Searching for a ringdown overtone in GW150914

We reanalyze the GW150914 postmerger data searching for quasinormal modes beyond the fundamental, quadrupolar mode. There is currently an ongoing disagreement in the literature about whether, and to what extent, the data contains evidence for a quasinormal mode overtone. We use a frequency-domain approach to ringdown data analysis that was recently proposed by the authors. Our analysis has several advantages compared to other analyses performed mainly in the time domain; in particular, the source sky position and the ringdown start time are marginalized over (as opposed to simply being fixed) as part of a Bayesian ringdown analysis. We find tentative evidence for an overtone in GW150914, but at a lower significance than reported elsewhere. Our preferred analysis, marginalizing over the uncertainty in the time of peak strain amplitude, gives a posterior on the overtone amplitude peaked away from zero at $\sim 1.8\sigma$.

Figure 1

Our ringdown inference is run initially using a flat, uniform prior on the ringdown start time, $t_0$, over the plot range $\pm 15\widetilde{M_f}$ relative to $t_{\mathrm{ref}}$ (Hanford frame). In post processing, the posterior samples can be reweighted to a different choice of prior on $t_0$ (see Sec. 2c). The different prior choices used in this paper are shown in this figure. We use a sequence of narrow Gaussian priors (with different means $\overline{t_0}$ defined relative to $t_{\mathrm{ref}}$ and fixed standard deviation, $\sigma =1\widetilde{M_f}$) as well as using the posterior on the time of peak strain from a full inspiral-merger-ringdown (IMR) analysis as a prior.

Figure 2

Posterior distributions on the remnant mass, $M_f$, and dimensionless spin, ${\chi }_f$, for different choices of $t_0$ prior (the colors and line styles correspond to those used in Fig. 1). Left: the results from the (2, 2, 0) fundamental-mode-only analysis (i.e., $N=0$). Right: the results from the overtone analysis including the (2, 2, 0) and (2, 2, 1) modes (i.e., $N=1$). Each line corresponds to a different choice of $t_0$ prior. Colored lines correspond to Gaussians with widths of $1\widetilde{M_f}$ and means $\overline{t_0}$ (see Fig. 1). The dashed black line corresponds to using the posterior on time of peak strain (from a full IMR analysis) as our prior, which marginalizes over uncertainty on the time of peak strain. Also shown for reference (dotted line) is the posterior from a full IMR analysis. The main panel shows the 90% confidence contours while the side panels show the one-dimensional marginalized posteriors.

Figure 3

Posteriors on the overtone amplitude, and Bayes’ factors in favor of the overtone model for different choices of $t_0$ prior (the colors and line styles correspond to those used in Fig. 1). Top: posterior on the time of peak strain in the Hanford frame, from a imrphenompv2 analysis, as in Fig. 1 (and originally from Ref. [28]). Middle: overtone amplitude posteriors for different choices of $t_0$ prior. The left panel corresponds to Gaussian priors with standard deviation $1\widetilde{M_f}$, centered at the time they are plotted. The dotted line indicates the expected exponential decay of the $A_1$ mode; this is included merely to guide the eye and was produced using the median mass and spin values from the full IMR analysis and the median value of $A_1$ from the $\overline{t_0}=0$ prior. The right panel corresponds to using the imrphenompv2 time of peak strain as a prior. For earlier start times the posteriors on the amplitude are peaked further away from zero; this is quantified in the inset plot where the ratio of the median to the standard deviation of the $A_1$ posterior is plotted. Bottom: the Bayes’ factor in favor of the overtone model for each prior choice; circles with error bars show the Bayes’ factor calculated from nested sampling (with errors estimated by the sampler) while the crosses show the results calculated using the Savage-Dickey density ratio.

Figure 4

Posteriors on the overtone amplitude from our $N=1$ overtone analysis, rescaled to a fixed reference time of $t_{\mathrm{ref}}$. The rescaling does not significantly affect the significance with which the posteriors are peaked away from zero. The colors and line styles indicate the prior used on $t_0$ and correspond to those used in Fig. 1.

Figure 5

Posteriors on the deviation parameter from the Kerr value for the real part of the overtone frequency. The colors and line styles distinguish different choices for the $t_0$ prior and correspond to those used in Fig. 1. The mode frequency is given by $f_{221}^{\mathrm{Kerr}}\exp (\delta f)$, so that ${\delta }_f=0$ is the expected result for the Kerr metric. For all choices of $t_0$ prior the data is consistent with $\delta f=0$.

Figure 6

The posterior on the dimensionless complex frequency of the second QNM (50% and 90% regions), assuming the first is the fundamental $(\ell ,|m|,n)=(2,2,0)$ mode. Lines indicate the Kerr frequencies parametrized by the remnant spin; dots and crosses indicate points with ${\chi }_f=0.7$ and 0 respectively. Lines are colored according to their $\ell$ and $n$ indices and the $m$ index increases left to right in each set. The frequency of the second QNM is consistent with the expected (2, 2, 1) overtone, but also with several other modes. However, all fundamental modes (those with $n=0$) are excluded.

Figure 7

Posteriors on the amplitude ratio $A_1/A_0$ from our $N=1$ overtone analysis. The 90% contours are plotted, with the colors and line styles indicating the $t_0$ prior and correspond to those used in Fig. 1. The solid gray curve shows the results of a two-QNM fit to the numerical relativity simulation SXS:BBH:0305 which has parameters consistent with GW150914. The dashed gray curve shows the results of a multi-QNM fit to SXS:BBH:0305 which follows closely the expected exponential decay rate for the amplitude ratio.

Figure 8

Posterior on the source sky position using geocentric coordinates in Mollweide projection. Shown in blue is the results from the $N=1$ overtone analysis using the IMRP $t_{\mathrm{peak}}$ reweighting for the prior on the ringdown start time. The LIGO skymap for this event is shown by the dashed black line for comparison. The inset plot shows an enlarged map plotted using right ascension and the sine of the declination. In both cases, 50% and 90% contours are plotted.

Figure 9

Posterior on the reconstructed whitened waveform. Shown in gray is the strain data from both LIGO interferometers (top: Hanford; bottom: Livingston) whitened according to the noise amplitude spectral density in the detector and bandpass filtered between 32 and 512 Hz for clarity. Shown in blue is the waveform reconstruction from the $N=1$ overtone analysis with the IMRP $t_{\mathrm{peak}}$ reweighting for the prior on the ringdown start time. The blue lines and shaded regions indicate median and the 90% credible interval. The signal is plotted as a function of time from $t_{\mathrm{ref}}$ using both SI and natural units on the upper and lower $x$ axis respectively.

Figure 10

Posteriors on the ringdown start time obtained from our initial analysis using a flat prior over the range shown in the plot. Results are shown for the fundamental only ($N=0$) and overtone ($N=1$) analyses. Vertical colored lines show the locations of the means $\overline{t_0}$ of the narrow Gaussian priors used for the subsequent reweighting (see Fig. 1). The $N=1$ posterior has ample support across the entire range of interest, as required for the reweighting to remain accurate. The $N=0$ posterior has enough support everywhere except the $\overline{t_0}=-2\widetilde{M_f}$ prior.

Figure 11

This is similar to the middle panel of Fig. 3 in the main text, but shows the posteriors on the overtone amplitude from the noise injection study. The different violin plots are for the different priors on the ringdown start time and the colors are the same as those used in Figs. 1 and 3. On the right-hand side of each set of violin plots, the filled posterior shows the result obtained using the real GW150914 data (this is the same as what is plotted in Fig. 3). On the left-hand side are all the posteriors from the injection campaign, which indicate the spread in results due to different noise realizations. Finally, the dashed lines on the right-hand side are the posteriors from the zero-noise injection. This plot is intended to be compared to Fig. 2 of Ref. [27], and Fig. 6 of Ref. [28].

Figure 12

Posteriors on selected parameters for the $W=3$ wavelets used in the $N=1$ overtone analysis, reweighted using the IMRP $t_{\mathrm{peak}}$ prior on the ringdown start time. Left: the wavelet central times, ${\eta }_w$. Middle: the wavelet widths, ${\tau }_w$. Right: the wavelet frequencies, ${\nu }_w$. The index runs over values $w=1$, 2 and 3, where the numbering of the wavelets is chosen to enforce the ordering ${\nu }_w<{\nu }_{w+1}$. All plots use SI units on the upper $x$ axis and natural units on the lower $x$ axis.