Synthetic Population of Binary Cepheids. II. The effect of companion light on the extragalactic distance scale

Classical Cepheids (hereafter referred to as Cepheids) are a vital component of the extragalactic distance scale. These young, massive (above $2.5M_{\sun}$), and luminous pulsators follow tightly the period-luminosity relation (PLR), which makes them prominent and potent standard candles up to distances of $\sim 40$ megaparsecs (Riess et al., 2016).

Still, the accuracy of the PLR can be improved with a better understanding of factors that affect the intrinsic or apparent brightness of Cepheids, e.g. the effect of metallicity or binarity. Unlike the effect of metallicity (Breuval et al., 2022), the effect of binarity has so far been only described qualitatively, estimated based on a simplistic model, or assumed negligible (e.g. Lanoix et al., 1999; Szabados & Klagyivik, 2012; Anderson et al., 2016b).

Diligent quantification of the effect of binarity is indeed a challenging task, because the census of binary Cepheids is far from complete. Out of more than 3000 Milky Way (MW) Cepheids known so far (Pietrukowicz et al., 2021) only about 200 have confirmed companions (Szabados, 2003a; Kervella et al., 2019, Shetye et al. in prep.), which constitutes $\sim 6$% of all known MW Cepheids, yet $60-80$% of them is expected to be binaries (Szabados, 2003b; Neilson et al., 2015; Mor et al., 2017). In the Large and Small Magellanic Clouds (LMC, SMC), hosting altogether around 10000 Cepheids (Soszyński et al., 2015), a total of about 80 binary Cepheids are known (Szabados & Klagyivik, 2012; Pilecki et al., 2021). Such scarcity of observed binary Cepheids relative to the expected ones indicates a strong selection bias, which originates from the limited applications of the various binary detection techniques.

The evidence is growing that Cepheids are not only members of binary systems, but also triples and possibly quadruples (Evans et al., 2005, 2020; Dinnbier et al., 2022). This is causing a paradigm shift in our thinking about Cepheids first as single, later as binary, and now as triple and quadruple systems, and calls for action to include the presence of companions in the calculations of Cepheids’ masses or apparent brightnesses, as neglecting the effect of companion(s) can lead to inaccurate estimations and constraints on stellar models.

Binary (and multiple) Cepheids appear brighter than their single counterparts, due to the extra—and unaccounted for—light of their companions. In the context of distance determination, Cepheids’ companions contaminate the PLR, causing the zero point to “shift upwards” (toward brighter, i.e. more negative magnitudes). Consequently, the distance determined from the PLR of binary Cepheid seems smaller than if determined from the PLR of only single Cepheids. Binary Cepheids could have an adverse effect also on the slope of the PLR (Szabados & Klagyivik, 2012), further distorting the calibration of the extragalactic distance scale.

Since the majority of binary Cepheids evade detection, let alone elimination from the calibration and application of the PLR, we have explored an alternative, simulation-based approach, in order to estimate their effect on the extragalactic distance scale. We employed the binary population synthesis method, and created a collection of synthetic populations of binary Cepheids, whose properties we described and compared with observational data (Karczmarek et al., 2022, hereafter Paper I). Synthetic data are by definition free from selection and completeness biases, unaffected by the reddening, and their crucial features, like metallicity and binarity, are established a priori. Therefore, synthetic data are advantageous in situations in which the real data are incomplete or inaccurate. In Paper I of this series we established the synthetic population of binary Cepheids as a reliable tool to investigate the effect of the companions on the PLR, and in this work we provide a quantitative description of this effect on the extragalactic distance scale.

In section 2 we recap the most important features of the synthetic populations that were extensively analyzed in Paper I. In section 3 we present the effect of binarity on the slope and zero point of Cepheid PLRs, within a chosen metallicity environment. In section 4 we quantify the effect of binarity between the relative distance moduli of galaxies of different metallicities. We discuss the implications of binary Cepheids on $H_{0}$ in section 5, and in section 6 we address other, more subtle factors that might affect the results from sections 3 and 4.

In Paper I we created a collection of 48 synthetic populations of binary Cepheids, by applying the StarTrack population synthesis code (Belczynski et al., 2002, 2008) to four different sets of initial parameter distributions (see Table 1 and Figure A1 in Paper I), two prescriptions for the shape and location of the instability strip (IS), two star formation histories (SFH), and three metallicities $Z=\{0.004,0.008,0.02\}$, which reflect the mean metal content of Cepheids in the SMC, LMC, and MW, respectively (see Section 2 in Paper I). The produced variants differed in a number of features. For instance, the relative fraction of Cepheids on their first instability strip crossing (IS1) is the largest for variants with uniform SFH, the number of hot Cepheid companion (spectral type BA) main sequence (MS) stars is the largest in variants with the SFH based on the observed age distribution of the Cepheids; for a detailed comparison between all the variants see Paper I.

By analyzing different variants of binary Cepheid populations, we can assess the relevance of input parameters on the results. The differences between the results, arising from different variants, can also be adopted as a kind of systematic error and included in the total error budget. Moreover, a variant that yields results which are in obvious disagreement with observations, can be discarded as unrealistic, further constraining the input values and the results. For example, we have established that variants with initial parameter distributions from set B yield nonphysically high estimation on the percentage of binary Cepheids in the LMC (Section 4.4 in Paper I), and variants with a uniform SFH show implausibly high percentage of IS1 Cepheids (Section 3.2 in Paper I). The populations created from set B of the initial parameter distributions and the uniform SFH very likely do not reflect the true populations of binary Cepheids, nevertheless, we keep them in our analysis for the sake of completeness.

As already introduced in Paper I, the effect of binarity is controlled by the binarity percentage, $f_{\mathrm{bin}}$, which is a free parameter that can range from 0% (only single Cepheids in the population), through 25, 50, 75, to 100% (i.e., all Cepheids are in binaries). Now we explain further how a synthetic population with a given $f_{\mathrm{bin}}$ is created and illustrate the results in Figure 1.

By default, every synthetic population in our collection consists of 10000 binary systems ($f_{\mathrm{bin}}=100\%$.) Every system has passed the filtering algorithm (described in detail in Paper I, Section 2), which selected only stars that fulfilled the requirements for Cepheid variables and had not experienced substantial mass transfer or a common envelope episode. The fact that Cepheids evolved without interactions with their companions allows us to ignore the existence of companions for the sake of creating a mixed population of single+binary Cepheids. In order to do that, we first select the binarity percentage, for instance $f_{\mathrm{bin}}=50\%$. Next, choose 50% of the systems randomly in our population for which we simply ignore the extra light from the companion. As a result, the total brightness of these systems comes solely from the Cepheids. For the remaining 50% of the population, for which the extra light from the companion is taken into account, we calculate the total magnitude of each system as:

where $m_{A}$ and $m_{B}$ are the magnitudes of the primary and secondary, respectively.

Figure 1 illustrates the PLRs of the LMC synthetic Cepheids with companions that constitute 25, 50, 75, and 100% of the sample. Plots present one variant of the synthetic population as an example, composed of binaries created from set D of the initial parameter distributions, filtered with the IS prescription of Anderson et al. (2016a) and scaled to the SFH based on the age distribution of the Cepheids. This variant was chosen because it seems to best emulate the true population of Cepheids in terms of the IS and SFH, while set D incorporates the most up-to-date research on the initial parameter distributions of binary stars.

Grey data points in Figure 1 represent single Cepheids, with the rest being binary Cepheids. The resulting fractional pulsation amplitude (diminished due to the light from the Cepheid companion) is encoded in the color of the points, with the darkest points being almost unaffected Cepheids due to the minimal contribution of the companions to the total light of the system. Meanwhile, the amplitude of the Cepheids denoted by the brightest points is diminished to just a few percent of their original values. A gap in the data at $\log(P/\mathrm{d})\approx 0.6$ separates IS1 from IS2+3 Cepheids. IS1 Cepheids are presented for the sake of completeness, since they met all the requirements for pulsating stars set in the filtering algorithm, but their counts, and properties demand further investigation, which is beyond the scope of this paper.

The scatter of the PLRs in Figure 1 is due to the finite width of the IS (the largest for the shortest wavelengths, the smallest for the longest wavelengths, and the Wesenheit index $W_{VI}$), but also due to the binary nature of Cepheids. The biggest contributors to that scatter are bright, giant companions located well above the main period-luminosity trend, and best visible in the near-infrared (NIR) and $W_{VI}$. They are easily detectable because they are 1.5-2 times brighter than single Cepheids of a similar pulsation period, and have already proven an unmatched observational opportunity to find Cepheids with giant companions (Pilecki et al., 2021). However, in the context of distance determinations, such systems could bias the results, and therefore are routinely removed from the sample in the process of data cleaning.

On the other hand, binary Cepheids with MS companions (color-coded with navy blue in Figure 1), which lie very close to the PLR, are virtually undetectable via photometric methods and therefore cannot be cleaned from the sample. The light contribution from a MS star to the total brightness of individual systems is minuscule, but because they outnumber companions of other evolutionary types, their cumulative effect on the PLR might not be negligible. This effect cannot be assessed from the visual inspection of Figure 1 alone, therefore we quantify it in Section 3.

In order to calculate the cumulative effect of binary Cepheids on the PLR, we followed a standard routine of distance determination: (i) cleaning the data from the outliers by applying $3\sigma$ clipping iteratively three times; (ii) least-squares linear fitting to the cleaned data to the Leavitt Law (Leavitt & Pickering, 1912):

with the slope ($sl$) and intercept (zero point, $zp$) as free parameters; (iii) determining the distance from the zero point of the PLR. The values and uncertainties of the slope and zero point were calculated using a bootstrap technique; the least-squares fit was performed on randomly chosen 50% of the sample (i.e. 5000 systems), the values of the slope and zero point were saved, and the routine was repeated 1000 times, producing normal distributions of the slope and zero point. Finally, a Gaussian fit yielded the mean and standard deviation, which we recognized as a true value and its uncertainty, respectively, of the PLR slope and the zero point. This calculation was performed for the five different adopted binary percentages $f_{\mathrm{bin}}$ = 0, 25, 50, 75, 100%, on all synthetic populations in our collection.

None of the environments we simulate has exactly 5000 Cepheids; the number of Cepheids in the LMC and SMC is 4620 and 4915, respectively, with the completeness of nearly 100% (Soszyński et al., 2015), and the number of Galactic Cepheids is 3352 with the completeness of about 88% down to a magnitude $G=18$ (Pietrukowicz et al., 2021). However, by keeping our populations equal in size, we assure that statistical errors affecting the samples remain the same, allowing a fair comparison.

The above calculation of the shift of the zero point has been achieved by varying the number of binary Cepheids within the simulated isolated environments of the SMC, LMC, and MW (left, middle, and right panels in Figure 5, respectively). However, extragalactic distances are usually calculated as differences of zero points between two galaxies. In order to properly assess the effect of binarity on the extragalactic distance scale, one needs to consider the shift of the PLR zero point not within one, but between two galaxies, which have different metallicities, and possibly different percentages of Cepheid binaries.

In Figure 6 we visualize the distance modulus differences between two exemplary galaxies of different metallicities (LMC and NGC 300) and unknown percentage of binary Cepheids (for the sake of argument we propose two values: 0 and 50%). When calculating the distance modulus between a target and reference galaxy, $\mu=zp^{\mathrm{tar}}-zp^{\mathrm{ref}}$, with an unknown percentage of binary Cepheids, we take into account three possibilities: (i) the reference (LMC) and target (NGC 300) galaxies have only single Cepheids, resulting in the distance modulus $\mu_{0}$; (ii) the reference galaxy has a larger binarity percentage than the target galaxy, resulting in $\mu_{1}>\mu_{0}$; (iii) the reference galaxy has a smaller binarity percentage than the target galaxy, resulting in $\mu_{2}<\mu_{0}$.

The shift in the distance modulus, $\Delta\mu$, is the difference between the distance modulus from the reference to the target galaxy, relative to $\mu_{0}$:

Since the binarity percentages for the reference and target galaxies do not have to be equal, $\Delta\mu$ can be either positive or negative.

In order to examine the dependence of $\Delta\mu$ on the binarity percentage in the target and reference galaxies, we follow the same procedure for generic distance determination as described in Section 3, but this time we assume the PLR slope is the same for all binary percentages and for all metallicities. This slope value is fixed to the LMC slope and $f_{\mathrm{bin}}=0\%$.

All possible $\Delta\mu$ values as a function of the binarity percentages of reference and target galaxies are presented in Figure 7 in a form of a matrix. As before, the analyzed synthetic population variant is: set D of the initial parameter distributions, the IS prescription of Anderson, the SFH based on the age distribution of the Cepheids, and the metallicities are $Z=0.008$ and 0.02 for the reference and target galaxy, respectively. Figure 7 shows the results for the $V$ and $K$ band; matrices for other bands and variants are available online (see Section DATA AVAILABILITY).

In each matrix, the bottom left cell represents the baseline value, explicitly $\Delta\mu=\mu_{0}-\mu_{0}=0$. In other words, in this cell the distance modulus between two galaxies (both having $f_{\mathrm{bin}}=0\%$) is calculated and subtracted from itself, resulting in a zero value. By contrast, in the bottom middle cell the value of distance modulus difference between the reference ($f_{\mathrm{bin}}=50\%$) and target ($f_{\mathrm{bin}}=0\%$) galaxies has been calculated and reduced by the baseline value; this cell corresponds to the value of $\mu_{1}-\mu_{0}$ from Figure 6.

Two most extreme values in the top and bottom cells on the antidiagonal ($\searrow$), i.e. $-0.0184$ and 0.0245 ($V$-band), and $-0.0021$ and 0.0031 ($K$-band), show the maximum systematic error of the distance modulus, i.e., the maximum value by which the distance modulus is shifted if the “assumed” binarity percentage is 0% while the “true” one is 100%, and vice versa. On the other hand, the diagonal going from the lower left corner to the upper right corner ($\nearrow$) shows the smallest shifts in the distance modulus for the cases of equal or almost equal binarity percentage in the target and reference galaxies. This result means that the effect of binarity on the extragalactic distance scale is virtually null if the galaxies have the same binary Cepheid fractions.

If the target and reference galaxies were identical²²2Two galaxies would be identical if they had the same chemical composition, SFH, proportions of Cepheids on 1st, 2nd and 3rd crossing, and Cepheid companions of the same distribution of physical characteristics (masses and radii)., the matrices would be symmetrical, with the diagonal ($\nearrow$) filled with zeros. However, in more realistic cases the two galaxies have different metallicities, which affects the shape and location of the IS, and physical characteristics (masses and radii) of Cepheid companions. Different SFHs affect the proportions of Cepheids on the 1st, 2nd, and 3rd IS crossings. This further contributes to the dissimilarities between the galaxies, and by extension, to the asymmetries of matrices in Figure 7.

The Hubble-Lemaître constant, $H_{0}$, has been determined by the SH0ES Program (Riess et al., 2022, and references therein) in three steps, and anchored in three galaxies: MW, LMC, and NGC 4258. These galaxies, apart from hosting classical Cepheids, have distances determined from purely geometric methods (Gaia parallaxes in the MW, detached eclipsing binaries in the LMC, and water masers in the nucleus of NGC 4258), which allows calibrating the zero point of the Cepheid PLR (step 1). Next, the calibrated PLR is used to determine distances to galaxies that host both Cepheids and type Ia supernovae (SNeIa), and by doing so, to calibrate SNeIa as distance indicators (step 2). Finally, the SNeIa at high redshifts, whose distances have been calculated in step 2, yield $H_{0}$ (step 3).

The calibration of the zero point of the Cepheid PLR (step 1) has been repeatedly refined by including factors like metallicity (Breuval et al., 2022), crowding, or reddening (Riess et al., 2022). However, the effect of binarity has not been taken into account yet. From Figure 5 we expect that the zero point of the Cepheid PLR in the LMC should be shifted by about 0.02 and 0.003 mag in the $V$ and $K$ bands, respectively, if the binarity percentage is $f_{\mathrm{bin}}=100\%$, and half of these values if $f_{\mathrm{bin}}=50\%$. In the MW, the shifts are 0.016 and 0.002 mag in the $V$ and $K$ bands, respectively ($f_{\mathrm{bin}}=100\%$). However, none of these values matter for the calibration of the PLR zero point at step 1. In principle, the calibration serves to establish a dependence between the geometrical distance and the value of the PLR zero point, thus the absolute value of the zero point is irrelevant as long as this very calibration is preserved in all Cepheid populations.

If we were able to exclude binary Cepheids or eliminate the extra light of the companions in the three anchor galaxies, and calibrate the zero point of the PLR based only on pure samples of single Cepheids, then such a calibration would prove useful only if Cepheids in the SNeIa host galaxies were also all single or if the adjustment for the extra light from the companions was introduced to the calibration. Since the existing census of binary Cepheids suffers from tremendous completeness biases, which render it impossible to remove the companions, the only viable alternative is to include binary Cepheids into the PLR calibration.

Binary Cepheids do not compromise the accuracy of the PLR calibration at step 1, but they might decrease the precision of this calibration if the percentages of binary Cepheids in the three anchor galaxies vary considerably. This might cause a scatter in the PLR calibration averaged from the three anchors. However, this statistical error at its maximum ($f_{\mathrm{bin}}=100\%$) is expected to be around 0.004 and 0.001 mag in the $V$ and $K$ bands, respectively, and most probably it is even smaller.

Binary Cepheids play a much bigger role at step 2 of the determination of $H_{0}$, when the already established PLR calibration is used to determine distances to SNeIa host galaxies. The premise is that Cepheids in anchor and SNeIa host galaxies follow the same PLR, and any significant differences in their brightnesses, attributed to crowding, reddening, or different metallicities, are already taken into account in the $H_{0}$ determination. Our study shows that different binarity percentages in anchor and SNeIa host galaxies cause a shift in the distance modulus between the two galaxies, which can be either positive (galaxies appear to be farther away from each other) or negative (galaxies appear to be closer to each other). The maximum value of the shift (if the values of $f_{\mathrm{bin}}$ in reference and target galaxies are 0 and 100%, or vice versa) is $|\Delta\mu|\approx 0.008$ mag in $W_{H}$, but is substantially minimized if the values of $f_{\mathrm{bin}}$ are similar in the two investigated galaxies.

Currently, step 2 of the $H_{0}$ determination involves 37 SNeIa host galaxies (Riess et al., 2022). If we assume $f_{\mathrm{bin}}=50\%$ in the reference galaxy, then the maximum shift in a distance modulus to any target galaxy is around $\pm 0.004$ mag in $W_{H}$. If we further assume that the binary percentages in the 37 target galaxies are distributed uniformly between 0 and 100%, then the 37 shifts in distance moduli will also create a uniform distribution, with a mean of 0 mag and a standard deviation of 0.002 mag. This means that the binarity of Cepheids does not change the value of $H_{0}$, but does increase the statistical error on $H_{0}$ by $0.07\,\mathrm{km\,s^{-1}Mpc^{-1}}$ or 0.1%⁴⁴4 We estimated the change in $H_{0}$ from Eq. 2.39 of Breuval (2021): $H_{0}^{\prime}/H_{0}=10^{-0.2\Delta\mu}$, where $H_{0}$ is the most recent value of the Hubble-Lemaître constant, $73.04\pm 1.04~{}\mathrm{km\,s^{-1}Mpc^{-1}}$ (Riess et al., 2022), and $H_{0}^{\prime}$ is the value of $H_{0}$ changed by $\Delta\mu=0.002$ mag. We consider the difference $|H_{0}^{\prime}-H_{0}|=0.07\,\mathrm{km\,s^{-1}Mpc^{-1}}$ as the statistical error on $H_{0}$. . The values presented above are much smaller than the uncertainty of the most accurate fit of the PLR zero point derived so far (0.017 mag in $W_{H}$, Cruz Reyes & Anderson, 2022), meaning that in $W_{H}$ the effect of binarity is insignificant and stays below the detection threshold set by the uncertainty of the PLR zero point fit.

Binary Cepheids are an inherent and irremovable part of the PLR calibration. Since they appear brighter than their single counterparts (due to the extra—and not accounted for—light of their companions), the zero point of the PLR is shifted toward brighter magnitudes. This shift scales linearly with the binary Cepheid fraction and decreases with increasing wavelength. We confirm the finding of Anderson & Riess (2018), that in the NIR domain and Wesenheit indices the effect of binary Cepheids is the smallest.

The shifts of the PLR zero points that occur in a pair of galaxies cause a shift in the distance modulus difference between them. The sign of this shift can be positive or negative, depending on the fractions of binary Cepheids in the two galaxies. If they have similar fractions, the shift in the distance modulus is consistent with zero. On the contrary, a maximum shift in the distance modulus is expected when the fractions of binary Cepheids in the two galaxies are 0 and 100% or vice versa.

The value of the shift depends on the variant of the synthetic population (a unique combination of metallicity, star formation history, shape and location of the instability strip, as well as the distribution of the initial parameters). In the $W_{H}$ index, used by Riess et al. (2022) to derive $H_{0}$, the distance modulus between a pair of galaxies is shifted due to binary Cepheids by $\pm 0.004$ mag at most. Considering the sample of 37 SNeIa host galaxies from Riess et al.’s study, whose distance moduli are altered by any value in range $[-0.004,0.004]$ mag, we concluded that the effect of binary Cepheids does not shift the value of $H_{0}$, instead it increases the statistical error on $H_{0}$ by $0.07\,\mathrm{km\,s^{-1}Mpc^{-1}}$ or 0.1%.

The effect of binary Cepheids on the extragalactic distance scale has proved minuscule yet not negligible, if one aims to achieve a sub-percent precision of $H_{0}$. The most important step that can be taken to minimize this effect is to observationally establish the percentage of binary Cepheids and the types of Cepheid companions in anchor and SNeIa host galaxies.

The data used to create Figures 7-10 are available at araucaria.camk.edu.pl/index.php/synthetic-population-of-binary-cepheids/. The collection of synthetic populations is available upon request.