TY - JOUR
AU1 - Shukla,, Hemant
AU2 - Bonissent,, Alain
AB - Abstract We present the parameterized simulation of an integral-field unit (IFU) slicer spectrograph and its applications in spectroscopic studies, namely, for probing dark energy with type Ia supernovae. The simulation suite is called the fast-slicer IFU simulator (FISim). The data flow of FISim realistically models the optics of the IFU along with the propagation effects, including cosmological, zodiacal, instrumentation and detector effects. FISim simulates the spectrum extraction by computing the error matrix on the extracted spectrum. The applications for Type Ia supernova spectroscopy are used to establish the efficacy of the simulator in exploring the wider parametric space, in order to optimize the science and mission requirements. The input spectral models utilize the observables such as the optical depth and velocity of the Si ii absorption feature in the supernova spectrum as the measured parameters for various studies. Using FISim, we introduce a mechanism for preserving the complete state of a system, called the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix, which allows for compression, reconstruction and spectrum extraction, we introduce a novel and efficient method for spectrum extraction, called super-optimal spectrum extraction, and we conduct various studies such as the optimal point spread function, optimal resolution, parameter estimation, etc. We demonstrate that for space-based telescopes, the optimal resolution lies in the region near R ∼ 117 for read noise of 1 e− and 7 e− using a 400 km s−1 error threshold on the Si ii velocity. techniques: spectroscopic, telescopes, surveys, supernovae: general, dark energy 1 INTRODUCTION With the advent of optical and near-infrared (NIR) telescopes with large fields of view, spectroscopic techniques have evolved from the traditional multi-exposure, single-slit spectroscopy of sources, to the highly efficient, single-exposure, simultaneous dispersion of the wide field that garners collective spectra of multiple objects within the field. This technique, known as integral-field spectroscopy (IFS), is implemented in the integral-field unit (IFU) spectrographs primarily using the slicer (Content 1997), micro lenses (lenslets; Barden & Wade 1988) or optical fibers (Bacon et al. 1995). All the implementations of the IFUs, at the outset, segment and rearrange the two-dimensional (2D) image plane into a continuous array that is subsequently channelled through the spectrograph collimator to the disperser for generating the simultaneous spectra. The spectra are finally integrated at non-overlapping locations on the detector. Thereafter, the spectra are combined to create a data cube made of two spatial coordinates and one wavelength (or velocity) coordinate. In contrast, multi-object spectroscopy (MOS) ignores the spatial information and only generates spectra of sources in the wide field. More recently, the two techniques of IFS and MOS have been combined to design a more flexible approach, called diverse field spectroscopy (DFS; e.g. Murray et al. 2010). The different IFS techniques have their merits and limitations; for a more analytical comparison, see the review by Allington-Smith et al. (2006). More importantly, it must be noted that, compared to the traditional single-slit spectroscopy, all the IFS methodologies are far more effective. IFS eliminates the errors associated with the slit (i.e. those resulting from differential slit-loss related to atmospheric refraction and source positioning). In addition, IFS methods allow for high signal-to-noise ratio (S/N) spectroscopy for wider bandwidth images, thus facilitating the extraction of physical parameters that otherwise cannot be derived easily. The flux variations in frequency are measured to generate velocity maps of extended sources in the aperture synthesis techniques used in radio astronomy. The output is a data cube with two spatial axes and a frequency axis. These data are rich in information about the spatial mapping of the kinematics of sources, chemical compositions, underlying physical processes, column densities, temperatures and magnetic fields. Similar observables are obtained using IFUs in the optical and IR domains. The scientific drivers for IFS cover a wide range from protoplanetary discs to cosmological surveys, and from extended sources to supernovae in the parent galaxy. Many existing and proposed IFUs, ground- and space-based, are leveraging the emerging detector and optical technologies to provide new insights and to significantly enhance the overall understanding of the physical phenomenon in the Universe. In this paper, we examine the simulation of the slicer IFU spectrograph and its applications to Type Ia supernova cosmology. In the slicer IFU spectrograph, the sky image is focused on a segmented (sliced) mirror. Each slice is tilted to reflect the section of the sky on the disperser with a different angle, thus yielding the spectra of the entire sky image from all the slices simultaneously on the detector. For a stable sky image on the field of view, the detector data are not affected by variations of the point spread function (PSF). In contrast to a slit spectrometer, the slicer IFU can be seen as a collection of multiple adjacent slits such that no light is lost, irrespective of the PSF width. A typical design would allow the PSF to cover two or three slices, permitting the collection of most of the photons under normal conditions while minimizing the number of pixels to consider – this might be important for space-based observations where the detector errors are dominant. Realistic simulations of IFUs have broad applications and are critical to study jointly the scientific requirements and technical specifications of ground- and space-based missions. The co-development of hardware prototyping with simulation feedback fosters robust designs and failure testing of the instrument. In addition, simulating a wide range of properties and their overall effects on the scientific goals helps to define the observational boundaries in order to directly optimize the instrument design before committing to the hardware development. The layout of the paper is as follows. In Section 2, we discuss the development of the IFU spectrograph simulator, SpecSim, and its associated challenges. In Section 3, we define what we call the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix, which is a novel method to save the state of the system, to compress data and to super-optimally extract the spectrum. In Section 4, we discuss the challenges of the traditional spectrum extraction techniques and we introduce the new super-optimal spectrum extraction method. The new fast and flexible simulator, the fast-slicer IFU simulator (FISim), is discussed in Section 5. The validation of FISim is performed in Section 6. The application of FISim for Type Ia supernova cosmology is examined in Section 7, followed by the conclusions in Section 8. In the Appendix, we discuss the advances in massively parallel computational architectures and associated programming models that offer opportunities for developing next-generation simulators. 2 MODELLING A SLICER IFU The co-development of the IFU software simulator suite, tentatively titled SpecSim, was aligned with the development of the IFU spectrograph prototype (Ealet et al. 2005, 2006; Aumeunier et al. 2006a, b; Prieto et al. 2008) designed for the Supernova Acceleration Probe (SNAP)/Joint Dark Energy Mission (JDEM) mission (Sholl et al. 2004; Aldering 2005; Bernstein et al. 2012). For an introductory description of the initial efforts of SpecSim, see the discussion by Tilquin et al. (2006) and Young et al. (2007). The title, SpecSim, should not be confused with another and completely unrelated effort by Lorente et al. (2008). The SpecSim simulator suite was co-designed to serve as the software testbed for the SNAP spectrograph. The design parameters of the IFU required delivering high S/N spectrometric identification of the Type Ia supernovae as far as z = 1.7 with an Si ii absorption-line velocity error tolerance of 400 km s−1. The primary science drivers for these requirements focused on mitigating a wide range of systematic errors. The simulations offer an effective framework to exhaustively study various effects that distort the signal and contaminate the parameters of interest. 2.1 Realistic IFU Simulator: SpecSim Fig. 1 depicts a schematic diagram of the full-scale end-to-end data flow of SpecSim. The data flow showcases the simulation of the spectrum of a Type Ia supernova with velocity and optical depth of the Si ii line as the parameters of interest. The simulator is agnostic to the input spectrum; however, the studies discussed in this paper are related to supernova cosmology. In addition, for the purposes of studies and testing, a single spectrum template is used as the input. In principle, a realistic compound spectrum of multiple sources is equally valid. Figure 1. Open in new tabDownload slide Data flow of SpecSim: end-to-end simulation of a Type 1a supernova spectrum with a slicer IFU. Figure 1. Open in new tabDownload slide Data flow of SpecSim: end-to-end simulation of a Type 1a supernova spectrum with a slicer IFU. From the top left, the simulator input is the rest-frame template of the source spectrum of a Type Ia supernova. The spectrum templates, along with the source magnitude, cosmology, dust models, etc., were reused from the collaborative conjoined simulation effort by Kim et al. (2006). The input spectrum is modified by the propagation effects, such as cosmology, redshift distortions, source magnitude, intervening dust and zodiacal light. The spectral flux (or number of incident photons) is scaled for the prescribed exposure time. To simulate the spectrum, a series of monochromatic PSFs for the discrete wavelength range, representative of the input spectrum, are generated. Note that the input spectrum is at a much higher resolution, and as a result we generate more PSFs than the wavelength bins in the extracted spectrum. Each monochromatic PSF is individually positioned on the slicer, propagated through the IFU optical train assembly (OTA) up to the detector plane, and integrated on the pixels. This process is repeated for all the monochromatic PSFs, thus yielding a simulated spectral image. There are two simulated detectors: the optical CCD detector and the HgCdTe IR detector. Collectively, the detectors cover the spectral features of interest, for dark energy research (Riess et al. 1998; Perlmutter et al. 1999), covering the wavelength range of 0.35–1.7 μm. The resolution of the spectrograph was set to R = λ/Δλ ≃ 100. The computations of the PSF for the telescope and spectrometer are performed separately, employing techniques involving ray-tracing and Fourier optics. The telescope spot diagrams are generated for the IFU location on the focal plane along with the Zernike coefficients. The Zernike polynomials (normalized using Gram–Schmidt orthonormalization; see Zernike 1934; Noll 1976) on a unit circle are used to depict the phase errors on the wavefront due to the propagation effects along the telescope at the entry pupil of the IFU. The aberrated wavefront is propagated through the IFU OTA, which includes the mirror pupils, slices and the finite aperture due to the prism. In the Fraunhofer far-field approximation, the Fourier transform of this wavefront with phase errors yields the monochromatic PSFs for a given wavelength (see Born & Wolf 1999; Goodman 2005). The final PSFs are integrated at a specified location on the detector. It is computationally expensive to simulate a monochromatic PSF at a single location on the slicer, which quickly turns prohibitive when repeated for more than 2000 wavelength samples of the input spectrum. A lookup table approach is adopted to mitigate the computational overhead by pre-computing monochromatic PSFs for all the input wavelengths on a 10 × 10 grid spatially covering a single slice (10 transversely and 10 vertically). Furthermore, the PSFs are decomposed using shapelets (Refregier 2003). Note that the 10 × 10 PSF grid is spatially discrete. For generating PSFs on off-grid positions, a neural network based method is developed for interpolating the shapelet coefficients necessary for PSF reconstruction (Schwindling 1999). This approach of pre-computing and decomposition allows for reassembling the PSFs without the exorbitant computational burden for subsequent simulations. Integrating the PSFs on the detector pixels yields the 2D image of the spectrum, as shown in the bottom right of Fig. 1. The detector image consists of two axes: wavelength (denoted by λ) and sky position (θ). The sky position (θ) includes both the sky coordinates. The dispersion is along the wavelength axis. Each slice illuminated with the telescope PSF creates its equivalent spectrum at different locations of the position axes of the detector. Fig. 1 shows five of 20 slices with tapered spectral images at five different locations on the detector. The intensity of the spectral image depends on the corresponding segmented PSF on that slice. For this reason, in the case shown with the telescope PSF falling at the centre of the slices, the central spectral image is brightest and diminishes, on either side, away from the centre (in the horizontal direction of Fig. 1). The pixel size defines the grid size for the integration. The zodiacal sky background, detector quantum efficiency, shot noise, readout noise, etc., contribute to the total noise in the detector image. The final output is the noisy spectrum image, in fits format. As a part of the spectral analyses, spectrum extraction and error estimation were also developed for SpecSim, and are shown as the bottom-middle and bottom-left panels of Fig. 1. The extraction and error analysis is discussed in Section 5. For initial development, see Tilquin et al. (2006). The IFUs provide a unique mechanism to identify diversity in Type Ia supernovae through high S/N spectral subclassification. However, amongst the many other requirements to accurately represent the hardware through simulations, these resulted in fixing the parameters, thus limiting the flexibility of SpecSim. 2.2 Spectrum extraction: traditional techniques We briefly review the traditional spectrum extraction methodologies and we highlight their limitations. For a polychromatic source, the task of reconstructing the spectrum from the detector data requires converting the pixels into wavelength bins; the correspondence is given by the dispersion relation. In addition, the evaluation of the pixel noise provides the input error, which has to be propagated to the wavelength bins of the extracted spectrum. In all the cases, the location of the star for a hypothetical wavelength of zero gives the origin of the dispersion scale (wavelength axis). For the studies presented here, we have assumed that the problem of finding this location would be solved separately, either by adequate use of an associated imager or by a separate analysis of spectroscopic data. Assuming that the dispersion is parallel to the pixel rows, there are two classes of spectral extraction algorithms (Horne 1986): linear extraction, in which the contents of all pixels in a given column (spatial axis) are added to build the content of one wavelength bin; optimal extraction, in which the count of each pixel is weighted according to the factor that is evaluated from the distance between the pixel centre and the dispersion line – this probability distribution function can either be computed a priori from the PSF of the instrument or evaluated from observations or calibrated data. The two approaches are equivalent when the shot noise is dominant. In more realistic cases, the optimal extraction assigns more weight to the pixels where the S/N is higher. In the case of a low-resolution slicer IFU, each object (star) is seen by several slices and the signals need to be combined. This is further complicated when two detectors (CCD and HgCdTe) are used to capture the spectrum in the overlapping region. In such scenarios, the wavelength bins no longer coincide with the pixels and the content of some pixels must be split between the two bins. The distribution between the two bins is a non-trivial problem. It is important to note that the size of a monochromatic PSF is close to the detector pixel size (or the slit width). In addition, the dispersion line is not always parallel to the pixel rows. The result is that one wavelength bin (resolution element) can deposit photons in several pixels, and that one pixel can receive photons from several wavelength bins. This induces correlations between the contents of wavelength bins in the extracted spectrum. These problems arise from the fact that the low-resolution SNAP spectrometer was designed to minimize the detector noise, one of the primary contributors to the error budget in the absence of sky noise. The optimal extraction does not account for these correlations, and it produces an extracted spectrum with errors that do not always cover the deviations between true and reconstructed. 2.3 Challenges Simulating the optical assembly employs numerical techniques primarily rooted in ray-tracing and Fourier optics. Both of these methods are computationally demanding. Furthermore, the methodologies developed to circumvent and/or mitigate the computational bottlenecks add to the inflexibility of the simulator. In SpecSim, the input spectrum is of higher resolution than the resolution of the output. For various studies with SpecSim, the given input spectrum template consisted of ∼2250 unique wavelengths. This is equivalent to the identical number of high-resolution 2D fast Fourier transforms times the number of pupils per slice times the total number of slices. Consequently, a single run of the simulation is computationally expensive. As mentioned in Section 2.1, to mitigate the computational bottlenecks, monochromatic PSFs were pre-computed and stored as decomposed shapelets (Refregier 2003). This sped up the simulation process relatively and was only possible because the hardware parameters of the IFU were fixed. However, the computational gains are negated for the scenarios where accurate reconstruction of the PSFs from the shapelets is needed or when generating PSFs for wavelengths that involve interpolation with neural networks. The accuracy of the PSF depends on the higher-order shapelet coefficients that increase the computational burden during reconstruction. For the studies that required variation of the hardware parameters (telescope optics, IFU optics, etc.), the approach of SpecSim is cumbersome and requires a review, as discussed further. 3 THE $\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$ MATRIX A more robust approach, which also forms the basis for our novel spectrum extraction method (discussed in Section 5), is introduced as the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix. The matrix is the detector image of monochromatic PSFs with normalized flux. The |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix is the Jacobian of the (linear) transformation between the pixel readout and wavelength bin amplitudes. The matrix is sparse and is stored in a dimensionally reduced array. In the following sections, we explore the details and merits of this matrix. 3.1 Monochromatic source As discussed earlier, a monochromatic source placed in the field of view of the telescope forms a PSF on the slicer (typically three to five slices) IFU and, upon dispersion, illuminates the detector in specific locations for each slice. Every segmented PSF, when deposited on the detector, covers several pixels (typically 3 × 3 to 5 × 5 pixels). All the other detector pixels remain dark (below a defined threshold). From the realistic assumption that the pixel counts are proportional to the input flux, there is a linear relationship between the pixel counts and the source flux, with zero intercept (there is no pixel count if there is no source in the field). The slope of the linear relation is the derivative of the pixel count p with respect to the source flux f. This forms the sparse |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix for a given wavelength. Note that in the case of a monochromatic PSF, there is contribution only from a single wavelength and therefore the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix is not a partial derivative but a single variable derivative |${\rm d}\mathbf{p}/{\rm d}\!\,\mathbf{f}$| instead. This matrix is dimensionally reduced to a vector of pixel objects, which contain the original indices of the detector and the pixel counts, thus encapsulating the complete information of the monochromatic noisy detector image for a given wavelength, source position in sky and unit amplitude. 3.2 Polychromatic source (spectrum) The polychromatic source yields the full continuous spectrum. This is simulated as an aggregation of monochromatic PSFs for high-resolution discrete wavelengths, which leads to a stack of detector images for corresponding wavelengths. This stack of sparse |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrices is dimensionally reduced to a matrix form for storage. It is important to note that sampling the input spectrum at the spectrometer resolution further reduces the number of wavelengths for the monochromatic PSF generation. The continuous spectrum from its discrete sampling is reconstructed using Whittaker–Shannon interpolation by convolving the Dirac comb of the sampled measurements with the appropriate sinc function. Thereby, the relevant parameter is the derivative of the pixel counts with respect to the fluxes of the sinc convolved discrete spectrum. 3.3 Use of the $\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$ matrix The |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix is used for both simulations and super-optimal spectrum extraction. 3.3.1 For simulations By storing the state of a system, the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix allows for the reconstitution of the full detector image for a given simulation without rerunning the entire process. For a monochromatic point source, once the wavelength and source position in the field of view are known, the image on the detector is readily rebuilt from the non-zero |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| values by de-indexing the compressed list of pixels and placing them at the proper locations defined in terms of detector pixel indices. Similarly, in the case of a polychromatic source, the procedure consists of co-adding all the images obtained in the above step for all the wavelengths in the spectral range. An extended source is defined as a collection of spectral point sources. The rebuilding of the detector image requires co-adding the images from each point. 3.3.2 For spectrum extraction The sky position of the object is determined from the imaging telescope and converted into spectrometer coordinates. The specific |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix for this position can be interpolated from the grid – recall that the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix is evaluated at only 10 × 10 positions of the telescope PSF within a slice. This provides the linear relationship between pixels and spectrum bins, which needs to be inverted. This system of equations is overdetermined and with errors. It can be solved by the least-squares fitting method. There are three steps involved in the solution, as follows. Build the matrix of the second derivatives of the chi-square with respect to the bin contents. This is accomplished by left (transpose) and right multiplication of the pixels inverse error matrix by the Jacobian |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$|⁠. Build the vector of first derivatives by a single multiplication by the Jacobian. This involves the residuals at each iteration. Solve the system, ∂χ2/∂f = 0. The third step requires the inversion of the matrix of the second derivatives with respect to the extracted flux in the wavelength bins. Because the system of equations is linear, the chi-square is parabolic and no iterations are needed. 3.4 $\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$ matrix size Because the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix allows for data compression, it is useful to examine its size. For an IFU of resolution 100, and spectrum wavelength ranging from 0.3 to 1.7 μm, we have a sampling at 0.01 μm. This results in 140 samples. For each sample, the telescope PSF covers a maximum of five slices and, for each slice, at most 5 × 5 = 25 pixels receive a significant number of photons. This gives a number, 5 × 25 × 140 = 17 500, of non-null elements in the matrix for a given position of the star in the focal plane. If we were to assume symmetry between the slices, then we would still have to evaluate the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix at 10 × 10 locations within the slice. This results in less than 16 Mb of data for double precision. This is a significant compression that saves all the instrument properties needed to perform any simulation or spectrum extraction. While the introduction of the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix provides an efficient recreation mechanism, the Fourier optics tied to the telescope and spectrograph optical configuration continue to limit the scope of the simulations with SpecSim. 3.5 Calibration of the $\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$ matrix We have shown earlier that the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix is an accurate description of the optical properties of the instrument, which can be stored in a reasonably compact form and enables both simulation and spectral extraction. In Section 3.1, we describe how it can be built from the result of optical simulations of a sufficient number of monochromatic point sources. For real-world observations, this would not achieve the required accuracy and it should be supplemented or replaced by a proper calibration procedure. The design of a complete calibration plan is beyond the scope of this paper. It would require a specific study for each instrument, based on simulation runs and experiments with prototypes prior to the construction of the final hardware. In practice, a precise modelling of the instrument would permit us to identify a smaller number of parameters and the equations that relate them to the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| elements. For example, only six parameters are sufficient to define the position in space of each piece of hardware, only some of which are more likely to vary over the lifetime of the instrument. Similarly, it might be possible to factorize between detector properties on one side and the parameters of the mechanical assembly, the latter being further decomposed into telescope (PSF) and specific geometric parameters that determine the distortion fields. In Section 3.3.2, we outline the estimation of the source spectrum from the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix and observed pixel data. The same fitting technique could be used to estimate the set of calibration parameters from the measured detector images of known input (reference) spectra. For a typical SNAP IFU-like instrument the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix comprises a few thousand non-null elements. This means that the fitting procedure would imply inverting the square matrix of second derivatives with the dimension of at most a few thousand. This is a tractable problem with modern available hardware, especially as this matrix would be highly sparse. This would be appropriate for monitoring of the evolution of the calibration constants along the lifetime of the mission. 4 SPECTRUM EXTRACTION Traditionally, the spectrum extraction from a 2D detector image is performed by methods that use two separate algorithms: linear and optimal techniques. For higher accuracy, however, a more robust method was developed called the super-optimal spectrum extraction method (see Tilquin et al. 2006). A similar technique was later developed by Bolton & Schlegel (2010). 4.1 Super-optimal spectrum extraction In a low-resolution spectrometer (R ∼ 100), the comparable sizes of the pixel, PSF and wavelength resolution lead to flux correlations between the adjacent wavelength bins. Failure to account for these correlations during spectrum extraction means that spurious effects are generated and errors are underestimated. The correlations between the adjacent wavelength bins result from the optical design of the instrument. Note that even for a point source, several (three to five) slices are involved in the process. Each slice has a different origin of wavelength, and consequently the slicer provides built-in spectral dithering, which compensates for the finite size of the PSF but correlates incident photons across pixels. Because of the finite sampling over pixels, the spectrum can only be reconstructed at a set of discrete wavelengths. The separation between these wavelengths reflects the pixel granularity. The photon count in each pixel of the detector is assumed to be the linear combination of the signal at all the wavelengths. The super-optimal spectrum extraction utilizes the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix to estimate the correlated flux in the extracted spectrum by inverting the system of linear equations defined as follows: \begin{equation} \mathbf{N} = \mathbf{f (\mathbf{\lambda })} \frac{\mathrm{\partial} \mathbf{p}}{\mathrm{\partial} \mathbf{f}}. \end{equation} (1) Here, |$\mathbf{N}$| is the vector of the number of photon counts in each pixel, and the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix is the derivative of the pixel counts with respect to the flux, |$\mathbf{f} (\mathbf{\lambda })$|⁠, at each discrete wavelength λi. The underlying assumption is that the position of the object in the field is known. The number of pixels is larger than the number of discrete wavelengths, and therefore the system is overdetermined and is solved by least-squares minimization. The covariance (correlation) matrix of the extracted spectrum is obtained as a byproduct of the chi-square minimization. It is the inverse of the matrix of second derivatives of chi-square with respect to the fitted parameters, namely the fluxes in bins. 4.1.1 Practical implementation In the implementation of the super-optimal extractor, a practical consideration stems from the fact that while processing the real data, the behaviour of the instrument can differ from the simulation. As a result, the instrument can be represented by two sets of properties (functions): the field of distortions and dispersion relation, which transforms a point source at a given wavelength into its image on the IFU focal plane; the integrated PSF over a pixel, as a function of wavelength and spatial coordinates x, y on the focal plane with respect to the centre of the PSF, as defined above. These functions, in principle, should result from the calibration procedures. In SpecSim, they are simulated. The first function is ‘measured’ on the simulator and approximated by second-order polynomials for the distortions, and by a spline function for the dispersion. The second function is tabulated by moving a single pixel detector in the focal plane of the simulator where a monochromatic point source was simulated. This provides a tabulation of the pixel response on a grid of wavelengths and positions on the sky for each slice and detector. Associated with a multidimensional linear interpolator, this provides an appropriate description of the instrument. It is important to note that, in typical cases, the final error matrix is not diagonal. Because the PSF covers several slices/pixels, the error matrix has non-zero terms on the first diagonal and its nearest neighbour descending diagonals. There is no direct correlation between the wavelength bins, which are far apart. Moreover, in the hypothetical scenario of a flat spectrum and constant resolution, all the elements in a given descending diagonal would be identical; recall that the error matrix is symmetric by definition. Under these conditions, the error matrix has a Toeplitz structure. Such a matrix has eigenvalues that depend on the few independent parameters that define its contents, and can be obtained analytically (Noschese, Pasquini & Reichel 2013). Depending on the value of these parameters, the matrix might have negative eigenvalues, which means that it is not invertible and the system of equations cannot be solved. Thus, the spectrum could not be extracted. In the real world, the geometry of the spectrograph will define the error matrix as Toeplitz. The confluence of three conditions determines the range of eigenvalues: (i) the same ratio between PSF size and pixel width (on the detector) at all wavelengths; (ii) the same incident flux at all wavelengths; (iii) the ratio between diagonal and off-diagonal terms. Some values can be null or negative. While it is highly unlikely that the three conditions will be satisfied exactly, there is always a chance of them almost being satisfied. In which case, instead of null eigenvalues we might see significantly smaller values compared to the others. This results in a poorly conditioned matrix. Having a large range of eigenvalues causes precision problems when trying to invert the matrix. A more careful study might be needed to explore the conditions when negative, null or very small eigenvalues can occur. 4.1.2 Results for the slicer Fig. 2 shows the result of the super-optimal procedure on a slicer simulated supernova. Using SpecSim, we simulated the observation of a Type Ia supernova at redshift 1.5 with a 2-m telescope of focal length 20 m with an exposure time of 8 h. The simulated errors include the zodiacal light, detector and shot noise. The high-resolution input spectrum, shown in blue, is the spectrum input to the slicer (theoretical Type Ia supernova spectrum scaled to magnitude, mirror area and exposure time). The red error bars display the 1σ errors computed by the super-optimal extraction. The choice of the exposure time yields S/N of ∼10.4 for the Si ii absorption feature at 1.53 μm. Figure 2. Open in new tabDownload slide SPECSim simulation of the Type Ia supernova spectrum and extraction with the super-optimal extractor. The input truth (blue) is simulated and reconstructed (red) with 1σ error for a supernova at redshift 1.5 observed with the slicer spectrograph on a 2.0-m space telescope of 20.0-m focal length. The exposure time is 8 h. The error bars estimated by the super-optimal extractor result from the zodiacal light, detector noise and shot noise. Figure 2. Open in new tabDownload slide SPECSim simulation of the Type Ia supernova spectrum and extraction with the super-optimal extractor. The input truth (blue) is simulated and reconstructed (red) with 1σ error for a supernova at redshift 1.5 observed with the slicer spectrograph on a 2.0-m space telescope of 20.0-m focal length. The exposure time is 8 h. The error bars estimated by the super-optimal extractor result from the zodiacal light, detector noise and shot noise. Fig. 3 shows the percentage error to compare the optimal (grey) and super-optimal (red) spectrum extraction methods highlighting the Si ii line region. The input spectrum and the observational parameters used are identical to those used in Fig. 2. The super-optimal extraction performs significantly better than the optimal extraction. At the redshifted Si ii absorption line at λ = 1.537μm, the super-optimal extraction performs 1.6 times better in the error percentage than the optimal extraction, and the residuals (deviation from theoretical) are also smaller, with their distribution compatible with the theory of chi-square fit. The goodness of fit can be tested by the pull distribution, which corresponds to the histogram of the ratio of the residuals and estimated error. Ideally, this should be a normal distribution centred at 0 and with rms of 1. This is the case for the super-optimal extraction and not for the optimal extraction. In our case, this is because the latter underestimates the errors. Figure 3. Open in new tabDownload slide Comparison of the optimal (grey) and super-optimal (red) spectrum extraction methods as applied to the Type Ia supernova input spectrum shown in Fig. 2. The percentage error is higher, especially around the Si ii absorption feature. Figure 3. Open in new tabDownload slide Comparison of the optimal (grey) and super-optimal (red) spectrum extraction methods as applied to the Type Ia supernova input spectrum shown in Fig. 2. The percentage error is higher, especially around the Si ii absorption feature. 5 FAST IFU SIMULATOR (FISim) Upon reviewing the limitations due to SpecSim and leveraging the effectiveness of the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix, we consider a faster version of the simulator. The fast IFU simulator, FISim, is designed to facilitate the exploration of wider parametric space for scientific and mission studies by avoiding computational bottlenecks. Various approximations and assumptions lead to a faster simulator. The telescope PSF is assumed to be an Airy function and diffraction in the slicer OTA is neglected. This circumvents the entire Fourier optics step in simulation. The validation results, as discussed below, demonstrate that the assumptions employed in FISim affect the accuracy of the simulations to a degree that is acceptable, depending on the considered study, as it is compensated by a shorter development cycle. 5.1 Implementation overview In FISim, the monochromatic telescope PSF is approximated by an Airy function whose size is determined by the mirror diameter and focal length of the telescope for a specific wavelength. The Airy function is segmented to represent the effect of the slicer. In a three-dimensional (3D) representation (two spatial and intensity), this is equivalent to a sliced 3D Gaussian/Bessel function. Another approximation assumes the absence of distortions and therefore aligns the dispersed PSFs along the pixel rows on the detector. The diffraction due to the disperser and other slicer components is also neglected, thereby leading to abrupt truncation of the monochromatic PSF slice on each edge. Because the resolution of the final extracted spectrum cannot be better than one wavelength bin per pixel width along the dispersion direction, we replace the continuous (high-resolution) input spectrum by its sampling at regular intervals of the wavelength. With these samplings, one can reconstruct the true continuous spectrum by replacing each sample with an equivalent sinc function. The width of the sinc function depends on the resolution expressed in pixel unit equivalent. For example, in the case of a single pixel per resolution element, this results in a convolution of the segmented PSF by the sinc function of width one in the dispersion direction. The convolved monochromatic segmented PSF is integrated on the detector pixels. The PSF amplitude is scaled by the mirror area and exposure time. The incident flux is further scaled by the OTA and detector efficiencies of 0.5 and 0.95, respectively. This yields the observed source spectral signal (without the noise) in corresponding detector pixels for one sample of the continuous spectrum. It is easily verifiable that the background zodiacal light is negligible, and therefore only the shot noise and detector noise are added (in quadrature) to the noise budget. It is worth noting that each pixel has some contribution of photons from the neighbouring spectral wavelengths; this is taken into account and the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix is estimated. The calibration is assumed to be perfect but in the case of a known procedure, the associated bias could easily be included. 5.2 FISim data flow The data flow of FISim is similar to that of the SpecSim, as outlined in Fig. 1. However, as mentioned in the previous section, the approximations in FISim affect the overall implementation. The data flow in FISim is summarized in the following steps: input spectral instantiation, propagation through the IFU optics, signal integration on the detector pixels, flux scaling due to exposure time and noise estimation. To avoid computational overheads for the studies discussed in this paper, we circumvent the image creation and separate spectrum extraction. Instead, we compute only the error matrix in the extracted spectrum (in this case, the resampled input spectrum) by the Jacobian conversion. We note that, unless specified, all the input spectra used for testing, validation and studies are of Type Ia supernovae reused from Kim et al. (2006). The input spectrum in FISim is scaled by the AB magnitude factor at the source redshift. The flux units are in erg mm−2 s−1 mm−1. The wavelengths are redshifted and depend upon the type of study. The input spectrum is either interpolated from the complete spectral range (e.g. 0.3–1.8 μm in ∼2000 bins) or undersampled at the spectrograph resolution. The spectrum flux values are used to scale the respective normalized monochromatic PSFs. Recall that the PSF in the FISim case is an Airy function that depends on the aperture diameter, telescope focal length and the monochromatic wavelength. For convenience, the PSF is imaged at the centre of the focal plane where it is segmented by the IFU slices. For example, as shown in Fig. 4, the simulated IFU has five slices. The input spectrum is for a Type Ia supernova at z = 1.5 scaled by the AB magnitude of 25.08. The monochromatic PSF is for the rest-frame wavelength of λ = 0.59 μm. Figure 4. Open in new tabDownload slide Top: the image of slice 2 of five depicting the segmented monochromatic PSF (Airy function) for rest frame λ = 5900 Å, redshifted to z = 1.5. Middle: detector image of the slice 2 PSF convolved with the sinc function. Bottom: the detector image of all five slices, each with the respective dispersed PSF located at different detector locations by the slicer optical system. Combining all the monochromatic PSFs for different wavelengths in the spectral range yields the spectrum. Figure 4. Open in new tabDownload slide Top: the image of slice 2 of five depicting the segmented monochromatic PSF (Airy function) for rest frame λ = 5900 Å, redshifted to z = 1.5. Middle: detector image of the slice 2 PSF convolved with the sinc function. Bottom: the detector image of all five slices, each with the respective dispersed PSF located at different detector locations by the slicer optical system. Combining all the monochromatic PSFs for different wavelengths in the spectral range yields the spectrum. The top image of Fig. 4 shows the image of the segmented PSF on a single slice. The centre of the Airy function falls on the centre of the five slices. The portion of the PSF that falls on slice number 2 is shown. The PSF is integrated on a specific location on the detector defined by the dispersion relation. The middle image of Fig. 4 shows the segmented monochromatic PSF corresponding to the slice shown on the top of the figure, convolved and integrated on the detector. It should be noted that the PSF suffuses the monochromatic flux to the neighbouring pixels, in the spatial direction due to the wings of the Airy function and in the dispersion direction due to convolution by the sinc of the spectrum sampling. This leads to a correlated signal across the pixels. The detector image of all five segmented monochromatic PSFs, as dispersed on the detector for the specified slice location, is shown at the bottom of Fig. 4. It is evident that the central slice, which contains the largest part of the PSF, has the most flux. For the detector image of the full spectrum of a supernova, see the bottom-right CCD image in Fig. 1. The tapering of the spectral image is a result of the decreasing size of the PSF radii with decreasing wavelengths. Because the PSFs are normalized, it is at this stage that the pixel values are multiplied by the flux of the source at the wavelength and the noise is evaluated. These elements enable the construction of the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix and error matrix on pixel readout. The noise on the pixel is the quadrature sum of the shot noise and the detector read noise. The super-optimal extraction is simulated by assuming that the output spectrum is identical to the input and by propagating the pixel readout errors to the extracted spectrum fluxes using the usual Jacobian multiplications. In order to speed up the process, we neglect the extra-diagonal error terms in the inverse Fisher matrix, which permits fast inversion. For a more realistic and more accurate procedure, the full matrix could be built and inverted for a minimal increase of complexity. Finally, depending on the study being performed, the extracted spectrum could be further smeared according to the estimated errors. This was not done in FISim. 6 VALIDATION OF FISim RESULTS To estimate the accuracy of FISim as compared with the fiducial SpecSim results, a set of baseline simulations were performed with both simulators. Table 1 lists the parameters for the simulations. Note that SpecSim provides higher accuracy. For the validation studies, the S/N is estimated at the continuum at rest-frame wavelength of 0.59 μm. The minima of the Si ii absorption feature coincides with the rest-frame wavelength of 0.615 μm. The input spectrum of a Type Ia supernova at redshift z = 1.5, is operated upon by the two simulation implementations incorporating the listed parameters. For an exposure time of 8 h, the S/N for the spectrum is estimated at wavelength bins ranging from 1.5 to 1.7 μm. The wavelength range is derived from the redshifted spectral feature of interest. The Type Ia supernovae are deficient in H and He i but have a prominent absorption line due to Si ii. The left panel of Fig. 5 shows the scatter plot of the S/N versus wavelength for the two simulators. The right panel of Fig. 5 shows the ratio of the S/N of FISim with SpecSim versus the observed wavelengths. Fig. 5 shows that there is a bias of ∼20 per cent in S/N estimated with FISim. This difference is attributed to the fact that, in FISim, the off-diagonal terms in the error matrix are not taken into account. This exclusion of the off-diagonal terms provided the added speed to the simulation while sacrificing some accuracy. In other words, the fast simulator can be trusted within 20 per cent of the true error, which is probably the level of accuracy at which we can expect to predict the performances of the final instrument and its data processing software. Figure 5. Open in new tabDownload slide Left: S/N for spectral features between 1.5 and 1.7 μm for the two simulators, SpecSim and FISim. Right: ratio of the S/N as a function of the observed wavelengths. Figure 5. Open in new tabDownload slide Left: S/N for spectral features between 1.5 and 1.7 μm for the two simulators, SpecSim and FISim. Right: ratio of the S/N as a function of the observed wavelengths. Table 1. Simulation parameters for the validation of FISim with SpecSim results. The read time is the time taken to readout the detector before it saturates. Every detector read leads to noise of 7 e−. There are roughly nine readouts in the 8 h of exposure time. Simulation parameters IFU slices 5 Slice size 18 μm Telescope diameter 1.5 m Focal length 20 m Exposure time 3600 × 8 s Pixel size 18 μm Read noise 7 e− Read time 3000 s Telescope efficiency 0.5 Detector efficiency 0.95 Supernova redshift 1.5 Simulation parameters IFU slices 5 Slice size 18 μm Telescope diameter 1.5 m Focal length 20 m Exposure time 3600 × 8 s Pixel size 18 μm Read noise 7 e− Read time 3000 s Telescope efficiency 0.5 Detector efficiency 0.95 Supernova redshift 1.5 Open in new tab Table 1. Simulation parameters for the validation of FISim with SpecSim results. The read time is the time taken to readout the detector before it saturates. Every detector read leads to noise of 7 e−. There are roughly nine readouts in the 8 h of exposure time. Simulation parameters IFU slices 5 Slice size 18 μm Telescope diameter 1.5 m Focal length 20 m Exposure time 3600 × 8 s Pixel size 18 μm Read noise 7 e− Read time 3000 s Telescope efficiency 0.5 Detector efficiency 0.95 Supernova redshift 1.5 Simulation parameters IFU slices 5 Slice size 18 μm Telescope diameter 1.5 m Focal length 20 m Exposure time 3600 × 8 s Pixel size 18 μm Read noise 7 e− Read time 3000 s Telescope efficiency 0.5 Detector efficiency 0.95 Supernova redshift 1.5 Open in new tab 7 APPLICATIONS With FISim validated to acceptable error budget, we now examine its flexibility for a variety of relevant studies in the following sections. The studies discussed are focused on retrieving the astrophysical properties that are relevant for Type Ia supernova cosmology for dark energy research. The key science driver for the IFU for supernova cosmology is the ability of the spectrograph to accurately (within the error budget) measure the Si ii spectral features. The studies are based on a simulated data set as discussed below. It is important to note that the underlying simulation methodologies used in FISim are not limited to supernova cosmology, and are thus applicable to other spectroscopic studies. 7.1 Simulated input data The higher accuracy measurements of the spectral features of Type Ia supernovae, such as the Si ii line, help to subclassify the supernovae species, enabling the development of separate Hubble diagrams and thus improving the constraints on dark energy parameters (Spergel et al. 2015). In all subsequent studies, we use the measurement of the Si ii absorption feature as the primary metric for the IFU performance. To explore the parametric space of the Si ii line itself, we define a 5 × 5 grid of Type Ia supernova spectra with varying absorption lines due to two parameters: the line velocity (v) and the opacity (τ). We used synow (Parrent, Branch & Jeffery 2010) to generate the spectral grid with Si ii line features in the ranges of v = 1.0–1.4 × 104 km s−1, and τ = 1.0–5.0. The following studies are based on these simulated data. 7.2 S/N versus PSF size We examine the effect of the size of the PSF on the S/N of the Si ii line. Conceptually, the increasing size of the PSF will distribute more signal to neighbouring wavelength bins, thereby decreasing the S/N. However, it is worth exploring the behaviour for varying opacity and velocity of the spectral lines. For the study, the mirror diameter of the telescope is fixed to 1.5 m and the focal length was varied from 5 to 40 m. Note that PSF size (in this study, the radius, defined as the distance to the first zero-crossing of the Airy function) is linearly proportional to the focal length. The other simulation parameters are the same as listed in Table 1. The simulations are run for two readout noise values, 1.0 and 7.0 e−. The S/N is estimated for a monochromatic PSF at the rest-frame wavelength of 0.59 μm. The flux associated with the PSF is scaled by the supernova spectrum at the redshifted wavelength. The total noise is quadrature sum of the shot noise and the detector noise. Fig. 6 shows the variation of S/N with the radius of the PSF for the two readout noise values. The 25 input spectra are from the grid (in the ranges for opacity τ = 1.0–5.0 and for line velocity v = 1.0–1.4 × 104 km s−1). The S/N on the Si ii feature in the extracted spectra are shown in blue and red groups for readout noise values of 1.0 e− and 7.0 e−, respectively. The decline in S/N with the increase in PSF size is seen for both the readout noises across the opacity and velocity ranges. The percentage difference in the S/N declines, with respect to the PSF size for all values of τ and v, is 11 and 78 per cent for both readouts. This is more pronounced for larger readout noise. In addition, as indicated by the arrows in Fig. 6, it is noticeable that the S/N decreases with the increasing opacity and velocity of the Si ii feature. If using the worst case (i.e. readout of 7 e−), S/N > 20 will require a PSF radius of size <30 μm, which corresponds to the focal length of ≤24.4 m for a 1.5-m aperture. In practice, PSFs are Nyquist sampled, which in turn sets the bounds to the detector pixel size. In addition, the study highlights that the S/N is sensitive to the readout noise, especially at larger PSF radii (longer focal lengths). Note that these results provide constraints on specific parameters that belong to a wider parametric space used for trade-offs in defining the optimal instrument. Figure 6. Open in new tabDownload slide The S/N versus the PSF radius for 25 spectra with varying opacity and line velocity of the Si ii feature in Type Ia supernovae, obtained for two separate readout noise values: 1.0 e− (top, blue) and 7.0 e− (bottom, red). Figure 6. Open in new tabDownload slide The S/N versus the PSF radius for 25 spectra with varying opacity and line velocity of the Si ii feature in Type Ia supernovae, obtained for two separate readout noise values: 1.0 e− (top, blue) and 7.0 e− (bottom, red). 7.3 Fitting Si ii line profile The high-resolution discrete input spectrum (or the continuous spectrum in the real case) is sampled at the rate of the resolution of the spectrograph. In the study shown in Fig. 7, the input spectrum of a Type Ia supernova at redshift z = 1.5 is sampled with IFU resolution R = 100. As a result, for example, the feature of interest, the Si ii absorption line is sampled roughly by nine discrete points. The extracted spectrum contains the estimated flux values and associated errors for corresponding wavelengths. The input spectrum is from the simulated grid with known parameters (i.e. optical depth and line velocity). The fitting study focuses on estimating how well the IFU performs in extracting these parameters. Figure 7. Open in new tabDownload slide The 1σ covariance confidence levels for errors in simulated observations of opacity and velocity of the Si ii feature in nine spectra in the ranges τ = 2–4 and v = 11.0–13 × 104 km s−1. The colours represent three opacity values with τ = 2.0 (blue), 3.0 (red) and 4.0 (black). Figure 7. Open in new tabDownload slide The 1σ covariance confidence levels for errors in simulated observations of opacity and velocity of the Si ii feature in nine spectra in the ranges τ = 2–4 and v = 11.0–13 × 104 km s−1. The colours represent three opacity values with τ = 2.0 (blue), 3.0 (red) and 4.0 (black). The fitting process takes the extracted spectrum and fits it with three parameters: opacity, velocity and scaling. The three parameters are fit using the similar methodology discussed in Section 4.1. A Jacobian matrix, |$\mathbf{J}$|⁠, with respect to the three parameters, |$\mathbf{\Theta }(\tau , v, s)$|⁠, is estimated. For the numerical differentiation, in order to evaluate the flux at the incremental values, interpolations are computed from a bicubic spline polynomial function (Flanagan 2008) built from the 25 spectra in the grid. The relationship for the scale parameter is linear. For the nine sampled data points for the absorption feature, |$\mathbf{J}$| is then a 3 × 9 matrix. The diagonal error matrix, |$\mathbf{e}$|⁠, is multiplied by the transpose, |$\mathbf{J}^{\rm T}$|⁠, on the left-hand side and |$\mathbf{J}$| on the right-hand side. This yields a 3 × 3 Fisher matrix, |$\mathbf{f}$|⁠, the inverse of which is the covariance matrix, |$\mathbf{C}$|⁠. Fig. 7 shows 1σ confidence level ellipses for the covariance in the optical depth, στ, and velocity, σv, for the spectra in the opacity range τ = 2.0–4.0 and the velocity range v = 11.0–13.0 × 104 km s−1. The uncertainty in στ increases for larger values of τ and v. The nine extracted spectra are more sensitive to the velocity parameter. The standard deviation in the area of the ellipses for τ = 2.0–4.0 are 23, 34 and 48, respectively. For the same exposure time, the size of the ellipses is proportional to the redshift. For low-redshift supernovae, the ellipses are relatively smaller. In a real mission, the exposure time would be adjusted for identical errors at all redshifts. 7.4 Optimal spectrograph resolution Using FISim, we address the question: what is an optimal resolution of a slicer IFU for a 1.5-m class space telescope? One of the primary challenges of detecting the Si ii absorption feature is the detector noise (readout + dark noise) floor. This study examines the errors in velocity estimation for the spectrograph resolution, λ/Δλ between two extreme cases: (i) where the resolution is high enough such that the signal is oversampled and is reduced to the noise floor in the pixels of interest; (ii) where the resolution is low such that the line feature is unresolvable. The acceptable error in velocity value, of the order σv = 400 km s−1, results from preliminary mission studies. The resolution values of λ/Δλ = 29–390 are explored. For the 40 λ/Δλ resolution values, a single input spectrum with opacity τ = 2.0 and line velocity of v = 13.0 × 104 km s−1 is used for the FISim parameters listed in Table 1. The procedure for estimating the error in the line velocity is the same as described in the previous section. In Fig. 8, the errors in the line velocity are shown against the IFU resolution for the two values of the read noise. The lower read noise performs relatively better. The two curves have a similar shape and showcase the effects of the two extreme cases mentioned above. It is noticeable that at low resolution the spectral features are unresolved, leading to lower S/N and resulting in larger errors in line velocity. At higher resolution, meanwhile, the oversampling reduces the flux and the noise dominates leading to larger velocity errors. The range of spectrograph resolutions where the threshold velocity error (σv = 400 km s−1) is reached is wider for read noise of 1 e− (R ∼ 58–263) than that for 7 e− (R ∼ 97–146). The lowest velocity error for both values of read noise is reached at R ∼ 117. This is consistent with the original JDEM IFU specification of resolution 100. However, it must be noted that these estimates provide an upper bound, as they are based on a pristine supernova spectrum with the assumption that the host galaxy subtraction, calibration, sky subtraction and the wider range of other propagation effects have been successfully corrected. Errors resulting from intermediate subtraction methods would only make reaching the mission objective of velocity error of below 400 km s−1 more difficult. Figure 8. Open in new tabDownload slide Velocity errors versus spectrograph resolution for two read noise values. The minimum error threshold is 400 km s−1. Figure 8. Open in new tabDownload slide Velocity errors versus spectrograph resolution for two read noise values. The minimum error threshold is 400 km s−1. 7.5 Whittaker–Shannon Reconstruction In the previous sections, the estimation of various parameters was model-dependent (simulated grid). Such models do not exist in the real world. It might be necessary to estimate the velocity only by the position of the minimum. This requires us to reconstruct the continuous spectrum from the extracted sampled spectrum. We apply the Whittaker–Shannon interpolation technique to achieve this reconstruction. In this study, only the region of the spectrum with the Si ii line is considered. The set of samples at discrete wavelengths is replaced by a summation of corresponding flux-scaled sinc functions. Sampling the spectral line at low resolution can introduce a bias in the determination of the minimum, depending on the shape of the real line and position in the wavelength of the sampled points. Following, we evaluate this bias by comparing the position of the minimum in the original realistic spectrum and the one reconstructed with the Whittaker–Shannon method. The minimum of the Si ii line is searched with a gradient method with an accuracy of 3.0 × 10− 5 μm (14 km s−1). Fig. 9 shows the model of the Si ii line reconstructed from the sampled spectrum for a spectrograph with R = 50 where the bias is more pronounced with a value of 300 km s−1. The bias is a systematic error, which will add to the statistical errors that we have estimated in previous studies. Figure 9. Open in new tabDownload slide Model-independent reconstruction (black dashed line) of extracted discrete spectrum (red plus symbols) for IFU with R = 50, superimposed over the input model (blue solid line). The bias in the minima of the model and reconstructed spectrum is |b| = 300 km s−1. Figure 9. Open in new tabDownload slide Model-independent reconstruction (black dashed line) of extracted discrete spectrum (red plus symbols) for IFU with R = 50, superimposed over the input model (blue solid line). The bias in the minima of the model and reconstructed spectrum is |b| = 300 km s−1. In Fig. 10, we use the spectra from the grid and plot the bias versus the opacity for two extreme cases of v = 1.0 and 1.4 × 104 km s−1 for separate IFU resolutions of R = 50, 100 and 200. The standard deviation of the bias for the entire grid for the resolutions R = 50, 100 and 200 is 305, 223 and 107 km s−1, respectively. This illustrates that the standard deviation in the bias decreases with higher resolution. This is expected, as the higher resolution results in more sampled data points and a more accurate approximation of the signal. For example, for the higher resolution R = 200, the standard deviation is 97.35. Figure 10. Open in new tabDownload slide Reconstruction bias versus the optical depth for spectrograph resolutions R = 50, 100 and 200, for velocities 1.0 × 104 km s−1 (upper limit of the shaded regions) and 1.4 × 104 km s−1 (lower limit of the shaded regions). Figure 10. Open in new tabDownload slide Reconstruction bias versus the optical depth for spectrograph resolutions R = 50, 100 and 200, for velocities 1.0 × 104 km s−1 (upper limit of the shaded regions) and 1.4 × 104 km s−1 (lower limit of the shaded regions). These results complemented the results in the optimal resolution study and many others collectively inform the design considerations for the optimal IFU. 8 CONCLUSION In this paper, we have examined the simulation methodologies and applications of slicer IFU designed for a space-based mission conducting dark energy research with Type Ia supernovae. We have addressed the following issues. The FISim: We demonstrate the efficacy of simulations as testbeds for examining the technical and scientific drivers for missions. We have shown that simulation designs tightly bound to hardware (e.g. in the case of SpecSim) are prone to inflexibilities, because in their attempt to mimic the hardware, they are hardwired with fixed parameters and accuracy yielding implementation that limit the exploration of the parametric space. We show that the trade-offs between accuracy and flexibility afford the development of code base, FISim, that widens the scope of exploring the parameter space. Inherent to the flexibility challenges of the simulations is the computation. We have identified various computationally intensive components where the accuracy is traded for flexibility within the margin of errors. For example, the Fourier optics allows for generating extremely realistic PSFs that are integrated on the detector at dispersion defined locations. We replaced the entire process of PSF generation by an equivalent Airy function in FISim. Generating monochromatic PSFs corresponding only to the detector sampling wavelengths further increased FISim's efficiency. We also validated FISim and showed that the new results lie within the 20 per cent error margins. The |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix. We have introduced a highly effective mechanism of saving the state of the instrument. This is the detector image for monochromatic sources of unit flux. Because this is a sparse matrix, it can be stored and rebuilt by efficient and well-known algorithms. Besides simulations, the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix is used for the spectrum extraction technique that we have developed, called the super-optimal spectrum extraction. The super-optimal spectrum extraction. The optimal extraction uses the |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix, which relates the pixel counts and incident flux. The |$\mathrm{\partial} \mathbf{p}/\mathrm{\partial} \mathbf{f}$| matrix is combined with the error matrix on pixel counts to build the second derivative of the chi-square with respect to the flux. This is inverted to yield the covariance in the extracted fluxes. This provides a robust and accurate method for error estimation in the extracted flux. We demonstrate that compared with optimal spectrum extraction, the super-optimal method performs 1.6 times better for the Si ii feature for a Type Ia supernova at z = 1.5. FISim applications. We have demonstrated the application of FISim for Type Ia supernova spectroscopy. The studies primarily focus on the Si ii absorption feature of the supernova spectrum. Two parameters, opacity (τ) and velocity (v), are measured in simulations to quantify performance. We show that for a Type Ia supernova at z = 1.5 the S/N, based on the velocity parameter >20 and read noise of 7 e−, will require a telescope of a focal length of 24.4 m or greater for a 1.5-m aperture. We also show the interplay between optical depth and velocity error. Larger optical depth results in a shallower and wider Si ii line, which in turn causes a larger error on velocity estimate. This also induces a higher correlation between the two errors. We also estimate the optimal resolution, for reaching a velocity error of below 400 km s−1, to be R ∼ 58–263 and 97–146 for read noise of 1 e− and 7 e− respectively. Whittaker–Shannon reconstruction. Finally, we examine the model-independent reconstruction of the Si ii feature from the sampled data using the Whittaker–Shannon interpolation. We show that a significant bias is introduced in the velocity parameter estimated from the Si ii absorption line due to the sampling. The studies shown here are the best-case scenarios and assume that the host galaxy and other effects have been successfully removed. Therefore, they provide the upper bound on the precision and the lower bound on the errors. For more rigorous Type Ia supernova cosmology studies, realistic input data and methodologies must be used on a simulation testbed. Studies such as the effects of host galaxy spectrum subtraction, dust and other propagation effects on the estimated parameters must be considered. In addition, other parameters and spectral features must also be examined. From this effort, an important lesson for the mission-driven simulators is that developing a flexible solution to help explore a wider parametric space should be the primary objective. A more realistic simulation solution (the ‘digital twin’) should be developed closer to the stage when the instrument specifications are frozen. 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This process could be independently estimated in parallel for all the wavelengths of the high-resolution input spectrum, thus removing the need for various assumptions introduced to avoid computational bottlenecks. The speed and accuracy afforded by parallel architectures enables the handling of larger volumes and additional computation, thereby making simulations more realistic and effective. © 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society
TI - Modelling space-based integral-field spectrographs and their application to Type Ia supernova cosmology
JF - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/stw3167
DA - 2017-04-11
UR - https://www.deepdyve.com/lp/oxford-university-press/modelling-space-based-integral-field-spectrographs-and-their-ahvZ8Fnca0
SP - 2352
VL - 466
IS - 2
DP - DeepDyve
ER -