I understand RNAV 1 to mean it will be within 1 mile of the desired track 95% of the time, but that is lateral accuracy, what about longitudinal accuracy, in front and behind the estimated position along the route. Does the RNAV specification give us any guarantees for that?
It does, but it's a different issue than just positional accuracy. The goal of RNAV is to keep the aircraft within the confines of the specified airspace. Longitudinally, you will always be within the airspace if you're meeting the cross-track and vertical containment as you're moving along the defined path.
What really matters longitudinally is when you reach a point along the path. Since airspeed, winds, and aircraft performance affect this it is very dynamic. And it's not just an absolute time, ATC can adjust these targets to keep aircraft properly spaced along the route. This is accomplished using 'Time of Arrival Control' requirements.
RTCA document DO-236 Minimum Aviation System Performance Standards: Required Navigation Performance for Area Navigation provides the following guidance:
1.3.2 Containment Methodology
An aircraft meeting the required Time of Arrival Control requirements will satisfy the lateral and vertical confines of RNP RNAV/VNAV airspace while meeting a required time of arrival. Application of this MASPS is intended to assure that systems will achieve required times of arrival within 30 seconds 95% of the time.
1.4.6 Time of Arrival Control (TOAC)
The CNS/ATM operating environment is expected to utilize aircraft capability to improve traffic management for route crossing and metering operations. The use of time of arrival control is intended to provide an increased certainty of an aircraft arriving at a fix at a required time. TOAC as the time element for RNP navigation will not change the system navigation performance. TOAC may alter, where allowable, the lateral and vertical paths in space, and associated containment/vertical path performance limits to reflect changes in aircraft path and speed.
Unfortunately in the US many of the regulations are written to make it difficult to separate accuracy/integrity in the along-track direction from the cross-track direction. DO-229D claims this is "since RNAV positioning accuracy is generally independent of the aircraft track.", but I have to scratch my head at that since using a circular window only makes sense in some circumstances. Latency effects and DME/DME accuracy for example can be much greater in one direction than another. As @JanHudec points out though, when considering changes of direction the circular accuracy is more useful and intuitive than, for example, separate measures of NS and EW accuracy.
For example DO-229D, "MASPS: Required Navigation Performance for Area Navigation" says:
The Horizontal Protection Level Fault Detection (HPLFD) is the radius of a circle in the horizontal plane (the local plane tangent to the WGS-84 ellipsoid), with its center being at the true position, that describes the region assured to contain the indicated horizontal position.
DO-236B and DO-229D have similar language for the Radius of Containment.
In this image protection level would be R1, or R2 if being conservative. This image is not regulatory, since this is just from a class discussing position integrity.
DO-229D does describe in its appendix some scenarios though where the position integrity is not necessarily circular:
For en route through LNAV approach, HPL_SBAS must bound horizontal radial position error with a probability of 1 - 10-7 per hour; i.e., the probability that horizontal radial position error exceeds HPL_SBAS must not exceed 10-7 in any hour, except possibly for brief periods less than the time-to-alert. For LNAV/VNAV, LP, and LPV approaches, the probability that horizontal cross-track error or vertical error or both exceed their respective protection levels must not exceed 2x10-7 per approach. Only one dimension is used for HPL_SBAS in LNAV/VNAV, LP, and LPV approaches, since the along-track tolerance is so much larger than the cross-track. The worst-case dimension is used.
Also, keep in mind that for climbs and descents, large along-track directional uncertainty will mean you're not sure how close to the vertical path you are. AC 90-101A says that for RNP AR approaches like the one at Aspen equipment should account for this coupling and warn the pilot if uncertainty about where you are along the path could lead your altitude to be too far above or below that path.