Cosmological Prediction of the CSST Ultra Deep Field Type Ia Supernova Photometric Survey

Type Ia supernova (SN Ia) plays a crucial role in the discovery of cosmic accelerating expansion and the study of the nature of dark energy (Riess et al., 1998; Perlmutter et al., 1999). In the past two decades, significant advancements have been made in calibrating and expanding SN Ia samples across various redshift ranges. In recent years, many SN Ia surveys have been performed, such as Dark Energy Survey (DES, Abbott et al. 2019; Möller et al. 2022; DES Collaboration et al. 2024), Panoramic Survey Telescope and Rapid Response System (PanStarrs, Rest et al. 2014), Supernova Legacy Survey (SNLS, Guy et al. 2010), Equation of State: SupErNovae trace Cosmic Expansion (ESSENCE, Miknaitis et al. 2007), and several Harvard-Smithsonian Center for Astrophysics (CFA) surveys (Riess et al., 1999; Jha et al., 2006; Hicken et al., 2009, 2012)). Besides, there are deeper SN surveys conducted with the Hubble Space Telescope (HST), such as Great Observatories Origins Deep Survey and Probing Acceleration Now with Supernova (GOODS+PANS, Riess et al. 2004, 2007), Supernova Cosmology Project (SCP, Suzuki et al. 2012), Cosmic Assembly Near Infra-Red Deep Extragalactic Legacy Survey and Cluster Lensing And Supernova survey with Hubble (Candels+CLASH, Riess et al. 2018), etc. Some combined SN Ia samples are also recalibrated and generated as extensive datasets, such as Joint Light-curve Analysis (JLA, Betoule et al. 2014), Supernova H0 for the Equation of State (SH0ES, Riess et al. 2019), and Pantheon+ (Scolnic et al., 2022). However, most of the current SN Ia samples are based on ground-based telescopes, covering relatively low redshift ranges. High-redshift SN Ia samples at $z\gtrsim 1$ are still extremely limited, which are important to accurately measure the cosmic expansion history and variation of the equation of state of dark energy.

In the next-generation ground- and space-based Stage IV surveys, such as the Vera C. Rubin Observatory (or LSST, Ivezić et al. (2019)), Euclid (Laureijs et al., 2011; Amendola et al., 2018), Nancy Grace Roman Space Telescope (RST) (or WFIRST, Akeson et al. (2019)), and China Space Station Telescope (CSST Zhan 2011; Gong et al. 2019; Zhan 2021), SN observation will be implemented as one of the main cosmological probes in these surveys. Thousands to millions of SNe Ia are expected to be detected by these surveys in the next decade, and high-redshift SN Ia samples will be greatly expanded for precise cosmological study. However, since most of these SNe will be discovered photometrically, it would be quite challenging for spectroscopic follow-up measurements to accurately identify SNe Ia from the core-collapse supernovae (CCSNe) and determine their redshifts, given the huge amounts of SN data covering large redshift ranges. Therefore, it is necessary to explore the method of SN Ia confirmation and redshift estimation from the SN photometric data obtained by these future surveys, and assess the impacts on the relevant cosmological analysis.

Here we focus on the CSST SN photometric survey to investigate its capability of detecting and identifying SN Ia, and the constraint power on the dark energy equation of state and other cosmological parameters. The CSST is an upcoming Stage IV space-based telescope as a major science project of China Manned Space Program, which is expected to be launched in 2026 (Zhan, 2011; Gong et al., 2019; Zhan, 2021). The diameter of its primary mirror is 2 m, and the field of view is 1.1 deg². It carries several astronomical instruments, including the survey camera, terahertz receiver, multichannel imager (MCI), integral field spectrograph (IFS), and coronagraph specialized in cool planet imaging. The CSST cosmological surveys performed by the survey camera plan to explore about 17,500 deg² and 400 deg² sky areas as the wide and deep field surveys, respectively, with multi-color photometric imaging and slitless spectroscopic observations in about ten years. Various important cosmological probes, such as weak and strong gravitational lensing, galaxy angular and redshift-space clustering, and galaxy clusters, will be carried out for studying the formation and evolution of cosmic large-scale structure and properties of dark energy and dark matter.

In addition, CSST also can perform an ultra-deep-field survey (CSST-UDF) covering $\sim$9 deg² sky area, which is planned to be located at high Galactic latitude. The single exposure time is 250 s with 60 exposures for each band in about two years. The magnitude limit in AB mag for point sources with 5$\sigma$ detection can reach $i\simeq 28.2$ in the CSST-UDF survey (Cao et al., 2022). For a single exposure, we can find that the magnitude limits can achieve 25.3, 25.7, 26.4, 26.1, 25.9, 25.4, 24.4 for the seven photometric bands covering $\sim$2500Å to 10000Å, i.e. $\it NUV$, $\it u$, $\it g$, $\it r$, $\it i$, $\it z$, and $\it y$. So CSST has great potential in SN Ia detection and photometric redshift (photo-$z$) estimation, especially at high redshifts, given the survey cadence and depth. Besides, as we discuss later, the host galaxy also can be detected with a certain probability by these bands, which can provide more information on the evaluation of redshift and dust extinction in the SN Ia analysis. In Li et al. (2023), they also explore the SN Ia detection and cosmological constraint by the CSST-UDF survey. Here we will consider the contamination from CCSNe, discuss the contamination reduction method, and adopt more realistic survey strategy.

In this work, we generate the light-curve mock data for both SN Ia and different types of CCSNe, based on the SN spectral energy distribution (SED) templates, the CSST instrumental design, and survey strategy of the CSST-UDF survey. We then use the SN Ia templates to fit the light-curve mock data at different CSST photometric bands, and try to distinguish and remove CCSNe according to the chi-square estimation. In order to further suppress the CCSN contamination, we also propose to use a method based on Chauvenet’s criterion for identifying and eliminating CCSNe based on the derived data of the distance modulus. We investigate the effectiveness of this method, and perform the cosmological constraints on the equation of state of dark energy. The baryon acoustic oscillation (BAO) mock data from CSST spectroscopic survey are also included to reduce the degeneracies between different cosmological parameters.

The structure of this paper is as follow: Section 2 provides a detailed description of the CSST-UDF observation, SN templates, mock data generation, and light curve fitting process and result. In Section 3, we discuss the estimation of distance modulus, and the relevant contamination removal method. We present the cosmological constraint results for the non-flat $\Lambda$CDM model and flat $w$CDM model using various SN datasets and CSST BAO mock data in Section 4. Finally, we give summary and discussion in Section 5. We assume a flat $\Lambda$CDM model as the fiducial cosmology with the dark matter energy density parameter $\Omega_{\text{M}}=0.3$ and the reduced Hubble constant $\text{h}=0.7$.

We employ SNCosmo (Barbary et al., 2016) as a tool to generate and fit the mock light curve data. SNCosmo is a supernova cosmology simulation Python package, which is similar to SNANA (Kessler et al., 2009). It includes functions such as light curve generation and parameter fitting, and is easy to modify and customize. Based on the SN SED templates, we first generate natural SN populations for both SN Ia and CCSN, and then derive a mock light curve SN catalog with flux and error for the CSST photometric bands, considering the CSST-UDF survey strategy. After applying selection criteria, we can obtain the final mock light curve data.

Using the fitting results of the light curve parameters, i.e. $m_{B}$, $x_{1}$ and $c$, we can calculate the distance modulus for the $i$th SN in the sample, which is given by

Here $\alpha$, $\beta$, and $M_{0}$ are nuisance parameters, and we set them as free parameters in the fitting process. The error of distance modulus consists of three parts, i.e.

Here $\sigma_{\mathrm{int}}=0.1$ is the intrinsic dispersion of the SN Ia absolute magnitude (Kenworthy et al., 2021; Vincenzi et al., 2024), $\sigma_{\mu,z}$ is the error from the photo-$z$ uncertainty $\sigma_{z}$, which has the relation as $\sigma_{\mu,z}=\frac{5}{\ln(10)}\frac{1+z}{z(1+z/2)}\sigma_{z}$ (Campbell et al., 2013), and $\sigma^{2}_{\mu,i}$ is the error accounting for the uncertainties of $m_{B}$, $x_{1}$ and $c$ for each SN, which can be derived by their covariance matix obtained in the light curve fitting process.

On the other hand, the theoretical distance modulus is related to the luminosity distance $d_{\rm L}$, which is determined by the cosmological model, and we have

Here $d_{\rm L}$ is given by

where $H(z)$ is the Hubble parameter that takes different forms under various cosmological models. In this work, it can be expressed as

Hence the cosmological parameter vector $\boldsymbol{\theta}_{\rm c}=(\Omega_{\rm M},\Omega_{\rm DE},w,H_{0})$, where $\Omega_{\rm M}$ and $\Omega_{\rm DE}$ are the energy density parameters of dark matter and dark energy, $\Omega_{\rm k}=1-\Omega_{\rm M}-\Omega_{\rm DE}$ is the curvature energy density parameter, $w$ is the dark energy equation of state, and $H_{0}$ is the Hubble constant. In this study, we mainly explore two cosmological models, i.e. the flat $w$CDM and nonflat $\Lambda$CDM model. In the flat $w$CDM model, since $\Omega_{k}=0$ and $\Omega_{\rm DE}=1-\Omega_{\mathrm{M}}$, we focus on $w$, $\Omega_{\mathrm{M}}$, $H_{\text{0}}$. In the nonflat $\Lambda$CDM model, we have $w$=-1, and the free parameters are $\Omega_{\rm DE}=\Omega_{\Lambda}$, $\Omega_{\mathrm{M}}$, and $H_{\text{0}}$. Besides the cosmological parameters, we also have $\alpha$, $\beta$ and $M_{0}$ as the nuisance parameters in the model.

In Figure 9, we show the Hubble diagram for the input redshifts. To show it clearly, the values of $\alpha$, $\beta$ and $M_{0}$ are fixed to be the fiducial values as shown in Section 2.2 when generating the data points, and the theoretical curve is obtained by assuming the fiducial flat $\Lambda$CDM model. We can find that most of the SN Ia data are overlapped with or closely around the the theoretical curve, which could correctly indicate the underlying cosmology. On the other hand, the rest of the CCSNe after the selection of the light curve fitting process are mainly at $0.4<z<0.8$ (CCSNe at low redshifts have good data quality and are easily rejected, and CCSNe at $z>0.8$ are hardly detected in the CSST-UDF survey), and most of them have large deviation from the SN Ia data or the theoretical curve as outliers. This provides an effective way to further remove the CCSNe from the SN dataset.

Here we propose to use a method based on Chauvenet’s criterion to remove the CCSN contamination by fitting the distance modulus with the MCMC (Thompson, 1998). Chauvenet’s criterion is widely used for outlier exclusion, and is commonly applied in various SN cosmological studies (e.g. Conley et al., 2011; Foley et al., 2018; Scolnic et al., 2022; Vincenzi et al., 2023). Here we employ the Median Absolute Deviation (MAD) method to derive the median $\tilde{X}$ and the dispersion of the sample $\sigma_{X}=1.4826\,{\rm median}\left(\left|X_{i}-\tilde{X}\right|\right)$ for removing the outliers. The MAD can naturally suppress the weighting of the outliers, and could be more effective to remove the CCSN contamination. Based on Chauvenet’s criterion, we remove all data points with $\left|X_{i}-\tilde{X}\right|>n\,\sigma_{X}$, where $n$ is found to be 3.68 for our sample size with total 2175 SNe. Besides, considering the error of the data point $E_{Xi}$, we will keep the data point when $E_{Xi}>\left|X_{i}-\tilde{X}\right|$.

Since $X_{i}=\mu_{i}$ here, as shown in Equation (4), it not only depends on the light curve parameters, but also the nuisance parameters $\alpha$, $\beta$ and $M_{0}$. These nuisance parameters should be determined or fitted along with the cosmological parameters in a cosmological model. We use the flat $w$CDM model for jointly fitting the cosmological and nuisance parameters, and deriving $\mu_{i}$ to further remove CCSNe. Note that the result of this removing process is not sensitive to a specific cosmological model, and we can obtain the same result by using the nonflat $\Lambda$CDM model.

We employed the MCMC method for parameter fitting using emcee package (Foreman-Mackey et al., 2013). MCMC can derive the parameter probability distribution by illustrating the posterior probability $P(\boldsymbol{\theta}\mid\boldsymbol{D})$ for the parameter set $\boldsymbol{\theta}$ given the observational dataset $\boldsymbol{D}$, and we have

Here $P(\boldsymbol{\theta})$ is the parameter priors assumed to be uniform, $P(\boldsymbol{D})$ is the normalization factor which does not impact our analysis here, and $\mathcal{L}(\boldsymbol{D}\mid\boldsymbol{\theta})\sim e^{-\frac{1}{2}\chi^{2}}$ is the likelihood function, where $\chi^{2}$ is the chi-square given by

The parameter ranges we set are $\Omega_{\rm M}\in(0,0.7)$, $\Omega_{\rm\Lambda}\in(0,2)$, $w\in(-2,0)$, $H_{0}\in(65,75)$, $M_{0}\in(-19.35,-19.15)$, and a range with $\pm 5\%$ relative error for $\alpha$ and $\beta$ in the flat $w$CDM and nonflat $\Lambda$CDM models, which are based on the current meansurements. We generate 80 chains, and each chain contains 10,000 chain points after burn-in.

After obtaining $\mu_{i}$ by the MCMC process, we perform the calibration for further removing the CCSN contamination using the method based on Chauvenet’s criterion. We find that 82 CCSNe and 8 SNe Ia are removed from the dataset. Finally, there are 2085 SNe in the calibrated sample, and 73 CCSNe are left in the sample with the contamination fraction $\sim 3.5\%$, compared to $\sim 7.1\%$ before the calibration. We find that the SNe Ia at $z>0.5$ and $z>1$ can take the fractions of 83% and 16% of the total sample, respectively, which indicates that the CSST-UDF SN sample contains a large number of SNe Ia at high redshifts.

We show the Hubble diagram of the calibrated sample in Figure 10. To compare with Figure 9, we use the fiducial values of the parameters to show the data points after calibration. We can find that the CCSN contamination is significantly reduced, and the remaining CCSN data are all close to the fiducial theoretical curve. The effectiveness of this method also can be indicated by the constraint results of the cosmological parameters as we discuss in the next section.

In addition, we also try other methods to further remove or reduce the CCSN contamination. For instance, we use the statistical method based on Bayesian estimation to reduce the contamination statistically on the cosmological parameter constraint (Kunz et al., 2007; Gong et al., 2010). Additional $\chi^{2}$ with new parameters are added to include the effect of CCSN contamination in the fitting process, and the fraction of contamination also can be derived in this method. However, we find that this kind of method is not quite effective when multi types of CCSNe are considered, probably due to the complexity of the CCSN samples, and the method based on Chauvenet’s criterion works better in this case. Besides, the machine learning method for supernova classification also can be used as an effective tool, given sufficient large and accurate training sets (e.g. Hložek et al., 2023). We will leave it for future study.

After obtaining the calibrated SN sample in the CSST-UDF survey, we investigate the constraints on the cosmological parameters. In Figure 11, we present the constraint results for the flat $w$CDM (left panel) and nonflat $\Lambda$CDM (right panel) models using the pure SN Ia sample (blue contours), uncalibrated SN sample (without calibration using the distance modulus data, in gray contours), and calibrated sample (i.e. UDF sample, in red contours). We can see that there are clear discrepancies for the constraint results of the uncalibrated SN sample from the fiducial values (green lines), which also indicates the necessity of the calibration after the light curve fitting process. On the other hand, the cosmological constraints using the pure SN Ia and calibrated UDF sample are in good agreement, and the contour maps and 1-D probability distribution functions (PDFs) are almost overlapped for some parameters. For the calibrated UDF sample, we find that the relative accuracies of the $1\sigma$ constraint on the cosmological parameters are $\sim 14\%$, $18\%$ and $3.2\%$ for $\Omega_{\rm M}$, $w$ and $H_{0}$ in the flat $w$CDM model, and $\sim 11\%$, $14\%$ and $3.2\%$ for $\Omega_{\rm M}$, $\Omega_{\Lambda}$ and $H_{0}$ in the nonflat $\Lambda$CDM model for the CSST-UDF SN survey. The details can be found in Table 4. Note that the constraint on $H_{0}$ could be worse if assuming a larger prior or range of $M_{0}$ in the fitting process, since there is a strong degeneracy between these two parameters. Compared to the current SN Ia surveys, e.g. Pantheon+ and DESSN-5YR (Brout et al., 2022; DES Collaboration et al., 2024), the CSST-UDF SN survey can improve the cosmological constraints by a factor of $\sim$2.

Besides, in order to explore the improvement of the constraint power, we also include the mock data of baryon acoustic oscillation (BAO) from the CSST spectroscopic wide-field galaxy survey, as another powerful measurement of the cosmic acceleration expansion history. Here we use the mock BAO data from Miao et al. (2023), and the angular diameter distance $d_{\rm A}$ and Hubble parameter $H(z)$ are derived at different redshift bins from $z=0$ to 1.5. The values of $d_{\rm A}$ and $H(z)$, and the relative errors are shown in Table 5. The expression of $d_{\rm A}$ is given by $d_{\rm A}=(1+z)^{-2}d_{\rm L}$. In the MCMC fitting process for the BAO data, we have $\chi_{\rm BAO}^{2}=\chi_{d_{\rm A}}^{2}+\chi_{H}^{2}$. More details of the generation of the CSST BAO mock data can be found in Miao et al. (2023). The BAO mock data we use are generated randomly from a Gaussian distributiaon based on the central values and errors shown in Table 5.

In Figure 12, the constraint results of the SN Ia, BAO, and joint data for the flat $w$CDM (left panel) and nonflat $\Lambda$CDM (right panel) models have been shown and compared. We find that the BAO data can be helpful to break the degeneracy between parameters, and make the constraints more stringent, especially for $H_{0}$. The constraint accuracies of the cosmological parameters after including the BAO data are $\sim 9\%$, $13\%$ and $1.2\%$ for $\Omega_{\text{M}}$, $w$ and $H_{0}$ in the flat $w$CDM model, and $\sim 9\%$, $10\%$ and $1.2\%$ for $\Omega_{\text{M}}$, $\Omega_{\Lambda}$ and $H_{0}$ in the nonflat $\Lambda$CDM model. Hence the constraints on the cosmological parameters can be improved by a factor of $\sim$1.4 for the SN Ia+BAO joint analysis, compared to the case with SN Ia data only.

In this work, we explore the SN Ia observation and cosmological constraint for the CSST-UDF survey. We generate light curve mock data for both SNe Ia and CCSNe, based on the SED templates given in SALT3 and V19 models, the SN natural generation rates, luminosity functions, CSST instrumental design and CSST-UDF photometric survey strategy. Then high-quality light curve data are selected for the light curve analysis. By using the MCMC algorithm embedded in SNCosmo, we fit the SN light curve data to derive the SN Ia light curve parameters (i.e. $z$, $t_{0}$, $m_{B}$, $x_{1}$ and $c$) and distinguish CCSNe as contamination. The SN samples that cannot match the SN Ia model, and the values of $x_{1}$ and $c$ exceeding the SALT3 model requirements are identified and excluded as CCSNe. The resulting SN sample shows a $\sim$7.1% contamination fraction.

To further reduce the contamination, we analyze the data of distance modulus, and adopt a method based on Chauvenet’s criterion and considering the data error. After applying this method, we find that the contamination fraction can be effectively reduced to $\sim 3.5\%$, resulting in a calibrated UDF sample with a SN Ia purity of $\sim$96.5%. This calibrated CSST-UDF SN sample has a large number of SNe Ia at high redshifts with $\sim$16% of SNe Ia are at $z>1$, which is expected to provide stringent constraints on cosmological parameter, such as dark energy equation of state.

Using this SN sample, the constraints on the cosmological parameters are investigated for the flat $w$CDM and nonflat $\Lambda$CDM models. We find that the constraint result of the calibrated sample is in good agreement with that derived from the pure SN Ia sample. This indicates the effectiveness of our method for reducing the CCSN contamination. The constraint accuracy of the cosmological parameters is $\sim$15% for $\Omega_{\rm M}$, $\Omega_{\Lambda}$, and $w$ in the CSST-UDF SN survey, which is better than the current SN Ia surveys by a factor of $\sim$2. When performing a joint analysis with the BAO data from the CSST spectroscopic galaxy survey, the accuracy can be improved by a factor of $\sim$1.4.

We note that there may be still room for improvement in the purity of the dataset. For example, the machine learning classifiers can be a good choice to further suppress the contamination. In the analysis of the DES data, i.e. DESSN-5YR (DES Collaboration et al., 2024), they can obtain a SN Ia purity above 98% by using the machine learning classifier (Möller & de Boissière, 2020; Vincenzi et al., 2023; Möller et al., 2022), which is better than our result ($\sim 96.5\%$). In the future we will try the machine learning method to further improve the dataset quality.

Comparing to other Stage-IV surveys, e.g. LSST, the magnitude limit of the CSST-UDF single exposure is about one magnitude deeper than the LSST Deep Drilling Field (LSST-DDF) (Lochner et al., 2018; Mitra et al., 2023). This means that the sample with a larger high-$z$ SN Ia fraction can be obtained in the CSST-UDF survey. Besides, since the CSST-UDF survey has a great depth in the ultraviolet (UV) band with $NUV\sim 25$ AB mag for a single exposure, the dust model can be well constrained, whcih would be quite helpful to calibrate this one of the biggest systematics in SN Ia cosmology.

We also notice that CSST can detect SNe Ia by the slitless spectroscopic observation, which is very helpful to identify SNe Ia and accurately measure the redshift. Although the magnitude limit of the CSST spectroscopic survey is lower than that in the photometric survey, most of SNe Ia at low redshifts could be measured by the spectroscopic survey. Hence, the current constraint results can be further improved by including the CSST spectroscopic data. Besides the CSST survey camera, CSST-MCI module also is capable to detect hundreds of SNe Ia, especially at high redshifts, which could effectively extend the CSST SN Ia sample. These imply that CSST has great potential in the SN Ia survey, and is expected to provide accurate measurements on cosmic distance and expansion history.

MW and YG acknowledge the support from National Key R&D Pro- gram of China grant Nos. 2022YFF0503404, 2020SKA0110402, and the CAS Project for Young Scientists in Basic Research (No. YSBR- 092). XC acknowledges the support of the National Natural Science Foundation of China through Grant Nos. 11473044 and 11973047, and the Chinese Academy of Science grants ZDKYYQ20200008, QYZDJ-SSW-SLH017, XDB 23040100, and XDA15020200. This work is also supported by science research grants from the China Manned Space Project with Grant Nos. CMS- CSST-2021-B01 and CMS-CSST-2021-A01.

The data that support the findings of this study are available from the corresponding author upon reasonable request.