

For a gravitationally unstable disc, kinematics offers a unique opportunity to
quantify the transport of angular momentum of a gravitationally unstable disc.
We show that gravitational instability in Elias 2-27 transports angular
momentum and the equivalent $\alpha-$viscosity parameter is
$\alpha_{\text{GI}}=0.038\pm 0.018$. Since the effective viscosity is
responsible for the accretion process, it is useful to predict the expected
accretion rate onto the central object, and compare it to the observed one.

According to the self-similar solution (Lynden-Bell & Pringle 1974), the
surface density and the accretion rate of the disc can be written as:

$\Sigma=\frac{M_{d}}{2\pi
R_{c}^{2}}\left(\frac{R}{R_{c}}\right)^{-1}\exp\left[-\frac{R}{R_{c}}\right],$
(18)
$\dot{M}=\frac{3M_{d}\nu_{c}}{2R_{c}^{2}}\exp\left[-\frac{R}{R_{c}}\right]\left(1-\frac{2R}{R_{c}}\right),$
(19)

where we have supposed that $\nu=\nu_{c}(R/R_{c})$, where the subscript $c$
means that the quantity is evaluated at the scale radius $R_{c}$. Within a
$\alpha-$viscosity framework, the kinematic viscosity, $\nu,$ is

$\nu=\alpha c_{s}H=\alpha\left(\frac{H}{R}\right)^{2}u_{k}R.$ (20)

We note that within our assumption that $T\propto R^{0.5}$, for a constant
$\alpha$ the kinematic viscosity scales as $\nu\propto R$. In this way, we can
write the accretion rate onto the central object as the limit for $R\to 0$ of
Eq. (19):

$\dot{M}_{\star}=-\frac{3\alpha}{2}\left(\frac{H}{R}\right)_{R_{c}}^{2}M_{d}\Omega_{c},$
(21)

where $\Omega_{c}=\Omega(R_{c})=\sqrt{GM_{\star}/R_{c}^{3}}$.

Using $\alpha_{\text{GI}}=0.038$ and the disc parameters described in the
previous paragraph, it is possible to compute the accretion rate onto the
central object by using Eq. (21). Thus, we obtain:

$\log_{10}\dot{M}_{\star}[\text{M}_{\odot}/\text{yr}]=-6.99\pm 0.17,$ (22)

where the error has been computed through propagation from the errors in
$\alpha_{\text{GI}}$. The model for the accretion rate of Elias 2-27
reproduces the one measured by Natta et al. (2006) very well, namely,
$\log_{10}\dot{M}_{\star}[\text{M}_{\odot}/\text{yr}]=-7.2\pm 0.5$. They used
J and K-band spectra to derive the mass accretion rate of objects in the
$\rho-$Ophiuchi star forming region from the intensity of the hydrogen
recombination lines. More recently, a new estimate of Elias 2-27 accretion
rate was provided by Testi et al. (2022), being
$\log_{10}\dot{M}_{\star}[\text{M}_{\odot}/\text{yr}]=-7.3$, which is
consistent with the previous measurement.

To constrain $\alpha_{\text{GI}}$ in Elias 2-27, we made the strong hypothesis
that this quantity, or equivalently the cooling $\beta$, is constant
throughout the disc. We are aware that, realistically, this is not the case.
However, we point out that our disc model is self consistent. Indeed, assuming
a self similar solution for the surface density with $\Sigma\propto R^{-1}$,
we are imposing that the kinematic viscosity scales as $\nu\propto R$. Since
$\nu=\alpha c_{s}H$, with $c_{s}\propto R^{-0.25}$ (Pérez et al. 2016) and
$H\propto R^{1.25}$, the viscosity coefficient, $\alpha,$ should be constant
with radius. We also point out that, in order to obtain an estimate of the
amount of angular momentum transported within the disc from the GI wiggle, an
assumption should be made with respect to how $\alpha$ varies with the radius.
A more realistic disc model for Elias 2-27 is not within the scope of this
work. We stress that the choice of a self-similar profile for the surface
density was made to maintain consistency with the work of Veronesi et al.
(2021).

Figure 3: Comparison between Elias 2-27 velocity map (left) and the best fit
model for the velocity field with $\alpha_{\text{GI}}=0.038$ (right).

##  4 Discussion

The ability of our model to correctly reproduce the observed accretion rate
points to the fact that gravitational instability is responsible for angular
momentum transport in this system. As a matter of fact, the value of the
$\alpha-$viscosity we get from the GI wiggle is the one required to explain
the observed accretion rate onto the central object, having fixed the density
structure of Elias 2-27. In addition, the strong hypothesis we made is that
viscous processes are responsible for accretion. The inferred value for the
$\alpha-$viscosity is higher than usually assumed $(\sim 10^{-3}-10^{-4})$.
This is not surprising, since the strength of the viscosity generated by
gravitoturbulent motions is higher than the expected in the non-self
gravitating state (Cossins et al. 2009).

###  4.1 Infall and interaction with the environment

Elias 2-27 is a young system, and its interactions with the surrounding
environment are possibly perturbing the disc. Specifically, the disc is
partially embedded within the molecular cloud, which absorbs 12CO and a
portion of the 13CO emission (Pérez et al. 2016; Paneque-Carreño et al. 2021),
then feeds the disc with mass. In particular, infall is an alternative way to
trigger gravitational instability (Kratter & Matzner 2006; Kratter et al.
2008; Kratter & Lodato 2016). Gravitational instability occurs when the Toomre
parameter is of the order of unity, and this threshold can be reached by
cooling the disc (i.e. decreasing the sound speed) or by adding mass (i.e.
increasing the surface density). When GI is triggered by infall, there is a
mechanism akin to the thermal saturation (Kratter & Lodato 2016).

Our $\alpha_{\text{GI}}$ estimate remains agnostic to the source of self-
regulation, whether it arises from cooling or from the addition of mass to the
disc. Indeed, what we are measuring through the wiggle is the amplitude of the
surface density perturbation, $\delta\Sigma/\Sigma$. It is intrinsically
linked to the efficiency of angular momentum that is transported by the
spiral, regardless of the origin of the instability.

###  4.2 Planet formation in Elias 2-27

The value of $\beta$ we measure is much higher than the threshold for disc
fragmentation into bound gas clumps (Gammie 2001; Deng et al. 2017), implying
that (while the disc is gravitationally unstable) direct planet formation
through gravitational instability is unlikely. However, many studies
(Paardekooper 2012; Young & Clarke 2015) showed that stochastic fragmentation
of gas in spiral arms can happen for high values of $\beta$. As for the solid
component, Longarini et al. (2023a, b); Rowther et al. (2024) investigated the
possibility of forming planetary cores in gravitationally unstable discs
through dust collapse. They found that for a sufficiently long cooling time,
$\beta>10,$ and high disc-to-star mass ratio, $M_{d}/M_{\star}\sim 0.2$, dust
efficiently collects inside spiral arms and its dispersion velocity is so low
to induce collapse into bound objects with a mass of $\sim
10\text{M}_{\oplus}$. The inferred disc-to-star mass ratio (Veronesi et al.
2021) and the cooling time for Elias 2-27 make it a perfect candidate for
planet formation through dust collapse.

Huang et al. (2018) characterised annular substructures in the discs within
the DSHARP sample and found that Elias 2-27 has a gap at $R_{g}=69.1\pm 0.4$au
with a width of $\Delta=14.3\pm 1.1$au. Although several mechanisms can
explain the origin of gaps in protoplanetary discs, a common explanation is
planet disc interaction. Under the planetary interpretation, the width of the
gap scales as the Hill radius of the planet, defined as:

$R_{h}=\left(\frac{M_{p}}{3M_{\star}}\right)^{1/3}R_{g},$ (23)

where $M_{p}$ is the mass of the protoplanet. The last relation has been
obtained by averaging results from hydrodynamical simulations. Following
Lodato et al. (2019), the relation between the gap width and the Hill radius
is $\Delta=5.5R_{h}$, that translates into

$M_{p}=3\left(\frac{\Delta}{5.5R_{g}}\right)^{3}M_{\star}.$ (24)

Using the gap width and location of Huang et al. (2018), and the star mass of
Veronesi et al. (2021), the inferred mass of the protoplanet is $M_{p}=24\pm
6\text{M}_{\oplus}$. This result is in good agreement with the mass range of
Longarini et al. (2023a, b). Another element that points towards the dust
collapse is the value of the Toomre parameter. As shown in Longarini et al.
(2023a), when the gravitational instability is driven by the cold component
(dust in this case), the critical value of the Toomre parameter is $>1$, as
observed in Elias 2-27 (see Fig. 1).

##  5 Conclusion

In this work, we investigate the kinematic signatures of gravitational
instability in the protoplanetary disc Elias 2-27. It is well known that
gravitational instability leaves clear kinematic perturbations in molecular
line emission (Hall et al. 2020) and their characteristics are related to the
spiral density wave (Longarini et al. 2021). There are multiple arguments
suggesting that Elias 2-27 is undergoing gravitational instability. Under the
hypothesis that angular momentum is transported through the GI spirals, we
estimate the $\alpha-$viscosity of the system, and link it to the accretion
rate onto the central object. We find $\alpha_{\text{GI}}=0.038\pm 0.018$ and
$\log_{10}\dot{M}_{\star}=-6.99\pm 0.17\text{M}_{\odot}/\text{yr}$. There is a
very good agreement between the observed accretion rate and the one estimated
from our model, pointing to the fact that gravitational instability is at play
in this system and that is indeed driving angular momentum transport. We
underline that the results obtained in this work are valid assuming a disc
model (as described in Sect. 3.3) and by fitting for the amplitude of the
velocity perturbation in the central channel of the velocity map. The range of
disc masses (Veronesi et al. 2021) and cooling factors inferred by our model
makes Elias 2-27 a perfect candidate for dust collapse and the formation of
planetary cores in spiral arms. The gap present in dust continuum emission at
$\sim 70$au points to the presence of a $\sim 20\text{M}_{\oplus}$
protoplanet, in agreement with the mass range of planets formed by collapse of
the dust component by Longarini et al. (2023b).

###### Acknowledgements.

This work has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Sklodowska-Curie grant agreement #
823823 (RISE DUSTBUSTERS project). CL and CJC have been supported by the UK
Science and Technology research Council (STFC) via the consolidated grant
ST/W000997/1. PC acknowledges support by the Italian Ministero dell
Istruzione, Università e Ricerca through the grant Progetti Premiali 2012 –
iALMA (CUP C52I13000140001). J.S. acknowledges financial support from the
Natural Sciences and Engineering Research Council of Canada (NSERC) through
the Canada Graduate Scholarships Doctoral (CGS D) program. The authors thank
Francesco Zagaria, Andrew Sellek and Myriam Benisty for useful discussions.

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# A Cosmological Model without Singularity and Dark Matter

Shi-Hao Chen  Institute of Theoretical Physics, Northeast Normal University,
Changchun 130024, China.
<EMAIL_ADDRESS>
###### Abstract

According to the cosmological model without singularity, there are s-matter
and v-matter which are symmetric and have oppose gravitational masses. In
V-breaking s-matter is similar to dark energy to cause expansion of the
universe with an acceleration now, and v-matter is composed of v-F-matter and
v-W-matter which are symmetric and have the same gravitational masses and
forms the world. The ratio of s-matter to v-matter is changeable. Based on the
cosmological model, we confirm that big bang nucleosynthesis is not spoiled by
that the average energy density of W-matter (mirror matter) is equal to that
of F-matter (ordinary matter). According to the present model, there are three
sorts of dark matter which are v-W-baryon matter (4/27), unknown v-F-matter
(9.5/27) and v-W-matter (9.5/27). Given v-F-baryon matter (4/27) and v-W-
baryon matter can cluster and respectively form the visible galaxies and dark
galaxies. Unknown v-F-matter and v-W-matter cannot cluster to form any
celestial body, loosely distribute in space, are equivalent to cold dark
matter, and their compositions are unknown. The number in a bracket is the
ratio of the density of a sort of matter to total density of v-matter. The
decisive predict is that there are dark celestial bodies and dark galaxies.
The energy of F-matter can transform into the energy of W-matter by such a
process in which the reaction energy is high enough.

Primordial nucleosynthesis, Dark matter, Cosmology of theories beyond the SM,
Dark energy, Physics of early universe

###### pacs:

95.30.Cq, 98.80.Es, 98.80.Cq, 95.35.+d, 11.10Wx

LABEL:FirstPage1 LABEL:LastPage#146

###### Contents

  1. I Introduction
  2. II The average density of W-matter is equal to that of F-matter
  3. III The sorts and average density of dark matter
  4. IV The interaction of F-scalar fields and W-scalar fields
  5. V Features and observation of dark matter in present model
  6. VI Conclusions

##  I Introduction

Mirror matter as dark matter has been presented${}^{[1]}.$ In order that the
model is consistent with the standard cosmological framework, the model
ascribes the macroscopic asymmetry of the universe to asymmetric initial
conditions of matter and mirror matter[2]. However, when temperature is high
enough, the expectation values of all Higgs fields are zero so that the static
masses of all particles are zero, and all particles can transform from one
into other. Consequently the energy densities of two sorts of matter must be
equal if both are symmetric so that their degrees of freedom are equal to each
other. Thus the asymmetric initial conditions make one to be uncomfortable.

A quantum field theory without divergence has been constructed which can
obtain all results of the given quantum field theory and in which there is no
divergence of loop-corrections and the energy of the vacuum must be zero
without normal product[3]. There must be two sorts of matter in this quantum
field theory. The two sorts of matter are called $F-matter$ and $W-matter$,
respectively. $F-matter$ and $W-matter$ are symmetric, and there are only
gravitation and very weak interaction by Higgs bosons in low energy between
both. $F-matter$ forms the given world and $W-matter$ is dark matter for a
$F-observer^{[4]}$. This model of dark matter is equivalent to the model of
mirror matter as dark matter.

A cosmological model without singularity has been constructed${}^{[5]}.$ The
model can well explain evolution of the universe, formation of large scale
structure and the features of huge voids, naturally determines the
cosmological constant to be zero, has no singularity and gives some
predictions. Based on the cosmological model, we will see in the present paper
that the average density of $W-matter$ may be equal to that of $F-matter$. The
average density of dark matter is the $23/4$ times of that of visible matter,
because dark matter is composed of $W-matter$ and a part of $F-matter$.

Second 2 discusses that the average density of W-matter is equal to that of
F-matter. Second 3 discusses the sorts and average density of dark matter.
Second 4 the interaction of F-scalar fields and W-scalar fields. Section 5 is
the conclusions.

##  II The average density of W-matter is equal to that of F-matter

According to the cosmological model without singularity, the space evolving
equations are[5]

$\overset{\cdot}{R}^{2}(t)+K=\eta\left[\rho_{v}+V_{v}(\varpi_{v})+V_{0}-\rho_{s}-V_{s}(\varpi_{s})\right]R^{2}\left(t\right)=\eta\left(\rho_{g}+V_{g}\right)R^{2}\left(t\right),$
(1)

where $\omega=\Omega,$ $\Phi,$ and $\chi,$ $\eta\equiv 8\pi G/3;$ $\rho_{v}$
and $\rho_{s}$ are the mass densities of $v-matter$ and $s-matter$
$\left(\text{here }c=1\right)$, respectively; $V_{v}$ and $V_{s}$ are the
densities of $v-Higgs$ potential energy and $v-Higgs$ potential energy,
respectively. the curvature factor $K$ is a function of the gravitational mass
density, and

$\rho_{g}\equiv\rho_{v}-\rho_{s},\text{ \ }p_{g}=p_{v}-p_{s},\text{ \
}V_{g}=V_{v}(\varpi_{v})+V_{0}-V_{s}(\varpi_{s}).$ (3) $1\geq K>0\text{ \ for
}\rho_{g}>0,\text{ }K=0\text{ \ for }\rho_{g}=0,\text{ \ }0>K\geq-1\text{ \
for }\rho_{g}<0.$ (4)

In $V-breaking$ in which the expectation values of $v-Higgs$ fields
$\langle\omega_{v}\rangle\equiv\varpi_{v}=0$, and the expectation values of
$s-Higgs$ fields
$\langle\omega_{s}\rangle^{\prime}\equiv\varpi_{s}^{\prime}\neq 0,$ the
gravitational mass of $v-matter$ is positive and the gravitational mass of
$s-matter$ is negative.

When temperature rises to the critical temperature $T_{cr}$ because space
contracts${}^{[5]},$ $\varpi_{v}=\varpi_{s}=0,$ $\rho_{v}=\rho_{s},$
$V_{v}(\varpi_{v})=V_{s}(\varpi_{s})=0$ so that

$\displaystyle\overset{\cdot}{R}^{2}(t)$ $\displaystyle=$
$\displaystyle-K+\eta V_{0}R^{2}\left(t\right),\TCItag{5}$ (2)
$\displaystyle\overset{\cdot\cdot}{R}(t)$ $\displaystyle=$ $\displaystyle-\eta
V_{0}R\left(t\right)<0.\TCItag{6}$ (3)

There is the highest temperature $T_{\max}$ at which $\overset{\cdot}{R}=0,$
$R\left(t\right)=R_{\min}\equiv\sqrt{K/\eta V_{0}}$ and space inflation must
occur.

$V-matter$ is composed of $v-F-matter$ and $v-W-matter$, $s-matter$ is
composed of $s-F-matter$ and $s-W-matter$. $F-matter$ and $W-matter$ are
symmetric so that divergence of loop corrections is eliminated and the energy
of the vacuum is determined to be zero. There is only the gravitation and the
interaction by Higgs bosons between $F-particles$ and $W-particles.$ The
interaction by Higgs bosons can be ignored when temperature is low because the
masses of the Higgs bosons are all very large. Consequently $W-matter$ is
regarded dark matter relative to $F-matter$, and vice versa. These are the
necessary inferences of the quantum field theory without divergence${}^{[3]}.$
$F-matter$ and $W-matter$ correspond to ordinary matter and mirror matter in
Ref${}^{[1,2]},$ respectively.

The state with $T\geq T_{cr}$ is such a state with the highest symmetry.
$\rho_{sF}=\rho_{sW}=\rho_{vF}=\rho_{vW}$ and $T_{sF}=T_{sW}=T_{vF}=T_{vW}=T$
in this state because $F-matter$ and $W-matter$ are symmetric and can
transform from one into another, $s-matter$ and $v-matter$ are symmetric and
can transform from one into another, and the thermal equilibrium can realized
due to $\overset{\cdot}{R}\sim 0$ when $T\geq T_{cr}.$

After space inflation, reheating process occurs. After reheating process, the
potential energy density transforms into $\rho_{v}^{\prime}=xV_{0}$ and
$\rho_{s}^{\prime}=\left(1-x\right)V_{0}.$
$\rho_{v}^{\prime}>\rho_{s}^{\prime}$ because
$\langle\omega_{v}\rangle=0\longrightarrow\langle\omega_{v}\rangle_{0}\neq 0,$
$\langle\omega_{s}\rangle=0\longrightarrow\langle\omega_{s}\rangle_{0}=0,$
$V_{v}=0\longrightarrow-V_{0}$ and $V_{s}=0\longrightarrow 0^{[5]}.$ Thus,
after reheating process, the evolving equations become

$\overset{\cdot}{R}^{2}(t)+K=\eta\left[\rho_{vF}+\rho_{vW}-\rho_{sF}-\rho_{sW}\right]R^{2}\left(t\right)=\eta\left[2\rho_{vF}-2\rho_{sF}\right]R^{2}\left(t\right),$
(7)
$\overset{\cdot\cdot}{R}(t)=-\eta\left[\left(\rho_{vF}+3p_{vF}\right)-\left(\rho_{sF}+3p_{sF}\right)\right]R\left(t\right),$
(8)

Here we still denote $\rho_{v}^{\prime}+\rho_{v}$ and
$\rho_{s}^{\prime}+\rho_{s}$ by $\rho_{v}$ and $\rho_{s},$ respectively, for
convenience.

It is seen from $\left(4\right)$ and $\left(7\right)$ that in contrast with
the conventional cosmological models, although $\rho_{vF}=\rho_{vW},$ the
$\overset{\cdot}{R}^{2}(t)+K$ cannot doubled because of
$\rho_{s}=\rho_{sF}+\rho_{sW}.$ Consequently the big bang nucleosynthesis
cannot be spoiled by $\rho_{vF}=\rho_{vW}$ provided $\rho_{s}$ is suitably
chosen. For example, if $\rho_{s}=\rho_{vW}$ in the stage of the big bang
nucleosynthesis, the evolving equations $\left(7\right)-\left(8\right)$ become

$\overset{\cdot}{R}^{2}(t)+K=\eta\rho_{vF}R^{2}\left(t\right)$
$\overset{\cdot\cdot}{R}(t)=-\frac{1}{2}\eta\left(\rho_{vF}+3p_{vF}\right)R\left(t\right),$

This is consistent with the conventional theory.

##  III The sorts and average density of dark matter

Recent astronomical observations show that the universe expanded with a
deceleration early and is expanding with an acceleration now. This implies
that there is dark energy[6]. $\rho_{de}/\rho_{tot}=0.73,$
$\rho_{M}/\rho_{tot}=0.27$, $\rho_{M}=\rho_{VM}+\rho_{DM}$,
$\rho_{VM}\sim\rho_{B},$ $\rho_{DM}/\rho_{tot}=0.23$ and
$\rho_{B}/\rho_{tot}=0.04$, here $\rho_{de}$ is the density of dark energy,
$\rho_{tot}$ is the density of the total energy of the universe
$\left(c=1\right)$, $\rho_{VM}$ is the energy density of visible matter,
$\rho_{DM}$ is the energy density of dark matter, and $\rho_{B}$ is the energy
density of visible baryon matter. According to the cosmological model without
singularity[5], in the $V-breaking$, the effects of $s-matter$ are equivalent
to that of the so-called dark energy, and $\rho_{v}=\rho_{M}$. According to
this dark-matter model $[3,4]$, because of the symmetry of $F-matter$ and
$W-matter$, we have

$\displaystyle\rho_{M}$ $\displaystyle=$
$\displaystyle\rho_{v}=\rho_{vF}+\rho_{vW}=2\rho_{vF},\text{ \
}\rho_{B}=\rho_{vFB},$ $\displaystyle\rho_{vF}$ $\displaystyle=$
$\displaystyle\rho_{vFB}+\rho_{vFu},\text{ \
}\rho_{vW}=\rho_{vWB}+\rho_{vWu},$ $\displaystyle\rho_{vFB}$ $\displaystyle=$
$\displaystyle\rho_{vWB},\text{ \ \ }\rho_{vFu}=\rho_{vWu},\text{ \
}\rho_{vD}=\rho_{vFu}+\rho_{vW}=\rho_{DM},\TCItag{9}$ (4)

where $\rho_{v}$ is the total energy density of $v-matter$, $\rho_{vF}$ and
$\rho_{vW}$ are respectively the energy density of $v-F-matter$ and the energy
density of $v-W-matter,$ $\rho_{vFB}$ and $\rho_{vWB}$ are respectively the
energy density of $v-F-baryon$ matter $(v-FBM)$ and the energy density of
$v-W-baryon$ matter $(v-WBM)$, $\rho_{vFu}$ is the energy density of unknown
$v-F-matter$ $(v-UFM),$ $\rho_{vWu}$ is the energy density of $v-W-matter$
$(v-UWM)$ corresponding to $v-UFM,$ and $\rho_{vD}$ is the total energy
density of invisible $v-matter$. Here $v-FBM$ is the given and visible matter
which contains given baryon matter, black holes and neutrinos etc., $F-matter$
contains $v-FBM$ and invisible and unknown $v-UFM$. Considering
$\rho_{vF}=\rho_{vW}$ because $F-matter$ and $W-matter$ are symmetric and can
transform from one into another when temperature is high enough, we can
determine the ratios of a density to another.

$\displaystyle\frac{\rho_{vF}}{\rho_{vW}}$ $\displaystyle=$
$\displaystyle\frac{0.27/2}{0.27/2}=1=\frac{\rho_{vFB}}{\rho_{vW}B}=\frac{\rho_{vFu}}{\rho_{vW}u},\TCItag{10a}$
(5) $\displaystyle\frac{\rho_{vFB}}{\rho_{v}}$ $\displaystyle=$
$\displaystyle\frac{\rho_{vWB}}{\rho_{v}}=\frac{0.04}{0.27}=\frac{4}{27},\TCItag{10b}$
(6) $\displaystyle\frac{\rho_{vFu}}{\rho_{v}}$ $\displaystyle=$
$\displaystyle\frac{\rho_{vWu}}{\rho_{v}}=\frac{0.27/2-0.04}{0.27}=\frac{9.5}{27},\TCItag{10c}$
(7) $\displaystyle\frac{\rho_{vD}}{\rho_{v}}$ $\displaystyle=$
$\displaystyle\frac{\rho_{vW}+\rho_{vFu}}{\rho_{v}}=\frac{23}{27},\TCItag{10d}$
(8) $\displaystyle\frac{\rho_{vD}}{\rho_{vFB}}$ $\displaystyle=$
$\displaystyle\frac{0.27/2+0.095}{0.04}=\frac{23}{4}.\TCItag{10e}$ (9)

Thus, according to the present model${}^{[3]},$ there are three sorts of dark
matter which are $v-UFM,$ $v-WBM$ and $v-UWM.$ Given $v-FBM$ can cluster and
form visible galaxies. $v-WBM$ can cluster and form dark galaxies. $v-UFM$ and
$v-UWM$ cannot cluster to form any celestial body, loosely distribute in
space, and their compositions are unknown.
$\rho_{vWu}/\rho_{vFu}=\rho_{vFB}/\rho_{vWB}=1$ is invariant because of the
symmetry of $W-matter$ and $F-matter$.

According to the cosmological model without singularity${}^{[5]},$ there are
$s-matter$ and $v-matter$ which are symmetric and repulsive each other. In
$V-breaking$, $v-SU(5)$ breaks into $v-SU(3)\times U(1),$ $v-FBM$ forms the
visible world, $v-WBM,$ $v-UFM$ and $v-UWM$ form the dark-matter world;
$s-SU(5)$ does not break, all $s-particles$ form $s-SU(5)$ color single states
which loosely distribute in whole space and cause space to expand with an
acceleration in the present stage. This is because there is no interaction
similar to the given electroweak interaction among the $s-SU(5)$ color single
states[5].

$\rho_{s}/\rho_{v}$ is changeable as space expands[5] because
$\rho_{vM}\propto R^{-3},$ $\rho_{v\gamma}\propto R^{-4},$ and
$\rho_{s}=\rho_{sM}\propto R^{-3}.$ Here $\rho_{vM}$ and $\rho_{v\gamma}$ are
respectively the energy density of massive $v-particles$ and the energy
density of massless $v-particles$, $\rho_{s}$ is the energy density of
$s-SU(5)$ color single states. The masses of all color single states are not
zero, hence $\rho_{s}\propto R^{-3}.$

##  IV The interaction of F-scalar fields and W-scalar fields

The Ref. $[7]$ given the interaction of $F-scalar$ fields and $W-scalar$
fields

$V=-\frac{2A}{225}Tr\Phi_{f}^{2}Tr\Phi_{w}^{2},$ (11)

where $\Phi_{f}$ and $\Phi_{w}$ are the $\underline{24}$ representation of
$SU(5)$ group. The breaking component is

$\Phi_{i}=Diagonal\left(1,1,1,-\frac{3}{2},-\frac{3}{2}\right)\left(\sigma_{i}+\varphi_{i}\right),$
(12)

where the subscript $i=f,$ $w.$ From $\left(11\right)-\left(12\right)$ we
obtain

$V_{fw}=-A\left(2\sigma_{f}\sigma_{w}\varphi_{f}\varphi_{w}+\sigma_{f}\varphi_{f}\varphi_{w}^{2}+\sigma_{w}\varphi_{w}\varphi_{f}^{2}+\frac{1}{2}\varphi_{f}^{2}\varphi_{w}^{2}\right).$
(13)

$\mid\sigma_{w}\mid=\mid\sigma_{f}\mid$ because of the symmetry of $s-matter$
and $f-matter$. Both $\sigma_{i}$ and $m\left(\varphi_{i}\right)$ are
functions of temperature $T$. When $T\geq T_{cr},$
$\sigma_{i}=m\left(\varphi_{i}\right)=0$. Consequently $f-particles$ and
$w-particles$ can easily transform from one to another so that
$\rho_{F}=\rho_{W}.$ When $T\sim 0,$ both $\mid\sigma_{i}\mid$and
$m\left(\varphi_{i}\right)$ are large enough. Consequently interaction between
$f-particles$ and $w-particles$ by the scalar bosons may be ignored. Thus
there is only the gravitation between $f-matter$ and $w-matter$ when
temperature is low.

There are the couplings of fermions (and gauge particles) and scalar
bosons${}^{[7]}.$ Hence there are the interactions of $f-fermions$ and
$w-fermions$ shown in figures 1-3 and the interactions of $f-gauge$ bosons and
$w-gauge$ bosons via the scalar bosons $\varphi_{f}$ and $\varphi_{w}.$ In the
figures the dotted lines with arrows denote $W-fermion$ field $\psi_{w}$, the
dotted lines without arrow denote $W-scalar$ field $\varphi_{w},$ the lines
with arrows denote $F-fermion$ field $\psi_{f}$, the lines without arrow
denote $F-scalar$ field $\varphi_{f},$ $M^{2}=-2A\sigma_{f}\sigma_{w}$,
$R_{f}=-A\sigma_{f}$, $R_{w}=-A\sigma_{w}$ and $S=-A/2.$

It can be seen from figure 1 and $\left(13\right)$ that when
$-A\sigma_{f}\sigma_{w}>0$ and $k^{2}-m^{2}<0$ or $-A\sigma_{f}\sigma_{w}<0$
and $k^{2}-m^{2}>0,$ $f-fermions$ and $w-fermions$ are repulsive each other;
when $-A\sigma_{f}\sigma_{w}>0$ and $k^{2}-m^{2}>0$ or
$-A\sigma_{f}\sigma_{w}<0$ and $k^{2}-m^{2}<0,$ $f-fermions$ and $w-fermions$
are attractive each other.

Figure 1: $f+\overline{f}\longrightarrow w+\overline{w}$ realized by the tree
digram.

Figure 2: $f+\overline{f}\longrightarrow w+\overline{w}$ realized by the two
one-loop digrams.

Figure 3: $f+\overline{f}\longrightarrow w+\overline{w}$ realized by the two-
loop diagram.

##  V Features and observation of dark matter in present model

According to the present model[3,4], $v-UFM$ and $v-UWM$ cannot form cluster,
loosely distribute in space and have positive gravitational masses, hence both
should be identified as cold dark matter. From $\left(10\right)$ we have

$\displaystyle\frac{\rho_{vFu}+\rho_{vWu}}{\rho_{v}}$ $\displaystyle=$
$\displaystyle\frac{0.095\times 2}{0.27}=\frac{19}{27},$
$\displaystyle\frac{\rho_{vFu}+\rho_{vWu}}{\rho_{vD}}$ $\displaystyle=$
$\displaystyle\frac{0.095\times 2}{0.23}=\frac{19}{23}.\TCItag{14}$ (10)

It is seen that the present model cannot differ from the cold dark matter
model by $v-UFM$ and $v-UWM.$ $W-baryon$ matter $v-WBM$ can form clusters of
dark matter and dark galaxies[3,4]. There possibly are such celestial bodies
which are composed of mixture of $F-matter$ and $W-matter$ because both have
positive gravitational masses. This is a decisive predict of the present model
and mirror dark matter model. Dark celestial bodies flying to the earth are
possibly detected by probing gravity-meters of clustering dark matter[4].

It can be seen from $\left(13\right)$ and figures $1-3$ that the energy of
$F-matter$ can transform into the energy of $W-matter$ by such a process in
which the reaction energy is high enough.

##  VI Conclusions

According to the cosmological model without singularity, there are $s-matter$
and $v-matter$ which are symmetric and have oppose gravitational masses. In
$V-breaking$ $s-matter$ is similar to dark energy to cause expansion of the
universe with an acceleration now, and $v-matter$ is composed of $v-F-matterv$
and $v-W-matterv$ which are symmetric and have the same gravitational masses.
The ratio of $s-matter$ to $v-matter$ is changeable.

Based on the cosmological model, we confirm that big bang nucleosynthesis is
not spoiled by that the average energy density of $W-matter$ (mirror matter)
is equal to that of $F-matter$ (ordinary matter).

According to the present model, there are three sorts of dark matter which are
$v-W-baryon$ matter ($4/27)$ and unknown $v-F-matterv$ ($9.5/27)$ and $v-W-
matter$ ($9.5/27).$ Given $v-F-bayon$ matter ($4/27)$ can cluster to form the
visible galaxies. $V-W-bayon$ matter can cluster to form dark celestial bodies
and dark galaxies. Unknown $v-F-matterv$ and $v-W-matter$ cannot cluster to
form any celestial body, loosely distribute in space, are equivalent to cold
matter, and their compositions are unknown. The number in a bracket is the
ratio of the density of a sort of matter to total density of $v-matter$.

The decisive predict is that there are dark celestial bodies and dark
galaxies. The energy of $F-matter$ can transform into the energy of $W-matter$
by such a process in which the reaction energy is high enough.

Acknowledgement

I am very grateful to professor Zhao Zhan-yue and professor Wu Zhao-yan for
their helpful discussions and best support.

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