The Physics of Gravitational Redshifts in Clusters of Galaxies

Home , Blue Shift (short story), Gravitational redshift

Nick Kaiser Institute for Astronomy, U. Hawaii Clusters of galaxies

14 15 • Largest bound virialised systems ~10 -10 Msun • Velocity dispersion σv~1000 km/s (~0.003c) 2 -5 2 • so grav. potential is φ ~ σv ~ 10 c • Centres - often deﬁned by the brightest galaxy (BCG) • Usually very close to peak of light, X-rays, DM Clusters in the Millenium Simulation (Y. Cai) Gravitational redshift & uRSD 5

0.8 Wojtek, Hansen & Hjorth stacked 7,800 0.7 0

• Δ [km/s] positions of the brightest cluster galaxies. The error bars represent Poisson noise. f(R) • “Conﬁrms GR, rules out TeVeS” -15 -20 TeVeS • Had been suggested before (Cappi 1995; -25 Broadhurst+Scannapiaco, ....) 0 1 2 3 4 5 6 R [Mpc]

Figure 2 Constraints on gravitational redshift in galaxy clusters. The effect manifests itself as a blueshift ∆ of the velocity distributions of cluster7 galaxies in the restframeoftheirBCGs.Velocity shifts were estimated as the mean velocity of a quasi-Gaussian component of the observed velocity distributions (see Fig. 1). The error bars represent the range of ∆ parameter containing 68 per cent of the marginal probability and the dispersion of the projected radii in a given bin. The blueshift (black points) varies with the projected radius R and its value at large radii indicates the mean gravitational potential depth in galaxy clusters. The red profile represents theoretical predictions of general relativity calculated on the basis of the mean cluster gravitational potential inferred from fitting the velocity dispersion profile under the assumption of the most reliable anisotropic model of galaxy orbits (see SI for more details). Its width shows therangeof∆ containing 68 per cent of the marginal probability. The blue solid and dashed lines show the profiles corresponding to two modifications of standard gravity: f(R) theory4 and the tensor-vector-scalar (TeVeS) model5, 6. Both profiles were calculated on the basis of the corresponding modified gravitational potentials (see SI for more details). The prediction for f(R) represents the case which maximises the deviation from the gravitational acceleration in standard gravity on the scales of galaxy clusters. Assuming isotropic orbits in fitting the velocity dispersion profile lowers the mean gravitational depth of the clusters by 20 per cent. The resulting profiles of gravitational redshift for general relativity and f(R) theory are still consistent with the data and the discrepancybetweenpredictionofTeVeS and the measurements remains nearly the same. The arrows showcharacteristicscalesrelatedto the mean radius rv of the virialized parts of the clusters.

8 SDSS Redshift Survey r 2r 3r 4 v v v

1 km/s] 3 0 [10

los -1 v -2

-3

-4 0 1 2 3 4 5 R [Mpc]

Supplementary Figure 1 Velocity diagram combined from kinematic data of 7800 galaxy clusters 11 detected in the SDSS Data Release 7. Velocities vlos of galaxies with respect to the brightest cluster galaxies are plotted as a function of the projected cluster-centric distance R.Bluelinesare the iso-density contours equally spaced in the logarithm of galaxy density in the vlos R plane. − The arrows show characteristic scales related to the mean virial radius estimated in dynamical analysis of the velocity dispersion proﬁle. Data points represent 20 per cent of the total sample.

20 Wojtak, Hansen & Hjorth (Nature 2011)

0.8 Wojtek, Hansen & Hjorth stacked 7,800 0.7 0

8 The physics of cluster gravitational redshifts • Einstein gravity • gravitational "time dilation" • Weak ﬁeld limit • δν/ν = -Φ/c2 • Measured by Pound & Rebka (Harvard '59)

Is that it?

cluster Equivalence principle & the Pound + Rebka experiment

• Einstein’s Equivalence Principle: Observers on earth (being accelerated by the stress in the ground under them imparting momentum to them) will see light being red-shifted (and all other local physics being modiﬁed) exactly as would a pair of astronauts in empty space being accelerated by a rocket motor. • Pound and Rebka (1959, 1960): He was right. • But if you replace non-inertial apparatus by freely falling kit with same instantaneous velocities then B&H say Doppler formula will apply. They are right too - almost exactly.... VOLUME ), NUMBER 9 PHYSICAL REVIEW LETTERS NovEMBER 1, 1959 the results of reference 1 required by these con- modify the conclusions of reference 1 concern- siderations is easily made. The matrix element ing the n =6 state in the following way. The de- appropriate to a collision in which the (K, p) sys- population of the P level in any collision is es- tem in a state g„ f is changed to a state g„ f sentially unaffected but the reshuffling of other which can be the products of the K - p interaction, states is much reduced and their direct depopula- is tion is largely forbidden. This greatly reduces the transfer into the P level and the average atomic lifetime is considerably increased, en- hancing the importance of radiative transitions. where the U are the plane wave functions repre- Calculations of the same kind as reported in senting the relative (K, p) atom and proton co- reference 1 lead then to the result that about 20 /p ordinates and H, the Hamiltonian, will be equal to of atoms in a n =6 state reach the 2P state in- (e'/iR-r I) -(e'/IR -rhl), where R is the vec- stead of the 1.4 %%d stated in reference 1. tor coordinate of the colliding proton and rp and The uncertainties involved in the estimates rI, are the coordinates of the proton and K meson made in this note, and also in reference 1, are in the atom. For IR l) I r I a multipole expan- quite large, and the conclusions reached in these sion of H can be made. Setting R =a, as in refer- calculations are not presented with the intention ence 1, the matrix element can be rewritten as of establishing that P-wave capture is large, or (V (a ) IV (a ))(g IH'lg ), that the Stark effect is unimportant. But we be- Pg p lieve that these results do indicate the necessity where, with appropriate averaging of geometric of a more detailed examination of the problem. factors, the second term is precisely that evalu- ated by Day. et al. In the first factor V represents the radial part of the wave functions U, and the Day, Snows, and Sucher, Phys. Rev. Letters 3, 61 square of this term has a value of 1/5 for S to P (1959). transitions which are the most favorable. Changes 2L. B. Okun' and I. A. Pomeranchuk, J. Exptl. of more than one unit of angular momentum are Theoret. Phys. U. S.S.R. 34, 997 (1958) [translation: much more strongly forbidden. These corrections Soviet Phys. JETP 34, 688 (1958)j.

GRAVITATIONAL RED-SHIFT IN NUCLEAR RESONANCE R. V. Pound and G. A. Rebka, Jr. Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts (Received October 15, 1959) It is widely considered desirable to check ex- where R is the radius of the earth and h is the perimentally the view that the frequencies of altitude measured in cm. Very high accuracy is electromagnetic spectral lines are sensitive to required of the clocks even with the altitudes the gravitational potential at the position of the available with artificial satellites. Although emitting system. The several theories of rela- several ways of obtaining the necessary frequen- tivity predict the frequency to be proportional to cy stability look promising, it would be simpler the gravitational potential. Experiments are if a way could be found to do the experiment be- proposed to observe the timekeeping of a "clock" tween fixed terrestrial points. In particular, if based on an atomic or molecular transition, when an accuracy could be obtained allowing the meas- held aloft in a rocket-launched satellite, relative urement of the shift between points differing as to a similar one kept on the ground. The fre- little as one to ten kilometers in altitude, the quency v& and thus the timekeeping at height h is experiment could be performed between a moun- related to that at the earth's surface p, according tain and a valley, in a mineshaft, or in a bore- to hole. Recently Mossbauer has discovered' a new b. = = v v„- v0 vugh/c'(1+h/R) aspect of the emission and scattering of y rays by nuclei in solids. A certain fraction f of y = v h x (1.09 x 10 i8), rays of the nuclei of a solid are emitted without Is there any more to the physics of cluster grav-z? • Is the Pound & Rebka (P&R) experiment relevant here? • equipment ﬁxed to the tower in Harvard phys. dept • Einstein rocket thought experiment: • observers on surface of earth (e.g. P&R) being accelerated by the earth see the same physics as accelerated observers in a rocket • but there is no gravity in the rocket • P&R measured effect of non-gravitational acceleration VOLUME ), NUMBER 9 PHYSICAL REVIEW LETTERS NovEMBER 1, 1959 the results of reference 1 required by these con- modify the conclusions of reference 1 concern- siderations is easily made. The matrix element ing the n =6 state in the following way. The de- appropriate to a collision in which the (K, p) sys- population of the P level in any collision is es- tem in a state g„ f is changed to a state g„ f sentially unaffected but the reshuffling of other which can be the products of the K - p interaction, states is much reduced and their direct depopula- is tion is largely forbidden. This greatly reduces the transfer into the P level and the average atomic lifetime is considerably increased, en- hancing the importance of radiative transitions. where the U are the plane wave functions repre- Calculations of the same kind as reported in senting the relative (K, p) atom and proton co- reference 1 lead then to the result that about 20 /p ordinates and H, the Hamiltonian, will be equal to of atoms in a n =6 state reach the 2P state in- (e'/iR-r I) -(e'/IR -rhl), where R is the vec- stead of the 1.4 %%d stated in reference 1. tor coordinate of the colliding proton and rp and The uncertainties involved in the estimates rI, are the coordinates of the proton and K meson made in this note, and also in reference 1, are in the atom. For IR l) I r I a multipole expan- quite large, and the conclusions reached in these sion of H can be made. Setting R =a, as in refer- calculations are not presented with the intention ence 1, the matrix element can be rewritten as of establishing that P-wave capture is large, or (V (a ) IV (a ))(g IH'lg ), that the Stark effect is unimportant. But we be- Pg p lieve that these results do indicate the necessity where, with appropriate averaging of geometric of a more detailed examination of the problem. factors, the second term is precisely that evalu- ated by Day. et al. In the first factor V represents the radial part of the wave functions U, and the Day, Snows, and Sucher, Phys. Rev. Letters 3, 61 square of this term has a value of 1/5 for S to P (1959). transitions which are the most favorable. Changes 2L. B. Okun' and I. A. Pomeranchuk, J. Exptl. of more than one unit of angular momentum are Theoret. Phys. U. S.S.R. 34, 997 (1958) [translation: much more strongly forbidden. These corrections Soviet Phys. JETP 34, 688 (1958)j.

Non-GRAVITATIONAL RED-SHIFT IN NUCLEAR RESONANCE R. V. Pound and G. A. Rebka, Jr. Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts (Received October 15, 1959) It is widely considered desirable to check ex- where R is the radius of the earth and h is the perimentally the view that the frequencies of altitude measured in cm. Very high accuracy is electromagnetic spectral lines are sensitive to required of the clocks even with the altitudes the gravitational potential at the position of the available with artificial satellites. Although emitting system. The several theories of rela- several ways of obtaining the necessary frequen- tivity predict the frequency to be proportional to cy stability look promising, it would be simpler the gravitational potential. Experiments are if a way could be found to do the experiment be- proposed to observe the timekeeping of a "clock" tween fixed terrestrial points. In particular, if based on an atomic or molecular transition, when an accuracy could be obtained allowing the meas- held aloft in a rocket-launched satellite, relative urement of the shift between points differing as to a similar one kept on the ground. The fre- little as one to ten kilometers in altitude, the quency v& and thus the timekeeping at height h is experiment could be performed between a moun- related to that at the earth's surface p, according tain and a valley, in a mineshaft, or in a bore- to hole. Recently Mossbauer has discovered' a new b. = = v v„- v0 vugh/c'(1+h/R) aspect of the emission and scattering of y rays by nuclei in solids. A certain fraction f of y = v h x (1.09 x 10 i8), rays of the nuclei of a solid are emitted without How do we understand the grav-z in clusters? • Cluster gravitational redshift is difference between redshift for centre galaxy and general cluster population • Equivalently, what is redshift of centre as seen by others? • But these are objects in free-fall • P&R analogy is questionable at best • GR: gravity is "transformed away" for freely-falling observer • How should we understand redshifts in astronomy? • Digression: • redshifts in cosmology • redshifts in general Redshift in homogeneous FLRW cosmology...

• Wavelength scales as a(t) - but why? • Analogy with expanding reﬂecting cavity • a) lots of little redshifts as photons bounce off walls • b) symmetry - standing waves - ﬁxed # of nodes • either way: accumulated effect: λ ~ a(t)

Misner, Thorne and Wheeler

redshift as an effect on standing waves....

But is this a standing wave? Expanding space and redshifts in textbooks..... • E.R. Harrison (2000)

1.0

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0.0 Wolfgang Rindler (1970) 0.0 0.2 0.4 0.6 0.8 1.0 • t

FIG. 2: The scale factor of a hypothetical “loitering” universe as a function of time (measured in units of the present time). At the time indicated by the dot, a galaxy emits radiation, whichisobservedatthepresenttime.Atthetimesofbothemission and observation, the expansion is very slow, yet the galaxy’sobservedredshiftislarge.

Let us begin by reviewing a standard derivation8,13 of the cosmological redshift. Consider a photon that travels from a galaxy to a distant observer, both of whom are at rest in comoving coordinates. Imagine a family of comoving observers along the photon’s path, each of whom measures the photon’s frequency as it goes by. We assume that each observer is close enough to his neighbor so that we can accommodate them both in one inertial reference frame and use special relativity to calculate the change in frequency from one observer to the next. If adjacent observers are separated by the small distance δr,thentheirrelativespeedinthisframeisδv = Hδr,whereH is the Hubble parameter. This speed is much less than c,sothefrequencyshiftisgivenbythenonrelativisticDoppler formula

δν/ν = −δv/c = −Hδr/c = −Hδt. (4)

We know that H =˙a/a where a is the scale factor. We conclude that the frequency change is given by δν/ν = −δa/a; that is, the frequency decreases in inverse proportion to thescalefactor.Theoverallredshiftisthereforegivenby ν(t ) a(t ) 1+z ≡ e = o , (5) ν(to) a(te)

where te and to refer to the times of emission and observation, respectively. In this derivation we interpret the redshift as the accumulated eﬀect of many small Doppler shifts along the photon’s path. We now address the question of whether it makes sense to interpret the redshift as one big Doppler shift, rather than the sum of many small ones. Figure 2 shows a common argument against such an interpretation. Imagine a universe whose expansion rate varies with time as shown in Fig. 2. A galaxy emits radiation at the time te indicated by the dot, and the radiation reaches an observer at the present time t0.Theobservedredshiftisz = a(t0)/a(te) − 1=1.5, which by the special-relativistic Doppler formula would correspond to a speed of 0.72c.Atthetimesofemissionandabsorptionoftheradiation,the expansion rate is very slow, and the speedar ˙ of the galaxy is therefore much less than this value. We can construct models in which botha ˙(te)and˙a(t0)arearbitrarilysmallwithoutchangingtheratioa(t0)/a(te)andhencewithout changing the redshift. Upon closer examination this argument is unconvincing, because the calculated velocities are not the correct velocities. We should not calculate velocities at a ﬁxed instant ofcosmictime(eithert = te or t = t0). Instead we should calculate the velocity of the galaxy at the time of light emission relative to the observer at the present time.Afterall, if a distant galaxy’s redshift is measured today, we wouldn’texpecttheresulttodependonwhatthegalaxyisdoing today, nor on what the observer was doing long before the age ofthedinosaurs. In fact, when astrophysicists talk about what a distant object is doing “now,” they often do not mean at the present value of the cosmic time, but rather at the time the object crossed our past light cone. For instance, when astronomers measure the orbital speeds of planets orbiting other stars, the measured velocities are always of this sort. There is an excellent reason for this convention: we never have information about what a distant object is doing (or if the object even exists) at the present cosmic time. Any statement in which “now” is used to refer to the present cosmic time at the location of a distant object is not about anything observable, because it refers to events far outside our light cone. In summary, if we wish to discuss the redshift of a distant galaxy as a Doppler shift, we need to be willing to talk about vrel,thevelocityofthegalaxythen relative to us now.Talkingaboutvrel is precisely the sort of thing that 9 Î v

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Ê Ê Ê Ã Ê Ê Ê { Ê ° Ê Ê £ The rubber balloon analogy redshift caused by expansion of space?

• Textbooks are correct • λ does increase with a(t) • But is it reasonable to say expansion causes the shift? • And is it obvious? • what is the mechanism by which space stretches light? • is space expanding in this room? • is space expanding in a cluster of galaxies? Expanding Space: the Root of all Evil?∗

Matthew J. Francis1,4,LukeA.Barnes1,2,J.BerianJames1,3 &GeraintF. Lewis1

1Institute of Astronomy, School of Physics, University of Sydney, NSW 2006, Australia 2Institute of Astronomy, Madingley Rd, Cambridge, UK 3Institute for Astronomy, Blackford Hill, Edinburgh EH9 3HJ,U.K. 4Email: [email protected]

Abstract: While it remains the staple of virtually all cosmological teaching, the concept of expanding space in explaining the increasing separation of galaxies has recently come under ﬁre as a dangerous idea whose application leads to the development of confusionandtheestablishmentofmisconceptions. In this paper, we develop a notion of expanding space that is completely valid as a framework for the description of the evolution of the universe and whose application allows an intuitive understanding of the inﬂuence of universal expansion. We also demonstrate how arguments against the concept in general have failed thus far, as they imbue expanding space with physical properties not consistent with the expectations of general relativity. Keywords: cosmology: theory

1Introduction is far from uncontroversial. Recent attacks on the physical concept of expanding space have centred on When the mathematical picture of cosmology is first the motion of test particles in the expanding universe; introduced to students in senior undergraduate or ju- Whiting (2004), Peacock (2006) and others claim that nior postgraduate courses, a key concept to be grasped expanding space fails to adequately explain the motion is the relation between the observation of the redshift of test particles and hence that it should be abandoned. of galaxies and the general relativistic picture of the ex- But what, exactly, is at fault? Crucially, these claims pansion of the Universe. When presenting these new rely on falsifying predictions made from using expand- ideas, lecturers and textbooks often resort to analo- ing space as a tool to guide intuition, to bypass the full gies of stretching rubber sheets or cooking raisin bread mathematical calculation. However, the very meaning toExpanding allow students to Space: visualise the how Root galaxies of are all moved Evil?∗ of the phrase expanding space is not rigorously defined, apart, and waves of light are stretched by the “expan- despite its widespread use in teaching and textbooks. sion of space”. These kinds of analogies are appar- Hence, it is prudent to be wary of predictions based entlyMatthew thought to J. be Francis useful1 in,4,LukeA.Barnes giving students a men-1,2,J.BerianJameson such a poorly1,3 defined&GeraintF. intuitive frameworks. tal pictureLewis1 of cosmology, before they have the ability In recent work, Barnes et al. (2006) solved the test to directly comprehend the implications of the formal particle motion problem for universes with arbitrary general1 relativistic description. However, the academic asymptotic equation of state w and found agreement Institute of Astronomy, School of Physics, University of Sydney, NSW 2006, Australia argument surrounding the expansion of space is not as between the general relativistic solution and the ex- 2 clearInstitute as standard of Astronomy, explanations Madingley suggest; Rd, an Cambridge, interested UK pected behaviour of particles in expanding space. We student3Institute and reader for Astronomy, of New Blackford Scientist Hill, may Edinburgh have seen EH9 3HJsuggest,U.K. that the apparent conflict between this work Martin4 Rees & Steven Weinberg (1993) state and others, for instance Chodorowski (2006b), lies pre- Email: [email protected] arXiv:0707.0380v1 [astro-ph] 3 Jul 2007 ...how is it possible for space, which is ut- dominantly in differing meanings of the very concept of terly empty, to expand? How can noth- But see alsoexpanding Weinberg, space. 1st This 3 Minutes, is unsurprising, given that it is aphraseandconceptoftenstatedbutseldomdefined Abstract:ing expand?While it The remains answer the stapleis: space of virtually does allp31: cosmological “One can te thinkaching, of thethe concept wave crests of expanding with any rigour. spacenot in expand. explaining Cosmologists the increasing sometimes separation talk of galaxiesbeing h aspulled recently farther come and under farther fire asapart a dangerous by ideaabout whose expanding application leadsspace, to but the development they should of confusiothe expansionnandtheestablishmentofmisconceptions.In of this the paper, universe.” we examine the picture of expand- In thisknow paper, better. we develop a notion of expanding space that ising completely space within valid as the a framework framework for of the fully general rela- whiledescription being told of the by Harrisonevolution of(2000) the universe that and whose applicationtivistic allows cosmologies an intuitive and understanding develop it into a precise def- of the influence of universal expansion. We also demonstrateinitionhow arguments for understanding against the the concept dynamical in properties of generalexpansion have failed redshifts thus far, are as produced they imbue by expanding the spaceFriedman-Robertson-Walker with physical properties not consistent (FRW) spacetimes. This withexpansion the expectations of space of general between relativity. bodies that framework is pedagogically superior to ostensibly sim- are stationary in space Keywords: cosmology: theory pler misleading formulations of expanding space — or What is a lay-person or proto-cosmologist to make of more general schemes to picture the expansion of the this apparently contradictory situation? Universe — such as kinematic models and approxi- Whether or not attempting to describe the obser- mations to special relativity or Newtonian mechanics, 1Introduction is far from uncontroversial. Recent attacks on the vations of the cosmos in terms of expanding space isphysical a since concept it is of both expanding clearer space and easier have tocentred understand on as well Whenuseful the goal, mathematical regardless picture of the of devices cosmology used is first to do so,the motionas being of test a particles more accurate in the expanding approximation. universe; In particu- introduced to students in senior undergraduate or ju- Whiting (2004), Peacock (2006) and others claim that nior postgraduate courses, a key concept to be grasped expanding space fails to adequately explain the motion is the relation between the observation of the redshift of test1 particles and hence that it should be abandoned. of galaxies and the general relativistic picture of the ex- But what, exactly, is at fault? Crucially, these claims pansion of the Universe. When presenting these new rely on falsifying predictions made from using expand- ideas, lecturers and textbooks often resort to analo- ing space as a tool to guide intuition, to bypass the full gies of stretching rubber sheets or cooking raisin bread mathematical calculation. However, the very meaning to allow students to visualise how galaxies are moved of the phrase expanding space is not rigorously defined, apart, and waves of light are stretched by the “expan- despite its widespread use in teaching and textbooks. sion of space”. These kinds of analogies are appar- Hence, it is prudent to be wary of predictions based ently thought to be useful in giving students a men- on such a poorly defined intuitive frameworks. tal picture of cosmology, before they have the ability In recent work, Barnes et al. (2006) solved the test to directly comprehend the implications of the formal particle motion problem for universes with arbitrary general relativistic description. However, the academic asymptotic equation of state w and found agreement argument surrounding the expansion of space is not as between the general relativistic solution and the ex- clear as standard explanations suggest; an interested pected behaviour of particles in expanding space. We student and reader of New Scientist may have seen suggest that the apparent conflict between this work Martin Rees & Steven Weinberg (1993) state and others, for instance Chodorowski (2006b), lies pre- arXiv:0707.0380v1 [astro-ph] 3 Jul 2007 ...how is it possible for space, which is ut- dominantly in differing meanings of the very concept of terly empty, to expand? How can noth- expanding space. This is unsurprising, given that it is ing expand? The answer is: space does aphraseandconceptoftenstatedbutseldomdefined not expand. Cosmologists sometimes talk with any rigour. about expanding space, but they should In this paper, we examine the picture of expand- know better. ing space within the framework of fully general rela- while being told by Harrison (2000) that tivistic cosmologies and develop it into a precise definition for understanding the dynamical properties of expansion redshifts are produced by the Friedman-Robertson-Walker (FRW) spacetimes. This expansion of space between bodies that framework is pedagogically superior to ostensibly sim- are stationary in space pler misleading formulations of expanding space — or What is a lay-person or proto-cosmologist to make of more general schemes to picture the expansion of the this apparently contradictory situation? Universe — such as kinematic models and approxi- Whether or not attempting to describe the obser- mations to special relativity or Newtonian mechanics, vations of the cosmos in terms of expanding space is a since it is both clearer and easier to understand as well useful goal, regardless of the devices used to do so, as being a more accurate approximation. In particu-

1 Peebles (’71) explanation of cosmological redshift • The redshift λrec/λem is the product of a lot of small shifts between a set of FOs along the look-back path In the vicinity of a neighouring pair of • FOs • space-time is locally ﬂat, so incremental redshifts are Doppler • shifts • Yields differential equation • dλ/λ = da/a with solution λ ∝ a(t) • So fractional change in proper separation is the same as the fractional change in λ • i.e. δlog(λ/D) = 0 Redshift and expansion in cosmology

• So in cosmology wavelength λ is tied to expansion a(t) • may or may not be caused by it • If observer/source are moving apart then λ increases • Exactly as for Doppler shift in empty space • So any gravitational component to redshift is somehow hidden • Mathematically: Δln(λ/D) = 0

• Is this a general principle? What about our lumpy universe?

• Bondi (1947): Spherical models: • for low-Z, redshift is product of Doppler and gravitational redshift • But Synge (1960) argued that all redshifts are Doppler shifts • “In attributing a cause to this spectral shift, one would say .... that the spectral shift was caused by the relative velocity of the source and the observer''. Synge, 1960; General Relativity

Observed (or emitted) energy • is dot product of observer 4- velocity and the photon 4- momentum. • → wavelength shift is given by Doppler’s formula with “relative velocity” being the l.o.s. component of the difference of the receiver 4- velocity and a parallel transported version of the emitter 4-velocity “Not a gravitational redshift as • the Riemann tensor does not appear in formula” Bunn & Hogg, 2009 Like Peebles they break photon path into a set of • intervals • set of intervening observers along line of sight • local ﬂatness →product of Doppler shifts • But intervening observers need not be freely falling Claim: Any incremental shift can be considered to be • either Doppler or gravitational • "gravitational redshifts are just Doppler shifts viewed from an unnatural coordinate system" "an enlightened cosmologist would never try to draw • any distinction" • All redshifts can (and should!) be considered to be Doppler, or ‘kinematic’ in nature. (much like Synge) • Again suggests Δln(λ/D) = 0 is universal? ideas about redshifts in astronomy - summary

• The redshift of light in cosmology • redshift is caused by the expansion of space? • standing waves in a cavity • Maxwell's equations in expanding space: • "Hubble damping" + the adiabatic invariant • Thermodynamics & photons as particles • Peebles' picture - lots of little Doppler shifts • The redshift of light in general • Synge ('60): redshifts "caused by the relative velocity..." • Bunn & Hogg ('09): "gravitational redshifts are just Doppler shifts viewed from an unnatural coordinate system" • 1st order "relativistic" redshift space distortion (Yoo+09) • Δz = Δr + .... is also purely a "Doppler" effect "what causes the redshift?" and why do we care?

• All the foregoing support the "kinematic picture" for astronomical redshifts. • redshifts come entirely from motions • in nice accord with Equivalence Principle • But clusters are not expanding! • and observers, sources are freely falling • so why would we see any gravitational redshift?

• At the very least one might have doubts about the Einstein/ Newton/Pound+Rebka picture • What additional physics might there be? Back to the Wojtak et al. measurement

0.8 Gravitational redshift for light 0.7 0

at ~2.5 sigma level 5 r 2r 3r • v v v Figure 1 Velocity distributions of galaxies combined from 7, 800 SDSS galaxy clusters. The line- 0 of-sight velocity (vlos)distributionsareplottedinfourbinsoftheprojectedcluster-centric distances Claimed to conflict with TeVeS R.Theyaresortedfromthetoptobottomaccordingtotheorderof radial bins indicated in the • upper left corner and offset vertically by an arbitrary amountforpresentationpurposes.Redlines -5 modified gravity present the histograms of the observed galaxy velocities in the cluster rest frame and black solid lines show the best fitting models. The model assumes a linear contribution from the galaxies which do not-10 belong to the cluster and a quasi-Gaussian contribution from the cluster members (see SI for more details). The cluster rest frames and centresaredefinedbytheredshiftsandtheGR

positionsΔ [km/s] of the brightest cluster galaxies. The error bars represent Poisson noise. descendent of Milgrom f(R) • -15 theory -20 TeVeS But OK with GR or e.g. f(R) • -25 0 1 2 3 4 5 6 modiﬁcations R [Mpc]

8 0.8 0.7 0

Figure 1 Velocity distributions of galaxies combined from 7, 800 SDSS galaxy clusters. The line- of-sight velocity (vlos)distributionsareplottedinfourbinsoftheprojectedcluster-centric distances R.Theyaresortedfromthetoptobottomaccordingtotheorderof radial bins indicated in the upper left corner and offset vertically by an arbitrary amountforpresentationpurposes.Redlines present the histograms of the observed galaxy velocities in the cluster rest frame and black solid lines show the best fitting models. The model assumes a linear contribution from the galaxies which do not belong to the cluster and a quasi-Gaussian contribution from the cluster members (see SI for more details). The cluster rest frames and centresaredefinedbytheredshiftsandthe positions of the brightest cluster galaxies. The error bars represent Poisson noise.

7 Jimeno, Broadhurst, Coupon, Imetzu, LazkosSDSS Clusters 20152005 Downloaded from http://mnras.oxfordjournals.org/ at University of California, Berkeley on September 14, 2016

Figure 6. GMBCG (top), WHL12 (middle) and redMaPPer (bottom) phase-space diagrams before (left) and after (right) removing statistically the foreground and background contribution of galaxies. Black contours represent iso-density regions. The asymmetry between the positive and negative vlos region can be particularly clearly seen in the redMaPPer case. This difference disappears after the statistical interloper removal. We also plot as red dashed lines the boundaries at 1, 2.5 and 4.5 Mpc that will determine the radial bins we will use in Section 3. In these diagrams, the position of the BCG is fixed at r 0Mpc 1 ⊥ = and vlos 0kms by definition, and the density is determined by the number of galaxies with spectroscopic redshift measurements around them. = − Also, this will help us identify which of the BCGs have the best proximation, where galaxies not belonging to clusters are not iden- spectroscopic measurements, so, taking a conservative approach, tified individually in each cluster, as in the direct method, but taken we will only work with those BCGs identified in our ‘high quality’ into account statistically once all the cluster information has been SDSS galaxy sample, discarding this way BCG redshift measure- stacked into one single distribution of galaxies. See Wojtak et al. ments obtained from ‘bad’ plates. This leaves us with a total sample (2007) for a detailed study of different direct and indirect foreground of 19 867 BCGs in the GMBCG catalogue, 52 255 in the WHL12 and background galaxies removal techniques. case, and 10 197 in the redMaPPer one. We compute the pro- In our case, we apply the following procedure: first, jected transverse distance r and the line-of-sight velocity vlos we bin the whole phase-space distribution in bins of size ⊥ = 1 c (zgal zBCG)/(1 zBCG)ofallSDSSgalaxieswithrespecttothe 0.04 Mpc 50 km s− . After that, we take all those bins lying in two − + × 1 1 BCGs, and keep those that lie within a separation of r < 7 Mpc stripes 4500 km s− < vlos < 6000 km s− , where we assume that 1 ⊥ | | and vlos < 6000 km s− from these. It should be noted that, as all the galaxies there belong either to the pure foreground or to the we are| working| mainly in a low redshift region, the impact of the pure background sample. Then, we fit a quadratic polynomial de- cosmological parameters used is not significant. Stacking all the ob- pendent of both vlos and r to the points in both stripes, and use the tained pairs into one single phase-space diagram, we get the density interpolated background⊥ model to correct the ‘inner’ phase-space distributions shown on the left-hand side of Fig. 6. region bins. We use a function that depends not only on r , but also ⊥ To remove the contribution of foreground and background galax- on vlos; this is because at high redshifts, and due to observational ies not gravitationally bound to clusters, we adopt an indirect ap- selection, we may have more spectroscopic measurements of those

MNRAS 448, 1999Ð2012 (2015) Jimeno, Broadhurst, Coupon, Imetzu, Lazkos 2015 Jimeno, Broadhurst, Coupon, Imetzu, Lazkos 2015 4 Sadeh et al 2014 3 ×10 + 0.2 + 0.3 + 0.4 + 0.5 gal rgc [Mpc] : 0.5 1.0 1.6 2.1 n 14 - 0.2 - 0.2 - 0.3 - 0.3 12 20 wsw = 0.5 × r200 GR only 10 wsw = 1.0 × r200 GR + kinematic [km/s] 3 gc w = 1.0 r (mix) WHH 10 sw × 200 × v 10 7 ∆ clst n 6

5 0

2.3 -10 clst

/ n 2.2 gal n 2.1 -20

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 r / r gc 200 rgc / r200 (a) (b)

FIG. 1. (a):Dependenceofthenumberofgalaxies,ngal,ofthenumberofassociatedclusters,nclst,andoftheratio,ngal/nclst, on the separation between BCGs and associated galaxies, rgc.Binsofrgc are defined by a sliding window with a width of 0.5r200,whereeachdata-pointisplacedinthecenter-positionofthe corresponding bin. (b):DependenceofthesignaloftheGRS,∆vgc,onrgc,wherethewidthoftheslidingwindowisdenotedbywsw.Theshaded areas around the two nominal results (circles and squares) correspond to the variations in the signal due to the systematic tests − +7 described in the text, combined with the uncertainty on the model-fit. On average, ∆vgc = 11−5 km/sfor1masses is based on extracting the GRS signal from the distri- used in our analysis is comparable to that of the WHH bution of the velocities of galaxies in the rest frame of sample. We therefore refer to the corresponding esti- corresponding BCGs. We focus on optimizing the selec- mate of Kaiser of 9 (GR only) or 12 km/s (including tion procedure of clusters and of galaxies, and take into kinematic effects)− [11]. Our results− are in good agree- account multiple possible sources of systematic biases not ment with this prediction for rgc >r200, while at smaller considered by WHH. values of rgc, the profile of ∆vgc is steeper in the data. We find an average redshift of 11 km/s with a stan- − Additionally, we observe that it is not possible to distin- dard deviation of +7 and 5km/sfor1Doppler effect: 1st order Doppler effect averages • to zero, but.... • to 2nd order <δz> = /2 can be understood as time • dilation - moving clocks run slow Generally of same order of magnitude • as gravitational redshift from virial theorem, Jeans eq... • (And this doesn’t really test GR • see also Bekenstein & Sanders, 2012 • more later.....) • Is that the full story? No - there is another effect of same order

• Light cone effect • we will tend to see more objects moving away from us than towards us in any observation made using light as a messenger • this gives an extra red-shift effect • again of the same order of magnitude as the gravitational redshift Light-cone effect • Light cone effect • we will see more particles moving away from us in a photograph of a swarm of particles • past light cone of event of our observation overtakes more galaxies moving away than coming towards us • just as a runner on a trail sees more hikers going the other way... • So not Lorentz-Fitzgerald contraction effect • phase space density contains a factor (1-v/c) 2 • <δz> = <(vlos/c) > • same sign as TD effect • 2/3 magnitude (for isotropic orbits) Quasar absorption lines Light-cone effect - more particles moving away! Another way to look at LC effect

• Particle oscillating in a pig-trough • r(t) = a cos(ωt + φ) • v(t)/c = -(aω/c) sin(ωt + φ) • v(t) averages to zero average could be over phase or • time • but vobs = v + (r/c) dv/dt + ... • where r/c is the look-back time and the extra term does not • average to zero ~ same as Einstein prediction for • Pound & Rebka • δz ≈ / c2. Yet another view of the light-cone effect

• Consider a particle oscillating in a square well potential and emitting pulses at a steady rate (2N per period) • Observer sees intervals between pulses red- or blue-shifted • N short intervals followed by N long intervals • In observation taken at a random time there is a greater chance to catch the particle when it is moving away • In an observation of an ensemble of particles more particles will be seen going away from the observer Why is the transverse Doppler effect a redshift?

• Transverse Doppler redshift effect: • first order Doppler shift ~v/c is large but averages to zero • residual is a quadratic ~(v/c)2 effect which caused randomly moving objects appear redshifted on average • can also be understood as a time dilation effect • But moving objects have more energy per unit mass (in the observer frame) • So if they convert their rest mass to photons we should see a blue-shift on average a thought experiment • bake cake, light candles, spin the cake up on a turntable and measure the energy of the photons in the lab frame • <1st order Doppler> = 0 • 2nd order transverse Doppler effect gives a redshift • but the candles are moving.... • so they have more energy (in our frame) per unit rest mass... How do we • so should there not be, on average, a transverse Doppler blueshift? resolve this? Transverse Doppler Effect: Redshift or Blueshift? • Averaging over objects vs averaging over photons • averaging over objects we will see a redshift • but objects emitting isotropically in their rest frame do not emit isotropically in the lab frame - more photons come out in the forward direction - and these have a blue shift on average in the lab frame • this flips the sign of the effect • e.g. unresolved objects show blue shift (e.g. stars in the BCG or low resolution 21cm radio for integrated cluster z) • here we have a hybrid situation: • redshifts measured for objects • but objects are selected according to flux density Surface brightness modulation

• Line of sight velocity changes surface brightness • relativistic beaming (aberration) plus change of frequency • but doesn’t change the surface area • so velocities modulate luminosity depends on SED: δL/L = (3 + α)v/c • Gravitational Redshifts in Clusters 3 • α ~= 2, so big ampliﬁcation • spectroscopic sample is ﬂux limited at r=17.8 dN/dZ

• Δn/n = - d ln n(>Llim(Z))/d ln L * ΔL/L • opposite sign to LC, TD effects, but larger because the sample here is limited to bright end of the luminosity function

Figure 1. Spectral index vs. redshift for representative galaxy Figure 2. The dot-dash curve is the logarithmic derivative of types observed in Sloan r-band the comoving density of objects above the luminosity limit asa function of redshift. The bell-shaped curve is dN/dZ = Z2n(Z) and the solid curve is that truncated at the minimum redshift imposed by the parent cluster catalogue. The mean of the log- derivative, averaged over the redshift distribution turns out to be ≃ 2.0. luminosity function does not, and the parameters are not very diﬀerent from the ﬁeld galaxy luminosity function, so we will use the latter, as determined by Montero-Dorta & Prada (2009), as a proxy. Their estimate of the LF obtained 4 EFFECT OF SECULAR INFALL from the r-band magnitudes K-corrected to Z =0.1 has

M∗ − 5 log10 h = −20.7 and faint end slope of α = −1.26. The discussion so far has focused mostly on the stable, viri- The resulting d ln n(>L)/d ln L, computed using the flux alised regions. Clusters, however, are evolving structures and limit r = 17.77 appropriate for the SDSS spectroscopic sam- the mass within a fixed physical radius M(L)/d ln L over the galaxies used. The 7,800 clus- which would reduce the mass and would have an associated ters used by WHH were selected by applying a richness outflow. limit to the parent GMBCG catalog (Hao, J., et al. 2010) The combination of infall and the associated M˙ will re- that contains 55,000 clusters extending to Z =0.55. These sult in a positive offset of the mean line of sight velocity clusters were derived from the SDSS photometric catalog since the density will be slightly higher in front of the clus- that is much deeper than the spectroscopic catalog. Con- ter where we see the galaxies later and these galaxies will sequently, at the low redshifts where the spectroscopically be moving against us. There is also a potentially larger ef- selected galaxies live, this parent catalog is essentially vol- fect from the fact that along any line of sight we observe ume limited for the clusters used, so the redshift distribu- galaxies that lie in a cone that will be wider at the back tion for the cluster members used is essentially the same of the cluster, and at the same order, we need to allow for as that for the redshift distribution for the entire spec- the bias caused by the fact that the more distant galaxies troscopic sample, save for the fact that the GMBCG cat- will be fainter. These geometric and flux limit effects, whose alog has a lower redshift limit Zlim =0.1, which is very effects on the foreground and background galaxies was dis- close to the redshift where dN/dZ = Z2n(Z) peaks. This is cussed by Kim and Croft (2004), will cause a back/front the bell shaped curve in figure 2. Combining these we find anisotropy in the number of galaxies within the clusters ⟨d ln n/d ln L⟩ = dZ Z 2n(Z)d ln n/d ln L/ dZ Z 2n(Z) ≃ ∆N/N ∼ 2H∆r(1 − δ(Z))/cz while the change in the phys- 2.0 with integrationR range 0.1 L)/d ln L⟩≃10. This may be a slight asymmetry ∆N/N ∼ (r/c)(M/M˙ ) ∼ Hr/c, where we have overestimate, as the cluster catalogue is not precisely vol- assumed that the mass within radius r for the ensemble av- ume limited and the actual dN/dZ may lie a little below the erage cluster is changing on a cosmological timescale. Evi- solid curve in figure 2 at the highest redshifts. dently, for low redshift clusters, the geometric and flux limit With this value, the surface brightness modulation ef- effects will tend to be the largers. fect is roughly a factor 10 larger in amplitude than the light- To order of magnitude, the mean offset induced is 2 2 2 cone effect, but has opposite sign. For isotropic orbits the ⟨βz⟩∼H r /c z. The gravitational potential, for compar- combination of these gives a blue-shift 6 times as large as ison, is Φ/c2 ∼ (δρ/ρ)H2r2/c2 where δρ/ρ ≃ 200 at the the TD effect so the overall effect is therefore similar in am- virial radius. Thus, within the virialized region, this geo- plitude to the TD effect but with opposite sign, so it causes metric term is small compared to the gravitational redshift, the total observed effect to be larger than the gravitational but further out at the turnaround radius where δρ/ρ ∼ 5, effect alone rather than smaller. this is a substantial correction.

⃝c 0000 RAS, MNRAS 000,000–000 More implications of the transverse Doppler red/ blue-shift dichotomy Contribution to cluster grav-Z from motions of stars in the • BCG velocity dispersions are smaller than in cluster, but not • negligible • stars are unresolved so we get a transverse Doppler blue- shift • 21cm radio observations of galaxies sees mostly galaxies falling into cluster for ﬁrst time as gas is • stripped within virial region • should have a large potential difference relative to BCG • but the prediction for δZ is highly dependent on whether one makes unresolved single dish (e.g. Aricebo) measurements or resolved (e.g. Westerbork, ASKAP) Corrected grav-z measurement

Fairly easy to correct for TD • 620 isotropic 7 +LC+SB effects anisotropic isotropic anisotropic 6 • TD depends on vel. disp. 600 5 v c

anisotropy [km/s] 580 los • LC+SB directly measured σ 4 560 3 • net effect is a blue-shift 540 2 ~-9km/s in centre, falling 0 0.2 0.4 0.6 0.8 1 1.2 -2.5 -2.4 -2.3 -2.2 -2.1 • R [Mpc] α to ~-6km/s at larger r 0.8 0.7 0

21 Gravitational Redshifts in Clusters 7

WHH figure 1 appear to decrease by about 15% between the inner-bin and the outer points. This is reasonably con- − sistent with the expected σ2 ∝ r 0.2 trend predicted if galaxies trace mass, but this is perhaps fortuitous since the outer points are well outside the virial radius. Regardless of whether the galaxies at large radius are equilibrated or not, we can use the change in the observed velocity dispersion with radius to obtain the differential TD+LC+SB effect which is shown, added to the GR effect, as the solid line in figure 3. The kinematic effects flatten out the predicted profile, so the prediction is quite different from the gravitational redshift alone. The situation is clearly rather complicated, especially when using BGCs as the origin of coordinates since the effects depend on things like the relative velocities of the top ranked pair of cluster galaxies, and on the BCG halo prop- Figure 3. Data points from figure 2 of WHH and prediction based erties, that are quite poorly known. However, those factors on mass-traces-light cluster halo profile and measured velocity only influence the prediction for the innermost data point. dispersions as described in the main text. The dashed line is the The empirically based theoretical prediction for the profile gravitational redshift prediction, which is similar to the WHH of the redshift offset for the hot population as a function model prediction. The dot-dash line is the transverse Doppler of impact parameter at r⊥ > 0.6Mpc is the most robust; effect. The dotted line is the LC effect. The triple dot-dash line if galaxies are reasonable tracers of the mass then profile is the surface brightness effect. The solid curve is the combined should be very flat, quite unlike the GR effect from a NFW effect. profile. The predicted GR and total effects are shown in figure 3. However, this analysis ignores the effect of secular would appear to be discrepant, but only at about the 1.5- infall and out-flow which we consider next. sigma level. − The NFW model predicts δz ≃−10 km s 1/c for the outer measurements r ≃ 3.3, 5.3Mpc, and the measurements straddle this value. While this model may provide a reason- 6 EFFECT OF INFALL AND OUTFLOW able description for isolated clusters in the virialised domain, it is not at all clear that it is appropriate to describe the com- The discussion so far has focused mostly on the stable, posite cluster being studied here. Tavio et al. (2008) have virialised regions. Clusters, however, are evolving structures claimed that beyond the virial radius the density in numer- and the mass within any fixed physical radius M(r)willin ical LCDM simulations actually falls off like ρ ∼ 1/r rather general be changing. Outside of the virial radius (generally than the ρ ∼ 1/r3 asymptote for the NFW profile, and the considered, inspired by the spherical collapse model, to be extended peculiar in-fall velocities found by Cecccarelli et the radius within which the mean enclosed mass density is al. (2011) also argue for shallow cluster profiles, but it is not 3π/Gt2) we expect to see net infall, and the enclosed mass at clear that these results are widely accepted. those radii will be increasing with time, while at still larger An alternative, and possibly more reliable, approach is radii there will be outflow tending asymptotically toward the to assume that galaxies trace the mass reasonably well, in Hubble flow. In the spherical collapse model the transition which case the density profile of the stacked cluster has the from inflow to outflow takes place at the turnaround radius 2 same shape as the cluster-galaxy cross correlation function where the mean enclosed mass density is ρt =3π/32Gt . (e.g. Lilje & Efstathiou, 1988; Croft et al, 1997). This has a This is for a matter dominated Universe; allowing for a cos- − power-law dependence ρ ∼ r γ with γ ≃ 2.2, i.e. intermedi- mological constant makes only a small change (Lokas & Hoff- ate between the NFW and Tavio et al. model predictions. man, 2001). −γ −γ For space density ρ(r)=ρ0(r/r0) ,wherer0 is an ar- For the empirically motivated ρ = ρ0(r/r0) 2−γ bitrary fiducial radius, the potential is Φ(r)=Φ0(r/r0) model the mean enclosed mass is ρ(r) = 3(γ − −1 2 γ−2 −γ and the 1-D velocity disperson, for isotropic orbits, is 1)(2πG) σ0 r0 r and the nominal virial radius is rvir = 2 2 2−γ 2 2 γ−2 2 2 1/γ σ (r)=σ0 (r/r0) with Φ0 = 2((1 − γ)/(2 − γ))σ0 . ((γ − 1)σ0 r0 t /2π ) ≃ 1.8Mpc using γ =2.2, r0 = −1 The projected velocity dispersion measured is related to 1Mpc, σ0 = 545 km/s and t ≃ 1/H =1/(70 kms /Mpc) 2 2 2−γ the 3-D velocity dispersion by σ (r⊥)/σ (r)= dy y (1+ and turnaround is at rt ≃ 8.7Mpc. − y2) γ/2 but the projected potential is relatedR to the 3- In the centres of clusters there may be softening of the D potential in the same way, so the projected quantities cores which would reduce the enclosed mass and would have 2 are related by Φ(r⊥) = 2((1 − γ)/(2 − γ))σ (r⊥). This an associated outflow. is the potential relative to infinity. The difference in pro- In any single cluster, the density may be changing jected potential between two projected radii r1 and r2 is rapidly — on the local dynamical timescale — especially 2 0.2 Φ(r2) − Φ(r1) ≃ 12σ (r1)(1 − (r1/r2) ) for γ =2.2. The during mergers and as clumps rain in, but for a compos- resulting GR effect is shown as the dashed line in figure 3 ite cluster such as considered here these rapid changes will and is actually quite similar to the shape of the profile for average out and the mass can only change on a cosmologi- the WHH NFW composite model. cal timescale: M˙ ∼ HM. For power law profile with γ ≃ 2 The FWHM of the bell-shaped velocity distributions in M ≃ 4πρr3 and M˙ ≃ 4πρr2v,wherev is the mean infall ve-

⃝c 0000 RAS, MNRAS 000,000–000 Gravitational redshift & uRSD 3 obtain the conformal velocity r˙ ≡ dr/dη = av), where conformal The Doppler shift (the redshift of the emitting galaxy as seen time us defined, up to a constant, by dη = dt/a(t).Letusalso by a co-located stationary observer) is (REF) assume that we are provided with the peculiar potential Φ and its √1+x ˙ 2 gradient g = ∇rΦ,againatsomegivenconformaltimeη = η0. (1 + z)Doppler = =1+˙x + v /2+..., (11) 1 − v2 We will use units such that c =1temporally, and put back c for the final expression of our derivation. If we set a =1at the but here x˙ is the peculiar velocity at the time of emission, which output time then v and r˙ are identical at that time and separations differs (at 2nd order) from the velocity at the output time. The equa- in r are proper separation in physical units. tion of motion for the peculiar velocity is Extending off the time-slice η = η0,agalaxywillhavetrajec- v˙ = g − Hv (12) tory Modelling gravitational-z in simulations (Cai+'06) where g is the peculiar acceleration and the second term is the − r(η)=r +(η η0)r˙ + ... (4) ‘Hubble drag’ term that arises because peculiar velocities are defined to be with respect to the expanding (constant r)observers. where r, r˙ without an argument indicates the value at η0 and ...indicates terms that are of second or higher order in conformal look-• NK'13Thus modelling the line of sight assumed velocity virialised appearing in (non-expanding) (11) is clusters back time ∆η ≡ η − η0. x˙(η)=x ˙(η0) − (gx − Hx˙)x (13) We will place the observer event at some large distance along • this breaks down at large r the (minus) x axis, and at the time such that the observer receives • needwhere to we allow have used for ∆infallt = ∆η = −x. photons that left the origin (which we will ultimately take tobethe asymmetryMultiplying gives (10) other and (11) biases and keeping up to 2nd order terms • and adding the peculiar gravitational redshift gives, for the redshift centre of the cluster) at time η0.Theequationofthesurfaceinr- Cai+2016 have used Millennium simulation to quantify this space that contains the points within the observer’s past light cone• of the galaxy with respect to that for a stationary emitter at the with conformal time η is • formalismorigin, for extracting observables from "snapshots": 2 − η = η0 − xˆ r(η)+... (5) cz =Hx + vx + v /2c Φ/c · (14) − xg + Hxv /c + H 2 − a/¨ (2a2) x2/c, where xˆ is the unit vector parallel to the x-axis and where we are x x ignoring the fact that the coordinate speed of light is not exactly where we have put back the speed% of light. The above& equation fully unity because of the metric perturbations (this introduces errors of • includesaccounts light-cone for the observed effects redshift relative to a stationary emitter order v × Φ which we may safely neglect). This formula gives validin the to past 2nd light order cone toin the velocity second order(Hubble (if the and/or potentials peculiar) are not the conformal emission time of a photon from a particle at relative • evolving). We call the total distortion to the Hubble term induced by position r that is received at the same time as a photon which leaves all the other terms the ultimate redshift-space distortions(uRSD). the origin at time η0. COMPLICATION: What is relevant is the redshift with re- For simplicity here we are making the ‘plane-parallel’ approx- spect to the cluster centre – which may be a centroid of the cluster imation, which is valid for sufficiently distant clusters. galaxies, with cluster membership suitably defined, or it maybe Substituting (4) in (5) yields the conformal look-back time in with respect to the brightest cluster galaxy. The relevant relative ′ ′ ′ terms of r and r˙: redshift is 1+δz = λobs/λobs =(1+z)/(1 + z ) where λobs is the observed wavelength for light received from the centre and z′ is ∆η = −xˆ r/(1 + xˆ r˙)=−xˆ r +(xˆ r)(xˆ r˙)+... (6) ′ · · · · · the corresponding redshift. Because z appears in the denominator, − ′ or, with x = xˆ r and x˙ = xˆ r˙ = vx we cannot simply take δz = z z .Rather... · · − The first two terms on the RHS is the Doppler shift from the ∆η = x + xx˙ + ... (7) • total (i.e. Hubble + peculiar) velocity. We may use this to calculate the (inverse) redshift associated We then have the transverse Doppler effect and the peculiar • with the expansion of the universe during this look-back interval: gravitational redshift. Next we have minus the product of the line-of-sight dis- −1 a(η) a˙ 1 a¨ 2 • (1 + z) = =1+ ∆η + (∆η) + ... (8) placement and the line-of-sight acceleration; these tend tobeanti- a(η0) a 2 a correlated for over-dense systems and is the (positive redshift) ef- or fect shown in (Kaiser 2013), but we will see that at large distance 2 from the cluster centre, the situation is different. a˙ a˙ 1 a¨ 2 1+z =1− ∆η + − (∆η) + ... (9) Next we have a second order term Hxvx that is the product a a 2 a • !" # $ of the Hubble and peculiar velocities. In the virialised region these or with ∆η given by (7) will be uncorrelated, but in the outskirts of a cluster will beanti- correlated so this should give a negative contribution to themean 2 redshift. Again, the situation in velocity space and furtherfromthe a˙ − a˙ a˙ − 1 a¨ 2 1+z =1+ x xx˙ + x + ... (10) cluster centre may be different. a a ! a 2 a $ " # We then have the quadratic (in x)termthatcomesfromthethe • Equation (10) is not the redshift of the galaxy with time-slice combination of the background gravitational redshift and Doppler position r and velocity r˙ (at the time η when it intercepts the ob- effects (it is present even if v and Φ are zero). In a situation where server’s past light cone), rather it is the redshift of a stationary (in the density of galaxies is constant in real space, this will introduce, conformal coordinates) galaxy co-located with that galaxy at that at leading order, a linear ramp in the density. However, in anal- time relative to the redshift of a stationary galaxy at the origin ysises of gravitational redshift like Wojtak et al. (2011); Jimeno r =0.Toobtainthetheredshiftoftheactualparticleofinterestwe et al. (2015) analysis this gets removed because they fit for the lo- need to multiply (10) by the appropriate Lorentz boost factorand cal large-scale gradient using the density of galaxies well separated we need to include the peculiar gravitational redshift. from the cluster in velocity. Similar effects arise from the fact that

⃝c 0000 RAS, MNRAS 000,000–000 Clusters in the Millenium Simulation (Y. Cai) Gravitational redshift & uRSD 5

Figure 1. Top row: particle distributions within 10 Mpc/h radius from the main halo centre projected along one major axises of the simulation box. n in the label of the colour bars is the number of dark matter particles in each pixel. Middle row: the same zones but showing the potential values of all particles. Sub-haloes and neighbouring structures induce local potential minima. Bottom row: the gravitational redshift profileswithrespecttotheclustercentres.The dashed lines shows the spherical averaged profile, Φiso,whichisthesameasequal-directionalweightingfromthehalo centres. Sub-haloes and neighbours cause the mass weighted profiles Φobs to be biased low compared to the spherical averaging. This is similar to observations where the observed profiles are weighted by galaxies. ⃝c 0000 RAS, MNRAS 000,000–000 Gravitational redshift & uRSD 3 Gravitational redshift & uRSD 5 obtain the conformal velocity r˙ ≡ dr/dη = av), where conformal The Doppler shift (the redshift of the emitting galaxy as seen time us defined, up to a constant, by dη = dt/a(t).Letusalso by a co-located stationary observer) is (REF) assume that we are provided with the peculiar potential Φ and its √1+x ˙ 2 gradient g = ∇rΦ,againatsomegivenconformaltimeη = η0. (1 + z)Doppler = =1+˙x + v /2+..., (11) 1 − v2 We will use units such that c =1temporally, and put back c for the final expression of our derivation. If we set a =1at the but here x˙ is the peculiar velocity at the time of emission, which output time then v and r˙ are identical at that time and separations differs (at 2nd order) from the velocity at the output time. The equa- in r are proper separation in physical units. tion of motion for the peculiar velocity is Extending off the time-slice η = η0,agalaxywillhavetrajec- v˙ = g − Hv (12) tory where g is the peculiar acceleration and the second term is the − r(η)=r +(η η0)r˙ + ... (4) ‘Hubble drag’ term that arises because peculiar velocities are defined to be with respect to the expanding (constant r)observers. where r, r˙ without an argument indicates the value at η0 and ...indicates terms that are of second or higher order in conformal look- Thus the line of sight velocity appearing in (11) is back time ∆η ≡ η − η0. x˙(η)=x ˙(η0) − (gx − Hx˙)x (13) We will place the observer event at some large distance along the (minus) x axis, and at the time such that the observer receives where we have used ∆t = ∆η = −x. photons that left the origin (which we will ultimately take tobethe ModellingMultiplying (10) gravitational-z and (11) and keeping upin to simulations 2nd order terms (Cai+'06) centre of the cluster) at time η0.Theequationofthesurfaceinr- and adding the peculiar gravitational redshift gives, for the redshift space that contains the points within the observer’s past light cone of the galaxy with respect to that for a stationary emitter at the with conformal time η is origin,Formalism: mapping from x to v (2nd order) 2 − η = η0 − xˆ r(η)+... (5) cz =Hx + vx + v /2c Φ/c · (14) − xg + Hxv /c + H 2 − a/¨ (2a2) x2/c, where xˆ is the unit vector parallel to the x-axis and where we are x x ignoring the fact that the coordinate speed of light is not exactly where we have put back the speed% of light. The above& equation fully unity because of the metric perturbations (this introduces errors of accounts for the observed redshift relative to a stationary emitter order v × Φ which we may safely neglect). This formula gives in the past light cone to the second order (if the potentials are not the conformal emission time of a photon from a particle at relative evolving). We call the total distortion to the Hubble term induced by position r that is received at the same time as a photon which leaves all the other terms the ultimate redshift-space distortions(uRSD). the origin at time η0. COMPLICATION: What is relevant is the redshift with re- For simplicity here we are making the ‘plane-parallel’ approx- spect to the cluster centre – which may be a centroid of the cluster imation, which is valid for sufficiently distant clusters. galaxies, with cluster membership suitably defined, or it maybe Substituting (4) in (5) yields the conformal look-back time in with respect to the brightest cluster galaxy. The relevant relative ′ ′ ′ terms of r and r˙: redshift is 1+δz = λobs/λobs =(1+z)/(1 + z ) where λobs is the observed wavelength for light received from the centre and z′ is ∆η = −xˆ r/(1 + xˆ r˙)=−xˆ r +(xˆ r)(xˆ r˙)+... (6) ′ · · · · · the corresponding redshift. Because z appears in the denominator, − ′ or, with x = xˆ r and x˙ = xˆ r˙ = vx we cannot simply take δz = z z .Rather... · · ∆η = −x + xx˙ + ... (7) Figure 1. TopThe row: particle first distributions two terms within on 10 Mpc the/h RHSradius from is the the main Doppler halo centre projected shift alongfrom one the major axises of the simulation box. n in the label• of the colour bars is the number of dark matter particles in each pixel. Middle row: the same zones but showing the potential values of all particles. Sub-haloestotal and (i.e. neighbouring Hubble structures + peculiar) induce local poten velocity.tial minima. Bottom row: the gravitational redshift profileswithrespecttotheclustercentres.The We may use this to calculate the (inverse) redshift associated dashed lines showsWe the then spherical have averaged the profile, transverseΦiso,whichisthesameasequal-directionalweightingfromtheh Doppler effect and the peculiaralo centres. Sub-haloes and neighbours cause the• mass weighted profiles Φobs to be biased low compared to the spherical averaging. This is similar to observations where the observed profiles are with the expansion of the universe during this look-back interval: weightedgravitational by galaxies. redshift. ⃝c 0000 RAS,Next MNRAS we000,000–000 have minus the product of the line-of-sight dis- −1 a(η) a˙ 1 a¨ 2 • (1 + z) = =1+ ∆η + (∆η) + ... (8) placement and the line-of-sight acceleration; these tend tobeanti- a(η0) a 2 a correlated for over-dense systems and is the (positive redshift) ef- or fect shown in (Kaiser 2013), but we will see that at large distance 2 from the cluster centre, the situation is different. a˙ a˙ 1 a¨ 2 1+z =1− ∆η + − (∆η) + ... (9) Next we have a second order term Hxvx that is the product a a 2 a • !" # $ of the Hubble and peculiar velocities. In the virialised region these or with ∆η given by (7) will be uncorrelated, but in the outskirts of a cluster will beanti- correlated so this should give a negative contribution to themean 2 redshift. Again, the situation in velocity space and furtherfromthe a˙ − a˙ a˙ − 1 a¨ 2 1+z =1+ x xx˙ + x + ... (10) cluster centre may be different. a a ! a 2 a $ " # We then have the quadratic (in x)termthatcomesfromthethe • Equation (10) is not the redshift of the galaxy with time-slice combination of the background gravitational redshift and Doppler position r and velocity r˙ (at the time η when it intercepts the ob- effects (it is present even if v and Φ are zero). In a situation where server’s past light cone), rather it is the redshift of a stationary (in the density of galaxies is constant in real space, this will introduce, conformal coordinates) galaxy co-located with that galaxy at that at leading order, a linear ramp in the density. However, in anal- time relative to the redshift of a stationary galaxy at the origin ysises of gravitational redshift like Wojtak et al. (2011); Jimeno r =0.Toobtainthetheredshiftoftheactualparticleofinterestwe et al. (2015) analysis this gets removed because they fit for the lo- need to multiply (10) by the appropriate Lorentz boost factorand cal large-scale gradient using the density of galaxies well separated we need to include the peculiar gravitational redshift. from the cluster in velocity. Similar effects arise from the fact that

⃝c 0000 RAS, MNRAS 000,000–000 What was wrong with the "kinematic picture"?

• "A gravitational redshift is just a Doppler shift viewed from an unnatural coordinate system"? • This confuses gravity and acceleration • In GR the gravitational field is the Riemann (curvature) tensor • just the tidal field in the Newtonian limit • can be measured from relative motion of test particles • So is there a truly gravitational component to the redshift? • and why does e.g. cosmological z appear kinematical? Why is the gravitational-z hidden in cosmology? • Consider expanding sphere of dust and source A sending photon to receiver B • Photon suffers gravitational red-shift climbing up the potential and then a Doppler red-shift on reception A • For source B sending to A the photon has a Doppler red-shift (as seen in our frame) then enjoys a gravitational B blue-shift • But the net effect is the same. • The opposite gravitational shifts are cancelled by the Doppler shift change • But this is a special situation The non-kinematic part of the redshift Consider pair of freely-falling observers 1,2 in arbitrary • gravitational field who exchange a photon. • Use rigid, non-rotating lattice picture to calculate changes in wavelength and proper separation (work in CoM frame) • work to 2nd order in v/c and 1st order in φ/c2 2 • Δλ/λ = n . (v1 - v2)t1 / c + ∫dr . (g2 - g(r)) / c (1) 2 • ΔD/D = n . (v1 - v2)t1 / c + Δr . (g2 - g1) / 2c (2) • Both are 1st order Doppler (with initial Δv) plus ‘tidal’ term • Spatially constant tidal field stretches λ just like D • includes Minkowski spacetime and FRW • but that's because of special symmetry of FRW • does not apply for a galaxy cluster • extra intrinsically gravitational term (gradient of tide) 8 Nick Kaiser it is if the relative velocity is taken to be in the rest-frame, W r −W ′ r ✻ 0( )= 1( ) or whether it also involves the gravity, as is the case if the ✲ velocity difference is at different times. This is at odds with r 8 NickSynge’s Kaiser statement that the curvature does not appear in the redshift. it is if theWhat relative about velocity the change is taken in to the be separation in the rest-frame, during the W r −W ′ r ✻ 0( )= 1( ) or whetherlight-propagational it also involves time? the Letting gravity, the as is centre the case of velocity if the ✲ frame separations, in the absence of gravity, at reception W r −W ′ r r velocity difference is at different times. This is at odds with ✻ 1( )= 2( ) Synge’sand statement emission that be D theR,D curvatureE =(D ± does∆D/ not2) we appear see from in the the caption8 of Fig.Nick 3 that Kaiser∆D/D is not precisely the same as redshift. +d the flat-space ∆λ/λ =2µβ.Butthedifference is of order ✲ What3 about the change in the separation during the − r β , so toit our is if precision the relative goal velocity we can is take taken them to be to in be the equal. rest-frame, d W (r)=−W ′ (r) light-propagational time? Letting the centre of velocity ✻ 0 1 Switchingor onwhether gravity, it the also fractional involves the change gravity, of the as separation is the case if the ✲ frame separations, in the absence of gravity, at reception W r −W ′ r betweenvelocity the particles difference as measured is at diff byerent lattice times. based This observers is at odds with ✻ 1( )= 2( ) r and emission– i.e. theSynge’s be changeDR statement,D inE the=( properD that± the∆ separationD/ curvature2) we in see does the from not centre appear the of in the W =(r2 − d2)θ(|r| − d)/2 captionvelocity of Fig.redshift. frame 3 that – is∆D/D is not precisely the same as ✻ 2 ✲ +d the flat-space ∆λWhat/λ =2 aboutµβ.Butthedi the changeff inerence the separation is of order during the r ✲ 3 2 −d r β , so to our∆light-propagationalD/D precision= n · ( goalv2 − wev1) time? cant1 /c take+ Letting∆r them· (g2 − the tog1 be) centre/2c equal. of(4) velocity Switching onframe gravity, separations, the fractional in the change absence of the of gravity, separation at reception W r −W ′ r ✻ 1( )= 2( ) betweenwhich the is particlesand also emission a simple as measured be NewtonianDR,D byE latticelooking=(D ± based result.∆D/2) observers Unsurpris- we see from the ingly, tocaption first order of Fig. in 3 the that relative∆D/D velocityis not precisely the fractional the same as – i.e. the change in the proper separation in the centre of 2 2 +d W2 =(r − d )θ(|r| − d)/2 ✲ velocitychanges framethe in – isflat-spacewavelength∆ andλ/λ separation=2µβ.Butthedi are identical.fference Both is of order ✻ ✲ 3 −d r containβ an, so additional to our precision gravitational goal we term can that take is, them to lowest to be equal.Figure 4. The upper plot shows, schematically, the dimensionless r order, aSwitching tidal effect. on gravity, In general, the fractional these gravitational change2 of the effects separationweight function W0(r)=θ+(r)θ−(r) − d(δ(r − d)+δ(r + d))/2 ∆D/D = n · (v2 − v1)t1 /c + ∆r · (g2 − g1)/2c (4) ′ are notbetween precisely the equal particles – g2 − asg measured1 is the integral by lattice of thebased tide observersthat, when multiplied by the gravity φ (r)givesthedifference – i.e. the change in the proper separation in the centre of∆λ/λ − ∆D/D.TheDiracδ-functions are shown as the narrow while the wavelength shift involves the integral of the grav- W =(r2 − d2)θ(|r| − d)/2 which is alsovelocity a simple frame Newtonian – is looking result. Unsurpris- box-cars at r = ±d and together have (minus)✻ 2 the same weight as ✲ ity – so the ‘cosmological relation’ that wavelengths vary r ingly,precisely to first orderin proportion in the to relative the source-observer velocity the proper fractional sepa- the central box-car. As described in the text, this difference can 2 also be computed as a weighted average of the tide φ′′(r)using changesration, in wavelength does not∆D/D hold and= inn general. separation· (v2 − v1)t1 are/c + identical.∆r · (g2 − Bothg1)/2c (4) the weighting function W1(r)showninthecentreplot,whichis contain anBut additional if the tide gravitational is spatially termconstant that – is, i.e. to the lowest poten- Figure 4. The upper plot shows, schematically, the dimensionless (minus) the integral of W0(r), and also has zero net weight. The order,tial a tidal haswhich no effect. spatial is also In derivativesgeneral, a simple these Newtonian higher gravitational than looking second result. eff –ects then Unsurpris-weight function W0(r)=θ+(r)θ−(r) − d(δ(r − d)+δ(r + d))/2 third way to compute the difference is averaging′ the gradient of are notthe precisely gravityingly, varies equal to first linearly – g order2 − withg1 inis the position the relative integral then velocity weof the can the tide write fractionalthat,the tide whenφ′′′ multiplied(r)withtheweightfunctionshowninthebottom by the gravity φ (r)givesthedifference − while theg(r)= wavelengthgchanges1 +(g2 − in shiftg wavelength1)|r involves− r1|/|r2 and the− r1 integral separation| and the of gravitational are the identical. grav- Both∆plot.λ/λ ∆D/D.TheDiracδ-functions are shown as the narrow ity – soterms the arecontain ‘cosmological readily an found additional relation’ to be gravitational identical. that wavelengths Thus term the that relation vary is, to lowestbox-carsFigure at r = ± 4.dTheand upper together plot have shows, (minus) schematically, the same the weight dimensionless as − − seen inorder, cosmology a tidal is of eff widerect. In generality, general, these and applies gravitational for an effectsthe centralweight box-car. function As describedW0(r)=θ+ in(r the)θ−( text,r) d this(δ(r diffderence)+δ(r can+ d))/2 precisely in proportion to the source-observer proper sepa- ′ ′′ arbitraryare pair not of precisely particles equal moving – g in2 − ag field1 is the that integral has a spa- of the tidealso beSecond, computedthat, taking when as multiplied the a weighted difference by average the of gravity (3) ofand theφ (4),(r tide)givesthedi weφ have(r)usingfference ration, does not hold in general. ∆λ/λ − ∆D/D.TheDiracδ-functions are shown as the narrow tially constantwhile the tide. wavelength This includes, shift involves as a special the integral case, a ofpair the grav-the weighting function W1(r)showninthecentreplot,whichis But if the tide is spatially constant – i.e. the poten- box-cars at r = 1±d and together have (minus) the same weight as of particlesity – with so the relative ‘cosmological motion along relation’ their thatseparation wavelengths in a vary(minus)∆ thelog( integralλ/D)= of W0(dr),n and· (g1 also+ g2 has) − zerodr net· g weight.(6) The tial has no spatial derivatives higher than second – then the central box-car.c2 „ As described in theZ text, this« difference can precisely in proportion to the source-observer proper sepa-third way to compute the difference is averaging the gradient′′ of quadratic potential as in a FRW model containing matter also′′′ be computed as a weighted average of the tide φ (r)using the gravity variesration, linearly does not with hold position in general. then we can write the tide φ (r)withtheweightfunctionshowninthebottom and/or dark energy. Note that there is no need for the parti- where dthe≡ weighting|r2 − r1|/ function2. TakingW1 the(r)showninthecentreplot,whichis origin of coordinates to g(r)=g1 +(g2 −Butg1)| ifr − ther1 tide|/|r2 is− spatiallyr1| and the constant gravitational – i.e. the poten-plot. cles to be co-moving with the matter density, though again lie at (r1(minus)+ r2)/2 the for integral simplicity, of W0 this(r), is and a weighted also has zero average net weight. of The ′ termsthis are result readilytial does has found apply no spatial to in be that identical. derivatives situation. Thus higher the than relation second – thenthe gravitythird− wayφ : to compute the difference is averaging the gradient of ′′′ seen in cosmologyThisthe is gravity one is of of the wider varies two linearlygenerality, main results with and of position this applies paper: then for a we con-an can write the tide φ (r)withtheweightfunctionshowninthebottom g r g g g r r r r 1 ′ arbitrarystant pair tide( of stretches)= particles1 +( wavelength2 moving− 1)| of in− radiation a1| field/| 2 − that just1| and has as it the a changes spa- gravitational Second,plot. ∆ takinglog(λ/D the)= differencedr W of0 (3)(r)φ and(r) (4), we have (7) c2 Z tiallythe constant separationterms tide. are of This test-particles. readily includes, found Arguably as to a be special identical. this case, ‘explains’ Thus a pair the the relation seen in cosmology is of wider generality, and applies for an 1 of particlesapparent with stretching relative of motion wavelength along of their light separation by the expansion in a with∆ dimensionlesslog(λ/D)= weightingd n · (g1 function+ g2) − Wd0r(r·)g ≡ (6) arbitrary pair of particles moving in a field that has a spa- Second,Subtracting takingc2 „ the diff erence(1) - of(2) (3)Z gives and (4), « we have quadraticof space potential in FRW as models. in a FRW model containing matter θ+(r)θ−(r)−d(δ(r−d)+δ(r+d))/2; where φ(r) ≡ φ(r = rn); tially constant tide. This includes, as a special case, a pair ± and/or darkTo energy. highlight Note the that differences there is between no need forseparation the parti- and whereand whered ≡ θ|r2(r−) ≡r1|θ/(2.±r Taking− d)with1 theθ( originr) and ofδ( coordinatesr) denoting to 8 Nick Kaiser∆ log(λ/D)= d n · (g1 + g2) − dr · g (6) of particles with relative motion along their separation inthe a Heaviside function and the2 Dirac delta function respec- or cles towavelength be co-moving changes with –the what matter one density, might though call the again ‘non- lie at (r1 + r2)/2 for simplicity,c „ this is a weightedZ average« of quadratic potential as in a FRW model containing matter ′ tively.it is The if the weight relative′ velocity function is takenW to0 be(r in) the is rest-frame, shown schematically W (r)=−W (r) this resultkinematic’ does apply component in that of the situation. redshift – and to see how this the gravity −φ : ✻ 0 1 and/or dark energy. Note that there is no need for the parti-as theor whetherwhere upper itd also plot≡ involves|r in2 − Fig.ther1 gravity,|/2. 4. Taking asThe is the product case the if the origin of Heaviside of coordinates to ✲ depends on the tide (and its derivatives) we note the follow- velocity difference is at different times. This is at odds with r This is onecles of to the be two co-moving main results with the of matter this paper: density, a con- though again lie at (r1 + r2)/2 for simplicity, this is a weighted average of ing: functionsSynge’s is statement zero for that the|r′| curvature>d1so does the not appear range in ofthe ′ integration is stant tide stretchesthis result wavelength does apply of radiation in that situation. just as it changes redshift.the gravity∆ log(λ−/Dφ :)= dr W0(r)φ (r) (7) First, we can write now unrestricted. The integralc2 ofZ W0(r) over all r vanishes, the separation ofThis test-particles. is one of the Arguably two main this results ‘explains’ of this paper: the a con- What about the change in the separation during the so welight-propagational can immediately time? Letting integrate the centre by1 parts of velocity to eliminate′ the apparent stretchingstant tide of stretches wavelength wavelength of light of by radiation the expansion just as it changeswithframe dimensionless separations, in∆ thelog( absence weightingλ/D of)= gravity,2 at function receptiondr W0(r)φW(r0)(r) ≡ (7) W r −W ′ r ∆r ′′ gravity and write (7) as an integral ofZ the tide: ✻ 1( )= 2( ) and emission be D ,D =(D ± ∆D/2) wec see from the the∆D/D separation= n · (v of2 − test-particles.v1)t1 /c − Arguablydrφ ( thisr) ‘explains’ (5) theθ+(r)θ−(r)−d(δ(rR−dE)+δ(r+d))/2; where φ(r) ≡ φ(r = rn); of space in FRW models. 2c2 Z caption of Fig. 3 that ∆D/D is not precisely the same as Redshifts and the Expansion of Space 9 1 ′′ +d To highlightapparent the stretching differences of wavelength between separation of light by the and expansionand wherethewith flat-spaceθ±(∆r dimensionless)λ/λ≡=2θ(µβ±.Butthedir − d)with weightingfference isθ( ofr) order and functionδ(r) denotingW0(r) ≡ ✲ 3 ∆ log(λ/D)= 2 dr W1(r)φ (r)−d (8) r of space in FRW models. β ,θ so+ to(r our)θ− precision(r)−d goal(δ( wer− cancd)+ takeZ δ( themr+d to)) be/2; equal. where φ(r) ≡ φ(r = rn); wavelengthwhere φ changes(r)=φ(rn –) is what the gravitational one might potential call the and ‘non- prime themore HeavisideSwitching by parts on gravity, function to express the fractional and the the change diff Diracerence of the separation delta of these function as a weighted respec- The perfect correlation of ∆ log(λ) and ∆ log(D) in ho- ± To highlight the differences between separation and betweenand the where particles′′′ θ as measured(r) ≡ θ by(± latticer − basedd)with observersθ(r) and δ(r) denoting kinematic’denotes component the operator of the∂r = redshiftn · ∇, i.e. – and the to spatial see how derivative this tively.whereaverage TheW1( ofr weight)=φ (r−): functiondr W0(rW)=0(r−)r isθ+ shown(r)θ−(r)whichis schematically mogeneous models can be considered to be a reflection of ′ – i.e. the change in the proper separation in the centre of wavelength changes – what one might call the ‘non- the Heaviside function and the Dirac delta function respec- W =(r2 − d2)θ(|r| − d)/2 dependsalong on the the photon tide (and path, its so derivatives)φ (r)=n · ∇ weφ( noter)= the−n · follow-g.Thus asshown thevelocity is upper the frame middle – plot is R in plot Fig. in Fig. 4. The (4). product But the integral of Heaviside of the✻ 2 symmetry of the✲ gravitational fields that are allowed in kinematic’ component of the redshift – and to see how this tively. The weight function1 W0(r) is′′′ shown schematically r ing: the gravitational contribution to ∆D/D is the average of functionsW1(r) also is vanishes zero∆ log( forλ (so,/D|r| )= as>d alreadyso thedr mentioned W range22(r)φ of( forr integration) spatially (9) is these models, in accord with the conjecture of Melia (2012). depends on the tide (and its derivatives)2 we note the follow- as∆D/D the= uppern · (v2 − v plot1)t1 /c + in∆cr2· Fig.Z(g2 − g 4.1)/2 Thec (4) product of Heaviside the tide along the photon path times (∆r/c) /2. nowconstant unrestricted. tide ∆λ/λ The= ∆ integralD/D) and of W we0(r can) over integrate all r vanishes, once First, weing: can write functions is zero for |r| >dso the range of integration is In the Introduction we asked what redshift would be so wewhich can is immediately also a simple Newtonian integrate looking result.2 by Unsurpris-2 parts to eliminate2 the First, we can write withingly, weightnow to first unrestricted. function order in theW relative2 The(r)=( velocity integralr − thed offractional)θW+(0r()rθ)− over(r)/2=( all r vanishes,r − seen by a pair of observers in a cluster who have the same 2 ⃝c 0000 RAS, MNRAS 000,000–000 ∆r ′′ gravityd )θchanges(| andsor| − we in writed wavelength) can/2 now (7)immediately and shownas separation an integral as integrate are the identical. bottom of the by Both plot parts tide: in to figure eliminate (4). the separation at emission as at reception. Our analysis shows contain an additionalThere gravitational is a non-kinematic term that is, to lowest component of the redshift: it is a ∆D/D = n · (v2 − v1)t1 /c − 2 drφ (r) (5) Figure 4. The upper plot shows, schematically, the dimensionlessnot 2c Z ∆r ′′ This isgravity the second and write main (7) result as an of integral this paper. of theweight tide: function W (r)=θ (rthat)θ−(r) − theyd(δ(r − dod)+δ(r +seed))/2 the effect of any local tidal field. What order, a tidal effect. In general, these gravitational effects′′ 0 + ∆D/D = n · (v2 − v1)t1 /c − drφ (r) (5) 1 ′ r 2c2 Z are not precisely∆ log( equalλ/D – g2)=−measurementg1 is the integraldr W of1 the(r tide) φof( rthe)that, whengradient multiplied (8) by of the happens gravitythe φtide( )givesthedi is thatfference any gravitational stretching of the wave- while the wavelength shift involvesc the2 Z integral1 of the grav- ∆λ′′/λ − ∆D/D.TheDiracδ-functions are shown as the narrow ∆ log(λ/D)= dr W1(r)φbox-cars(r) at r = ±d and together(8) length have (minus) that the would same weight be as perceived in the frame of the rigid non- where φ(r)=φ(rn) is the gravitational potential and prime ity – so the ‘cosmological relation’ that wavelengths2 Z vary precisely in proportion to the source-observerc proper sepa- the central box-car. As described in the text, this difference can where φ(r)=φ(rn) is the gravitational potential and prime −also be computed as a weightedinertial average of observers the tide φ′′(r)using is counteracted by the red- or blue-shifts denotes the operator ∂r = n · ∇, i.e. the spatial derivative whereration,W1( doesr)= not hold− in general.dr W0(r)=−rθ+(r)θ (r)whichis ′ 3 DISCUSSION the weighting function W1(r)showninthecentreplot,whichis denotes the operator ∂r = n · ∇, i.e. the spatial derivative whereBut if theW tide1(r is)= spatially− constantdr W –0 i.e.(r the)= poten-−rθ+(r)θ−(r)whichisin boosting from the observers’ frames to the rigid frame along the photon path, so φ (r)=n · ∇′ φ(r)=−n · g.Thus shown is the middleR plot in Fig. (4). But the(minus) integral the integral of of W0(r), and also has zero net weight. The along the photon path, so φ (r)=n · ∇φ(r)=−n · g.ThusWetial haveshown has nomade spatial is various the derivatives middle simplifying higherR plot than in second assumptions. Fig. – then (4). Butthird The the way results to integral compute the of differenceand is back averaging again the gradient (if, o forf example they were moving apart at the gravitational contribution to ∆D/D is the average of W1(r)the also gravity vanishes varies linearly (so, with as position already then we mentioned can write the for tide spatiallyφ′′′(r)withtheweightfunctionshowninthebottom the gravitational contribution to 2∆D/D is the average of W1(r) also vanishes (so, as already mentioned for spatially g(r)=g1 +(g2 − g1)|r − r1|/|r2 − r1| and the gravitational plot. the emission time they will be moving together again by the the tide along the photon path times (∆r/c) /2. 2 constantare only tide valid∆λ up/λ to= 2nd∆D/D order) andin velocity we can and integrate 1st order once in the tide along the photon path times (∆r/c) /2. termsconstant are readily tide found∆ toλ be/λ identical.= ∆ ThusD/D the) relationand we can integrate once time of reception). theseen potential, in cosmology but is of wider this generality, is adequate and applies to for encompass an the phe- arbitrary pair of particles moving in a field that has a spa- Second, taking the difference ofIn (3) an and inhomogeneous (4), we have system such as the solar system, nomena that have proved⃝c 0000 controversial.⃝ RAS, MNRAS We have000,000–000 ignored tially constant tide. This includes, as a specialc 0000 case, RAS, a pair MNRAS 000,000–0001 anyof time particles variation with relative motion of the along potential, their separation and in a we have∆ alsolog(λ/D ig-)= d nor· (g a1 + galaxy,g2) − dr cluster· g (6) or supercluster, the tidal field neces- c2 „ Z « noredquadratic any potential bending as in of a FRW light model rays. containing These matter are higher order ef- sarily varies with position. There is then a non-kinematic and/or dark energy. Note that there is no need for the parti- where d ≡ |3r2 − r1|/2. Taking the origin of coordinates to fects;cles to the be co-moving Rees-Sciama with the matter (1968) density, effect though is again of orderlie at (v/c (r1 +)r2)/for2 for simplicity,component this is a weighted to average the redshift of that violates (1) and which is es- ′ instance.this result We does apply can in also that situation.ignore spatial curvature.the In gravity a FRW−φ : sentially gravitational in nature. Combining this with the This is one of the two main results of this paper: a con- 1 ′ model,stant tide for stretches example, wavelength the of distance radiation just between as it changes two FOs as mea-∆ log(λ/D)= kinematicdr W0(r)φ ( redshiftr) (7) component, if any, one obtains complete c2 Z suredthe separation by observers of test-particles. on a Arguably ruler thisis not ‘explains’ precisely the the same as consistency with the conventional view of the gravitational apparent stretching of wavelength of light by the expansion with dimensionless weighting function W0(r) ≡ theof distance space in FRW as models. determined by a chain of FOsθ+ lying(r)θ−(r)− alongd(δ(r−d)+δ(r+redshiftd))/2; where inφ(r clusters) ≡ φ(r = r ofn); galaxies and other gravitating systems. a geodesicTo highlight in the the 3-space differences of between constant separation proper and timeand where sinceθ±( ther) ≡ θ(±r − dNote)withθ that(r) and ourδ(r) denotingdescription of components of the redshift is wavelength changes – what one might call the ‘non- the Heaviside function and the Dirac delta function respec- big-bang,kinematic’ but component the fractional of the redshift di –ff anderence to see how is at this mosttively. of order The weight the function diWff0(rerent) is shown from schematically the terminology of Chodorowski (2011) who depends on the tide (and its derivatives) we note the follow- as the upper plot in Fig.was 4. The considering product of Heaviside the gravtiational component of the redshift squareing: of the separation in units of the curvaturefunctions radius is zero (or for |r| >dso the range of integration is 2 of orderFirst, (v/c we can) ) write so any effect on ∆D/D is of highernow unrestricted. order. The integralin of FRWW0(r) over models all r vanishes, that arises if the ‘kinematic’ component is so we can immediately integratetaken by parts to be to eliminate the relative the velocity at emission or reception Similarly the asynchronicity∆ betweenr ′′ clocks carriedgravity and by write our (7) as an integral of the tide: ∆D/D = n · (v2 − v1) 1 /c − drφ (r) (5) t 2c2 Z non-inertial grid-based observers is negligible at our stated 1rather than′′ the average velocity. Here we consider redshifts ∆ log(λ/D)= dr W1(r)φ (r) (8) c2 Z levelwhere ofφ precision.(r)=φ(rn) is the gravitational potential and prime between observers in FRW models to be purely kinematic denotes the operator ∂r = n · ∇, i.e. the spatial derivative wheretimeW1(r)=− dr W0(r)=−rθ+(r)θ−(r)whichis More fundamentally′ while matter tells space- how in the sense that (1) is obeyed. along the photon path, so φ (r)=n · ∇φ(r)=−n · g.Thus shown is the middleR plot in Fig. (4). But the integral of to curve,the gravitational it is contribution only the to curvature∆D/D is the averageof time of thatW1(r tells) also vanishes non- (so, as alreadyFor mentioned emitter/receiver for spatially pair separation that is small com- relativisticthe tide along matter the photon how path to times move, (∆r/c and)2/2. is is also onlyconstant the tide time-∆λ/λ = ∆D/Dpared) and we to can the integrate size once of the gravitating system the difference time part of the metric perturbation that is relevant for the ⃝c 0000between RAS, MNRAS the000 fractional,000–000 change in the wavelength and sep- calculation of the redshift. So at the specified level of preci- aration is on the order of the gravitational potential well sion we can ignore the spatial part of the metric. depth times the cube of the separation in units of the over- We have asked: what is the domain of validity of the re- all system size. This is seen most easily from (9), and the 2 lationship between wavelengths and emitter/receiver proper fact that W2(r) ∼ d , which together imply ∆ log(λ/D) ∼ separation (1) that we see for FOs in homogeneous mod- (d/R)3φ/c2. So, just as the local gravity is invisible to freely- els? We have shown that a spatially constant tide stretches falling observers, as far as the ratio of wavelength to separa- wavelength in exactly the same way it affects the observers’ tion is concerned, the local tide is also invisible. But if the separation, but if the tide varies with position the relation- path length has a similar size to the entire system the error ship between wavelength and separation is modified. is on the order of the gravitational potential. From this perspective, the perfect correlation seen be- To see better how this relates to Synge’s result that tween changes in wavelengths ∆λ/λ of light exchanged be- the redshift is always given by the Doppler formula, con- tween FRW FOs and the change ∆D/D in the space (be- sider the case of an emitter at the centre of the poten- tween said FOs) is not a causal relationship, rather both tial for a small uniform spherical distribution of matter of the change in the wavelength and the change in the space mass M and radius r and a receiver outside at distance between the observers are ‘caused’, or determined, by a com- D ≫ r who happens to be at rest at the moment of re- bination of the observers’ initial velocities and the tidal field ception. In this situation, the redshift is just the static in which they and the photons propagate. Echoing Whiting gravitational redshift: ∆λ/λ = dr g/c2 ∼ GM/c2r.But (2004), the expansion rate defined by the matter content the more distant the receiver,R the smaller any fractional of the universe is irrelevant (which is a jolly good thing if change in the emitter/receiver proper separation during the universe has a cosmological constant or a scalar field the time of flight: ∆D ! (GM/D2)(∆t)2/2whichimplies to realise dark energy since neither defines either a frame of ∆D/D ! GM/c2D ≪ ∆λ/λ. Evidently the kinematic re- motion or a state of expansion). Rather, on the parabolic po- lation does not apply here. But, following Peebles, we can tential generated by gravitating matter and dark energy one still break the net wavelength ratio down into the product can have emitter/receiver pairs that recede from each other of ratios between a set of pairs of neighbouring particles. or pairs that approach each other, and what determines both We can take these to be particles on a set of radial orbits the changes in the wavelengths and proper separations is a such that each particle is at apogee at the time the photon combination of initial conditions and the curvature, or tidal passes (see Fig. 5). Thus the nth particle has zero velocity field. as the photon passes it, as does the (n+1)th particle. In the

⃝c 0000 RAS, MNRAS 000,000–000 Why we observe a gravitational z in clusters

• The "kinematic picture" is wrong • redshifts are not solely determined by change of separation of observer, source • there is an additional, intrinsically gravitational, effect • but the gravitational-z comes from gradients of the tide • that's why it's not seen in FRW cosmology • a consequence of symmetry • Total z is kinematic plus an integral involving grad(tide) • sums to give naive (P&R) gravitational redshift • but we also need TD, LC and SB effects.. Future prospects...

• Can expect immediate improvements in measurement • 3x increase in number of redshifts available (BOSS) • and more to come: • optical: big-BOSS • radio: FAST, ASKAP-Wallaby+WNSHS • interesting to compare unresolved radio and optical • Extension to larger scales. Bright-faint cross correlation Gaztanaga++2015, Alam++2016.. • Lots of rich material in the front-back asymmetry of the galaxy correlation function. • Lots of interesting scope for modelling: Redshift space distortions (symmetric) What else does it mean? • Probe of curvature of space in GR? • matter tells space how to curve • space tells matter how to move.... • Like how lensing tests gravity? • Not quite: • motion of galaxies & grav-z are determined only by gtt • It is really a test of the equivalence principle Provides a test of theories with long-range non-gravitational • forces in the “dark sector” • e.g. Gradwohl & Frieman 1992; Farrar & Peebles 2004; Farrar & Rosen 2007; Keselman, Nusser & Peebles 2010; and many, many more.... and (maybe) f(R) gravity. though such theories are already constrained by X-ray • temp. vs galaxy motions in clusters.... Scalar ﬁelds, "Fifth forces" & Violation of the EP • a common feature of modiﬁed gravity theories

• string theory inspired: dilaton ﬁeld - couples to matter

• also interacting DE & DM models where m = m(φ)

• f(R) gravity etc. etc.

• extra long-range (1/r potential) force augmenting gravity

• must be suppressed/small on solar system scale

• or only coupling to DM

• Violations of the Equivalence Principle (foundation of GR)

• interesting - and testable - consequences

• lensing - galaxy clustering - gravitational redshifts - BHs see different g - dynamics in clusters (gas vs *s vs DM) Conclusions • Gravitational redshifts in clusters of galaxies have been measured! • Technically challenging but apparently real and prospects for better measurements and extension to larger scales is promising. • Potentially useful test of alternatives to GR & 5th forces • But also interesting as a "sand-box" that illustrates some subtleties of simple special relativity + Newtonian gravity • Effect raises some questions of principle about how to think about redshifts in cosmology and astronomy in general. • Redshifts are not purely kinematic - there is an truly gravitational component - but it is hidden in cosmology