Accurate mapping of the cosmological expansion requires challenging observations over a broad redshift range with precision measurements. Technological developments such as large format CCDs, large telescopes, and new wavelength windows make such surveys possible. In addition to obtaining large data sets of measurements, we must also address systematic uncertainties in measurements and astrophysical source properties. Beyond that, for accurate mapping we must consider systematics in the theoretical interpretation and the data analysis. Here we present some brief illustrations of the impact of such systematics on mapping the expansion.
5.1. Parameterizing dark energy
In extracting cosmological parameters from the data, one wants parameters that are physically revealing, that can be fit precisely, and that accurately portray the key physical properties. For exploration of many different parametrizations and approaches see  and references therein. Any functional form for the expansion, e.g. the dark energy equation of state w(z), runs the risk of limiting or biasing the physical interpretation, so one can also consider approaches such as binned equations of state, defined as tophats in redshift, say, or decomposition into orthogonal basis functions or principal component analysis (see, e.g., ). However, for a finite number of components or modes this does not solve all the problems, and introduces some new ones such as model dependence and uncertain signal-to-noise criteria (see  for detailed discussion of these methods).
Indeed, even next generation data restricts the number of well-fit parameters for the dark energy equation of state to two , greatly reducing the flexibility of basis function expansion or bins. Here, we concentrate on a few comments regarding two parameter functions and how to extract clear, robust physics from them.
For understanding physics, two key properties related to the expansion history and the nature of acceleration are the value of the dark energy equation of state and its time variation w' = dw / d ln a. These can be viewed as analogous to the power spectral tilt and running of inflation. The two parameter form
based on a physical foundation by , has been shown to be robust, precise, bounded, and widely applicable, able to match accurately a great variety of dark energy physics. See  for tests of its limits of physical validity.
One of the main virtues of this form is its model independence, serving as a global form able to cover reasonably large areas of dark energy phase space, in particular both classes of physics discussed in Section 2.3 - thawing and freezing behavior. We can imagine, however, that future data will allow us to zero in on a particular region of phase space, i.e. area of the w - w' plane, as reflecting the physical origin of acceleration. In this case, we would examine more restricted, local parametrizations in an effort to distinguish more finely between physics models.
First consider thawing models. One well-motivated example is a pseudoscalar field , a pseudo-Nambu Goldstone boson (PNGB), which can be well approximated by
where F is inversely proportional to the PNGB symmetry breaking energy scale f. Scalar fields, however, thawing in a matter dominated universe, must at early times evolve along the phase space track w' = 3(1 + w) , where w departs from -1 by a term proportional to a3. One model tying this early required behavior to late times and building on the field space parametrization of  is the algebraic model of ,
with parameters w0, p (b = 0.3 is fixed). Figure 14 illustrates these different behaviors and shows matching by the global (w0, wa) model, an excellent approximation to w(z) out to z 1. More importantly, it reproduces distances to all redshifts to better than 0.05%. The key point, demonstrated in , is that use of any particular one of these parametrizations does not bias the main physics conclusions when testing consistency with the cosmological constant or the presence of dynamics. Thus one can avoid a parametrization-induced systematic.
Figure 14. Within the thawing class of physics models one can hope to distinguish specific physics origins. A canonical scalar field evolves along the early matter dominated universe behavior shown by the "matter limit" curve, then deviates as dark energy begins to dominate. Such thawing behavior is well fit by the algebraic model. A pseudoscalar field (PNGB) evolves differently. Either can be moderately well fit by the phenomenological (w0, wa) parametrization for the recent universe.
For freezing models, we can consider the extreme of the early dark energy model of , with 3% contribution to the energy density at recombination, near the upper limit allowed by current data . This model is specifically designed to represent dark energy that scales as matter at early times, transitioning to a strongly negative equation of state at later times. If future data localizes the dark energy properties to the freezing region of phase space, one could compare physical origins from this model (due to dilaton fields) with, say, the H phenomenological model of  inspired by extra dimensions. Again a key point is that the global (w0, wa) parametrization does not bias the physical conclusions, matching even the specialized, extreme early dark energy model to better than 0.02% in distance out to z 2.
Parametrizations to be cautious about are those that arbitrarily assume a fixed high redshift behavior, often setting w=-1 above some redshift. These can strongly bias the physics [133, 135]. As discussed in Section 2, interesting and important physics clues may reside in the early expansion behavior.
Bias can also ensue by assuming a particular functional form for the distance or Hubble parameter . Even when a form is not assumed a priori, differentiating imperfect data (e.g. to derive the equation of state) leads to instabilities in the reconstruction [142, 143]. To get around this, one can attempt to smooth the data, but this returns to the problems of assuming a particular form and in fact can remove crucial physical information. While the expansion history is innately smooth (also see Section 6.2), extraction of cosmological parameters involves differences between histories, which can have sharper features.
5.2. Mirage of Lambda
As mentioned in Section 3.3, interpreting the data without fully accounting for the possibility of dynamics can bias the theoretical interpretation. We highlight here the phenomenon of the "mirage of ", where data incapable of precisely measuring the time variation of the equation of state, i.e. with the quality expected in the next five years, can nevertheless apparently indicate with high precision that w = -1.
Suppose the distance to CMB last scattering - an integral measure of the equation of state - matches that of a CDM model. Then under a range of circumstances, low redshift (z 1) measurements of distances can appear to show that the equation of state is within 5% of the cosmological constant value, even in the presence of time variation. Figure 15 illustrates the sequence of physical relations leading to this mirage.
Figure 15. Matching the distance to CMB last scattering between dark energy models leads to convergence and crossover behaviors in other cosmological quantities. The top left panel illustrates the convergence in the distance-redshift relation, relative to the CDM case, for models with w0 ranging from -0.8 to -1.2 and corresponding time variation wa. The top right panel illustrates the related convergence and crossover in the dark energy density DE(a), and the bottom panel shows how the CMB matching necessarily leads to a crossover with w = -1 at the key redshift for sensitivity of low redshift experiments. This crossover in w(z) leads to the mirage of , and is impelled by the physics not the functional form.
Matching the distance to last scattering imposes a relation between the cosmological parameters, here shown as a family of (w0, wa) models holding other parameters fixed. The convergence in their distances beginning at a 0.65 is associated with a similar convergence in the fractional dark energy density, and a matching in w(z) at a 0.7. Note that even models with substantial time variation, wa 1, are forced to have w(z 0.4) = -1, i.e. look like the cosmological constant. This is dictated by the innate cosmological dependences of distance and is robust to different parameter choices; see  for more details. (Note the matching in DE(a) forced at a 0.5 has implications for the linear growth factor and nonlinear power spectrum, as explored in .)
To see beyond the mirage of , or test its reality, requires measurements capable of directly probing the time variation with significant sensitivity (hence extending out to z 1.7 as shown in Section 3.1). Current and near term experiments that may show w -1 to a few percent can induce a false sense of security in . In particular, the situation is exacerbated by the pivot or decorrelation redshift (where the equation of state is measured most precisely) of such experiments probing to z 1 being close to the matching redshift imposed on w(z); so, given CMB data consistent with , such experiments will measure w=-1 with great precision, but possibly quite inaccurately. See Figure 16 for a simulation of the possible data interpretation in terms of the mirage of from an experiment capable of achieving 1% minimum uncertainty on w (i.e. w(zpivot)). Clearly, the time variation wa is the important physical quantity allowing us to see through the mirage, and the key science requirement for next generation experiments.
Figure 16. If CMB data are consistent with CDM, this can create a mirage of for lower redshift distance data even if the dark energy has substantial time variation. The curves show simulated 68% confidence regions for w(z) for two different CMB-matched models. The value of the equation of state w(z) = -1 necessarily, for each one at a redshift close to the "sweet spot" or pivot redshift. Experiments insufficiently precise to see time variation will think w = -1 to high precision (the width of the narrow waist at z 0.38, here 1%) even if the true behavior is drastically different.
5.3. Inhomogeneous data sets
Turning from theory to data analysis, another source of systematics that can lead to improper cosmological conclusions are heterogeneous data sets. This even holds with data all for a single cosmological probe, e.g. distances. Piecing together distances measured with different instruments under different conditions or from different source samples opens the possibilities of miscalibrations, or offsets, between the data.
While certainly an issue when combining, say, supernova distances with baryon acoustic oscillation distances, or gravitational wave siren distances, we illustrate the implications even for heterogeneous supernova samples. An offset between the magnitude calibrations can have drastic effects on cosmological estimation (see, e.g., ). For example, very low redshift (z < 0.1) supernovae are generally observed with very different telescopes and surveys than higher redshift (z > 0.1) ones. Since the distances in the necessary high redshift (z 1-1.7) sample require near infrared observations from space then the crosscalibration with the local sample (which requires very wide fields and more rapid exposures) requires care. The situation is exacerbated if the space sample does not extend down to near z 0.1-0.3 and a third data set intervenes. This gives a second crosscalibration needed to high accuracy.
Figure 17 demonstrates the impact on cosmology, with the magnitude offset leading to bias in parameter estimation. If there is only a local distance set and a homogeneous space set extending from low to high redshift, with crosscalibration at the 0.01 mag level, then the biases take on the values at the left axis of the plot: a small fraction of the statistical dispersion. However, with an intermediate data set, the additional heterogeneity from matching at some further redshift zmatch (with the systematic taken to be at the 0.02 mag level to match a ground based, non-spectroscopic experiment to a space based spectroscopic experiment) runs the risk of bias in at least one parameter by of order 1. Thus cosmological accuracy advocates as homogeneous a data set as possible, ideally from a single survey.
Figure 17. Heterogeneous datasets open issues of imperfect crosscalibration, modeled here as magnitude offsets m. One scenario involves calibration between a local (z < 0.1) spectroscopic set and a uniform survey extending from z 0.1-1.7. This imposes cosmological parameter biases given by the intersection of the curves with the left axis. Another scenario takes the high redshift data to consist of two, heterogeneous sets with an additional offset m at some intermediate matching redshift zmatch. (When zmatch = 0.1 this corresponds to the first scenario with no extra offset.)
Similar heterogeneity and bias can occur in baryon acoustic oscillation surveys mapping the expansion when the selection function of galaxies varies with redshift. If the power spectrum shifts between samples, due for example to different galaxy-matter bias factors between types of galaxies or over redshift, then calibration offsets in the acoustic scale lead to biases in the cosmological parameters. Again, innate cosmology informs survey design, quantitatively determining that a homogeneous data set over the redshift range is advantageous.
5.4. Miscalibrated standard
Miscalibration involving the basic standard, i.e. candle luminosity or ruler scale, has a pernicious effect biasing the expansion history mapping. This time we illustrate the point with baryon acoustic oscillations. If the sound horizon s is improperly calibrated, with an offset s (for example through early dark energy effects in the prerecombination epoch [146, 147]), then every baryon acoustic oscillation scale measurement (z) = d(z) / s and (z) = sH(z) will be miscalibrated. Due to the redshift dependence of the untilded quantities, the offset will vary with redshift, looking like an evolution that can be confused with a cosmological model biased from the reality.
To avoid this pitfall, analysis must include a calibration parameter for the sound horizon (since CMB data does not uniquely determine it [148, 147]), in exact analogy to the absolute luminosity calibration parameter required for supernovae. That is, the standard ruler must be standardized; assuming standard CDM prerecombination for the early expansion history blinds the analysis to the risk of biased cosmology results.
The necessary presence of a standard ruler calibration parameter, call it , leads to an increase in the w0 - wa contour area, and equivalent decrease in the "figure of merit" defined by that area, by a factor 2.3. Since we do not know a priori whether the high redshift universe is conventional CDM (e.g. negligible early dark energy or coupling), neglecting for BAO is as improper as neglecting for supernova standard candle calibration. (Without the need to fit for the low redshift calibration , SN would enjoy an improvement in "figure of merit" by a factor 1.9, similar to the 2.3 that BAO is given when neglecting the high redshift calibration .)
For supernovae, in addition to the fundamental calibration of the absolute luminosity, experiments must tightly constrain any evolution in the luminosity . This requires broadband flux data from soon after explosion to well into the decline phase, and spectral data over a wide frequency range. Variations in supernova properties that do not affect the corrected peak magnitude do not affect the cosmological determination.
5.5. Malmquist bias
Distance measurements from the cosmological inverse square law of flux must avoid threshold effects where only the brightest sources at a given distance can be detected, known as Malmquist bias. Suppose the most distant, and hence dimmest, sources were close to the detection threshold. We can treat this selection effect as a shift in the mean magnitude with a toy model,
Consider a data set of some 1000 supernovae from z=0-1, with the Malmquist bias setting in at z* = 0.8 (where ground based spectroscopy begins to get quite time intensive and many spectral features move into the near infrared). The bias in a cosmological parameter relative to its uncertainty is then
for p = (m, w0, wa). Thus, the Malmquist bias must be limited to less than 0.05 mag at z = 0.95 to prevent significant bias in the derived cosmological model. In fact, this is not a problem for a well designed supernova survey since the requirement of mapping out the premaximum phase of the supernova ensures sensitivity to fluxes at least two magnitudes below detection threshold.
5.6. Other issues
In addition to the theory interpretation and data analysis systematics discussed in this section, recall the fundamental theory systematics of Section 4.4.3 and Table 2. We finish with a very brief mention of some other selected data and data analysis systematics issues of importance that are often underappreciated and that must be kept in mind for proper survey design and analysis.
Figure 18. Assuming a prior on dust properties (AV and RV) in order to reduce extinction errors can cause systematic deviations in the magnitudes. These give strong biases to the equation of state, leading to a false impression of a transition from w < -1 to w > -1. To avoid this systematic, one can use samples with minimal extinction (elliptical galaxy hosted supernovae) or obtain precise multiwavelength data that allows for fitting the dust and color properties. Based on .