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Like vs. Like: Strategy and Improvements in Supernova Cosmology Systematics

Control of systematic uncertainties in the use of Type Ia supernovae as standardized distance indicators can be achieved through contrasting subsets of observationally-characterized, like supernovae. Essentially, like supernovae at different redshifts reveal the cosmology, and differing supernovae at the same redshift reveal systematics, including evolution not already corrected for by the standardization. Here we examine the strategy for use of empirically defined subsets to minimize the cosmological parameter risk, the quadratic sum of the parameter uncertainty and systematic bias. We investigate the optimal recognition of subsets within the sample and discuss some issues of observational requirements on accurately measuring subset properties. Neglecting like vs. like comparison (i.e. creating only a single Hubble diagram) can cause cosmological constraints on dark energy to be biased by 1\sigma or degraded by a factor 1.6 for a total drift of 0.02 mag. Recognition of subsets at the 0.016 mag level (relative differences) erases bias and reduces the degradation to 2%.


I. INTRODUCTION
Distance-redshift measurements of Type Ia supernovae (SN) provide direct mapping of the cosmic expansion history. The peak brightness of most SN have tighter dispersion than any other cosmological object and this can be standardized with a simple light curve amplitude-width relation, first established by [1] in the early 1990s. This allows a SN to be calibrated to 0.15 mag or about 7% in distance, and provided the technique to discover the accelerated cosmic expansion in the late 1990s [2,3]. See [4] for a review as of 2001. For revealing the nature of the physics causing the acceleration, generically called dark energy, SN have continued to play a central role (e.g. [5,6,7,8,9,10,11,12,13]).
The limits to the standardization for SN are not known; a second parameter to further reduce the intrinsic dispersion is actively sought among the SN observables (see, for example, [14]), and more detailed measurements in spectroscopy and a wider range of wavelength bands may turn up new observables and correlations. The uncertainty on cosmological parameters improves as the intrinsic scatter decreases, both more rapidly than linearly as the reduced dispersion further improves color and dust corrections, and less rapidly as measurement uncertainties remain.
Reduction in scatter can also be achieved by characterizing each supernova with a detailed array of measurements, expecting that supernovae with identical observed properties must also have identical intrinsic luminosities. These empirical observations can define subsets of SN. Note that the converse does not necessarily hold -SN that differ in some property, e.g. position in the host galaxy or its metallicity, may not diverge in luminosity (see [15] for one recent study). This was referred to as the mapping of subsets (empirical differences) to subclasses (intrinsic luminosity differences) in [16].
Mere differences in luminosities are not sufficient to affect cosmological parameter estimation, since they will be absorbed into the "nuisance" fit parameter for the intrinsic luminosity (which will be impacted). A further ingredient must be present: population drift, or evolution of the relative fraction of each subclass with redshift. Note that SN are not per se aware of the Hubble expansion: the explosions and radiation transport take place on scales 10 −13 times smaller than the Hubble length. So SN should not evolve in a cosmic sense; rather they may be affected by their immediate environment and progenitor conditions. Since the full diversity of environments from higher redshifts also exists at low redshifts (e.g. stars and galaxies continue to form today), only the proportion of different environments changes, population drift is a more accurate description of possible changes in SN luminosity.
It is important to note here that while subsets of SN have been recognized, and the proportion of some subsets has been seen to change with redshift, current data show no definite indication that SN luminosity evolvesother than is automatically corrected for in using a standard single parameter light curve amplitude-width relation. That is, we know of no subsets that are subclasses.
This article looks to the future when suites of observations on large samples of SN, more detailed measurements than we have on any individual SN today, may show that indeed some subset, defined through those observational characteristics, is a subclass having a different luminosity. The basic method of comparing subsets of like SN -likes vs. likes, or SN demographics -was explained clearly in [17], and we follow this approach while extending it to calculating detailed effects on cosmological parameter determination.
Systematics is emphatically the name of the game in accurate science. Understanding the level of control is essential: without an intrinsic floor, SN are only limited by cosmic variance (from the number of SN within a Hubble volume) to 0.003% in distance precision. And of course a biased answer can be worse than an imprecise one.
For the reader wanting a quick conclusion, see Fig. 3. In §II we establish the formalism of subclass luminosity functions and calculate the effects of population drift on the mean and variance of the full sample luminosity function. Using this in §III, we identify three distinct impacts on cosmology determination, and show that the bias is a major effect. In §IV we examine the interplay of bias and uncertainty as we investigate strategies for controlling systematics, such as adding fit parameters for observationally recognized subclasses. We address aspects of the observational requirements for identifying subclasses in §V, and conclude in §VI.

II. SUBCLASSES, POPULATION DRIFT, AND MAGNITUDE EVOLUTION
We begin by considering an observed sample of SN to be composed of a set of subsamples that we may or may not distinguish. The intrinsic luminosity, or magnitude, distribution of the overall SN population at some redshift is a sum over all the individual subset luminosity functions. That is where φ i is an individual subset luminosity function, L i the mean luminosity of that subset, and f i (z) the fraction of the total population sample that subset represents at redshift z. Note again that L i is the mean luminosity: we are not imposing that the subset is a subclass with standard luminosity 1 , only that the mean luminosity is independent of redshift. This still places a strong burden on excellence of observations and requires something in between an empirically defined subset (since we need some knowledge of the luminosity behavior, i.e. the subset of SN discovered on a Tuesday is insufficient) and a subclass. We discuss this challenge further, and how to handle deviations, in §V D. For now we continue to call it a subset, and effectively each subset's L i represents a different absolute magnitude M i . Given Eq. (1) for the probability distribution function we can calculate whichever moments of the total luminosity distribution desired, in terms of moments of the individual subset luminosities, without requiring any assumption of, say, a Gaussian form. The drift of the mean luminosity of the sample, relative to the value at some redshift z 0 , is where we define the offset of each subset mean luminosity as 1 Of course more tightly defined subsets should have smaller dispersion about the mean, and poorly characterized subsets, or those defined through variables unrelated to the luminosity, may have larger dispersion such that the difference between subsets' luminosities is smeared out, but this does not affect the formalism. See §V where we return to discussion of these points.
We are free to evaluate the subset offset relative to some other redshift z ⋆ , though generally we will take z 0 = z ⋆ = 0. The result for the variance of the total sample luminosity is The first term is a subset-weighted dispersion, where σ 2 i is the luminosity variance of subset i, and the final two terms are contributions from the offset of the mean subset luminosity relative to the mean sample luminosity. If the offsets δL i are zero (if they are equal, they must be zero by the delta function in Eq. 1), then these bias terms vanish.

III. EFFECTS OF THE MAGNITUDE DISTRIBUTION ON SUPERNOVA COSMOLOGY
Recognizing subclasses of SN can have three effects on the calculation of cosmological parameter uncertainties: it might 1) reduce the dispersion of the sample used in the Hubble diagram, 2) reduce the residual systematic error, 3) reduce cosmology parameter bias if analyzed in the proper way.
For the first effect, let us first consider the influence of the offset terms in Eq. (5). The following argument indicates they likely do not have a substantial impact. Consider two subsets, offset in absolute magnitude by δm 12 . (For the remainder of the article we phrase the analysis in terms of magnitudes rather than luminosities; for small differences in subset luminosities one can make a direct substitution in the formulas.) Then where f 1 is fraction of the population in subset 1 (and 1 − f 1 is in subset 2). Since the maximum of f 1 (1 − f 1 ) is 1/4, and the magnitude offset should be (much) less than the dispersion, the last term is unlikely to be important. For σ 1 ∼ σ 2 , we simply have that the dispersion of the sample is nearly the dispersion of the subsets. Next within effect 1 we consider when the dispersion in a subsample is reduced. While this has a mild effect on the variance of the full sample, we can imagine a Hubble diagram formed only from the subsample (as suggested for example for elliptical galaxy hosted SN, though this reduces the external systematic of dust extinction not the internal luminosity variation). This will have fewer data points, decreasing the cosmological leverage in opposition to the lesser dispersion. In the statistical error regime, the error is effectively , so the subset must account for a sizeable fraction of the population over a wide range of redshifts in order for this subdiagram to improve in precision over the full Hubble diagram. For example, if σ full = 0.15 and σ 1 =0.1, one requires f 1 > 0.44. Data from ongoing large surveys, such as the Supernova Legacy Survey (SNLS [18]), the Nearby Supernova Factory (SNf [19]), and the CfA Supernova Archive [20], that characterize SN properties in detail may lead to such reduced-dispersion subsets. However, future surveys are likely to be fundamentally limited by systematics. In the systematic error regime, while one has supernovae to spare and could use only those from the reduced-dispersion population, the systematics dominate over the statistical dispersion and the subset Hubble diagram does not help, unless effect 2 enters, reducing systematics.
As we obtain more incisive measurements of the supernova sample, characterizing each SN in more detail, we can potentially reduce the systematic uncertainties. Recall that systematics refers to the uncertainties remaining after correction procedures have been applied, so the more information about a SN, the better chance for corrections to work to a deeper level. In the systematics dominated regime, an improvement by a factor of two in systematics leads to a factor of two tighter constraints on cosmological parameters. This is a strong reason for gathering a large suite of measurements to recognize subsets. However, not all systematics in an experiment arise from source properties -instrumental errors such as from filters and calibration also enter. The effect of subset recognition on residual systematics requires detailed, experiment-specific simulations for quantitative answers. Thus, although such data holds considerable promise for improving supernova probes, in this paper we concentrate on the third effect, cosmological parameter bias and degradation, treated in the next section.

IV. MINIMIZING SYSTEMATICS IMPACT ON COSMOLOGY DETERMINATION
Without recognition of subclasses, population drift among them will appear as evolution in the mean absolute magnitude of the sample. Again, the differential population demographics is key -mere constant differences between subclasses are absorbed completely into the fit parameter for the absolute magnitude, M, and do not impact the cosmological parameters. The bias induced by the magnitude evolution on the cosmological parameters can be evaluated at the same time as the parameter estimation uncertainties with standard Fisher, or information, matrix techniques.
Specifically, the bias on parameter p i is where m k is a supernova magnitude, ∆m k the offset due to the effective evolution, and F is the Fisher matrix over parameters p. For simplicity, we write this for a diagonal error matrix with entries σ k ; see [21] for the generalization.
As an example of the importance of accounting for bias, note that for two populations differing in absolute magnitude by 0.02, with one dominating at low redshift and the other at high redshift, the cosmological parameter bias can amount to greater than one statistical sigma.
Only two ingredients are required for determining the impact on cosmology: the mean absolute magnitude of each population, M i , and the population fractions or demographics f i (z), which combine together to form the offset ∆m through Eq. (2), where ∆M = −2.5 log δL i (0). The first ingredient is of course not known from observations, while one could measure the populations f i from the data itself (we discuss this further in §V). We consider several different models for each and examine the range of cosmology impacts.
To remove a bias induced by different M i , one could introduce additional fit parameters for them (or equivalently for M i = M i −5 log h, where h is the dimensionless Hubble constant). This of course only applies to those subsets that are recognized , i.e. empirically distinguished by the values of a certain set of measurements (for example, high line velocity, elliptical galaxy host, strong ultraviolet flux, etc.). The more subsets recognized, and fit parameters introduced, the less bias in the cosmology determination, but the more uncertainty in the estimation of the cosmology parameters due to the larger parameter space.
Explicitly, if there are N subsets and we have the observational acuity to recognize R of them, then we can fit for M (representing the gross subsample of unrecognized subsets) and M 1 ,. . . M R and suffer a cosmology bias due to the N − R unrecognized subsets. The question then is simply which wins out: improved precision from fitting for fewer parameters, or improved accuracy from reducing bias. That is, what is the optimum value for R (given the properties M i , f i (z) of the subsets).
To take into account both the dispersion and bias in parameter estimation, a standard statistical tool is the risk [22], the square root of the quadratic sum of the two terms, i.e.
We analyze the risk as a function of magnitude offsets, population model, and subsets recognized, seeking the optimal strategy for supernova cosmology -is it better to have a single Hubble diagram of all supernovae, which will have tight but biased parameter constraints, or to divide the sample into the maximum number of recognized subsets, giving looser but less biased cosmology determination. For the population model we adopt the form (See §V D and the Appendix for generalizations.) This is subject to the constraint that the populations sum up to the total sample, f i (z) = 1 for all z, which is easiest to implement if B i = B. We consider B = 1/3, 1, 3 to cover a range of behaviors. Figure 1 illustrates these population evolutions, giving respectively a high rate of change at low redshift, even weighting, or a high rate at high redshift. While one could consider scale factor or cosmic time as the independent variable, this is not qualitatively different from changing the value of B. Also, as a concrete example note that the population drift in the mean stretch parameter seen by [23] follows a linear redshift dependence (B = 1) from z = 0.03 − 1.12. For the absolute magnitudes of the individual subsets we take them to differ from the mean absolute magnitude by ±X, ±2X, considering four subsets. A constant shiftM in the magnitudes is simply absorbed into the absolute magnitude nuisance parameter -if all subsets are accounted for. The mean absolute magnitude is relevant when only some subsets are recognized, since then the sum of those populations can be redshift dependent (i.e. not unity, or zero). Explicitly, where a prime denotes the sum runs over unrecognized subsets. But the last term is zero only when ′ f i (z) = 1 for all z, i.e. all subsets are included in the sum (all unrecognized), or the sum is trivially zero (all subsets recognized).
The cosmology bias will scale with the subset magnitude offsets so we can express the results as a function of the effective magnitude evolution in the full sample. That is, we can phrase the offset amplitude X in terms of ∆m(z = 1.7), say. In the specific example treated below, we take {f i (z = 0)} = {1/4, 1/4, 1/4, 1/4} and {f i (z = 1.7)} = {1/2, 3/8, 1/8, 0}, with the population drift rate determined by the value adopted for B, as in Eq. (11). In this case, ∆m(z) = (5X/4)(z/1.7) B .
To analyze the cosmological impact, we must take into account both the cosmology parameter estimation and the bias. To do this compactly, we adapt the "area figure of merit" to the full risk. Here, the dark energy equation is the cosmic expansion factor, and the area of some likelihood contour in the w 0 -w a plane is taken as the area figure of merit. In practice, one equivalently quotes 1/[σ(w a ) × σ(w p )], where w p is the pivot value, the value of w at the redshift where the uncertainties in w 0 and w a are uncorrelated. To incorporate parameter biases δp, we define the risk figure of merit 2 from Eq. (10) as Now we can quantify to what extent it is advantageous or not to rigorously define subsets through detailed observations. Figure 2 illustrates the effect on the dark energy parameter determination. This combines simulated high quality data from 2300 SN between z = 0 − 1.7 with Planck CMB information to estimate the cosmological parameters. If we somehow knew that all subsets had the same mean absolute magnitude, i.e. that no magnitude evolution were possible, then the figure of merit is simply the usual area of merit and is shown by the horizontal line labeled "ideal". If we use only a single Hubble diagram, making no effort to, or failing to, recognize subsets, then the degradation in figure of merit is severe, shown by the solid, black curves. These show the best and worst cases of the values of B used in the population drift. For a total effective evolution to z = 1.7 of 0.02 mag, the single Hubble diagram approach degrades the cosmology constraint by a factor of 1.3-1.6. No increase in the number of SN can fully make up for this degradation, assuming a systematic floor of dm sys = 0.02(1 + z)/2.7. Even worse, while larger numbers of SN will tighten the precision they will increase the relative bias on the cosmological parameters.
FIG. 2: Recognition of like SN subsets has significant impact on the dark energy figure of merit incorporating the trade off between precision and bias. Unrecognized population drift induces evolution in the SN magnitude, ∆m(z), and bias in the cosmological parameters, while adding a fit parameter for recognized subsets costs in precision. For a single (full sample) Hubble diagram, the degradation in figure of merit due to bias can be substantial, as shown by lowest solid curve. For each case we plot the envelope of worst and best results scanning over the population evolution and permutation of subsets recognized. Maximizing the number of subsets recognized is the optimum strategy except for very small drifts, and even then the cost is less than 2%.
Recognizing 1 (dotted, red curves) or 2 (short dashed, magenta curves) of the 4 subsets acts to improve the situation. (The case of 3 subsets recognized is equivalent to that of all recognized, since the remainder of the sample is simply the fourth subset.) Here the upper and lower curves represent the best and worst of not only variation over B, but also the permutations of which of the 4 subsets are recognized. That is, identifying the subset with the most extreme magnitude offset is most useful, while one with an offset little different from the mean is of marginal effect. Indeed, if we recognize the two most extreme subsets, we approach the perfect situation, while finding the two least extreme ones only improves by 14% over the worst case of the single Hubble diagram (also see §V A).
Finally, we consider sufficiently good observations to recognize all subsets (long dashed, blue curves). In this case we must fit for 4 different M parameters, and the key question was whether the elimination of bias was worth the loss in precision due to the expanded parameter set. The answer is emphatically yes -the figure of merit is only 1.7% below the ideal case. This represents up to a 55% improvement over using exactly the same SN in a single Hubble diagram (see Fig. 3 for a clear view of these essential points). Moreover, the answer obtained represents the true cosmology without a bias. Only when the evolution is extremely small, ∆m(z = 1.7) < 0.005, which we do not know a priori, do we fail to gain by employing the likes vs. likes approach, where again the highest cost is less than 2%.

V. OBSERVATIONAL REQUIREMENTS ON DEFINING SUBCLASSES
While the optimum survey design would be to obtain a full suite of observations that enables recognition of all subclasses, this cannot always be realized. In this section we examine three cases of less than perfect observations and investigate the implications for cosmology determination. Finally, while the issue of exactly how to define subclasses is complex and largely unknown, we discuss generically some possible routes toward this.
The ability to recognize subsets depends on the acuity of the observations. This in turn depends on the instrumentation, exposure time, types of data collected, etc.
While this is too complex an area to explore here, we can get an idea of the effect on cosmology through toy models in the next three subsections, exploring respectively the degree of difference between subsets, overlap and confusion, and a continuum of subclass properties.

A. Separated Subclasses
Given disjoint subsets, with absolute magnitudes offset from the mean, one is most likely to recognize those subsets that are most discrepant. Since these also tend to induce the greatest cosmological parameter bias (depending on the evolution of the subset fraction f i (z)), this can mean that recognizing merely the most extreme subsets helps substantially toward removing bias.
For example, in the results of Fig. 2 we find that recognizing the two most discrepant subsets gives a 28-52% improvement over recognizing none (for a drift of 0.02 mag out to z = 1.7), while recognizing the two least discrepant only improves by 8-14%. The absolute level from recognizing the two most discrepant subsets approaches 2.5% below the ideal case. For recognizing a single subset, the improvements are 17-29% and 4-7% for the most and least discrepant, respectively, and recognizing the most discrepant subset brings the figure of merit within 11% of the ideal case.
To examine this further, we consider the effect of increasing the ability to resolve the subsets. For example, if we believe that some observable such as line velocity correlates with luminosity, then we need to have the capability to make sufficiently accurate measurements of this variable. As a toy model we take subsets with absolute magnitudes distributed at X, −(3/4)X, −X/2, and X/4 relative to the mean at z = 0 (recall from §III the sum of the ∆M's is taken to be zero). We then consider experiments with varying ability to resolve discrepancies from the mean and ask at what degree of difference does the figure of merit degrade by a certain percent.
As the resolution degrades past the smallest degree of difference of a subset from the mean, that subset is no longer recognized per se, but can still be identified as the "leftover" from all the other subsets. Once the next subset threshold is passed, however, then the cosmology determination degrades, and so on as the resolution coarsens, until no subsets can be recognized. Figure 4 shows the behavior for the case of four subsets as before, though with the absolute magnitude distribution as above.
The resolution required for no more than 10% degradation in dark energy figure of merit is at the level of 0.019, 0.028, 0.042 mag for total drifts ∆m(z = 1.7) = 0.01, 0.02, 0.03 respectively. To limit the degradation to 20% one requires resolution of 0.037, 0.042 for ∆m(z = 1.7) = 0.02, 0.03 (for only a 0.01 evolution in magnitude, the bias does not become large enough to reduce the figure of merit by 20%). In fact, these numbers are too optimistic in that one would need to see the subset deviate from FIG. 4: Finite resolution in measuring supernova characteristics translates into a finite resolution for distinguishing subsets from the mean sample luminosity. For three levels of magnitude evolution we plot the degradation in dark energy figure of merit as resolution coarsens and subsets become unrecognized. A rough rule of thumb is that an observational resolution less than the total evolution sensitivity to be probed, σ(∆M) < ∼ ∆m(z = 1.7), is needed.
the mean at 2-3σ for robust recognition and suppression of normal outliers. In general, if the survey aims to be sensitive to an evolution at some level ∆m(z = 1.7) (with the associated cosmological bias and leverage), then the observational resolution should be designed to be somewhat finer, σ(M ) ≈ (1/2) ∆m(z = 1.7). Of course when the experimental resolution weakens and makes discrimination of subsets from the mean difficult, the uncertainty effectively broadens the subsets and could cause them to overlap. This is a situation distinct from straightforward recognition or not, and we discuss it next.

B. Overlapping Subclasses
So far we have discussed the subsets as either recognized or unrecognized, and assumed that the recognized subsets are distinct. Subsets however can have some luminosity function distribution, as mentioned in §II, and the recognition can be fuzzy. For example, two subsets may possess overlapping luminosity functions and a member of one subset might be misassigned to another. This will change the fraction in each subset away from the true value, inducing a cosmology bias where δf i is the misestimated population fraction. Note that here the sum runs over recognized subsets, unlike in Eq. (9); if the subsets are not recognized to begin with, then the mixing has no effect. Here ∆M i represents the true (though unknown) mean absolute magnitude of each subset, which we take to be unaffected by the misassignment (we consider fuzziness in both subset population and absolute magnitude in §V C). We examine the consequences for a model with a fraction f 12 of the total sample, belonging to subset 1 but overlapping with subset 2, having a probability f 1→2 of these being misassigned. In general, the misestimation gives Summing over all subsets (including the unrecognized ones) enforces that δf i = 0, i.e. a supernova lost from one subset shows up in another, or in the main undifferentiated group.
First consider the case of two overlapping subsets, and two extremes. If the misassignment leads to a complete swap of one subset with another, so that f ji = f j and P j→i = 1, then dm(z) = X (A 2 − A 1 ))(z/1.7) B where the absolute magnitudes of the two subsets differ by X. For the parameters of §IV, this amounts to dm(z = 1.7) = −0.00125, which is insignificant. As the other extreme, if instead of a swap, a transfer occurs, i.e. a one-sided loss, with P 1→2 = 1, P 2→1 = 0, f 12 = f 1 then dm(z = 1.7) = −X A 1 = −0.0025.
These bias effects from recognized, but overlapping and confused, subsets will add to the bias due to the unrecognized subsets. The overlap contribution is small, however, because one is not mistaking the absolute magnitude by the full deviation from the sample mean but only by the amount to the nearest subset's magnitude; furthermore, the biases from each subset are not additive but are differenced during an exchange. Because the confusion is due to intrinsic luminosity function width then the observational resolution does not play a major role as in the previous section. Indeed, with fine resolution one might be tempted to subdivide the sample into more subsets, which could lead to more overlaps, but such multiplicity of subsets further reduces the fractions A i and so the overlap biases are even smaller.

C. Continuous Subclasses
Many supernova properties are not discrete, but continuous, and the subset categorizations may not be well separated, as discussed in the last two subsections. The limit of fuzziness in supernova properties is a continuous subclass distribution. Here the sample is a cloud in some multidimensional observational data space and the absolute magnitude is a function of the location in that space. We assume this is deterministic, so improved knowledge of the properties leads to a tighter distribution for the absolute magnitude. This means that bias due to unrecognized subsets is replaced by bias due to unpinpointed properties. In the completely unlocalized case (where observations are too weak to determine the location within the cloud, e.g. missing some type of observation can make the uncertainty in some dimension span the entire range), this is equivalent to the case of no recognized subsets, i.e. a single Hubble diagram.
We can adapt the formalism of the discrete subsets by taking the sum over subsets to an integral over continuous variables. If π represents the multidimensional parameter set over properties x 1 ,. . . x N (e.g. metallicity, velocity decline rate, silicon line ratio, etc.), then For simplicity, we first consider a one dimensional space over a continuous property parametrized by x.
Both ∆M and f will be functions of x. Taking the mean value of the parameter x drifts from This drift causes a change in the mean absolute magnitude of the full sample (recall there are no individual subsets in this approach), ∆M( x (z)). To evaluate Eq. (15) further, we must adopt forms for ∆F (x) and ∆M(x). Suppose so the values x F , x M define the standards. For example, if x represents metallicity, then a supernova with x = x M defines the baseline in absolute magnitude, and supernovae with x = x F maintain a constant population fraction, i.e. do not drift (the value x F does not have to actually be realized in the sample). As the value of x deviates from the standards, the demographics changes according to Eq. (18), with lower metallicity supernovae becoming more common at high redshift, say, and their luminosity changing according to Eq. (19), with lower metallicity supernovae being brighter, say. The completeness conditions are dx ∆F (x) = dx ∆M(x) = 0, i.e. every supernova lies somewhere in the parameter space and there is a mean absolute magnitude for the sample. Then over some finite range x ∈ [x − , x + ], and the equivalent for x M with p substituting for n. The magnitude offset generating bias in the cosmological parameter estimation then takes the form for the continuous case For n = p = 1 this gives We recognize the maximum drift for the sample is ∆F max = ∆ F (x + − x − ), with a similar expression for ∆M, so ∆m(z) = (1/12)∆F max ∆M max (z/1.7) B . An analogous expression holds for other values of n, p.
To generalize to a multidimensional parameter space over x 1 , . . . , x N , we have where V π = dx 1 . . . dx N is the parameter space volume and we have assumed the parameters x i are independent of each other. So far we have considered that we can measure the observational parameters x with perfect accuracy and with this possibly determine the demographics f (x). Of course if knowing x allows us to predict ∆M(x) as well, then we can compute ∆m(z) and there will be no cosmology bias. But now we consider the case where the measurements are not perfect but have some uncertainty δx, which will propagate through the demographics and absolute magnitude into the magnitude offset. This is similar to the "fuzzy" philosophy of §V B.
The measurement imprecision will lead to an magnitude uncertainty This can be viewed as taking place in a multidimensional parameter space of x as well. If there is no uncertainty in some x i , or if f and M are independent of x i then this dimension does not contribute to the magnitude systematic. For simplicity we will write the expressions in terms of a single continuous parameter x.
Using the forms of Eqs. (16), (18), (19), we can evaluate the magnitude systematic in Eq. (24) given some observational input for the uncertainty δx(x). As the simplest case, we take δx constant. For n = p = 1 the completeness conditions ensure that the systematic is zero. The general result is of the form The range ∆x can be defined through either theoretical model limits on the variation of x (e.g. metallicity) or as some weighted range that captures 90%, say, of the magnitude drift. Thus, perfect measurements give no uncertainty in this situation where the functional dependences are assumed known, but as the observations become more imprecise, i.e. δx increases, the magnitude uncertainty grows. Eventually the perturbative formalism used here breaks down, but when δx becomes comparable to ∆x then this approach should reduce to the single Hubble diagram case.
One can remove the bias due to the lack of observational resolution by fitting for the form of the magnitude offset, e.g. Eq. (25). Two fit parameters are the evolution power index B and the prefactor, call it C. If no prior is placed on these quantities, then the degradation on dark energy parameters is severe, reducing the figure of merit to less than 2. In particular, B is poorly determined and covariant with the dark energy variables, so that even an overidealized prior of 0.002 on C gives a figure of merit of only 62. We therefore fix B = 1 and investigate the degradation as a function of the prior on C, essentially equivalent to observational resolution δx/∆x. Figure 5 shows the results as a function of this resolution. When δx/∆x = 1 then the observations are blurred over the entire sample, making this equivalent to using only a single Hubble diagram, but with an added fit parameter for the drift amplitude, reducing the figure of merit by a factor of ∼ 3.
The figure of merit is rapidly degraded as the observational acuity decreases. The effective total magnitude offset here is δm(z = 1.7) = 0.05 (δx/∆x), so a resolution of 0.4 corresponds to 0.02 mag evolution. To defend against degradation of more than 20% in the figure of merit requires a resolution of 0.27. Of course as more fit parameters are added, the requirements will tighten. Thus, lack of observational resolution leads directly to unpinpointed or confused subclasses and loss of cosmological information.

D. Defining Subclasses
A central issue mentioned in §II is the consequence when a subclass fails, i.e. when a carefully characterized subset does not have an unevolving mean luminosity. (Recall we don't require the luminosity distribution to be independent of redshift, only that the mean stays constant.) If the subset is not a true subclass then we can absorb the residual luminosity evolution into an effective population driftf i (z) via So a drift in L i (z) because the subset i is not a true subclass can be viewed as an uncertainty inf i (z). We can then try to account for this by fitting forf i (z). (Note that now the quantityf i (z) is not directly observable.) As a fitting function we consider an expansion in Chebyshev polynomials over the range z = 0 − 1.7. We include terms through second order so as to allow the possibility for non-monotonic behavior, with where we normalize the polynomials to the interval [0, 1]. Adding such freedom degrades the figure of merit unless tight priors are placed on the amplitude of magnitude evolution allowed within the subset. Figure 6 explores the effect of fitting for a residual magnitude evolution of amplitude ∆m(z = 1.7) = 0.02, considering two subsets that fiducially linearly evolve from equal fractions at z = 0 to 100% in one subset at z = 1.7 (i.e. the fiducial case is α 0 = 0.5, α 1 = 0.5, α 2 = 0). We have further simplified the situation by takingf 2 (z) = 1 −f 1 (z), which will not be true in general sincef i no longer represent physical fractions of the sample. If this were relaxed or more subsets were used, the number of fit parameters increases and the degradation worsens.
Without priors on either Chebyshev coefficient, α 1 or α 2 , the figure of merit plunges by a factor 100 to a value of 2. Even freely fitting one coefficient lowers the figure of merit to 30. Priors on each α of 0.5, corresponding to a maximum evolution uncertainty of 0.02 mag from each, degrades the figure of merit by a factor 2.2 (1.5 if only allowing linear evolution).
FIG. 6: Unrecognized magnitude evolution can be accounted for by fitting for an effective population drift f (z). Here the evolution is expanded to second order in Chebyshev polynomials and the fiducial parameters correspond to ∆m(z = 1.7) = 0.02 mag. We show the dark energy figure of merit as a function of the priors placed on the subset evolution expansion coefficients. The solid, black curve corresponds to using only a first order Chebyshev expansion while the dashed, blue curve uses a second order expansion. In the first case the prior refers to that on α1 while in the second case we take the priors on α1 and α2 to be equal. When no priors are used, the value of the figure of merit is shown by the x's at the right axis. The magnitude of the priors can be related to an additional evolution by δmmax = 0.02 [σ(α)/0.5] for either coefficient αi.
The size of the effect due to residual uncertainty in whether the subset is truly a subclass points up the importance of having a comprehensive suite of precision observations. Exactly what these should be is not yet known. The supernova spectrum should contain the required information (see for example [24,25,26,27,28]); broad band photometry may not be sufficient. Recall that ∼60% of the bolometric flux is emitted in the rest frame BV R bands, so relying only upon rest frame ultraviolet or near infrared measurements leaves open the possibility that the tail does not move in the same way the dog does. Similarly, another area of active research involves the use of particular spectral features [29,30,31,32]. While any of these may prove robust, global analysis of the supernova spectrum appears less subject to such uncertainties. One way to implement this could be through principal component analysis (PCA) for example (see [33,34] for early steps).
PCA could effectively tell us whether the defining subclass variables involve, e.g., line ratios, velocities, velocity changes, etc. While the amount of degradation was significant when adding only two extra fitting parameters in the Chebyshev polynomial case, PCA by its nature focuses on the most relevant combinations of variations, and so may prove a tractable analysis approach in combination with spectral observations. Indeed, preliminary indications point to the first two PCs accounting for 85% of the spectral variation [34].

VI. CONCLUSIONS
Without any systematics, Type Ia supernovae would be statistically the most powerful tool for probing the accelerating expansion of the universe. One of the key approaches for controlling systematics is that of likes vs. likes, or supernova demographics, carefully comparing sample properties through a suite of observational characterizations. The simple concept is that like supernovae at different redshifts accurately reveal the cosmology, while supernovae at the same redshift, differing in essential ways, can define subsets giving clues to reining in systematics.
The issue is not one of evolution, but uncorrected evolution and unrecognized evolution. This article examines techniques for evaluating the cosmological consequences of systematic control or the lack of it, and strategies for implementing such control. The main pitfall is bias of the cosmological parameters -this is a bad thing, not just because it degrades the effective dark energy figure of merit calculated in terms of the risk, but because physics is lost. One may end up with an impressively precise but simply inaccurate conclusion.
Having high observational acuity and using all this information to define robust subsets is the optimal strategy. We quantify this and demonstrate that this holds even at the price of additional subset parameters in the fit. Analyzing the data in a single Hubble diagram can lead to biases of order a full statistical sigma and (secondarily) loss of figure of merit by a factor 1.6. To avoid these consequences, one uses the recognized subsets to add fit parameters; this restores essentially all the cosmological leverage, as long as the subsets are sufficiently well defined by the observations that these subset mean luminosities do not evolve.
The observational requirements to define the subsets is a complex subject but we consider three categories, of separated, overlapping, and continuous, or unrecognized, confused, and unpinpointed, subsets, and quantify some requirements within simplistic models. We further briefly consider subsets whose luminosities do in fact evolve and speculate that principal component analysis applied to supernovae spectra may prove the best path to robust control. In the appendix we illustrate how combining separate data sets, especially from different redshift ranges, can act similarly to evolution and have significant deleterious effects.
A supernovae survey designed without the controls en-abled by high observational acuity is taking a gamble on the astrophysics and supernova properties being kind. While we do not yet know exactly what subsets to define, the capability and flexibility to do so, as measured quantitatively along the lines of the simple calculations here, are required to ensure confidence in the cosmological results.
The redshift dependence of the population drift, or more generally the subset distribution, convolves with the Fisher sensitivity derivatives ∂m/∂p j in Eq. (7) in a complicated manner to lead to parameter bias. One cannot in general predict analytically how a given form for f k (z) leads to a bias. The offset ∆m(z) beats against ∂m/∂p j but the set of ∂m/∂p j do not form a complete basis, nor even an orthogonal one. Even if ∆m(z) had exactly the same functional form as some ∂m/∂p j , the offset propagates not just to the parameter p j but to all the parameters (unless the inverse Fisher matrix in Eq. 7 is formed purely from SN magnitude information, without CMB information or priors). The one exception is a redshift independent ∆m(z), which induces a pure shift in M since this parameter enters only into SN magnitudes.
Thus we must calculate the effect of various forms of f (z) numerically, and it is important to consider a variety of behaviors as we do. In general, we find that nearly linear redshift evolution, B ≈ 1, has the greatest impact on the risk figure of merit.
However we could consider another type of magnitude offset, not intrinsic to the SN populations, but rather the measurement process. If different surveys are combined, a miscalibration between the magnitudes can ensue, even if the SN absolute magnitudes are equal, due to filter or instrumental zeropoint offsets. The redshift dependence of the samples, taking the place of population drift f (z), can be particularly sharp, for example when combining a lower redshift ground-based sample with a high redshift space-based sample. As one example, if these sets are matched at z = 0.8 with an unrecognized miscalibration of 0.02 mag, then the cosmological parameter bias causes the risk figure of merit to be degraded by a factor 2.7 (with parameters biased by up to 1.9σ). See Fig. 17 of [35] for other matching scenarios. Overlap between the sets needs to be substantial to ameliorate the degradation.
In general one would want to define new fit parameters for possible offsets when using multiple samples, to eliminate bias, but these additional parameters tend to increase the dispersion substantially. For example, with a single offset fit parameter the area figure of merit de-grades by a factor 2.4 without a prior on the offset; the factor is still 1.6 with a prior of 0.02 mag. So to add to the other strategies for controlling systematics, a homogeneous sample over the full redshift range, or substantial overlap between sets, strongly improves the cosmological accuracy.