Radio Properties

Chapter II: A systematic survey of luminous, extragalactic radio transients in

2.6 Radio Properties

field that have SDSS spectra. This spectroscopic completeness is not exact, as it is appropriate for only those galaxies with SDSS spectra. The true spectroscopic completeness of the full sample will be higher than 𝑓_𝑆 = 0.47, so our measured 𝑛¤ will be overestimated. The true completeness is also likely distance dependent, as nearby, bright galaxies are slightly more likely to have a spectrum. We ignore these effects, as statistical uncertainties dominate our results: if we restrict our analysis to only those transients with SDSS spectroscopic hosts, our measured rates and slopes change by< 1𝜎. Future, larger transient samples will need to consider the spectroscopic selection function more carefully.

We assume the time between FIRST and VLASS E1 for all sources is Δ𝑡 = 20 years. Δ𝑡 is, in reality, a function of position on the sky and ranges from∼10−20 years. It is possible to account for this when modelling the luminosity function by assuming a form for the typical radio lightcurve and, assuming random, uniformly distributed launch dates for each transient, calculating the exact number of transient expected as a function of sky location, lightcurve parameters, and luminosity func- tion parameters, which can then be fit to the observations. This method would be particularly powerful for surveys where there are multiple epochs of detections for each transient, allowing the light curve evolution to be directly constrained. Be- cause we have a relatively small sample with only two epochs of detections (VLASS E1/E2), which show little flux evolution for most transients (see Section 2.6), it is unlikely that the more complex fit including lightcurve parameters would result in any reliable information. Hence, we simplify our analysis by assuming a constant Δ𝑡 =20 years. The implicit assumption here is thus that each transient has a constant lightcurve that lasts for∼Δ𝑡 years. ReducingΔ𝑡 would increase the measured rate

¤ 𝑛.

With the definitions above, the total number of radio transients within the field- of-view considered and with a spectroscopic host is given by

𝑁₀= 4 3𝜋 𝑑³

𝑀𝑓_𝐴𝑓_𝑆𝑛¤Δ𝑡

𝐿^1−Γ

min

(10^27(1−Γ) −10^30(1−Γ)), (2.5) where 𝐿_min is in units of erg s⁻¹ Hz⁻¹. The final term involving 𝐿_min scales the number of transients expected in the luminosity range 10²⁶⁻³⁰ erg s⁻¹ Hz⁻¹ to the total number we should observe, which includes all those transients with 𝐿 > 𝐿_min.

This equation does not account for our incompleteness due to the VLASS flux density limit 𝑓_lim = 0.7 mJy. As described in Appendix 2.9, we can calculate the fraction of sources with 𝑓 > 𝑓_lim, which we write 𝑃(𝑓 > 𝑓_lim). The observed number of sources is then𝑁₀𝑃(𝑓 > 𝑓_lim).

We can use this value and the probabilities of obtaining a given luminosity,𝑃_𝐿(𝐿), and distance, 𝑃_𝑑(𝑑), to calculate the Poisson log-likelihood for a given set of pa- rameters log𝑛¤ andΓ(Equation 2.6):

logL =−𝑁₀𝑃_𝑓(𝑓 > 𝑓_lim) +

𝑁_obs

∑︁

𝑖=0

log 𝑁₀

𝑁_obs

𝑃_𝐿(𝐿_𝑖)𝑃_𝑑(𝑑_𝑖)

. (2.6)

Here, 𝑖 sums over each transient with luminosity 𝐿_𝑖 and distance 𝑑_𝑖. 𝑁_obs is the observed number of transients, in contrast to 𝑁₀, which is the predicted number of transients for a given𝑛¤ andΓ.

We fit this log-likelihood forΓand log𝑛¤. We perform the fit using thedynestydy- namic nested sampler (Higson et al., 2019; Skilling, 2006; Speagle, 2020; Skilling, 2004; Feroz, Hobson, and Bridges, 2009) with uninformative tophat priors:

log𝑛¤ ∈ [−100,100], Γ∈ [1,5]. (2.7) We perform this fit for all the transients and the transients in each class (AGN flares, TDEs, stellar explosions). Because we only detect two strong TDE candidates, we fix the slope atΓ =2. We choseΓ =2 because, as we will discuss, we obtain values consistent withΓ =2 for the fits to every other class and the full transient sample.

In Table 2.1, we summarize the results of ourdynestyfits. We report the results separately for all transients, AGN flares, TDEs, and stellar explosions. Note that some events are not included in any of these subsamples, so we do not expect the rates of these two subsamples to sum to the total rate. We also show the luminosity functions in Figure 2.13 for the full sample and each subsample.

In Table 2.1, we include the luminosity function index Γ, the volumetric rate, the areal rate, and the rate within 200 Mpc per year. We also include the rate within 200

Table 2.1: Luminosity function and rate parameters measured as described in Sec- tion 2.6. Reported uncertainties are 1𝜎. The rate per galaxy is determined by normalizing the observed rate for each transient class to the number of galaxies with the corresponding BPT classification.

Selection Γ log Rate log Rate log Rate log Rate

Gpc⁻³yr⁻¹ deg⁻²yr⁻¹ yr⁻¹ Galaxy⁻¹yr⁻¹

All transients 1.91⁺₋⁰₀^._.⁰⁸₀₈ 3.81⁺₋⁰₀^._.⁰⁹₁₀ −1.66⁺₋⁰₀^._.⁰⁹₁₀ 2.34⁺₋⁰₀^._.⁰⁹₁₀ −2.42⁺₋⁰₀^._.⁰⁹₁₀ SMBH Flares 1.97⁺⁰₋₀^._.¹⁴₁₃ 3.56⁺⁰₋₀^._.¹⁵₁₅ −1.91⁺⁰₋₀^._.¹⁵₁₅ 2.09⁺⁰₋₀^._.¹⁵₁₅ −2.05⁺⁰₋₀^._.¹⁵₁₅ TDEs 2 (fixed) 2.62⁺₋₀^0._.²⁸₃₈ −2.85⁺₋₀^0._.²⁸₃₈ 1.15⁺₋₀^0._.²⁸₃₈ −2.53⁺₋₀^0._.²⁸₃₈ Stellar Explosions 1.99⁺⁰₋₀^._.¹⁴₁₄ 3.57⁺⁰₋₀^._.¹⁵₁₆ −1.91⁺⁰₋₀^._.¹⁵₁₆ 2.09⁺⁰₋₀^._.¹⁵₁₆ −2.48⁺⁰₋₀^._.¹⁵₁₆

SDSS transients 1.87⁺⁰₋₀^._.⁰⁹₁₀ 3.64⁺⁰₋₀^._.¹¹₁₁ −1.83⁺⁰₋₀^._.¹¹₁₁ 2.17⁺⁰₋₀^._.¹¹₁₁ −2.59⁺⁰₋₀^._.¹¹₁₁ SDSS SMBH Flares 1.98⁺⁰₋₀^._.¹⁹₁₉ 3.36⁺₋₀^0._.²²₂₂ −2.12⁺₋₀^0._.²²₂₂ 1.88⁺₋₀^0._.²²₂₂ −2.26⁺₋₀^0._.²²₂₂ SDSS TDEs 2 (fixed) 2.61⁺⁰₋₀^._.²⁹₃₆ −2.87⁺⁰₋₀^._.²⁹₃₆ 1.13⁺⁰₋₀^._.²⁹₃₆ −2.54⁺⁰₋₀^._.²⁹₃₆ SDSS Stellar Explosions 1.92⁺₋₀^0._.¹⁵₁₄ 3.45⁺₋₀^0._.¹⁷₁₇ −2.03⁺₋₀^0._.¹⁷₁₇ 1.97⁺₋₀^0._.¹⁷₁₇ −2.61⁺₋₀^0._.¹⁷₁₇

Mpc per galaxy, where the total number of galaxies is calculated from SDSS. For the all transients category, we divide the total rate within 200 Mpc by the total number of SDSS spectroscopic galaxies within 200 Mpc, corrected for spectroscopic and areal completeness. For the AGN flares category, we divide by the number of BPT AGN galaxies in SDSS within 200 Mpc. For TDEs, we divide by the number of quiescent galaxies in SDSS within 200 Mpc. Finally, we divide the stellar explosion rate by the number of star-forming galaxies within 200 Mpc to calculate the rate per galaxy.

We also report the results for the full sample and each subsample when only consid- ering those events with SDSS spectroscopic hosts. The results for the SDSS-only samples are, in all cases < 1𝜎 consistent with the full samples, supporting our hypothesis that we are dominated by statistical uncertainty rather than uncertainty in our assumptions about spectroscopic completeness.

For all subsamples, the slope of the luminosity function is Γ ∼ 2 within 1𝜎.

10⁻² 10⁻¹ 10⁰ Luminosity [10²⁹erg s⁻¹Hz⁻¹] 10⁻¹

10⁰ 10¹ 10² 10³ 10⁴ 10⁵

N(L)

All

10⁻² 10⁻¹ 10⁰

Luminosity [10²⁹erg s⁻¹Hz⁻¹] All AGN Flare

10⁻² 10⁻¹ 10⁰

Luminosity [10²⁹erg s⁻¹Hz⁻¹] 10⁻¹

10⁰ 10¹ 10² 10³ 10⁴ 10⁵

N(L)

All

Stellar Explosions

10⁻² 10⁻¹ 10⁰

Luminosity [10²⁹erg s⁻¹Hz⁻¹] All TDE

Figure 2.13: The observed luminosity function for all transients (top left), AGN flares (top right, stellar explosions (bottom left) and TDEs (bottom right). The best- fit model is shown as a shaded region with 1𝜎 uncertainties. The median best-fit model for the full transient sample is overlaid on each panel in yellow.

This consistency of the luminosity function slope between all subsamples is in- triguing. Interpretation of this result is complicated: our luminosity function is not measured using, e.g., the peak luminosity of each transient. We expect that most of our transients are detected at peak because VLASS is flux limited and the volume in which a given source is detectable will be maximized at its light curve peak.

With the limited sample size and poorly sampled radio light curves available, we cannot confirm this expectation. If we are not observing each source at peak, the luminosity function is determined by both the peak luminosities and the lightcurve shapes. For example, suppose we are observing sources that all have the same peak luminosity but are decaying as 𝐿_𝜈(𝑡) ∝ 𝑡⁻¹, where 𝑡 is measured relative to the transient launch date. This assumed lightcurve is consistent with the lightcurve expected for a spherical outflow shocking against a dense medium. If the launch dates are uniformly distribution between FIRST and VLASS, then the luminosity

function will be related to the probability distribution of 𝑡⁻¹, i.e., the probability distribution of the inverse of a uniform variable. Thus, ^𝑑𝑁_𝑑𝐿 ∝ 𝐿⁻², as observed. Our constraint of a constant luminosity function shape for all sources could thus be ex- plained by taking into account the fact that we are not observing each source at peak.

The total transient rate is∼10^3.81Gpc⁻³yr⁻¹, or∼200 transients within 200 Mpc per year in the luminosity range 10²⁷⁻³⁰erg s⁻¹Hz⁻¹. This rate includes roughly equal contributions from SMBH flares and stellar explosions. Stellar explosions occur at a rate of∼10^3.56Gpc⁻³yr⁻¹, or 123 events per year. SMBH flares occur at roughly the same rate. Normalized to the number of galaxies, stellar explosions occur a factor of∼3 more frequently than SMBH flares, although the rates per galaxy are consistent within uncertainties. TDEs are the least frequent event, at 10^2.62 Gpc⁻³ yr⁻¹, or 400 events per year.

Timescales

As briefly discussed in the previous section, we do not have sufficient data to perform a detailed fit to the transient timescales. We can, however, qualitatively comment on the typical timescales using the change in flux between VLASS E1 and E2. In Figure 2.14, we show histograms of the ratio of VLASS Epoch 1 to Epoch 2 flux.

Those sources that are not significantly detected in VLASS E2 are shown as upper limits. Seven of the sources rose between E1 and E2 (14%) and fourty-four of the sources faded (86%). This immediately suggests that the transients in our sample fade on much slower timescales than those on which they rise.

If we split the transients by classification (SMBH flare or stellar explosion, it is clear that SMBH flares are more strongly biased toward fading sources than the general population: 91% of SMBH flares are fading in contrast to 9% rising. This SMBH flare sample is likely contaminated with GPS sources, which could bias our result as these sources would not evolve between VLASS E2 and E1. However, re- moving sources with similar E1 and E2 fluxes (within uncertainties) only increases the ratio of fading to rising flares.

These percentages are difficult to interpret directly as we are most sensitive to sources near the peak of their lightcurves since VLASS is flux-limited, so we expect cases where VLASS E1 was observed during the flare rise and VLASS E2 was observed during the flare decay. The fact that the large majority of SMBH flares

are decaying tells us that the lightcurve shape is such that≳ 91% of the time during which the flare is above the VLASS sensitivity, the E1 observation was either taken (a) before the lightcurve peak but sufficiently close in time to the lightcurve peak that the flare appears to have faded ∼2 years later or (b) the E1 observation was taken post-peak.

Stellar explosions are also biased toward fading sources, but not as significantly.

18% of stellar explosions are rising and 82% are fading. This suggests that the ratio of the rise-to-fade time for stellar explosion flares is higher than that for SMBH flares.