SearcharxivSearch

arXiv subjects

J. Bolmont

Publications and source records attributed to J. Bolmont.

At least 19 recordsLinked to original sources

Evidence for a spectral steepening of the gamma-ray emission from the Galactic Center ridge

Very-high-energy (VHE) $\gamma$-ray emission detected from the central molecular zone (CMZ) hints at diffusive propagation of cosmic rays (CRs) injected continuously near the Galactic center (GC). Using H.E.S.S. VHE $\gamma$-ray observations, we aim to construct a multi-component description of the region in order to derive the spatial and spectral distributions of CRs and discuss their potential origin. We rely on a spectro-morphological analysis, in which spectral and spatial parameters of different components are fit simultaneously, and on a three-dimensional description of the CMZ and CR distributions. The GC ridge emission is well reproduced with the assumed gas distribution model weighted by a $1/r^{\alpha}$ CR density profile with $\alpha = 1.10 \pm 0.05_{\rm{stat}} \pm 0.10_{\rm{syst}}$. We detect no significant spectral variations across the CMZ, which supports a continuous injection from the GC. We report a significant (> 3 $\sigma$) curvature in the GC ridge $\gamma$-ray spectrum, implying a parent proton break (or cutoff energy) of $E_b \sim 15-50$ TeV ($E_{\rm{cut}} \sim 60-160$ TeV). We also derive the best-fit spectral parameters of the other sources in the region and report a significant cutoff at $E \sim 5$ TeV in the spectrum of the pulsar wind nebula G0.9+0.1. Finally we constrain the position of the injection site to be close to the GC (ruling out the Arches and Quintuplet clusters as major contributors to the emission). We discuss possible origins for the GC ridge emission, potential mechanisms for causing the observed curvature, a possible contribution of the gas content to the emission of HESS J1745-290 and derive qualitative constraints on the line-of-sight positions of the clouds shaping the CMZ.

astro-ph.HE

Search for gamma-ray spectral lines from dark matter annihilation with the H.E.S.S. Inner Galaxy Survey

Spectral gamma-ray line features are expected as key signatures from dark matter (DM) annihilations of TeV-scale particle DM. Observations of the Galactic Centre with atmospheric Cherenkov telescopes are unique to probe thermal-relic TeV particle DM, well beyond the reach of direct detection and collider searches. We report here on the search for line signals in very-high-energy gamma rays using data from the Inner Galaxy Survey, consisting of 546 hours of H.E.S.S. observations of the inner few degrees of the Galactic Centre. No significant signal is detected. We then compute the exclusion limits on the annihilation line cross section $\langle \sigma v \rangle_{\rm line}$, with a two-dimensional log-likelihood ratio test statistics, exploiting spectral and spatial features of the DM signal. Assuming an Einasto DM density profile for the Milky Way, our results provide the most constraining limits so far, reaching $\langle \sigma v \rangle_{\rm line} = 2.3$ $\times$ $10^{-28}$ and $2.4 \times$ $10^{-27}$ cm$^3$s$^{-1}$ for DM masses of 1 and 10 TeV, respectively. The present limits are used to constrain the widely searched Wino, Higgsino and Quintuplet models. For the first time, thermal Higgsino DM is probed for DM Milky Way models.

astro-ph.HE

Gamma-Ray Constraints on Heavy Axion-Like-Particle Decays from Fermi-LAT and H.E.S.S. Blazar Spectra

The propagation of very-high-energy (VHE; $E_{\gamma} \geq 100$ GeV) gamma rays from extragalactic sources is affected by interactions with photons of the extragalactic background light (EBL), resulting in pair production that attenuates the intrinsic gamma-ray flux. This interaction renders the Universe increasingly opaque to VHE photons at high energies and redshifts. New physics scenarios involving axion-like particles (ALPs) could modify this expected optical depth. In particular, ALPs with masses $m_a \sim 10$ eV can decay into two photons over cosmological timescales, thereby contributing to the diffuse EBL. If such ALPs constitute a significant fraction of the dark matter density, their decay would enhance the EBL intensity and consequently increase the gamma-ray optical depth. In this study, we investigate this scenario using a large sample of gamma-ray spectra observed with the High Energy Stereoscopic System (H.E.S.S.) and the Fermi Large Area Telescope. We model the contribution of decaying ALPs to the EBL and assess their impact on the spectra of blazars across redshifts. By comparing these observations with standard EBL models, we place constraints on the properties of heavy ALPs, specifically their mass and photon coupling, and evaluate their viability as a dark matter candidate capable of modifying the gamma-ray transparency of the Universe. From the combined analysis, and under the assumption that ALPs constitute the entire dark matter density, we derive 95% confidence exclusion limits on the photon-ALP coupling down to $g_{a\gamma} \sim 7 \times 10^{-12}$ GeV$^{-1}$ for masses $m_a\sim 15$ eV. These constraints are competitive with existing astrophysical bounds and provide complementary sensitivity to other techniques, closing a previously unconstrained region of parameter space in the $m_a \sim 2.5$-$20$ eV range.

astro-ph.HE

Search for long-term variability of HESS J1745-290

At the center of our Galaxy lies the bright γ-ray point-like source HESS J1745-290, which is compatible in position with Sgr A star, although an association between the two remains uncertain. Using data obtained between 2004 and 2019 with the High Energy Stereoscopic System (H.E.S.S.) on the Galactic center region, we studied the variability of HESS J1745-290 over 353 hours of observations collected over 16 years, representing the largest dataset gathered yet on this region at TeV energies. We performed a 3D maximum-likelihood analysis of the central source and the diffuse γ-ray emission in the Galactic center region. This analysis allowed us to extract the spectral and morphological intrinsic behavior of the two components. By performing this analysis on an annual basis, we derived the light curve of HESS J1745-290 and the diffuse emission over the past 16 years. The 3D maximum-likelihood analysis method allowed us to separate the central source from the overlapping diffuse emission, enabling a recalibration of the former by the latter and alleviating some of the systematic effects. We find no long-term or yearly variability. We also provide an estimate of the sensitivity of H.E.S.S. to variation of this specific source over 16 years. We rule out any yearly gamma-ray flux variation of this source larger than 30 percent, as well as any linear flux variation exceeding 30% over this time period.

astro-ph.HE

H.E.S.S. observations of composite Seyfert-starburst galaxies

Context: Composite galaxies that contain both Seyfert and starburst components may produce very high-energy (VHE; >100 GeV) gamma-ray emission at a wide range of spatial scales, from a few Schwarzschild radii of a supermassive black hole to dimensions of kiloparsec-size jet-driven outflows. In addition to supernova remnants, various sources have been suggested to explain data collected on composite galaxies, including multi-messenger neutrino and ultra-high-energy cosmic-ray data. Aims: The closest composite Seyfert-starburst galaxies (NGC 1068, the Circinus galaxy, and NGC 4945) are observed with the High Energy Stereoscopic System (H.E.S.S.) to provide constraints on cosmic-ray populations in these systems. Methods: Data obtained in H.E.S.S. observations have been analyzed to search for VHE gamma-ray counterparts to the GeV gamma-ray signals detected with Fermi-LAT and for potential spectral components in the VHE range. Results: No significant signals have been found in these H.E.S.S. data. Upper limits on the VHE gamma-ray fluxes were applied to constrain theoretical models involving different spectral components.

astro-ph.HE

The second H.E.S.S. gamma-ray burst catalogue: 15 years of observations with the H.E.S.S. telescopes

Recent observational efforts using imaging atmospheric Cherenkov telescopes (IACTs) have led to firm detections of very-high-energy (VHE) signals from bright gamma-ray bursts (GRBs), often at moderate redshifts. This work presents 15 years of H.E.S.S. GRB observations and examines their implications through population comparisons and selected modelling cases. GRBs are a key science target of the High Energy Stereoscopic System (H.E.S.S.). With a low-energy threshold ($\lesssim$100 GeV) and rapid repointing capabilities, H.E.S.S. can begin follow-up observations within tens of seconds after a GRB trigger, covering the late prompt or early afterglow phases. We report GRB follow-up observations with H.E.S.S. from 2004 to 2019, which resulted in no significant VHE signals (aside from the detections of GRB~180720B and GRB~190829A). The resulting upper limits comprise the largest set available for GRBs at VHE. A subset of bursts with favourable conditions were selected for X-ray analysis and emission modelling. Population studies were performed to compare detected and non-detected GRBs. The results indicate that VHE-detected GRBs are not a distinct population, but tend to feature luminous X-ray emission and favourable redshift and observing conditions. This highlights the potential of next-generation IACTs such as the Cherenkov Telescope Array Observatory (CTAO), whose lower energy threshold will enhance the detection of fainter and more distant GRBs.

astro-ph.HE

Scrutinizing the 2020 multiwavelength outburst of PKS 0903-57 through observations with H.E.S.S

The blazar PKS 0903-57 has recently been classified as a flat spectrum radio quasar at a redshift of $z=0.2621$. In March and April 2020, Fermi-LAT and AGILE reported tremendous activity in high-energy $γ$ rays with the flux increasing by $\sim$2 orders of magnitude compared to quiescence. The flare was observed with H.E.S.S. in very-high-energy $γ$ rays for six nights with a total observation time of 13.1 h, resulting in the discovery of PKS 0903-57 in this energy band with an average flux of $1.5\times 10^{-10}$ ph cm$^{-2}$s$^{-1}$ above an energy threshold of $\sim 180$ GeV corresponding to $60\%$ of the Crab Nebula flux above the same threshold. The very-high-energy $γ$-ray flux was strongly variable. X-ray and optical data were collected with Swift and ATOM, and also indicate significant variability. The observed multiwavelength flux and spectral variability during the H.E.S.S. observation window suggest variability time scales on the order of a few hours and reveal complex correlation patterns. The lack of absorption beyond that of the extragalactic background light in the $γ$-ray domain suggests that the emission region was located outside of the broad-line region. A leptonic one-zone modeling of the six H.E.S.S. observation nights using the dusty torus as seed photons for the inverse-Compton scattering, results in a low magnetization of the emission region. This implies that shock acceleration is likely the main driver during the event.

astro-ph.HE

H.E.S.S. detection of the PSR J0855-4644 nebula

HESS J0852-463 is a TeV γ-ray source located in the Galactic plane. The region consists of a supernova remnant (SNR, RX J0852.0-4622) with a shell-like morphology, commonly referred to as Vela Junior, and a pulsar denoted PSR J0855-4644. Pulsars are among the most efficient leptonic accelerators in our Galaxy, making this region particularly interesting to study. We utilise the most recent data taken by the High Energy Stereoscopic System (H.E.S.S.), to investigate any γ-ray emission associated with the pulsar in this region, PSR J0855-4644. We applied a full forward folding method on the H.E.S.S. data. Utilising 3D modelling techniques, we evaluated the TeV γ-ray emission towards the various components of this complex system. The distinct energy-dependent morphology observed in our data motivates further investigation of this source. We resolved the emission in the Vela Junior region into various components, several of which correspond to the SNR itself. In particular, we find a new extended component which is coincident with the position of PSR J0855-4644. The spectrum follows a power-law distribution with a best-fit index of ΓE = 1.81 \pm 0.07stat which differs from the properties of the surrounding γ-ray emission of the Vela Junior SNR. A one-zone leptonic joint fit between the X-rays (from XMM-Newton) and γ-rays (from H.E.S.S.) leads to a lower limit on the magnetic field of 1.6μG and a spectral index of α = 1.88 \pm 0.01, in line with expectations of pulsar wind nebulae (PWNe). In this paper, we report the first detection of the PWN of PSR J0855-4644 at TeV energies with the H.E.S.S. experiment at a significance of 12.2σ. This is attributed to the advanced techniques of the 3D analysis. Based on the pulsar's characteristics, its PWN is consistent with the known TeV PWNe population in the Galaxy.

astro-ph.HE

Prompt Searches for Very-High-Energy γ-Ray Counterparts to IceCube Astrophysical Neutrino Alerts

The search for sources of high-energy astrophysical neutrinos can be significantly advanced through a multi-messenger approach, which seeks to detect the gamma rays that accompany neutrinos as they are produced at their sources. Multi-messenger observations have so far provided the first evidence for a neutrino source, illustrated by the joint detection of the flaring blazar TXS 0506+056 in highenergy (HE, E > 1 GeV) and very-high-energy (VHE, E > 100 GeV) gamma rays in coincidence with the high-energy neutrino IceCube-170922A, identified by IceCube. Imaging atmospheric Cherenkov telescopes (IACTs), namely FACT, H.E.S.S., MAGIC, and VERITAS, continue to conduct extensive neutrino target-of-opportunity follow-up programs. These programs have two components: followup observations of single astrophysical neutrino candidate events (such as IceCube-170922A), and observation of known gamma-ray sources after the identification of a cluster of neutrino events by IceCube. Here we present a comprehensive analysis of follow-up observations of high-energy neutrino events observed by the four IACTs between September 2017 (after the IceCube-170922A event) and January 2021. Our study found no associations between gamma-ray sources and the observed neutrino events. We provide a detailed overview of each neutrino event and its potential counterparts. Furthermore, a joint analysis of all IACT data is included, yielding combined upper limits on the VHE gamma-ray flux.

astro-ph.HE

Constraining the nature of the most extreme Galactic particle accelerator. H.E.S.S. observations of the microquasar V4641 Sgr

Microquasars have emerged as promising candidates to explain the cosmic-ray flux at petaelectronvolt energies. LHAASO observations revealed V4641~Sgr as the most extreme example so far. Using $\approx$100~h of H.E.S.S. data, we performed a spectro-morphological study of the gamma-ray emission around V4641~Sgr. We employed HI and dedicated CO observations of the region to infer the target material for cosmic-ray interactions. We detected multi-TeV emission around V4641~Sgr with a high significance. The emission region is elongated. We found a power-law spectrum with an index $\approx$1.8, and together with results from other gamma-ray instruments, this reveals a spectral energy distribution that peaks at energies of $\approx$100~TeV for the first time. We found indications (3$σ$) of a two-component morphology, with indistinguishable spectral properties. The position of V4641~Sgr is inconsistent with the best-fit position of the single-component model and with the dip between the two components. We found no significant evidence of an energy-dependent morphology. No dense gas was found at any distance towards V4641~Sgr. The peak of the SED at $\approx$100~TeV identifies V4641~Sgr as a candidate cosmic-ray accelerator beyond the so-called knee. The absence of dense target gas places stringent energetic constraints on hadronic interpretations, however. The H.E.S.S. measurement requires an unusually hard ($\approx 1.5$) spectral index for the protons. A leptonic scenario faces fewer obstacles if the particle transport is fast enough to avoid losses and to reproduce the observed energy-independent morphology. The absence of bright \xray emission across the gamma-ray emission region requires a magnetic field strength $\lesssim3$~$μ$G, however. Our findings favour a leptonic origin of the gamma-ray emission. This conclusion does not exclude hadron acceleration in the V4641~Sgr system.

astro-ph.HE

H.E.S.S. programme searching for VHE gamma rays associated with FRBs

Fast Radio Bursts (FRBs) are highly energetic, extremely short-lived bursts of radio flashes. Despite extensive research, the exact cause of these outbursts remains speculative. The high luminosity, short duration, and high dispersion measure of these events suggest they result from extreme, high-energy extragalactic sources, such as highly magnetized and rapidly spinning neutron stars known as magnetars. The number of detected FRBs, including repeating ones, has grown rapidly in recent years. Except for FRB 20200428D, and FRB-like radio burst that is associated to Galactic magnetar SGR 1935+2154, no multi-wavelength counterpart to any FRB has been detected yet. The High Energy Stereoscopic System (H.E.S.S.) telescope has developed a {program} to follow up FRBs searching for their gamma-ray counterparts, helping to uncover the nature of FRBs and FRB sources. This paper provides an overview of the searches for FRB sources conducted by H.E.S.S., including follow-up observations and simultaneous multi-wavelength campaigns with radio and X-ray observatories. Among the FRB sources observed by H.E.S.S., nine are localized with redshifts ranging between 0.11 and 0.492 from 2015 to 2022. No significant very high energy (VHE) emission was detected during these observations. We report constraints on the VHE luminosity ranging from $10^{44}$ erg s$^{-1}$ and $10^{48}$ erg s$^{-1}$, placing limits on the FRB's region persistent VHE emission and potential FRB afterglow emission across timescales from hours to years.

astro-ph.HE

The H.E.S.S. extragalactic sky survey with the first decade of observations

The results of the first extragalactic gamma-ray survey by the High Energy Stereoscopic System (H.E.S.S.) are presented. The survey comprises 2720 hours of very high-energy gamma-ray observations of the extragalactic sky, recorded with H.E.S.S. from 2004 up to the end of 2012. These data have been re-analysed using a common consistent set of up-to-date data calibration and analysis tools. From this analysis, a list of 23 detected objects, predominantly blazars, was obtained. This catalogue was assessed in terms of the source class populations that it contains. The level of source parameter bias for the blazar sources, probed by this observational dataset, was evaluated using Monte-Carlo simulations. Spectral results obtained with the H.E.S.S. data were compared with the \textit{Fermi}-LAT catalogues to present the full gamma-ray picture of the detected objects. Lastly, this unique dataset was used to assess the contribution of BL Lacertae objects and flat-spectrum radio quasars to the extragalactic gamma-ray background light at several hundreds of giga-electronvolts. These results are accompanied by the release of the high-level data to the astrophysical community.

astro-ph.HE

Discriminating between different modified dispersion relations from gamma-ray observations

The fact that the standard dispersion relation for photons in vacuum could be modified because of their interaction with the quantum nature of spacetime has been proposed more than two decades ago. A quantitative model [Jacob \& Piran, JCAP 01, 031 (2008)], has been tested extensively using distant highly energetic astrophysical sources, searching for energy-dependent time delays in photon arrival times. Since no delay was firmly measured, lower limits were set on the energy scale $Λ$ related to these effects. In recent years, however, different but equally well-grounded expressions beyond the Jacob \& Piran model were obtained for the photon dispersion relation, leading to different expressions for the dependence of lag versus redshift. This article introduces a general parameterization of modified dispersion relations in cosmological symmetry, which directly leads to a general parameterized lag versus redshift dependence encompassing both existing and new models. This parameterization could be used in the future to compare the predicted time lags of the different models and test them against observations. To investigate this possibility, realistic data sets are simulated, mimicking different types of extragalactic sources as detected by current and future instruments. When no lag is injected in the simulated data, each lag-redshift model leads, as expected, to a different value for the limit on $Λ$, and the Jacob \& Piran model gives the most stringent bound. When a lag at $Λ\sim E_P$ in the Jacob \& Piran model is injected, it is detected for all the other lag-redshift relations considered, although leading to different values. Finally, the possibility to discriminate between several lag-redshift models is investigated, emphasizing the importance of an evenly distributed sample of sources across a wide range of redshifts.

astro-ph.HE

Detection of very-high-energy gamma-ray emission from Eta Carinae during its 2020 periastron passage

The colliding-wind binary system $η$ Carinae has been identified as a source of high-energy (HE, below $\sim$100\,GeV) and very-high-energy (VHE, above $\sim$100\,GeV) gamma rays in the last decade, making it unique among these systems. With its eccentric 5.5-year-long orbit, the periastron passage, during which the stars are separated by only $1-2$\,au, is an intriguing time interval to probe particle acceleration processes within the system. In this work, we report on an extensive VHE observation campaign that for the first time covers the full periastron passage carried out with the High Energy Stereoscopic System (H.E.S.S.) in its 5-telescope configuration with upgraded cameras. VHE gamma-ray emission from $η$ Carinae was detected during the periastron passage with a steep spectrum with spectral index $Γ= 3.3 \pm 0.2_{\mathrm{stat}} \, \pm 0.1_{\mathrm{syst}}$. Together with previous and follow-up observations, we derive a long-term light curve sampling one full orbit, showing hints of an increase of the VHE flux towards periastron, but no hint of variability during the passage itself. An analysis of contemporaneous Fermi-LAT data shows that the VHE spectrum represents a smooth continuation of the HE spectrum. From modelling the combined spectrum we conclude that the gamma-ray emission region is located at distances of ${\sim}10 - 20$\,au from the centre of mass of the system and that protons are accelerated up to energies of at least several TeV inside the system in this phase.

astro-ph.HE

White Paper and Roadmap for Quantum Gravity Phenomenology in the Multi-Messenger Era

The unification of quantum mechanics and general relativity has long been elusive. Only recently have empirical predictions of various possible theories of quantum gravity been put to test, where a clear signal of quantum properties of gravity is still missing. The dawn of multi-messenger high-energy astrophysics has been tremendously beneficial, as it allows us to study particles with much higher energies and travelling much longer distances than possible in terrestrial experiments, but more progress is needed on several fronts. A thorough appraisal of current strategies and experimental frameworks, regarding quantum gravity phenomenology, is provided here. Our aim is twofold: a description of tentative multimessenger explorations, plus a focus on future detection experiments. As the outlook of the network of researchers that formed through the COST Action CA18108 ``Quantum gravity phenomenology in the multi-messenger approach (QG-MM)'', in this work we give an overview of the desiderata that future theoretical frameworks, observational facilities, and data-sharing policies should satisfy in order to advance the cause of quantum gravity phenomenology.

gr-qc

High-Statistics Measurement of the Cosmic-Ray Electron Spectrum with H.E.S.S

Owing to their rapid cooling rate and hence loss-limited propagation distance, cosmic-ray electrons and positrons (CRe) at very high energies probe local cosmic-ray accelerators and provide constraints on exotic production mechanisms such as annihilation of dark matter particles. We present a high-statistics measurement of the spectrum of CRe candidate events from 0.3 to 40 TeV with the High Energy Stereoscopic System (H.E.S.S.), covering two orders of magnitude in energy and reaching a proton rejection power of better than $10^{4}$. The measured spectrum is well described by a broken power law, with a break around 1 TeV, where the spectral index increases from $Γ_1 = 3.25$ $\pm$ 0.02 (stat) $\pm$ 0.2 (sys) to $Γ_2 = 4.49$ $\pm$ 0.04 (stat) $\pm$ 0.2 (sys). Apart from the break, the spectrum is featureless. The absence of distinct signatures at multi-TeV energies imposes constraints on the presence of nearby CRe accelerators and the local CRe propagation mechanisms.

astro-ph.HE

Separating source-intrinsic and Lorentz invariance violation induced delays in the very high energy emission of blazar flares

Aims: The aim of the present study is to explore how to disentangle energy-dependent time delays due to a possible Lorentz invariance violation (LIV) at Planck scale from intrinsic delays expected in standard blazar flares. Methods: We first characterise intrinsic time delays in BL Lacs and Flat Spectrum Radio Quasars in standard one-zone time-dependent synchrotron self-Compton or external Compton models, during flares produced by particle acceleration and cooling processes. We simulate families of flares with both intrinsic and external LIV-induced energy-dependent delays. Discrimination between intrinsic and LIV delays is then investigated in two different ways. A technique based on Euclidean distance calculation between delays obtained in the synchrotron and in the inverse-Compton spectral bumps is used to assess their degree of correlation. A complementary study is performed using spectral hardness versus intensity diagrams in both energy ranges. Results: We show that the presence of non-negligible LIV effects, which essentially act only at very high energies (VHE), can drastically reduce the strong correlation expected between the X-ray and the VHE gamma-ray emission in leptonic scenarios. The LIV phenomenon can then be hinted at measuring the Euclidean distance $d_{E}$ from simultaneous X-ray and gamma-ray flare monitoring. Large values of minimal distance $d_{E,min}$ would directly indicate the influence of non-intrinsic time delays possibly due to LIV in SSC flares. LIV effects can also significantly modify the VHE hysteresis patterns in hardness-intensity diagrams and even change their direction of rotation as compared to the X-ray behaviour. Both observables could be used to discriminate between LIV and intrinsic delays, provided high quality flare observations are available.

astro-ph.HE

Dark Matter Line Searches with the Cherenkov Telescope Array

Monochromatic gamma-ray signals constitute a potential smoking gun signature for annihilating or decaying dark matter particles that could relatively easily be distinguished from astrophysical or instrumental backgrounds. We provide an updated assessment of the sensitivity of the Cherenkov Telescope Array (CTA) to such signals, based on observations of the Galactic centre region as well as of selected dwarf spheroidal galaxies. We find that current limits and detection prospects for dark matter masses above 300 GeV will be significantly improved, by up to an order of magnitude in the multi-TeV range. This demonstrates that CTA will set a new standard for gamma-ray astronomy also in this respect, as the world's largest and most sensitive high-energy gamma-ray observatory, in particular due to its exquisite energy resolution at TeV energies and the adopted observational strategy focussing on regions with large dark matter densities. Throughout our analysis, we use up-to-date instrument response functions, and we thoroughly model the effect of instrumental systematic uncertainties in our statistical treatment. We further present results for other potential signatures with sharp spectral features, e.g.~box-shaped spectra, that would likewise very clearly point to a particle dark matter origin.

hep-ph