Excavating black hole continuum spectrum: Possible signatures of scalar hairs and of higher dimensions

Continuum spectrum from black hole accretion disc holds enormous information regarding the strong gravity regime around the black hole and hence about the nature of gravitational interaction in extreme situations. Since in such strong gravity regime the dynamics of gravity should be modified from the Einstein-Hilbert one, its effect should be imprinted on the continuum spectrum originating from the black hole accretion. To explore the effects of these alternative theories on the black hole continuum spectrum in an explicit manner, we have discussed three alternative gravitational models having their origin in three distinct paradigms—(a) higher dimensions, (b) higher curvature gravity, and (c) generalized Horndeski theories. All of them can have signatures sculptured on the black hole continuum spectrum, distinct from the standard general relativistic scenario. Interestingly all these models exhibit black hole solutions with tidal charge parameter which in these alternative gravity scenarios can become negative, in sharp contrast with the Reissner-Nordström black hole. Using the observational data of optical luminosity for eighty Palomer Green quasars we have illustrated that the difference between the theoretical estimates and the observational results gets minimized for negative values of the tidal charge parameter. As a quantitative estimate of this result we concentrate on several error estimators, including reduced ${\chi }^2$, Nash-Sutcliffe efficiency, index of agreement etc. Remarkably, all of them indicates a negative value of the tidal charge parameter, signaling the possibility of higher dimensions as well as scalar charge at play in those high gravity regimes.

Figure 1

The above figure illustrates the variation of the theoretically derived luminosity from the accretion disc with frequency for two different masses of the supermassive black hole and for seven different values of $q$, both positive and negative. The case $q=0$ corresponds to the standard Schwarzschild scenario, which is depicted by the thick black line in the middle. Further, a clear distinction between positive and negative $q$ values is evident from the above figure, for positive values of $q$ the optical luminosity is larger than the Schwarzschild value, while for negative values of $q$ it is smaller. The accretion rate assumed is $1M_{\odot }{\mathrm{yr}}^{-1}$ and $\cos i$ is taken to be 0.8. See text for more discussions.

Figure 2

Theoretical optical luminosities along with the observed luminosity values for high mass PG quasars have been plotted with black hole mass $M$ for different negative values of the tidal charge parameter $q$. The differences between theoretical and observational values have also been depicted. It is quite clear that some intermediate value of negative $q$ should minimize the difference between theoretical estimates and observed data. See text for more discussions.

Figure 3

Theoretical optical luminosities associated with the electromagnetic radiation emanating from the accretion discs around the most massive black holes among the eighty PG quasars have been plotted against the black hole mass $M$ for different positive values of $q$. The same plot also depicts the corresponding observed values. The difference between the two values have been depicted in red lines. It is obvious that as one increases the value of $q$ the difference between the theoretical values and observed data also increases. See text for more discussions.

Figure 4

Theoretical optical luminosity for different negative values of $q$, along with observed luminosity values have been plotted for the high mass black holes among the eighty PG quasars against the accretion rate $\dot{M}$. The difference between theoretical and observational values have also been illustrated. In this case also it is clear that for some intermediate negative value for $q$ the difference between theoretical values and observed data are minimum.

Figure 5

Theoretical optical luminosities for different positive values of $q$, along with the observed luminosity values have been plotted for high mass black holes among the eighty PG quasars against the mass accretion rate. The difference between theoretical and observational values can be clearly seen. The increase in the difference between theoretical values and observed data as $q$ is being increased, is also present here.

Figure 6

The figure demonstrates variation of ${\chi }_{\text{Red}}^2$ against the tidal charge parameter $q$ obtained using a sample of eighty supermassive black holes. Surprisingly, ${\chi }_{\text{Red}}^2$ attains minimum value for a negative value of $q\sim -0.4$, suggesting that the errors between theoretical estimates with observations are minimum when one considers either higher dimensional models or scalar hairs with a specific coupling to gauge fields. See text for more discussions.

Figure 7

The above figure depicts variation of the Nash-Sutcliffe efficiency with the tidal charge parameter $q$. In this case the maximum of $E$ minimizes the theoretical deviations from observational estimates. It is clear that once again negative value of $q$ maximizes the efficiency, in particular $E$ gets maximized at $q\sim -0.4$, close to the value which minimizes ${\chi }_{\text{Red}}^2$.

Figure 8

Plot of index of agreement $d$ against the tidal charge parameter $q$. The plot shows that for negative values of $q$, the index of agreement is larger than for positive $q$ values. In particular for $q\sim -0.4$ it depicts a maximum, pointing to the fact that for this value of $q$, theoretical predictions are much closer to the observational one and thus it is more preferred compared to the Schwarzschild scenario. See text for more discussions.

Figure 9

The above figures depicts variation of the modified error estimators with tidal charge parameter. The left one illustrates modified index of agreement $d_1$ while the right one corresponds to the modified Nash-Sutcliffe efficiency $E_1$. Both of them follows an identical trend of being maximum at some negative values of tidal charge parameter as fit with our earlier discussions.