On the all-India rainfall index and sub-India rainfall heterogeneity

Spencer A. Hill1, Adam H. Sobel1,2, Michela Biasutti1, and Mark A. Cane1
1Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York
2Department of Applied Physics and Applied Mathematics and Lamont-Doherty Earth Observatory, Columbia University, New York, New York
Spencer A<EMAIL_ADDRESS>
* We examine Indian summer monsoon rainfall interannual variability and teleconnections 1901-2020

* All-India Rainfall Index (AIRI) closely tracks spatial and temporal extent of wet anomalies

* ENSO teleconnection projects onto AIRI but Indian Ocean Dipole onto a known tripole pattern

We revisit long-standing controversies regarding relationships among the all-India rainfall index (AIRI), sub-India summer rainfall variations, El Niño-Southern Oscillation (ENSO), and the Indian Ocean Dipole (IOD) using 120-year sea surface temperature and high-resolution rainfall datasets.  AIRI closely tracks with the spatial extent of wet anomalies and with the average across gridpoints in rainy day count.  The leading rainfall variability mode is a monopole associated primarily with rainy day count and ENSO.  The second mode is a tripole with same-signed loadings in the high-rainfall Western Ghats and Central Monsoon Zone regions and opposite-signed loadings in Southeastern India between.  The IOD projects onto this tripole and, as such, is weakly correlated with AIRI.  However, when the linear influence of ENSO is removed, the IOD rainfall regressions become quasi-homogeneously more positive, making the ENSO-residual IOD and AIRI timeseries significantly correlated.

§ PLAIN LANGUAGE SUMMARY

The Indian summer monsoon generates copious rainfall each June through September, but more in some years than others, and more in some Indian sub-regions than others.  We use observation-based datasets spanning 1901-2020 to reconcile past disagreements about these fluctuations.  There has long been concern that the rainfall rate averaged over the whole summer and whole of India—the All-India Rainfall Index—doesn't necessarily track with the spatial extent of wet or dry anomalies within India which is more relevant for many societal purposes, but we show that the two measures vary closely with one another.  There has been disagreement about how the all-India average relates to rainfall in fixed sub-regions of India, and we show using high-resolution rainfall data that the issue stems in part from the use of coarse datasets in the past.  Finally, we reconcile disagreements about whether or not the Indian Ocean Dipole (IOD) teleconnection influences all-India rainfall by showing that, when the El Niño-Southern Oscillation signal that influences both the IOD and the monsoon rains is removed, the IOD significantly influences the all-India average and with a distinct sub-India spatial pattern.

§ INTRODUCTION

Rainfall averaged across India and over June-July-August-September (JJAS)—the All-India Rainfall index (AIRI)—is a widely used, physically intuitive, and societally relevant <cit.> bulk indicator of the summer monsoon.  But its use is controversial, largely owing to summer monsoon rainfall's pronounced sub-India and sub-seasonal heterogeneity.  Summer rainfall is high within a broad band of northern-central India, higher still within a narrow band in the southwest between the Arabian Sea and the Western Ghats mountains, and much lower in southeastern India between these two bands. And throughout the subcontinent JJAS rainfall variability is much higher on daily than interannual timescales <cit.>.  In this study we address three long-running controversies regarding AIRI's interpretation in light of this spatiotemporal variability, attempting to synthesize and reconcile past arguments made with differing and often short and/or coarse datasets by using state-of-the-art, 120-year datasets of Indian rainfall at high resolution and of sea surface temperature (SST).

First is the relationship between AIRI and the overall spatiotemporal extent of wetting or drying across India.  Spatially, in most summers there are parts of India that experience drought and others excess rainfall, a heterogeneity in summer-mean rainfall anomalies that led gadgil_climate_1995 to describe AIRI as “not very meaningful.”  Indeed, for rain-fed agriculture quite plausibly the spatial extent of drought relative to local rainfall normals is more relevant than the average rainfall anomaly in / across India <cit.>, leading to past attempts to define bulk indices in terms of the fraction of the Indian surface area experiencing rainfall anomalies exceeding specified thresholds <cit.>.

Temporally, a recent series of studies <cit.> employ a useful decomposition of JJAS-mean rainfall at any given point into the product of the frequency of rainy days and the mean rainfall intensity on rainy days, arguing that rainy day frequency tends to be more spatially coherent across India and thus potentially predictable.  We will make use of this decomposition to construct and compare alternative indices to AIRI.

Second is the relationship between AIRI and rainfall anomalies over geographically fixed sub-India regions.  The literature on dividing India into so-called “coherent” or “homogeneous” sub-regions—/ over which rainfall interannual variability tends to be similar—dates to the 19th century (see shukla_interannual_1987,parthasarathy_homogeneous_1993,gadgil_coherent_1993 and references therein).  Subseasonally, it is well established that the predominant variability mode—the so-called active-break cycle—comprises a tripole: in active periods, rainfall is enhanced over the Western Ghats (henceforth WG) and Central Monsoon Zone (henceforth CMZ) bands but suppressed over southeastern India (henceforth SEI) <cit.>.  And because the CMZ and WG anomalies are larger than the SEI anomalies in absolute terms, AIRI is enhanced (the same results hold with signs reversed for break periods).

Interannually, the predominant mode is instead a quasi-uniform wetting or drying across the subcontinent that strongly projects onto AIRI <cit.>.  A second seasonal-mean variability mode has been identified <cit.> but characterized by mishra_prominent_2012 as a north-south dipole rather than the CMZ-SEI-WG tripole, with generally same-signed anomalies across peninsular India including both the WG and SEI, and opposite-signed conditions to the north including the CMZ.  This would seem consistent with the argument put forth separately that rainfall anomalies in the CMZ and WG are weakly correlated, and that therefore AIRI effectively amounts to the superposition of two quasi-independent processes <cit.>.  However, we will argue that the \(\geq\)1\(\times\)1 rainfall datasets used regarding both points lead to erroneous or incomplete conclusions.

Third is the relationship between both AIRI and sub-India rainfall anomalies to teleconnection modes emanating from the tropical Pacific and Indian Oceans.
The link between the Indian summer monsoon and the / (ENSO) is well established, with drought more likely in / summers and excess rain in / summers <cit.>, both at the all-India scale and most individual points (again excluding the far northeast).  Nevertheless, ENSO indices have always left the majority of AIRI variance unexplained <cit.>, even before a weakening of the ENSO-AIRI lag-zero correlation in recent decades compared to the mid 20th century <cit.>.  The Indian Ocean Dipole (IOD)—the zonally oriented oscillatory mode spanning the equatorial Indian ocean <cit.>—has been argued to modulate ENSO's influence on AIRI <cit.>, with the IOD-AIRI correlation relatively high in decades when the ENSO-AIRI correlation is relatively low and vice versa.  While it has been argued that the IOD is in fact largely controlled by ENSO <cit.>, ashok_impacts_2007 argue that the IOD spatial imprint on Indian summer monsoon rainfall is much more heterogeneous, concentrated  roughly over the CMZ region.

Meanwhile, other studies focused on the atmospheric counterpart to the IOD, the Equatorial Indian Ocean Oscillation (EQUINOO), have argued that the correlation between the IOD and AIRI is effectively zero <cit.>.  We will attempt to reconcile these arguments by examining the spatial imprint of the IOD and ENSO on rainfall amount, rainy-day frequency, and mean rainfall intensity (c.f. moron_spatial_2017) and by isolating the IOD and ENSO signals from one another (c.f. <cit.>).

After describing the datasets and methods used (Section <ref>), we argue the following (Section <ref>): that AIRI is an excellent indicator of the spatial extent of wet anomalies across the subcontinent; that the second leading mode of interannual variability is the familiar CMZ-SEI-WG tripole; that rainfall anomalies within the CMZ and WG regions are well correlated; that while ENSO strongly imprints on the leading monopole pattern and thus AIRI, the IOD imprints on the tripole pattern and thus weakly on AIRI; and finally that when the ENSO signal is linearly removed the IOD influence becomes more positive at most points such that the ENSO-residual IOD and AIRI timeseries are significantly correlated.  We then conclude with summary and discussion (Section <ref>).

§ METHODS

For rainfall, we use the Indian Meteorological Department (IMD) daily, gridded 0.25\(\times\)0.25 dataset spanning 1901-2020, which is derived from a dense but time-varying network of rain gauges across India comprising 6955 in total and an average of around 2,600 operational in each year <cit.>.   Daily values from 1 June to 30 September are averaged to form a JJAS seasonal mean for each year.  We have compared the IMD dataset with the TRMM 3B42v7 <cit.> daily dataset over the period 1998-2014.  While on any given day the differences calculated from the two datasets over India can be substantial (not shown), the JJAS AIRI timeseries between the two datasets are very highly correlated (\(r=0.94\)).

As common in studies of the summer monsoon <cit.>, we exclude points in far north and northeast India, in our case by applying the “monsoonal India” mask of gadgil_new_2019.  In far northern India the terrain makes rain gauges less reliable and representative, and the vicinity to the Himalayas and extratropical influences make interannual rainfall variability not clearly related to the summer monsoon circulation.  For Northeast India there appear to be spurious data points in a cluster of grid cells on the border of Bangladesh, with the JJAS mean and variance jumping in 1971 to implausible values over the remainder of the record (not shown).  Nonetheless, the mean JJAS rainfall restricted by the monsoon mask is very highly correlated with the all-India rainfall average computed with these spurious points included, (\(r=0.95\)).  For simplicity henceforth we will refer to the monsoonal-India average as AIRI.

We use standard SST-based indices of ENSO and the IOD—the NINO3.4 and Dipole Mode Index (DMI; saji_dipole_1999) respectively—computed with monthly SSTs from the NOAA Extended Reconstruction SST <cit.> dataset (ERSST) version 5, 1901-2020.

NINO3.4 is defined as the average SST anomaly spanning 120-170W, 5S-5N, and DMI is defined as the difference between the SST anomalies averaged over a western equatorial Indian Ocean box (90-110E, 10S-0) and over an eastern equatorial Indian Ocean box (50-70E, 10S-10N).  Anomalies are defined as the deviation at each gridpoint from the local climatology over 1901-2020 for that month, with June through September anomalies then averaged to yield JJAS values for each index.

To focus on interannual variability, for all analyses we first subtract off the linear trend computed from simple least squares regression of the 1901-2020 timeseries of JJAS-mean values.  Because 1901-2020 trends are weak at the vast majority of gridpoints as well as for the teleconnection indices (not shown), all major results we present are insensitive to this choice.

When decomposing JJAS-mean rainfall into the product of the number of rainy days times the mean rain intensity on rainy days, we use a 1 / threshold to define rainy days, but results are very similar if 0 / is used instead (not shown).  For all detrended JJAS timeseries analyzed the interannual autocorrelations are sufficiently weak and the distributions are sufficiently normal (not shown) that a simple t-test yields a reasonable measure of statistical significance; in particular, using a conservative estimate of 100 degrees of freedom yields significance at the 5% level for correlations with \(|r|>0.195\).  Though it should be noted that for spatially gridded data such a point-by-point approach likely overstates significance to some degree due to spatial autocorrelations <cit.>.

§ RESULTS

§.§ AIRI relationship to spatiotemporal extent of rainfall anomalies

Fig. <ref>(a) shows the climatological JJAS rainfall and its interannual standard deviation for each gridpoint within monsoonal India.  The standard deviation largely scales with the local climatological mean, and both are heterogeneous—high within CMZ, highest within WG, and much lower in SEI.  Fig. <ref>(b) shows the fraction of annual-mean rainfall that occurs in JJAS as well as surface elevation contours.  The JJAS rainfall fraction exceeds 95% in parts of the far western CMZ, exceeds 80% over most of the WG and CMZ, and does not exceed 50% in most of SEI, where in the far southeast the rainy season occurs during the northeast monsoon in boreal autumn <cit.>.  The elevation contours highlight how southwesterly low-level summer monsoon flow largely perpendicular to the Western Ghats mountain range axis results in a concentrated rainfall band on the windward side and a much broader and drier regime on the leeward side over the Deccan Plateau <cit.>.
Fig. <ref>(c) shows the climatological number of rainy days in JJAS and the mean rainfall rate on those rainy dates.  Rain falls nearly every summer day within most of WG, most days within most of the CMZ, and less frequently within most of SEI.  Mean rainfall intensity is highest within WG directly along the coast, with a sharp gradient moving inland up the mountain slope, with smoother spatial variations elsewhere.

(a) Climatological JJAS rainfall mean in color shading and standard deviation in contours in the IMD 0.25\(\times\)0.25 gridded dataset, 1901-2020.  Standard deviation contours from white to black are 1, 2, 3, 4, and 5 / (maximum standard deviation is 14.2 /).  (b) Climatological fraction of annual-mean rainfall that occurs during JJAS in color shading and surface elevation in contours.  Elevation contours from white to black are 100, 300, 500, 700, and 900 m (maximum elevation is 1028 m).  (c) Climatological mean number of rainy days in JJAS in color shading and mean rainfall intensity on rainy days in contours.  Mean rainfall intensity contours from white to black are 10, 15, 20, 25, and 30 / (maximum mean rainfall intensity is 44.8 /.  Blue outlines in panel a and in subsequent figures show the borders of the three sub-India regions we analyze in detail, namely the Central Monsoon Zone (CMZ) in the north, Western Ghats (WG) in the southwest, and Southeastern India (SEI) in the southeast.

Given this heterogeneity, in principle AIRI need not be a useful indicator of the spatial extent of wet or dry anomalies: an average across regions with high variance and regions with low variance will depend primarily on the former and weakly on the latter.  We have therefore constructed six alternative all-India measures of JJAS rainfall variability: first, the average standardized rainfall anomaly (/ the raw anomaly divided by the standard deviation, rather than just the raw anomaly as for AIRI); second, the number of gridpoints in which the local JJAS rainfall anomaly is positive; third, the average across all gridpoints of the number of rainy days; fourth, the average across all gridpoints of the rainfall rate on rainy days; fifth, the mean rainfall anomaly restricting to those gridpoints for which the rainfall anomaly is positive; and sixth, the mean rainfall anomaly restricting to those gridpoints for which the rainfall anomaly is negative.  The first, third, and fourth variants were introduced by moron_spatial_2017.

Fig. <ref> provides a cross-correlation matrix of AIRI and these six alternative measures (along with numerous additional fields below the first horizontal line to be discussed below).  Among AIRI and the first three variants—which characterize the spatial or temporal extent of wet anomalies—all correlation coefficients exceed 0.9.  Among AIRI and the last three variants, which characterize the intensity of wet or dry anomalies, no correlation coefficient exceeds 0.77.  Thus, AIRI depends more on the spatiotemporal extent of rainfall across India and JJAS than on the severity of rainfall anomalies when and where they occur.  This agrees with moron_spatial_2017, who also note that this contrasts with interannual variability at the gridpoint scale: for most gridpoints of monsoonal India the JJAS rainfall amount depends more on the intensity of rain events than the number of rainy days.

Cross-correlation matrix showing the correlation coefficients (unitless) among AIRI, six variants thereof as defined in the text, four PC timeseries as described in the text, regional rainfall averages over each of the CMZ, SEI, and WG regions and a linear combination of CMZ and SEI as described in the text, and the NINO3.4 (with sign reversed) and DMI timeseries, all for JJAS and over the 1901-2020 period.
For the final row only, labeled “DMI, N34 resids”, both the DMI timeseries and all target timeseries are the residuals after linearly removing the NINO3.4 signal.  Horizontal and vertical lines separate the timeseries into categories: AIRI and variants thereof, PC timeseries, sub-region averages, and teleconnection indices, respectively from top to bottom and left to right.

We therefore conclude that AIRI is an excellent indicator of the overall spatial extent of wet or dry anomalies, as well as of the average anomaly in rainy day occurrence across India.  Insofar as the latter two measures are more directly relevant to agricultural and thus economic outcomes, this helps explain the high correlations found by previous studies of AIRI itself with agricultural yields and gross domestic product on interannual timescales <cit.>.

§.§ Rainfall relationships among fixed sub-India regions

We have repeated and extended the empirical orthogonal function (EOF) analysis of moron_spatial_2017 for JJAS rainfall amount, rainy day frequency, and mean rainfall intensity, by including more recent years and including the second rainfall amount EOF / mishra_prominent_2012.  Fig. <ref> shows the first EOF for each and the second EOF for rainfall amount, and Fig. <ref> reports correlation coefficients among the corresponding principal component (PC) timeseries and AIRI.  The first EOF of the rainfall amount accounts for 18.2% of the total variance and consists of same-signed loadings at the vast majority of gridpoints <cit.>.  This resembles the first EOF of rainy day frequency, which explains 32.4% of the rainy day frequency variance and exhibits an even more homogeneous pattern, though sharing the two local minima located east of the WG mountains axis and in the far east.  The PC timeseries corresponding to these amount and frequency first EOFs are each highly correlated with AIRI (0.96 and 0.93, respectively).

Both the first EOF of mean rainfall intensity (7.2% of variance) and the second EOF of rain amount (9.3% of variance) comprise a CMZ-SEI-WG tripole pattern.  The corresponding PCs are highly correlated with one another (0.85) but weakly with AIRI.  Unlike for rainfall amount and rainy day frequency, the first EOF for mean rainfall intensity is not well separated in variance explained from the remaining EOFs (the second EOF, not shown, accounts for 5.5%).  But its physical relevance is supported by the correspondence of its pattern to the rainfall EOF2 and to the IOD teleconnection results we will present below.

(a) First and (b) second EOF of JJAS rainfall amount, and first EOF of (c) rainy day frequency, and (d) rainfall intensity, each with the percentage of variance explained by that EOF printed in the top-left.    In each panel, values are normalized by that EOF's maximal value, and solid gray contours signify that EOF's zero contour.

We compute averages over each of the CMZ, WG, and SEI sub-regions using border definitions as follows (and are shown as blue contours in Fig. <ref>(a)).  The Central Monsoon Zone (CMZ) boundaries follow gadgil_new_2019.  We define the non-coastal borders of WG to align with the sharp rainfall gradient separating the coastal and inland peninsular regimes,

and define (SEI) to be all gridpoints west thereof and south of 18N.

Pointwise correlation maps (not shown) indicate that rainfall in points within the CMZ tends to vary positively both with other CMZ points and with points within WG, but weakly or negatively with those in SEI; the region-mean JJAS rainfall correlations are 0.23, 0.63, and 0.33 respectively for CMZ-SEI, CMZ-WG, and SEI-WG.

Using the CMZ, SEI, and WG regional averages, we have constructed one, two, and three-region linear regression models for AIRI and for the tripole mode as indicated by the rain amount PC2 (results are similar if the intensity PC1 is used instead; not shown).  For AIRI, at 0.93 the CMZ alone explains most of the variance as shown by gadgil_new_2019, leaving little additional skill to be gained by including either of the other regions.  For the tripole mode, by contrast, the highest-magnitude single-region correlation is with SEI at -0.70.  The CMZ-SEI two-region model, whose relative weights are 0.6 and -1 respectively and whose timeseries is included in Fig. <ref> along with the three individual region averages, yields 0.92 with the tripole mode.  This is effectively unchanged by including the WG as well.

Thus, despite WG being home to the highest JJAS-mean rainfall variances throughout monsoonal India, it adds little predictive power to CMZ and SEI for these bulk measures, due to it being well correlated with CMZ (0.63).  This is considerably higher than two similar-but-different regions (\(r\approx0.2\)) reported by vecchi_interannual_2004.  That study's WG region (borders overlain in Fig. <ref>d), based on coarse 2.5\(\times\)2.5 gridded CMAP rainfall data, spans into peninsular India and thus includes points for which rainfall anomalies are anti-correlated with those along the coast.  The correspondence between WG and CMZ rainfall anomalies has also been established through cluster analysis of seasonal-mean rainfall anomalies by moron_spatial_2017 using 0.25\(\times\)0.25 data.

We conclude that the sharp rainfall gradients in the southwest—which have dictated the border definitions of subdivisions of the Indian Meteorological Department station network since the 19th century <cit.>—are insufficiently resolved in gridded datasets with resolutions of 1 or more.  With higher resolution the second interannual variability mode after the quasi-uniform first mode is a CMZ-SEI-WG tripole.  A semblance of the tripole is apparent in Fig. 1 of mishra_prominent_2012 based on 0.5\(\times\)0.5 data, but it is described in their text as a north-south dipole.  As such, the CMZ and WG mean rainfall anomalies cannot be considered quasi-independent <cit.>.

§.§ ENSO and IOD teleconnections

Fig. <ref> includes the -1\(\times\)NINO3.4 and DMI timeseries, the sign of NINO3.4 flipped so that positive values correspond to positive AIRI anomalies.  The correlation between -1\(\times\)NINO3.4 and AIRI is 0.54, between \(-1\times\)NINO3.4 and DMI is -0.29, and between DMI and AIRI nearly zero, 0.01, all broadly consistent with past studies.  Using partial correlation analysis to remove the linear NINO3.4 signal from both rainfall and the DMI timeseries, the resulting DMI and AIRI residual timeseries become significantly positively correlated, 0.21.  Conversely, removing the DMI signal hardly affects the ENSO-AIRI relationship (0.56), and mechanisms through which ENSO influences the IOD are well documented <cit.>.  We therefore now analyze the spatial imprints of ENSO and the IOD via the frequency-intensity decomposition, in the case of the IOD focusing on the NINO3.4 residuals (/ last row of Fig. <ref>).

Fig. <ref>(a-c) shows the correlation of each of JJAS rainfall amount, rainy day frequency, and mean rainfall intensity on \(-1\times\)NINO3.4.  For rainfall amount, the correlation is positive at most gridpoints, most positive within SEI and the central portion of CMZ, and least positive (including some slightly negative values) in the far east of the CMZ and along the Western Ghats mountain axis.

The rainy day frequency correlation is more homogeneously positive and with higher magnitudes in most areas, especially along the Arabian Sea coast within WG and the central and western CMZ.  The mean rainfall intensity correlation is noisier and has generally weaker magnitudes, with a considerable fraction of gridpoints below the approximate statistical significance threshold of \(|r|\gtrsim0.2\) described in Section <ref>.  It takes the reverse sign over an appreciable fraction of gridpoints including much of the CMZ, and of the three sub-regions it is most consistently positive over SEI.  These three spatial patterns are broadly comparable to the corresponding first EOFs for JJAS rainfall amount (mostly same-signed), frequency (more homogeneous than amount), and intensity (heterogeneous with a CMZ-SEI-WG tripole; the weaker statistical significance notwithstanding).

Correlation coefficient between JJAS -1\(\times\)NINO3.4 and gridpoint local (a) rainfall amount, (b) frequency of rainy days, and (c) mean rainfall intensity.  Overlain in thin gray are the -0.2 and +0.2 contours, which give a reasonable estimate of statistical significance as described in Section <ref>.  Dashed box in (d) shows the WG region border used by vecchi_interannual_2004.

Fig. <ref>(d-f) show the correlations between the NINO3.4-residual timeseries of DMI and JJAS rainfall amount, rainy day frequency, and mean rainfall intensity at each gridpoint.  For amount and frequency, the predominant signal is a WG-SEI-CMZ tripole, with positive IOD events moderately correlated with excess rain over most of the WG and CMZ and deficient rain over SEI.  Unlike for NINO3.4, the patterns in rainfall amount, rainy day frequency, and mean rainfall intensity are all grossly similar—to first order, all three are CMZ-SEI-WG tripoles, just with more positive values overall for rainy day frequency and more negative values overall for mean rainfall intensity.  Though the values over most gridpoints for all three are insignificant on statistical grounds, the correspondence of all three patterns to the physically well-grounded tripole mode suggests to us that they are unlikely artifacts.  Moreover, the tripole structure helps explain why the full, non-residual DMI timeseries correlation with AIRI is very small (0.01) but significantly larger for the tripole indicators (0.31 for amount PC2, 0.34 for intensity PC1).

We conclude that, while it is the case that DMI is weakly correlated with AIRI, the IOD imprints a familiar spatial pattern, the WG-SEI-CMZ tripole.  Moreover, when the quasi-uniform drying influence of ENSO is linearly removed, the DMI correlations become quasi-uniformly more positive (even where they were negative), and as such the component of AIRI interannual variance not associated with ENSO is signficantly related to the IOD.

§ CONCLUSIONS

We use observational datasets of SST and high-resolution gridded rainfall spanning 1901-2020 to revisit three controversies in the literature involving relationships among the June-July-August-September (JJAS) seasonal averages of the All-India Rainfall Index (AIRI), the sub-India spatial distribution of rainfall variability, ENSO, and the Indian Ocean Dipole.

The first is the extent to which AIRI is representative of the spatiotemporal extent of rainfall anomalies.  Despite past concerns raised to the contrary, AIRI is found to be an excellent indicator of the spatial extent of wet anomalies across monsoonal India, and additionally of the average anomalous number of rainy days.  It is less closely related to the severity of rainfall anomalies when and where they occur.

The second is how AIRI variability relates to rainfall in fixed sub-India regions, and the third is how variability in AIRI and the subregions relate to ENSO and the IOD.  We confirm prior analyses of the leading variability mode via EOF analysis: it is a quasi-homogeneous wetting or drying that results primarily from an even more homogeneous and stronger signal in rainy day frequency, and it is strongly influenced by ENSO.  The second mode is a tripole with same-signed loadings in the high-rainfall Central Monsoon Zone and Western Ghats regions and opposite loadings in Southeastern India between.  This mode is associated more with rainfall intensity than with rainy day frequency and is effectively unrelated to ENSO.  The IOD imprints appreciably on the tripole mode and, with respect to the residual fields with the ENSO signal removed, imprints significantly on AIRI as well.

Why is it that, across India, ENSO affects the number of rainy days strongly and fairly homogeneously?  This could reflect the increased static stability throughout the tropics driven by upper-tropospheric warming due to / <cit.>: convection becomes harder to initiate, resulting all else equal in fewer rainy days.  Why is the mean rainfall intensity increased over much of the CMZ with / by contrast?  With weaker rainfall suppressed, more instability can build up in regions where it already is predilicted to, so that when convection comes it is intensified relative to neutral conditions.  This is analogous to the behavior of convection in coarse-resolution numerical models in which the convective parameterization is removed: large, very intense gridpoint storms emerge but much less frequently <cit.>.

One controversy we have elided is the extent to which intraseasonal rainfall variability—from which in principle the seasonal-mean signal is removed—may nonlinearly rectify onto the seasonal-mean and thereby influence interannual variability <cit.>.
This is largely orthogonal to the questions relating to lag-zero, JJAS-mean relationships of our present focus.  Nevertheless, the concern raised by this literature that the intraseasonal component is inherently chaotic and thus unpredictable <cit.> is worth keeping in mind, since the lag-zero correlations we have presented are partly means toward the end of improving the skill of seasonal forecasts of the summer monsoon rains.  At the same time, the well-documented relationships at the seasonal-mean timescale between the summer monsoon and ENSO, for example, convince us that clarifying the relationships between the monsoon and remote teleconnection modes is worthwhile, even if ultimately some portion of summer monsoon rainfall variance proves effectively unpredictable.

All authors acknowledge support from the Monsoon Mission Project under India’s Ministry of Earth Sciences.  S.A.H. acknowledges funding from the Columbia University Earth Institute Postdoctoral Fellowship.  We thank Sulochana Gadgil for many valuable discussions and comments.  We thank P. Mukhopadhyay for helpful comments.
IMD rainfall data used is available at <https://www.imdpune.gov.in/Clim_Pred_LRF_New/Grided_Data_Download.html>.

ERSST data is available at <https://www.ncei.noaa.gov/pub/data/cmb/ersst/v5/netcdf/>


# Lorentzian wormholes in an emergent universe.

Rikpratik Senguptaa, Shounak Ghoshb, B C Paulc, M Kalama111∗Corresponding
author.
E-mail addresses<EMAIL_ADDRESS>(RS),
<EMAIL_ADDRESS>(SG<EMAIL_ADDRESS>(BCP),
<EMAIL_ADDRESS>(MK)  aDepartment of Physics, Aliah University,
Kolkata 700 160, West Bengal, India.
bDepartment of Physics, Indian Institute of Engineering Science and
Technology, Shibpur, Howrah 711103, West Bengal, India.
cDepartment of Physics, University of North Bengal, Siliguri 734 013, West
Bengal, India..

###### Abstract

A non-singular Emergent Universe (EU) scenario within the realm of standard
Relativistic physics requires a generalization of the Equation of State (EoS)
connecting the pressure and energy density. This generalized EoS is capable of
describing a composition of exotic matter, dark energy and cosmological dust
matter. Since the EU scenario is known to violate the Null Energy Condition,
we investigate the possibility of presence of static, spherically symmetric
and traversable Lorentzian wormholes in an EU. The obtained shape function is
found to satisfy the criteria for wormhole formation, besides the violation of
the NEC at the wormhole throat and ensuring traversability such that tidal
forces are within desirable limits. Also, the wormhole is found to be stable
through linear stability analysis. Most ${importantly}$, the numerical value
of the emergent universe parameter $B$ as estimated by our wormhole model is
in agreement with and lies within the range of values as constrained by
observational data in a cosmological context. Also, the negative sign of the
second EU parameter $A$ as obtained from our wormhole model is in agreement
with the one required for describing an EU, which further indicates on the
existence of such wormholes in an emergent universe ${without}$ accounting for
any additional exotic matter field or any modification to the gravitational
sector.

###### keywords:

Wormhole, Emergent Universe

##  1 Introduction

It is well known today that the standard big bang cosmology is plagued by the
singularity problem. The beginning of time, or the early stage of the universe
cannot be described in the standard relativistic context as singularity
appears in the Einstein field equations (EFE’s). In fact, for describing the
universe, as long as its radius does not exceed the Planck scale, a theory of
Quantum Gravity (QG) is required. However, till date there is no single
consistent QG theory that is fully developed, although plenty of work is going
on in this direction and Loop Quantum Gravity (LQG) [1, 2] along with the
higher dimensional Superstring and M-theories [3, 4] can be said to be the two
main contenders in this aspect. The later also attempts to unify the four
natural interactions. The problem with such QG theories is that they can be
tested only in very extreme physical conditions, situations involving very
high spacetime curvature, believed to be found inside the event horizon of
black holes, or in the very early universe-at the moment of, or just following
its creation.

Two decades back, Ellis and Maartens [5] proposed a cosmological model known
as the “Emergent Universe”, which attempted to resolve the long standing
initial singularity problem within the classical context of Einstein’s General
Relativity (GR). As a positive curvature universe is not ruled out by
observation, they argued that the role of a positive curvature term may be
significant in the early universe, though it can be neglected at the late
times. The consideration of a such a term results in a non-singular cosmology,
with the universe originating as an Einstein’s static universe (ESU) and is
also free of the horizon problem. This static universe undergoes an
inflationary phase and later reheating, to give the standard big bang era.
Moreover, if the initial radius of the static universe is above the Planck
scale, then a purely QG regime can be avoided altogether in this cosmological
model.

This emergent universe scenario has attracted much attention from cosmologists
in the last two decades [6, 7, 8, 9, 10, 11, 12]. In a later work, Ellis [13]
extended the first model by considering the early universe to be dominated by
a minimally coupled scalar field $\phi$, described by a physically interesting
potential $V(\phi)=\bigg{(}Ae^{B\phi}-C\bigg{)}^{2}+D$, such that the
constants $A$, $B$, $C$ and $D$ may be determined from specific properties of
the emergent universe. In another work [14], it was shown that such a
potential for the emergent universe can be reproduced by modifying the
Lagrangian, adding a quadratic term of the scalar curvature, such that
$L=R+\alpha R^{2}$, where the coupling parameter $\alpha$ turns out to be
negative and the field can be identified with a negative logarithmic function
of the curvature. A very important extension to the emergent universe (EU)
scenario was done by Mukherjee et al. [15], where they obtained an EU even for
a flat spacetime, in the context of the semi-classical Starobinsky [16] model.
A one parameter family of solutions describing the EU were obtained, such that
parameter was determined by the number and species of the primordial fields.

In a following work, Mukherjee et al. [17] obtained EU solution within the
relativistic context, for a flat spacetime. They considered the composition of
matter, such that the generalized Equation of State (EoS) is given as
$p=A\rho-B\rho^{\frac{1}{2}}$. Such an EoS serves a three fold purpose. It
accommodates the scope for description of the matter or the source of gravity
sector by quantum field theory, besides the scope for including exotic matter
capable of violating the energy conditions of GR in a cosmological context and
also accommodating the present late time acceleration of the universe as
inferred from supernova observations. For such an EoS, the universe is large
enough initially to avoid a QG regime. For obtaining an emergent universe, we
must have $B>0$. It may also be noted that an emergent universe is permitted
with $A\leq 0$ [17, 18]. For realistic values of the parameter $B$ constrained
from observations [19], the universe is found to contain dark energy, exotic
matter and cosmological dust (matter).

The idea of wormholes on a serious mathematical context date back to 1935,
with the Einstein-Rosen bridge [20] being proposed as a solution to the field
equations of GR, which connected two different spacetime points located within
the same universe, or even possibly in two different universes. Two decades
later Misner and Wheeler [21] revisited the idea and coined the term
“wormhole”. However, such a wormhole was not traversable and found their
existence from quantum fluctuations taking place in the spacteime foam. They
are basically microscopic. The idea of Schwarzschild and Kerr wormholes cannot
be considered seriously for traversability, as such wormholes contain event
horizons. It is known that beyond an event horizon, the tidal forces are
extremely large and also there is the presence of curvature singularity. The
first formal, modern take on wormholes and their traversability was provided
by Morris and Thorne in 1988 [22]. In this view, a wormhole is considered as a
topological as well as geometrical object characterized by a compact spacetime
region which has a trivial boundary with a non-trivial interior and connects
two spacetime points in the same universe or in two different universes. It
has both local and global structure [23]. Such wormholes are well described by
a static and spherically symmetric spacetime metric. Any closed time-like
curve is known to violate causality and such a possibility may arise from
traversable wormholes [24].

Morris and Thorne [22] laid down a prescription for constructing a traversable
wormhole. There are certain essential features that the radial metric
component of a wormhole must follow in order to ensure traversability. We
shall discuss these criteria in the context of our wormhole model in the
concluding section. From a physical point of view, the tidal acceleration must
be small enough so that any traveller attempting to traverse the wormhole does
not get ripped apart. This can be ensured from the absence of horizons in the
wormhole. Also, they suggested that the wormhole throat must be kept open so
that it does not ‘pinch off’, in order to ensure traversability through it.
This is possible by the violation of the Null Energy Condition (NEC) at the
throat. There are two ways in which the NEC may be violated. Firstly, by
introduction of exotic matter at the throat within the relativistic context
and secondly, by modifying the gravitational action, such that the additional
terms in the modified field equations contribute effectively to the violation
of NEC at the throat in the presence of ordinary matter. The possibility of a
wormhole existing in the outer galactic halo region has been investigated
recently [25]. Wormholes have been studied in the modified gravity context
[26, 27, 28, 29, 30, 31, 32] and also in the relativistic context with exotic
matter [33, 34, 35, 36, 37, 38, 39, 40]. However, recently a successful
traversable wormhole model has been proposed in the relativistic context [41]
where the wormhole is supported by a Maxwell and two Dirac fields in the
absence of any exotic coupling and exotic matter and the wormhole is
asymmetric about the throat, such that there is no $Z_{2}$ or mirror symmetry.

As we have already discussed, in order to obtain an emergent universe solution
with a flat spacetime, we have to either modify the Lagrangian describing the
gravitational action or consider a generalized EoS for the matter source,
within the relativistic context. Such an EoS is well known to accommodate
exotic matter and also accounts for the late-time acceleration of the
universe. So, it could be natural that an emergent universe described by such
an EoS accommodates traversable wormholes as well, due to such a composition
of the universe. Lorentzian wormholes are the ones which exist on a Lorentzian
spacetime manifold. Since experimental physics seems to favour a Lorentzian
signature, here we will investigate the possibility of existence of static,
traversable Lorentzian wormholes in the background of an emergent universe. In
our previous papers, we had successfully constructed static and traversable
Lorentzian wormholes involving higher dimensional braneworld gravity [26] or
by introduction an additional tachyon field as the matter source [33]. Time
evolving Euclidean wormholes have been found to be present in an EU in the
massive gravity context [42]. So, it would be worth an attempt to investigate
the possibility of static and traversable Lorentzian wormholes being present
in an EU without considering any additional matter field or modifying the
Einstein-Hilbert action leading to standard relativistic GR.

In the following sections we have obtained the solutions for different
physical parameters of the wormhole under the framework of the EU and
investigated them different features of the wormhole that may be analyzed from
the solution and consider its stability. A detailed analysis should throw
light on whether a stable, traversable wormhole can exist automatically in an
emergent universe setup.

##  2 Mathematical model of the wormhole

In this section we shall make use of the Einstein field equations (EFE’s) to
obtain the shape function for the wormhole and check whether it satisfies the
criteria of being a well-defined function to describe the wormhole. We shall
also check the validity of the null energy condition. The traversability
condition can be verified by computing the tidal acceleration. The essential
model parameters can by estimated from the Darmois-Israel junction conditions,
from which we shall also compute the surface density and surface pressure of
the wormhole. We shall also perform a linearized stability check in order to
ensure that the wormhole is stable.

###  2.1 The field equations

For a static, spherically symmetric matter distribution, the spacetime is
characterized by the line element

$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}).$ (1)

For such a wormhole, the metric potential
$e^{\lambda}(r)=\frac{1}{1-\frac{b(r)}{r}}$, where $b(r)$ represents the shape
function of the wormhole. The Einstein Field Equations (EFEs) for such a
wormhole in the relativistic context may be written down as (taking $G=$$c=$1)

$\displaystyle\frac{b^{\prime}}{r^{2}}=8\pi\rho,$ (2)
$\displaystyle\left(1-\frac{b}{r}\right)\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=8\pi
p,$ (3)
$\displaystyle\left(1-\frac{b}{r}\right)\left(\nu^{\prime\prime}+{{\nu^{\prime}}^{2}}+\frac{\nu^{\prime}}{r}\right)-\frac{b^{\prime}-b}{2r}\left({\nu^{\prime}}+\frac{1}{r}\right)=8\pi
p,$ (4)

As we are performing our analysis in the background of an Emergent Universe,
we shall consider the static Lorentzian wormhole to be composed of dark
energy, exotic matter and dust, represented by the generalized EoS [17].

$p\left(r\right)=A\rho\left(r\right)-B\sqrt{\rho\left(r\right)}.$ (5)

Here the parameters $A$ and $B$ characterize the Emergent Universe. For
simplicity of our analysis we take the redshift function $\frac{\nu}{2}$ to be
a constant which implies $\nu^{\prime}=0$. This simplification can be found in
case of other wormhole constructions with exotic matter in the relativistic
context [33, 43].

###  2.2 Solution for the shape function

Using the EoS in the matter conservation equation with a constant redshift
function, the energy density of the wormhole is obtained as

$\rho\left(r\right)=\frac{B^{2}}{4A^{2}}.$ (6)

We note that the energy density is expressed in terms of the EU parameters $A$
and $B$ as expected.

The shape function can be obtained by plugging in the energy density into Eq.
(2) and turns out to be

$b(r)=\frac{2\pi B^{2}r^{3}}{3A^{2}}+C_{1}.$ (7)

Figure 1: Variation of shape function with respect to $r$.

We plot the shape function against $r$ in Figure 1. We have assumed that the
throat radius of the wormhole is at $r=r_{0}=0.1$. As we can see from the
Figure 1, the shape function satisfies the necessary criteria to describe a
traversable wormhole as per the Morris Thorne prescription [22]. Firstly at
the throat radius $r=r_{0}$, as we can see the shape function
$b(r_{0})=r_{0}$. Secondly, as $r$ becomes greater than $r_{0}$, at every
point the shape function $b(r)<r$ for that point. So, the obtained shape
function is satisfactory for describing a wormhole.

###  2.3 Null Energy condition

The null energy condition (NEC) in GR can be written in a simplified form as
$\rho+p\geq 0$. According to the Morris-Thorne prescription for construction
of a traversable wormhole [22], one of the most essential conditions for
ensuring the traversability must be the violation of the NEC at the throat, so
that the throat does not pinch off due to the gravitational attraction. In
order to ensure this within the relativistic context, the matter constructing
the wormhole must violate the NEC. So, for our model we must check whether the
composite matter described by the generalized EoS in Eq. (6) can is capable of
violating the NEC.

By solving the EFE in Eq. (4), we obtain the pressure to be given as

$p(r)=-\frac{2\,{B}^{2}\pi r^{3}+3C_{1}A^{2}}{24\pi A^{2}r^{3}}.$ (8)

As expected, the pressure has a dependence on the EU parameters $A$ and $B$.

Summing up the above obtained pressure and energy density, we have

$p+\rho={\frac{4B^{2}\pi r^{3}-3A^{2}C_{1}}{24\pi A^{2}r^{3}}}.$ (9)

Figure 2: Variation of the NEC with respect to $r$.

We plot $p+\rho$ as a function of $r$ in Figure 2. We see that the NEC is
violated at the throat as $p+\rho$ turns out to be negative at the throat and
as we further increase $r$, it continues to stay negative but with a
vanishingly small value. It appears from the figure that $p+\rho$ turns zero
after a value of $r$ close to 1.5, but on enlarging we can see that it
continues to be negative with a vanishingly small value. This ensures that the
flare out condition is satisfied which implies that the first derivative of
the shape function with respect to r, at the throat is less than unity. Hence,
traversability is ensured.

###  2.4 Darmois Israel Junction Condition

Outside the wormhole we consider a vacuum spacetime. So, the exterior to which
the wormhole spacetime is to be matched can either be described by a
Schwarzschild or deSitter metric depending on whether the vacuum has vanishing
energy density and pressure ($\rho_{vac}=p_{vac}=0$) or constant energy
density and pressure given by $p_{vac}=-\rho_{vac}=-\frac{\Lambda}{8\pi}$,
respectively. However, for our analysis there is no need for considering a
$\Lambda$-term (cosmological constant) as the composite matter described by
the EoS can describe the late time acceleration effectively even in the case
of $\Lambda=0$. So, we consider a Schwarzschild exterior. Thus, the spacetime
exterior to the surface of the wormhole is described by the well known
Schwarzschild metric given as

$ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (10)

As a consequence of matter being present on the wormhole surface, there is an
extrinsic discontinuity which produces an intrinsic surface energy density and
surface pressure. The wormhole surface acts as the boundary between the
interior and the exterior, as a result of which the wormhole structure is a
geodesically complete manifold characterized by the EoS used in our model,
which describes a composition of dark energy, exotic matter and cosmological
dust. Being a geodesically complete manifold, the continuity of the metric
coefficient at the wormhole surface yields a matching condition. We shall now
obtain the components for the intrinsic stress-energy $S_{ij}$ using the
Darmois-Israel prescription [44, 45].

From the Lanczos equation [46, 47, 48, 49] we have

$S_{j}^{i}=-\frac{1}{8\pi}(\kappa_{j}^{i}-\delta_{j}^{i}\kappa_{k}^{k}).$ (11)

The discontinuity in extrinsic curvature as a result of presence of matter on
the wormhole surface is expressed as

$\kappa_{ij}=\kappa_{ij}^{+}-\kappa_{ij}^{-},$ (12)

and the extrinsic curvature is defined as

$\kappa_{ij}^{\pm}=-n_{\nu}^{\pm}\left[\frac{\partial^{2}X_{\nu}}{\partial\xi^{i}\partial\xi^{j}}+\Gamma_{\alpha\beta}^{\nu}\frac{\partial
X^{\alpha}}{\partial\xi^{i}}\frac{\partial
X^{\beta}}{\partial\xi^{j}}\right]|_{S},$ (13)

where $n_{\nu}^{\pm}$ is the unit normal vector which may be expressed in the
form

$n_{\nu}^{\pm}=\pm\left|g^{\alpha\beta}\frac{\partial f}{\partial
X^{\alpha}}\frac{\partial f}{\partial
X^{\beta}}\right|^{-\frac{1}{2}}\frac{\partial f}{\partial X^{\nu}},$ (14)

such that $n^{\nu}n_{\nu}=1$. $\xi^{i}$ denotes intrinsic coordinate of the
wormhole surface and is described by the parametric equation
$f(x^{\alpha}(\xi^{i}))=0$. Here $+$ and $-$ denotes the wormhole spacetime at
the exterior and interior, respectively.

The surface stress energy tensor for the spherically symmetric line element
has the components expressed as $S_{i}^{j}=diag(-\Sigma,\mathcal{P})$. Here
$\Sigma$ and $\mathcal{P}$ denotes the surface energy density and surface
pressure respectively, at the wormhole surface $r=R$ which may be obtained as

$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle-\frac{1}{4\pi
R}\bigg{[}\sqrt{e^{\lambda}}\bigg{]}_{-}^{+}$ (15) $\displaystyle=$
$\displaystyle\frac{1}{4\pi
R}\left[\sqrt{\left(1-\frac{2M}{R}\right)}-\sqrt{1-\frac{b(r)}{r}}\right]=\frac{1}{4\pi
R}\left[\sqrt{\left(1-\frac{2M}{R}\right)}-\sqrt{1-\frac{2\pi
B^{2}r^{2}}{3A^{2}}-\frac{C_{1}}{r}},\right]$ $\displaystyle\mathcal{P}$
$\displaystyle=$ $\displaystyle\frac{1}{16\pi
R}\bigg{[}\bigg{(}\frac{2f+f^{\prime}R}{\sqrt{f}}\bigg{)}\bigg{]}_{-}^{+}$
(17) $\displaystyle=$ $\displaystyle{\frac{1}{4\pi
R}\left(\sqrt{1-{\frac{2M}{R}}}-\sqrt{{1-\frac{C_{1}}{R}-\frac{2B^{2}\pi
R^{2}}{3A^{2}}}}\right)}+{\frac{M}{4R^{3}\pi\sqrt{{1-\frac{2M}{R}}}}}+\frac{\left(\frac{B^{2}}{6A^{2}}-\frac{C_{1}}{8R^{3}\pi}\right)}{{\sqrt{{1-\frac{C_{1}}{R}-\frac{2B^{2}\pi
R^{2}}{3A^{2}}}}}}$

As we are considering the construction of a static wormhole which does not
evolve with time itself and is a local object in the EU, so at the surface of
the wormhole the surface energy density and surface pressure must vanish
($\Sigma=\mathcal{P}=0$)[26, 40]. $\Sigma=0$ leads to the condition

$b(r)|_{r=R}=2M,$ (18)

which yields

$-\left(-{\frac{2\pi B^{2}{R}^{2}}{3A^{2}}}-{\frac{{C_{1}}}{R}}\right)R=2M.$
(19)

In addition, there is an additional boundary condition which involves the
continuity of the metric potential $g_{rr}$ and its derivative $\frac{\delta
g_{rr}}{\delta r}$ at the wormhole surface $r=R$. It is to be noted that the
same boundary condition obtained in Eq. (18) an Eq. (19) can also be arrived
at by making use of the matching condition involving the continuity of the
metric potential $g_{rr}$ to ensure smooth matching between the wormhole and
exterior Schwarzschild spacetimes. Another matching condition involving the
coninuity of the derivative of the radial derivative of the metric potential
at the wormhole surface $r=R$, again in order to ensure smooth matching of the
two spacetimes yield the second boundary condition which can be written as

$\displaystyle\frac{\partial g_{rr}}{\partial r}|_{int}=\frac{\partial
g_{rr}}{\partial r}|_{ext}$ (20) $\displaystyle\Rightarrow$
$\displaystyle-\frac{4\pi
B^{2}R}{3A^{2}}+\frac{C_{1}}{R^{2}}=\frac{2M}{R^{2}}$

As there are three unknown model parameters (the EU parameters $A$ and $B$ and
the integration constant $C_{1}$), we shall requaire an additional boundary
condition. The third and final boundary condition comes from the vanishing of
the surface pressure at the wormhole boundary, which is a consequence of the
wormhole being static as already discussed. So, we can put Eq. (17) as zero
which yields the third boundary condition. Now for the three unknown model
parameters $A$, $B$ and $C_{1}$ we have three equations in the form of the
three boundary conditions and we have evaluated the parameters by solving
these equations using MAPLE.

These boundary conditions allow us to evaluate the unknown model parameters
$A$, $B$ both of which are the EU parameters in addition to the integration
constant $C_{1}$. We had previously assumed the throat radius to be $0.1$ (in
$km$). As the wormhole contains exotic matter as one of its ingredients, so it
is appropriate to minimize the mass of the wormhole in order to minimize the
mass of exotic matter used. However, it may be noted here that we are not
introducing any exotic matter by hand. The EoS which supports an EU within the
relativistic context is itself generalized to describe exotic matter as a
constituent. We assume the mass of the wormhole to be $0.8M_{\odot}$, which
appears to be a realistic choice. The surface or the boundary of the wormhole
is taken at $R=5$ (in $km$).

Matching the junction conditions at the wormhole surface and using the values
of the physical model parameters mentioned above, we evaluate the unknown
model parameters to be $A=-0.054$, $B=0.003784713921$ and
$C_{1}=0.09998799979$. These obtained values have been used for the plots, and
we shall justify their physical significance in the concluding section.

###  2.5 Tidal acceleration

For a traversable wormhole model, we must ensure that the tangential and
radial components of the tidal acceleration for an observer passing through
the throat of the wormhole must be lesser than the acceleration due to gravity
on earth, so that the observer does not get ripped apart while traversing the
throat due to large tidal gravitational forces.

The condition that has to be satisfied by the radial component of the tidal
acceleration $|R_{rtrt}|$ in order to ensure traversability at the throat is

$|R_{rtrt}|=|(1-\frac{b}{r})\big{[}\frac{\nu^{\prime\prime}}{2}+\frac{\nu^{\prime
2}}{4}-\frac{b^{\prime}r-b}{2r(r-b)}.\frac{\nu^{\prime}}{2}\big{]}|\leq
g_{earth}.$ (21)

For our model, it has been considered that $\nu^{\prime}(r)=0$. A constant
redshift function is a convenient choice in case of many wormhole models with
exotic matter in the context of standard GR [33, 43]. As a result, the radial
component of the tidal acceleration vanishes and the condition of
traversability is automatically satisfied for this component.

However, for the tangential component, the condition has to be evaluated at
the throat as the tangential component is non-trivial and a constraint on the
velocity of the traveller traversing the throat can be obtained on evaluating
this condition. The condition for traversability is expressed by the
inequality

$\gamma^{2}|R_{\theta t\theta t}|+\gamma^{2}v^{2}|R_{\theta r\theta
r}|=|\frac{\gamma^{2}}{2r^{2}}\big{[}v^{2}(b^{\prime}-\frac{b}{r})+(r-b)\nu^{\prime}\big{]}|\leq
g_{earth},$ (22)

where the left hand side represents the tangential tidal acceleration.

Plugging in the expression for $b(r)$ and using $\nu^{\prime}(r)=0$ we get

$\frac{{\gamma}^{2}}{2}\left|{\frac{v^{2}}{r}\left(-\frac{4}{3}{\frac{\pi
B^{2}r}{A^{2}}}+{\frac{C_{1}}{r^{2}}}\right)}\right|\leq g_{earth}.$ (23)

Putting the values of the model parameters $B=0.003784713920$, $A=-0.054$,
$M=0.8M_{\odot}$, $C_{1}=0.09998799990$ obtained from the matching conditions
and taking $\gamma\approx 1$ (as the velocity of the traveller $v<<1$ so
$\gamma=\frac{1}{\sqrt{1-v^{2}}}\approx 1$), the condition for traversability
reduces to

$\frac{1}{2}\left|{\frac{v^{2}}{r}\left(-0.02400019199r+0.09998799990{r}^{-2}\right)}\right|\leq
g_{earth}.$ (24)

At the throat of the wormhole, the abobe inequality reduces to

$v\leq 0.1414468192\sqrt{g_{earth}}.$ (25)

Since the upper limit on the velocity of the traveller traversing the throat
of the wormhole turns out to be a realistic one for our wormhole, we claim
that the tidal forces at the throat are not too strong to disrupt the
traveller and hence traversability is ensured.

###  2.6 Linearized Stability Analysis

In order to perform a qualitative stability analysis of our obtained wormhole,
it is assumed that the throat radius of the wormhole is a function of proper
time. The throat radius is represented by $r_{0}=x(\tau)$. The energy density
has the form

$\sigma=-\frac{1}{2\pi x}\sqrt{f(x)+\dot{x}^{2}},$ (26)

and the pressure may be expressed as

$p=\frac{1}{8\pi}\frac{f^{\prime}(x)}{\sqrt{f(x)}}-\frac{\sigma}{2},$ (27)

such that $f(x)=1-\frac{2M}{x}$. Here $M$ denotes the wormhole mass.

Making use of the conservation equation, an equation of motion

$\dot{x}^{2}+V(x)=0,$ (28)

can be obtained, such that the potential $V(x)$ can be expressed as

$V(x)=f(x)-[2\pi x\sigma(x)]^{2}.$ (29)

The objective now is to perform stability analysis considering linearization
around $x_{0}$, which is an assumed static solution to Eq.(27).

Taylor expanding the potential around the assumed static solution $x_{0}$
yields

$V(x)=V(x_{0})-V^{\prime}(x_{0})(x-x_{0})+\frac{1}{2}V^{\prime\prime}(x_{0})(x-x_{0})^{2}+O[(x-x_{0})^{3}],$
(30)

’prime’ indicating derivative with respect to x.

Since the wormhole spacetime considered by us is static, we have $V(x_{0})=0$
and $V^{\prime}(x_{0})=0$. Thus to ensure stability of the wormhole, it is
necessary that $V^{\prime\prime}(x_{0})>0$. A parameter $\beta$ is introduced,
physically representing the sound speed which is defined as

$\beta=\frac{\delta p}{\delta\sigma}.$ (31)

The second derivative of the potential can be expressed in terms of the newly
defined physical parameter $\beta$ as

$V^{\prime\prime}(x)=f^{\prime\prime}(x)-8\pi^{2}[(\sigma+2p)^{2}+\sigma(\sigma+p)(1+2\beta).$
(32)

Expressing the stability condition for the wormhole $V"(x_{0})>0$ in terms of
$\beta$, we have

$\beta<\frac{\frac{f^{\prime\prime}(x_{0})}{8\pi^{2}}-(\sigma+2p)^{2}-2\sigma(\sigma+p)}{4\sigma(\sigma+p)}.$
(33)

Plugging in the expressions for $\sigma$ and $p$, the stability criterion for
the wormhole in terms of $\beta$ takes the final form

$\beta<\frac{x_{0}^{2}(f_{0}^{\prime})^{2}-2x_{0}^{2}f_{0}^{\prime\prime}f_{0}}{4f_{0}(a_{0}f_{0}^{\prime}-2f_{0})}-\frac{1}{2}.$
(34)

The parameter $\beta$ can be calculated for our wormhole model as

$\displaystyle\beta={\frac{-2{x_{0}}^{6}+\left(-12\pi+9\right)m{x_{0}}^{5}+20{m}^{2}\left(\pi-\frac{1}{2}\right){x_{0}}^{4}}{8{x}^{5}\left(-x_{0}+2m\right)\pi\left(-x_{0}+3m\right)}}.$
(35)

We plot $\beta$ for different values of $x_{0}$ in Figure 3. The regions
denoted as 1, 2 and 3 satisfy the stability criterion. Thus, we can claim that
the wormhole model obtained by us in the background of an EU is stable.

Figure 3: Variation of $\beta$ with respect to $x_{0}$.

##  3 Discussion and Conclusion

In this paper, we explore the possibility of constructing static, traversable
and spherically symmetric Lorentzian wormholes in the background of an
Emergent Universe (EU). As we have discussed in the introductory section, EU
cosmological models have been widely explored by cosmologists due to the
absence of the initial singularity within the relativistic context. It is the
most well established non-singular cosmological picture which is consistent
even without modifying General Relativity. However, in order to accommodate EU
in the relativistic context, a generalized EoS is introduced which describes
an universe composed of exotic matter, dark energy and cosmological dust for
suitable values of the model parameters. Again, it is known that in order to
have an EU, it is necessary that the NEC must be violated [50]. Also, this is
a necessary condition for the successful construction of a traversable
wormhole [22]. In the relativistic context, exotic matter also becomes
effective in constructing traversable wormholes. Therefore, the common grounds
of requirement for exotic matter and violation of the NEC make it worth an
attempt to explore the possibility of existence of traversable wormholes in an
Emergent Universe. Also, the existence of time evolving Euclidean wormholes
have been previously found in EU in the context of massive gravity [42]. This
provides an additional motivation for the present investigation.

We have considered a constant redshift function to simplify our analysis.
Using the generalized EoS used to describe an EU, we obtain a solution for the
shape function and on plotting the shape function, it turns out that all the
essential criteria that are required for a shape function to describe a
traversable wormhole have been obeyed. At the throat radius $r_{0}$, the shape
function takes the value of the throat radius itself. Moreover, for all values
of $r>r_{0}$, the ratio remains $\frac{b(r)}{r}<1$ (from Fig. 1).

The energy density and pressure inside the wormhole are obtained making use of
the field equations and the conservation equation. They are found to depend on
the EU parameters. Plotting the variation of $p+\rho$ along $r$, we find that
the NEC is violated at the throat of the wormhole. This ensures that the
derivative of the shape function $b(r)$ with respect to $r$ must be less than
unity at the throat, which is known as the flaring-out condition. Satisfying
this condition physically means that the throat of the wormhole does not get
pinched off due to the gravitational attraction, thus preventing the wormhole
structure from collapsing at the throat. This can be justified by the presence
of exotic matter and dark energy components in the EU, described by the
generalized EoS.

The discontinuity in the extrinsic curvature resulting from the presence of
matter at the surface of the wormhole, in turn results in the origin of a
surface stress-energy term containing surface energy density and surface
pressure terms as its components. These components have been obtained and are
also found to be dependent on the EU parameters. Since we have performed our
analysis for a static wormhole metric, so we consider the surface density and
pressures to vanish at the junction. This results in yielding the first
boundary condition. Other boundary conditions can be obtained from the
continuity of the metric potential and its derivative at the junction.

These matching conditions at the surface of the wormhole enable us to obtain
an estimate of the unknown model parameters, namely the EU parameters $A$ and
$B$ and also the unknown integration constant $C_{1}$, by choosing realistic
values of the parameters associated with the wormhole. The throat radius and
the boundary of the wormhole are considered to be at $0.1km$ and $5km$,
respectively. These are not unique choices but the values we have considered
are reasonable. Since, exotic matter is one of the components of an EU, the
wormhole in consideration will also contain some exotic matter as one of its
constituents. So, it is justified to minimize the mass of the wormhole
although dust matter is also a constituent. We make a reasonable assumption
that the mass of the wormhole is $0.8M_{\odot}$. Using the chosen values of
the above mentioned wormhole parameters, we get the integration constant
$C_{1}=0.09998799979$. This is of relatively less physical significance.

However, it is interesting to check the values we obtain for the EU
parameters, specially the parameter $B$ as this parameter has been constrained
observationally in a cosmological context [19]. If the value we obtain for the
parameter $B$ from our wormhole model happens to lie within or in close
approximate to the constrained observational range, it would provide strong
support in favor of existence of static traversable Lorentzian wormholes in an
EU. Also, for EU, the parameter $A$ must be negative and $B$ must be slightly
positive [17]. However when we applied the EoS containing these two parameters
to the field equations to obtain the wormhole solution, we did not put any
constraint on the sign of these parameters by hand. So the obtained negativity
and positivity of $A$ and $B$, respectively is an indication in favor of the
possibility of existence of such wormholes in an EU. The observationally
constrained value for the parameter $B$ for an EU constituted of exotic
matter, dark energy and matter is $0.003<B<0.5996$ [19]. From our model, the
obtained value from the matching condition is $B=0.003784713921$, which is
within the constrained range. This, along with the small negative value
obtained for the parameter $A$, essential to construct an EU, provides strong
support in favor of the existence of wormholes in an EU.

It is also essential to check that the tidal force at the wormhole throat is
not strong enough, such that an observer attempting to traverse the throat of
the wormhole gets ripped apart due to the tidal forces. The radial component
of the tidal force becomes insignificant with the vanishing of the first
derivative of the redshift function. However, the tangential component has a
contributing term even for constant $\nu$. Considering the upper limit for the
tidal acceleration to be equal to the acceleration due to gravity on the
earth, we obtain a reasonable upper limit on the velocity with which the
traveller can attempt to traverse the wormhole throat safely, without getting
ripped apart due to excess tidal forces. Finally, we perform a linearized
stability analysis to check whether our obtained wormhole solution in an EU is
stable. Plotting the parameter $\beta$ with respect to $x_{0}$, we show three
regions of stability which justifies that the wormhole we have obtained is a
stable one. The stable regions are obtained imposing constraint on the sound
speed in terms of the wormhole solution parameters.

There have been some recent investigations on the possibilities of detecting
wormholes [51, 52, 53, 54]. As we know, wormholes are asymptotically flat
tubular structures that connect two different spacetimes where the individual
fluxes might not be conserved, as a result of which there shall be mutual
observable effects on objects present within close distances of both the
wormhole mouths [51]. It may be possible to detect such effects on the orbits
of stars in close approximately to the black hole present at the centre of our
galaxy. Wormholes may also produce some micro-lensing effects which might
appear identical to gamma ray bursts and these effects can lead to obtaining a
higher limit on the mass density of wormholes making use of the BATSE data
[52]. The scattering properties near a traversable and rotating wormhole can
be explored considering the quasinormal ringing of black holes. Asymmetric
wormholes shall be characterized by super radiance, while the wormholes
exhibiting symmetry differ from black holes due to non-identical simultaneous
ringing at a number of dominant multipoles [53]. If the redshift function of
the wormhole is variable, the radial tidal forces do not vanish, resulting in
possible detection of long-lived quasinormal modes dubbed as ’quasi-
resonances’ in the background of the wormhole [54].

We may conclude that not only time evolving Euclidean wormholes can exist in
the EU scenario in the massive gravity context, but also static, traversable
Lorentzian wormholes can also be accommodated in an EU within a relativistic
context. The EoS describing an EU in the relativistic context is capable of
producing stable, static, traversable wormhole solutions violating the NEC.
Moreover, the sign of the EU parameter $A$ is obtained as desired for an EU
and also the numerical value of the EU parameter $B$ appearing for a
consistent wormhole model, is estimated to be within the range of values
constrained from an observational basis in a cosmological context. Therefore,
it can be concluded that Lorentzian wormholes are naturally present in an
Emergent Universe without taking into account either any additional matter
field (like our previous wormhole model supported by a tachyonic field [33])
or any modification to standard GR (like our previous wormhole model in the
braneworld gravity context [26]).

##  4 Acknowlwdgement

MK and BCP is thankful to the Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Pune, India for providing the Visiting Associateship
under which a part of this work was carried out. RS is thankful to the Govt.
of West Bengal for financial support through SVMCM scheme. SG is thankful to
the Directorate of Legal Metrology under the Department of Consumer Affairs,
West Bengal for their support.

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# Cauchy, normal and correlations versus heavy tails

Hui Xu  Center for Applied Mathematics
Cornell University<EMAIL_ADDRESS>,  Joel E. Cohen  Laboratory of
Populations, The Rockefeller University & Columbia University; Earth Institute
& Department of Statistics, Columbia University; Department of Statistics,
University of Chicago<EMAIL_ADDRESS>,  Richard Davis  Department of
Statistics
Columbia University<EMAIL_ADDRESS>and  Gennady Samorodnitsky∗
School of Operations Research and Information Engineering
Cornell University<EMAIL_ADDRESS>
###### Abstract.

A surprising result of Pillai and Meng (2016) showed that a transformation
$\sum_{j=1}^{n}w_{j}X_{j}/Y_{j}$ of two iid centered normal random vectors,
$(X_{1},\ldots,X_{n})$ and $(Y_{1},\ldots,Y_{n})$, $n>1$, for any weights
$0\leq w_{j}\leq 1$, $j=1,\ldots,n$, $\sum_{j=1}^{n}w_{j}=1$, has a Cauchy
distribution regardless of any correlations within the normal vectors. The
correlations appear to lose out in the competition with the heavy tails. To
clarify how extensive this phenomenon is, we analyze two other transformations
of two iid centered normal random vectors. These transformations are similar
in spirit to the transformation considered by Pillai and Meng (2016). One
transformation involves absolute values: $\sum_{j=1}^{n}w_{j}X_{j}/|Y_{j}|$.
The second involves randomly stopped Brownian motions:
$\sum_{j=1}^{n}w_{j}X_{j}\bigl{(}Y_{j}^{-2}\bigr{)}$, where
$\bigl{\\{}\bigl{(}X_{1}(t),\ldots,X_{n}(t)\bigr{)},\,t\geq 0\bigr{\\}},\
n>1,$ is a Brownian motion with positive variances; $(Y_{1},\ldots,Y_{n})$ is
a centered normal random vector with the same law as
$(X_{1}(1),\ldots,X_{n}(1))$ and independent of it; and $X(Y^{-2})$ is the
value of the Brownian motion $X(t)$ evaluated at the random time $t=Y^{-2}$.
All three transformations result in a Cauchy distribution if the covariance
matrix of the normal components is diagonal, or if all the correlations
implied by the covariance matrix equal 1\. However, while the transformation
Pillai and Meng (2016) considered produces a Cauchy distribution regardless of
the normal covariance matrix. the transformations we consider here do not
always produce a Cauchy distribution. The correlations between jointly normal
random variables are not always overwhelmed by the heaviness of the marginal
tails. The mysteries of the connections between normal and Cauchy laws remain
to be understood.

###### Key words and phrases:

heavy tails, dependence, Cauchy, normal

###### 1991 Mathematics Subject Classification:

Primary

∗The corresponding author. This research was partially supported by NSF grants
DMS-2015379 (Davis) at Columbia and DMS-2015242 (Samorodnitsky) at Cornell.

##  1\. Introduction

The normal distribution and the Cauchy distribution are among the most
fundamental probability distributions with numerous applications in many areas
where uncertainty is present. They both arise as limits of suitably normalized
sums of independent copies of random quantities. They both have nearly magical
invariance properties. They contrast strongly in that the normal distribution
has finite moments of all orders (even all exponential moments are finite),
while the Cauchy distribution does not even have a finite mean. It is well
known that normal and Cauchy random variables are connected in simple ways.
These connections sometimes appear to have mysterious properties as well. This
note aims to shed additional light to some of these mysteries.

By definition, the centered normal distribution with standard deviation
$\sigma>0$ has the density

$\phi(x;\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\exp\bigl{\\{}-x^{2}/(2\sigma^{2})\bigr{\\}},\
x\in{\mathbb{R}},$

while the Cauchy distribution with scale $\sigma>0$ has the density

(1.1) $f(x;\sigma)=\frac{\sigma}{\pi(x^{2}+\sigma^{2})},\ x\in{\mathbb{R}}.$

For both distributions, the case $\sigma=1$ is called standard. The invariance
properties of these distributions say that, if $X_{1},\ldots,X_{n}$ are iid
copies of a centered normal $X$ and $\overset{d}{=}$ denotes equality in
distribution, then

(1.2) $\sum_{j=1}^{n}w_{j}X_{j}\overset{d}{=}X$

if $\sum_{j=1}^{n}w_{j}^{2}=1$, while if $X_{1},\ldots,X_{n}$ are iid copies
of a Cauchy $X$, then (1.2) holds if $\sum_{j=1}^{n}|w_{j}|=1$.

Versions of (1.2) may hold when the $X_{1},\ldots,X_{n}$ are not independent.
For example, recall that $X_{1},\ldots,X_{n}$ are jointly Cauchy if there is a
finite symmetric measure $\Gamma$ (the spectral measure) on the unit sphere
$S^{n}$ such that

(1.3)
$E\exp\biggl{\\{}i\sum_{j=1}^{n}\theta_{j}X_{j}\biggr{\\}}=\exp\biggl{\\{}-\int_{S^{n}}\biggl{|}\sum_{j=1}^{d}\theta_{j}s_{j}\biggr{|}\,\Gamma(ds_{1},\ldots,ds_{n})\biggr{\\}}.$

Because $(X_{1},\ldots,X_{n})$ has a symmetric distribution, the
characteristic function is real-valued on the right side of (1.3). Then
$X_{1},\ldots,X_{n}$ have identical marginals if
$\int|s_{j}|\,\Gamma(d\mathbf{s})$ is the same for all $j$; see Samorodnitsky
and Taqqu (1994). In particular, if the spectral measure $\Gamma$ is
concentrated on the positive and negative quadrants of the unit sphere and
$X_{1},\ldots,X_{n}$ have the same marginals, then (1.2) still holds as long
as all $(w_{j})$ are also of the same sign. In this scenario,
$X_{1},\ldots,X_{n}$ are generally dependent since their independence requires
$\Gamma$ to be concentrated at the points where the unit sphere intersects the
axes. In particular, if $\Gamma$ is concentrated at the points
$\pm(1,\ldots,1)/n^{1/2}$, then $X_{1}=\ldots=X_{n}$ a.s. This special case
shows that a convex linear combination of iid Cauchy random variables has the
same law as a convex linear combination of almost surely equal Cauchy random
variables!

##  2\. The Pillai and Meng result

An elementary calculation shows that, if $X,Y$ are iid centered normal and $W$
is a standard Cauchy, then

(2.1) $X/Y\overset{d}{=}W.$

If $X$ and $Y$ are bivariate normal, then the probability density function of
the ratio $X/Y$ is derived in Pham-Gia et al. (2006) and earlier sources they
cite. The ratio has a Cauchy distribution if and only if $X$ and $Y$ are
centered and uncorrelated. For the remainder of this note, we will take $X$
and $Y$ to be centered and uncorrelated.

Pillai and Meng (2016) proved an amazing extension of (2.1). Let
$(X_{1},\ldots,X_{n})$ and $(Y_{1},\ldots,Y_{n}),\ n>1,$ be iid multivariate
centered normal random vectors with positive variances and a covariance matrix
$\Sigma$. Then for any weights $0\leq w_{j}\leq 1$,
$j=1,\ldots,n,\,\sum_{j=1}^{n}w_{j}=1$,

(2.2) $\sum_{j=1}^{n}w_{j}\frac{X_{j}}{Y_{j}}\overset{d}{=}W,$

where $W$ is a standard Cauchy. Earlier, Drton and Xiao (2016) proved (2.2)
when $n=2$.

What is amazing in (2.2)? Each ratio $X_{j}/Y_{j}$ in the sum has the standard
Cauchy distribution by (2.1). If the covariance matrix $\Sigma$ is diagonal,
then these Cauchy random variables are independent, and (2.2) is simply the
invariance property (1.2). If, on the other hand, all the correlations implied
by the covariance matrix $\Sigma$ equal 1, then all terms in (2.2) are equal
almost surely, and (2.2) is equivalent to the above-mentioned fact that a
convex linear combination of equal Cauchy random variables has the same law as
a convex linear combination of iid Cauchy random variables. The amazing part
is that (2.2) holds for all other covariance matrices $\Sigma$.

The only comparable property of non-independent Cauchy random variables that
appears to be known at this point is the property of jointly Cauchy random
variables with the spectral measure $\Gamma$ concentrated on the positive and
negative quadrants of the unit sphere. It is straightforward to show that the
terms in (2.2) are jointly Cauchy if and only if the normal random vectors
consist of independent blocks of equal normal random variables.

Therefore, it is a special feature of the multivariate normal distribution
that the ratios of the components of iid centered normal random vectors have
the invariance property (1.2), regardless of the correlations among the normal
components. Pillai and Meng (2016) speculate that “the dependence among them”
(i.e., among the ratios of centered, dependent normal random variables) “can
be overwhelmed by the heaviness of their marginal tails” (of the ratios) “in
determining the stochastic behavior of their linear combinations.” This
speculation continues to generate interest and new results. For example,
Cohen, Davis, and Samorodnitsky (2020) give another example where heavy tails
overwhelm correlations among normal random variables.

Here we shall show, by two examples, that the correlations between jointly
normal random variables are not always overwhelmed by the heaviness of the
marginal tails. In the two subsequent sections, we analyze two other natural
transformations, in the spirit of Pillai and Meng (2016), from the normal
world to the Cauchy world.

##  3\. Transformation 1: Absolute Values

The symmetry of a centered normal random variable and (2.1) imply that, if
$X,\ Y$ are iid centered normals and $W$ is a standard Cauchy, then

(3.1) $X/|Y|\overset{d}{=}W.$

Following Pillai and Meng (2016), let $(X_{1},\ldots,X_{n})$ and
$(Y_{1},\ldots,Y_{n}),\ n>1$, be iid multivariate centered normal random
vectors with positive variances and a covariance matrix $\Sigma$. Is it true
that, for any weights $0\leq w_{j}\leq 1$,
$j=1,\ldots,n,\,\sum_{j=1}^{n}w_{j}=1$,

(3.2) $\sum_{j=1}^{n}w_{j}\frac{X_{j}}{|Y_{j}|}\overset{d}{=}W,$

where $W$ is a standard Cauchy? As was the case with (2.2), the claim (3.2)
holds if the covariance matrix $\Sigma$ is a diagonal matrix, and also when
all the correlations implied by the covariance matrix $\Sigma$ equal 1.
However, we will show that (3.2) is not true for every covariance matrix
$\Sigma$. The argument uses a simple property of Cauchy densities that follows
immediately from (1.1):

(3.3) $f(x;\sigma)=\frac{f(0;\sigma)}{\pi^{2}f^{2}(0;\sigma)x^{2}+1},\
x\in{\mathbb{R}}.$

It is enough to show that (3.2) does not, in general, hold when $n=2$. A non-
singular covariance matrix $\Sigma$ with equal variances can be scaled so that
its inverse matrix is

(3.4) $\Sigma^{-1}=\left[\begin{matrix}1&\theta\\\
\theta&1\end{matrix}\right]\ \ \text{for $-1<\theta<1$.}$

Let $R_{i}=X_{i}/|Y_{i}|,\,i=1,2$. An elementary calculation for the joint
density of $(R_{1},R_{2})$ gives, in the parametrization (3.4),

$\displaystyle f_{R_{1},R_{2}}(r_{1},r_{2})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{(1-\theta^{2})|y_{1}y_{2}|}{(2\pi)^{2}}\cdot$
$\displaystyle\quad\quad\quad\quad\quad\exp\bigg{\\{}-\frac{1}{2}(y_{1}^{2}r_{1}^{2}+2\theta|y_{1}y_{2}|r_{1}r_{2}+y_{2}^{2}r_{2}^{2})-\frac{1}{2}(y_{1}^{2}+2\theta
y_{1}y_{2}+y_{2}^{2})\bigg{\\}}dy_{1}dy_{2}$ $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\int_{0}^{\infty}\frac{(1-\theta^{2})xy_{2}^{3}}{(2\pi)^{2}}\exp\bigg{\\{}-\frac{1}{2}\bigg{(}x^{2}y_{2}^{2}(r_{1}^{2}+1)+2\theta
xy_{2}^{2}(r_{1}r_{2}+1)+y_{2}^{2}(r_{2}^{2}+1)\bigg{)}\bigg{\\}}dxdy_{2}$
$\displaystyle+$ $\displaystyle
2\int_{0}^{\infty}\int_{0}^{\infty}\frac{(1-\theta^{2})xy_{2}^{3}}{(2\pi)^{2}}\exp\bigg{\\{}-\frac{1}{2}\bigg{(}x^{2}y_{2}^{2}(r_{1}^{2}+1)+2\theta
xy_{2}^{2}(r_{1}r_{2}-1)+y_{2}^{2}(r_{2}^{2}+1)\bigg{)}\bigg{\\}}dxdy_{2}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{(1-\theta^{2})x}{\pi^{2}\big{(}x^{2}(r_{1}^{2}+1)+2\theta
x(r_{1}r_{2}+1)+(r_{2}^{2}+1)\big{)}^{2}}dx$
$\displaystyle+\int_{0}^{\infty}\frac{(1-\theta^{2})x}{\pi^{2}\big{(}x^{2}(r_{1}^{2}+1)+2\theta
x(r_{1}r_{2}-1)+(r_{2}^{2}+1)\big{)}^{2}}dx.$

Define the convex combination of $R_{1}$ and $R_{2}$ to be
$V:=w_{1}R_{1}+w_{2}R_{2}$, where $w_{1}\in[0,1]$ and $w_{2}=1-w_{1}$. Then
the density $g_{V}$ of $V$ is given by

$\displaystyle g_{V}(v)=$
$\displaystyle\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{(1-\theta^{2})w_{2}^{3}x}{\pi^{2}\big{(}w_{2}^{2}x^{2}(r^{2}+1)+2\theta
w_{2}x(r(v-w_{1}r)+w_{2})+((v-w_{1}r)^{2}+w_{2}^{2})\big{)}^{2}}dxdr$ (3.5)
$\displaystyle+\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{(1-\theta^{2})w_{2}^{3}x}{\pi^{2}\big{(}w_{2}^{2}x^{2}(r^{2}+1)+2\theta
w_{2}x(r(v-w_{1}r)-w_{2})+((v-w_{1}r)^{2}+w_{2}^{2})\big{)}^{2}}dxdr$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{(1-\theta^{2})(w_{2}^{2}x^{2}-2\theta
w_{1}w_{2}x+w_{1}^{2})x}{2\pi\big{(}(w_{2}^{2}x^{2}-2\theta
w_{1}w_{2}x+w_{1}^{2})(x^{2}+2\theta
x+1)+(1-\theta^{2})v^{2}x^{2}\big{)}^{3/2}}dx$
$\displaystyle+\int_{0}^{\infty}\frac{(1-\theta^{2})(w_{2}^{2}x^{2}-2\theta
w_{1}w_{2}x+w_{1}^{2})x}{2\pi\big{(}(w_{2}^{2}x^{2}-2\theta
w_{1}w_{2}x+w_{1}^{2})(x^{2}-2\theta
x+1)+(1-\theta^{2})v^{2}x^{2}\big{)}^{3/2}}dx.$

For $w_{1}\in(0,1)$ fixed, we will prove that

(3.6) $\lim_{v\to\infty}v^{2}g_{V}(v)=1/\pi,$

for all $\theta\in(0,1)$. Assuming that (3.6) is true, if $g_{V}$ were a
Cauchy density for any $\theta\in(0,1)$ it would follow from (3.3) that

$g_{V}(0)=1/\pi\ \text{for any $\theta\in(0,1)$.}$

We will show that, in fact,

(3.7) $g_{V}(0)>1/\pi$

for all $\theta\in(0,\epsilon)$ for some $\epsilon>0$. This precludes the
possibility that $g_{V}$ is a Cauchy density for all such $\theta$.

To establish (3.7), it suffices to show that for any $w_{1}\in(0,1)$, the
derivative of the function $g_{V}(0)$ with respect to $\theta$ is positive at
$\theta=0$. Indeed, by (3.5)

$\displaystyle g_{V}(0)=$
$\displaystyle\int_{0}^{\infty}\frac{(1-\theta^{2})(w_{2}^{2}x^{2}-2\theta
w_{1}w_{2}x+w_{1}^{2})x}{2\pi\big{(}w_{2}^{2}x^{2}-2\theta
w_{1}w_{2}x+w_{1}^{2}\big{)}^{3/2}\bigl{(}x^{2}+2\theta x+1\big{)}^{3/2}}dx$
$\displaystyle+\int_{0}^{\infty}\frac{(1-\theta^{2})(w_{2}^{2}x^{2}-2\theta
w_{1}w_{2}x+w_{1}^{2})x}{2\pi\big{(}w_{2}^{2}x^{2}-2\theta
w_{1}w_{2}x+w_{1}^{2}\bigr{)}^{3/2}\bigl{(}x^{2}-2\theta x+1\big{)}^{3/2}}dx,$

It is elementary that one may differentiate with respect to $\theta$ inside
the integrals to obtain

$\frac{\partial
g_{V}(0)}{\partial\theta}\bigg{|}_{\theta=0}=\int_{0}^{\infty}\frac{w_{1}w_{2}x^{2}}{\pi(w_{1}^{2}+w_{2}^{2}x^{2})^{3/2}(x^{2}+1)^{3/2}}\,dx>0,$

as promised. We conclude that (3.7) holds and, hence (3.2) cannot be valid for
$\theta$ in a neighborhood of zero.

It remains to prove (3.6). We use the last equality of (3.5) and write in the
obvious notation

$\displaystyle
v^{2}g_{V}(v)=v^{2}\left[\int_{1}^{\infty}+\int_{1}^{\infty}\right]+v^{2}\left[\int_{0}^{1}+\int_{0}^{1}\right]=I_{1}(v)+I_{2}(v).$

We have, as $v\to\infty$,

$\displaystyle I_{1}(v)\sim$
$\displaystyle(1-\theta^{2})w_{2}^{2}\int_{1}^{\infty}\frac{x^{3}v^{2}}{\pi\bigl{(}w_{2}^{2}x^{4}+(1-\theta^{2})v^{2}x^{2}\bigr{)}^{3/2}}\,dx$
$\displaystyle=$ $\displaystyle
w_{2}\int_{w_{2}/(v(1-\theta^{2})^{1/2})}^{\infty}\frac{1}{\pi(x^{2}+1)^{3/2}}\,dx\to
w_{2}\int_{0}^{\infty}\frac{1}{\pi(x^{2}+1)^{3/2}}\,dx=w_{2}/\pi.$

Since $I_{2}$ can be reduced to $I_{1}$ by a variable change $x\to 1/x$ and
switching $w_{1}$ and $w_{2}$, this means that $I_{2}(v)\to w_{1}/\pi$ as
$v\to\infty$. Because $w_{1}+w_{2}=1$, this proves (3.6).

In summary, it is not true that, for any choice of $(w_{1},w_{2})$ with
$0<w_{1}<1,\,w_{1}+w_{2}=1$, the sum on the left side of (3.2) has a Cauchy
distribution for all $\theta$ in an arbitrarily small neighborhood of the
origin.

##  4\. Transformation 2: Randomly Stopped Brownian Motions

Recall that an ${\mathbb{R}}^{n}$-valued Brownian motion
$\bigl{\\{}\bigl{(}X_{1}(t),\ldots,X_{n}(t)\bigr{)},\,t\geq 0\bigr{\\}}$ is a
continuous centered Gaussian process that starts at the origin at time zero
and has stationary and independent increments. The law of such a process is
completely determined by the law of its values at time 1,
$\bigl{(}X_{1}(1),\ldots,X_{n}(1)\bigr{)}$. The covariance function of a
Brownian motion is

(4.1) ${\rm cov}(X_{i}(s),X_{j}(t))=\min(s,t)\,{\rm cov}(X_{i}(1),X_{j}(1)),\
s,t\geq 0,\ i,j=1,\ldots,n.$

A Brownian motion is self-similar in the sense that, for any $c>0$,

(4.2) $\bigl{\\{}\bigl{(}X_{1}(ct),\ldots,X_{n}(ct)\bigr{)},\,t\geq
0\bigr{\\}}\overset{d}{=}\bigl{\\{}\bigl{(}c^{1/2}X_{1}(t),\ldots,c^{1/2}X_{n}(t)\bigr{)},\,t\geq
0\bigr{\\}}.$

If $\bigl{\\{}X(t),\,t\geq 0\\}$ is a one-dimensional Brownian motion, $Y$ is
an independent copy of $X(1)$ and $X(Y^{-2})$ is the value of the Brownian
motion $X(t)$ evaluated at the random time $t=Y^{-2}$, then (4.2) implies that

(4.3)
$X\bigl{(}Y^{-2}\bigr{)}\overset{d}{=}X(1)/|Y|\overset{d}{=}X(1)/Y\overset{d}{=}W$

by (2.1), where $W$ is a standard Cauchy.

Again following Pillai and Meng (2016), let
$\bigl{\\{}\bigl{(}X_{1}(t),\ldots,X_{n}(t)\bigr{)},\,t\geq 0\bigr{\\}},\
n>1,$ be a Brownian motion with positive variances and $(Y_{1},\ldots,Y_{n})$
a centered normal random vector with the same law as
$\bigl{(}X_{1}(1),\ldots,X_{n}(1)\bigr{)}$ and independent of it. Let their
common covariance matrix be $\Sigma$. Is it true that for any weights $0\leq
w_{j}\leq 1$, $j=1,\ldots,n,\,\sum_{j=1}^{n}w_{j}=1$, and any $\Sigma$,

(4.4) $\sum_{j=1}^{n}w_{j}X_{j}\bigl{(}Y_{j}^{-2}\bigr{)}\overset{d}{=}W,$

where $W$ is a standard Cauchy? If $\Sigma$ is diagonal, or if all the
correlations implied by $\Sigma$ equal 1, then (4.4) holds. However, we will
show that (4.4) is not true in general. The argument is similar to the one
used in Section 3 and relies on the property (3.3) of the Cauchy distribution.
Again we let $n=2$ and parametrize the inverse of $\Sigma$ as in (3.4).

Let
$V=w_{1}X_{1}\bigl{(}Y_{1}^{-2}\bigr{)}+w_{2}X_{2}\bigl{(}Y_{2}^{-2}\bigr{)}$,
with $w_{1}\in[0,1]$ and $w_{2}=1-w_{1}$. Let $g_{V}$ be the density of $V$.
If $f_{Y_{1},Y_{2}}$ is the joint density of $(Y_{1},Y_{2})$, then by (4.1),

(4.5)
$g_{V}(v)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}h_{y_{1},y_{2}}(v)f_{Y_{1},Y_{2}}(y_{1},y_{2})\,dy_{1}dy_{2},$

where $h_{y_{1},y_{2}}$ is the density of a centered normal random variable
with variance

$\frac{w_{1}^{2}}{y_{1}^{2}(1-\theta^{2})}-\frac{2w_{1}w_{2}\theta}{\max(y_{1}^{2},y_{2}^{2})(1-\theta^{2})}+\frac{w_{2}^{2}}{y_{2}^{2}(1-\theta^{2})}.$

The part of the integral in (4.5) over the range $|y_{1}|<|y_{2}|$ can be
written in the form

$\displaystyle\int_{-1}^{1}\int_{-\infty}^{\infty}\frac{(1-\theta^{2})|x|y_{2}^{2}}{(2\pi)^{3/2}(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})^{1/2}}\cdot$ $\displaystyle\hskip
36.135pt\exp\bigg{\\{}-\frac{(1-\theta^{2})x^{2}y_{2}^{2}l^{2}}{2(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})}-\frac{1}{2}(x^{2}y_{2}^{2}+2\theta
x|y_{2}|y_{2}+y_{2}^{2})\bigg{\\}}dy_{2}dx$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{(1-\theta^{2})(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})x}{2\pi\big{(}(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})(x^{2}+2\theta
x+1)+(1-\theta^{2})x^{2}v^{2}\big{)}^{3/2}}dx$
$\displaystyle+\int_{0}^{1}\frac{(1-\theta^{2})(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})x}{2\pi\big{(}(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})(x^{2}-2\theta
x+1)+(1-\theta^{2})x^{2}v^{2}\big{)}^{3/2}}dx.$

Computing in an identical manner the part of the integral in (4.5) over the
range $|y_{2}|<|y_{1}|$ gives

(4.6) $\displaystyle g_{V}(v)=$
$\displaystyle\int_{0}^{1}\frac{(1-\theta^{2})(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})x}{2\pi\big{(}(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})(x^{2}+2\theta
x+1)+(1-\theta^{2})x^{2}v^{2}\big{)}^{3/2}}dx$
$\displaystyle+\int_{0}^{1}\frac{(1-\theta^{2})(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})x}{2\pi\big{(}(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})(x^{2}-2\theta
x+1)+(1-\theta^{2})x^{2}v^{2}\big{)}^{3/2}}dx$
$\displaystyle+\int_{0}^{1}\frac{(1-\theta^{2})(w_{2}^{2}-2w_{1}w_{1}\theta
x^{2}+w_{1}^{2}x^{2})x}{2\pi\big{(}(w_{2}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{1}^{2}x^{2})(x^{2}+2\theta
x+1)+(1-\theta^{2})x^{2}v^{2}\big{)}^{3/2}}dx$
$\displaystyle+\int_{0}^{1}\frac{(1-\theta^{2})(w_{2}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{1}^{2}x^{2})x}{2\pi\big{(}(w_{2}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{1}^{2}x^{2})(x^{2}-2\theta
x+1)+(1-\theta^{2})x^{2}v^{2}\big{)}^{3/2}}dx.$

We will prove that (3.6) still holds, independently of $\theta$ in (3.4) and
independently of the weights $(w_{1},w_{2})$. Assuming that this is the case,
the claim that $g_{V}$ cannot be a Cauchy density for all choices of $\theta$
and $(w_{1},w_{2})$ will, as in Section 3, follow once we show that, for any
$0<w_{1}<1$, the function $g_{V}(0)$ has a strictly positive derivative in
$\theta$ at $\theta=0$. By (4.6),

$\displaystyle g_{V}(0)=$
$\displaystyle\int_{0}^{1}\frac{(1-\theta^{2})(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})x}{2\pi(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})^{3/2}(x^{2}+2\theta x+1)^{3/2}}dx$
$\displaystyle+\int_{0}^{1}\frac{(1-\theta^{2})(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})x}{2\pi(w_{1}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{2}^{2}x^{2})^{3/2}(x^{2}-2\theta x+1)^{3/2}}dx$
$\displaystyle+\int_{0}^{1}\frac{(1-\theta^{2})(w_{2}^{2}-2w_{1}w_{1}\theta
x^{2}+w_{1}^{2}x^{2})x}{2\pi(w_{2}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{1}^{2}x^{2})^{3/2}(x^{2}+2\theta x+1)^{3/2}}dx$
$\displaystyle+\int_{0}^{1}\frac{(1-\theta^{2})(w_{2}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{1}^{2}x^{2})x}{2\pi(w_{2}^{2}-2w_{1}w_{2}\theta
x^{2}+w_{1}^{2}x^{2})^{3/2}(x^{2}-2\theta x+1)^{3/2}}dx.$

It is elementary that one may differentiate with respect to $\theta$ inside
the integrals to obtain

$\frac{\partial
g_{V}(0)}{\partial\theta}\bigg{|}_{\theta=0}=\int_{0}^{1}\frac{w_{1}w_{2}x^{3}}{\pi(w_{1}^{2}+w_{2}^{2}x^{2})^{3/2}(x^{2}+1)^{3/2}}\,dx+\int_{0}^{1}\frac{w_{1}w_{2}x^{3}}{\pi(w_{2}^{2}+w_{1}^{2}x^{2})^{3/2}(x^{2}+1)^{3/2}}\,dx>0,$

as promised.

It remains to prove (3.6). We use (4.6) and we write in the obvious notation

$v^{2}g_{V}(v)=I_{1}(v)+I_{2}(v)+I_{3}(v)+I_{4}(v).$

We have, as $v\to\infty$,

$\displaystyle I_{1}(v)\sim$
$\displaystyle(1-\theta^{2})w_{1}^{2}\int_{0}^{1}\frac{xv^{2}}{2\pi(w_{1}^{2}+(1-\theta^{2})v^{2}x^{2})^{3/2}}\,dx$
$\displaystyle=$ $\displaystyle
w_{1}\int_{0}^{v(1-\theta^{2})^{1/2}/w_{1}}\frac{x}{2\pi(x^{2}+1)^{3/2}}\,dx\to
w_{1}\int_{0}^{\infty}\frac{x}{2\pi(x^{2}+1)^{3/2}}\,dx=w_{1}/(2\pi).$

Similarly, as $v\to\infty$,

$I_{2}(v)\to w_{1}/(2\pi),\ \ I_{3}(v)\to w_{2}/(2\pi),\ \ I_{4}(v)\to
w_{2}/(2\pi),$

and (3.6) follows from the fact that $w_{1}+w_{2}=1$.

In summary, (4.4) does not hold for all values of the parameters. We prove the
stronger contrary result that, for any choice of $(w_{1},w_{2})$ with
$0<w_{1}<1,\,w_{1}+w_{2}=1$, the sum on the left side of (4.4) does not have a
Cauchy distribution for all $\theta$ in an arbitrarily small neighborhood of
the origin.

##  5\. Conclusions

Two natural transformations of multivariate normal vectors considered here and
the similar transformation Pillai and Meng (2016) considered all result in a
Cauchy distribution if the covariance matrix of the normal ingredients is
diagonal, or if all the correlations implied by the covariance matrix equal 1.
However, the transformation (2.2) Pillai and Meng (2016) considered produces a
Cauchy distribution regardless of the normal covariance matrix. By contrast,
the transformations (3.2) and (4.4) do not always produce a Cauchy
distribution. This shows that the mysteries of the connections between the
normal laws and the Cauchy laws remain to be understood. The result of Pillai
and Meng (2016) exhibits a family of laws in ${\mathbb{R}}^{n}$ with standard
Cauchy marginals that share with a subclass of multivariate Cauchy laws (1.3)
the property that any convex linear combination of the coordinates is again a
standard Cauchy. Since the two transformations we have considered lack this
property, the correlations between jointly normal random variables are not
always overwhelmed by the heaviness of the marginal tails. Future work must
clarify the extent of the phenomenon described by Drton and Xiao (2016) and
Pillai and Meng (2016).

## References

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  * Samorodnitsky and Taqqu (1994) G. Samorodnitsky and M. Taqqu (1994): Stable Non-Gaussian Random Processes.  Chapman and Hall, New York.