Neutrino masses and beyond-CDM cosmology with LSST and future CMB experiments
Cosmological measurements over the next decade will enable us to shed light on the content and evolution of the Universe. Complementary measurements of the Cosmic Microwave Background (CMB) and Baryon Acoustic Oscillations are expected to allow an indirect determination of the sum of neutrino masses, within the framework of the flat CDM model. However, possible deviations from CDM such as a non-zero cosmological curvature or a dark energy equation of state with would leave similar imprints on the expansion rate of the Universe and clustering of matter. We show how future CMB measurements can be combined with late-time measurements of galaxy clustering and cosmic shear from the Large Synoptic Survey Telescope to alleviate this degeneracy. Together, they are projected to reduce the uncertainty on the neutrino mass sum to 30 meV within this more general cosmological model. Achieving a 3 measurement of the minimal 60 meV mass (or 4 assuming ) will require a five-fold improved measurement of the optical depth to reionization, obtainable through a large-scale CMB polarization measurement.
Cosmological data in the coming decade will be used to tackle fundamental questions about the physical make-up of the Universe. The currently favored model is CDM, describing a flat universe with a cosmological constant, cold dark matter, and a near-zero mass of neutrino particles (Ade et al., 2016). Upcoming data measuring the evolution of cosmic structures will be used to search for deviations from this model. The only detectable deviation that we have strong reason to expect is the non-zero neutrino mass, known to be at least meV from oscillation experiments (Maltoni et al., 2004; Gonzalez-Garcia and Nir, 2003; Mohapatra and Smirnov, 2006; Feldman et al., 2013), but departures from a cosmological constant or a flat universe are not theoretically excluded (Ade et al., 2016; Moresco et al., 2016).
Each departure from CDM imprints a unique signature on cosmological observables including the growth rate of structure and the expansion rate of the Universe, but at any given cosmic epoch their effects will be partly degenerate. Previous forecasts have shown how measuring the amplitude of matter fluctuations can lead to a detection of neutrino mass, since a higher mass suppresses structure growth. For example, improving the growth rate measured by CMB lensing data, supplemented with measurements of the baryon acoustic oscillation (BAO) scale, should give an uncertainty on the mass sum of meV (Allison et al., 2015; Abazajian et al., 2016)111Or meV with an improved measurement of the optical depth to reionization, which better determines the primordial amplitude of fluctuations.. Galaxy clustering and lensing data from the Large Synoptic Survey Telescope (LSST) have been forecast to measure the mass to similar precision (Font-Ribera et al., 2014). However, Allison et al. (2015) found that the mass uncertainty when using CMB lensing can triple when allowing or geometry to vary, and Font-Ribera et al. (2014) found that constraints on the dark energy equation of state from the optical surveys are also significantly degraded when the sum of neutrino masses is varied as a free parameter.
In this paper we examine the issue of how complementary datasets, measuring structure formation over a range of cosmic epochs and with different systematic effects, will be able to distinguish between the various departures from CDM. For example, if a non-zero neutrino mass is preferred by the data, will we be able to exclude a time-varying dark energy component, or a non-zero geometry? We present forecasts considering CMB lensing data to be measured from next generation surveys (e.g., Abazajian et al., 2016), galaxy lensing and galaxy clustering from the Large Synoptic Survey Telescope (Abell et al., 2009), cross-correlations between these datasets, and BAO distance scales from the DESI experiment (Aghamousa et al., 2016). These are combined with measurements of primordial CMB fluctuations. We do not explore other cosmological datasets including redshift-space distortions or supernova measurements (de Putter et al., 2009; Das et al., 2013).
The paper is organized as follows. In Sec. II we review the cosmological neutrino mass signal and describe the main physical degeneracies. In Sec. III we describe the datasets, systematic effects and nuisance parameters considered. In Sec. IV we present forecasts for , curvature, and a time-varying dark energy equation of state. We show the impact of improved constraints on the optical depth to reionization, and of BAO measurements, and explore the impact of possible systematic effects and assumptions about the optical data. We conclude in Sec. V.
Ii Physical degeneracies
The effects of massive neutrinos on the Cosmic Microwave Background (CMB) and Large Scale Structure (LSS) have been extensively studied in the literature (Allison et al., 2015; Ade et al., 2016; Dolgov, 2002; Elgaroy and Lahav, 2005; Hannestad, 2005; Fukugita, 2006; Lesgourgues and Pastor, 2006; Hložek et al., 2017; Komatsu et al., 2009; Abdalla and Rawlings, 2007; Tegmark, 2005; Hu and Dodelson, 2002; Abazajian et al., 2016; Moresco et al., 2016; Banerjee et al., 2018). Broadly, neutrinos transition from being a relativistic gas behaving as radiation in the early Universe to being a non-relativistic fluid that behaves like Cold Dark Matter (CDM). This transition establishes a natural scale corresponding to the horizon size at which neutrinos began free-streaming, and power appears suppressed at scales smaller than this.
Neutrinos therefore contribute to the total matter density, proportional to the sum of their masses, at late times, but with suppressed clustering below the free-streaming scale. This is unlike CDM, which clusters strongly on all scales at low redshifts to aid in structure formation. These features can be measured through a relative amplitude measurement, or a measurement of the change in shape of the matter power spectrum at small scales. Currently, growth measurements from Planck combined with baryonic acoustic oscillations (BAO) from low-redshift surveys (Drinkwater et al., 2010; Ross et al., 2015; Anderson et al., 2014) constrain the total neutrino mass to be eV at 95% confidence (Ade et al., 2016). Here the CDM model is assumed, with a cosmological constant and no curvature.
The effect of massive neutrinos on CMB and LSS observables, in particular the suppression of clustering, can be mimicked by non-minimal extensions to CDM cosmology, in particular through changes in dark energy and its dynamics, or changes in the spatial curvature, as shown in (Allison et al., 2015). Canonically in CDM the ratio of dark energy pressure to its density is , but many models predict deviations from this constant value (e.g., Doran and Robbers, 2006; Linder, 2003), and is consistent with current data (Ade et al., 2016; Moresco et al., 2016). We consider a description of possible dynamical dark energy using the standard Taylor expansion in the scale factor (Linder, 2003):
as well as non-zero spatial curvature. Due to the accelerated expansion, growth in dark matter structures and induced structure formation slows down in a dark energy-dominated universe. Varying the equation of state of the dark energy affects both the growth of structure and the expansion rate of the Universe, impacting both the amplitude of the power spectrum and the angular position of the baryon acoustic oscillations. Curvature also affects the same observables: varying the geometry modifies the angular diameter distances to a given redshift, and to hold fixed the primary CMB peak positions requires changing the matter density which affects the growth.
If we had a measurement of the growth at only one single effective redshift, for example through CMB lensing, the degenerate effects of neutrino mass, dark energy equation of state, and curvature lead to a degradation in forecast sensitivity when these parameters are jointly varied. In Fig. 1 we show the forecast sensitivity on with the same combination Planck+CMB-S4+DESI considered in (Allison et al., 2015), showing the effect of adding in dark energy and its linear evolution as well as curvature as free parameters to the forecast neutrino mass. Here is degraded to an uncertainty of meV compared to meV when these are not varied (see Tab. 2 and (Allison et al., 2015) for details). The addition of baryon acoustic oscillation measurements only partially breaks the degeneracies.
Accurate measurements of the late-time matter power spectrum at multiple epochs provide a path for disentangling the neutrino mass signal from non-minimal cosmological scenarios. Gravitational lensing of background galaxies by foreground large-scale structure measures the growth of the projected matter distribution over several redshift bins, which can directly probe the suppression of structure at late times. Including the clustering of galaxies over cosmic time also adds both growth and distance information.
Iii Projected data and forecasting method
We use a standard Fisher forecasting method to project the expected constraints on cosmological parameters, using, as observables, angular power spectra of the primary CMB anisotropies, the CMB lensing convergence and projected LSST shear and galaxy clustering measurements. We use the GoFish code222Code available at https://github.com/damonge/GoFish. The results of the present analysis may be reproduced using the configuration files and Jupyter notebooks in the LSST_nu folder. originally described in (Alonso and Ferreira, 2015), which has been refined and extended for this analysis and for internal forecasts for the CMB-S4 and LSST collaborations.
We forecast an extended set of ten free cosmological parameters:
The first six are core CDM parameters: baryon density, cold dark matter density, Hubble constant, amplitude and slope of the primordial spectrum of metric fluctuations, and optical depth to reionization. The extension parameters are the neutrino mass sum , dark energy with equation of state and the curvature parameter . In places we will consider a limited subset of this model with fixed , or fixed . Models with more complicated equations of state, or with modifications to general relativity, in particular those that more closely mimic neutrinos or curvature, can be expected to introduce additional degeneracies (Lorenz et al., 2017).
Below we describe the specifications used for each experiment, and the associated observables and nuisance parameters. Where experiments are still in their design or proposal stages, we use nominal values proposed by the respective collaborations. Both collaborations have carried out their own forecasts, or are in the process of producing their own science requirement studies, and our results should not be interpreted as official forecasts for either experiment.
iii.1 Cosmic Microwave Background
We use the proposed CMB Stage-4 (S4) experiment (Abazajian et al., 2016), which will aim to observe half the sky in intensity and polarization, with anticipated white noise levels of 1 K-arcmin in intensity. We assume that 40% of the sky will be useable for analysis, and that the CMB lensing field will be reconstructed in the angular range . We also consider experiments with higher noise levels increasing up to 10 K-arcmin, which includes the level anticipated for the Simons Observatory444https://simonsobservatory.org/. Since CMB-S4 is a ground-based experiment, characterized by a large, non-white noise contribution on large scales, we use a minimum multipole , and add large-scale temperature and polarization information from Planck (Ade et al., 2016) below that scale. The noise levels have been calibrated to reproduce the errors reported in the 2015-16 Planck papers, with the large-scale polarization noise tuned to reproduce an optical depth uncertainty of (Aghanim et al., 2016). In cases where we do not use CMB-S4 in forecasts, we extend the Planck multipole range up to . We also consider the impact of adding forecast constraints from proposed space-based experiments (e.g. LiteBIRD (Matsumura et al., 2013)) that better measure the optical depth from large-scale polarized -modes. The specifications are summarized in Tab. 1. The polarization noise level of 4 K-arcmin is conservatively higher than anticipated from LiteBIRD, but still able to give a cosmic-variance limited measurement of .
iii.2 Large-scale structure from LSST
The Large Synoptic Survey Telescope (LSST) (Abell et al., 2009) is a Stage-IV photometric survey that will map out the galaxy distribution on half the sky down to magnitude . LSST will pursue 5 main cosmological observables: weak lensing, galaxy clustering, clusters of galaxies, supernovae and strong lensing. Of these, we focus here on the first two. We assume that the baseline data vector for LSST will be in the form of a “-pt” analysis (Krause et al., 2017), based on the combination of weak lensing and galaxy clustering, both in auto-correlation and cross-correlation. The specifications of the lensing and clustering samples used here follow closely those assumed in (Alonso and Ferreira, 2015), which we describe briefly below.
We divide the galaxy clustering sample into two populations of “red” and “blue” galaxies. The red sample is characterized by better photometric redshift (photo-) uncertainties (see below), lower number density, a higher galaxy bias and shallower redshift coverage. The blue sample, on the other hand, has a much higher number density and reaches to higher redshifts, at the cost of increased photo- uncertainties. The redshift distribution of these samples is based on the measurements of the luminosity functions presented by (Faber et al., 2007) (red sample) and (Gabasch et al., 2006) (all galaxies), and assuming a magnitude limit , corresponding to the so-called LSST “gold” sample (Abell et al., 2009). The -corrections needed to transform the luminosity functions into redshift distributions were computed using kcorrect (Blanton and Roweis, 2007). We model the photo- distribution of both samples as a Gaussian with a redshift-dependent standard deviation , with and for the red and blue samples respectively. Finally, we split the full redshift range covered by each sample into bins distributed such that the width of each bin in photo- space is three times the value of at the bin centre. This results in 15 bins for the red sample and 9 bins for the blue sample. For more details on the clustering specification, see (Alonso et al., 2015). Our fiducial constraints will only include the blue sample, but we will also explore the impact of adding the red sample in Section IV.3.
The distribution of inhomogeneities in the cosmic density field produces perturbations on the trajectories of photons emitted by distant sources, an effect known as gravitational lensing Bartelmann and Schneider (2001). This effect can be traced by the correlated distortions it produces in the observed shapes of galaxies, labelled “cosmic shear” or “weak lensing”, and is the same effect probed by the lensing of the CMB. As a direct probe of the density fluctuations, cosmic shear is a tremendously useful cosmological probe, particularly in combination with galaxy clustering. The LSST weak lensing sample is modeled here using the “fiducial” redshift distribution estimated by (Chang et al., 2013). As in the case of galaxy clustering we assume a tomographic analysis in which the sample is split into redshift bins. For this, we model the photo- uncertainty of the lensing sample as Gaussian with , and define 9 top-hat bins in photo- space with a width corresponding to .
Angular power spectra.
The data vector considered for LSST is the collection of maps of the galaxy overdensity or the cosmic shear field associated with the samples described above. Let be the harmonic coefficients of the -th map, corresponding for instance to one of the galaxy clustering redshift bins, and let us collect all maps for a given multipole order into a vector . The power spectrum is defined as the covariance of this vector . For both galaxy clustering and cosmic shear, the cross-correlation between two maps can be directly related to the 3D matter power spectrum as
where is a transfer function associated to the -th map.
where is the selection function of the -th bin, is the linear galaxy bias, is the growth rate, is the spherical bessel function of order , is the radial comoving distance to redshift and is the matter power spectrum.
For cosmic shear555Note that the cosmic shear is a spin-2 field, which can be decomposed into and modes. We only include modes in our analysis, which is the only non-zero contribution for the cosmological models explored here, the transfer function for the -th redshift bin is given by
We also include all the angular cross-spectra between CMB lensing, cosmic shear, and galaxy clustering.
The noise power spectra for clustering and shear are associated to the discrete sampling of the underlying density field and the intrinsic scatter of galaxy shapes. They are zero for disjoint redshift bins, and their contribution to the auto-correlation is given by and , where is the projected number density of sources (in units of srad) and is the intrinsic shape noise per ellipticity component.
Both galaxy clustering and cosmic shear suffer from a number of sources of systematic uncertainties that introduce extra nuisance parameters that must be marginalized over. We review these here:
Galaxy bias: The relation between the galaxy and matter power spectra is expected to be well-approximated by a linear, scale-independent, factor on large scales. Our forecasts therefore marginalize over the value of this quantity defined, for each galaxy sample, at a discrete set of nodes in redshift (with the full function reconstructed by interpolating between these nodes). See (Alonso et al., 2015) for further details.
Baryonic effects: We use the prescription in (Schneider and Teyssier, 2015) to account for the effects of baryons in the angular power spectra using their three-parameter correction model. The parameters and their fiducial values are the mass dependence of the halo gas fraction (log), the corresponding ejection radius as a fraction of the virial radius () and the scale of the stellar component (), which we additionally marginalize over.
Multiplicative bias: Estimating the shear from galaxy shapes may lead to redshift-dependent multiplicative biases (Schaan et al., 2017; Huterer et al., 2006; Massey et al., 2013). The shear multiplicative bias is degenerate with the amplitude of the signal and its time evolution can hide the true evolution of the growth of structure, which probes dark energy as well as massive neutrinos. We apply an overall multiplicative factor in each shear redshift bin , marginalizing over each .
Intrinsic alignments: Intrinsic alignments (IA) of galaxies associated to structure formation can contaminate the cosmic shear signal by up to 1-10% (Krause et al., 2016; Schaan et al., 2017; Troxel and Ishak, 2014; Kiessling et al., 2015; Kirk et al., 2015; Joachimi et al., 2015). Evidence suggests that IA are caused to a large extent by the alignment of galaxies with the direction of tidal forces in the cosmic web. We model this effect according to the so-called non-linear alignment model (Hirata and Seljak, 2004), and marginalize over 4 values describing the redshift evolution of the overall amplitude of the IA contribution.
Photo- uncertainties: Inaccuracies in the characterization of the photo- distribution of individual sources lead to uncertainties in the overall galaxy redshift distribution in each redshift bin, needed to estimate theoretical predictions for the expected power spectrum. We therefore marginalize over two additional parameters in each clustering redshift bin – an overall bias in the determination of photo-s as well as over the scatter of redshifts (Schaan et al., 2017). We impose a prior of 0.005 on the photo- bias following (Abbott et al., 2017) and with the expectation that LSST will be able to achieve better sensitivity.
Scale cuts: Theoretical uncertainties in the effect of baryons on the matter power spectrum, and on the details of the galaxy-matter connection prevent us from using the smallest scales of the shear and galaxy power spectrum respectively. For this reason we must drop all multipoles beyond a given . For cosmic shear, where the main effect is that of baryons, we choose a fiducial cut , independent of redshift. This corresponds to a comoving scale at . Since galaxy clustering is affected by complicated non-linear, non-local and scale-dependent bias terms, we use a more conservative, redshift-dependent cut. In this case, at the median redshift of each bin , we compute a minimum comoving scale defined by requiring that matter perturbations up to that wavenumber have a standard deviation . We then translate into an angular scale . The resulting scale cuts can be seen in the left panel of Fig. 2, which shows the auto-power spectra of all redshift bins for clustering and lensing considered here.
iii.3 Baryon Acoustic Oscillation from DESI
Besides the CMB and large-scale structure datasets described above, we will also include in some cases the expected constraints from the measurements of the redshift-distance relation made with the future Dark Energy Spectroscopic Instrument (DESI) due to begin in 2018 (Levi et al., 2013). We use the forecast uncertainties on and provided in Aghamousa et al. (2016), in 13 redshift bins between and 1.85 and 5 redshift bins between and 0.45 with bin width , expected to be achievable by DESI and the DESI Bright Galaxy Survey respectively covering .
In this section, we study the sensitivity of LSST and a future CMB-S4 experiment to the sum of neutrino masses, the dark energy equation of state and cosmological curvature. We consider the following combinations:
Setup 1: Planck + S4 (+ DESI BAO, as in Allison et al. (2015))
Setup 2: Planck + LSST-shear
Setup 3: Planck + LSST-clustering
Setup 4: Planck + LSST-clustering + LSST-shear
Setup 5: Planck + S4 + LSST-shear + LSST-clustering
We also consider in this section the impact of including a cosmic variance (CV)-limited measurement of the optical depth to reionization , and describe the effect of including BAO measurements from DESI as an additional tracer of late-time clustering.
iv.1 Forecasts with LSST and CMB-S4
Figure 3 shows forecast constraints for the error on , the dark energy parameters , and the cosmological curvature obtainable with shear and clustering measurements from LSST and CMB-S4. These results are shown in Tab. 2. In all these cases Planck is included as described in Sec III. Individually, CMB-S4, LSST clustering and LSST shear can achieve forecast constraints of 111, 91 and 120 meV respectively, strongly degraded with respect to the case where the flat CDM fixed- model is assumed (73, 69 and 41 meV respectively). In combination, however, the three probes are able to achieve an error of meV. This would be an almost 4 measurement of the minimal mass in the inverted hierarchy, and for the normal hierarchy. By combining these datasets, the degradation is only with respect to the fixed-CDM case.
It is worth pointing out that, while a free equation of state represents a more complex extension of the standard CDM model, in which the accelerated expansion is driven by something other than a simple cosmological constant, the spatial curvature is a core parameter needed to fully describe the Friedmann-Robertson-Walker metric. It is therefore interesting to investigate the possible degeneracy on from freeing while fixing . This case is shown in the last row of Tab. 2: is significantly less degenerate with , and therefore the uncertainty on the latter parameter remains unchanged after freeing up the former.
The improvement on from the combination of CMB and LSS probes is a result of the high sensitivity of future CMB data. Experiments including Advanced ACTPol and the Simons Observatory will measure the CMB lensing over large sky areas to higher noise levels than S4, so we explore how the forecast constraint on neutrino masses depends on the CMB noise level, keeping the sky area fixed to 40%. Figure 4 shows the relative degradation in the 1 uncertainty on , , and as a function of the CMB noise level in temperature , with respect to the fiducial case . We find that could be improve by with respect to the constraints achievable with a Stage-3 experiment such as Advanced ACTPol (De Bernardis et al., 2016) (), but the curvature and dark energy parameters would benefit little ( or less) from the higher sensitivity. Interestingly, we also find that the improvement in uncertainty on with is much more modest when are fixed, consistent with Allison et al. (2015).
|( + DESI BAO)||29||76||0.48||0.13||0.41|
iv.2 Impact of extended datasets
Optimal measurement of the optical depth
In Table 3 we show the forecast constraint on , the dark energy parameters , and the cosmological curvature obtainable when adding a cosmic-variance limited measurement of the optical depth to reionization with . In this case, LSST and CMB-S4 together are projected to achieve an error of meV, enough for a measurement of the minimal neutrino mass at significance even within this broader cosmological model. These results show that an improved measurement of is vital to break the degeneracy with the amplitude of scalar perturbations, not only for CMB-based measurements as found in Allison et al. (2015), but also for large-scale structure surveys aiming to constrain neutrino mass. We also note that, in the absence of S4, LSST alone would benefit less from a better measurement of , projecting only a minimal improvement on . Finally, we find that improving the optical depth measurement has little impact on the , and forecast constraints.
Additional BAO measurements
Although the projected clustering of galaxies as measured by LSST will implicitly measure the BAO scale, the large radial kernels associated with photometric redshift uncertainties will inevitably degrade any BAO-related constraints. It is therefore interesting to explore whether including measurements of the distance-redshift relation from spectroscopic surveys would lead to a significant improvement in the final constraints. Table 4 shows forecast uncertainties on , the dark energy parameters , and the cosmological curvature obtainable with an additional BAO measurement from DESI. By comparison with Tab. 2, we observe that, although the additional constraining power of DESI’s BAO measurements could significantly help some individual probes (e.g. LSST shear or clustering), the final combined constraints on are only marginally improved by by the additional information.
iv.3 Impact of systematics
Ultimately, given their sensitivity, Stage-IV surveys will be limited by systematic effects. Understanding the impact of these systematics is therefore crucial to identify the most critical ones and prioritize their modeling and calibration. In this section we explore the effect of a set of these systematics in turn, as well as the dependence of our final results on the choice of specifications we have made for LSST. Our findings are summarized in Fig. 5, which we will refer to in what follows.
Scale cuts: in the baseline case we include shear power spectra up to . Here we show forecasts including only scales up to . This has a small effect on the forecasts, and results in a slight degradation of the dark energy parameter predictions. This is understandable since, although small scales are much less affected by cosmic variance, weak lensing measurements from LSST become strongly dominated by noise beyond . This result also shows that the constraints on derived here are associated mostly with the effect of neutrino masses on the growth of structure, rather than their effect on the shape of the matter power spectrum on small scales.
Multi-tracer clustering: including an extra clustering sample made up of lower-redshift red galaxies with higher-quality photo-s can improve the final constraints in a number of ways. These include the possible cosmic-variance cancellation through the multi-tracer effect (Seljak, 2009), the improved coverage of the - plane afforded by the narrower photo- distributions and the possible self-calibration of photo- uncertainties in the more numerous blue sample through their cross-correlation with the red galaxies (Krause et al., 2017). As we can see in Fig. 5 , we find that including this sample (‘Red sample’) results in a significant improvement in our ability to measure (with a % improvement in both and errors), in agreement with our assessment of the impact of photo- uncertainties, described below. We find the impact on neutrino masses to be much milder, with only a % improvement. This agrees with the results of (Schmittfull and Seljak, 2017) that neutrino masses only benefit mildly from the presence of multiple tracers.
Photo- uncertainties: although photometric redshift surveys cannot obtain precise redshift measurements, their success relies on their ability to trace the growth and geometry of the galaxy and matter distributions over different cosmic times, and therefore systematic uncertainties on the redshift distributions of all redshift bins must be kept under control. This is particularly relevant for the measurements of the distance-redshift relation made with galaxy clustering since, unlike shear, this probe provides a measurement of the clustering pattern at the redshift of the sources (and not integrated over a broad kernel). Figure 5 shows the effect of fixing the bias and scatter parameters for the photo- distributions. This reduces the dark energy equation of state parameter uncertainties in particular, due to their impact on the distance-redshift relation. We find that the effect on neutrino masses is negligible however, and therefore photo- calibration requirements are driven by their impact on the dark energy constraints.
Baryonic effects: since baryonic effects affect the shape of the matter power spectrum at high-, using the same argument as for the scale cuts, the low signal-to-noise of the high- shear power spectrum, combined with the conservative scale cuts applied to the clustering samples imply that the final constraints on do not suffer much from the possible degeneracy with baryonic parameters. When we hold the baryonic parameters fixed, we find only a small improvement in the dark energy parameters, and no effect on the neutrino mass sum and curvature. Note however that this result may depend on the choice of parametrization for these baryonic effects, and that a more general treatment (e.g., allowing for a redshift dependence of the associated parameters or a more general functional form able to accommodate results from different hydrodynamical models (Mead et al., 2015; Chisari et al., 2018)) could affect this degeneracy.
Multiplicative bias: the multiplicative bias associated with shape measurement systematics can severely limit the ability of lensing shear to constrain the growth of structure as a function of cosmic time. However, we find that there is no significant degradation of or curvature due to this systematic effect, as shown in Fig. 5 (‘no m-bias’ holds fixed the bias parameters, while they are varied in the fiducial case), and little degradation for the and dark energy parameters. This agrees with the results of (Schaan et al., 2017) – CMB lensing is able to calibrate the multiplicative bias well within the LSST requirement, with forecast constraints shown in Fig. 6, and even in the absence of such calibration, low-redshift parameters only suffer mildly from this systematic (Huterer et al., 2006; Massey et al., 2013).
Intrinsic alignments: the observed shapes of galaxies can be affected, not only by the distortion in the photon path caused by gravitational lensing, but also by local physical forces that alter their actual shapes in a correlated manner. This contaminates the shear power spectrum by up to a few percent, and is an effect that must be taken into account to avoid significant biases in final cosmological parameters. We model IAs using the non-linear alignment model, in which galaxy shapes are proportional to the local tidal field. Uncertainties in, for instance, the alignment amplitude as a function of redshift and magnitude, can lead to significant degradation in cosmological parameters Krause et al. (2016). Our results show a mild, but significant, degradation for the dark energy equation of state parameters of up to 20%, in rough agreement with the results of Krause et al. (2016)666The authors of Krause et al. (2016) find a quantitatively larger impact for IAs on dark energy constraints. We have studied the impact of IAs for more complicated models, with higher IA amplitudes and a larger number of free parameters (e.g. sampled more finely in redshift), finding consistent results in all cases. We therefore ascribe the differences with Krause et al. (2016) to the different modelling assumptions for IAs used there.. On the other hand, the impact of IAs on the final uncertainty on is negligible. This can be understood as the IA contamination to the shear power spectrum being self-calibrated through the cross-correlation with galaxy clustering. In fact, a non-negligible degradation of caused by IAs on is obtained when considering shear measurements only, in the absence of clustering.
Finally, although we have shown that our results do not change significantly when altering our assumptions about the main sources of systematic uncertainty, it is worth reminding the reader that the full extent of these effects can only be fully quantified through more sophisticated forecasts. These should include more detailed modelling of all relevant astrophysical uncertainties, as well as a more robust examination of the likelihood function in the presence of systematics than is achievable through a Fisher forecast. It is also worth mentioning that our forecasts assume a Gaussian covariance matrix for the large-scale structure tracers. Non-Gaussianity caused by the non-linear gravitational collapse couples different modes, effectively reducing the total number of degrees of freedom that can be used to constrain cosmological parameters. Thus, although the effect of the non-Gaussian terms have been quantified to be relatively small Barreira et al. (2017), the absolute forecast uncertainties presented here should be interpreted with care.
Future cosmological measurements will significantly improve on our ability to constrain non-minimal cosmological scenarios beyond CDM. In this paper we have studied the degradation in our ability to constrain the sum of neutrino masses with cosmological data associated with allowing for departures from a perfect cosmological constant or a flat Universe. Besides quantifying this degradation, we have also assessed the degree to which it can be mitigated through the combination of data from future CMB experiments, such as CMB-S4, and low-redshift large-scale structure data from galaxy surveys such as LSST.
Within CDM, CMB-S4, together with measurements of the BAO scale from DESI should be able to constrain to the level of within CDM, enough for a 2 measurement of the minimal neutrino mass sum of . However, this measurement would get degraded to when allowing for the dark energy equation-of-state parameters to vary freely. Combining S4 with tomographic measurements of the growth of structure from a Stage-IV optical galaxy survey such as LSST can, however, bring back the constraining power to the same level as CDM-only uncertainties by breaking the degeneracies between dark energy and neutrino parameters.
More importantly none of these two experiments would be able measure the minimal beyond the level independently, and combining their datasets will be necessary to reach this sensitivity. In particular, the noise level achievable by CMB-S4 is important to achieve this goal, with a improvement in from Stage-3-like noise. This improvement with noise level, however, is contingent on the extended parameter space, and is significantly less important if we limit ourselves to CDM, as noted by Allison et al. (2015). Once S4 and LSST are combined, we do not find the DESI BAO measurements to significantly improve the figure of merit , although they could have a significantly positive impact on the dark energy parameters when combined with some individual datasets.
A better measurement of the optical depth to reionization, , is an important limitation that future experiments will face when constraining the neutrino mass, a fact that has been noted before in the literature Allison et al. (2015); Calabrese et al. (2017) in the context of CDM. This is still the case when confronting an extended parameter set – we find that both S4 and LSST would benefit significantly from a cosmic-variance limited measurement of , and that this is a necessary requirement to go beyond the -level measurement of the minimal mass sum.
Although the focus of this paper is the measurement of , the formalism used here also allows us to explore the possible constraints on the dark energy equation of state. We find that, unlike in the case of neutrino masses, the measurement of small-scale fluctuations in the CMB from a Stage-4 experiment do not significantly help constrain beyond what would already be achievable with LSST alone. This result is dependent on the severity of some of the systematic effects that LSST will have to confront, some of which have been described in the present work. CMB data could help calibrate some of these systematics (e.g. Schaan et al. (2017)), and therefore the value of combining both datasets cannot be underestimated for any science case.
Finally, we have considered a fairly comprehensive list of systematic effects in our analysis – photo- uncertainties, baryonic effects, intrinsic alignments, scale cuts and shear multiplicative biases. We find that the impact of most of the systematic uncertainties that LSST is sensitive to is only marginal in the joint constraints of CMB-S4 and LSST on . This is due to the self-calibration of several of these systematics, and the conservative scale cuts included in our fiducial analysis. On the other hand, we have also shown that is significantly more sensitive to some of these systematics, especially in the case of photo- uncertainties. Although we have tried to make a conservative treatment of these systematic effects, the true impact of these will only be quantified through more elaborate forecasts beyond the Fisher matrix formalism and ideally using simulated data with a level of realism commensurate with the expectation of these two experiments. Our forecasts are therefore limited in this sense, and should be interpreted with care. This should not affect the main qualitative message of this work: an optimal measurement of the sum of neutrino masses that can be safely distinguished from the effects of dark energy and non-zero curvature will require the combination of both CMB and large-scale structure data in the Stage-IV era, and cannot be achieved individually by either probe.
We thank Rupert Allison, Jonathan Blazek, Elisa Chisari, William Coulton, Elisabeth Krause and Emmanuel Schaan for useful conversations. DA acknowledges support from the Science and Technology Facilities Council, the Leverhulme and Beecroft trusts and Christ Church college. This research made use of the Astropy (Robitaille et al., 2013) and IPython (Pérez and Granger, 2007), software packages.
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